J Appl Physiol 117: 586–592, 2014.
First published June 19, 2014; doi:10.1152/japplphysiol.00072.2014.
A micromechanical model for estimating alveolar wall strain in mechanically
ventilated edematous lungs
Zheng-long Chen,1,2 Ya-zhu Chen,1 and Zhao-yan Hu2
1
Biomedical Instrument Institute, School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, China;
and 2Department of Precise Medical Device, Shanghai Medical Instrumentation College, Shanghai, China
Submitted 31 January 2014; accepted in final form 22 May 2014
pulmonary edema; stress and strain; ventilator-induced lung injury;
surface tension; alveolar wall
PULMONARY EDEMA is a characteristic radiographic feature of acute
lung injury (ALI) and acute respiratory distress syndrome
(ARDS)(12, 42, 44, 45)1. At the early stage of pulmonary edema,
fluid begins to pass into alveolar lumen where it first appears as
crescents in the angles between adjacent septa. As edema gradually progresses to the stage of alveolar flooding, fluid accumulates
in the alveoli to some extent such that a critical radius of curvature
is reached when surface tension effects sharply increase the
transudation pressure gradient. This effect leads to the all-or-none
liquid filling of each individual alveolus. As a result, alveolar
flooding is quantally characterized: some alveoli are completely
flooded while others, frequently adjacent, show only crescentic
filling or normally air-filled (17).
The management of mechanical ventilation in patients with
pulmonary edema has been a matter of medical concern.
Mechanical ventilation at high tidal volume and plateau airway
pressure may cause alveolar overdistension, in turn resulting in
ventilator-induced lung injury (VILI). Therefore, to effectively
prevent VILI, it is necessary to get a clear understanding of
mechanics of edematous lungs.
1
This article is the topic of an Invited Editorial by Dhananjay T. Tambe (36a).
Address for reprint requests and other correspondence: Z. Chen, Biomedical
Instrument Institute, School of Biomedical Engineering, Shanghai Jiao Tong
Univ., Shanghai, China (e-mail: [email protected]).
586
For a long time, our knowledge of mechanics of pulmonary
edema has been based on the interpretations of CT scans and
P-V curves (11, 18). Recently, Perlman and Bhattacharya et al.
have applied real-time confocal microscopy to observe the
micromechanics of alveolar perimeter expansion in the isolated, perfused rat lung (23, 24, 49). They instilled liquid into
one alveolus of a pair of juxtaposed alveoli and found unexpected micromechanical effects. At constant alveolar air pressure Palv, filling the alveolus with liquid produced a meniscus
that changed the septal curvature and consequently the pressure
difference across the septum. As a consequence, the air-filled
alveolus bulged into its liquid-filled neighbor even at functional residual capacity (FRC). Given the feature that liquidfilled and air-filled alveoli are focal or diffuse or patchy in
pulmonary edema, their findings may provide a novel understanding of segmental heterogeneities and alveolar overdistension during mechanical ventilation.
In another interesting study Protti and colleagues (26) mechanically ventilated 29 healthy pigs with a tidal volume
causing a volumetric strain (the ratio between tidal volume and
FRC) between 0.45 and 3.30. Their results demonstrated that
there existed a critical strain interval, reasonably ranging from
1.5 to 2, above which mechanical ventilation invariably induced edema formation in healthy lungs, whereas lower strains
proved to be normal or safe without any increase in lung
weight. Based on similarity of total lung capacity (TLC) and
FRC between pigs and humans, the authors suspected that the
same threshold phenomenon may occur in human lungs.
Along this line, we develop a two-alveoli model in the
present paper to quantitatively estimate deformation of alveolar
walls observed in Perlman and colleagues’ experiments. Assuming the alveolus to be in the shape of a tetrakaidecahedron,
the model includes elastic properties of lung parenchyma,
change in thickness of alveolar walls, and the effect of surface
tension. Theoretical calculations not only agree well with
experimental results but predict different patterns of alveolar
expansion. Our model is a preliminary step in picturing a
global topographical distribution of stress and strain on the
scale of the lung as a whole to guide ventilation parameter
settings for the prevention of VILI.
