Vol. 89, Issue 5, 1859-1867, November 2000
A human acinar structure for simulation of realistic alveolar
plateau slopes
Brigitte
Dutrieue1,
Frederique
Vanholsbeeck1,
Sylvia
Verbanck2, and
Manuel
Paiva1
1 Biomedical Physics Laboratory, Université Libre de
Bruxelles, 1070 Brussels, Belgium; and 2 Department of
Pneumology, Akademisch Ziekenhuis, Vrije Universiteit Brussel, 1090 Brussels, Belgium
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ABSTRACT |
We simulated the intra-acinar contribution to phase III
slope (Sacin) for gases of differing
diffusivities (He and SF6) by solving equations of
diffusive and convective gas transport in multi-branch-point models
(MBPM) of the human acinus. We first conducted a sensitivity study of
Sacin to asymmetry and its variability in
successive generations. Sacin 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 SF6, 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 Sacin values
obtained in normal subjects for He, N2, and
SF6. Ten variants of such a MBPM were constructed to
estimate intrinsic Sacin variability owing to
peripheral lung structure. The realistic simulation of
Sacin in the normal lung and the understanding
of how asymmetry affects Sacin for different
diffusivity gases make Sacin a powerful tool to
detect structural alterations at different depths in the lung periphery.
multi-branch-point models; multiple-breath washout; gas mixing; diffusion-convection interdependence; airway asymmetry; helium; SF6
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INTRODUCTION |
PHASE III SLOPE OF THE
SINGLE-BREATH 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
intra-acinar airways, where the balance of diffusion and convection
establishes a quasi-stationary diffusion front during inspiration. At
this level, the asymmetry of the acinar structure generates
intra-acinar 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
convection-dependent 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 intra-acinar part of the phase III slope
(Sacin).
By using a multiple-breath 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 isolate
Sacin. The purpose of the present work was to
quantitatively reproduce the experimental Sacin
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 intra-acinar 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 intra-acinar gas transport (9, 11) were
based on the morphometric data reported by Hansen and Ampaya
(6) that accounted, to some extent, for intra-acinar
asymmetry. A more detailed description of the human acinus by
Haefeli-Bleuer and Weibel (5) prompted the development of
a more realistic model of intra-acinar 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 Sacin depended on the diffusivity of
the gas. For He, N2, and SF6, phase III slopes
were underestimated by 85%, 74%, and 28%, respectively
(13). The present study reexamines the intra-acinar
morphometry, makes a quantitative evaluation of the critical
characteristics of the intra-acinar branching pattern, and
quantitatively reproduces the experimental
Sacin, i.e., the acinar contribution to the
phase III slope, for gases with different diffusivities.
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METHODS |
Model of the human acinus.
The equations used to describe gas transport in the human acinus are
given in the APPENDIX. The anatomic basis for all acinar model geometries considered here is the morphometric study by Haefeli-Bleuer 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 multi-branch-point model
(MBPM) (13) that will be considered here as a
reference model, MBPMref. The branching pattern of
MBPMref is depicted in Fig.
1A, and the volume asymmetry
at each branch point of the model is plotted in Fig. 1B.
Each branch point subtends two intra-acinar units of volume,
V1 and V2, respectively. Assuming
V1 
V2, the asymmetry (Asym) at any given
branch point is defined as
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(1)
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In such a way, Asym = 0 corresponds to a symmetrical branch
point, i.e., subtending two units of identical volume.
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Fig. 1.
Reference multi-branch-point model (MBPM;
MBPMref) from a previous study (13).
A: branching pattern. B: volume asymmetry in
successive intra-acinar generations.  , Asymmetry (Asym) values
computed from Eq. 1 for each MBPM branch point; solid lines,
mean Asym per generation (µAsym).
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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 intra-acinar 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
(µAsym, SDAsym) of the Asym distribution
attributed to each generation. One example of a MBPM with an Asym
distribution characterized by µAsym = 0.4 and
SDAsym = 0.3 in each successive intra-acinar
generation is shown in Fig.
