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1Department of Biomedical Engineering, Northwestern University, Chicago, Illinois 60208; 2Department of Biomedical Engineering, Marquette University, Milwaukee, Wisconsin 53201; and 3Department of Physiology, University of Tennessee Health Science Center, Memphis, Tennessee 38163
Submitted 14 May 2003 ; accepted in final form 1 October 2003
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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spreading; migration; proliferation; mitomycin C; cytochalasin D; 5-bromo-2'-deoxyuridine
The airways undergo cyclic deformation at a frequency dependent on the respiration rate, and we have recently shown that these forces impact cellular motility, spreading, and proliferation (17, 29). Others have demonstrated that mechanical deformation of alveolar epithelial cells causes increased cell death (25, 26) and plasma membrane stress failure (27, 28). We used an in vitro model to study the repair mechanisms of wounded monolayers of human and cat AEC cultured on elastic membranes and subjected to cyclic strain. Owing to the radially dependent strain profile in the older Flexercell plates, we were able to compare the effects of cyclic elongation and compression in the same culture well (17), and we verified these findings in a custom-built stretching device that applied uniform biaxial strain or compression (29). We have shown that both cyclic elongation and compression significantly attenuate wound closure in a frequency-dependent manner, in part by limiting spreading and migration (17). To gain more insight into these mechanisms and to develop tools to aid in our interpretation of experimental results, we developed a mathematical model of the wound closure process.
Several mathematical models of wound closure have been developed that contain nonlinear reaction and diffusion terms, with the simplest and most cited model developed by Fisher (6)
 | (1) |
represents a partial derivative, u is cell density, t is time, and x is position. This partial differential equation describes the evolution of a cell density function u(x,t). The change in the density (u/t) is influenced by the second-order influx of particles or cells represented by the 2u/x2 term and by a generation of particles [u(1 - u)]. This equation has also been used with constants added to define a diffusion term (D) and a growth term
 | (2) |
Most previous models of wound healing were developed with the contributions of proliferation and migration alone and provide a framework for understanding the relationship between wound healing and chemical stimulation or inhibition. None of the previous models has specifically addressed the contribution of cellular spreading or the influence of cyclic mechanical strain. In addition, very few studies have compared model predictions with experimental measurements of cell density and wound closure rate. In this paper, we describe a new modification of Fisher's equations to describe wound healing of AECs. New modifications include the addition of a term that describes spreading and the incorporation of a time delay into the mitotic term. Model parameters were estimated by comparing model predictions with experimental data from several AEC types.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Cell culture. Cat tracheal epithelial cells (CTE) were isolated as described previously (17). Isolated cells were seeded on six-well Flex-I plates, 2.5-3 x 105 cells/well, and cultured in CTE media with 10% FBS (Ham's F-12 medium with 10 µg/ml insulin, 7.5 µg/ml endothelial cell growth supplement, 0.5 µg/ml transferrin, 0.4 µg/ml hydrocortisone, 2 µg/ml triiodothyronine, 1% antibiotic/antimycotic solution, 2% sodium bicarbonate). The medium was changed every 2 days, and the cells were used in experiments on days 6 or 7 of culture.
Calu 3 cells (American Type Culture Collection, Rockville, MD) are derived from a human lung adenocarcinoma and resemble tracheal epithelial cells (18). Cells were maintained in T-150 culture flasks in Calu 3 media (MEM, 1 mM sodium pyruvate, 1 mM nonessential amino acids, 0.1% gentamycin, and 10% FBS) before experiments. Cells between passages 18 and 26 were used in experiments 4-5 days after plating into Flex-I six-well plates or manufactured stretcher membranes at 3-3.5 x 105 cells/well (
350,000 cells at confluence). These cells retain constant properties over repeated passages. The medium was changed every 2 days.
