Stochastic simulation of benign avascular tumour growth using the Potts model

We simulate the growth of a benign avascular tumour embedded in normal tissue, including cell sorting that occurs between tumour and normal cells, due to the variation of adhesion between different cell types. The simulation uses the Potts model, an energy minimisation method. Trial random movements of cell walls are checked to see if they reduce the adhesion energy of the tissue. These trials are then accepted with Boltzmann weighted probability. The simulated tumour initially grows exponentially, then forms three concentric shells as the nutrient level supplied to the core by diffusion decreases: the outer shell consists of live proliferating cells, the middle of quiescent cells, and the centre is a necrotic core, where the nutrient concentration is below the critical level that sustains life. The growth rate of the tumour decreases at the onset of shell formation in agreement with experimental observation. The tumour eventually approaches a steady state, where the increase in volume due to the growth of the proliferating cells equals the loss of volume due to the disintegration of cells in the necrotic core. The final thickness of the shells also agrees with experiments.