Modeling Soluble Gas Exchange in the Airways and Alveoli

Annals of Biomedical Engineering,Vol. 31, pp. 1402–1422, 2003 0090-6964/2003/31~11!/1402/21/$20.00Printed in the USA. All rights reserved. Copyright © 2003 Biomedical Engineering Society

Modeling Soluble Gas Exchange in the Airways and Alveoli

JOSEPHC. ANDERSON,1 ALBERT L. BABB,1 and MICHAEL P. HLASTALA 2,3

1Department of Chemical Engineering,2Department of Medicine, and3Department of Physiology and Biophysics,University of Washington, Seattle, WA

(Received 25 April 2003; accepted 16 September 2003)

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Abstract—A mathematical model of heat, water and solubgas exchange in the airways and alveoli was used to prediclocation of soluble gas exchange in the lung. A previoupublished model of heat, water and soluble gas exchange inairwayswas improved by incorporating anatomical data onairway wall to better describe the bronchial circulation aexpanding the model to include a time varying descriptionsoluble gas concentration in the alveoli. Next, the model wvalidated using two experimental data sets from the literatu~1! ethanol expirograms and~2! the uptake of seven solublgases. Then, the model simulated the excretion of ten solgases whose blood:air partition coefficient (lb:a), a measure ofblood solubility, ranged over 5 orders of magnitude. We fouthat gases withlb:a,10 exchange almost solely in the alveoand gases withlb:a.100 exchange almost exclusively in thairways. Gases withlb:a between 10 and 100 have significainteraction with the airways and alveoli. These results suggthat the airways play a larger role in pulmonary gas exchathan previously assumed and may require a reevaluationpulmonary tests that involve exhaled samples of gaseslb:a.10. © 2003 Biomedical Engineering Society.@DOI: 10.1114/1.1630600#

Keywords—Mathematical model, Bronchial circulation, Inegas.

INTRODUCTION

Over the past five decades, the major focus of pulmnary gas exchange research has rightly been on gaschange in the alveoli. From the simple gas exchanmodel developed by Farhi10 that described the lungs aone large alveolar unit to the more complex multipinert gas elimination technique proposed by Waget al.46 that helped to quantify gas exchange heterogeity among alveolar units, the models of alveolar gexchange were developed and refined because the gmost important to life, O2 and CO2, exchange thereFrom this multitude of studies, respiratory physiologislearned that alveolar gas exchange is dependent ontilation, pulmonary perfusion, and the blood:air partitio

Address correspondence to Michael P. Hlastala, Ph.D., DivisionPulmonary and Critical Care Medicine, Box 356522, UniversityWashington, Seattle, WA 98195-6522. Electronic [email protected]

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coefficient,lb:a . Of these three parameters, the blood:partition coefficient,lb:a , is arguably the most importanbecause it determines that low blood-soluble ga(lb:a,10) must exchange in the alveoli.

Over the past thirty years, some pulmonary physiogists have also been investigating another avenue ofmonary gas exchange—the airways. However, the gathat exchange in the airways are not respiratory gasessolvent gases that are very water soluble and have ablood solubility (lb:a.1000). In this laboratory, a mathematical model describing the exchange of water, hand a soluble gas in the upper airways was develope42

and fitted with experimental temperature profiles42 andethanol data.13 Using this model, our laboratory has determined that gas exchange in the airways is similaralveolar gas exchange in that gas exchange in bothgions are highly dependent on ventilation andlb:a .However, unlike alveolar gas exchange, diffusion of gthrough the airway tissue and bronchial blood flow aalso important factors determining airway gas exchan

While the exchange mechanisms for gases at eachof the blood solubility spectrum have been closely eamined, the location and the exchange mechanismsgases residing between these two extremes (10,lb:a

,1000) has not been quantified. In this study, we toour previously published model of heat, water, asoluble gas exchange in theairways and improved thedescription of the bronchial circulation by incorporatinanatomical data on the bronchial circulation and airwwall. Then, the model was expanded to include a tivarying description of soluble gas concentration in talveoli and to allow soluble gas to enter the lungs froeither or both the inspired air or/and circulating blooTo provide accurate predictions of pulmonary gas echange, the mathematical model was validated usingsets of experimental data. The first validation usedhaled ethanol profiles measured by Georgeet al.13 Themodel was matched to these data by adjusting bronc

blood flow, Qbr . The second validation used data froa short 10 min washin of seven soluble gases wblood:air partition coefficient ranging from 90 t

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1403Gas Exchange in the Airways and Alveoli

12 000.29 Matching the model with the end-expired anmixed-expired data from that experiment was alsocomplished by adjustingQbr . Then, the validated pulmonary ~airway and alveolar! model of gas exchangwas used to predict the exchange of ten soluble ga(0.09,lb:a,2709) and thereby address the primaquestion of this study. How does the blood:air partiticoefficient affect the location of soluble gas exchangethe lung?

It was hypothesized that the location and extentsoluble gas exchange in the lungs was highly dependon the blood:air partition coefficient. Gases with a loblood:air partition coefficient exchange completely in talveoli while high blood-soluble gases exchange copletely in the airways. Intermediate blood-soluble gaexchange partially with the airways and the alveoli.

MATHEMATICAL MODEL

The original mathematical model described the simtaneous exchange of heat, water, and a soluble gasthe airways. The initial detailed model derivation avalidation with airway temperature profiles was pulished by Tsuet al.42 Validation with exhaled ethanoprofiles and further model modifications were publishby Georgeet al.13 The current model, described hernow includes the bronchial circulation, pulmonary circlation and a time-dependent alveolar compartment. Althese modifications are presented here with a briefscription of the airway wall parameters~including bron-chial circulation! and model equations~Appendix!.

Axial Structure. The lung model has a symmetricalbifurcating structure through eighteen Weibel genetions. The dimensions~lengths and diameters! for theupper respiratory tract~nasal and oral! are taken fromHanna and Scherer.16 The lower respiratory tract dimensions are taken from Weibel.47 Because the dimensionof Weibel’s model were determined at a volume of threfourths total lung capacity~4800 ml!, these dimensionswere scaled. For model simulations involving tidbreathing, the airway lengths and diameters were scby a factor ~2650/4800!1/3 to represent a functional residual capacity~FRC! of 2650 ml.21 When the modelsimulated a single exhalation maneuver, an exhalafrom TLC to RV, the lung was not scaled~i.e., airwaydimensions5 3

4 TLC!. The airway dimensions for tidabreathing or a single exhalation maneuver were scaleaccount for the size of the subject. Since the vital capity of the Weibel lung model is;5000 ml, the dimen-sions were scaled by a factor~VC/5000!1/3 where VC isvital capacity. Axially, the model is divided into 48control volumes ~discussed in the following section!

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from the mouth to the 18th generation. The alveoli arespiratory bronchioles are lumped into a single alveounit ~i.e., 481st control volume!.

Previous versions of this model13,42 only allowedsoluble gas to enter the lung from the bronchial apulmonary circulations and assumed the alveolar airalways be fully saturated with water and soluble gasbody temperature (37 °C). These two restrictions wrelaxed in this model version. First, the current modallows soluble gas to enter the lung from the inspiredfrom the circulating blood or from both the air anblood. Second, the mole fraction of soluble gas in talveolar air is allowed to vary with time as describeda mass balance on the alveolar compartment~see Appen-dix for a brief description!. This time-dependent alveolacompartment will not affect the exchange of high bloosoluble gases such as ethanol (lb:a51756) but this com-partment will be important when simulating the echange of gases withlb:a,100. Pulmonary blood flow,Qp , is assumed to be 100 ml s21 which approximatelymatches the ventilation rate for the studies modeled hThe pulmonary blood volume,Vp , is assumed to be 100ml resulting in a pulmonary capillary residence time ofs.2 The alveolar volume,VA , at FRC was assumed to b2650 ml which is characteristic of a FRC of a healtyoung man.21

Since the mole fraction of soluble gas in the alveocompartment can vary with time, the mole fractionsoluble gas in the pulmonary artery is not necessaequal to the mole fraction of soluble gas in the bronchartery. In the current model, the soluble gas mole frtion in the bronchial artery,Xa , is assumed to be equato the soluble gas mole fraction leaving the pulmonacirculation. This assumption will not affect the exchanof high blood-soluble gases such as ethanol but it willnecessary when simulating the exchange of gaseslb:a,100.

Radial Structure: General.In earlier models,13,42 the ra-dial structure of the airway control volume consistedfour compartments:~1! the airway lumen;~2! a thin layerof mucus;~3! a nonperfused tissue layer that represethe respiratory epithelium, basement membrane, andconnective tissue before reaching~4! the capillary bed ofthe bronchial circulation. This capillary bed was consered to be an infinite source or sink for heat andsoluble gas; that is, the temperature and mole fractionthe soluble gas in the bronchial circulation were costant. In a recent version of this model,6 the structure ofthis capillary bed was modified to allow the temperatuand soluble gas mole fraction to vary with time and axposition. Thus, the effects of airway perfusion on airwgas exchange could be studied.

