The reduction of tidal volume during mechanical ventilation has been shown to reduce mortality of patients with acute respiratory distress syndrome, but epithelial cell injury can still result from mechanical stresses imposed by the opening of occluded airways. To study these stresses, a fluid-filled parallel-plate flow chamber lined with epithelial cells was used as an idealized model of an occluded airway. Airway reopening was modeled by the progression of a semi-infinite bubble of air through the length of the channel, which cleared the fluid. In our laboratory’s prior study, the magnitude of the pressure gradient near the bubble tip was directly correlated to the epithelial cell layer damage (Bilek AM, Dee KC, and Gaver DP III. J Appl Physiol 94: 770–783, 2003). However, in that study, it was not possible to discriminate the stress magnitude from the stimulus duration because the bubble propagation velocity varied between experiments. In the present study, the stress magnitude is modified by varying the viscosity of the occlusion fluid while fixing the reopening velocity across experiments. This approach causes the stimulus duration to be inversely related to the magnitude of the pressure gradient. Nevertheless, cell damage remains directly correlated with the pressure gradient, not the duration of stress exposure. The present study thus provides additional evidence that the magnitude of the pressure gradient induces cellular damage in this model of airway reopening. We explore the mechanism for acute damage and also demonstrate that repeated reopening and closure is shown to damage the epithelial cell layer, even under conditions that would not lead to extensive damage from a single reopening event.
- acute respiratory distress syndrome
- ventilator-induced lung injury
- surfactant replacement therapy
- lung epithelial cells
treatment of acute respiratory distress syndrome (ARDS) has been greatly revised recently with the recognition that ventilator-induced lung injury can result from mechanical ventilation by using conventional settings (2, 9, 32). The use of high-tidal volume ventilation has been implicated in causing damage to the small airways and alveoli by tissue stretch (volutrauma) (4, 7–11, 24, 26, 30, 31). To reduce this damage, low-tidal volume ventilation has been used to reduce patient morbidity (3) and mortality (9).
Unfortunately, low-volume ventilation can lead to airway damage, as demonstrated by Muscedere et al. (23). Ventilation at low lung volumes and pressures may cause airway and alveolar fluid-structure instabilities that can lead to cyclic opening and closing (recruitment and derecruitment) of small airways and alveoli (13, 18). The pulmonary epithelium is particularly at risk of being damaged by mechanical stresses associated with this behavior (5, 26). Recruitment of closed areas of the lung can result from a bubble propagating through edematous regions of the lung (with little deflection of the airway wall), or by the separation of airway walls that are held shut by a thin layer of lining fluid (12, 14). In either case, the complex mechanical stress field may damage the airway tissue (22). It is known that surfactant protects the lung from damage; however, even mild surfactant dysfunction can lead to severe lung injury (28, 29). The physicochemical behavior related to this protection is a current topic of study (15–18, 21, 25).
We have recently investigated the mechanical stresses that induce epithelial cell damage in a model of airway reopening (5). The results of that study strongly suggest that the pressure gradient (dP/dx) [not shear stress (τs)] is the primary determinant of mechanical damage. That and the present study used a fluid-filled parallel-plate chamber lined with epithelial cells as an idealized model of an occluded airway. Airway reopening was modeled by the steady progression of a semi-infinite bubble of air down the length of the channel, which cleared the fluid. A computational model was developed to determine the mechanical stimuli applied to the cells during the bubble progression (as described in detail in the discussion). This analysis indicates that the reduction of bubble velocity (U) decreases the τs, τs gradient (dτs/dx), and pressure, while increasing the dP/dx. Because cell damage increased with reduced velocity, it was concluded that the most mechanically damaging element of the stress cycle was the steep dP/dx along the cell in the region of the bubble tip (Fig. 1). This causes an intracell pressure variation that might damage sensitive tissue (discussed below).
