Embed Size (px)
Citation preview
Biomedical Mass Transport andChemical Reaction
Biomedical Mass Transport andChemical Reaction
Physicochemical Principles and Mathematical Modeling
James S UltmanHarihara BaskaranGerald M Saidel
Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved
Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada
No part of this publicationmay be reproduced stored in a retrieval system or transmitted in any form or by anymeanselectronic mechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108of the 1976 United States Copyright Act without either the prior written permission of the Publisher or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood DriveDanvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web at wwwcopyrightcom Requests to thePublisher for permission should be addressed to the Permissions Department John Wiley amp Sons Inc 111 RiverStreet Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions
Limit of LiabilityDisclaimer ofWarranty While the publisher and author have used their best efforts in preparing thisbook they make no representations or warranties with respect to the accuracy or completeness of the contents of thisbook and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose Nowarranty may be created or extended by sales representatives or written sales materials The advice and strategiescontained herein may not be suitable for your situation You should consult with a professional where appropriateNeither the publisher nor author shall be liable for any loss of profit or any other commercial damages including butnot limited to special incidental consequential or other damages
For general information on our other products and services or for technical support please contact our CustomerCare Department within the United States at (800) 762-2974 outside the United States at (317) 572-3993 orfax (317) 572-4002
Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not beavailable in electronic formats For more information about Wiley products visit our web site at wwwwileycom
Library of Congress Cataloging-in-Publication Data
Names Ultman James S author | Baskaran Harihara author | SaidelGerald M author
Title Biomedical mass transport and chemical reaction physicochemicalprinciples and mathematical modeling James S Ultman Harihara Baskaranand Gerald M Saidel
Description Hoboken New Jersey John Wiley amp Sons 2016 | Includes indexIdentifiers LCCN 2015048303| ISBN 9780471656326 (cloth) | ISBN 9781119184652 (epub)Subjects LCSH Biological transport | Biomedical engineeringClassification LCC QH509 U48 2016 | DDC 61028ndashdc23LC record available at httplccnlocgov2015048303
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
We dedicate this book to our wivesmdashDeena Ultman Lakshmi Balasubramanyan andMina Saidelmdashwhose love support and continual encouragement sustained us duringthis endeavor that has lasted a decade
Contents
Preface xviGuidance to Instructors xviiMethods for Solving Model Equations xixAcknowledgments xxAbout the Companion Website xxi
Part I Introduction 1
1 Biological Structure and Function 311 Cell Energy Related to Whole-Body Function 4111 Energy Generation 5112 Energy Transfer 612 Tissue and Organ Systems 8121 Circulation of Extracellular Fluid 9122 Lungs 10123 Kidneys 12124 Small Intestine 14125 Liver 1513 Cell Structure and Energy Metabolism 16131 Cell Composition 17132 Cellular Organelles 19133 Mechanism of Cellular Energy Metabolism 19
2 Modeling Concepts for Biological Mass Transport 2121 Representation of Biological Media 21211 Continuum Point of View 21212 Homogeneous and Heterogeneous Materials 22213 Composition Variables 2222 Mechanisms of Mass Transport 25221 Convection and Diffusion 25222 Transport through Cell Membranes 27223 Transport across Cell Sheets 2923 Formulation of Material Balances 3024 Spatially Lumped and Distributed Models 32241 Spatially Lumped Models 33
|vii
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Biomedical Mass Transport andChemical Reaction
Biomedical Mass Transport andChemical Reaction
Physicochemical Principles and Mathematical Modeling
James S UltmanHarihara BaskaranGerald M Saidel
Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved
Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada
No part of this publicationmay be reproduced stored in a retrieval system or transmitted in any form or by anymeanselectronic mechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108of the 1976 United States Copyright Act without either the prior written permission of the Publisher or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood DriveDanvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web at wwwcopyrightcom Requests to thePublisher for permission should be addressed to the Permissions Department John Wiley amp Sons Inc 111 RiverStreet Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions
Limit of LiabilityDisclaimer ofWarranty While the publisher and author have used their best efforts in preparing thisbook they make no representations or warranties with respect to the accuracy or completeness of the contents of thisbook and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose Nowarranty may be created or extended by sales representatives or written sales materials The advice and strategiescontained herein may not be suitable for your situation You should consult with a professional where appropriateNeither the publisher nor author shall be liable for any loss of profit or any other commercial damages including butnot limited to special incidental consequential or other damages
For general information on our other products and services or for technical support please contact our CustomerCare Department within the United States at (800) 762-2974 outside the United States at (317) 572-3993 orfax (317) 572-4002
Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not beavailable in electronic formats For more information about Wiley products visit our web site at wwwwileycom
Library of Congress Cataloging-in-Publication Data
Names Ultman James S author | Baskaran Harihara author | SaidelGerald M author
Title Biomedical mass transport and chemical reaction physicochemicalprinciples and mathematical modeling James S Ultman Harihara Baskaranand Gerald M Saidel
