Keynote speech : Analytical inversion approach for magnetic resonance electrical properties tomography

  Keynote Speaker: Prof. Takaaki Nara, The University of Tokyo

  Prof. Takaaki Nara obtained his B.S. and M.S. degrees in Mathematical Engineering and Information Physics in 1995 and 1997, respectively, and a Ph.D. in Advanced Interdisciplinary Studies in 2000 from the University of Tokyo, Japan. Since 2017, he has been serving as a Professor at the Graduate School of Information Science and Technology at the University of Tokyo. His research interests include mathematical algorithms and sensors for inverse problems. His dedication to his field has been recognized through various accolades, including the IEEE Virtual Reality Best Paper Award in 2000 and the Highlights of Inverse Problems in 2011. He serves as an Associate Editor for the SICE Journal of Control, Measurement, and System Integration. He is a Director of the Japan Society for Industrial and Applied Mathematics. He is a member of IEEE and SIAM. 

  Abstract: Inverse analysis is essential for diagnosing abnormal states in living tissues within the human body. For inverting the law of causality that connects unknown states to their resultant data, the physical law should be considered as a constraint to ensure a valid and interpretable solution. This report introduces an inverse problem focused on identifying the electrical conductivity and permittivity of tissues. Because these electrical properties differ between cancellous and normal tissues, three-dimensional, in vivo imaging can serve as a complementary modality to conventional structural imaging. Recently, magnetic resonance electrical properties tomography has garnered significant attention owing to its ability to use radio frequency magnetic field data measured inside the body—not on its surface—with MRI, thereby alleviating the ill-posed nature of the inverse problem. Mathematically, this inversion involves determining the coefficients of Maxwell's equations from the measured magnetic field. Initially, we focus on a two-dimensional problem and develop an explicit reconstruction formula for the electrical properties in terms of the measured magnetic field using a complex analysis method. This formula offers the unique solution to the inverse problem in a comprehensible manner. Subsequently, we extend this method to a three-dimensional scenario and establish an iterative algorithm with guaranteed convergence. The proposed algorithm is validated through numerical simulations and in vivo experiments. If time permits, we will delve into its potential extension to magnetic resonance elastography.