METHODS
Model introduction. The geometric model of the single alveolus
used in the present study was a 14-sided polyhedron with six square
and eight hexagonal faces as indicated in Fig. 1 (5, 6, 10, 14). We
assumed that one hexagonal face which has its wall removed serves as
the alveolar mouth, which is reinforced with elastin and collagen
connective tissue fibers. A most important property of the tetrakaidecahedron is that it can be surrounded by many other tetrakaidecahedrons until filling space, leaving no void. With this single alveolar
model, we subsequently developed the model of a pair of juxtaposed
8750-7587/14 Copyright © 2014 the American Physiological Society
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Chen Z, Chen Y, Hu Z. A micromechanical model for estimating
alveolar wall strain in mechanically ventilated edematous lungs. J
Appl Physiol 117: 586 –592, 2014. First published June 19, 2014;
doi:10.1152/japplphysiol.00072.2014.—To elucidate the micromechanics of pulmonary edema has been a significant medical concern,
which is beneficial to better guide ventilator settings in clinical
practice. In this paper, we present an adjoining two-alveoli model to
quantitatively estimate strain and stress of alveolar walls in mechanically ventilated edematous lungs. The model takes into account the
geometry of the alveolus, the effect of surface tension, the lengthtension properties of parenchyma tissue, and the change in thickness
of the alveolar wall. On the one hand, our model supports experimental findings (Perlman CE, Lederer DJ, Bhattacharya J. Am J Respir
Cell Mol Biol 44: 34 –39, 2011) that the presence of a liquid-filled
alveolus protrudes into the neighboring air-filled alveolus with the
shared septal strain amounting to a maximum value of 1.374 (corresponding to the maximum stress of 5.12 kPa) even at functional
residual capacity; on the other hand, it further shows that the pattern
of alveolar expansion appears heterogeneous or homogeneous,
strongly depending on differences in air-liquid interface tension on
alveolar segments. The proposed model is a preliminary step toward
picturing a global topographical distribution of stress and strain on the
scale of the lung as a whole to prevent ventilator-induced lung injury.
A Model for Estimating Alveolar Wall Strain
•
587
Chen Z et al.
Fig. 1. Assemblage of tetrakaidecahedrons
modeling the alveoli.
␭⫽
L
(1)
L0
As shown in Fig. 2, the initial length L0 is defined to equal alveolar
radius Ra0 when Palv is 5 cmH2O, nearly corresponding to alveolar
volume at FRC. It is worth noting that in reality the length of septum
SS= is greater than Ra0 even at FRC due to its protrusion toward the
edematous alveolus. According to the geometric relationship illustrated in Fig. 2, Eq. 1 can be written as
PliqA
PliqB
␭⫽
L
L0
⫽
L
Ra0
⫽
␳·␪
Ra0
⫽
PliqA ⫺ PliqB ⫽ PE
Ra
B
O
Palv
Palv
␴T ⫽
Fig. 2. Two-alveoli model describing deformations of alveolar walls in edematous lungs. See text for abbreviations.
(2)
(3)
␳ · PT
(4)
2t
Next we derive the pressure equilibrium relation in the above
two-alveoli model. For alveoli A and B, we have Eqs. 5 and 6,
respectively
Palv ⫽ PliqA ⫹ P␥A
(5)
Palv ⫽ PliqB ⫹ P␥B
(6)
where P␥A and P␥B are pressure produced by the curved air-liquid
interfaces. They can be calculated using the Laplace formula stated in
Eq. 7 and Eq. 8
P␥B ⫽
2␥A
(7)
␳
2␥B
(8)
Rb
where ␥A and ␥B are the surface tension coefficient. Substituting Eq.
7 and Eq. 8 into Eq. 5 and Eq. 6, respectively, then subtracting Eq. 6
from Eq. 5, yields the following equation
S’
PE
兲
where PE is recoil pressure generated by alveolar wall elasticity (48).
Histological evidence (38a) indicates that thickness of alveolar
walls is less than radius of the alveolus almost one order of magnitude.
