  Mathematical models of diffusive transport underpin our understanding of
chemical, biochemical and biological transport phenomena. Analysis of such
models often focusses on relatively simple geometries and deals with diffusion
through highly idealised homogeneous media. In contrast, practical applications
of diffusive transport theory inevitably involve dealing with more complicated
geometries as well as dealing with heterogeneous media. One of the most
fundamental properties of diffusive transport is the concept of mean particle
lifetime or mean exit time, which are particular applications of the concept of
first passage time, and provide the mean time required for a diffusing particle
to reach an absorbing boundary. Most formal analysis of mean particle lifetime
applies to relatively simple geometries, often with homogeneous
(spatially-invariant) material properties. In this work, we present a general
framework that provides exact mathematical insight into the mean particle
lifetime, and higher moments of particle lifetime, for point particles
diffusing in heterogeneous discs and spheres with radial symmetry. Our analysis
applies to geometries with an arbitrary number and arrangement of distinct
layers, where transport in each layer is characterised by a distinct
diffusivity. We obtain exact closed-form expressions for the mean particle
lifetime for a diffusing particle released at an arbitrary location and we
generalise these results to give exact, closed-form expressions for any
higher-order moment of particle lifetime for a range of different boundary
conditions. Finally, using these results we construct new homogenization
formulae that provide an accurate simplified description of diffusion through
heterogeneous discs and spheres.
