Forward solvers for photon migration in the presence of highly and totally absorbing objects embedded inside diffusive media

In this paper, after a critical review of the literature, we present two forward solvers and a new methodology for description of photon migration in the presence of totally absorbing inclusions embedded in diffusive media in both time and CW domains. The first forward solver is a heuristic approach based on a higher order perturbation theory applied to the diffusion equation (DE) [denoted eighth-order perturbation theory (EOPT)]. The second forward solver [denoted eighth-order perturbation theory with the equivalence relation (EOPTER) ] is obtained by combining the EOPT solver with the adoption of the equivalence relation (ER) [J. Biomed. Opt. 18, 066014 (2013)]. These forward solvers can possibly overcome some evident limitations of previous approaches like the theory behind the so-called banana-shape regions or exact analytical solutions of the DE in the presence of highly or totally absorbing inclusions. We also propose the ER to reformulate the problem of a totally absorbing inclusion in terms of another inclusion having a finite absorption contrast and a re-scaled volume. For instance, we have shown how this approach can indeed be used to simulate black inclusions with the Born approximation. By means of comparisons with the results of Monte Carlo simulations, we have shown that the EOPTER solver can model totally absorbing inclusions with an error smaller than about 10%, whereas the EOPT solver shows an error smaller than about 20%, showing a performance largely better than that observed with solvers proposed previously.