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Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215
ABSTRACT
INTRODUCTION
MODEL DEVELOPMENT
METHODS
RESULTS
DISCUSSION
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Suki, Béla, Huichin Yuan, Qin Zhang, and Kenneth R. Lutchen. Partitioning of lung tissue response and inhomogeneous airway constriction at the airway opening. J. Appl.
Physiol. 82(4): 1349-1359, 1997.
During a
bronchial challenge, much of the observed response of lung tissues is
an artifactual consequence of inhomogeneous airway constriction.
Inhomogeneities, in the sense of time constant inequalities, are an
inherently linear phenomenon. Conversely, if lung tissues respond to a
bronchoagonist, they become more nonlinear. On the basis of these
distinct responses, we present an approach to separate real tissue
changes from airway inhomogeneities. We developed a lung model that
includes airway inhomogeneities in the form of a continuous
distribution of airway resistances and nonlinear viscoelastic tissues.
Because time domain data are dominated by nonlinearities, whereas
frequency domain data are most sensitive to inhomogeneities, we apply a
combined time-frequency domain identification scheme. This model was
tested with simulated data from a morphometrically based airway model
mimicking gross peripheral airway inhomogeneities and shown capable of
recovering all tissue parameters to within 15% error. Application to
our previously measured data suggests that in dogs during histamine infusion 1) the distribution of
airway resistances increases widely and
2) lung tissues do respond but less
so than previously reported. This approach, then, is unique in its
ability to differentiate between airway and tissue responses to an
agonist from a single broadband measurement made at the airway opening.
lung resistance; lung elastance; tissue resistance nonlinearity; distribution of resistances; Wiener structure; Horsfield model
INTRODUCTION THE IMPORTANCE of the contribution of lung parenchymal
tissue resistance (Rti) to total lung resistance
(RL) at physiological frequencies (0.1-1 Hz) has recently become evident (1-5, 9, 11-13, 19-22, 27). Several studies have reported that in
response to bronchochallenge Rti and lung tissue elastance (Eti) are
significantly elevated (1, 2, 7, 9, 12, 13, 16, 18, 19, 21, 22, 28, 31)
and show an increased nonlinear inverse dependence on tidal volume
(VT) delivered (1, 4, 22, 26, 30, 32-34). Simple partitioning of
RL to tissue and airway
components has been attempted in control as well as in challenged
conditions (1, 5, 11-13, 16, 19-23, 27). Bates et al.
(2), however, suggested that in dogs, inhomogeneous airway constriction
contributes significantly more to the increases in dynamic lung
elastance (EL) than lung
tissues. Therefore, the partitioning of
RL to tissue and airway
components can be subject to large errors that may then lead to false
conclusions about the responsiveness of parenchymal tissues. Indeed,
the influence of inhomogeneities on the partitioning has been
confirmed experimentally (21) and inferred via modeling analysis in two other recent studies (16, 20). Nevertheless, to our
knowledge there is no method available to reliably partition airway and
tissue responses from pressure-flow data measured noninvasively at the
airway opening in the presence of inhomogeneous airway constriction and
nonlinear tissue viscoelasticity.
In a recent study, we addressed one aspect of the above issue.
Identifying tissue nonlinearities from data taken at the airway opening, we examined the correlations between the changes in the frequency and amplitude dependences of lung tissue by using a nonlinear
block-structured modeling approach (33). We found that the best lung
structure was a linear airway compartment consisting of an airway
resistance (Raw) and inertance
(Iaw) in series with a nonlinear model for the
lung tissues. The nonlinear tissue model was a so-called "Wiener
structure," which is a cascade connection of a linear viscoelastic
block and a nonlinear zero-memory block (Fig.
1). This model provided excellent fits to
the pressure-flow data in both control and in the late response to
histamine that occurred 15-20 min after the histamine infusion was
started. Moreover, our analysis showed that the primary cause for the
increased VT dependence during
constriction is not a change in the nonlinear mechanisms. Rather, it
results from an exacerbation of the preexisting nonlinear mechanisms
through an increase in the magnitude of the linear tissue impedance.
This conclusion is also in agreement with some recent histological
findings (17).
