Efficient numerical modelling of time-domain light propagation in curved 3D absorbing and scattering media with...

An efficient approach is introduced for modelling light propagation in the time domain in 3D heterogeneous absorbing and scattering media (e.g. biological tissues) with curved boundaries. It relies on the finite difference method (FDM) in conjuction with the Crank-Nicolson method for accurately solving the optical diffusion equation (DE). The strength of the FDM lies in its simplicity and efficiency, since the equations are easy to set up, and accessing neighboring grid points only requires simple memory operations, leading to efficient code execution. Owing to its use of Cartesian grids, the FDM is generally thought cumbersome compared to the finite element method (FEM) for dealing with media with curved boundaries. However, to apply the FDM to such media, the blocking-off method can be resorted to. To account for the change of the refractive index at the boundary, Robin-type boundary conditions are considered. This requires the computation of surface normals. We resort here for the first time to the Sobel operator borrowed from image processing to perform this task. The Sobel operator is easy to implement, fast, and allows obtaining a smooth field of normal vectors along the boundary. The main contribution of this work is to arrive at a complete numerical FDM-based model of light propagation in the time domain in 3D absorbing and scattering media with curved geometries, taking into account realistic refractive index mismatch boundary conditions. The fluence rate obtained with this numerical model is shown to reproduce well that obtained with independent gold-standard Monte Carlo simulations.