Parameter Search to Find Ranges of Activation and Inhibition of Wound Healing Rate in a Mathematical Model with Introduced Photobiomodulation

Abstract

When light stimulation is used for wound healing therapy, a biphasic dose-response curve is observed, where cells are activated below and inhibited above a treatment dose threshold. Light treatment-dose responses are not yet incorporated into mathematical models of wound healing—yet these relationships would support optimization of wound healing treatment protocols. This work adapts an existing wound healing mathematical model by exploring parameter values and introducing exogenous photobiomodulation treatment inputs for future applications in model-based experimental research. A wound healing mathematical model, created by Sherratt and Murray in 1990, includes proliferation, migration, and activating and inhibitory chemical terms. This model was implemented and discretized by Forward Euler (FE) in time and the Central Difference Method (CDM) in space in 1D. Travelling wave solutions of cell density and chemical concentration were obtained and used to plot wound closure in time and to estimate the wound healing rate. A parameter search was conducted to identify ranges where model simulations resulted in activation, inhibition, saturation, or numeric instability of wound healing. Published results of photobiomodulation treatment-control studies reporting a percentage change in proliferation were used to scale proliferation terms, thus serving as a proxy for light stimulation. Results showed the inhibition model was more sensitive to parameter variation than the activation model. Changes in the cell migration parameter are most sensitive overall. Most model parameters were bounded by saturation or numeric instabilities, while otherwise demonstrating activating and/or inhibitory effects on the rate of wound healing. Light stimulation simulations were consistent with expectations that increasing the proliferation term increased wound healing rate. To support photobiomodulation model-based experimental wound healing research, the model parameter search identified threshold values categorizing activation or inhibition of wound healing rate and this work also adapted a model proliferation term consistent with photobiomodulation biological effects.