A simplified model of wound healing (with particular reference to the critical size defect)

This paper is an attempt to construct a simple mathematical model of wound healing/tissue regeneration which reproduces some of the known qualitative features of those phenomena. It does not address the time development of the wound in any way, but does examine conditions (e.g., wound size) under which such healing may occur. Two related one-dimensional models are examined here. The first and simpler of the two corresponds to a “swath” of tissue (or more realistically in this case, bone) removed from an infinite plane of tissue in which only a thin band of tissue at the wound edges takes part in tissue/bone regeneration. There is no tissue or bone in the interior. The second model has a similar geometric structure, except that not all the tissue in the interior has been removed: it is a “gouge” or “graze” rather than a hole or puncture. In each model, there is a thin layer of tissue (e.g., the epidermis) or bone (depending on the context) that is responsible for increased mitotic activity at the edges of the wound by manufacturing a generic growth stimulator of concentration C(x, t) small, where x is the direction of wound closure and t is time. Using a combination of results from these two models, we have been able to predict the size of the critical size defect, which is defined as the smallest intraosseous wound that does not heal by bone formation during the lifetime of the animal being studied. We have also been able to isolate parameter ranges that will give reasonable values for both the thickness of the active region and the critical size defect, and in addition, establish that the models discussed here have the sensitivity to place reasonable bounds on such parameter values.