Aspects of modelling solid tumours

This thesis considers aspects of modelling solid tumours. We begin by considering the common assumption that nutrient or drug concentrations in avascular tumour spheroids are radially symmetric. We derive a simple Poisson equation for biomolecular diffusion into an avascular tumour, but with highly oscillatory boundary conditions due to the surrounding capillary network. We find that the assumption of radial symmetry is legitimate for biomolecules that are taken up in sufficient quantities by proliferating cancer cells; however radially symmetric profiles need not be observed otherwise. We then investigate how the gap between an avascular tumour and the neighbouring vasculature varies as the tumour grows. This is explored by (i) using scaling arguments based on ordinary differential equations, (ii) coupling the rate of oxygen flux from the vasculature to oxygen evolution within the tumour, and (iii) deriving a system of six coupled non-linear partial differential equations modelling the tumour evolution. It is found that as the tumour grows any initial gap between the tumour and neighbouring vasculature closes since there is no mechanism which would sufficiently up-regulate non-cancerous cell proliferation. This is in contrast to the intra-cornea implantation observations, upon which several mathematical models are based. Finally, we study the growth and treatment of a vascular tumour subjected to chemotherapies, particularly when the therapies can exhibit an anti-angiogenic effect and resistance to the therapy is incorporated. A multi-compartment model is derived for the evolution of a tumour undergoing treatment and parameters are estimated, with extensions to incorporate numerous different therapy protocols in the literature. We find that anti-angiogens can be effective, though the appropriate scheduling is counter-intuative and contradicts many standard therapy rules. We conclude that chemotherapy protocol design is very sensitive to the mode of action of the drug and simple general strategies will, in many ...