In this current model~Fig. 1!, the airways are dividedinto six radial compartments:~1! the airway lumen,~2! a

1404 ANDERSON et al.

FIGURE 1. Model control volume. The source of soluble gas for this diagram is the blood „pulmonary venous and bronchialarterial …. Blood enters the bronchial bed compartment with mole fraction, Xa . Soluble gas is transported by secretion anddiffusion across a series of annular layers before entering the passing airstream. The ratio of gas solubility between layers isspecified by a partition coefficient, l. The shaded region in the airway lumen represents the boundary layer resistanceassociated with the air–mucus interface. The thickness of the connective tissue compartment, ø t , and the thickness of theadventitial layer, øad , represent recently measured parameters that characterize the newly incorporated bronchial circulationand adventitial layers.

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thin mucous layer,~3! a connective tissue layer,~4! thebronchial circulation,~5! the adventitial layer, and~6! thepulmonary circulation. Within a control volume, the suface area for exchange between radial layers,As , isassumed to be constant and equal top d Dz whered isthe diameter of the airway lumen andDz is the axiallength of the control volume. The soluble gas mole fration, water content, and temperature are assumed touniform within each radial layer. Gas transfer betwelayers occurs by radial diffusion and secretion. Axdiffusion of energy and mass between similar layersadjacent control volumes is neglected except for masthe airway lumen. A brief derivation of the energy anmass balances for each layer is in the Appendix.

Airway Lumen. Within this layer, the air consists oftertiary system of soluble gas, water and dry air. Maand energy are transported between control volumesbulk convection and axial diffusion for mass as describby Twedt.44 The diffusivity of soluble gas through aiwas calculated using the Wilke–Lee modification of tHirschfelder–Bird–Spotz method.41 Between the air andmucus, the heat transfer coefficient is described byempirical correlation derived by Ingenito22 and the masstransfer coefficient is calculated from the ChiltonColburn analogy.7 Near the air–mucus interface, locvapor–liquid equilibrium is assumed. This equilibriumdescribed by Raoult’s law for water and the water:

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partition coefficient,lw:a , for the soluble gas. The exchange of O2 and CO2 is assumed to have a negligibeffect on the soluble gas mole fraction.

Mucus. The mucous layer surrounds the airway lumand is treated as a binary mixture of water and dilusoluble gas. Because the mucus is;95% water,31 thephysical properties of the mucus are assumed to be eto those found in water. The mucous layer thicknevaries with local hydration and dehydration. A minimuthickness~10 mm in the trachea! is maintained by secretions from the bronchial bed. A minimum thicknessmaintained in each generation such that the volumemucus across a generation is equal to the minimum vume in the trachea.

Connective Tissue.The tissue layer is treated as a bnary mixture of dilute soluble gas and water~i.e., thephysical properties of tissue are assumed to be equivato water!. The diffusivity of soluble gas through thconnective tissue is assumed to be 33% of its diffusivin water.12 The diffusivity of soluble gas through watewas calculated using the Wilke–Chang method.50 Thesolubility of gas in tissue is described by two partitiocoefficients. For most gases, the blood:tissue partitcoefficient,lb:t , is assumed to equal one but for ethancyclopropane, halothane, ether, and acetone,lb:t is equalto 1.18, 1.63, 1.27, 1.53, and 1.38, respectively.53 The

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tissue:mucus partition coefficient,l t:m , is calculated byrelating the partition coefficient between radial layersone another:44

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The mucus:air partition coefficient is assumed toequal to the water:air partition coefficient (lm:a

5lw:a). To maintain a constant tissue volume, fluidfiltered from the bronchial bed to replace the fluid thwas lost maintaining a minimum mucous thickness. Tthickness of this layer for each airway generation wdetermined from bronchial vessel measurementssheep3 that were scaled to humans~see below in ModelDimensions for Airway Wall!.

Bronchial Capillary Bed. In the earliest versions of thimodel,13,43 the bronchial bed was described as an infinsource for heat and mass implying airway gas exchato be dominated by diffusion through the connective tsue. However, more recent experimental39,40 and theoret-ical investigations6,14 demonstrated that airway gas echange depends strongly on the tissue diffusing capaand perfusion of the airway wall. Thus, the curremodel reflects these dependencies through the devement of a bronchial bed layer that uses new measments of the bronchial circulation.3

The bronchial bed is modeled as an axial seriesannular sheets of blood that surround the tissue laThe blood entering each control volume has a solugas mole fraction,Xa , and a temperature,Tb , equal tobody temperature (37 °C). Within each control volumblood exchanges heat and mass with the tissue layfluid is filtered from the capillary bed to the connectivtissue layer exactly replacing the fluid lost from the tsue by secretion, and the blood exits the bronchialwith a soluble gas mole fraction,Xc , and temperatureTc . The blood is considered well mixed both axially anradially. Each control volume contains an individual vesel that is characterized by three parameters.

The first parameter is the total blood flow rate to tairways,Qbr(ml s21), and is defined as the sum of all othe control volume blood flow rates:

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where I is the total number of control volumes anddefined to be

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where N is the total number of compartments in thmodel, i (n) is the number of control volumes in compartmentn, and g(n) is the airway generation numbeAlthough total blood flow can be specified, local blooflow to a control volume at axial positionz is calculatedusing the following relationship:

qbr~z!5v t~z! qbrF~z!. ~4!

The variablev t(z) is the volume of tissue (cm3) sur-rounding the airway at axial positionz, qbr is the meanblood flow per unit volume of tissue, andF(z) is aweighting factor given by Bernardet al.5 These param-eters are defined by the following relationships:

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The second and third parameters are the bronccapillary surface area and capillary volume, respectiveThese two parameters and the connective tissue thness,, t , were recently measured in sheep.3 These threeparameters provide a stronger quantification of bronchcapillary diffusing capacity and residence time than pvious models of airway gas exchange.

Adventitial Tissue. The adventitial layer resides betweethe bronchial bed and the pulmonary bed where the pmonary capillary conditions are assumed to exist. Tphysical properties of this layer are assumed toequivalent to water with the exception of solubility andiffusivity ~see above in Connective Tissue!. The thick-ness of this layer was calculated by subtracting the cnective tissue thickness~see below in Model Dimensionfor Airway Wall! from the lumen-alveolar tissue thickness. The lumen-alveolar tissue thickness was calculas the sum of the airway wall tissue thickness andepithelial thickness~Table 1!. In the current model, thislayer is present for generations three and greater,intraparenchymal airways. For airway generations ththrough ten, cartilage resides in the adventitial tissueis thought to impede the diffusion of soluble gas throuthis tissue.15 As a result, the molecular diffusivity osoluble gas through this layer was assumed to be 10%the molecular diffusivity through the connective tiss~i.e., ;3% of the molecular diffusion coefficient osoluble gas through water!.


TABLE 1. Bronchial vasculature measurements and airway wall measurements from sheep andhumans used to estimate the airway wall and bronchial bed dimensions.

Analysis Best fit line Reference

Sheep measurementsEpithelial thickness y50.00148d10.0022 3

Connective tissue thickness y50.0012d10.004 3Wall tissue thickness y50.0847d10.0056 3

A Capillary cross-sectional area y50.0118d10.003 3A Wall area y50.554d 3

Human measurementsEpithelial thickness y50.0052d10.0016 11

A Wall area y50.306d10.014 30

All variable dimensions are in centimeters. The variable d is the airway diameter.

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Pulmonary Capillary Bed. The pulmonary bed layer isassumed to be an infinite source or sink of heat amass. The temperature is assumed to be constantbody temperature of 37 °C. The mole fraction of solubgas in this layer,Xpb , is assumed to be the averagethe soluble gas mole fraction entering the pulmoncirculation,Xv , and bronchial circulation,Xa ~i.e., leav-ing the pulmonary circulation!. For extremely blood-soluble gases like ethanol,Xpb'Xv . For a soluble gasvery much less soluble than ethanol~e.g., ether!, Xa

,Xpb,Xv . In the current model, this layer is presefor airway generations three and greater, namely,intraparenchymal airways.

Boundary Conditions.Boundary conditions are applieat the mouth and alveoli. During inspiration, the ambietemperature and relative humidity of the inspired air aset equal to 23 °C and 50%, respectively. The mole frtion of soluble gas in the inspirate can be set equazero to study the elimination of soluble gas from tbody or to a finite value to study the uptake of solubgas from the ambient air. For all times, the alveolar airassumed to be fully saturated with water and to havtemperature equal to a mean body core temperatur37 °C. The mole fraction of soluble gas in the alveocompartment varies with time based on a mass balaIn the respiratory tract, local vapor–liquid equilibriumassumed at the air–mucus interface. This equilibriumdescribed by Raoult’s law for water and by the mucuspartition coefficient,lm:a , for soluble gas. Between radial layers, partition coefficients are used to definesolubility of gas in one layer relative to another.

Solution of Model Equations

Mass and energy balances around each controlume produce twelve differential equations in time,t, andspace,z. The equations are solved simultaneously forfollowing twelve dependent variables: the mole fracti

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of soluble gas in the air, mucous, connective tissue, brchial bed, and adventitial tissue layers (Ya , Xm , Xt , Xc ,Xad); the temperature of the air, mucous, connectivesue, bronchial bed, and adventitial tissue layers (Ta , Tm ,Tt , Tc , Tad); the mole fraction of water in the air (Yw);and the mucous thickness (,m). The twelve differentialequations are solved numerically on a Pentium IV coputer runningDIGITAL VISUAL FORTRAN. The spatial di-mensions are solved by upstream finite difference whthe time derivatives are handled usingLSODE, an inte-grating software package developed by Hindmarsh.18 Themass balance around the alveolar compartment is igrated separately from the airway control volumes usLSODE. On inspiration, the mole fraction and temperatuof gas leaving the last control volume in generationbecome the input conditions for the alveolar compament. On exhalation, the mole fraction and temperatof gas leaving the alveolar compartment become theput conditions for generation 18.

