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Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Lutchen, Kenneth R., and Heather Gillis. Relationship
between heterogeneous changes in airway morphometry and lung resistance
and elastance. J. Appl. Physiol.
83(4): 1192-1201, 1997.
We present a dog lung model to predict
the relation between inhomogeneous changes in airway morphometry and
lung resistance (RL) and
elastance (EL) for frequencies
surrounding typical breathing rates. The
RL and
EL were sensitive in distinct
ways to two forms of peripheral constriction. First, when there is a
large and homogeneous constriction, the
RL increases uniformly over the
frequency range. The EL is
rather unaffected below 1 Hz but then increases with frequencies up to
5 Hz. This increase is caused by central airway wall
shunting. Second, the RL and
EL are extremely sensitive to mild inhomogeneous constriction in which a few highly constricted or
nearly closed airways occur randomly throughout the
periphery. This results in extreme increases in the levels
and frequency dependence of RL
and EL but predominantly at
typical breathing rates (<1 Hz). Conversely, the
RL and
EL are insensitive to highly inhomogeneous airway constriction that does not produce any nearly closed airways. Similarly, alterations in the
RL and
EL due to central airway wall
shunting are not likely until the preponderance of the periphery
constricts substantially. The RL
and EL spectra are far more
sensitive to these two forms of peripheral constriction than to
constriction conditions known to occur in the central airways. On the
basis of these simulations, we derived a set of qualitative criteria to
infer airway constriction conditions from RL and
EL spectra.
airway resistance; inhomogeneities; lung impedance; lung
resistance; lung elastance
INTRODUCTION INCREASED LUNG RESISTANCE
(RL) and elastance
(EL) at the breathing
frequencies can compromise the mechanical function of the lung and make
breathing more difficult. The levels of
RL and
EL for frequencies surrounding
typical breathing rates (0.1-5 Hz) depend on several mechanisms,
including direct constriction of airways, the heterogeneity of airway
constriction, airway closure, and explicit changes in the tissue
viscoelastic properties. It is fundamental to appreciate how structural
changes in the lung contribute to each of these mechanisms, especially
in manners producing clinically relevant elevations in
RL and
EL. Much of the recent focus has
been on how disease, particularly bronchoconstriction, alters the
tissue properties (12, 14-16, 27). Unfortunately, the role of the
airway tree structure per se has received little attention. There has
recently been considerable advancement of practical techniques to
reliably measure low-frequency
RL and EL in situ (14, 29). Thus this
study addresses two key questions, What is the relation between how the
airway network constricts and changes in
RL and
EL at typical breathing rates,
and can we infer the structural status of the airways and tissues from
abnormalities in the RL and
EL spectra?
To address these questions investigators have advocated stochastic (1)
and/or morphometrically consistent models in which the airway
tree is an asymmetrical branching system of distinct airway orders
(Fig. 1) (10, 17, 30). The latter modeling approach requires a minimum number of ad hoc parameter assignments. The
RL and
EL of the system at any
frequency are governed by the geometric properties of the airways, the
physical properties of the airway walls and of the gas in the airways,
the connectivity of the tree system, and the viscoelastic properties of
lung tissue, all of which are consistent with lung structure and
independent experiments (6, 17). Our previous model permitted
inhomogeneous airway constriction (17) but only in a manner in which a
subpopulation of diseased airways constricted by an identical amount.
However, there is inferential (2, 22) and direct (9, 11, 19, 23)
evidence that airway constriction itself can be quite heterogeneous. Thorpe and Bates (30) recently proposed a model that permitted inhomogeneous airway constriction by stochastically altering the degree
of smooth muscle shortening throughout the airway tree. They predicted
the time course of RL and
EL at three discrete frequencies
during infusion of a bronchoagonist but did not explicitly examine the
relation between structural changes in the airway network and the
overall frequency-dependent features of
RL and EL.
In this paper we present an advanced version of our own model so that
airway constriction diameter changes are imposed via a designated
statistical distribution. By using this approach, we predict the
relative importance of mean level vs. the spread (i.e., inhomogeneity)
of peripheral airway constriction. We further examine the relative
sensitivity of RL and
EL to central vs. peripheral constriction. Our focus is on the specific peripheral airway structural alterations that can cause important clinical changes in
RL and EL. Our model suggests that it
is not the inhomogeneity of constriction per se that is critical but
whether the constriction produces a few closed or nearly closed
peripheral airways. On the basis of these simulations, we have derived
a set of criteria from which one can infer the presence of specific
mechanisms and airway structural status from
RL and
EL measurements alone.
