Most quantitative studies on wound healing have been done on animals. There is a basic difference between animal and human wound healing. Animals possess a skin organ (the panniculus carnosus) under the dermis which is absent in humans. It confers greater mobility to the skin in response to stress. In animals up to 100% of animal wound area may be closed by contraction; values of 20 to 40% are more the norm in humans. Nevertheless, the belief is that the mechanisms of wound healing are similar. Animal wounds are studied also to try and determine the wound geometry which closes the fastest and with least scarring. This is often done by measuring the wound area following an exci-sional wound, typically using tatoo marks to delineate the wound boundary and distinguish contraction from epithelialisation (see, for example, the article by Bertolami et al. 1991). Among the best quantitative studies for comparing our results with experiment are those by McGrath and Simon (1983) who studied dermal wounds on rats. They showed there is a rapid retracting phase, then a plateau phase followed by a contracting phase, which can be described by a simple exponential dependence on time given by
where A0 is the wound area when contraction begins, Af is the area remaining after contraction is complete, both areas being scaled to the excised area and kc is the contraction rate constant. Figure 10.2 shows the experimental results from McGrath and Simon (1983).
Murray et al. (1988) showed that this base model can produce a stable nonuniform steady state solution which evolves from the initial wounded state. However, it was not qualitatively consistent with the extant data on a contracting wound for the parameter ranges investigated and particularly the data shown in Figure 10.2. Typically they found a transient retraction of the wound boundary outwards (that is, in one dimension u > 0) and a relaxation inwards back to the undisturbed state u = 0 rather than any contraction of the wound boundary inward (u < 0); this is certainly a crucial deficiency. It appears therefore that what we considered to be a minimal set of properties applicable to dermal wound healing are inadequate for simulating wound contraction.
We certainly made some gross assumptions in deriving the model system (10.5), such as a small strain approximation, no production of ECM and so on. Tranquillo and Murray (1992) discuss in more detail the deficiencies of this model and investigate amended versions. Tranquillo and Murray (1993) discuss the clinical implications of the modelling. Here we briefly describe these amendments for pedagogical reasons. In the following sections we discuss more realistic versions but with the some of the same basic concepts. The question Murray et al. (1988) addressed was how to amend the above model to make it more biologically realistic. All of the amendments added to the cell functions and the ECM properties. With the exception of a term for the biosynthesis of the ECM they were reasonably motivated by the recognised influence of inflammatory-derived biochemical mediators on fibroblast functions. These can be extremely complex and much is still unknown about them. One possibility to incorporate some aspects of
this is to include a generic equation for such biochemical mediators. Instead, based on experimental evidence, Murray et al. (1988) and Tranquillo and Murray (1992, 1993) incorporated a mediator dependence in the cell traction term by setting
where tf is a traction enhancing parameter and tq is the base cell traction. They then took the biochemical mediator to be a given function, namely, c = cQ exp (—x2fa), where c0 is the concentration at the wound centre, x measures distance from the wound centre and a is a parameter specifying the spatial influence of the mediator from the wound centre. Tranquillo and Murray (1992) studied this modified system in detail and found that for appropriate parameter ranges the solutions for the tissue displacement agreed reasonably well with experiment. They went on to discuss other possible variations such as including chemotaxis, ECM biosynthesis and cell growth variation.
A major criticism of these models is the prescription of the mediator concentration gradient in lieu of it being autonomously determined from a model of inflammation. It makes the comparison of the results with experimental data somewhat unconvincing. It also circumvents the whole question of the causal factors and the ultimate regulatory control of wound contraction. What it does show, however, is how the modulation of fibroblast function by an inflammatory mediator could result in the wound healing scenario we described above. Tranquillo and Murray (1992) investigated the effect of cell traction forces within this modelling scenario.
There are several obvious deficiencies in the above formulation other than that associated with the inflammatory response, such as the omission of biosynthesis of the cells and the ECM which are not known. Also the theory is based on a small strain vis-coelastic formulation when in vivo wound contraction involves finite strains. So, more realistic stress-strain constitutive relations are required, in particular one that accounts for aniosotropic fibre orientation, a topic we discussed above in the chapter on embryonic epidermal wound healing. It is much more complex in full-depth wounds. In spite of these difficulties the simple models discussed within this framework provided a means of considering the effects of known and speculated cell proliferation, migration, and traction responses and ECM rheological properties both individually and collectively. In the following sections we describe some of the more realistic models which have been proposed and studied based on the above modelling concepts.