2A; the individual Asym values
are plotted in Fig. 2B.
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Fig. 2.
An example of a MBPM with Asym distribution characterized
by µAsym = 0.4 and Asym standard deviation per
generation (SDAsym) = 0.3 in intra-acinar generations
15-23. A: branching pattern. B: Asym in
successive intra-acinar generations; same representation as in Fig.
1.
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The acinar volume of all MBPM constructed here was set to 187 × 10
3 ml, i.e., the mean acinar volume measured by
Haefeli-Bleuer 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 5.5-liter
lung volume was considered to be in the acini, and, with an acinar
volume of 187 × 103 ml, this resulted in 28,785 acini (5,500 
119 ml)/(187 × 103
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 multiple-breath washout tests including He and SF6 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, SF6 washin, and N2 washout
tracings were reanalyzed to fully comply with the method of phase III
slope analysis for Sacin computation. The
detailed description and underlying theory for the computation of
Sacin can be found elsewhere (15).
Briefly, inspired gases (He, SF6) are first rescaled to
represent lung resident gases (such as N2) to obtain
positive phase III slopes for all gases under study. Then, phase III
slopes are computed by linear regression on N2, He, or
SF6 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 obtain
Sacin.
In this way, the following experimental Sacin
values were obtained for He, N2, and SF6 gases
(means ± SD): 0.054 ± 0.009 liter1 (He),
0.080 ± 0.007 liter1 (N2), and
0.109 ± 0.005 liter1 (SF6),
representing, respectively, 91.3, 91.6, and 92.0% of corresponding phase III slopes of the first breath. These
Sacin values constitute the experimental basis
for comparison with simulations.
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RESULTS |
Intra-acinar asymmetry and diffusion front.
Figure 3 shows simulated He and
SF6 diffusion fronts (mean gas concentrations) at the end
of a 1.23-liter inspiration using a flow equal to 0.5 l/s and
diffusion coefficients 0.6 cm2/s and 0.1 cm2/s
for He and SF6, respectively. Dashed lines are the He and
SF6 concentrations for a symmetrical MBPM
(µAsym = 0; SDAsym = 0 in all
generations), whereas solid lines correspond to the most asymmetrical MBPM (µAsym = 0.6; SDAsym = 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 SF6 in generations 17 and 19, irrespective of MBPM asymmetry (at least in the µAsym
range 0-0.6).
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Fig. 3.
Simulated diffusion fronts at end-inspiration for He
(light lines) and SF6 (heavy lines), using a 0.5 l/s flow,
with a symmetrical MBPM (dashed lines) and an asymmetrical MBPM with
µAsym = 0.6 and SDAsym = 0 in
generations 15-23 (solid lines). The vertical dashed line
corresponds to the acinar entrance.
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Sensitivity of phase III slope to intra-acinar asymmetry.
We conducted a systematic study of He and SF6 phase III
slope sensitivity to µAsym. First, a set of simulations
was considered in which µAsym = 0.4 was imposed
individually in successive branching generations of the MBPM by
considering µAsym = 0 in all MBPM generations, except for one generation in which µAsym = 0.4 and
SDAsym = 0. An example of the resulting Asym
distribution in MBPM generations is shown in Fig.
4A, in which µAsym = 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 slope
Sµ=0.4,i. A corresponding slope increase
Sµ=0(i) with respect to the
symmetrical MBPM (with simulated slope Sµ=0)
was then defined as
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(2)
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Figure 4B shows the slope increases
Sµ=0 as a function of i,
indicating that maximal He and SF6 phase III slope
increases were obtained when the branching asymmetry
µAsym = 0.4 was introduced in generations 18 and 21, respectively.
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Fig. 4.
