Effects of common dead space on inert gas exchange in mathematical models of the lung JOHN B. FORTUNE AND PETER D. WAGNER Department of Medicine, University of California, San Diego, La Jolla,

FORTUNE, JOHN B., AND PETER D. WAGNER. Effects of common dead space on inert gas exchange in mathematical models of the Lung. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 47(4): 896-906, 1979.-Theoretical gas exchange is compared in lung models having two different types of dead space. In one, the dead space of a lung unit is “personal” and contains gas equivalent in composition to its own alveolar gas; in the other, the dead space is “common” and contains mixed gas from all gas-exchanging units. Formal algebraic analysis of tracer inert gas exchange in two-compartment models shows that values of compartmental ventilation and perfusion can be found that establish one and only one personal dead-space model equivalent for every common dead-space model. When the total dead space and distribution of blood flow and ventilation in the two models are the same, common dead space will always result in improved inert gas elimination. Under these conditions, the amount of improvement is usually greatest when the partition coefficient of the inert gas is between 0.1 and 1.0 and when there is greatest disparity in the ventilation-perfusion ratios #A/@. In the inert gas elimination technique that analyzes all dead space as personal, the presence of common dead space consistently causes the recovered VA/Q distributions to be narrower than the actual distributions, but the resultant error is small. ventilation-perfusion

ratio

OF RESPIRATORY gas exchange and lung models using distributions of ventilation-perfusion ratios (VA/Q) consider the anatomic dead space to be only a dilutional volume with no effect on gas exchange other than to reduce alveolar ventilation. This assumption is correct if every alveolus rebreathes only its own expired gas from a “personal” dead-space compartment anatomically extending to the source of the inspired gas, thereby guaranteeing that the dead-space gas will be identical in composition to the alveolar gas. If the dead-space gas entering an alveolus has a different composition than alveolar gas at end expiration, this will influence its gasexchanging capabilities. In normal lungs, the upper bronchial tree, trachea, pharynx, and oral-nasal passagesprovide a common mixing chamber from which alveoli rebreathe dead-space gas; in the clinical situation this volume may be increased by the addition of external breathing devices. In 1960, Ross and Farhi (9) presented an analysis of the reinspiration of “common” dead space on oxygen and carbon dioxide exchange using the single-compartment mathematical models available at the time. They concluded that the presence of common dead space resulted

EQUATIONS

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in an alteration of the composition of average inspired gas which led to a limitation of the range of alveolar gas compositions, and a shift in the position of the Oz-CO2 line when compared to the classical analysis of Riley and Cournand (8) and Fenn, Rahn, and Otis (4). For multicompartment models they implied that a buffering effect would be exerted on alveolar gas composition tending to improve gas exchange in low \;TA/Q compartments and impair gas exchange in high VA/Q compartments. The magnitude of this effect could not be demonstrated, but they offered the final admonition that common dead space must be considered as a factor when working with the gas-exchange equations of the VA/Q theory. While developing a method of measuring continuous distributions of ventilation-perfusion ratios using the steady-state exchange of several inert gases, we have questioned the influence of the reinspiration of common dead space on the measured excretions and retentions and derived distributions. The unknown nature and magnitude of this effect has prompted a mathematical analysis of the potential influence of common dead space on the exchange of both inert and respiratory gases.In this paper the influence of common dead space initially will be examined algebraically (formally) using simple twocompartment inert gas models of the lung and subsequently by numerical analysis for more complicated multicompartmental models considering inert gasesand the effect of common dead space on recovery of ventilationperfusion ratio distributions. METHODS

Summary of Approach Throughout this study, gas exchange will be examined in mathematical models of the lung containing either or both of two types of dead space. The first is termed personal, which means that each functional gas-exchanging unit has a separate anatomic dead space from which it inspires dead-space gas that is identical to its alveolar gas. With personal dead space, effective alveolar ventilation can be determined by subtracting the dead space ventilation (VD) from the expired ventilation (VE). The second type of dead space, termed common, constitutes that part of conducting airway volume which, at end expiration, consists of mixed alveolar gas from all units in proportion to their expired ventilation. Subsequent inspiration of this dead-space gas will change the composition of gas in each lung unit and will therefore modify gas exchange.

0161-7567/79/oooO-~$~1.25

Copyright

0 1979 the American

Physiological

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Society

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This paper has been divided into two sections. In the first, a rigorous algebraic analysis is given for the effect of common dead space in two-compartment models of the lung using inert gases. In the second section, a numerical analysis is performed to study the influence of common dead space on inert gas exchange in multicompartment models. For the algebraic analysis, the principles of conservation of mass (Fick principle) are applied to two-compartment lung models containing dead space. The compartments will differ with respect to the fractional distribution of total ventilation and perfusion that determines the compartmental ventilation-perfusion ratios. A variable fraction of the total dead space is common, with the remainder being personal. Algebraic expressions for the individual compartmental inert gas partial pressures are derived and then, by use of standard mixing theory, the general equations for the mixed arterial partial pressure of each inert gas are formulated. These expressions will be used in two ways. In the fust, the mixed arterial partial pressure of a lung model in which all of the dead space is personal will be equated to that in which a variable amount of the same total dead space is common. Along much the same lines as reported elsewhere (12), the conditions of ventilation and perfusion distribution that guarantees such equivalence will be established. In this way it will be possible to determine whether the perturbation of gas exchange caused by common dead space can be analyzed in terms of a unique model containing only personal dead space. If such equivalence and uniqueness are established, then the quantitative relationships between the distributions of ventilation and blood flow in the two models with equivalent gas exchange can be determined and the magnitude of the effect of common dead space assessed. These algebraic expressions will also be used to compare overall gas exchange in a model with common dead space to that occurring in an otherwise identical model in which all of the dead space is personal. With such a comparison, general statements concerning the relative effects of common and personal dead space on gas exPERSONAL

