Cytotechnology 10: 225-250, 1992. @ 1992KluwerAcademic Publishers. Printed in the Netherlands.

Tissue engineering science: Consequences of cell traction force Robert T. Tranquillo, Mohammed A. Durrani and Alice G. Moon Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA Received 11 May 1992; acceptedin revisedform 25 September 1992

Key words: cell traction, collagen gel, contact guidance, fibroblast-populated, mathematical model, wound healing

Abstract Blood and tissue cells mechanically interact with soft tissues and tissue-equivalent reconstituted collagen gels in a variety of situations relevant to biomedicine and biotechnology. A key phenomenon in these interactions is the exertion of traction force by cells on local collagen fibers which typically constitute the solid network of these tissues and gels and impart gross mechanical integrity. Two important consequences of cells exerting traction on such collagen networks are first, when the cells co-ordinate their traction, resulting in cell migration, and second, when their traction is sufficient to deform the network. Such cell-collagen network interactions are coupled in a number of ways. Network deformation, for example, can result in net alignment of collagen fibers, eliciting contact guidance, wherein cells move with bidirectional bias along an axis of fiber alignment, potentially leading to a nonuniform cell distribution. This may govern cell accumulation in wounds and be exploited to control cell infiltration of bioartificial tissues and organs. Another consequence of cell traction is the resultant stress and strain in the network which modulate cell protein and DNA synthesis and differentiation. We summarize, here, relevant mathematical theories which we have used to describe the inherent coupling of cell dynamics and tissue mechanics in cell-populated collagen gels via traction. The development of appropriate models based on these theories, in an effort to understand how events in wound healing govern the rate and extent of wound contraction, and to measure cell traction forces in vitro, are described. Relevant observations and speculation from cell biology and medicine that motivate or serve to critique the assumptions made in the theories and models are also summarized.

Abbreviations: ECM - Extracellular Matrix; FPCL - Fibroblast-Populated Collagen Lattice; FPCM Fibroblast-Populated Collagen Microsphere

Introduction The mechanical interaction of motile cells with fibers in the surrounding extracellular matrix (ECM) is fundamental to cell behavior in soft

tissues and tissue-equivalent reconstituted gels, and thus to many problems in tissue engineering spanning from biomedicine to biotechnology. Examples are wound healing (Ehrlich, 1988; Montandon et al., 1977), tissue regeneration (Ma-

226 dri and Pratt, 1986; Trinkaus, 1984), bioartificial skin and organs (Yannas et al., 1982; Hu et al., 1991), and mammalian cell immobilization in bioreactors (Nilsson and Mosbach, 1987; Nilsson et al., 1983). Here, we introduce the nature and consequences of this mechanical interaction through a detailed description of wound contraction and a relevant in vitro analogue, the fibroblast-populated collagen lattice assay. In the sections that follow, we summarize existing and newly proposed theories for ceI1-ECM mechanical interactions and then apply them to mathematical models of wound contraction and the in vitro analogue. W o u n d healing and contraction

Wound repair is a basic and vital process for homeostasis. It is also complex: in full-thickness cutaneous wounds, where loss of dermis has occurred, there are characteristic phases of immediate inflammation, short-term granulation tissue formation (simultaneous fibroplasia and angiogenesis), and long-term wound matrix remodeling (Clark, 1988). Wound contraction is another hallmark in the healing of these wounds (Montandon et al., 1977). It often occurs during fibroplasia, the repopulation of the fibrin clot by dermal fibroblasts and wound fibroblasts (phenotypically modified dermal fibroblasts called myofibroblasts) (Guber and Rudolph, 1978; Montandon et al., 1973; Skalli and Gabbiani, 1988). Compelling evidence now suggests that these fibroblastic cells, which secrete the collagen needed to transform the weak fibrin clot to a collagenous matrix with mechanical integrity, are also responsible for the forces underlying wound contraction (Rudolph, 1979). However, the migration and growth signals which determine the spatial and temporal distribution of fibroblastic cells in and around a wound, as well as the factors which govern the magnitude of force transmitted by the cells to local ECM fibers, are poorly understood. These issues underlie the different hypotheses about the locus of cell force during wound contraction (Rudolph, 1979). Two popular views are the "picture frame" hypothesis, where fibroblastic

cells are thought to pull the wound boundary in as they migrate inward from the wound edge, and the "pull" hypothesis, where fibroblastic cells already within the wound are proposed to generate a tension which causes the inward movement of the wound boundary. Even the nature of the cell force has been debated (Ehrlich and Rajaratnam, 1990; Ehrlich, 1988), with it being considered either a "contraction force" associated with the hypothesized shortening of the entire cell body of a smooth muscle cell-like myofibroblast (Skalli and Gabbiani, 1988), or a "traction force" associated with the continuous pseudopodal activity of stationary or migrating fibroblasts cultured in a reconstituted collagen gel (Harris, 1982, 1984; Harris et al., 1980, 1981, 1984). In either case, however, alignment and packing of local fibers around the cell results, termed "tractional structuring" (to economize, we use fiber in many instances where fibril may be more appropriate). Thus, the motility mechanism by which fibroblasts and/or myofibroblasts cause wound contraction is still an open question. Here, however, we shall refer to the operative force solely as traction force, reflecting our bias, and use "fibroblast" to mean any fibroblastic cell exerting traction. Nonetheless, the traction which presumably allows a fibroblast to migrate from the surrounding dermis into the developing granulation tissue concurrently causes some local reorganization of ECM fibers. This tractional structuring causes a local deformation of the ECM. The macroscopic manifestation of all the local deformations due to the traction forces exerted by all the fibroblasts involved in wound healing is wound contraction. Of course, other forces, such as viscous, elastic, and osmotic forces also contribute to the actual rate and extent of wound contraction. There is great motivation for understanding the interplay between the complex biochemical, cellular, and biomechanical phenomena which conspire to result in wound contraction, as contraction can be either a beneficial or deleterious consequence of wound healing (Rudolph, 1980). It can be beneficial by enhancing the rate of wound closure beyond that associated with re-

227 epithelialization. However, the disfigurement of the adjacent skin can be cosmetically unacceptable and even render a proximal joint nonfunctional. If a predictive mathematical model of wound healing (which accounted for the salient phenomena) were available, it might provide a rationale for the design and application of pharmacologic modulators of cell behavior and functional modifiers of granulation tissue in order to augment or mitigate wound contraction, as appropriate for the nature of the wound being managed. The implication of understanding the interplay between these phenomena goes far beyond improved and novel therapies for full-thickness cutaneous wounds, however (Seemayer et al., 1981; Skalli and Gabbiani, 1988). Mechanisms related to fibroblast-dfiven dermal wound contraction are believed to be operative in other pathologies involving inflammatory and wound healing components (e.g., liver cirrhosis, burn contracture, kidney fibrosis, granuloma, fibrous encapsulation of implants, hepatic fibrosis, and pulmonary fibrosis), in fibromatoses, where a prominent inflammatory component is not evident (e.g., Dupuytren's contracture and dermatofibroma), and in the stromal response to neoplasia, where solid tumor growth is known to involve induction of a wound healing response (Dvorak, 1986). The same mechanisms, although evidently contributing to different degrees, are operative in fetal wound healing of dermal wounds, where tissue regeneration rather than contraction and scarring results (Mast et al., 1992). Many of these mechanisms are also important in the transformation of artificial skin substitutes which initially contain a dermal layer comprised of a type I collagen-based gel in which cultured fibroblasts are dispersed (Bell, 1983). A mathematical model of dermal wound contraction based on a monophasic theory of cellECM mechanical interactions (described in the next section) has been recently proposed (Tranquillo and Murray, 1992). Because dermal wound healing involves fibroblast population of the initial fibrin clot via migration and mitosis, as well as tissue contraction resulting from fibroblast traction on local ECM fibers, any such theory

