Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium

A linear perturbation model for reconstructing images of absorption ((Sigma) _a ) and scattering ((Sigma) _s ) cross sections of a highly scattering medium is presented. Two factors limiting the accuracy of image reconstructed from linear perturbation models are described. These are the self-shadowing effect of a single perturbation and the mutual coupling effect of two perturbations. A relaxation method to numerically solve the diffusion equation for a slab geometry and to compute the flux of diffuse light crossing both surfaces of an initially nonabsorbing ((Sigma) _s equals 1.0, (Sigma) _a equals 0.00 slab, as the (Sigma) _a in one or two cells of the medium is increased. When a single voxel was perturbed, it was found that: 1) for all voxel locations considered, a plot of change in light flux versus change in (Sigma) _a deviates significantly from a straight line when the additional (Sigma) _a exceeds approximately 0.1; 2) the rate at which the flux perturbation approaches its limiting value as (Sigma) _a increases is independent of the detector location; 3) with the exception of voxels in the immediate vicinity of the source, the rate at which the flux perturbation approaches its limiting value as (Sigma) _a increases is independent of the location of the perturbed voxel. When two voxels were perturbed simultaneously, it was found that: 1) the distance separating two voxels is the most important determinant of the maximal mutual coupling effect they can have; 2) the maximal mutual coupling effect falls rapidly as the distance between two voxels increases; 3) if both perturbed voxels are lie in the same layer (i.e., depth), the rate at which the mutual coupling effect approaches its limiting value as the (Sigma) _a perturbations increase is independent of the detector locations; 4) when the perturbed voxels are in different layers, there is a small but significant difference between the effects of mutual coupling on the diffuse transmission and on the diffuse reflectance. Low-order rational functions are sufficient for modeling both the self-shadowing and mutual coupling effects. Methods for modifying image reconstruction algorithms to incorporate corrections for these two effects are discussed.