Chandan Sen, a research professor in the surgery department at Ohio State University, teamed up with Avner Friedman, professor of mathematicians at the university, and Chuan Xue, a postdoc at Ohio State’s Mathematical Biosciences Institute, to create a mathematical model for ischemic wounds. This new computational tool should provide predictive guidance on how a given wound might progress, allowing researchers to develop more precise protocols to deal with wounds and dehiscences.
The mathematical model, to date, simulates both non-ischemic wounds – those typical of wounds in healthy people with good circulation – and ischemic wounds. The current model produced results that generally match pre-clinical expectations: that a normal wound will close in about 13 days, and that 20 days after the development of an ischemic wound, only 25 percent of the wound will be healed.
The model also showed that normal wounds have higher concentrations of proteins and cells expected to be present during the healing process, while ischemic wounds lack oxygen and remain in a prolonged inflammatory phase that interferes with the subsequent cascade of events required to begin wound closure.
“Wound geometry is complicated because it is three-dimensional,” said Avner Friedman, a senior author of the paper and a Distinguished University Professor at Ohio State. “It would be infeasible to perform our computations within the framework of this geometry. However, we used some mathematical ideas to reduce the problem to a simpler geometry without giving up any of the important aspects of the process.”
It is not just the wound that is three-dimensional, the researchers noted. The complexity of this process is compounded by the fact that the wound-healing model must take into account both the total space occupied by the wound and the time required for the healing process.
In developing the mathematical model, Friedman worked with first author Chuan Xue, a postdoctoral researcher in Ohio State’s Mathematical Biosciences Institute, to assign values to variables in the first two stages of wound healing. These included oxygen concentration, concentration of growth factors, density of white blood cells that fight pathogens, density of fibroblasts that perform part of the repair, and density of tips and sprouts of tiny new blood vessels.
The two also modeled the extracellular matrix – the bed on which cells work to close the wound – in a way that allows for the matrix to change the way it functions over time. This part of the model also allowed for simulation of the exertion of pressure – a characteristic of certain types of ulcers that people with diabetes are prone to develop.
Xue noted that the equations were borrowed from the mathematical theory of homogenization by manipulating a single parameter – called parameter alpha – to draw the distinction between ischemic and nonischemic wounds in the model. This is one example, Friedman noted, of simplifying the model without leaving out important biological details.