Download Effect of alveolar wall shape on alveolar water stability To the Editor

Effect of alveolar wall shape on
alveolar water stability
To
To the
the Editor:
Editor: In
In the
the paper
paper in
in which
which he
he described
described the
the
low
low surface
surface tension
tension of
of bubbles
bubbles formed
formed of
of fluid
fluid taken
taken from
from
the
the lung,
lung, Pattle
Pattle (5)
(5) mentioned
mentioned that
that a low
low surface
surface tension
tension
at
at the
the lung
lung surface
surface is
is required
required to
to keep
keep the
the lung
lung dry.
dry.
Clements
Clements (1) modeled the alveolus as a spherical surface
and made quantitative estimates of the effect of surface
tension on the fluid balance of the alveolus. Reifenrath
(6) pointed out that this equilibrium would be unstable
if the liquid-air interface were spherical and surface
tension were constant, and he suggested that the polyhedral shape of the alveolus has a regulating and stabilizing effect. He argued as follows. The lowest pressure in
the liquid lining occurs above alveolar wall intersections
where the radius of curvature of the wall R is least and
the pressure difference across the liquid-air interface T/
This total water flux must be zero for equilibrium. If
R due to surface tension T is greatest. Fluid that enters
the
wall permeability, interstitial pressure, and surface
a polyhedral alveolus would collect to form pools in the
tension
are all uniform over the surface, equilibrium is
corners. This would increase the radius of curvature of
the surface over the pool, increase the pressure in the possible only if
liquid lining at that point, and reduce the water flux.
(PIS - PALV)
c + 2n T = 0
(2)
This description of the stabilizing effect of alveolar shape
It is unlikely that this particular relation between the
has been repeated in recent review articles (3, 4).
There is a question about this logic that is difficult to parameters would be satisfied, and it is especially unlikely
answer by qualitative reasoning. As water is added to a that this relation would be satisfied at all states of lung
pool in a corner, the pressure in the pool rises, but the inflation and vascular pressure.
If there is a net flux of water into the alveolus, water
area covered by the pool also increases. To what extent
does the increase in the area that is exposed to low would collect at the points of greatest wall curvature
pressure near the corner offset the decrease in pressure where the pressure in the liquid lining is least. The gasliquid interface would not coincide with the alveolar wall
at the corner? An analysis of a simple two-dimensional
model in which wall permeability and interstitial pressure surface, and the contribution of the surface tension term
to the integral of Eq. (I) would be slightly different. A
are assumed to be uniform leads to the conclusion that
detailed representation of a pool of water covering a
the fluid balance is unstable for a wall, such as Reifenrath
region around a point of maximum wall curvature is
pictured, that is concave toward the alveolus. However,
the fluid balance for a wall that is convex toward the shown in Fig. 2. For convenience, the coordinate s is
alveolar lumen, such as that near a fold where fluid pools
ALVEOLAR
WALL
CIRCUMFERENCE
WITH
C
Ah
Wa
INTERSTITIAL
PRESSURE
Ts
FIG. 1. An idealized
Z-dimensional
2-dimensional
representation
of an alveolus.
Net flux of water into the alveolus is zero only if (PIS - PALV)C
+ 2nT
= 0.
222
FIG. 2. A pool of fluid covering
a region around a point of maximum
wall curvature.
Flux of water into the alveolus through
segment of the
wall covered
by the pool is greater than it would be if there were no
pool and increases
with increasing
pool size. A concave
alveolus
is
therefore
unstable.
It either dries out or fills with fluid.
0161-7567/81/oooO-oooO$O1.25
Copyright
0 1981
the American
Physiological
Society
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are observed in electron micrograph, is stable.
An idealized two dimensional representation of an
alveolus is shown in Fig. 1. The driving force for the flow
of water into the alveolus is the pressure difference
between interstitial pressure PIS and the pressure in the
liquid lining of the alveolus. The pressure in the liquid
lining layer is alveolar gas pressure PALv minus surface
tension T times the curvature of the gas-liquid interface.
The curvature is equal to dO/ds where 0 is the inclination
of the alveolar wall and s is a coordinate along the
alveolar wall.
The total flux of water into the alveolus Q is the
integral over the circumference C of the driving force
times the permeability of the wall K.
c
d6’
=
K PIS-PALV+T~
ds
Q
(1)
I0
(
>
LETTERS
TO THE
223
EDITOR
measured from the point of maximum curvature, and for
small values of s the inclination 8, relative to the inclination at the origin, can be described by the equation
8 = as - bs2
(3)
where a and b are constants. The pressure within the
pool must be nearly uniform since the resistance to flow
between points within the pool is relatively small. Therefore, the curvature l/R of the gas-liquid interface covering the pool is constant. The gas-liquid interface and the
wall are assumed to be tangent at the edges of the pool.
The pressure in the liquid layer is reduced below alveolar
pressure by T/R, but the water flux occurs across the
wall surface element ds. In the interval from -sl to sl,
the surface tension term in Eq. 1 should be replaced by
(T/R) ds.
” T
2T
ds
E
=P
= 2T81 + 2T[(sl/R)
- O,]
(4
ALVEOLAR
SPACE
An idealized representation of the geometry near a fold
is shown in Fig. 4. The alveolar wall is represented as
two circular arcs of radius c that meet to form a cusp at
the origin. The liquid-air interface is again assumed to
be an arc of constant curvature that is tangent to the
wall at the edge of the pool.