would necessarily include aspects of cell mitosis, migration, and traction in a viscoelastic medium. The monophasic continuum-mechanical theory first proposed by Oster, Murray, and Harris (Oster et al., 1983) was originally developed to demonstrate the possibility of pattern and form generation in morphogenesis. It describes the traction-driven aggregation of motile cells in an idealized homogeneous, viscoelastic medium due to mechanical instability of the dispersed state. Here we present an extension of Tranquillo and Murray's wound contraction model (1992), based on this theory, to account for two biomechanical feedback phenomena which may occur upon contraction of the wound and in turn directly influence the rate and extent of contraction: haptotaxis, the biased migration of cells up an adhesion gradient, and contact guidance, the biased migration of cells in a bidirectional manner along an axis of fiber alignment. The feedback arises because cell traction drives wound contraction, which can create adhesive ligand gradients and fiber alignment (haptotactic and guidance cues, respectively), resulting in a new distribution of fibroblasts in the wound and thus the redistribution of traction. While a mathematical model for dermal wound contraction allows for direct comparison to actual wound contraction data, thereby helping to illuminate the biochemical, cellular, and biomechanical interplay, dermal wounds are difficult to control experimentally. There is a clear need for an in vitro assay that mimics dermal wound healing and allows the interplay to be quantitatively manipulated and characterized in the context of a predictive mathematical model. Figure 1 illustrates several key features of a novel in vitro wound healing and contraction assay (fibroblast-populated collagen microsphere (FPCM) wound assay) that we have developed to fill this need. A microsphere of collagen gel is formed which initially has a core that is devoid of cells (cf., the initial fibrin clot), and an outer concentric sphere containing a uniform distribution of cultured fibroblasts (cf., fibroblasts in the adjacent unwounded dermis). Compaction of the microsphere as the cells populate the inner core via proliferation and migra-

228

incubation

Fig. 1. Schematic of novel in vitro wound healing and contraction assay (FPCM wound assay). Initial state of the collagen gel microsphere (left) and a subsequent state (right) following incubation in medium. Filled ellipses represent cells, lines represent collagen fibrils, and filled regions depict the concentration of a diffusible stimulatory factor. The wound assay microsphere can be prepared by microinjecting a cell-free collagen solution into a partially gelled fibroblast-populated collagen microsphere (FPCM) at the appropriate time. Alternatively, the microsphere can be made from a fibrin gel-forming fibrinogen solution.

tion (c.f., fibroplasia and wound contraction) is measured. With the additional determination of the radial distribution of cells in the sphere over time, this assay offers the possibility of relating the traction-induced gel compaction (i.e., mechanics) with the cell migration and proliferation as the fibroblasts populate the core (i.e., cell dynamics). As will become evident, this assay is a natural extension of our adaptation of the fibroblast-populated collagen lattice assay (the FPCM traction assay) which we use to quantify cell traction. The fibroblast-populated collagen lattice assay

As discussed above, traction exerted by fibroblasts on ECM fibers in the developing granulation tissue and in the adjacent dennis, leading to their reorganization, is the driving force behind wound contraction and thus merits fundamental understanding. The pioneering work of Harris and co-workers in the documentation of traction in vitro, and its morphogenetic implications in vivo, is only recent. They first visualized traction force by observing the tension wrinkles developed in a silicon rubber sheet on which various cells were cultured, with fibroblasts generating the greatest wrinkling among the various cell types examined (Harris et al., 1980). They then documented the extensive tractional structuring of collagen fibrils which occurred over time around cells dispersed in a type I collagen gel (Harris et al., 1981;

Stopak and Harris, 1982). The macroscopic manifestation of this phenomenon was documented earlier by Bell and co-workers, who conducted a study based on their seminal fibroblast-populated collagen lattice (FPCL) assay (Bell et al., 1979). They exploited the ability to assess cell behavior in the physiologically-relevant environment of a type I collagen gel, made by restoring a dilute aqueous solution of collagen to physiological conditions in order to initiate collagen fibrillogenesis (as popularized by Elsdale and Bard (1972)). They observed that fibroblasts cultured in a small floating disk of collagen gel can dramatically compact the gel, displacing medium from the gel while organizing and concentrating the collagen fibrils - a cell-induced syneresis of the gel. The floating cell-populated disk is made by pouring a cold suspension of cells in dilute collagen solution into a well of a tissue culture dish and then restoring physiological temperature. After gelation, the disk is transferred to a dish with culture medium. Based on measurements of the disk diameter with time, Bell et al. observed that the rate and extent of gel compaction decreased as a function of collagen concentration and increased as a function of cell concentration over the ranges tested. The FPCL assay and its variants quickly became established as the primary means of investigating the ability of normal (Bellows et al., 1981; Ehrlich et al., 1986; Grinnell and Lamke, 1984) and pathological cells (Buttle and Ehrlich, 1983;

229 Delvoye et al., 1983) to mechanically interact with the ECM, including assessment of the effects of various pharmacological mediators (Danowski and Harris, 1988; Leader et al., 1983), serum factors (Gillery et al., 1986; Montesano and Orci, 1988) and other ECM components (Guidry and Grinnell, 1987) on type I collagen gel compaction. More recent studies have focused on the compositional and microstructural alterations of the gel and morphological and functional alterations in the cells which occur during gel compaction. Bellows and co-workers (1981, 1982) observed stages of cell attachment to the collagen fibrils, spreading, motility and migration (leading to rearrangement of the local fibrillar network), and eventually, the establishment of cell-cell contacts and the development of a three-dimensional cellular network. Grinnell and co-workers have conducted a series of investigations to elucidate the mechanism of fibril reorganization and gel compaction (Grinnell and Lamke, 1984; Guidry and Grinnell, 1985, 1986, 1987). Several important observations and conclusions have been made: fibrils in the gel interior are rearranged even when the cells reside only at the surface, and disrupting the network connectivity inhibits compaction, both implying the transmission of traction force through a connected fibrillar network of collagen; only 5% of the collagen is degraded even though the gel volume may decrease by 85% or more,'implying compaction involves primarily a rearrangement of existing collagen fibrils rather than degradation and replacement; few covalent modifications of the collagen occur; a partial re-expansion of compacted gels occurs after treatment of the cells with cytochalasin D or removal of the cells with detergent; and cell-free gels compacted under centrifugal force exhibit a partial re-expansion similar to the fibroblast-compacted gels. Based on these observations, a twostep mechanism was hypothesized for the mechanical stabilization of collagen fibrils during gel compaction: cells pull collagen fibrils into proximity via traction-exerting pseudopods, and over a longer time scale, the fibrils become noncovalently crosslinked, independent of cell-secreted factors (Guidry and Grinnell, 1986).

