  We consider a mathematical model for the evolutionary dynamics of tumour
cells in vascularised tumours under chemotherapy. The model comprises a system
of coupled partial integro-differential equations for the phenotypic
distribution of tumour cells, the concentration of oxygen and the concentration
of a chemotherapeutic agent. In order to disentangle the impact of different
evolutionary parameters on the emergence of intra-tumour phenotypic
heterogeneity and the development of resistance to chemotherapy, we construct
explicit solutions to the equation for the phenotypic distribution of tumour
cells and provide a detailed quantitative characterisation of the long-time
asymptotic behaviour of such solutions. Analytical results are integrated with
numerical simulations of a calibrated version of the model based on
biologically consistent parameter values. The results obtained provide a
theoretical explanation for the observation that the phenotypic properties of
tumour cells in vascularised tumours vary with the distance from the blood
vessels. Moreover, we demonstrate that lower oxygen levels may correlate with
higher levels of phenotypic variability, which suggests that the presence of
hypoxic regions supports intra-tumour phenotypic heterogeneity. Finally, the
results of our analysis put on a rigorous mathematical basis the idea,
previously suggested by formal asymptotic results and numerical simulations,
that hypoxia favours the selection for chemoresistant phenotypic variants prior
to treatment. Consequently, this facilitates the development of resistance
following chemotherapy.
