## Abstract

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 length-tension 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.

- 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.

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 P_{alv}, 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 liquid-filled 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 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, P_{alv} is pressure in the alveolar air. P_{liqA} and P_{liqB} are liquid pressures in the alveolus A and B, respectively. The radius *R*_{a} of alveoli is defined as the circumradius of the regular hexagon. *R*_{b} represents radius of meniscus developed in the edematous alveolus B. As Perlman et al. observed, the intervening septum SS′ was deformed when alveolar pressure P_{alv} 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 P_{alv} increases, the alveolar wall is subjected to the uniform and spherical stretch with segment SS′ from the initial reference length *L*_{0} (P_{alv} = 5 cmH_{2}O) to current length *L*. Here, we introduce the stretch ratio λ stated in *Eq. 1* to quantitatively describe the extent of stretching of septum SS′.
(1)

As shown in Fig. 2, the initial length *L*_{0} is defined to equal alveolar radius *R*_{a0} when P_{alv} is 5 cmH_{2}O, nearly corresponding to alveolar volume at FRC. It is worth noting that in reality the length of septum SS′ is greater than *R*_{a0} 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
(2)

where represents the length of the chord , ρ is the air-liquid interface radius, and θ is the central angle subtended by circular arc . For septum SS′, the pressure equilibrium has the form (3)

where P_{E} 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 σ_{T} within the spherical membrane and pressure difference P_{T} acting across the membrane (thickness *t*) have the following relationship
(4)

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
(5)
(6)

where P_{γA} and P_{γB} are pressure produced by the curved air-liquid interfaces. They can be calculated using the Laplace formula stated in *Eq. 7* and *Eq. 8*
(7)
(8)

where γ_{A} and γ_{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
(9)

*Equation 9* implies that only when the condition of γ_{B}/*R*_{b} > γ_{A}/ρ is satisfied will the septum SS′ bulge into the liquid-filled alveolus. Conversely, if γ_{B}/*R*_{b} < γ_{A}/ρ, then the septum SS′ will bulge into the air-filled alveolus. That is to say, in the edematous lungs, septal deformation not only depends on tissue elastic properties but also on surface tension as well as on alveolar size.

Given that recoil pressure P_{E} equals membrane pressure difference P_{T} in magnitude, combining *Eqs. 3* and *4* with *Eq. 9* we obtain
(10)

If stress-strain curve and thickness of the alveolar wall, surface tension coefficient γ_{A} and γ_{B} are known, then we can approximately estimate the change of radius ρ as meniscus radius *R*_{b} 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*
(11)

The unit for σ_{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 γ_{B} in the edematous alveolus in three situations where *1*) the surfactant layer is entirely disturbed or replaced by edema fluid, then γ_{B} > γ_{A} and γ_{B} being unchanged; *2*) the surfactant layer keeps intact or simply slightly agitated by edema fluid, then γ_{B} = γ_{A} and γ_{B} being variable; and *3*) the alveolus B is completely flooded and air-liquid interface disappears, namely, γ_{B} = 0.

Next we will determine the dynamic value of surface tension γ_{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 (γ) 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% 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 γ_{max} = 28.1 mN/m and γ_{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 ∝ area^{1/2} ∝ volume^{1/3}, we further fitted data of Schürch et al. as shown in Fig. 4. The relationship between surface tension (γ_{A}) and percent increase in alveolar radius (ε) (%baseline) is given by *Eq. 12*.
(12)

#### 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 μm was found in the collapsed lung at a given gas/tissue volume ratio (V_{r}) of 0.6∼0.7. Wall thickness diminished curvilinearly during initial inflation, and then became nearly constant at about 4 μm close to TLC (V_{r} 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* = *K*V_{r}^{−0.44}.

We measured thickness of alveolar walls in rat from sections under confocal microscopy at alveolar pressure P_{alv} of 5, 15, and 25 cmH_{2}O, corresponding to 0%, 18%, and 19.8% increase in radius of alveolus *R*_{a} (baseline P_{alv} of 5 cmH_{2}O), respectively (24, 49). These data was fitted to the following exponential curve:
(13)where *t*_{0} is thickness of alveolar walls at P_{alv} of 5 cmH_{2}O (here taking the typical values of *t*_{0} as 7 μm), and ε is the increase in alveolar radius (%baseline).