Therefore, we consider the alveolar wall as an elastic membrane that
only produces tensile stresses unable to bear bending moment. According to Young-Laplace force balance across a curved interface,
internal stresses ␴T within the spherical membrane and pressure
difference PT acting across the membrane (thickness t) have the
following relationship
S
Rb
Ra0
៮⬘ represents the length of the chord SS
៮⬘, ␳ is the air-liquid
where SS
interface radius, and ␪ is the central angle subtended by circular arc
២. For septum SS=, the pressure equilibrium has the form
SS=
P␥A ⫽
A
共
៮’ ⁄ 2␳
␳ · 2arcsin SS
PliqA ⫺ PliqB ⫽
2␥B
Rb
⫺
2␥A
␳
(9)
Equation 9 implies that only when the condition of ␥B/Rb ⬎ ␥A/␳ is
satisfied will the septum SS= bulge into the liquid-filled alveolus.
Conversely, if ␥B/Rb ⬍ ␥A/␳, then the septum SS= will bulge into the
air-filled alveolus. That is to say, in the edematous lungs, septal
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alveoli, one (alveolus A) of which is air-filled, the other (alveolus B)
liquid-filled, as shown in Fig. 2. For the sake of simple illustration, we
here analyzed the alveolar stress and strain in two dimensions.
In Fig. 2, Palv is pressure in the alveolar air. PliqA and PliqB are
liquid pressures in the alveolus A and B, respectively. The radius Ra
of alveoli is defined as the circumradius of the regular hexagon. Rb
represents radius of meniscus developed in the edematous alveolus B.
As Perlman et al. observed, the intervening septum SS= was deformed
when alveolar pressure Palv increased, causing the air-filled alveolus A
to bulge into its liquid-filled neighbor B. To evaluate this septal
deformation quantitatively, we first consider the simplest situation in
which the alveolus A is juxtaposed with only one edematous alveolus
and all other septa but segment SS= are kept planar during inflation. As
Palv increases, the alveolar wall is subjected to the uniform and
spherical stretch with segment SS= from the initial reference length L0
(Palv ⫽ 5 cmH2O) to current length L. Here, we introduce the stretch
ratio ␭ stated in Eq. 1 to quantitatively describe the extent of
stretching of septum SS=.
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A Model for Estimating Alveolar Wall Strain
•
Chen Z et al.
TLC the values for alveolar surface tension are equal regardless of
alveolar size and location (29, 30, 31). Assuming that the points of
intersection between PV curves for air-filled lungs and those for test
liquid lungs with constant interfacial tensions define states of equal
surface tension, Smith and Stamenovic (33) obtained values of surface
tension for normal air-filled lungs ranging from 7 mN/m at 30% TLC
to 28 mN/m at TLC. These data are in close agreement with measurements by Lu et al. of the surface tension of bovine lipid extract
surfactant during different degrees of dynamic compression, with
␥max ⫽ 28.1 mN/m and ␥min ⫽ 1.0 mN/m. The relation between
relative area and surface tension can be fitted to a second-order
polynomial during inflation from FRC to TLC (1, 16). Using the
generally accepted relationship length ⬀ area1/2 ⬀ volume1/3, we
further fitted data of Schürch et al. as shown in Fig. 4. The relationship
between surface tension (␥A) and percent increase in alveolar radius
(ε) (%baseline) is given by Eq. 12.
8
7
6
5
4
3
2
␥A ⫽ 305.5ε2 ⫺ 548.1␧ ⫹ 250.5
1
0
1
1.05
1.1
1.15
1.3
1.25
1.2
1.35
1.4
1.45
deformation not only depends on tissue elastic properties but also on
surface tension as well as on alveolar size.
Given that recoil pressure PE equals membrane pressure difference
PT in magnitude, combining Eqs. 3 and 4 with Eq. 9 we obtain
2t · ␴T
␳
⫽
2␥B
Rb
⫺
2␥A
␳
(10)
If stress-strain curve and thickness of the alveolar wall, surface
tension coefficient ␥A and ␥B are known, then we can approximately
estimate the change of radius ␳ as meniscus radius Rb increases during
inflation. We will determine these parameters in the following text.
