Asymmetric zonal shim coils for magnetic resonance applications

Abstract

A method is presented for the systematic design of asymmetric zonal shim coils for magnetic resonance applications. Fourier-series methods are used to represent the magnetic field inside and outside a circular cylinder of length 2L and radius a. The current density on the cylinder is also represented using Fourier series. Any desired field can be specified in advance on the cylinder's radius, over some nonsymmetric portion pL<z<qL of the coil's length (-1<p<q<1). Periodic extension of the field is used, in a way that guarantees the continuity of the field, and therefore gives good convergence of the Fourier series. All that is required is to calculate the Fourier coefficients associated with the specified desired field, and from these, the current density on the coil and the magnetic field components then follow automatically. The method is illustrated by applying it to three sample zonal coil designs, namely, linear, quadratic, and cubic fields located asymmetrically in the coil. Current densities and corresponding coil winding patterns are shown for these three illustrative cases. Field calculations directly from the coil patterns and spherical harmonic deconvolutions of these fields indicate that the example designs match the theory well. Asymmetric shim coils can be used in conventional symmetric MRI magnets, particularly those architected for "head-only" studies. One of their major applications is expected to be in the newly developed asymmetric magnet systems.

A cylindrical basis set for shim coil design in magnetic resonance imaging

“…Substituting the equation for B z,out , as evaluated at ρ = c 2 , into (16), and applying orthogonality relations yields a set of five equations relating the current on the shield coil to the current on the primary coil. Substituting the equation for B z,out , as evaluated at ρ = c 2 , into (16), and applying orthogonality relations yields a set of five equations relating the current on the shield coil to the current on the primary coil.…”

“…Also, note that each of these functions is orthogonal to all the others over the surface of a cylinder running from −L to L. To design coils that will produce functions from the basis set, we apply a modified version of Turner's target field approach. Note that other methods could also be applied to design coils for this basis set, including numerical approaches 15 and those incorporating asymmetric imaging regions, 16,17 for example.…”

“…and B z;out ðc 2 ; ϕ; zÞ ¼ 0: Here, B z,in refers to the sum of Equation (10) as it is applied to each of the primary and shield coils, and B z,out refers to the sum of the equation analogous to (10) but applicable for the ρ > a case. Substituting the equation for B z,out , as evaluated at ρ = c 2 , into (16), and applying orthogonality relations yields a set of five equations relating the current on the shield coil to the current on the primary coil. Notably these equations are independent of c 2 , so satisfying them should yield a zero field for all values of ρ > a 2 and any choice of c 2 ≥ a 2 will give the same winding pattern.…”

“…with coefficients α k,0 , α k,l and β k,l as given previously in (14). Convergence analysis of these series shows that the target field must be specified on the primary coil to guarantee convergence, but, as in the unshielded case, when the target field is from the cylindrical basis set (5) there will only be one non-zero coefficient in each sum, so convergence is not a concern.…”

Theory and development of biplanar active shim coils for a permanent NMR analyzer

“…This method has no ill-conditioned problem because Fourier transforms have unique inverses, and it has been used to design cylindrical surface shim coils for superconducting magnets for magnetic resonance imaging (MRI). Forbes and Crozier[7–9] advanced the TF method, considering the finite coil length using a Fourier series technique. Wen Tao Liu[10–12] proposed an alternative current density expansion, rather than Tikhonov regularization, and Xiao Fei You[13] combined this method with Forbes and Crozier’s.…”