In this case, the integral over the region covered by
the pool is again 2T81 + 2T[(sJR) - 811.The additional
term 2T[(sJR) - &] is again positive. In terms of the
parameters of the wall
7T
2T[(sl/R
l
micrograph
of a rabbit
point to fluid pools in
h/4
- >
1 - COS(Sl/C) 5
(5)
This additional term is a decreasing function of R and s1
so that, as the pool size increases, the tendency for fluid
to enter the alveolus decreases, and a stable equilibrium
is possible.
If there are n folds around the circumference C, the
equation for equilibrium is
(PIS
-
PALV)C
(6)
h/a)
77
1 - cos(sl/a) - Z
=0
The radius of curvature of the surface of the pool can lie
between zero and infinity. As the radius of curvature and
sl approach zero, the pool shrinks into the cusp of the
fold. As R becomes large and s1 approaches 42, the
pool fills the space between adjacent points of maximum
wall height. A value of R and the corresponding sl/c that
satisfies Eq. 6 can therefore be found for any value of PIS
which satisfies the inequality
PIS < PALV
FIG. 3. Drawing
of a transmission
electron
lung at 40% of total lung capacity
(2). Arrows
cusp-shaped
crevices in alveolar surface.
- 811= 2T
T
C
- -
The alveolar space can never dry out. A small pool of
fluid exists in the cusp; no matter how negative interstitial pressure is, the pools cover the region of convex wall
curvature, the stabilizing effect of convex curvature is
lost, and the alveolus floods.
The surface of the pools shown in Fig. 3 have a
radius of about 0.5 ,um. The radius c of the wall near the
fold is approximately 5 pm. From the relation, c/R =
cos(sl/c)], sl/c is found to be 0.34. The
cash/d
/ [l
-
Downloaded from http://jap.physiology.org/ by 10.220.33.1 on June 17, 2017
The first term 2T& is the value the integral would have
if the liquid-air interface coincided with the wall. The
additional term 2T[ (sl/R) - 011 is positive, since the
length of wall covered . bY the pool is greater than the
liquid-air interface on the surface of the pool. Furthermore, this term, which is approximately equal to
Ta2bs14/6,increases with the size of the pool.
The presence of water in the alveolus increases the
tendency for water to enter the alveolus. If surface tension is constant, a stable equilibrium for water in the
alveolar space is not possible for a concave alveolus. If
(PIS - PALV)C + 2nT < 0, the alveolus dries out. If (PIS
- PALV)C
+ 277T > 0, water enters the alveolus and
forms pools that augment the water flux. If (PIS PALV)C
+ 27~T = 0, a slight perturbation that forms a
pool of water leads to alveolar flooding.
Transmission electron micrographs show folds and
crevices in the alveolar walls that are partially filled with
fluid. A drawing, patterned after an electron micrograph
presented by Gil et al. (2) is shown in Fig. 3. Arrows point
to fluid pools in the folds where the walls are convex
inward.
FIG. 4. A model for the geometry
of the wall near a fold. Wall is
represented
as 2 circular
arcs of radius c. Pool surface has a radius of
curvature
R and is tangent
to the wall at the edge. Force for water flow
into the alveolus
decreases with increasing
pool size. This geometry
provides
a stable fluid equilibrium
for the alveolus.
224
folds occur on each side of a capillary and hence C/n is
approximately
10 pm. If C is taken to be 200 r_crnand T is
assumed to be 1 dyn/cm, PIS, as computed from Eq. 6,
is 9 cmH20 less than PALL The value of Prs for alveolar
flooding would be 0.4 cmHz0 below PALV. The contribution of the regions near the folds is significant.
Osmotic pressure has not been included in the equations. If the concentrations
of osmotically effective solutes are different on the two sides of the alveolar wall,
the osmotic pressure difference should appear in Eq. 6.
Interstitial
pressure has been assumed to be uniform.
Since the stresses are probably not uniform in the alveolar wall, fluid pressures may be nonuniform also and this
would affect the analysis. Only the effect of alveolar wall
geometry on fluid equilibrium has been included in the
model. The objective of the letter is to point out that the
regions of convex wall curvature provide a geometry that
can produce a stable alveolar fluid balance.
LETTERS
This work was supported
Institutes
of Health.
by Grant
HL-21584
TO
from
THE
EDITOR
the National
REFERENCES
J. A. Pulmonary
edema and permeability
of alveolar
1. CLEMENTS,
membranes.
AMA Arch. Environ.
HeaZth 2: 280-283, 1961.
2. GIL, J., H. BACHOFEN,
P. GEHR, AND E. R. WEIBEL,
Alveolar
volume-surface
area relation
in air- and saline-filled
lungs fixed by
vascular
perfusion.
J. Appl. Physiol.:
Respirat.
Environ.
Exercise
Physiol.
47: 990-1001,
1979.
3. HOPPIN, F.B., JR., AND J. HILDEBRANDT.
Mechanicalpropertiesof
the lung. In: Bioengineering
Aspects of the lung, edited by J. B.
West. New York: Dekker,
1977, vol. 3, chapt. 2, p. 83-162.
4. MACKLEM,
P. T. Respiratory
mechanics.
Am. Rev. Physiol.
40: 157184, 1978.
5. PATTLE, R. E. Properties,
function,
and origin of the alveolar
lining
layer. Proc. Ry. Sot. London
148: 217-240, 1958.
R. The significance
of alveolar geometry
and surface
6. REIFENRATH,
tension in the respiratory
mechanics
of the lung. Respir. Physiol.
24: 115-137, 1975.
Downloaded from http://jap.physiology.org/ by 10.220.33.1 on June 17, 2017
Theodore A Wilson
Department
of Aerospace,
Engineering, and Mechanics
University
of Minnesota
Minneapolis, Minnesota 55455