J Appl Physiol 115: 64–70, 2013.
First published April 18, 2013; doi:10.1152/japplphysiol.00112.2013.
Parenchymal mechanics, gas mixing, and the slope of phase III
Theodore A. Wilson
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota
Submitted 28 January 2013; accepted in final form 11 April 2013
nonuniform ventilation; Marangoni flow; mixing efficiency
NONUNIFORM VENTILATION HAS physiological and clinical significance, both intrinsically and because the ventilation distribution is an underlying component of the ventilation-perfusion
ratio (V̇A/Q̇) distribution. Mixing inefficiency, deviations from
simple exponential plots of expired concentration as a function
of breath number for multibreath washin or washout of a test
gas, and the slope of phase III (SIII) of expiration of a test gas
are taken as signs of nonuniform ventilation. Matching of
multicompartment models to data on mixing efficiency or data
on expired gas concentration vs. breath number yield the result
that logSDv̇, where v̇ is regional specific ventilation, is 0.5– 0.6.
Thus ventilation is quite nonuniform; the ratio of v̇ for a region
that lies 1 SD above the mean to that for a region that lies 1 SD
below the mean is ⬃3. Both functional and anatomic studies
show that the bulk of this variability in v̇ occurs at small scale.
This raises the question: what regional differences in parenchymal microstructure could produce this variance in v̇?
To investigate this question, a model of parenchymal mechanics (27) is revisited. The effects of variations in the values
of the parameters of the model on specific ventilation are
described. Particular examples of parameter values for two
regions fed by the daughter branches at a bifurcation are
chosen, and the volume oscillations of the two compartments
are calculated. The magnitude of the oscillations in surface
tension (␥) that accompanies the volume oscillations is different for the two compartments. Differences in ␥ drive Marangoni flows that equilibrate ␥. This relaxation process is then
Address for reprint requests and other correspondence: T. A. Wilson, 4132
Aldrich Ave. S, Minneapolis, MN 55409 (e-mail: [email protected]).
64
included in the model, the volume oscillations are recalculated,
and the concentration of N2 in the expired gas after an inspiration of pure oxygen is calculated. Because of the phase
difference between the volume oscillations of the compartments that is introduced by ␥ relaxation, the concentration of
nitrogen in mixed expired gas increases during the course of
expiration. Thus the modeling generates a new hypothesis
about the source of the SIII.
MODELING AND ANALYSIS
The conceptual basis of the model of parenchymal mechanics (27)
is the following. Total lung recoil pressure is assumed to consist of
two components: recoil of the saline-filled lung, and the additional
recoil of the air-filled lung. The recoil of the saline-filled lung is
provided by the connective tissue framework of the lung, consisting of
the pleural membrane, the bronchial tree, and the interconnected
network of interlobular membranes that extend between the bronchial
tree and the pleural membrane. The parenchyma is pictured as a
collection of polyhedral alveoli that open on the lumens of the
alveolar ducts. Cables, composed of connective tissue, form the free
edges of the alveolar walls. The outward pull of ␥ on the two faces of
the alveolar wall extending outward from a cable is balanced by the
combination of tension carried by the cable and the curvature of the
cable. It is assumed that tissue tension in the alveolar wall is negligible. This assumption is based on the observation that the alveolar
walls in saline-filled lungs are undulating and appear to be flaccid. In
air-filled lungs, excess wall material collects in pockets at the alveolar
intersections, and the surfaces are flat. This assumption is also consistent with later anatomical measurements reported by Oldmixon and
Hoppin (18). They found that 80% of the elastin and collagen in
parenchyma are located in features that they call “ends,” cables that
form the free edges of alveolar walls, and “bends,” cables from which
two alveolar walls extend outwards. Thus, mechanically, the alveolar
walls are pictured as merely platforms for ␥, and the additional recoil
of the air-filled lung is provided by ␥ and cable tension.
Basic Equations
Pressure. Total recoil pressure (P) is the sum of the recoil pressure
in the saline-filled lung (Psal) and the additional recoil of the air-filled
lung (P␥). The tissue component of recoil pressure is taken to be
linearly proportional to lung volume (V) with an elastance of 2⫻103
dyn/cm2 per total lung capacity (TLC) (Vo).
