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1 Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455; and 2 Department of Physics and Biophysics, Massey University, Palmerston North, New Zealand
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
ACKNOWLEDGEMENTS
FOOTNOTES
REFERENCES
ABSTRACT 
Hill, Mark J., Theodore A. Wilson, and Rodney K. Lambert.
Effects of surface tension and intraluminal fluid on the mechanics
of small airways. J. Appl. Physiol.
82(1): 233-239, 1997.
Airway constriction is accompanied by
folding of the mucosa to form ridges that run axially along the inner
surface of the airways. The muscosa has been modeled (R. K. Lambert.
J. Appl. Physiol. 71: 666-673,
1991
[Medline]
) as a thin elastic layer with a finite bending stiffness, and the
contribution of its bending stiffness to airway elastance has been
computed. In this study, we extend that work by including surface
tension and intraluminal fluid in the model. With surface tension, the
pressure on the inner surface of the elastic mucosa is modified by the
pressure difference across the air-liquid interface. As folds form in
the mucosa, intraluminal fluid collects in pools in the depressions
formed by the folds, and the curvature of the air-liquid interface
becomes nonuniform. If the amount of intraluminal fluid is small,
<2% of luminal volume, the pools of intraluminal fluid are small, the air-liquid interface nearly coincides with the surface of the
mucosa, and the area of the air-liquid interface remains constant as
airway cross-sectional area decreases. In that case, surface energy is
independent of airway area, and surface tension has no effect on airway
mechanics. If the amount of intraluminal fluid is >2%, the area of
the air-liquid interface decreases as airway cross-sectional area
decreases, and surface tension contributes to airway compression. The
model predicts that surface tension plus intraluminal fluid can cause
an instability in the area-pressure curve of small airways. This
instability provides a mechanism for abrupt airway closure and abrupt
reopening at a higher opening pressure.
mathematical model; airway compliance; airway closure; airway
opening pressure
INTRODUCTION REPORTS THAT DESCRIBE ridges that run axially
along the inner surface of airways can be traced back to the last
century. Until recently, this feature of airway geometry has been
largely ignored, and the airways have been modeled as smooth-bored
tubes that expand or contract uniformly around their circumference. For
example, in their theoretical studies of surface tension and
intraluminal fluid, Halpern and Grotberg (5, 6), Johnson et al. (10), and Otis et al. (17) modeled the airway lumen as circular. The work of
James et al. (8, 9) brought new attention to the folding of the airway
epithelium. They showed that the length of the perimeter of the
basement membrane remains constant as airway area decreases, even for
the large area decreases that are caused by smooth muscle contraction,
and they used basement membrane length to characterize the intrinsic
size of airways that were fixed during bronchoconstriction. The
constant perimeter is accommodated inside a smaller
outer circumference by folding of the mucosa to form ridges that
penetrate into the airway lumen. Yager et al. (20) reported
observations of airways in which the interstices between the folds were
filled with fluid, and this observation added to the interest in
mucosal folding. Lambert (12) then suggested that the basement
membrane, which contains elastin and collagen, might resist bending and
that its bending stiffness might be an important component of airway
elastance. Lambert et al. (13) used this model to interpret data that
showed correlations between number of folds, airway area, and wall
thickness in sheep.
