J Appl Physiol 99: 458-471, 2005.
First published March 31, 2005; doi:10.1152/japplphysiol.00796.2004
8750-7587/05 $8.00
Epithelial cell deformation during surfactant-mediated airway reopening: a theoretical model
Shailesh Naire1 and
Oliver E. Jensen2
1Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh; and 2Centre for Mathematical Medicine, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, United Kingdom
Submitted 27 July 2004
; accepted in final form 24 March 2005
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ABSTRACT
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A theoretical model is presented describing the reopening by an advancing air bubble of an initially liquid-filled collapsed airway lined with deformable epithelial cells. The model integrates descriptions of flow-structure interaction (accounting for nonlinear deformation of the airway wall and viscous resistance of the airway liquid flow), surfactant transport around the bubble tip (incorporating physicochemical parameters appropriate for Infasurf), and cell deformation (due to stretching of the airway wall and airway liquid flows). It is shown how the pressure required to drive a bubble into a flooded airway, peeling apart the wet airway walls, can be reduced substantially by surfactant, although the effectiveness of Infasurf is limited by slow adsorption at high concentrations. The model demonstrates how the addition of surfactant can lead to the spontaneous reopening of a collapsed airway, depending on the degree of initial airway collapse. The effective elastic modulus of the epithelial layer is shown to be a key determinant of the relative magnitude of strains generated by flow-induced shear stresses and by airway wall stretch. The model also shows how epithelial-layer compressibility can mediate strains arising from flow-induced normal stresses and stress gradients.
recruitment; atelectrauma; volutrauma; fluid-structure interaction; surface tension
SUCCESSFUL RECRUITMENT OF liquid-filled airways is a process of fundamental importance in human respiration, from the first breath onward, and is critical in the treatment of conditions such as infant or acute respiratory distress syndrome (ARDS). It is of particular current interest in the context of ARDS, where atelectrauma [damage to airway epithelium through the repeated opening and closing of airways and alveoli (9, 13, 43)], volutrauma [airway overdistension (56, 58)], and biotrauma (inflammatory airway insult secondary to mechanical injury) are candidate mechanisms of ventilator-induced lung injury (49, 53). Mechanisms of atelectrauma, for example, involve processes interacting over widely varying length scales: at the scale of the whole lung (involving forced inflation of a heterogeneous airway network); at the scale of an individual airway (where surface tension, airway compliance, and airway liquid viscosity are significant); at the scale of an airway epithelial cell (deforming in response to its local mechanical environment); and at the scale of individual molecules (for example epithelial-cell mechanosensors such as stretch-activated ion channels). Mathematical and computational models provide powerful tools with which to integrate descriptions of processes operating across such disparate scales. This paper seeks to contribute to the development of a theoretical framework for airway recruitment and specifically aims to provide insights into the mechanism of epithelial cell deformation during surfactant-mediated airway reopening, a process of relevance to atelectrauma and volutrauma.
We wish to develop an improved theoretical description of the inflation of an individual airway, building on Macklem et al.'s (38) description of the reopening process as the peeling apart of the walls of a collapsed liquid-lined airway. Through a series of bench-top experiments (16, 47, 48), animal studies (45, 62), and theoretical models (15, 33, 44, 61), Gaver and coworkers have made significant advances in this direction by investigating the mechanism whereby a bubble of air propagates into a flexible-walled airway, modeled physically as a compliant liquid-filled channel or tube, when viscous forces and surface tension initially hold the walls of the airway in apposition. Experiments and theoretical models involving bubble motion in a flexible channel or tube [in both 2 and 3 dimensions (26)] demonstrate that a critical pressure must be exceeded for the bubble to advance steadily along an initially liquid-filled tube. As the tube walls are peeled apart, with the tube under large longitudinal tension, the largest stresses on the tube wall can be transient normal (rather than shear) stresses (15, 33). Parallel studies have considered an alternative mode of reopening, following occlusion of the airway by a short liquid plug (22, 46). Displacement of the plug by an imposed pressure gradient can cause the moving plug to deposit fluid on the airway wall and thereby shrink in volume (28, 60) until it either ruptures or forms a foam-like lamella. The relative importance of these different modes of airway recruitment is an area of current debate (30, 50).
Surfactant is important clinically in facilitating airway recruitment, particularly during surfactant replacement therapy for infant respiratory distress syndrome (20); likewise, surfactant dysfunction in ARDS can induce lung injury (54, 55). However, the biomechanical function of surfactant at the airway level is not fully understood. The one previous theoretical study addressing the role of surfactant in a deformable reopening airway predicted only modest reductions in reopening pressures (61), partly because the technical challenges of coupling descriptions of surfactant physical chemistry, complex flow fields, and deforming interfaces necessitated some restrictive modelling assumptions. We seek to relax these assumptions here. We will also exploit significant advances that have recently been made in assessing the effects of artificial lung surfactants on bubble motion in rigid tubes or channels (17–19). These experimental and theoretical studies have established relationships between surfactant properties (bulk and surface diffusion, adsorption and desorption rates, surface-tension reduction) and physical flow parameters (such as the thickness of the film deposited behind an advancing bubble and the pressure drop required to displace a bubble along the tube). Complementing these studies are the first in vitro experiments directly relating epithelial cell damage to the passage of a meniscus along a rigid channel (4, 35). These studies employed a parallel-plate flow chamber, on one wall of which was a layer of cultured epithelial cells. Cell injury was shown to be more prominent at low bubble speeds (an effect that correlated with pressure gradient, not exposure duration), and injury was reduced in the presence of surfactant. Because these studies took no account of wall flexibility, they did not allow either for cell-substrate stretching or for the large normal stresses that may arise when a deformable airway is peeled open.