Despite the significant understanding which has been obtained from the FPCL assay about the mechanical interactions between cells and the ECM, there is a fundamental limitation in characterizing compaction simply in terms of rate or extent of disk diameter reduction. Both of these measures of compaction are dependent on assay properties which complicate the interpretation of the traction being exerted by the cells, such as the initial cell and collagen concentrations, or properties which are completely irrelevant, such as the geometry of the gel. The viscoelastic properties of the gel, for example, are highly dependent on the collagen concentration, so the measured tractiondriven compaction is also directly dependent on the collagen concentration. Consequently, the usual reported measures of compaction in the FPCL assay are not objective indices of traction and are thus unsatisfactory from a scientific viewpoint. There is a clear need, then, to identify an objective index of traction which can be measured in the FPCL assay, another type of population assay, or a single cell assay. In addition to the usual statistical disadvantages of single cell measurements, the dimensions of a cell and the magnitude of its traction are far too small to be within the operating range of conventional mechanical testing devices, even if the normal cell-gel environment could be created and maintained. Therefore, it is necessary to develop a continuummechanical theory, or validate an existing one, which includes all the necessary parameters to describe the state of stress in a cell-collagen gel continuum, in particular, a traction parameter reflecting the intrinsic force transmitted through a cell-fiber mechanical interaction. This theory could then be used to model a population assay and deduce a value for the traction parameter. Similar to the wound contraction model, this theory would necessarily include aspects of cell dynamics (e.g., proliferation, migration, and convection) coupled to the mechanics of the cell-gel composite via cell traction. We present here a mathematical model for the FPCM traction assay, our spherical analogue of the FPCL assay, based on the same monophasic theory used in the wound healing model. In order

230 to account for the significant cell-induced gel syneresis that occurs during compaction in this assay, we also outline an extension of the monophasic model based on biphasic theory similar to that used previously for modeling cartilage mechanics (Mak, 1986; Mow et al., 1980, 1986) and cytomechanics (Dembo and Harlow, 1986). The two significant benefits of a validated predictive model for the FPCM traction assay are the possibilities of defining and measuring an objective parameter of cell traction and extending the model to the related FPCM wound assay (Fig. 1).

Mathematical theories of celI-ECM mechanical interactions

Monophasic theory The phenomenological theory developed by Oster, Murray, and co-workers (Murray and Oster, 1984; Murray et al., 1983; Oster et al., 1983) is a continuum theory which accounts for known cell behavioral properties, such as migration and mitosis, and the coupling of cell traction forces to the mechanical state of the ECM. This coupling leads to a predicted traction-induced deformation of the ECM. The theory is summarized below in their notation. In its simplest form, there are three variables in the formulation: the local cell concentration, n(x,t), the local ECM concentration, p(x,t), and the displacement vector for the cell/matrix composite, u(x,t) (u is the vector connecting a material point in the composite located at position _x at time t with its initial position, x0, i.e., x0(x,t) + _u(_x,t) = x_. It measures the magnitude and direction of displacement of a material point due to acting forces). These variables satisfy equations deriving from the fundamental laws of species conservation, yielding equations describing n and p, and linear momentum conservation, yielding the equation describing u. The specific terms of these equations depend, of course, on both the phenomena presumed operative as well as how they are modeled. The species conservation equations express the

requirement that the rate of accumulation of a species in a control volume equals the net flux of the species through the boundaries of the control volume plus the net rate of generation within the control volume. For the cell conservation equation, this translates into a specification of terms reflecting net cell flux due to active migration, j_n, and proliferation, R n (Segel, 1980). In general form, the cell conservation equation describing the local cell concentration, n(x,t), is: ~-+V-

I0ul

n ~-t = V ' J n + R n

(1)

The convective term is a contribution to the net cell flux by virtue of convection of the ECM, in which the cells reside, with velocity 0u_]0t (i.e., a time-dependent displacement). The simplest case for specifying J-n is when the ECM environment is isotropic. A cellular analogy of Fick's Law ( ~ = - D Vn, where D is equivalent to a molecular diffusivity) is known to accurately model the resultant random migration (i.e., persistent random walk) of blood and tissue cells in isotropic environments, assuming sufficiently low cell concentrations such that the cells move independently. This has been documented for embryonic fibroblasts migrating in a type I collagen gel (Noble and Shields, 1989). Discussion of the occurrence of biased migration and its modeling is deferred, but essentially amounts to additional terms in J-a (e.g., Dickinson and TranquiUo, 1993; Tranquillo and Lanffenburger, 1991). A reasonable choice for modeling the net rate of fibroblast proliferation is the logistic rate law, or cell concentration-regulated growth (growth here indicates cell division), which has been demonstrated for fibroblasts cultured in type I collagen gel (Schor, 1980). With these assumptions, (1) becomes: 0---~n+V'ot n

)-V. DoVn+kon(No-n)

(2)

The cell random motility coefficient, D o, growth rate constant, k 0, and maximum concentration, N 0, are all subscripted with "0" to indicate that these are base values in the absence of any external fields (e.g., growth factor concentration or ECM stress).

231 For the E C M conservation equation, terms reflecting transport and net synthesis of the ECM, R o, must be specified. In this monophasic theory, " E C M " refers to the fiber network and surrounding fluid considered together as a single, homogeneous phase (i.e., a compressible solid). Also, unlike in the momentum conservation equation below, the presence of cells is essentially ignored in the E C M conservation equation. Because the E C M is assumed to be constituted of a fibrillar network, there is only a convective component to its flux (the network can not diffuse). With these assumptions, the E C M conservation equation describing the local ECM concentration, p(x,t), in analogy to (1), is simply:

The simplest relevant form for (YECM is an isotropic, linear viscoelastic stress tensor. This assumption is valid only at sufficiently small strains, such that material and geometrical nonlinearities, and anisotropy associated with straininduced fiber alignment, are not significant. Following Oster, Murray, and co-workers, we use the following constitutive expression for the E C M (Landau and Lifshitz, 1970):

-~-+V.

where [.t1 and g2 are related to the shear, g, and bulk, K, viscosities (gl = 2g, g2 = K - 2/3 g), E is Young's modulus, "o is Poisson's ratio, and I is the unit tensor. The form proposed by Oster, Murray, and co-workers for GCell/ECM, which we have also adopted, views cell traction as a "negative pressure" proportional to the product n.p (implying a rapid equilibrium of pseudopod-fiber anchorage interactions or receptor-mediated adhesions), where the proportionality constant, "c, reflects the traction stress generated per unit cell and E C M concentrations. In order to accommodate the in vitro observation of contact inhibition of motility, and therefore presumably of traction, of cells in physical contact on planer substrata, Oster et al. assume "c to be a monotonic decreasing function of cell concentration: "fin) = "%(1 + )me) -l, where "c0 is the traction parameter and )~ is a contact inhibition parameter. These assumptions lead to:

P~-t

=Rp

(3)

The equation of motion deriving from momentum conservation, which describes the local displacement of a material point of the cell/ECM composite, u(x,t), is simplified given the fact that inertial forces are negligible for cell/ECM mechanics (Odell et al., 1981), yielding a mechanical force balance between traction exerted by the cells and forces associated with the ECM physicochemical properties. In its most general form, the mechanical force balance involves the total stress tensor for the cell/ECM composite, ~, and the net body force acting on a volume element of the composite, f : V.o+pf=O

(4)

Oster, Murray, and co-workers assume that the traction exerted by cells resident within the ECM can be modeled as an active stress, so that 6 can be written as the sum of the stress tensor describing the ECM alone, CrECM, and that associated with the active (traction) stress, IJCeI1/ECM:

(5) At this stage, the treatment of momentum conservation is quite general. However, in order to proceed, constitutive forms for GECM and GCelI/ECM must be specified.