Lambert (12) modeled the airway as a composite structure consisting of
an elastic inner layer, which represents the mucosa, an intermediate
fluid layer, which represents the submucosa, and an outer layer of
smooth muscle, which imposes a compressive stress on the inner layers
when it contracts. Surface tension was not included in the model. He
computed luminal area as a function of transmucosal pressure for this
model. Yager et al. (20) discussed the effect of intraluminal fluid on
luminal area and estimated the effect of surface tension on the airway
area-transmural pressure curve in the case where the interstices of the
folds were completely filled and the air-liquid interface formed a
cylindrical surface inside the folds. In this study, we
describe a model in which surface tension and intraluminal fluid are
added to the Lambert model. The effect of surface tension is analyzed
for the case where fluid partially fills the folds. We find that
surface tension has no effect on airway area if the amount of
intraluminal fluid is small and the fluid lining layer is thin. In that
case, the air-liquid interface coincides with the epithelial surface,
and the length of the air-liquid interface is constant because the perimeter of the mucosa is constant. Therefore, surface energy is
independent of luminal area, and surface tension does not affect airway
mechanics. Thus one of the principal consequences of mucosal folding is
to remove the compressive effect of surface tension if the airway is
relatively dry. However, with fluid volumes on the order of a few
percent of luminal volume, fluid pools form in the folds, and the
length of the air-liquid interface changes as airway area changes and
the shape of the folds changes. In that case, the model predicts
that surface tension can cause area-pressure instabilities and
fluid-balance instabilities.
METHODS
Psub, was uniform.
For negative values of Pair Psub, the mucosa is
compressed, and at a critical negative transmucosal pressure,
Pcr = (D/R3)(n2 1), where D is the bending
stiffness of the mucosa and n is the
number of folds in the bending mode, the mucosa buckles. He analyzed
the buckling of the mucosa for
Pair Psub < Pcr.
) at the interface between
the fluid and the air in the lumen. If the cylinder is circular and the
fluid layer is thin, the curvature of the air-liquid interface is
~1/R and
Pfl = Pair /R, where
Pfl is the pressure applied to the
inner surface of the mucosa. Therefore, with intraluminal fluid and
surface tension, buckling occurs at a less negative value of
Pair Psub. If the fluid layer is thin,
the folds that form when the cylinder buckles quickly penetrate the
fluid layer. When that occurs, the curvature of the air-liquid
interface is no longer uniform around the circumference of the wall,
and the pressure applied to the inner surface is no longer uniform. We have analyzed the buckling of the cylinder under this nonuniform loading.
Governing equations.
A segment of the folded elastic wall is shown in Fig.
1. The elastic wall carries a
shear force (V), a tensile force
(T), and a bending moment
(M) that vary with position
(s) measured along the perimeter of
the cylinder. The inclination of the tangent to the wall is denoted as
. The equations of equilibrium for the wall are
|
(1) |
|
(2) |
|
(3) |
/ds) and
the undeformed curvature (1/R)
|
(4) |
, angle
between tangent to sheet and horizontal; a, radius of
curvature of surface of pool; 
, angle subtended by arc that forms
surface of pool; Pair, Pfl, and
Psub: pressures in air in lumen of airway, in fluid
layer, and in submucosa that surround mucosa, respectively. A
pool of fluid covers mucosa to point s = s*.
Pfl that appears in Eq. 2 differs from the uniform Pair by the pressure difference across the air-liquid interface. The pressure difference across the air-liquid interface is given by the product of surface tension and interface curvature. Surface tension is assumed to be uniform, but the curvature of the air-liquid interface varies around the circumference. The curvature of the interface above the fluid pool must be uniform if Pfl is uniform within the pool. The radius of curvature of this surface is denoted as a. Beyond the boundary of the pool (s*), the air-liquid interface coincides with the inner surface of the wall, and the interface curvature is the same as wall curvature (d
/ds). Therefore,
Pfl is given by
|
(5) |
R/n), where R is the
radius of the undeformed cylinder. The point s = 0 is taken
to be the point of maximum wall curvature at the depth of the fold, and
the point s = R/n is the point of
minimum curvature. The boundary conditions for V and at
these points are
|
(6) |
|
(7) |
must meet the wall and be tangent to the wall
at s = s*. This requires that the
solution satisfy the following conditions
|
(8) |
|
(9) |
|
(10) |
Psub,
Vfl, and
n, were integrated numerically from s = 0 to
s = R/n.
Because the values of T and
d/ds at
s = 0 are not imposed by the
boundary conditions and the values of
a and s* are not known a
priori, the solution method required a search through values of
T(0),
d/ds(0),
a, , and s* to find solutions that satisfied the boundary conditions at
s = R/n, given by Eqs. 6 and 7, and the conditions at s*
given by Eqs. 8-10. Solutions
were obtained for a series of values of
Pair Psub up to the value of
Pair Psub at which the sides of the
folds touched at the midplane of the fold.