The present study has two specific objectives. First, we will present a new theoretical model for the reopening of a pulmonary airway that integrates key findings from a computational model of surfactant transport around a bubble in a rigid tube (17, 18) into an existing theoretical model for bubble propagation in a flexible tube (33) in a systematic but tractable fashion. The resulting model predicts substantial reductions in airway reopening pressures for a physiologically relevant surfactant (Infasurf) and clarifies some of the factors (such as the degree of initial airway collapse) that enable this to occur. Second, complementing experimental studies (4, 35), we will extend our theoretical model to predict the likely time- and space-dependent strains experienced by epithelial cells lining a deformable airway as it is inflated by an advancing bubble. This allows us to assess the relative effects of airway stretch (potentially leading to volutrauma) and flow-induced stresses (potentially leading to atelectrauma) and to assess the roles of surfactant and cell compressibility in mediating the resulting cellular strains.
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METHODS
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We consider the physical model illustrated in Fig. 1A, which is a representation of an idealized airway. A long air bubble is blown at prescribed pressure (pb) into a flexible tube that is collapsed and liquid-filled at one end. We assume that the bubble advances at a steady speed (U), leaving a thin film of liquid on the wall of the inflated section of tube. The tube in the physical model is assumed to remain axisymmetric, whereas a real airway may buckle into a nonaxisymmetric configuration when compressed (36). We bypass such geometric complications for the present by regarding physical quantities {such as the tube radius [R(x)], where x measures distance along the axis of the tube} as representative of the mean airway radius averaged around the tube cross section. [We shall account explicitly below for wall buckling by assuming that the airway perimeter is unchanged when the airway is under compression, although we recognize that, in the interests of obtaining a tractable model, other aspects of buckling must be neglected. However, previous studies of airway reopening that accounted explicitly for buckling (26) show that cross-sectionally averaged models capture well the dominant features of reopening.] The transmural pressure (p) (the liquid pressure at the tube wall minus that outside) is assumed to depend on R(x) through
 | (1) |
where E is a stiffness parameter, R1 is the radius of the unstressed tube, and T is the longitudinal tension in the tube wall. The nonlinear function F(z) = z5 – z–0.8 (see Fig. 1B) is a "tube law" that represents the overall mechanical properties of the airway wall and accounts for variations in airway compliance with radius and particularly for airway stiffening on inflation (34). In the final term in Eq. 1, representing longitudinal tension, the subscript x denotes a derivative. This relatively crude representation of three-dimensional airway mechanics is ad hoc, but its effectiveness in the present context is supported by good qualitative agreement between previous two-dimensional (15) and three-dimensional (26) airway-reopening studies. For simplicity, we consider here a single airway and ignore geometric effects such as bifurcations and distal gas trapping.
We work in the frame of reference of the advancing bubble tip, which we assume lies at x = 0, so that the airway wall moves past it with speed U in the –x direction. Far ahead of the bubble, the tube is collapsed with R = R0 < R1 and the liquid is stationary with respect to tube walls; far behind the bubble tip, the airway is inflated and the liquid is again assumed stationary with respect to the tube walls. The degree of initial tube collapse is measured by the parameter 
= R0/R1. The bulk surfactant concentration far ahead of the bubble is assumed to have a uniform steady value C0. In the absence of surfactant, the air-liquid interface has a surface tension 
clean. The surface tension of the interface, if it is allowed to come into equilibrium with C0, is denoted eq (
clean). When C0 exceeds a critical bulk concentration CCBC, eq takes its minimum value sat. We will make use of the relationship (given in Ref. 19 and illustrated in Fig. 2) between eq and C0 for Infasurf, a natural surfactant containing surfactant-associated proteins B and C.
We wish to determine the relationship between pb and U in terms of the airway's mechanical properties (represented by E, T, and F in Eq. 1), fluid properties (represented by a viscosity µ) and the physicochemical properties of the surfactant. Our modeling framework (described in the APPENDIX) rests on the assumption that airway wall slopes are uniformly small, i.e., |Rx| << 1, and allows us to exploit the recent experimental and theoretical results of Ghadiali and Gaver (17–19) for the motion of a bubble in a rigid tube in the presence of a range of surfactants, including Infasurf. The complex surfactant properties may be distilled into the dependence of two functions on C0: the thickness 
RT of the film deposited on the tube wall behind the advancing bubble, where RT [equal to R(0); see Fig. 1A] is the tube radius at the bubble tip; and the corresponding pressure drop eq
/RT across the bubble tip. This pressure drop has two dominant contributions: one arises from the capillary pressure drop across a curved interface (as described by the Young-Laplace equation) and is present when the bubble is static; the other is associated with flow-induced pressure variations in the neighborhood of the bubble tip and arises only when the bubble is moving. Both the static and dynamic contributions are affected by the presence of surfactant in complex ways. By reducing surface tension, surfactant can reduce the static capillary pressure drop across a curved interface. However, by adsorbing nonuniformly onto the air-liquid interface, surfactant generates gradients of surface tension. These interfacial forces are transmitted to the bulk liquid by viscosity, generating so-called "Marangoni" flows. Because the surfactant distribution is coupled to the bulk flow through a range of physical effects, the net effects of the surfactant are governed by multiple parameters and are hard to predict without detailed calculations. For example, under certain circumstances, the net effect of Marangoni flows is to rigidify the air-liquid interface, which increases the dynamic contribution to and thickens the deposited film. Under alternative conditions, Marangoni flows can drive flows that thin the deposited film (18). These effects are illustrated for Infasurf in Fig. 3, A and B, which shows the dependence of and on C0 at three fixed bubble speeds (see the curves marked 1–3). There is a clear transition in behavior as C0 passes through the critical bulk concentration. In Fig. 3, C and D, we plot the corresponding dependence of and on the dimensionless speed parameter Ub = µU/eq (a capillary number, representing the relative strength of viscous to surface tension forces). Here, the relationship shown in Fig. 2 has been used to determine eq for increasing C0 over the range of concentrations shown in Fig. 3, A and B. Because eq is constant for C0 > CCBC (see Fig. 2), the three curves at constant U in Fig. 3, C and D, terminate in vertical segments along which C0 increases. For later reference, the solid continuous curves in Fig. 3, C and D (using data from Ref. 18), illustrate the dependence of and on Ub arising when surface tension is assumed to be uniform, as it would be for a pure liquid. Figure 3D, for example, shows how surface tension gradient-driven (Marangoni) flows create additional viscous dissipation that causes the Infasurf bubble-tip pressure drop to be greater than it would be were surface tension uniform along the bubble interface. Because Infasurf adsorbs slowly, at high bubble speeds the bulk and surface concentrations do not come into equilibrium at high bulk concentration; the nonlinear Langmuir equation of state used to relate the dynamic surface tension to the subsurface concentration (17, 18) allows Marangoni flows to persist for large C0. Notice also that the deposited film thickness is elevated at low speeds but reduced at high speeds (Fig. 3C) relative to the uniform surface-tension case.
