On forward and inverse models in fluorescence diffuse optical tomography
Institute for Mathematics and Scientific Computing, Karl-Franzens University Graz, Heinrichstraße 36, 8010 Graz, Austria
Institute of Medical Engineering, Graz University of Technology, Kronesgasse 5/II, 8010 Graz, Austria
Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany
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.
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