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

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. 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

http://www.jappl.org

A Model for Estimating Alveolar Wall Strain

•

587

Chen Z et al.

Fig. 1. Assemblage of tetrakaidecahedrons modeling the alveoli.

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=. ␭⫽

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

J Appl Physiol • doi:10.1152/japplphysiol.00072.2014 • www.jappl.org

588

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

1.2

1.25

1.3

1.35

1.4

1.45

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.

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%

(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 ␮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.

J Appl Physiol • doi:10.1152/japplphysiol.00072.2014 • www.jappl.org

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

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 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.

J Appl Physiol • doi:10.1152/japplphysiol.00072.2014 • www.jappl.org

590

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

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°. 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.

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

J Appl Physiol • doi:10.1152/japplphysiol.00072.2014 • www.jappl.org

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

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.

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 55

72 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

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. REFERENCES 1. Andreassen S, Steimle KL, Mogensen ML, Serna JB, Rees S, Karbing DS. The effect of tissue elastic properties and surfactant on alveolar stability. J Appl Physiol 109: 1369 –1377, 2010. 2. Bachofen H, Schurch S, Urbinelli M, Weibel ER. Relations among alveolar surface tension, surface area, volume, and recoil pressure. J Appl Physiol 62: 1878 –1887, 1987. 3. Bilek AM, Dee KC, Gaver DP. Mechanisms of surface-tension-induced epithelial cell damage in a model of pulmonary airway reopening. J Appl Physiol 94: 770 –783, 2003. 4. Brower RG, Lanken PN, MacIntyre N, Matthay MA, Morris A, Ancukiewicz M, Schoenfeld D, Thompson BT; National Heart, Lung, and Blood Institute ARDS Clinical Trials Network. Higher versus lower positive end-expiratory pressures in patients with the acute respiratory distress syndrome. N Engl J Med 351: 327–336, 2004. 5. Dale PJ, Matthews FL, Schroter RC. Finite element analysis of lung alveolus. J Biomech 13: 865–873, 1980. 6. Denny E, Schroter RC. A model of non-uniform lung parenchyma distortion. J Biomech 39: 652–663, 2006. 7. dos Santos CC, Slutsky AS. The contribution of biophysical lung injury to the development of biotrauma. Annu Rev Physiol 68: 585–618, 2006. 8. Dreyfuss D, Saumon G. Ventilator-induced lung injury: lessons from experimental studies. Am J Respir Crit Care Med 157: 294 –323,1998. 9. Eichacker PQ, Gerstenberger EP, Banks SM, Cui X, Natanson C. Metaanalysis of acute lung injury and acute respiratory distress syndrome trials testing low tidal volumes. Am J Respir Crit Care Med 166: 1510–1514, 2002. 10. Fung YC. A model of the lung structure and its validation. J Appl Physiol 64: 2132–2141, 1988. 11. Gattinoni L, Caironi P, Pelosi P, Goodman LR. What has computed tomography taught us about the acute respiratory distress syndrome? Am J Respir Crit Care Med 164: 1701–1711, 2001. 12. Gluecker T, Capasso P, Schnyder P, Gudinchet F, Schaller MD, Revelly JP, Chiolero R, Vock P, Wicky S. Clinical and radiologic features of pulmonary edema. Radiographics 19: 1507–1531, 1999. 13. Grasso S, Stripoli T, De Michele M, Bruno F, Moschetta M, Angelelli G, Munno I, Ruggiero V, Anaclerio R, Cafarelli A, Driessen B, Fiore T. ARDSnet ventilatory protocol and alveolar hyperinflation: role of positive endexpiratory pressure. Am J Respir Crit Care Med 176: 761–767,2007. 14. Hoppin FG, Hildebrandt J. Mechanical properties of the lung. Bioeng Aspects Lung 3: 83–162, 1977. 15. Hubmayr RD. Perspective on lung injury and recruitment: a skeptical look at the opening and collapse story. Am J Respir Crit Care Med 165: 1647–1653, 2002. 16. Lu JY, Distefano J, Philips K, Chen P, Neumann AW. Effect of the compression ratio on properties of lung surfactant (bovine lipid extract surfactant) films. Respir Physiol 115: 55–71, 1999. 17. Lumb AB. Pulmonary vascular disease. In: Nunn’s Applied Respiratory Physiology. Elsevier Health Sciences, 2012, chapt. 29, p. 419 – 425. 18. Martin-Lefevre L, Ricard JD, Roupie E, Dreyfuss D, Saumon G. Significance of the changes in the respiratory system pressure-volume curve during acute lung injury in rats. Am J Respir Crit Care Med 164: 627–632, 2001. 19. McGee KP, Mariappan YK, Hubmayr RD, Carter RE, Bao Z, Levin DL, Manduca A, Ehman RL. Magnetic resonance assessment of parenchymal elasticity in normal and edematous, ventilator-injured lung. J Appl Physiol 113: 666 –676, 2012.

•

Chen Z et al.

