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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 beta (TGFb).

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 TGFb 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.

MATHEMATICAL MODELLING OF CONTRACTION, FIBROPLASIA AND
ANGIOGENESIS IN DERMAL WOUND HEALING
Luke Olsen, Centre for Mathematical Biology, Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB
Website address: http://www.maths.ox.ac.uk/cmb/Research/Abstracts/Olsen/1996_thesis.html

The healing of full-thickness excisional wounds in adult mammalian skin is conventionally subdivided into three temporal phases: ‘inflammation’, ‘proliferation’ and ‘remodelling’. We formulate and investigate deterministic continuum models for the proliferative phase of repair, with the aim of improving biological understanding of the mechanisms underlying normal and pathological dermal wound healing.

Following biological and mathematical modelling reviews, we develop a mechanochemical model for wound contraction and fibroplasia. This model reflects fibroblast cell migration and proliferation, growth factor secretion and extracellular matrix (ECM) deposition. We also explicitly include contractile myofibroblasts. Conservation laws govern the dynamics of these variables. Traction stresses generated by cell-ECM interactions and the intrinsic tissue response, modelled as a linear, isotropic, viscoelastic medium, are described by a force-balance equation. Analysis and simulations of the model predict a healed, contracted state evolving during the proliferative phase of repair, in concordance with experimental findings.

The balance between production and removal of the regulatory chemical in our model is a key determinant of whether a wound heals normally or with a pathological fibro-proliferative response. We present a caricature model of the cell-chemical dynamics, which yields insight into the pathogenic mechanisms of fibrocontractive diseases such as keloids and hypertrophic scars, suggesting novel therapeutic strategies.

Angiogenesis, wherein a vascular network is re-established in the wound, is the other vital event during the proliferative phase. The effects of diffusible chemical mediators have been intensively studied, but the roles of fibrillar ECM components such as collagen and fibronectin are unclear. We develop a simple model focusing on ECM-mediated capillary endothelial cell migration and proliferation. Solutions exhibit profiles of cells and ECM propagating towards the wound centre, as observed biologically.

Motivated by the correlation between scarring and ECM fibre organization during wound repair, we extend the model to account for cell flux- and stress-induced fibre alignment. In the former, we investigate alignment by travelling wave analysis, highlighting the roles of the parameters and of contact guidance. In the latter case, we study the coupling between anisotropic cell traction and fibre alignment: with a single ECM type, the stress mechanism predicts either an isotropic or fully-aligned outcome, according to the feedback sensitivity, whereas a range of anisotropic equilibria are possible with multiple ECM types.