Abstract
Li, M. H., J. Hildebrandt, and M. P. Hlastala.Quantitative analysis of transpleural flux in the isolated lung.J. Appl. Physiol. 82(2): 545–551, 1997.—In this study, the loss of inert gas through the pleura of an isolated ventilated and perfused rabbit lung was assessed theoretically and experimentally. A mathematical model was used to represent an ideal homogeneous lung placed within a box with gas flow (V˙box) surrounding the lung. The alveoli are assumed to be ventilated with room air (V˙a) and perfused at constant flow (Q˙) containing inert gases (x) with various perfusateair partition coefficients (λ_{p,} _{x}). The ratio of transpleural flux of gas (V˙pl_{x}) to its total delivery to the lung via pulmonary artery (V˙ v̅), representing fractional losses across the pleura, can be shown to depend on four dimensionless ratios:1) λ_{p,} _{x},2) the ratio of alveolar ventilation to perfusion (V˙a/Q˙), 3) the ratio of the pleural diffusing capacity (Dpl_{x}) to the conductance of the alveolar ventilation (Dpl_{x} /V˙aβ_{g}, where β_{g} is the capacitance coefficient of gas), and 4) the ratio of extrapleural (box) ventilation to alveolar ventilation (V˙box/V˙a). Experiments were performed in isolated perfused and ventilated rabbit lungs. The perfusate was a buffer solution containing six dissolved inert gases covering the entire 10^{5}fold range of λ_{p,} _{x} used in the multiple inert gas elimination technique. Steadystate inert gas concentrations were measured in the pulmonary arterial perfusate, pulmonary venous effluent, exhaled gas, and box effluent gas. The experimental data could be described satisfactorily by the singlecompartment model. It is concluded that a simple theoretical model is a useful tool for predicting transpleural flux from isolated lung preparations, with known ventilation and perfusion, for inert gases within a wide range of λ.
 inert gas exchange
 pleural diffusing capacity
 diffusion
isolated lung and openchest lung preparations have been widely used in studies of gas exchange (28, 15). Although transpleural loss has been considered negligible, the large pleural surface area combined with often large partial pressure gradients across the pleural membrane and high water solubilities could result in significant transpleural flux. Indeed, Anderson and Madsen (2) concluded that the transpleural diffusive loss of gases from isolated mouse lungs was a source of error even in studies of lung compliance. Furthermore, Magnussen et al. (6) also showed that transpleural gas loss led to underestimates of the pulmonary blood flow measured by the acetylene rebreathing technique. Shepard et al. (12) reported that persistence of gas exchange in lungs deprived of pulmonary and bronchial arterial blood could be explained by the combined mechanisms of tissue metabolism and transpleural diffusion. Actual fluxes were measured by Anderson and Madsen, Magnussen et al., and Plewes et al. (9) for lowsolubility gases such as He, Ar, N_{2}, and N_{2}O. However, only Rehder and Marsh (10) reported quantitative data on transpleural flux for the six multiple inert gas elimination technique (MIGET) gases that are infused via the femoral vein. They measured the flux by ventilating the pleural space via catheters inserted through the chest wall. The partial pressure ratios of gas in the pleural space to the gas in the mixed venous blood was close to zero for sulfur hexafluoride (SF_{6}), ethane, cyclopropane, and enflurane. However, for the more soluble gases (diethyl ether and acetone), mean excretions were 0.08 and 0.6, respectively. Transpleural gas loss in this preparation was limited by the thick chest wall in contact with the lung over a large fraction of the lung surface. Transpleural flux would increase in isolated lung preparations because of the increased available surface area. Furthermore, no quantitative model or theory has been reported to predict the dependence of transpleural flux on solubility. We therefore undertook to evaluate transpleural flux with a mathematical model and then to test the model against experimental data.
THEORY
Model of Transpleural Flux
An isolated homogeneous lung (“single alveolus”) is considered within a box with flow V˙box (Fig.1). Ventilation of the lung (V˙a) is constant and continuous, as is perfusate flow (Q˙). Test gas (x) is infused into the pulmonary arterial perfusate at a constant rate, achieving a steadystate arterial (systemic mixed venous) pressure and pulmonary venous partial pressure, Pa_{x} and
Movement of gas through the pleura between the alveoli and the box is a passive process governed by the laws of diffusion. According to Fick’s first law, transpleural flux can be defined as
By mass balance in the steady state,V˙pl_{x} is equal to the flux of gas x out of the box (V˙box_{x}), which can be expressed as
Again, by mass balance, the total amount of gasx gained by the lung from the pulmonary circulation per minute is the amount per minute lost from the circulation, i.e., delivery via artery
V˙e
_{x}can be represented in either of two forms
The total amount of gas transferred from the vessels to the alveolar spaces must in the steady state be equal to the gas exhaled plus the transpleural flux
Pleural leak (V˙pl_{x}) can also be expressed as a fraction of the gas extracted from the perfusate (V˙
Another ratio of interest is that ofV˙pl_{x} toV˙e
_{x}, the volume of gas x exhaled (=V˙aβ_{g}Pa). This ratio can be transformed using Eqs.3
and
9
to become
This ratio is useful in quickly assessing the extent to which pleural loss of gas will impact mixed expired gas data. Note the similarity of this ratio to Eq. 9 and its dependence on the same two dimensionless ratios.
