Nonlinear effects of localized absorption perturbations on the light distribution in a turbid medium

A theoretical model of photon propagation in a scattering medium is presented, from which algebraic formulas for the detector-readings perturbations (ΔR) produced by one or two localized perturbations in the macroscopic absorption cross section Δμ_a are derived. Examination of these shows that when Δμ_a is “titrated” from very small to large magnitudes in one voxel, the curve traced by the corresponding ΔR values is a rectangular hyperbola. Furthermore, while  is dependent on the location of the detector with respect to the source and the voxel, the ratio ΔR/ΔR^∞ is independent of the detector location. We also find that when Δμ_a is varied in two voxels simultaneously, the quantity ΔR(Δμ_a,1 ∧ Δμ_a,2) is a bilinear rational function of the Δμ_as. These results apply not only in the case of steady-state illumination and detection but extend to time-harmonic measurements as well. The validity of the theoretical formulas is demonstrated by applying them to the results of selected numerical diffusion computations. Potential applications of the derived expressions to image-reconstruction problems are discussed.