A Type 1 Diabetic Model

A Thesis

Submitted to the Faculty

of

Drexel University

by

Brian Ray Hipszer

in partial fulfillment of the

requirements for the degree of

Master of Science

September 2001

iii

Dedication

To my Laurel…

iv

Acknowledgments

I wish to thank Laurel for her love and support. Her assistance in the preparation of my

thesis was invaluable. I wish to thank Moshe for his consummate friendship and advice.

I wish to thank Jeff for his patience and guidance.

I would to acknowledge the love and devotion of my family and friends.

I would also wish to thank Dr. Onaral and the School of Biomedical Engineering,

Sciences and Health Systems for granting me the Calhoun Fellowship that continues to

sustain me during my research.

v

Table of Contents

List of Tables....................................................................................................................vii

List of Figures .................................................................................................................viii

Abstract .............................................................................................................................ix

Chapter 1. Introduction ....................................................................................................1

Chapter 2. Diabetes Mellitus ............................................................................................4

2.1. Type 1 Diabetes .........................................................................................................6

2.2. Complications of Diabetes ........................................................................................7

2.2.1. Acute Complications...........................................................................................8

2.3. Conventional Insulin Replacement Therapies .........................................................10

2.4. Social and Economic Impact of Diabetes................................................................12

Chapter 3. Diabetic Patient Model.................................................................................13

3.1. Review.....................................................................................................................13

3.1.1. “Minimal” Model of Glucose Metabolism .......................................................16

3.1.2. Three Compartment Model of Insulin Kinetics ................................................19

3.2. Model Construction.................................................................................................21

3.3. Model Equations ......................................................................................................22

3.3.1. Calculation of Glucose Subsystem Parameters.................................................26

3.3.2. Parameter Values from the Literature ...............................................................27

vi

Chapter 4. Model Validation..........................................................................................29

4.1. Absolute Insulin Deficiency....................................................................................29

4.2. Insulin Half-Life ......................................................................................................32

4.3. Basal Insulin Infusion..............................................................................................33

4.4. Insulin Sensitivity....................................................................................................33

4.5. 100 gram Oral Glucose Tolerance Test ...................................................................37

Chapter 5. Discussion ......................................................................................................41

Chapter 6. Future Work .................................................................................................43

List of References.............................................................................................................45

Appendix A : Abbreviations ...........................................................................................48

Appendix B : Figure 81 from Sorensen’s doctoral thesis.............................................49

vii

List of Tables

Table 1: Anabolic effects of insulin.....................................................................................5

Table 2: Models of glucose- insulin dynamics ...................................................................15

Table 3: Description of variables and parameters for the minimal model of Bergman et al.

(1979).....................................................................................................................17

Table 4: Parameter values for the “minimal” model from the literature [6]. ....................19

Table 5: Description of variables and parameters for Sherwin et al. (1974) insulin kinetics

model .....................................................................................................................20

Table 6: Values for the fractional transfer rates of Sherwin et al. (1974) insulin kinetics

model .....................................................................................................................21

Table 7: Description of diabetic model variables and inputs ............................................24

Table 8: Description of diabetic model parameters and values .........................................28

Table 9: Summary of tests for diabetic model validation through simulation ..................30

Table 10: The relationship between basal insulin requirements and insulin sensitivity....35

Table 11: Type 1 diabetic model parameter values used in the 100g OGTT simulations .38

Table 12: Parameter values used in a search for "best- fit" of the model’s blood glucose

and insulin trajectories to clinical data from a 100g OGTT ..................................39

Table 13: A summary of the abbreviations used throughout the text. ...............................48

viii

List of Figures

Figure 1: Typical variations in plasma glucose concentrations in patients with type 1 and

type 2 diabetes versus non-diabetic individuals over a two day period [31]. ..........6

Figure 2: Physiological interpretation of the “minimal” model of glucose metabolism...18

Figure 3: Construction of a model of type 1 diabetes........................................................23

Figure 4: Simulation of an absolute insulin deficiency (onset of diabetic ketoacidosis) ..32

Figure 5: The drop in blood glucose in response to an injection of 1U of insulin for given

basal insulin infusion rates.....................................................................................36

Figure 6: The performance of a type 1 diabetic model to simulate blood glucose

concentration (upper) and plasma insulin concentration (lower) for a 100g OGTT

...............................................................................................................................40

ix

Abstract A Type 1 Diabetic Model

Brian Hipszer Moshe Kam, Ph.D. and Jeffrey Joseph, D.O.

Diabetes mellitus is a chronic metabolic disease characterized by the uncoupling of blood

glucose levels and insulin secretion, causing abnormal glycemic excursions. Type 1

diabetes is characterized by a complete insulin deficiency. Treatment consists of

attempting to match exogenous insulin delivery to the metabolic needs of the patient

during meals, exercise, and sleep. The objective is to minimize the frequency and severity

of glycemic excursions and, more importantly, avoid hypoglycemic episodes.

In this study, we present a model of glucose and insulin dynamics aimed at

assisting in the management of type 1 diabetes. This nonlinear model is a system of four

differential equations that estimate glycemic changes in response to intravenous insulin

infusion and glucose absorption from the gut. The model represents the integration of two

existing models proposed earlier by Bergman et al. (1979) and Sherwin et al. (1974).

Model parameter values were determined from data in the literature, and

optimized through performance analysis in simulation. The resulting model appears to

correspond to physiological values (e.g., pharmakokinetic insulin half- life and basal

insulin requirements). In computer simulations, the model was capable of describing the

time course of glucose and insulin in blood during an oral glucose tolerance test.

1

Chapter 1. Introduction

Diabetes mellitus is a chronic metabolic disease characterized by the inability to maintain

blood glucose concentrations within physiological limits. The study described in this

thesis will focus on type 1 diabetes, a particular manifestation of diabetes mellitus.

Type 1 diabetes is characterized by a loss of pancreatic beta-cell (B-cell) function and an

absolute insulin deficiency. Since insulin is the primary anabolic hormone that regulates

blood glucose level, type 1 diabetics require a continuous supply of insulin for survival.

Conventional therapy for type 1 diabetics involves insulin replacement through multiple

daily injections (MDI) or a continuous subcutaneous insulin infusion (CSII) guided by

daily blood glucose measurements.

The complications of diabetes necessitate modes of treatment that minimize the

frequency and severity of glycemic excursions. The importance of intensive management

of glycemia was demonstrated in the Diabetes Control and Complications Trial (DCCT).

The risks and benefits of ‘tight’ glycemic control (narrow target range for blood glucose)

achieved through an aggressive insulin replacement regimen and diligent blood glucose

monitoring were evaluated in a large-scale study. Diabetics who achieved ‘tight’

glycemic control were able to reduce significantly their risk of developing the chronic

complications associated with the disease [2]. However, even in the controlled

environment of this clinical study – with the most advanced technology, patient

2

education, and professional assistance available – only a small percentage of the

participants could successfully maintain ‘tight’ control for an extended period of time.

The problem with conventional diabetes management is that insulin therapy

remains “open loop”. For example, in CSII therapy, predefined basal insulin infusion

rates are supplemented with patient- initiated adjustment and pre-meal boluses in an

attempt to achieve or maintain normal glycemia. The goal of this study is to identify a

model of type 1 diabetes, which can aid in the development of a “closed loop” system

where there is automatic feedback control of insulin infusion from a glucose sensor

(‘artificial endocrine pancreas’) [4, 23, 30, 38, 39].

