Monte Carlo Method For Photon Transport - Implementation of Photon Transport in A Scattering Medium

Implementation of Photon Transport in A Scattering Medium

Presented here is a model of a photon Monte Carlo method in a homogeneous infinite medium. The model is easily extended for multi-layered media, however. For an inhomogeneous medium, boundaries must be considered. In addition for a semi-infinite medium (in which photons are considered lost if they exit the top boundary), special consideration must be taken. For more information, please visit the links at the bottom of the page. We will solve the problem using an infinitely small point source (represented analytically as a Dirac delta function in space and time). Responses to arbitrary source geometries can be constructed using the method of Green's functions (or convolution, if enough spatial symmetry exists). The required parameters are the absorption coefficient, the scattering coefficient, and the scattering phase function. (If boundaries are considered the index of refraction for each medium must also be provided.) Time-resolved responses are found by keeping track of the total elapsed time of the photon's flight using the optical path length. Responses to sources with arbitrary time profiles can then be modeled through convolution in time.

In our simplified model we use the following variance reduction technique to reduce computational time. Instead of propagating photons individually, we create a photon packet with a specific weight (generally initialized as unity). As the photon interacts in the turbid medium, it will deposit weight due to absorption and the remaining weight will be scattered to other parts of the medium. Any number of variables can be logged along the way, depending on the interest of a particular application. Each photon packet will repeatedly undergo the following numbered steps until it is either terminated, reflected, or transmitted. The process is diagrammed in the schematic to the right. Any number of photon packets can be launched and modeled, until the resulting simulated measurements have the desired signal-to-noise ratio. Note that as Monte Carlo modeling is a statistical process involving random numbers, we will be using the variable ξ throughout as a pseudo-random number for many calculations.