Analysis of a Mathematical Model of Ischemic Cutaneous wounds

TL;DR

This paper analyzes a mathematical model of ischemic skin wounds, focusing on how ischemia affects wound healing dynamics through a coupled PDE system with a moving boundary.

Contribution

It establishes global existence and uniqueness of solutions for the model and examines how ischemia level influences wound boundary movement.

Findings

Proves global existence and uniqueness of the free boundary problem.

Shows dependence of wound boundary on ischemia parameter gamma.

Provides insights into ischemia's impact on wound healing dynamics.

Abstract

Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Ischemia represents a serious complicating factor in wound healing. In this paper we analyze a recently developed mathematical model of ischemic dermal wounds. The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The extracellular matrix (ECM) is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary of the open wound. The model equations involve the concentrations of oxygen, cytokines, and the densities of several types of cells. The ischemic level is represented by a parameter which appears in the boundary conditions, 0 <= gamma < 1; gamma near 1 corresponds to extreme ischemia and gamma = 0 corresponds to normal non-ischemic conditions. We establish global existence and uniqueness of the free boundary problem and study the dependence of the free boundary on gamma.