Once the relation between pb and U has been determined, our model allows us to predict the corresponding shear and normal stresses (
s and n, respectively; see Fig. 1A) exerted by the fluid on the tube wall and the shape of the deformed tube R(x). We then suppose that lining the idealized airway in Fig. 1A is a thin layer of epithelial cells. By treating the cell layer as a uniformly thin homogeneous linearly elastic medium, having effective Young's modulus G, Poisson ratio 
P, and undisturbed thickness He, we use predictions of s, n, and R(x) to predict likely strains experienced by the cell layer during an airway reopening event. Our approximation exploits the fact that the airway wall is a composite structure, with a relatively stiff outer layer (including the basement membrane) that accommodates the fluid loading (modeled through the tube law, Eq. 1) and that is assumed to inhibit significant axial wall motion, and a much softer thinner lining (the epithelium), which is mechanically passive but nevertheless deforms in response to the forces imposed on it.
In the APPENDIX, we explain how we integrate models for the flow, surfactant, and cell layer. To describe the interaction between the flow of airway liquid, deformation of the tube wall and motion of the bubble, we derive a third-order nonlinear ordinary differential equation for the tube radius R(x) in the region ahead of the advancing bubble, supplemented by four boundary conditions (see Eqs. A2–A4). The reduction of the problem to such a simple mathematical form relies on the assumption that the wall tension is large compared with surface tension, an assumption that is likely to be increasingly justified in the presence of surfactant. The boundary conditions in Eq. A4 require us to include a functional relationship between , , and Ub; the relationships shown in Figs. 2 and 3 allow us to include data specific to Infasurf (Fig. 3, C and D). This approach combines for the first time two existing and well-established models, one for the flexible-tube flow (33) and one for the surfactant (18). In the APPENDIX, we also outline briefly a new approximate continuum model for the epithelial layer that captures deformations due to flow-induced forcing acting on the apical cell surface and substrate stretch acting on the basal surface. This model, described in detail in the online supplementary material (at http://jap.physiology.org/cgi/content/full/00796.2004/DC1), enables us to predict the three dominant strains experienced by the epithelial layer, the azimuthal strain 
, the radial strain rr, and the shear strain rx (illustrated in Fig. 7A below).
Baseline parameter values chosen to simulate a peripheral airway are listed in Table 1. We focus on a peripheral airway of radius 2 mm, collapsed initially to a mean radius of 1 mm, containing liquid of viscosity comparable to that of water. The airway model is governed by two dimensionless parameters, Ea = ER0/eq and Ta = T/eq, measuring the stiffness and tension of the tube wall relative to surface tension forces. In the absence of surfactant, when eq = clean, we take Ea = 1 and Ta = 100, values that enable us clearly to illustrate possible modes of behavior, given the limited available surfactant data (Fig. 3). The corresponding value of E (0.7 cmH2O) is low compared with values in the literature for dog (10) and rat (51, 52) airways, from which we estimate E 
6 and 10 cmH2O, respectively; reliable estimates for T are harder to determine, requiring us to exercise caution in interpreting results. The cell layer is substantially thinner than the airway radius (we assume the epithelial thickness Hc is of the order of 1 µm), so that terms in the cell model of magnitude (where 
= Hc/R1) or smaller are neglected. Cell strains resulting from flow-induced stresses scale with the parameter 1/Gc, where Gc = GR1/eq. There is considerable variation in estimates of G, depending for example on the strength of the applied force and the measurement technique. Estimates from experiments on cultured airway and alveolar epithelial cells using magnetic twisting cytometry give G between 300 (3) and 2,000 dyn/cm2 (37) for example; measurements with atomic force microscopy give Gc 104 dyn/cm2 or higher at high frequencies (1). In our simulations, we take G =50 (when eq = sat), corresponding to G = 6,850 dyn/cm2, to stay within the small-strain assumption of the model. [This high value of G is still substantially smaller than the effective Young's modulus of the stiff component of airway wall, Ew say, for the following reason: treating the wall as a cylindrical shell of thickness Hw, where Hw << R1, implies that Ew is a factor (R1/Hw)3 greater than E (36); taking R1/Hw = 10 then ensures that Ew is 
100 times greater than G.] We anticipate that larger strains are likely, although strain-hardening effects may protect the cell from large deformations. We also explore the role of cell compressibility by taking the Poisson ratio to be either P = 0.5 or P = 0.25.
Background.
To put later results into context, we first briefly review simulated airway reopening using our theoretical model of the physical system illustrated in Fig. 1A, assuming that surface tension is uniform along the air-liquid interface.
Reopening in the absence of surfactant: pushing and peeling.