OecM= gl ~0~+g2-~ -I+ 1

e = -~ ( Vt2 + V u T ) = strain tensor

~+ (1--2a~) 0I

(6)

0 = V 9u = dilation

pn _ I

Ocett / ECM= ZO 1 + )~n2

(7)

However, the true form of (YCelI/ECM is still an open question. In fact, several effects may motivate the future inclusion of a p-dependent term in the denominator that reduces (YCelI/ECM as the E C M concentration, p, increases. The saturation of available E C M binding sites on cell pseudopods and physical inhibition of pseudopod extension and retraction would both warrant such a

232 modification. In the absence of any guiding data, we use the original form for 9 proposed by Oster et al. in this work. Note that there is no explicit relationship between active migration and traction stress in the model presented (i.e., the cell random motility coefficient, Do, and the traction parameter, c0, are unrelated). In support of this, motile but nontranslocating ceils, as well as migrating cells, can be observed to locally restructure collagen fibers in vitro, although it is not yet known whether they exert different magnitudes of traction. However, Ehrlich and Rajaratnam (1990) suggest that fibroblasts observed in the periphery of an FPCL assay disk, with pronounced cytoplasmic stress fibers and cell-cell attachments (characteristic of myofibroblasts), presumably resulting in less migration (smaller Do), exert less force than the fibroblasts with ordinary phenotype at the center. This is an indication that "c0 may be different for a myofibroblast subpopulation. (It should be noted that Harris et al. (1980) observed the general trend that cell types exerting the greatest traction on a silicon rubber sheet are the least efficient in migration (e.g., fibroblasts) and vice versa (e.g., neutrophils), and a similar trend can be observed when cells are embedded in a reconstituted collagen gel.) At this stage, an explicit relationship between D O and "c0 cannot be proposed. If such a relationship is posed in the future, the isotropic form of '~Cell/ECMwould be consistent only with the random migration case, as biased migration would be associated with net directional orientation of cells and the generation of anisotropic traction stress. The possibility of oriented, motile but nontranslocating cells would also need to be considered, following from the discussion above. It should be understood that the local concentration of E C M f i b e r s is really the relevant quantity for p in CrCelI/ECM(Eqn. 7), not p as defined (ECM fiber network plus fluid as a single phase). This is one limitation of the monophasic postulate, although it affords considerable simplification relative to the biphasic theory, as will be seen. It is also important to note that the possibility of differential solvation or osmotic forces associated with fiber concentration gradients at

large compaction are neglected here, consistent with the small strain assumption. This may be an important effect in wound healing because of the presence of glycosaminoglycans with associated fixed charges in dermis and wound ECM, as known to be the case in cartilage (e.g., Eisenberg and Grodzinsky, 1985; Grodzinsky, 1983). Equations 1, 3 and 5 comprise the most general starting form of the monophasic theory considered here, with the key term being the active traction stress, ~CelI/ECMwhich couples cell and ECM dynamics with deformation of the cell/ ECM composite. Biphasic theory There is considerable literature concerning the flow of fluid through connective tissue as it is deformed, largely due to investigations of articular cartilage mechanics. A model designed to describe fluid flow through a fibrillar network must make explicit account for conservation of mass and momentum for both the fluid and network phases, to be distinguished from the single "fluid/network ECM" phase as in the monophasic theory. Mow and co-workers (Kwan et al., 1990; Mow et al., 1980, 1986) and others (Farquhar et al., 1990; Holmes, 1986; Mak, 1986) have developed a class of models for describing cartilage mechanics based on the theory of mixtures (Bowen, 1976; Crane et al., 1970; Green and Naghdi, 1970). The main interest is to predict fluid permeation and the accompanying stress field in cartilage due to applied loads. Because cartilage is essentially acellular, there is no account for cell traction in these models. Thus, there is no possibility (with the current form of these models) of predicting permeation and the traction-induced stress field in granulation tissue during wound contraction, or the same for collagen during compaction in the FPCL assay. One approach would be to introduce ~CelI/ECM into the mechanical force balance (derived by Mow and co-workers) for the network. However, these expressions are rigorously derived from conservation laws and constraints imposed by mixture theory. Consequently, it may not be valid to simply introduce an

233 active stress term. Rather, we adopt the biphasic model developed by Dembo and Harlow (1986) for cytomechanics based on the multi-phase averaging theory of Drew and Segel (Drew, 1971; Drew and Segel, 1971), which allows an active stress term to be introduced in a natural way. The equations of Dembo and Harlow for modeling cytomechanics are based on the view of cortical cytoplasm as a reactive, contractive, interpenetrating two-phase fluid. Here reactive refers to a dynamic assembly and disassembly of actin filaments, contractive refers to contraction forces associated with actomyosin complexes in the dynamic cortical actin network, interpenetrating indicates the two phases are finely interspersed relative to the length scale of the system, and two-phase fluid indicates the cytosol and the filamentous actin network are considered as distinct phases. The model accounts for contractive forces due to actomyosin complexes distributed within an actin filament network and implicit activation of the sliding filament mechanism. These cytomechanical phenomena can be thought of as analogous to the contractive forces generated by cells distributed within a collagen fibril network and their implicit tractional structuring activity. However, there are clear differences between the biphasic cytomechanical model developed by Dembo and Harlow and one which would be relevant for the FPCM assays. Namely, the dynamic' assembly and disassembly of actin filaments (r.e., collagen fibrils) is not relevant (a simplification) and, therefore, the view that the actin network (r.e., collagen network) can be considered as a fluid without elastic properties does not follow (a complication). A biphasic wound contraction model would entail the complication without the simplification (unlike the stable collagen network during compaction in the FPCM assays, there is significant degradation and synthesis of collagen during wound contraction). Here we provide an outline of the simplest, general set of biphasic modeling equations used by Dembo and Harlow (1986) for describing cytomechanics, which are proposed above to be relevant for wound contraction and compaction in our FPCM assays. Defining the average fractional

volumes of network and solution as 0n(X_,t) and 0s(X,t) respectively, the excluded volume relation requires: On (x,t) + 0s (x_,t)= 1

(8)

As in the monophasic model, the presence of the cells is ignored in mass conservation. Defining the average velocities of network and solution as v(x,t) and _w(_x,t), respectively, mass conservation can be expressed in terms of volume fractions when the intrinsic phase densities are constant and equal (a good approximation for protein networks and biological media, as Dembo and Harlow note): 3t

