Implementation of the equation of radiative transfer on block-structured grids for modeling light propagation in tissue

Abstract

We present the first algorithm for solving the equation of radiative transfer (ERT) in the frequency domain (FD) on three-dimensional block-structured Cartesian grids (BSG). This algorithm allows for accurate modeling of light propagation in media of arbitrary shape with air-tissue refractive index mismatch at the boundary at increased speed compared to currently available structured grid algorithms. To accurately model arbitrarily shaped geometries the algorithm generates BSGs that are finely discretized only near physical boundaries and therefore less dense than fine grids. We discretize the FD-ERT using a combination of the upwind-step method and the discrete ordinates (SN) approximation. The source iteration technique is used to obtain the solution. We implement a first order interpolation scheme when traversing between coarse and fine grid regions. Effects of geometry and optical parameters on algorithm performance are evaluated using numerical phantoms (circular, cylindrical, and arbitrary shape) and varying the absorption and scattering coefficients, modulation frequency, and refractive index. The solution on a 3-level BSG is obtained up to 4.2 times faster than the solution on a single fine grid, with minimal increase in numerical error (less than 5%).

Decoupled fluorescence Monte Carlo model for direct computation of fluorescence in turbid media

“…This is because the dfMC model is directly derived from the RTEs. [11][12] Also, the dfMC ensures that the fluorescence statistical quantities are unbiased, which means that the dfMC model can be as accurate as the RTEs if there are sufficient simulated photons. The discrepancies between the dfMC simulations and measurements mainly arise from statistical and measurement noise.…”

“…The present models for the excitation-to-emission conversion and the transport of the fluorescence in turbid media are derived from the radiative transport equations (RTEs), 11,12 including the simple analytical model, 13 diffusion approximation model, [14][15][16][17][18][19] and Monte Carlo (MC) model. [20][21][22][23][24][25][26][27] The analytical model can be applied only in relatively simple configurations, such as the slab model with homogeneous media.…”

Renormalization of the highly forward-peaked phase function using the double exponential formula for radiative transfer

“…This is because in highly forwardpeaked scattering, the value of the phase function of the RTE at the small polar scattering angle tends to increase exponentially, resulting in slow convergence of the numerical calculation of the scattering integral. For the quadrature set, the extended trapezoidal rule (ETR) has been employed [16,19]. The advantage of the ETR set is that the total number of the directions can be chosen according to a computer memory size, so that the numerical calculation of the scattering integral can converge with a sufficiently large number of the directions and enough computer memory size.…”

Simulating photon-transport in uniform media using the radiative transport equation: a study using the Neumann-series approach

“…The discrete ordinates method has been implemented with finite difference [16,19,24] and boundary element [25] methods. Also, the spherical harmonics method has been used along with finite element methods [26,27].…”