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change will be made. The manner in which these relative effects vary with gases of different partition coefficients and with different distributions of ventilation and perfusion among the compartments will also be examined. In the subsequent numerical analysis of multicompartment models, the effects of introducing common dead space into the model while keeping all other factors unchanged will be studied for the inert gasesused in the measurement of continuous distributions of VA/& ratios. Various representative distributions of ventilation and blood flow will be generated, and the internal consistency of the effects on inert gas exchange and on 02 and CO2 exchange examined. A specific question to be asked is whether the perturbation of inert gas exchange caused by common dead space in multicompartment distributions sufficiently alters retention-solubility curves (15) to cause substantial changes in the recovered pattern of ventilation-perfusion inequality as determined by the multiple inert gas-elimination technique. The Model Two-compartment approach. The anatomic basis for common and personal dead space as used in the algebraic analysis is shown in Fig. 1. The symbols for the partition of ventilation and blood flow between the compartments and the partial pressures are defined in the caption. Note that the common dead-space model allows for a variable portion of the dead space to be common, fractionally defined by p. If p = 1, then all of the dead space will be common; conversely, if p = 0, all of the dead space is personal. To formulate the expressions for gas exchange, several assumptions are made. a) Ventilation and blood flow are continuous rate processes. b) Gas exchange is in a steady state, and the Fick principle can be applied. c) In each compartment, the gas tensions are homogeneous thereby assuming perfect mixing. d) In each compartment there is complete diffusion equilibration between alveolar gas and end capillary blood. COMMON

Pa common

FIG. 1. Anatomic representation of two gas-exchanging alveoli with personal (p) and common (c) dead space. P1 and Pz are the compartmental partial pressures for each test gas. Fractions, F and G, specify the proportion of total blood flow (QT) and total expired ventilation (\j~) directed to each compartment. In the personal dead-space model gas tension in the dead space is equal to alveolar gas tensions, while in the common deadspace model, gas tension in the common dead space is equal to mixed expired gas from both alveoli. In the common deadspace model it is possible to specify only a portion of dead space as common (p) with the rest being personal (1 - p). Vinyl, and VZ,H,, refer to inspired ventilation to the left (VI) and right (VI*) compartments, respectively, and are related to expired ventilation by characteristics of the resident gas. Distribution of VD is the same as VE and is determined by value of F.

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e) In the common dead-space model there is complete mixing in the dead-space compartment, and the gas tensions here reflect the mixed expired gas tensions of all compartments weighted according to their expired ventilation. Implicit in this assumption is the absence of serial emptying of alveoli. f) The dead space is distributed to the individual compartments in proportion to their alveolar ventilation. While this may not be the case in some physiological situations, arbitrary distribution of dead space not in proportion to ventilation presents considerable mathematical difficulties and will not be treated here. An inert “test” gas that has a linear dissociation curve with blood defined by its partition coefficient, X, will be present in trace quantities such that its exchange will have no volumetric influence on the gas-exchanging properties of the unit. Also, in each unit will be a “resident” gas that because of its assigned solubility will necessarily influence the exchange of the test gas by determining relationships between total expired and total inspired gas volumes. In all the examples, \jE will be the assigned variable and from this inspired ventilation (VI) can be determined. When including dead space in the calculations the total ventilation of the unit must be considered; alveolar ventilation can be derived by subtraction of the dead-space ventilation from the total ventilation. The assigned global variables that are identical in each model are \j~ (total expired ventilation), QT (total pulmonary blood flow), PV (mixed venous test gas partial pressure), \jD (total expired or inspired dead-space ventilation), and PI (partial pressure of the inspired test gas). The venous partial pressure (PV,) and partition coefficient (h,) of the resident gas will also be identical in each example. For simplicity, the PV of all test gases will be assumed to be constant and equal to one. The total gas tension in each compartment and arterial blood must sum to 713 Torr (barometric pressure (PB) minus water vapor pressure (P&O)). MuLticompartment approach. Unimodal and bimodal distributions of alveolar ventilation and perfusion are generated in a log normal fashion in a 50-compartment model (17) such that the degree of mismatching in each mode is determined by the log standard deviation of the distribution of ventilation (o\jE); as C&E becomes larger, the distributions become broader with ventilation and perfusion being more mismatched. Dead-space ventilation is then distributed in a desired fashion to the compartments. If the dead space is only personal then the gas tension in the reinspired dead space is equal to the gas tension in the compartment to which it is directed, but if it is common then it is the same as the mixed expired alveolar gas from all compartments. For simplicity in this model the dead-space gas is either all common or personal (i.e., p = 1 or 0) and the exchange of inert gases with different partition coefficients is examined. Factors such as oxygen and carbon dioxide dissociation curves, hemoglobin, hematocrit, temperature, acid-base status, and barometric pressure are each the same for the two models when dealing with the physiologic gases. Equations of Gas Exchange Two-compartment approach.

With

the total ventila-

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P. D. WAGNER

tion, blood flow, dead-space ventilation, and the inspired and mixed venous partial pressures of the test gas being the same in both models, algebraic expressions defining the behavior of the test gas in terms of the Fick principle can be formulated for each compartment. The only real difference in the equations for the models is the inclusion of the common dead space as part of the inspired gas in the common dead-space model. Since a relationship between the distributions of ventilation and perfusion in the personal (p) and common (c) models is the goal of this analysis, the fractional ventilations (F, and F,) and blood flows (G,, and G,) have been independently defined in this section. Because of their complexity, the actual equations have been relegated to the APPENDIX. By rearrangement of the equations of mass balance, general expressions for the compartmental test gas partial pressures are determined in each model. Each compartmental partial pressure is defined generally in terms of the test gas partition coefficient. If there is no diffusion impairment (assumption d, above), then the partial pressure of the test gas in the effluent blood from each compartment will equal the compartmental partial pressure of that gas. The general expression for the partial pressure of the test gas in the mixed arterial blood for each model (Pa,, and Pat) is determined as the blood flow-weighted mean of the compartmental partial pressures. These expressions, which are again defined in terms of X, will be used as a measure of the gas-exchanging capacity of each model. To determine the existence and definition of the conditions of equivalence, the general expressions of Pa, and Pa, are equated. If a unique solution for F, and Gc in terms of F, and G, can be found in this equivalence statement, then it will have been shown that a single personal dead-space equivalent model exists for each set of conditions in common dead-space models. Next, the fractional ventilation and blood flow will be set identical in each of the models (F,. = F,,; G, = G,,) and these equations will be used to determine the relative influence of common dead space on gas exchange when compared to personal dead space in otherwise identical models. To accomplish this, the rate of elimination of the test inert gas from the venous blood of each model will be analyzed by examining its arteriovenous difference since the total gas transfer is proportional to this difference. The magnitude of the relative effects of common and personal dead space will be determined by comparing the arteriovenous test gas differences in each of the models for different test gases and various distributions of ventilation and blood flow. Multicompartment approach. Both the complexity of the calculations and the nonlinearity of the 02 and CO:! dissociation curves dictate the use of numerical (as opposed to the above algebraic) analysis in multicompartment models utilizing the respiratory gases. Previously published subroutines (18) were modified so that personal and common dead space could be included using the same logic as in the two-compartment analysis above. The common dead space is again distributed in proportion to alveolar ventilation as in the algebraic model. The multicompartment common dead-space model presents a particular numerical problem in that alveolar

