Biomechan Model Mechanobiol DOI 10.1007/s10237-007-0075-0
O R I G I NA L PA P E R
Three-dimensional simulation of anisotropic cell-driven collagen gel compaction Toshiro K. Ohsumi · Joseph E. Flaherty · Michael C. Evans · Victor H. Barocas
Received: 19 July 2005 / Accepted: 30 December 2006 © Springer-Verlag 2007
Abstract Tissue equivalents (TEs), formed by entrapping cells in a collagen gel, are an important model system for studying cell behavior. We have previously (Barocas and Tranquillo in J Biomech Eng 117:161–170, 1997a) developed an anisotropic biphasic theory of TE mechanics, which comprises five coupled partial differential equations describing interaction among cells and collagen fibers in the TE. The model equations, previously solved in one or two dimensions, were solved in three dimensions using an adaptive finite-element platform. The model was applied to three systems: a rectangular isometric cell traction assay, an otherwiseacellular gel containing two islands of cells, and an idealized tissue-engineered cardiac valve leaflet. In the first two cases, published experimental data were available for comparison, and the model results were consistent with the experimental observations. Fibers and cells aligned in the fixed direction in the isometric assay, and a region of strong fiber alignment arose between the T. K. Ohsumi Department of Computer Science, Colgate University, Hamilton, NY, USA J. E. Flaherty Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY, USA M. C. Evans Department of Materials Science, University of Minnesota, Minneapolis, MN, USA V. H. Barocas (B) Department of Biomedical Engineering, 7-105 Hasselmo Hall, University of Minnesota, 312 Church St SE, Minneapolis, MN 55455, USA e-mail:
[email protected]
two cell islands. For the valve problem, the alignment predicted by the model was generally similar to that observed experimentally, but an asymmetry in the experiment was not captured by the model.
1 Introduction Since the advent of the fibroblast-populated collagen lattice (Bell et al. 1979), “tissue equivalents” (TEs) formed by entrapping contractile cells in a collagen matrix have been demonstrated to be an efficient and easily controlled system for studying cell behavior. TEs have been used extensively to study contractility of fibroblasts (Brown et al. 1998; Ehrlich et al. 2000; Guidry and Grinnell 1986; Piscatelli et al. 1994; Taliana et al. 2000; Zhu et al. 2001) and other cell types (Chen et al. 1991; Eschenhagen et al. 1997; Smith-Thomas et al. 2000). Volume change of a compacting TE provides a quantifiable measure of cell contractility. Converting compaction rate to cellular stress (or force per individual cell) is complicated by the need to understand the mechanics of the collagen matrix. Compaction is an indirect measure of cell contractility, requiring knowledge of the gel mechanics to calculate the cell stress. The mechanics of collagen gels are highly complex (Barocas et al. 1995; Knapp et al. 1997; Ozerdem and Tozeren 1995; Roeder et al. 2002), making interpretation difficult. An alternative strategy involves direct measurement of the cell force by attaching the TE to a force gauge (Eastwood et al. 1996; Kolodney and Wyslomerski 1992). The direct approach has the advantage that the total force is easily quantifiable, but the disadvantage that unlike the indirect methods above, it necessarily involves an inhomogeneous deformation
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field. Inhomogeneous compaction presents two challenges. First, the stress and cell density are not uniform, so interpretation of the results is difficult. Second, the inhomogeneous deformation leads to anisotropy of the fiber matrix, which in turn causes anisotropy in the cells. Thus, the number of cells aligned (and exerting traction primarily) along the measurement axis increases over time. The Anisotropic Biphasic Theory (ABT) of TE mechanics (Barocas and Tranquillo 1997a) has the potential to elucidate the interactions among inhomogeneous deformation, cell alignment, and force measurement. Specifically, the ABT accounts for
• • • •
•
• •
•
Contractile stress arising from cell traction on the surrounding collagen matrix. Cell spreading and the increase in cell traction as cells adapt to the new environment (Barocas et al. 1995). Cell migration, which is similar to Fickian diffusion in most cases of interest (Noble and Shields 1989). Cell proliferation, represented as exponential in the 1997 paper but also representable using logistic growth when appropriate (Moon and Tranquillo 1993; Ohsumi et al. 2000). Mechanical properties of the gel, including collagen matrix viscoelasticity and permeability to water because the gel compacts by exuding interstitial water (Knapp et al. 1997), and the deformation of the gel that arises due to contraction of the cells. Contact guidance, the tendency of cells to align with an anisotropic collagen matrix (Guido and Tranquillo 1993). Alignment of the collagen fibers and fiber reorientation due to anisotropic deformation of the matrix and/or prealignment of the matrix (e.g., by gelation in a strong magnetic field; Torbet and Ronziere 1984). The theory accounts for an average fiber represented by an orientation matrix. Fiber orientation and contact guidance are crucial features of tissueequivalent compaction and have been exploited in the synthesis of bioartificial blood vessels (Barocas et al. 1998; L’Heureux et al. 1993). Contact inhibition of cell contractile stress, the tendency of cells to exert less contractile stress once their concentration reaches a certain level (Oster et al. 1983). This effect was not in the original theory but was added later (Ohsumi et al. 2000). Although the contact inhibition hypothesis is not proven, this effect represents a convenient way to account for other phenomena that limit the extent of compaction of cell-populated gels.
