School of Mathematics and Statistics - Theses
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ItemMathematical and computational models of re-epithelialisation( 2023-06)Wound healing is a complex process occurring across multiple temporal and spatial scales. In this thesis we use individual-based computational and mathematical models to explore the epidermal re-epithelialisation stage of wound healing. We focus specifically on the overlapping spheres, Voronoi tessellation and vertex dynamics models in two dimensions and the computational choices available for each model that may impact healing dynamics. We find that the overlapping spheres and Voronoi tessellation models are highly sensitive to model choice, whereas the vertex dynamics model is insensitive to model choice. Therefore, we use the vertex dynamics model to investigate six re-epithelialisation mechanisms in small, acute wounds: cell compression, proliferation, cell edge contractility, the purse-string mechanism, cell crawling in response to local cues and cell crawling towards a point source. The scale of the mechanism can affect not only the time scale of re-epithelialisation, but also the shape of the wound over time. The most significant mechanisms for larger wounds are cell proliferation and crawling in response to local cues. We incorporate these mechanisms into a model of a larger wound to study the interplay between cell migration and proliferation during re-epithelialisation. Drawbacks of the individual-based approach include the computational costs and limited mathematical frameworks for model analysis. Hence, we convert our two-dimensional individual-based model of a larger wound into a one-dimensional model and derive a partial differential equation continuum approximation that can provide further insights into the re-epithelialisation process. From the one-dimensional model we can investigate the relationship between mechanical relaxation and tissue growth and derive an upper bound on tissue growth over time. We conclude that increasing the amount of cell proliferation or migration alone is insufficient to promote re-epithelialisation and that there is an interdependence between migration and proliferation. Overall, this thesis contributes to the understanding of the mechanisms regulating re-epithelialisation of the epidermis. A better understanding of the re-epithelialisation process allows for more informed studies of wound treatments, with the aim of reducing the global burden of wounds.
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ItemUnderstanding the regulation of epidermal tissue thickness by cellular and subcellular processes using multiscale modelling( 2020)The epidermis is the outermost layer of the skin, providing a protective barrier for our bodies. Two important aspects to the barrier function of the epidermis are maintenance of its barrier layer and constant cell turnover. The main barrier layer in the epidermis is the outermost layer, called the stratum corneum. This layer blocks both the entry of antigens and the loss of internal water and solutes. If antigens do enter the system, cell turnover has been hypothesised to propel them out the system by providing a constant upwards velocity of cells which carry the toxins with them. The majority of severe diseases of the epidermis relate to a reduction in thickness of the stratum corneum. Decreased thickness reduces the barrier function of the layer, causing discomfort and inflammation. Due to its importance to barrier function, the maintenance of stratum corneum thickness, and consequently overall tissue thickness, is the focus of this thesis. In order to maintain both stratum corneum thickness and overall tissue thickness it is necessary for the system to balance cell proliferation and cell loss. Cell loss in the epidermis occurs when dead cells at the top of the tissue are lost to the environment through a process called desquamation. Cell proliferation occurs in the base, or basal, layer. As the basal cells proliferate, cells above them are pushed upwards through the tissue, causing constant upwards movement in the tissue. Not only does this contribute directly to the barrier function through the cell turnover as discussed above, but the velocity of the cells is likely to be key in regulating the tissue thickness. Assuming the cell loss occurs at a fairly constant rate, the combination of the velocity and the loss rate determine tissue thickness. In order to investigate these processes we develop a three dimensional discrete, multiscale, multicellular model, focussing on maintenance of cell proliferation and desquamation. Using this model, we are able to investigate how subcellular and cellular level processes interact to maintain a homeostatic tissue. Our model is able to reproduce a system that self-regulates its thickness. The first aspect of this regulation is maintaining a constant rate of proliferation in the epidermis, and consequently a constant upwards velocity of cells. The second aspect is a maintained rate of desquamation. The model shows that hypothesised biological models for the degradation of cell-cell adhesion from the literature are able to provide a consistent rate of cell loss which balances proliferation. An investigation into a disorder which disrupts this desquamation model shows reduced tissue thickness, consequently diminishing the protective role of the tissue. In developing the multiscale model we have begun to delve deeper into the relationship between subcellular and cellular processes and epidermal tissue structure. The model is developed with scope for the integration of further subcellular processes. This provides it with the potential for further experiments into the causes and effects of behaviours and diseases of the epidermis, with much higher time and cost efficiency than other experimental methods.