Numerical solution for an inverse MRI problem using a regularised boundary element method - Marin, Liviu; Power, Henry; Bowtell, Richard W.; Cobos Sanchez, Clemente; Becker, Adib A.; Glover, Paul; Jones, Arthur (Engineering analysis with boundary elements, 2008)
We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils used in magnetic resonance imaging (MRI) and is formulated using its corresponding integral representation according to potential theory. A constant boundary element method (BEM) which satisfies the continuity equation for the current density, i.e. divergence-free BEM, and was originally proposed by Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005;26B:67–80], is presented based on geometrical arguments with respect to the linear (flat) triangular boundary elements employed in order to emphasise its possible extension to further higher-order divergence-free interpolations. Since the discretised BEM system is ill-posed and hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularisation method is employed in order to retrieve accurate and physically correct solutions. A rigorous numerical approximation for the calculation the magnetic energy, which reduces the errors induced by employing the approach of Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005;26B:67–80], is also proposed.