Stress-strain curve. Due to the lack of available data concerning
length-tension properties of alveolar walls in rat, stress-strain relationships of human lung parenchyma read from Sugihara et al. (36)
are used in our computation. Their experimental data and the fitted
curve are shown in Fig. 3. The equation for the curve fit is stated in
Eq. 11
␴T ⫽ 4,256.6 ⫹ 3,876.6 e2.9145␭ ⫺ 4,840 e2.75␭
(11)
The unit for ␴T in this equation is Pa. Strictly speaking, Eq. 11 is a
stress-stretch ratio relationship not a stress-strain relationship because
in linear elasticity tensile strain is defined as the elongation with
respect to unit length.
Surface tension. The composition of edema fluid commonly includes water, the electrolyte, and large molecules such as plasma
proteins (35, 42). Water and serum display much higher surface
tensions for each surface area compared with the pulmonary surfactant. Moreover, unlike the lung surfactant, water does not show
dynamic surface tension behavior, namely, its surface tension does not
vary with surface area (39). Therefore, we are to consider the surface
tension ␥B in the edematous alveolus in three situations where 1) the
surfactant layer is entirely disturbed or replaced by edema fluid, then
␥B ⬎ ␥A and ␥B being unchanged; 2) the surfactant layer keeps intact
or simply slightly agitated by edema fluid, then ␥B ⫽ ␥A and ␥B being
variable; and 3) the alveolus B is completely flooded and air-liquid
interface disappears, namely, ␥B ⫽ 0.
Next we will determine the dynamic value of surface tension ␥A in
the normal air-filled alveolus A during inflation. Schürch et al.
measured alveolar surface tension directly using test fluid droplets.
They reported that alveolar surface tension (␥) in the excised rat lungs
was about 29.7 mN/m at TLC and 7 mN/m at FRC. They also found
that at any given lung volume in the range between 70% and 40%
Thickness of alveolar walls. Tsunoda et al. (38a) measured average
thickness of alveolar walls in both air-filled and liquid-filled cat lungs
using light microscopy. A maximum wall thickness of approximately
10.7 ␮m was found in the collapsed lung at a given gas/tissue volume
ratio (Vr) of 0.6⬃0.7. Wall thickness diminished curvilinearly during
initial inflation, and then became nearly constant at about 4 ␮m close
to TLC (Vr about 8). Log-log plots of data points showed a power-law
function description between wall thickness (t) and lung volume ratio,
which has the form t ⫽ KVr⫺0.44.
We measured thickness of alveolar walls in rat from sections under
confocal microscopy at alveolar pressure Palv of 5, 15, and 25 cmH2O,
corresponding to 0%, 18%, and 19.8% increase in radius of alveolus
Ra (baseline Palv of 5 cmH2O), respectively (24, 49). These data was
fitted to the following exponential curve:
t ⫽ 13.66t0e⫺2.6145ε
(13)
where t0 is thickness of alveolar walls at Palv of 5 cmH2O (here taking
the typical values of t0 as 7 ␮m), and ε is the increase in alveolar
radius (%baseline).
Finally, substituting in Eq. 10 for ␴T, ␥A, and t from Eq. 11, Eq. 12,
and Eq. 13, respectively, and rewriting Eq. 10, yields the following
30
25
20
15
10
5
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
Fig. 4. The relationship between surface tension and percent increase in
alveolar radius (%baseline). The data (*) are from Schürch et al. (29, 30). Solid
curve is simulated surface tension.
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Fig. 3. Stress-strain curve of human alveolar walls. The data (⫹) are from
Sugihara et al. (36). Solid line has been fitted to the data.
(12)
A Model for Estimating Alveolar Wall Strain
relationship between stretch ratio ␭ and percent increase in alveolar
radius (ε):
4,256.6 ⫹ 3,876.6e2.9145␭ ⫺ 4,840e2.75␭ ⫽
⫺
␳␥B
Rb13.66t0e⫺2.6145ε
305.5ε2 ⫺ 548.1 ε ⫹250.5
13.66t0e⫺2.6145ε
•
589
Chen Z et al.
