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Department of Physics, Massey University, Palmerston North, New Zealand 5320; and Respiratory Health Network of Centres of Excellence, University of British Columbia Pulmonary Research Laboratory, St. Paul's Hospital, Vancouver, British Columbia, Canada V6Z 1Y6
ABSTRACT
INTRODUCTION
MODEL
SIMULATIONS
DISCUSSION
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Lambert, Rodney K., and Peter D. Paré. Lung
parenchymal shear modulus, airway wall remodeling, and bronchial
hyperresponsiveness. J. Appl. Physiol.
83(1): 140-147, 1997.
When airways narrow, either through the
action of smooth muscle shortening or during forced expiration, the
lung parenchyma is locally distorted and provides an increased
peribronchial stress that resists the narrowing. Although this
interdependence has been well studied, the quantitative significance of
airway remodeling to interdependence has not been elucidated. We have
used an improved computational model of the bronchial response to
smooth muscle agonists to investigate the relationships between airway
narrowing (as indicated by airway resistance), parenchymal shear
modulus, adventitial thickening, and inner wall thickening at lung
recoil pressures of 4, 5, and 8 cmH2O. We have found that, at low
recoil pressures, decreases in parenchymal shear modulus have a
significant effect that is comparable to that of moderate thickening of
the airway wall. At higher lung recoil pressures, the effect is
negligible.
dose-response; airway mechanics
INTRODUCTION THE PRIMARY FUNCTION of the lung parenchyma is gas
exchange. However, its elastic properties play an important role in
maintaining the patency of the conducting airways. Parenchymal tissue
transmits pleural pressure to the peribronchial region, thus preventing the collapse of the bronchi under the action of their own wall tension.
The Laplace relationship can be used to estimate this wall tension from
the transmural pressure and the radius of the airway. The radius can be
estimated from the airway area-pressure relationship. When airway
smooth muscle contracts and the airway narrows, the parenchyma is
locally distorted and provides an additional stress that opposes the
muscle contraction. The magnitude of this stress can be calculated from
the following relationship (19)
where
r is the increase in radial
peribronchial stress [that is, transbronchial pressure difference
(Ptb)] maintaining airway patency, µ is the elastic shear
modulus of the parenchyma, and
ro is the
decrease in the airway outer radius
(ro). The shear modulus has been shown to be proportional to the recoil pressure of the
lung (PL). For dog lungs it is
equal to 0.7PL (14), and for
rabbit lungs it is 0.5PL (34).
During chronic inflammatory disease processes such as asthma, the airway wall thickens (10). Morphometric examination of airway cross sections has allowed quantification of the thickening of the airway wall subdivisions (2, 13). The manner in which the thickening of the inner wall, which decreases airway luminal cross-sectional area, increases the resistance to flow and amplifies the effect of airway smooth muscle shortening has been well described (9, 22). It has also been shown that this thickening can contribute to bronchial hyperresponsiveness (9, 18, 22, 36-38). However, the potential effect of thickening of the adventitia has only recently been studied quantitatively (18, 21, 25). There are two mechanical consequences of adventitial thickening. First, the airway smooth muscle is less tightly coupled to the parenchyma. This is a geometrical effect and results in the muscle having to generate less tension for a given amount of shortening. That is, the muscle experiences a decreased afterload. Second, the increased thickness of the airway causes a local reduction in peribronchial stress through the airway encroaching into the parenchyma. Thus there is less preload on the airway muscle. As a result of the altered peribronchial pressure, the effective Ptb is reduced and the airways narrow. At low values of PL, the peribronchial pressure can even become positive with respect to luminal pressure, as noted by Macklem (21). Thus thickening of the adventitia should result in an increase in both baseline resistance (Rawbase) and in responsiveness to muscle stimulants.