-

b0s 0t

V. (0nv)+Rn V.(0sw)-R,,

(9) (10)

where R n is now the net rate at which volume is transferred from solution to network phase due to all reactions. The assumption of equal intrinsic phase densities is unnecessary for writing (9) and (10) in terms of volume fractions when reactions are absent. Because the network will be considered a viscoelastic solid, we define v = 0u_/&, where u is the displacement vector defined previously, but now referring solely to the network phase, not the "single phase gel". The sum of (9) plus (10) yields the overall incompressibility relation: 0 = V . [0nV+ 0sW]

(11)

Of course, only two of the three forms of mass conservation equations are independent. The key result of the averaging theory of Drew and Segel (Drew, 1971; Drew and Segel, 1971) as adapted by Dembo and Harlow (1986), after omitting inertial terms (as in the monophasic theory), is the following set of coupled mechanical force balances:

V'(O,,on)+PVOn+~p(w-L')+OnPnf=O

(12)

V.(Os~s)+PVOs+~p(v-w)+OsPsf=O

(13)

234 where 9n and 9s are the intrinsic phase densities of network and solution, respectively. The same form of the stress tensor defined in (6) for (YECM can be assumed for the viscoelastic component of (Yn, but now Poisson's ratio (for the network) is well-defined. Comparison to (4) reveals that intrinsic phase stress tensors and net body forces are now weighted by the local volume fractions of the phases. Two new terms are also apparent, both of which represent interphase forces. P is the interphase pressure existing at the hypothetical interface between the interspersed phases, and is generally not equal to the intraphase pressures, Pn and Ps- Pn is associated with the traction-induced tension in the network and Ps is associated with hydrostatic pressure of the solution (a component of Pn associated with a reaction force due to dilation of the network phase is retained in the form of On). r is a drag coefficient which may depend on some local measure of porosity of the network (c.f., Mansour and Mow, 1976). Using (8), note that the total interphase force defined by (12) and (13) is equal in magnitude and of opposite sign, as required. Defining = Ps - Pn = "contraction stress"

(14a)

f" = Ps - P = "solvation stress"

(14b)

P f = Ps + Fln0s = "effective pressure"

(14c)

(Y~.= cYi + Pi I

(14d)

then (12) and (13) can be written V'(On Oiz) -- O n V P f + FVln0s + V(0nW) +

(15)

cp(w_-v) + 0,,on s = 0 V.(O s 0~) - O s V P f + ~ ( v - w )

+ OsPsf- = 0

(16)

by assuming that F is a constant independent of On (justified by Dembo and Harlow on the basis that solvation forces are of nonspecific nature) but that may depend on On as well as other variables (in our case the local cell concentration, n). Upon defining constitutive forms for J_n, (Y'n, (Y's and % the general biphasic model for cell traction problems is comprised of (1), (8), (9) or (10), (11),

(15) and (16). These six equations define the local cell concentration n(_x,t), the local average fractional volumes of network and solution, 0n(X,t) and 0s(X,t), respectively, the local average velocity of solution, _w(x,t), the local average displacement of network, u(x,t), and the local average effective pressure, Pf(x_,t).

Models and in vitro assays of cell traction force: the F P C M assay We have devised a protocol for a spherical version of the FPCL assay called the fibroblastpopulated collagen microsphere (FPCM) assay (Fig. 2). The spherical geometry of this assay has several key advantages over the traditional disk geometry. First, it closely resembles our FPCM wound assay (c.f., Fig. 1), which must be of spherical geometry to most conveniently and predictably establish concentration gradients of cellular mediators by diffusion, and thus allows a desirable close correspondence between the two assays in terms of protocols, conditions, and measurement techniques. Second, the spherically symmetric set of equations associated with either the monophasic or biphasic theories are far simpler to solve than those applicable to the original disk geometry (i.e., inherently one- rather than two-space dimensional). Third, the FPCM preparation protocol yields collagen fibril orientation that is relatively isotropic initially, consistent with the simplest models outlined above and unlike the traditional FPCL and related assays where fibrillogenesis in the presence of bounding surfaces can lead to significant local anisotropy (Modis, 1991). Finally, the microspheres can be prepared at arbitrarily small diameters and still compact uniformly, which allows diffusion gradients of nutrients and metabolites to be minimized. This is not the case with the thin disk geometry of the FPCL assay. As will be seen, these advantages of the FPCM configuration come with only a very modest increase in experimental effort. Before presenting the monophasic model predictions of the change in FPCM diameter with time, we state our experimental protocols and some preliminary

235

Fig. 2. In vitro traction force assay (FPCM traction assay). Similar to the FPCM wound assay (Fig. 1), but cells are initially uniformly dispersed throughoutthe microsphere.The initial state of a FPCM (first panel) and several subsequent states during the time course of compaction (final three panels) are shown. Here, the initial FPCM diameter is 1.3 mm and the initial cell concentration is 3.5 x 105 cells/ml. data. A complete presentation of the model predictions and experimental data can be found in M o o n and Tranquillo (1993a,b).

ways for passing and discarded prior to the tenth passage.

Collagen solution Materials and methods Cell cultures H u m a n skin fibroblast cultures are first initiated from neonatal foreskins. Cells are subsequently grown in 75 c m 2 tissue culture flasks containing Dulbecco's modified Eagle medium ( D M E M ) supplemented with 20% fetal calf serum (FCS), penicillin/streptomycin, and Fungizone. Cultures are maintained at 37~ in a humidified incubator with an atmosphere o f 10% CO2-90% air. The fibroblasts are harvested prior to confluency with a 0.25% trypsin solution. Cultures are split four

Vitrogen 100 collagen (Celtrix Laboratories) is a sterile solution of 99.9% pure pepsin-digested bovine dermal collagen dissolved in 0.012 N HC1 at a concentration o f 3.0 mg/ml. It consists o f 9 5 - 9 8 % type I collagen (largely monomeric and without telopeptides) with the remainder being type III collagen.

FPCM preparation Cold Vitrogen 100 solution (730 ~tl) is adjusted to physiological ionic strength and pH (7.4) with the addition of 20 ~tl of 1 M H E P E S buffer solution,

236 90 ].tl 10X M e d i u m 199 (M199), 30 gl FBS, and 130 I.tl o f 0.1 M NaOH. A small volume of a concentrated cell suspension in growth medium is added to yield the desired final cell concentration. The final collagen concentration is 2.1 mg/ml as standard. The cells are added after the solution pH is adjusted and suspended in the mixture immediately after the Vitrogen 100 is added to ensure the cells are entrapped during fibrillogenesis when the solution is heated to 37~ After 3 minutes, about 0.5 gl of the partially gelled preparation is micropipetted into a test tube containing silicone fluid (Harwick SF 1250) at 37~ to yield an F P C M of around 1 m m diameter. After 7 additional minutes the F P C M is sufficiently gelled to allow isolation. The silicone fluid is drawn off and replaced by M199 with 20% FBS in a multiwell culture dish for microscopic observation. F P C M diameter measurements The F P C M diameters are measured using a Zeiss Axiovert 10 inverted light microscope in brightfield, a Hamamatsu C2400 Newvicon camera system, and a Kontron Electronics IBAS image analysis system. The F P C M are maintained under atmospheric conditions at 37~ with an air stream incubator. The microscope is focused to the image plane with the largest image area, representing the center cross-section of the FPCM. A digitizing pad and mouse are used to mark the endpoints of line segments defining the diameter of the F P C M displayed on the IBAS image monitor. Eight or more measurements are taken at different orientations for each FPCM, and the diameter is taken as the average of these values. The F P C M diameter is measured immediately after preparation and at prescribed time points.