The long-term goal of this research is the development of an optimal controller

that will use blood glucose measurements and past insulin infusion rates to compute the

required insulin dose. The model of glucose metabolism and insulin kinetics described

here is expected to aid in this endeavor. The model estimates glycemic behavior in a

type 1 diabetic. It will be integrated into a model-predictive control scheme to manage

glycemia.

The model described in this thesis combines two existing models originally

investigated in the 1970s. They are: a one-compartment model of glucose dynamics

developed by Bergman et al. [7]; and a three-compartment model of insulin dynamics

developed by Sherwin et al. [37].

In Chapter 2, normal glucose metabolism and the pathophysiology of type 1

diabetes is described. Conventional therapies for type 1 diabetes and the social impact of

the disease are discussed as well. In Chapter 3, a model for type 1 diabetes is introduced.

3

In Chapter 4, simulation results and analysis to validate the accuracy and realism of the

model are presented. Conclusions and future work are discussed in Chapter 5 and Chapter

6, respectively.

4

Chapter 2. Diabetes Mellitus

Glucose is one of the body’s main sources of energy. In normal physiology, the body

maintains blood glucose levels within a narrow range (70-130mg/dl). Blood glucose is

balanced between endogenous appearance from the liver (through glycogenolysis and

gluconeogenesis) and kidneys, exogenous appearance from the intestines (following a

meal), and utilization of glucose by all tissues. Two gross metabolic conditions exist.

When fasting, the body relies primarily on glucose stored in the form of glycogen and

fatty acids stored in the form of triglycerides to fuel its metabolic needs. After a meal,

glucose absorbed from the gut is used to replenish glycogen and fat stores diminished

while fasting.

The body regulates the processes that control the production and storage of

glucose by secreting the endocrine hormone, insulin, from the pancreatic B-cells. Insulin

facilitates anabolic metabolism throughout the body (Table 1). An increase in insulin

above basal concentrations (2-12 mU/l) will decrease the release of glucose from the liver

and increase glucose uptake into insulin-receptive tissues. This has the net effect of

decreasing endogenous blood glucose appearance [31]. There are many substances in the

body that promote and inhibit insulin secretion, refining the detail to which the B-cells

react to changes in the body’s metabolic state. Glucose is by far the dominant stimulus

for insulin secretion, establishing a direct relationship between insulin secretion and the

blood glucose level in the body. When glucose concentrations increase, insulin

5

concentrations will increase as well – a classical negative feedback system that keeps

glycemia within very narrow range.

In diabetes, there is an uncoupling of blood glucose levels and the concentration

of insulin that prevents the proper regulation of glycemia (Figure 1). Instead of a narrow

glycemic range, blood glucose deviations can extend from hypoglycemia (less than

60 mg/dl) into hyperglycemia (fasting blood glucose greater than 126 mg/dl, post-

prandial blood glucose greater than 200 mg/dl). This can be the result of a complete

insulin deficiency, which is classified as insulin-dependent diabetes mellitus (type 1

diabetes). However, the predominant form of diabetes is non-insulin-dependent diabetes

mellitus (type 2 diabetes). Those afflicted with type 2 diabetes are commonly overweight

Table 1: Anabolic effects of insulin

Tissue Anabolic Effects

Liver Glucose uptake and storage increased - glycogen synthesis increased - gluconeogenesis decreased

Muscle Protein synthesis increased - amino acid transport increased - protein synthesis increased Glycogen synthesis increased - glucose transport increased - glycogen synthase activity increased - phosphorylase activity decreased

Adipose Triglyceride storage increased - lipoprotein lipase activated - triglyceride hydrolysis increased - glucose transport increased - intracellular lipase inhibited

6

with a sedentary lifestyle. An abnormally high resistance to insulin causes sustained

hyperglycemia, especially following meals. A third class of diabetes, gestational diabetes,

presents itself during pregnancy and is a health concern for the mother and the

developing fetus.

8 1612 20 24 4 8 1612 20 24 4

400

300

200

100

0

time [hrs]

bloo

d gl

ucos

e [m

g/dl

]

non-diabetictype 1 diabetictype 2 diabetic

400

300

200

100

0

type

1 d

iabe

ticty

pe 2

dia

betic

non-

diab

etic

rang

e of

gly

cem

icex

curs

ions

[mg/

dl]

Figure 1: Typical variations in plasma glucose concentrations in patients with type 1 and type 2 diabetes versus non-diabetic individuals over a two day period [31].

2.1. Type 1 Diabetes

Type 1 diabetes is characterized by absolute insulin deficiency. Insulin replacement is

required for survival. The etiology of type 1 diabetes is complex. Environmental factors

coupled with a genetic predisposition for diabetes trigger an autoimmune response that

results in destruction of the pancreatic B-cells over several years [31]. Significant

glycemic excursions are common occurrences. Patients rely on physician advice, glucose

measurements, and intuition to determine the proper insulin dose.

7

There are close to one million children and adults with type 1 diabetes in the

Unites States [1]. Since there is no cure, a diabetic is required to cope with the physical

and psychological consequences of the disease for the balance of his/her life. In a study

of 95 diabetic children age 8-13 years, 36% had developed sufficient psychological stress

to be considered a diagnosable mental disorder [24]. Acute complications are almost a

daily occurrence. The number of factors that a diabetic must consider (in addition to

many unknown variables) makes optimal treatment very difficult to achieve.

Hypoglycemic and hyperglycemic excursions alter mood and disrupt the lifestyle of the

patient as well as family, friends, and coworkers. The risk of developing a chronic

complication (retinopathy, neuropathy and nephropathy) has been shown to correlate

closely with the length of time an individual has been diabetic and the long-term degree

of glycemic control. The severity and frequency of hypoglycemic episodes tends to

increase as patients practice intensive therapeutic regiments (‘tight’ control).

2.2. Complications of Diabetes

Diabetic patients are predisposed to many problems associated with their disease. These

complications are classified as acute (quick manifestation and correctable) or chronic

(taking years or decades to develop). Complications may result in hospitalizations,

permanent disabilities, and death. Acute complications include hypoglycemia,

hyperglycemia, and diabetic ketoacidosis (DKA). Macrovascular (atherosclerosis) and

microvascular (nephropathy, neuropathy, and retinopathy) diseases are the most common

chronic complications of diabetes.

8

2.2.1. Acute Complications

Acute complications result primarily from improper insulin replacement therapy.

Hypoglycemia and hyperglycemia may be a daily occurrence for a type 1 diabetic.

Clinical hypoglycemia is defined as blood glucose levels below 60 mg/dl. Excessive

insulin, insufficient or delayed carbohydrate intake, and extreme physical exertion may

cause hypoglycemia.

Hypoglycemia may result when an excessive amount of insulin has been

administered. This can be from a misjudgment by the patient in determining the correct

insulin dose. Changes in eating habits – the time, quantity, and content of a meal – can

predispose a patient to hypoglycemia. Reduced or delayed carbohydrate intake (caused

by diabetic gastroparesis, celiac disease, or chronic pancreatitis) can lead to a mismatch

between insulin bioavailability and glucose absorption from the gut. Fluctuations in

insulin bioavailability and insulin sensitivity may also cause hypoglycemia. Exercise

increases peripheral blood flow, which increases insulin absorption from subcutaneous

injection sites (increased insulin bioavailability). Exercise will also enhance insulin

action, decreasing a patient’s daily insulin requirements (increased insulin sensitivity). If

not accounted for in the determination of insulin dose, increases in either insulin

bioavailability or insulin sensitivity will lead to hypoglycemia.

