# Identification of the Functional Form of Nonlinear Effects of Localized Finite Absorption in a Diffusing Medium

### H. L. Graber *et al.* (1995)

#### Summary

A linear perturbation model for reconstructing images of absorption (Σ_{a}) and scattering
(Σ_{s}) cross sections of a highly scattering medium is presented. Two factors limiting the accuracy of
images reconstructed from linear perturbation models are described. These are 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 intially
nonabsorbing (Σ_{s}= 1.0, Σ_{a}= 0.0) slab, as the Σ_{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 vs. change in Σ_{a} deviates significantly from a straight line when the
addtional Σ_{a} exceeds ~0.1; 2) the rate at which the flux perturbation approaches its limiting value as
Σ_{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 Σ_{a} perturbations increase is independent
of the detector location; 4) when the perturbed voxels are 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 incoporate corrections for these two effects are discussed.