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Computers and Biomedical Research 32, 355–390 (1999)Article ID cbmr.1999.1515, available online at http://www.idealibrary.com on

Theoretical Study of Inspiratory Flow Waveforms duringMechanical Ventilation on Pulmonary Blood Flow and

Gas Exchange

S. C. Niranjan,*,†,‡ A. Bidani,*,‡ F. Ghorbel,§ J. B. Zwischenberger,*,¶

and J. W. Clark, Jr.*,†

*Biomedical Engineering Center, ‡Department of Internal Medicine, and¶Department of Thoracic Surgery, University of Texas Medical Branch, Galveston, Texas 77555;

and †Department of Electrical and Computer Engineering and§Department of Mechanical Engineering, Rice University, Houston, Texas 77251

Received October 15, 1998

A lumped two-compartment mathematical model of respiratory mechanics incorporating gasexchange and pulmonary circulation is utilized to analyze the effects of square, descendingand ascending inspiratory flow waveforms during mechanical ventilation. The effects on alveolarvolume variation, alveolar pressure, airway pressure, gas exchange rate, and expired gas speciesconcentration are evaluated. Advantages in ventilation employing a certain inspiratory flowprofile are offset by corresponding reduction in perfusion rates, leading to marginal effects onnet gas exchange rates. The descending profile provides better CO2 exchange, whereas theascending profile is more advantageous for O2 exchange. Regional disparities in airway/lungproperties create maldistribution of ventilation and a concomitant inequality in regional alveolargas composition and gas exchange rates. When minute ventilation is maintained constant, foridentical time constant disparities, inequalities in compliance yield pronounced effects on netgas exchange rates at low frequencies, whereas the adverse effects of inequalities in resistanceare more pronounced at higher frequencies. Reduction in expiratory air flow (via increasedairway resistance) reduces the magnitude of upstroke slope of capnogram and oxigram timecourses without significantly affecting end-tidal expired gas compositions, whereas alterationsin mechanical factors that result in increased gas exchanges rates yield increases in CO2 anddecreases in O2 end-tidal composition values. The model provides a template for assessingthe dynamics of cardiopulmonary interactions during mechanical ventilation by combiningconcurrent descriptions of ventilation, capillary perfusion, and gas exchange. q 1999 Aca-

demic Press

Key Words: ventilatory mechanics; pulmonary circulation; convective–diffusive gas transport.

INTRODUCTION

Mechanical ventilation is one of the major supporting therapeutic modalitiesused to manage critically ill patients with acute or chronic respiratory failure

355

0010-4809/99 $30.00Copyright q 1999 by Academic Press

All rights of reproduction in any form reserved.

356 NIRANJAN ET AL.

(1, 37, 38). The implicit goal of mechanical ventilation is to assure or restoreappropriate levels of gas exchange at the lungs with minimal alterations of otherphysiological functions (5). Because of the interactions between the patient and theventilator, physiological quantities (e.g., minute ventilation, alveolar gas exchange,pulmonary blood flow, cardiac output, work of breathing) can be significantlyaffected by the choice of ventilatory mode (37). Rational application of clinicaltherapies during ventilator management therefore requires an understanding ofthe underlying physiological mechanisms at play during mechanical ventilation,particularly when different ventilatory patterns are employed (44).

Experimental studies examining the effects of mechanical ventilatory mode ongas exchange (2, 6, 26, 34) and pulmonary hemodynamics (46, 47) have beenreported. Numerous studies using mechanical lung models (3, 21), animal models(29, 34), and human studies (2, 24, 25) concur in their assessment that the descend-ing waveform yields improved distribution of ventilation (39). Opinion is divided,however, over the relative effectiveness of individual waveforms on gas exchange.Reports varied from gas exchange being insignificantly affected (6, 25, 34) to thatfavoring the descending waveform (2) while the ascending waveform was pickedto perform better elsewhere (26). Comparison was difficult in the clinical studies,since changing the inspiratory waveform also induced changes in tidal volume andduty cycle. A theoretical study predicted that changes in both inspiratory andexpiratory flow patterns can result in different steady state blood gas tensionsand gas exchange (17). Significant differences in carbon dioxide elimination, asmeasured by end-tidal carbon dioxide partial pressure, were reported with inspira-tory waveforms; higher end-tidal values were obtained with the ascending waveformthan with the square and descending waveforms (39). A relatively recent surveyof the literature (27) has concluded that no definitive statement could be maderegarding the superiority of either flow pattern over the other. The only consistentresult across all studies was that cardiovascular performance was not affecteddifferently by either flow pattern. The inability to make concrete inferences resultedfrom: (i) inconsistent ventilator settings across the experiment set (such as lowtidal volumes, short duty cycles and use of postinspiratory pause, insignificantchanges in peak flow); (ii) presence of morphological differences in the flowpatterns used; (iii) design employed, such as the absence of randomized trials orrepeated-measures; (v) small size of sample population studied; and (vi) failure tosegregate samples by disease process. In that report (27), major factors attributedto the inability to maintain standardized settings were the number of dynamic andconfounding variables encountered (e.g., changes in pulmonary perfusion, sedationeffects, inotropes, and diuretics) and unstable patient states that could in themselveshave accounted for inconsistencies in the conclusions reported.

This paper provides a template for quantitatively characterizing the dynamiceffects of intermittent positive pressure mechanical ventilation on capillary perfu-sion and ensuing gas exchange. A lumped two-compartment mathematical modelof respiratory mechanics incorporating descriptions of gas exchange and pulmonaryblood flow is employed for this purpose. The generalized framework facilitatesthe investigation of problems pertaining to the interaction of mechanical ventilation

STUDY OF INSPIRATORY FLOW WAVEFORMS 357

on cardiovascular dynamics and gas exchange. An idealized flow generator (time-cycled) with prespecified tidal volume and duty cycle is applied to generate agiven flow pattern. The categories of flow patterns assessed (square, ascending,and descending) are those available in modern microprocessor controlled ventilators(4). The objectives of this study were to: (a) assess the effects of choice of inspiratoryflow waveforms on gas exchange as well as on alveolar volume variation, alveolarpressure, airway pressure, expired gas species concentration (e.g., oxigram andcapnogram), and regional pulmonary blood flow for a given minute ventilationrate; (b) evaluate quantitative indices such as the ventilation, regional blood-gasconcentrations, and pulmonary blood flow distribution ratios; and (c) investigatethe effect of cycling frequency on regional ventilation, perfusion, and gas exchange.

METHODS

Model Development

1. Patient-ventilator model. The respiratory system was considered as a lumpedsystem composed of a rigid anatomic dead space region in series with two parallelcompliant alveolar compartments (Fig. 1). Time-varying volume changes producedby deformation of the elastic alveolar compartments were characterized using linearpressure–volume relationships with compliances denoted by CA1 and CA2, respec-tively. Viscous dissipation caused by motion of the elastic alveolar structure (42,43) was ignored, however. Air flow through the peripheral conduits of the bronchialtree feeding the individual alveolar compartments was characterized by lumpedairway resistances (R1 and R2, respectively). The alveolar compartments communi-cated only through the branch point. Nonlinear flow-dependent resistance to airflowoffered by the upper airways (40) was ignored. The subject was assumed to beintubated; hence the resistance to airflow offered by an endotracheal tube wasmodeled by a fixed airway resistance (RUAW ). All driving frequency changesconsidered herein do not exceed 1 Hz. Hence the inertial effects of gas accelerationwere ignored in the present formulation.

