Mathematical approaches to modelling healing of full thickness circular skin wounds

Wound healing is a complex process, in which a sequence of interrelated events at both the cell and tissue levels interact and contribute to the reduction in wound size. For diabetic patients, many of these processes are compromised, so that wound healing slows down and in some cases halts. In this thesis we develop a series of increasingly detailed mathematical models to describe and investigate healing of full thickness skin wounds. We begin by developing a time-dependent ordinary differential equation model. This phenomenological model focusses on the main processes contributing to closure of a full thickness wound: proliferation in the epidermis and growth and contraction in the dermis. Model simulations suggest that the relative contributions of growth and contraction to healing of the dermis are altered in diabetic wounds. We investigate further the balance between growth and contraction by developing a more detailed, spatially-resolved model using continuum mechanics. Due to the initial large retraction of the wound edge upon injury, we adopt a non-linear elastic framework. Morphoelasticity theory is applied, with the total deformation of the material decomposed into an addition of mass and an elastic response. We use the model to investigate how interactions between growth and stress influence dermal wound healing. The model reveals that contraction alone generates unrealistically high tension in the dermal tissue and, hence, volumetric growth must contribute to healing. We show that, in the simplified case of homogeneous growth, the tissue must grow anisotropically in order to reduce the size of the wound and we postulate mechanosensitive growth laws consistent with this result. After closure the surrounding tissue remodels, returning to its residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is used to predict the outcome of rewounding experiments as a method of quantifying the amount of stress in the tissue and the application of pressure treatments to control tissue synthesis. The thesis concludes with an extension to the spatially-resolved mechanical model to account for the effects of the biochemical environment. Partial differential equations describing the dynamics of fibroblasts and a regulating growth factor are coupled to equations for the tissue mechanics, described in the morphoelastic framework. By accounting for biomechanical and biochemical stimuli the model allows us to formulate mechanistic laws for growth and contraction. We explore how disruption of mechanical and chemical feedback can lead to abnormal wound healing and use the model to identify specific treatments for normalising healing in these cases.