Particular values of the parameters were used in the calculations. On
the basis of the data of Codd et al. (2), Young's modulus
E was taken to be 200 cmH2O, the thickness-to-radius
ratio (t/R)
was taken as 0.04, and the bending stiffness
D/R3
was taken to be 0.0015 cmH2O. The
value of R was chosen to be 0.5 mm.
Solutions were obtained for two values of , 10 and 20 dyn/cm,
respectively; for a series of values of
Vfl, ranging from 1 to 4% of the
area of the undeformed cylinder,
R2; but for
only one value of n,
n = 16. This value of
n was chosen because it lies in the
middle of the range of n observed by
Lambert et al. (13).
RESULTS
The area-pressure curves that were computed from the model are shown in
Fig. 2. In these curves, airway area
(A), normalized by the undeformed
area (Ao), is
plotted against the pressure difference across the liquid lining and
mucosa, Pair Psub. For pressures greater than
the buckling pressure, the cylinder remains circular. The specific
elastance,
Ao[d(Pa Psub)/dA]
of the circular cylinder is
Et/2R.
Therefore, for Pair Psub greater than the critical buckling pressure, the curves shown in Fig. 2 have slope
2R/Et. At the buckling pressure, area decreases sharply with decreasing pressure. The curves for fluid volumes <1% nearly coincide with Lambert's curve for n = 16 (12). With
intraluminal fluid, the buckling pressure is increased by
/R. However, if the amount of fluid
is small, the fluid pools are small, the air-liquid interface nearly
coincides with the wall surface, and the area-pressure curve is nearly
the same as the area-pressure curve for the dry airway. If the amount
of intraluminal fluid is greater, the area-pressure curves depart from
those for dry airways. The shapes of a half-fold at various points
along the area-pressure curves are also shown. The pressure in the
fluid pool (Pfl) relative to gas
pressure (Pair) is plotted vs.
Vfl at fixed values of
Pair Psub in
Fig. 3.
), namely, = 10 (A) and 20 dyn/cm
(B), respectively. Ratio of luminal area (A) to undeformed area
(Ao;
A/Ao)
is shown vs. pressure difference across mucosa
(Pair Psub) for different volumes of
intraluminal fluid, expressed as %undeformed luminal volume. Shapes of
mucosal fold and fluid pool are shown for 3 cases
(a-c).
Psub = 0.3
cmH2O vs. volume of fluid
(Vfl) in airway lumen as
%luminal volume.
DISCUSSION
Psub for a given deformation
scales with n2.
The value of in the airways is also uncertain. Alveolar surface tension has been measured and is known to change as lung volume changes, but the distribution of along the airways is unknown. The
effect of scales with /R. Thus,
for a given surface tension, the effects of intraluminal fluid and
surface tension are greater in smaller airways than in larger airways.
If the value of R had been chosen to
be 1 mm instead of 0.5 mm, the curves for = 20 dyn/cm for that
airway would be the same as the curves for = 10 dyn/cm that are
shown. We chose to analyze the deformation of a small airway to
emphasize the effect of surface tension. Finally, we show results for
intraluminal fluid volumes of 1-4% of undeformed luminal area.
Yager et al. (21) found intraluminal fluid volumes of ~2% of luminal
area at both high and lower lung volumes. For fluid volumes greater
than ~6%, the fluid simply floods the space between the folds, and
the air-liquid interface is circular. In that case, transmucosal
pressure is uniform, and the effect of surface tension on mucosal
folding can be obtained from Lambert's results (12).
