Understanding the regulation of chronic wounds by tissue inhibitors of matrix metalloproteinases through mathematical modelling

Understanding the biochemistry and pharmacodynamics of chronic wounds is of key importance, due to the millions of people in the UK affected and the significant cost to the NHS. Chronic wounds are characterised by elevated concentrations of matrix metalloproteinases (MMPs) that destroy the surrounding extracellular matrix (ECM). However, fibroblasts can produce tissue inhibitors of MMPs (TIMPs) in order to regulate wound healing. Therefore, the role of TIMPs in both acute and chronic wounds needs to be properly understood in order to develop therapeutic treatments. In this work, we propose a reaction-diffusion system of four partial differential equations that describe the interaction of the ECM, fibroblasts, MMPs, and TIMPs in a wound. We observe that, subject to parameter sets corresponding to both acute and chronic wound healing, this mathematical model gives rise to travelling wave solutions. Using bifurcation analysis, we demonstrate that excessive degradation of the ECM results in the emergence of chronic wounds, and the reversal of these chronic wounds is prohibited for lower TIMP production values. These results are replicated within a simplified model obtained via a parameter sensitivity analysis. This model is further extended to more realistic spatial domains where we demonstrate the effectiveness of a therapeutic hydrogel containing TIMPs as a treatment for chronic wounds.