Discrete and continuum models for the evolutionary and spatial dynamics of cancer: a very short introduction through two case studies

Abstract: We give a very short introduction to discrete and continuum models for the evolutionary and spatial dynamics of cancer through two case studies: a model for the evolutionary dynamics of cancer cells under cytotoxic therapy and a model for the mechanical interaction between healthy and cancer cells during tumour growth. First we develop the discrete models, whereby the dynamics of single cells are described through a set of rules that result in branching random walks. Then we present the corresponding continuum models, which are formulated in terms of non-local and nonlinear partial differential equations, and we summarise the key properties of their solutions. Finally, we carry out numerical simulations of the discrete models and we construct numerical solutions of the corresponding continuum models. The biological implications of the results obtained are briefly discussed.