Simulations of phase III slope sensitivity to
µAsym imposed at different levels of a MBPM, in the
absence of variability in asymmetry at any MBPM level (see text for
details). A: example of MBPM Asym distribution. In this
case, a constant Asym of 0.4 is imposed in generation i = 18, with µAsym = 0 in all other generations. Solid
lines, µAsym. B: Phase III slope increases
(Sµ=0) with respect to the symmetrical MBPM
(Eq. 2) as a function of generation i in which
µAsym = 0.4 is imposed (µAsym = 0 in all other generations; SDAsym = 0 in all
generations at all times). Triangles, He; circles, SF6;
open symbols correspond to the MBPM described in A.
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Figure 5 extends the results of Fig.
4B by applying any given µAsym 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 (Sµ=0)
is obtained for each µAsym value. The resulting
Sµ=0 values for He and SF6 are
plotted as a function of the µAsym in Fig. 5, showing an
exponential-like increase with µAsym in the range
0-0.6. By comparison of selected data points in Figs.
4B and 5, it can be inferred that the He and SF6
phase III slope increases obtained by imposing µAsym = 0.4 on all MBPM generations simultaneously roughly correspond to the
summation, over all generations, of He and SF6 phase III slope increases obtained by imposing µAsym = 0.4 in each
individual generation (considering all data points in Fig.
4B). Indeed, the Sµ=0
value for µAsym = 0.4 in Fig. 5, yielding 0.022 liter1 (He) and 0.031 liter1
(SF6), almost equals the sum of
Sµ=0(i) for i = 15-23 (open data points in Fig. 5), which amounts to 0.021 liter1 (He) and 0.028 liter1
(SF6).
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Fig. 5.
Simulations of phase III slope sensitivity to constant
asymmetry imposed in MBPM generations 15-23 simultaneously
(SDAsym = 0 in all generations at all times).
Sµ=0 with respect to the symmetrical MBPM
is plotted as a function of the µAsym value.
Triangles, He; circles, SF6; open symbols
(Sµ=0 values for µAsym = 0.4) are the summation of the
Sµ=0(i) values in Fig.
4B for i between 15 and 23.
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Sensitivity of phase III slope to the variability of intra-acinar
asymmetry.
In analogy to Fig. 4, we conducted a systematic study of He and
SF6 phase III slope sensitivity to Asym variability between parallel branch points (i.e., nonzero SDAsym) in another
set of MBPM simulations. In this case, we considered
µAsym = 0.4 in all MBPM generations, and
SDAsym = 0 was imposed on all intra-acinar generations
except for one generation i in which SDAsym = 0.3 (variable asymmetry only among branch points of generation
i). An example of the resulting Asym distribution in MBPM
generations is shown in Fig.
6A, in which the variability
of asymmetry was imposed in generation i = 18. For each
MBPM simulation with SDAsym = 0.3 in generation
i, a corresponding slope increase
SSD=0(i) was defined, this time
with respect to a MBPM with constant asymmetry (µAsym = 0.4), as
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(3)
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where SSD=0.3,i is the simulated
slope in a MBPM with SDAsym = 0.3 in generation
i (SDAsym = 0 in all other
generations and µAsym = 0.4 in all generations) and
SSD=0 is the simulated slope in a MBPM with
µAsym = 0.4 and
SDAsym = 0 in all generations between 15 and
23. Both for He and SF6, the resulting
SSD=0(i) are shown in Fig.
6B for i = 16-23 (generation 15 has
only one branch point). Maximal phase III slope increases were obtained when the variability in asymmetry (SDAsym = 0.3) was
introduced in generations 17 and 20, for He and SF6,
respectively. Figure 7 extends the
results of Fig. 6B by applying any given
SDAsym value ranging between 0 and 0.35 to all
generations between 15 and 23 simultaneously (while µAsym
remains 0.4 throughout). In this case, a global slope increase with
respect to the MBPM with constant asymmetry
(SSD=0) is obtained for each
SDAsym value and plotted against SDAsym in Fig.
7. Note that, in this figure, the He and SF6 simulated data
points corresponding to SDAsym = 0.3 were obtained
with the MBPM shown in Fig. 2 (SDAsym = 0.3 and
µAsym = 0.4 in all generations).
