Exploration of prostate cancer cell dynamics

In this thesis we want to investigate the use of delay and diffusion in modelling prostate cancer cell dynamics while undergoing treatment; to study any limitations in their use; and to provide insights into optimal therapeutic strategies. We first analyse a discrete delay system to represent the phenomenon of neuroendocrine trans differentiation, which is the ability of cells to change their intrinsic nature in response to a hostile environment. For this system, we prove some basic properties of its solutions, which ensure the biological significance of the model, and the local and global stability properties of the tumour free equilibrium. We perform sensitive and bifurcation analyses, and produce numerical simulations that reflect the discovered behaviour. Based on the results of this analysis, we propose a new distributed delay system, and as in the previous case we analytically derive basic properties of the solutions, and prove the existence of a tumour free and a tumour present equilibrium. Local stability properties of the equilibria are also analysed, and numerical simulations using the uniform and exponential distributions are produced that show the stability switch of the tumour present equilibrium with the uniform distribution. The final model we consider is a hybrid cellular automata system to incorporate the diffusion of cells and chemicals in the in vivo representation introduced in the second system. We explore the effect of different drug therapies on the prostate cancer tumour, which we consider to have two phenotypes, one which is sensitive to one of the three considered drugs, and one resistant to the same drug.