The curve marked eq = clean in Fig. 4A plots the predicted bubble pressure Pb = pb/(clean/R1) vs. bubble speed U/(clean/µ) assuming that the interface is free of any surfactant. Here, Pb measures the bubble pressure (relative to that outside the tube) scaled with respect to a fixed reference pressure clean/R1, equivalent to one-half of the capillary pressure drop across a static meniscus in the undeformed tube. This pressure scale has a magnitude of 350 dyn/cm2 0.35 cmH2O using baseline parameters in Table 1; the corresponding velocity scale is large, so that µU/clean = 0.1 when U = 7 m/s. In this simulation, = R0/R1 = 0.5, implying that the tube is collapsed at its downstream end under transmural pressure of –1.7E. As shown previously (15, 26, 33), two types of behavior for a steadily propagating bubble are represented by the two branches of the solid pressure-speed curve either side of the minimum (which is marked with a bullet). For Pb in excess of a critical value Pbcrit (3.2), the bubble can advance either in a steady peeling motion, for which Pb increases with U (the curve to the right of the minimum continues to rise monotonically for µU/clean > 0.35) or a steady pushing motion, for which Pb increases with decreasing U. The insets illustrate schematically the difference between the two types of behavior of this nonlinear system: in peeling motion, the bubble tip is pointed and the membrane has a sharp bend ahead of the bubble tip; in pushing motion, the bubble acts like a piston and the tube is expanded ahead of the bubble tip. (In peeling motion, the liquid pressure is very low near the bend in the membrane. This low pressure provides the adhesive force that holds the tube walls together; the associated pressure gradient drives viscous flows within the tube.) For a bubble advancing with Pb prescribed, provided Pb > Pbcrit, the peeling solution has been shown to be stable to time-dependent disturbances both theoretically (23, 44) and experimentally (16, 47, 48); steady pushing motion, in contrast, is unstable to perturbations at long times if either the bubble pressure or the bubble volume flux is prescribed (23) and is therefore not accessible experimentally, although unsteady pushing motion may occur transiently at low Pb and low U. In summary, with pb prescribed, steady reopening is predicted to occur only for sufficiently large pb and it occurs preferentially at high bubble speeds in peeling mode; unsteady pushing motion may occur at low pressures and speeds (23), but this falls outside the framework of the present (steady) model.
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Fig. 4. A: pbR1/clean = Pb, where pb is prescribed bubble pressure, is plotted against µU/clean for steady airway reopening. The long solid curve represents a clean interface [with eq = clean, stiffness parameter (Ea) = 1, longitudinal tension (Ta) = 100, = 0.5]; the dashed curve represents the equivalent case with uniformly lower surface tension (eq = sat, Ea = 2.5, Ta = 250). Bullets denote the minimum of each curve, marking the transition between pushing and peeling behavior, illustrated schematically in insets, which show the air-liquid interface and the tube wall shape. The vertical lines (marked 1–3) show the effect of increasing C0 for three fixed values of U. B: pbR1/clean is plotted against C0/CCBC, corresponding to cases 1–3 in A; labels A, B, and C identify low or high values of C0.
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Effect of a uniform reduction in surface tension.
Also shown in Fig. 4A is the curve marked eq = sat, which shows the pb vs. U relationship obtained by assuming that surface tension has been uniformly reduced to sat (which is effectively equivalent to replacing a surfactant-free liquid with a different pure liquid having a lower surface tension). Changing the surface tension in this manner changes the dimensionless wall parameters used in the model characterizing the mechanical properties of the wall; thus Ea and Ta are increased by a factor 1/
( = sat/clean 0.4 for Infasurf) to account for the relative stiffening of the tube wall relative to surface tension forces. In peeling motion, for which much of the tube is collapsed and therefore under a negative transmural pressure, elastic restoring forces in the tube wall can do work on the liquid inside the tube as the bubble advances, so that stiffening the tube promotes bubble motion, an effect previously identified in Ref. 26. This effect, combined with the net reduction in surface-tension forces, leads to an overall reduction in the pressure required to inflate the tube. The 60% reduction in surface tension in going from clean to sat leads in the example shown in Fig. 4A to an 80% reduction in Pbcrit, and also lowers the minimum speed for which peeling can occur. Overall, a uniform reduction in surface tension increases the range of pressures and speeds over which stable peeling reopening can occur.
We can regard the lower curve in Fig. 4A as representative of a "perfect" surfactant, i.e., one that is able to reduce surface tension without inducing any surface-tension gradients. For real surfactants, of course, Marangoni flows cannot be avoided. We now demonstrate the unavoidable effect of such flows by introducing spatially variable surface tension to the problem.
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RESULTS
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Reopening in the presence of surfactant.
The data reproduced in Fig. 3 allow us to simulate reopening at fixed bubble speed U for increasing values of the far-field bulk surfactant concentration C0 for a fixed surfactant species (Infasurf). Increasing C0 causes a reduction in eq (Fig. 2). This is therefore equivalent to an increase in the dimensionless speed parameter Ub = µU/eq, implying that viscous forces become increasingly important relative to capillary forces. We have used our model to compute the corresponding change in Pb = pbR1/clean: this is shown on Fig. 4A with vertical lines for three different values of µU/clean (corresponding to cases 1–3 of Fig. 3). In cases 2 and 3, moving from points A to B, Pb falls monotonically as C0 increases, as shown in Fig. 4B. For comparison, we may compare the large-C0 limit of these curves (points B of cases 2 and 3) to the case of a "perfect" surfactant for which the surface tension is uniformly lowered to eq = sat (Fig, 4A). Even for very large C0, the bubble pressure in the presence of Infasurf is higher than that for a perfect surfactant (reflecting the behavior of the function in Fig. 3D) because of nonequilibrium normal stresses (arising for example from adsorption of surfactant that is insufficient to reduce surface tension to equilibrium levels). In contrast, Fig. 4B shows that at low speed (case 1 of Fig. 4A) the pressure drop Pb falls significantly as C0 increases (along segment AB) before rising for very large C0 (segment BC). This nonmonotonic behavior can be attributed to the effect of surfactant on the deposited film thickness (Fig. 3C, case 1): for low Ub, with reopening in pushing mode, Pb is proportional to 1/ (15), and so Marangoni effects that increase can cause a corresponding fall in Pb below the curve eq = sat (Fig. 4A, case 1). As C0 increases further, however, the bubble speed is low enough for the bulk surfactant to come into equilibrium with that on the surface, allowing Pb to approach the curve eq = sat (point C). In summary, substantial pressure reductions are achieved in low-speed (unstable steady pushing) reopening, whereas slow adsorption limits the effectiveness of Infasurf in achieving maximal pressure reductions for high-speed (stable steady peeling) reopening.
Effect of the degree of downstream collapse.
