The orthogonal shim coil for 3-Tesla brain imaging

Abstract

Purpose

We designed and implemented an orthogonal shim array consisting of shim coils with their planes perpendicular to the planes of neighboring radio-frequency (RF) coils. This shim coil improves the magnetic field homogeneity with the minimal interference to RF coils.

Methods

Using realistic off resonance maps of the human brain, we first evaluated the performance of shim coils in different orientations. Based on simulations, we developed a 7-channel orthogonal shim array, whose coil plan was perpendicular to neighboring RF coils, at the forehead. A programmable open-source current driver supplied shim currents.

Results

The 7-channel orthogonal shim array caused only marginal SNR loss to the integrated 32-channel RF-shim array. The 7-channel orthogonal shim array itself improved the magnetic field homogeneity by 30% in slice-optimized shimming, comparable to the baseline shimming offered by the scanner’s second order spherical harmonic shimming.

Conclusion

Orthogonal shim coils can improve the field homogeneity while maintaining good image SNR.

Introduction

Among different MRI scanning protocols, the spatial homogeneity of the main magnetic field, B₀, is always required to generate high quality measurements. An inhomogeneous B₀ field (ΔB₀) can cause several types of artifacts (1), such as geometrical image distortions, signal voids, warping artifacts, incomplete fat suppression or other issues related to frequency optimized excitation, magnetization refocusing, or signal reception.

Modern MRI magnets are shimmed to a high tolerance (<1ppm), so the majority of magnetic field inhomogeneity inside the body is caused by anatomy and physiology (2). Time-invariant inhomogeneities can be found at the boundary between tissues with distinct susceptibilities. For example, inferior temporal lobes and orbitofrontal cortex are typically susceptible to off-resonance artifacts, because they are in the proximity of the nasal cavity and auditory passages(3). These artifacts are more serious in higher field strength data acquisitions with low bandwidth and long readout time.

One approach to improve the B₀ homogeneity is to generate a compensating magnetic field to counteract the off-resonance. In practice, this has been done by spherical harmonic (SH) shimming (4), which generates a magnetic field with the spatial distribution described by spherical harmonic functions up to the second or the third order. While helpful, the 0^th-2^nd order shims available on commercial MRI scanners cannot effectively shim localized field inhomogeneity (5,6). Different from global shimming, localized shimming using an array of multiple small shim coils (7,8) around the imaging object has been demonstrated as a promising approach to reduce B₀ inhomogeneity in the prefrontal cortex and the temporal lobes (9,10).

To generate a sufficiently large shim field with as small current as possible, it is advantageous to place shim coils close to the imaging object. At the same time, RF coils are typically placed around the imaging object in order to maximize the sensitivity (10). The spatial proximity between shim and RF coils can cause undesirable interactions which degrade MRI quality. RF and shim coils should have minimal coupling between each other in order to ensure the optimal function of both coil sets. To address the challenge of competing for space, integrated RF and shim coils have been recently proposed (11,12). Prototypes of the integrated RF-shim coil have been demonstrated at 3T (13–16) and 7T (17). Specifically, inductive chokes were used to bridge tuning capacitors of an RF coil to create shim current loops. Thus, RF and shim coils share the same conductor for RF signal detection and shimming, respectively. Preliminary results show that an integrated RF-shim coil array can be used for parallel MRI acquisition and localized B₀ shimming (13,14).

However, in some realizations, RF-shim coil integration can cause non-negligible interference to RF coils and consequent image quality degradation. This coupling has been previously reported (11) to account for 10%-15% SNR loss (13,14) at 3T. Besides the SNR loss, the design of an integrated RF-shim coil has another disadvantage: the wire pattern for the shim currents is pre-determined by the RF coil geometry that is optimal for MRI signal reception. This constraint can limit the shimming performance, because the locations for RF coils may not be the best places for shim coils to create the desired shim field distribution. With sub-optimal coil locations, RF-shim coils may require stronger currents to correct a target off-resonance distribution.

Here we purpose a new design of shim coil array, where the shim current is arranged to flow on a plane perpendicular to the plane of neighboring RF coils. The orthogonal arrangement minimizes the coupling between RF and shim coils, while multiple shim coils in the array provides higher degree of freedom in generating a shim field. The reduced coupling between RF and shim coils allows for high SNR afforded by RF coils and homogeneous magnetic field brought by multiple shim coils at the same time. We used both simulations and in vivo experiments to demonstrate the feasibility of this proof-of-concept multi-coil shim array design.