Air forced into the lungs during positive pressure ventilation performs work onthe lung and chest wall, as well as on the air already present in the system (17).This is characterized by including the compressibility of air (Cg1, Cg2, and Cgu inFig. 1; derived using the ideal gas law) at each nodal pressure site. A subjectundergoing mechanical ventilation was assumed to be sedated; hence the diaphragmand chest wall were assumed to behave passively and their motion was describedusing a passive lumped compliant element (Cth). Changes in pressure across thelumped thoracic cavity (viz., intrapleural pressure, PPL) were then characterizedby a constant value for this compliance (23, 35). Viscous resistance to flow wasalso neglected in considering chest wall dynamics; it is generally considered to besmall both in health and in disease (16, 19, 23).

2. Pulmonary circulation model. A lumped system composed of compliantpulmonary arterial, capillary, and venous regions (denoted by Ca , Cc1, Cc2, and Cv ,respectively) was employed. Resistance to blood flow offered in each of the regions(denoted by Ra , Rc1, Rc2, and Rv , respectively) subsequently dictated regional

358 NIRANJAN ET AL.

FIG. 1. Pneumatic representation of lumped two-compartment respiratory mechanics and gasexchange model.

perfusion (Fig. 2). Constant pressure sources of 15 and 5 mm Hg with respect toPPL (30) were used to denote the mean pulmonary arterial (Ppa) and venous (Ppv)pressures, respectively. This implied that instantaneous changes in Ppa and Ppv dueto the pumping action of the right ventricle and the left atrium (auricle) wereneglected. The resulting variation in mean blood volume in each of the capillaryregions was presumed to be governed by the modulation of the transmural pressurein each region. In the extraalveolar pulmonary arterial and venous regions, the


FIG. 2. Hydraulic representation of the lumped pulmonary circulation model.

perivascular pressure was assumed to be the intrapleural pressure; hence volumechanges in these regions were dictated by the variation in PPL. On the other hand,volume variation in the intraalveolar regional capillary compartments was governedby the changes in the regional alveolar pressure. Blood flow through each alveolarregion was then dictated by the pressure drop exhibited across the regional nodes;this regional blood flow rate was combined to yield the total mean blood flow(cardiac output) through the pulmonary circulation.

3. Gas-exchange model. Oxygen, carbon dioxide, and nitrogen were assumedto be transported across the lumped alveolar–capillary barrier solely via diffusion.

360 NIRANJAN ET AL.

Oxygen was taken up by the blood and carbon dioxide was excreted while nitrogen(a relatively inert gas) diffused in either direction dependent on the instantaneousregional overall ventilation–perfusion ratio (49). All compartments were assumedto be well-mixed with the movement of gases in the respiratory system and bloodin the pulmonary circulation being primarily through bulk axial convection (consid-ered one-dimensional ignoring all radial variations). Blood was considered as auniform, single-phase medium with instantaneous chemical reactions; hence speciescontents were related to the equilibrium tensions using empirical dissociation curves(15, 28, 32, 41) (refer to Appendix C for details). Mixed venous inlet blood gastensions were held constant throughout the simulations and physiological shuntingof blood was ignored.

Employing the model described above (dynamic equations for which are fur-nished in Appendix A), the objective of this study was to evaluate the effects ofdifferent input driving airflow waveforms and frequency changes on respiratorymechanics, pulmonary circulation, and gas exchange for a given set of modelparameters. Parameter values describing the compliant and resistive elements inthe airway/lung mechanics and pulmonary circulation models for nominal operatingconditions (fully recovered patient with endotrachael tube) are listed in Table 1.

On the air side, the model structure was modified and parameter values wereadopted from descriptions published elsewhere (22, 23). Heterogeneous lungs wereemulated using parallel alveolar regions connecting through a branch point to acommon resistive airway. Nominal compliance values for the individual alveolarand thoracic regions in Table 1 were chosen to yield a net compliance of 0.1 L/cm H2O, a value reported for normal subjects (35). Thoracic compliance value of0.1 L/cm H2O was adopted from (23). Nominal peripheral airway resistance valueswere adopted from (22). Inspiratory and expiratory resistances were chosen to beunequal, consistent with reported observations (7, 13). Upper airway resistancewas chosen to be 10 cm H2O/L/s, an average value for resistance offered to flowby a size 7 endotrachael tube for flow rates between 0.5 and 2 L/s under constantflow conditions (50). On the blood side, the model structure adopted was a modifiedform of that presented in (12). Parameter values for the pulmonary circulationwere chosen to yield nominal values of total cardiac output, total pulmonaryvolume, and pulmonary arterial and venous volumes. A total pulmonary bloodvolume of approximately 440 ml was assumed as indicated in (33) for man. Thedistribution of volumes obtained was different from that reported in (33), however,but was comparable to data reported in (51).

Inspiratory flow waveforms employed in this study were the ascending, descend-ing, and square waveforms. No postinspiratory pause (protoexpiratory pause) wasset and the expiratory phase was considered to be purely passive. The total volumeof gas introduced per breath at the mouth, cycling frequency (and hence minuteventilation), and duty cycle were unchanged during the comparative study of thefour modes of ventilation (at 600 ml/breath, 10 breaths/min, and 0.333, respec-tively). While investigating the effects of frequency changes the minute ventilationand duty cycle were maintained invariant.


TABLE 1

Nominal Model Parameters Describing Respiratory Mechanics and Pulmonary Circulation

Parameter Symbol Value Units Reference

Endotrachael tube resistance RUAW 10.0 50cm H2OL/s

Peripheral airway resistancea Rj 2.0,3.0b 22cm H2OL/s

Pulmonary compliancea CAj0.1 11L

cm H2O

Chest wall compliance Cth 0.2 11Lcm H2O

Inspired O2 mole fraction FAIO2 0.2089 35Inspired CO2 mole fraction FAICO2 0.0004 35Inspired N2 mole fraction FAIN2 1 2 FAIO2–FAICO2 L

mm HgPulmonary arterial compliance Ca 4 3 1023 33L

mm HgPulmonary capillary compliance Ccja 4.167 3 1024 33

Lmm HgPulmonary venous compliance Cv 4 3 1023 33

mm HgL/s

0.035 10.150Va

22

Pulmonary arterial resistance Ra This study

mm HgL/s

0.042 10.0375VBj

22

Pulmonary capillary resistance This studyRcja

Pulmonary venous resistance Rv This studymm HgL/s

0.035 10.200Vv

22

Mean pulmonary arterial pressure PPA 15 mm Hg 30Mean pulmonary venous pressure PPV 5 mm Hg 30Mixed venous O2 tension pbO2v

40.0 mm Hg 49Mixed venous CO2 tension pbCO2v

46.0 mm Hg 49Mixed venous N2 tension pbN2v

575.0 mm Hg 49

2–3 DPG Shift factor DP50 0 This studygm

100mlHemoglobin concentration Hb 14.4604 This study

a j 5 1, 2 and denotes the individual alveolar regions. j 5 1 denotes the control alveolus andj 5 2 designates the affected alveolus throughout this paper.

b Rj takes on different values during inspiration and expiration (22) with the larger value duringexpiration.

