and so we obtain the expression for the diffusion coefficient matrix D as

Cook (1995) derives various forms for the cell flux, J, again from the microscopic properties of the fibres, in situations where the cells are in an environment with a distribution of fibre orientation and also when they do not follow the fibres. He uses probability distributions in which he defines a probability of cells moving in a given direction, then derives forms for the flux and hence a relevant diffusion equation for N. In the case of cells which do not follow the fibre directions he obtains the following diffusion form, d N

Again, a uniform cell density is not a steady state solution of the equation. Another point about this form is that the variable diffusion coefficient appears inside both derivatives. This form of the diffusion terms is used in the first model mechanism below and in numerical simulations discussed later.

10.9 Model System for Dermal Wound Healing with Tissue Structure

We now give the governing equations for Cook's (1995) finite strain model mechanism. We then give the small strain approximation. A key element in the modelling, as it was in the Tracqui et al. (1993) model in Sections 10.4 and 10.5, is the structure of the ECM and in particular its influence on cell movement. Although it is the underlying basis of this discussion on deformations and stress fields that it is cell traction which generates them, we must also keep in mind that there is passive convection with velocity v; we discuss this aspect further below.

The equations (the specific details of which we discuss immediately below) for cell density (N), ECM density (S), displacement (ui), residual strain (Zj) and force equilibrium are quite different in detail to those used before and are given by

dS dt

The expressions for the actual deformation gradient (with respect to the deformed state), effective strain, stress, diffusion matrix and traction are here:

Mij = Stj - Ui,j, €tj = 1 (I - MTZM), ay = Sx [2etj + Sijeaa] D

Tr = Trace[MT ZM], A = Det [MT ZM], where the repeated a implies summation; here eaa = e11 + e22 with M the deformation matrix defined by (10.19) and the strain matrix, Z, defined by (10.23). Before discussing the various functions in (10.56), and T (N, S) in (10.57), let us recall what the various terms in the equations represent.

Equation for the Cell Density N

The first term, involving the diffusion matrix, incorporates the effect of contact guidance by the fibres and is a function of the strain and is of the form used in (10.55). If we consider the contact guidance to be along fibre directions this term would be different: the diffusion matrix would then be given by (10.48) with (10.49). The second term on the right is the contribution from the passive convective flux of cells as the cellmatrix continuum deforms with material velocity v. The last term, f (N), is the cell proliferation contribution and includes cell death, differentiation and dedifferentiation. It is also probably a function of the strain since it is well known that cell shape can influence mitosis (for example, if a cell is too flat it tends not to divide).

The evolution equation for N is based on a conservation law and as written is in an Eulerian frame of reference (as opposed to a Lagrangian frame of reference); that is, the coordinate system is fixed and the material flows past it. This is the usual way of writing an equation for a conservation law. There is a difference between v, which is the material velocity, and the term d u/d t which is the derivative of the displacement for a point in the fixed frame of reference. When we consider finite strains these are not the same and we need a further equation, namely, the third equation in (10.56). In the case of a small strain approximation, however, they are equal.

Equation for the Matrix Density, S

The tissue is also convected since it is part of the cell-matrix continuum. We have separated the matrix secretion, g1, and degradation, g2 terms since in the model only the secretion term affects the mechanical plasticity via the strain matrix Z.

Equation for the Residual Strain Matrix, Z

This equation, which shows how the residual strain tensor, Z, changes, uses the specific form we derived above for the plastic remodelling. The function q(S) describes the fraction of newly secreted matrix that forms new fibres in the tissue. The last term, the convection term, is like that in the cell equation and is the contribution from moving from a Lagrangian frame of reference to the Eulerian frame. The second definition in (10.57) determines the effective strain once we have obtained the residual strain Z and the deformation matrix M.

Force Balance Equation

This is the now customary equation which says that the forces are in quasi-equilibrium at all times. The form of the stress-strain relation we use is given by the third equation in (10.57) which defines the stress tensor aij. The specific form we have chosen for the cell traction, Tij is given in (10.57). We discuss the body force, bi and T(N, S) below.

Let us now consider the specific functions and some reasons for the forms chosen in (10.57). A crucial ingredient is the cell traction. We first assume that it is directly related to cell movement in the presence of fibre orientation in a similar way that the diffusion matrix is associated with fibre orientation. There is considerable experimental justification on cell-matrix interaction for this in that cells tend to align themselves with their fibrous environment (see, for example, the experimental work by Vernon et al. 1992,1995 and the discussion in Chapter 8). Other factors no doubt play a role but we do not yet have enough experimental evidence to incorporate these in the formulation; the effect we do include is certainly a major one. We thus model the (nonisotropic) cell traction tensor by

Tij = T(N, S)D, where the magnitude of the traction, T (N, S), has to be specified.

Cell traction is certainly a function of cell density but its dependence is such that outside of the wound the traction is essentially zero. We do not have any biological data on how the traction depends on matrix density, S but we assume a major effect on a cell's traction is the proximity of its neighbours. We thus consider the traction to be given by

where 0 is a positive (even number) constant and t0 is the traction force per cell. With this form, when N = N the traction force is zero.

As with so much about wound healing the synthesis and degradation of the ECM is also not well understood so we take very simple linear forms in which production is proportional to the cell density and the degradation is proportional to the matrix density S. We thus take gi(N, S) = K1N, g2(N, S) = K2NS. (10.59)

The first of these means that we assume the cells in the intact dermis (away from the wound) continue to make ECM at the same rate as within the wound. However, as the ECM matures we assume it has a weaker effect and so we use the matrix density S as a measure of tissue maturity. The effect of maturity is reflected in the production function q(S) which is a decreasing function of the newly secreted collagen that makes up the new fibres. As q (S) ^ 0 the only effect of ECM turnover is then to maintain the density at a constant level. A simple function that reflects this is i 1 - S when S < p q (S) = p , (10.60)

[ 0 otherwise where p is another parameter. Note that from the second of (10.56) the forms in (10.59) imply the uniform steady state matrix density S = k\/k2. This implies that we do not consider any difference between the matrix density within the wound and that outside. The orientation of the fibres that make up the ECM via the effective strain is the sole measure of scar quality.

Finally we have to model the subdermal attachment forces reflected in bi: these resist the tissue deformation. We assume they are proportional to the displacement and so we take bi (S, u) = aSui,

where a is a constant. With the form of the force balance equation where ECM response and traction are positive, this means that a > 0. Although we have taken a to be constant it no doubt depends on the wound depth in three-dimensional versions of the models.

Finally, gathering all these functions together, the finite deformation Cook (1995) model for dermal wound healing is given by (10.56) and (10.57) where the functions for cell proliferation, matrix secretion, matrix degradation, traction magnitude, body force and new fibre fraction are defined by

0 0

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