Nicolas Arango^{1}, Jason Stockmann^{2,3}, Elfar Adalsteinsson^{1,4}, and Jacob White^{1}

Simulations providing an upper bound on ∆B0 shimming of 1096 human brains from the human connectome project with currents outside the target volume were performed and used to construct optimal n-channel shim fields. Optimal truncated shim basis performance was evaluated suggesting 70 optimal channels are required to achieve 95% of ultimate performance. Comparisons with arrays of regularly spaced circular loops suggests that under realistic current constraints, regular loop arrays with hundreds of elements only achieve 85% of ultimate performance. The ultimate ∆B 0 shim and optimal n-channel coils will be useful tools in the analysis and comparison of shim array designs.

Significant effort has been made in the design of wire-patters for local ∆B_{0} shim arrays for human brain imaging. Coil designs improved performance through the careful design of wire patterns [1, 2, 3, 4]. However,the gap in performance between current shim coil geometries and the ultimate shim performance, that of a field generated from an arbitrary current density on a surface outside of the target volume has not been explored. Further, the performance gap between current designs and an optimal equivalent-channel-count-coil are unknown. Previous work has characterized the low number of magnetic field modes needed to describe the ∆B_{0} inhomogeneity over the full human brain [5], designed fields to match dominant modes of ∆B_{0} variation in heads, not the dominant producible ∆B_{0} modes.

We compute the ultimate ∆B_{0} shim of randomly sampled slabs from 1096 ∆B_{0} maps of human heads from the Human Connectome Project [6]. These ultimate shims are used to construct optimal n-channel shim bases. Performance of regular n-channel arrays of loops on a cylinder are compared to the performance of the n-channel optimal field patterns producible on that cylinder.

To compute the ultimate shim from arbitrary currents on a a specified surface, we first construct a basis of the fields producible from divergence free currents on that surface. By Amper’s law, a dipole distribution M produces the same magnetic field field as a current density J if M = ∇ × J. To compute the ultimate shim basis for the specified surface, we can calculate the set of fields producible from dipoles scattered on and normal the specified surface. An ultimate minimum σ ∆B_{0} field can be computed for a ∆B_{0} map by shimming the volume using the ultimate shim basis.

Optimal n-channel shim fields for a 200mm radius 200mm tall cylinder are calculated by computing the dipole distributions with the most energy in the set of dipole distributions used to generate a collection of target fields. The method used is illustrated in figure 1. Random thickness axial, sagital and coronal slabs from a set of ∆B_{0} maps are sampled and shimmed jointly with a combination of the second order spherical harmonics and the ultimate shim basis. The left singular vectors of the matrix formed for the collection of ultimate shim dipole moments form an orthogonal set of dipole distributions ordered by their energy in the population. The top n singular dipole moments are taken as the optimal n-channel shim basis. Discrete wire pattern approximations of the dominant dipole distributions may be constructed [7].

We evaluate the performance of the ultimate ∆B_{0} homogeneity shim as compared to a baseline 2nd orders pherical harmonic shim through histograms of off-resonances for 1000 randomly sampled axial, sagital,and coronal slices. The 2nd order spherical harmonic shim achieves a 20.8Hz standard deviation across all slabs while the ultimate shim achieves 12.1Hz. Figure 2 shows a normalized histogram of the standard deviation of off resonances for the one thousand random slabs used in the test set for 2nd order spherical harmonic shimming and the ultimate shim. These curves provide lower and upper bounds respectively for the random-slab-sampled off-resonance histograms of finite coils constructed on the same surface.

N-channel optimal coils and coils constructed from regular arrays of loops are evaluated with the relative improvement of each shimmed volume between the upper and lower bound of the ultimate and 2nd order shim baseline. The simulated regular 100-turn loop array was evaluated with and without 1A current limits. Figure 3 shows performance of the unconstrained optimal and regular-circular coils converging in performance to the ultimate, while current constrained regular array is limited to 85% of the ultimate performance.

To achieve above 80% mean performance of the ultimate array 30 optimal channels are necessary while, as shown in figure 4, equivalent performance is achieved with 45 regular loops without current constraints and 123 regular loops with 1A current constraints.

Performance close to the n-channel optimal shim array is achievable with regular arrays of circular loops. However, under realistic maximum current constraints, performance is limited to to that of an 35-channelultimate coil.

We find that neither the ultimate n-channel shim coil nor the n-coil regular array are practical approaches for achieving near ultimate shim performance. The former is impractical because of wiring complexity, and the latter impractical because of the huge currents required. Finding a practical alternative remains an open question, one that the tools described herein can help answer.

The authors graciously acknowledge the support of:

Skolkovo Institute of Technology Next Generation Program

MIT-MGH seed grant.

NIH NIBIB R01EB018976

Data were provided in part by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University

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Flowchart of proposed procedure for computing optimal n-channel shim coils. Slabs are randomly sampled from a collection of brain off resonance maps. Optimal dipole moment distributions arec omputed for each coil. Dipole distributions are collected in a matrix and the singular value decomposition is computed. The top n singular vectors form the n-channel optimal shim basis.

a) histogram of second order and ultimate shim performance for 1000 randomly sampled axial, sagital, and coronal slabs. Representative b) 2nd order spherical harmonic c) ultimate ∆B 0 maps for a representative whole-brain shim.

Relative shim performance compared to ultimate shim of n-channel a) optimal shim and b)regular circular array. Mean and worst-case performance of n-channel shim basis compared to n-channel orthogonal shim basis. Red star and yellow circle illustrate the mean and worst-case performance for 30channel optimal shim array.

Left: illustration of regular array with an increasing number of channels. Right: (Blue) Number of channels in a regular loop array to achieve the same mean performance as an n-channel optimal ∆B 0shim array. (Orange) Number of 1A 100 turn regular loops to achieve same mean performance as an n-channel optimal ∆B 0 shim

Wire pattern discretization of top two optimal shim modes and their ∆B 0 field patterns.