Mathematical modelling of tumour evolution and radiation response: The impact of heterogeneity

This thesis seeks to use mathematical and computational models to develop measures of clinically available data to deepen our understanding, and improve our treatments, of cancer. We consider two broad characteristics of cancer: heterogeneity, in the form of differences in cellular phenotype, and the physical microenvironment; and evolution, which has become accepted as a driver of tumour progression. To ensure that the conclusions drawn are as translatable as possible, we will attempt to use data types that are clinically available. Using a hybrid discrete-cell-based model in two spatial dimensions, we focus on these fundamental aspects of cancer, with the hope of generating new understanding and useful hypotheses to benefit current patients and oncologists. First, we model a tumour growing under the rules of the cancer stem cell hypothesis and a neutral model of evolution, and ask if we can infer the underlying biological proliferative structure. Specifically, we work toward predicting the symmetric division probability of our simulated tumours from clincally relevant observables, as this is a key driving parameter of tumour progression and therapeutic response. We focus on measures of clonal diversity, group size and shape, and a suite of statistical measures of the phylogenetic trees resulting from the tumour's evolution in different regions of parameter space. We find strikingly different patterns in these measures for changing symmetric division probability which hinge on the inclusion of spatial constraints. These results give us insight into differences between solid and liquid tumours, and also generate a number of actionable clinical and biological hypotheses. Second, we explicitly consider the physical microenvironment of tumours invading into healthy tissue, and model oxygen transport, uptake and cellular competition. We then explore the effect of spatial organisation of blood vessels within the tumour on tumour growth kinetics and cellularity. Finding wide variability in the distribution of oxygen across tumours dependent on both vascular organisation and density, we proceed to explore the utility of spatial measures of vessels on radiation efficacy. Our results offer a novel hypothesis as to the failure of vascular normalisation therapy and radiation, and a possible clinical solution.