1.4
6000
1.38
5500
strain
1.36
(14)
5000
stress
4500
1.34
4000
1.32
RESULTS
3500
3000
1.28
2500
1.26
1.24
2000
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1500
1.18
Fig. 5. Stress-strain of alveolar septa on inflation when ␥B ⬎ ␥A. Y-axis on the
left side represents strain, and on the right side represents stress. ␭, stretch
ratio; ε, percent increase in alveolar radius; ␴T, internal stress in alveolar walls;
␥, surface tension coefficient.
of 31.4 ␮m at volume corresponding to ε ⫽ 1.12; then ␳ varies
inversely as ε increases up to near TLC. Moreover, the central
angle ␪ presents similar nonmonotonic changes, indicating
alveolar heterogeneous distension. It is worth noting that in this
situation interface radius ␳ (circle) is always less than alveolar
radius Ra (diamond) throughout alveolar inflation from FRC to
near TLC. This result is consistent with the illustration in Fig.
2, where ␳ ⬍ Ra.
2) In the second simulation, ␥B ⫽ ␥A with ␥B being variable.
In striking contrast to Fig. 5, in Fig. 8 both strain and stress are
monotonically increasing as ε varies from 1.00 to 1.18. Furthermore, the strain of alveolar walls is linearly related to the
percent increase in alveolar radius. The minimum strain and
stress are 1.066 and 0.12 kPa, respectively, both arising at
volume corresponding to FRC. The maximum values of 1.239
and 1.64 kPa, for strain and stress, respectively, occur not at
FRC but near TLC.
The air-liquid interface radius ␳, which is less than alveolar
radius Ra at FRC, gradually approaches Ra when volume
increases up to near TLC, as shown in Fig. 9. The changes in
central angle ␪ (square) also differ from those in the case of
␥B ⬎ ␥A, and ␪ appears to monotonically decrease from the
maximum 71° at FRC to the minimum 62°near TLC. Another
interesting finding shown in Fig. 9 is that the angle SOS=
(cross) remains constant at 60° during the whole alveolar
inflation. The above results congruously demonstrate that in
the case of ␥A ⫽ ␥B the alveoli show homogeneous distension.
3) In the third simulation, ␥B ⫽ 0, namely, the alveolus B is
completely flooded. In this case, Eq. 14 is not applicable to
compute alveolar wall strain because its premise ␥B/Rb ⬎ ␥A/␳
does not hold true any more. However, this situation might
well occur in reality, as shown in Fig. 6C. In fact, if small
airways were occluded by liquid bridges interspersed with
trapped gas, pressure in alveolar air Palv would be partly
dissipated over liquid bridges and trapped gas leading to PliqB
less than PliqA (3, 15). As a result, the septum sector SS= would
bulge into the flooded alveolus.
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Now we consider the simultaneous equations consisting of
Eqs. 2 and 14 with one independent variable ε ranging from
៮⬘. It is an
1.00 to 1.18 and three unknown variables ␳, ␭, and SS
underdetermined system. To estimate the dynamic change of ␳
with ε on alveolar inflation, we made the following assump៮⬘ was always greater than or
tions. First, the length of chord SS
equal to the alveolar radius Ra during inflation from FRC to
៮⬘ ⱖ Ra. Second, for each equilibrium state of the
TLC, i.e., SS
internal structure, tension within the septum SS= was mini៮⬘ being minimum. For each fixed value of ε
mum, namely, SS
within the interval [1.00,1.18], we calculated the air-liquid
interface radius ␳ through Eq. 2 and Eq. 14 with initial chord
៮⬘ ⫽ εRa0. If the resulting ␳ was not a real solution, the
length SS
calculation procedure was repeated with new chord length of
៮⬘ ⫽ SS
៮⬘ ⫹ ⌬ where ⌬ is a small increment, say ⌬ ⫽ 0.001,
SS
until the real solution ␳ was obtained. By trial and error, we can
determine the approximate values of unknown variables ␳ and
␭ within the allowable precision range.
We took radius of the alveolus A and radius of meniscus at
alveolar air pressures Palv of 5 cmH2O (near FRC) as Ra0 ⫽ 41
␮m and Rb0 ⫽32 ␮m, respectively. Increasing Palv from 5
cmH2O to 15 cmH2O (near TLC) increased both Ra and Rb by
about 18% (23). As discussed in Surface tension, we have
performed simulations in the following three situations:
1) In the first simulation, ␥B ⬎ ␥A with ␥B taking a constant
value of 36 mN/m for different surface areas on inflation (39, 45).
