Partial Differential Equation Models of Collective Migration During Wound Healing
I will discuss the collective migration of epithelial cells. As epithelial cells are known to remain physically connected to their neighbors during migration, we begin with a study into the role of cell-cell adhesions on keratinocyte migration during wound healing. We next turn our efforts to the influence of biochemical pathways on cell phenotype. We do so by the derivation of a biochemically-structured version of Fisher's Equation that provides a framework to study how patterns of biochemical activation influence migration into the wound. In this equation, cells are simultaneously activating along some biochemical pathway and migrating in space into the wound area. As numerical methods are often used to approximate solutions to complicated equations in practice, I will conclude with an investigation into the role of numerical error on an inverse problem methodology. We show that the numerical cost function in an inverse problem for hyperbolic advection equations is dominated by either numerical or experimental error in its computation, which leads to different rates of convergence as numerical precision decreases. We also demonstrate how numerical error influences residual computation when numerical error dominates experimental error and derive an autocorrelative statistical model to improve parameter estimation.