Here we briefly recap the review given by Murray et al. (1998). Since the first wound healing models were published mathematical modelling has greatly increased. More complicated versions of the model described above include multiple cell types (with differentiation between types) and multiple types or phases of ECM (which interact and possess different mechanical properties and have different effects on cells). Other models include additional equations for chemicals (such as growth factors) which modulate cell proliferative, motile and contractile behaviour. When the mechanical properties of the evolving ECM are modelled with greater realism further equations are required.
Maini, Sherratt and their coworkers have presented some very interesting results which include the following. Olsen et al. (1995) extended the Murray-Tranquillo model to include both explicit interconversion between fibroblasts and traction-enhancing my-ofibroblasts and more complex inflammatory mediator dynamics. The model captures the experimentally observed temporal changes in the densities of myofibroblasts and growth factors. A wound remains transiently contracted while ECM remodelling (slow) is in progress. Olsen et al. (1996) related pathological scarring to the existence of alternative steady states using a mechanical model and a simpler caricature. A high rate of growth factor production, for example, could cause a spread of the pathological state across the wound and into surrounding tissue. Complementary to this study, Dale et al. (1996) used a non-mechanical model to investigate the ratio of collagen I to collagen III in scar tissue (this ratio is known to be related to fibre thickness and scar quality) focusing on the regulation by TGF isoforms (the latter growth can be topically applied).
Recent work by the Maini-Sherratt group (Dallon et al. 1999) on extracellular matrix dynamics is an important contribution to the study of tissue regeneration and reorganization and scar formation in general. They consider the cells to be discrete and the matrix to be a continuum: cell and fibre orientation dominate the process. The work of Edelstein-Keshet and her colleagues (see Edelstein-Keshet and Ermentrout 1990,
Mogilner and Edelstein-Keshet 1996 and other references there) on cell orientation is particularly apposite to this approach. Dallon et al. (1999; see other references there) show how cell movement is directed by the matrix substrate and in turn how they in turn reorient the extracellular matrix. They are able to quantify, by extensive numerical simulation of their models under different biological assumptions, a variety of effects such as the rate of movement of the cells, the influence of cell contact guidance, the original orientation of the matrix fibres, fibre production and degradation and so on. They present visually graphic results which show the effect of these different factors on the final fibre alignment pattern. They conclude that in wound healing cell flux is a particularly important factor in tissue alignment.
Tracqui et al. (1995) extended the Murray-Oster mechanical formulation to include two distinct types of extracellular matrix, an early provisional matrix being replaced by the collagenous matrix characteristic of mature scars. A key feature of this model is that ECM turnover results in plastic behaviour whereby the wound remains permanently contracted after the inflammatory response has subsided. Barocas and Tranquillo (1994, 1997a) accounted separately for the interstitial fluid and the fibrous network making up the ECM (this is potentially important in collagen gel assays which can contract much more rapidly than wounds). In their wound healing models they use the limit of zero drag which effectively reduces the system back to a single phase model. An important extension remains however: they were able to account for matrix anisotropy via an orientation tensor which is based on the strain tensor. The fibre orientation tensor governs cell movement (contact guidance) and traction (greatest in the direction of greatest fibre alignment). In circular wounds, contact guidance can reduce the extent of wound contraction as cells align (and pull) circumferentially (Barocas and Tranquillo 1997).
Cook (1995) extended the Murray et al. (1988) and Tranquillo and Murray (1992) model primarily by introducing more realistic mechanics and structure of an evolving, anisotropic ECM, effective strain and by introducing a fibre orientation tensor which is related to it. In addition to providing a measure of scar quality, fibre orientation feeds back to affect (i) cell movement (the flux of cells due to contact guidance depends upon effective strain in a similar way to the Barocas and Tranquillo 1997a model, although Cook's derivation was based on a biased random walk) and (ii) cell traction (greatest contractile force in the direction of fibre orientation). Cook (1995) also allowed for or-thotropic skin tension (and wound orientation relative to skin tension lines). In contrast to previous models contractile forces were assumed to be zero outside the wound and late in scar formation. Finally, this was the first study to attempt numerical solution of the equations in two dimensions (radially symmetric solutions are quite different).
Cook's results (1995) included the following: (i) a study of changes in wound shape, contraction rates, and fibre alignment (wound orientation proves important); (ii) true plasticity: wounds remain contracted (and fibre alignment is anisotropic) in the absence of inflammatory mediators or any other externally imposed differences between skin and mature scar; (iii) strong contact guidance: this can give rise to instabilities (pattern formation) suggestive of nodules of aligned collagen fibres in pathological scars. Finally, a highly simplified model based on linear elasticity suggested that circular wounds should contract most slowly (agreeing with experiments). The role of wound shape is yet to be addressed completely in the full model.
