Abstract
We simulated the intraacinar contribution to phase III slope (S _{acin}) for gases of differing diffusivities (He and SF_{6}) by solving equations of diffusive and convective gas transport in multibranchpoint models (MBPM) of the human acinus. We first conducted a sensitivity study ofS _{acin} to asymmetry and its variability in successive generations. S _{acin} increases were greatest when asymmetry and variability of asymmetry were increased at the level of the respiratory bronchioles (generations 17–18) for He and at the level of the alveolar ducts (generations 20–21) for SF_{6}, corresponding to the location of their respective diffusion fronts. On the basis of this sensitivity study and in keeping with reported acinar morphometry, we built a MBPM that actually reproduced experimental S _{acin} values obtained in normal subjects for He, N_{2}, and SF_{6}. Ten variants of such a MBPM were constructed to estimate intrinsic S _{acin} variability owing to peripheral lung structure. The realistic simulation ofS _{acin} in the normal lung and the understanding of how asymmetry affects S _{acin} for different diffusivity gases make S _{acin} a powerful tool to detect structural alterations at different depths in the lung periphery.
 multibranchpoint models
 multiplebreath washout
 gas mixing
 diffusionconvection interdependence
 airway asymmetry
 helium
 SF_{6}
phase iii slope of the singlebreath washout, which is generally referred to as a marker of small airway alterations, can be generated by various mechanisms. In the human lung, two major mechanisms can be distinguished (10). One is thought to be operational at the level of the intraacinar airways, where the balance of diffusion and convection establishes a quasistationary diffusion front during inspiration. At this level, the asymmetry of the acinar structure generates intraacinar concentration differences that cause a phase III slope upon expiration. The other mechanism involves purely convective gas transport, generating concentration differences between lung units on inspiration and sequential emptying during expiration, also leading to a sloping phase III. The most familiar example of the latter mechanism results from sequential emptying between top and bottom lung regions with different specific ventilation. However, such a convectiondependent mechanism can also be operational within small lung regions. In recent years, several studies have been aimed at assessing the convective component of the phase III slope, in particular the contribution from gravity (4, 12). The present study focuses on the intraacinar part of the phase III slope (S _{acin}).
By using a multiplebreath washout analysis that was initially developed for physiological studies of ventilation distribution (2) and was adapted in recent years for clinical application (14, 15), it is possible to isolateS _{acin}. The purpose of the present work was to quantitatively reproduce the experimental S _{acin}values obtained from the existing literature simply by using equations that describe diffusive and convective gas transport in a realistic structure of the human acinus.
Previous simulation studies of gas transport at the acinar level (8) have indicated that intraacinar concentration differences occur when 1) transport due to convection and diffusion are of the same order of magnitude and 2) the acinar structure is asymmetrical. The initial model geometries for simulation of intraacinar gas transport (9, 11) were based on the morphometric data reported by Hansen and Ampaya (6) that accounted, to some extent, for intraacinar asymmetry. A more detailed description of the human acinus by HaefeliBleuer and Weibel (5) prompted the development of a more realistic model of intraacinar transport (13). All these studies highlighted the complex interactions between serial and parallel branch points and confirmed the principle of the diffusion convection mechanism by which a phase III slope could be generated, also mimicking the dependence of phase III slope on lung volume, inspired volume, flow, and gas diffusivity (11). However, these studies did not actually succeed in quantitatively reproducing the experimental phase III slope, and the degree of underestimation of absolute S _{acin} depended on the diffusivity of the gas. For He, N_{2}, and SF_{6}, phase III slopes were underestimated by 85%, 74%, and 28%, respectively (13). The present study reexamines the intraacinar morphometry, makes a quantitative evaluation of the critical characteristics of the intraacinar branching pattern, and quantitatively reproduces the experimentalS _{acin}, i.e., the acinar contribution to the phase III slope, for gases with different diffusivities.
