John Dallon: Paper Abstract
A Mathematical Model for Spatially Varying Extracellular Matrix
AUTHOR:
John C. Dallon 1 and Jonathan. A. Sherratt 2
1: Department of Mathematics, Brigham Young University,
Provo, UT 84602
2: Department of Mathematics, Heriot-Watt University,
Edinburgh EH14 4AS, UK.
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.
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