METHODS The concept of our structural model is shown in Fig. 1. There is a
baseline airway tree as originally established by the morphometric studies on dog lungs by Horsfield et al. (10). The tree is categorized by a branching pattern of distinct airway orders. The highest order
(order 47 in the dog) is the trachea,
and terminal airways are order 2. Each
order is characterized by a length and diameter, and the branching
pattern is established through a set of order-specific recursion
indexes
Fig. 1.
Depiction of dog lung model based on morphometric data
of Horsfield et al. (10). Tree has 47 airway orders, each with a defined length and diameter. Branching pattern is asymmetrical. Impedance (Z) of a given order (n)
is calculated via an acoustic transmission line analysis, which
accounts for shunting into gas compression in the tube
[Cg(n)] and into
nonrigid airway walls (Zw). Walls are modeled as the parallel
combination of soft tissue and cartilage tissues according to Suki et
al. (28). Finally, an alveolar-tissue element is attached to the
terminal airways in the tree. There is gas compression corresponding to
volume of the alveolus (Cg), and tissue element is viscoelastic,
containing a tissue damping (G) coupled to elastance (H) to ensure a
constant tissue hysteresis (for more detail see Refs. 6, 17, and 18). R, resistance, 
, recursion index;
j, imaginary unit = 
; Iti, tissue inertance.
[View Larger Version of this Image (21K GIF file)]
(i),
i = 1,...,47 (3, 4, 13) (Table 1). Each bifurcation consists
of a parent branch and two daughter branches. The ordering at a
bifurcation is as follows: for parent branch
i, the daughters are
i 
1 and
i 1 (i). Thus for
(i) > 0, the branching is
asymmetrical.
Table 1.
Scaled airway dimensions at FRC in dogs
Order
Recursion Index
(i) Length, mm
Diameter, mm
Thickness, mm
47
2
196.0
17.0
0.75
46
2
7.4
17.0
0.75
45
2
17.6
16.4
0.72
44
10
9.9
9.0
0.43
43
10
9.4
8.4
0.41
42
10
8.9
7.8
0.38
41
10
8.5
7.2
0.36
40
10
8.0
6.7
0.34
39
10
7.6
6.2
0.32
38
10
7.2
5.8
0.30
37
10
6.8
5.4
0.28
36
10
6.5
5.0
0.26
35
10
6.1
4.7
0.25
34
10
5.8
4.3
0.24
33
10
5.5
4.0
0.22
32
10
5.2
3.8
0.21
31
10
4.9
3.5
0.20
30
10
4.7
3.2
0.19
29
10
4.4
3.0
0.18
28
10
4.2
2.8
0.17
27
10
4.0
2.6
0.16
26
10
3.8
2.4
0.15
25
4
3.6
2.3
0.14
24
4
3.4
2.1
0.13
23
4
3.2
1.9
0.13
22
4
3.0
1.8
0.12
21
4
2.9
1.7
0.11
20
4
2.7
1.6
0.11
19
4
2.6
1.5
0.10
18
4
2.5
1.3
0.10
17
4
2.3
1.3
0.09
16
4
2.2
1.2
0.09
15
4
2.1
1.1
0.08
14
4
2.0
1.0
0.08
13
4
1.9
0.9
0.08
12
4
1.8
0.8
0.07
11
4
1.7
0.7
0.07
10
4
1.6
0.7
0.06
9
4
1.5
0.6
0.06
8
4
1.4
0.5
0.06
7
4
1.4
0.5
0.05
6
0
1.3
0.4
0.05
5
0
1.2
0.4
0.05
4
0
1.2
0.3
0.05
3
0
1.1
0.3
0.04
2
0
1.0
0.2
0.04
FRC, functional residual capacity; i, order;
,
recursion index parameter. Data are from Ref. 10.