20. Mead J, Takishima T, Leith D. Stress distribution in lungs: a model of pulmonary elasticity. J Appl Physiol 28: 596 –608, 1970. 21. Mercer RR, Russell ML, Crapo JD. Alveolar septal structure in different species. J Appl Physiol 77: 1060 –1066, 1994. 22. Muscedere JG, Mullen JBM, Gan K, Bryan AC, Slutsky AS. Tidal ventilation at low airway pressures can augment lung injury. Am J Respir Crit Care Med 149: 1327–1334, 1994. 23. Perlman CE, Bhattacharya J. Alveolar expansion imaged by optical sectioning microscopy. J Appl Physiol 9: 1037–1044, 2007. 24. Perlman CE, Lederer DJ, Bhattacharya J. Micromechanics of alveolar edema. Am J Respir Cell Mol Biol 44: 34 –39, 2011. 25. Plataki M, Hubmayr RD. The physical basis of ventilator-induced lung injury. Expert Rev Respir Med 4: 373–385, 2010. 26. Protti A, Cressoni M, Santini A, Langer T, Mietto C, Febres D, Chierichetti M, Coppola S, Conte G, Gatti S, Leopardi O, Masson S, Lombardi L, Lazzerini M, Rampoldi E, Cadringher P, Gattinoni L. Lung stress and strain during mechanical ventilation: any safe threshold? Am J Respir Crit Care Med 183: 1354 –1362, 2011. 27. Roan E, Waters CM. What do we know about mechanical strain in lung alveoli? Am J Physiol Lung Cell Mol Physiol 301: L625–L635, 2011. 29. Schürch S. Surface tension at low lung volumes: dependence on time and alveolar size. Respir Physiol 48: 339 –355, 1982. 30. Schürch S, Goerke J, Clements JA. Direct determination of volume- and time-dependence of alveolar surface tension in excised lungs. Proc Natl Acad Sci USA 75: 3417–3421, 1978. 31. Schürch S, Goerke J, Clements JA. Direct determination of surface tension in the lung. Proc Natl Acad Sci USA 73: 4696 –4702, 1976. 32. Schwenninger D, Runck H, Schumann S, Haberstroh J, Guttmann J. Locally measured shear moduli of pulmonary tissue and global lung mechanics in mechanically ventilated rats. J Appl Physiol 113: 273–280, 2012. 33. Smith JC, Stamenovic D. Surface forces in lungs. I. Alveolar surface tension-lung volume relationships. J Appl Physiol 60: 1351–1350, 1986. 34. Stamenovic D, Smith JC. Surface forces in lungs. III. Alveolar surface tension and elastic properties of lung parenchyma. J Appl Physiol 60: 1358–1362, 1986. 35. Staub NC. Pulmonary edema. Physiol Rev 54: 678 –811, 1974. 36. Sugihara T, Martin CJ, Hilderbrandt J. Length-tension properties of alveolar wall in man. J Appl Physiol 30: 875–878, 1971. 36a.Tambe DT. Building a theoretical framework to quantify alveolar injury. J Appl Physiol; doi:10.1162/japplphysiol.00525.2013. 37. Terragni PP, Rosboch G, Tealdi A, Corno E, Menaldo E, Davini O, Gandini G, Herrmann P, Mascia L, Quintel M, Slutsky AS, Gattinoni L, Ranieri VM. Tidal hyperinflation during low tidal volume ventilation in acute respiratory distress syndrome. Am J Respir Crit Care Med 175: 160–166, 2007. 38. Tschumperlin DJ, Oswari J, Margulies AS. Deformation-induced injury of alveolar epithelial cells: effect of frequency, duration, and amplitude. Am J Respir Crit Care Med 162: 357–362, 2000. 38a.Tsunoda S, Fukaya H, Sugihara T, Martin CJ, Hildebrandt J. Lung volume, thickness of alveolar walls, and microscopic anisotropy of expansion. Respir Physiol 22: 285–296, 1974. 39. Verbrugge SJ, Lachmann B, Kesecioglu J. Lung protective ventilatory strategies in acute lung injury and acute respiratory distress syndrome: from experimental findings to clinical application. Clin Physiol Funct Imaging 27: 67–90, 2007. 40. Vlahakis NE, Hubmayr RD. Cellular stress failure in ventilator-injured lungs. Am J Respir Crit Care Med 171: 1328, 2005. 41. Ware LB, Matthay MA. The acute respiratory distress syndrome. N Engl J Med 342: 1334 –1349, 2000. 42. Ware LB, Matthay MA. Acute pulmonary edema. N Engl J Med 353: 2788 –2796, 2005. 43. Warren WE, Kraynik AM. Linear elastic behavior of a low-density Kelvin foam with open cells. J Appl Mech 64: 787–794, 1997. 44. Wheeler AP, Bernard GR. Acute lung injury and the acute respiratory distress syndrome: a clinical review. Lancet 369: 1553–1564, 2007. 45. Wilson TA, Anafi RC, Hubmayr RD. Mechanics of edematous lungs. J Appl Physiol 90: 2088 –2093, 2001. 46. Wilson TA. Surface tension-surface area curves calculated from pressurevolume loops. J Appl Physiol 53: 1512–1520, 1982. 47. Wilson TA, Bachofen H. A model for mechanical structure of the alveolar duct. J Appl Physiol 52: 1064 –1070,1982. 48. Wilson TA. The relations among recoil pressure, surface area and surface tension in the lung. J Appl Physiol 50: 921–926, 1981. 49. Wu Y, Perlman CE. In situ methods for assessing alveolar mechanics. J Appl Physiol 112: 519 –526, 2012.

J Appl Physiol • doi:10.1152/japplphysiol.00072.2014 • www.jappl.org