In our previous study, we fit data during the late response to
histamine challenge (33). In the late response, the distribution of
alveolar pressures (assessed by the alveolar capsule technique) suggested that airway inhomogeneities had decreased compared with the
peak response that occurred in 1-3 min after infusion (22). Nevertheless, we cannot rule out that inhomogeneities also contributed to the estimates of Rti and Eti. Moreover, during the peak response the
capsules displayed significant inhomogeneities. This paper addresses
the following question: Is it possible to distinguish from
pressure-flow measurements made at the airway opening whether the lung
tissues underwent real changes, or have
RL and
EL increased only due to the
presence of inhomogeneities? From the input-output point of view,
parallel inhomogeneities are an inherently linear phenomenon, whereas
real tissue changes have been found to be accompanied by increased
VT dependence (33, 34). Thus we
hypothesized that new model classes that include different types of
airway inhomogeneities as well as tissue nonlinearities will allow us to differentiate between airway and tissue responses. We tested our
hypotheses by developing and fitting such models to pressure-flow data
measured previously in histamine-challenged dogs and simulated via an
anatomically based structural model.
MODEL DEVELOPMENT
Fig. 1.
Wiener structure, which consists of a linear dynamic block (L) and
nonlinear zero-memory block (N) connected in cascade. Input to model is
x(t),
u(t) is a temporary signal,
and output of L and
y(t)
is final output of nonlinear system, where
t is time.
[View Larger Version of this Image (3K GIF file)]
where
PLaw is the linear pressure
drop across the airways,
(1)
is the volume
flow rate, 
is the volume acceleration, and t is time. Accordingly, the linear
airway impedance
(ZLaw) is
where
j is the imaginary unit, and
(2)
is
the angular frequency.
|
(3) |
governs the degree of the frequency dependence of tissue
resistance, Rti = G/
,
and elastance, Eti = H1
.
This model is called the constant-phase model because the phase of the
ZLti is independent of frequency.
The hysteresivity coefficient 
, introduced by Fredberg and
Stamenovic (9), is
G/H in this model
representation. A mathematical framework for Eq. 3 and a proposed mechanistic basis with respect to
molecular theories of polymer viscoelasticity have recently been
offered (31). The combination of Eqs.
2 and 3 provides a
homogeneous linear lung model (HL) with four parameters
(Raw, Iaw,
G, H) that has been used in several previous studies (11-13, 20-22, 27, 31, 33) and features a single airway compartment in series with a linear
viscoelastic tissue compartment.
For the nonlinear tissue block Nti we found that the following
third-order polynomial provided appropriate balance between quality of
fit and number of free parameters
|
(4) |
t,
airway opening flow, input to model; R1...
RN, parallel
distribution of airway resistances in linear block, each terminated in
identical impedance; Z, defined in
Eq. 6;
PLin(t),
output of distributed linear subsystem and input to nonlinear tissue
block (Nti); Pao,m(t), pressure drop
across system and output to Nti.
The variation in airway resistance from path to path (Fig. 2) is implemented in a probabilistic manner. If we distribute the pathway resistances according to a distribution function n(Ri), we can calculate the total admittance of the linear part of the system (YL), which is the summation of the conductance of the individual branches
|
(5) |
|
(6) |
R (R > 0). Note that Iaw and
ZLti do not depend on
the pathway i. As the number of pathways increases, we can invoke a continuous distribution of Ri so that
|
(7) |
|
(8) |
0, the mean resistance is not equal to the
arithmetic mean of Ra and Rb. An average
airway resistance, defined as the expected value of the random variable
R, E[R], can be obtained by
![E[R ] = <LIM><OP>∫</OP><LL>R<SUB><IT>a</IT></SUB></LL><UL>R<SUB><IT>b</IT></SUB></UL></LIM> R<IT>n</IT>(R ) dR](/content/vol82/issue4/fulltext/1349/img010.gif)
|
(9) |
1, 0, and 1. When µ = 1,
n(R)
increases linearly with R; when µ = 0, R is uniformly distributed over all the
branches; and when µ = 1, n(R)
decreases hyperbolically with R. Finally,
the total impedance of the linear system
ZL can be calculated by
taking the inverse of the admittance
YL given in
Eq. 7. Accordingly,
ZL for the three different µ values are obtained as following
|
|
(10) |
|
(11) |
|
(12) |
. These models
now account for linear tissue viscoelasticity as well as parallel
inhomogeneity with five free parameters
(Ra,
Rb,
Iaw, G,
H) and, hence we denote them as IHL
models. Finally, when the ZL is in
cascade with the nonlinear block (Fig. 2), we obtain the inhomogeneous nonlinear models (IHNL) with seven parameters
(Ra,
Rb, Iaw, G,
H,
a2,
a3). The IHNL
models, therefore, contain only one additional parameter to be
estimated compared with the HNL model, and yet they permit a
distribution of inhomogeneous parallel airway constriction.