The impedance of a specific airway is calculated using the acoustic impedance equations (cf. Ref. 18) with airway wall properties incorporated according to Suki et al. (28). The model assumes that the airway wall is comprised of soft tissue and cartilage with the relative amounts of each a function of the specific order as derived from the studies of Gunst and Stropp (5). The wall thickness is also a function of the order number and depends on the airway diameter and cross-sectional wall area (31). The effective wall properties for each order depend on the rheological properties of soft tissue and cartilage, the relative amounts of both, and the wall thickness. For constant wall area, WA, a decrease in airway diameter will automatically increase wall thickness and correspondingly alter these wall properties. Note that the WA from (31) ignores the adventitia (outside the smooth muscle). Thus our simulations slightly underestimate wall thickness. The terminal airways lead to an alveolar-tissue element comprised of a shunt gas compression compliance (Cg) for the alveolus and a viscoelastic tissue model. The tissue model is constant phase (8) in which there are coefficients for tissue damping (G) and tissue elastance (H) such that the tissue resistance (Rti) and tissue elastance (Eti) are
|
(1) |
= 2
f,
f is frequency in Hz, and 
= (2/)tan
1(H/G). Thus
there is a hyperbolic-like frequency-dependent decrease in Rti and a
small frequency-dependent increase in Eti. This model assumes that
tissue hysteresivity (
= G/H) is constant with frequency.
Previous studies (3, 4, 6, 13, 17) using morphometric models calculated input impedance (Zin) by combining the impedance of each branch in the proper serial and parallel fashion and imposing self-similarity on the network. To incorporate realistic inhomogeneous constriction, we impose a constriction distribution function such that the diameter of a particular order is no longer the same everywhere in the tree but is derived from a random draw of the designated distribution function. Self-similarity can no longer be invoked. We devised an efficient computational approach that traverses the entire tree (see APPENDIX).
Distribution Functions
Airway constriction inhomogeneities were simulated by applying Gaussian (PG) or lognormal (PL) constriction probability distribution functions to an order with airway diameters of d
|
(2) |
|
(3) |
are the mean and SD.
The degree of spread can be quantified by the percent coefficient of
variation (CV = 100/µd). The mean
constricted diameter, µd, is equal to 1 µ where µ is the mean constriction level specified as a fraction of
the baseline value, and the CV is specified relative to this µ factor. Then, for each occurrence of a particular airway order, a
random draw is performed from Eqs. 2
or 3, and the baseline diameter
(dhealthy) is
multiplied by this constriction factor (see
APPENDIX). Each order that is
subject to constriction in the entire tree will then have a
distribution of diameters with mean diameter being (1 µ)dhealthy.
Figure 2 shows an example of the Gaussian
and lognormal distribution functions applied to an order with a
baseline diameter that was 0.4 mm (e.g., an order
6, Table 1) and that is going to be subject to a µ of
50% with a CV = 30 or 60%. Note that when the CV is high, airway
closure and dilation may occur. Airway closure occurs because the
random draw on the distribution function can produce negative diameters
that are then set to zero. In the Gaussian distribution, as the CV
increases so does the number of closed airways. Dilation is more
prominent with the lognormal distribution while closure is more
prominent with the Gaussian distribution.
Simulation Studies
Baseline diameters and lengths in the original Horsfield model were established at total lung capacity. Our simulations were run at functional residual capacity (FRC). The lengths were all scaled equally (by 0.98 for the dog). The diameters were scaled by an order-dependent sigmoidal scaling function (6), which allows the peripheral airways to narrow more than the central ones in a manner consistent with experimental data and their relative higher wall compliances. The corresponding FRC values of diameters, lengths, wall thicknesses, and recursion indexes for all orders are shown in Table 1. The baseline healthy tissue properties were derived from previous dog studies (i.e., for the whole lung the G = 3 cmH2O/l and H = 18 cmH2O/l) (20). The simulation studies distinguish between peripheral airway constriction defined as orders 2-22 (i.e., baseline diameters < 1.9 mm) and central airway constriction (orders 22-44).There are two major themes to our simulation studies. The first theme examines peripheral constriction alone. The emphasis is on the relative impact of mean constriction level vs. spread (variance) in constriction. We focus on whether particular forms of inhomogeneous peripheral constriction provide distinct signatures in their effect on the RL and EL spectra. The second theme examines the sensitivity of RL and EL to central vs. peripheral constriction. For central constriction we impose a mean and variance of constriction that are consistent with the recent imaging studies of Mitzner and Brown (23). For the sake of clarity, the exact conditions of each of these simulation studies are presented together with the results in the next section.
We point out here that while the diameters are drawn from a distribution function of known mean and CV, with peripheral constriction it was not necessary to average multiple simulations to arrive at a mean response for any one condition. Because of the complexity of the tree, as long as the same mean and CV were used, any one run looked very similar to another with a separate set of random draws. With central constriction, we did have to ensure against using a run with an unusually constricted proximal airway. Hence we chose from among several simulations those in which the mean diameters of the generations were closer to the population means.