phology. The F P C M are also quite spherical before the onset of compaction, with less than 1% standard deviation in the initial diameter measurements. Over a period of about twelve hours, the cells develop a characteristic stellate morphology. Compaction proceeds and the gel gradually becomes opaque due to the exclusion o f medium and tight packing of collagen fibrils by the cells. The viability is >95% across the bead over the time-course of the assay, indicating that there is no toxic effect o f the silicon fluid used in the preparation procedure and that adequate nutrient and metabolite transfer is taking place. No compaction is observed for control microspheres prepared with fibroblast-conditioned medium instead of cells. A typical time-course of compaction is presented in Fig. 3 for three different initial cell concentrations, n 0, with the same initial F P C M diameter, Do. As previously reported for the FPCL assay (Bell et al., 1979), the rate and extent of gel compaction increases with increasing initial cell concentration (within the range tested) in

q

.~ e.: L

a o J

o

n, = 5xl0*eells/ml [ no = txI06eells/ml l n a = 3xl0'eells/ml ]

Experimental results The compaction behavior of the fibroblast-populated microspheres in our F P C M assay is qualitatively similar to that reported for the fibroblastpopulated disks in the traditional F P C L assay (Fig. 2). Initially, the cells are distributed uniformly throughout and exhibit a spherical mor-

o.o

l

I

5.0

loo

I

I

~ 5 . o zo.o

I

I

I

~.o

30.0

~.o

400

Time (hours) Fig. 3. Observed dependence of compaction in the FPCM traction assay on initial cell concentration,no. In each case, the initial FPCM diameter, D0, is close to 0.9 mm.

237 our FPCM assay. We have evidence that the lag time preceding the active phase of compaction may be explained, at least in part, by the initial period required for cells to "spread" from the rounded state observed when the FPCM is prepared to a stellate morphology (expression of motility and traction typically accompanies cell spreading). Note that only the initial part of the compaction curve is relevant for comparison to the model predictions given the small strain assumption implicit in (6). The four compaction curves in Fig. 4, which derive from duplicate trials at two different values of D 0, each with n o = 3.0 • 105 cells/ml, are shifted along the time axis so that t = 0 corresponds to the beginning of the active phase of compaction. The rate and extent of compaction for all four spheres in Fig. 4 are roughly the same. The consistency between duplicates demonstrates excellent reproducibility. Although sphere diameter, as it determines surface area for syneresing medium, has been proposed to play a role in the chemically-induced syneresis of gels (Scherer,

oi a o

i!

D,=0.90mm D,=0.88mm D o = 1.14 m m

* Do=1.16mrn

1989), there appears to be a negligible effect over the range of diameters tested here (with smaller diameters, the continuum assumption implicit in defining a cell concentration becomes less valid, and with larger diameters, the assumption of non-negligible gradients of nutrients and metabolites becomes less valid). M o d e l predictions: m o n o p h a s i c theory

As noted earlier, the collagen fibril orientation is initially quite isotropic in the FPCM. Thus, we assume that (2), (3) and (5)-(7) apply initially. In addition, we take f = 0 (the free-floating sphere experiences negligible body force) and R# = 0 (in accord with the observation that fibroblasts in a type I collagen gel do not extensively degrade the fibrils nor secrete new collagen that is incorporated into the network over the time scale of interest (Guidry and Grinnell, 1985; Nusgens et al., 1984)). Because of the approximating spherical symmetry of the assay, only the radial component of u, u = Ur, needs to be considered (i.e., u0 = u~ = 0 by assumption). It is necessary to prescribe initial and boundary conditions on the variables n, p, and u. From symmetry, we require the spatial derivative of n to be zero at the origin. We also assume that cells are confined within the FPCM. Combined with the previous assumption implicit in (2) that cells migrate only randomly, this requires that the spatial derivative of n be zero on the FPCM surface. These lead to the following boundary conditions on n: ~n

49

~--~= 0, r = 0

~r I

0.0

5.0

I

10.0

I

15.0

I

I

I

20.0

25.0

30.0

35.0

Time (hours)

Fig. 4. Observed dependence of compaction in the FPCM traction assay on initial microsphere diameter, D 0. In each case, the initial cell concentration, n 0, is 3.0 • 105 ceUs/ml.

- O , r = S (t )

(17a)

(17b)

where S(t) denotes the position of the moving FPCM surface during compaction. Again from symmetry, we require that there be no displacement at the origin. We also assume a zero normal stress condition at the moving FPCM surface (i.e., the normal component of the differential equation of interfaciaI statics under the

238 assumption of negligible surface tension at the interface between the FPCM and suspending medium). Using (6) and (7), these lead to the following boundary conditions on u: u = 0, r = 0

(18a)

~2u 292 r . E ( 1 - ~) (Ire = ( ~1 + ~2 ) ~ - ~ ' t - 7 " ~ ' - t - ( 1 + 1 ) ) ( 1 - - 2 V )

[

~u

v

( 2u )]

-aTr+17~(~7 -

pn

+'co 1 +9~n2 =0, r=S(t)

(18b)

Because a linear viscoelastic constitutive expression, (6), is assumed, it can be shown that a self-consistent approximation in the small strain limit is to replace (3) with its linearization (Lin and Segel, 1974): p=l

1 3u 1.2 Dr

0 0 and e0 = g/2 for ~xx < 0) and p is an empirical scaling factor. This form of the Von Mises distribution requires that I ~xx I/P be small. When ~xx = 0, the distribution is uniform, and when ~xx ~e 0, the distribution concentrates at e0. Evaluating the integral in (26) using (27) gives the following relationship between the extent of fiber orientation and the uniaxial strain: < P (x,t) > =

~ (x,t) 4p

(28)

Having defined

and its variation in time and space during wound healing, we now consider its quantitative relationship to fibroblast contact guidance. Matthes and Gruler (1988) showed in their investigation of neutrophil contact guidance that the value of D in a guidance field is direction-dependent, with D being largest in the direction of substratum alignment. The constitutive cell flux equation based on generalizing Fick's law to anisotropic diffusion for the assumed linear geometry takes the form (Crank, 1975): bn

Jx = - D xx ~

(29)

where Dxx = D0(1 + [3)

(30)

is consistent with the data of Matthes and Gruler

(1988). Dxx denotes the component of the cell random motility tensor associated with a cell flux in the x-direction due to a cell concentration gradient in the x-direction. [3 is a dimensionless sensitivity constant, which measures the cell guidance response to a given extent of fiber alignment. Matthes and Gruler (1988) also showed that the direction-dependence of D for neutrophils is due largely to direction-dependent speed. An extended cell flux expression motivated by this finding is that derived for the case where cell speed, c, varies with the local concentration of a chemical stimulus, termed orthokinesis for the general case where the stimulus is any scalar variable (Tranquillo and Lauffenburger, 1991): _~n

1 ~D

J x = - O ~ x - ~ fffx n

(31)

The new term arises because D o~ c 2 and c varies in the x direction due to its dependence on a scalar variable (describing the stimulus) which also varies in x. A consequence of orthokinesis as modeled by (31) is that cell accumulation can occur at steadystate where c is a minimum. Because of the symmetry that follows from the assumed linear geometry of the wound, we rewrite (31) in terms of Dxx and its dependence on c via the scalar by analogy: Jx=-Dxx