METHODS
1 · min1).
Airway-opening pressure (Pao) with reference to atmospheric pressure
was measured via a side tap of the tracheal tube with a Validyne MP-45
(±30 cmH2O) pressure
transducer. Tracheal flow (tr) was measured with a
heated screen pneumotachograph (28 mm ID) and another Validyne MP-45
(±5 cmH2O) transducer. All
signals were low-pass filtered at a cutoff frequency of 5 Hz, then
sampled at 20 Hz. Apparent lung input impedance
(Zapp) at the NSND input frequencies was
calculated as the ratio of the cross-power spectrum of
tr and Pao and the autopower spectrum of
tr.
Simulation studies.
To test the abilities of our IHNL models, we carried out the following
simulation studies. The pressure-flow relationship of the entire lung
was simulated by using a model that includes an asymmetric branching
airway system based on the Horsfield studies (15) modified to allow for
inhomogeneous changes in airway diameters (20). The airway tree
terminates in alveolar tissue elements that were also modified to
incorporate nonlinearity. Briefly, the alveolar tissue model contains
an alveolar compartment that is the parallel combination of the
constant-phase tissue model (Eq. 3) and alveolar gas
compressibility. The tissue properties are uniformly distributed over
all terminal airway elements. The total input impedance of the model is
then determined by combining the impedance of the alveolar tissue
elements and the airways in the proper parallel and serial fashion (8).
The model also accounts for airway wall viscous, elastic, and inertial
properties as described previously (10). Inhomogeneous peripheral
airway disease was simulated by decreasing the airway diameters in
various fractions of the last five generations of the peripheral
airways. We chose to constrict all airways subtended by order 46 in the dog lung (20). This simulated a lung disease with 72% of the airways
constricted and 28% kept at the baseline diameters. The constriction
levels were set to 30, 40, 50, and 60% decrease in diameters from
their baseline values at functional residual capacity (FRC). After the
input impedances were obtained, the time domain pressure output of this
linear model was calculated and Eq. 4 was used to add nonlinearity. This is equivalent to replacing the
parallel set of airway pathways terminated by the constant phase model
in Fig. 2 with the above Horsfield structure. Zapp of the
nonlinear model at the NSND frequen-cies were then calculated as described below. These data were fit with our HNL and IHNL models of
the previous section and used the model-fitting approach described
below. The tissue parameters (G,
H,
a2,
a3) were kept constant in the simulations and were evaluated as a function of airway
constriction as obtained from the fittings.
Computations.
According to Fig. 2, the airway opening flow
(t) is the input
signal to the model. The predicted output pressure of the model,
Pao,m(t), is calculated as follows.
The
(t) is
subject to the linear subsystem to obtain a temporary pressure output, PLin(t).
First, PLin() = ()ZL()
is calculated, and then
PLin(t) is obtained by taking the inverse Fourier transform of
PLin(). The input impedance of
the linear subsystem ZL
is defined in Eqs. 10-12. The
PLin(t)
signal is then passed through the nonlinear subsystem with a
third-order polynomial in Eq. 4 to
obtain the total pressure drop across the model,
Pao,m(t). The apparent input
impedance for both fitting measured data and simulating data (i.e.,
inverse and forward models, respectively)
Zapp,m was also calculated at the NSND input
frequencies (i.e., the frequencies at which the input flow contained
energy).
Model-fitting approach.