Measurement of Transpleural Diffusing Capacity
Combination of Eqs. 1
and
2
and Pa
_{x} = Pe
_{x} (1 − Vd/Vt) yields
METHODS
Isolated lungs were obtained from 28 rabbits [3.7 ± 0.1 (SE) kg body wt] of either sex after anesthesia with pentobarbital sodium (60–90 mg/kg iv). Tracheostomy was performed, and the animals were ventilated with room air. After midsternal thoracotomy, heparin (500 U/kg) was injected into the right ventricle for anticoagulation, and blood (100–200 ml) was withdrawn from the right atrium. Catheters were tied into the pulmonary artery and left atrium, the lungs were flushed with perfusate (modified Ringer buffer solution plus 4% dextrose and 2% albumin), and the catheters were connected to the perfusion system. Lungs were isolated and enclosed in a Plexiglas chamber (15 × 15 × 20 cm) through which warmed saturated air (38°C) flowed at fixed rates. Lungs were ventilated with room air, maintaining positive endexpiratory pressure of 3 cmH_{2}O. Twenty minutes after stable perfusion with the buffer solution, six inert gases with a solubility range of ∼10^{4} and dissolved in isotonic saline (Table 1) were mixed into the arterial reservoir. Also shown in Table 1 are partition coefficients (λ) for these six gases against air for water, perfusate, rabbit blood, bloodfree rabbit lung tissue, and rabbit lung tissue not flushed free of blood. Data were obtained in this laboratory by following a standard doubleextraction procedure (13).
Static pressures in the pulmonary artery, pulmonary vein, and trachea were monitored with smalldiameter tubing connected to pressure transducers. Perfusate samples (
The measurement of partial pressures of various inert gases in the samples has been described by Wagner et al. (13). Extraction of the gases dissolved in the perfusate is carried out by equilibration with N_{2} in a shaking water bath (38°C). The concentration of each of the trace inert gases extracted from the liquid is measured on a gas chromatograph (Varian 3300) equipped with flame ionization and electron capture detectors.
From the partial pressures of the six inert gases measured at these four sites, diffusing capacity can be calculated fromEq. 11 . Transpleural flux is calculated from Eq. 8 as the fraction of total delivery to the lung via pulmonary artery (V˙v̅ _{x}) or from Eq. 10 as the ratio relative to the amount exhaled (V˙e _{x}).
Two experimental series were carried out: For group 1 (n = 8), ventilation, perfusion, and transbox flow were maintained constant at 1.2, 0.1, and 1.2 l/min, respectively, respiration rate was 40 min^{−1}, tidal volume was 30 ml, and the samples were withdrawn at 45, 60, 75, 90, and 120 min after onset of perfusion with the inert gas mixture. Forgroup 2(n = 12), ventilation and perfusion were held constant at 1.2 and 0.1 l/min, respectively, while transbox flow rate was varied among 0.6, 1.2, and 2.4 l/min, in random order.
Values are means ± SE. Linear regression was applied for best linear fit. Multiple regression and Fstatistics were used for statistical comparison between experimental data and theoretical values.
RESULTS
Predicted Ratio of Transpleural Flux to Total Delivery
Equation 8
shows that for any given value of λ_{x}, the transpleural flux ratio depends onV˙a/Q˙,V˙box/V˙a, and Dpl_{x} / V˙aβ_{g}. For purposes of illustration, letV˙a/Q˙ andV˙box/V˙abe unity; then Eq. 8
reduces to
Inspection of Eq. 8
shows thatV˙a/V˙box andV˙aβ_{g} /
Experimental Data
Mass balance.
In the absence of experimental errors, the delivery (V˙
Pleural diffusing capacities.
Dpl calculated using Eq. 11
are shown in Fig. 3. Numerically, they were as follows: SF_{6} = 0.0050 ± 0.0002, ethane = 0.019 ± 0.004, cyclopropane = 0.047 ± 0.012, halothane = 0.12 ± 0.03, ether = 0.56 ± 0.14, and acetone = 5.4 ± 2.1 (SE) ml ⋅ min^{−1} ⋅ Torr^{−1}. Because diffusional transport of gas xdepends on the solubility β_{x}(and therefore λ _{x}
), there should be a nearly linear relationship between Dpl_{x}
and λ_{x}. Figure 3 does show linear relationships, but on loglog coordinates, with slopes of ∼0.6 – 0.7 and not 1.0
Dependence of flux ratios on λ.