In contrast to hypoglycemia, hyperglycemia results when there is insufficient

insulin available to maintain blood glucose concentrations below 130 mg/dl. Many of the

factors precipitating hypoglycemia are also involved in hyperglycemia (e.g., insulin

therapy mismanagement, and fluctuations in eating habits and exercise routine). If

insufficient insulin is prescribed, glucose appearance in the blood will outpace glucose

9

uptake by insulin-sensitive peripheral tissues. Poor patient health or lack of exercise will

decrease insulin sensitivity (causing greater insulin requirements). A patient’s insulin

regimen must be acutely adjusted to prevent hyperglycemia.

The symptoms of hyperglycemia include polyuria (passage of large amounts of

urine), polydipsia (excessive fluid intake), polyphagia (excessive hunger), weight loss,

and fatigue. Above a certain threshold (180 mg/dl), glucose cannot be fully recovered

during the filtration process in the kidneys and spills into the urine. An obligate diuresis

occurs as water follows glucose leading to an increased frequency of urination (polyuria).

This leads to an increased thirst and fluid intake as the body attempts to replenish lost

fluids (polydipsia). Dehydration, electrolyte imbalance, and the non-physiological

accumulation of certain metabolites cause fatigue.

Diabetic ketoacidosis (DKA) is a prominent cause of morbidity and mortality in

the diabetic population [1]. DKA is typified by an insulin deficiency and elevated

concentrations of catabolic hormones. Hyperglycemia and ketogenesis cause dehydration

and acidosis. This leads to further release of catecholamines and cortisol. Tissue

sensitivity to insulin is decreased; glucose production in the liver and kidneys is

increased; and blood glucose levels increase further. What separates DKA from

hyperglycemia and hyperosmolar non-ketotic syndrome is the accumulation of ketone

bodies in the blood. The combination of hypoinsulinemia and excess catabolic hormone

promote lipolysis. In the liver, free fatty acids fuel ketogenesis, which results in the

formation of ketone bodies (acetoacetate, acetone and 3-hydroxybutyrate). Ketone bodies

are moderately strong organic acids and their accumulation results in ketoacidosis.

10

The symptoms of hyperglycemia are also present in DKA. In addition, the

increase in catecholamines is evident through hypotension (peripheral vasodilatation) and

tachycardia. The respiratory rate increases as the body attempts to eliminate acetone and

excessive carbon dioxide through the lungs.

2.3. Conventional Insulin Replacement Therapies

Insulin replacement therapy is necessary for the survival of patients with type 1 diabetes.

In order to determine and administer the proper insulin dose, an otherwise healthy type 1

diabetic may choose from a myriad of glucose measurement and insulin delivery devices.

Commercial blood glucose monitors measure the glucose oxidase reaction to determine

the concentration of glucose in whole blood. Capillary blood samples are commonly

obtained by lancing the fingertip. Glucose monitors differ primarily in size, measurement

time, and functionality (e.g., memory and computer connectivity). Insulin delivery

devices are more diverse. Single-use syringes, multi-use insulin pens, needle- less micro-

jet devices, external portable pumps, and implantable pumps are all currently available

[27, 32]. However, in conventional therapy, the final determination of the insulin dosage

rests upon the patient.

Insulin preparations have evolved since the discovery of insulin by Banting and

Best in 1921. Currently, human insulin is mass-produced using recombinant DNA

technology, which has reduced the severity and frequency of immune responses

(compared to prior insulin preparations obtained from animals). Through amino acid

manipulation or the addition of a buffering solution, insulin preparations with various

11

characteristics (onset time and duration) have been developed for use in subcutaneous

insulin injection therapies.

In order to achieve ‘tight’ glycemic control, a diabetic’s therapeutic regimen must

also focus on proper diet and exercise. Carbohydrate counting (e.g., 1U of short-acting

insulin will cover a meal consisting of 15 grams of carbohydrates) and a sliding scale

insulin delivery method (e.g., 1U of short-acting insulin will lower blood glucose by

50mg/dl) have been used to aid diabetics in the determination of the proper insulin dose

[19]. These generalizations are modified, based upon an individual patient’s insulin

sensitivity. A diet with a low fat content is a precautionary measure, due to the increased

risk of atheriosclerosis and heart disease [28]. Reducing the amount of dietary starch and

simple sugars in a meal will limit post-prandial glycemic excursions (i.e., subcutaneous

insulin delivery cannot mimic the profile or portal route of endogenous insulin that

allows healthy individuals to maintain euglycemia when faced with large glucose loads).

Exercise normalizes insulin sensitivity and reduces insulin resistance, affording diabetics

the knowledge that the administered insulin will respond more consistently.

The most important strategy in the prevention of diabetic complications is the

implementation of frequent blood glucose monitoring coupled with a well-planned

regimen of insulin delivery. The degree of success noted in the DCCT trial in long-term

glucose control translated into a significant reduction (50-70%) in the clinical incidence

and progression of retinopathy, nephropathy, and peripheral neuropathy [2].

Unfortunately, the goals of ‘tight’ long-term glucose control (fasting blood glucose less

than 126mg/dl, 2hr. post-prandial blood glucose less than 200mg/dl, Hg A1c less than

12

7.0%, no hypoglycemia) cannot be achieved in the majority of type 1 diabetics despite

frequent blood glucose monitoring and MDI or CSII. In fact, only 25% of the type 1

diabetic patients practicing ‘tight’ control in the DCCT were able to achieve normal

Hg A1c levels for a period of one year. However, this group experienced a three-fold

increase in the incidence of hypoglycemia, demonstrating the contradictory requirements

in type 1 diabetes therapy.

2.4. Social and Economic Impact of Diabetes

An estimated 16 million Americans – or 5.9% of the U.S. population – have diabetes.

Close to 1 million Americans have type 1 diabetes and require insulin for survival. The

economic impact of diabetes is staggering. Estimated direct and indirect costs of diabetes

care approached $98 billion in 1997 in the US [5]. Long-term normalization of blood

glucose levels has been shown to decrease significantly the risk for the development and

progression of the microvascular complications of the disease [2]. The data that long-

term tight glucose control prevents the development and progression of macrovascular

disease (coronary artery disease, cerebral vascular disease, and peripheral vascular

disease) is not as strong [41]. Despite improvements in home glucose monitoring, human

insulin preparations, and insulin delivery techniques, diabetes continues to be the most

significant risk factor for the development of blindness, kidney failure, and neuropathy

[1].

13

Chapter 3. Diabetic Patient Model

A good diabetic patient model would accurately predict blood glucose levels using

current blood glucose measurements and exogenous insulin infusion rate data. The

amount of infused insulin will be transformed into a change in glycemia, which requires a

model of glucose metabolism and insulin kinetics. Unfortunately, insulin-glucose

dynamics are nonlinear and of high order. The challenge is to employ a model that is

“simple, but not too simple.” In this chapter, a diabetic model of a mathematically

tractable size (fourth order) is introduced.

In this chapter we briefly review existing models of glucose metabolism and

propose a new type 1 diabetic model that combines two existing models. The model is in

the form of a system of differential equations and its accuracy depends on a host of

parameters whose values were collected from the literature. In Chapter 4, we present

simulation results of the model in conditions of insulin deficiency, fasting, and exogenous

glucose load.

3.1. Review

Models have been exploited for understanding, prediction, and control. Diabetic models

are capable of estimating the time-course of blood glucose levels in diabetics. Some

models have been used to create educational diabetes software such as AIDA (Automated

Insulin Dosage Advisor) [25]. Diabetic models have been used to characterize patient

14

physiology such as classification of insulin sensitivity through the identification of the

values of key parameters [6, 7, 9]. Finally, models of glycemia in diabetic patients have

been and continue to be used to develop control schemes for automated determination of

insulin dosing from glucose data measurements [16, 23, 29, 38, 39].

