## A Slough/Wound Interaction Model Accounting for Cellular Diffusion

International Conference on Mathematics and Mathematics Education, Ankara, Turkey, 16 - 18 September 2021, pp.71-72

• Publication Type: Conference Paper / Summary Text
• City: Ankara
• Country: Turkey
• Page Numbers: pp.71-72
• Karadeniz Technical University Affiliated: Yes

#### Abstract

In this study we consider Slough/Wound interaction model, called SWODE, a nonlinear Ordinary Differential Equation, ODE system, developed by Jones et. al in 2003 [1] that describes evolution of wound and slough area.

We first nondimensionalize the model and estimate a larger domain of attraction for the zero equilibrium solution with an appropriately formulated Liapunov function. Thus, we determine the region where the wound heals and slough tissue disappears in terms of nondimensional model parameters.

Next, we extend the SWODE model to account for one dimensional cellular diffusion resulting in a model, called SWPDE1, a one dimensional nonlinear system of Partial Differential Equations. Evolution of wound and slough interactions is described and equilibrium solutions with lower and upper solutions are determined. The uniqueness of corresponding stationary system is also handled numerically and qualitative behaviour of equilibrium solutions are analysed.

Finally, we further extend SWPDE1 to account for two dimensional cellular diffusion, resulting in a model, SWPDE2, a two dimensional time-dependent nonlinear PDE system. In this model, we consider rectangular wound and slough within a rectangular region and investigate wound-slough interaction as the healing process goes on. Method of lines is used to obtain a nonlinear system of ODEs and MATLAB ODE solvers [2] are used to integrate the resulting system.

Key Words: Wound healing, nonlinear ordinary differential equation system, nonlinear partial differential equation system, ODE solvers of MATLAB.

REFERENCES
[1]
M.A. Jones, B. Song and D.M. Thomas, Controlling Wound Healing Through

Debridement, Mathematical and Computer Modelling 40 (2004), 1057-1064.

[2] L.F. Shampine, M.W. Reichelt, The Matlab ODE Suite, SIAM Journal on Scientific Computing, 18(1997), 1-22.