Solutions for diffuse optical tomography using the Feynman-Kac formula and interacting particle method

In this paper, we propose a novel method to solve the forward and inverse problems in diffuse optical tomography. Our forward solution is based on the diffusion approximation equation and is constructed using the Feynman-Kac formula with an interacting particle method. It can be implemented using Monte-Carlo (MC) method and thus provides great flexibility in modeling complex geometries. But different from conventional MC approaches, it uses excursions of the photons' random walks and produces a transfer kernel so that only one round of MC-based forward simulation (using an arbitrarily known optical distribution) is required in order to get observations associated with different optical distributions. Based on these properties, we develop a perturbation-based method to solve the inverse problem in a discretized parameter space. We validate our methods using simulated 2D examples. We compare our forward solutions with those obtained using the finite element method and find good consistency. We solve the inverse problem using the maximum likelihood method with a greedy optimization approach. Numerical results show that if we start from multiple initial points in a constrained searching space, our method can locate the abnormality correctly.