A mathematical treatment of the fluorescence capillary-fill device

Rebelo, Magda and Diogo, Teresa and McKee, Sean (2012) A mathematical treatment of the fluorescence capillary-fill device. SIAM Journal on Applied Mathematics, 72 (4). pp. 1081-1112. ISSN 0036-1399

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A mathematical model in the form of two coupled diffusion equations is provided for a competitive chemical reaction between an antigen and a labelled antigen for antibody sites on a cell wall; boundary conditions are such that the problem is both nonlinear and nonlocal. This is then re-characterized first as a pair of coupled singular integro-differential equations and then as a system of four Volterra integral equations. The latter permits a proof of existence and uniqueness of the solution of the original problem. Small and large time asymptotic solutions are derived and, from the first characterization, a regular perturbation solution is obtained. Numerical schemes are briefly discussed and graphical results are presented for human immunoglobulin.