METHODS
Model of the human acinus.
The equations used to describe gas transport in the human acinus are given in the
. The anatomic basis for all acinar model geometries considered here is the morphometric study by HaefeliBleuer and Weibel (5), who measured airway dimensions from generation 15 (first respiratory bronchiole) down to the alveolar sacs with the longest pathways extending into generation 27. These authors also provided the branching pattern of a single acinus that was previously used to construct a multibranchpoint model (MBPM) (13) that will be considered here as a reference model, MBPM_{ref}. The branching pattern of MBPM_{ref} is depicted in Fig.1
A, and the volume asymmetry at each branch point of the model is plotted in Fig. 1
B. Each branch point subtends two intraacinar units of volume, V_{1} and V_{2}, respectively. Assuming V_{1} ≤ V_{2}, the asymmetry (Asym) at any given branch point is defined as
For the purpose of the present study, we developed an algorithm to build different MBPM geometries, following predefined Asym patterns distributed over serial and parallel branch points. The algorithm starts by subtracting the volume of the parent duct from the total acinar volume and partitioning the remaining volume into two subunits so as to obtain the predefined Asym value for that first branch point. In the next branching generation, each subunit undergoes the same procedure, and subdivision continues until all pathways end in subunits that fall within the range of alveolar sac volume. The number of intraacinar generations imposes a constraint on the range of Asym values that can be achieved to obtain a realistic MBPM of the human acinus. For instance, if one assumes a constant Asym = 0.7 throughout the acinar branching tree, this would require subdivisions down to generation 28. In fact, considering a MBPM with a constant asymmetry in generations 15 through 23, the maximum asymmetry that can be achieved is Asym = 0.6.
Most MBPM branching patterns presented here are obtained by assigning to each successive MBPM generation a normal distribution of Asym values over all parallel branch points of that generation. The MBPM can then be characterized in terms of the mean and standard deviation (μ_{Asym}, SD_{Asym}) of the Asym distribution attributed to each generation. One example of a MBPM with an Asym distribution characterized by μ_{Asym} = 0.4 and SD_{Asym} = 0.3 in each successive intraacinar generation is shown in Fig.2 A; the individual Asym values are plotted in Fig. 2 B.
The acinar volume of all MBPM constructed here was set to 187 × 10^{−3} ml, i.e., the mean acinar volume measured by HaefeliBleuer and Weibel (5) with a lung inflated to 5.5 liters (corresponding to 90% of its total lung capacity). The conductive airway dimensions from Weibel's symmetrical model A (17), between generations 0 and 14, were rescaled isotropically from 4.8 to 5.5 liters (5) to obtain a conductive airways volume of 119 ml. The remainder of the 5.5liter lung volume was considered to be in the acini, and, with an acinar volume of 187 × 10^{−3} ml, this resulted in 28,785 acini (5,500 − 119 ml)/(187 × 10^{−3}ml). Finally, an additional dead space of 50 ml was added to account for the pharyngolaryngeal and mouth cavity volume (7).
Comparison between simulations and experimental results.
The experimental data used for actual comparison with simulations were those obtained from three subjects performing multiplebreath washout tests including He and SF_{6} tracer gases (12). These washout maneuvers involved 12 breaths with a mean tidal volume of 1.23 liters (flow rate was ∼0.5 l/s), starting from functional residual capacity that averaged 3.33 liters. The original He, SF_{6} washin, and N_{2} washout tracings were reanalyzed to fully comply with the method of phase III slope analysis for S _{acin} computation. The detailed description and underlying theory for the computation ofS _{acin} can be found elsewhere (15). Briefly, inspired gases (He, SF_{6}) are first rescaled to represent lung resident gases (such as N_{2}) to obtain positive phase III slopes for all gases under study. Then, phase III slopes are computed by linear regression on N_{2}, He, or SF_{6} concentration between 0.7 and 1.2 liters expired volume and normalized by the corresponding mean expired concentration in all 12 expirations. By plotting these normalized slopes as a function of lung turnover (TO, cumulative expired volume divided by the subject's functional residual capacity), the conductive component to ventilation inhomogeneity can be determined. By defining the conductive component as the regression slope of the normalized slope vs. TO for TO ≥ 1.5, its contribution to the normalized slope of the first breath can be subtracted (in proportion to TO of the first breath) to obtainS _{acin}.
