Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions

In this paper we consider a mathematical model of cancer cell invasion of tissue (extracellular matrix). Two crucial components of tissue invasion are (i) cancer cell proliferation, and (ii) over-expression and secretion of proteolytic enzymes by the cancer cells. The proteolytic enzymes are responsible for the degradation of the tissue, enabling the proliferating cancer cells to actively invade and migrate into the degraded tissue. Our model focuses on the role of nonlocal kinetic terms modelling competition for space and degradation. The model consists of a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. First of all we prove results concerning the local existence, uniqueness and regularity of solutions. We then prove global existence. Using Green's functions, we transform our original nonlocal equations into a coupled system of parabolic and elliptic equations and we undertake a numerical analysis of this equivalent system, presenting computational simulation results from our model showing the effect of the nonlocal terms (travelling waves we observed have the shape closely linked to the nonlocal terms). Finally, in the discussion section, concluding remarks are made and open problems are indicated.