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Fig. 6.
Simulations of phase III slope sensitivity to variability
of asymmetry (SDAsym) imposed at different levels of a MBPM
in the presence of a constant µAsym of 0.4 in all MBPM generations (see text for details). A: example
of MBPM Asym distribution. In this case, a variable Asym of
SDAsym = 0.3 is imposed in generation
i = 18, with µAsym = 0.4 in
generations 15-23. Solid lines, µAsym.
B: SSD=0 with respect to the MBPM
with constant asymmetry (Eq. 3) as a function of the
generation i in which SDAsym = 0.3 is
imposed (SDAsym = 0 in all other generations;
µAsym = 0.4 in all generations at all times).
Triangles, He; circles, SF6; open symbols correspond to the
MBPM described in A.
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Fig. 7.
Simulations of phase III slope sensitivity to variable
asymmetry imposed in MBPM generations 15-23 simultaneously
(µAsym = 0.4 in all generations at all
times). SSD=0 with respect to the MBPM with
constant asymmetry is plotted as a function of the SDAsym
value that is imposed.
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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 µAsym
0.
For He and SF6 simulations, µAsym0 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 µAsym0, 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
SF6 slope increases compared with the symmetrical model (Sµ=0) are also depicted in
Table 1 and show that Sµ=0 obtained with
b were indeed larger than those obtained with a, and that Sµ=0 obtained from MBPM
configurations c and d add up exactly to that
obtained with b.
Sacin: Simulations vs. experiments.
Figure 8 shows the experimental
Sacin data points (leftward shifted horizontal
bars with SD) for He, N2, and SF6 compared with
simulated slopes. First, the reference model
MBPMref-simulated Sacin
(
) are reported. Then, on the basis of the sensitivity study of He
and SF6 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 Sacin for the gases of different diffusivity. We first considered the phase III slopes simulated with the MBPM of Fig. 2, corresponding to
µAsym = 0.4 and SDAsym = 0.3 in all
generations, and another MBPM with the same µAsym = 0.4 but with SDAsym = 0 in all generations. The
resulting Sacin are shown by the solid squares
connected by dotted lines in Fig. 8 (upper dotted lines,
SDAsym = 0.3; lower dotted lines,
SDAsym = 0). Because the experimental
Sacin values for all gases fell within these
limits, we decided to keep µAsym = 0.4 in all
generations and to vary SDAsym values in successive generations.
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Fig. 8.
Comparison of experimental Sacin
values (horizontal bars; means ± SD) for He, N2, and
SF6 with simulations.  Connected by dotted
lines, Sacin obtained with a MBPM where
µAsym = 0.4 and SDAsym = 0 in all generations (lower symbols) or where
µAsym = 0.4 and SDAsym = 0.3 in all
generations (upper symbols); with error bars,
Sacin (means ± SD) simulated with 10 equivalent models in terms of µAsym and
SDAsym (see MBPM* in Table 2); ,
Sacin obtained with the reference model (see
MBPMref in Table 2).
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The SDAsym values per generation that produce a slope in
good agreement with experimental Sacin for He,
N2, and SF6 are given in Table
2 (MBPM*), where µAsym and
SDAsym values corresponding to MBPMref of Fig.
1 are also shown for comparison. In fact, MBPM* refers to a set of 10 MBPM with the same µAsym and SDAsym (each MBPM was built by random generation of Asym distributions). The mean
and SD of the Sacin simulations obtained with
MBPM* are shown in Fig. 8 (solid squares with error bars).
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DISCUSSION |
The main goal of the present study was to obtain quantitative
agreement between Sacin 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 experimental
Sacin values for He, N2, and
SF6 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
intra-acinar 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, SF6)
can be indexes of airway structural change at different lung depths
within the acinus.
Sensitivity of phase III slope to intra-acinar asymmetry.
