Computational Approaches to Solving Equations Arising from Wound Healing

Abstract

In the wound healing process, the cell movement associated with chemotaxis generally outweighs the movement associated with random motion, leading to advection-dominated mathematical models of wound healing. The equations in these models must be solved with care, but often inappropriate approaches are adopted. Two one-dimensional test problems arising from advection-dominated models of wound healing are solved using four algorithms--MATLAB's inbuilt routine pdepe.m, the Numerical Algorithms Group routine d03pcf.f, and two finite volume methods. The first finite volume method is based on a first-order upwinding treatment of chemotaxis terms and the second on a flux limiting approach. The first test problem admits an analytic solution which can be used to validate the numerical results by analyzing two measures of the error for each method: the average absolute difference and a mass balance error. These criteria as well as the visual comparison between the numerical methods and the exact solution lead us to conclude that flux limiting is the best approach to solving advection-dominated wound healing problems numerically in one dimension. The second test problem is a coupled nonlinear three species model of wound healing angiogenesis. Measurement of the mass balance error for this test problem further confirms our hypothesis that flux limiting is the most appropriate method for solving advection-dominated governing equations in wound healing models. We also consider two two-dimensional test problems arising from wound healing, one that admits an analytic solution and a more complicated problem of blood vessels growth into a devascularized wound bed. The results from the two-dimensional test problems also demonstrate that the flux limiting treatment of advective terms is ideal for an advection-dominated problem.

Numerical and experimental analysis of the sedimentation of spherical colloidal suspensions under centrifugal force

“…A very fine spatial mesh is therefore required near the front and in the sediment to accurately capture the solution behaviour and prevent numerical instabilities. 44 For regions where the volume fraction is relatively uniform, a coarser spatial mesh is sufficient. An adaptive spatial mesh is therefore used to follow the time-dependent nature of the solution, which increases efficiency by reducing the number of spatial mesh points required.…”

Mathematical Modeling in Wound Healing, Bone Regeneration and Tissue Engineering

“…Numerical methods for PDE models arising from wound healing and bone regeneration can vary from using built-in solvers designed in programs like Matlab to implementing finite volumes methods in order to obtain accurate solutions, and is an active area of research (for a review regarding wound healing models, see Thackham et al (2009), and for models of bone regeneration see Gerisch and Geris (2007)). Numerical challenges arise because wound healing and bone regeneration are advection-dominating processes as the chemotactic effects that growth factors have on cells is more dominant than the effects of random motility (Lauffenburger 1983;Stokes et al 1991).…”

A method for the analysis of thermal tweezers for manipulation and trapping of nanoparticles and adatoms on crystalline surfaces

“…Therefore, it can be effectively solved by the numerical finite-volume solution method ͑FVM͒. 11 In this approach, the computational window is divided into grid cells which are referred to as control volumes ͑Fig. 1͒.…”

“…11 In this paper, we choose to use the Crank-Nicolson method because it is inherently more accurate and has better convergence. 11 Standard first order approximations are used to discretize the derivatives of at the control volume faces. For example,…”

“…1,2 Therefore, the aim of this paper is to introduce and develop a new approach for the analysis of thermal tweezers, based on the direct determination and investigation of probability distributions for diffusing particles or adatoms on a surface in the presence of strong temperature gradients. This will be done by means of numerical finite-volume solution 11 of the Smoluchowski diffusion equation. The local values of the diffusion constant will be determined by solution of the Fokker-Planck equation for the considered crystalline potential of the substrate and local temperature.…”