Computational Aspects

The resulting ordinary differential equation system was implemented in the “C”programming language. Numerical integration was performed using CVODE, an

362 NIRANJAN ET AL.

implementation of the variable coefficient Adam’s method in Nordsieck form,attributed to Brown et al. (8). CVODE was written by S. D. Cohen and A. C.Hindmarsh at the Lawrence Livermore National Laboratory. A maximum stepsizeof 5 3 1023 s and a relative error tolerance of 1 3 1025 were employed. A steadystate was achieved after 30 breaths and subsequently used.

RESULTS

Time Courses during Symmetric Regional Ventilation

Computed variables describing respiratory mechanics and pulmonary circulationare depicted in Fig. 3 for the case where the resistive and compliant properties inthe individual alveolar regions are set equal (i.e., R1 5 R2 and CA1 5 CA2). Parametervalues are as listed in Table 1. Time course of all variables during the inspiratoryphase of the ventilation cycle varies significantly for all ventilatory modes; tracesfor the square pattern (solid lines) lie between the corresponding traces for ascending(dash-dot line) and descending (dash lines) patterns, in general.

1. Airway/lung mechanics. When the resistive and compliant characteristicsrepresenting the individual alveolar regions are identical, ventilation at the mouthis equally distributed between the alveolar regions leading to a symmetric expansionin alveolar volume and corresponding buildup of alveolar pressure in the individualregions (Fig. 3). The volume and pressure waveforms depend on the specificalveolar ventilation profile. Higher mean airway pressure is predicted with thedescending flow profile. At equivalent tidal volumes and duty cycle, the descendingwaveform yields the highest cycle-averaged branch-point, intrapleural, and alveolarpressure. In contrast, the ascending waveform provides the lowest values. Peakinflation pressures are highest with the ascending waveform and lowest with thedescending waveform. The square profile yields intermediate values. These modelspredicted that effects on distribution of tidal ventilation and pressure in the airwayare similar to those reported previously (5, 22, 23).

A similar comment may be made regarding the qualitative changes in alveolarvolume. The descending waveform provides the largest change in initial alveolarvolume from the end-expiratory equilibrium value compared to the ascendingwaveform with the square profile yielding intermediate changes. Since the alveolarcompartmental compliances are assumed to be linear, tracings of alveolar pressurechanges mirror those of alveolar volume changes. Furthermore, a linear thoraciccompliance leads PPL changes to qualitatively mirror alveolar volume changes.Intrapleural pressure rises during the inspiratory phase of IPPV. This is quite unlikespontaneous breathing where PPL is subatmospheric throughout the cycle. Bothalveolar and mouth pressures are positive during IPPV, whereas mouth pressureis zero (atmospheric) and alveolar pressure subatmospheric during the inspiratoryphase of spontaneous breathing.

2. Pulmonary circulation. During the inspiratory phase, the ensuing rise in PPL

and regional PA tends to decrease the transmural pressure across the compliantelements representing the lumped pulmonary artery-arterioles, regional pulmonarycapillaries, and pulmonary venules–vein (PTMa, PTMb1&2, and PTMv, respectively)


FIG. 3. Time course of airway mechanics variables. Regional alveolar resistive and compliantproperties are identical. Minute ventilation rate is set at 6 L(BTPS)/min with a cycling frequency of10 breaths/min and a duty cycle of 0.333. Ascending, descending, and square waveforms are denotedby a, d, and f, respectively in the legend in all the panels.

364 NIRANJAN ET AL.

while simultaneously elevating the pulmonary arterial, regional capillary, and ve-nous nodal pressures, Pa, Pc1&2, and Pv, respectively, in Fig. 2. Elevation in Pa

reduces the forward pressure gradient, Ppa 2 Pa , which then tends to lower totalinlet blood flow rate, Qa , on the arterial side. In contrast, the nodal pressure gradientat the outlet, Pv 2 Ppv, increases, thereby favoring increased outlet blood flowrate, Qv , on the venous side. The resulting inequality of inlet and outlet bloodflow rates points to a reduction in the blood volume stored in the pulmonarycirculation at all the compliant elements during the inspiratory phase of mechanicalventilation. Depicted in Fig. 4, this trend is qualitatively consistent with observationsmade during experimental investigations using a porcine model wherein a reductionin pulmonary blood volume was observed during the inspiratory phase of mechani-cal ventilation (47, 48).

Blood flow in the arterial, regional capillary, and venous regions (Qa ,Qin

b1,2, Qoutb1,2, and Qv) are affected by the choice of inspiratory waveform and can

be explained by the related effects on intrapleural and alveolar pressures. Thedescending profile causes the greatest perturbation in the early part of the inspiratoryphase, thereby creating significant effects on arterial and venous perfusion ratesearly on. The early rise in alveolar pressure while employing the descendingwaveform similarly results in a steeper fall in capillary perfusion rate comparativelybecause of the concomitant effects on capillary nodal pressure, Pcj. In contrast, theascending waveform mainly influences the latter portion of the inspiratory phaseand yields the slowest rate of variation. The square profile again grades intermedi-ately. Acting in synchrony, these changes point to a reduction in stored bloodvolume (Va , Vb1&2, and Vv) during the inspiratory phase.

3. Gas exchange. As indicated in Fig. 5, variables describing gas exchangealso show markedly different temporal behavior during the inspiratory phasefor the three flow ventilatory modes considered. The greater initial airflow encoun-tered using the descending ventilatory mode (Fig. 3, top row), facilitates betterconvection in the airway compartments, thereby resulting in higher O2 and lowerCO2 cycle-averaged partial pressures in the alveolar and dead space regions. Alveo-lar composition for the ascending profile initially lags in comparison with thedescending and square profiles. This is due to the initial mixing of the dead spaceair with the alveolar gas combined with gas exchange across the capillary. Gascomposition in the expirate corresponds directly with ambient air and is relativelyunchanged during the inspiratory phase. Alveolar species partial pressures arepresumed to be equal to the end-capillary species blood–gas tensions. No diffusionallimitations are considered and the gas exchange across the capillary is assumedto be instantaneous.

During the course of a respiratory cycle, the ensuing variation in the instantaneousO2 and CO2 exchange across the capillary (fO2 and fCO2, respectively) is propor-tional to the difference in perfusion rates and the difference in the end-capillaryblood species tension (i.e., referenced to a respiratory cycle, Dfij 5 Cin

bij DQinbj 2

Coutbij DQout

bj 2 Qoutbj DCout

bij for a priori known constant mixed venous conditions).During the inspiratory phase, with the descending profile, the rapid initial increasein O2 tension accounts for the increase in fO2. As time elapses, the decrease in


FIG. 4. Time course of pulmonary blood flow descriptors. Regional alveolar resistive and compli-ant properties are identical. Minute ventilation rate is set at 6 L(BTPS)/min with a cycling frequencyof 10 breaths/min and a duty cycle of 0.333. Ascending, descending, and square waveforms aredenoted by a, d, and f, respectively, in the legend in all the panels.