Solving the simultaneous Eq. 2 and Eq. 14 in Matlab (Mathworks,
Natick, MA) obtained the following results. As shown in Fig. 5,
even at FRC the presence of liquid-filled alveolus B preextends
the adjoining air-filled alveolus A with alveolar septum SS=
stretch ratio amounting to a maximum value of 1.374, corresponding to maximum stretch stress of 5.12 kPa. This prediction is in
agreement with findings by Perlman et al. (Fig. 6). Of note, on
alveoli inflation from Palv of 5 cmH2O to 15 cmH2O, the ratio ␭
doesn’t show monotonic changes; instead, it first gradually decreases to the minimum value of 1.264 (minimum stress 2.07 kPa)
where the percent increase of alveolar radius Ra corresponds to
about 1.15, then slightly increases with inflation reaching 1.266 at
volume near TLC.
It can be seen from Fig. 7 that as alveolar radius Ra
(diamond) gradually enlarges on inflation, the angle SOS=
(cross) subtended by alveolar septum SS= gradually decreases
from 78° at FRC to 60° at a point corresponding to ε ⫽ 1.13,
and thereafter, the angle SOS= remains unchanged until ε ⫽
1.18. Figure 7 also shows that as the shared septum sector
protrudes into the adjacent flooded alveolus B the air-liquid
interface radius ␳ (circle) slowly reduces to the minimum value
1.3
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A Model for Estimating Alveolar Wall Strain
•
Chen Z et al.
Fig. 6. High-power images of single alveolar
edema at transpulmonary pressures (Palv) of
5 cmH2O (A, B) and 15 cmH2O (C). Alveoli
are air-filled (Control) or liquid-filled with
albumin solution (Edema), as indicated.
White arrows indicate septum separating the
two alveoli. Yellow scalemarks indicate alveolar diameters at Palv of 5 cmH2O in
control condition. Aedem, cross-sectional area
of liquid-filled alveolus. Aadj, cross-sectional
area of air-filled alveolus. [Reprinted with
permission of the American Thoracic Society. Copyright © 2014 American Thoracic
Society (24).]
DISCUSSION
55
100
95
50
90
45
85
80
40
75
35
70
central angle
angel SOS’
interface radius
alveolar radius Ra
30
25
1
1.02
1.04
1.06
65
60
1.08
1.1
1.12
1.14
1.16
55
1.18
Fig. 7. Central angle ␪ (square), angle SOS= (cross), air-liquid interface radius
␳ (circle), and alveolar radius Ra (diamond) as a function of percent increase
in alveolar radius (ε) during inflation when ␥B ⬎ ␥A. Y-axis on the left side
represents radius and on the right side represents angle.
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In this paper we have presented a model for quantitatively
describing deformation of alveolar walls in pulmonary edema.
On the one hand, our model successfully reproduced the
experimental phenomenon in Ref. 24 that the air-filled alveolus
bulged into its neighboring liquid-filled alveolus even at FRC.
On the other hand, the model further indicated that, besides
variations in thickness and compositions of alveolar segments,
differences in air-liquid interface tension on alveolar segments
have a major impact on the pattern of alveolar segmental
distension. More specifically, if surface tension in liquid-filled
alveolus is much greater than in air-filled alveolus, then, as
shown in Figs. 5 and 7, alveolus expansion is heterogeneous:
for a pair of juxtaposed alveoli, at low alveolar pressure, the
septum shared by the liquid-filled alveolus is overdistended
with the maximum strain of 1.374, and angel SOS= opens up to
a maximum of 78°; on the contrary, as alveoli inflate to near
TLC, the strain of the shared septum decreases, and angel SOS=
narrows down. And if surface tension in two adjacent alveoli is
identical, then alveolus expansion shows homogeneous (Figs. 8
and 9): the strain of alveolar walls appears to be a linear
increase as alveolar volume varies from FRC to near TLC, and
angle SOS= remains unchanged at 60°.