RESULTS
Peripheral Airway Constriction
We simulated RL and EL from 0.1 to 5 Hz after creating the following four peripheral airway constriction conditions. Condition 1: Mild homogeneous (low mean and low variance). Here we imposed a low mean diameter reduction from baseline (µ = 20%) with a low variance (CV = 10%). This represents a mild amount of airway constriction occurring fairly homogeneously throughout the periphery. Condition 2: Severe homogeneous (high mean and low variance). We imposed a large mean constriction (µ = 50%) but one that still occurred fairly homogeneously (CV = 10%). Condition 3: Mild inhomogeneous (low mean and high variance). This represents a mild mean diameter reduction (µ= 20%) but highly inhomogeneous constriction (CV = 50%) condition. If imposed via a Gaussian constriction distribution (Fig. 2), a few peripheral airways will become highly constricted and/or closed. Condition 4: Severe inhomogeneous (high mean and high variance). This represents both a large mean diameter reduction (µ = 50%) with a large amount of spread (CV = 50%). Hence, most of the airways experience substantial constriction but with greater spread than in condition 2. Figure 3 shows the comparisons of all these conditions when a Gaussian constriction distribution function is imposed while Fig. 4 compares the use of a Gaussian vs. a lognormal function for conditions 1-3 only (and on a more expanded scale).
We first describe the results for the Gaussian-imposed constriction (Figs. 3 and 4A). The healthy baseline RL and EL spectra are consistent with published data (8, 20, 26) and show a distinct frequency-dependent drop in RL from 0.1 to 1 Hz. The EL shows a slight frequency-dependent increase over the same frequency range and then a decrease as the airway inertance became more dominant. Between 2 and 5 Hz the RL reaches a constant plateau. As described previously (17, 20, 25), the value of RL at this plateau represents airway resistance (Raw) alone. Here the baseline Raw = 0.44 cmH2O · l
1 · s.
Recall (17) that the baseline frequency dependence in RL and
EL from 0.1 to 1.0 Hz is
entirely due to the viscoelastic tissue properties. The inherent
baseline asymmetry of the tree does not produce any additional
frequency dependence.
What is the impact of homogeneous constrictions? For a mild homogeneous
constriction (µ = 20%, CV = 10%) there is a small uniform increase
in RL at all frequencies (Fig.
3). There is no noticeable increase in
EL. With more severe homogeneous
constriction (µ = 50%, CV = 10%), the
RL becomes substantially
elevated at all frequencies. The Raw
(RL at 5 Hz) increases from 0.44 to 3.02 cmH2O · l1 · s.
More dramatically, the EL shows
a large (~75%) frequency-dependent increase from 16.4 cmH2O/l at 0.1 Hz to 28.6 cmH2O/l at 5 Hz. However, most of
this increase occurs after 1 Hz (Fig.
4A). This additional frequency
dependence is a consequence of a significant shunting of flow into the
airway walls due to the uniformly large peripheral impedance (21). This
airway wall shunting will occur only if the mean constriction is high
and uniform enough. We found that airway wall shunting did not induce a
frequency-dependent increase in
EL until the mean diameter
reduction was 40% or more. Moreover, we found that the frequency
dependence in EL for this severe
homogeneous constriction no longer existed if we imposed rigid airway
walls everywhere.
What is the impact of inhomogeneous Gaussian constriction? A mild
inhomogeneous constriction (µ = 20%, CV = 50%) produced dramatic
changes in RL and
EL features, greater than either
of the previous homogeneous conditions (Fig.
4A). The value of
RL at 0.1 Hz increased by a
factor of four from baseline and is 114% greater than that which
occurred with severe homogeneous constriction. There is a large
frequency-dependent decrease in
RL that continues until ~3 Hz.
The effective Raw is a elevated from baseline (0.44 vs. 2.30 cmH2O · l1 · s)
but by a lesser amount than with the severe homogeneous constriction
(Fig. 4A). The impact on
EL is even more noticeable. First, the EL at 0.1 Hz has
nearly doubled, from 16 to 30 cmH2O/l. Also, the
EL now displays a substantial
frequency-dependent increase, but most of the increase occurs below 2 Hz. The EL increased from 30 cmH2O/l at 0.1 Hz to 57 cmH2O/l at 2.0 Hz (see Fig.
4A). Note the distinction in the
shape and frequency extent of this
EL increase compared with that
due to airway wall shunting, as occurs with severe homogeneous
constriction.