A limitation of our laboratory’s prior study (5) resulted from the direct link between stimulus magnitude and exposure duration (i.e., the period of time that an individual cell experiences a damaging stress component). For example, a modification of the U led to a simultaneous modification of the stress magnitudes and the period of time an individual cell was exposed to the traveling stress field near the bubble tip. Because both the exposure time (texposure) and dP/dx are inversely related to the U (see discussion), the dP/dx was increased during experiments in which the texposure was the longest. So our prior study (5) did not permit the disassociation of the stimulus magnitude from the duration of exposure to the applied stimulus. This coupling confounds our understanding of the cell response because it is possible that cell membrane damage might be induced not only by the magnitude, but also by the length of time that the cell is exposed to the stress. Conceptually, a low-magnitude stress exerted over a prolonged period of time could be more damaging than a high-intensity stress exposed to the cell for only a short time. So because the link between U and cell damage was evident in our previous study (5), we could not be certain that the damage was due solely to the dP/dx. Alternatively, it was possible that the damage was induced by the low-magnitude τs or dτs/dx, and that the damage occurred only if these stress components were exerted for a significant duration of time. In this case, even though the magnitudes were lower with reduced velocity, the damage could be a result of the commensurate increase in exposure duration.
To decouple the stress magnitude from the duration, in the present experiments we modify the fluid viscosity (μ) without any variation in the U. Because the stress magnitudes are functions of μU (see discussion), a 10-fold increase in μ results in precisely the same change in stress magnitude as a 10-fold increase in velocity. However, whereas an increase in U decreases the texposure, an increase in μ causes an increase of the texposure (see discussion for an explanation of this effect). Thus, by comparing the damage to computed stress predictions calculated in Ref. 5, we can discern the relative influence of stress magnitude to stress exposure duration in the system.
A human pulmonary epithelial cell line (A549, American Type Culture Collection, Manassas, VA) was maintained in a culture medium of Ham's F-12K with 10% fetal bovine serum and 1% antibiotic-antimycotic solution (Invitrogen, Carlsbad, CA) and used at passage 89. Before experiments, cells were enzymatically lifted from flasks by using a 1.5% trypsin solution (Invitrogen) and cultured on a small circular region (3.8 cm2) of a glass microscope slide (25 × 75 mm). Specifically, circular sections of the microscope slides were isolated by using polycarbonate cylinders held onto the slides with silicone rubber (Regent Pet Products, Moorpark, CA). Epithelial cells were then enzymatically suspended and seeded at ∼100 × 103 cells/cm2 inside the circular region. The slides were incubated in sterile 100-mm petri dishes under standard culture conditions (humidified, 37°C, 5% CO2, 95% air) and cultured to confluence (3 days), providing a density of adherent cell of ∼65 × 103 cells/cm2 with an average cell diameter of ∼40 μm. Because the cells were cultured under static conditions, no preferred orientation existed. The polycarbonate cylinder that defined the cell-seeded region of the domain was removed immediately before experiments were conducted.
A parallel-plate chamber (Fig. 2) designed by Bilek et al. (5) was used as an idealized model of a collapsed segment of an airway in which the walls are held in opposition by a viscous fluid. The upper and lower walls of the parallel-plate chamber were formed by two glass microscope slides. The lower wall of the parallel-plate chamber consisted of the glass microscope slide cultured with pulmonary epithelial cells, as described above. The cell-free upper wall of the parallel-plate chamber consisted of a larger glass slide (38 × 75 mm) seated over the separation wall with a 5-mm-wide, 0.4-mm-thick Silastic (Pharmelast, SF Medical, Hudson, MA) gasket to form a tight seal. The channel height of this model was 0.17 cm [channel half-height (H) = 0.085 cm], which is equivalent to the diameter of airways that are susceptible to fluid obstruction, according to the study by Burger and Macklem (6). We address the issues related to this choice of scale below in Limitations.
Generation of reopening conditions.
Two occlusion fluids were examined as model airway lining fluids: phosphate-buffered saline, including 0.1 mg/ml CaCl2 and MgSO4 (PBS) with a viscosity of 8.0 × 10−3 g·cm−1·s−1; and PBS supplemented with 14.1 wt% clinical grade dextran (Sigma; average molecular weight, 68,800) with a viscosity of 8.0 × 10−2 g·cm−1·s−1. CaCl2 and MgSO4 was included in the PBS formulation to be consistent with Ref. 5. However, the Ca2+ and Mg2+ are not necessary, unless surfactant is included in the lining fluid. Viscosities were measured by using a Cannon-Fenske glass viscometer (Cole-Parmer, Vernon Hills, IL). Both occlusion fluids were warmed to 39°C before use. The surface tension (γ) was not modified by the addition of dextran, as confirmed by measuring the capillary rise in a 1-mm-diameter vertical glass tube.
