Computer simulation of radial immunodiffusion. I. Selection of an algorithm for the diffusion process

Theories of diffusion with chemical reaction are reviewed as to their contributions toward developing an algorithm needed for computer simulation of immunodiffusion. The Spiers-Augustin moving sink and the Engelberg stationary sink theories show how the antibody-antigen reaction can be incorporated into boundary conditions of the free diffusion differential equations. For this, a stoichiometric precipitate was assumed and the location of precipitin lines could be predicted. The Hill simultaneous linear adsorption theory provides a mathematical device for including another special type of antibody-antigen reaction in antigen excess regions of the gel. It permits an explanation for the lowered antigen diffusion coefficient, observed in the Oudin arrangement of single linear diffusion, but does not enable prediction of the location of precipitin lines. The most promising mathematical approach for a general solution is implied in the Augustin alternating cycle theory. This assumes the immunodiffusion process can be evaluated by alternating computation cycles: free diffusion without chemical reaction and chemical reaction without diffusion. The algorithm for the free diffusion update cycle, extended to both linear and radial geometries, is given in detail since it was based on gross flow rather than more conventional expressions in terms of net flow. Limitations on the numerical integration process using this algorithm are illustrated for free diffusion from a cylindrical well.