Modelling aspects of cancer growth: insight from mathematical and numerical analysis and computational simulation

In this chapter we present a variety of reaction-diffusion-taxis(i.e. macroscopic) models of several key stages of solid tumour growth - avascular growth, the immune response to solid tumours and invasive growth. The basis for all of the models is deriving the continuum PDE using a conservation of mass argument. In the model for avascular growth we examine the potential role prepattern theory (diffusion-driven instability `a la Turing) may play in the generation of spatio-temporal heterogeneity of mitotic activity on the surface of multicell spheroids. In the model for the immune response to cancer, working from an initial“microscale” cell interaction scheme, we derive a system of PDEs which are used to predict the capacity of the immune system to eradicate cancer (or not). In the final model of cancer invasion, once again we initially focus on “microscale” activity of matrix degrading enzymes and their binding to cell-surface receptors to derive a PDE model of the process of cancer invasion of the local tissue. For each system, we carry out mathematical and numerical analyses of the model and perform computational simulations and attempt to draw relevant biological conclusions and make experimentally-testable predictions from the results.