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gas concentrations and the dead-space gas concentrations directly influence each other. A solution is found by successive iterations in the following manner. First, the gas compositions in each compartment are calculated as if the dead-space compartment is personal and contains no inert gas. Then, the gas compositions in the common dead-space compartment are determined as the expired ventilation-weighted mean of each of these compartmental gas tensions. From this value, the compartmental gas tensions are recalculated, in turn, allowing calculation of a common dead-space composition. The above procedure is repeated until the mixed expired oxygen and carbon dioxide partial pressure (POT and Pc0.J or inert gas partial pressure varies by less than a certain tolerance in two successive iterations. This tolerance is selected to be less than 0.5 Torr for POT and PCO~ and less than 1.0 X lo-" (relative to PV = 1.0) for inert gases. In general, convergence is fairly rapid with the average number of iterations being 3 or 4. Initially, several representative distributions will be used to generate retention- and excretion-solubility diagrams for the cases when the dead space is first personal and then common in otherwise identical models. This will be performed for the elimination of inert gases with solubilities equal to those used in the six inert gas method for measuring ventilation-perfusion distributions (13). These retentions and excretions will then be used to recover the distribution by the method of enforced smoothing (Z), which considers the lung to have only personal dead space. Any difference found in the recovered distributions for the common and personal dead space models will reflect consistent influences of common dead space on this laboratory method.

To show that these relationships are independent of the solubility of the gas being considered, these conditions of equality were evaluated on a model that used 02 and CO2 as the test gases in physiological concentrations and N2 as the resident gas. Digital computer subroutines for O2 and CO2 gas exchange in compartments of varying VA/($ were modified so that they could be used in a two compartment system. Pairs of G, and F, values were generated for the personal dead-space model and the appropriate G, and F, values for the equivalent common dead-space model were calculated from Eqs. 21 and 22 (APPENDIX) as if tracer inert gas were present. Using these models, values of mixed arterial POT and PCO~ were calculated for each set of values and compared (mixed venous PO:! and PCO~ of 40 and 45 Torr, respectively). Figure 2 shows the results of the comparison and demonstrates that the conditions that are calculated for the inert gases in trace concentrations also hold for the respiratory gases at physiological concentrations even though the physiologic gases have complex blood dissociation curves. The calculated values differed by less than 1 Torr in all cases, which is well within the accuracy of the numerical analysis procedure used here. Relative Effect of Common and Personal Dead Space on Inert Gas Exchange With identical distributions of blood flow and ventilation, a ratio of the gas transfer (Rg) of personal and common dead-space models can be formulated as

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RESULTS

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The general equations derived for the mixed arterial partial pressures of the inert gases (Eqs. 10 and 11, APPENDIX) are set equal. The result of the analysis is the formal demonstration that for any inert gas, and distribution of ventilation (F,) and blood flow (G,) in the personal dead-space model, one and only one distribution of ventilation (F,) and blood flow (G,) could be found for the common dead-space model that equated the gasexchanging characteristics. Specifically, a set of equations exists that define the fractional compartmental ventilation and blood flow for the common dead-space model in terms of fractional compartmental ventilation and blood flow for the personal dead-space model and the other global variables (QT, VE, VD, and the characteristics of the resident gas). This relationship is independent of the test gas partition coefficient and therefore should hold for all gases of any solubility. A parallel analysis shows that it is possible to determine the values of F, and G, when F, and G, are known. Therefore, in analyzing lung function in the presence of common dead space, a unique representation can be found in a personal dead-space model. The interaction of common dead space on gas exchange, however, will be represented in a personal dead-space model as a perturbation of the compartmental ventilation and perfusion distribution.

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ARTERIAL PO2, mmHg COMMON DEADSPACE FIG. 2. Similarity in Pao, for models with common and personal dead space. Conditions of perfusion and ventilation that established equivalence of inert gas exchange for models with common and personal dead space were calculated and appear in insets. These results demonstrate that conditions of equivalence for inert gas exchange also apply for physiologic gases in spite of nonlinearity and interdependence of dissociation curves. Similar agreement was found for Pa 1 or PI = 0.0, then it can be shown that inert gas transfer in the common dead-space model will always be greater than in the personal dead-space model (Rg > 1.0) when