In this way, the following experimental S _{acin}values were obtained for He, N_{2}, and SF_{6} gases (means ± SD): 0.054 ± 0.009 liter^{−1} (He), 0.080 ± 0.007 liter^{−1} (N_{2}), and 0.109 ± 0.005 liter^{−1} (SF_{6}), representing, respectively, 91.3, 91.6, and 92.0% of corresponding phase III slopes of the first breath. TheseS _{acin} values constitute the experimental basis for comparison with simulations.
RESULTS
Intraacinar asymmetry and diffusion front.
Figure 3 shows simulated He and SF_{6} diffusion fronts (mean gas concentrations) at the end of a 1.23liter inspiration using a flow equal to 0.5 l/s and diffusion coefficients 0.6 cm^{2}/s and 0.1 cm^{2}/s for He and SF_{6}, respectively. Dashed lines are the He and SF_{6} concentrations for a symmetrical MBPM (μ_{Asym} = 0; SD_{Asym} = 0 in all generations), whereas solid lines correspond to the most asymmetrical MBPM (μ_{Asym} = 0.6; SD_{Asym} = 0 in generations 15–23). Note that the asymmetrical MBPM extends into four more peripheral generations than the symmetrical MBPM. The steepest slope in concentration curves, indicating the branch points contributing the most to the phase III slope generation, are located, respectively, for He and SF_{6} in generations 17 and 19, irrespective of MBPM asymmetry (at least in the μ_{Asym}range 0–0.6).
Sensitivity of phase III slope to intraacinar asymmetry.
We conducted a systematic study of He and SF_{6} phase III slope sensitivity to μ_{Asym}. First, a set of simulations was considered in which μ_{Asym} = 0.4 was imposed individually in successive branching generations of the MBPM by considering μ_{Asym} = 0 in all MBPM generations, except for one generation in which μ_{Asym} = 0.4 and SD_{Asym} = 0. An example of the resulting Asym distribution in MBPM generations is shown in Fig.4A, in which μ_{Asym} = 0.4 was imposed in generation 18. For each successive generation i in which the asymmetry was introduced (i = 15–23), this led to a slopeS
_{μ=0.4,i}. A corresponding slope increase ΔS
_{μ=0}(i) with respect to the symmetrical MBPM (with simulated slope S
_{μ=0}) was then defined as
Figure 5 extends the results of Fig.4 B by applying any given μ_{Asym} value ranging between 0 and 0.6 to all generations between 15 and 23 simultaneously. In this case, one global slope increase with respect to the symmetrical MBPM (ΔS _{μ=0}) is obtained for each μ_{Asym} value. The resulting ΔS _{μ=0} values for He and SF_{6} are plotted as a function of the μ_{Asym} in Fig. 5, showing an exponentiallike increase with μ_{Asym} in the range 0–0.6. By comparison of selected data points in Figs.4 B and 5, it can be inferred that the He and SF_{6}phase III slope increases obtained by imposing μ_{Asym} = 0.4 on all MBPM generations simultaneously roughly correspond to the summation, over all generations, of He and SF_{6} phase III slope increases obtained by imposing μ_{Asym} = 0.4 in each individual generation (considering all data points in Fig.4 B). Indeed, the ΔS _{μ=0}value for μ_{Asym} = 0.4 in Fig. 5, yielding 0.022 liter^{−1} (He) and 0.031 liter^{−1}(SF_{6}), almost equals the sum of ΔS _{μ=0}(i) for i = 15–23 (open data points in Fig. 5), which amounts to 0.021 liter^{−1} (He) and 0.028 liter^{−1}(SF_{6}).
