Introduction

Halbach arrays are an appealing approach for low field MRI due to low cost and weight [1,2] but typically suffer from poor B₀ homogeneity. B₀ imperfections can be introduced by variations in the magnetization strength and direction of the individual magnets[3] or manufacturing tolerances in the magnet formers. Arrays typically need several shimming iterations, involving the addition of magnetic material inside the magnet, with positioning often calculated using stochastic algorithms.
We present a new shimming approach that exploits the field patterns created by higher-order Halbach shim-arrays to correct for distinct spherical harmonic B₀-inhomogeneity terms. A homogenous B₀ field (degree (l) = 0, order (m) = 0 spherical harmonic) is generated using a cylindrical dipolar (k=1) Halbach array where the magnetization direction rotates twice over circumference of the cylinder. A transverse linear gradient (l=1, m=|1| spherical harmonics) can be created using a quadrupolar (k=2) (see figure 1) Halbach array comprising three rotations over the surface, etc. For spherical harmonics where l > m, the field can be approximated by varying the magnetization of the Halbach array along its length. For example, to create a linear gradient along the bore (l=1, m=0) one could use a k=1 Halbach array with the sign of the magnetization reversing (smoothly) along its length. As such, given a sufficiently long Halbach array, a k=1 shim can be used to generate all spherical harmonics of order 0. We demonstrate this concept on a new custom-built 47 mT, head system where we improve the homogeneity of the magnet by 50% by only correcting for the m=2 spherical harmonics.

Method

A 50 cm long, 31 cm diameter bore Halbach array based magnet was designed using the variable diameter homogeneity optimisation described previously [6]. The magnet consists of 25 rings, with 20 mm centre-to-centre spacing, of 12 mm cuboid N48 magnets: the central 19 rings have two layers of magnets and the three rings at each end have 3 layers. The magnetic field was mapped over a 250 mm diameter spherical volume (DSV) with a Teslameter attached to a 3D positioning robot. The acquired field maps were fitted to up to 20^th order spherical harmonics to reduce noise. The inhomogeneity over a 25 cm DSV was measured to be 11723 ppm (figure 3). The l=2 , m=2 spherical harmonic term was the most significant contributor with a magnitude of 10245 ppm and as such we designed a k=3 Halbach shim array.
Shim magnets (up to 6 mm) are placed in 12 holders located on the outside of the magnet (see figure 2): each holder can accommodate 49 trays of magnets with 10 mm spacing between the trays. For the first shim iteration we filled only 13 trays (40 mm spacing) to allow for later iterations. Magnets were placed on a cylindrical radius with up to seven magnets per tray segment (84 in total for each position along the magnet). The field distribution generated by each ring of magnets was calculated, and a least squares fit of the field performed to calculate the total magnetization needed in each ring. This was then approximated by varying the size and number of magnets in each ring. An experimental field map was acquired after shimming and we simulated a correction of up to 5^th order spherical harmonics based on this map.
To show the impact of the shim insert, spectra and images of an 8 centimeter spherical phantom were acquired. In both cases the field was shimmed using the linear gradients prior to data acquisition. Images were acquired using a 3D TSE sequence with the following scan parameters: FoV: 150x150x150 mm, resolution: 1.5x1.5x1.5 mm, TR/TE: 600ms/15ms, ETL: 20, bandwidth: 20kHz, duration: 5 minutes

Results

Figure 2c shows the magnet configuration after optimisation with a single populated shim tray shown in 2b. The initial field (figure 3) shows the characteristic Z² – Y² spherical harmonic field. The correction field created by a k=2 Halbach shim array closely matches this inhomogeneity. There is good agreement between the simulated (6156 ppm) and measured (5879 ppm) fields after shimming. Simulations of corrections show the B₀ homogeneity can be improved to 2771 ppm using shim arrays up to k=6 with only high frequency perturbations at the outer bore remaining. Over a 20 cm DSV, sufficient for full brain coverage, homogeneity improves to 282 ppm using the same correction. Figure 5 shows that the acquired spectra and images are significantly improved after the shims are inserted.

Discussion

In this work we show a new approach to shimming Halbach-array magnets using higher order Halbach shim arrays to correct for spherical harmonic inhomogeneities. We achieve a 50 % reduction in inhomogeneity by correcting only for the m=2 order, with excellent agreement between simulations and measurements, showing promise for using this approach for correcting a larger set of spherical harmonics. There are practical limits to the highest order spherical harmonics one can correct due to discretization errors associated with the very high number of rotations needed. While we have applied this method to shimming an existing magnetic field, the basic principle could also be used to optimize the field distribution of new magnets designs.

Acknowledgements

This work is supported by ERC Advanced Grant PASMAR and Simon Stevin Meester Award