We assumed that the mucosa folded in a 16-lobe mode. The buckling
pressure for this mode is larger than the buckling pressure for lower
values of n, and one would expect that
buckling with lower values of n would
be favored. Nonuniformities in mucosal properties around the
circumference of the airway could affect the buckling mode. Lambert
(12) suggested that the higher value of
n is imposed by the geometric
condition that the folded mucosa fit within the surrounding ring of
smooth muscle, and Mitzner and Wagner (15) suggested that folds form
over blood vessels that impose an initial deformation on the mucosa.
Thus, although the mechanism is unknown, multifold buckling occurs, and
we have chosen a value of n that lies
in the middle of the observed range. However, the computed curves do
not take account of the mechanisms that produce multifold buckling, and
the sharp change in the slope of the curves shown in Fig. 2 seems
unrealistic. It seems likely that the transitions between the segment
of the curve for
A/Ao >1 to the segments for
A/Ao
<1 are smoother than those shown in Fig. 2.
Smooth muscle contraction simultaneously imposes both a compressive
pressure difference across the mucosa and maintains a circular outer
boundary that confines the mucosa. If there were no smooth muscle tone
and transmural pressure were negative, the entire airway wall, mucosa,
submucosa, and smooth muscle could buckle in the lower energy two-lobe
mode. Excised airways with no muscle tone collapse in this two-lobe
mode. However, in intact airways, the surrounding parenchyma also tends
to maintain a circular outer boundary for the airway. Certainly,
reports of ridges in the surface of airways are not restricted to
preparations in which smooth muscle has been activated by
bronchoconstrictive agents. In the Appendix, we analyze the effect of
the surrounding parenchyma on the critical buckling pressure and
conclude that, in the absence of smooth muscle tension, the critical
buckling pressure for 16-lobe buckling of the mucosa within a circular
outer boundary is smaller than the critical pressure for two-lobe
buckling of the entire airway wall if transpulmonary pressure is >1
cmH2O. Because passive or active
smooth muscle tension and parenchymal stiffness both contribute to
maintaining a circular outer boundary, the outer boundary of intact
airways may remain circular, and the model described here may be an
appropriate model for elastance of the mucosa of normal airways.
Finally, our analysis describes the configuration of the mucosa and
intraluminal fluid in static equilibrium. The equilibrium configuration
depends on Pair Psub, and if
Pair Psub changes, the system must
relax from the old to the new configuration. The relaxation time for
redistribution of intraluminal fluid can be estimated by using the
equations that describe two-dimensional viscous fluid flow. The volume
flow rate (
) in the fluid layer is given by
= (h3/12µ)(
p/d)
where h is the thickness of the layer, µ is fluid viscosity, and p is the difference between the
pressures at two points separated by a distance (d). To change the
shape of the fluid layer, a volume of the order of
dh must collect, and the time required
to transport this volume of fluid is
dh/. For a value of µ equal to the viscosity of water, p equal to the difference
between pressure in the pool and pressure in the thin layer, and
h and d equal to pool dimensions, the
relaxation time is 0.1 s. Thus, for changes of configuration that occur
over time scales that are >0.1 s, our assumption that pressure is
uniform in the pool seems justified. However, is
proportional to
h3. An initially
uniform layer of fluid would redistribute on a time scale of 0.1 s, but
as the fluid layer on the ridge thins, flow becomes smaller and the
rate of thinning decreases. In addition, our solution contains a
discontinuity in the curvature of the air-liquid interface and hence, a
discontinuity in fluid pressure, at the boundary between the pool and
the thin layer. Our model does not address the complex mechanics of the
fluid layer that covers the ridge; we have simply assumed that this
layer is relatively thin.
Airway mechanics.