A severe inhomogeneous constriction (µ = 50%, CV = 50%) produced
similar changes as did the mild-inhomogeneous case but with a more
amplified response (Fig. 3). The severe nature of the
constriction caused a greater increase in Raw. Unlike the mild
inhomogeneous case, the frequency-dependent decrease in
RL and increase in
EL began at 0.1 Hz and extended
to 5 Hz. This reflects the combined influence of airway wall shunting
and inhomogeneous constriction.
In summary, when airway constriction is imposed via a Gaussian
distribution of diameters the "signature" of the changes in RL and
EL from 0.1 to 5 Hz is markedly
distinct for mild inhomogeneous constriction compared with severe
homogeneous constriction (Fig. 4A).
The former can induce huge increases in
RL and
EL at the lower and more typical
breathing frequencies. The latter induces large, uniform increases in
RL, no increase in
EL at the lower breathing
frequencies, and, because of airway wall shunting, large increases in
EL at the higher and
nonbreathing frequencies.
How sensitive are the above results to the form of the inhomogeneous
constriction? From Fig. 4, we see that the severe homogeneous constriction cases are nearly identical when created via a Gaussian vs.
a lognormal constriction distribution function. However, the mild
inhomogeneous constrictions are remarkably different. Unlike the
Gaussian case, the lognormal case showed almost no sensitivity in any
features of the RL or
EL. This difference occurs
because the Gaussian constriction produced a small but finite number of closed or highly constricted airways occurring randomly in the periphery while the lognormal produced none (Fig. 2). The implication is that the large changes in the levels and frequency dependence of
RL and
EL that occur at very low
frequencies (0.1-1 Hz) require extreme constriction of only a few,
but randomly dispersed, peripheral airways. If there is a mild and very
inhomogeneous constriction that still does not produce some highly
constricted peripheral airways (e.g., the lognormal case of Fig.
4B), there will be no noticeable
changes in RL and
EL.
During our simulations the tissue properties were not altered. The
increase in EL at 0.1 Hz is
strictly a consequence of airway closure occurring in a dispersed
manner throughout the periphery. This results in an increase in the
effective elastance of the whole lung but not a change in the elastic
properties of lung tissue. For our specific simulation conditions (µ = 20%, CV = 50%), we calculated that only 8.7% of the peripheral
airways actually closed, but this resulted in an 83% increase in
EL at 0.1 Hz. The real issue is
which 8.7% of the airways close. Recall that if the random draw for a
individual airway diameter is below zero, the simulation simply closed
the airway. If a higher order airway closes, communication (from the
airway opening) is lost to a greater amount of lung tissue than if
lower airway order closes. Hence, by closing only a small percentage of
the airways of orders 2-22, communication is lost to a substantially greater percentage of order 2 airways and their associated
tissue elements.
In Fig. 5 we evaluate whether complete
closure is needed to induce this increase in
EL at 0.1 Hz. Here we repeated
the simulations for mild inhomogeneous constriction but now placed a
maximum constriction for any airway to be 90, 80, or 70% of the
baseline diameter. With an 80% diameter constriction limit, the effect
of closure on EL at 0.1 Hz is
abolished, but the large increase in the frequency dependence of
EL and
RL due to airway inhomogeneities
remain. Thus complete airway closure need not occur to produce a large increase in EL at 0.1 Hz, but
the constriction must be rather severe (>80% diameter reduction).
Alternatively, if closure is limited to 70% diameter constriction,
there is a substantial drop in the
RL and
EL at all frequencies,
particularly EL at the lower frequencies. Thus, again, it appears that the
RL and
EL spectra are rather
insensitive to inhomogeneous constriction unless a few highly
constricted airways are included.
Central vs Peripheral Constriction
Recently, Mitzner and Brown (23) used high-resolution computer tomography to measure the maximal diameter responses to methacholine or histamine of dog airways having baseline diameters at FRC that were 1.7 mm or larger. They imaged 13-14 airways per dog in 5 dogs. Relative to FRC, the average maximum methacholine response for all airways was a mean constriction of 47 ± 19%. These data are further refined by grouping the airways by size, and the results are shown in Table 2. All airways >4 mm in diameter at FRC constricted by a similar mean and variance. Smaller airways with diameters between 1.7 and 4 mm tended to show slightly less mean constriction but more variance. At present, this imaging technique cannot provide reliable data for very small airways. Nevertheless, these data provide a framework for imposing two kinds of Gaussian central airway constrictions. The first used a µ = 50% and CV = 15% and closely represents the maximum constriction conditions from the data of Mitzner and Brown. For contrast, the second used a µ = 20% and CV = 50%, which correspond to very heterogeneous central airway constriction. These simulations were performed with and without concomitant peripheral airway constrictions (Fig. 6).