Most diabetic models have a compartmental structure. They describe the flow of

material (e.g., insulin and glucose) between biological tissues (e.g., liver and muscle).

Models have been often characterized in terms of their realism, precision, and generality,

mathematical tractability, robustness, and complexity.

Since type 1 diabetics rely solely on exogenous insulin to manage their glycemic

levels, there is no endogenous source of the hormone to model (literature that focused on

modeling insulin secretion was not used in this study). The study of insulin kinetics in

type 1 diabetes reduces to the identification of those tissues that bind and/or degrade

insulin, and the identification of the appropriate route of insulin administration. The main

targets of insulin action and degradation are the liver, muscle and adipose tissues. A small

portion of plasma insulin is degraded or eliminated unchanged by the kidneys. The

movement of insulin between (and the elimination of insulin from) these biological

tissues is reflected in the compartmental nature of the models.

Sorensen's doctoral thesis (1985) provides an excellent review of glucose

metabolism models [40]. Table 2 summarizes the models reviewed by Sorensen as well

as the model proposed in his doctoral thesis. The table provides a description for each

model and its reference(s). The table does not include the “minimal” model, which is

explained in detail in section 3.1.1.

15

Table 2: Models of glucose- insulin dynamics

Description Reference

A 22nd order nonlinear system of equations simulating glucose, insulin, and glucagon concentrations in normal and diabetic man (with modifications to model structure). The body was divided into a number of physiological compartments and mass balance equations were written for each compartment.

22, 40

Linear two-compartment model of the glucose/insulin relationship in nondiabetic man with a single compartment representing (plasma) glucose kinetics. Acknowledged simplifications to maintain linearity prevented the addition processes such as renal excretion of glucose and limited model precision.

3, 9, 20, 34

Model of glucose metabolism with multihormonal (insulin and glucagon) controls and nonlinear auxiliary equations in nondiabetic man. Three differential equations represented changes of plasma glucose, insulin, and glucagon concentrations; and hyperbolic tangent functions determining the appearance rates for these quantities.

12

Multi-compartmental (plasma, muscle and liver) model of glucose kinetics with multiple hormonal controls (insulin, glucagon and free fatty acids). 6 differential equations representing these quantities were augmented by auxiliary equations, which introduced nonlinearities into the system.

17, 18

A single differential equation of glucose dynamics modified by rates of exogenous glucose input, renal excretion, hepatic production, and peripheral tissue uptake. Three additional differential equations simulate insulin kinetics in the plasma, interstitial fluid and an unidentified compartment (similar to [37]).

11

The model is a multi-compartment structure of glucose kinetics. The liver was divided into two separate compartments to explicitly model the storage of glucose as glycogen. Insulin, glucagon and adrenalin were modeled as effectors of glucose metabolism.

15

The system of five differential equations describes the behavior of glucose, insulin, and glucagon. Three of the equations represent insulin kinetics using a model developed by [37]. Nonlinear auxiliary functions determined the net hepatic glucose balance, renal glucose threshold, and peripheral glucose utilization.

13, 14

16

3.1.1. “Minimal” Model of Glucose Metabolism

Bergman et al. (1979) identified a model that could estimate glycemia for the purpose of

quantifying an individual’s insulin sensitivity. In their study, seven mathematical models

were investigated. In each model, glucose was represented as an absolute value and

insulin as the deviation from the preinjection basal value. The disappearance of glucose

via renal excretion was not included.

Thirteen sets of glucose and insulin data were collected in IV glucose tolerance

tests on mongrel dogs. The experimental protocol consisted of a single injection of

glucose in the jugular vein following an overnight fast. After each glucose dose was

administered (either 100, 200, or 300 mg/kg), blood was sampled and assayed for glucose

and insulin. Nine datasets collected from two animals were used for model parameter

identification using a nonlinear recursive estimation procedure. The remaining four

datasets from three additional animals were used for subsequent analysis. For the

purposes of parameter estimation, it was assumed that insulin and glucose returned to

basal values 120 minutes after the IV glucose tolerance test began.

Each model was evaluated in terms of parameter identifiability, meaning of

parameter values, and goodness-of- fit. Comparisons between the proposed structures

identified one model that was unequivocally superior to the others. Given that more

simplistic models could not accurately estimate the clinical data, the selected model was

deemed “minimal” (i.e., it was the least complex structure needed to describe glucose

dynamics).

17

Figure 2 illustrates the physiological basis for the formulation of the minimal

model. Plasma insulin I(t) enters a “remote” compartment where it stimulates the uptake

of glucose by insulin sensitive tissues, hence its removal from the plasma. The variable

X(t) is proportional to the insulin in this remote compartment and the authors suggest that

it represents the receptor pool for insulin in the periphery. Table 3 summarizes the

variables, input, and parameters of the minimal model. The mathematical equations are

given in (1) and (2).

[ ] bdtd GptGtXptG 11 )()()( ++−= (1)

)()()( 32 tIptXptXdtd +−= (2)

Bergman et al. (1979) also defined the insulin sensitivity index SI as the ratio

p3/p2, which quantified the fraction of glucose removed from the blood per insulin

concentration unit (min-1/µU/ml). The identification of the minimal model structure

Table 3: Description of variables and parameters for the minimal model of Bergman et al. (1979).

Symbol Description Units

G(t) Plasma glucose concentration mg/dl

X(t) Insulin-dependent fractional transfer rate min-1

I(t) Plasma insulin concentration above basal value. µU/ml

p1, p2 Fractional transfer rates min-1

p3 Fraction transfer rate and conversion factor min-2/µU/ml

Gb Basal glucose concentration mg/dl

Ib Basal insulin concentration µU/ml

18

provided a means to standardize the measurement of an individual’s insulin sensitivity.

The glucose and insulin data collected in an IV glucose tolerance test was used to

estimate the parameter values – thus obtaining the values needed to compute SI. In a later

study, this technique was employed to calculate the insulin sensitivity in 18 non-diabetic

subjects [6]. This study estimated the model parameter values for each subject. For the

current study, the mean and standard deviation for the parameters (Table 4) was

computed for those subjects with average (lean) body-to-weight index. Data from obese

subjects was discarded to limit the model description to lean (and hence, nominally

insulin sensitive) type 1 diabetics.

Plasma Insulin

Insulin in acompartment"remote" from

plasma

Plasmaglucose

PeripheralTissueLiver

IrreversibleInsulin Loss

net h

epat

icgl

ucos

e ba

lanc

e

gluc

ose

upta

ke in

to th

epe

riph

ery

Figure 2: Physiological interpretation of the “minimal” model of glucose metabolism

19

Table 4: Parameter values for the “minimal” model from the literature [6]. 2

1 10×p 22 10×p 6

3 10×p Ib Gb

2.74 ± 1.19 2.49 ± 1.43 10.46 ± 5.94 9.13 ± 5.0 93.75 ± 4.50

3.1.2. Three Compartment Model of Insulin Kinetics

The model developed by Sherwin et al. (1974) is based on the evidence that systemic

insulin disappearance is multi-exponential [37]. The delivery of insulin into the system

IDR(t) represents the amount of insulin delivery to the blood plasma after its first pass

through the liver. Continued hepatic extraction in subsequent passes through the liver is

reflected in the irreversible loss constant, L01. Insulin loss occurs from compartment 1.

Its volume of distribution (45 ± 3 ml/kg) is similar to plasma volume. Compartment 2

has a volume of 17 ± 6 ml/kg and compartment 3 has a volume of 95 ± 8 ml/kg. The

authors hypothesized that compartment 2 represents the blood-perfused organs such as

the kidneys, heart, brain, and gut, and compartment 3 represents muscle and adipose

tissue.