366 NIRANJAN ET AL.

FIG. 5. Time course of gas exchange during mechanical flow ventilation. Resitive and compliantproperties of the individual alveolar regions are identical. Minute ventilation rate is set at 6 L(BTPS)/min with a cycling frequency of 10 breaths/min and a duty cycle of 0.333. Ascending, descendingand square waveforms are denoted by a, d, and f, respectively, in the legend in all the panels.


perfusion rate compromises the increased O2 tension; hence fO2 falls accordingly(Fig. 5, bottom left). Analogous arguments can be made to explain the predictedresults for the other flow waveforms. Performance is intermediate for the squareprofile. Exchange rate for CO2 is more sensitive to the inspiratory waveform,compared to that for O2, as the relative change in CO2 content is much greaterthan for O2. Note that the direction of changes in gas exchange rate for O2 andCO2 are in opposite directions. This is due to the concurrent effects of changingperfusion rate and change in species content. In a separate study (results not shownhere), when capillary blood flow rate was presumed to be invariant with time(achieved by decoupling the circulation model), the time course of O2 and CO2

gas exchange rates qualitatively varied in a similar manner, both increasing inmagnitude during the inspiratory phase and decreasing during the passive expiratoryportion of the cycle.

4. Change in lumped thoracic compliance. Variations in thoracic volume duringa respiratory cycle modulate the intrapleural pressure which, in turn, dictate gasexchange rates by manipulating inlet and outlet capillary perfusion rates. Figure6 shows the effect of modifying assumed thoracic compliance on resulting cycle-averaged mixed arterial blood-gas values and total O2 and CO2 gas exchange rates.Decreasing thoracic compliance exaggerates variations in intrapleural pressure forsimilar excursions in alveolar inflation. An elevation in the mean intrapleuralpressure ensues which, in turn, increases the nodal pressures in the pulmonarycirculation leading to a concomitant reduction in blood perfusion rates through thepulmonary capillaries (Fig. 6, top). Although end-capillary species blood tensionsdo not change significantly with lowered Cth (scale exaggerates the slight differencescalculated in Fig. 6, middle), the overall gas exchange rate falls with loweredthoracic compliance as a direct result of lowered perfusion rates Fig. 6, bottom).In all cases, a gradation based on the choice of the inspiratory waveforms isproduced. As explained earlier, reduced perfusion during descending waveformovercomes increased oxygenation, resulting in O2 exchange rates lower than thatobtained during ascending waveform. For the case of CO2 exchange, however, thechange in CO2 content (due to the higher slope of the CO2 dissociation curve)effectively overrides the adverse effects of decreased perfusion. The net result thenis an increase in CO2 exchange with the descending waveform with increasingthoracic compliance.

5. Averaged effects. The previous sections indicate that the time courses areaffected by the choice of the specific input inspiratory (and resulting expiratory)flow ventilatory waveforms. As depicted in Fig. 6, cycle-averaged quantities indeedresult in observed differences, albeit to a smaller extent. End capillary bloodoxygenation grades downward with descending, square, and ascending inspiratoryflow waveforms, respectively. The effect on gas exchange is complex and isdictated by the concurrent effects of perfusion rates and extent of blood arterization.Perfusion and O2 exchange are adversely affected by the descending waveformwhile proving favorable to blood gas composition and CO2 excretion.

The ratio of regional ventilation to perfusion with the choice of inspiratorywaveform was approximately the same for the various inspiratory waveforms. If

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FIG. 6. Effect of thoracic compliance. Resitive and compliant properties of the individual alveolarregions were again identical. Minute ventilation rate was set at 6 L(BTPS)/min with a cycling frequencyof 10 breaths/min and a duty cycle of 0.333. Ascending, descending, and square waveforms aredenoted by a, d, and f, respectively, in all the panels.

the cycle-averaged value is defined as the ratio of the mean ventilation to meanperfusion rates, then the descending waveform had the highest value while theascending waveform had the lowest value with an intermediate value obtained forthe square waveform. During the inspiratory phase values of 3.45, 3.26, and 3.35were obtained for the descending, ascending, and square waveforms, respectively.


The corresponding magnitudes during the expiratory phase were calculated to be1.561, 1.558, and 1.559. The weighted mean average over the entire ventilatorycycle subsequently evaluated to 2.194, 2.124, and 2.156, respectively. These mar-ginal differences in the averaged regional ventilation to perfusion ratios amongthe waveforms directly reflect the relatively small differences calculated for thenet gas exchange rates of O2 and CO2. Experimental studies conducted on healthyanesthetized subjects concluded that any respiratory pattern that could providesufficient alveolar ventilation would be adequate during artificial ventilation andthe provision for using different respiratory waveforms seemed unnecessary (6).

Disparate Regional Ventilation

The effects of alterations in regional airway resistance or lung complianceparameters were studied to examine pathophysiological conditions. Maldistributionin ventilation results from temporal and spatial differences in the distribution ofgas flow in the various units of the lung and leads to disparate regional gas exchangerates. Differences in regional ventilation were studied by altering either the R orC parameter values in one of the branches (affected alveolus) while maintainingthe other branch (control alveolus) unmodified. When one of the branch resistancesis increased, air flow through that branch is impeded during both the inspiratoryand the expiratory phases, thereby prolonging both the filling and the emptyingtimes. In due course this results in gas trapping, leading to CO2 buildup withconcomitant O2 dilution in the affected alveolar region. Impeded airflow in theaffected region, however, is compensated to some extent by hyperinflation in thecontrol branch, leading to a better dilution of alveolar composition. When thecompliances in the two branches are different, however, the more compliant alveolarregion exhibits a larger volume change (alveolar hyperinflation) for similar changesin transmural pressure. Dilution of the alveolar gas with ambient air creates favor-able transcapillary partial pressure gradients for both O2 and CO2.

Distribution of regional tidal volumes, peak alveolar pressures, mean perfusionrates, alveolar composition, and gas exchange rates is affected by spatial differencesin resistance or compliance. One can track the distribution of ventilation by em-ploying a compartmental dimensionless tidal volume ratio (VTR), defined as thequotient of the tidal volumes of the control and affected branches. A value of unityrepresents uniform regional ventilation. Peak alveolar pressure ratio (PPR) canalso be similarly defined. Maldistribution in ventilation is then reflected in thedistribution ratios diverging from unity. At low cycling rates, the divergence causedby compliance disparities is more significant when compared to that caused byresistive disparity. As depicted in Fig. 7, when branch resistances are unequal, useof the decelerating flow results in a more even distribution of tidal volumes andpeak alveolar pressures when compared to that obtained with the ascending andsquare waveform. The decelerating waveform delivers the majority of ventilationduring the early part of the respiratory cycle, thus favoring a more uniform distribu-tion through extended equilibration times even when resistance was unilaterallyincreased (9). This would suggest that flow tapering inspiratory flows should bepreferred in order to match regional resistive differences. On the other hand, with