The three flux ratios in Eqs.810 each depend on Dpl, and as shown byEq. 13 , Dpl is a function of the λ appropriate to the experimental circumstance. To illustrate trends, we have chosen λ_{p}(λ_{ti,p} or λ_{ti,b} would result in minor differences).
By substituting Dpl from Eq. 13
inEq. 8
, the fraction of gasx delivered via the circulation, which was lost via transpleural diffusion, can be expressed as
Equations 9 and 10 can be similarly rewritten. The dependence of the three flux ratios on λ_{p} is shown in Fig.4 for various fixed values of the other parameters.
The first transpleural flux ratio (Fig.4 A) has the property of a clear maximum near λ = 10, but the ratio approaches zero when solubilities are extremely high or low. The curves shift slightly to the right and upward with increasing transbox flow rate. Experimental data confirm the presence and location of the predicted maxima and are in good agreement with predicted effects of varying the turnover rate of air surrounding the lung (V˙box). AsV˙box approaches zero, as in the closed chest,Eq. 14 predicts that the flux ratio would likewise approach zero. RaisingV˙a would lower the flux ratio by lowering alveolar gas concentrations and thus diffusive losses.
The dependence of the ratio of transpleural flux to gas extracted from the perfusate [V˙pl_{x} /V˙(
DISCUSSION
Condition of Preparation
The gross appearance of lungs after 90 min of inert gas infusion was that of a normal healthy lung. There were no signs of hemostasis, edema, or atelectasis. The pulmonary arterial pressure, pulmonary venous pressure, and peak airway pressure remained stable over the course of the experiments. Pulmonary arterial pressure changed from 10.0 ± 0.7 cmH_{2}O at 0 min to 13.0 ± 0.9 cmH_{2}O at 70 min. Peak airway pressure changed from 9.2 ± 0.5 cmH_{2}O at 0 min to 12.3 ± 0.7 cmH_{2}O at 70 min.
Critique of Model
Our analytic model posits homogeneous diffusion across a uniform visceral pleura. We also assume for the purpose of calculating Dpl that alveolar pressure (Pa) in the alveolar space is uniform right up to the pleural membrane. We employed a high respiratory rate to maximize mixing. Because Pa is also dependent on uniformity ofV˙a/Q˙, we applied the 50compartment analysis of Wagner et al. (14) to our inert gas data (5). The general pattern ofV˙a/Q˙distribution shows a high degree of homogeneity (log SD_{V˙} = 0.59 ± 0.04, log SD_{Q˙} = 0.65 ± 0.03).
Pleural Diffusing Capacity for Inert Gases
There are few data available concerning Dpl, especially for highly soluble gases. Comparison can be made with data only in the lower solubility range, where agreement with findings reported by Magnussen et al. (6) are shown (Fig. 3, ▪). For dog lower lobe (lung surface = 177 cm^{2}, estimated to be in the range of that of our rabbit lung, 182 ± 10 cm^{2}), they found Dpl for He (λ = 0.0092) to be 0.0025, for Ar (λ = 0.0276) 0.0019, and for N_{2}O (λ = 0.41) 0.0332 ml ⋅ min^{−1} ⋅ Torr^{−1}. Their data are generally close to our bestfit line (Fig. 3,curve 1). As predicted by the definition of diffusing capacity, Dpl should be linearly proportional to the lung tissuegas λ (λ_{ti}). Figure 3 does show a strong linear correlation (r = 0.99,P < 0.01) between Dpl and λ_{p} on loglog coordinates but a slope of only ∼0.66. Possible explanations for a slope <1.0 are as follows: 1) inaccurate values of λ_{ti} for inert gases and2) underestimation of Dpl due to inaccurate values of Pa for inert gases, especially for highsolubility gases.
We have measured the solubilities (λ_{ti}) of six MIGET gases in rabbit lung homogenate diluted with various amounts of saline at 37°C (Table 1). The solubilities in undiluted rabbit lung tissue were then obtained by extrapolation to zero dilution, which entails some uncertainty. It is clear that the solubilities are different among species and tissues (1, 13). The solubilities in pleura may therefore also differ from those in the lung homogenates that we measured. The slope increased from 0.66 to 0.74 when Dpl was regressed against λ_{ti,b} (Fig. 3,curve 2), in which the homogenate included blood retained after isolation.
Solubilities also vary with temperature (1, 13). In our preparation, the temperature of pulmonary artery, vein, and box were 36.7 ± 0.3, 36.1 ± 0.6, and 34.2 ± 0.80°C, respectively. The temperature of pleura may be close to that in the box. Therefore, the solubilities in pleura in our preparation should be higher than that in lung homogenate measured at 37°C. If we assume that the relationship between temperature coefficient and solubility in lung is equal to that in water, the corrected λ will increase. However, the increase in λ after temperature correction was insufficient to explain the slope <1.0 in our data.
