| Abstract: | This paper is concerned with the elastodynamics of embryonic epidermal wound closing. Underlying the recovery process for this type of wounds is a mechanism of wound recognition through directed cell-to-cell signaling. The observed actin filament realignment induced by the biological signals leads to a purse-string effect and the resulting (unknown) "active stresses." The circumferential contraction of the epidermis surrounding the wound is then determined by the laws of mechanics and propagation properties of the relevant cell–cell signaling that decays with distance. With the wound known to retract for a short period immediately after infliction, the quasi-equilibrium configuration reached during this initial phase serves as the initial condition for the dynamic wound closing phase. A small strain variation of the Murray–Sherratt model of the quasi-equilibrium problem will be formulated for speedy computation of this initial state at the inception of the wound closure phase, with the latter problem being the main concern of this paper. Some theoretical developments are found to be instrumental to an efficient algorithm for the otherwise time-consuming task of calculating the effect of the biological signals generated by the presence of a wound. Application of our elastodynamic model to the case of a circular wound suggests that the propagation range of our choice of cell–cell signaling mechanism must be above a certain minimum fraction of the wound radius for wound closure. As expected, stress concentration occurs adjacent to the edge of the remaining small wound near the end of the wound closing process. At that point, the present model is not expected to be adequate and more appropriate expressions of elastic strain and active stress induced by actin filaments may be in order. Other biological processes such as cell proliferation and differentiation may be involved. |