AECs transformed with the SV40 virus (16HBE14o-) were obtained from Dr. D. Gruenert and have been characterized by his laboratory (9). Cells were grown in MEM containing 10% FBS, 2 mM L-glutamine, 100 µg/ml streptomycin, and 100 U/ml penicillin G. Cells were seeded on six-well Flex-I plates at 2-3 x 105 cells/well and used on days 4 or 5 of culture (350,000 cells/well at confluence).
Normal human bronchial epithelial (NHBE) cells were obtained from Clonetics (San Diego, CA) and maintained in bronchial epithelial medium (0.5 µg/ml hydrocortisone, 0.5 ng/ml human recombinant epidermal growth factor, 0.5 µg/ml epinephrine, 10 µg/ml transferrin, 5 µg/ml insulin, 0.1 ng/ml retinoic acid, 6.5 ng/ml triiodothyronine, 50 µg/ml gentamycin, 50 ng/ml amphotericin B, and bovine pituitary extract). Passage 2-4 NHBE cells were seeded onto six-well Flex-I plates at 1.75 x 104 cells/well. The medium was changed every 2 days, and the cells were used in experiments on days 4 or 5 of culture.
Experimental protocol. Cells were grown to confluence on Flex-I six-well plates, and a metal spatula was used to create linear wounds of 500 µm width across the diameter of the well. Before the initiation of the experiment, the cells were rinsed twice with PBS to remove cellular debris. Two milliliters of complete growth medium were added to the wells; this medium was not removed during the course of the experiment, unless otherwise indicated. Cytochalasin D (CYD, 1 µg/ml) was dissolved in DMSO before dilution (wound healing was not affected by low concentrations of DMSO, data not shown). CYD was applied for 2 h before wounding and was replaced with unsupplemented growth medium at the time of wounding. CYD was not kept in the media during wound closure because preliminary studies showed that cell detachment began to occur after several hours. Mitomycin C (MMC, 10 µg/ml) was dissolved in growth medium. Images were obtained at the initial time of wounding and at various times up to 72 h postwounding.
Cyclic strain. The original Flexercell strain unit was used in these studies and has been described in detail (8). The system utilizes vacuum pressure regulated by a solenoid valve to deform a silicone rubber substrate on which the cells have been cultured. When the vacuum is applied, the culture plate bottom deforms downward to a known percentage elongation, which is translated to the cells. On release of the vacuum, the Silastic substrate returns to its original conformation. The frequency, duration, and magnitude of applied strain can be varied in this system. When cells were stretched at 30 cycles per min, the membrane was stretched for 1 s and relaxed for 1 s. Finite-element analysis of the membrane strain has shown that, on deformation at -15 kPa, the Flex-I plate yields a nonhomogeneous radial strain profile that is maximum at the periphery (20%) and minimal at the center, with 1-2% compression seen in the center of the stretched wells (8). The newer versions of this device have been designed to reduce the nonuniform strain distribution, but we utilized the older system in these studies to examine both elongation and compression within the same culture well. There was no appreciable cell death over the 48-h time course in any cell type as measured by lactate dehydrogenase release or Trypan blue uptake (data not shown).
Imaging the wound. Images of the wounds were collected at specified times with a Nikon Diaphot 300 inverted microscope equipped with a Hamamatsu integrating charge-coupled device camera, an Argus-20 real-time digital image processor, and a Pentium DataStor computer with a frame grabber (Fryer, Huntley, IL). Images were analyzed by using the Metamorph image analysis program (Universal Imaging, West Chester, PA). After the image was acquired, it was converted from pixels to micrometers by using a calibration image. Using Metamorph, an outline of the wound was manually drawn, and mean wound width w was tabulated by a Metamorph defined parameter. The w is given by
 | (3) |
5-Bromo-2'-deoxyuridine staining. 5-Bromo-2'-deoxyuridine (BrdU) labeling was performed to measure new DNA synthesis as an indication of cell proliferation. For the BrdU studies, cells were grown to confluence, treated, and wounded as described. BrdU (10 µM) was added for 60 min at the end of the incubation period. The cells were washed with a PBS-based wash buffer, fixed in 70% ethanol (in glycine buffer, pH 2.0) for 20 min at -20°C, and washed again. The cells were subsequently incubated with anti-BrdU antibodies (1:10 dilution, Boehringer Mannheim, Indianapolis, IN) for 30 min at 37°C. The cells were rinsed, incubated with anti-mouse-IgG-alkaline phosphatase solution for 30 min at 37°C, and processed according to the manufacturer's instructions. The number of stained cells was determined by visual examination in at least eight high-power fields.