If the hypothesized representations of these volumes are correct, the model

ignores the physiological role of the skeletal muscle and adipose tissue in the removal of

insulin from circulation by assuming irreversible losses from compartment 1 only. The

study had an interesting observation. The profile of glucose (infused to maintain normal

glycemia) was proportional to the time-course of insulin in compartment 3 – similar to

(and preceding) the conclusions of Bergman et al. (1979).

20

Data were collected through euglycemic-clamp studies on healthy human

subjects. Insulin was introduced into the subject by primed step infusions (insulin

injection followed by constant rate insulin infusion) and single injections. Plasma insulin

and glucose concentrations were measured. Glucose was infused to maintain a normal

fasting blood glucose concentration. It was assumed that endogenous insulin secretion

remained at basal levels during the study using the technique of glucose clamping.

Insulin behavior was modeled using linear three-compartment kinetics. It was

hypothesized that insulin removal is a linear process. If nonlinearities existed, they would

appear as discrepancies in the fit of the model to data collected from a variety of

individuals. Table 5 summarizes the model variables, input, and parameters.

Table 5: Description of variables and parameters for Sherwin et al. (1974) insulin kinetics model

Symbol Description Units

G(t) Plasma glucose concentration at time t mg/dl

i1(t) Insulin mass in the blood mU

i2(t) Insulin mass that is in fast equilibrium with compartment 1 mU

i3(t) Insulin mass that is in slow equilibrium with compartment 1 mU

IDR(t) Intravenous insulin delivery rate after its first pass through the liver mU/min

L01 Insulin elimination constant from compartment 1 min-1

Lij Transfer rates to compartment i from compartment j min-1

21

The equations for insulin kinetics model are

[ ] )()()()()( 31321213121011 tIDRtiLtiLtiLLLtidtd +++++−= (3)

)()()( 2121212 tiLtiLtidtd −= (4)

)()()( 3131313 tiLtiLtidtd −= . (5)

The mean and standard deviation for the parameter values in (3)-(5) are given in

Table 6.

Table 6: Values for the fractional transfer rates of Sherwin et al. (1974) insulin kinetics model

L01 L12 L13 L21 L31 0.251 ± 0.041 0.394 ± 0.055 0.021 ± 0.007 0.142 ± 0.030 0.042 ± 0.011

3.2. Model Construction

A type 1 diabetic patient model has been constructed by integrating two models that were

investigated previously and published separately. These are (i) the nonlinear minimal

model of glucose metabolism identified by Bergman et al. [7] and (ii) the linear 3rd order

model of insulin kinetics developed by Sherwin et al. [37]. They will be referred to as the

glucose subsystem and insulin subsystem, respectively. The combined model estimates

the behavior of blood glucose in response to exogenous IV insulin infusion and a meal

disturbance.

As depicted in Figure 3, the two subsystems were combined by assuming the

variable X(t) from the glucose subsystem was related to the variable i3(t) in the insulin

22

subsystem. The statement that “the variable X(t) is proportional to insulin in the remote

compartment… where it is active in accelerating glucose disappearance” by Bergman et

al. [6] is very similar to the statement that “insulin in compartment 3 was found to

correlate remarkably with… glucose [appearance]” by Sherwin et al. [37]. As such, it

was assumed that X(t) and i3(t) are related through an affine transformation given in (6)

for the purpose of transforming the parameters of the glucose subsystem (see section

3.3.1).

βα += )( )( 3 titX (6)

Finally, modifications to the combined model structure allowed for (i) an

irreversible insulin loss from compartment 3 and (ii) the addition of glucose appearance

from the gut following a meal.

3.3. Model Equations

The combined (diabetic) model is a continuous-time 4th order nonlinear system of

equations, (7)-(10). G(t) represents blood glucose concentrations and i1(t) represents

plasma insulin mass. The state variables G(t) and in(t), n = 1…3, of the glucose and

insulin subsystems are identical to those in (1) and (3)-(5), respectively. The two model

inputs are the IV insulin infusion rate RI(t) and the rate of glucose absorption from the

gut following a meal RG(t). The convention for the model parameters is that kn, n = 1…3,

refer to the glucose subsystem parameters and an, n = 1…7, refer to the insulin subsystem

parameters. A summary of the state variables, model inputs and parameters is given in

Table 7.

23

i2(t)

i1(t)

i3(t) I(t)

plasma

X(t)

G(t)

p1GbIDR(t)

i2(t)

i1(t)

i3(t)

plasma

plasma

G(t)

plasma

i2(t)

i1(t)

i3(t)

plasma

IR(t)

G(t)

RG(t)

plasma

Selection of two models from the literature

insulin kinetics model glucose metabolism model

Combination of the models

Modifications to the combined model

L01

L21L31

L13L12

p3 p1

p2

X(t)

a7

a4a6

a5a2

a3

a1

k1

k2

k3

Figure 3: Construction of a model of type 1 diabetes

24

The model is described by the following equations.

Glucose subsystem

[ ] )()()()( 3321 tRGktGtikktGdtd +++−= (7)

Insulin subsystem

)()()( 12313 tiatiatidtd +−= (8)

)()()()()( 72534131 tRIatiatiatiatidtd +++−= (9)

)()()( 16252 tiatiatidtd +−= (10)

Table 7: Description of diabetic model variables and inputs

Symbol Description Units

G(t) Plasma glucose concentration mg/dl

i3(t) Insulin mass in a remote compartment in slow equilibrium with insulin in the blood

mU

i1(t) Insulin mass in the blood mU

i2(t) Insulin mass in a remote compartment in fast equilibrium with insulin in the blood

mU

RI(t) Intravenous insulin delivery rate mU/min

k1 Fractional transfer rate min-1

k2 Conversion factor and factional transfer rate [mU•min]-1

k3 Rate of glucose appearance mg/dl/min

a1- a6 Fractional transfer rates min-1

a7 Fractional first-pass hepatic removal of insulin %

RG(t) Rate of appearance of exogenous glucose following a meal mg/dl/min

25

Glucose appearance is described using a single compartment representing the

blood plasma. Equation (7) represents multiple processes within this compartment.

Hepatic glucose balance, peripheral tissue glucose uptake, and glucose absorption from

the gut determine the rate of change of glucose. Insulin in a remote compartment i3(t)

increases glucose utilization in both the liver and peripheral tissue. The parameter k2

modulates the effect of insulin on the rate of change of glucose. This parameter is a

measure of “insulin sensitivity,” describing the effect of peripheral insulin on blood

glucose levels.

Equations (8)-(10) represent the time-varying distribution of insulin in the body.

Parameters an, n = 1...7 determine the amount of insulin that is transferred between the

three compartments in(t), n = 1...3, and irreversible lost. Compartment 1 is identified as

the blood plasma. Compartment 2 may represent the highly perfused organs such as the

kidneys, heart, brain, and gut. Compartment 3 may represent muscle and adipose tissue.

The delivery of insulin into the systemic circulation IR(t) is multiplied by a7, which

represent the percentage of insulin that is degraded in the liver after its first pass through

the organ. Continued hepatic extraction in subsequent passes through the liver is

reflected in parameter a3. The model originally proposed by Sherwin et al. (1974) had

a1 = a4. However, if compartment 3 truly represents the skeletal muscle and adipose

tissue, there would be notable insulin degradation in this compartment. Two separate

parameters have been assigned to represent an irreversible loss of insulin from

compartment 3. As such, the fraction of insulin leaving compartment 3 should be greater

than the amount of insulin entering compartment 1 from compartment 3, or a1 > a4.

