Transport through diffusive and nondiffusive regions, embedded objects, and clear layers

This paper deals with the diffusion approximation of transport equations in diffusive domains with nondiffusive inclusions, such as embedded objects and clear layers, where classical diffusion is not valid. Generalized diffusion equations with suitable nonlocal interface conditions at the boundary of the inclusions are derived. The asymptotic convergence of the transport solution to the generalized diffusion solution is shown for two types of inclusions. Numerical experiments assess the accuracy of the generalized diffusion equation. Applications include the modeling of thin clear layers filled with cerebrospinal fluid within the human head. Near-infrared spectroscopy is increasingly being used in medical imaging for monitoring certain properties of human tissues. The presence of thin clear layers hampers the use of classical diffusion. This difficulty can be overcome by using the generalized diffusion model presented in this paper.