The ABT equations have been solved previously (Barocas and Tranquillo 1997b; Ohsumi et al. 2000) for conditions of axisymmetry, and qualitative analysis of the direct cell traction measurement experiment was performed for a cylindrical assay. The actual experiments (e.g., Kolodney and Wyslomerski 1992), however, are performed almost exclusively in the slab geometry, lacking axisymmetry and requiring full 3D analysis. A similar need arises in the design of engineered bioartificial tissues. An artery, being cylindrical, can easily be modeled axisymmetrically (Barocas et al. 1998), but a more complex structure, such as a cardiovascular valve (Neidert et al. 1999), obviously requires a 3D platform for any meaningful analysis. The objective of this work was to solve the ABT equations in three dimensions, creating a platform for analysis of various 3D cell compaction problems. Model problems were based on published experimental studies (Kolodney and Elson 1993; Sawhney and Howard 2004; Neidert and Tranquillo 2006) with the intention of exploring and, if possible, quantifying the observations made therein.
2 Methods 2.1 Anisotropic Biphasic Theory The Anisotropic Biphasic Theory has been published previously (Barocas and Tranquillo 1997a), so only a brief summary is presented here. The theory consists of four conservation equations—cell concentration (c), matrix volume fraction (θ), matrix velocity (v), which is a vector, and pressure (p)—and a constitutive equation for the stress (σ ) in the matrix, a second-order tensor, which is modeled as an upper-convected Maxwell fluid. The five equations, representing, in order, cell conservation, collagen matrix conservation, mechanical equilibrium of the matrix, mechanical equilibrium of the interstitial fluid, and the constitutive law for the network, are as follows: Dc = k(cmax − c)c + ∇ • [D0 C • ∇c] − c(∇ • v) Dt Dθ = −θ(∇ • v) Dt τ0 cθ ∇ • θσ + C − ∇p = 0 1 + λc2 1−θ ∇p − ϕv = 0 ∇• θ 1 ν 1 1 σ˙ + σ = ∇v + (∇v)T + (∇ • v)I 2G 2µ 2 1 − 2ν
(1) (2) (3) (4) (5)
Three-dimensional simulation of anisotropic cell-driven collagen gel compaction
where I is the second-order unit tensor, and the parameters are D0 (cell migration coefficient), k (cell proliferation rate constant), cmax (logistic growth maximum cell concentration), G (matrix shear modulus), µ (matrix shear viscosity), λ (contact inhibition coefficient), and ϕ (interstitial drag coefficient) are taken as constant. The dot in Eq. (5) represents the upper-convected derivative, and the upper-convected Maxwell model has been shown to be accurate for tissue equivalents up to 10– 15% strain (Knapp et al. 1997). As reported previously (Barocas et al. 1995), we found that the theory matched the experiment better if τ0 (cell traction coefficient) was allowed to vary with time to account for the initial period of cell spreading. The cell traction and migration terms are anisotropic according to the cell orientation tensor C , which is determined from a fiber orientation tensor, which is in turn determined from the deformation of the matrix. An additional parameter, κ, is introduced to account for the degree of contact guidance, that is the amount of cell orientation that occurs in response to a given amount of fiber orientation; more information on C and κ is provided in the Appendix.
