The unidirectional flux transient as a tool for quantifying parallel diffusional pathways through membranes. Exact solution for two pathways

If a plane membrane consists of patches, each with a given area and a given diffusion coefficient, then the transient of the total unidirectional flux of a diffusing substance (as defined experimentally by Ussing) is predictable. Here the inverse problem is studied: given only the observed transient of the total unidirectional diffusion flux, the unknown membrane heterogeneity transverse to the flux is to be quantified. The ratio of the arithmetic and of the harmonic means (both area-weighted) of the diffusion coefficients, evaluated over the membrane, is expressed in terms of the observed transient alone and is used to characterize the heterogeneity. A unique exact solution of the inverse problem for two kinds of patches is obtained in closed form. A singular limit of this solution pertains to currently postulated models of endothelial membranes, for which a characteristically shaped transient is predicted.