Characterisation of glioblastoma sub-populations using mathematical modelling and inference

Glioblastomas (GBMs) are the most aggressive primary brain tumours and have no known cure. Each individual tumour comprises multiple sub-populations of genetically-distinct cells that may respond differently to targeted therapies and may contribute to disappointing clinical trial results. Image-localized biopsy techniques allow multiple biopsies to be taken during surgery and provide information that identifies regions where particular sub-populations occur within an individual GBM, thus providing insight into their regional genetic variability. These sub-populations may also interact with one another in some way; it is important to ascertain the nature of these interactions, as they may have implications for responses to targeted therapies. In this work, we combine genetic information from image-localised biopsies with a mechanistic model of interacting GBM sub-populations to characterise the nature of interactions between two commonly occurring GBM sub-populations, those with EGFR and PDGFRA genes amplified. Firstly, we develop a mathematical model using a PDE-based formalism and explore the dynamics of our model under a variety of interaction types (Chapter 2). Following on from this, we study population levels found across image-localized biopsy data from an initial cohort of patients and compare this to model outputs under competitive, cooperative and neutral interaction assumptions (Chapter 3). We explore other factors affecting the observed simulated sub-populations, such as selection advantages and phylogenetic ordering of mutations, and conduct a sensitivity analysis, as these factors may also contribute to the levels of EGFR and PDGFRA amplified populations observed in biopsy data. The patient dataset is then expanded to include image-localised biopsies from additional patients and we examine the intra- and inter-tumoural heterogeneity in EGFR and PDGFRA amplification observed in this data (Chapter 4). We then proceed to explore the inferability of the model parameters using synthetic datasets. Finally, we perform inference for the patient dataset, where we are able to gain some insights into the dynamics of and nature of interactions between these amplified sub-populations