2.2 3D adaptive finite element method The ABT equations were solved using a modified version of the h-adaptive object-oriented finite element method framework Trellis (Beall and Shephard 1999), which includes an algorithm-oriented mesh database (Remacle and Shephard 2003) and mesh motion (Ohsumi 2003). Trellis was modified further to compute C at each integration point, based on the local deformation gradient. The initial domain was tessellated into tetrahedral elements using Gmsh (Geuzaine and Remacle 2004). Trellis automatically enriches the mesh by using the ZZ error indicator (Zienkiewicz and Zhu 1987), which computes a local approximation of the gradient of the solution using a least squares fit over a patch of elements, refining the mesh where the gradient is large, such as stress concentration near the boundary where the no-slip and free conditions meet. It has been shown that enriching the mesh has reduced computation time by an order of magnitude while maintaining numerical accuracy (Ohsumi et al. 2000). Trellis solves the temporal component using the method of lines. Since the split between spatial and temporal domains results in a system of differential algebraic equations, the backward-difference-formula-based package DASPK (Brenan et al. 1996) was incorporated into Trellis. DASPK adjusts the time step and method order based on an error estimate, similar to mesh enrichment. Trellis
enriches the history vector in DASPK at the same time as the mesh, allowing DASPK to restart without loss in a method order, thus further reducing computation time by a factor of four for test problems (Ohsumi 2003). Simulations were performed on an SGI Altix; most simulations required between 24 and 48 h of CPU time to complete.
2.3 Case 1: Isometric cell traction assay (Kolodney and Wyslomerski 1992) The first case studied was a rectangular slab of cellpopulated gel with one pair of opposite faces fixed. The isometric slab geometry has been used by various researchers (Eastwood et al. 1996; Kolodney and Elson 1993) to monitor cell force generation by attaching the fixed ends to a force transducer. The simulations here were based on the work of Kolodney and Wysolmerski (KW, 1992). KW reported a thickness of 8 mm and a width (set by their mounting posts) of 70 mm. They did not report sample length, but based on the 25-mm trough in which the sample was formed and the 3-mm diameter of each post, we estimated a sample length of 19 mm. The ends attached to the posts were assumed to allow no slip, and the other surfaces were assumed free. By symmetry, only one-eighth of the slab was modeled. Symmetry planes were assumed to have no shear stress or normal displacement. Kolodney and Wysolmerski used 107 chick embryo fibroblasts in a 13-ml sample, corresponding to an initial cell density of 7.75 × 105 cells/ml. KW did not report spreading data; since their cell density was higher than any studied by Barocas et al. (1995), we took the largest cell density case and used a spreading half-time of 8.7 h. KW used 1 mg/ml collagen, corresponding roughly to an initial collagen volume fraction θ0 = 0.001. Other model parameters were based on our previous studies. We took G = 1.5 × 104 Pa, µ = 7.4 × 108 Pa·s, ν = 0.2, k = 5.3 × 10−6 s−1 , and D0 = 1.7 × 10−10 cm2 /s (Barocas and Tranquillo 1997a). The interstitial drag coefficient was set to 3.2 × 108 Pa·s/cm2 (Knapp et al. 1997). The contact inhibition coefficient λ was set to 6 × 10−12 cm6 /cell, based on a best fit of the data of Barocas et al. (1995). The long-time (i.e., after spreading was complete) value of the cell traction parameter τ0 was set to 0.0114 Pa/(cell/cm3 ) based on the KW’s reported stress coefficient, adjusting the units to account for differences in the definition of the coefficient. The value of κ was set to 5, based on Girton and Tranquillo (2002). Two different cases were studied to assess the role of geometry and cell alignment in determining the cell
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traction stress. The geometry of the sample was varied by changing which coordinate axis was isometric. That is, in addition to simulating the actual KW experiment, we also simulated the experiments that would be done if instead of a “short” 19 × 70 × 8 mm sample, a 70 × 19 × 8 mm sample was used (the first direction being the axis along which the sample is held isometrically). The “long” sample was expected to exhibit greater lateral compaction and stronger axial alignment, which would have been undesirable in the KW experiment but has been exploited in other tissue engineering applications (e.g., Shi and Vesely 2003, 2004). For each case, 28 h of compaction was simulated. The measured isometric cell force was calculated by integrating the total stress (sum of the pressure, matrix stress, and cell traction stress terms) over the fixed surface.
