Cellular automata coupled with steady-state nutrient solution permit simulation of large-scale growth of tumours

In this thesis, I present a hybrid computational algorithm for simulating the complete growth of an avascular tumour. In particular, I employ cellular automaton to model the growth of cells and steady-state diffusion equation to describe the distribution of oxygen in the tumour volume. While the cellular automaton model is a discrete model, the oxygen diffusion equation, which is a partial differential equation, forms the continuum model thus making the complete model a hybrid discrete-continuum model. I show that, in the case of a brain tumour, oxygen distribution in the tumour volume may be sufficiently described by a time-independent steady-state equation without losing the characteristics of a time-dependent diffusion equation. This makes the solution of oxygen concentration in the tumour volume computationally faster and more efficient which consequently makes the simulation of tumour growth on a large scale possible. In the simulation of the complete growth of avascular tumour, I solve the steady-state equation using the central difference method to describe the distribution of oxygen in the tumour volume. My hybrid discrete-continuum model takes into account the types of cells that compose the tumour volume as well as inter-cellular adhesion in addition to processes involved in cell cycle, namely, proliferation, quiescence, apoptosis and necrosis. More importantly, I incorporate into my model cell mutation that gives rise to different phenotypes and therefore, a tumour with a heterogeneous population of cells. A new phenotype is probabilistically chosen and has the ability not only to survive at lower levels of nutrient concentration but also to reproduce faster.