The effects of some structural alterations in the airways on airway responsiveness have been modeled (18). The model produces simulated dose-response curves, in which the bronchial tree responds to a smooth muscle agonist and narrowing is indicated by increased airway resistance (Raw). The model incorporates the geometrical effects of airway wall thickening described by Moreno et al. (22) and the simulated computational dose-response curve introduced by Wiggs et al. (36-38). In a previous paper, morphometric data concerning the amount of airway smooth muscle were used to determine the limiting stress in each generation of airways (18). In this report we have quantified the effects of different transpulmonary pressures, adventitial thickness, and parenchymal shear modulus on the mechanics of airway narrowing. The results show that adventitial thickening and changes in parenchymal shear modulus have little effect at higher transpulmonary pressures but a substantial effect at lower pressures. Macklem (21) has reached similar conclusions. The results also show that the same fractional increase in adventitial thickness and inner wall thickness do not have equal effects on Raw. Increases in inner wall thickness have a larger effect on both Rawbase and maximal Raw (Rawmax).
MODEL
The model (18) airways are based on the symmetrically bifurcating bronchial tree described by Weibel (35). The passive compliance of the airways is described by the equations developed by Lambert et al. (20). The cross-sectional areas of airway wall subdivisions are specified generation by generation by using morphometrically derived relationships that give area as a function of airway internal perimeter (13). It is assumed that these areas stay constant during muscle contraction. Shortening of airway smooth muscle is calculated from a simulated dose-response curve as a result of a step in dose of simulated muscle agonist. Generation-by-generation airway luminal area (Ai) and Raw are then calculated. Shortening is halted in any generation when the muscle in that generation reaches its stress limit. This stress limit is defined by the maximally activated length-stress relationship of porcine airway smooth muscle acting against elastic loads (7). The plateau of the dose-response curve is reached when all airway smooth muscle has reached its limiting stress.
Previously, the model took no account of encroachment of increased (above control) adventitial tissue thickness into the surrounding parenchyma. We have now corrected this by calculating the altered peribronchial pressure caused by the parenchymal encroachment by using the continuum mechanics approach used previously (18, 19)
|
(1) |
P is the change in Ptb
resulting from a change in the radius of the outer boundary of the
adventitia
(ro), and µ represents the shear modulus of the parenchyma. In the unchallenged airway, Ai is
determined by Ptb and the pressure-area relationship of that airway.
Thus when Ptb is reduced by a thickening of the adventitia,
Ai must also be
reduced. We calculated this new area by deflating the airway until Ptb
was just balanced by passive wall tension. Both passive tension and Ptb
change during this deflation.
The model was further corrected by allowing encroachment of thickened inner wall into the lumen. We did this by assuming that the radius of the outer boundary of the smooth muscle (rmo) is unchanged by the changes in the inner wall. Ai is calculated by subtracting inner wall area (WAi) from the area inside the outer boundary of smooth muscle in the control case
|
The increase in inner wall thickness observed in chronic asthma is a result of edema, cellular infiltration, tissue cell hypertrophy and hyperplasia, and connective tissue deposition (9, 10). It has been suggested that the increase in the thickness of the lamina propria that accompanies thickening of the inner wall causes an increase in airway stiffness by increasing the airway's resistance to folding (17, 26). This theoretical contention is supported by histological observations that airways with thicker submucosas are less narrowed in both passively deflated lungs (17) and in lungs that have been challenged with a muscle agonist (27). However, the data that would allow a proper incorporation of the effects of a stiffened folded membrane have not yet been obtained. We, therefore, chose to disregard this potential stiffening.
In previous work with this model, Raw was calculated by using a formula developed by Pedley et al. (28). This formula provides a reasonable estimate of pressure losses in the large airways but a poor estimate in the peripheral airways. In fact, it yields pressure losses smaller than those predicted for fully developed laminar flow, the smallest possible pressure loss. Because a major part of Raw in narrowed airways occurs in the peripheral airways, this is an unsatisfactory state of affairs. Therefore, we have substituted the relationship developed by Reynolds (30), which gives a much better estimate of peripheral pressure losses and has been used before in modeling respiratory airflow (16).
As before, the control values for airway cross-sectional areas were
taken from the work of Kuwano et al. (13). The model was programmed in
a spreadsheet on a personal computer under the control of a macro. In
use, the sheet is initialized to set up the correct starting values at
zero dose of muscle stimulant. The model is then run by incrementing
the dose of simulated stimulant. This controls percent muscle
shortening through a dose-response relationship. The muscle stress is
deduced by calculating the muscle tension for a given level of
shortening from the force balance for the airway wall.
