Mathematical Modelling of Tumour Invasion and Metastasis

Sandy Anderson

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) and matrix-degradative enzymes associated with the tumour cells.

The first model focuses 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 two space dimensions and compared qualitatively with experimental and clinical observations.

The second type of model focuses 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 individual cells and hence it is possible to include micro-scale processes such as proliferation and cell/cell adhesion. Which in turn allow us to examine the implications for metastatic spread.

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