Because of the design of the NSND frequencies, nonlinearity in
Zapp at the NSND frequencies is minimized
(32). Nevertheless, to identify nonlinearities from an OVW-NSND input
flow signal, we can fit the time domain pressure. The time domain
recording does retain the effects of all nonlinear harmonic influence
from which the the nonlinear coefficients can be identified (33). During lung disease, however, the dominant change in
ZL is the increased
frequency dependence of RL and
EL, which can be because changes
in tissue properties and/or a consequence of increased airway
constriction inhomogeneity. A two-compartment linear impedance model
mimicking inhomogenity has been found to provide much better fit to
lung input impedance data than the single-compartment model during
constriction in frequency domain (20). This suggests that inhomogeneity
may be better identified in frequency domain. Thus we hypothesized
that, to identify both inhomogeneity and nonlinearity, we must
simultaneously fit our models to both time domain pressure data and
apparent input impedance at the NSND frequencies. Accordingly, the
optimal parameters of the model are obtained by minimizing the
following mean square
error
![&egr;<SUP>2</SUP> = <IT>W<SUB>t</SUB></IT> <FR><NU><LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>N</IT></UL></LIM> [Pao, <IT>d</IT>(<IT>t<SUB>i</SUB></IT>) − Pao, <IT>m</IT>(<IT>t<SUB>i</SUB></IT>)]<SUP>2</SUP></NU><DE><LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>N</IT></UL></LIM> [Pao,<IT>d</IT>(<IT>t<SUB>i</SUB></IT>)]<SUP>2</SUP></DE></FR> +](/content/vol82/issue4/fulltext/1349/img015.gif)
|
|
(13) |
was minimized by using a global-optimization
approach (6). To match the error units, the time and frequency domain
errors were normalized with the sum of the measured pressure and
impedance magnitude squares, respectively. The model can be fit only to input impedance when
Wt = 0 and
Wf = 1, or, conversely, it can be fit only to the time domain pressure by
setting Wt = 1 and Wf = 0. These weighting factors are chosen to account for the difference in the relative contributions of the first and second terms
in Eq. 13 to . In preliminary
fitting exercise, we found that
Wt = 0.1 and
Wf = 0.9 gave
appropriate weights to the time and frequency domain errors,
respectively; that is, the smaller time domain weight was still
sufficient to estimate the nonlinear coefficients.
RESULTS
We next examined whether inhomogeneities can be more reliably identified by including the frequency domain data in the identification. We repeated the above model fitting by using the combined domain-fitting technique (i.e., Eq. 13) with Wt = 0.1 and Wf = 0.9. The nonlinear coefficients a2 and a3 obtained from the combined domain fitting (
0.0395 and 0.00138,
respectively) were very close to those obtained from the time domain
fitting (0.0364 and 0.00113, respectively). Figure
4 demonstrates that now the prediction of
the RL and
EL spectra is excellent but only
with the inhomogeneous models, and this is true regardless of whether
nonlinearities were included. It is worth appreciating that fitting a
nonlinear model only in the frequency domain at selected NSND
frequencies (26) will result in biased parameters. The reason is that,
at the NSND frequencies, the higher order harmonic distortions and
cross-talk are minimized and only certain cross-talk terms contribute
to the data (32). Hence, the estimated nonlinear coefficients may not
capture the essence of the nonlinearity. Overall, we can draw the
following conclusions. When OVW-NSND flow input is applied at the
airway opening, identification of both airway inhomogeneities and
tissue nonlinearities requires the combined time-frequency domain
fitting- approach.
Model form and parameters. Among all models (the inhomogeneous models with the three different airway resistance distributions), the inhomogeneous model with a hyperbolic resistance distribution (µ = 1) had the best fit to the peak response as well as in control and the late response. The inhomogeneous models are always superior to the homogeneous models. The errors of the different models are compared in Table 1. In peak response, and even in control, there are large error reductions for the inhomogeneous models over the homogeneous model (>30%). Moreover, F-ratio tests show that the IHNL (with µ = 1) model is statistically superior to the HL model for all four dogs in control, peak, and late response, and it is also statistically superior to the HNL model for all dogs in peak response, three dogs in late response, and only one dog in control. Thus, while the inclusion of nonlinearity was essential for all conditions, inhomogeneity was needed mostly in the peak and late responses.