Another possible explanation for the lower slope is underestimation of Dpl, especially for gases of high solubility. Dpl is calculated for the transpleural flux and partial pressure gradient between subpleural alveoli and Pbox (Eq. 1 ) . However, the partial pressure in subpleural alveoli cannot be measured directly. We made the assumption that partial pressures in subpleural alveoli are homogeneous and equal to that of the average of all (Pa). It is likely that the subpleural alveolar partial pressure is at least slightly lower than that of averaged Pa because of the transpleural gas loss. The overestimation of the actual gradient by using mean Pa will reduce the calculated Dpl and could decrease the slope of the bestfit line. For example, if we corrected Pa by 10% for each gas, Dpl would increase 10% for SF_{6} and 50% for acetone, raising the slope of the bestfit line to 0.83.
Pa was calculated from Pe, the partial pressure (of some gas specie) in expired gas, and dead space volume (Vd) according to Bohr’s equation. Pe is measured in our preparation, and Vd is derived using the MIGET algorithms. Vdis underestimated by ∼4–5% because of transpleural flux (5), resulting in an ∼3% underestimate of Pa. After correction, Dpl and the slope of the regression decrease slightly. The underestimation of Vd due to transpleural flux does not account for the slope that is already too low. Schrikker et al. (11) suggested that Vd for a highsolubility gas such as acetone is smaller than Vd measured by MIGET because of the uptake and release by the conducting airways. Dpl for acetone increases after correction for airway acetone exchange, and the slope increases to 0.87. Therefore, uncertainties regarding Vd, and thus the value of Pa used in Eq.1 , may best account for the slope being <1.0.
Transpleural Flux Ratios
Equations 810
, along with the corresponding graphs in Fig. 4, illustrate the manner in which the various ratios depend on λ and the three other dimensionless variables. The first ratio (V˙pl_{x} /V˙
The fact that the transpleural flux ratio can exceed 1.0 for acetone implies huge losses through the pleura for gases of high solubility and major impact on gas exchange data inferred fromV˙e. In contrast, for gases of low solubility, all the transpleural flux ratios are close to zero, and little impact on lowsolubility gas exchange data is expected.
Agreement was found between the experimental data and the values predicted from the mathematical model over most of the range of solubilities that are encountered in practice. Therefore, the model should be applicable to calculate the transpleural flux ratio for any given gas in other isolated lung preparations and to predict the possible impact of transpleural leak on gas exchange data. In openchest preparations, lungs are exposed to outside air but with very low V˙box, such that transpleural losses could be small. When the air around the lungs is changed rapidly (i.e., V˙box is large), losses are maximized. If one takes a lung preparation withV˙a/Q˙= 1, V˙a = 500 ml/min (close to theV˙a of a 4kg rabbit), and V˙box ≫V˙a, thenEq. 8
predicts a maximum of ∼9% located near λ = 1.0. TheV˙pl_{x} /V˙e
_{x}ratio (Eq. 10
) is inversely related to increasing λ, becoming surprisingly large for gases with high λ. For example, in the case of acetone (λ = 327),V˙pl_{x} /V˙
The perturbation by transpleural flux of gas exchange data has been reported by Magnussen et al. (6) and Shepard et al. (12). MIGET uses the retention in blood and the elimination in exhaled air of six inert gases with varying solubility to assess theV˙a/Q˙distribution in the lung. The transpleural leak will reduce the expired air partial pressure (Pe) and the pulmonary venous partial pressure (Pa), especially for highly soluble gas, as discussed above. Thus transpleural flux will perturb MIGETderived gas exchange data, such as recoveredV˙a/Q˙distribution and indexes of heterogeneity. The transpleural flux would result in the underestimation of Q˙ and Vd but have only minor effects on estimates of shunt.
In summary, a mathematical model predicting transpleural gas fluxes has been applied to experimental data from isolated perfused lungs. This model predicts that such fluxes can be expressed as a function of four dimensionless variables: λ,V˙a/Q˙, Dpl/V˙aβ_{g}, andV˙box/V˙a. The dependence of fluxes on λ was calculated and illustrated graphically for various fixed values of the other variables, especially as tissue solubility increases, and can even become the dominant mode of gas exchange. The theoretical expectation that Dpl should be simply proportional to λ was not borne out by the data, and possible explanations are offered.
Footnotes

Address for reprint requests: M. P. Hlastala, Div. of Pulmonary and Critical Care Medicine, Box 356522, University of Washington, Seattle, WA 981956522 (Email:mike{at}colossus.pulmcc.washington.edu).
 Copyright © 1997 the American Physiological Society