Ai and thus flow
resistance in each generation of airways are calculated from each new
value of percent muscle shortening. An airway stops constricting when
its muscle reaches the stress limit for elastically loaded airway
smooth muscle. The plateau of the dose-response curve occurs when the muscle in every airway has reached its stress limit. The dose step was
controlled such that the increment in Raw, for a step change in dose,
never exceeded 5 cmH2O · l
1 · s.
This was done to better define the sharp increase in resistance that
preceded the development of the plateau.
Provision was made in programming the spreadsheet for easy alteration of PL, µ, and a generation-by-generation change of outer wall area (WAo) and WAi. Any desired combination of these parameters can be selected. For the results reported here, any fractional change in airway dimensions was applied uniformly throughout the bronchial tree.
SIMULATIONS
The predicted dependence of Raw on
PL for the control case is not
qualitatively different from that reported previously (18) (Fig.
1). At
PL = 2 cmH2O, the plateau resistance
exceeds 300 cmH2O · l1 · s,
whereas Rawbase is <2
cmH2O · l1 · s.
At PL = 8 cmH2O, the plateau resistance is
only twice the baseline value.
The parenchymal shear modulus is programmed into the spreadsheet as
|
1 · s.
On the other hand, changes in k at
PL = 8 cmH2O cause only minor changes in
Rawmax.
Alterations to Rawbase over a range of values of PL for several values of WAo and WAi are shown in Fig. 3. It is apparent that doubling of WAi increases Rawbase more than does doubling of WAo. However, in neither case is the resulting resistance abnormal.
To investigate the significance of thickening of the adventitia, the
model was run with values of
WAo up to four times
the control value, with PL set
to 5 cmH2O (Fig.
4). Quadrupling
WAo resulted in an
increase in Rawmax to 130 cmH2O · l1 · s.
However, there was no leftward shift in the onset of the plateau with
increasing WAo.
To provide a comparison with the effects of adventitial thickening, the
model was also run with control values of all parameters except
WAi, which was
increased up to double the control value (Fig.
5). A doubling of
WAi resulted in
Rawmax exceeding 300 cmH2O · l1 · s.
This was accompanied by a small leftward shift in the onset of the
plateau. It is apparent that the model airway is more sensitive to
changes in WAi than to
changes in WAo.
The interaction among changes in
WAo,
WAi, and
PL was investigated by running
the model at values of
WAi up to twice the
control value and, at each value, varying
WAo by up to four times
the control value at three values of
PL. Isopleths of constant
Rawmax (at values of 25, 50, and
100 cmH2O · l1 · s)
were deduced from these results and are shown in Fig.
6.
), 50 (
), and 100 cmH2O · l1 · s
(
)] at 3 values of PL
[4 (dashed lines), 5 (solid lines), and 8 cmH2O (dotted lines)].
Isopleths show combination of
WAi and
WAo that will give 1 of
3 plateau resistances at chosen value of
PL.
DISCUSSION
1 · s
by reducing the step in dose until this is true. When the increment in
resistance is small, the dose step is increased.
Use of the Reynolds expression for Raw results in an increase in
Rawbase of 7% and in plateau
resistance of ~40% in the control case over that calculated with the
Pedley expression. The zero-dose resistance in the control case is 1.07 cmH2O · l1 · s
as opposed to 1.0 cmH2O · l1 · s
in the previous study, and the plateau resistance is 5.3 cmH2O · l1 · s
as opposed to 3.8 cmH2O · l1 · s
previously. We decided that this was of no great significance because
these results also fall within the bounds of "normal" response.
The "control" airway morphometry is that of Kuwano et al. (13).
In the central airways, airway dimensions were obtained by
extrapolation from the data for the peripheral airways.
The simulations.
All conclusions drawn from a model study such as this must be tempered
by the knowledge that it is a model study and not reality. Nonetheless,
the previous version of this model has been shown to give results that
are in qualitative accord with experimental data. The modifications
discussed above are designed to incorporate some features of real
airways in a more realistic manner. We will now discuss how the model
predictions compare with experimental data.