The cell-seeded slide of the parallel plate chamber was placed into the apparatus, and the channel was flooded with an occlusion fluid. The apparatus was assembled and transferred into a warming bath at 39°C. A small volume of air (2.7 ml) was then infused into the upstream end of the chamber at a rate of 7 ml/min, setting a U of 0.34 cm/s, which “reopened” the channel by removing the occlusion fluid. Thus the bubble propagated through the parallel plate flow chamber over a period of ∼30 s, but passed over the cell-seeded layer for ∼6 s. The period of time an individual cell experiences the traveling stress wave is very short (∼5 × 10−2 s) and is estimated in the discussion.
Experiments were conducted at 39°C to be consistent with our prior investigations (5), in which the slight increase in temperature allowed the surfactant micelles to disperse more uniformly on dilution, which maintained the uniformity of the surfactant solution. This was also important to correct for cooling effects as the solution entered into the chamber, which was originally at room temperature. The time for assembly, experimental trial, and disassembly was ∼5 min.
As a control condition, cell-seeded slides were rinsed with PBS and placed in a petri dish filled with either 39°C PBS or PBS/dextran for 5 min on the benchtop. Whereas an optimal control would have the slides processed in an identical manner without exposure to bubble progression, this is not possible because the action of disassembly without prior bubble progression introduces an uncontrolled expanding air-liquid interface when the slides are separated. These effects could not be discriminated from the purposefully applied reopening stimulus.
Quantification of cellular injury.
After removal from the apparatus (or petri dish for the control), each slide was gently rinsed with 37°C PBS. A 250-μl aliquot of a solution containing 1.2 μl ethidium homodimer-1 (Eth-1) and 1.2 μl calcein AM (Live/Dead Kit, Molecular Probes, Eugene, OR) in 1 ml PBS was gently applied to the surface of the cells. The slide was then incubated at 37°C for 10–30 min. These two dyes are supplied in the commercially available “Live/Dead” kit used to differentiate “live” from “dead” cells. If injury or death compromises a cell membrane, Eth-1 enters the cell and binds to DNA, producing a red fluorescent nucleus. Uninjured cells are marked by the calcein AM binding to active intracellular esterases, producing green fluorescence at the cell membrane.
To assess the magnitude of damage, the numbers of injured (red) cells (Eth-1 stained) in each of five random fields were counted manually by using fluorescence micrographs (Fig. 3), with the average number of injured cells expressed either as “injured cells” or cells per centimeters squared of slide surface area. The data are reported as means ± SE for five slides per condition. Statistical significance was set at P < 0.01, and differences between means were statistically evaluated by using Duncan's multiple-range test after model adequacy checking verified the normal distribution of the data. Statistical significance was determined for the reopening experiments compared with the control and for the damage induced by low-viscosity (PBS) reopening compared with the damage from the high-viscosity (PBS/dextran) fluid for the same U.
This investigation of epithelial cell damage provided insight into mechanisms responsible for cellular damage during the airway reopening cycle. Figure 3 shows images of cells from control and experimental groups for both occlusion fluids, with injured cells demonstrating red (Eth-1)-stained nuclei. Figure 4 shows the average number of injured cells per centimeter squared for each testing group (n = 5). The control slides for both PBS and PBS/dextran show low numbers of Eth-1-stained nuclei (2.75 ± 0.7 and 3.86 ± 2.2 × 103 injured cells/cm2, respectively). There was no significant difference between the two control groups.
For occluded channels, bubble progression (U = 0.34 cm/s) over the cells produced a significant increase in the number of Eth-1-stained nuclei compared with the controls for both occlusion fluids. The low-viscosity PBS (μ = 8.0 × 10−3 g·cm−1·s−1) produced 34.05 ± 3.13 × 103 injured cells/cm2, which was significantly (P < 0.01) more damage than the PBS control. The high-viscosity (μ = 8.0 × 10−2 g·cm−1·s−1) PBS resulted in 14.16 ± 6.89 × 103 injured cells/cm2. Thus an increase in viscosity at the same reopening speed resulted in a reduction of cell damage.
