Application of Variational Methods for Field Mapping in Magnetic Resonance Imaging

Over the years, magnetic resonance imaging (MRI) has evolved to one of the leading imaging modalities in medical diagnosis, because of its high soft tissue contrast without deposing ionizing radiation. Besides pure morphological imaging, MRI can also provide quantitative or semi-quantitative information about physical and physiological processes or the micro-structure of the tissue, which can serve as biomarker for several diseases. Among others, these are blood flow, diffusion, perfusion, quantitative susceptibility, fat content or tissue relaxation times. The main drawback of MRI is the inherently long acquisition time. To overcome this problem, several strategies were proposed over the years, by improving imaging sequences and data acquisition trajectories to increase the acquired data per time. With this development in combination with improved hardware, the physiological limits in terms of peripheral nerve stimulation and RF energy deposition were reached.

Another way to accelerate the MR data acquisition is to reduce the amount of acquired data below the Nyquist limit and to reconstruct the image using mathematical methods. Because of the ill-posedness of this inverse problem, regularization becomes necessary to stabilize the solution of the reconstruction. The theory of variational methods is perfectly suited for this purpose.

Some applications have very high requirement according the homogeneity of the fields necessary to obtain an MRI signal. These are the static magnetic field B0 and the radio frequency (RF) field B1. Careful coil design, low manufacturing tolerances, and shimming approaches for B0 and B1 lead already to very homogeneous field distributions, but the magnetic and electric properties of biological tissue lead to field distortions which are specific for a certain patient. To correct for influences arising from that, filed mapping becomes necessary.

This thesis covers the physical background of field inhomogeneities in B0 and B1, as well as the most important mapping methods and shimming approaches to increase their homogeneity. Moreover, the mathematical background of image reconstruction is described, as well as a deviation of the most important regularization functionals and their numerical solution.
This thesis also considers the questions, how variational methods can be applied to either increase the accuracy of an acquired field map or to reconstruct highly accurate field maps from highly undersampled data. For this purpose, two different algorithms are described, one to gain highly accurate B0 maps dedicated for the separation of fat and water signal components and the other one for the reconstruction of B1+ field maps from highly accelerated Bloch-Siegert data. Several examples are shown for phantom and in-vivo measurements at 3T and 7T.