Rmn

Strain, e

Figure 6.5. The effect of straining the extracellular matrix is to align the fibres and stiffen the material. If we think of a one-dimensional situation the strain from (6.9) is e = du/dx and the dilation 9 = du/dx. The effective elastic modulus E is the gradient of the stress-strain curve. It increases with strain until the yield point whereupon it levels off and drops for large enough strains as the material tears. The ECM is in compression when e < 0 (also 9 < 0). Because a given amount of material (cells + matrix) cannot be squeezed to zero, there is a lower limit of e = —1 (also 9 >—1) where the stress tends to —to.

Figure 6.5. The effect of straining the extracellular matrix is to align the fibres and stiffen the material. If we think of a one-dimensional situation the strain from (6.9) is e = du/dx and the dilation 9 = du/dx. The effective elastic modulus E is the gradient of the stress-strain curve. It increases with strain until the yield point whereupon it levels off and drops for large enough strains as the material tears. The ECM is in compression when e < 0 (also 9 < 0). Because a given amount of material (cells + matrix) cannot be squeezed to zero, there is a lower limit of e = —1 (also 9 >—1) where the stress tends to —to.

Now consider the contribution to the stress tensor from the cell tractions, that is, a cell. The more cells there are the greater the traction force. There is, however, experimental evidence indicating cell-cell contact inhibition with the traction force decreasing for large enough cell densities. This can be simply modelled by assuming that the cell traction forces, t (n) per unit mass of matrix, initially increase with n but eventually decrease with n for large enough n. Here we simply choose

where t (dyne-cm/gm) is a measure of the traction force generated by a cell and X is a measure of how the force is reduced because of neighbouring cells; we come back to this below. Experimental values for t are of the order of 10—3 dyne/^m of cell edge, which is a very substantial force (Harris et al. 1981). The actual form of the force generated per cell, that is, T(n)/n, as a function of cell density can be determined experimentally as has been done by Ferrenq et al. (1997).

Even though cell traction plays such a central role in pattern formation in development it has proved very difficult to quantify the cellular forces involved because of the complexity of the cell-matrix interactions and the difficulty of separating out the various mechanical effects in real biological tissue. Ferrenq et al. (1997) describe a new experimental technique and general approach for quantifying the forces generated by endothelial cells on an extracellular matrix. They first developed a mathematical model, based on the Murray-Oster mechanochemical theory described in detail in this chapter, and in which different forms for the cell generated stress are proposed. They then used these as the basis for a novel experimental device in which cells are seeded on a biogel of fibrin (the matrix) held between two holders one of which can move and is attached to a force sensing device. By comparing the displacement of the gel calculated from the model expressions with the experimental data recorded from the moving holder they were able to justify specific expressions for the cell traction stress; they did this for var ious experimental setups. They were then able to compare different plausible analytical expressions for the cell traction stress with the corresponding force quantification and to compare the results with experiment and other reported measurements for different kinds of cells by similar and different experimental devices. They show how experimentally justifiable forms for the cell-gel traction stress can be derived and give estimates for each of the parameters involved. They found that the expression where t is the cell traction and the parameter N2 controls the inhibition of cell traction as the cell density increases was validated by experiments and estimates given for their values. This paper is an excellent example of genuine interdisciplinary mathematical biology research with theory and experiment each playing an important role in the outcome. This interdisciplinary approach was exploited by Tranqui and Tracqui (2000) in their investigation of mechanical signalling in angiogenesis. They again used the mechanical theory with the viscous stress tensor given by (6.12) and the elastic stress tensor given by (6.13) which includes long range elastic effects.

If the filopodia, with which the cells attach to the ECM, extend beyond their immediate neighbourhood, as they probably do, it is not unreasonable to include a nonlocal effect analogous to the long range diffusion effect we included in the cell conservation equation. For our analysis we take the contribution acell to the stress tensor to be where y > 0 is the measure of the nonlocal long range cell-ECM interactions. The long range effects here are probably more important than the long range diffusion and haptotaxis effects in the cell conservation equation.

If the cells are densely packed the nonlocal effect would primarily be between the cells and in this case a more appropriate form for (6.15) should perhaps be

There are various possible forms for the cell traction all of which might reasonably be used. One way of resolving the issue might be to use a molecular method developed by Sherratt (1993) in his derivation of the actin generated forces involved in embryonic wound healing; we discuss his technique in Chapter 9.

Finally let us consider the body force F in (6.10). With the applications we have in mind, and discussed below, the matrix material is attached to a substratum of underlying tissue (or to the epidermis) by what can perhaps best be described as being similar to guy ropes. We model these restraining forces as body forces proportional to the density of the ECM and the displacement of the matrix from its unstrained position and therefore take

where s > 0 is an elastic parameter characterising the substrate attachments.

In the model we analyse we shall not include all the effects we have discussed but only those we feel are the more essential at this stage. So, the force equation we take for the mechanical equilibrium between the cells and the ECM is (to be specific) (6.10), with (6.11)-(6.17), which gives

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