To briefly explore the impact of repeated closure and reopening, we exposed rat distal airway tissue L2 cells (CCL-149, American Type Culture Collection, Manassas, VA) to multiple passages of a bubble at a velocity of 4 cm/s. To culture these cells, a 1-cm2 section of a microscope slide was isolated by using a 0.4-mm-thick Silastic gasket (Pharmelast, SF Medical, Hudson, MA). Pulmonary epithelial cells were suspended in a culture medium of Ham's F-12K medium with 10% fetal bovine serum and 1% antibiotic-antimycotic solution (Invitrogen, Carlsbad, CA) and plated at 50 × 103 cells/cm2 in the isolated region. The slides were incubated in 100-mm petri dishes under standard culture conditions (humidified, 37°C, 5% CO2/95% air) for 6 h. The gaskets were then removed, and 15 ml of culture medium were added to the petri dishes. The pulmonary epithelial cells were cultured to confluence. Cells were exposed to the multiple passages of the bubble and then fixed with formaline and stained with Coomassie brilliant blue. At the 4 cm/s reopening rate, little (if any) membrane damage would be expected during a single bubble passage, according to the results of Bilek et al. (5). Figure 5 shows that a single passage resulted in little obvious damage, but multiple passages over the same cells can cause significant damage, with 20 passages resulting in a nearly complete ablation of the epithelial layer. This result indicates that multiple closure and reopening events can result in severe damage to epithelial tissues, even if a single reopening event induces little membrane damage. Note that this study was performed in a surfactant-free system, and thus Mg2+ and Ca2+ were not explicitly included in the model-lining fluid. However, the PBS was still osmotically balanced.
The results of our studies demonstrate that increasing the occlusion viscosity serves to protect the epithelial cells from damage due to bubble progression in a parallel-plate flow chamber. Below, we approximate the magnitudes of stress and the exposure duration and relate these to the observed cell membrane damage.
To investigate the stress magnitudes in this system, we use the regression formulas provided by Bilek et al. (5) to calculate the maximum τs, dτs/dx, and dP/dx that the cells experience. These relationships were calculated in dimensionless form, which exploits the fact that the fundamental physical interactions depend on the ratio of viscous to surface tension forces. The dimensionless velocity, also known as the capillary number, (1) represents the ratio of viscous tension to surface tension effects and determines the dynamic response of the system. The stress relationships (accurate for Ca < 10−2) in dimensionless (a) and dimensional (b) form are as follows:
τs (2a) or (2b) dτs/dx, (3a) or (3b) and dP/dx, (4a) or (4b) From these relationships, it is clear that all stress magnitudes depend directly on the product (μU); thus an increase in μ has precisely the same effect on the stress magnitude as an increase in the U. It is this relationship that allows us to vary the stress magnitudes using μ instead of U, which has a different implication for exposure duration, as will be explained below.
The relationships provided in Eqs. 2–4 are potentially counterintuitive, and thus it is important to understand the physical processes that cause this behavior. Figure 6A shows a schematic representation of the interface propagating through the flow chamber, with Fig. 6, B and C, representing the magnified view of the domain and pressure field, respectively. Figure 6B shows that an increase in Ca causes the film around the bubble to thicken. In the limit as Ca → 0, the bubble contacts the wall at a contact line, spanwise across the channel. The pressure drop between the interior (air) and exterior (liquid) is approximated by the Laplace–Young relationship, ΔPtot = γ/H, where ΔPtot is the change in total pressure. Therefore, as Ca → 0, a step-jump in pressure occurs at the contact line (Fig. 6C), and because ΔPtot is established over an infinitesimal region, dP/dx → −∞. As Ca increases, the bubble leaves a minuscule layer of fluid (“lubrication film”) along the wall (Fig. 6B). This lubrication film grows in depth with increasing Ca and reduces the magnitude of the dP/dx (Fig. 6C). So, although the dP/dx remains large at small Ca, it is reduced by an increase in μ or U. In contrast, as Ca is increased, the τs increases because an increasing volume of fluid is squeezed over the cell surface through the lubrication film. For this reason, an increase in Ca results in a decrease in the dP/dx and an increase in the τs.
Stress exposure duration.
As demonstrated above, a change in Ca results in a modification in the slope of the pressure wave that travels across the cell, which directly relates to the texposure. To determine the texposure for a cell as the stress traveling wave of length Lwave sweeps over the cell surface, consider the representation of the system provided in Fig. 6. We approximate ΔPtot = γ/H for our studies because the majority of the pressure drop is due to capillarity, not viscosity (i.e., Ca << 1). Using the relationship for dP/dx (Eq. 4), the extent (Lwave, see Fig. 6C) of the traveling wave region is approximately (5) From this relationship, and The texposure is determined by the length of time required for the entire width of the traveling wave region to propagate past a point on the wall occupied by a cell (6) From this relationship, the texposure in the present experiments is approximately and
This calculation indicates that, even though U is held constant in the present experiments, the increase in μ extends the length of the traveling wave and thus increases Δtexposure. In contrast, the experiments of Ref. 5 modified Ca by increasing U, such that an increase in Ca simultaneously decreased Δtexposure.