Description Hoboken New Jersey John Wiley amp Sons 2016 | Includes indexIdentifiers LCCN 2015048303| ISBN 9780471656326 (cloth) | ISBN 9781119184652 (epub)Subjects LCSH Biological transport | Biomedical engineeringClassification LCC QH509 U48 2016 | DDC 61028ndashdc23LC record available at httplccnlocgov2015048303
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
We dedicate this book to our wivesmdashDeena Ultman Lakshmi Balasubramanyan andMina Saidelmdashwhose love support and continual encouragement sustained us duringthis endeavor that has lasted a decade
Contents
Preface xviGuidance to Instructors xviiMethods for Solving Model Equations xixAcknowledgments xxAbout the Companion Website xxi
Part I Introduction 1
1 Biological Structure and Function 311 Cell Energy Related to Whole-Body Function 4111 Energy Generation 5112 Energy Transfer 612 Tissue and Organ Systems 8121 Circulation of Extracellular Fluid 9122 Lungs 10123 Kidneys 12124 Small Intestine 14125 Liver 1513 Cell Structure and Energy Metabolism 16131 Cell Composition 17132 Cellular Organelles 19133 Mechanism of Cellular Energy Metabolism 19
2 Modeling Concepts for Biological Mass Transport 2121 Representation of Biological Media 21211 Continuum Point of View 21212 Homogeneous and Heterogeneous Materials 22213 Composition Variables 2222 Mechanisms of Mass Transport 25221 Convection and Diffusion 25222 Transport through Cell Membranes 27223 Transport across Cell Sheets 2923 Formulation of Material Balances 3024 Spatially Lumped and Distributed Models 32241 Spatially Lumped Models 33
|vii
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Biomedical Mass Transport andChemical Reaction
Physicochemical Principles and Mathematical Modeling
James S UltmanHarihara BaskaranGerald M Saidel
Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved
Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada
No part of this publicationmay be reproduced stored in a retrieval system or transmitted in any form or by anymeanselectronic mechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108of the 1976 United States Copyright Act without either the prior written permission of the Publisher or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood DriveDanvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web at wwwcopyrightcom Requests to thePublisher for permission should be addressed to the Permissions Department John Wiley amp Sons Inc 111 RiverStreet Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions
Limit of LiabilityDisclaimer ofWarranty While the publisher and author have used their best efforts in preparing thisbook they make no representations or warranties with respect to the accuracy or completeness of the contents of thisbook and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose Nowarranty may be created or extended by sales representatives or written sales materials The advice and strategiescontained herein may not be suitable for your situation You should consult with a professional where appropriateNeither the publisher nor author shall be liable for any loss of profit or any other commercial damages including butnot limited to special incidental consequential or other damages
For general information on our other products and services or for technical support please contact our CustomerCare Department within the United States at (800) 762-2974 outside the United States at (317) 572-3993 orfax (317) 572-4002
Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not beavailable in electronic formats For more information about Wiley products visit our web site at wwwwileycom
Library of Congress Cataloging-in-Publication Data
Names Ultman James S author | Baskaran Harihara author | SaidelGerald M author
Title Biomedical mass transport and chemical reaction physicochemicalprinciples and mathematical modeling James S Ultman Harihara Baskaranand Gerald M Saidel
Description Hoboken New Jersey John Wiley amp Sons 2016 | Includes indexIdentifiers LCCN 2015048303| ISBN 9780471656326 (cloth) | ISBN 9781119184652 (epub)Subjects LCSH Biological transport | Biomedical engineeringClassification LCC QH509 U48 2016 | DDC 61028ndashdc23LC record available at httplccnlocgov2015048303
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
We dedicate this book to our wivesmdashDeena Ultman Lakshmi Balasubramanyan andMina Saidelmdashwhose love support and continual encouragement sustained us duringthis endeavor that has lasted a decade
Contents
Preface xviGuidance to Instructors xviiMethods for Solving Model Equations xixAcknowledgments xxAbout the Companion Website xxi
Part I Introduction 1
1 Biological Structure and Function 311 Cell Energy Related to Whole-Body Function 4111 Energy Generation 5112 Energy Transfer 612 Tissue and Organ Systems 8121 Circulation of Extracellular Fluid 9122 Lungs 10123 Kidneys 12124 Small Intestine 14125 Liver 1513 Cell Structure and Energy Metabolism 16131 Cell Composition 17132 Cellular Organelles 19133 Mechanism of Cellular Energy Metabolism 19
2 Modeling Concepts for Biological Mass Transport 2121 Representation of Biological Media 21211 Continuum Point of View 21212 Homogeneous and Heterogeneous Materials 22213 Composition Variables 2222 Mechanisms of Mass Transport 25221 Convection and Diffusion 25222 Transport through Cell Membranes 27223 Transport across Cell Sheets 2923 Formulation of Material Balances 3024 Spatially Lumped and Distributed Models 32241 Spatially Lumped Models 33
|vii
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved
Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada
No part of this publicationmay be reproduced stored in a retrieval system or transmitted in any form or by anymeanselectronic mechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108of the 1976 United States Copyright Act without either the prior written permission of the Publisher or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood DriveDanvers MA 01923 (978) 750-8400 fax (978) 750-4470 or on the web at wwwcopyrightcom Requests to thePublisher for permission should be addressed to the Permissions Department John Wiley amp Sons Inc 111 RiverStreet Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 or online at httpwwwwileycomgopermissions
