Mathematical modelling of tumour growth and treatment effect of hyperthermia and radiotherapy.

My research aims to investigate different continuum mathematical models that describe the growth and response of solid tumours to treatment with hyperthermia and radiotherapy.

I extended a spatially-resolved mathematical model for tumour growth proposed by Greenspan (Stud. Appl. Math., 1972), incorporating the effect of hyperthermia treatment alone and combined with radiation. The model is a moving boundary-value problem where a radially-symmetric tumour is developed according to the concentration profile of a fixed oxygen source on the tumour's boundary. Tumour composition consists of a proliferating rim, a hypoxic annulus and a necrotic core, and these layers' boundaries are determined by local oxygen levels. The extended Greenspan model helped me address the two main goals of my project. I first compared the model's predictions to experimental data and data generated by a cellular automaton (CA) model developed by Bruningk et al. (J. Royal Soc. Interface, 2018). Using a computational approach, I ran model simulations that did not align with the data, suggesting that this model cannot accurately predict the effect of hyperthermia on tumour growth. I believe altering the simplifying biological assumptions made could improve the modelling approach. Secondly, I explored the benefits of a combination therapy of hyperthermia and radiation. Our model predicted that treatment efficacy differences between uni- and multi-modal therapies increase linearly with hypoxic tumoural volume at the time of treatment. I had finalised this analysis when I received the data that established the inadequacy of the model. I still discussed the results obtained to highlight an interesting question : can an appropriate mathematical model demonstrate that tumour composition is key for treatment efficacy ?

My DPhil project will build upon the work described above through the development of more detailed continuum models of tumour growth that account for differences in the processes regulating cell death due to hyperthermia and radiotherapy.
Radiation-induced cell death is not an instantaneous biological process as most irradiated cells die after attempting and failing mitosis. Therefore, a key feature to include in our models is a time delay between radiation and cell death. Cells can alternatively become senescent, i.e. non-proliferative but viable, which should also be incorporated in the models. The aforementioned CA model accounts for these details and I would be interested in comparing it with our models.
Motivated by published experimental results, I previously assumed that heat reduces the resistance of hypoxic cells to radiation. However, there is experimental evidence supporting other hypotheses about the symbiotic action of hyperthermia and radiotherapy. For example, heat preferentially targets cells in hypoxic environments due to the lower pH, rather than a lack of oxygen, or heat radiosensitises the tumoural environment by reoxygenating it. I think it would be infomative to ensure the new models include a more detailed mathematical description of hyperthermia-induced cell death that accounts for the different mechanisms at play.
Another factor to consider is the fate of cells upon their death : are they expelled into the extratumoural environment ? Do they remain within the tumour ? If so, do they join the necrotic core or not ? The assumptions made can significantly alter the model, which is why it is important to trial the different options.
New models will be assessed and validated using in vitro data from experiments on 3D avascular tumour spheroids. Approved models can then be compared and used to predict the treatment response of in vivo vascular tumours. Using these predictions, I could investigate various therapeutic protocols combining hyperthermia and radiotherapy and establish whether there is an optimal method.

This project falls within the EPSRC Mathematical Biology research area.