Modelling Solid Tumour Invasion: The effects of Mutation

The development of a primary solid tumour (e.g., a carcinoma) begins with a single normal cell becoming transformed as a result of mutations in certain key genes, this leades to uncontrolled proliferation. An individual tumour cell has the potential, over successive divisions, to develop into a cluster (or nodule) of tumour cells consisting of approximately $10^{6}$ cells. This avascular tumour cannot grow any further, owing to its dependence on diffusion as the only means of receiving nutrients and removing waste products. For any further development to occur the tumour must initiate angiogenesis - the recruitment of blood vessels. After the tumour has become vascularised via the angiogenic network of vessels, it now has the potential to grow further and invade the surrounding tissue. There is now also the possibility of tumour cells finding their way into the circulation and being deposited in distant sites in the body, resulting in metastasis.

In this talk we present two types of mathematical model which describe the invasion of host tissue by tumour cells. In the models, we focus on three key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix, ECM) and matrix-degradative enzymes (MDE) associated with the tumour cells. The first model focusses on the macro-scale structure (cell population level) and considers the tumour as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix and the migratory response of the tumour cells. Numerical simulations are presented in one and two space dimensions and compared qualitatively with experimental and clinical observations. The second type of model focusses on the micro-scale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of discrete cells whilst still allowing the chemicals (e.g. MDE, ECM) to remain continuous. Hence it is possible to include micro-scale processes both at the cellular, such as proliferation, cell/cell adhesion and sub cellular such as cell mutation properties. This in turn allows us to examine the effects of such micro-scale changes upon the overall tumour geometry and subsequently the potential for metastatic spread.