Keywords

alignment, integrodifferential equations, perturbation theory, collagen, fibroblast, wound healing

Abstract

Orientation of extracellular matrix fibers in the skin is a key ingredient of tissue appearance and function, and differences in fiber alignment are one of the main distinctions between scar tissue and normal skin. In this paper, the authors develop a mathematical model for alignment of collagen fibers and the fibroblast cells that remodel them; the model extends previous work in which spatial variation was excluded. Numerical simulations of the model are presented, which show spatial variations in alignment over long transients, but with spatially uniform behavior in the long term. This is investigated further via asymptotic analysis, using the angular diffusion coefficient as a small parameter. This method enables calculation of the form of the steady state orientation peaks observed numerically; by considering behavior at large times, the rate of approach to these peaks is shown to be exponential. Extension of this analysis to the spatially varying model confirms that long-time behavior will be spatially uniform except in one special, and biologically unrealistic, case. The authors conclude that the spatially varying alignment patterns observed in skin are in fact slow transients, and biological implications of the modeling are discussed.

Original Publication Citation

J.C. Dallon and J.A. Sherratt: A Mathematical Model for Spatially Varying Extracellular Matrix. SIAM Journal of Applied Mathematics. 61(2): 56-527 (2).