The results above were computed for = R0/R1 = 0.5, implying that the liquid-filled end of the tube is moderately collapsed. For larger , when a greater proportion of the tube is at a higher transmural pressure, tube inflation by an advancing bubble requires more work to be done by the airway liquid against elastic forces in the tube wall; however, for smaller , elastic forces in the tube wall can do more work on the liquid during reopening (effectively, very low external pressure can release stored elastic energy to pull open the tube walls, which helps draw the bubble into the tube). To understand this effect, we evaluated data equivalent to that shown in Fig. 4 for a range of values of , determining in each case the minimum pressure (Pbcrit) for steady reopening when eq = clean and eq = sat. Figure 5 plots Pbcrit = pbcritR1/clean vs. both without surfactant (solid) or allowing for the maximal uniform surface tension reduction (dashed). Lowering surface tension leads to a uniform reduction in the critical reopening pressure for all values of . Furthermore, Pbcrit is negative for sufficiently low , a condition implying that rapid spontaneous reopening is possible (48), with the airway being pulled open by tethering forces. For a range of values near 0.4, for example, Pb must exceed a positive threshold pressure when eq = clean to allow steady (peeling) reopening, whereas the addition of sufficient surfactant (lowering eq uniformly, or at least almost uniformly, to sat) allows spontaneous reopening to occur.
Effect of surfactant on wall stresses.
Figure 6, A and B, shows the predicted normal stress n and shear stress s exerted on the tube wall with parameters corresponding to case 3 of Fig. 4, with (point B) and without (point A) surfactant. The stresses are plotted with respect to the reference stress scale clean/R1. The stresses, as predicted by our model, are discontinuous near the bubble tip at x = 0, reflecting the fact that the model does not describe the bubble-tip region explicitly. In practice, the stresses will vary smoothly over a short distance near x = 0. Detailed stress distributions near the bubble tip are not available for exactly these parameters from other studies. However, computational simulations at speeds characteristic of peeling motion (15) indicate that both the normal and tangential stress vary monotonically near x = 0 over a length-scale comparable to R1. In Fig. 6, A and B, we therefore smoothed predicted stress distributions using a function representative of simulations (see APPENDIX) to illustrate the large stress gradients that arise near the bubble tip. A significant reduction in n in the bubble region is observed for large bulk surfactant concentrations (Fig. 6A). However, the stress magnitudes and gradients are otherwise largely unchanged by increasing the surfactant concentration (holding the bubble speed constant), although there is a significant reduction in the jump in normal stress across the bubble tip; because at these capillary numbers the length-scale of variation is unlikely to change significantly in the two cases, there is consequently a drop in the normal stress gradient across the bubble tip. Figure 6, C and D, shows the corresponding stresses in case 1 of Fig. 4, with (dashed line) and without (solid line) surfactant (we made no attempt to smooth stress distributions across the bubble tip in this case). With pushing rather than peeling reopening, surfactant has a much more dramatic effect, substantially reducing both the distance over which fluid is displaced by the advancing bubble and stress magnitudes. Furthermore, nonmonotonic shear-stress distributions can be expected to arise over short distances near the bubble tip for such low bubble speeds (4). However, the predicted stress distributions extend an unrealistic distance (many airway radii) ahead of the bubble, implying that neglected finite-length effects will undoubtedly be significant. Finally, comparing the dashed curves in Fig. 6, A and C, demonstrates the effect of a reduction in bubble speed with pb held (roughly) constant: the corresponding transition from peeling to pushing motion leads to a reduction in the magnitude and gradient of the stresses in the region ahead of the bubble. However, large stress variations across the bubble tip remain.
Cell deformation during airway reopening by peeling.
We now use the wall stresses shown in Fig. 6, A and B, to determine the likely strains experienced by a thin layer of epithelial cells. We characterize the deformation of the cell layer by using the three dominant strains (illustrated in Fig. 7A), the radial strain rr, the shear strain rx, and the azimuthal strain . is positive when the airway is dilated; stretching due to dilation also compresses the cell layer radially (causing rr < 0), as does elevated internal pressure; rx arises from viscous shear stresses due to airway liquid flow. (Recall that the stiffer airway wall is load bearing, whereas the softer and thinner cell layer is mechanically passive.) We assume initially that the cell layer is incompressible (see Fig. 7B). Our model then predicts that the layer is almost insensitive to the normal stress distribution n (see Eq. A6) because it is thin, responding instead to wall stretch and shear stress. The tube is dilated in most of the inflated-bubble region (R > R1 in x/R1 < –5), stretching the cell layer azimuthally and (by incompressibility) compressing it radially, giving > 0 and rr < 0. Elsewhere, the tube is under compression (R < R1), and we anticipate that here a real airway is likely to buckle to a nonaxisymmetric configuration, preserving constant perimeter, so shielding the epithelium from significant compression. We model this (very crudely) by setting = rr = 0 for R < R1; the dominant strain is then assumed to be the shear strain rx induced by the shear stress s. (Both quantities will, of course, vary azimuthally if the tube is buckled; we treat them here as approximate averages around the cross section.) Just as the predicted jump in shear stress s across x = 0 (Fig. 6B) is smoothed over a length scale comparable to R1, so the shear strain (which is proportional to s/Gc; Fig. 7B) likewise varies smoothly but steeply near the bubble tip. We have used a large value of the cell's elastic modulus (Gc = 50) to ensure that predicted shear strains are uniformly small (no more than 2%); experimental estimates suggest smaller values of Gc may be appropriate (1, 3, 37), implying that proportionally larger shear strains may arise in practice. As in Fig. 6, A and B, we show in Fig. 7B the strains arising both with and without surfactant in peeling motion, assuming the bubble speed is constant. Surfactant reduces substantially the azimuthal and radial strains in the region where the tube is inflated but does not have a large effect elsewhere. Figure 7C shows the corresponding strain distributions when the cell layer is assumed to be compressible (taking P = 0.25). The radial strain rr now has contributions both from azimuthal stretching (rr < 0 where the tube is dilated) and from flow-induced normal stresses (notably across and ahead of the bubble tip, where rr > 0 due to low fluid pressures). The magnitude of the flow-induced radial strain is once again limited by the large value of Gc used in the simulations. As before, surfactant reduces radial and azimuthal strains arising from wall stretching, but it has only a limited effect on shear strains; surfactant also reduces the radial strain gradient across the bubble tip. In the context of the present model, it is only by allowing the thin cell layer to be compressible that flow-induced normal stresses generate appreciable cellular strains.