Simulating Steady-State Gas Exchange

Before the model simulates any tidal or prolongbreathing maneuvers, the model first must reach breto-breath steady-state conditions. In general, the tempture, water concentrations, and soluble gas concentratwithin the mathematical model were brought to steastate by simulating 30 tidal breaths from FRC.43 A res-piratory rate of 12 breaths/min, a sinusoidal flow wavform, a tidal volume of 500 ml, and a vital capacity o5000 ml were used. Additionally, the inspired air temperature and relative humidity was set at 23 °C and 50respectively. Depending on the simulation, soluble genters the lung from either the inspired air or the pmonary circulation.

Model Dimensions for Airway Wall

A major improvement to the current airway gas echange model is the use of anatomical measurement

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the bronchial circulation and thickness of airway tissuFour measurements, made in sheep,3 provide a means toaccurately describe the dependence of airway gaschange to diffusion through the airway tissue and persion of the airways via the bronchial circulation. Twothese measurements, the capillary perimeter~i.e., surfacearea! and the connective tissue thickness, completspecify the diffusing capacity of the bronchial circultion. The diffusion coefficient of heat and mass in tissand the tissue solubility of a soluble gas are reasonawell known. The third measurement, capillary volumcan be calculated from the capillary cross-sectional aand provides a description of the perfusion dependeof airway gas exchange on the bronchial circulation. Tfourth measurement, the wall tissue thickness, is thetance between the airway lumen and the alveoadventitial boundary. Subtraction of the connective tissthickness from the wall tissue thickness provides a msure of the adventitial layer thickness.

The measurements of tissue thickness and capilcross-sectional area were scaled to humans usingfollowing method. Each sheep parameter was normaliby a second sheep tissue measurement. This secondsurement had two characteristics:~1! a similar function-ality with airway diameter as the first and~2! a corre-sponding measurement in humans. For example,connective tissue thickness was normalized by epithethickness in sheep~Table 1!, and the capillary crosssectional area was normalized by the airway wall areasheep ~Table 1!. To determine the connective tissuthickness and capillary cross-sectional area in humathe normalized measurements were multiplied by the ethelial thickness in humans and the airway wall areahumans, respectively~Table 1!. The bronchial capillaryvolume,vc , was calculated by multiplying the bronchiacapillary cross-sectional area by the thickness of the ctrol volume, Dz. The bronchial capillary perimeter~i.e.,surface area! was also measured in sheep but wasscaled for humans as it was found to equal 58% ofairway lumen surface area for all airway generatioThe capillary area fraction is equal to half of the ratiocapillary-to-lumen surface area,Fc50.29, as half thesurface is available for exchange with the airway lumand the other half with the pulmonary capillary.

Gas Flow From Airway Wall to Lumen

When a subject with soluble gas in their bloobreathes fresh air, soluble gas will be absorbed from

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mucus during inspiration and desorbed back to the mcus during expiration. During inspiration each airwageneration will contribute a small quantity of soluble gfrom the airway wall to the gas stream. The reveroccurs during expiration. Per airway generation or mocompartment, the molar flow of soluble gas can be maematically described by the following expressions finspiration, j insp, and expiration,j exp:

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where g is the generation number,i is the number ofcontrol volumes in the compartment,x is the controlvolume number,ky,a is the local mass transfer coefficienfor the soluble gas,Vinsp is the volumetric flow rate of airduring inspiration,Vexp is the volumetric flow rate of airduring expiration,Vee is the lung volume at end expiration, andVei is the lung volume at end inspiration. Thdifferential volume,dv, is equal to the differential timeelement,dt, multiplied by the volumetric flow rate,V.Equations ~9! and ~10! confirm that the main drivingforce for soluble gas transfer is the difference betwethe soluble gas mole fraction in the airway lumen,Ya ,and the mole fraction of soluble gas at the air–mucinterface,Ywall,a .

Determination of Gas Exchange Location

To determine where the majority of gas exchanoccurs ~i.e., airway versus alveolar!, an exchange ratio~ER! can be defined as the ratio of airway gas exchato total pulmonary gas exchange@Eq. ~10!#:

ER5net airway gas transfer

net airway gas transfer1net alveolar gas transfer. ~10!

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As a measure of airway gas exchange, the net mtransfer of inert gas from the airway was calculatedthe sum of the inert gas flow from the mucus to tairway lumen on inspiration and the inert gas flow frothe mucus to the airway lumen on expiration. Formeasure of alveolar gas exchange, the net mass traof inert gas from the alveoli was calculated as the sumthe inert gas flow from the alveolar gas to the pulmonblood on inspiration and the inert gas flow from thalveolar gas to the pulmonary blood on expiration. Topulmonary gas exchange is defined as the sum of airnet gas transfer and net transfer of inert gas fromalveoli.

MODEL VALIDATION

Ethanol Expirograms

Experimental Single Exhalation Maneuvers.The detailsof the experimental protocol can be found elsewher13

The pertinent results are presented here. Six male voteers ingested enough alcohol to achieve a blood alcoconcentration of approximately 0.09 gm% accordingthe Widmark formula.48 The mean age, weight, and vitacapacity for the single-exhalation maneuver were610 yr, 78614 kg, 54006740 ml, respectively. Follow-ing ingestion of the alcohol, the subjects waited appromately 1 h for absorption to take place. Each subjeperformed two different breathing maneuvers, repeafive times each. Maneuver A consisted of a single inlation to total lung capacity~TLC! followed by an exha-lation at a slow constant flow rate to residual volum~RV!. The second breathing maneuver, maneuver B,identical to the first except that the exhalation was arapid constant flow rate. Ethanol concentration, exhavolume, and exhaled flow rate were measured dynacally throughout the prolonged exhalation. Quiet tidbreathing was done for three minutes between manvers. The expired concentration of ethanol was normized by the concentration of ethanol in the alveolar gThen, each group of expirograms within a subject amaneuver type was condensed into a single represetive exhalation maneuver by averaging the normalizalcohol concentration in the exhaled air at intervalsone-tenth the total exhaled volume.

Simulated Single Exhalation Maneuvers.Prior to simu-lating a breathing maneuver, the airways were broughsteady-state conditions as outlined in theSimulatingSteady-State Gas Exchangeexcept the tidal volume waapproximated to be 10% of the subject’s vital capacit17

and ethanol entered the lung via the pulmonary circution. To simulate a single exhalation maneuver, the

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spired volume, expired volume, and exhalation time mbe specified. The large inspired volume was appromated to be 65% of each subject’s vital capacity.17 Theexperimentally measured expired volume for each exlation maneuver was used in the model. The exhalattime for each maneuver and subject was calculateddividing the smallest expired volume by the mean florate for each group of maneuvers. The single exhalamaneuver was simulated at a lung volume equal to thfourths TLC.

The bronchial blood flow,Qbr , was used as an adjustable parameter to optimize the fit of the model to texperimental data for each subject and maneuver.accuracy of the model prediction was evaluated usingmethod of least squares and summarized with a corrtion coefficient (R2). The best-fitQbr was determined towithin 0.1 ml s21.

Short Duration Soluble Gas Washin

Experimental Washin.Detailed results of the experimental protocol have been previously published;29 hence,only the salient results will be presented here. The mage, weight, height, and tidal volume for the four sujects were 46.563.2 (SD) yr, 60.5613.5 kg, 167.068.8 cm, and 500630 ml, respectively. The mean respiratory rate for inhalation of each soluble gas w13.061.0 min21. After inhaling room air to stabilize endogenous substance in their exhaled air, subjects inhtest air containing one of ten organic soluble gases formin and then switched to inhaling room air for 5 miThroughout the 15 min experiment, mixed exhaledwas collected for an entire minute using a Tedlar bag athen analyzed. The bags were exchanged for anoevery minute. Breath-to-breath end-exhaled gas conctrations were measured and averaged over each 2period throughout the 15 min experiment. Both tmixed exhaled and end-exhaled gas concentrations wnormalized by the inspired soluble gas concentratiAfter 8 min of washin, the mixed-exhaled and enexhaled concentrations of these soluble gases reachsteady-state plateau.

Simulated Steady-State Washin.Of the ten gases studieby Kumagai et al.,29 the steady-state uptake of sevewere modeled. They are listed with their blood:air awater:air partition coefficients in Table 2. Three solubgases were not modeled in this investigation becausewere metabolized within the airways and another gcaused throat irritation during exposure. The final 2 mof mixed-exhaled,Emix , and end-exhaled,Eend, washindata from Kumagai29 were modeled as if


TABLE 2. Literature values for blood:air and water:air partition and temperature coefficients„%Õ°C… of twelve soluble gases at 37 °C.

