Generating unilateral field modulation for MRI using a pyrolytic-graphite-based Halbach array
Richard Bowtell1

1University of Nottingham, Nottingham, United Kingdom

### Synopsis

A flat Halbach array consisting of an array of long, thin permanent magnets whose magnetization orientation varies linearly with position, has the interesting property of generating a unilateral field perturbation. Such a pattern of field variation could be usefully employed in MRI, for example for attenuating signals from surface structures. Here we show that a Halbach array can be formed by exposing appropriately oriented strips of material with anisotropic magnetic susceptibility to a strong static field, and also validate the predicted behaviour in experiments carried at 3T using a 40-element structure formed from pieces of pyrolytic graphite sheet.

### Introduction

The flat Halbach array1, which consists of a planar array of permanent magnets of thickness, $t$ , in which the magnetization in the x-y plane varies as $\textbf M=m\left(\textbf i \cos cx+\textbf k \sin cx \right)$ (Figure 1a), generates a spatially varying magnetic field $$\textbf B = \mu_{0} mct\left(\textbf i\cos cx+\textbf k \sin cx \right)e^{cx} \: \: \: Eq. 1$$ on one side of the array,where z<-t/2, and zero field on the other, where z>t/2. This interesting behaviour has led to a diverse range of applications of planar Halbach arrays which include uses in fridge magnets, free electron lasers and magnetic levitation.

Planar arrangements that produce a one-sided local magnetic field variation are also of potential value in MRI where they could for example be used for selectively eliminating signals2,3 from surface structures (e.g. subcutaneous fat). Here we show that a Halbach array can be formed by exposing appropriately oriented strips of material with anisotropic magnetic susceptibility to a strong static field, and also validate the predicted behaviour in experiments carried at 3T using a 40-element structure formed from pieces of pyrolytic graphite sheet (PGS).

### Theory

A planar Halbach array can be formed by arranging long, thin pieces of material with anisotropic magnetic susceptibility, characterised by susceptibility tensor $$\begin{bmatrix}\ \chi_{a}& 0 & 0 \\0 & -\frac{\chi_{a}}{2} & 0\\0 & 0 & -\frac{\chi_{a}}{2}\ \end{bmatrix}$$

in a static magnetic field ,so that the angle, $\theta$ , that the principal axis of the susceptibility tensor makes with the field varies linearly with $x$-position (Figure 1b) . When a magnetic field, $B_{0}$, is applied along the $x$-direction and $\theta = c x/2$, the magnetization of the array follows Eq. 1 with $$m=\frac{3 \chi_{a}B_{0} }{2 \mu_{0}}\: \: \: Eq. 2$$thus generating a one-sided field perturbation of the form shown in Eq. 1. Similar behaviour obtains if the magnetic field is directed along $z$..

### Methods

A planar array comprising 40 rotatable Perspex elements (each 4 x120x1 mm3 in size) was constructed (Fig.2), with one of the 4 mm faces of each element covered with a 70μm-thick layer of pyrolytic graphite sheet (EYGS121807–Panasonic), which has a highly anisotropic magnetic susceptibility. The array was immersed in a 27x19x9 cm3, saline-filled container and imaged at 3T using a 32 channel head Rx array. Field maps were acquired using a dual-GE sequence (1 mm isotropic resolution, TE = 4.8 ms, $\Delta$TE = 2 ms), along with coarser resolution (2 mm) gradient echo images. Data were acquired with the spatial period of element rotation by angle π set equal to (16, 32 and 64 mm), giving c-values of 393, 196 and 98 m-1, and also with all the elements parallel with one another (infinite period). Most images were acquired with the array lying in a coronal plane with the element orientation changing along the $B_{0}$-field direction, but data were also acquired with the array lying in an axial plane. Image data were processed in Matlab. Linear fits to the field maps were subtracted to remove the effect of large length-scale field variation.

### Results and Discussion

Figure 3 shows maps of the field variation in coronal planes located 6mm above (and below) the centre of the array for the different array modulations. A sinusoidally varying field following the array modulation period is evident below the array, but not above. Figure 4 shows that the amplitude of field modulation decreases exponentially with distance below the array in agreement with Eq. 1 with the rate of decay varying inversely with the period of modulation. Fitting the measured data to Eqs. 1 and 2 gave $\chi_{a}$≈175ppm which is consistent with previous measurements of PGS anisotropy4. Additional measurements indicated that field modulation was produced above the array and zero field below when it was turned through 180o –about the L-R axis, so that the sense of rotation of magnetization was reversed and we also found that a one-sided spatially varying field was also produced when it lay in an axial plane. Figure 5 shows axial magnitude images acquired with short (5 ms) and long (50 ms) TE with the array flipped in the coronal plane. It shows that the signal is more significantly attenuated above the array as a result of the greater signal dephasing due to field inhomogeneity. In general the magnitude of the gradient of the field produced by the array structure scales as $\frac{3 }{2}\chi_{a}B_{0} c^2e^{cx}$ so that a > 1 mTm-1 gradient was generated at 3T a distance of 5 mm from the array using less than 8 mg of PGS per cm2.

None

### References

1. Mallinson. JC. One-sided fluxes - magnetic curiosity. IEEE Transactions on Magnetics 1973;MAG9:678-82.

2. Boer VO, van de Lindt T, Luijten PR, Klomp DWJ. Lipid suppression for brain MRI and MRSI by means of a dedicated crusher coil. Magnetic Resonance in Medicine 2015;73:2062-8.

3. Jehenson P, Bloch G. Elimination of surface signals by a surface-spoiling magnetic-field gradient - theoretical optimization and application to human invivo NMR-spectroscopy. Journal of Magnetic Resonance 1991;94:59-72.

4. Wilson JL, Jenkinson M, Jezzard P. Optimization of static field homogeneity in human brain using diamagnetic passive shims. Magnetic Resonance in Medicine 2002;48:906-14.

### Figures

Figure 1 a) Schematic diagram of a Halbach array showing the variation of the magnetization vector orientation with position. This arrangement generates a sinusoidal field variation below the array and zero field above. b) Realisation of the Halbach array using pyrolytic graphite elements with anisotropic magnetic susceptibility.

Figure 2 40 element array structure constructed from rotatable 4 x 120 x 1 mm3 perspex pieces, with one of the 4mm width faces of each piece covered with a 70 μm thick layer of pyrolytic graphite sheet. Array modulation period is set to 32 mm in this picture.

Figure 3 Coronal maps (FOV =120x160mm2) of the frequency variation in Hz (all maps equally scaled) 6 mm below the PGS array set with (a) 64mm (c) 32mm (d) 16mm (e) infinite period of modulation and (b) 6 mm above the array with 64mm modulation period.

Figure 4 Variation of the relative amplitude of sinusoidal field modulation with distance from the array (over the range 4 - 9 mm). The rate of decay scales inversely with the modulation period as predicted by Eq. 1.

Figure 5 Axial gradient echo images (2mm isotropic resolution) acquired with (a) TE=5 ms; (b) TE=50 ms, with the array set to a modulation period of 32mm and oriented with zero-field below. The greater attenuation of the signal above the array due to greater dephasing is evident in (b).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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