Methods

Simulation

Off-resonance field maps were measured from seven healthy participants. Each participant provided written informed consents approved by the Institutional Review Board of National Taiwan University Hospital before they joined the experiment. We took 65 slices of dual-echo gradient-echo images (2×2×2 mm³ voxel; TR = 10ms; TE₁ = 2.00 ms; TE₂ = 4.46 ms; flip angle = 15°) on a 3T scanner (Skyra, Siemens, Erlangen, Germany) after applying the system’s second-order spherical harmonic shimming. An off-resonance field map, ΔB₀ (x, y, z), was calculated by measuring the phase accrued between two echo times at each image voxel indexed by x, y, and z (18). The off-resonance field map was first processed by unwrapping the phase at each image voxel and then registered to a selected target participant using PRELUDE and FLIRT (19,20) in the FMRIB Software Library (FSL) package(21).

The magnetic field from each shim coil was calculated by the Biot-Savart’s law. We simulated the shimming from seven different shim array geometries: 1) “32-channel orient 1” array: a 32-channel shim array with circular shim coils covering the head surface and arranged in a “soccer ball” pattern. This shim array has been previously implemented as an integrated RF-shim array (13). 2) “32-channel orient 2” array: a 32-channel shim array with rectangular shim coils, each of which has the shim coil center coincident with the center of each circular shim coil in the “32-channal orient 1” array in the “32-channel orient 1” array. This array was meant to investigate the difference in shimming when a shim coil was placed either at the local tangential or perpendicular plane of the array surface. 3) “38-channel face loop orient 1” array”: a 38-channel shim array, which consisted of 32 circular shim coils in the “32-channel orient 1” array and additional 6 “face loops”, which were circular shim coils distributed around face, adding more shim coils to shim the prefrontal cortex region. This array was meant to study the effect of additional face loops. 4) “39-channel orthogonal” array: a 39-channel shim array, which consisted of 32 circular shim coils in the “32-channel orient 1” array and additional 7 rectangular shim coils, which were located at the “brim” of the “32-channel orient 1” shim array. This array was meant to evaluate the shimming brought by 7 rectangular shim coils. 5) “64-channel orient 1+2” shim array”: a 64-channel shim array, which consisted of 32 circular shim coils like those in the “32-channel orient 1” array with additional 32 rectangular shim coils like those in the “32-channel orient 2” array. This array was meant to understand the shimming performance of combined “32-channel orient 1” array and “32-channel orient 2” array. 6) “64-channel orient 1” array: a 64-channel shim array, which consisted of 64 circular shim coils distributed over the head surface. We were interested in the shimming difference between this array and “32-channel orient 1” array as they differed in the number of shim coils. 7) “7-channel orthogonal” array: this array was the 7-channel rectangular shim coils at the “brim” of the “39-channel orthogonal” array. Note that the normal directions of these seven coils were in parallel with the B₀ field direction. We use this array to study what shimming can be achieved by this array without modifying the RF-receiver array over the head. Note that “39-channel orthogonal” array is the combination of “32-channel orient 1” array and “7-channel orthogonal” array. These shim array geometries were shown in Figure 2.

Figure 2.

Shim array geometries and specifications. A magenta circle indicates an integrated RF-shim coil. A blue circular and rectangular coil indicates an orientation 2 shim coil. A gray circle indicates an RF receive-only coil. Shim coil diameter is defined as the diameter of a circular coil and the length of a square coil. Three array geometries inside a dot-frame were further tested experimentally.

The standard deviation of the off-resonance was quantified by σB₀,

$$\sigma B_0={\Sigma }_{\mathit{voxel}}\sqrt{\left|B_0\right(\stackrel{\rightharpoonup }{r})-\stackrel{\_}{B_0\left(\stackrel{\rightharpoonup }{r}\right)}{|}^2},$$ [1]

where $\stackrel{\rightharpoonup }{r}$ is a location vector. $B_0\left(\stackrel{\rightharpoonup }{r}\right)$ represented the magnetic field strength at location $\stackrel{\rightharpoonup }{r}$. $\stackrel{\_}{B_0\left(\stackrel{\rightharpoonup }{r}\right)}$ represented the average of $B_0\left(\stackrel{\rightharpoonup }{r}\right)$ over the slice for slice-optimized shimming or the volume in global shimming.

Shim currents were estimated by the MATLAB function ‘quadprog’ (Mathworks, Natick, MA) (22) to minimize a cost, which was defined as the sum of the residual magnetic field:

$$\mathrm{Cost}={\Sigma }_{\mathit{voxel}}\left|B_0\right(\stackrel{\rightharpoonup }{r})-B_0^{shim}(\stackrel{\rightharpoonup }{r}){|}^2,$$ [2]

where $B_0^{shim}\left(\stackrel{\rightharpoonup }{r}\right)$ denotes the magnetic field created by shim coils. To avoid ill-condition solutions in slice-optimized shimming, we used a “Tikhonov” regularization implemented in the objective function to limit the current amplitudes. Specifically, the regularization term was the sum of the square of the current across all shim channels, and the regularization parameter was heuristically set to 100 for purposes of limiting the shim current changes between adjacent slices (caused by noise in field maps) and overall current amplitudes without significant reduction in shim performance.