Sensitivity of phase III slope to the variability of intraacinar asymmetry.
In analogy to Fig. 4, we conducted a systematic study of He and SF_{6} phase III slope sensitivity to Asym variability between parallel branch points (i.e., nonzero SD_{Asym}) in another set of MBPM simulations. In this case, we considered μ_{Asym} = 0.4 in all MBPM generations, and SD_{Asym} = 0 was imposed on all intraacinar generations except for one generation i in which SD_{Asym}= 0.3 (variable asymmetry only among branch points of generationi). An example of the resulting Asym distribution in MBPM generations is shown in Fig.6
A, in which the variability of asymmetry was imposed in generation i = 18. For each MBPM simulation with SD_{Asym} = 0.3 in generationi, a corresponding slope increase ΔS
_{SD=0}(i) was defined, this time with respect to a MBPM with constant asymmetry (μ_{Asym} = 0.4), as
To further explore the role of variability of asymmetry, as evidenced by Figs. 6 and 7, we considered additional sets of simple simulations (Table 1). We considered MBPM that are symmetrical except in one generation in which μ_{Asym}≠0. For He and SF_{6} simulations, μ_{Asym}≠0 was considered in generations 18 and 21, respectively (because, in these generations, phase III slope of the gas under consideration had been shown to be particularly sensitive to asymmetry). In the generation in which μ_{Asym}≠0, four configurations were considered:a) Asym = 0.4 on all branch points of that generation;b) Asym = 0.6 on half the number of branch points and Asym = 0.2 on the other half; c) Asym = 0.6 on half the number of branch points and Asym = 0 on the other half; and d) Asym = 0.2 on half the number of branch points and Asym = 0 on the other half. The resulting He and SF_{6} slope increases compared with the symmetrical model (ΔS _{μ=0}) are also depicted in Table 1 and show that ΔS _{μ=0} obtained withb were indeed larger than those obtained with a, and that ΔS _{μ=0} obtained from MBPM configurations c and d add up exactly to that obtained with b.
S_{acin}: Simulations vs. experiments.
Figure 8 shows the experimentalS _{acin} data points (leftward shifted horizontal bars with SD) for He, N_{2}, and SF_{6} compared with simulated slopes. First, the reference model MBPM_{ref}simulated S _{acin}(□) are reported. Then, on the basis of the sensitivity study of He and SF_{6} phase III slope to asymmetry (Figs. 4 and 5) and variability of asymmetry (Figs. 6 and 7), we attempted to build a MBPM that could reproduce these experimental S _{acin}for the gases of different diffusivity. We first considered the phase III slopes simulated with the MBPM of Fig. 2, corresponding to μ_{Asym} = 0.4 and SD_{Asym} = 0.3 in all generations, and another MBPM with the same μ_{Asym} = 0.4 but with SD_{Asym} = 0 in all generations. The resulting S _{acin} are shown by the solid squares connected by dotted lines in Fig. 8 (upper dotted lines, SD_{Asym} = 0.3; lower dotted lines, SD_{Asym} = 0). Because the experimentalS _{acin} values for all gases fell within these limits, we decided to keep μ_{Asym} = 0.4 in all generations and to vary SD_{Asym} values in successive generations.
The SD_{Asym} values per generation that produce a slope in good agreement with experimental S _{acin} for He, N_{2}, and SF_{6} are given in Table2 (MBPM*), where μ_{Asym} and SD_{Asym} values corresponding to MBPM_{ref} of Fig.1 are also shown for comparison. In fact, MBPM* refers to a set of 10 MBPM with the same μ_{Asym} and SD_{Asym} (each MBPM was built by random generation of Asym distributions). The mean and SD of the S _{acin} simulations obtained with MBPM* are shown in Fig. 8 (solid squares with error bars).
