A three phase model to investigate the effects of dead material on the growth of avascular tumours

In vivo tumours are highly heterogeneous entities which often comprise intratumoural regions of hypoxia and widespread necrosis. In this paper, we develop a new three phase model of nutrient-limited, avascular tumour growth to investigate how dead material within the tumour may influence the tumour's growth dynamics. We model the tumour as a mixture of tumour cells, dead cellular material and extracellular fluid. The model equations are derived using mass and momentum balances for each phase along with appropriate constitutive equations. The tumour cells are viewed as a viscous fluid pressure, while the extracellular fluid phase is viewed as inviscid. The physical properties of the dead material are intermediate between those of the tumour cells and extracellular fluid, and are characterised by three key parameters. Through numerical simulation of the model equations, we reproduce spatial structures and dynamics typical of those associated with the growth of avascular tumour spheroids. We also characterise novel, non-monotonic behaviours which are driven by the internal dynamics of the dead material within the tumour. Investigations of the parameter sub-space describing the properties of the dead material reveal that the way in which non-viable tumour cells are modelled may significantly influence the qualitative tumour growth dynamics.