Solvent

Partition and Temperature Coefficient

lb:a ab:a lw:a aw:a

Ethane 0.088 25.0 0.036a 22.1a

Cyclopropane 0.36b 26.2b 0.23a 21.7a

Halothane 2.4b 23.7b 0.7b 25.0b

Ether 15.2c 23.8c 16c 24.1c

Methyl isobutyl ketone1 90d 25.0 79d 26.7d,e

Methyl propyl ketone1 150d 25.0 166d 25.7d,e

Acetone1 341f 25.5f 279f 25.5f

isoPropanol1 848g 25.0 1500g 23.2g,h

Ethanol 1756i 26.7i 2132i 26.4i

Methanol1 2709j 25.9j 3400j 22.9j

Ethylene glycol monobutyl ether1 7970k 25.0 7050k 25.0Propylene glycol monomethyl ether1 12380k 25.0 12280k 25.0

All values not referenced were assumed.1Soluble gas used in the Kumagai et al.29 study.aWilhelm.49

bLowe.32

cEger.9dSato and Nakajima.34

eYaws et al.52

fWagner.45

gKaneko et al.27

hYaws et al.51

iJones.24

jJones.26

kJohanson and Dynesius.23

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the data were in a steady state because the time courtheEmix andEend data show these two measures reacha plateau at approximately the eighth minute of expsure. Additionally, the venous concentration of inert gwas assumed to be zero because of the highlb:a of thegases used in the experiment.28 Incorporating these twoassumptions into the current model, the temperatureconcentrations within the mathematical model webrought to steady state as outlined in theSimulatingSteady-state Gas Exchangeexcept a respiratory rate o13 breaths per minute was used and a single solubleentered the lung via the inspired air. The end-exhaand mixed-exhaled concentrations of soluble gas wcalculated from the 31st simulated breath. The broncblood flow, Qbr , was used as an adjustable parametefit the model predictedEmix andEend to the experimentavalues. The accuracy of this fit was determined by cculating the percent difference between the model pdicted values and the experimental values for esoluble gas. The optimal fit and associatedQbr was es-tablished by minimizing the sum of the absolute percdifference for Emix and the absolute percent differenfor Eend. The optimal blood flow was determined twithin 0.1 ml s21 .

of

s

PREDICTING LOCATION OF PULMONARY GASEXCHANGE

While investigations have shown experimentally amathematically that low blood-soluble gases (lb:a,10)exchange largely in the alveoli and high blood-solubgases (lb:a.1000) exchange in the airways, an expemental or theoretical study has not demonstrated hintermediate blood-soluble gases (10,lb:a,1000) par-tition their exchange between the airways and alveHere, this newly expanded and validated model of pmonary gas exchange was used to predict the locaand extent of low, intermediate, and high blood-solubgas exchange in normal lungs during tidal breathing.

The model simulated the exchange of ten gases wlb:a ranging between 0.088 and 2709~Table 2! fromlungs breathing tidally. Before simulating a tidal breathe mathematical model was brought to steady stateoutlined in theSimulating Steady-State Gas Exchang.For each simulation, a single soluble gas entered the lvia the pulmonary circulation and bronchial blood flowas set at 1 ml s21. To evaluate where each gas echanged in the lung and to provide insight into tmechanisms of their exchange, three model outputs w

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examined for each soluble gas:~1! the ratio of the netairway gas transfer to total pulmonary gas transfer@Eq.~10!#; ~2! the axial distribution of soluble gas flux; an~3! the exhaled expirograms.

RESULTS

Validation–Ethanol Expirograms. The best-fit modelpredictions of the experimental ethanol expirografrom a representative subject~subject No. 5! for the slowexhalation and rapid exhalation maneuver are shownFig. 2. For this subject, the model required a bronchblood flow of 1.3 and 0.7 ml s21 to accurately predict theexpirograms for maneuver A and B, respectively. Feach maneuver and subject, the best-fit values ofQbr andR2 are summarized in Table 3. For all six subjects,

FIGURE 2. Experimental and model-predicted exhaled etha-nol profiles for maneuver A and maneuver B for a represen-tative subject „subject 5 …. Experimental data are designatedby solid circles „d… for maneuver A and open circles „s… formaneuver B. Model-predicted profiles are shown using asolid line for maneuver A and a dashed line for maneuver B.The model fit to the data was optimized by adjusting thebronchial blood flow rate, Qbr .

TABLE 3. Bronchial blood flows „ml s À1… corresponding to the

best-fit model predictions of the experimental exhaled ethanoldata collected by George et al. 13

Subject

Maneuver A Maneuver B

Qbr R2 Qbr R2

1 0.1 0.990 0.1 0.8222 0.1 0.950 0.1 0.8883 0.8 0.981 0.5 0.9794 1.5 0.992 0.6 0.9465 1.3 0.991 0.7 0.9686 1.6 0.996 1.1 0.994

Mean 0.9§ 0.983 0.5 0.933SD 0.7 0.017 0.4 0.066

§Statistical difference between maneuver A and maneuver B (p,0.05).

meanQbr and R2 were 0.9 ml s21 and 0.983 for maneu-ver A and 0.5 ml s21 and 0.933 for maneuver B. Between maneuvers, the meanR2 was statistically similarbut the meanQbr was statistically different. When bothmaneuvers were combined, the meanQbr and R2 were0.7 ml s21 and 0.958.

The axial distribution of ethanol transport from thairway wall to the lumen as predicted by the curremodel are shown in Fig. 3~a!. This transport of ethanoper generation during inspiration,j insp, and expiration,j exp, are presented for maneuver A from subject 5.positive value indicates absorption of ethanol from tmucus to the airstream. A negative value denotes destion of ethanol from the airstream to the mucus. Indpendent of maneuver, the axial distribution of ethantransport on inspiration shows a bimodal distributiowith a small peak in the trachea and larger peak at12th generation. After the 12th generation,j insp for eth-anol decreases until the alveolar region where it is neazero. Similar to the results of a previous study,6 almost100% of ethanol exchange occurs completely withinairways.

In the current model, between 50% and 70% of tethanol absorbed from the mucous layer on inspirationdeposited back to the airways on exhalation. Tabsorption–desorption phenomenon is similar to predtions made by previous airway gas echange models6,13,43 and is the mechanism responsibfor the phase III slope of the ethanol expirogram~seeDiscussion!.

The axial distribution of water and heat transport frothe airway wall to the lumen are shown in Figs. 3~b! and3~c!, respectively. Like the distribution of ethanol tranport, the distribution of water and energy transportbimodal with a large peak in the trachea and a smapeak at the sixth generation. Heating and humidificatof inspired air is nearly complete at generation 14 whalmost no transport of water and energy between the wand lumen occurs. Notice that the shape of the axdistribution of heat transport is almost identical to tdistribution of water transport. These distributions asimilar because of the highly coupled nature of these tquantities. In fact, 80% of the total energy transportedbecause of the latent heat effects of water.43

Validation–Uptake of Soluble Gas.The current modelwas able to accurately predict the values ofEmix andEend as measured by Kumagaiet al.29 by adjusting thebronchial blood flow rate. These results have been sumarized in Table 4. The blood flow for ethylene glycmonobutyl ether was removed from the analysis aswas greater than two standard deviations from the meOn average for the gases, the model overpredictedEmix

by 11% but predictedEend to less than 0.1% accuracy

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The model was unable to predict the exchange ofextreme blood-soluble gas ethylene glycol monobuether and had difficulty predictingEmix for the otherextreme blood-soluble gas propylene glycol monometether.

Predicting Pulmonary Gas Exchange Location.Becauseof the mathematical nature of our model, parametersmay be impossible to measure experimentally can

FIGURE 3. Axial distribution of ethanol „panel a …, water„panel b …, and energy „panel c … transport from the airwaywall to lumen over one inspiration „black columns … and ex-piration „gray columns … for maneuver A. Ethanol transport„panel a … has a bimodal distribution with peaks in the tra-chea and generations 10–12. The shape of the axial distribu-tion of heat transport is very similar to the distribution ofwater transport since 80% of the total energy transportedresults from latent heat effects. 43

t

calculated. The ratio of airway gas exchange to topulmonary gas exchange can be calculated to determthe location of soluble gas exchange within the lung@Eq.~10!#. The ratio of airway gas exchange to total pulmnary gas exchange as a function of the blood:air partitcoefficient has been plotted in Fig. 4. This figure deonstrates where the majority of gas exchange takes pfor a range of blood-soluble gases. Soluble gases wiblood:air partition coefficient less than ten (lb:a,10)exchange almost solely with the alveoli whereas gawith a blood:air partition coefficient greater than 10(lb:a.100) exchange almost exclusively with the aways. Gases in between these two extremes (10,lb:a

,100) exchange partially with the airways and partiawith the alveoli.

FIGURE 4. Ratio of airway gas exchange to total pulmonarygas exchange †Eq. „10…‡ is plotted vs. lb :a for ten solublegases as calculated from the airway–alveolar gas exchangemodel during tidal breathing. Greater than 95% of ethane,cyclopropane, and halothane gas exchange in the alveoli.The exchange of ether is partitioned equally between theairway and alveoli. Acetone and gases with a lb :a greaterthan acetone exchange completely in the airways.