2.4 Case 2: “Strap” formation between cell islands (Sawhney and Howard 2004)
Fig. 1 Tissue-engineered cardiovascular valve. a Typical geometry used in leaflet design. b idealized model
y − y1 2 z − z1 2 x − x1 2 c = c0 exp − − − δx δy δz
y − y2 2 z − z2 2 x − x2 2 − − + exp − δx δy δz (6)
Sawhney and Howard (2004) conducted studies in which two small fibroblast-populated regions were created within an otherwise acellular gel. Over time, SH observed “ligament-like straps” to form between the cell-populated regions, even though there were no cells in the region, highlighting the importance of long-range interactions among cells mediated by the interconnected collagen network. Although SH used a different cell type (human periodontal ligament fibroblasts), we used the same cell parameters as in case 1 because periodontal ligament fibroblast parameters were not available. The SH experiment consisted of three steps. First, an initial acellular collagen gel was formed by putting 525 µL of solution in a 35 mm dish, which corresponds to a gel thickness of 0.55 mm. Next, cells were seeded onto the gel in two pipet drops roughly 0.5 mm in diameter. Based on the images in SH (e.g., their Fig. 1a), we estimate that the edges of the drops were 0.5–0.6 mm apart. Finally, Sawhney and Howard added an additional 475 µL of solution, corresponding to an upper layer 0.49 mm thick. The two layers also had slightly different collagen densities, but for convenience we set both layers to the upper-layer concentration of θ0 = 0.0015 and to be 0.5 mm thick. Sawhney and Howard assumed that the two collagen layers fused to form a continuous gel, and that assumption was made in the model. Because of the very large size of the SH disk relative to the important regions, we truncated the SH domain and considered a disk of diameter 5 mm. The cell-populated regions were constructed by introducing an initial cell distribution function of the form
where c0 is a scaling constant, (xi, yi , zi ) is the center of cell-populated region i, and δj is the decay distance for the concentration in direction j. For the SH system, we centered the regions at (± 0.5 mm, 0, 0) and set δ = (0.2 mm, 0.2 mm, 0.1 mm). The concentration in the center of the gel (0, 0, 0) is thus approximately 0.4% that at the center of a cell-populated region; this was assumed to be negligibly small. The constant c0 was set to 106 cells/cm3 , and simulations were run for 8 h of simulated compaction (the amount of time SH allowed their gels to compact before adding cytochalasin D). 2.5 Case 3: bioartificial heart valve leaflet (Neidert et al. 1999; Neidert and Tranquillo 2005) The third case studied was compaction of an idealized model of a tissue-engineered cardiovascular valve of Neidert and Tranquillo (Neidert et al. 1999; Neidert and Tranquillo 2006). The valve consists of two neonatalhuman-dermal-fibroblast-seeded leaflets attached to a circular root; a CAD design of one leaflet shown in Fig. 1. In the creation of the valve, the compaction of the root is restricted by an inner mandrel (cf. Girton et al. 1999; L’Heureux et al. 1993); the restriction was modeled by fixing the nodes at the base of the leaflet. The surface of the mold surrounding the leaflet is nonadhesive but impenetrable, so a rigorous model would include solving the contact problem between the mold and the leaflet surfaces (cf. Donzelli et al. 1999). As an initial exploration of the problem, we considered a single flat leaflet consisting of a semicircular “belly” region
Three-dimensional simulation of anisotropic cell-driven collagen gel compaction
and a rectangular “cusp” region. The belly region was specified to be 2.5 cm diameter and 1 mm thick initially, and the edges were assumed to be fixed (i.e., attached to the root). The upper and lower surfaces were free. The cusp region, representing the unconstrained portion of the leaflet, was 1 cm long and free on all surfaces except for the edges of the leaflet. These edges were treated as symmetry boundaries to represent the fact that the leaflet is attached to another leaflet during the incubation (cf. Driessen et al. 2005). Cell and collagen parameters were as in case 1. The initial collagen volume fraction was set to 0.002, and the initial cell concentration was set to 106 cells/cm3 . Compaction was allowed to proceed for 10 h, representing the initial compaction period in which fibril orientation is set. The remaining culture time (weeks) involves significant biological changes in the leaflet, as well as stiffening due to glycation (Girton et al. 1999), but the collagen orientation is relatively unchanged after the initial period.
