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d y2

d U2

For example, under this transformation a circle dxTIdx = 1 is mapped (deformed) into an ellipse dyT MT M dy = 1. We can further find a rotation matrix P satisfying PTP = I that diagonalises the symmetric MTM, which is the 'strain' matrix. If we now let dz = PT dy we then have

(dz)TPTMtMP dz = dzT A dz = 1, where the matrix A = diag (Ai, A2) is diagonal where the Ai are the eigenvalues of the strain matrix MT M and P is made up of normalised eigenvectors.

Cook (1995) derived the orientation and stretch distributions, obtained under a general deformation, by an initially uniform distribution of relaxed fibres. He went on to derive evolution equations for the fibre density, fibre orientation distribution and stretch distribution based on the (Lagrangian) velocity v = Du/Dt (the total derivative of u), the material velocity, with a linear approximation v = d u/d t.

Plasticity, Zero Stress State and Effective Strain

The problem with realistic modelling of dermal wound healing, as we have mentioned before, is that we must include plasticity of the ECM. Mechanical plasticity involves the formation and breaking of chemical bonds during deformation in such a way that the tissue does not return to its original state before deformation. More relevant to wound healing, however, is remodelling plasticity which results from the cells remodelling the tissue not only while it is being deformed but also afterwards. In the model discussed in the last section we had new matrix formed and provisional matrix (which originally filled the wound area) continuously being degraded. We have already mentioned that there are well-recognised skin tension lines3 in the human body (and exploited by plastic surgeons) so the skin does not completely relax after wound healing is finished. Wound healing models therefore, have to allow for stresses and a nonuniform distribution of fibres to exist at equilibrium. At this stage of our knowledge of tissue plasticity in vivo and in vitro it is necessary to derive evolution equations for the ECM structure variables which rely on many simplifying approximations. The fine details of stretch and orientation distributions are not included directly but rather implied by the effective strain and plastic effects via the evolution of the zero stress state.

It is possible locally to deform the matrix such that there is no stress which means that the zero stress state is the deformation state in which the local matrix is stress free. This is at each point; it does not imply that we can construct a deformation that makes the whole tissue return to a stress free state.

In the reconstruction of the wounded area there is matrix production and matrix degradation and so, after a while, there is little connection with the original reference state. We introduce the concept of an effective strain which is the strain relative to the zero stress state. We further introduce the residual strain which is the zero stress

3The anatomy of these skin tension lines was studied in considerable detail by Langer (the German papers date from 1862; see the translations in Langer 1978a,b,c,d). He talked about cleavability and tension lines and mapped the directions on the human body with some precision; they are known as Langer lines. These cleavage lines reflect collagen fibre orientation and are conserved throughout adult life except under extended stress, such as by pregnancy.

zero stress state, dz current (deformed) state, dy zero stress state, dz current (deformed) state, dy

reference state, dx

Figure 10.7. Schematic deformations of a tissue element in the reference state to the current deformed state and from the zero stress state. The deformation matrices are denoted by M and N as explained in the text.

reference state, dx

Figure 10.7. Schematic deformations of a tissue element in the reference state to the current deformed state and from the zero stress state. The deformation matrices are denoted by M and N as explained in the text.

state relative to the original configuration. The effective strain is computed from the residual strain and the actual deformation with respect to the original reference state as schematically shown in Figure 10.7.

Rather than having to deal with fibre densities we make a further approximation and assume, not unreasonably, that the fibres are uniformly distributed and unstretched when the tissue is in the zero stress state. Cook (1995) discussed the more sophisticated approach in which this assumption was relaxed.

Refer now to Figure 10.7 and let points in the reference configuration have coordinates denoted by Xi, those in the zero stress state by zi and those in the current deformed state by yi. Consider a small element, dx, about a particular point in the reference state with the corresponding elements dz and dy in the zero stress and deformed states respectively as schematically shown in Figure 10.7. We write the linear transformations in the form dz = N dx, dx = M dy, (10.20)

where, if yi = xi + ui and xi = zi + vi the matrices M and N are given by (cf. (10.19))

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