Psal ⫽ 2 ⫻ 103 · V ⁄ Vo
(1)
The additional recoil in the air-filled lung is the sum of the contributions of ␥ and tension in the line elements that form the free edges of
the alveolar walls around the circumference of the alveolar duct. For
randomly oriented surface normals, the average stress contributed by
␥ is 2·␥·S/3·V, where S is surface area (12). For randomly oriented
cable tangents, the stress contributed by tension in the cables (␶) is
␶·L/3·(V ⫹ Vti), where L is cable length, Vti is tissue volume, and
(V ⫹ Vti) is total V (27).
P␥ ⫽
2␥·S
3 V
⫹
1
␶·L
3 共V ⫹ Vti兲
(2)
Surface area. S is assumed to depend on V and L. This function,
S(V,L), is governed by two constraints. The first is the condition of
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Wilson TA. Parenchymal mechanics, gas mixing, and the slope of
phase III. J Appl Physiol 115: 64 –70, 2013. First published April 18,
2013; doi:10.1152/japplphysiol.00112.2013.—A model of parenchymal mechanics is revisited with the objective of investigating the
differences in parenchymal microstructure that underlie the differences in regional compliance that are inferred from gas-mixing studies. The stiffness of the elastic line elements that lie along the free
edges of alveoli and form the boundary of the lumen of the alveolar
duct is the dominant determinant of parenchymal compliance. Differences in alveolar size cause parallel shifts of the pressure-volume
curve, but have little effect on compliance. However, alveolar size
also affects the relation between surface tension and pressure during
the breathing cycle. Thus regional differences in alveolar size generate
regional differences in surface tension, and these drive Marangoni
surface flows that equilibrate surface tension between neighboring
acini. Surface tension relaxation introduces phase differences in regional volume oscillations and a dependence of expired gas concentration on expired volume. A particular example of different parenchymal properties in two neighboring acini is described, and gas
exchange in this model is calculated. The efficiency of mixing and
slope of phase III for the model agree well with published data. This
model constitutes a new hypothesis concerning the origin of phase III.
Slope of Phase III
internal equilibrium between the outward pull of ␥ and the inward
Laplace stress in the cables. At equilibrium, the stored energy in the
system is minimum, and the net work done by a virtual displacement of
the cables is zero. This principle yields the following equation governing
S(V,L).
␥
⭸S
⭸L
⫹␶⫽0
(3)
•
65
Wilson TA
The variables V, L, and S in Eqs. 10 –13 are extensive quantities for
the whole lung. The variables are transformed into intensive variables
by normalizing V by Vo, L by Lo, and replacing S by the surface-tovolume ratio. The new variables are defined as v ⫽ V/Vo, l ⫽ L/Lo,
s ⫽ S/V, and the governing equations, expressed in terms of these
variables are the following.
P ⫽ 2 ⫻ 103v ⫹
The second is a similar statement; the change in internal energy during a
virtual volume displacement equals the incremental work done by pressure.
␥
⭸S
⭸V
⫽ P␥
(4)
Combining Eqs. 2, 3, and 4 yields the following partial differential
equation for S(V,L).
⭸S
⭸V
⫹
1
L
⭸S
⫺
3V
⫽0
(5)
The general solution to Eq. 5 is given by Eq. 6, where F is an arbitrary
function of its argument.
S ⫽ V2⁄3 · F关L2 ⁄ 共V ⫹ Vti兲2⁄3兴
(6)
The following form is chosen for S(V,L), where C1 and C2 are
constants.
冋
L2
S ⫽ V2⁄3 · C1 · 1 ⫺ C2 ·
共V ⫹ Vti兲2⁄3
册
(7)
The first term on the right side of Eq. 7 describes the dependence of
S on V for a foam in which geometry remains similar as air volume
changes. The second describes the reduction of S due to missing
surfaces in the volume occupied by the lumen of the duct with lumen
radius proportional to L.
Cable tension. The properties of the cable are described by Eq. 8.
In this equation, C3 and C4 are constants that describe the elastance of
the cable and the strength of the nonlinearity, respectively, and Lo is
the resting length of the cable.
