Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation.

The accuracy of results obtained from the diffusion equation (DE) has been investigated for the case of an isotropic point source in a homogeneous, weakly absorbing, infinite medium. The results from the DE have been compared both with numerical solutions of the radiative transfer equation obtained with Monte Carlo (MC) simulations and with cw experimental results. Comparisons showed that for the cw fluence rate, discrepancies are of the same order as statistical fluctuations on MC results (within 1%) when the distance r from the source is > 2/mu(s)', (mu(s)' is the reduced scattering coefficient). For these values of r, discrepancies for the time-resolved fluence rate are of the same order of statistical fluctuations (within 5%) when the time of flight is t > 4t0 with to time of flight for unscattered photons. For shorter times the DE overestimates the fluence discrepancies are larger for larger values of the asymmetry factor. As to the specific intensity, for small values of r the MC results are more forward peaked than expected from the DE, and the forward peak is stronger for photons arriving at short times. We assumed r > 2/mu(s)' and t > 4t0 for the domain of validity of the DE and we determined the requirements for which the simplifying assumptions necessary to obtain the DE, expressed by two inequalities, are fulfilled. Comparisons with cw experimental results showed a good agreement with MC results both at high and at small values of r mu(s)', while the comparison with the DE showed significant discrepancies for small values of r mu(s)'. Using MC results we also investigated the error made on the optical properties of the medium when they are retrieved using the solution of the DE. To obtain accuracy better than 1% from fitting procedures on time-resolved fluence rate data it is necessary to disregard photons with time of flight < 4t0. Also from cw data it is possible to retrieve the optical properties with good accuracy: by using the added absorber technique discrepancies are < 1%, both on mu(s)' and on mu(a), if the absorption coefficient is small (mu(a)/mu(s)' < 0.005).