3 Results 3.1 Case 1: isometric cell traction assay Figure 2 shows the initial conditions of the gel for case 1. To simplify the interpretation, the symmetry planes have been reflected, and the figure thus shows a volume eight times the computational volume. The shape (Fig. 2a) was a rectangular slab, with the fixed ends noted on the figure. The mesh (Fig. 2b) was laid out uniformly, with some stretching of the elements in the long direction so as to keep a manageable problem size. The initial cell density was uniform, as was the collagen volume fraction (for the parameters used, cell growth and migration are slow, so collagen and cell densities track each other for all cases). Finally, the system was initially isotropic. These initial conditions were the same for all cases. Figure 3 shows the same information as Fig. 2 for a TE after 28 h of simulated compaction. The fixed ends did not move, but the free surfaces were pulled in by the action of the cells, resulting in a slight indentation of the slab (Fig. 3a). As was seen in our axisymmetric studies (Ohsumi et al. 2000), stress concentration occurred as the sharp corner developed between the free and fixed surfaces, and the adaptive finite element routine responded by inserting more elements there (Fig. 3b). The concentration (Fig. 3c) was highest near the center of the sample’s free edge and lowest near the corners, where the constraint on the system prevented as much compaction from occurring. Orientation was most
pronounced at the free edges and less so near the center of the sample (Fig. 3d). The “long” version (Fig. 4) showed qualitatively similar but quantitatively different behavior. Since much of the sample was far from the fixed-edge constraint, there was more compaction (6% vs. 4.4% area reduction at the midplane), resulting in higher cell concentration and stronger alignment of the collagen network. The effect was quantified in Fig. 5, which shows the area reduction and the average stress (i.e., the measured force divided by the cross-sectional area) for the “short” and “long” versions. Although the cells are essentially the same, the force per unit area is roughly 10% higher in the “long” case because of compaction-induced-alignment. 3.2 Case 2: “Strap” formation between cell islands Figure 6 shows the initial and final cell concentration profiles in the simulated SH experiment. The peak cell concentration goes down over time due to the small but significant migration of the cells, especially in the thickness direction. Figure 6c shows the very strong fiber alignment that developed between the two isolated regions even though virtually no cells were present there, consistent with the “ligament-like strap” reported by SH. 3.3 Case 3: bioartificial heart valve leaflet The concentration profile (Fig. 7a) and orientation (Fig. 7b) of the simulated valve behaved as expected. Concentration was highest in the cusps, where the compaction of the free surface (the upper surface in the figures) allowed more cells to accumulate. As the surface moved toward the root, the anisotropic strain field in the valve caused circumferential alignment of the collagen and cells. There was a slight reduction in cell concentration near the root as the gel tried to pull away. Figure 7a also shows spontaneous non-uniformity in cell concentration near the edges of the leaflet, an observation to which we will return in Sect. 4. Alignment in the upper portion of the leaflet was strongly horizontal (corresponding to circumferential alignment in the valve). In the lower portion, the alignment tended to be circumferential and to run commissure-to-commissure.