26

3.3.1. Calculation of Glucose Subsystem Parameters

To transform the parameters p1, p2, and p3 from equations (1) and (2) into the parameters

k1, k2, and k3 in equation (7), equation (1) needs to have the same form and state variables

as equation (7). To perform the transformation, the relationship in equation (12) between

the state variable I(t) from the minimal model and the state variable i1(t) from the insulin

kinetics model was used. The plasma insulin concentration about the basal concentration

I(t) can be equated to the plasma insulin mass i1(t) by equation (11), where Vi,1 is the

volume of distribution for plasma insulin (3.520 l) for i1(t) [37] and Ib is the fasting, or

basal, plasma insulin concentration (9.13 ± 5.00 µU/ml) [6]. We get

1,1b

1ml1000

lmU

µU1000)()(

iVtiItI ×××=+ (11)

b1i,1

)(1

)( ItiV

tI −= . (12)

In order to find the coefficient values α and β of the affine transformation, we

differentiate equation (6) and substitute equation (8) for d/dt i3(t).

[ ]44 344 2143421

)(

112

)(

1

121

1231

3

32

)()()()(

)()(

)()(

tIptXp

dtd

dtd

atiatXatiatXa

tiatia

titX

βααβ

αα

α

++−=+−−=

+−=

=

−

(13)

From (13), a1 and p2 must be equal. From [32] a1 = 0.021 ± 0.007 and, from [6],

p2 = 0.0249 ± 0.0143. The data collected in both studies agrees since a1 ~ p2.

27

Also, from equation (13)

.)()(

)()(

3

11

3

1

1123

pa

tipa

tI

atiatIpβα

βα

+=

+= (14)

We equate the coefficients in equation (6) and equation (14).

b1

3

2i,1

3 and Iap

aVp

−== βα . (15)

We substitute equation (6) with the coefficient values in equation (15) into equation (1)

[ ]

.)()(

)()(

)()()(

3

21

13i,12

3b

2

31

1b2

33

i,12

31

11

32132143421 k

b

kk

b

bdtd

GptGtiVap

Ipp

p

GptGIpp

tiVap

p

GptGtXptG

+

+−−=

+

−+−=

++−=

(16)

The values for kn, n = 1…3 using the values in Table 4 and Table 6 are:

. 2.567

,10074.7

,0236.0

13

5

i,12

32

b2

311

==

×==

=−=

−

bGpk

Vap

k

Ipp

pk

and (17)

3.3.2. Parameter Values from the Literature

The diabetic model in equations (7)-(10) contains 10 parameters. The value obtained

from the literature for each parameter is given in Table 8.

28

The values of the glucose subsystem parameters were determined in section 3.3.1.

The insulin subsystem parameters are directly related to the parameters of Sherwin et al.

[37] insulin kinetics model as follows:

. 12.047.07

216

125

134

3121013

312

131

±===

=++=

=

=

aLaLa

LaLLLa

La

La

(18)

Since diabetes does not alter insulin distribution or removal in the body, the model

parameter values identified in [37] have been used in the present study. The parameter

values were taken directly from the literature as the averaged fractional transfer rates.

Table 8: Description of diabetic model parameters and values

Symbol Value Description Reference Units k1 0.0236 Fractional transfer rate 6 min-1 k2 7.074e-5 Conversion factor 6 [mU•min]-1 k3 2.570 Rate of glucose appearance 6 mg/dl/min a1, a2, a3, a4,

a5, a6

0.021 0.042 0.435 0.021 0.394 0.142

Fractional transfer rates 37 min-1

a7 0.47 Fractional first-pass hepatic removal of insulin 37 %

Vg 154† Distribution volume for glucose 26 dl

Vi,1 3.52 Plasma-equivalent volume of distribution for insulin 37 l

† Plasma-equivalent volume for glucose is dependent on body weight. Value computed for a weight of 70kg. This parameter does not explicitly appear in the model equations but it is required for the calculation of exogenous glucose appearance.

29

Chapter 4. Model Validation

The proposed type 1 diabetic model is comprised of two models developed independently

under different clinical conditions. To evaluate the realism of the model, various aspects

of the model were analyzed. For example, a simulation was performed in which insulin

was not supplied to the model. The model’s glucose output was then compared to

existing physiological information about the time course and magnitude of glycemic

excursions following a prolonged and absolute insulin deficiency (early onset of diabetic

ketoacidosis). Further, the ability of the model to simulate actual blood glucose and

insulin data collected from an oral glucose tolerance test was evaluated. The model

parameters were tuned to reduce the sum of the squared error between the physiological

profiles of plasma insulin and glucose and the model’s prediction. The tests that were

conducted in order to validate the model are summarized in Table 9.

4.1. Absolute Insulin Deficiency

Without exogenous insulin, the body cannot use available glucose to meet its metabolic

needs. In the absence of insulin (and its anabolic effects), the liver will increase

production of glucose through glycogenolysis and gluconeogenesis. In parallel with

glucose production, the liver will also produce ketone bodies (through ketogenesis of

fatty acids mobilized from adipose tissue). The heart, skeletal muscle, and even portions

of the brain will rely more heavily on ketone bodies for their metabolic needs even

30

though glucose is in abundance. The acidic metabolites cause a decrease in the body’s

pH, leading to a condition known as diabetic ketoacidosis (see section 2.2.1, page 8).

It takes 6-10 hrs after insulin is initially withdrawn for blood glucose

concentrations to increase to a plateau of 250-350 mg/dl [31]. If there is no insulin

present, the insulin subsystem equations are identically zero, or in(t) = 0, n = 1…3. Given

fasting conditions, RG(t) = 0, the glucose subsystem (7) is reduced to

3DKA1DKA )()( ktGktGdtd +−= , (19)

Table 9: Summary of tests for diabetic model validation through simulation

Condition Description

Insulin Deficiency

The time-course of blood glucose was examined when no exogenous glucose or insulin was provided resulting in a complete insulin deficiency.

Insulin Half-Life

The physiological quantity for the half- life of insulin is compared to the model.

Basal Insulin Infusion Rate

The basal insulin infusion rate that would maintain normal glycemic during fasting conditions was compared to physiological data.

Insulin Sensitivity

Under initial hyperglycemic steady-state conditions, a bolus of 1U of insulin was provided to the model to examine insulin sensitivity (the drop in blood glucose per 1U of insulin).

100g OGTT Using data from [40], the model behavior was compared to actual clinical data from a 100 g oral glucose tolerance test.

31

which can be solved analytically for the initial condition G(0) = Gb, to give

)exp()( 1b1

3

1

3DKA tkG

kk

kk

tG −

−−= . (20)

From (20), we can determine estimates for k1 and k3 from physiological data. At time

1/k1, the glucose level will be one-fifth of its steady-state value, which is k3/k1

(GDKA(1/k1) = 1/5 k3/k1). Thus, choosing 8hrs as the time to reach steady-state and

300mg/dl as the steady-state value,

.min

mg/dl125.3mg/dl300)(lim

min0104.0hrmin60

hr8511

31

3DKA

11

1

=⇒==

=⇒××=

∞→

−

kkk

tG

kk

t

(21)

Figure 4 shows the trajectory of blood glucose for two simulations of the type 1 diabetic

model. One simulation was performed using the literature set of parameter values in

Table 8. In this simulation, the performance of the model is poor. Blood glucose

stabilizes at 108 mg/dl. Thus, normal glycemia is maintained without any insulin (which

conflict with physiological information). The second simulation replaced the values of k1

and k3 with those calculated in (21). The set of parameter values was “locally optimized”

for the specific condition describing absolute insulin deficiency. This parameter set is

consistent with physiological knowledge detailing the onset of diabetic ketoacidosis

(DKA).