Limit of LiabilityDisclaimer ofWarranty While the publisher and author have used their best efforts in preparing thisbook they make no representations or warranties with respect to the accuracy or completeness of the contents of thisbook and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose Nowarranty may be created or extended by sales representatives or written sales materials The advice and strategiescontained herein may not be suitable for your situation You should consult with a professional where appropriateNeither the publisher nor author shall be liable for any loss of profit or any other commercial damages including butnot limited to special incidental consequential or other damages
For general information on our other products and services or for technical support please contact our CustomerCare Department within the United States at (800) 762-2974 outside the United States at (317) 572-3993 orfax (317) 572-4002
Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not beavailable in electronic formats For more information about Wiley products visit our web site at wwwwileycom
Library of Congress Cataloging-in-Publication Data
Names Ultman James S author | Baskaran Harihara author | SaidelGerald M author
Title Biomedical mass transport and chemical reaction physicochemicalprinciples and mathematical modeling James S Ultman Harihara Baskaranand Gerald M Saidel
Description Hoboken New Jersey John Wiley amp Sons 2016 | Includes indexIdentifiers LCCN 2015048303| ISBN 9780471656326 (cloth) | ISBN 9781119184652 (epub)Subjects LCSH Biological transport | Biomedical engineeringClassification LCC QH509 U48 2016 | DDC 61028ndashdc23LC record available at httplccnlocgov2015048303
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
We dedicate this book to our wivesmdashDeena Ultman Lakshmi Balasubramanyan andMina Saidelmdashwhose love support and continual encouragement sustained us duringthis endeavor that has lasted a decade
Contents
Preface xviGuidance to Instructors xviiMethods for Solving Model Equations xixAcknowledgments xxAbout the Companion Website xxi
Part I Introduction 1
1 Biological Structure and Function 311 Cell Energy Related to Whole-Body Function 4111 Energy Generation 5112 Energy Transfer 612 Tissue and Organ Systems 8121 Circulation of Extracellular Fluid 9122 Lungs 10123 Kidneys 12124 Small Intestine 14125 Liver 1513 Cell Structure and Energy Metabolism 16131 Cell Composition 17132 Cellular Organelles 19133 Mechanism of Cellular Energy Metabolism 19
2 Modeling Concepts for Biological Mass Transport 2121 Representation of Biological Media 21211 Continuum Point of View 21212 Homogeneous and Heterogeneous Materials 22213 Composition Variables 2222 Mechanisms of Mass Transport 25221 Convection and Diffusion 25222 Transport through Cell Membranes 27223 Transport across Cell Sheets 2923 Formulation of Material Balances 3024 Spatially Lumped and Distributed Models 32241 Spatially Lumped Models 33
|vii
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
We dedicate this book to our wivesmdashDeena Ultman Lakshmi Balasubramanyan andMina Saidelmdashwhose love support and continual encouragement sustained us duringthis endeavor that has lasted a decade
Contents
Preface xviGuidance to Instructors xviiMethods for Solving Model Equations xixAcknowledgments xxAbout the Companion Website xxi
Part I Introduction 1
1 Biological Structure and Function 311 Cell Energy Related to Whole-Body Function 4111 Energy Generation 5112 Energy Transfer 612 Tissue and Organ Systems 8121 Circulation of Extracellular Fluid 9122 Lungs 10123 Kidneys 12124 Small Intestine 14125 Liver 1513 Cell Structure and Energy Metabolism 16131 Cell Composition 17132 Cellular Organelles 19133 Mechanism of Cellular Energy Metabolism 19
2 Modeling Concepts for Biological Mass Transport 2121 Representation of Biological Media 21211 Continuum Point of View 21212 Homogeneous and Heterogeneous Materials 22213 Composition Variables 2222 Mechanisms of Mass Transport 25221 Convection and Diffusion 25222 Transport through Cell Membranes 27223 Transport across Cell Sheets 2923 Formulation of Material Balances 3024 Spatially Lumped and Distributed Models 32241 Spatially Lumped Models 33
|vii
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Contents
Preface xviGuidance to Instructors xviiMethods for Solving Model Equations xixAcknowledgments xxAbout the Companion Website xxi
Part I Introduction 1
1 Biological Structure and Function 311 Cell Energy Related to Whole-Body Function 4111 Energy Generation 5112 Energy Transfer 612 Tissue and Organ Systems 8121 Circulation of Extracellular Fluid 9122 Lungs 10123 Kidneys 12124 Small Intestine 14125 Liver 1513 Cell Structure and Energy Metabolism 16131 Cell Composition 17132 Cellular Organelles 19133 Mechanism of Cellular Energy Metabolism 19
2 Modeling Concepts for Biological Mass Transport 2121 Representation of Biological Media 21211 Continuum Point of View 21212 Homogeneous and Heterogeneous Materials 22213 Composition Variables 2222 Mechanisms of Mass Transport 25221 Convection and Diffusion 25222 Transport through Cell Membranes 27223 Transport across Cell Sheets 2923 Formulation of Material Balances 3024 Spatially Lumped and Distributed Models 32241 Spatially Lumped Models 33
|vii
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
242 One-Dimensional Spatially Distributed Model 35References 39
Part II Thermodynamics of Biomedical Processes 41
3 Basics of Equilibrium Thermodynamics 4331 Thermodynamic Systems and States 4332 Heat Work and the First Law 4433 Enthalpy and Heat Effects 4534 Entropy and the Second Law 4635 Gibbs Free Energy and Equilibrium 46351 Gibbs Free Energy Changes in Closed Systems 46352 Chemical Potential Changes in Open Systems 47353 GibbsndashDuhem Equation 49354 Spontaneous Processes and Electrochemical Equilibrium 4936 Properties of the Chemical Potential 51361 Constitutive Equations 51362 Temperature and Pressure Dependence 52363 Composition Dependence 53
References 53
4 Interfacial and Membrane Equilibria 5441 Equilibrium Criterion 5442 Interfacial Equilibria 56421 Immiscible Liquid Phases 56422 GasndashLiquid Interfaces 57423 Multiphase Equilibrium 6143 Membrane Equilibria 62431 Electrochemical Equilibrium 62432 Osmotic Pressure 64433 Colloid Osmotic Pressure 6844 Electrical Double Layer 71
References 75
5 Chemical Reaction Equilibrium 7651 Equilibrium Criterion 7652 Equilibrium Coefficients 78521 Gas Phase 78522 Liquid Phase 7853 Acid Dissociation 80531 Monovalent Acids 80532 Complex Acids 8154 LigandndashReceptor Binding 83541 Monovalent Binding 83542 Competitive Binding 85
Contentsviii |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