The primary purpose of this work is to investigate the effects of
surface tension and intraluminal fluid on the folding of the mucosa
that occurs when airways are compressed. A primary, and perhaps
surprising, result is that surface tension has no effect on airway
compression if the amount of intraluminal fluid is small. This result
can be understood immediately by considering the effect of the folds on
surface energy. By folding, the perimeter of the airway remains nearly
constant as airway area decreases. If the amount of fluid is small so
that the pools of intraluminal fluid are small, the air-liquid
interface lies near the inner surface of the airway, and the area of
the air-liquid interface and hence the surface energy also remain
constant. If surface energy is independent of airway area, surface
tension cannot affect the pressure-area curve. This result can also be
obtained from Eq. 2. If the air-liquid
interface coincides with the airway wall, the value of
Pfl is
Pair (d/ds). Another term in
Eq. 2,
T(d/ds), has the same form as the term
(d/ds). Therefore, a change in can be balanced by an equal and opposite change in
T, leaving all other equations
unchanged. Thus surface tension can be balanced by wall compression
with no change in the area-pressure curve, and Lambert's original
model (12), in which surface tension was neglected, is
valid for any value of surface tension if the airway is relatively dry.
Conversely, changes in luminal area would have little effect on surface
tension in that case. In the parenchyma, surface tension changes
because alveolar surface area changes with lung volume. In the airways,
if the amount of intraluminal fluid is small, airway perimeter is
nearly independent of airway luminal area, and airway wall area changes
only by the change in airway length as lung volume changes. Therefore,
the direct effect of lung volume on surface tension in the airways is
weak.
The combination of intraluminal fluid and surface tension affect the
area-transmucosal pressure curves as shown in Fig. 2. Before buckling
of the mucosa, the fluid forms a thin layer around the
circumference of the lumen, and the pressure in the fluid layer is
lower than luminal pressure because of the pressure drop across the
air-liquid interface. Therefore, the compressive pressure applied to
the mucosa is >Pair Psub by
/R or 0.2 and 0.4 cmH2O for
R of 0.5 mm and of 10 and 20 dyn/cm, respectively. Thus, with intraluminal fluid, buckling occurs at
values of Pair Psub that are greater than the
value for the dry tube. For 1% fluid volume, the fronts of the folds
quickly push through the thin fluid lining, and fluid collects in the
depths of the folds. However, the curvature of the surface of the pools
nearly equals the curvature of the mucosa, and the region covered by
the pools is small so that the area-pressure curves are displaced only
slightly to the right along the pressure axis. For 2% fluid, the
area-pressure curve lies a bit farther to the right down to
A/Ao
of ~0.2. At that value of
A/Ao,
fluid fills most of the space within the fold and the air-liquid
interface is near the neck between adjacent folds. The neck is narrow
and the radius of curvature of the air-liquid interface is small as it
passes throught the neck. As a result, pressure in the fluid pool is
much more negative than luminal pressure, and the region covered by the
pool is subjected to a large compressive stress. This buckled
configuration is maintained by the negative pressure in the pool, and
Pair Psub is more positive. In fact,
this configuration occurs at positive values of
Pair Psub for both = 10 and = 20 dyn/cm. After the air-liquid interface passes through the neck, its
radius of curvature increases, pressure in the fluid pool is less
negative, and more negative values of
Pair Psub are required to decrease area
further. The same sequence occurs as
A/Ao
decreases with 3% fluid. However, the air-liquid interface passes
through the neck at a slightly higher value of
A/Ao,
the minimum radius of curvature as the interface passes through the
neck is larger, and the maximum value of
Pair Psub is smaller than for 2%
fluid. The sequence is illustrated by diagrams that show three
configurations, all of which occur at
Pair Psub = 0.5.
The combination of surface tension and intraluminal fluid can produce
convoluted area-pressure curves. The segments of the curves with
positive slope are locally stable branches. That is, if
Pair Psub is held constant, more work
is required to deform the tube along the equilibrium curve than would
be provided by the given Pair Psub. The segments of the
curve with negative slope are locally unstable. Thus
configuration b shown in Fig. 2 is
unstable. If Pair Psub were held constant, the tube
would jump to either configuration a or
c. The combination of stable and
unstable segments leads to hysteresis in airway closing and opening.