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The maximum homogeneous central airway constriction condition (µ = 50%, CV = 15%) produced a uniform increase in RL and almost no change in low-frequency EL. The mild inhomogeneous constriction (µ = 20%, CV = 50%) enhanced the frequency dependence of RL and EL. Nevertheless, when either of these conditions occurred simultaneously with peripheral airway constriction, the changes in RL and EL were far more dramatic. Thus changes in RL and EL are more sensitive to peripheral airway constriction than to central airway constriction, especially if the peripheral constriction includes a few closed or nearly closed airways. Also, the increased frequency dependence in EL characteristic of airway wall shunting is more likely a consequence of homogeneous constriction of the periphery than of the central airway constriction.
DISCUSSION
We know that after certain bronchial interventions, the frequency dependencies and levels of RL and EL can be drastically altered. The motivation of this study was to better understand how specific structural changes in the airways are coupled to mechanisms that can cause these changes. A morphometrically based modeling approach incorporates realistic anatomic features of the airway tree. This allows us to correlate explicit anatomic and geometric changes in the airways to changes in RL and EL. Self-similarity was not required in our model, which permitted us to impose virtually any constriction distribution condition desired. As a consequence, this study predicts the structural airway changes necessary for inhomogeneous constriction, airway walls, and airway closure to induce important changes in RL and EL.
Airway Mechanisms Influencing RL and EL
Increased levels of RL and EL at low frequencies may have fundamental clinical implications for breathing capabilities. The presence of a few highly constricted or closed peripheral airways will cause a huge increase in the levels and frequency dependence of RL and EL between 0.1 and 2 Hz (Figs. 3, 4, 5). Note that this can occur even if the mean constriction is low. The increased frequency dependence is due to a few extremely high airway-tissue mechanical time constants (24). The key term here is "highly constricted." Our simulations indicate that this corresponds to diameter reductions >80%. Randomly dispersing just a few highly constricted airways can also produce a rather large increase in EL at 0.1 Hz (Figs. 3 and 4A). Such an increase is often interpreted as an increase in the elastance of the lung tissue. Indeed, several previous studies have reported postbronchoconstrictor data very similar to our CV of 50% simulation conditions (Refs. 15, 16, 20, 26, 27 and in particular Figs. 3 and 4A compared with Fig. 2 in Ref. 20). These studies routinely conclude that there is tissue stiffening occurring simultaneously with airway constriction. Our results raise the alternative hypothesis that in these previous studies there was random severe constriction or near closure in the lung periphery rather than increased tissue elastance. Finally, if constriction is limited to 70% of the baseline diameter, much of the increase in RL and EL at the breathing frequencies is abolished. This means that for subjects suffering from acute bronchoconstriction, there is potentially a large clinical benefit to achieve even mild hyperinflation such that mild airway dilation occurs (from nearly closed state to 70% closed).It is important to emphasize that RL and EL are highly tolerant to inhomogeneous constriction as long as the constriction does not include a few highly constricted airways. This was displayed in the results of Fig. 4B in which the highly inhomogeneous (µ = 20%, CV = 50%) but lognormal constriction condition had nearly negligible impact on RL and EL. Recall that these lognormal distributions produced a wider range of diameters than did the Gaussian distributions, but they did not include any highly constricted airways.
Homogeneous bronchoconstriction will, of course, increase Raw and produces a uniform increase in RL (Fig. 3). However, such constriction must be rather severe with a diameter constriction of 40% or more throughout the entire periphery to affect to a measureable extent the RL and EL spectra. Such severe homogeneous peripheral constriction provokes the mechanism of central airway wall shunting that will cause a substantial rise in EL with frequency but will not increase EL at 0.1 Hz (Figs. 3 and 4). Moreover, the "signature" of this EL spectrum is quite distinct from that due to parallel constriction inhomogeneities (Fig. 4A). In particular, the increase in EL due to shunting is not distinct until frequencies are above 1.5 Hz and extends monotonically over a greater frequency range.