543 Allosteric Binding 8755 Equilibrium Models of Blood Gas Content 90551 Blood Chemistry 90552 Oxygen Content 92553 Carbon Dioxide Content 97
References 101
Part III Fundamentals of Rate Processes 103
6 Nonequilibrium Thermodynamics and Transport Rates 10561 Transport Velocities and Fluxes 105611 Molar and Mass Average Velocity 105612 Convective Flux 106613 Diffusive Flux 10762 StefanndashMaxwell Equation 10963 Diffusion of Uncharged Substances 111631 Binary Diffusion 111632 Multicomponent Diffusion 112633 Pseudo-binary Diffusion 11564 Diffusion of Electrolytes 11665 Transport across Membranes 117651 Entropy Generation Function for Uncharged Solutes 117652 Chemical Potential Driving Forces 119653 KedemndashKatchalsky Equations 120654 Starling Equations 120
References 123
7 Mechanisms and Models of Diffusion 12471 Transport Rates in Homogeneous Materials 12572 Diffusion Coefficients in Gases 125721 Kinetic Theory 125722 Ideal Gas Model 12773 Diffusion Coefficients in Liquids 128731 Einstein Model 128732 Diffusion Coefficients of Nonelectrolytes 130733 Diffusion Coefficients of Electrolytes 13274 Transport in Porous Media Models of Tissue 134741 Representative Volume Element and Volume Averaging 134742 Hydrodynamic Model of a Porous Medium 136743 Renkin Model of Solute Diffusion 140744 Hydraulic and Solute Permeabilities 14175 Transport in Suspension Models of Tissue 144751 Fiber Matrix Model 144752 Spheroidal Suspension Models 146
References 151
Contents | ix
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
8 Chemical Reaction Rates 15281 General Kinetic Models 152811 Reaction Rates in a Closed System 152812 Single-Step Reactions 15382 Basis of Reaction Rate Equations 154821 Equilibrium Constraint on Reaction Rate Expressions 154822 Transition State Theory 15783 Multi-Step Reactions 15884 LigandndashReceptor Kinetics 161841 Monovalent Binding 161842 Competitive Binding 16385 Enzyme Kinetics 166851 Enzyme Behavior 166852 MichaelisndashMenten Kinetics 168853 Enzyme Inhibition 17086 Urea Cycle as a Reaction Network 173861 Reaction Rate Equations 173862 Material Balances 175863 Dimensional Analysis and Simulations 176
References 178
Part IV Transport Models in Fluids and Membranes 179
9 Unidirectional Transport 18191 Unidirectional Transport Equations 181911 Species Fluxes 181912 Rectilinear Transport 182913 Radial Transport 18492 Steady-State Diffusion 186921 Rectilinear Diffusion 186922 Radial Diffusion 18793 Diffusion with Parallel Convection 19194 Diffusion with Chemical Reaction 194941 Metabolic Demand of a Cell 194942 Augmented Diffusion by Protein Binding 19795 Unsteady-State Diffusion 201
References 203
10 Membrane Transport I Convection and Diffusion Processes 204101 Ordinary Diffusion 2041011 Nonequilibrium Thermodynamics 2051012 Mechanistic Models 2051013 Selectivity 210102 Diffusion with Parallel Convection 2111021 Nonequilibrium Thermodynamics 2111022 Mechanistic Models 212
Contentsx |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
1023 Selectivity and Sieving 213103 Cell Membrane Channels 2161031 Electrodiffusion Model 2161032 Resting Potential 2201033 Voltage Clamp Measurements 221
References 223
11 Membrane Transport II Carrier-Mediated Processes 224111 Facilitated Transport of a Single Substance 224112 Cotransport of Two Substrates 227113 Simulation of Tracer Experiments 2301131 Cotransport of a Labeled and Unlabeled Solute 2301132 Inhibition of Carrier-Mediated Transport 235114 Primary Active Transport 2371141 A Model of Primary Active Transport 2371142 ATP Concentration Constraint 2391143 Limiting Solute Flux 240115 Electrical Effects on Ion Transport 242
References 244
12 Mass Transfer Coefficients and Chemical Separation Devices 245121 Transport Through a Single Phase 2451211 Individual Mass Transfer Coefficient 2451212 Stagnant Film Model 2471213 Penetration Model 2471214 Dimensional Analysis 2501215 Hydraulically Permeable Surfaces 254122 Transport Through Multiple Phases 2561221 Diffusion at a Two-Phase Interface 2561222 Diffusion Through a Membrane 2571223 Parallel Convection and Diffusion Through a Membrane 2611224 Concentration Polarization 262123 Design and Performance of Separation Devices 2651231 Blood Oxygenation by Membrane Devices 2651232 Blood Purification by Hemodialysis 2711233 Hemodialysis with Negligible Plasma Filtration 2741234 Hemodialysis with Uniform Filtration 277
References 279
Part V Multidimensional Processes of Molecules and Cells 281
13 Fluid Mechanics I Basic Concepts 283131 Application of Conservation Principles 2831311 Mass Conservation in a Flowing System 2831312 Momentum Balance in a Flowing System 2861313 Relation of Contact Forces to the Stress Tensor 287
Contents |xi
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
132 Mechanical Properties and Rheology of Fluids 2891321 Fluid Deformation 2891322 Newtonian Fluids 2901323 Non-Newtonian Fluids 291133 Model Formulation and Scaling of Fluid Flow 2931331 Elements of Model Formulation 2931332 Interface Relationships 2941333 Dimensionless Flow Equations 298134 Steady Flow Through A Tube 2991341 Flow of Newtonian and Power-Law Fluids 2991342 Two-Phase Annular Flow 303
References 306
14 Fluid Mechanics II Complex Flows 307141 Boundary Layer Flows 3071411 Flow Development over a Flat Plate 3071412 Flow Induced by a Rotating Disk 313142 Creeping Flow Through a Leaky Tube 319143 Periodic Flow Along a Tube 323
Reference 329
15 Mass Transport I Basic Concepts and Nonreacting Systems 330151 Three-Dimensional Mass Balances 330152 Special Cases 3321521 Constant Mass Density 3331522 Constant Molar Density 334153 One-Dimensional Transport Equations 3341531 Cross-Sectional Averaging 3341532 Generalized One-Dimensional Transport 337154 Model Formulation and Scaling of Mass Transport 3391541 Elements of Model Formulation 3391542 Interface Relationships 3401543 Dimensionless Concentration Equation 343155 Diffusion and Convection in Nonreacting Systems 3441551 Unsteady-State Diffusion in a Finite Domain 3441552 Concentration Boundary Layer over a Flat Plate 3481553 Dispersion of an Inert Tracer Flowing in a Tube 353
References 357
16 Mass Transport II Chemical Reacting Systems 358161 Single-Phase Processes 3581611 Reactive Gas Transport in the Lung Mucus Layer 3581612 Urea Uptake by an Encapsulated Enzyme 363162 Multiphase Processes 3681621 Reactive Gas Transport in a Lung Airway Wall 368
Contentsxii |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