For example, in the instance where = 10 and there is 3% fluid
volume, area would decrese smoothly along the upper
segment of the area-pressure curve down to
A/Ao = 0.25 as Pair Psub decreased to 0.6. For
a further decrease in Pair Psub,
A/Ao would drop to the value
A/Ao = 0.1 on the lower segment of the curve. If
Pair Psub were then increased, area
would increase along the lower segment until
Pair Psub reached 0.3. For any further increase,
A/Ao
would jump from 0.15 to 0.8. For some values of and fluid volume,
namely, = 10 dyn/cm and 2% fluid or = 20 dyn/cm and 2-4%
fluid, positive values of
Pair Psub are required to reopen the
airway.
Recent studies (3, 16, 18) have shown that a positive transmural
pressure is required to open airways that have closed. The airways
appear to open abruptly at a critical positive transmural pressure. In
some modeling studies, the positive opening pressure has been ascribed
to the pressure required to drive a miniscus axially into the closed
segment of the airway (3). Our model describes a different mechanism
that could maintain the airway nearly closed at small positive
transmural pressures and produce a sudden opening at a critical
pressure. The mechanism does not require an axial nonuniformity between
open and closed segments of the airway; it is an instability in the
opening of an axially uniform tube with fluid-filled folds.
Fluid instabilities.
In addition to the mechanical instability that can occur for certain
values of and fluid volume, a fluid instability can also occur. The
mechanism is described by Fig. 3, in which pressure in the fluid pool,
relative to luminal pressure, is plotted against fluid volume at a
fixed value of Pair Psub. This figure shows that at
Pair Psub = 0.3, fluid pressure
decreases as fluid volume increases because the radius of curvature of
the air-liquid interface decreases as the volume of the pool increases.
In that situation, a uniform distribution of fluid along the axis of
the airway is unstable. If fluid volume were slightly greater at one point along the axis, pressure would be lower at that point and fluid
would flow toward it. Pressure would fall further at the point where
volume was high and would rise at other points, driving further flow.
For = 20 dyn/cm, fluid accumulation would be terminated by a
mechanical instability that occurs at a fluid volume of 3%.
Both the mechanical and the fluid instabity mechanisms are driven by
the decrease in radius of curvature as the air-liquid interface
approaches the neck between adjacent folds. Folds that are one-half or
more filled with fluid are susceptible to these instabilities.
Therefore, it would be unlikely to find airways in which folds are
one-half filled with fluid; it would be more likely to observe airways
that are nearly dry or airways in which the folds are filled past the
neck.
Comparison with airway elastance.
Our modeling of the effects of surface tension and intraluminal fluid
describes potential mechanisms for instabilities that may play a role
in airway closure and reopening, and we have argued that this model of
mucosal bending is pertinent both to airway compression driven by
smooth muscle contraction and compression by transmural pressure
differences due to flow. However, there is a fundamental question about
the significance of these results; it is not clear that the magnitudes
of mucosal bending stiffness and surface tension are large enough for
these mechanisms to be significant. Because surface tension affects the
area curves shown in Fig. 2, surface tension is significant, relative
to mucosal bending stiffness, in 1-mm airways. However, it is not clear
that the magnitudes of both bending stiffness and surface tension in small airways are significant. Figure 2 shows the relationship between
area and transmucosal pressures. Transmucosal pressure is important if
it is significant compared with the pressure generated by smooth muscle
tension or if it is a significant fraction of transmural pressure. The
range of pressures shown in Fig. 2 is small, on the order of 1 cmH2O. Maximum smooth muscle
contraction generates compressive pressures of 30-40
cmH2O (4). Therefore, mucosal
bending stiffness offers little resistance to maximum smooth muscle
contraction. However, the bending stiffness of the mucosa may have been
underestimated. Also, airway wall thickening occurs in asthma, and
bending stiffness is proportional to the cube of thickness. A doubling
of mucosal thickness would produce an increase of nearly a factor of
ten in bending stiffness.