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DISCUSSION
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Computerized tomography imaging of the ARDS lung provides evidence that airway recruitment occurs along the pressure-volume curve (12) and the associated distribution of reopening pressures indicates multiple types of atelectasis. Evidence exists that liquid plugs and foam lamellae in conducting airways (rather than collapse) cause loss of aeration in the ARDS lung (40, 41). Other studies provide evidence of the importance of recruitment at the alveolar level (24). Here, we have focused on a third candidate mechanism of airway recruitment, namely the inflation of an initially flooded and collapsed airway by an advancing bubble of air. We have represented the airway with an idealized physical model (Fig. 1A), the relevance of which has been debated extensively elsewhere (14, 16, 47). Accepting its potential limitations, we have developed a theoretical description of the physical model by exploiting approximations allowing a semianalytical (rather than a fully computational) approach. In going from the real airway to the physical model and in then going from the physical model to its theoretical description, we have been obliged to focus on a subset of key processes that we believe are of biomechanical and physiological significance. We must now assess the insights arising from our predictions in light of the approximations we have made. First, however, we review our main findings.
Biomechanics of airway reopening.
Previous theoretical studies and experiments indicate that the primary mechanism of reopening an airway that is initially collapsed and flooded along its length, if the airway is sufficiently deformable, is for an advancing bubble to peel apart the airway walls, overcoming the adhesive effects of surface tension and viscosity. Steady peeling motion is represented by the right-hand solution curves in Fig. 4A, computed for two different values of surface tension. Theory (23) predicts that the left-hand steady "pushing" solutions are unstable (at sufficiently long times) and therefore unrealizable experimentally when either the bubble pressure or the bubble volume flux is prescribed, although slow unsteady pushing may be an accessible mode of reopening transiently, particularly at low bubble pressures. Under low fixed volume flux, an oscillatory instability is predicted to develop at large times (23) for which the tube switches abruptly between slow pushing motion (when the bubble enlarges predominantly by transverse inflation without advancing appreciably) and rapid peeling motion (when the bubble enlarges by elongating rapidly while also shrinking in width); the longer the bubble, the greater the propensity to exhibit this unsteady oscillatory behavior. In conjunction with unpredictable avalanche effects operating throughout the airway network (2), such an instability would make airway recruitment under external ventilation difficult to control in the sense that imposing a steady volume flux can lead to strongly time-dependent motion. Stable quasi-steady pushing motion can, however, arise if the airway wall is sufficiently permeable (32) or, presumably, for finite-length liquid plugs. For airways in the configuration shown in Fig. 1A, however, there is a minimum pressure and a minimum speed (represented by the bullets in Fig. 4A) at which a bubble can steadily reopen a flexible tube.
Surfactant facilitates airway reopening.
The only previous theoretical study examining surfactant effects in the reopening of a flexible airway (61) considered a spatially uniform bulk surfactant concentration and a linear relationship between surface tension and surfactant concentration. As a result, this study predicted only marginal changes in reopening pressure due to the presence of surfactant. Here, by integrating recent theoretical results on the effects of surfactant on bubble propagation in rigid tubes (17, 18) into an established theoretical description of airway reopening (33), we have been able to capture the effects of a fully nonlinear equation of state and nonequilibrium adsorption/desorption kinetics for a surfactant with physiologically relevant properties (Infasurf). Our results demonstrate that surfactant can lead to substantial reductions in the pressure required to reopen a collapsed airway (Fig. 4A) and that the effectiveness of this reduction is dependent on the mode of reopening and on the degree of initial airway collapse.
We find that in the presence of surfactant the two generic modes of reopening (slow pushing and rapid peeling) are preserved (Fig. 4A). In pushing motion (case 1), the bubble advances slowly enough that high concentrations of surfactant in the bulk can come into equilibrium with that on the air-liquid interface, enabling a full and uniform reduction in surface tension to its minimum value (sat) to be achieved. In peeling motion (case 3), the bubble travels too fast for the interface to be fully remobilized at high concentrations: the beneficial effect of a large global reduction in surface tension (through the nonlinear equation of state; Fig. 2) is offset by Marangoni flows that induce increased viscous dissipation. Nevertheless, the reductions in reopening pressure predicted for Infasurf are appreciable, and the range of pressures and bubble speeds at which steady peeling motion can occur is enhanced in the presence of surfactant.
Airway collapse mediates surfactant effects.
For an initially collapsed flooded airway, we predict that pb must exceed a threshold pbcrit for steady reopening to occur. [Assuming peribronchial pressure is close to pleural pressure, we can interpret pb as the transpulmonary pressure (62).] Situations in which pbcrit > 0 can be interpreted in two ways: either an external ventilator must increase the airway pressure above a critical level to allow reopening, with the airway remaining otherwise passive; or active lung inflation must increase the tethering forces on the airway, thus lowering the effective external pressure below a critical value. The degree of initial airway collapse is an important parameter in determining pbcrit, because airways that are initially more collapsed have stronger elastic restoring forces that help draw the bubble into the airway (26, 48). For highly collapsed airways, pbcrit < 0 (see Fig. 5), implying that elastic forces in the airway wall are large enough to draw the bubble spontaneously into the airway (a situation likely to be of significance during an infant's first breath). Addition of surfactant to an initially collapsed airway can therefore have two possible outcomes: either pbcrit is reduced to a lower but positive value (the reduction is approximately uniform for all degrees of collapse; see Fig. 5) or pbcrit is reduced below zero, leading to spontaneous reopening through the release of elastic energy stored in the airway wall.
Cells deform in response to stretch and shear stress; compressibility mediates the response to normal stress.
