Mathematical modelling of avascular tumour growth based on diffusion of nutrients and its validation

In this paper, a mathematical model based on the diffusion of nutrients is developed by considering the physiological changes accompanying the growth of avascular tumour. Avascular tumour growth involves the formation of three different zones namely proliferation, quiescent and necrotic zones. The main processes on which avascular tumour growth depends are: (i) diffusion of nutrients through the tumour from the contiguous tissues, (ii) consumption rate of the nutrients by the cells in the tumour, and (iii) cell death by apoptosis and necrosis. In the model, we consider the tumour to be spherical and the principal nutrients responsible for its growth are oxygen and glucose. By solving for the concentration profiles using the model developed, we are able to compute the radii of the quiescent and necrotic zones as well as that of the tumour. The proposed model is also validated using in vitro tumour growth data and Gompertzian empirical relationship parameters available in the literature. Our model is also successful in capturing the saturated volume of the avascular tumour for different nutrient concentrations at the tumour surface.