According to observations made by Wilson and Bachofen, the
increase in angle SOS= is due to the two possible mechanisms: 1)
the stretching of the septal tissue; and 2) the unfolding of septal
pleating located in the corners of the alveoli (27, 47). The increase
in angle SOS’ suggests that tissues between neighboring alveoli
might produce shear stress owing mechanical interdependence
(20, 25). However, in the current tetrakaidecahedron model we
did not consider shear moduli of lung tissue. Kelvin’s tetrakaidecahedron including effects of stretching, bending and twisting
remains a future research direction (43).
Many randomized clinical trials have produced disparate
results in testing the ARDS network protocol limiting tidal
volume to 6 ml/kg predicted body weight and plateau pressure
to 30 cmH2O in ALI and ARDS (9, 22, 37), and overinflation
may occur even with normal tidal volumes (8). Therefore, our
present prediction may provide a possible explanation for these
discrepancies. If pulmonary edema in ARDS patients were
progressing in its early stage (12), normal aerated alveoli might
be surrounded by edematous alveoli to some extent, which
predisposed them to overdistension injury even at low or
normal tidal volumes, according to our calculation. In addition,
in the present model, we took the alveolus radius and meniscus
radius as 41 and 32 ␮m, respectively. It is obvious from Eq. 10
that the calculated maximum strain and stress would vary with
values of radii.
Of note, overdistension occurring in aerated alveoli is not
the only determinant of VILI. In fact, the extent of injury is
associated not only with the magnitude of strain, but also with
the rate of strain (38). Ventilation at low volume and pressure
may lead to repetitive reopening and closure of small airway
and alveoli, generating shear stress on the epithelial cells
(atelectrauma). In contrast to the alveolar epithelial cells,
pulmonary endothelial cells are more sensitive to the effects of
cyclic stretch, which is often exacerbated by mechanical ventilation. The fracture of liquid bridge and movement of airliquid interfaces with respiration are also likely to damage the
alveolar lining cells (40). Recently dos Santos and Slutsky (7)
mentioned a novel mechanism of lung injury termed biotrauma
that may occur even in the absence of overt structural damage.
Important as these mechanisms of injury are, they are not
included in the present model.
It should be recognized that there are several limitations in
the proposed model, the most notable being changes in the
thickness of alveolar walls. In Eqs. 13 and 14, we described
thickness of septum (t) as a function of percent change in
radius of alveolus (ε) rather than lung volume, as indicated by
Tsunoda et al. (38a). This relationship presupposes that all
A Model for Estimating Alveolar Wall Strain
1.26
1700
1.24
1500
1.22
strain
stress
1.2
1300
1.18
1100
1.16
900
1.14
1.12
700
1.1
500
1.08
300
1.06
1.04
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
100
1.18
septa comprising an alveolus experienced the same change in
thickness. However, this is not always the case. We ignored the
approximation error and made this transformation due mainly to
the following two reasons: 1) a direct comparison of linear
distension between alveolar septal segment and radius, and 2) the
fact that measurements by Perlman and Bhattacharya were about
the length of the perimeter segments and alveolar diameter.
Another limitation is that stress-strain properties of human
lung parenchyma were used to approximate those of alveolar
walls in rats. Considering inter- and intraspecies differences in
elastic properties of alveolar walls (21), this approximation
definitely introduced an error in calculated stress and strain;
however, this specific parameter selection would not weaken
the rationality and validity of our model as a whole in estimating lung tissue overdistension in pulmonary edema.
Positive end-expiratory pressure (PEEP) has been widely
used in mechanically ventilated patients with ALI and ARDS
to improve arterial oxygenation and prevent high shear stress
associated with cyclic opening and closing atelectatic alveoli.
Mechanical stimuli may be transformed by cells or tissues into
biochemical and biomolecular alterations termed mechanotransduction. Tschumperlin et al. (38) found that small cyclic
deformations superimposed on a tonic deformation significantly reduced injury of alveolar epithelial cells compared with
large-amplitude deformations with the same peak deformation.
Consequently, the mechanotransduction responses of lung tissues that are inflated from zero end-expiratory pressure (ZEEP)
to a given end-inspiratory pressure are quite different from
those of lung tissues that are prestressed by PEEP, then
expanded to the same end-inspiratory pressure. When increases
in mean airway pressure caused by applying PEEP or by
increasing VT are identical, the lungs ventilated under PEEP
developed less edema, indicating that large cyclic changes in
lung volume boost edema (8).
