Radiative Transfer Equation and Diffusion Theory For Photon Transport in Biological Tissue - Diffusion Theory Solutions Vs. Monte Carlo Simulations

Diffusion Theory Solutions Vs. Monte Carlo Simulations

Monte Carlo simulations of photon transport, though time consuming, will accurately predict photon behavior in a scattering medium. The assumptions involved in characterizing photon behavior with the diffusion equation generate inaccuracies. Generally, the diffusion approximation is less accurate as the absorption coefficient μ_a increases and the scattering coefficient μ_s decreases. For a photon beam incident on a medium of limited depth, error due to the diffusion approximation is most prominent within one transport mean free path of the location of photon incidence (where radiance is not yet isotropic) (Figure 3).
Among the steps in describing a pencil beam incident on a semi-infinite medium with the diffusion equation, converting the medium from anisotropic to isotropic (step 1) (Figure 4) and converting the beam to a source (step 2) (Figure 5) generate more error than converting from a single source to a pair of image sources (step 3) (Figure 6). Step 2 generates the most significant error.

• Figure 3: Diffuse reflectance vs. radius from an incident pencil beam as determined by a Monte Carlo simulation (red) and diffuse reflectance vs. radius from two isotropic point sources as determined by the diffusion theory solution to the RTE (blue). The transport mean free path is 0.1 cm.

• Figure 4: Diffuse reflectance vs. radius from incident pencil beam for an anisotropic (blue) and isotropic (red) medium.

• Figure 5: Diffuse reflectance vs. radius from photon source for a pencil beam (blue) and an isotropic point source (red).

• Figure 6: Diffuse reflectance vs. radius from the photon source for an isotropic point source as characterized by the solution to the RTE (blue) and a Monte Carlo simulation (red).