According to our model, the time-scale for reopening an individual airway in steady peeling motion is of the order of a few milliseconds [taking the minimum speed for peeling motion to be 7 m/s, based on assuming µU/clean = 0.1 in Fig. 4A and assuming a typical airway length of a few millimeters; slower pushing motion may arise but it will be unsteady (23) and is therefore outside the framework of this model]. Such rapid reopening may correspond to "popping" behavior reported experimentally (45). Because this time scale is very short compared with measured epithelial cell viscoelastic relaxation times, we modeled the epithelial cell layer as an elastic material. For simplicity, we assumed that the cell layer is uniform, homogeneous, and isotropic; there is little doubt that the cell's complex and dynamic internal architecture ensures that these three characteristics are very approximate, and our results must therefore be interpreted cautiously. To keep our model simple, we also worked in the small-strain framework of linear elasticity, an assumption consistent with the qualitative nature of the model. We also exploited the composite structure of the airway wall, whereby the stiff basement membrane balances the fluid loading, shielding the softer epithelial layer from excessive strains. Motivated by evidence from other studies indicating that some cells may exhibit compressible properties (8, 39), we considered both compressible and incompressible behavior in the epithelium. Finally, we assumed that the cell layer is thin compared with the airway radius, allowing a simplified description of the elastic response to be developed. We then identified the dominant strains resulting from stretching of the airway wall and hydrodynamic forcing. These are radial and azimuthal strains (arising largely from airway-wall stretching) and shear strain (driven by viscous shear stress). In the incompressible limit, our model predicts that strains arising from normal stresses or normal stress gradients are negligible compared with those arising from shear and stretch. Assuming that the basement membrane preserves its perimeter by buckling when the airway is collapsed, protecting the epithelium from severe compressive forces, we found that the strains generated by shear forces are smaller than those arising from wall stretching when we assumed a large value of the cell layer's elastic modulus G (Fig. 7B). Smaller (and possibly more realistic) values of G would lead to larger shear-induced strains. Large shear-strain gradients are predicted near the bubble tip. Crucially, the model shows that the cell layer exhibits a significant response to fluid normal stresses (pressures) only if it is compressible (Fig. 7C). Our simulations also showed how the addition of surfactant (at fixed bubble speed) reduced stretch-induced axial and radial strains significantly, but not flow-induced shear strains. For a compressible cell layer, surfactant reduced the radial strain gradient across the bubble tip.
Limitations.
There are numerous approximations that limit the validity of the present study. For example, we did not provide a consistent treatment of geometric effects such as wall buckling (26), airway bifurcations, gas trapping, or instabilities of the liquid lining in the inflated section of airway (22, 46), all of which may have a significant effect either at the scale of the whole airway or at the scale of individual cells. We did not model explicitly the delivery of exogenous surfactant to a surfactant-deficient airway but assumed instead that the surfactant was present in situ before reopening. We described only steady bubble motion and did not consider time-dependent modes of airway recruitment. [This is a particularly severe limitation, since the rapid peeling motion on which we have focused here, which is a stable steady solution of our model, is not representative of much slower unstable pushing motion that can occur transiently (23); such unsteady behavior merits further investigation.] We decoupled surfactant effects near the bubble tip (region II in Fig. 1A) from the flow-structure interaction region (III) ahead of the bubble; we cannot be confident that recirculation zones extending into region III may not disturb the bulk surfactant concentration from its assumed uniform value nor that nonaxisymmetric effects associated with airway buckling introduce important flow features not present in Refs. 17 and 18. We used idealized descriptions of airway wall mechanics, cell mechanics, and airway liquid rheology. However, despite these and other deficiencies, we believe the model captures some important qualitative features of airway recruitment.
Physiological implications.
The biological responses of epithelial cells to deformation are numerous: for example, cyclic deformation of rat primary alveolar epithelial cells increases the level of cell death compared with static deformation (56); cyclic stretch induces apoptosis and necrosis in alveolar type II cells (25); stretch increases paracellular permeability by disrupting tight-junction structure and function (6); and deformation induces inflammatory signaling (59). Mechanical events such as stress failure of the plasma membrane (11, 58) underlie potential mechanisms of damage in ventilator-induced lung injury (57). Although our model does not seek to describe the biological impact of cell deformation, nor does it predict the deformation of subcellular components, it does illustrate the likely strain experienced by epithelial cells during airway recruitment, and it is reasonable to hypothesize that damage correlates with some measure of strain at the cellular scale. The model predicts, for example, that airway stretch and (for soft cells) flow-induced shear can induce strains in the plane of the cell membrane (Fig. 7) well in excess of the 2–3% range sufficient to cause transient rupture of the plasma membrane (58). Although cells have the capacity to reseal damaged plasma membranes very rapidly (42), sufficiently large strains are likely to induce permanent cell damage. Recent experiments (4, 35) showed that the damage to epithelial cells in a flow chamber due to the passage of a bubble at very low capillary numbers correlates with normal-stress (pressure) gradients rather than shear stress, shear-stress gradients, or the time of exposure to stress. Our model predicts (Eq. A6) that a thin layer of incompressible cells is unlikely to respond significantly to normal stress or to normal-stress gradients. [Even though the normal stress in the experiments varied on a length scale comparable to the length of a highly spread epithelial cell (100 µm), the high aspect ratio of the cell still argues against a significant strain response if the cell layer is incompressible.] This indicates that compressibility or possibly another effect neglected in our model [e.g., anisotropy, inhomogeneity, or nonuniform topology (31)] may instead be responsible for generating the strain leading to mechanical damage in the flow-chamber experiments.
Role of surfactant in mediating epithelial damage.
The operating conditions under which we were able to simulate surfactant-mediated reopening (at fixed bubble speed, varying the bulk surfactant concentration) give limited insight into the full spectrum of conditions likely to arise in practice. However, an effective surfactant that can both reduce surface tension substantially and adsorb rapidly at high concentrations enables the steady reopening of an airway by peeling apart the airway walls to occur at reduced pressures and lower speeds (Fig. 4). The reduction in reopening pressure lowers the stretch-induced radial and azimuthal strains in airway epithelial cells in the inflated section of the airway (Fig. 7, B and C); thereby, surfactant can potentially limit damage via volutrauma. However, the peeling mode of reopening is a dynamic process that cannot occur at very low speeds so that, even in the presence of an effective surfactant, viscous shear and normal forces can generate significant shear (and, for compressible cell layers, large radial) strains. In this sense, surfactant does not appear to provide such substantial protection against this potential mechanism of atelectrauma. The protective effects of surfactant noted in flow-chamber experiments (4) at low capillary numbers characteristic of pushing motion may not be relevant to the more rapid peeling behavior illustrated in Fig. 7. Nevertheless, assuming that the cell layer is compressible, our model shows how surfactant can reduce radial strain gradients at the bubble tip (Fig. 7B), an effect that correlates with lower normal stress gradients.