TABLE 4. Bronchial blood flows corresponding to the best-fitmodel predictions of the steady-state uptake of seven soluble

gases. 29

Soluble gas Qbr

% DifferenceEmix

% DifferenceEend

Methyl isobutyl ketone 4.0 22.6 0.2Methyl propyl ketone 1.1 211.5 0.1Acetone 1.6 211.1 0.3isoPropanol 2.2 210.5 20.1Methanol 1.7 1.7 20.2Ethylene glycolmonobutyl ether

*

Propylene glycolmonomethyl ether

3 234.5 0.0

Mean 2.3 211.4 0.0SD 1.1 12.5 0.2

*Data point deleted from analysis as outlier (.2 SD from themean).

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Another perspective on location and extent of gexchange can be achieved by examining the axial disbution of gas transport from the airway lumen to mucModel predictions of these gas fluxes during inspiratioj insp, and expiration,j exp, are presented for cyclopropane, ether, and acetone in Figs. 5~A!, 5~B!, and 5~C!,respectively. For cyclopropane, the flux distributio

FIGURE 5. Axial distribution of gas transport during inspira-tion „black columns … and expiration „gray columns … for atidal breath of cyclopropane „panel A …, ether „panel B …, andacetone „panel C …. Each flux has been normalized by thetotal inspiratory soluble gas flux. As the blood solubility ofgas increases from cyclopropane to acetone, the distributionshifts from a sharp concentrated peak in the alveolar regionfor cyclopropane to a wider distribution that spreadsthroughout the airways for acetone.

shows a large peak in the alveoli that is an ordermagnitude larger than the next largest flux@Fig. 5~A!#.Ether exchanges both in the airways and the alveoli~Fig.4! and its flux profile illustrates this result. The largeflux of ether occurs in the alveoli; however, this fluxsimilar in size to the next largest flux occurring in geeration 18@Fig. 5~B!#. Ether has a greater spread in iflux distribution than cyclopropane. The flux profile foacetone@Fig. 5~C!# shows a bimodal distribution similato that for ethanol~Fig. 3! with a small peak in thetrachea and a large peak at generation 15. Unlike cypropane but similar to ether, all of the fluxes in thacetone distribution are within the same order of magtude. However, the spread in the acetone flux distributis larger than that for ether. In general, the results frFig. 5 show that as the blood:air partition coefficieincreases the flux profile shifts from being a concentraspike in the alveoli to a more uniform distribution ofluxes spread throughout the airways.

The model’s performance and its mechanisms ofexchange were qualitatively evaluated by examiningpirograms from tidal breathing. The expirograms of eiggases have been grouped and plotted according tolocation of exchange. Gases with (lb:a,15) exchangemostly in the alveoli and are plotted in Fig. 6~a! whilegases with (lb:a.15) exchange mainly in airways anare plotted in Fig. 6~b!. Each expirogram has three distinct phases. These phases have been labeled oncyclopropane expirogram@Fig. 6~a!#. Phase I representthe delay in the rise of soluble gas concentration asvolume of dead space is emptied. Phase II is the tration from dead space to exchange space, and phasrepresents the emptying of the exchange space.

Three trends are predicted by the model as basedthe soluble gas expirograms. First, phase I volumemains at;20% of the exhaled volume for gases wilb:a,15. For gases withlb:a.15, the phase I volumedecreases to;10% for acetone (lb:a5341) and;5%for methanol (lb:a52709) aslb:a increases. Second, thmodel predicts that the phase III slope is slightly positifor soluble gases withlb:a,15. For gases with 15,lb:a,341, the phase III slope is more positive bdecreases in value aslb:a increases abovelb:a5341.Third, the model predicts thatEend increases aslb:a

increases to 15 and thenEend decreases aslb:a increasespast 15.

DISCUSSION

Model Validation

The current model of heat, mass, and solubletransport in the airways and alveoli was fit to two setsdata by usingQbr as the free parameter. The modaccurately predicted 12 ethanol expirograms us

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mean best-fit values ofQbr andR2, 0.7 ml s21 and 0.958.The model was further optimized with a short duratiwashin of seven soluble gases. The model predictedEmix

and Eend for six of the seven gases to within 11% an0.1% of the experimental measurements and requiremeanQbr of 2.3 ml s21.

Although different, theQbr required by the model tooptimally predict the two sets of experimental data copares favorably with measured values in the literaturethe average cardiac output is assumed to be 6 l mi21

(100 ml s21), the mean bronchial blood flow for the ethanol data and the uptake data as predicted by the cumodel is approximately 0.7% and 2.3% of cardiac oput, respectively. TheQbr predicted by our current modeis within the range reported in the literature, name1%–3% of cardiac output1,8 or 0.25– 2 ml s21.4,8,33 Be-cause of this variability in the measured value ofQbr ,the bronchial blood flow rate was chosen as the adjable parameter to fit the model predicted values toexperimental data as described above. Additionally,using Qbr as the adjustable parameter for fitting the e

FIGURE 6. Expirograms for tidal breaths of soluble gas thatexchange mainly in the alveoli „panel A … and airways „panelB…. The curves in panel A show È30% dead space and end-exhaled concentrations that increase with blood solubility.Panel B curves show that both dead space and end-exhaledconcentration decrease with increasing blood solubility.

t

perimental data, the model implicitly demonstratedexperimental observation: airway gas exchange is depdent on airway perfusion.39

A best-fit Qbr was found to fit the model predictionto the ethanol expirograms for each subject and man

ver. The mean best-fitQbr across subjects for maneuveA was found to be significantly different (p50.045) than

the mean best-fitQbr for maneuver B as determined bytwo-tailed t test ~Table 3!. However within each subject

the difference inQbr between maneuvers was relative

small. To determine if this difference inQbr betweenmaneuvers resulted in decreased accuracy of predictmade by the current model, we calculated the aver

best-fit Qbr for each subject. For example, the avera

best-fit Qbr for subject 5 was 1.0 ml s21. Then, with thecurrent model, we simulated both exhalation maneuv~A and B! for each subject using a single bronchial blo

flow, the subject’s average best-fitQbr . Using these con-ditions, the current model predicted the exhalation pfiles from subjects 3–6 with slightly less accuracy~asmeasured byR2) than the best-fit predictions. The aveage change inR2 from the best-fit predictions was 0.0for subjects 3–6. For all six subjects, the meanR2 was0.978 for maneuver A and 0.925 for maneuver B. W

concluded that the difference in the meanQbr betweenmaneuvers, although statistically significant, was retively insignificant in terms of the predictive ability othe current model.

To predict the uptake of soluble gas, the modelquired an averageQbr of 2.1 ml s21 to accurately predictEend and Emix of the first five soluble gases up to anincluding methanol. Using the optimalQbr , the modelpredictedEend to within 0.1% andEmix to within 11%.For all the fitting exercises, the model always predictEend more precisely thanEmix becauseEend was moresensitive to changes inQbr . As a result, minimizing thedifference between the predicted and experimentalEend

was very important to fit the model. The model hadifficulty predicting the experimental data of the etremely soluble gases: ethylene glycol monobutyl et~EGME! and propylene glycol monomethyl ethe~PGME!. From examining the data of Kumagai, itpossible that these two gases were not in a steady sBecause their blood:air partition coefficient is 3–5 timlarger than methyl alcohol which only just reachedsteady-state plateau within 10 min, these gases may hneeded to be inhaled for a few more minutes in orderreach a steady state. Additionally, the lung volumeeach subject was not reported by Kumagai but mayimportant for predicting the exchange of these gasHlastala19 has hypothesized that the lung volume ofsubject may affect the exhaled concentrations of solugases that exchange solely in the airways. Even w

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these concerns, it is important to remember thatmodel was able to accurately predict the exchangegases with 90,lb:a,2709. This validation positionedthe model to make accurate predictions of gas exchalocation ~airway versus alveolar! within the lung.

Diffusing Capacity

Airway gas exchange has been shown to be depenon the molecular diffusion of soluble gas through tairway tissue.13,40 This diffusion dependence is best uderstood by examining the diffusing capacity of the away tissue,DL :

DL5DtFcAs

, t.

This diffusing capacity is a function of the moleculdiffusion coefficient through tissue,Dt , the surface areafor diffusion between the capillary and tissue whichrepresented by the capillary area fraction,Fc , multipliedby the airway surface area,As , and divided by the con-nective tissue thickness,, t . In previous model studies oairway gas exchange, the diffusing capacity was incopletely specified. One goal of the current study wasaccurately specify the tissue diffusing capacity andcorporate it into our model of airway gas exchange.

One factor, the molecular diffusion coefficient, dscribes how easily a molecule can diffuse throughmedium. The larger the coefficient the faster a moleccan travel through the medium. Because of the hetegeneous structure of tissue, many small molecules hlower diffusion coefficients in tissue relative to wateThe diffusion coefficient of ethanol in the respiratotissue was measured and found to be 33% of its valuwater.12 As a result, the diffusion coefficient for ethanis fairly well quantified.

Values for the other two factors, the capillary surfaarea,FcAs , and connective tissue thickness,, t , wereunknown. Recently, these values were measured in shby Andersonet al.3 Since As can be calculated easilyonly the capillary area fraction,Fc , needed to be determined. The capillary area fraction was found to be indpendent of airway diameter and equal to 0.58. Howevonly half of that, 0.29, participates in exchange with tairway lumen. The connective tissue thickness,, t , wasmeasured, scaled to simulate human airways, andfound to range from 160mm in the trachea to 30mm inthe bronchioles. Accurately specifying the diffusing cpacity is an important reason why the model was ableaccurately predict the experimental data using realivalues ofQbr .