Cell spreading studies. As a measure of cell spreading and motility, the area of cells was measured in images taken of the center and periphery of stretched and unstretched wells over time. Cells were outlined, and the area was calculated with Metamorph imaging software. The accuracy of the cell size measurements was confirmed by measuring cell size of the same field at higher magnifications. For each experimental condition, at least four sets of cells at the leading edge of the wound were traced in each experiment.
Cell density was measured by counting the number of cells contained within consecutive 20 x 100 µm boxes, starting within the wound (0 cells) up to 200 µm from the edge. Density was normalized by the cell number in a box far from the wound edge and was measured in two different images from six different experiments.
Model development. We considered the simplest case of epithelial cell movement controlled mainly by spreading, migration, and later by proliferation. The governing equation takes the following form: rate of increase of cell density = cell migration + cell spreading + mitotic generation.
We did not include a term for cell loss, because it is unlikely that there is appreciable cell death during the time course of examination (24 h). Cell movement was modeled as Fickian diffusion to capture cells moving down a gradient in cell density due to the sudden loss in contact inhibition. In addition to migration and proliferation terms, we included a term for cellular spreading, because the spreading of cells at the wound edge also contributes to wound closure, separately from migration. In the model, parameters are based solely on the existing mechanical environment and not on the production or inhibition of production of autocrine factors or the presence of growth factors, because most experiments were done in basal media (with serum). We also modeled each of the components to have maximal effect at the wound edge, decreasing to a minimal contribution far back from the wound edge. After previous studies, we used a logistic growth form for the mitosis term (20, 22, 24, 32). However, we also incorporated a time delay, because proliferation does not begin until several hours after wounding (5, 12).
The thickness of the monolayer is much smaller than the wound length and width, so we can treat the wound as two-dimensional, and, by considering only a linear wound geometry, we can simplify the spatial domain to only one dimension, x. We solved the equations on the semi-infinite domain 0 
x < 
, with 0 x wo representing one-half of the original wound, where 0 is the center of the wound and wo is one-half of the initial wound width. Because the monolayer did not contain fibroblasts, we can assume that 0 is a fixed boundary and that the wound does not contract due to force generated by the cells. Our mathematical adaptation of the diffusion equation for cell density n(x,t) at position x and at time t is initially
 | (4) |
 | (5) |
Boundary conditions. The biologically relevant initial conditions considered in this model are
 | (6) |
 | (7) |
are zero.
The model results for cell density evolve as diffusive wave fronts to assume the same shape of a front of cells spreading into the denuded area, and, to achieve this set of curves, we solved Eqs. 4 and 5 numerically. Using nondimensionalized parameters, n is normalized by no: n* = n/no; t is normalized by the final time (tfinal): t* = t/tfinal; position x is normalized by wo: x* = x/wo; kp and ks are normalized by tfinal: k* = ktfinal; and the D is normalized by tfinal and wo: 
.
Substituting into Eqs. 4 and 5 and dropping asterisks for convenience, we have the dimensionless model for 0 t 1.