P. D. WAGNER

personal dead space. Possible situations where this might occur include conditions of carbon dioxide exchange when small amounts of carbon dioxide are present in the inhaled gas. It is important to note from the above equations that X,.(1 - FV,)*PI all of the above conditions which cause common dead xtest gas > (PI - PV) space to affect gas exchange are independent of the where X, is the partition coefficient of the resident gas particular distribution of ventilation and perfusion and and FV, is the fractional concentration of resident gas in the amount of dead space. venous blood. For simple conditions, the generality and power of this Effect of Blood-Gas Partition Coefficient statement is evident. When the test gas is absent from on Relative Gas Elimination the inspired gas (PI = O.O), then the expression to the The magnitude of the difference in gas transfer when right of the inequality is always zero, therefore making common dead space is imposed on a two-compartment Rg greater than one for every value of test gas X, proving model is not necessarily the same for all values of X and that the presence of common dead space improves gas there will exist a X such that for a given F and G the exchange for all gases. This is especially important for magnitude of the improvement will be maximum. Bethe inert gas method, which utilizes only gas elimination cause this analysis becomes prohibitively complicated (PI = 0.0). Another situation where common dead space when dealing with a soluble resident gas and an inspired will always improve gas exchange is in cases where the test gas tension greater than zero, the following analysis resident gas is insoluble (h, = O.O), regardless of PI. will be done only for test gas elimination (Pr/PV = 0) Conditions that offer the greatest improvement in gas when X, = 0 (i.e., VI = \jE). Under these conditions, elimination by the common dead space under these concommon dead space will always improve inert gas elimiditions will be discussed in later sections. nation (as demonstrated in the previous section). When PI # 0 and PI/PV > 1, the situation becomes To find the X for which there is maximal improvement much more complex and the range of values of h in which of gas elimination when common dead space is imposed common dead space improves gas exchange depends on on a system, the equation for Rg was differentiated with the relative values of all variables in the equation above. For a single relatively insoluble resident gas, the term respect to A and the resulting value was set to zero. The single positive root of this equation will give the X that is 1 - FV, will be near zero and X, will be quite small, thus affected most by common dead space under a given pair giving conditions similar to those above where the presof values of F and G. To graphically demonstrate the ence of common dead space will benefit ‘gasexchange for range of these values, an array of F and G values was almost all test gas solubilities. When the resident gas is created over all possible combinations from zero to one soluble then the term 1 - FV, becomes closer to one. If in increments of 0.05 (thus giving 400 pairs of VA/Q PI is much lower than PV for the test gas in this situation, ratios). From each of these pairs the “most affected X” the lower end of the possible range of X that will result in was calculated from a model with \~D/VE = 0.5 and the common dead space benefiting gas exchange will be close dead space being either all personal or all common. The to the value of the partition coefficient of the resident result was graphed as a point for each F and G value on gas. If PI is near the value of PV and the gasesremain in Fig. 3, thus given a three-dimensional plot. Note that Fig. trace concentrations, the lower end of this possible range 3 places the most affected h always in the decade of of X will be higher than the X of the resident gas. This X = 0.1-1.0. Figure 3 has roughly a saddle shape (with an condition will apply under conditions of gas uptake. For absent diagonal portion that corresponds to a homogeoxygen exchange under physiological conditions, N:! is neous lung, in which there will be no effect of the common the major resident gas (h = 0.01212) and the partial dead space when compared to personal dead space). pressure of oxygen is much greater in the inspired gas These values of most affected X correspond roughly to than in the venous blood; all but the most insoluble of ethane (from the inert gas method) and to oxygen. gases would allow co&non dead space to improve O2 exchange when compared to personal dead space. Inequality on Improvement In the situation when PI/PV < 1 but PI # 0, gas Effect of VA/Q) transfer in the common dead-space model will exceed of Gas Exchange by Common Dead Space that in the personal dead-space model whenever Not only does the solubility of the gas affect the magnitude of the improvement caused by imposition of XJl - FV,)~PI x test gas < common dead space on a system involved in inert gas PI - PV elimination, but also the relative ventilation and perfusion of each compartment. For a given h, the percentage where PI/PV < 1. In these particular conditions the denominator of the improvement of gas elimination caused by common dead expression to the right of the inequality will always be space is plotted against compartmental blood flow and negative. Because there can exist no test gas having a h ventilation in Fig. 4. In this case, arterial retentions of that is negative, this expression is always false and, under test gaseswere determined using Eqs. 10 and 11 (APPENthese conditions, the presence of common dead space will DIX) and the difference between the values with common always result in less gas exchange than the presence of and personal dead space were determined as a percentage

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of the personal dead-space value. In the upper left and lower right extremes of the partition axes (corresponding to extremely high and low VA/Q compartments) the percentage differences in gas exchange between the models is markedly larger. In areas where the blood flow and ventilation are more evenly matched, the change is much less to the point where there is no difference when the V and Q are identical. Thus, the more mismatched the ventilation and perfusion in a lung, the more improvement in gas elimination would be afforded by the presence of common dead space. In models of extreme VA& inequality, the percentage difference in arterial test gas values can be as high as 80%.

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“N PARTITION OF BLOODFLOW 3. Numerical example of partition coefficients of the gas in which the presence of common dead space most affects gas exchange in a two-compartment model. A series of combinations of fractional blood flows (G) and ventilations (F) have been generated in the horizontal plane and the partition coefficient of the gas most affected by common dead space is plotted on vertical axrs. This is done under conditions of inert gas elimination (test gas PI = 0) in which the presence of common dead space will always improve gas exchange when compared to an identical model with personal dead space. In this example, VE = 6.0 l/min, VD = 3.0 l/min (i.e., VD/VE = 0.5), and QT = 6.0 l/min. Diagonal corresponding to homogeneous lung is absent, since. under these conditions, gas exchange in models with personal and common dead space is equivalent. FIG.

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PARTITION OF BLOODFLOW FIG. 4. Numerical example of amount of improvement offered by common dead space when compared to identical model with personal deadspace in a two-compartment model. Horizontal plane is the same as in Fig. 3. Vertical axis demonstrates percentage improvement in gas exchange with common dead space with respect to distribution of ventilation and perfusion among compartments. Values of VE, VD, and QT are the same as in Fig. 3.

Effect of Common Dead Space on Recovery of Distributions with the Six Inert Gases Method

VA/Q

In the inert gas-elimination technique for obtaining distributions of ventilation perfusion ratios, all data are processed “as if” there is only personal dead space in the lung. Since we have shown inert gas elimination is improved maximally by common dead space among a set of gases with X in the 0.1-1.0 decade, asymmetric changes in the retention-solubility curves caused by common dead space might distort the recovered distributions. In the upper panels of Fig. 5,. a broad unimodal distribution with a EVE = 1.5 and VD/VE = 0.3 was generated and the retentions and excretions of the six inert gases were calculated with the dead space being all common or all personal. The resulting retentionand excretion-solubility curves show that the maximal difference between the curves is indeed in the range of h = 0.1-1.0. Although among the set of gases with low solubilities the percentage difference in the common and personal dead space curves may be greatest, it has been shown the absolute magnitude of the difference in retentions is most important in altering the recovered distributions (15). The recovered blood flow distributions from these data are shown on the right; both have been processed as if all dead space is personal. The replacement of personal dead space by common dead space in the model results in a narrower distribution from both ends of the VA/&$ spectrum; the recovered & for the personal dead space model is 1.47 and that for the common dead space model is 1.33. Common dead space in the model is therefore represented as a better matching of the ventilation and perfusion in the inert gas analysis. The lower panels of Fig. 5 demonstrate the same finding in a distribution that is bimodal and similar to those found in patients with asthma (11). Again, the improvement in gas elimination with common dead space in only a limited range of gas partition coefficients slightly distorts the recovered distribution and appears to represent better matching of the ventilation and perfusion in the lung units. However, in both cases, the general pattern and information content of the recovered distributions are similar. Figure 6 shows the effect of common dead space on the inert gas-elimination technique in a broader sense. In this case, the standard deviation of recovered unimoda1 ventilation distributions are compared for situations in which the retentions and excretions were calculated

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FIG. 5. Effect of common dead space on results obtained from inert gas-elimination technique for representative unimodal and bimodal distributions. Left panels show retentionand excretionsolubility plots for distributions with both types of dead space. Right panels show recovered distributions from these data. Presence of common dead space results in a narrowing of distributions from both high and low VA& directions.