The early wound healing models were derived from the Murray-Oster mechanical models of Chapter 6 for developmental processes such as the positioning of skin primor-dia (hairs, feathers, scales) and pre-cartilage patterning in the developing limb. Wound healing can be seen as just one of many morphogenetic processes which involve tissue growth and remodelling both in development and in adults. In a comprehensive review Taber (1995) describes how related processes occur in bone, skeletal muscle, the heart and arteries. He also describes a number of theoretical advances which have allowed the biomechanics of growth and remodelling to be brought into a continuum-mechanical framework.
In classical elasticity theory a strain field characterizes the mapping between material points in a reference state and locations in a deformed state. The mechanical properties of the material are incorporated in a constitutive equation which relates stresses to strains. Finally, applied body and boundary forces determine the global deformation via the equations of equilibrium which constrain the stresses in the material. Although finite strains and nonlinear constitutive equations provide complications, applying classical viscoelasticity theory to biological materials is no different from applying it to nonliving materials. Once experiments have been devised to determine the appropriate constitutive equations for each mechanically distinct component (a highly nontrivial task) a whole range of biomechanical problems can be solved (Fung 1993 gives many examples).
However, there are two qualities of living tissues which require that classical continuum mechanics be extended. Firstly, biomaterials can support residual stresses: even when all external forces are removed stresses may exist within the tissue. Residual stresses can be indirectly observed by cutting out sections of tissue and recording deformations (for example, Fung and Liu 1989). The existence of residual stresses violates the assumption of classical elasticity theory that the reference state is globally stressfree. The second problem is that living tissues can change their form, their structure and even their material properties, either as part of the natural developmental process (growth is an example) or in response to other signals (such as injury or applied stress). Thus, classical mechanics must be extended to accommodate tissue growth and tissue remodelling.
A number of scientists have addressed these problems including Cook (1995) in the particular case of dermal wound healing. Rodriguez et al. (1994) described, using the language of tensors, a general finite strain theory for elastic biomaterials which incorporates both residual stresses and volumetric tissue growth and remodelling (surface growth, such as occurs in bone, has been described by Skalak et al. 1982; see Taber 1995 for other references). The Rodriguez theory has application to a wide range of important biomedical processes, including dermal wound healing.
It is not possible to discuss all of these interesting advances on wound healing modelling. For anyone seriously interested in modelling wound healing the above articles are required reading. We restrict our discussion primarily to some of the work of Tracqui et al. (1995) and particularly Cook (1995) since most of it remains unpublished.
10.4 Model for Fibroblast-Driven Wound Healing: Residual Strain and Tissue Remodelling
In this section we discuss the mechanical model of Tracqui et al. (1993) for the contraction and relaxation phases of healing dermal wounds based on the extracellular control of the traction forces exerted by the migrating fibroblastic cells. We assume that these cell-extracellular medium interactions are controlled by the viscoelastic properties of both the provisional matrix formed initially in the lesion and the newly synthesised col-lagenous matrix secreted at the same time. In addition, and importantly, we include the plastic response of the collagenous matrix to the cell traction. This mechanical model accounts for the different experimental phases of the wound boundaries' movement with time. Furthermore, it provides a quantification of the residual strain and stress resulting from the plastic response and remodelling of the extracellular matrix which could help to characterise the scar tissue formation in wound healing.
In the model we describe here we concentrate on wound contraction and the related displacement of the wound margins with time. The model tries to characterise some of the major biological factors which, through their interactions, are sufficient at least to qualitatively take into account the dynamics of the wound margins. In this section we consider only the one-dimensional version.
Let us briefly summarize the wound healing process we consider: further biological details are given below. The movement of the original wound boundaries toward the centre of the lesion (wound contraction) results from the mechanical interaction between the fibroblastic cells (fibroblasts and myofibroblasts) and the surrounding extracellular matrix: a good overview is given by Jennings et al. (1992). In cutaneous wounds, this reduction in the wound area takes place after an initial retraction of the wound boundaries, due to the intrinsic tension that is usually present in skin. Next a latent phase occurs, during which no gross movement takes place. The contraction phase is then characterised by an exponential decay of the wound area as shown in Figure 10.2. Later, an incomplete relaxation phase takes place, leading to a sustained contraction of the wound. Figure 10.2, taken from the data of McGrath and Simon (1983) illustrates each of these phases. The latent phase lasts about a week with the exponential phase lasting until around the beginning of the 6th week. The epithelialisation starts around the beginning of the 3rd week eventually merging with the dermal curve about week 8.