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Adventitial Layer

A major improvement to the current model was tinclusion of experimental measurements on the airwwall and bronchial circulation. These measurementslowed the addition of an adventitial layer and an adjacpulmonary layer to the model. However, we felt thelayers should not be included at every airway generatfor the following reasons.

In the extrapulmonary airways~mouth to second generation!, the adventitial and pulmonary layers were nincluded because anatomical measurements and phological data for analogous structures in the extrapulmnary airways were unavailable. For the current moand study, the extrapulmonary airways play a significrole in the exchange of energy and water@Figs. 3~b! and3~c!#. We feel the essence of this exchange has bcaptured by the current model without the inclusionthese adventitial and pulmonary layers:~1! our axial dis-tributions of energy and water transport during a plonged exhalation@Figs. 3~b! and 3~c!# compare nicelywith a similar model of airway exchange;6 ~2! the meanend-exhaled temperature for our model simulatio(35.6 °C) is very similar to that measured under simiconditions (34.6 °C);25 and ~3! the current model accurately predicts the exchange of ethanol which hasblood and water solubility that is strongly dependenttemperature.24

For airways beyond the second generation~intraparen-chymal airways!, the adventitial and pulmonary layerwere included. For airway generations three through tcartilage is present in the adventitial layer.20 Gerdeet al.15 in their model of toxicant absorption by the aiway wall assumed the cartilage was an impermeabarrier to gas transport. As a result, we assumedcartilage in this layer presented a significant barriersoluble gas transport. Data on the diffusivity of solubgas through cartilage were unavailable. Consequently,assumed that the molecular diffusivity of soluble gthrough the adventitial layer to be 10% of the molecudiffusivity through the connective tissue~i.e., ;3% ofthe molecular diffusion coefficient of soluble gas throuwater!. Without this assumption, the model overpredictthe ethanol expirograms of subjects 1 and 2. Sincetilage is assumed to be a major barrier to the diffusionsoluble gas from the pulmonary layer to the airwaymen in the current model, the major function of thadventitial and pulmonary layers in airway generatiothree through ten for this model is thermoregulatioFigures 3~b! and 3~c! show that these airway generationparticipate significantly in the exchange of energy awater which is a result of the pulmonary layer’s particpation in energy exchange. This finding that the pulmnary circulation participates in airway energy exchanreinforces an experimental investigation demonstrat

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the prominent role of the pulmonary circulation in ththermoregulation of the airways.38

For airway generations greater than ten, the advetial and pulmonary layers were included without adeviation from that presented in the model descriptand shown in Fig. 1. Here, the alveoli lie in close proimity to the intraparenchymal airways. These layers pticipate mostly in soluble gas exchange with the brochial circulation or connective tissue as the inspired aicompletely heated and humidified by the time it reachthese airways.

Solubility Dependence on Temperature

We know thatlb:a of a soluble gas is the most important factor for determining the location of pulmonagas exchange. However,lb:a depends strongly on temperature~Table 2! which can range from 23 °C in thmouth to 37 °C in the alveoli. Over this range, thblood–air partition coefficient can vary considerably. Fexample, ethanol’s temperature coefficient forlb:a is26.7%/°C.24 As a result,lb:a for ethanol at the mouth(T523 °C) can be 150% larger thanlb:a for ethanol inthe alveoli (T537 °C). The dependence oflb:a on tem-perature primarily affects soluble gas exchange that taplace in the same airways that heat and humidifyspired air. In these airways, a large axial temperatgradient is present that will change the value oflb:a .When breathing room air at rest, this dependency wmost likely only affect soluble gases withlb:a.800 likeethanol that primarily exchange in the first ten airwagenerations. However, during exercise or breathing codry air, a greater number of airway generations partpate in heating and humidifying inspired air. As a resuthe exchange of soluble gases with much smalb:a(T537 °C) values than isopropanol will be significantly influenced by the axial temperature gradient.account for this temperature effect on airway gaschange, the current model of pulmonary gas exchaincludes a description of energy exchange which, in tuplaces the model in a strong position to study the effeof inspired air conditions and exercise on pulmonary gexchange.

Location of Gas Exchange

Figure 6~a! shows that the model predicted expirgrams of some low blood-soluble gases (lb:a,15) havea phase I volume that is;20% of the expired volume, aslightly positive phase III slope, andEend that increasesas blood solubility increases. These trends can bedicted from classical respiratory physiology becauthese gases exchange solely in the alveoli and dointeract with the airways. On exhalation, air in the repiratory tract does not contain soluble gas and leaveslung before the alveolar air. This volume of nonexchan

s

-

t

ing air is represented by phase I on the expirograHere, the concentration of soluble gas is zero in the fi;20% of the expirate. In addition to phase I, the phaIII slope of a low blood-soluble gas is thought to resufrom ventilation/volume effects, a longitudinal stratification of soluble gas in the alveolar region, and continuigas exchange.37 Because the model has a symmetricairway structure and the alveolar space is modeled awell-mixed compartment, the small positive phaseslope for these gases as predicted by the model mresult from continuing gas exchange, the transportsoluble gas from the pulmonary blood into the contraing alveolar compartment. At the end of phase III isEend

which increases as blood solubility increases.Eend isrepresentative of the alveolar gas concentration andsuch, this trend can be predicted by the alveolarexchange kernel developed by Farhi10 @E5lb:a /(lb:a

1VA /Qp)#.Before examining the model predicted expirograms

the high blood-soluble gases (15.lb:a.2700) that pri-marily exchange with the airways@Fig. 6~b!#, it is help-ful to briefly describe the mechanisms of airway gexchange. As fresh air is inspired, this air absosoluble gas from the mucous layer thereby depletingsoluble gas concentration in the airway wall. Becausesmall bronchial blood flow and the significant tissue brier between the bronchial circulation and mucus laythe mucus is not replenished with soluble gas befexpiration begins. During expiration, the air encounterlower concentration of soluble gas in the mucus atherefore a larger driving force for the redepositionsoluble gas on the mucus. This large air-to-mucus graent promotes recovery of soluble gas by the mucolayer and delays the rise in soluble gas concentrationthe mouth, thus accounting for the steep phase III sloThese absorption–desorption phenomena decreaseamount of soluble gas leaving the lung late in exhalatand is the major mechanism of pulmonary gas exchawhen lb:a.100.

Examining the expirograms from Fig. 6~b!, the modelpredicts the following for these high blood-soluble gasthe phase I volume decreases with increases inlb:a , thephase III slope is strongly positive, andEend decreases asblood solubility increases. First, the air in the conductiairways contains soluble gas and thus the airwaysconsidered part of the exchange space. As a result,volume of nonexchanging air~phase I! which originatesin the airways is less than that for gases withlb:a

,15. As blood-solubility increases, the phase I volumdecreases because more proximal airways participatelarger portion of gas exchange. This result is nicely deonstrated in Fig. 5 where the axial distribution of gexchange moves towards the proximal airways aslb:a

increases from cyclopropane to acetone. In additionthis model prediction, experimental observations ha

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,l to

ason,


reported that these soluble gases appear in the expearlier than gases of low-blood solubility.13,36 Second,the positive phase III slope of these gases reflectstemporal exchange mechanism described above andGeorgeet al.13 The model predicts the phase III slopdecreases for gases withlb:a;800 ~isopropanol! andlarger. This result coincides with the appearance olarge soluble gas flux in the trachea@compare Fig. 3~a! to5~c!# and might simply imply that the phase III slopdecreases as the majority of gas exchange occurs inmore proximal airways. Third, the model predictsEend ofthese gases to be inversely proportional to the blosolubility. This trend is caused by the absorptiondesorption phenomena which decrease the amounsoluble gas leaving the lung late in exhalation and cauthe end-exhaled concentration of soluble gas to bethan its alveolar concentration. This finding was sugested by Schrikkeret al.35

Intermediate Soluble Gas Exchange

Figure 4 shows that soluble gases with 10,lb:a

,100 exchange partially with the airways and partiawith the alveoli. The gases that lie within these twextremes, the intermediate soluble gases, are encoundaily and monitored as a result of their importance inbiological, medical and occupational venues. Toluediethyl ether, and methyl isobutyl ketone are all solvebut are also used as a gasoline component, anesthand lacquer, respectively. Because these intermedsoluble gases interact with both the airways and alvethe exchange of these intermediate soluble gases is lidependent on both alveolar and airway gas exchamechanisms: alveolar ventilation, pulmonary perfusibronchial perfusion, and the diffusing capacity of tairway tissue. The influence of these factors on intermdiate soluble gas exchange may vary depending onspatial heterogeneity of these factors within a subjectthe temperature and water concentration of the inspair. Consequently, we may have to reevaluate howexchange of these gases are interpreted. In particularmultiple inert gas elimination technique relies on diethether to quantify heterogeneity of alveolar ventilatiand perfusion.46 This measure of heterogeneity mayaffected by changes in airway tissue thickness, broncblood flow, or inspired air conditions—factors that impact airway gas exchange. Conversely, an understanof the exchange of these gases in normal subjectsconjunction with our current knowledge of low and higsoluble gas exchange should provide new insightssecondary factors that determine pulmonary gaschange. Perhaps such new information on intermedsoluble gas exchange will lead to tests that focusthese gases to more accurately predict the exchangeficiency and health of the airways and alveoli.

e

y

e

fss

d

c,e

y

e

l

g

f-

CONCLUSIONS

A mathematical model of heat, water, and soluble gexchange in the airways was improved by incorporatanatomical data on the airway wall and expandingmodel to include a time varying alveolar compartmeThe model was tested using experimental ethanol expgrams and the uptake of seven soluble gases from husubjects. To accurately predict these data, the modequired a bronchial blood flow that is similar to the eperimentally measured values of bronchial blood flowhumans. The model was used to predict gas exchalocation in the lung~airways versus alveoli! and foundthat gases with a blood:air partition coefficient,lb:a ,less than 10 exchange almost solely in the alveoli agases withlb:a.100 exchange almost exclusively in thairways. Gases withlb:a between 10 and 100 have significant interaction with the airways and alveoli. Manmore soluble gases than previously thought exchaprimarily in the airways. The exchange of these gamay be affected by inspired air conditions and bronchblood flow. The exchange of soluble gases that interwith both the airways and alveoli will be affected tvarying degrees by all pulmonary gas exchange mecnisms. In general, these results suggest that the airwplay a larger role in pulmonary gas exchange than pviously assumed and we may need to reevaluateimplications of pulmonary tests employing soluble gaswith lb:a.10.

