Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system.

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A simple model of wound healing angiogenesis is presented, and investigated using numerical and asymptotic techniques. The model captures many key qualitative features of the wound healing angiogenic response, such as the propagation of a structural unit into the wound centre. A detailed perturbative study is pursued, and is shown to capture all features of the model. This enables one to show that the level of the angiogenic response predicted by the model is governed to a good approximation by a small number of parameter groupings. Further investigation leads to predictions concerning how one should select between potential optimal means of stimulating cell proliferation in order to increase the level of the angiogenic response.

(1) Background: Adaptive therapy aims to tackle cancer drug resistance by leveraging intra-tumoural competition between drug-sensitive and resistant cells. Motivated by promising results in prostate cancer there is growing interest in extending this approach to other cancers. Here we present a theoretical study of intra-tumoural competition during adaptive therapy, to identify under which circumstances it will be superior to aggressive treatment, and how it can be improved through combination treatment; (2) Methods: We study a 2-D, on-lattice, agent-based tumour model. We examine the impact of different micro-environmental factors on the comparison between continuous drug administration and the adaptive schedule pioneered in the first-in-human clinical trial. (3) Results: We show that the degree of crowding, the initial resistance fraction, the presence of possible resistance costs, and the rate of tumour cell turnover are key determinants of the benefit of adaptive therapy. Subsequently, we investigate whether combination with treatments which amplify proliferation or which target cell turnover can prolong tumour control. While the former increases competition, we find that only the latter can robustly improve time to progression; (4) Conclusion: Our work helps to identify selection factors for adaptive therapy and provides stepping stones towards the rational design of multi-drug adaptive regimens.