Mathematical models for wound healing lymphangiogenesis and other biomedical phenomena
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In this thesis we explore the mathematical modelling of wound healing lymphangiogenesis, tumour neoneurogenesis and Drosophila courtship behaviour. We begin by focussing on the mathematical modelling of lymphatic regeneration in wound healing. Indeed, several studies suggest that one possible cause of impaired wound healing is failed or insu cient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. In Chapter 2 a model of five ordinary differential equations is presented to describe lymphangiogenesis in a skin wound. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from experimental and clinical data. The system output is compared with the available biological literature and, based on parameter sensitivity analysis, new therapeutic approaches are suggested to enhance lymphangiogenesis in diabetic wounds. Chapter 3 extends the aforementioned work to two PDE systems aimed at describing two possible hypotheses for the lymphangiogenesis process: 1) lymphatic capillaries sprout from existing and interrupted capillaries at the edge of the wound, in analogy to the blood angiogenesis case; 2) lymphatic endothelial cells first collect together in the wound region through following the lymph flow and then begin to form a network. Furthermore, we include the effect of advection from both background interstitial flow and additional lymph flow from the open capillaries, andaddress the question of their relative importance in the lymphangiogenesis process. Malignant tumours induce not only the formation of a lymphatic and a blood vascular network, but also innervation around themselves. However, the relationship between tumour progression and the nervous system is still poorly understood. In Chapter 4 we study the interactions between the nervous system and tumour cells through an 8-dimensional ODE model. The model confirrms experimental observations that a tumour promotes nerve formation around itself, and that high levels of nerve growth factor (NGF) and axon guidance molecules (AGMs) are recorded in the presence of a tumour. Our results also reflect the observation that high stress levels (represented by higher norepinephrine release by sympathetic nerves) contribute to tumour development and spread, indicating a mutually bene cial relationship between tumour cells and neurons. In Chapter 5 a preliminary model for courtship behavioural patterns of Drosophila melanogaster is suggested. Drosophila courtship behaviour is considered a good model to investigate neurodegenerative diseases (such as Parkinson's) in humans. The present chapter illustrates the biological and health-care related background to this topic, and then presents a possible modelling approach based on Pasemann's work on neural networks. We conclude with a brief discussion that summarises the main results and outlines directions for future work.