Magnetic resonance imaging (MRI) is an imaging modality that noninvasively measures magnetic fields by selectively exciting the magnetization of protons inside the body. When combined with an understanding of electromagnetic theory, MRI can be used in a novel way to provide a powerful tool for measuring the electromagnetic fields and electrical properties of biological tissues. This thesis presents the analytical, numerical, processing and experimental components of a successful implementation of Low-Frequency Current Density Impedance Imaging (LF-CDII), an impedance imaging method based on MRI measurements. The accuracy, stability and noise tolerance of this technique are examined. The first in-vivo LF-CDII experiment was conducted with a clinical MRI scanner, and the conductivity distribution of the heart of a live piglet was obtained. Both the simulation and experimental results show that LF-CDII can be used as a reliable tool for accurate noninvasive, quantitative imaging of tissue conductivities. This thesis also presents new data processing algorithms, imaging procedures and hardware development for the measurement of electromagnetic fields at radio frequencies, based on Polar Decomposition Radio Frequency Current Density Imaging (PD-RFCDI). The method was tested on both numerical models and experiments on phantoms. The results show that the techniques presented here are able to successfully image current density fields without the strict restrictions on the direction and magnitude of the currents required in previous versions of RFCDI.

Magnetic resonance imaging (MRI) is an imaging modality that noninvasively measures magnetic fields by selectively exciting the magnetization of protons inside the body. When combined with an understanding of electromagnetic theory, MRI can be used in a novel way to provide a powerful tool for measuring the electromagnetic fields and electrical properties of biological tissues. This thesis presents the analytical, numerical, processing and experimental components of a successful implementation of Low-Frequency Current Density Impedance Imaging (LF-CDII), an impedance imaging method based on MRI measurements. The accuracy, stability and noise tolerance of this technique are examined. The first in-vivo LF-CDII experiment was conducted with a clinical MRI scanner, and the conductivity distribution of the heart of a live piglet was obtained. Both the simulation and experimental results show that LF-CDII can be used as a reliable tool for accurate noninvasive, quantitative imaging of tissue conductivities. This thesis also presents new data processing algorithms, imaging procedures and hardware development for the measurement of electromagnetic fields at radio frequencies, based on Polar Decomposition Radio Frequency Current Density Imaging (PD-RFCDI). The method was tested on both numerical models and experiments on phantoms. The results show that the techniques presented here are able to successfully image current density fields without the strict restrictions on the direction and magnitude of the currents required in previous versions of RFCDI.

We study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity from the magnitude of one current density field inside. The method now known as Current Density Impedance Imaging is based on solving boundary value problems for the 1-Laplacian in an appropriate Riemann metric space. We consider two types of methods: the ones based on level sets and a variational approach, which aim to solve specific boundary value problem associated with the 1-Laplacian. We will address the Cauchy and Dirichlet problems with full and partial data, and also the Complete Electrode Model (CEM). The latter model is known to describe most accurately the voltage potential distribution in a conductive body, while taking into account the transition of current from the electrode to the body. For the CEM the problem is non-unique. We characterize the non-uniqueness, and explain which additional measurements fix the solution. Multiple numerical schemes for each of the methods are implemented to demonstrate the computational feasibility.