370 NIRANJAN ET AL.

FIG. 7. Effect of disparate regional resistive and compliance properties on regional distributionbased on inspiratory waveform. Total minute ventilation rate was set at 6 L(BTPS)/min with a cyclingfrequency of 10 breaths/min and a duty cycle of 0.333 in all cases. [3R,C] denotes the case whenR2 5 3R1 while CA1

5 CA2. [R,3C] denotes the case when R2 5 R1 while CA2

5 3CA1.


a disparity in compliance, the ascending waveform provides a slightly better tidalvolume distribution. Decelerating flow causes a larger portion of the deliveredventilation to be diverted to the more compliant region, further aggravating themaldistribution of ventilation. These predictions are qualitatively consistent withexperimental observations conducted using a mechanical lung model (21) and inhuman patients under standard anaesthesia undergoing respirator therapy withoutheart or chest disease (25). Model calculations indicate marginal differences inregional perfusion with the descending profile, creating the greatest diversion inall cases. The effects on alveolar composition and total gas exchange rate arecomparatively exaggerated in the case when compliance disparities exist whencompared with resistive differences. This effect is solely the result of the cyclingfrequency as explained later.

Effects on overall gas exchange are summarized in Figs. 8 and 9. The decrementin perfusion rate caused by elevation in mean airway pressures during descendingflow ventilation results in lower overall O2 exchange rates comparatively. Theascending flow facilitates better O2 exchange rate with the square waveform per-forming intermediate (Figs. 8 and 9, bottom left). In contrast, the descending flowbest promotes better overall CO2 exchange. In one experimental study, 10 adultpatients without known respiratory or cardiac disease were studied during anesthesiaand artificial ventilation before elective surgery (26). Results indicated improvedgas distribution with a decelerating flow pattern, but when the effects of gasexchange (measurements of oxygenation) were judged, greater benefits were ob-tained with accelerating flow.

Effect of Cycling Frequency

When the regional mechanical properties of the airways are dissimilar (as mani-fested by differences in the regional Rj and/or Cj values), the behavior of the overallsystem can be expected to vary with cycling frequency (36). Predicted effects onthe distribution ratios are depicted in Fig. 10. When the regional mechanicalproperties are similar, all ratios are independent of frequency and trivially equalunity. Marked difference in the qualitative behavior of the system with frequencydue to resistive (denoted by 3RC in Fig. 10) and compliance disparities (denotedby R3C) are observed. Increasing cycling frequency leads to an attenuation of theinfluence of compliance disparities (dashed-dot lines in Fig. 10). Resistive disparit-ies, in contrast, are manifest at higher frequencies and lead to the distributionratios deviating from unity at increasing frequencies. These results suggest that ifachieving equilateral lung performance is the goal, then it is beneficial to operate atlow cycling frequencies in the presence of regional differences in airway resistance,whereas high-frequency ventilation would be the preferred mode when compliancedisparities prevail.

Cycling frequency in fact governs the “effective” resistive and compliant load(or equivalently, “effective impedance”) offered to the ventilator by the respiratorysystem. Details about the derivation are furnished in Appendix B. In general, botheffective resistance and compliance fall with increasing frequency, an observationconsistent with earlier reports (14, 18, 36, 45). The general formulae for overall

372 NIRANJAN ET AL.

FIG. 8. Effect of disparate regional resistances. Total minute ventilation rate was set at 6 L(BTPS)/min with a cycling frequency of 10 breaths/min and a duty cycle of 0.333 in all cases.


FIG. 9. Effect of disparate regional compliances. Total minute ventilation rate was set at 6L(BTPS)/min with a cycling frequency of 10 breaths/min and a duty cycle of 0.333 in all cases.

374 NIRANJAN ET AL.

FIG. 10. Effect of cycling frequency on regional distribution ratios during square wave inspiratoryflow. Minute ventilation and duty cycle were held constant at 6 L [BTPS]/min and 0.333, respectively.The case when R2 5 3R1 is denoted by the dashed line, whereas the case when CA2

5 3CA1is

denoted by the dashed-dot line in all the panels.


impedance pertaining to the model presented herein reduce to those reported byOtis et al. (36) when simplifying assumptions about gas incompressibility andthoracic compliance are invoked as demonstrated in Appendix B.

During homogeneous ventilation (figures not shown here), for the same minuteventilation rate, increasing cycling frequency results in smaller tidal volume dis-placement, increased perfusion rates, and lower arterial O2 and higher CO2 bloodtensions. This results in a decrease in O2 and CO2 gas exchange performance withincreasing frequency. The gradation based on inspiratory waveforms qualitativelyresembles that observed in Fig. 6. Effects of increasing cycling frequency whilemaintaining the minute ventilation unchanged are summarized in Fig. 11 duringsquare wave ventilation. Despite the higher mean perfusion rate obtained at in-creased cycling frequency, the significant inefficiency in arterialization (low O2

and high CO2 blood tensions depicted in the Fig. 11 (middle)) results in poorerperformance with respect to overall gas exchange rates.

Effect of imposed (extrinsic) PEEP

Figure 12 depicts the effects of externally applying progressive levels of PEEP(viz., 5 and 10 cm H2O, respectively). Elevation in the levels of PEEP lead to anincrease in mean alveolar and airway pressures and functional residual capacitywhich, in turn, lead to a reduction in the rate of perfusion through the capillarywhen Pa and Pv are maintained constant. Reduction in perfusion with PEEP leadsto a significant decrement in overall gas exchange rate of both O2 and CO2.Increasing alveolar volumes result in diluting alveolar contents with ambient air,ultimately leading to improvements in arterial oxygenation of blood as indicatedby the gradation in alveolar composition.

Effect on Expired Gases

The time course of expired gas concentrations is generally insensitive to thechoice of the inspiratory flow profile, all other conditions being the same. Allresults presented here are for the square inspiratory flow waveform. Figure 13depicts contrasting behavior observed in the tracings of the expired gas concentra-tion under different conditions. In the top panels, increasing endotrachael tuberesistance mainly affects the initial upstroke slope of the capnogram (or correspond-ingly, the initial downstroke on the oxigram). Increasing airway resistance mainlyhinders flow during expiration, affecting the rate of emptying of alveolar contents.This is directly reflected in the observed slopes in the capnogram. End-tidal compo-sitions are marginally affected as the alveolar contents are effectively exhaustedto the ambient at the end of the expiratory phase. In contrast, in the bottom panels,increasing the level of external PEEP affects gas exchange rates significantly. Gasexchange rate and hence alveolar composition is directly affected. This is nowdirectly reflected in the gas composition obtained at end-tidal conditions. Note thegradation with levels of external PEEP depicted in Fig. 13.

DISCUSSION

The model presented in this paper includes a concurrent description of respiratorymechanics, pulmonary blood flow, and gas exchange to examine the effects of

376 NIRANJAN ET AL.

FIG. 11. Effect of cycling frequency on overall gas exchange during square wave inspiratoryflow with branch resistive disparities. Minute ventilation and duty cycle were held constant at 6 L[BTPS]/min and 0.333, respectively.


