Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion

The ability to invade tissue is one of the hallmarks of cancer. Cancer cells achieve this through the secretion of matrix degrading enzymes, cell proliferation, loss of cell–cell adhesion, enhanced cell–matrix adhesion and active migration. Invasion of tissue by the cancer cells is one of the key components in the metastatic cascade, whereby cancer cells spread to distant parts of the host and initiate the growth of secondary tumours (metastases). A better understanding of the complex processes involved in cancer invasion may ultimately lead to treatments being developed which can localise cancer and prevent metastasis. In this paper we formulate a novel continuum model of cancer cell invasion of tissue which explicitly incorporates the important biological processes of cell–cell and cell–matrix adhesion. This is achieved using non-local (integral) terms in a system of partial differential equations where the cells use a so-called “sensing radius” R   to detect their environment. We show that in the limit as R→0R→0 the non-local model converges to a related system of reaction–diffusion–taxis equations. A numerical exploration of this model using computational simulations shows that it can form the basis for future models incorporating more details of the invasion process.