~n 3x

10Dxx 2 Ox n

(32)

where

~D xx dD xx ~

~

Ox - d < P > O x - 1 3 bx

(33)

The value for 13 is approximately constant for the experimental data generated by Matthes and Gruler (1988) for neutrophils on various two-dimensional substrata, and assumed so here for fibroblasts in three-dimensional fiber networks. Note that this mathematical theory of contact guidance based solely on orthokinesis, as motivated by Matthes and Gruler's (1988) observation of direction-dependent speed, may not be consistent with their accompanying observation of biased

244 orientation in the direction where speed is maxim u m , and a general theory of contact guidance for the case of oriented fibers remains to be derived. It is not obvious a p r i o r i whether alteration of the evolving E C M gradient presented in Fig. 6c will lead to fibroblast accumulation in the wound, and therefore contraction, w h e n contact guidance is added to the base m o d e l (with "Co constant, fibroblast accumulation in the w o u n d (i.e., n > N) is required for the traction imbalance leading to contraction). I f any contraction (i.e., negative strain within the wound) were elicited by the altered time-evolution of the cell distribution in the contact guidance case, the resultant alignment of fibers normal to the x-direction might lead to cell accumulation in the w o u n d due to orthokinesis (or "trapping" due to decreased cell speed associated with contact guidance as modeled by (28), (30) and (32)) and thus augmented contraction. Unfortunately, the partial differential equation s y s t e m b e c o m e s extremely difficult to solve numerically in this case. H o w e v e r , the boundary value p r o b l e m defining the steady-state solutions is feasible to solve numerically (this approach requires the linearized f o r m of the matrix conservation equation, similar to (3'), which for linear g e o m e t r y is p = 1 - au/Ox). Note that a uniform steady-state, ns(X) = N, ps(X) = P0, and Us(X) = 0 (i.e., no contraction), is a solution in this case (the base case a u g m e n t e d with contact guidance). Although nonuniform solutions m a y exist, it is doubtful that any consistent with contraction would arise, as there would be no initial contraction to trigger the orthokinetic accumulation of cells (associated with contact guidance) and augmentation o f contraction. H o w e v e r , w h e n contraction is provided by the steady, spatially-varying traction assumption of the "variable traction case", such contact guidance-driven cell accumulation occurs and is found to enhance the extent of contraction above that for traction variation alone. As seen in Fig. 7, the concentration of fibroblasts in the w o u n d exceeds that predicted in the absence of contact guidance (i.e., ns > N (or ns > 1, scaled) for 0 < x < 1), leading to a greater negative displacement, or

m o v e m e n t of E C M toward the wound center, at the wound boundary (i.e., Us decreases near x = 1). Although the increases in fibroblast concentration and w o u n d contraction are modest (necessarily so in order to maintain validity o f the small strain assumption), these results certainly suggest that fibroblast accumulation caused by contact guidance effects developed during wound contraction can lead to its enhancement. o

I-

,

o.o

4'.o

,;.o

.

l:Lo

.

.

,6.0

z6.o

~ [

.

N 0.0

'l,,O

8.0

12-0

. 16.0

20.0

i[.oV

~o

~.o

1~.o

1;,.o

26.o

o.o

.,.o

;.o

i;.o

,6.o

26.0

Fig. 7. Comparison of steady-state predictions for the "variable traction" extension of the base model alone (dashed line) and in combination with contact guidance (solid line). Spatial profiles are shown of the (a) traction parameter, ~o; (b) fibroblast concentration, n; (c) displacement of the cell/ECM composite, u (u < 0 is consistent with contraction); and (d) strain of the celI/ECM composite, du/dx (du/dx < 0 is consistent with fibers aligned perpendicular to the direction of strain). Parameters are the same as in Fig. 6 with [$ = 2, but k0 = 0 to focus on cell dynamics due entirely to migration. The effect of contact guidance is seen to be cell accumulation in the wound region (-5% increase) and greater negative displacement in the region of the initial wound boundary, i.e., contraction (-5% increase).

245 Haptotaxis

Haptotaxis refers to the directed migration of cells in a gradient of adhesion. Although there are several reports of haptotaxis in nonbiological adhesion gradients (e.g., Harris, 1973), this has not yet been directly observed in the case of a concentration gradient of a cell adhesion molecule (it has been indirectly inferred many times (e.g., Brandley and Schnaar, 1989)). Nonetheless, it is expected that the proper type and steepness of substratum-bound adhesive ligands will eventually be shown to elicit haptotaxis. Oster et al. (1983) proposed a constitutive cell flux term for haptotaxis in analogy to that which others have derived and validated for chemotaxis (Tranquillo and Lauffenburger, 1991): J = - D V n + c~nVa

(34)

where o~ is a haptotaxis coefficient measuring the drift of cells up a concentration gradient of adhesive ligand, denoted by Va. Recently, (34) was derived from a stochastic model of cell migration based on traction exerted through bound adhesion receptors in the limit of a shallow adhesion ligand gradient (Dickinson and Tranquillo, 1993b). In the context of the wound healing model, if it is assumed that cell adhesion ligands are present at uniform density on ECM fibers in the dermis and wound, then Va o~ Vp and J = - D V n + o~'nVp

(35)

Thus, any change in the ECM concentration profile that the fibroblasts create due to traction will feed back and influence the cell concentration profile via the coupling of ECM and cell behavior through haptotaxis. As for the case of contact guidance, it is not obvious a priori whether alteration of the evolving ECM gradient presented in Fig. 6c will lead to cell accumulation in the wound and, therefore, contraction when haptotaxis is added to the base model. With haptotaxis as the only addition, and with all parameters the same as in Fig. 6, we again observed only the expansion/relaxation outcome for values of o~'

consistent with the diffusion approximation underlying (35) (larger values of r increasingly invalidate (35) because the drift term dominates over the diffusion term). However, in combination with steady, spatially-varying traction, as described above, haptotaxis leads to cell accumulation and can enhance contraction over that obtained with traction variation alone. Interestingly, this was only observed with the full form of the matrix conservation equation (3), not with the linearized form (see (3')).