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with common or personal dead space as above. The broadness of the distributions was increased and the amount of dead space was altered to determine the relative effects of these variables on the influence of common dead space; the alveolar ventilation remained constant in all examples (6 l/min). Figure 6 shows that, in all cases, distributions with common dead space tend to be represented as a narrower distribution than is actually present. As the distribution becomes broader the lines fall away from the identity line showing that the presence of common dead space results in a further discrepancy in the recovered distributions. The line for VD/VE = 0.5 is even farther to the right of the identity line showing increased volumes of common dead space tend to further aggravate the accuracy of the recovered distributions. It is well worth noting that distributions of VA/Q ratios with standard deviations of 1.5 only appear in lungs with greatly abnormal gas exchange. Standard deviations of 0.3-0.5 are closer to normal and Fig. 6 shows minimal effects of common dead space in this range (14). The result recovered from lungs with common dead space using this technique will always show some qualitative difference from actual distributions and in all cases provides a distribution that is somewhat more homogeneous. It is impossible to determine in practice the extent to which common dead space is influencing the recovered distributions, but from the examples shown it is justified to assume the actual distribution from a lung with common dead space is somewhat broader than that repre-

RATIO

sented by the inert gas technique; the basic pattern, however, is unlikely to be affected. DISCUSSION

Previous investigators have demonstrated the need to consider common dead space so as to fully explain gas exchange among lung units with uneven ventilation-perfusion ratios. Because all of the parameters necessary to define dead space are not realistically obtainable, full consideration of the effect of common dead space on gas exchange in all calculations is impossible. For this reason, assumptions have been made about the nature of the inspired dead space that make it easier to handle conceptually and mathematically; it is the relationship between the behavior of actual common dead space and the assumed behavior of personal dead space that has been analyzed in this paper. Mathematically, we have shown that values of blood flow and ventilation (i.e., compartmental VA/Q ratios) can be found in models with common and personal dead space which will equate the models’ gas-exchanging characteristics. The condition of an inert gas exchanging in a soluble inert resident gas used in this analysis was chosen to have the greatest applicability to physiological situations and yet be amenable to algebraic manipulation. A similar analysis of series and parallel ventilation (12) demonstrated that further definition of the conditions (i.e., two soluble inert resident gases,or physiologic resi-

Downloaded from www.physiology.org/journal/jappl at Midwestern Univ Lib (132.174.254.157) on February 14, 2019.

E FFECTS

OF

COMMON

DEAD

SPACE

ON

INERT

GAS

l-6

C

-2

q4

-6

PERSONAL

-8 GV,

I-0

l-2

l-4

903

EXCHANGE

l-6

DEADSPACE

6. Effect of increases in size of dead-space compartment and degree of homogeneity of ventilation and perfusion on recovery of distributions using inert gas-elimination technique. In this case, retentions and excretions were calculated from models with unimodal distributions of VA/~ as log standard deviation of the distribution is increased from 0.3 to 1.5 Recovered distributions from data generated with personal and common dead space is compared. In this example, TE increases as \j~/\j~ becomes VA = 6.0 l/min and QT = 6.0 l/min; larger to maintain a constant VA. FIG.

dent or tracer gases) resulted in greater mathematical complexity without appreciably adding to the applicability of the solution. From Fi .g. 2 it can be see n that the values of blood flow and ventilation that d etermine equivalence of gas exchange for our model can also be applied to the physiologic gases with complex and interdependent dissociation curves. This was predicted from the equations of equivalence in which the relative values of blood flow and ventilation are determined independent of the partition coefficient of the exchanging tracer gas. For functional analysis, the establishment of values of equivalence means that even under compl .ex conditions gas exchange in lungs with common dead space can be analyzed as if all dead space is personal. This will be represented as minor alterations of the perceived VA/Q ratios in the recovered data. This is especially useful when dealing with the inert gas-elimination technique, which does indeed analyze data as if all dead space were personal. We have demonstrated a small systematic effect that should be present in all ventilation and perfusion distributions obtained from mathematical and physiological models having common dead space but using the standard recovery techniques. In spite of this, however, there is remarkably little change in the general shape and informational content of the distributions. Another consequence of the establishment of equivalence is the inherent inability to distinguish common from personal dead space by steady-state techniques. Under nonsteady conditions, a similar equivalence has been demonstrated by Nye (7). In an analysis similar to ours but examining nitrogen washout, he showed that, in two-compartment models with common and personal dead space, conditions of volume and flow could be found that equated the washout curves in the two models. In his study, it was also demonstrated that the effect of the

common dead space would tend to impose consistent changes in the equilibration curves analyzed under personal dead-space conditions. In models with common dead space, he demonstrated that the volumes of wellventilated units would tend to be underestimated while those of poorly ventilated units would tend to be overestimated. Evans and co-workers (1) further demonstrated that, if the size of the dead-space compartment is known, the actual ventilation to volume parameters can be determined from the washout data in a nonhomogeneous lung. To proceed with the formal algebraic analysis, several assumptions were made, most of which have been discussed in other analyses of gas exchange. An assumption particular to this analysis, however, is that the deadspace ventilation is distributed in proportion to the alveolar ventilation of each unit. Some experimental work in this area has been published by Suwa and Bendixen (10) who determined a series of equations to predict the effect of increasing common dead space on gas exchange. They also assumed that the dead space was distributed in proportion to alveolar ventilation. Close matching of experimental values and predicted values with increasing common dead space seemed to support this assumption. Grant et al. (5), studying the topographical distribution of radioactive gases in the trachea at end expiration, concluded that during normal tidal breathing the tracheal gas is distributed preferentially to the apical regions of the lung. If we assume a gravitational dependence of VA/Q ratios with higher VA/Q ratios in the apex (16), this might predict that the dead space is distributed preferentially to areas of lung with higher VA/Q ratios. The extent to which this occurs has not been quantitated and would present enormous mathematical complexities in our model. Distribution of dead space not in proportion to alveolar ventilation would also result in a myriad of different examples in which resultant information would be less generalized and more difficult to interpret. In the sections of this paper dealing with the comparative effects of common and personal dead space on gas exchange, it is important to note that in the models being compared all the dead space is either common or personal. In studies on cadavers and living humans (6), it has been shown that about 50% of the total dead space is found in the extrathoracic airways (measured from a point 6 cm above the carina). With a tracheal diameter of 2 cm, it can be inferred that perhaps as much as 75% of the dead space (loo-125 ml) lies proximal to the carina. If there is similar gas exchange in each lung, then the volume of the main bronchi can be included in the common dead-space compartment making this value even larger. Therefore, although the assumption that all dead space is common cannot hold, the anatomic data show that the absolute effects of common dead space demonstrated in our comparative analysis are not greatly diminished in life. The exchange of oxygen and carbon dioxide under conditions of personal and common dead space must be handled using mathematical methods of numerical analysis; however, some general conclusions can be implied from the data presented here. It can be calculated that oxygen behaves as if it has an effective partition coeffi-

Downloaded from www.physiology.org/journal/jappl at Midwestern Univ Lib (132.174.254.157) on February 14, 2019.