Finally, substituting in *Eq. 10* for σ_{T}, γ_{A}, and *t* from *Eq. 11, Eq. 12*, and *Eq. 13*, respectively, and rewriting *Eq. 10*, yields the following relationship between stretch ratio λ and percent increase in alveolar radius (ε):
(14)

## RESULTS

Now we consider the simultaneous equations consisting of *Eqs. 2* and *14* with one independent variable ε ranging from 1.00 to 1.18 and three unknown variables ρ, λ, and . It is an underdetermined system. To estimate the dynamic change of ρ with ε on alveolar inflation, we made the following assumptions. First, the length of chord was always greater than or equal to the alveolar radius *R*_{a} during inflation from FRC to TLC, i.e., ≥ *R*_{a}. Second, for each equilibrium state of the internal structure, tension within the septum SS′ was minimum, namely, being minimum. For each fixed value of ε within the interval [1.00,1.18], we calculated the air-liquid interface radius ρ through *Eq. 2* and *Eq. 14* with initial chord length = ε*R*_{a0}. If the resulting ρ was not a real solution, the calculation procedure was repeated with new chord length of = + Δ where Δ is a small increment, say Δ = 0.001, until the real solution ρ was obtained. By trial and error, we can determine the approximate values of unknown variables ρ and λ within the allowable precision range.

We took radius of the alveolus A and radius of meniscus at alveolar air pressures P_{alv} of 5 cmH_{2}O (near FRC) as *R*_{a0} = 41 μm and *R*_{b0} =32 μm, respectively. Increasing P_{alv} from 5 cmH_{2}O to 15 cmH_{2}O (near TLC) increased both *R*_{a} and *R*_{b} by about 18% (23). As discussed in *Surface tension*, we have performed simulations in the following three situations:

*1*) In the first simulation, *γ*_{B} *> γ*_{A} with *γ*_{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 P_{alv} of 5 cmH_{2}O to 15 cmH_{2}O, the ratio λ 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 *R*_{a} 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 *R*_{a} (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 ρ (circle) slowly reduces to the minimum value of 31.4 μm at volume corresponding to ε = 1.12; then ρ varies inversely as ε increases up to near TLC. Moreover, the central angle θ presents similar nonmonotonic changes, indicating alveolar heterogeneous distension. It is worth noting that in this situation interface radius ρ (circle) is always less than alveolar radius *R*_{a} (diamond) throughout alveolar inflation from FRC to near TLC. This result is consistent with the illustration in Fig. 2, where ρ < *R*_{a}.

*2*) In the second simulation, γ_{B} = γ_{A} with γ_{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 ρ, which is less than alveolar radius *R*_{a} at FRC, gradually approaches *R*_{a} when volume increases up to near TLC, as shown in Fig. 9. The changes in central angle θ (square) also differ from those in the case of γ_{B} > γ_{A}, and θ 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 γ_{A} = γ_{B} the alveoli show homogeneous distension.

*3)* In the third simulation, γ_{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 γ_{B}/*R*_{b} > γ_{A}/ρ does not hold true any more. However, this situation might well occur in reality, as shown in Fig. 6*C*. In fact, if small airways were occluded by liquid bridges interspersed with trapped gas, pressure in alveolar air P_{alv} would be partly dissipated over liquid bridges and trapped gas leading to P_{liqB} less than P_{liqA} (3, 15). As a result, the septum sector SS′ would bulge into the flooded alveolus.

## DISCUSSION

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 cmH_{2}O 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 μm, 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 air-liquid 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 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 V_{T} 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 pressure, it is very likely that the routine use PEEP of 5–12 cmH_{2}O (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 μ/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 cmH_{2}O continuous positive-airway pressure (CPAP) to 5.8 ± 2.4 kPa at 30 cmH_{2}O, 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 cmH_{2}O, 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 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.

- Copyright © 2014 the American Physiological Society