In conclusion, using a theoretical description of an idealized model of airway recruitment, incorporating data relevant for a physiologically relevant surfactant (Infasurf), we have demonstrated how surfactant lowers the critical transmural pressure required to push a bubble of air into an initially collapsed flooded airway. Depending on the degree of initial airway collapse, the addition of surfactant can promote spontaneous reopening. Following Refs. 17–19, we found that the effectiveness of Infasurf in mediating airway reopening is limited at high bubble speeds by slow adsorption kinetics, even at high concentrations. We used the model to predict the stresses imposed on a layer of epithelial cells during reopening and related these to cell strains arising both from airway wall stretch and from liquid flows. Surfactant was shown to have a greater effect in reducing strains arising from wall stretch than from flow-induced shear. Cell compressibility was invoked to reconcile our predictions with experimental studies (4, 35) showing that cell damage correlates with normal-stress gradients.
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APPENDIX
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We describe here how we integrate three theoretical models operating at distinct length scales in Fig. 1A, one describing the liquid flow in the deformable tube, one describing the effects of surfactant operating near the bubble tip, and one describing deformation of the thin layer of cells lining the airway wall.
Flow model.
We represent the liquid in the airway as a single-phase incompressible Newtonian liquid with viscosity µ. For simplicity, we ignore gravitational, viscoelastic, and inertial effects, assuming the dominant resistance to motion comes from viscous forces. The effects of inertia and viscoelasticity in this problem have been described elsewhere (for example, see Refs. 27, 29). The liquid carries a surfactant that adsorbs to the air-liquid interface of the bubble where it reduces the surface tension in a manner described below.
We need to compute (in principle) the air-liquid interface location, the tube wall shape, the internal flow, and the surfactant distribution. However, instead of solving the full problem computationally, which is feasible but challenging, we reduce the problem to a more tractable form by exploiting some systematic approximations developed previously (15, 23, 33, 44). One necessary condition allowing this approach is that the wall tension T should be large compared with , which ensures that wall slopes are uniformly small. The flow may then be decomposed into three distinct regions (illustrated in Fig. 1A) in which different forces dominate: the inflated bubble (region I), the bubble tip (region II), and the liquid-filled tube (region III). We now outline briefly how we reduce the problem to a nonlinear ordinary differential equation for the tube radius R(x) in region III (Eq. A3 below), subject to boundary conditions capturing physical effects operating in regions I and II.
It is convenient first to recast the problem in nondimensional variables, i.e., variables scaled on characteristic length and time scales of the problem. Given the far-field bulk surfactant concentration C0, a corresponding equilibrium surface tension eq, and the radius R0 of the tube where it is collapsed far ahead of the bubble, we choose as a unit of length R0, pressure eq/R0, speed eq/µ, volume flux (eq/µ)R02, and surface tension eq. We can then define a set of dimensionless parameters that describe the relative importance of different physical effects. Describing the wall mechanics are Ea = ER0/eq (which compares spring stiffness to surface tension), Ta = T/eq (which compares longitudinal tension to surface tension) and = R0/R1 (which measures the degree to which the tube far ahead of the bubble is collapsed relative to its unstressed state). Additional parameters characterizing the surfactant are discussed further below. We seek the relationship between the dimensionless bubble speed Ub = µU/eq and the dimensionless bubble pressure p*b = pbR0/eq (where * denotes a dimensionless quantity).
We now describe the dominant forces acting in each of the three regions illustrated in Fig. 1. In region I, the tube wall is in equilibrium with its shape determined by Eq. 1, and the liquid layer sits passively on the wall with approximately uniform thickness and uniform surface tension. Because T >> eq, the pressure in the liquid is that of the bubble minus the capillary pressure drop across the approximately cylindrical air-liquid interface (p pb – eq/R), and so Eq. 1 {which we reexpress in dimensionless variables using R(x) = R0R*(x*), x = R0x*, and p = [(eq/R0)p*]} becomes
 | (A1) |
This equation may be integrated once to derive a relation between the tube radius R*(0) and its slope R*x*(0) at the bubble tip. Specifically, if the tube radius far behind the bubble in region I is R
= R0R* (see Fig. 1A), so that p*b = EaF(R*)+(1/R*), then
 | (A2) |
where G(z) = z6/6 – 5z0.2 [so that G'(z) = F(z)].
In region II, the large longitudinal tension means that the tube radius is almost uniform in the neighborhood of the bubble tip, so that the flow locally resembles that near the tip of a bubble advancing in a uniform rigid tube. This problem has been studied intensively by numerous workers, either neglecting or including the effects of surfactants (5, 17, 18). For our purposes, the effects of the complex flow near the bubble tip can be represented though two quantities: the ratio of the thickness of the uniform film deposited on the tube wall far behind the bubble tip to the tube radius [so the dimensional film thickness is RT where RT = R0R*(0), see Fig. 1A]; and the additional pressure drop (corresponding to a dimensional pressure drop eq/RT) arising from surface tension and viscous forces acting in the neighborhood of the bubble tip. Both parameters are functions of Ub, the dimensionless bubble speed, and parameters characterizing the surfactant. We use data for and shown in Fig. 3 from Refs. 17 and 18, as described below. The film thickness parameter is significant because it captures the manner in which the competing effects of surface tension and viscosity act as a valve that limits the flux of liquid that can pass behind the bubble tip. Surface-tension-gradient-driven (Marangoni) flows can both increase or reduce this flux relative to the surfactant-free case (see curves 1 and 3, respectively, of
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