In this section, we present the data from our experiments and demonstrate that the dP/dx, not the duration of stress exposure, is responsible for the damage to the cell layer. Figure 4 shows that the experimental model with low-viscosity PBS as the occlusion fluid (and hence the low Ca) exhibited the greatest amount of membrane disruption. An increase in μ with no change in U caused a reduction of the cell damage. In both the low- and high-viscosity cases, the damage was significantly greater than the control. Calculations of stress components (Table 1) show that cells in low-viscosity experiments are subjected to lower magnitudes of the τs and the dτs/dx, but to larger magnitudes of dP/dx compared with their high-viscosity counterparts. In addition, calculations of the texposure show that the present low-Ca experiments introduce a shorter texposure than the large Ca experiments. The relationship between texposure and Ca is converse to that of Bilek et al. (5).
Table 2 provides a synopsis of the experimental observations and trends from our analysis. Table 2 shows that the present low-Ca experiments demonstrate increased damage and have both a larger pressure-gradient magnitude and shorter duration than the large Ca experiments. In contrast, the low-Ca experiments of Ref. 5 (which varied U) had increased damage and a larger pressure-gradient magnitude and longer duration than the large-Ca experiment counterpart. Because in both cases the damage is increased with decreasing Ca, this provides compelling evidence that the magnitude of the dP/dx on the cell is the factor that induces membrane damage. In addition, the present study demonstrates a similar reduction of damage with a 10-fold increase in Ca, as was reported by our prior variable-U experiments (58.4 ± 26 vs. 69.7 ± 27.6%). Thus the duration of a single bubble passage is, at best, only a minor contributor to the overall membrane damage during a single reopening event.
We hypothesize that the dP/dx may create an imbalance in the pressure that acts on the cell membrane over the length of the cell (Lcell) (Fig. 1). We speculate that this induces a fore-aft pressure difference on the cell body (ΔPcell), approximated as ΔPcell = (dP/dx)Lcell, where ∼40 μm is the approximate Lcell. Table 1 shows that ΔPcell increases with reduced Ca, which is consistent with the observed damage pattern (Fig. 4). The pressure imbalance could result in nonuniform cell compression, leading to “pinching” of the cell and rupturing of the cell membrane.
From the present studies, as well as those of Ref. 5, as synopsized in Tables 1 and 2, significant cell membrane damage occurs when ΔPcell ∼ 300 dyn/cm2, which is reduced when ΔPcell ∼ 120 dyn/cm2. However, little membrane rupture was observed for ΔPcell ∼ 80 dyn/cm2 (5). Clearly, the propensity for membrane disruption decreases with decreasing ΔPcell; however, it is not yet evident that a specific critical level exists that induces damage.
As with all model experiments, the characteristics of these studies deviate from actual in vivo airway reopening conditions and may influence the validity of our results. One of the fundamental differences relates to the lack of mechanical flexibility of the experimental system. True pulmonary airways are compliant vessels in which airway reopening can cause separation of the airway walls if the liquid lining is not too voluminous. This separation and bending of the walls can induce large inward-directed normal stresses, which may cause additional damage to the tissue (5, 12, 20). The present experimental design also lacks a collagen substrate beneath the epithelial cells. Within the in vivo system, this collagen may act as a cushion beneath the cell that could protect the tissue during reopening.
The multiple-passage experiments (Fig. 5) were conducted by using a different cell line (L2) than the remainder of the studies (A549). The adhesion of these cells to the substrate is not known, and the relationship between reopening velocity and detachment may differ from that of membrane damage. The exposure to the Ca2+-Mg2+-free PBS over the short time period of the experiment would not be expected to significantly affect detachment of the cells, based on our observations of rinsing cell monolayers with Ca2+-Mg2+-free PBS, which did not remove cells from surfaces or begin to detach them from substrates, as confirmed with microscopy. Future studies should examine detachment as a function of different velocities and should include Ca2+-Mg2+ in the PBS because of the potential loosening of cells in the absence of these ions. Nevertheless, the sole purpose of this brief undertaking was simply to demonstrate that multiple reopening events may be far more deleterious than a single reopening event, which it does aptly.
