A Mathematical Model for Ischemic Wound Healing

Certain complications can slow or prevent the process of wound healing. Ischemia, which is the abnormally low supply of oxygen to tissue caused by poor blood flow, can occur in wounded tissue, and cause poor healing. Ischemia of wound tissue often occurs in patients with vascular disease, diabetes, or prolonged immobilization. In a paper published this week in the SIAM Journal on Mathematical Analysis, Avner Friedman, Bei Hu, and Chuan Xue use a mathematical model to investigate ischemic dermal wound healing.

Wound healing is the process by which an organ or part of the human body repairs and heals itself after injury. The body’s response to an injury is almost instantaneous: a cascade of interrelated biological events is set in motion, first containing and then repairing the wound. These events involve biochemical interactions among blood cells, chemical mediators, and the extracellular matrix in blood tissue.

The first stage of a body’s automatic response, inflammation, involves clot formation to prevent further bleeding, and removal of bacteria and other debris from the site of injury. The area then goes into a proliferative phase where repair of damaged tissue occurs. Collagen is deposited around the wound and a new layer of skin is formed. New blood cells are formed by a process called angiogenesis. During the final maturation stage, the skin layer is strengthened with permanent collagen. Unwanted blood vessels are removed, and the dark coloration of the wound scar eventually disappears.

Obstruction of the normal sequence of events in wound healing can lead to chronic wounds, a major public health concern in the US. It is estimated that chronic wounds cost several billion dollars in lost productivity and medical costs in the country annually.

The level of oxygen in tissues is one of the key factors in injury repair, and has been incorporated in several mathematical models of wound healing. Hypoxia--or a state of low oxygen—is in itself an important precursor to wound healing. However, severe hypoxia can obstruct the growth of new blood vessels, a process called angiogenesis, which is critical to injury repair.

In this paper, the authors analyze a previously proposed model, which uses a system of partial differential equations in the partially healed region with a free boundary surrounding the open wound. As healing occurs, the edges of a wound gradually move together until the edges are joined and the wound is healed.  The position of the wound edges area or "free boundary" in time must be determined. This is called the free boundary or moving boundary problem.

The authors provide a solution to this free boundary problem in ischemic wound healing.  Their equations are formulated based on the levels of oxygen, immunochemicals and cell densities in the wound region. By using this model, they show that under extreme ischemic conditions, the open wound does not close in finite time. On the other hand, non-ischemic wounds are capable of healing with time, as shown by simulations. Parameters of the system are set based on biological experiments. Further studies are needed to address specific situations, such as pressure and diabetic conditions in ischemic wound healing.

A Mathematical Model for Ischemic Wound Healing
Avner Friedman, Bei Hu, and Chuan Xue
SIAM Journal on Mathematical Analysis, 42 (2010), pp 2013-2040
Pub date 26 August 2010
© Society for Industrial and Applied Mathematics

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