Year

2019

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

In this thesis, we study the application of curvature flow to problems in wound healing, specifically to epithelial healing events occurring in idealised embryonic contexts. We tackle a generic situation, by proposing and studying a flow we term the “embryonic epithelial wound healing flow” as well as a well-studied specific healing event in Drosophila, dorsal closure. The generic flow is realised as a curvature flow of closed plane curves where the normal speed is an affine function of the scalar curvature. We establish local existence and uniform estimates for convex initial data. Furthermore, by studying a variety of rescalings, we investigate the asymptotic shape of the flow. The solution is always singular in finite time, and by adapting the famous monotonicity formula of Huisken and using Hamilton’s Entropy, we are able to prove the convex flow is asymptotically round. We are additionally able to obtain results for non-convex initial data where the concave regions are “small” in a certain sense. This employs techniques inspired by the work of Kuwert-Schaetzle for the Willmore flow and McCoy-Williams-Wheeler for the surface diffusion and related flows. The dorsal closure event has singular geometry and this affects the results we obtain. Analysis of the flow in the case of graphical initial data allows us to prove that the solution eventually converges to a horizontal line segment, which qualitatively resembles the physical situation. This is done through the use of a combination of techniques, including maximum principle/Hopf lemma and integral/energy methods. For dorsal closure, we also study the notion of weak solution. Finally, we finish the thesis by completing a numerical study of solutions to the dorsal closure flow. We investigate the standard dorsal closure event as well as the flow for mutated or modified Drosophila. In every case we see that the curvature flow closely mimics the observed data under the microscope, adding quantitative weight to the theoretical qualitative results already established.

FoR codes (2008)

010102 Algebraic and Differential Geometry, 010110 Partial Differential Equations, 010202 Biological Mathematics

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Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.