The mathematical model for chemoattractant diffusion in the Boyden chamber assay was developed using the C++ programming language. Initial implementation of the analytical solution was performed in Mathematica and then used to verify the solutions (both analytical and numerical) programmed in C++. The model consists of the partial differential equation describing diffusion of a biochemical gradient and was solved using finite difference methods for various experimental conditions to obtain parameter estimates for diffusion for input into the overall hybrid model. The partial differential equation was then solved by the explicit (forward) and implicit (backward) finite difference methods. These approximations were then compared to the C++ program for the analytical solution (using Crank’s equation for diffusion in a finite system). The Microsoft Visual C++ program was the user-interface implemented and lacked a comprehensive library of functions. In addition to programming the analytical solution, it was necessary to write programs for both the error function (erf(x)) and integration. A two thousand iteration Riemann sum was used to approximate integration. Both the explicit finite difference approximation and the analytical solution were simple algebraic problems. The implicit finite difference approximation required systems of equations to solve and required matrix inversion to solve. It was found that the approximations gave sufficiently accurate (within two to four decimal places) results for concentration of chemoattractant, with the explicit method being the least accurate. Both approximation methods were, however, found to be less accurate than the analytical solution. The analytical solution modeled what would be expected; that is, it presented the concentration of chemoattractant moving towards equilibrium throughout the height of the Boyden chamber as time moved foraward.
Equation for Diffusion of Chemoattractant: