Modelling the impact of heterogeneity in tumour composition on the response to fractionated radiotherapy
Keywords: Mathematical biology

Current protocols for delivering radiotherapy are based primarily on tumour stage and nodal and metastases (TNM) status, even though it is well known that tumours and their microenvironments are highly heterogeneous. Mathematical models have the potential to inform protocol selection on a more patient-specific basis. However such applications require early prediction of tumour response and, thus, calibration of the model parameters from pre-treatment data which is typically limited to two CT scans - one at diagnosis and one at the treatment planning stage. In this thesis we aim to determine whether pre-treatment tumour volume data may be used to accurately predict tumour response to radiotherapy. We further aim to provide a mechanistic description of the different qualitative responses to radiotherapy observed in clinical data and the influence of intratumoural heterogeneity on these dynamics.
Mathematical models for clinical applications typically contain few model parameters for ease of calibration with the sparse available data. Such models often take the form of single or multiple compartment ordinary differential equations (ODEs). We use parameter estimation and model selection techniques to perform both synthetic and clinical data studies and assess the predictive power of such models. We show that, while the models considered may be adequate in some cases, the range of qualitative dynamics exhibited by the clinical data extends beyond that which may be captured by these simple models.
To address this discrepancy, we develop a suite of increasingly complex models that account for spatial heterogeneity in tumours, and use a range of methods to assess their validity and predictive power. In particular, we use a combination of numerical simulation and asymptotic analysis to investigate the dynamics of each model and the impact of increasing spatial heterogeneity on tumour response to radiotherapy.
In vitro avascular tumour spheroids have a well-defined spatial structure, providing a simple geometry that is often exploited in tumour growth models. We extend one such model (Greenspan, 1972) to develop a new spatially-resolved model for tumour response to radiotherapy. Model simulations reveal a rapid transient increase in hypoxia upon regrowth of the tumour spheroid post-irradiation, a factor known to inhibit radiotherapy efficacy. We also investigate the response to different fractionation schedules and identify a tumour-specific relationship between inter-fraction time and dose per fraction to achieve cure.
We further use the framework of multiphase mixtures to develop new models of both tumour growth and radiotherapy response that allow for the greater degree of heterogeneity typically observed in vivo. The multiphase models describe a wider range of spatial tumour compositions and consider mechanical interactions between different constituents of the tumour microenvironment. They reproduce the qualitative radiotherapy response dynamics observed in clinical data, including behaviours not captured by simpler models. We characterise a range of dynamics exhibited by the model and demonstrate the important influence of the distribution of dead cellular material within the tumour on both the tumour growth dynamics and response to radiotherapy. In particular, we highlight that radiotherapy induces a significant change in the tumour composition and, as such, intratumoural heterogeneity is likely to be a significant factor in determining treatment response.