A key feature of intra-acinar gas transport is the diffusion front
(Fig. 3), which is quasi-stationary 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
(SF6) 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, diffusion-convection 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. 4B and 6B
with Fig. 3 confirms that this is applicable to He and SF6
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 SF6 phase III slope.
Sensitivity studies to intra-acinar 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 µAsym
per generation is highlighted by MBPM a-d
simulations. 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 exponential-like 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 intra-acinar generations (Fig. 4B) 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 experimental Sacin.
Simulations of experimental Sacin values.
The extent of phase III slope increases (S) with
µAsym (Figs. 4 and 5) or SDAsym (Figs. 6 and
7) suggested that some combination of µAsym and
SDAsym should yield a MBPM that could reproduce the experimental Sacin values. The effect of
variability of asymmetry even suggested that several such MBPM could be
constructed within the same µAsym and
SDAsym constraints. When we consider ten such MBPM, corresponding to the µAsym and SDAsym
values of MBPM* (Table 2), the resulting Sacin
(solid squares with SD bars in Fig. 8) compare well with the
experimental Sacin values. The
Sacin 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 well-documented healthy subjects, with raw data available on
all three gases (He, N2, and SF6), from which
Sacin could be accurately computed.
Nevertheless, partial comparisons, e.g., with the N2 data
obtained in different laboratories, show overall consistency. Verbanck
et al. (15) obtained
Sacin(N2) = 0.075 ± 0.022 liter1 in 10 healthy subjects, and, from the normalized
slope curves of five normal subjects reported in Crawford et al.
(2), we computed
Sacin(N2) = 0.052 ± 0.032 liter1 (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 Sacin with
MBPM* (solid squares with SD bars in Fig. 8), which amounted to 0.012 liter1 for N2.
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% of
Sacin can be accounted for by variability of
asymmetry (for a given µAsym = 0.4 in all models).
This could partly explain the better performance of MBPM* equivalent
models with respect to the most recent previous one
(MBPMref), which underestimated Sacin by 85% for He and by 28% for
SF6. Indeed, in MBPMref, SDAsym was
generally lower than in MBPM* (Table 2). Nevertheless, part of the
discrepancy between simulations obtained with MBPM* and MBPMref can also be explained on the basis of
µAsym per generation in MBPM* and MBPMref
(Table 2). In generations 17-18, in which He slope is most
sensitive to asymmetry, µAsym is considerably smaller in
MBPMref (0.13-0.16) than in MBPM* (0.40). In
generations 20-21, in which SF6 slope is most
sensitive to asymmetry, µAsym is moderately smaller in
MBPMref (0.26-0.38) than in MBPM* (0.40). This could
explain why the underestimation of experimental
Sacin was more marked for He than for
SF6 with MBPMref simulations. In summary, to
reproduce experimental Sacin for gases of
different diffusivity, the Asym distribution in MBPM* with respect to
MBPMref was modified in two steps: first,
µAsym was set to 0.4 uniformly, and, second, it was
necessary to increase SDAsym throughout all generations.
The fact that overall asymmetry is an important contributor to phase
III slope has previously encouraged simulation studies to artificially
increase µAsym 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 MBPMref (Table 2), we have checked whether the increased µAsym in the first
MBPM* generations (with respect to MBPMref) 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 Haefeli-Bleuer and Weibel (5), we
computed Asym values (Eq. 1) for all two-by-two combinations
of acinar volume. This yielded Asym = 0.35 ± 0.21 (mean ± SD), which would correspond to a set of
µAsym and SDAsym values for
generation 14. This computation at least provides an indication that
µAsym values in generations 15 and 16 of
MBPMref (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 realistic
Sacin, seems reasonable.