DISCUSSION
The main goal of the present study was to obtain quantitative agreement between S _{acin} in the normal human lung and phase III slopes simulated with a model of the acinus (MBPM). This was essentially done by modifying MBPM geometry within the constraints of available anatomic data. We successfully reproduced experimentalS _{acin} values for He, N_{2}, and SF_{6} obtained in healthy subjects. More importantly, the sensitivity study that led up to this final goal provided major insights into the relation between acinar structure and phase III slope. It was shown that, besides the mean volume asymmetry between any two lung units subtended by all parallel branch points of a given intraacinar generation, the variability in asymmetry among parallel branch points has a crucial impact on phase III slope. The study also confirmed that, even in a complex MBPM structure, phase III slopes of gases with a sixfold difference in diffusivity (He, SF_{6}) can be indexes of airway structural change at different lung depths within the acinus.
Sensitivity of phase III slope to intraacinar asymmetry.
A key feature of intraacinar gas transport is the diffusion front (Fig. 3), which is quasistationary over the course of an inspiration and results from a balance of convective and diffusive gas transport. This implies that the diffusion front of the least diffusive gas (SF_{6}) is more peripherally located than that of a more diffusive gas (He). Of particular relevance is the fact that MBPM asymmetry has virtually no impact on the actual location of the diffusion front (Fig. 3), which greatly facilitates interpretation of phase III slope changes for different diffusivity gases. Indeed, diffusionconvection interdependence theory (8) predicts that branch points situated on the steepest part of the diffusion front of a given gas will contribute the most to the overall phase III slope of that gas. The comparison of Figs. 4 B and 6 Bwith Fig. 3 confirms that this is applicable to He and SF_{6}in a MBPM structure. Consequently, structural alterations affecting asymmetry at the level of the respiratory bronchioles (generations 17–18) will preferentially modify He phase III slope, whereas changes in asymmetry at the level of the alveolar ducts (generations 20–21) will preferentially affect SF_{6} phase III slope.
Sensitivity studies to intraacinar asymmetry indicate the crucial importance of variable Asym compared with constant Asym in generating a phase III slope. The reason a variable asymmetry produces larger phase III slopes than a constant asymmetry for a given μ_{Asym}per generation is highlighted by MBPM a–dsimulations. Indeed, if phase III slope had shown a linear dependence on Asym, the slopes generated with b should equal the slopes generated with a, but this is not the case, at least in part because phase III slope shows an exponentiallike dependence on Asym (Fig. 5).
Another interesting feature of the MBPM simulations is that the sum of slopes generated at the MBPM entrance by imposing a given Asym in successive intraacinar generations (Fig. 4 B) approximately corresponds to the slope generated at the MBPM entrance when the same Asym is imposed on all MBPM generations simultaneously (Fig. 5). Indeed, in relatively simple MBPM examples such as these, the superposition principle holds for phase III slopes generated at serially distributed branch points. The superposition principle holds also for parallel distributed branch points in simple MBPM such as illustrated from simulations with MBPM b–d(Table 1). Indeed, the phase III slopes generated by imposing either Asym = 0.6 (c) or Asym = 0.2 (d) on half the number of branch points of a given generation, and Asym = 0 (symmetry) on the other half, added up to the slope generated with Asym = 0.6 on half the number of branch points and Asym = 0.2 on the other half (b). Both superposition principles, together with results from sensitivity studies, were essential guidelines to build the MBPM* to reproduce experimentalS _{acin}.
Simulations of experimental S_{acin} values.