32

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8

time [hr]

Blo

od

Glu

cose

[m

g/d

l]

literature parameter valuesinsulin deficient parameter values

Figure 4: Simulation of an absolute insulin deficiency (onset of diabetic ketoacidosis)

4.2. Insulin Half-Life

The anabolic action of insulin is the chief source of glycemic changes. Therefore, it is

important to model accurately the time-course of insulin within the body. In humans, the

half- life of insulin in the blood is 3-5 minutes and the diagnosis of diabetes does not alter

this quantity [31]. Computing the half- life of insulin as described by the type 1 diabetic

model is a task of identifying the elimination coefficient, or fractional rate of irreversible

insulin loss. The elimination of insulin from the body is related to its half- life by the

equation, t½ = ln(2)/ke, where t½ is the half- life of insulin and ke is the insulin elimination

rate. From the literature, the parameter value L01 of the insulin kinetics model is the

insulin elimination constant. Thus, ke equals L01 = 0.251 ± 0.041 min-1 and t½ is between

2.4 and 3.3 minutes, which is in agreement with physiological data.

33

4.3. Basal Insulin Infusion

Ideally, insulin replacement for type 1 diabetes would mimic the normal (non-diabetic)

insulin profile during fasting and meals. The normal pattern consists of background, or

basal, insulin with large peaks, or boluses, directly following ingestion of a meal.

Typically, basal insulin concentrations in non-diabetic individuals are between 2-12 mU/l

and post-prandial insulin concentrations peak at 60-120 mU/l [31]. To establish

physiological basal insulin concentrations, a type 1 diabetic using CSII would set his/her

predefined basal rate to 0.5-1.0 U/hr [19]. Since the bioavailability of subcutaneously

injected insulin is 100%, this corresponds to a basal IV insulin infusion rate of

8-20 mU/min.

In order to test the realism of the diabetic model, we computed the basal insulin

requirements needed to maintain blood glucose of 100 mg/dl during fasting (steady-state)

conditions. Setting the differential equations (7)-(10) equal to zero, we solved for the

basal insulin infusion rate, RI(t)=RIb.

( )b13b272

426131b Gkk

Gkaaaaaaaa

RI −−−

= (22)

The basal insulin infusion rate for the parameter values in Table 8 is RIb = 7.93 mU/min,

which agrees with physiological quantities.

4.4. Insulin Sensitivity

The sensitivity of target tissues to the actions of insulin varies with time. Resistance to

insulin may be an acute response due to elevation in the hormone levels (e.g., glucagon,

epinephrine, growth hormone, and glucocorticoids oppose the actions of insulin), a

34

chronic response due to the production of circulating antibodies, or a change in the post-

receptor actions of insulin (phosphorylation/dephosphorylation cascade) [31]. For

example, surging growth hormone levels prior to waking causes a diurnal change in

insulin sensitivity. In the treatment of type 1 diabetes using CSII, the effect of the “dawn

phenomenon” is managed by increasing the preprogrammed basal insulin rates during the

early morning hours. In addition, exercise can significantly increase insulin sensitivity.

Stress, illness, pregnancy, and drugs such as the thiazide diuretics and β-blockers (whose

use coincides with insulin replacement therapy) can decrease insulin sensitivity [31].

It is very important to understand the effect of the “insulin sensitivity” parameter

k2 on changes in blood glucose. A local search of the parameter space of k2 will better

define the range of this parameter. As a further measure of model validity, the

relationship between the basal infusion rate RIb from (22) and the drop in blood glucose

in response to the IV injection of 1U insulin was evaluated. The “1500 Rule” associates

the total daily insulin dose with the estimated glucose drop per 1U insulin in type 1

diabetics using CSII [8]. Combined with the knowledge that 50% of the total daily insulin

dose is used for basal insulin infusion [19], we were able to compute the desired relation

from physical observations taken from the literature (Table 10).

For example, suppose that patient takes 30U insulin per day (the sum of the basal

requirements over 24 hours and pre-meal boluses equals 30U insulin). Divide 1500 by

30, which equals 50. As a rule of thumb, 1U insulin would lower the patient’s blood

glucose by 50 mg/dl. Next, to compute the patient’s basal insulin infusion rate, divide 30

by 2, which equals 15. 15U insulin is used every day just to supply the basal insulin

requirements of the body. Divide 15U by 24hr, which is 0.625U/hr, and then covert to

35

mU/min by multiplying by 16.67. The basal insulin infusion rate is 10.4mU/min. The

data pair (50mg/dl, 10.4mU/min) appears in Table 10.

Table 10: The relationship between basal insulin requirements and insulin sensitivity

Total Daily Insulin Requirements [U]

Basal Infusion Rate [mU/min]

Drop in Glucose per 1U Insulin Bolus [mg/dl]

15 5.2 100 20 6.9 75 25 8.7 60 30 10.4 50 35 12.2 43 40 13.9 38 45 15.6 33 50 17.4 30 65 22.6 23

Figure 5 compares the data in Table 10 to data collected through two simulations

with the parameter set of parameter values discussed in section 4.1. In these simulations,

k2 was varied between 10-4 and 10-2 while all other parameters remained constant. One

simulation established steady-state conditions at 150 mg/dl (hyperglycemic) by

calculating the appropriate insulin infusion rate RI(t) to maintain a blood glucose value of

150 mg/dl in the absence of any disturbances (RG(t) = 0). The other simulation

established steady-state conditions at 100 mg/dl (euglycemic). A 6hr simulation was

completed for each value of k2 under both hyper- and euglycemic conditions. One hour

into the simulation, 1U of insulin was infused and the drop in blood glucose was

36

measured as the difference between the steady-state value and the lowest blood glucose

value observed in the simulation.

3.0E-04 k 2 [min-1]4.0E-04

5.0E-04

6.0E-04

7.0E-04

9.0E-04

8.0E-04

1.0E-03

0

20

40

60

80

100

120

4 8 12 16 20 24

Basal Insulin Infusion Rate [mU/min]

Dro

p in

Blo

od

Glu

cose

[m

g/d

l] clinical datasimulation - euglycemic clampsimulation - hyperglycemic clamp

Figure 5: The drop in blood glucose in response to an injection of 1U of insulin for given basal insulin infusion rates

In both simulations, 5e-5 < k2 < 1e-3 produced physiological meaningful values

for basal insulin infusion rates and drops in blood glucose for the infusion of 1U insulin.

However, both sets of simulated data were inconsistent with the “1500 Rule” dataset at

low basal insulin infusion rates. The fact that the simulated datasets are of the same

magnitude and, for the most part, they bound the “1500 Rule” data is encouraging.

37

4.5. 100 gram Oral Glucose Tolerance Test

A final measure of the validity of the type 1 diabetic model was its ability to simulate

clinical data from an oral glucose tolerance test (OGTT). An OGTT is usually performed

to verify diagnosis when diabetes is suspected but symptoms are absent, or for research

purposes. Data from an OGTT can be used to determine if an individual has an impaired

glucose tolerance, which is a prerequisite for diabetes. Variations in the protocol can

affect the outcome. Even with a standard protocol, subjects classified as having a

particular glucose tolerance are likely to fall into another category on subsequent testing.