Initially
 | (8) |
 | (9) |
1 x 10-11 cm2/s, and ks and kp were estimated to be 1/h. To account for the possibility of localized minima in the solution space, we often changed the starting point of the parameters. In the case of experimental treatments intended to knock out a specific process (such as treatment with MMC to block proliferation), we selected initial values of 0.0, but allowed the parameter to vary with the optimization. Also, in cases in which we were comparing experimental conditions such as control vs. elongated or compressed, the initial parameters were set at the same values for all groups. If solutions were constrained to a local minima, then it is unlikely that we would have observed changes in the parameter values. In the optimizations involving transformed human bronchial epithelial (16HBE) cells, the time delay parameter was allowed to vary. In the optimizations involving other cell types, there were fewer data points, and the time delay parameter was fixed to reduce the number of parameters to be fit.
Parameter sensitivity. Parameter sensitivity was evaluated to determine whether our model predictions were sensitive to small changes in parameters. Because we wished to evaluate wound-healing parameters by comparing model predictions to the experimental data, we determined the sensitivity functions for each parameter. We calculated Bode's sensitivity function (7) over time using wound width model simulations with nominal parameter values. For example, the sensitivity (S) of predictions of wound width (W) to changes in the D would be given by
 | (10) |
W/D represents the change in predicted wound width when the parameter D is changed from a nominal value to one increased by 20%. Similar sensitivity functions were generated for the other parameters. In general, a nonzero function indicates sensitivity of the model prediction to the parameter, with larger magnitudes indicating more sensitivity. As shown in Fig. 1, there is sensitivity for the measurement of D and ks that increases over the entire time course. In the case of kp and time delay, an increase in sensitivity is seen only after the time delay. These plots indicate that parameter estimations from experimental wound width data will be sensitive to each of the four parameters.
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RESULTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Wound widths. Based on the predictions of cell density, we also predicted the wound width, defined as the position where n is <1%, the last of the points in the spatial vector in which there are no cells. In other experiments, we measured the wound width (not cell density), and Fig. 3 shows experimental wound closure data from four different types of AEC compared with model predictions. We optimized the fits by varying parameters using a nonlinear least squares algorithm, the Levenberg-Marquardt method. The best fit parameters used are shown in Table 1. The time course of wound closure differed for each cell type. For example, wounds in Calu3 cells closed much more slowly than in other cells. In all cases, both cyclic elongation and cyclic compression inhibited wound closure compared with static cells, and these changes are reflected in the parameters (Table 1).
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Effect of cyclic strain. We previously showed that cyclic elongation and compression (30 cycles/min, 20% maximum elongation, -2% compression) significantly inhibited cellular spreading and migration in AEC monolayers (17). As indicated in Table 1, parameters for cells undergoing cyclic strain were altered relative to static cells. For each cell type, the D, indicative of migration velocity, was decreased by cyclic strain. This is consistent with previous measurements of cell migration velocity during wound healing (30). For each cell type, the ks in Table 1 was either unchanged or decreased by cyclic strain. To test this prediction, we measured cell spreading at the wound edge as shown in Fig. 4. The decrease in ks in Table 1 correlates with experimental measurements indicating that elongated cells do not spread compared with static cells, and that compressed cells actually decrease in size. The decrease in size and the corresponding increase in cell density were not predicted well by the model simulations in Fig. 2. Table 1 also suggests that cyclic strain increased cellular proliferation in 16HBE, CTE, and NHBE cells, whereas Calu 3 cells exhibited negligible proliferation. To verify this prediction, we measured the incorporation of BrdU into 16HBE cells at the wound edge. As shown in Fig. 5A, incorporation of BrdU occurred earlier and more extensively in cells undergoing cyclic strain, in agreement with the model predictions.