4 Discussion The three-dimensional ABT simulations were consistent with experimental observation and demonstrate the potential of the ABT for prediction of complex
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Fig. 2 Initial state for isometric cell traction assay. a Geometry; b finite-element mesh. The end of the sample is circled in anticipation of Fig. 3b Fig. 3 Final state for isometric cell traction assay. a Geometry; b finite-element mesh. Adaptive refinement is seen at the free end (circled). c Cell concentration (cells/cm3 ); d orientation state. Orientation is visualized by showing the eigenvectors associated with the two largest eigenvalues of C , with the length normalized by subtracting the smallest eigenvalue (leading to length zero for an isotropic state)
behaviors and for design of tissue engineering conditions. Numerical solution by an adaptive finite-element framework provided improved efficiency over fixedmesh techniques. In the cases studies here, the restriction to small strain (and therefore to small gradients) led to relatively little mesh refinement. Thus, r-adaptivity (moving the nodes to track material deformation) was more important than h-adaptivity (inserting and removing nodes to maintain uniform accuracy). As we refine the ABT’s constitutive equations to apply to larger strains, we anticipate the need for greater adaptive refinement, which is within the capability of the current software (Ohsumi 2003; Ohsumi et al. 2000). Our simulations suggest that the “short” geometry of the KW isometric assay successfully minimizes artifacts from spontaneous alignment, with a majority of the sam-
ple remaining roughly isotropic. The “long” geometry provides an example of what could occur if the experiment were poorly designed — a 10% overestimation of the cell traction parameter even at very small compactions. If the experiment were allowed to run longer, the effect would be greater, leading to a serious error in cell traction force calculation. The long version of the KW model can be compared to the early stages of the bioengineered chordae tendinae of Shi and Vesely (2003, 2004). In their extremely thorough characterization of the engineered chordae, Shi and Vesely (2004) reported that cells and collagen fibrils near the edge were straight, elongated, and aligned with the edge. Near the center of the sample, however, the cells and fibrils were wavier. Shi and Vesely suggested, “because the distance between the anchors
Three-dimensional simulation of anisotropic cell-driven collagen gel compaction
Stress (dyn/cm2,
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Fig. 4 Final state for isometric cell traction assay in “long” configuration. a Cell concentration (cells/cm3 ); b orientation state
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Fig. 5 Quantitative comparison of ICTA results. Average stress (dashed lines) and midplane compaction (solid lines) for KW (squares) and “long” geometry (circles). Even at small strains, the greater compaction in the long geometry leads to larger stresses because of cell alignment in the measurement direction
along the edge of the constructs is greater than that along the central fibers, there is an uneven strain or stress distribution: low in the center and increasing toward the edges.” This qualitative suggestion was consistent with our simulations of a similar geometry, especially near the fixed ends of the sample. The SH case study showed the ability of the model to predict collagen gel reorganization even in cell-free regions, provided that they are in contact with cellpopulated regions. Continued studies could explore the subsequent effects of cytochalasin D, butanedione monoxide, nocodazole, and deoxycholate addition, as performed by SH, but doing so would require a more sophisticated model of the contribution of different cytoskeletal components to the traction stress and/or passive mechanics of the TE (Zahalak et al. 2000). The idealized leaflet showed circumferential and commissure-to-commissure alignment. This is comparable to the strong commissural alignment reported by NT (Fig. 4 of Neidert and Tranquillo 2006). NT reported
the weakest alignment in the belly region (bottom center of the leaflet), which is consistent with our model results. NT also reported a significant symmetry break, however, with a positive primary orientation angle (i.e., fiber alignment from lower left to upper right) in all regions of the leaflet. Since our model was symmetric, we did not reproduce that result. Further experiments on an idealized system with a fixed and a free surface are currently under way in our group. The spontaneous nonuniformity of concentration shown in Fig. 7a is attributed to two factors. Initially, small inhomogeneities can arise due to numerical errors. Subsequently, we believe that an instability arises comparable to that reported by others (Cruywagen et al. 1997; Oster et al. 1983) for similar models. The ABT did not exhibit such behavior in our previous spherically symmetric (Barocas et al. 1995) or axisymmetric (Barocas et al. 1998; Barocas and Tranquillo 1997a; Ohsumi et al. 2000) studies, presumably because the imposed symmetry quashed any instability. The ABT, although capturing many important phenomena, is restricted to short compaction times (hours) by a number of limitations. The upper-convected Maxwell model accounts for nonlinear kinematics, but material nonlinearities in mechanical response of the collagen network become significant above 10–15% strain (Knapp et al. 1997). Equations (1)–(5) do not account for synthesis or degradation of collagen, nor do they account for synthesis of other ECM components, both of which occur during long-term culture, especially as observed recently in fibrin-based constructs (Grassl et al. 2003; Long 2002). Finally, the model does not incorporate the contribution of cells to the mechanical properties of the tissue equivalent, an effect that has been recognized to be important at high cell concentrations (Wakatsuki et al. 2000; Zahalak et al. 2000). In our opinion, these are the primary challenges to be overcome in creating a broad model of tissue-equivalent mechanics. In the particular applications here, it was assumed that the collagen gels had identical properties up to a