␶ ⫽ C3 · 共L ⁄ L0 ⫺ 1兲 · exp兵C4 · 关共L ⁄ L0兲 ⫺ 1兴2其
(8)
Surfactant dynamics. For small S oscillations, surfactant acts like
an elastic material with a specific elastance determined by its physical-chemical properties. The value of specific elastance reported by
Schurch et al. (22) is 200 dyn/cm. Thus an incremental change in ␥
(d␥) is related to an incremental change in S (dS) by Eq. 9.
S · d␥ ⫽ 200 · dS
(9)
Summary of basic equations. The variable ␶ can be eliminated by
substituting for ␶ from Eq. 3 into Eqs. 2 and 8. The remaining
equations reduce to the following.
P ⫽ 2 ⫻ 103 · V ⁄ Vo ⫹
2
3
· C1 ·
冋
␥
V1⁄3
S ⫽ C1 · V2⁄3 · 1 ⫺ C2 ·
2 · C1 · C2 ·
冋
1 ⫺ C2 ·
Vti · L2
共V ⫹ Vti兲5⁄3
L2
共V ⫹ Vti兲2⁄3
册
册
(10)
(12)
v1⁄3
冋
1 ⫺ c2 ·
s · ␯ ⫽ c1 · ␯2⁄3 1 ⫺ c2 ·
␥ · l · ␯2⁄3
共␯ ⫹ ␯ti兲2⁄3
vti · l2
共v ⫹ vti兲5⁄3
l2
共␯ ⫹ ␯ti兲2⁄3
册
⫽ c3 · 共l ⫺ 1兲 · exp关c4 · 共l ⫺ 1兲2兴
s · ␯ · d␥ ⫽ 200 · d共s · ␯兲
册
(14)
(15)
(16)
(17)
The lower case constants in Eqs. 14 –17 are related to the upper case
2
2/3
constants by the relations: c1 ⫽ C1/V1/3
o , c2 ⫽ C2·Lo/Vo , c3 ⫽
C3/2·C1·C2·Lo, and c4 ⫽ C4.
Parameter Values and Mechanical Behavior
Base case. The values of the four parameters in Eqs. 14 –17 were
chosen so that the model matches the anatomic and mechanics data for
human lungs. For a foam with spherical cells, the surface-to-volume
ratio ⬇ 3/r, where r is the radius of the cell. The value of c1 was taken
as 210 cm⫺1 to match the values of alveolar radius reported by Weibel
(25). The value of c2 was taken to be 0.22 so that the second term in
the brackets in Eq. 15 is 0.4, consistent with the observation that the
lumen of the duct occupies 40% of the volume of the acinus (25). The
values of c3 and c4 were chosen rather arbitrarily so that the
specific elastance of the model is ⬇ 4P (15). These values are listed
in Table 1 as parameter values for the base case. The value of vti
was taken to be 0.12 (26).
Equations 14 –17 were solved numerically for an imposed sinusoidal pressure oscillation with mean value 2.5 ⫻103 dyn/cm2, an
amplitude of 1.0 ⫻ 103 dyn/cm2, and with ␥ ⫽ 10 dyn/cm at
midoscillation. The resulting v-P and relation is shown in Fig. 1.
Effect of varying parameter values. For a larger value of c1,
corresponding to a smaller alveolar size, the value of v at midoscillation is smaller, but the compliance is essentially unchanged. Also,
the rate of change of ␥ with P is larger. For larger values of c2,
corresponding to a larger duct lumen, the changes in v and ␥ are
opposite those for larger c1. For larger values c3 and c4, corresponding
to stiffer alveolar entrance rings, parenchymal elastance is larger, and
the ␥-P curve is essentially unchanged.
Equations 14 –17 are rather opaque, but some understanding of the
changes in mechanical properties with changes in parameter values
can be obtained from the form of these equations. The effect of c1 can
be explained from Eq. 14. The right side of Eq. 14 is a weak function
of v. Therefore, a relatively small increase in c1 requires a relatively
large decrease in v to match a given value of P at a given ␥. Also, for
a given change in pressure, the change in ␥ must be smaller if c1 is
larger. The effect of greater cable stiffness is determined by Eqs.
