Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function III Frequency-domain and time-domain results

Abstract

We study the performance of a previously proposed perturbation theory for the diffusion equation in frequency and time domains as they are known in the field of near infrared spectroscopy and diffuse optical tomography. We have derived approximate formulas for calculating higher order self- and mixed path length moments, up to the fourth order, which can be used in general diffusive media regardless of geometry and initial distribution of the optical properties, for studying the effect of absorbing defects. The method of Padé approximants is used to extend the validity of the theory to a wider range of absorption contrasts between defects and background. By using Monte Carlo simulations, we have tested these formulas in the semi-infinite and slab geometries for the cases of single and multiple absorbing defects having sizes of interest (d=4-10 mm, where d is the diameter of the defect). In frequency domain, the discrepancy between the two methods of calculation (Padé approximants and Monte Carlo simulations) was within 10% for absorption contrasts Deltamu(a)<or=0.2 mm(-1) for alternating current data, and usually to within 1 degrees for Deltamu(a)<or=0.1 mm(-1) for phase data. In time domain, the average discrepancy in the temporal range of interest (a few nanoseconds) was 2%-3% for Deltamu(a)<or=0.06 mm(-1). The proposed method is an effective fast forward problem solver: all the time-domain results presented in this work were obtained with a computational time of less than about 15 s with a Pentium IV 1.66 GHz personal computer.