Figures 1 and 2 appear to have some interesting implications for
respiratory disease. It is apparent from Fig. 1 that a decrease in
PL at which
Rawmax is calculated results, in
itself, in a heightened susceptibility to muscle stimulants, as has
been observed experimentally in human subjects (3) and in animals (1).
Reduction in PL results in a
narrowing of the airways and thus in an increase in Raw, regardless of
whether there is bronchoconstriction. However, because µ is
proportional to PL, the support
of the parenchyma for the airway during muscle shortening is also
lowered and this, too, causes an elevation of Raw but only during
bronchoconstriction. Additionally, the compliance of the airway (in
this model) increases with decreasing
PL until
PL reaches zero. This is
significant because part of the load on the shortening smooth muscle is
calculated from the airway compliance curve. Values for
Rawmax are given in Fig. 1 to
indicate that, in the model, there is a plateau. In a real experiment,
plateaus for values of PL < 3.5 cmH2O would not be observed
because the experiment would be stopped before the required dose was
reached. Thus, in terms of experiments with live subjects, the model
predicts an abolition of the dose-response plateau at
PL values of <3.5
cmH2O in accord with the
experiments of Ding et al. (3). That is, the model predicts that normal subjects breathing at low lung volumes respond to muscle stimulants in
a similar manner to asthmatic subjects breathing at normal functional
residual capacity (FRC), whereas, at lung volumes greater than FRC,
normal subjects show only a small elevation of the plateau above
baseline.
The results shown in Fig. 2 are an attempt to evaluate the importance
of µ, on its own, in supporting the airway against muscle shortening.
The elasticity of the parenchymal tissue keeps the airways open. This
tissue also stretches locally near an airway in which the muscle is
shortening. The curves in Fig. 2 are concerned with only the latter
effect. It is assumed for this analysis that the airway
Ai in the control state
at the particular value of PL is
not influenced by the change in µ. Thus the intersections of the
curves with the Rawmax axis
indicate the effect of the total absence of parenchymal resistance to
local deformation caused by airway narrowing; that is, this is
analogous to the response of an excised airway. At high values of
PL, removal of parenchymal support is of little consequence because it results in only an approximate doubling of Rawmax at
a PL of 8 cmH2O compared with the control
value. However, at FRC (assumed to be the volume corresponding to
PL = 5 cmH2O) and lower volumes, the lack
of parenchymal support has serious consequences, with
Rawmax at FRC going from 5.3 cmH2O · l1 · s
in the control condition to ~90
cmH2O · l1 · s
when µ is zero. Thus we conclude that the parenchymal shear modulus
plays a significant part in opposing the narrowing of airways. A
reduced shear modulus results in an increased response to muscle
agonists.
In trying to confirm these model results by comparison with
experimental data in the literature, we must first decide what is being
modeled when we alter k. An increase
in k implies a stiffened parenchyma,
that is, one in which the parenchymal elastance
(EL) has been increased. There
is evidence that the administration of aerosolized methacholine (MCh)
in rabbits increases EL (23). Concomitant increases in whole lung resistance
(RL) and tissue viscance were
also observed. However, the increases in Raw failed to reach
statistical significance. The model predicts that when k is increased, as it appears to have
been in the above experiments and possibly is in experiments with live
human subjects when MCh is used, the measured value of Raw should be
smaller than would be produced by an agonist that had no effect on
other lung tissues. This is qualitatively consistent with the
experimental results.
In another set of experiments by the same group, but one using dogs
instead of rabbits, EL was again
observed to increase with increasing doses of MCh (32). Raw of the
maximally constricted airways fell dramatically when the level of
positive end-expiratory pressure (PEEP) was raised from 5 to 7.5 cmH2O and more than doubled when
PEEP was reduced from 5 to 2.5 cmH2O. These results are
consistent with the model predictions, but it is not possible to deduce
from the experimental results what contribution the stiffened
parenchyma played in moderating the changes in Raw. The observed
increase in Rawmax with the
decrease in PEEP is considerably less than the model predicts or than
is seen in humans (3).
