Modelling a competitive antibody/antigen chemical reaction

A fluorescent capillary-fill device is a biosensor based on antibody-antigen technology for determining whether a patient is suffering from a particular pathogen. The specific antibody is affixed to a side wall in a small container. On the other side is the antigen with a fluorescent label which dissolves upon entry of the bulk fluid (usually urine) containing the antigen. A competitive reaction then takes place on the side wall for the antibody sites. In this talk we consider the development of a mathematical model of this competitive chemical reaction within a small cell which occurs in a biosensor. The model consists of two coupled diffusion equations with nonlinear boundary conditions which can be expressed equivalently as a system of two integro- differential equations. Through this reformulation an asymptotic result is derived, a perturbation solution is developed and a product integration method is presented. Finally, an alternative formulation is presented in the form of a system of four Volterra integral equations, which provides local existence and uniqueness of the solution to the original diffusion problem. Two product integration methods are applied to this second reformulation. We present several numerical results, using real data, that illustrate the performance of the methods.