A fast numerical method for time-resolved photon diffusion in general stratified turbid media

Efficient forward models of photon migration in complex geometries are important for noninvasive imaging of tissue in vivo with diffuse optical tomography (DOT). In particular, solving the inverse problem requires multiple solutions of the forward model and is therefore computationally intensive. We present a numerical algorithm for the rapid solution of the time-dependent diffusion equation in a semi-infinite inhomogeneous medium whose scattering and absorption coefficients are arbitrary functions of depth, given a point source impulsive excitation. Such stratified media are biomedically important. A transverse modal representation leads to a series of one-dimensional diffusion problems which are solved via finite-difference methods. A novel time-stepping scheme allows effort to scale independently of total time (for fixed system size). Tayloring to the DOT application gives run times of order 0.1 s. We study convergence, computational effort, and validate against known solutions in the case of 2-layer media. The method will be useful for other forward and inverse diffusion problems, such as heat conduction and conductivity measurement