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2)
was overestimated by both models but to a much lesser extent by the
IHNL model (errors 60 vs. 90%). The
E[R]
in the IHNL model increased much more than parameter
R in the HNL model. The
Iaw parameter decreased in both models but
to a lesser extent when the IHNL model was fit to the simulated data.
DISCUSSION
The primary purpose of this study was to develop analytic tools that can better distinguish between real tissue changes and virtual tissue changes due to heterogeneous airway constriction after an agonist-induced bronchochallenge. The basis for our approach was the finding that frequency and amplitude dependences of Rti and Eti are coupled during challenge (33). Additionally, we also hypothesized that airway inhomogeneities contribute to the frequency dependence of RL and EL and that this is primarily a linear phenomenon. Thus models that simultaneously account for nonlinear tissue viscoelasticity and airway inhomogeneities could have the potential to separate the contributions of tissues and airway inhomogeneities to RL and EL. Our findings indicate that in bronchochallenged dogs, although inhomogeneities are significant in both peak and late responses, there are also real changes in lung tissue properties. Before discussing the physiological implications, we must first examine to what extent our model selection is appropriate when fitting measured data and what the limitations are of the our modeling approach on the basis of analyzing simulated data.
Model selection. With regard to the inhomogeneous airway compartment, we note that our purpose was to develop a reasonably simple model that still captures the distributed parallel nature of inhomogeneities. We only dealt with small integer values of the exponent µ in the airway resistance distribution (Eq. 8) because this led to closed-form solutions of the probabilistic airway compartment. The resultant model complexity thus allowed iterative minimization of the error function in Eq. 13, while maintaining identifiability with respect to available data. Indeed, in most cases the global optimization (6) provided a single unique set of parameters. More importantly, we hypothesized that it is the alterations in the distribution of pathway resistances that contributed most to the increased inhomogeneities during constriction. This is a key point because the underlying assumption of all data analysis is that nonlinearity comes exclusively from the tissues and that airways only contribute to time constant inequalities. To test this, we included a Rohrer-type airway pressure-flow nonlinearity in the IHNL model. Similar to our previous study (33), this added complexity did not significantly improve the fits in any of the conditions and often resulted in nonuniqueness. This finding thereby supports our key assumption that inhomogeneities contribute to the data primarily as a linear phenomenon. In our previous study, we found that, among six different nonlinear block structures (e.g., a Hammerstein structure, which is an inverse cascade connection of the L-N structure in Fig. 1, i.e., N-L), a Wiener structure for the lung tissues in series with a linear single-compartment airway model was the most appropriate for describing the pressure-flow behavior in tidal-ventilated dog lungs in control and late response to histamine infusion (33). In this study, we combined the previous model with a distributed airway compartment. This IHNL model provided a significantly improved fit to the data in both the peak and late responses compared with the HNL model. To verify that the best lung model was indeed a Wiener-type structure, we also fit the Hammerstein structure with and without airway inhomogeneities. Although not shown, these additional modeling efforts during both peak and late responses remained inferior to the Wiener structure regardless of the airway compartment included in the model (i.e., single-compartment or distributed model). Indeed, on average, the errors by using the Wiener structure were 45% smaller than the errors with the Hammerstein structure. The linear part of the lung tissue model was the constant-phase model given by Eq. 3. This model has been shown in several previous studies to provide excellent fits to small- and tidallike amplitude tissue impedance data (3,11-13, 20-22, 27, 31, 33) as well as pressure and stress relaxation data (3, 31). To verify that this was the case in the peak response, we replaced the constant-phase model by a Kelvin model, which did not improve the fits despite the increased number of parameters (i.e., 2 vs. 3). Furthermore, on the basis of our previous study (33), here we combined the various airway compartments with only a third-order zero-memory nonlinear block (N block in Fig. 1). As an additional check on the model validity, we also verified that the third-order polynomial nonlinearity was still sufficient. Although an arctangent form or a second-order polynomial nonlinearity gave worse fits, including a fourth-order polynomial did not improve the fits statistically significantly. Limitations of the modeling approach. We established the applicability of the model-fitting approach by examining its ability to recover the tissue parameters (G, H, a2 