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Mathematical models of diffusive transport underpin our understanding ofchemical, biochemical and biological transport phenomena. Analysis of suchmodels often focusses on relatively simple geometries and deals with diffusionthrough highly idealised homogeneous media. In contrast, practical applicationsof diffusive transport theory inevitably involve dealing with more complicatedgeometries as well as dealing with heterogeneous media. One of the mostfundamental properties of diffusive transport is the concept of mean particlelifetime or mean exit time, which are particular applications of the concept offirst passage time, and provide the mean time required for a diffusing particleto reach an absorbing boundary. Most formal analysis of mean particle lifetimeapplies to relatively simple geometries, often with homogeneous(spatially-invariant) material properties. In this work, we present a generalframework that provides exact mathematical insight into the mean particlelifetime, and higher moments of particle lifetime, for point particlesdiffusing in heterogeneous discs and spheres with radial symmetry. Our analysisapplies to geometries with an arbitrary number and arrangement of distinctlayers, where transport in each layer is characterised by a distinctdiffusivity. We obtain exact closed-form expressions for the mean particlelifetime for a diffusing particle released at an arbitrary location and wegeneralise these results to give exact, closed-form expressions for anyhigher-order moment of particle lifetime for a range of different boundaryconditions. Finally, using these results we construct new homogenizationformulae that provide an accurate simplified description of diffusion throughheterogeneous discs and spheres.

Diffusion in heterogeneous discs and spheres: new closed-form expressions for exit times and homogenization formulae