15–17. From Eq. 15, it can be seen that S increases with V, and that
this increase is buffered by increases in L with increasing ␥. For a
Table 1. Parameter values for the base case and for the two
regions in the two-compartment model
⫽ C3 · 共L ⁄ Lo ⫺ 1兲 · exp关C4 · 共L ⁄ Lo ⫺ 1兲2兴
S · d␥ ⫽ 200 · dS
冋
␥
(11)
␥ · L · V2⁄3
共V ⫹ Vti兲2⁄3
3
· c1 ·
(13)
For a given pressure history, P as a function of time, and an initial
value of ␥, Eqs. 10 –13 constitute a complete set of equations for the
variables, V, L, S, and ␥, as functions of time.
Base case
Compartment 1
Compartment 2
c1, cm⫺1
c2
c3
c4, dyn/cm
210
220
185
0.22
0.24
0.16
80
30
120
6
2
12
c1–c4, constants 1– 4.
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3 共V ⫹ Vti兲 ⭸ L
2S
2
66
Slope of Phase III
A
v
s1 · ␯1 ·
2
0.4
0.3
base
1
0.2
18
16
γ (dynes/cm)
14
12
d␥1
dt
⫽ 200 ·
冋共
d s 1 · ␯ 1兲
dt
⫹ k · 共 ␥ 2 ⫺ ␥ 1兲
册
(18)
Terms with opposite signs are added to the equation for compartment
2. The value of k was taken as 3 cm⫺1·s⫺1, and approximate
numerical solutions for Eqs. 14, 15, 16, and 18 for a breathing
frequency of 12 breaths/min were obtained by the methods described
in the APPENDIX.
Gas exchange. An N2 washout maneuver was simulated. The initial
concentration of N2 in both compartments was taken as 80%. The
concentrations of N2 in the two regions after an inspiration of pure O2
were calculated, assuming a dead space of 0.14 liter, and the concentration of N2 in the mixed expired gas during the subsequent expiration was calculated. For this simulation of a single-breath washout, a
line was fit to the curve of N2 vs. expired volume (Vexp) in the range,
0.4 liter ⬍ Vexp ⬍ 0.8 liter, to obtain the SIII, and SIII was divided by
mean expired concentration to obtain the normalized SIII (Sn).
Two other single-breath maneuvers were simulated. The first
was for a 50% larger pressure oscillation with a breathing frequency of 8 breaths/min, and the second was a breath hold of 4 s
at constant pressure at end inspiration followed by expiration. In
addition, the values of Sn were calculated as a function of breath
number for a multibreath washout. In this calculation, the concentrations of N2 in the compartments were calculated, taking account
of penduluft that occurs at the beginning and end of inspiration and
the reingestion of gas from the dead space at the beginning of
inspiration.
10
RESULTS
8
1
6
4
2
2
0
P (cm H2O)
Fig. 1. A: volume (v) vs. pressure (P) for the base case (dashed line) and each
of the two compartments (solid lines). B: surface tension (␥) vs. P for the two
compartments.
stiffer cable, the increase in l with increasing ␥ is smaller (Eq. 16), and
thus the increase in S for a given change in v is larger. The change in
␥ is proportional to the change in S (Eq. 17), and since the change in
␥ is fixed by the change in P (Eq. 14), the change in v must be smaller.
Two-compartment Model
Mechanics. The two-compartment model follows the familiar
scheme of two compartments fed by the daughter branches of a
common airway. The parameter values for each compartment are
listed in Table 1, and the v-P and ␥-P curves for the two compartments
are shown in Fig. 1. It can be seen from Fig. 1 that differences in ␥ are
generated during oscillatory pressure changes. These differences
would drive Marangoni flow of the liquid lining layer from the
compartment with low ␥ into the compartment with high ␥. The
remaining surface expands in the compartment with low ␥ and
contracts in the compartment with high ␥, thereby tending to equilibrate ␥s. This effect is described by adding a term to the right side of
The calculated v-P curves for the two compartments and the
curve of ␥2-␥1 vs. P are shown in Fig. 2. The volume loops
shown in Fig. 2 are not pure sinusoids, but, roughly speaking,
v1 leads P by 8° and v2 lags P by 23°. Thus, the v-P curve for
compartment 1 rotates clockwise, and that for compartment 2
rotates counterclockwise. For TLC ⫽ 7 liters, end-expiratory
volume in Fig. 2 corresponds to 3 liters, and the volume
excursions correspond to a tidal volume of 1 liter. Compartmental volumes during expiration, normalized by their volume
at end inspiration, are shown as functions of total volume,
normalized by total end-inspiratory volume, in Fig. 3.