1622 Nutrient Transport and Reaction in Perfused Tissuethe Krogh Model 370
1623 Oxygenation of Pulmonary Capillary Blood 375163 Processes with Interfacial Reaction 3801631 Solute Transport to a Rapidly Rotating Disk with
Surface Reaction 3801632 Solute Transport with Surface Reaction in a Blood Vessel 385
References 387
17 Cell Population Dynamics 388171 Cell Number Balances 388172 Cell Transport and Fate Processes 3891721 Cell Movement 3891722 Cell Division and Proliferation 3901723 Cell Death 3931724 Cell Differentiation 393173 Single Cell Population Dynamics 3941731 Axon Growth by Haptotaxis 3941732 Endothelial Cell Migration 397174 Multiple Cell Population Dynamics 3991741 Tumor Vascularization and Growth 3991742 Chemotaxis with an Inflammatory Response 4031743 Stem Cells for Cartilage Tissue Engineering 406
Reference 409
Part VI Compartmental Modeling 411
18 Compartment Models I Basic Concepts and Tracer Analysis 413181 Compartmental Modeling Concepts 4131811 Pool Models and Physiologically Based Models 4131812 Tracer Inputs to a Flow-Through Model 4161813 Dynamic Responses of a Single-Compartment Model 419182 Multiple-Compartment Models 4211821 Two Compartments in Series 4211822 Multiple Compartments in Series 4231823 Parallel Compartments without Interaction 4251824 Parallel Compartments with Flow Interaction 4271825 Parallel Compartments with Diffusion Interaction 429183 Nonideal Inputs and Moment Analysis 4301831 Moments of Dynamic Inputs and Outputs 4301832 Relationship of Transfer Function to Impulse-Response Function 4311833 Moment Relationships for a Nonideal Input Response 432
Reference 438
Contents |xiii
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
19 Compartment Models II Analysis of Physiological Systems 439191 Open-Loop Models 4391911 Multipool Model of Glucose Metabolism 4391912 Multibreath Lung Washout 4421913 Pulmonary Ventilation Diffusion and Perfusion 4461914 Urea Dynamics with Hemodialysis 450192 Models with Feedback and Recirculation 4521921 Cardiovascular Recirculation of a Tracer 4521922 Control of Ventilation by Carbon Dioxide 4561923 Perfusion-Controlled Ethanol Metabolism 462
References 466
Part VII Advanced Biomedical Applications 467
20 Therapies for Tissue and Organ Dysfunction 469201 Dynamics of Urea Clearance in a Patient During Hemodialysis 469202 Hemodialyzer Performance with Varying Filtration 474203 Gas Exchange in an Intravascular Lung Device 480204 Separation of Blood Components by Apheresis 486205 Epidermal Regeneration in Tissue-Engineered Skin 490
References 497
21 Drug Release Delivery and Distribution 498211 Drug Release From an Agglomerated Tablet 498212 Drug Release From an Osmotic Pump Device 504213 Intestinal Drug Transport 509214 Drug Distribution in Ablated Tissues 515215 Intracranial Drug Delivery and Distribution 520216 Whole-Body Methotrexate Distribution 526
References 534
22 Diagnostics and Sensing 535221 Chemical Monitoring of Tissue by Microdialysis 535222 Dual-Electrode Measurement of Blood Flow and Oxygen 541223 Detection of Ethanol in Blood from Exhaled Gas 546224 Oxygen Uptake and Utilization in Exercising Muscle 552225 Tracer Analysis with Pet Imaging 562226 Cancer Cell Migration with CellndashCell Interaction 569
References 576
Appendix A Units and Property Data 577A1 American National Standard for SI Units 577A2 Definitions of Concentration 579A3 Thermodynamic Properties 580A4 Transport Properties 583
References 586
Contentsxiv |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Appendix B Representing Transport Processes in Complex Systems 587B1 Vector and Tensor Operations 587B11 Algebraic Operations 587B12 Derivative Operations 589B13 Key Theorems 590B14 VectorndashTensor Calculus 591B2 Nonequilibrium Thermodynamics 592B21 Entropy Generation Rate 592B22 GibbsndashDuhem Equation 596B3 Spatially Averaged Balances for Heterogeneous Tissue 596B31 Interstitial and Macroscopic Volume Averages 597B32 Solution Balances 598B33 Solute Balances 599B34 ConvectionndashDiffusion Equations 600B4 Tables for Fluid Motion in Common Coordinate Systems 602
References 604
Appendix C Mathematical Methods 605C1 Dimensionless Forms and Scaling 605C11 Dimensionless Representation of a Spatially Lumped Model 605C12 Dimensionless Representation of a Spatially Distributed Model 607C2 Inversion of Square Matrices 608C3 Initial-value Problems 609C31 Classification 609C32 Reduction of Order 610C33 Solution of a Linear First-Order Initial-Value Problem 611C4 Laplace Transforms 613C5 Alternative Representation of a Point Source 614C6 Similarity Transform of a Partial Differential Equation 615
Nomenclature 619Index 624
Contents |xv
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Preface
The impact of engineering on medicine and biology continues to grow significantly Not onlyhas this resulted in an impressive worldwide increase in educational biomedical engineeringprograms but many traditional chemical and agricultural engineering departments havechanged their names to include ldquobio-rdquo Recognizing the importance of biomedical engineer-ing research and development to human welfare and the global economy we have written thisbook to enhance the education of those students who will establish the biomedical technol-ogies of the futureEngineers who work in ldquobiordquo areas use analytical methods and quantitative modeling of
physical chemical and mathematical sciences that distinguish them from those who aretrained primarily in biological and medical sciences This textbook is designed for studentswhose educational emphasis involves physicochemical aspects of biomedical systems Thisrequires instruction in principles of thermodynamics mass transfer chemical reactionkinetics and fluid mechanicsA major objective of this textbook is to integrate engineering principles with relevant bio-
medical applications at the cellular tissue organ and whole-body levels These applicationsincorporate basic as well as more sophisticated and complex concepts which are appropriatefor graduate as well as advanced undergraduate engineering students Another major goal ofthis book is to teach students how to develop mathematical models and analyses associatedwith medical diagnostics and therapeuticsIn order to accomplish this the book is divided into seven parts The chapters in Part