Sorensen used clinical data from a 100g OGTT in his doctoral thesis [40]. Blood

glucose and insulin concentrations from the OGTT were used to develop rates of glucose

and insulin appearance in the blood, which are the inputs RI(t) and RG(t) in our model in

this simulation. Sorensen chose a time course for both of these rates so as to minimize

the error between the clinical data and the output of his model. The mean clinical data

and the calculated rates of glucose and insulin appearance were transcribed from a figure

in his thesis (see Appendix B). Sorensen’s model is more accurate and complex than the

type 1 diabetic model in this study. As such, his computed rates of glucose appearance

(from the gut following the ingestion of 100 grams of glucose) and insulin appearance are

assumed to be physiologically accurate. Using these rates, the ability of the type 1

diabetic model to estimate the clinical data was evaluated. These simulations used three

separate sets of parameters (see Table 11). The results were plotted on the same graph as

the clinical data (Figure 6).

In the first OGTT simulation, parameter values from the literature were used

(Table 8). These values were modified as a result of the analysis completed in sections

38

4.1-4.3 to produce a set of parameter values (i.e., the values of k1, k2, and k3 were changed

to 0.0104, 5.0e-4, and 3.125, respectively while all other parameters were equal to the

values in Table 8) that was used in a second OGTT simulation. Finally, a search over a

specified subspace of the parameter space (Table 12) yielded a third set of parameter

values that optimized model performance in the sense that it minimized the sum squared

error between the time-course of the model’s blood glucose and insulin and the clinical

data for the 100g OGTT.

Table 11: Type 1 diabetic model parameter values used in the 100g OGTT simulations

Set k1 k2 k3 a1 a2 a3 a4 a5 a6 a7

1 0.0236 7.074e-5 2.570 0.021 0.042 0.435 0.020 0.394 0.142 0.47

2 0.0104 5.000e-4 3.125 0.021 0.042 0.435 0.020 0.394 0.142 0.47

3 0.0400 1.292e-4 4.000 0.025 0.042 0.435 0.020 0.394 0.142 0.60

Set 1 – values collected from the literature (literature parameter set)

Set 2 – values derived from simulation (insulin deficient parameter set)

Set 3 – values obtained through a guided search (OGTT parameter set)

39

Table 12: Parameter values used in a search for "best- fit" of the model’s blood glucose and insulin trajectories to clinical data from a 100g OGTT

Parameter Values

k1 0.0100, 0.0175, 0.0250, 0.0325, 0.0400

k2 (0.0100, 0.0167, 0.0278, 0.0464, 0.0774, 0.1292, 0.2154, 0.3594, 0.5995, 1.0000) x 10-3

k3 1.0000, 1.7500, 2.5000, 3.2500, 4.0000

a1 0.0200, 0.0225, 0.0250, 0.0275, 0.0300

a2 0.0420

a3 0.4350

a4 0.0200

a5 0.3940

a6 0.1420

a7 0.4500, 0.4875, 0.5250, 0.5625, 0.6000

40

20

40

60

80

100

120

140

160

180

0 30 60 90 120 150 180 210 240 270 300 330 360

Time [min]

Blo

od

Glu

cose

[m

g/d

l]

0

20

40

60

80

100

120

0 30 60 90 120 150 180 210 240 270 300 330 360Time [min]

Pla

sma

Insu

lin [

mU

/l]

clinical datasimulation - OGTT parameter valuessimulation - literature parameter valuessimulation - insulin deficient parameter values

Figure 6: The performance of a type 1 diabetic model to simulate blood glucose concentration (upper) and plasma insulin concentration (lower) for a 100g OGTT

41

Chapter 5. Discussion

The type 1 diabetic model developed here appears to be realistic and accurate. It is

capable of describing the time course of glucose and insulin in blood during diabetic

ketoacidosis, infusion of 1U insulin in normal and hyperglycemic conditions (insulin

sensitivity analysis), and a 100g OGTT; and it uses physiologically meaningful parameter

values. As such, it is a good candidate to predict glucose and insulin dynamics in a type 1

diabetic.

The model is able to simulate the time course of glucose in the onset of DKA

(section 4.1, page 29). This condition was marked by an increase in the ratio of the

parameters k3/k1. From two sets of parameter values, one from the literature and the other

from the exhaustive search, the ratio was found to be k3/k1 ~ 100 mg/dl. This ratio must

increase by a factor of 2-3 to model physiologic data. Since the diabetic model is a

simplification of the physiological processes involved in glycemic regulation, these

parameter values may vary for different conditions.

In the insulin sensitivity analysis (section 4.4, page 33), the value of k2 in each of

the three sets of parameter values is physiologically accurate (Figure 5, page 36). The

value of k2 from the insulin-deficient parameter set is associated with a relatively high

insulin sensitivity – blood glucose drops 45 mg/dl and basal insulin requirements are

11.3 mU/min – while the value of k2 from the literature and optimized OGTT parameter

sets describes a lower insulin sensitivity.

42

Inspection of the plots of glucose and insulin concentrations versus time (Figure

6, page 40) demonstrates that the type 1 diabetic model can simulate the to the OGTT

clinical data. Since the standard protocol in an OGTT is to administer the glucose load in

the morning after a 12 hour fast, glycogen stores will be depleted. Therefore, there would

be increased hepatic uptake of insulin (as well as glucose) to replenish glycogen stores.

In the model, this is reflected in an increase in the value of the parameter a7. This

explanation is compatible with the exhaustive search parameter value of a7 – with

a7 = 0.60, 60% of the insulin delivered to the model is removed after its first pass through

liver (and will not be available for subsequent acceleration of glucose uptake).

43

Chapter 6. Future Work

Our long-term goal is to develop a controller that will maintain normal glycemia in

individuals with type 1 diabetes. To continue along this track, the adaptive model-

predictive control (MPC) algorithm must be developed and tested. This task will be

divided into the development of (1) a recursive parameter estimation algorithm and (2) a

MPC algorithm. The development of the parameter estimation algorithm will involve

updating the model parameters to enhance its ability to predict glycemia. The success of

the algorithm will be determined by its ability to track the glycemia using computer-

generated and clinical datasets. It will be deemed successful if the model with its updated

parameters is able to predict the next blood glucose within 10% of the actual value.

Failure will require modifications to either the recursive parameter estimation technique

or the model structure itself.

The MPC algorithm will require the incorporation of the model developed in this

study and refined in the development of the recursive parameter estimation algorithm.

The complete adaptive MPC algorithm will be evaluated in simulation. Specifically, I

will evalua te its ability to control another model (acting as the patient), namely the

diabetic model developed by Sorensen [40]. The simulation will include standard meals

and variable insulin sensitivity (modification of the Sorensen model is required). The

success of the adaptive MPC algorithm will be gauged by its rank against other published

control algorithms (e.g., [16], [23], and [29]) using the indices such as mean amplitude of

44

glycemic excursions (MAGE) [36], M-value [33, 36], low blood glucose index (LBGI)

[35], and mean indices of meal excursions (MIME).

45

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28. Malmberg K. McGuire DK. Diabetes and acute myocardial infarction: the role of insulin therapy. Am Heart J. 138(5 Pt 1):S381-6, 1999.

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Appendix A: Abbreviations

Table 13: A summary of the abbreviations used throughout the text.

Abbreviation Description

ADA American Diabetes Association

CSII Continuous Subcutaneous Insulin Infusion

DCCT Diabetes Control and Complications Trial

DKA Diabetic Ketoacidosis

Hg A1c Hemoglobin A1c

The substance of red blood cells that carries oxygen to the cells and sometimes joins with glucose. Because the glucose stays attached for the life of the cell (about 4 months), a test to measure hemoglobin A1C correlates to average blood glucose level was for that period of time.

IV Intravenous

MDI Multiple Daily Injections

MPC Model-Predictive Control

OGTT Oral Glucose Tolerance Test

VLDL Very Low Density Lipids

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Appendix B: Figure 81 from Sorensen’s doctoral thesis