The concentrations of N2 in compartments 1 and 2, which
were initially 80%, are 54 and 68%, respectively, at end
inspiration. The curve of mixed expired concentration of N2 as
a function of Vexp is shown in Fig. 4. Also shown in Fig. 4 is
the concentration of N2 for ideal mixing. The efficiency of
mixing is 93%. Because of the phase difference between the
volume oscillations, the expired concentration of N2 rises
during expiration. The slope of the line fit to the curve is
3.9%/liter, and the Sn ⫽ 0.068 liter⫺1.
For the tidal volume of 1.5 liters, the end-inspiratory concentrations in both regions are lower than for the tidal volume
of 1.0 liter, the difference in concentrations is slightly lower,
and the phase angles are about the same. Therefore, the
increase of expired concentration with phase angle is about the
same, and the SIII is smaller primarily because the volume
excursion is larger. For this maneuver Sn ⫽ 0.046 liter⫺1.
For the breath hold at end inspiration, the difference in ␥
decreases from 1.1 dyn/cm at end inspiration to 0.3 dyn/cm at
the end of the 4-s breath hold. Volume decreases in region 1
and increases in region 2 during breath hold. During expiration
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P (cm H2O)
B
Wilson TA
Eq. 17 for region 1, to obtain Eq. 18 where the superscripts denote
compartment number.
0.6
0.5
•
Slope of Phase III
A
•
1.1
0.7
0.6
67
Wilson TA
1
2
2
vi/viei
0.9
v
0.5
0.8
0.4
0.7
0.3
1
1
0.6
vtot/vtotei
0.2
B
2
DISCUSSION
The study of gas mixing contains two schools. The objective
of the first school is to determine the magnitude of the variance
of the distribution of regional ventilation. This distribution is
physiologically significant because it, together with the distribution of perfusion, forms the basis for the V̇A/Q̇ distribution.
Despite its significance, it has received relatively little attention.
The functional studies of this school include studies of
efficiency of mixing, fitting of multiexponentials to washin
data, and matching predictions of multicompartment models to
washin data. This school includes anatomic studies of variable
ventilation. These began with the work of Hubmayr et al. (13).
They found that parenchymal expansion was variable, that the
variability increased with increasing resolution, and that this
variability at small scale was largely independent of gravity.
Subsequent studies, using new imaging techniques and aerosol
deposition, yielded results that are consistent with those of
1
γ2-γ1 (dynes/cm)
Fig. 3. Compartmental v during expiration (vi), normalized by compartmental
i
v at end inspiration (vei
) vs. total v (vtot), normalized by total v at end
tot
inspiration (vei
). Line of identity is shown by dashed line.
0
-1
-2
P (cm H2O)
Fig. 2. A: compartmental v vs. P with Marangoni flow between compartments
1 and 2 during breathing, with tidal volume of 1 liter (solid lines). B: difference
in ␥ values between the two compartments (␥2-␥1) vs. P for breathing (solid
lines). v (A) and ␥ difference (B) during breath hold (vertical dashed lines) and
subsequent expiration (curved dashed lines) are shown.
62
60
%N2
after breath hold, ␥ differences and surface flows are reestablished, and the expiratory volume trajectories are curved. The
changes in volume and ␥ difference during breath hold and
subsequent expiration are shown by the vertical and curved
dashed lines in Fig. 2. The difference between the gas concentrations in the two compartments is reduced by the penduluft
that occurs during breath hold, the curvatures of the volume
trajectories are smaller than for normal breathing, and the
resulting value of Sn, 0.044 liter⫺1, is smaller.
The plot of Sn vs. breath number is shown in Fig. 5. Sn
increases with breath number and reaches an asymptote of
⬃0.4 liter⫺1 for breath number ⬎ 20.
64
58
56
54
52
50
Expired volume (L)
Fig. 4. Concentration of N2 in expired gas after a single breath of oxygen (solid
line) and expired gas concentration for ideal mixing (dashed line) vs. expired
volume.
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P (cm H2O)
68
Slope of Phase III
0.5
0.4
Sn (L-1)
0.3
0.2
0.1
0
Hubmayr et al. To date, the magnitude of the variability
measured anatomically is smaller than that inferred from
washin studies, perhaps because of the limits on the resolution
of the imaging methods. This school has generated no hypotheses about the source of small-scale ventilation inhomogeneity.
