Mathematical method in optical molecular imaging

Optical molecular imaging is an important technique of studies at molecular level and provides promising tools to non-invasively delineate in vivo physiological and pathological activities at cellular and molecular levels, and it has been widely used for diagnosing, managing diseases, metastasis detection and drug development. From a mathematical perspective, this paper mainly focuses on the forward problem and inverse problem in biological tissues based on the radiative transfer equation (RTE). The forward problem is accustomed to describing photon propagation in biological tissues and the inverse problem is used to reconstruct internal source distribution from the signal detected on the external surface. We also introduce the detailed derivation of the RTE and Robin boundary condition and discretization of the forward problem, along with the reconstruction methods and iterative solution algorithms summarized for the inverse problem. Finally, the current and future challenges of optical molecular imaging are discussed. This survey aims to construct a mathematical method, a state-of-the-art framework for optical molecular imaging, from which future research may benefit.