Although it appears that the mechanisms described here would not be
significant during maximum muscle contraction, they may be significant
at lower levels of smooth muscle tension. To determine whether they
contribute significantly to normal airway elastance, the theoretical
curves shown in Fig. 2 should be compared with experimental curves of
airway area vs. transmural pressure. A precise comparison is not
possible at present for a number of reasons. First, the pressures shown
in Fig. 2 are the difference between gas pressure in the lumen and
pressure in the submucosa. The pressure difference across the muscle
and other elastic components of the airway wall must be added to the
pressure differences shown in Fig. 2 to obtain transmural pressure.
Therefore, plots of airway area vs. transmural pressure would be
expected to be shifted to the right along the pressure axis, and the
slopes of the curves would be reduced because of the elastance of the
smooth muscle and other components of the airway wall, but the
magnitudes of these shifts are unknown. Also, the relationship between
Ao, the area of
the undeformed mucosa, and maximum airway area for large transmural
pressures would be needed to align the ordinate in Fig. 2 with the
ordinate of experimental area-pressure curves. At present, the
transmural pressure at which mucosal folding begins is not known.
Although a precise comparison between the theoretical curves shown in
Fig. 2 and experimental data is not possible, an approximate comparison
can be made between the magnitude of the elastance of the model mucosa
and the magnitude of airway elastance. The value of specific elastance,
Ao
d(Pair Psub)/dA,
for the model curves is ~1
cmH2O. Gunst and Stropp (4) report
area-transmural pressure curves of fluid-filled 4-mm canine airways. To
obtain a value of specific elastance that is comparable to the model elastance, an approximate
Ao value for the
canine airways is needed. The report of Yager et al. (21) shows that
the epithelium of airways of guinea pig lungs fixed at a transpulmonary
pressure of 5 cmH2O is fairly
smooth. The mucosa of sheep airways fixed at zero transpulmonary
pressure (13) have deep folds. We will assume that folding begins at a
transmural pressure of ~2 cmH2O and take this value of A as the value
of Ao. Then, the
data of Gunst and Stropp (4) yield a value of specific elastance of 2-3 cmH2O. The model mucosa
would provide a significant fraction of the elastance of these 4-mm
airways, and because airway elastance decreases with decreasing airway
size (7, 14), mucosal elastance would provide a larger component of the
elastance of smaller airways.
ACKNOWLEDGEMENTS
We thank Wayne Mitzner for pointing out the early literature on mucosal folding.
FOOTNOTES
Address for reprint requests: T. A. Wilson, 107 Akerman Hall, 110 Union St. SE, Minneapolis, MN 55455.
Received 13 March 1996; accepted in final form 9 September 1996.
APPENDIX
The effect of the surrounding parenchyma on the buckling of an airway can be modeled by representing the surrounding parenchyma as an elastic continuum and adding this feature to the model for a thin-walled elastic tube. This model can be analyzed by combining two results from classic elasticity theory. The first is the analysis of buckling of a thin-walled tube supported by a Winkler foundation (1). A Winkler foundation provides a restoring stress that is proportional to the local displacement. The critical buckling pressure (Pcr W) for a thin-walled tube supported by a Winkler foundation is given by the equation
|
(A1) |
) where is the azimuthal angle around the axis of the tube. The ratio of the two is
k. The effective
k of an elastic continuum can be
obtained from a second result from classic elasticity theory (19). The
stress and displacements in an elastic body loaded by normal stresses acting at a circular inner boundary can be expressed as series that
reduce to Fourier series in the azimuthal angle at the boundary. The
ratio of normal stress to radial displacement for each term in the
series provides k for that order. The
ratio depends on the order, or buckling mode
(n), and is given by
|
(A2) |
|
(A3) |
The objective here is to compare the buckling pressures for two
different buckling geometries, buckling of the entire wall in the
n = 2-lobe mode and buckling of the
mucosa in the n = 16-lobe mode, with
the outer boundary of the airway remaining circular. If
Pcr W, the buckling pressure
given by Eq. A3 for
n = 2, is more negative than
0.4, the critical pressure for mucosal buckling shown in Fig. 2,
mucosal buckling will occur. Thus we seek the value of
Epar for which
Pcr W <0.4
cmH2O
|
(A4) |
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