ACKNOWLEDGMENT

This work was supported, in part, by GranNo. HL24163 from the National Heart, Lung, and BlooInstitute.

APPENDIX

The equations describing the transport of massenergy in each radial compartment are briefly describand presented below. Derivation of the equations arepresented as the methodology is similar to the convolume approach used by Tsuet al.42 For each compart-ment, the volume is assumed to be constant withexception of the mucous layer. The gas in the airwlumen is assumed to behave like an ideal gas soCg5Pt /(RTABS) wherePt is the total pressure,R is thegas constant, andTABS is the airway temperature inKelvin (5Ta1273.15). The total molar concentrationC, in each radial compartment is assumed to be equathat of water and can be expressed asC5r/Mw wheregr is the density of water andMw is the molecularweight of water. The molar concentration of soluble gin each radial layer is expressed as the mole fracti

rt-.g.,

hesoflar

and

can

eby

ed

u

onerheof

re-eick-


X5Ca /C. The mole fraction between radial compaments can be related using a partition coefficient, eXt5l t:mXm .

Airway Lumen

Material Balance. The molar balance of a speciesi ,either soluble gas or water, for a control volume~C.V.!in the airway lumen was described by Tsuet al.42 A termhas been added to account for axial diffusion in tairway. Equation~A1! shows the accumulation of masin the airway lumen to be equal to the convectionmass by ventilation, axial diffusion of mass, and moevaporation of species i from the mucosal surface:

]

]t~ntotYi !U

C.V.

52V]

]z~CgYi !U

Dz

Dz

1]

]z S ADa,i

]~CgYi !

]z D UDz

Dz1ni .

~A1!

The molar evaporation rate of speciesi can be ex-pressed in terms of a local mass transfer coefficientthe mole fraction difference:

ni5Asky,i@Ywall,i2Yi #. ~A2!

Substituting Eq.~A2! into ~A1! and assumingntot andCg have ideal gas behavior, the governing equationbe written for gas speciesi @i.e., water~subscriptw) orsoluble gas~subscripta)#:

]Yi

]t5S Yi

TABSD ]Ta

]t2S 4V

pd2D ]Yi

]z1S 4VYi

pd2TABSD ]Ta

]z

1S 4RTABS

dPtD ky,i@Ywall,i2Yi #1Daxial,i , ~A3!

where

Daxial,i5Da,iF]2Yi

]z2 22

TABS

]Yi

]z

]Ta

]z1

2Yi

TABS2 S ]Ta

]z D 2

2Yi

TABS

]2Ta

]z2 G .Energy Balance. The accumulation of energy in thcontrol volume is equal to the convection of energyventilation, conduction of energy from the mucus~i.e.,sensible heating! and transportation of energy associat

with water vaporization/condensation~i.e., latent heating!and was initially described by Tsuet al.:42

]Etot

]t UC.V.

52]~ nHg!

]zU

Dz

Dz1Qsens1(i 51

2

ni H ig .

~A4!

After simplification and substitution of Eq.~A3! intoEq. ~A4!, the governing equation is similar to that by Tset al.42 with an additional term (Daxial,i) resulting fromaxial diffusion of soluble gas and water:

]Ta

]t52

4V

pd2

]Ta

]z1

4RTABS

dPth~Tm2Ta!

273.15

TABSCp,da

g 1(i 51

2

YiDCp,ig

2

(i 51

2

Daxial,i~DCp,ig Ta1DH i

vap!

273.15

TABSCp,da

g 1(i 51

2

YiDCp,ig

1

4RTABS

dPt(i 51

2

~Cp,dag Ta2Cp,i

, Tm!ky,i~Ywall,i2Yi !

273.15

TABSCp,da

g 1(i 51

2

YiDCp,ig

,

~A5!

where

DCp,ag 5Cp,a

g 2Cp,dag , DCp,w

g 5Cp,wg 2Cp,da

g

and

DHavap5Ha

vap2Hdavap, DHw

vap5Hwvap2Hda

vap.

Mucus

Thickness of Layer.The governing equation for thethickness of the mucous layer is the same as thederived by Tsuet al.42 The thickness of the mucous layeis assumed to be independent of airway curvature. Tmucous layer thickness is decreased by evaporationliquid but a minimum thickness is maintained by section of fluid from the bronchial bed into the layer via thconnective tissue. The general equation describing thness of the mucous layer can be written

hatcond

ini-ohe

be

n a

ime

-yerthek-

tedive

-witrgy

en-uc-id

ns-a-

dif-ialmporte

leat

viaofis-


AsC],m

]t5As~S2ky,a@Ywall,a2Ya#2ky,w@Ywall,w2Yw# !.

~A6!

This general equation is subject to two conditions tdepend on the thickness of the mucous layer. Theseditions result in the following two specific solutions usein the mathematical model.

When the mucous layer thickness is less than a mmum thickness,,m,,m,min , the secretion rate is equal tthe evaporation rate of liquid from the mucus to tlumen and the change in mucous layer per time canwritten

],m

]t50. ~A7!

When the mucous layer thickness is greater thaminimum thickness,,m.,m,min , the secretion rate isequal to zero and the change in mucous layer per tcan be written

],m

]t52

1

C~ky,a@Ywall,a2Ya#1ky,w@Ywall,w2Yw# !.

~A8!

Material Balance. The accumulation of mass in the mucous layer is equal to the diffusion of mass to the lafrom the connective tissue, and transfer of mass fromlayer to the airway lumen. When the mucous layer thicness falls below a minimum thickness, fluid is secrefrom the bronchial bed into the layer via the connecttissue to maintain a minimum mucous thickness.

]Xm

]t5

Dm

12~,m1, t!,m

S Xt

l t:m2XmD

21

C,mS Xj

l j :mky,w@Ywall,w2Yw#

2S 12Xj

l j :mD ky,a@Ywall,a2Ya# D , ~A9!

where j 5mucous layer (m) when S50 ~i.e., ,m

.,m,min) and j 5tissue layer (t) when S.0 ~i.e., ,m

,,m,min).

Energy Balance. The accumulation of energy in the mucous layer is equal to the energy change associatedthe changing mucous thickness, the conduction of eneto the layer from the connective tissue, transport ofergy to the lumen resulting from convective and condtive effects, transport of energy associated with flu

-

h

secretion from the connective tissue, and energy traport resulting from evaporation and condensation of wter and soluble gas:

]Tm

]t52

Tm

,m

],m

]t1

kth

CCp,w, ,m

12~,m1, t!

~Tt2Tm!

2h

CCp,w, ,m

~Tm2T!2SCp,tTt

CCp,w, ,m

21

CCp,w, ,m

3(i 51

2

~DHivap2Cp,i

, Tm1Cp,ig Ta!ky,i~Ywall,i2Yi !.

~A10!

Connective Tissue Layer

Material and Energy Balances.The accumulation ofmass/energy in the connective tissue is equal to thefusion of mass/energy to the layer from the adventitlayer and bronchial bed, diffusion of mass/energy frothe layer to the mucous layer, and mass/energy transvia secretion of fluid into and out of the layer from thbronchial bed and to the mucous layer, respectively.

]Xt

]t5

~12Fc!Dt

12~,ad1, t!, t

~Xad2Xt!1FcDt

12, t

2 S Xc

lb:t2XtD

2Dm

12~,m1, t!, t

~Xt2Xml t:m!

1S

, tCS Xc

lb:t2XtD , ~A11!

]Tt

]t5

kth

12CCp,t

, , t

F ~12Fc!~Tad2Tt!

~,ad1, t!1

Fc~Tc2Tt!

, t

2~Tt2Tm!

~,m1, t!G1

S

CCp,t, , t

~Cp,bTc2Cp,tTt!.

~A12!