Acute changes in the frequency dependence of RL and EL between 0.1 and 2 Hz are not likely due to central airway constriction (Fig. 6). We base this statement on how consistently with the imaging studies of Mitzner and Brown (23) our model responded to central airway constrictions. At the maximal bronchoconstriction dose, central airways showed some inhomogeneity of constriction (CV of ~15%) and a mean diameter decrease of ~50% from baseline FRC values, and none of these airways exhibited closure. Therefore, studies that have previously observed increased frequency dependence in RL and EL at very low frequencies most likely reflected a lung condition that produced a few highly constricted peripheral airways as described in Peripheral Airway Constriction. This is consistent with the conclusions of previous studies as well (e.g. Refs. 11, 24).
Classification of Airway Structure from RL and EL Data
Our simulations predict that measurements of RL and EL are sensitive to a variety of important mechanisms that are altered by lung disease. Is it possible to infer the structural airway status and relative presence of these mechanisms from a set of RL and EL data? This question is often addressed via formal systems identification tools with simpler lumped models (e.g., Refs. 7, 8, 19, 20, 25). We can, however, offer the alternative approach of developing qualitative criteria to associate airway and lung tissue conditions consistent with measured RL and EL data. Our simulation results suggest four patterns of peripheral airway constriction with respect to the changes they induce in RL and EL spectra. We propose the following set of criteria. Small uniform elevation in RL at all frequencies with normal EL. This condition would correspond to mild-to-moderate homogenous constriction in the periphery and/or central airways and may be hardly noticeable in actual data. The term moderate is employed because our model predicts that a mean constriction as great as 40% may occur and still produce only mild changes in RL and EL. Large uniform elevation in RL and normal EL at frequencies below 1 Hz followed by a frequency-dependent increase in EL continuing beyond 4 Hz. These features would be indicative of moderate-to-severe homogeneous constriction in peripheral airways with the frequency dependence in EL arising from airway wall shunting. The lack of an increase in EL at low frequencies again suggests no concomitant change in tissue properties because it would be unusual for tissue damping to increase without any change in tissue elastance (15). Data on isolated tissue strips support this notion (12). Severe increase in RL at frequencies below 1.5 Hz with relatively small increase in RL above 2 Hz; also, a large frequency-dependent increase in EL predominantly from 0.1 to 2 Hz and, perhaps, an increase in EL at frequencies below 0.2 Hz. These features would indicate mild-to-moderate inhomogeneous constriction with a few highly constricted or nearly closed airways scattered throughout the periphery. It is important to appreciate that only a few of the peripheral airways are highly constricted. If most of the airways were highly constricted, the features of condition 2 above would occur. The relatively low increase in the RL at higher frequencies (i.e., the Raw value) is consistent with a low mean level of constriction. The excessive frequency dependence in both RL and EL that occurs over a narrow and low- frequency range is consistent with parallel inhomogeneities that include some very constricted pathways. If the above features occur without the increase in EL at very low frequencies, less severe constriction is occurring with no airway closure. Changes in tissue rheology such as an increase in tissue stiffness and/or tissue damping can also result in an increase in EL at 0.1 Hz. To determine whether this increase is due to the tissues or airways, one can use RL and EL measurements taken with the lungs equilibrated on two gases with distinct viscosities (19). Severe increase in RL at frequencies below 1.5 Hz with a large increase in RL above 2 Hz and the decrease in RL continuing to 5 Hz; also, a large frequency-dependent increase in EL predominantly from 0.1 to 2 Hz but continuing to as high as 5 Hz; and, finally, perhaps an increase in EL at frequencies below 0.2 Hz. These features are indicative of moderate-to-severe inhomogeneous constriction involving the periphery and perhaps more central airways simultaneously. The features are essentially an accentuated version of condition 3, but now the Raw is highly elevated from a healthy value. The more severe increase in Raw reflects the greater mean level of constriction and the potential involvement of central airway constriction. Similar ambiguous issues of whether there are also concomitant changes in tissue viscoelasticity exist as in the previous case, and the possible solutions are as described. There is room for much overlap between these constriction conditions, and we do not suggest that classification of the state of airway constriction will now be unambiguous by applying our criteria. Nevertheless, we believe this qualitative approach is quite powerful as a means for developing first-line hypotheses on airway function from low-frequency impedance data.Summary
We have found that changes in RL and EL from 0.1 to 5 Hz are most sensitive to two distinct forms of peripheral constriction. First, when there is a large and fairly homogeneous constriction, the RL tends to increase uniformly over the frequency range. The EL is rather unaffected below 1 Hz but then increases significantly almost monotonically up to frequencies of 5 Hz. The altered frequency dependence of EL is a direct consequence of airway wall shunting. Alternatively, the RL and EL are extremely sensitive to inhomogeneous constriction for which a few highly constricted or nearly closed airways randomly dispersed throughout the very periphery of the lung occur. Such airways cause extreme increases in the levels and frequency dependence of RL and EL predominantly below 1-2 Hz and in a manner that is likely to significantly impact breathing capability. The increase in the EL at 0.1 Hz can falsely suggest stiffer tissues in the absence of rheological alteration in lung parenchyma. It is important to appreciate that RL and EL are quite tolerant of very inhomogeneous changes in airway diameters as long as there are no airways nearly or fully closed. Similarly, alterations in the frequency dependence of RL and EL due to central airway wall shunting are not likely until the preponderance of the periphery undergoes substantial constriction. We also found that RL and EL are far more sensitive to these two forms of peripheral constriction than to constriction conditions known to occur in the central airways. On the basis of these simulations we derived a set of qualitative criteria to infer airway constriction conditions from RL and EL spectra.ACKNOWLEDGEMENTS
We are grateful to Drs. Robert Brown and Wayne Mitzner for providing us with their recent high-resolution computer tomography data on the more-central airways of five dogs before and after maximum methacholine constriction. These data are from a recent paper from their laboratory group (23). We also thank Dr. Bela Suki for helpful comments during preparation of this manuscript.
