Analytical Methods For Certain Inverse Problems In Medical Imaging

A variety of medical imaging techniques are based on the solution of a class of mathematical problems, called inverse problems. In particular, in the medical imaging techniques of Positron Emission Tomography (PET) and Single Photon Emission Computed Tomography (SPECT), the associated inverse problems involve the inversion of the Radon transform and of the so-called Attenuated Radon transform, respectively. Although an analytical formula for the former transform is well known, an analogous formula for the latter transform was obtained only recently. We will develop further analytical algorithms based on the above formulae and on non-smooth-norm de-noising strategies. Furthermore, by employing a variety of clinical data, we will compare the new algorithms with commercial software.The medical techniques of Magnetoencephalography (MEG) and Electroencephalography (EEG), also involve the solution of certain inverse problems. In a recent breakthrough, the PI has presented a comprehensive solution of these inverse problems. We will develop further the relevant algorithms, we will use the techniques of reproducing kernels for their numerical implementation, and we will compare them with commercial software using a variety of real data.

The PI has been working for more than 15 years on inverse problems appearing in medical imaging. Some of the relevant results on electromagnetoencephalography were obtained in the framework of ``BRAIN , which was one of only 15 projects funded in Europe in 2005 for a Marie Curie Chair of Excellence. The PI has benefited from his collaboration with P. Barbano, who is currently funded by an HOPE grant of MRC. The PI has established a fruitful collaboration with Dr. Olaf Hauk of the Cognition and Brain Sciences Unit, MRC, Cambridge and with V. Michel, University Professor at the University of Siegen, Germany. It appears that this collaborative effort can provide the basis for a comprehensive approach to the important problem of obtaining an effective MEG-EEG reconstruction algorithm. Indeed, the recent analytical results of the PI yield the most effective known formulae for the ``visible part of the current, Dr. O. Hauk has a vast experience with acquisition and analysis of real data, and Prof. V. Michel has developed and implemented a powerful numerical technique which has already yielded practical results for the inverse seismic problem (this problem has certain similarities with the inverse MEG problem). P.E. Barbano will refine the relevant numerical techniques and will also deal with the problem of de-noising the data. In addition, the PI has established close contacts with Prof. A. Papanicolaou of the University of Texas (one of the world experts in the clinical application of MEG) and with Dr. C. Stamoulis of the University of Harvard (an Applied Mathematician who has now an appointment in the Neurology Department working on MEG). Taking into consideration the clinical significance of MEG-EEG, particularly in epilepsy, as well as the impact of real time imaging techniques for deciphering the functional properties of the brain, our work on MEG-EEG could have a significant impact on neurology, cognitive psychology and neuroscience. The University of Cambridge is at a close distance with UCL Hospital, where the PI has established a close collaboration with the group of Prof. B. Hutton. In addition, the PI has a close collaboration with Prof. Datseris of Evangelismos Hospital, Greece, where they perform approximately fifty PET scans per week. Furthermore, the PI and P.E. Barbano have established a close collaboration with Dr. G. Kastis of the Medical-Biological Institute of the Academy of Athens, Greece, which houses its own CT-PET. The above collaborators and contacts will provide us with clinical data for PET and SPECT. Taking into consideration the speed advantage of the analytical algorithms, as well as the crucial role played by PET and SPECT in several areas of medicine and in particular in oncology, neurology and cardiology (for example SPECT is part of the common stress-testing procedures for the evaluation of chest pain), the success of the proposed research for PET, SPECT could have an important benefit in health services. The PI has been the mentor for several post-graduate students and more than a dozen post-doctoral fellows. The PI has delivered more than 250 invited talks and colloquia at international conferences and major universities including Harvard, Stanford, Princeton, Yale, Berkeley, MIT, Caltech, Columbia, Oxford and Tokyo. Among his recent presentations are the opening address of the 45th Mathematical Olympiad, Greece, 2004, an invited address at the celebration of the Royal Irish Academy for the Bicentennial of W.R. Hamilton, Ireland, 2005, the SIAM Invited Address at the Annual meeting of AMS and MAA, USA, 2006, the opening plenary address of the international conference Nonlinear Waves - Theory and Applications , China, 2008, the opening plenary address at the 2nd World Congress of Controversies in Neurology, 2008 and a plenary address at the 2009 World Mental Health Congress, Athens, Greece.