Perturbation theory for the diffusion equation by use of the moments of the generalized temporal point-spread function II Continuous-wave results

Abstract

We investigate the performance of the method proposed in Part I of this paper in several situations of interest in diffuse optical imaging of biological tissues. Monte Carlo simulations were extensively used to validate the approximate scaling relationship between higher-order and first-order self moments of the generalized temporal point-spread function in semi-infinite and slab geometry. More specifically we found that in a wide range of cases the scaling parameters c1, c2, c3 [see Eq. (36) of Part I] lie in the intervals (1.48, 1.58), (3.1, 3.7), and (8.5, 11.5), respectively. The scaling relationships between higher-order and first-order self moments are useful for the calculation of the perturbation of a single defect in a straightforward way. Although these relationships are more accurate for inclusions of linear size less than approximately 6mm, their performance is also studied for larger inclusions. A good agreement, to within approximately 10%, was found between the perturbations of single and multiple defects calculated with the proposed method and those obtained by Monte Carlo simulations. We also provide formulas for the calculation of the moments up to the fourth order for which it is clear how lower-order moments can be used for the calculation of higher-order moments.