Mathematical modelling and investigation of tumour-immune interactions: the importance of metabolism

Abstract

In this Thesis, we derive and investigate a class of ODE and PDE models for metabolic competition between different types of tumour cells, and between tumour cells and immune cells inside the tumour microenviornment. We focus on macrophages, since these immune cells are one of the most abundant cell population that infiltrates solid tumours. Moreover these immune cells are very heterogeneous, with the two extremes being the anti-tumour M1 macrophages and the pro-tumour M2 macrophages. We start with ODE models for tumour-glucose interactions and tumour-immune-glucose interactions that lead to tumour growth and decay. Then we assume that some tumour cells and immune cells can move randomly, while nutrients can diffuse, and thus we develop PDE models that we use to investigate the invasion of tumour and immune cells into the surrounding tissue. Finally, we investigate the possibility that the PDE models derived in this study exhibit pattern formation, which could lead to the appearance of aggregations of tumour cells (or aggregations of tumour and immune cells) at different positions in space. Throughout this Thesis, we study the role of glucose influx as well as various metabolic parameters associated with tumour and macrophage consumption of glucose, on the growth and spread of cancer cells. Some of the steady-state results for the ODE models support the hypothesis of hyperglycemic memory, characterised by an increase in tumour size even after the reduction in the glucose influx. In regard to tumour spread, this is not influenced directly by any of the metabolic parameters. However, the metabolic parameters can influence the shape of the travelling wave solutions of the PDE models (solutions which can connect 2 steady states or even 3 steady states, whose values and stability depend on the model parameters). Finally, we show that the models developed and investigated in this Thesis do not exhibit diffusion-driven instabilities (at least for the parameter ranges investigated here). Biologically, this means that these models cannot describe the formation of new tumour and tumour-immune aggregations at different positions in space. Overall, the results in this Thesis help us gain a better understanding (at a theoretical level) of the mechanisms of tumour-glucose and tumour-glucose-immune interactions that are involved in tumour growth and spread. Analytical and numerical investigations of the transient and asymptotic dynamics of the ODE and PDE systems developed here allow us to propose hypotheses regarding the wide range of temporal and spatio-temporal behaviours exhibited by these models.