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Effects of CYD and MMC. We also determined model parameters for wound-healing data from cells treated with CYD (1 µg/ml), which prevents actin polymerization. CYD disrupts the actin cytoskeleton, which is necessary for locomotion and spreading and, in this manner, can be used to decrease or eliminate the contributions of spreading and migration to wound closure. Because cells detached with prolonged exposure to CYD, we pretreated the cells for 2 h before wounding and then removed CYD. Wound healing was substantially inhibited by CYD treatment, as shown in Fig. 6A, and the model parameters in Table 2 show that the ks was essentially zero. The kp did not change dramatically with cyclic strain, although experimental BrdU incorporation was increased with mechanical strain with CYD treatment (Fig. 5B). The increased BrdU labeling could, however, be due to incorporation of DNA during strand repair without proliferation. Because there was little closure in CYD-treated wells and model fits to these data resulted in very low values for the ks, this supports our earlier hypothesis that closure in untreated AEC is dominated by spreading and migration.
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We also determined parameters for the experiments in which we used the DNA cross-linking inhibitor MMC (10 µg/ml) on 16HBE14o- cells to eliminate the contribution of proliferation. We verified that MMC blocked DNA synthesis by demonstrating negligible BrdU incorporation in 16HBE cells (Fig. 5C). MMC treatment resulted in some decrease in wound closure at later times (Fig. 6B), particularly in the compressed and elongated cells, and this was reflected by the changes in parameters in Table 2. MMC effectively lowered the kp to negligible values in all three groups. The effect of MMC on D and ks was small, except in the case of elongated cells, where ks increased twofold. This again supports our findings that wound closure in AEC is dominated by spreading and migration and that the model fits utilize parameters that reflect this.
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DISCUSSION |
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Because spreading and migration are the primary mechanisms of closure in AEC wounds, we added a parameter to the basic wound closure equation to account for spreading, ks. The spreading term contributes to wound closure immediately and has its maximal effect at the wound edge and minimal contribution elsewhere. Although there is a mechanistic difference between spreading and motility (10), no previous models have accounted for this phenomenon. During the initial stages of wound healing, cells flatten and spread, and migration begins. These processes may be indistinguishable initially, but, at later times, cell spreading can be determined by changes in cell area (Fig. 4) and cell migration can be determined by measuring cell velocity (30). Experimental wound closure data suggest that we use a representation of the migration term that is highest at the wound edge and rapidly decays behind the wound edge. An increase in cell density resulting from cell proliferation and wound closure is accompanied by a decrease in average cell speed (34); there should be little spreading or migration once the wound is closed, so all of the contributions (from spreading, migration, and proliferation) should decline, even at the wound edge and back from it when the density approaches no.
We fit the curves to the experimental data by using a least squares routine to optimize the parameters. Parameter optimization yielded biologically relevant parameters, which reflected changes in wound closure processes after application of cyclic strain. For example, in most AEC types, strain induced an early increase in proliferation and decreased spreading and migration. This was reflected by increased kp. The spreading/migration term (D) for cells in static wounds was often twice as large as D in the compressed case and more than triple that in the elongated case. There were also decreases in ks with strain compared with ks in the static case. This indicates that our model is sensitive to alterations in strain and that it can recognize changes in wound width and density data. One potential complication of our experimental approach is that we examined both elongation and compression in the same culture well. Because cellular responses to stretch and compression may differ, the presence of both factors in the same well may limit the ability to distinguish the effects of each. However, we demonstrated nearly identical inhibition of wound closure, cell spreading, and cell migration when we compared cells stretched at different locations in the old Flex-I system with cells strained with the corresponding magnitude of uniform stretch or compression using our custom-made biaxial stretching device (29). Also, we were careful to obtain measurements at the same radial location within the wells so that the strain amplitude would be consistent. For example, maximal compression (2%) occurred at the center of the wells, and we consistently measured wound widths at the center location. One limitation of our model is that it did not predict the increase in cell density in the compressed cells; however, the overall wound closure was predicted by the diffusive wave solution.