FIG. 12. Effect of extrinsically imposed PEEP on alveolar composition, perfusion and gas exchangewhile employing a flat inspiratory flow ventilation. A cycling frequency of 12 breaths/min (withminute ventilation of 6 L [BTPS]/min and a duty cycle of 0.4) was maintained in all cases. Regionalresistance and compliance values were the same and equal to the nominal values as indicated in Table 1.

altering ventilation on capillary blood flow rate and ensuing gas exchange. Thiscoupled interaction has not been previously addressed in theoretical studies onmechanical ventilation.

Model predictions indicate that when symmetric regional ventilation is consid-ered, the choice of the pattern of inspiratory flow affects the time course of O2

378 NIRANJAN ET AL.

FIG. 13. Effect of changing conditions on expired gas composition while employing a squarewave inspiratory flow ventilation. A cycling frequency of 12 breaths/min (with minute ventilation of6 L [BTPS]/min and a duty cycle of 0.4 was maintained in all cases. Regional resistance and compliancevalues were the same and equal to the nominal values as indicated in Table 1. (A) Effect of endotrachaeltube resistance. (B) Effect of externally imposed PEEP.


and CO2 alveolar composition, capillary perfusion, and gas exchange rates (Figs.3, 4, and 5). Descending waveform yields the highest cycle-averaged airway,intrapleural and alveolar pressures while the ascending waveform yields the lowestvalues. In contrast, mean capillary perfusion rate is greatest during ascending flowand lowest during descending flow. Mean end capillary blood gas tension donot significantly change with choice of the inspiratory waveform. However, thecombined effect due to perfusion discrepancies subsequently result in changes ingas exchange rates. The descending waveforms favor CO2 exchange while theascending waveform favors O2 exchange. Reduced thoracic compliance exaggeratesvariations in intrathoracic pressures, which significantly decrease capillary perfu-sion rates, leading to deleterious effects on O2 and CO2 gas exchange rates (Fig.6). The square waveform grades in between throughout the study.

Parameters affecting the mechanical behavior of the lungs and/or regional ventila-tion (such as airway resistance and alveolar compliance) determine the extent towhich ventilation affects gas exchange. Alterations in inspiratory waveforms wouldbe expected to significantly affect the instantaneous V

˙/Q ratios with concomitant

effects on regional alveolar gas partial pressures and end-capillary blood tensions.Computed results presented here indicate that regional alveolar partial pressurestrack the cycle-averaged V

˙/Q, thereby attenuating the effects of significantly larger

intracycle V˙/Q variations. This is, in large part, due to the capacitance of the

alveolar compartments. Our finding that descending flow facilitates the most evendistribution of ventilation is in line with other reports (21, 23, 25, 27). The peakalveolar pressure distribution ratio, PTR, is the lowest with descending flow evenduring heterogeneous alveolar ventilation (Fig. 7).

Differences in regional distribution ratios vary significantly with cycling fre-quency. Inequalities in compliance yield pronounced effects on net gas exchangerates at low frequencies of operation, whereas the adverse effects of inequalitiesin resistance are more pronounced at higher frequencies (Fig. 10). One convenientway of evaluating the effects of airway mechanics on distribution of ventilationhas been to employ RC time constants (3). While alterations in regional timeconstants as a consequence of differences in regional airway resistance and/oralveolar compliance do affect distribution of ventilation, they do not include theeffects of airway mechanics on regional perfusion and thus are inadequate toevaluate the effects of different inspiratory waveforms on gas exchange.

Application of extrinsic PEEP elevates mean intrathoracic pressures, therebyadversely affecting overall pulmonary perfusion. This results in reduced gas ex-change performance in spite of dilution of alveolar contents because of increasedalveolar volume (Fig. 12). Time course of expired gas concentration is dictatedby the combined effects of gas exchange rate and expiratory air flow rate. Factorsthat lead to a reduction in expiratory airflow rate alone result in lowering of theinitial upstroke of the capnogram (or, correspondingly the initial downstroke ofthe oxigram). End-tidal values for CO2 change proportionally with CO2 exchangerates and inversely with O2 exchange rates (Fig. 13).

The model on the air-side comprised an upper airway compartment connectedto the alveolar region through a serial-parallel arrangement of peripheral airways.

380 NIRANJAN ET AL.

Alternate descriptions using a nonlinear model including the use of a mid airwaycollapsible segment have been employed to describe the mechanics of airflow (5).A collapsible segment is necessary to describe airway closure due to large positiveintrapleural pressures occurring during forced expiration. In a recent study, describ-ing the dynamics of the forced vital capacity maneuver necessitated the use of acollapsible segment (31). In the study presented in this paper, a linear pressure–volume relationship (thereby implying a constant alveolar compliance) was em-ployed to characterize the comparatively small excursions in lung volume andintrapleural pressures encountered during mechanical ventilation.

Model Limitations

The analysis presented must be considered within the context of severalmodel limitations:

1. The use of a lumped model precludes an accurate depiction of time delaysin the capnogram.

2. The description of pulmonary blood flow ignores gravitationally inducedchanges in hydrostatic pressures. The use of constant pressure sources to depictpulmonary arterial and venous pressures ignores the pulsatile blood flow causedby pumping action of the right ventricle and left atrium as well as compensatorychanges in arteriolar and venous pressures. Compensatory effects on pulmonaryarterial pressure resulting from changes in pulmonary resistance are not considered.Furthermore, modulation of pulmonary vascular resistance by O2 tension in blood(33, 35) has also been neglected.

3. The potential effects of PEEP on alveolar recruitment as would be expectedin patients with atelectasis (or shunt) are ignored.

4. Absence of inertial elements in the model limit its consideration to relativelylow frequencies. Investigation of higher frequencies must necessarily include in-ertance elements.

5. Changes in alveolar and blood gas composition are assumed to be causedby the effects of ventilation and perfusion at the level of the alveolar capillary.Mixed venous composition of blood entering the capillary bed is assumed to beconstant. Hence the predicted results are strictly applicable only for the evaluationduring open-loop conditions.

CONCLUSIONS

Countering effects describe the interaction between respiratory mechanics andperfusion on gas exchange during mechanical ventilation. Lung inflation decreasesblood flow through the pulmonary capillaries. This is an important factor whileevaluating the potential benefit of a certain type of gas flow pattern during respira-tory treatment. The cumulative effect on gas exchange rate is mixed because ofthe opposing outcomes of ventilation on perfusion. For instance, although thedescending waveform promotes blood arterialization, the ensuing reduction inperfusion rate dominates, thereby tending to decrease O2 exchange rate. In contrast,with the ascending flow pattern, a higher mean perfusion rate prevails, thus yielding


a higher O2 exchange rate across the capillary in comparison. The effect on CO2

exchange rate is dictated by the nature of the CO2 blood dissociation curve; in theregime of normocapnic operation, the larger slope of the dissociation curve man-dates large changes in CO2 content for slight changes in blood CO2 tension. Thisoverrides the benefits of improved perfusion during ascending flow. The descendingwaveform promotes better exchange rates with ascending flow trailing the listamong the waveforms investigated.