The focus of the second school is the SIII. Although the SIII
has little intrinsic functional significance, this school has been
more prolific than the first. Experimental studies include measurements of the SIII for both single-breath and multibreath N2
washouts and measurements of slopes for different test gases
and different breathing maneuvers.
This school has included hypothesizing and modeling from
the beginning. In his seminal paper, Fowler stated that the
slope of the alveolar plateau was the result of both a spatial and
a temporal inhomogeneity of ventilation, with well-ventilated
regions emptying early in expiration and poorly ventilated
regions emptying later (11). Studies of the washin and washout
of radioactive gases showed that ventilation was nonuniform,
with ventilation increasing from apex to base in upright subjects. Milic-Emili et al. (17) provided a mechanistic explanation for the observed gravitational gradient in ventilation. They
argued that, because of the gravitational gradient in pleural
pressure, regions at different heights in the gravitational field
were situated at different points on the P-V curve, with transpulmonary pressure higher at the apex than the base. With this
model, he obtained plots of regional volume, as a fraction of
volume at TLC, vs. total V as a fraction of TLC. The slopes of
these lines increased from apex to base. To explain his observations on the effect of posture on the slope of the alveolar
plateau of radioactive Xe, Anthonison et al. (1) concluded that
these lines must be curved: concave downward for apical
regions with lower than average slope and concave upward for
basal regions with greater than average slope. This diagram,
similar to Fig. 3, with lines curved toward the line of identity,
was referred to as the onion-skin diagram.
Observations that were contrary to this argument were reported. Piiper and Scheid (21) pointed out that the magnitude
Wilson TA
of the gravitational ventilation gradient was too small to
explain the inefficiency of mixing. Most telling, sampling of
gas concentrations from small airways showed that the bulk of
the source of the slope was located in regions that were fed by
airways of 3 mm diameter or less and were too small to be
significantly distorted by gravity (10). Later, modeling of
gravitational lung deformation predicted that regional volume
curves have the opposite curvature from those postulated by
Anthonisen et al. (1), and that the diagram is trumpet-shaped
(26). From these curves, one would predict a negative SIII (14).
Despite this contrary evidence, the onion-skin diagram with its
gravitational origin remained the conventional explanation for
the SIII into the early 1980’s.
Conventional ideas shifted with the work of Bowes et al. (4)
and Paiva and Engel (19, 20), beginning in the late 1970’s.
They and their colleagues used computational methods to
analyze convection and diffusion in either continuous or nodal
models of the peripheral airways. In these models, the geometries of the branches extending from a bronchiole are assumed
to be asymmetrical, so that the relative magnitudes of convective and diffusive transport are different in different branches.
As new morphometric data became available, the models
became more complex, and, at this point, it is difficult for the
reader to come away with a qualitative understanding of the
results. Experimental studies by Crawford and colleagues provide valuable data on the SIII for multibreath N2 washouts and
their dependence on tidal volume (6), gas diffusivity (8), and
breath hold (7) in humans. Usually, only the SIII normalized by
mean expired concentrations (Sn) are reported. Other data, such
as N2 concentration as a function of Vexp or mean concentration as a function of breath number are usually not reported or
compared with model predictions. Paiva, Engel, and colleagues
give particular attention to the plot of Sn vs. breath number for
washouts extending to 25–30 breaths. The curve predicted by
the convection-diffusion model does not match the observed
curve (8), and an interregional convective spatial-temporal
inhomogeneity is postulated to explain the difference (23). The
convection-diffusion model is the current conventionally accepted explanation for the SIII (24).
There has been little interaction between the two schools.
The first reports mean or end-expiratory concentrations and
ignores the slope. The second usually reports data on the slope,
normalized by mean concentration, and ignores the data on
concentration.
The key feature in the modeling of the first school is
nonuniform specific volume expansion and diffusive transport
is neglected, whereas, in the second school, diffusive transport
is crucial and specific volume expansion is usually assumed to
be uniform.
Here, a new mechanism for the SIII is proposed. The various
aspects of that mechanism will be discussed in the following
paragraphs.