An additional simplification arises from the steady motion imposed in the present investigations. Suki and colleagues (1, 27) have observed “avalanche” behavior that results in the rapid reopening of many segments of airways, followed by a period of time in which few airways reopen. This behavior may be related to a viscous pseudo stick-slip behavior that is evident in models of unsteady airway reopening (D. Halpern, S. Naire, O. E. Jensen, D. Gaver, unpublished calculations). These simulations indicate that airways that initially open slowly under a fixed upstream flow rate begin to spontaneously “jump” to a new, rapid reopening velocity and then return to the slow reopening rate. This periodic instability arises fundamentally from nonlinear behavior related to the fluid-structure interactions in compliant systems, which are not included in the present experimental model.
Our experiments were conducted in H = 0.085 cm, which is equivalent to the diameter of airways that are susceptible to fluid obstruction, according to the study by Burger and Macklem (6). However, the principles investigated in this study are general and relate to the stresses imposed due to bubble progression in any airway generation. Of course, the stress magnitudes will depend on the geometry of the airway in question. This includes the size of the airway and whether the airway retains a circular cross-sectional geometry or flattens to a channel. In a flattened channel, ΔPtot ∼ 8γ/R, where R is radius, which is much greater than the reopening pressure expected for a circular tube (14). Additionally, geometric differences, which include the morphology of the pulmonary epithelium cells and features such as airway bifurcations, could dramatically increase stresses to values greater than those predicted for our simplified models. This increase in stress would be caused by interfacial curvature variation and thin-film dynamics, as the bubble deforms to slide over the topographical variations in the surface, or splits as it divides into the daughter branches of the bifurcation.
Surfactant-free conditions were studied to accurately establish the stress magnitudes in these models; however, prior studies have established the importance of surfactant in reducing the stress magnitude (15, 16). Surfactant was not incorporated into our model because we were not interested in surfactant protection in the present study. Instead, we simply focused on the determination of stress magnitude as the primary stimulus for cell injury. The γ of the air-liquid interface in the absence of surfactant is ∼70 dyn/cm rather than static equilibrium γ of 25 dyn/cm of pulmonary surfactant. Whereas our prior experiments demonstrated the protective effect of high concentrations of surfactant, it is not feasible to vary the concentration of surfactant in the present study because of the complexities of dynamic γ effects. For example, even during dynamic equilibrium, the γ is not uniform over the surface of the bubble, which could confound our results due to surfactant physicochemical hydrodynamic effects (15), although we expect that those effects would be minimized with highly concentrated surfactant (5). At present, the γ and viscosity of lining fluid in a patient suffering from ARDS are not known; however, the γ is likely to be elevated due to protein leakage from the vascular system that has been shown to deactivate surfactant (19).
Although there are clearly a number of limitations to our investigations, the idealized models used for this study have allowed us to elucidate the importance of mechanical stress on epithelial cell damage during airway reopening. Additional components will be incorporated in these experimental systems to create better models of the physiological environment and thus to isolate critical aspects related to airway damage.
In conclusion, the goal of this paper was to establish the relative importance of mechanical stress magnitude vs. duration in causing damage to pulmonary epithelial cells during the opening of collapsed pulmonary airways. The results of this idealized study confirm that the progression of a semi-infinite bubble in a narrow channel lined with pulmonary epithelial cells inflicts significant injury to an epithelial cell population. In addition, by using the same U with occlusion fluids of varying viscosities, we have established that the magnitude of the dP/dx, and not the duration of bubble exposure, best predicts the degree of injury to the cell population in airway reopening. Finally, we have demonstrated that multiple passages of a bubble across the epithelial-cell surface can exacerbate the damage and lead to ablation of the cell layer. Future studies should investigate the critical surfactant concentration necessary to protect the airway from damage and the importance of airway wall flexibility.
This work was supported by National Science Foundation Grant BES-9978605 and National Aeronautics and Space Administration Physical Sciences Research Division Grant NAG3–2734. Computational facilities were provided by the Center for Computational Sciences at Tulane and Xavier universities.
We appreciate the insightful comments of the reviewers. We thank Lorraine McGinley for research administrative assistance.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
- Copyright © 2004 the American Physiological Society