FOOTNOTES
Address for reprint requests: K. R. Lutchen, Boston Univ., Dept. of Biomedical Engineering, 44 Cummington St., Boston, MA 02215.
Received 26 August 1996; accepted in final form 6 May 1997.
APPENDIX
Computational Approach
The Zin is the impedance looking into the entire branching tree network and involves combining the impedance of each branch in the proper serial and parallel fashion. To compute the equivalent impedance looking into order i [Zeq(i)] three impedances must be known: Zeq(i 1), Zeq[i 1 (i)], and the
impedance of the parent order alone
[Z(i)]
![Zeq(<IT>i</IT>) = Z(<IT>i</IT>) + <FR><NU>[Zeq(<IT>i</IT> − 1)]{Zeq[<IT>i</IT> − 1 − &Dgr;(<IT>i</IT>)]}</NU><DE>Zeq(<IT>i</IT> − 1) + Zeq[<IT>i</IT> − 1 − &Dgr;(<IT>i</IT>)]</DE></FR>](/content/vol83/issue4/fulltext/1192/img005.gif)
|
(A1) |
1) and
Zeq[i 1 (i)]. Actually, Eq. A1 is valid only at low
frequencies where the parent order can be modeled as a lumped impedance
element. At higher frequencies (approximately >20 Hz) the impedance
is distributed along the parent similar to a transmission line and
Eq. A1 must be modified as in Ref.
18.
A stack-based algorithm was devised that permits each branch of the
tree to be traversed in an organized fashion. Specifically, impedance
calculations begin at a terminal branch and continue up the longest
pathway (i.e., from 1 to 47). Two Z(2) are calculated and
combined in parallel with the parent
(i = 3), to give Zeq(3). The
Zeq(3) now becomes the higher order daughter leading to Zeq(4). Then to
compute Zeq(4) we must calculate the Z(4) and the parallel combination
of Zeq(3) and Zeq[4 1 (4)]
(Eq. A1). The Zeq[4 1 (4)] is Zeq(3) again because (4)= 0. Note, however,
that this is really a distinct Zeq(3) from the one previously
calculated because the diameters and lengths of the airways leading to
this Zeq(3) were not necessarily the same. We cannot continue up the tree yet. In general, the index
[i 1 (i)] will always be for a
Zeq(i) that has been calculated at
least once before, but which now must be recalculated separately for
the specific airways that the new
Zeq(i) subtends. Thus we now store
the order of the needed Zeq (in this case 3), and the impedance
calculations once again start at the terminal airways. When the order
of Zeq being calculated equals the stored value of the order needed
(i.e., 3), we can proceed with calculating the Zeq for the parent
[i.e., Zeq(4)]. This procedure continues to transverse the
tree until Zeq is the equivalent impedance looking into the highest
order and results in Zin. This computational procedure, then,
effectively builds a tree from the bottom up, which is consistent with
the set of recursion indexes but which allows
Z(i) to be different in every
pathway in which it occurs. This stack-based procedure is efficient
primarily from a memory perspective. We do not have to store all
possible pathways. In fact, the maximum number of distinct impedances
stored at any one time is equal to the number of orders in the model
(47 for the dog).
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