In the opposite aspect, our prediction supports the notion
that application of high PEEP to the patients with a focal
distribution of loss of aeration may increase the risk of alveolar
hyperinflation (22). Because the predicted maximum strain of
alveolar walls arises at low lung volume and transpulmonary
591
Chen Z et al.
pressure, it is very likely that the routine use PEEP of 5–12
cmH2O (4) just falls within the pressure range enough to
induce the large alveolar strain. When PEEP produces additional overdistension, conversely, there is greater edema (8). In
other words, any beneficial effect of PEEP may be offset by the
consequences of lung overdistension. Therefore, how to set the
optimal level of PEEP has been a subject of ongoing research
and debate. Grasso et al. (13) have applied in the clinical
setting the stress index strategy to titrate the optimal PEEP,
which reduced the risk of alveolar hyperinflation compared
with the ARDSnet strategy-guided ventilation.
Before getting a whole picture of the topographical distribution of parenchymal stress and strain, it is very important to
first measure elastic properties of lung parenchymal. Stamenovic and Smith (34) reported that for the normal air-filled
rabbit lungs, the ratios of bulk modulus to transpulmonary
pressure k/P and shear modulus to transpulmonary pressure
␮/P are 3 to 9 and 0.9 ⫾ 0.15, respectively, as volume changes
from 50 to 90% TLC. Furthermore, the bulk modulus changes
roughly exponentially with transpulmonary pressure, whereas
the shear modulus is nearly proportional to the transpulmonary
pressure over a wide range of volumes. Using a newly developed endoscopic system, Schwenninger et al. (32) measured
shear moduli in vivo in mechanically ventilated rats. The shear
modulus in healthy animals increased from 3.3 ⫾ 1.4 kPa at 15
cmH2O continuous positive-airway pressure (CPAP) to 5.8 ⫾
2.4 kPa at 30 cmH2O, whereas the shear modulus was 2.5 kPa
at all CPAP levels in the lung-injured animals.
Recently, McGee et al. (19) have assessed parenchymal
elasticity in normal and edematous, ventilator-injured lung by
virtue of magnetic resonance elastography (MRE). They found
that shear stiffness was equal to 1.00, 1.07, 1.16, and 1.26 kPa
for the injured and 1.31, 1.89, 2.41, and 2.93 kPa for normal
lungs at transpulmonary pressures of 3, 6, 9, and 12 cmH2O,
respectively. Their measurements are roughly consistent with
the above results. Therefore, MRE provides a method to
spatially resolve shear modulus of both normal and edematous
lungs; once the topographical distribution of elastic properties
of lung parenchyma is known, we could quantify parenchymal
72
55
central angle
angle SOS’
interface radius
alveolar radius Ra
50
70
68
45
66
64
40
62
35
60
30
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
58
1.18
Fig. 9. Central angle ␪ (square), angle SOS= (cross), air-liquid interface radius
␳(circle), and alveolar radius Ra (diamond) as a function of percent increase in
alveolar radius (ε) during inflation when ␥A ⫽ ␥B. Y-axis on the left side
represents radius, and on the right side represents angle.
J Appl Physiol • doi:10.1152/japplphysiol.00072.2014 • www.jappl.org
Downloaded from http://jap.physiology.org/ by 10.220.33.5 on June 18, 2017
Fig. 8. Stress-strain of alveolar septa on inflation when ␥A ⫽ ␥B. Y-axis on the left side
represents strain, on the right side represents stress. ␭, stretch ratio; ε, percent increase
in alveolar radius; ␴T, internal stress in alveolar walls; ␥, surface tension coefficient.
•
592
A Model for Estimating Alveolar Wall Strain
deformation using the present model, and further picture a map
of the topographical distribution of lung stress and strain to
avoid VILI as much as possible.
GRANTS
This research was supported by Shanghai Medical Instrumentation College
Grant No. E102001400131.
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
Author contributions: Z.-l.C. conception and design of research; Z.-l.C.
drafted manuscript; Z.-l.C. edited and revised manuscript; Y.-z.C. approved
final version of manuscript; Z.-y.H. analyzed data.
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