We have already discussed various hypotheses to account for the inward wound boundary movement; see also Welch et al. (1990). Experimental results seem to favour the pull theory, in which inward movement of the wound margins is due to forces lying in the regenerated tissue in the wound bed. In this section, we describe a model for wound contraction based on the modulation of the traction force exerted by the fibrob-lastic cells through the difference in the properties of the different wound extracellular matrices.
Various experimental data support this conceptual framework. First, the distortion by fibroblast traction of various substrata, including collagen, is well documented in the classic paper by Harris et al. (1981) which has been frequently referred to in this book (see also Guidry 1992). Also several steps in the reorganisation of the extracellular me dium in the early stages of the wound healing have been characterised by Guidry and Grinnell (1986). Coagulation activates fibrin, which cross-links to form an initial matrix, the fibrin clot. In addition, various plasma proteins are trapped within this porous gel-like network. Fibronectins, initially deposited in a soluble form by the plasma and later supplied locally by fibroblasts, can bind to fibrin, collagen, hyaluronic acid (HA) and fibroblast surface receptors and thus provide some anchoring for fibroblast movement and traction (for example, Jennings et al. 1992). Alternatively, the binding of HA to fibrin stabilises and increases the volume of the fibrin gel, creating a more porous medium as suggested by Stern et al. (1992) and Tranquillo and Murray (1992). The constructed HA-fibrin matrix provides within the wound a viscoelastic deformable medium in which fibroblasts and myofibroblasts can migrate and proliferate. At the same time, these cells degrade this provisional matrix and secrete a new collagenous matrix with different properties (Welch et al. 1990).
It is this biological scenario just described that was considered in the modelling framework developed in Chapter 6 for dealing with cell condensations in a deformable extracellular medium and used above in Section 10.1. We pointed out some of the drawbacks in that formulation, prime among which is the externally imposed biochemical gradient. Here we investigate how the wound contraction dynamics could result from the nonlinear intrinsic mechanical properties of the wound extracellular matrices.
This model again takes into account cell migration, mitosis and cell traction forces on the ECM. There are, however, some fundamental differences to those models we have discussed up to now. There are four dependent variables, namely, the fibroblastic cell concentration, n(x, t), the collagenous extracellular matrix concentration, p(x, t), the provisional matrix concentration inside the wound, m(x, t), and the displacement u(x, t) of the ECM at position x at time t. We consider here a one space-dimension wound along the x-axis where x = 0 is the wound centre and x = 1 defines the original wound margin inside a domain of half-size L.
The model assumes the customary conservation and force equilibrium equations. For the fibroblasts we assume there is random migration, passive convection and logistic mitotic growth. The inclusion of an equation for the provisional matrix is based on the above discussion and the experimental literature. For it we assume that it degrades, is a viscoelastic isotropic medium with distributed cell traction stress and without subdermal attachments. For the collagenous matrix, the ECM of earlier models, we consider biosynthesis and assume it is an isotropic, nonlinear elastoplastic viscous medium with distributed cell traction stress and elastic subdermal attachments.
The crucial incorporation of elastoplastic behaviour in the collagenous matrix in response to forces generated by cell traction is mainly supported by experiments in which fibroblasts contract hydrated collagen gels. Guidry and Grinnell (1986) have shown that the removal of cells from contracted gels holds in place the framework of the bundles of collagen fibrils that has been organised around and in between the cells. The same behaviour can be observed in the absence of cells, when the gel is centrifuged. It is the modelling of this behaviour that is quite different to the above force equilibrium equations.
Cell-ECM Mechanical Interactions
The general force balance equation of the matrix is again given by the divergence of the stresses set equal to the external forces; that is,
V ■ [<?ECM(x, t) + Wcell(X, t)] = spu(x, t), (10.7)
where aECM(x , t) denotes the stress tensor of the ECM and ff-cell(x, t) is the traction stress exerted by the cells on the ECM. The term spu (x, t) models the restriction of the collagenous matrix by its connections to the external substratum via subdermal attachments (essentially like a simple spring): the form is analogous to (10.5) with the positive parameter s reflecting the strength of the attachments.