These changes were also reflected when proliferation and spreading and migration were isolated pharmacologically. The agent CYD was used experimentally to isolate the contribution of spreading and migration. Wounds treated with CYD closed only slightly, and the parameters reflected these changes. D and ks were significantly decreased, and, to account for the healing that did occur in these wells, kp was increased and the time delay was decreased (Table 2). However, experimental measurements of BrdU incorporation in Fig. 5B do not suggest that proliferation was increased substantially, and an increase in proliferation within 1 h is unlikely. The reduced time delay in the parameter estimation results in ks and kp being indistinguishable. Because there was little closure in CYD-treated wells and model optimization, using these data resulted in lower values for the ks. This supports our hypothesis that closure in these cells is dominated by spreading and migration. Because prolonged exposure to CYD caused detachment of cells, we treated the cells for only 2 h before wounding. It is likely that CYD diffused out of the cells during the time course of wound healing, and actin repolymerization may have occurred during the later stages of the time course. Thus, because proliferation is unlikely to occur in <10 h, the lower time delay more likely reflects the recovery of the cells after CYD diffusion out of the cells. However, even this pretreatment profoundly affected cell spreading and migration during wound closure, and the parameters reflect this. Although the D for compressed and elongated cells was unaffected by CYD, in the static cells CYD treatment substantially decreased D.
Significant cell division as a mechanism of wound healing is not likely to occur in cells before 15-24 h (5), but may contribute to wound closure after this time. We observed in Fig. 5A (BrdU data) both elongation and compression stimulated early (4 h) and increased DNA synthesis while BrdU staining was not apparent in static wells until 10 h postwounding. BrdU staining was more abundant in strained wells than in static wells. Despite this increase, only 20% of the total cells were ever labeled. Although there may be basal proliferation occurring before the monolayer is wounded, it is unlikely that substantial proliferation occurs immediately. To account for this, our model incorporates a time delay before which spreading and migration are solely responsible for closure. After the time delay, the proliferation term is defined similarly to spreading; the term has the largest contribution at the wound edge and falls off to zero rapidly behind the edge. This scheme is more biologically consistent for AECs. Zahm et al. (34) showed that maximal proliferation occurs at the wound edge and at an intermediate distance behind it; in examining BrdU-labeled wounded monolayers, there was little to no staining observed more than 100 µm back from the edge. In the absence of significant spreading and migration, these cells increased proliferation as a mechanism to promote wound closure. The model reflected this with larger parameters for proliferation with mechanical strain. The contribution of proliferation was similarly investigated with MMC. The largest differences from the untreated case are shown with a lower kp compared with untreated (Table 2). This is consistent with the effect that MMC has on these cells; they should not proliferate in the presence of MMC, hence kp should be minimal. These data also demonstrate that model fits to the data are consistent with independent biological data (Figs. 4 and 5). Although the time delay parameter increased with MMC treatment, there is probably little sensitivity to the estimation of this parameter when kp is very low.
Previous models of wound closure have not incorporated a time delay, nor a separate parameter for spreading, although studies have noted that migratory and proliferative behavior are highly dependent on time and location (34). Moreover, very few models of wound healing have been directly compared with experimental data. While spreading and migration are two separate mechanisms, previous models have not accounted for this fact. The time delay for the proliferation term is important because we have shown that proliferation does not begin for several hours in AEC; our representation of proliferation accounts for this. We were able to make several improvements over the most sophisticated existing model (3) with the use of biologically relevant boundary conditions and parameters. Our model serves as a first step toward defining parameters that can be used to characterize wound closure under different conditions and to determine parameters that are affected. Furthermore, our model predicted changes in parameters from wound-healing data that were then used to design experiments to verify these changes independently.
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FOOTNOTES |
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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.
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), elongated (
), and compressed (
) cells, and curves represent model predictions. Values are means ± SE; n = 6. Fits were determined by fixing the tdelay at 9.6 h for CTE and NHBE and 33.6 h for Calu3 cells (note the difference in time scale). The 3 other parameters were varied; parameters are listed in 
), elongated (
) cells. Cells treated with cytochalasin D (CYD) exhibited little change in area for static (
), and smaller changes in area with compression (
25 (n is different for each time point). *Significant difference from static or between treated and untreated, P < 0.05.