The role of variability of asymmetry also has implications for the use
of Sacin in lung pathology. Indeed, if mild
structural changes occur at a given lung depth, with marginal
influence on µAsym in that generation, these
could nevertheless be reflected in an increased
Sacin through the effect of increased
SDAsym. Mean Sacin values for
N2 yielded 0.107 liter1 in hyperresponsive
subjects, 0.195 liter1 in asthmatic subjects, and 0.443 liter1 in chronic obstructive pulmonary disease patients
(14, 15). In the above studies, only N2 could
be used. The importance of using gases of different diffusivity is
illustrated by a recent follow-up study of heart-lung 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 intra-acinar 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 intra-acinar 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 Sacin 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 Sacin
solely on the basis of diffusion and convection in the lung periphery.
| |
APPENDIX |
Gas transport in the human lung was simulated by solving a
one-dimensional partial differential equation describing convective and
diffusive gas transport along each pathway down to the alveolar sacs
(10)
|
(A1)
|
where C is concentration and z is cumulative longitudinal
distance along the airway axes with its origin situated at the lung
model entrance (in our simulations, z = 0 corresponds to the
trachea); s and S are the airway cross sections without and with
alveoli, respectively; D is the binary molecular diffusion coefficient; and 
is convective flow (piston-type; positive and
constant during inspiration; negative and constant during expiration).
Finally, all volume changes (
V/t) were
considered isotropic, homogeneous, and in phase (16).
Initial condition was C(z,t = 0) = 0 and boundary
conditions were C(z = 0,t) = 1 during inspiration,
C/z(z = 0,t) = 0 during expiration, and C/z(t) = 0 on all peripheral boundaries at all times.
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. A1 was
![−<FR><NU>C(K)</NU><DE>Hom</DE></FR> <FR><NU><A><AC>V</AC><AC>˙</AC></A><SC>l</SC></NU><DE>V<SC>l</SC> (<IT>t=0</IT>)</DE></FR><IT> +</IT><FENCE><AR><R><C><FR><NU><IT>1</IT></NU><DE>S</DE></FR> <FR><NU>[C(K)<A><AC>V</AC><AC>˙</AC></A>(K)<IT>−</IT>C(K<IT>−1</IT>)<A><AC>V</AC><AC>˙</AC></A>(K<IT>−1</IT>)]</NU><DE><IT>&Dgr;</IT>z(K)</DE></FR> inspiration or</C></R><R><C><FR><NU><IT>1</IT></NU><DE>S</DE></FR> <FR><NU>[C(K<SUB><IT>1</IT></SUB>)<A><AC>V</AC><AC>˙</AC></A>(K<SUB><IT>1</IT></SUB>)<IT>+</IT>C(K<SUB><IT>2</IT></SUB>)<A><AC>V</AC><AC>˙</AC></A>(K<SUB><IT>2</IT></SUB>)<IT>−</IT>C(K)<A><AC>V</AC><AC>˙</AC></A>(K<IT>−1</IT>)]</NU><DE><IT>&Dgr;</IT>z(K)</DE></FR> expiration</C></R></AR></FENCE>](/content/vol89/issue5/fulltext/1859/img008.gif)
|
(A2)
|
View larger version (14K):
[in this window]
[in a new window]
|
Fig. 9.
Schematic representation of a discretized airway
bifurcation for the numerical solution of the gas transport equations
(Eq. A2), showing airway nodes K-1, K1, and
K2 around branch point node K and the axial distances
involved [Z(K), ZM(K1),
ZM(K2) and ZT(K)]; note that
all nodes are not necessarily equidistant.
|
|
where subscripts M and T, respectively, refer to entities
peripheral and proximal to a given node K, and where
zM(Ki) = [z(K) + z(Ki)]/2 (for i = 1,2) and
zT(K) = [z(K) + z(K 1)]/2;
VL (t = 0) is total lung volume at
t = 0, L is the total flow at lung
entrance, and Hom is the homogeneous expansion coefficient equaling
1 + L × t/VL(t = 0). Changes with
respect to the previously used discretization of Eq. A1 over
a bifurcation (published in Ref. 16) are highlighted in bold and allow
for different lengths of any two daughter ducts (represented by
K1 and K2).
| |
ACKNOWLEDGEMENTS |
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 Research-Flanders.
