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Classical receptor theory is largely built on assumptions of monomeric receptors. In this thesis, we contribute to receptor theory by considering the now widely accepted cases of dimeric receptors. The implications of dimerisation for drug discovery and therapeutics remain unclear. Therefore, a theoretical consideration of ligand binding and signalling via receptor dimers is warranted. Here, we develop mathematical models for ligand binding at dimerised and dimerising receptors. A key factor in developing these theoretical models is cooperativity across the dimer, whereby binding of a ligand to one protomer affects the binding of a ligand to the other protomer. The effects of cooperativity on binding dynamics are a primary point of interest. The first models we present focus on G protein-coupled receptors, where we assume that all receptors are pre-dimerised. Ligand binding models give linear systems of differential equations which we use to analyse time course behaviours. At equilibrium, these models may exhibit multi-phasic log dose response curves, critically depending on co-operativity factors. When considering receptor activation, we see dose response curves that are indicative of non-standard ligand-receptor interactions, giving a quantitative and qualitative platform for analysing and interpreting data when dimers are suspected. A ligand induced model for vascular endothelial growth factor receptors is developed, whereby receptors exist constitutively as monomers and dimerise in response to ligand binding. The resulting nonlinear system of differential equations is investigated using numerical computations and perturbation methods. We see an excellent fit to published data, validating the model. The utility of our models in parameter estimation is explored theoretically using structural identifiability analysis. This determines which parameters can be theoretically estimated from fitting. This analysis is valuable but often overlooked when fitting to ligand-receptor interaction models. We explore the identifiability of some canonical ligand binding models, and our dimer binding models, providing a tutorial and results to contribute to the receptor theory toolbox.

A multiscale, agent-based mathematical framework is here used to capture the multiscale nature of solid tumours. Tumour dynamics and treatment responses are modelled and simulated in silico. Details regarding cell cycle progression, tumour growth and oxygen distribution are included in the mathematical framework. Treatment responses to conventional anti-cancer therapies, such as chemotherapy and radiotherapy, as well as to more novel drugs, such as hypoxia-activated prodrugs and DNA-damage repair inhibiting drugs, are studied. Uncertainty and sensitivity analyses techniques are discussed in order to mitigate model uncertainty and interpret model sensitivity to parameter perturbations. This thesis furthermore discusses the role of mathematical modelling in current cancer research.