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

We model complete growth of an avascular tumour by employing cellular automata for the growth of cells and steady-state equation to solve for nutrient concentrations. Our modelling and computer simulation results 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 more efficient, thus enabling simulation of tumour growth on a large scale. We solve this steady-state equation using a central difference method. We take into account the composition of cells and intercellular adhesion in addition to processes involved in cell cycle--proliferation, quiescence, apoptosis, and necrosis--in the tumour model. More importantly, we consider cell mutation that gives rise to different phenotypes and therefore a tumour with heterogeneous population of cells. A new phenotype is probabilistically chosen and has the ability to survive at lower levels of nutrient concentration and reproduce faster. We show that heterogeneity of cells that compose a tumour leads to its irregular growth and that avascular growth is not supported for tumours of diameter above 18 mm. We compare results from our growth simulation with existing experimental data on Ehrlich ascites carcinoma and tumour spheroid cultures and show that our results are in good agreement with the experimental findings.