Reductions in the value of k below the
nominal normal value of 0.7 might be expected to mimic the effects of
emphysema. Both elastase and papain have been used in animals to
destroy some parenchymal tissue to model human emphysema (1, 12, 24, 29). In the earlier experiments (12, 24, 29), muscle agonists were not
administered. The unchallenged airways showed a reduced maximal forced
expiratory flow at any given lung volume (12, 24) and an increased
peripheral resistance (29) after the induction of emphysema. The model
under discussion here does not predict maximal flows, but it does
predict an increase in peripheral resistance with decreasing values of
k in accord with experimental data.
However, it does not predict the observed decrease in central resistance observed experimentally (29). We do not understand this
experimental finding. Bellofiore et al. (1) used elastase to induce
emphysema in rats and then measured airway responsiveness to MCh.
Induction of emphysema increased the responsiveness of the lung to MCh
as measured by RL. Because MCh
appears to stiffen the parenchymal tissue, the effect of the emphysema
in reducing Raw would appear to be being offset by the increase in
k from the MCh. However, in these
experiments only RL was
measured, not Raw. Thus it is not clear how much of the response is
attributable to airway narrowing. Nonetheless, the change in resistance
is in the same direction as that predicted by the model when there is a
reduction in k.
We performed the rest of the simulations shown in Figs. 3, 4, 5, 6 in an
attempt to assess the significance of changes in shear modulus in
relation to the effects of thickening of the inner and outer airway
wall. We excluded changes in muscle thickness from these considerations
because the importance of the muscle in this model has already been
pointed out (18, 25). Also, the maximal length-stress characteristic of
the muscle is that of porcine airway smooth muscle. There is some
evidence that human airway smooth muscle is unable to shorten as much
as porcine muscle (8). Thus any further investigation of
the sensitivity of the model to muscle mass could be misleading.
Encroachment of the wall into the lumen (as a result of inner wall
thickening) and encroachment of the wall into the parenchyma (as a
result of outer wall thickening) both result in small increases in
Rawbase as a result of luminal
narrowing (Fig. 3). This is in contrast to the earlier study in which
changes in wall areas were constrained from producing changes in
Rawbase (18). It is apparent from
Fig. 3 that increases in
WAi have a greater
effect on Rawbase than increases
in WAo, although
neither increase is significant in absolute terms. At
PL = 5 cmH2O, doubling of
WAi results in a 14%
increase in Rawbase, whereas
doubling of WAo results
in only a 4% increase. At lower values of
PL, the fractional increase in
Rawbase is greater. Whereas these
changes in Rawbase are not
functionally significant, the changes in the airway wall that cause
them are significant in determining airway response to muscle agonists,
as shown in Figs. 4 and 5.
It can be seen from Fig. 3A that the
increase in Rawbase induced by
adventitial thickening is greater at low values of
PL. This is a result of
resistance increasing strongly with reductions of
Ai and because the
airway becomes more compliant at lower values of
PL, and thus the effect of
adventitial thickening on
Ai is increased.
Figures 4 and 5 show that increases in either
WAo or
WAi result in increased
Rawmax. The model is much more
sensitive to changes in
WAi than to changes in
WAo. For instance, a
quadrupling of WAo
results in Rawmax increasing to
~130
cmH2O · l1 · s
(experimentally, this is an abolition of the plateau), whereas the same
increase in Rawmax is achieved
with a little more than a 60% increase in
WAi. The other
difference between the two sets of results is the slight leftward shift
of the dose-response curve with increasing
WAi but not with
increasing WAo.
The increase in Rawmax caused by a
quadrupling of WAo is
almost exactly the same as the increase caused by the reduction of PL to 2.5 cmH2O. This is very similar to the
result predicted by Macklem (21).
To show the interaction among
WAo,
WAi, and
PL, we calculated three
isopleths of Rawmax (at values of
25, 50, and 100 cmH2O · l1 · s)
while varying WAo and
WAi at three values of
PL (4, 5, and 8 cmH2O). These values are ~5, 10, and 20 times Rawmax in the control
case. The results are shown in Fig. 6. There is a remarkably linear
relationship between the values of
WAo and
WAi necessary to
achieve a given Rawmax at all
values of PL. The coordinates of
any point on any of the lines give the values of
WAi and
WAo which, together,
cause the model to yield the plateau resistance pertaining to that
line. Thus, at a PL of 4 cmH2O, a plateau resistance of 100 cmH2O · l1 · s
can be obtained by a 20% increase in
WAi combined with a 60% increase in WAo.
