Partial Differential Equation Models of Collective Migration During Wound Healing

Nardini, John Thomas

Graduate Thesis Or Dissertation

Partial Differential Equation Models of Collective Migration During Wound Healing Public Deposited

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Citeable URL: https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/1544bp08d

Abstract

This dissertation is concerned with the derivation, analysis, and parameter inference of mathematical models of the collective migration of epithelial cells. During the wound healing process, epidermal keratinocytes collectively migrate from the wound edge into the wound area as a means to re-establish the outermost layer of skin. This migration into the wound is stimulated by the presence of epidermal growth factor. Accordingly, this dissertation focuses on the migratory response of epidermal keratinocytes in response to this growth factor. Such studies will suggest suitable clinical treatments to consider for chronic wounds and invasive cancers.We begin with a study into the role of cell-cell adhesions on keratinocyte migration during wound healing. We use an inverse problem methodology in combination with model validation to show that cells use these connections to promote migration by pulling on their follower cells as they migrate into the wound. We next derive a biochemically-structured version of Fisher's Equation that provides a framework to study how patterns of biochemical activation influence migration into the wound. We prove the existence of a self-similar traveling wave solution. In considering a more complicated scenario where cell migration depends on biochemical activity levels, we show numerically that the threshold parameter where all cells in the population become activated yields the simulations that migrate farthest into the wound. Lastly, we consider the role of numerical error on an inverse problem methodology. The numerical approximation of a cost function is dominated by either numerical or experimental error in computations, which leads to different rates of convergence as numerical resolution increases. We use residual analysis to derive an autocorrelative statistical model for cases where numerical error is the main source of error for first order schemes. This autocorrelative statistical model can correct confidence interval computation for these methods and hence improve uncertainty quantification.

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