904

J. B. FORTUNE

cient of approximately 0.6 and carbon dioxide an effective partition coefficient of approximately 6.0. From Fig. 3, it can therefore be seen that the exchange of oxygen would be improved much more than CO2 in the presence of common dead space. From Fig. 4 it might also be expetted that common dead space would improve both 02 and CO2 exchange more as the inhomogeneity of ventilation and perfusion in the lung is increased. APPENDIX The derivations of the equations describing gas exchange in the models shown in Fig. 1 are quite cumbersome, but are important to the understanding of the principles of the analysis. Since the major portion of this analysis is a formal proof that lungs with common dead space can be analyzed with unique personal dead-space models, the step-bystep details are important and are presented below. and Mixed Pressures

Arterial

Test

-

Solving

dead

common

- PH~o)

space FJl

C)

- F,*\~D)/(PB

dead

- ~)J?D.P~/(PB

-

PH~o)

VP

- PH~o)

+ ((1 - F,)*P&PB

- F,.$o~~o[F,oP1

- ,8)*3D.P, - PI-[%?*

- F,J?D]

= AoG,mQ~o(PC

(1)

- PI)

It is important to note at this time that, in the formulation of this mass balance equation, fractional concentrations of the test gases were used to yield the actual volumes of gas transferred. As the equation appears here, the value PB - PH?O would appear as a divisor in every term and therefore has canceled out of the expression. This is possible on the right side of the equality by the use of the partition coefficient, h, in place of the Bunsen’s solubility coefficient. Although these equations all appear to be in terms of partial pressure, definitions in terms of fractional contents and therefore volume (i.e., mass) have preceded all of them. For the right compartment of the common dead-space model, the equation describing the behavior of the test gas is similar to Eq. I (1 - F,.)J?E.P~

-

(1 - F&(1

- p)*iTD.P”

(1 - F,)

-

+ (1 - F,) . Pz] - PI.

+?J?D*[F,.~P,

(%‘I,

= A.(1

-

(2)

(1 - F,.) irD) l

- G,.)&T*(PV

- F,,J?D*P,

-

PI+?I*

-

For the right (1 - F,,)*~E~P~

compartment

the expression

- (1 - F,,)J?o.P2

-

Solving

PI~[%

-

F,,)&‘D] (4)

= ho(l

- G,)*QT*(PV

expression

for the right

- F)*~E

- FV,)

+ (1 - G)oQ~A,~(l

(7)

Eq. 2 for PJPV

A,X2 + BJ

+ Cl

(8)

AIX2 + D,X + El

gives A.LX2 + BJ + Cp Azh2 + DzX + Eg

(9)

The

coefficients of X for each expression are defined in Table 1. Assuming no diffusion impairment across the blood gas membrane (assumption d, see text), the compartmental test gas partial pressure will be equal to the test gas partial pressure in the effluent blood draining the units. The test gas partial pressure in the mixed arterial blood (divided by PV) for the common dead-space model then becomes

1. Coefficients for compartmental test gas pressures- common dead-space model

TABLE

Left Compartment

A,X2 + BJ AIX2 + DJ

-=P* PV

Right

+ C, + El

A1 = G,.*(l

- G,)

B, = T*[(l

- F,)(G,

AJ2 AJ2

-=P2 I+

+

Y)

+ C2 +Ez

- G,.)

B:! = T-[F,*(l

+ IV

- G,.) + Y&=(1 PI

- F,.) + GceK.

(( 1 - F,)

+ We (1 - G,))]

Gdl Cl = T’*(l

+ BJ + DJ

AZ = G,-(1 + F,..Y)

Compartment

PI

- F,)+F,.+(l

+

= T*[Fc.(l - G,) + G,..(l F,.) + Y*F,..(l - F,)]

El

= T’e(l

+ Y).(l

F,) + W.((l (1 - F,.))]

+ We (G, + F,.Y)]

D1

PI

C2 = T2.FC#(l

Dz = T.[FJl - G,) + G,.(l F,) + Y*F,*(l - F,)]

-

- Pz)

where

- F,)mF,.(l

T = (VE

- \~D)/QT;

Ep = T”eF,*(l

+ Y)

W =

A,.(1

- FV,).QT i7E

-

- F&(1 ;

y=-

3D

Downloaded from www.physiology.org/journal/jappl at Midwestern Univ Lib (132.174.254.157) on February 14, 2019.

-

- G,) + Ye

P,)

is - (1 -

equivalent

is

P2 R2 = -= PV

F,,=~D] = AoG,,aQ~m(PV

The

(6)

- FV,)

+ GoQ~&(l

- PH~o).

(1 - G&$F,

(3)

- 713)

It is important to note that these equations for VI and VI* apply equally in both the common and personal dead-space models, inasmuch as the terms VD or p do not appear in the final solution. To determine compartmental partial pressure in each model, Eqs. 1-4 (with appropriate substitutions from Eqs. 6 and 7) are rearranged to solve for P1 and Pp in each model. Since PV appeared in many terms, the expressions were divided by PV with the results (P,/PV and P,/PV) being the retentions (RI and R2) for the gas in each compartment. Solving Eq. 1 in this manner for P,/PV (left compartment) yields

- Pa)

These equations apply to any test gas with a partition coefficient, X, and with partial pressure in inspired gas and venous blood of PI and PV, respectively. For the personal dead-space model, the equations of gas exchange in each compartment are identical to Eqs. I and 2 except that /3 = 0. Thus the equation for the left compartment becomes F,,*\~E*P~

F?I, = PV,/(PB

- PH~o))]