and a3) from simulated data in which the tissue parameters were kept constant and inhomogeneous airway constriction was gradually increased. The model used in the simulations is an anatomically consistent structural model based on Horsfield's morphological data (15). In several previous studies, we extended this model to be able to simulate inhomogeneous airway constriction (20). By adding nonlinear features to lung tissue properties, we have created, to our knowledge, the most faithful description of oscillatory pressure-flow relationships in the lung. Our best model with a hyperbolic distribution of parallel airway resistances is able to recover all tissue properties within 15% error when the constriction in 72% of the peripheral airway diameters in the last five generations is <60%. At 60% constriction, most parameters are still reasonably recovered, the largest errors occurring in G and a3, which are overestimated by 11 and 57%, respectively. Note, however, that the HNL model provided much higher errors in G and a3 (194 and 104%, respectively). At constriction levels above a 60% diameter reduction, the situation becomes worse, and other parameters (e.g., H and a2) will also become biased. The reason for the increased error in the estimated parameters is due to the fact that as peripheral airway constriction increases the impedance of the peripheral airways increases significantly. When peripheral airway impedance becomes comparable to central airway wall impedance, less flow enters the periphery and more flow is shunted into airway wall motion. Indeed, such a mechanism has been proposed to account for lung impedance data during severe bronchoconstriction (2, 24). Another limitation of the model analysis is the sharp serial distinction between airways and tissues. It is conceivable that when small airways constrict they will distort the surrounding parenchyma and that, as a consequence, tissue mechanical properties will change. Although correlations between airway resistance and tissue resistance or compliance during challenge have been reported (19), to our knowledge, such an airway-parenchymal interaction has not been implemented in a model of the oscillatory pressure-flow relationship of the lung. We examined this possibility by calculating correlations between the changes in the airway and tissue parameters in our model. No significant correlations were found between airway and tissue parameters. However, G and H showed some correlation (correlation coefficient = 0.63). This indicates that because this model takes into account the heterogeneities of pathway resistances, the tissue parameters are correlated with each other more than with the airway parameters. Correlation between G and H conforms with the structural damping hypothesis that postulates that dissipative and elastic phenomena are related at the elementary level (9). Physiological implications. Our IHNL model produces much smaller values for tissue damping G and and slightly smaller values for tissue elastance H than the HL model fit to the same data in the frequency domain (22). For example, in peak response, the IHNL model produced reduced estimates of G and H compared with the HL model by 44 and 20%, respectively. We attribute this to the fact that the inhomogeneous model can account for artificial increases in tissue resistance and elastance. Furthermore, for both control and late responses, the values of G obtained from the IHNL model were smaller that those obtained from the HNL model in our previous modeling study (33): the percent decrease in G was 14.6% in control and 25.6% in late response. The greater decrease in G is consistent with a stronger influence of inhomogeneities in late response. Additionally, both the HNL and the IHNL models in this study produced very close values for H as well as for a2 and a3, with their values in good agreement with those found in our earlier study in which the data were only fit in the time domain (33). The implications are that the hysteresivity coefficient, = G/H, is also
reduced by ~20% compared with that predicted by the HL or HNL models
in control, late and peak responses.
By using the IHNL model, G and
H in the late response were elevated
compared with their control values (Table 2). Although the magnitudes
of these elevations were similar to those previously found (33), these
increases reached a statistically significant level for only
H. This indicates that in our previous study
(33), part of the increase in G in the late
response may have been attributed to inhomogeneities. Additionally,
parameters a2 and
a3 slightly decreased in the late response but again not statistically
significantly. Because these results were obtained by a model that
accounts for inhomogeneities, they suggest the existence of real tissue
changes in the late response to histamine infusion. Thus, consistent
with our previous finding, we conclude that increased nonlinearity in
late response after histamine challenge as measured by the VT dependence of apparent
impedance is not due to changes in tissue nonlinearities but to an
increase in the magnitude of the linear tissue impedance (33). This is
also supported by the fact that H did not
correlate with the nonlinearity parameters
a2 and
a3.
