Light transport in two-layer tissues

We study theoretically light backscattered by tissues using the radiative transport equation. In particular we consider a two-layered medium in which a finite slab is situated on top of a half space. We solve the one-dimensional problem in which a plane wave is incident normally on the top layer and is the only source of light. The solution to this problem is obtained formally by imposing continuity between the solutions for the upper and lower layers. However, we are interested solely in probing the top layer. Assuming that the optical properties in the lower layer are known, we remove it from the problem yielding a finite slab problem by prescribing an alternate boundary condition. This boundary condition is derived using the theory of Green's functions and is exact. Hence, one needs only to solve the transport equation in a finite slab using this alternate boundary condition. We derive an asymptotic solution for the case when the slab is optically thin. We extend these results to the three-dimensional problem using Fourier transforms. These results are validated by comparisons with numerical solutions for the entire two-layered problem.