+ (1 - F&P21

= A,aGoQ~o(PV,

&i7~).713

R RI Z-E PV

Thus in terms of the variables in Fig. 1, gas exchange in the left compartment of the common dead-space model can be represented by the following mass balance equation - F,.*(l

-

= F.~E

space

F,+\~D*[F,.~P,/(PB

F,.J?E*P~

(VI*

yields

VI=(l

PI@I

b) personal

for VI*

compartment

Referring to the common dead space model in Fig. 1, gas enters the alveolar compartments with the venous blood and inspired air and is eliminated with arterial blood and expired gas. Beginning with left compartment, the inspired gas to the unit comes from three sources. a) gas outside the lungs

P. D. WAGNER

As described in the text, each compartment will also contain only one resident or carrier gas which has the main function of determining the relationship between inspired and expired gas volumes in each compartment. Up to this point, inspired gas volume has been defined in terms of %‘I* (left compartment) and VI (right compartment). To define VI in terms of the characteristics of the resident gas and the other variables, a solution for the gas exchange equation for the resident gas must be found. Since the test gas is present in only trace amounts, the resident gas must have a partial pressure of 713 Torr (PB - PH~o) in each compartment, in the inspired gas and in the arterial blood. The equation for the exchange of the resident gas in the left compartment is F*\j~*713 - F-(1 - p)4%713 - F+iTo*713 (5)

where

Compartmental Gas Partial

AND

+ Y)

PV ‘D l

i7E

-

3D

-

EFFECTS

OF

COMMON

DEAD

Pa,/PV which

through

SPACE

= G,*PI/PV

substitution

from

Pa,/PV

=

The coefficients

are defined

=

INERT

GAS

EXCHANGE

905

+ (1 - G,)*PJPG

A,,=G,=

(G, - F,) - A,*G,=(l

- Fc)

Eqs. 8 and 9 becomes

= A,*G,*(G,

A,h2 + B,X + Cc AJ2

(10)

+ D,X + E,

The coefficients are defined in Table 2. Following the same logic for the personal /? = 0, the resulting mixed arterial partial test gas retention, is Pa,/PV

ON

+ CD

in Table

= G,,41

-

= (1

x4

2.

of Equivalence

(18)

- Pr/PV).[L*(F,.

xr, = L=(Fc

Equating Eqs. 10 and II, cross-multiplying from the other yields a quartic polynomial equal zero for the equality to hold x,*x4

+x,.x3

For this to be true, neously or X, = ApA,

+ x&h2

X2 = B,*A,

(12)

= 0

of X must

be zero simulta-

= 0

- G,)‘-

M=(l

- B,*A,

+ A,.D,

- A,*D,

- E,) - A,*(C,

= 0

+ B,*E,

Xc, = CpE,,

- C,,*E,

= 0

- B,*E,

- BP-D,

TaA,,+(G,.=

a(1 + W) - G,.(l

= Ap(B,,

GJ

- F,)]

+ G,e(l through

(15)

l

- FP) + W) - G&l

like terms

Model

- FP)

pa, -= PV

- GJ

4

B, = T=[G,. (G, -F,.)+F,*(lG,.) + Y.F,.(l - F,.) + Gp(l

- G+-

p1 (I+

lt, G, =

B

-

4+B 2

II-

(22)

(M+K)*(l+W)-L

‘L

M*L*(l+

W)

I

s2 = To

AJ2 + BJ AJ2+D,h+E,,

[

Model

+ C,

s:j = T2.

(Common

Dead-Space/Personal

Dead

Space)

[M

+ K + 1M.Y

W*M*Ee ‘I

(G,, - F,,) + I?,,* (1 -

l

G,) +

G,,-(1

PI

1

=

Fa(l

Z*K*[M+L+K+L*(l+Y)]-K~W~M*pv

+ (K + L).(l

+ Y)]Z=L

-

[

- G,,)

B, = To [G,,

Ratio

Si = Z.K2

Dead-Space

= G&l

Gas Exchange

11

(M+K+L)

234 = T3.

ZL2.(1

+ Y) - W*MeL*E

ss =

ZLe[M

+ K + [(K

PI

p1 (1 - G,J •~0 + L).(l

+ Y)]]

- W.M.E.

1

Y)*F,*(l

-

C, = T” +F&

F,) + W. [G,. (G, - F,) + F,.(l - G,.) + Y*F,a(l -

Wll D,. = T.[G,.(l - F,) + F,*(l G,) + YmF,.(l - FJ] EC! = T’.(l

and

s4

+ WI

WI1 PI Cc = T2 &(‘+

(21)

2

R S,h3+ &X2+ SJ + = SJ” + &X2 + SJ + ss

gives

Personal

A,X2 + B,X + C, A,h2 + D,.X + E,

A, = G,-(1

4+A

TABLE 3. Coefficients for equation of gas exchange ratio (Eq. 24)

(PI/PV)

2. Coefficients for mixed arterial test gas expressions for common and personal dead-space models

Pa, -= PV

A

-

J

F, =

A

- G,,).(Pr/Pv).(l

and eliminating

Dead-Space

It

where

TABLE

Common

( 20)

where M = (F, - G& K = G,*(l - Gp), and L = F,,*(l - FP). The expressions for Xtj and X5 appear internally in the expression for X4. Therefore, if the conditions X:3 = X:, = 0 are satisfied, the conditions for X4 = 0 likewise will be satisfied. There remain two equations in two unknowns for which there will be a simple solution. The only physiologically feasible solution to this set of equations are the single positive roots of the quadratic expressions

- Dp)

= T*A,+G,.(G,

(19)

= 0

- F,.) = 0

(16) in Eq. 13. is obtained

- F,)] - G,.)‘]

2 gives

(G, - F,) + G,- (1 -

Multiplying

+ Y)*F,*(l

(17)

A,. (B, - 0,) Table

= 0

= 0

X1 = 0. This condition is always satisfied as is evident x2 = 0. After rearrangement, the following expression from Eq. 14

from

-M-(1

(14)

- EP) + B,=D,

- C,.D,.

Substituting

- G,)’

+ Y).F,.(l

- G,) - K(F,.

