Time heals all wounds? : mathematical models of epithelial and dermal wound healing
The mechanisms responsible for the healing of corneal surface wounds are the subject of biological controversy. In particular, the role and source of the regulatory chemical epidermal growth factor (EGF) is an area of intense debate. In the first part of this thesis, we propose a reaction-diffusion model which focuses on the stimulus for increased mitotic and migratory activity due to secretion of EGF. A detailed numerical study of various possible models, with parameter values based on biological data, reveals that, for realistic healing times, EGF must be released by the underlying layers of the cornea, in addition to the tear film source. The model exhibits travelling wave solutions and further analysis elucidates the interaction and role of the parameters in determining the speed of healing. Furthermore, we consider the effect of topical application of EGF and investigate the effect of curvature of the eye. We show that our model is consistent with many of the key features of corneal wound healing. Adult dermal wounds, in contrast to foetal wounds, heal with the formation of scar tissue. A crucial factor in determining the nature of the healed tissue is the ratio of collagen 1 to collagen 3, which indicates the fibril diameter. We develop a reaction-diffusion model which focuses on the stimulus for collagen synthesis due to the secretion of the different isoforms of the regulatory chemical transforming growth factor β (TGFβ). Numerical simulations of the model without diffusion lead to a value of this ratio consistent with that of healthy tissue for the foetus but corresponding to scarring in the adult. The model equations evolve to waves moving into the wound, but addition of TGFβ only has a transient effect on the final collagen levels. We investigate this effect by developing a caricature model. The model indicates that the main source of the fibroblasts is the underlying subcutaneous tissue and we determine key parameters which explain the difference between adult and foetal wound healing. Furthermore we make clinically testable predictions on the effects that topical application of various chemicals will have on scar formation.