Studies in optimization for magnetic resonance imaging

Methods for designing coils with optimal performance for use in MRI systems are presented here. A method called ‘supershielding’ is described and utilized to design actively shielded gradient coils. In solving these problems, a functional optimization subject to constraints has been successful using Lagrange multipliers. An iterative approach based on a variational approach has also been very successful, as well. These techniques help one to design a gradient coil that minimizes image distortions due to field inhomogeneities caused by eddy currents on surrounding metal surfaces. The absolute limits of shielding using an open shield are also considered and discussed. The supershielding approach is applied to the gradients in an open-MRI system.; For rf coil design, the problem of determining the normal mode frequencies of some LC network is paramount. A method for finding closed-form solutions for the normal mode frequencies of systems with circulant ($2pn$) symmetry is investigated. This method is particularly useful for the questions of degeneracy that arise when one considers parallel imaging techniques like SENSE and SMASH in MRI. It is applicable to systems that include birdcage coils as well as planar coils with the appropriate rotational symmetry. A proof is given that complete degeneracy of all normal mode frequencies is impossible when all mutual inductive couplings are included. We compare the theoretic expressions against measurements made on both a planar coil array and an 8-element birdcage coil. The inclusion of the co-rotating end-ring mode changes the fundamental symmetry of the system from circulant to ‘bordered circulant.’ Closed-form solutions for the normal mode frequencies of a bordered circulant system are also given. Group theory is then used to find the normal modes of systems that do not have circulant or bordered circulant symmetry yet still possess some symmetry. As an example, closed-form solutions for the normal mode frequencies of a three-ring birdcage coil have been obtained. These theoretic expressions have been tested against experimental measurements made on an 8-element three-ring birdcage coil.