I present basic biological and mathematical modeling concepts Part II provides an overviewof the thermodynamics that relate to interfacial membrane and chemical reaction equilibriaIn Part III rate equations are developed to analyze the mass diffusion and chemical reactionthat take place in homogeneous and heterogeneous media The application of convection-diffusion and reaction equations to membrane transport and chemical separation devicesare discussed in Part IV In Part V multidimensional transport of molecules and cell popu-lation dynamics are presented in the context of complex biomedical problems Part VI devel-ops general compartment models and analyses to represent dynamic and nonlinear responsesof biomedical systems More detailed mathematical models related to treatment of tissue andorgan dysfunction distribution and delivery of drugs and interpretation of biomedical mea-surements are developed in Part VII Keymathematical aspects related tomodel developmentand analyses are presented in appendices
xvi |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Guidance to Instructors
This textbook is especially intended for students in chemical and in biomedical engineeringParts IndashIV are presented mostly at an undergraduate level assuming knowledge of basic phys-ics chemistry and mathematics (including calculus differential equations and elements oflinear algebra) Parts VndashVII include more advanced physical chemical and mathematicalconcepts (eg vectorndashtensor representations)With its diversity of material this book can serve as a basis for various university courses
(i) a single course for students with different backgrounds (ii) distinct courses for undergrad-uate and graduates students or (iii) a sequence of lower- and higher-level courses In design-ing a particular course instructors can choose from the wide variety of topics in differentchapters to best serve specific student groupsChapter 1 provides students who have limited biological and physiological knowledge with
a context for the applications found in later chapters The basics of mass transport analysis inChapter 2 with simple biomedical applications are worthwhile for all students even as a par-tial review Those who have studied chemical thermodynamics may skip Chapter 3 but thematerial on electrochemical potential and equilibrium should be reviewed Most of the devel-opment of interfacial and membrane equilibrium in Chapter 4 and ligandndashreceptor bindingand bloodndashgas relationships in Chapter 5 provide a basis of topics in later chapters for allstudents Concepts of nonequilibrium thermodynamics in Chapter 6 may be of more interestto advanced students but their application to membrane transport should interest all Withthe exception of diffusion through multiphase materials the theory of diffusion mechanismsin Chapter 7 is primarily aimed at advanced students Students at every level would benefitfrom the sections in Chapter 8 on chemical reaction rates with biomedical applications buttheir theoretical basis would mainly interest some graduate studentsThe general presentation of one-dimensional transport in Chapter 9 which is not com-
monly found elsewhere is intended for all students Also recommended for all studentsare the early sections of Chapter 10 on membrane transport and of Chapter 11 on facilitatedand secondary active transport More complex aspects of membrane processes in the latersections of these two chapters are intended for graduate students Mass transfer coefficientsand their application to blood oxygenators and dialyzers which are covered in Chapter 12 areparticularly valuable to students interested in device developmentThe topics on multidimensional transport of molecules in Chapters 13ndash16 require a higher
level of sophistication expected of graduate students The early sections on cell populationdynamics in Chapter 17 and on compartment models in Chapter 18 are appropriate forall students The more complex models in the later sections of these two chapters and themore comprehensive compartmental modeling in Chapter 19 would be appreciated by
|xvii
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
graduate students especially in relation to the complex biomedical applications in Chapters20ndash22 These final chapters address three distinct application areasmdashmedical treatment drugdelivery and diagnosismdashthat provide a variety of choices for the instructorFor convenient reference symbols and notations in this book are defined where introduced
and common symbols are also defined in a final nomenclature section Symbols in italicizedfonts represent dimensionless quantities whereas bolded symbols refer to vectors and ten-sors Standard international units (specified in Appendix A) are used in computationsHomework problems related to each chapter are available from a supplementary website
(httpengineeringcaseeduBMTR) These problems provide practice in basic computa-tions model development and simulations using analytical and numerical methods
Guidance to Instructorsxviii |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Methods for Solving Model Equations
Whenever possible the model equations developed in this book are solved using standardanalytical techniques Models of the more realistic systems presented in Parts V VI andVII require numerical methods for computer simulation and quantitative characterizationThese models are composed of linear and nonlinear ordinary differential equations(ODE) partial differential equations (PDE) andmixed algebraic-differential equations Giventhis diversity of model structure we used several commercial software packages that are com-mon for engineering education and research Since instructors may prefer other packagesthis book does not provide any instruction related to software applicationsTo solve initial-value problems we applied MathCAD (viz rkfixed) or Matlab (viz ode15s
for stiff ODE) For interacting dynamic compartmental models we used Matlab and Simu-link Models of dynamic one-dimensional boundary-value problems represented by PDEswere solved with Matlab (viz pdepe) For models involving a one-dimensional PDEndashODEcombination we discretized the spatial derivatives to transform the model into an initial-value problem (ie method of lines) For numerical solution of models involving a hyperbolicPDE with a first-order spatial derivative we avoided instability by adding a second-order spa-tial derivative multiplied by an arbitrarily small coefficient COMSOLMultiphysics was espe-cially useful for model simulations involving a combination of fluid flow and mass transportin complex geometries Mathematica was used to solve complex algebraic functions Param-eter estimation for analysis of data and simple algebraic functions was accomplished with theSolver function in Microsoft ExcelWhen performing simulations to assess system dynamics or demonstrate device design