Discussion There has been significant research conducted towards quantifying and understanding how cell metabolism and growth are affected by hydrodynamic forces imposed by an abnormal flow environment (associated with bioreactors or cardiopathologies) and from the natural flow environment in vivo (Nerem and Girard, 1990; Nollert et al., 1991). There are only very recent reports of analogous studies for the case of mechanical forces imposed on the cell via external loads applied to the surrounding ECM (Jain et al., 1990) or via traction applied to the surrounding ECM by the cells themselves (Delvoye et al., 1991; Mochitate et al., 1991). This delay reflects the greater difficulty in designing mechanical loading experiments with structurally complex, deformable tissues and tissue-equivalent collagen gels which contain cells (not to mention identifying the force(s) to which the cells actually respond). However, most tissue engineering applications involve environments in which cellECM mechanical interactions occur, and the consequences must therefore be considered. We have described here two continuum theories that account for traction forces exerted by cells on the ECM (vs externally applied forces) and the resultant ECM deformation, and allow for the traction-driven deformation to "feedback" and influence cell migration behavior. The mathematical model for our FPCM traction assay based on the monophasic theory of Oster et al. (1983) allows us to make a preliminary estimate of the

246 traction parameter, "c0, appearing in the active stress term for the cell/ECM composite (Moon and Tranquillo, 1993a,b). Considerable effort must be expended to independently measure the other parameters in the theory in order to do so, but "~0 is the first objective measure of traction to be obtained in an FPCL-type compaction assay. The measured values of "Cocan then be legitimately compared for various cell types in a standard protocol, and the effect of cell stimuli (e.g., growth factors) on ~0 for a single cell type can also be investigated. It is of interest to compare our preliminary estimate of "c0 (0.0001-0.005 dyne.cm4/(mg collagen.cell) for human skin fibroblasts), based on Fig. 5, with a recently reported value of 4.5 x 104 dynes/(cm2 cell area) for the steady-state isometric force generated by chick embryo fibroblasts in a compacted slab of collagen gel contained between the plates of a force transducer (Kolodney and Wysolmerski, 1992). Using data reported in that paper, we estimate that the fractional cell area in the slab was N 0.0665 and the collagen concentration at the compacted steadystate was - 8.7 mg/cm 3. Using these two values, we can convert 4.5 x 104 dynes/(cm2 cell area) to 0.0002 dyne-cm4/(mg collagen.cell), which is right in the range of our estimate of "~0 obtained from the FPCM assay for human skin fibroblasts. Because the monophasic theory used in our initial model of the FPCM assay cannot account for the significant syneresis which occurs as cells compact the collagen fibrils and displace medium from the microsphere (manifested also as an ill-defined Poisson's ratio for the gel), we describe the adaptation of a biphasic theory, which again includes the active traction stress term, for a refined model of our FPCM traction assay. Use of this refined model will require measuring the frictional drag coefficient between solution and network phases and the related hydraulic conductivity, but should significantly increase the range of validity (in terms of compaction) of the model predictions. Although our suspension-prepared microspheres appear quite isotropic initially, the in-

crease in anisotropic fiber orientation that develops with increasing compaction is another limitation of the monophasic model. Overcoming this involves more than simply an anisotropic extension of the mechanical modeling, for which models of articular cartilage under loading are relevant (Schwartz et al., 1993; Farquhar et al., 1990; Mow et al., 1986). At a minimum, two effects of cell contact guidance must be considered. One is an anisotropic traction stress associated with cell orientation. The other is biased cell migration. A methodology for correlating contact guidance with anisotropic fiber orientation has recently been described which should guide the necessary theoretical extensions (Guido and Tranquillo, 1993). There are also a number of likely influences of stress and strain in the ECM on cell metabolism, which in the context of the model would be manifested, for example, as logistic growth rate parameters which depend in some fashion on ECM stress and strain. However, the nature and extent of ECM stress and strain transmitted to cells in tissues is poorly understood, and only a single model has been proposed to date, which computes the stress concentration on and in cells modeled as elastic inclusions in an elastic medium (Jain et al., 1990). We have also described the extension of a monophasic theory-based model of dermal wound contraction (Tranquillo and Murray, 1992) to assess the consequences of two possible "biomechanical feedback" mechanisms based on biased fibroblast migration in response to traction-driven development of fiber concentration gradients (i.e., adhesive ligand gradients), termed haptotaxis, and anisotropic fiber orientation, termed contact guidance. Assuming that contact guidance could be modeled as an anisotropic diffusion with cell speed being greater in the direction of fiber alignment (as suggested by experimental data for neutrophils (Matthes and Gruter, 1988)), and that fiber orientation is determined by the macroscopic strain, we examined the effects of this extension on the base model of wound contraction for a linear wound geometry. We did not observe any contraction with this as the only mechanism for generating nonuniform

247 traction. In order to examine the effect of haptotaxis on the base model, we assumed that the "convective" component of cell flux due to drift up an adhesive ligand concentration gradient can be modeled as proportional to the gradient (as supported by a recent theoretical model of adhesion receptor-mediated cell migration (Dickinson and Tranquillo, 1993)). We again did not observe any contraction. The apparent explanation for these results is that individually, neither haptotaxis nor contact guidance is capable of generating an ECM displacement field (and associated adhesive ligand gradients and fiber alignments, respectively) to promote the development of a nonuniform cell distribution and associated nonuniform traction distribution that would drive wound contraction, at least for the range of parameter space investigated. In contrast, we observed that both haptotaxis and contact guidance enhanced the contraction elicited by an imposed cell traction profile (~0 decreasing outward from the wound center) which represents the influence of an inflammation-derived diffusible mediator of cell traction. These model predictions are consistent with the fact that the majority of cases of wound contraction involve an inflammatory phase. Fibrocontractive pathologies, however, do not. Perhaps predictions of contraction exist in another region of parameter space that is appropriate for those cases, but that we did not examine. Alternatively, perhaps more accurate modeling of haptotaxis and contact guidance beyond the idealized treatment assumed here, or other biomechanical feedback phenomena, such as stress or strain modulation of cell proliferation and ECM synthesis, will accommodate those cases. There are recent studies which document the alteration of mitosis and protein synthesis in fibroblast-populated collagen gels under different mechanical loading conditions. Nakagawa et al. (1989) report a ten-fold increase of DNA synthesis in gels that compact while attached (cells are bipolar and oriented along with collagen fibrils parallel to the plane of attachment) over that found in free-floating gels (cells are largely stellate and oriented randomly, as are collagen fibrils). Cells in the attached gels

also synthesized collagen at a two-fold greater rate and, unlike those in floating gels, were responsive to most growth factors. Mochitate et al. (1991) describe an immediate decrease in DNA synthesis and protein synthesis in cells cultured in an attached gel that is detached. In the case of an externally applied uniaxial tensile load, where cells and fibrils presumably become aligned (which also occurs in the compacting attached gels of Nakagawa et al. (1989), although the loading is compressive), Jain et al. (1990) report an increase in DNA synthesis but a decrease in protein synthesis compared to unloaded controls. Thus, accounting for stress or strain modulation of ECM synthesis and proliferation in future modeling is indicated, albeit difficult. Ultimately, our goal is to gain mechanistic insight into wound contraction by systematically manipulating and quantitatively characterizing the cell redistribution and microsphere compaction in our in vitro analogue, the FPCM wound assay (Fig. 1). Although much simpler than developing a predictive mathematical model for a wound, doing so for our in vitro analogue is a challenging task, even without inclusion of a diffusible cellular mediator which mimics the inflammation phase. However, with careful and, where possible, independent validation of each element of the model, this goal appears feasible.

Acknowledgements This work has been supported by National Science Foundation PYI (BCS-8957736) and National Institutes of Health FIRST (1R29-GM4605201) AWards to Robert T. Tranquillo. Laboratory facilities provided by Drs. David Knighton and Michael Caldwell, Department of Surgery, and technical advice of Vance Fiegal are gratefully acknowledged.

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Address for offprints: Robert T. Tranquillo, Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA

Tissue engineering science: consequences of cell traction force.

Blood and tissue cells mechanically interact with soft tissues and tissue-equivalent reconstituted collagen gels in a variety of situations relevant t...
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