The same resistance can be obtained at 8 cmH2O by an 80% increase in
WAi and a quadrupling
of WAo. By way of
comparison, at a PL of 4 cmH2O, the same elevation of
plateau resistance is obtained with a reduction of shear modulus to
0.3PL (compared with the control
value of 0.7PL), whereas at a
PL of 8 cmH2O, total removal of
parenchymal support from the airway during airway narrowing (that is,
assuming that the airway has a constant distending transmural pressure
difference of 8 cmH2O) has almost
no effect.
These results appear to correlate with clinical observations. Patients
with chronic obstructive pulmonary disease do not have very marked
airway wall thickening, but they do breathe at lower PL despite an increased
end-expiratory lung volume (31). The decreased
PL, which is caused by
proteolytic lung destruction, will be accompanied by a decrease in
parenchymal shear modulus. The results in Fig. 2 show that a
combination of a 1-cmH2O reduction in PL to 4 cmH2O and a reduction in µ to 0.4PL results in a >10-fold increase in Rawmax over the
control case (PL = 5 cmH2O, µ = 0.7PL). Quantitative
comparison with experimental data is difficult, as discussed above,
because it is not clear how to associate
k with severity of emphysema. However,
elastase-induced emphysema in rats produced qualitatively similar
behavior to that of the model (1). Interestingly, after induction of
emphysema in the rats, the volume dependence of the dose-response curve
for RL disappeared. This could
be a reflection of the altered compliance of the lung, in that a
significant increase in volume was caused by a very small change in
PL.
It is not certain whether some of the increase in
Rawmax in asthmatic subjects is
caused by changes in lung parenchyma. Although these
subjects may breathe at increased lung volumes,
PL values (and, presumably,
shear modulus) are often normal, although they can be reduced in cases
of severe childhood asthma. However, the inner and outer walls of
asthmatic airways are markedly thickened. These changes can have
profound effects on Rawmax,
especially at low PL values
(Fig. 6).
There is evidence that the heightened response at low
PL values seen in asthmatic
subjects is seldom reversible by a deep inspiration in the way that
this model appears to predict (4, 33). It has also been reported that
normal subjects who are prohibited from taking a deep inspiration for a
prolonged period of time after inhaling MCh (33) also show a lack of
reversibility, at least on the first two or three deep inspirations.
The model cannot be used to study these effects because they appear to
be time dependent. Also, the model cannot distinguish between competing explanations of this effect, such as the surface-driven instability described by Hill et al. (6) and the latch-bridge phenomenon described
by Fredberg et al. (5).
We have not addressed the problem of gas trapping and the effect that
it has on increased Raw. This model is homogeneous in that every airway
of each generation has the same airflow, the same transmural pressure
difference, and the same mechanics. Airways in the model do not close
(although they can be driven arbitrarily close to closure). Gas
trapping is inherently a nonhomogeneous effect and is best addressed
with a model that has a built-in allowance for nonhomogeneity.
The results of this study show the potential importance of modest
changes in airway wall dimensions and illustrate again the critical
role of small changes in PL that
can accompany changes in end-expiratory lung volume. Although
hyperinflation puts the inspiratory muscles at a mechanical
disadvantage and increases the elastic work of breathing, the gains in
protecting against excessive airway narrowing may more than offset this
disadvantage.
At low PL values, reductions in
the parenchymal shear modulus cause significant increases in bronchial
responsiveness, comparable in magnitude with those caused by moderate
thickening of the airway wall. At high
PL values, the shear modulus
does not play a significant role in resisting airway narrowing.
ACKNOWLEDGEMENTS
This work was supported in part by grants from the Medical Research Council of Canada and the New Zealand Lottery Grants Board.
FOOTNOTES
Address for reprint requests: R. K. Lambert, Dept. of Physics, Massey Univ., Private Bag 11222, Palmerston North, New Zealand 5320 (E-mail: R.Lambert{at}massey.ac.nz).
Received 18 July 1996; accepted in final form 6 March 1997.
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