numerical values for model parameters were frequently obtained from the literature Whenthis was not possible a range of dimensionless parameter values was used to explore modelbehavior under a variety of circumstances
|xix
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Acknowledgments
Many aspects of this book have benefited from our interactions with numerous colleaguesand students JSU is especially indebted to his undergraduate mentor Herbert Weinsteinwho first turned him on to biomedical engineering research to his doctoral mentor MortonDenn who further encouraged him to apply his chemical engineering training to biomedicalproblems and to his postdoctoral mentor Kenneth Keller who inspired him to create bio-engineering courses HB is grateful to his high school mentor Swarupanthan for introductionto mathematical methods and to his graduate school mentor JSU for research training in bio-transport applications GMS is particularly thankful to his postdoctoral mentor the late Pro-fessor Stanley Katz of the City College of New York whose guidance in mathematicalmodeling and analysis was very influential for career development To our special familiesand friends who shared our enthusiasm for this project we also express our sincereappreciation
James S UltmanUniversity Park PAHarihara Baskaran
Cleveland OHGerald M SaidelCleveland OH
xx |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
About the Companion Website
This book is accompanied by a companion website
httpengineeringcaseeduBMTR
The website includes
bull Homework problems
bull Solution to homework problems for instructors
bull Tables of data and equations
bull Powerpoint slides of figures and tables for instructors
|xxi
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Part I
Introduction
|1
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
Chapter 1
Biological Structure and Function
11 Cell Energy Related to Whole-Body Function 412 Tissue and Organ Systems 813 Cell Structure and Energy Metabolism 16
Living organisms operate much like a complicated chemical factory in which raw materialsfrom the surrounding environment are distributed to chemical reactors that produce desiredproducts along with waste that must be discarded back to the environment And just like achemical factory a living organismmust be capable of purifying raw materials and separatingdesired products from wastes In humans nutrients are separated from food in the upper gas-trointestinal tract oxygen is separated from air in the lungs and many of the critical reactionsthat utilize these raw materials occur in the liver Waste products are eliminated through thelower gastrointestinal tract and the kidneys as well as the lungs It is the collection of allthese chemical processes that enable an organism to maintain itself perform work growand reproduceThe human body consists of about 100 trillion cells each bathed in its own fluid microen-
vironment In an adult there is about 40 L of fluid a third of which are extracellular (locatedoutside of cells) and two-thirds of which are intracellular (located within cells) Variouschemical species are nonuniformly distributed between the extracellular and intracellularfluids (Fig 10-1) and there is a constant movement of ions nutrients waste productsand other substances between these fluid compartments A function required of all cells isthe regulation of these dynamics such that the chemical and energy needs of an organismare metNature has provided for this by enclosing cells in a specialized membrane that supports a
variety of transport processes some of which are passive and others that are active in natureDuring passive diffusion the movement of a substance across a membrane occurs spontane-ously in the direction of decreasing chemical potential The uptake of O2 from a relativelyhigh concentration in extracellular fluid to a lower concentration in intracellular fluid isan example of passive diffusion During active transport energy from an independent chem-ical source is harnessed allowing a substance to cross a membrane in the direction of increas-ing chemical potential The maintenance of a low intracellular sodium level for example
|3
Biomedical Mass Transport and Chemical Reaction Physicochemical Principles and Mathematical ModelingFirst Edition James S Ultman Harihara Baskaran and Gerald M Saidelcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
requires active transport of sodium out of cells to compensate for the passive leakage ofsodium into cellsSeveral energy-requiring processes in addition to active transport are necessary if an organ-
ism is to function properly and maintain its structural integrity Energy is consumed by manyof the metabolic reactions that synthesize essential molecules within cells Energy is neededfor muscular contraction in the heart lungs and limbs Also energy dissipation as heat isnecessary to maintain a normal body temperature of about 37 C The ultimate source ofenergy for all these tasks is the controlled oxidation of nutrients In the remainder of thischapter we will discuss how metabolic reactions and chemical transport are coordinatedat different levels of structural organization beginning at the whole-body level progressingto the organ level and ending at the cellular level
11 Cell Energy Related to Whole-Body Function
Food entering the mouth is degraded to simpler molecules by hydrolytic reactions that occurin the oral cavity and the stomach Additional chemical reactions occur downstream in thesmall intestine where the digested nutrients are absorbed into the bloodstream ultimatelyreaching the intracellular space where they are oxidized to liberate energy Humans eat foodscontaining a variety of different carbohydrates fats and proteins However the chemicalselectivity of transport and reaction processes in the gastrointestinal tract produces a limited
Extracellular Intracellular
137 mM 10 mM
K+
Na+
5 mM 141 mM
Ca+ + 3 mM 0 mM
Mg+ + 2 mM 31 mM
Clndash 103 mM 4 mM
HCO3ndash 28 mM 10 mM
Phosphates 4 mM 75 mM