Bronchial Bed

Material Balance. The mass balance describing solubgas accumulation in the bronchial bed is similar to thderived by Buiet al.6 The accumulation of material inthe bronchial bed is equal to the convection of massbronchial perfusion, diffusion of material into and outthe layer from the adventitia and to the connective t

tionnce

:

yme

fol-

if-ryndrit-

as-fu-ole

i-al-htheithgh-berehe

often

as

ir


sue, respectively, and transport of material via secreto the connective tissue layer. A general material balacan be written

]~vcCXc!

]t5qbrCXa2~ qbr2SAs!CXc2SAsCXc

1AsFcCDt

12,ad

~Xadlb:t2Xc!

2AsFcCDt

12, t

~Xc2Xtlb:t!. ~A13!

After simplification, the final form of the equation is

]Xc

]t5

qbr

vc~Xa2Xc!1

DtAsFc

12,advc

~Xadlb:t2Xc!

2DtAsFc

12, tvc

~Xc2Xtlb:t!. ~A14!

Energy Balance. The governing equation for the energbalance in the bronchial bed can be derived the saway as the material balance and may be written aslows:

]Tc

]t5

qbr

vc

~Tb2Tc!1kthAsFc

12vcCCp,b

,

3F ~Tad2Tc!

,ad

2~Tc2Tt!

, tG . ~A15!

Adventitial Layer

Material and Energy Balances.The accumulation ofmass/energy in the adventitial layer is equal to the dfusion of mass/energy into the layer from the pulmonabed, and from the layer to the connective tissue abronchial bed via diffusion. These equations can be wten as:

]Xad

]t5

DtAs

12,advad

S Xp1Xc

2lb:t2XadD

2DtAsFc

12,advad

S Xad2Xc

lb:tD

2DtAs~12Fc!12~,ad1, t!vad

~Xad2Xt!, ~A16!

]Tad

]t5

kthAs

12CCp,ad

, vad

F ~Tbody2Tad!

,ad

2Fc~Tad2Tc!

,ad

2~12Fc!~Tad2Tt!

~,ad1, t!G . ~A17!

Alveolar Compartment

The alveolar gas and pulmonary blood spaces aresumed to be separate, well mixed, and without a difsion barrier between the spaces. On inspiration, the mfraction of soluble gas entering the alveolar space,YA,I ,is equal to theYa leaving the 18th generation. On expration, the mole fraction of soluble gas leaving theveolar space,YA , is the boundary condition for the 18tgeneration. The mole fraction of soluble gas leavingblood compartment is assumed to be in equilibrium wthe soluble gas mole fraction in the gas space. Throuout the breathing cycle, the alveolar air is assumed tofully saturated with water and to have a temperatuequal to the mean body core temperature, 37 °C. Tfinal form of the equation describing conservationsoluble gas in the alveolar compartment can be writfor both inspiration @Eq. ~A18!# and expiration @Eq.~A19!#:

dYA

dt5

dVA /dt

~lb:aVp1VA!F ~YA,I2YA!1

Qplb:a

dVA /dt~Yv2YA!G ,

~A18!

dYA

dt5

dVA /dt

~lb:aVp1VA!F Qplb:a

dVA /dt~Yv2YA!G , ~A19!

where

Yv5RT

Pt

r

MWw

Xv

lb:a.

NOMENCLATURE

A cross-sectional area of the airway lumen (cm2)As airway surface area5pdDz(cm2)C molar concentration of airway wall~properties of

water, assumed! (mol cm23)Ca molar concentration of soluble gas (mol cm23)Cg molar concentration of airway lumen~properties

of ideal gas, assumed! (mol cm23)Cp,a

g average volumetric heat capacity of soluble gin the vapor phase (J mol21 °C21)

Cp,dag average volumetric heat capacity of dry a

(J mol21 °C21)

or

as

ial

d

yer

ive

er

s

s

n-

ar-p-

nd

as

nn

ern

as

r

e

il-

-ss


Cp,wg average volumetric heat capacity of water vap

(J mol21 °C21)

Cp,a, average volumetric heat capacity of soluble g

in the liquid phase (J mol21 °C21)

Cp,ad, average volumetric heat capacity of adventit

tissue (J mol21 °C21)

Cp,b, average volumetric heat capacity of bloo

(J mol21 °C21)

Cp,m, average volumetric heat capacity of mucous la

(J mol21 °C21)

Cp,t, average volumetric heat capacity of connect

tissue (J mol21 °C21)

Cp,w, average volumetric heat capacity of wat

(J mol21 °C21)

DCp,ag Cp,a

g 2Cp,dag (J mol21 °C21)

DCp,wg Cp,w

g 2Cp,dag (J mol21 °C21)

d airway diameter~cm!Da,i molecular diffusion coefficient of gas species,i ,

through air (cm2 s21)DL diffusing capacity of the airway tissue (cm3 s21)Dm molecular diffusion coefficient of soluble ga

through mucus (cm2 s21)Dt molecular diffusion coefficient of soluble ga

through tissue (cm2 s21)F weighting factor that describes the axial depe

dence of bronchial blood flow~Ref. 5!FRC functional residual capacity (cm3)Fc capillary area fraction, ratio of surface area p

ticipating in exchange between the bronchial caillary and airway

g generation numberh heat transfer coefficient between the mucus a

airway lumen (J s21 °C21 cm22)

Havap latent heat of vaporization of a soluble g

(J mol21)

Hdavap latent heat of vaporization for dry air (J mol21)

Hwvap latent heat of vaporization for water (J mol21)

DHavap Ha

vap2Hdavap (J mol21)

DHwvap Hw

vap2Hdavap (J mol21)

i number of control volumes in a compartmentnI total number of control volumesj exp molar flux of soluble gas into the airway lume

within an airway generation during expiratio(mol breath21)

j insp molar flux of soluble gas into the mucous laywithin an airway generation during inspiratio(mol breath21)

ky,a local mass transfer coefficient for soluble g~mol soluble gas s21 cm22)

ky,i local mass transfer coefficient for speciesi~mol i s21 cm22)

ky,w local mass transfer coefficient for wate(mol water s21 cm22)

kth thermal conductivity of tissue (J cm21 s21 °C21),ad adventitial layer thickness~cm!,m mucous layer thickness~cm!, t connective layer thickness~i.e., distance from the

bronchial capillary to the airway lumen! ~cm!lb:a blood:air partition coefficientlb:t blood:tissue partition coefficientlm:a mucus:air partition coefficientl t:m tissue:mucus partition coefficientMw molecular weight of water (g mol21)n compartment numberntot total number of moles in the airway lumen~mol!ni molar evaporation rate of speciesi from the mu-

cus to airway lumen (mol s21)N total number of compartmentsPt total ambient pressure~mmHg!

qbr mean blood flow per unit volume of tissu(ml blood s21 ml tissue21)

qbr airway blood flow within a control volume(ml s21)

Qbr total airway blood flow (ml s21)

Qp pulmonary blood flow (ml s21)R universal gas constant

(62 360 mmHg cm3 mol21 K21)r density of water (g cm23)

S total flux of fluid via secretion (mol s21 cm22)t time ~s!Ta temperature of the airway lumen (°C)Tad temperature of the adventitial layer (°C)TABS absolute temperature of the airway lumen~K!Tb Body temperature (37 °C)Tc temperature of blood leaving the bronchial cap

lary bed (°C)Tm temperature of the mucous layer (°C)Tt temperature of the tissue layer (°C)TLC total lung capacity (cm3)vad adventitial tissue layer volume (cm3)vc bronchial capillary volume (cm3)v t volume of tissue surrounding an airway from lu

men to alveolar-adventitial boundary of thickneDz (cm3)

VC vital capacity (cm3)Vee lung volume at end expiration (cm3)Vei lung volume at end inspiration (cm3)VA alveolar gas volume (cm3)Vp pulmonary blood volume (cm3)Vt total volume of airway tissue (cm3)

V volumetric flow rate of air (cm3 s21)

VA alveolar ventilation (cm3 s21)

Vexp volumetric flow rate of air during expiration(cm3 s21)

ic

al

ial

erry

e

odn

as

lar

n

us

ce

t ofa-p.

n.ri-

b,n-

ialr.

l,n-

mi-

ialrip-

uid

J.te

in

essn.

P.a:

ofion

ala.tra-

w-

al

ira-.

ia:

n.

ofo-

.

ns-h-

ffi-

tsd

on-ged

eta-og-of

ofoil,

ngow

S.rt-

.ma


Vinsp volumetric flow rate of air during inspiration(cm3 s21)

Xa mole fraction of soluble gas in the systemblood

Xad mole fraction of soluble gas in the adventitilayer

Xc mole fraction of soluble gas leaving the bronchbed

Xm mole fraction of soluble gas in the mucous layXpb mole fraction of soluble gas in the pulmona

bed layer,5(Xa1Xv)/2Xt mole fraction of soluble gas in the connectiv

tissueXv mole fraction of soluble gas in the venous bloYa mole fraction of soluble gas in the airway lumeYA mole fraction of soluble gas in the alveolar g

spaceYA,I mole fraction of soluble gas entering the alveo

gas space from generation 18Yv mole fraction of soluble gas in air that is i

equilibrium with XvYw mole fraction of water in the airway lumenYwall,a mole fraction of soluble gas at the air–muc

interfaceYwall,w mole fraction of water at the air–mucus interfaz axial direction~cm!Dz control volume length~cm!

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Modeling Soluble Gas Exchange in the Airways and Alveoli