However, if the gas exchange rate that is time-averaged over the course of arespiratory cycle is to be considered as the criterion for defining ventilator perfor-mance, no discernable (hence clinically significant) difference can be reportedwhile comparing the various forms of inspiratory airflow. This remark holds trueeven for the scenario when RC mismatches were considered. Generally, the perfor-mance of the flat and sinusoidal waveforms graded between that of descendingand ascending flows. In addition, regional pressures and ventilatory dynamics causesignificant effects on regional gas exchange rates.

Modulation of cycling frequency yields contrasting behavior. Time constantdiscrepancies yield proportionate changes in the impedances of the separate path-ways. The volume-elastic and flow-resistive properties are different, however.At lower frequencies, effective branch impedances are mainly influenced by thecompliant properties, manifested by compliant disparities significantly affectingregional ventilation, alveolar species composition, and ensuing gas exchange at lowfrequencies. Resistive disparities are more apparent at higher frequencies, however.

The importance of considering the concomitant effects of perfusion on gasexchange is readily apparent. Increasing intrapleural pressure (due to either extrin-sic/auto PEEP or impeded movement of the chest wall) results in decreased and,hence, unfavorable perfusion rates leading to decrements in gas transfer.

APPENDIX A

Model equations are developed by employing macroscopic balances on theoverall mass, overall linear momentum and species mass in predefined controlvolumes representing the anatomic dead space and the individual alveolar regions.Air is assumed to behave ideally with its overall mass density (r) being definedby a (constitutive) ideal gas law equation of state, P 5 rRT/M, where R is theuniversal gas constant. Changes in density of air throughout the system are thereforeaccounted for and governed by the corresponding changes in the total pressure inthe various nodal point regions. The molecular weight of air, M, is presumed tobe invariant to changes in air composition in the ensuing formulation.

(I) Mechanics

Air-side. The dynamic equations for the total pressure and volume in the Nalveolar regions (PAj and VAj for the jth region respectively) and the resultingintrapleural pressure (PPL) can be written compactly as

382 NIRANJAN ET AL.

1 ??? 0 ??? 0 0 C1VA1

PA1

dVA1

dt

??? ??? ??? ??? ??? ??? ??? ??? ???

0 ??? 1 0 ??? 0 CNVAN

PAN

dVAN

dt

??? 0 1 ??? 0 21 = 0 , [1]21

CA1

dPA1

dt

??? ??? ??? ??? ??? ??? ??? ??? ???

0 ??? 0 ??? 1 21 021

CAN

dPAN

dt

??? 0 ??? 0 1 021

Cth2

1Cth

dPPL

dt

where CAj is the compliance of the alveolar region j and Cth is the compliance ofthe thorax. In general, these compliances can be nonlinear. The above representationlends itself to easy extension for the general case of considering N parallel alveolarregions in the model. The form of Cj for region j differs during inspiration andexpiration and is given as

Cj 5PD

PAj

QDAj 2Ps TB

TsPAj

F*TOTj during inspiration [2a]

Cj 5 2QADj 2PsTB

Ts PAj

F*TOTj during expiration, [2b]

where PD is the total pressure in the dead space. F*TOTj is the total rate of gasexchange of O2, CO2, and N2 across the alveolar–capillary barrier in alveolar spacej and is given as

F*TOTj 5 oNTOT

i51(Qout

bj Coutbij 2 Qin

bj Cinbij). [3]

The summation represents the total transfer rate (STPD) of all species i across thecapillary wall in alveolar region j. The corresponding dynamic equation in thedead space region is written as

dPD

dt5

1VDFPE QED 1 o

j5N

j51(QADj PAj 2 QDAj PD)G during inspiration [4a]


dPD

dt5

1VDFo

j5N

j51(QADj PAj 2 QDAj PD) 2 QDE PDG during expiration. [4b]

The summations in Eqs. [4a] and [4b] account for the cases when inspiration andexpiration in the individual alveolar regions do not occur in synchrony, therebyresulting in pendelluft flow between the alveolar regions. At any point in time inany alveolar branch j, only one of the flows (QDAj or QADj) will be nonzero. Allflows listed above are derived from the balance of overall linear momentum in eachregion and are evaluated at body temperature. Ignoring the inertial contributions (20)the expressions for flow in the various regions are given as

QED 5PE 2 PD

RUAW[5a]

QDAj 5PD 2 PAj

Rj[5b]

QDE 5PD 2 PE

RUAW[5c]

QADj 5PAj 2 PD

Rj. [5d]

The resistances, RUAW and Rj , in general, can take different values during theinspiratory and expiratory phases.

Blood-side. The pulmonary vasculature is partitioned into (i) extraalveolar pul-monary arterial and venous compartments whose volumes are modulated by theintrapleural pressure and (ii) intraalveolar regional capillary compartments whosevolumes are modulated by the corresponding regional alveolar pressure. The resis-tance to flow through each of the compliant compartments is described using aStarling resistor type relationship and is a function of the corresponding compart-mental volume. Mean pulmonary arterial and venous hydrostatic pressures (Ppa

and Ppv, respectively) are assumed to be constant and are denoted by constantpressure sources of 15 and 5 mm Hg relative to the intrathoracic pleural pressure,respectively. Pulsatile effects resulting from the pumping of the right ventricleduring the cardiac cycle are hereby ignored. The corresponding dynamic equationsare given as

Pa 5 PTMa 1 PPL [6a]

Pcj 5 PTMbj 1 PAj [6b]

Pv 5 PTMv 1 PPL [6c]

384 NIRANJAN ET AL.

Ra 5 Ra0 1V maxa

Va2

2

[6d]

Rcj 5 Rcj0 1V maxbj

Vbj2

2

[6e]

Rv 5 Rv0 1V maxv

Vv2

2

[6f]

dVa

dt5

(Ppa 2 PTMa)

Ra2 o

j5N

j51

(Pa 2 Pcj)

Rcj /2[6g]

dVbj

dt5

(Pa 2 Pcj)

Rcj /22

(Pcj 2 Pv)

Rcj /2[6h]

dVv

dt5 o

j5N

j51

(Pcj 2 Pv)

Rcj /22

(PTMv 2 Ppv)

Rv, [6i]

where the subscript j again indicates the region perfusing alveolar compartment j.In Equations [6g], [6h], and [6i] blood flow through the pulmonary circulation isimplicitly assumed to be incompressible. Perfusion through the capillary bed(Qin

bj and Qoutbj , respectively) is governed by the gradient in the regional nodal

pressures listed in Eqs. [6a] through [6c]. Blood flow entering and leaving thecapillary bed (Qa and Qv, respectively) is similarly dictated by the correspondingnodal pressure gradients. Ignoring inertial contributions once again, expressionsfor the regional blood flow rates are listed below:

Q inbj 5

(Pa 2 Pcj)

Rcj /2[7a]

Q outbj 5

(Pcj 2 Pv)

Rcj /2[7b]

Qa 5(Ppa 2 PTMa)

Ra[7c]

Qv 5(PTMv 2 Ppv)

Rv. [7d]

(II) Gas Exchange

Species mass balance equations describing the change in partial pressure ofspecies i in the airways (assumed well-mixed) can be written separately for theinspiratory and expiratory phases as (subscript denoting alveolar region j )


Inspiration

dPDi

dt5

1VDFQED PEi 1 o

j5N

j51(QADjPAij 2 QDAjPDi)G [8a]

dPAij

dt5

1VAj

FQDAjPDi 2 PAij

dVAj

dt2 1PSTB

TS2 fijG [8b]

Expiration

dPDi

dt5

1VDFo

j5N

j51(QADjPAj 2 QDAjPDi) 2 QDEPDiG [9a]

dPAij

dt5

1VAj

F2QADjPAij2 PAij

dVAj

dt2 1PSTB

TS2 fijG, [9b]

where fij, given by [Q outbj C out

bij 2 Q inbj C in

bij], represents the gas exchange rate ofspecies i across the lumped capillary perfusing alveolar region j. End-capillaryequilibration with alveolar composition is assumed; hence the species tension ofblood leaving the capillary, Pecij, is assumed to equal the alveolar species partialpressure, pAij. The formulation for gas exchange as presented herein then cannotbe applied to address the scenario of diffusion impairment across the alveolarcapillary barrier.

