A Continuum Mechanics Model of Enzyme-Based Tissue Degradation in Cancer Therapies

Abstract

We propose a mathematical model to describe enzyme-based tissue degradation in cancer therapies. The proposed model combines the poroelastic theory of mixtures with the transport of enzymes or drugs in the extracellular space. The effect of the matrix-degrading enzymes on the tissue composition and its mechanical response are accounted for. Numerical simulations in 1D, 2D and axisymmetric (3D) configurations show how an injection of matrix-degrading enzymes alters the porosity of a biological tissue. We eventually exhibit numerically the main consequences of a matrix-degrading enzyme pretreatment in the framework of chemotherapy: the removal of the diffusive hindrance to the penetration of therapeutic molecules in tumors and the reduction of interstitial fluid pressure which improves transcapillary transport. Both effects are consistent with previous biological observations.

Appendix A: Formulation of the Model in \(\varOmega _0\) in the General Case

The calculus in the general case gives the final system of Eq. (76). Recall that matrix B is defined as the inverse of matrix A given by (46). As,

$$\begin{aligned} (B^{-1})_{i,j}=A_{i,j} = \left( \frac{\partial \varPhi (t,\mathbf{X })}{\partial \mathbf{X }} \right) _{i,j} = \delta _{ij} + \frac{\partial u_i}{\partial X_j}(t,\mathbf{X })= \delta _{ij} + \frac{\partial {\overline{u}}_i}{\partial {\overline{\mathbf{X }}}_j}({\overline{t}},{\overline{\mathbf{X }}}), \quad \end{aligned}$$

(74)

we kept the notation B to refer to $B({\overline{\mathbf{u }}})$ in system (76). Note that we equally dropped bars on the dimensionless variables but kept them on the dimensionless parameters. Let us denote

$$\begin{aligned} J_{\scriptscriptstyle {\mathrm {enz}}}^B = \frac{1}{f} {\overline{\varvec{\kappa }}} \, B\nabla P - \overline{\mathbf{D }_{\scriptscriptstyle {\mathrm {enz}}}^0} B\nabla f \quad \text {and} \quad J_{\scriptscriptstyle {\mathrm {drug}}}^B = \frac{1}{f} {\overline{\varvec{\kappa }}} \, B\nabla P - \overline{\mathbf{D }^0_{\scriptscriptstyle {\mathrm {drug}}}} B\nabla f. \end{aligned}$$

(75)

The equivalent system in $\varOmega _0$ in dimensionless form reads

(76a)

(76b)

(76c)

(76d)

(76e)

(76f)

(76g)

where we get (76a) from Assumption 1, (76b) from Eq. (20), (76c) from (15), (76d) from (24), (76g) from (28), (76f) from (13a) and (76e) from (13b), using (50), (51) and (52).