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Fig. 6 Simulated SH experiment. a Initial cell concentration in the midplane, showing the two “islands.” b Final cell concentration (cells/cm3 , color) and fiber orientation (vectors) in the midplane. c Close-up of final state near and between the two islands
Fig. 7 Dimensionless cell concentration and orientation in the midplane of the compacted idealized leaflet. a The leaflet has compacted from its initial length, which is shown by the black line at the top of the figure. The concentration (cells/cm3 ) is highest at the commissures and lowest at the root, but the overall variation in concentration is small due to the logistic growth term in Eq. 1. b Alignment develops in the free leaflet due to radial compaction (vertical in the image). Arrow length corresponds to degree of alignment. Thickness (i.e., out-of-plane) compaction also occurred but is not visible in the planar view
scale factor for collagen volume fraction. Collagen gels made under different conditions exhibit different structural and mechanical properties (Roeder et al. 2002), so our model must be viewed as a representative but not exact with respect to collagen gel properties. Another concern is that the model parameters for cell behav-
ior, τ and κ, are potentially cell-line dependent. In the current study, we used one set of cell parameters even though the fibroblasts in the experimental studies were from chick embryo (KW), human periodontal ligament (SH), and neonatal human dermis (NT). The quantitative results would likely be affected by the differences between cell lines, but qualitative behavior is expected to be captured adequately using the same properties for all cell types. There are also limitations to the numerical implementation of the model. The current ABT code does not allow for contact boundaries (i.e., those that obstruct motion into the boundary but do not impede motion away from or along the boundary), which will be important in engineering complex structures (e.g., heart valves). Contact boundary conditions are well within the general capabilities of Trellis, and we are working on applying them to the ABT equations. Also, the current implementation cannot handle a change in boundary condition for a given surface during the course of compaction. Finally, although parallelization of the software is not strictly necessary, it would dramatically reduce computation time and may prove a practical necessity, especially for optimization and design applications requiring multiple simulations to identify the optimal parameter set.
Three-dimensional simulation of anisotropic cell-driven collagen gel compaction Acknowledgments This work was supported by the National Science Foundation’s VIGRE Program (grant DMS-9983646) and by the National Institutes of Health (1R01 HL071538-01). Simulations were made possible by a resources grant from the University of Minnesota Supercomputing Institute.
Further details are available in Barocas and Tranquillo (1997a).
Appendix: Supplementary ABT Equations The equation set (1)–(4) contains the basic conservation laws for the ABT, and Eq. (5) provides the mechanical constitutive equation for the network. The full specification of the model, however, requires a constitutive equation for the cell orientation tensor C , which appears in Eqs. (1) and (3). The key assumptions of the ABT’s constitutive description are •
A second-order fiber orientation tensor F , capturing the average orientation of a local distribution of fibers, is defined by F ≡ 3
nF (θ, φ)⊗nF (θ, φ)P(θ, φ)dθdφ
(A-1)
θ,φ
•
where nF is the vector corresponding to a fiber with spherical orientation angles θ and φ, P is the probability distribution function, and ⊗ denotes dyad product. The factor of 3 normalizes the system so F is the identity for a uniform fiber distribution. If, for example, all of the fibers were aligned in the x direction, then F,xx would be 3, and all other components of F would be zero. The probability distribution P is based on the deformation ellipsoid, which is calculated from the finger deformation tensor B = (∇x)(∇x)T , where the gradient of position x is with respect to initial position. Although it is possible to introduce initial orientation by assigning F a non-identity value initially (Barocas et al. 1998), for the problems here, the initial value of F was always the identity. A cell orientation tensor C is defined similarly, but rather than tracking individual cells, C is assumed to be a monotonic function of F to represent contact guidance. That is, the cells tend to align in the direction of the fibers with a positive parameter, κ, that captures the strength of cellular alignment response. Specifically, C =
sors are the same. If κ = 0, the cells are isotropic and independent of any fiber alignment, causing (1)–(5) to reduce to an isotropic model. If κ > 1, the cells are very sensitive to fiber alignment.
3 (F )κ tr(F )κ
where, as above, the scalar prefactor normalizes the system. If κ = 1, the cell and fiber orientation ten-
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