chapter 10, Comparisons of Analytical Solutions with Monte Carlo Results

Excerpt

10.1 Introduction

In this chapter, we detail comparisons between the solutions of the RTE described in Chapters 4–7 and the reference MC results reported in the enclosed CD-ROM. All of the solutions based on the diffusion approximation used for comparisons have been calculated using the code and the subroutines included in the CD-ROM. Comparisons with MC results thus provide a validation of the enclosed software. Furthermore, since almost all the analytical solutions presented are approximate solutions of the RTE, comparisons also provide information on their accuracy and range of applicability. As validation of the Diffusion&Perturbation software, comparisons are shown in Sec. 10.2 for the homogeneous infinite medium, the homogeneous slab, and the homogeneous slab with an internal inhomogeneity. Comparisons for the layered slab and the hybrid models are reported in Secs. 10.4 and 10.5, respectively. For the infinite medium, both the MC and the analytical solutions pertain to a point-like isotropic source. For the slab, MC results pertain to a pencil beam. The corresponding analytical results, according to the discussion in Sec. 4.2, pertain to an isotropic point-like source inside the medium at a distance z_s = 1∕μ_s^′ from the entrance point of the beam. All the figures of this chapter, apart from Fig. 10.31 (the MC results used for this figure have not been reported in the CD-ROM), can be reproduced using the software and the MC results provided with the enclosed CD-ROM.

Comparisons of analytical models with MC results are shown through the ratio MC∕MODEL. The results are presented without any filtering even though some TR curves are very noisy. We preferred this representation because the noise gives information on the accuracy of the reference MC results, and because a filtering procedure can severely distort the response at early times where the larger discrepancies between the results from analytical models and reference results are expected.