SO4ndash ndash 1 mM 2 mM
Glucose 90 mg 0ndash20 mg
Amino acids 30 mg 200 mg
Lipids 05 mg 2ndash95 mg
pO247 kPa 27 kPa
pCO261 kPa 67 kPa
Figure 10-1 Homeostatic concentration conditions
1 Biological Structure and Function4 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
number of digestive products such as glucose fructose and galactose resulting fromcarbohydrate breakdown triglycerides from fat breakdown and amino acids from proteinbreakdown Most of the galactose and fructose absorbed by the intestines are rapidlyconverted to glucose in the liver Thus glucose triglycerides and amino acids are theprincipal substrates for energy metabolism and chemical synthesis in the bodyDuring resting conditions the minimal power requirement of an adult is about 70W
This rate of energy production is provided primarily by the complete oxidation of glucoseand triglycerides into CO2 and water These combustion reactions require about 250 mlmin of O2 uptake through the respiratory tract Even when a personrsquos energy requirementis greater than 70W a suitable O2 supply is usually available to sustain this aerobicmetabolism When a person is involved in very strenuous exercise however the energydemand can be so great that glucose is incompletely oxidized forming a waste productlactic acid by anaerobic metabolism Whether energy metabolism is aerobic oranaerobic amino acids can never be completely oxidized Rather they are partially oxi-dized and form nitrogenous waste products urea and creatinine that are excreted by thekidneys
111 Energy Generation
Suppose we burn a nutrient with O2 in a closed vessel at an initial temperature of 37 C and wemeasure the heat that must be removed to reach a final temperature of 37 C According tothermodynamics the heat extracted from this calorimeter is identical to the total energyextracted as heat and work when the same reaction is carried out in a person at a constantbody temperature of 37 C Thus the thermal information obtained from an inanimate cal-orimeter experiment is directly applicable to energy metabolism in a living organism eventhough the mechanisms of nutrient oxidation are quite differentThe complete combustion of glucose with a stoichiometric amount of O2 carried out in a
calorimeter at atmospheric pressure and body temperature conditions yields the followinginformation about carbohydrate metabolism
C6H12O6 + 6O2 6CO2 + 6H2O
RQ=6 6 = 1 00
ΔHr 310 K 101 kPa = minus15 6 kJ g glucose
CE= 21 0 kJ LO2
1 1-1
The respiratory quotient (RQ) defined as the molar output of CO2 relative to the molarinput of O2 (equivalent to the CO2 production volume relative to the O2 consumption vol-ume) is a direct result of the reaction stoichiometry The heat of combustion ΔHr is definedas heat that must be added per gram of glucose that is consumed Because heat must actuallybe removed from a calorimeter to maintain a fixed pressure and temperature ΔHr is a neg-ative quantity The calorific equivalent (CE) represents the value of minusΔHr relative to the vol-ume of O2 consumed Therefore the combustion of 1 g of glucose produces 156 kJ of energyburns 156210 = 0743 L of O2 and produces 0743(100) = 0743 L of CO2Although many triglycerides participate in energy metabolism we can model these reac-
tions by focusing on a single triglyceride with a relative number of carbonndashhydrogenndashoxygen
11 Cell Energy Related to Whole-Body Function |5
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
atoms similar to most other triglycerides Calorimetric measurements of the completecombustion of triolein (C57H104O6) one such model of triglyceride result in the follow-ing data
C57H104O6 + 80O2 57CO2 + 52H2O
RQ=57 80 = 0 713
ΔHr 310 K 101 kPa = minus16 0 kJ g triolein
CE= 18 5 kJ LO2
1 1-2
We see from these values that 1 g of triolein produces 160 kJ of energy burns 160185 =0865 L of O2 and produces 0865(0713) = 0617 L of CO2 Thus on a per gram basis fatmetabolism produces slightly more energy requires substantially more O2 and produces sig-nificantly less CO2 than carbohydrate metabolismBecause of diversity in the structure of amino acids we cannot model their energy metab-
olism with a specific compound However calorimetric measurements of a typical mix offoodstuffs have established representative parameter values for the oxidation of ingested pro-teins RQ = 081 ΔHr = minus18 4kJ g protein and CE = 192 kJL O2 Although these valuesindicate that proteins are a favorable energy source the amino acids created during proteindigestion are normally more important for synthesizing new proteins than for producingenergyIndirect calorimetry is a convenient procedure for evaluating energy metabolism from the
rates at which a person excretes CO2 to and extracts O2 from the surroundings The RQ asso-ciated with this CO2ndashO2 exchange can identify the types of nutrients being metabolizedValues of CE can then be used to predict energy production and nutrient consumption ratecan be estimated from ΔHr Consider the example of a person who consumes 300 mlmin ofO2 and excretes CO2 at 300 mlmin as determined from respired gas measurements WithRQ = 300300 = 1 it is likely that carbohydrates are the primary substrates being consumedIt follows that energy is produced at (21000 JL)(0300 Lmin) = 6300 Jmin and the personmetabolizes carbohydrates at a rate of (6300 Jmin)(1440minday)(15600 Jg) = 582 gdayasymp1 lbday
112 Energy Transfer
By virtue of their high-energy phosphate groups several nucleotides act as intermediatesbetween the chemical reactions that generate energy and those that utilize energy The mostabundant of these nucleotides is adenosine triphosphate (ATP) At the fairly neutral pH con-ditions in physiological systems ATP has a valence of minus4 and a structure that is essentiallyC10H12N5O4minusPOminus
3 POminus3 POminus2
3 Two of the three terminal phosphate groups are linkedto the molecule by high-energy bonds (~) that can be broken in sequence to form adenosinediphosphate (ADP) (C10H16N5O4minusPOminus
3 POminus23 ) and adenosine monophosphate (AMP)
(C10H16N5O4minusPOminus23 ) according to the following reversible reactions
ATPminus4 +H2O ADPminus3 +HPOminus24 +H+
ADPminus3 +H2O AMPminus2 +HPOminus24 +H+
1 1-3a b
where HPOminus24 is the hydrogen phosphate ion When these reactions proceed in the forward
direction the hydrolysis of either ATP or ADP cleaves one high-energy phosphate bond
1 Biological Structure and Function6 |