The stress wecm is made up of an elastoplastic and a viscous part:
It is the modelling of the tensor aelastoplastic that is different and crucial. This stress is attributed to the passive elastic properties of the provisional matrix and to the nonlinear elastic and plastic characteristics of the collagenous matrix. To describe the nonlinear elastic behaviour of the matrix, we use the hypoelastic formulation based on the incremental form of the stress-strain relationship. In this formulation, the state of stress depends on the current state of strain as well as the stress path followed up to that state. A general form of the constitutive equation in tensor notation is given by Chen and Mizuno (1990), as daij = Ctjki(epq) dea, (10.9)
where Cijkl is usually called the tangential stiffness tensor of the material.
We use the simplest class of hypoelastic models where the incremental stress-strain relationships are formulated directly as an extension of the isotropic linear elastic model by replacing the elastic constants by variable tangential moduli which are taken to be functions of the strain invariants. We consider the incremental (one-dimensional) relationship between the stress a and the strain e to be given by da(x, t) = Et(e,p, m) de(x, t), (10.10)
where the tangential stiffness modulus ET (e, p, m) is a function of the strain e and of the time-varying density of ECM. In addition, experiments on collagen gels (Guidry and Grinnell 1986) suggest that the plasticity effect of the collagenous matrix depends on the matrix density p and the provisional matrix, m. We thus describe the spatiotemporal variation of the tangential stiffness modulus ET (e, p, m) by the relation dEt / d|e| \
The constants E0p and E0m are the characteristic elasticity moduli of the collagenous and provisional matrix respectively. They can be related to the Lame coefficients X and
H of a normalised density of each material (see, for example, Chen and Mizuno 1990 or Landau and Lifshitz 1970). The terms S(n, p,m) and P(n, p, m) correspond to the variation in the amount of collagenous matrix secreted and degraded respectively, while Q(n,p,m) is the amount of provisional matrix degraded. The coefficient a is a positive constant defining the amplitude of the plastic response of the collagenous matrix under increasing stretching or increasing compression (loading). When stretching or compression decreases (unloading), a is zero.
Under these hypotheses, the expression of the stress oelastoplastic is ft de(x, t') ,
Oelastoplastic (x, t) =/ Et (e,p, m)-—-dt- Or (x, t). (10.12)
The increase in the stiffness modulus ET (e, p, m) generates within the ECM a residual stress oR (x, t) which can be evaluated from (10.10) at the beginning of the unloading stage.
The stress tensor oviscous is given by the general form used before (as in (10.3), for example):
d t d t where and y2 are the shear and bulk viscosity respectively, and e is the dilation. The viscosity coefficients and y2 depend on the ECM density and on the strain e. For simplicity, we assume that these coefficients are proportional to the elastic coefficients through a single parameter. This approximation, which is often used, takes advantage of the symmetric structure in e and de/dt of the elastic and viscous stress respectively (Landau and Lifshitz 1970).
We model the traction force exerted by the cells on the matrix by taking it to be proportional to the wound matrix densities p(x, t) and m(x, t), according to the relationship:
where the constants t0 and t1 monitor the strength of the traction, while y measures the saturation of the traction stress as the cell density increases.
Matrix and Cell Conservation Equations
We model the collagenous matrix density conservation by dp did u\
in which we consider the rate of the collagenous matrix biosynthesis by the cells to be self-regulated around a saturation threshold p0 through a positive proliferation rate b.
The equation for the provisional matrix initially present in the wound is convected and degraded by the cells according to a first-order removal given by d m 3/3 u\
where w is the positive decay constant. We assume that there is no continuous creation of the provisional matrix; it is only formed in the very early stages after the wound injury.
The cell conservation consists of the usual terms, namely, diffusion with coefficient D, convection along with the ECM at velocity d u/dt together with inhibition of cell mitosis for high cell density which is qualitatively again modelled by a logistic type growth curve rn(n0 — n), where r is the mitotic rate. The velocity, d u/d t, is an approximation to the convection velocity; see the discussion below. The equation is then d n 3/3 u\ d 2n
The model mechanism is then given by the four equations (10.7), (10.15)-(10.17) for the four dependent variables n, p, m and u with the various terms in (10.7) given by (10.8)-(10.14).
10.5 Solutions of the Model Equations and Comparison with Experiment
As usual it is first necessary to nondimensionalise the equations and decide on the appropriate boundary and intitial conditions. The nondimensionalisation is left as an exercise; it is quite standard. We take the initial state of the wound (0 < x < 1) to be filled with the provisional matrix which is devoid of fibroblasts: this is why we do not have a production term in the provisional matrix equation (10.16). The surrounding dermis is approximated as a medium with size L > 1. We assume that there are initially cells and collagenous matrix outside the wound only. As boundary conditions we take
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