The parameter values for the base case are supported by
data, as described in the modeling section. The differences in
the values of the c’s for compartments 1 and 2 can be interpreted as follows. The differences in the values of c1 and c2
from the values for the base case imply modest differences in
alveolar radii and lumen radius. The differences in c3 and c4 are
more extreme. Variations in these parameters are justified by
the observations of Oldmixon and Hoppin (18) that cable
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breath number
Fig. 5. Normalized slope of phase III (Sn) vs. breath number for multibreath
washout.
•
Slope of Phase III
diameter and the relative proportion of elastin and collagen in
the cables is quite variable.
The value of k used in the calculation, 3 cm⫺1·s⫺1, was
chosen because the SIII was maximum for this value. Thus this
choice for the value of k is crucial and must be justified, and
this justification yields new information about the mechanism.
The term k(␥2 ⫺ ␥1) in Eq. 18 describes the surface flux
between compartments divided by Vo. Surface velocity (u) is
related to the gradient in ␥ (d␥/dx) by Eq. 19, where ␮ is fluid
viscosity, and t is the thickness of the fluid layer.
u⫽
冉冊冉 冊
t
␮
·
d␥
dx
(19)
k⫽
␲·d·t
2 · ␮ · b · Vo
(20)
The thickness of the liquid lining layer is ⬃0.01 d (28, 5), and
the viscosity of the fluid is 0.01 dyn·s·cm⫺2. The value of k is
inversely proportional to Vo because the change in ␥ for a
given flux of S is inversely proportional to indigenous S. For
TLC ⫽ 7 ⫻ 103 cm3, Vo ⫽ 7 ⫻ 103/2n cm3, where n is the
generation number of the branches. For generation 19 in the
Weibel model of the airways, b ⫽ 0.1 cm, d ⫽ 0.05 cm, and for
these values of the parameters on the right side of Eq. 20, k ⫽
3 cm⫺1·s⫺1. Thus the branches are identified as terminal
bronchioles, and the regions are identified as acini with a value
of Vo of 1.4 ⫻ 10⫺2 cm3. It should be noted that the geometry
of the peripheral airways is variable, and that the value of t was
obtained from one study of the liquid lining of peripheral
airways in guinea pigs. For comparison, the value of k for
generation 18 is 1.2 cm⫺1·s⫺1, and the value of Sn for this value
of k is 0.04 liter⫺1. If t were twice as big, k would be 2.4
cm⫺1·s⫺1 in generation 18, and this would be the crucial
generation.
The model described here shares a weakness with the convection-diffusion models in that both describe a particular
bifurcation of the peripheral airways; they do not provide
comprehensive models for the distribution of parenchymal
properties for the lung. In addition, the parameter values that
are listed in Table 1 have a particular correlation. For compartment 1, alveolar radius is smaller than the base case (c1 is
larger) and the entrance ring is more compliant (c3 and c4 are
smaller). For compartment 2, the relations are the opposite. If
the values of c3 and c4 were switched between the compartments, and the values of c1 were unchanged, compartment 1
would still lead pressure, but its specific compliance would be
smaller than that of compartment 2. As a result, the concentration of N2 in compartment 1 would be higher than that in
compartment 2, and the predicted SIII would be negative. A
correlation between alveolar size and entrance ring compliance
is required to obtain a positive SIII.
69
Wilson TA
For the parameter values shown in Table 1, the predictions
of this model agree well with a number of observations. The
mixing efficiency, 93%, is slightly higher than the 90% reported by Cumming and Guyatt (9). The predicted value of Sn,
0.068 liter⫺1, agrees with the observed values (6 – 8). Because
the mechanism is purely convective, equal slopes would be
predicted for gases of different diffusivities. Meyer et al. (16)
found no significant difference in the slopes for SF6 and He
washouts in dogs, whereas Crawford et al. (8) found that, in
humans, the value of Sn for SF6 is 40% higher than the values
for He. The predicted value of Sn for the larger tidal volume of
1.5 liters, 0.046, agrees with observation (6). The predicted
value of Sn for a 4-s breath hold at end inspiration, 0.044
liter⫺1, is smaller than the value for no breath hold, but larger
than the observed value of 0.025 liter⫺1 (7). The shape of the
curve of Sn vs. breath number is like the observed shape. The
values Sn for large breath number reported by Crawford et al. (8)
vary widely among the six subjects. The predicted value of Sn for
the 25th breath, 0.40 liter⫺1, falls within the range of observed
values, but is higher than the average value, 0.28 liter⫺1.