(13)

= Cc.0,

x4

+x5

+x4-x

all of the coefficients

- ApA,

X3 = A&C,

and subtracting one side in terms of A which must

- G,))

X;,=M.(G,.(l-G,))-(F,-G$*K=O

- ~.(PI/PV).[M*G,.(~ Conditions

G,))/(G,-(1

With reference to Table 2, it can be easily seen therefore that X2 = 0 is always satisfied. xj = x4 = x5 = 0. After substitution from Table 1, the conditions for the other components of Eq. 12 to be equal to zero are

(11)

A,h2 + D,h + E,,

- F,,)

out A, and A, yields A,/A,:

dead-space model in which pressure, expressed as the

A,X2 + BJ

Factoring

- FP) - AcoG&

+ Y).F,.(l

- F,)

-

(Gp

(M+K+L*(l+Y))

- F,,) + We l

Gp -

Fp)

+

Fp*(1

-

WI D, = T*[F&l

1

- GP) + G&l

-

S

6

=

where

T3.

z.L2,(l

+

y)

-

.&l

W.M.L

‘I

[ M

=

(F

-

G)“;

K

=

G-(1

-

+ Y)

1

G);

L

-

F);

J-VI E, = T’$F,,*(l

- F,)]

. (1 + W)

; Y, W, and T are defined

Downloaded from www.physiology.org/journal/jappl at Midwestern Univ Lib (132.174.254.157) on February 14, 2019.

as in Table

1.

906

J. B. FORTUNE

and

A>(kc-K)*(l+vv)+L M=K*(l

B= Existence and uniqueness fore formally demonstrated. ReZative

Effect

of Common

1 ‘L

+ w)2

of the conditions

and PersonaZ

for equivalence

Dead

When F and G are the same for the models then a ratio of gas transfer can be defined as

Rg

Vg, =y= Vg,,

are there-

is possible when divisor (& - S5) will not change. range of test gas space improving

Space

(PV - Pa),,

(23)

A>

is made into this from is obtained

Eqs. 10 and II, the following that

S,h3 + &X2 + S:J + S.1 Rg = SIX” + &X2 + SJ + ss

(24

The

coefficients of X in this expression are defined in Table 3. The coefficients of Xl’ and X” are identical in the numerator and denominator, therefore for Rg > 1 (i.e., common dead space always exceeds personal dead space gas transfer), then

SC; - sq s3 - sr,

X,*(1

- FV,)*PI PI - PV

(25)

When Pr/PV < 1 but PI = 0.0, the value of & - S, becomes so that dividing by it reverses the distribution of the inequality the expression x < X,.(1

When substitution general expression

P. D. WAGNER

PI/PV > 1 or PI = 0.0, since, for this condition, the will always be positive and the direction of inequality Upon substitution into this expression, the possible partition coefficients that will result in common dead gas exchange when compared to personal is

(i.e., F, = F[,, G, = G,,),

(PV - Pa),

AND

defines

- FV,)*PI PI - PV

the conditions

(where

PI/PI;

negative yielding

(26)

< 1)

for Rg > 1.

This work was supported by the California Lung Association, the Francis North Foundation, and National Institutes of Health Grants HL-17731, HL-00111, and HL-07212. Received

10 October

1978; accepted

in final

form

29 May

1979.

REFERENCES 1. EVANS, J., D. G. CANTER, AND J. R. NORMAN. The dead space in a compartmental lung model. BUZZ. Math. Biophys. 29: 711-718, 1967. 2. EVANS, J., AND P. D. WAGNER. Limits on VA/Q distribution from the analysis of experimental inert gas elimination. J. AppZ. Physiol.: Respirat. Environ. Exercise PhysioZ. 42: 889-898, 1977. 3. FARHI, L. E. Elimination of inert gas by the lung. Respir. Physiol. 3: l-11, 1967. 4. FENN, W. O., H. HAHN, AND A. B. OTIS. A theoretical study of composition of alveolar air at altitude. Am. J. Physiol. 146: 637653, 1946. 5. GRANT, B. J. B., H. A. JONES, AND J. M. B. HUGHES. Sequence of regional filling during a tidal breath in man. J. AppZ. Physiol. 37: 158-165, 1974. 6. NUNN, J. F., J. M. CAMPBELL, AND B. W. PECKETT. Anatomical subdivisions of the volume of respiratory dead space and effect of position of the jaw. J. AppZ. Physiol. 14: 174-176, 1959. 7. NYE, R. E. Theoretical limits to measurement of uneven ventilation. J. AppZ. Physiot. 16: 1115-l 124, 1961. 8. RILEY, R. L., AND A. COURNAND. ‘Ideal’ alveolar air and the analysis of ventilation-perfusion relationships in the lung. J. AppZ. Physiol. 1: 825-847, 1949. 9. Ross, B. B., AND L. E. FARHI. Dead-space ventilation as a determinant in the ventilation-perfusion concept. J. AppZ. Physiol. 15: 363-371, 1960. 10. SUWA, K., AND H. H. BENDIXEN. Changes in Pac.+ with mechanical

11.

12.

13.

14.

15.

16. 17.

18.

dead space during artificial ventilation. J. AppZ. Physiol. 24: 556563, 1968. WAGNER, P. D., D. R. DANTZKER, V. E. IACOVONI, W. C. TOMLIN, AND J. B. WEST. Ventilation-perfusion inequality in asymptomatic asthma. Am. Rev. Respir. Dis. 118: 511-524, 1978. WAGNER, P. D., AND J. W. EVANS. Conditions for equivalence of gas exchange in series and parallel models of the lung. Respir. PhysioZ. 31: 117-138, 1977. WAGNER, P. D., P. F. NAUMAN, AND R. B. LARAVUSO. Simultaneous measurement of eight foreign gases in blood by gas chromatography. J. AppZ. Physiol. 36: 600-605, 1974. WAGNER, P. D., P. F. NAUMAN, R. R. UHL, AND J. B. WEST. Continuous distributions of ventilation-perfusion ratios in normal subjects breathing air and 100% 02. J. CZin. Invest. 54: 54-68, 1974. WAGNER, P. D., H. A. SALTZMAN, AND J. B. WEST. Measurement of continuous distributions of ventilation-perfusion ratios: theory. J. AppZ. Physiol. 36: 588-599, 1974. WEST, J. B. Regional differences in gas exchange in the lung of erect man. J. AppZ. Physiol. 17: 893-898, 1962. WEST, J. B. Ventilation-perfusion inequality and overall gas exchange in computer models of the lung. Respir. Physiol. 7: 88-110, 1969. WEST, J. B., AND P. D. WAGNER. Pulmonary gas exchange. In: Bioengineering Aspects of the Lung. New York: Dekker, 1977, p. 361-457.

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