The extent of phase III slope increases (ΔS) with μ_{Asym} (Figs. 4 and 5) or SD_{Asym} (Figs. 6 and7) suggested that some combination of μ_{Asym} and SD_{Asym} should yield a MBPM that could reproduce the experimental S _{acin} values. The effect of variability of asymmetry even suggested that several such MBPM could be constructed within the same μ_{Asym} and SD_{Asym} constraints. When we consider ten such MBPM, corresponding to the μ_{Asym} and SD_{Asym}values of MBPM* (Table 2), the resulting S _{acin}(solid squares with SD bars in Fig. 8) compare well with the experimental S _{acin} values. TheS _{acin} data in Fig. 8 were obtained from three subjects of the experiments of Prisk et al. (12); one of the four subjects in this study was discarded because of the small tidal volume (0.92 liter) used. This small data set was chosen because these were welldocumented healthy subjects, with raw data available on all three gases (He, N_{2}, and SF_{6}), from whichS _{acin} could be accurately computed. Nevertheless, partial comparisons, e.g., with the N_{2} data obtained in different laboratories, show overall consistency. Verbanck et al. (15) obtainedS _{acin}(N_{2}) = 0.075 ± 0.022 liter^{−1} in 10 healthy subjects, and, from the normalized slope curves of five normal subjects reported in Crawford et al. (2), we computedS _{acin}(N_{2}) = 0.052 ± 0.032 liter^{−1} (mean ± SD). Note that at least part of the intersubject variability within each study group can be explained on the basis of Asym variability, more specifically, from the standard deviation obtained on the simulated S _{acin} with MBPM* (solid squares with SD bars in Fig. 8), which amounted to 0.012 liter^{−1} for N_{2}.
In Fig. 8, the difference between simulations obtained with MBPM* and with a MBPM with no variability in asymmetry (solid squares connected by the lower dotted line) shows that 40–60% ofS _{acin} can be accounted for by variability of asymmetry (for a given μ_{Asym} = 0.4 in all models). This could partly explain the better performance of MBPM* equivalent models with respect to the most recent previous one (MBPM_{ref}), which underestimatedS _{acin} by 85% for He and by 28% for SF_{6}. Indeed, in MBPM_{ref}, SD_{Asym} was generally lower than in MBPM* (Table 2). Nevertheless, part of the discrepancy between simulations obtained with MBPM* and MBPM_{ref} can also be explained on the basis of μ_{Asym} per generation in MBPM* and MBPM_{ref}(Table 2). In generations 17–18, in which He slope is most sensitive to asymmetry, μ_{Asym} is considerably smaller in MBPM_{ref} (0.13–0.16) than in MBPM* (0.40). In generations 20–21, in which SF_{6} slope is most sensitive to asymmetry, μ_{Asym} is moderately smaller in MBPM_{ref} (0.26–0.38) than in MBPM* (0.40). This could explain why the underestimation of experimentalS _{acin} was more marked for He than for SF_{6} with MBPM_{ref} simulations. In summary, to reproduce experimental S _{acin} for gases of different diffusivity, the Asym distribution in MBPM* with respect to MBPM_{ref} was modified in two steps: first, μ_{Asym} was set to 0.4 uniformly, and, second, it was necessary to increase SD_{Asym} throughout all generations.
The fact that overall asymmetry is an important contributor to phase III slope has previously encouraged simulation studies to artificially increase μ_{Asym} in models of the acinus to match experimental phase III slope values. For instance, Bowes et al. (1) contended to “have deliberately chosen a model with a degree of asymmetry greater than that demonstrated by a number of studies of acinar anatomy in the human lung.” A major problem indicated by these authors was that no detailed anatomic data and branching pattern of an “average” acinus existed at the time. Although the present study provides an interesting new perspective on the role of variability of asymmetry over parallel branch points, it also reiterates the demand for more morphometric data in terms of average asymmetry and variability of asymmetry in successive generations of an average human acinus.
