On forward and inverse models in fluorescence diffuse optical tomography

This paper investigates forward and inverse problems in fluorescence optical tomography, with the aim to devise stable methods for the tomographic image reconstruction. We analyze solvability of a standard nonlinear forward model and two approximations by reduced models, which provide certain advantages for a theoretical as well as numerical treatment of the inverse problem. Important properties of the forward operators, that map the unknown fluorophore concentration on virtual measurements, are derived; in particular, the ill-posedness of the reconstruction problem is proved, and uniqueness issues are discussed. For the stable solution of the inverse problem, we consider Tikhonov-type regularization methods, and we prove that the forward operators have all the properties, that allow to apply standard regularization theory. We also investigate the applicability of nonlinear regularization methods, i.e., TV-regularization and a method of levelset-type, which are better suited for the reconstruction of localized or piecewise constant solutions. The theoretical results are supported by numerical tests, which demonstrate the viability of the reduced models for the treatment of the inverse problem, and the advantages of nonlinear regularization methods for reconstructing localized fluorophore distributions.