During peak response, both G and
H increased statistically significantly
compared with their control values (Table 2). The nonlinear
coefficients decreased again, but their decrease did not reach a
statistically significant level. It is possible that part of the
increases in G and
H was still a result of inhomogeneous airway
constriction or central airway wall shunting. However, G obtained from the IHNL model decreased
considerably by 26% compared with that of the HNL model. In addition,
the IHNL model was able to recover G and
H within 15% when the Horsfield structure
undergoes a 60% inhomogeneous peripheral constriction (Fig. 6,
A and
B). Thus this presents indirect
evidence that in dogs real tissue changes do occur at the level of the
parenchyma in both peak and late responses during
bronchochallenge. This is in agreement with some recent
findings (13, 23). Nevertheless, our study suggests that the
tissue response may be considerably less than previously thought (1,
18, 19, 22, 26, 28).
The airway structure in our model consists of a set of parallel
pathways. The most appropriate distribution function for the parallel
airway resistances was a hyperbolically decreasing distribution with a
long tail toward the higher resistances. Interestingly, this implies
that airway conductance is uniformly distributed. A uniform
distribution of airway conductances has been used to predict the
effects of peripheral inhomogeneities on input impedance by Hantos et
al. (12). The expected value of the distribution E can be quite different from the arithmetic
mean of its left and right limits, i.e.,
(Ra + Rb)/2. As
constriction level increases in the simulated data (Fig. 6),
E becomes increasingly higher than the
parameter R from the HNL model. One might
expect that the width of the distribution of resistances in this model correlates with the degree of inhomogeneities. The simulation results
show that by using the Horsfield structure for the control case, the
estimated width of the distribution is negligible and hence the model
predicts a fairly homogeneous system (Fig.
7A). When constriction is increased in 72% of the peripheral airways of the
Horsfield structure, the width of the estimated airway resistance
distribution also increases. This extended tail of the distribution
increases the frequency dependence of both
RL and
EL at low and higher
frequencies, respectively (see Fig. 4). Hence the IHNL model can
partition RL differently from
any homogeneous model, and the error in G is
much smaller than that provided by the HNL model (Fig.
6A). A similar tendency can be seen
from the fits to the measured data: both the expected value (Table 2) and the width of the estimated distribution functions increase from
control to peak response and then decrease from peak to late response
(Fig. 7B). Experimentally, the
quantification of airway inhomogeneities seems quite arbitrary by using
alveolar capsules because of the inevitable undersampling of alveolar
pressures. On the other hand, our approach provides an average behavior
of the airway-tissue structure, and thus the resistance distribution should reflect changes in peripheral time constant inequalities. Indeed, the IHNL model seems to be sensitive to inhomogeneous peripheral constriction simulated by the Horsfield structure.
In summary, we presented an approach to distinguish between lung tissue changes and airway inhomogeneities by using a model that features nonlinear tissue viscoelasticity as well as airway inhomogeneities. For this approach to provide sensible results, we had to develop a combined time and frequency domain-fitting procedure because time domain data are dominated by nonlinearity, whereas fitting in frequency domain is necessary for the estimation of the contribution of airway inhomogeneities. Another method to distinguish tissue changes from airway inhomogeneities would be to use gases of different viscosities before and after steady-state bronchoconstriction (21). However, this presents a substantial increase in experimental and protocol complexity and may certainly prove difficult in asthmatic subjects. On the other hand, our approach is based on a single measurement made at the airway opening and does not require the use of the invasive alveolar capsule technique or different gases. Therefore, this technique offers great potential for use in human subjects in dynamic challenge studies. Nevertheless, future studies should examine and possibly include additional mechanisms such as tissue inhomogeneities, airway tissue interdependence, and central airway wall shunting.
ACKNOWLEDGEMENTS
This work was supported by National Science Foundation Grants BCS-9309426 and BES-9503008 and National Heart, Lung, and Blood Institute Grant HL-50515.
FOOTNOTES
Address for reprint requests: B. Suki, Dept. of Biomedical Engineering, Boston Univ., 44 Cummington St., Boston, MA 02215 (E-mail: bs{at}enga.bu.edu).
Received 20 May 1996; accepted in final form 4 December 1996.
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