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Epithelialization

The process where epithelial cells migrate across the wound bed, covering the defect. This migration is guided by chemotactic gradients and cell-to-cell adhesion mechanisms.
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The statement of the theorem

Let v(t,x)v(t, \textbf{x}) be the velocity of the migrating epithelial sheet boundary. The migration is modeled using a chemotaxis framework, where the velocity is proportional to the gradient of a chemoattractant CchemC_{chem} (e.g., growth factors):\nv=dxdt=β∇Cchem+χ1+e−k(Cchem−Cthreshold)n\mathbf{v} = \frac{d\textbf{x}}{dt} = \beta \nabla C_{chem} + \frac{\chi}{1 + e^{-k(C_{chem} - C_{threshold})}} \textbf{n} \nwhere β\beta is the chemotactic coefficient, χ\chi is the random motility component, and n\textbf{n} is the normal vector to the wound edge.