The quantitative differences between predictions and observation involve gas diffusivity or maneuvers with extended
times. This suggests that some diffusional equilibration is
occurring. The relaxation time for diffusional equilibration
between compartments fed by airways of the 19th generation
(with no flow) is 7 s. Thus diffusional equilibration could lower
Sn for He, lower Sn for breath hold, and decrease the asymptotic value of Sn for the multibreath washout.
In some ways, the proposed mechanism is similar to the
original onion-skin explanation for the SIII. However, the
mechanism responsible for the curvatures of the compartmental volume curves is different, and the scale of the compartments is quite different. It also differs in that compartmental
volume trajectories are loops: the expiratory trajectory does not
retrace the inspiratory trajectory. In addition, Marangoni flows
are dissipative, and this mechanism may contribute to the
viscoelastic properties of the lung. Furthermore, because of the
phase difference between compartmental volume oscillations,
the mechanism may contribute to aerosol dispersion.
APPENDIX
Approximate solutions to Eqs. 14, 15, 16, and 18 were obtained by
the following numerical methods.
For breathing with a tidal volume of 1 liter, the driving pressure
was expressed by the following equation, where t denotes time and ␶
denotes the period of a breath, 5 s.
P ⫽ 2.5 ⫻ 103 ⫹ 103 sin共2␲ · t ⁄ ␶兲
(A1)
The variables v, l, s, and ␥, for the two compartments were denoted xip,
where p denotes a particular variable, and i denotes the compartment
number. Each variable was expressed by a truncated Fourier series.
xip ⫽ aip ⫹ bip sin共2␲ · t ⁄ ␶ ⫹ ␾ip兲 ⫹ cip sin共4␲ · t ⁄ ␶ ⫹ ␺ip兲 (A2)
Time was discretized to t ⫽ n·␶/80, where n ranges from 0 to 80. The
equations were then written in the form of Eq. A3, where j denotes a
particular equation, LHSij(n) denotes the left side of the equation at t ⫽
n·␶/80, and RHSij(n) denotes the right side at that time.
LHSij共n兲 ⫺ RHSij共n兲 ⫽ ␦ij共n兲
(A3)
The value of a for the variable ␥ was set at 10 dyn/cm. Then the values
of the remaining 39 parameters, aip, bip, ␾ip, cip, and ␺ip were obtained by
finding the values that minimized the following objective Q.
1
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If the two compartments are assumed to be fed by the daughter
branches at a bifurcation in the bronchial tree, and ␥ is assumed to
be uniform in each compartment, the difference in ␥ occurs
between the distal ends of the two daughter airways. Then, the ␥
gradient is (␥2 ⫺ ␥1)/(2b), where b is the length of the airway, and
surface flux is u times the circumference of the tubes ␲·d, where
d is the diameter of the airway. It follows that k is given by the
following equation.
•
70
Slope of Phase III
Q⫽
兺 关␦ij共n兲兴2
(A4)
i,j,n
The root mean square value of ␦ij(n) was 0.04.
The solution for breathing with a tidal volume of 1.5 liters was
obtained by the same method with the values of the parameters
describing P(t) and a1 for ␥ as given in the text.
For breath hold, the initial values of the variables were taken as the
values of the variables at t ⫽ ␲·␶/2 for the solution for breathing with a
tidal volume of 1 liter. Time was discretized to values of n/5, where n
ranges from 1 to 20. The value of P for all times was set equal to the
initial value, and the values of the remaining variables at the discrete
times which minimized Q were determined. In calculating Q, the derivative in Eq. 18 was approximated by using finite differences. For the
subsequent expiration, the initial values of the variables were constrained
to equal the values at the end of breath hold, and the values during
expiration were found by the method used for breathing, but applied to
the time interval, ␶/4 ⬍ t ⬍ 3␶/4.
DISCLOSURES
AUTHOR CONTRIBUTIONS
Author contributions: T.A.W. conception and design of research; T.A.W.
prepared figures; T.A.W. drafted manuscript; T.A.W. edited and revised
manuscript; T.A.W. approved final version of manuscript.
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No conflicts of interest, financial or otherwise, are declared by the author(s).
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