APPENDIX B

The driving point impedance on the air side for the system presented in thispaper defines the effective “load” to the ventilator and can be derived easily bydetermining the equivalent admittance and subsequently carving out the real andimaginary parts. The process yields the equivalent resistance and compliance as

Requiv 5 RU AW 11

v(l2a 1 l2

b)(mala 1 mblb) [10a]

Cequiv 52(l2

a 1 l2b)

(mbla 2 malb), [10b]

where the terms la,b and ma,b themselves are implicit functions of C1, C2, Cth, Cgu,Cg1, Cg2, t1, t2 and the cycling frequency, v (in radians). The regional time constants,tj are defined as tj 5 RjCj. The terms la,b and ma,b indicated in Eqs. [10a] and[10b] are defined as

386 NIRANJAN ET AL.

ma 5 vC1C2[t2C1 1 t1C2 1 Cth(t1 1 t2)]

1 v(C1 1 C21Cth) [t1C2Cg1 1 t2C1Cg2] [11a]

mb 5 2C1C2(C1 1 C2 1 Cth) 1 v2t1t2[C1C2(Cg1 1 Cg2) 1 Cg1Cg2(C1 1 C2)]

1 v2t1t2Cth(C1 1 Cg1)(C2 1 Cg2) [11b]

la 5 C1C2[(C1 1 C2 1 Cth)(Cgu 1 Cg2) 1 Cth(C1 1 C2)]

2 v2t1t2CguCth(C1 1 Cg1) (C2 1 Cg2)

2 v2t1t2Cgu[Cg1Cg2(C1 1 C2) 1 C1C2(Cg1 1 Cg2)] [11c]

lb 5 vCguCth[t1C2Cg1 1 t2C1Cg2 1 C1C2(t1 1 t2)]

1 vCgu[C1C2(t2C1 1 t1C2 1 t1Cg1 1 t2Cg2) 1 t2C21Cg2 1 t1C2

2Cg1]

1 vCth[C21 t2(C2 1 Cg2) 1 C2 t1(C1 1 Cg1)(C2 1 Cg2)]

1 vC2Cg2[(t2C1 1 t1C2)(C1 1 Cg1)]. [11d]

If gas flow is assumed to be incompressible, the above expressions are simplifiedby evaluating the limiting expressions as Cg1, Cg2, and Cgu all → zero. This wouldresult in the following modified expressions

Requiv 5 RU AW 1(t1C1 1 t2C2) 1 v2t1t2(t1C2 1 t2C1)

(C1 1 C2)2 1 v2(t1C2 1 t2C1)2 [12a]

Cequiv 5(C1 1 C2)2 1 v2(t1C2 1 t2C1)2

(C1 1 C2) 1 v2 (t21C2 1 t2

2C1) 11

Cth[(C1 1 C2)2 1 v2(t1C2 1 t2C1)2]

.

[12b]

Furthermore, if the thoracic compliant element is completely ignored (by settingCth → `), Eqs. [12a] and [12b] may be further simplified. The resulting expressions(when RUAW 5 0) would then correspond with those reported in Otis et al. [36]and elsewhere [10]. The equivalent resistance and compliance diminish with cyclingfrequency and approach specific limits at the extremes of frequency.

APPENDIX C

Empirical dissociation curves based on previously reported studies (15, 28, 32,41) were employed to relate species contents to the equilibrium gas tensions inthe equations describing gas exchange in Appendix A. The relationship betweenthe gas tensions and species contents is complex and nonlinear. The first stage ofthe calculation procedure involves the following series of calculations:


pH* 5 7.59 2 0.2741 log1PCO2

20 2 [13a]

dT 5 37 2 T [13b]

dpH 5 7.4 2 pH* [13c]

dpCO2 5 40/PCO2 [13d]

V 5 0.024 dT 2 0.4 dpH 1 0.06 log10(dpCO2) [13e]

X 5 PO2 10V 26.826.8 1 DP50

[13f]

SO2 5X 3 1 150X

X 3 1 150X 1 23400[13g]

Y 5 3 3 1025 Hb (1 2 SO2) [13h]

pH 5 7.59 1 Y 2 0.2741 log1PCO2

20 2. [13i]

Equation [13a] is used to first guesstimate pH. Next, for a given temperature(T in Eq. [13b]) and known values of hemoglobin concentration (Hb), 2–3 DPGshift factor (DP50) and base excess (BE), Eqs. [13c] through [13i] are iterativelysolved till a convergent value for pH is obtained. The resulting oxygen saturation(SO2) so obtained is then used to determine blood O2 content as explained below.X denotes virtual PO2 used to calculate oxygen saturation levels using a standardizeddissociation curve correlation. In the second stage, the following set of calculationsare performed:

pKp 5 6.086 1 0.042 dpH 1 (38 2 T)(0.00472 1 0.00139 dpH) [14a]

aCO2 5 0.0307 1 0.00057 dT 1 2 3 1025 dT 2 [14b]

D1 5 0.02226aCO2PCO2(1 1 10(pH2pKp)) [14c]

D2 5 1 20.02924 Hb

(2.244 2 0.422 SO2)(8.74 2 pH). [14d]

Finally, blood species contents (in units of ml (STPD) per ml blood) are deter-mined using

CbO2 5 1.39 HbSO2

1001 3 3 1025 PO2 [15a]

CbCO2 5 D1 3 D2. [15b]

388 NIRANJAN ET AL.

Throughout this study, all variations in blood-gas tensions were assumed to bedirectly attributed solely to respiratory effects with no compensatory metabolicmechanisms. Hemoglobin concentration was assumed to be equal to 14.46 g/100ml blood and DP50 was assumed to be zero.

ACKNOWLEDGMENTS

The authors are grateful for the financial support provided by the Biomedical Engineering Centerat the University of Texas Medical Branch, Galveston, Texas. Additional support to Dr. A. Bidani bythe Moody Foundation, Galveston, Texas (Grant 94-48), The Shriner Hospital for Crippled Children,Galveston, Texas (Grant 15859), and The Constance Marsili Shafer Research Fund, and to Dr. J. B.Zwischenberger by The Shriner Hospital for Crippled Children, Galveston, Texas (Grant 15859), isalso deeply appreciated.

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