Address for reprint requests and other correspondence: B. Dutrieue, Laboratoire de Physique Biomédicale, Route de Lennik, 808, CP 613/3, B-1070 Brussels, Belgium (E-mail:
bdutrieu{at}ulb.ac.be).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 21 March 2000; accepted in final form 16 June 2000.
| |
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J APPL PHYSIOL 89(5):1859-1867
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Copyright © 2000 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![<FR><NU>∂C(K)</NU><DE><IT>∂t</IT></DE></FR><IT>=</IT><FR><NU><IT>D</IT></NU><DE><IT>&Dgr;</IT>z(K)</DE></FR> <FR><NU>s(K)</NU><DE>S(K)</DE></FR> <FENCE><FR><NU><IT>1</IT></NU><DE>(<B>S</B>(<B>K<SUB>1</SUB></B>)<IT>+</IT><B>S</B>(<B>K<SUB>2</SUB></B>))</DE></FR> <FENCE><FR><NU>[<B>C</B>(<B>K<SUB>1</SUB></B>)<IT>−</IT><B>C</B>(<B>K</B>)]<B>S</B>(<B>K<SUB>1</SUB></B>)</NU><DE><B>&Dgr;z<SUB>M</SUB></B>(<B>K<SUB>1</SUB></B>)</DE></FR><IT>+</IT><FR><NU>[<B>C</B>(<B>K<SUB>2</SUB></B>)<IT>−</IT><B>C</B>(<B>K</B>)]<B>S</B>(<B>K<SUB>2</SUB></B>)</NU><DE><B>&Dgr;z<SUB>M</SUB></B>(<B>K<SUB>2</SUB></B>)</DE></FR></FENCE><IT>−</IT><FR><NU>[C(K)<IT>−</IT>C(K<IT>−1</IT>)]</NU><DE><IT>&Dgr;</IT>z<SUB>T</SUB>(K)</DE></FR></FENCE>](/content/vol89/issue5/fulltext/1859/img005.gif)
![+<FR><NU>D</NU><DE>S(K)</DE></FR> <FR><NU><IT>1</IT></NU><DE><IT>2</IT></DE></FR> <FENCE><FR><NU><FENCE>s(K<SUB><IT>1</IT></SUB>)<IT>−</IT>s(K) <FR><NU>s(K<SUB><IT>1</IT></SUB>)</NU><DE>s(K<SUB><IT>1</IT></SUB>)<IT>+</IT>s(K<SUB><IT>2</IT></SUB>)</DE></FR></FENCE>[C(K<SUB><IT>1</IT></SUB>)<IT>−</IT>C(K)]</NU><DE><B>&Dgr;z</B><SUP><B>2</B></SUP><SUB><B>M</B></SUB>(<B>K<SUB>1</SUB></B>)</DE></FR> + <FR><NU><FENCE>s(K<SUB><IT>2</IT></SUB>)<IT>−</IT>s(K) <FR><NU>s(K<SUB><IT>2</IT></SUB>)</NU><DE>s(K<SUB><IT>1</IT></SUB>)<IT>+</IT>s(K<SUB><IT>2</IT></SUB>)</DE></FR></FENCE> [C(K<SUB><IT>2</IT></SUB>)<IT>−</IT>C(K)]</NU><DE><B>&Dgr;z</B><SUP><B>2</B></SUP><SUB><B>M</B></SUB>(<B>K<SUB>2</SUB></B>)</DE></FR></FENCE>](/content/vol89/issue5/fulltext/1859/img006.gif)
![<FENCE>+<FR><NU>[s(K)<IT>−</IT>s(K<IT>−1</IT>)][C(K)<IT>−</IT>C(K<IT>−1</IT>)]</NU><DE><IT>&Dgr;</IT>z<SUP><IT>2</IT></SUP><SUB>T</SUB>(K)</DE></FR></FENCE>](/content/vol89/issue5/fulltext/1859/img007.gif)
