Optimization magnetic resonance imaging shim coil using second derivative discretized stream function

Abstract

In Magnetic Resonance Imaging (MRI) equipment, a set of shim coils are designed to generate specific magnetic fields, thus eliminating harmonic components of magnetic field to obtain a high level homogeneous magnetic field within the region of interesting (ROI). In the electromagnetic design process, in order to produce the desired magnetic field, the deviation between the calculated magnetic field of shim coil and the theoretical magnetic field is treated as a kind of traditional objective functions to optimize the distribution of current density on the surface of shim coil skeleton. However, such function is ill-posed because of the overdetermined or underdetermined system of equations. The regularization method is commonly used to solve such problem by constructing the regularization term. This article proposes a new iterative optimization method for the design of shim coils in MRI. Based on the boundary element method (BEM), the discretized stream functions can be obtained by discretizing the surface of coil skeleton using a set of triangular elements. As the regularization term, the second derivative stream function is included in the minimization of the deviation between calculated magnetic fields and target magnetic fields. The distribution of coil which meets the design requirements can be obtained by using the BroydenFletcher-Goldfarb-Shanno (BFGS) algorithm. At last, the cubic spline interpolation is used to make lines as smooth as possible to be processed. In this article, the proposed method was employed to design two kinds of room temperature shim coils for cylindrical and/or biplanar MRI shim coil system. The simulation results demonstrate that the proposed method is effective and practical.

A theoretical framework of gradient coil designed to mitigate eddy currents for a permanent magnet MRI system

“…In the continuity equation of current density, J can be expressed as J = J r e r + J θ e θ for the planar coil design [18]. By defining the potential function ψ, and it has following relation to the current density [19]:…”