Besides the branching pattern of the single acinus reported in Ref. 5, which led to MBPM_{ref} (Table 2), we have checked whether the increased μ_{Asym} in the first MBPM* generations (with respect to MBPM_{ref}) could be accounted for, to some extent, by the available anatomic data. On the basis of the volume distribution of the 209 acini obtained from the human lung study of HaefeliBleuer and Weibel (5), we computed Asym values (Eq. 1 ) for all twobytwo combinations of acinar volume. This yielded Asym = 0.35 ± 0.21 (mean ± SD), which would correspond to a set of μ_{Asym} and SD_{Asym} values for generation 14. This computation at least provides an indication that μ_{Asym} values in generations 15 and 16 of MBPM_{ref} (Table 2), obtained on the basis of the Asym of, respectively, one and two branch points of a single acinus (5), may have led to a severe underestimation of actual asymmetry in the first acinar generations. Therefore, the Asym distribution chosen for MBPM*, which led to realisticS _{acin}, seems reasonable.
The role of variability of asymmetry also has implications for the use of S _{acin} in lung pathology. Indeed, if mild structural changes occur at a given lung depth, with marginal influence on μ_{Asym} in that generation, these could nevertheless be reflected in an increasedS _{acin} through the effect of increased SD_{Asym}. Mean S _{acin} values for N_{2} yielded 0.107 liter^{−1} in hyperresponsive subjects, 0.195 liter^{−1} in asthmatic subjects, and 0.443 liter^{−1} in chronic obstructive pulmonary disease patients (14, 15). In the above studies, only N_{2} could be used. The importance of using gases of different diffusivity is illustrated by a recent followup study of heartlung transplant recipients by Estenne and Van Muylem (3), who found that preferential He phase III slope increases reflect episodes of bronchiolitis obliterans, on average 329 days before the decline of forced expiratory volume in 1 s.
In summary, the systematic study of the phase III slope sensitivity to intraacinar asymmetry showed that each test gas is most sensitive to changes in asymmetry in the generations coinciding with the steepest part of its diffusion front. In addition, the study highlighted the role of variability in intraacinar asymmetry in bringing about considerable phase III slope increases. On the basis of these observations, we built a MBPM that allowed the simulation of realistic S _{acin} values for each test gas. In the absence of conclusive data about acinar branching pattern asymmetry, we have indicated the degree of asymmetry and variability of asymmetry that can reproduce experimental S _{acin}solely on the basis of diffusion and convection in the lung periphery.
Acknowledgments
We thank Yves Verbandt and Alain Van Muylem for stimulating discussions and Frederic Marteau for computational efforts in a preliminary study.
Footnotes

M. Paiva was supported by contract Prodex with the Belgian Federal Office for Scientific Affairs, and S. Verbanck was supported by the Federal Fund for Scientific ResearchFlanders.

Address for reprint requests and other correspondence: B. Dutrieue, Laboratoire de Physique Biomédicale, Route de Lennik, 808, CP 613/3, B1070 Brussels, Belgium (Email:bdutrieu{at}ulb.ac.be).

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 Copyright © 2000 the American Physiological Society
Appendix
Gas transport in the human lung was simulated by solving a onedimensional partial differential equation describing convective and diffusive gas transport along each pathway down to the alveolar sacs (10)
For the purpose of the numerical solution of the gas transport equation, the human lung structure was discretized into nodes, and on a bifurcation node K, as the one depicted in Fig.9, the finite difference equation corresponding to Eq. EA1 was
where subscripts M and T, respectively, refer to entities peripheral and proximal to a given node K, and where Δz_{M}(K_{i}) = [Δz(K) + Δz(K_{i})]/2 (for i = 1,2) and Δz_{T}(K) = [Δz(K) + Δz(K − 1)]/2; Vl (t = 0) is total lung volume att = 0, V˙l is the total flow at lung entrance, and Hom is the homogeneous expansion coefficient equaling 1 + V˙l ×t/Vl(t = 0). Changes with respect to the previously used discretization of Eq. EA1 over a bifurcation (published in Ref. 16) are highlighted in bold and allow for different lengths of any two daughter ducts (represented by K_{1} and K_{2}).