In global shimming, the cost was evaluated over all voxels in the brain. In slice-optimized shimming, the cost was evaluated at chosen slices. Specifically, global shimming was calculated over 65 slices and slice-optimized shimming was evaluated within a 6-mm slab centered at the slice-of-interest. During the optimization of different shim array geometries, we constrained the strength of the shim current. Specifically, the maximal shim current on each coil was limited to 3 A and the maximal total shim current across all shim coils was limited to 50 A for potential heating during the experiment. Choosing 3A as the maximum current was based on our previous experience with heating on both coil and current driver amplifier during fast updating in slice-optimized shimming We did not constrain the maximal strength of the current on spherical harmonic shimming.

Note that the magnetic field was differently shimmed in global and slice-optimized shimming in multi-coil shim scenarios. Global shimming was initially shimmed by the 2^nd-order SH shimming, while slice-optimized shimming was initially shimmed by the 1^st-order SH basis sets. The reason for different initial settings between global and slice-optimized shimming was based the feasible performance of most clinical scanners in the world. While there has been exciting prototype 2^nd-order slice-optimized shimming (23–25), today most of clinical scanners hardware can only perform the 2^nd-order global shimming and the 1^st-order slice-optimized shimming. Therefore, the initial setting in simulations in this study for global and slice-optimized shimming was set for the 2^nd-order and 1^st-order spherical harmonic shimming, respectively (26). For comparison, the field was also shimmed without using SH shimming, in order to understand the limit of multi-coil shimming alone. Also, we simulated the SH shimming up to 6^th-order. An important difference between scanner’s 2^nd-order SH shimming and our simulated shimming was that our simulated SH shimming was restricted to a volume-of-interest within the brain.

2.2. Coil construction

We implemented “7-channel orthogonal” array, which was also added to a “32-channel RF-Shim” array to make a “39-channel orthogonal” array. Seven rectangular shim coils at the “brim” of the 32-channel RF-shim array were built (13). These 7 rectangular shim coils were built on a plastic holder (acrylonitrile butadiene styrene) created by a 3D printer (Dimension SST 1200es, Dimension, Inc., Eden Prairie, MN). The 3D printed shim coil substrates were encircled by 4 turns of AWG22 copper wire, connected to chokes to isolate RF signal. The chokes were self-shielding toroidal inductors (32-turn, AWG22, 16 mm O.D.; 9 mm I.D.) with a self-resonance at approximately 170 MHz. The RF impedance was higher than 3 kΩ at the Larmor frequency (123.25 MHz).

Shim currents were supplied by a digitally-programmable, open-source current driver(27,28), which was connected to each shim coil via a twisted pair AWG18 copper wire. Each shim coil had its own returning current path. High-impedance blocking elements were built to prevent the wires from picking up transmitted RF signals.

2.3. Experiment

Single coil SNR maps were measured by a GRE sequence (TR = 462 ms; TE = 10 ms; flip angle = 25°, FOV: 196 × 196 mm, image matrix: 280×280, slice thickness = 5 mm;) on a 3T scanner, using spherical phantom. The coil noise map were measured by using the same sequence with the flip angle set to 0 degree. Orthogonal shim coils with different turns have been added to the same receiver coil, without re-tuned during the modification.

SNR maps (29) for “32-channel orient 1” array and “39-channel orthogonal” array were acquired using a head phantom using a gradient echo sequence (TR = 30 ms, TE = 6 ms, flip angle = 15°, image matrix: 226 × 226, FOV: 384 × 384mm, slice thickness = 7mm). The adjustment voltages were recorded manually before and after converting the “32-channel orient 1” array to the “39-channel orthogonal” array by adding a “7-channel orthogonal” array to evaluate the interference caused by the “7-channel orthogonal” array. Further, S11 reflection of loaded eye receiver loops on “32-channel orient 1” array have been measured before and after adding a “7-channel orthogonal” array.

The calculation of shim currents requires the knowledge of the fields produced by individual shim coils. Therefore, the magnetic field generated by each shim coil in either “32-channel orient 1” or “7-channel orthogonal” array was measured by supplying 700 mA to one shim coil at one time. A basis field map for each shim coil was measured using a water-filled balloon phantom (diameter = 20 cm) with a dual-echo gradient-echo sequence (transverse slices, TE₁ = 5ms, TE₂ = 7.46 ms, TR = 630 ms, flip angle = 50°, image matrix: 110×110×40, FOV: 220×220×80 mm, slice thickness = 2 mm with 100% slice gaps, TA = 90 s). The background magnetic field inhomogeneity in the balloon phantom (without supplying any shim current to the shim array) was also first measured and then subtracted from the magnetic field map measured for each shim coil. Totally 39 basis maps were measured (32 for “32-channel orient 1” array and 7 for “7-channel orthogonal array).

Two healthy volunteers were recruited to test the performance of the “39-channel orthogonal” shim array and the “7-channel orthogonal” shim array. Both participants provided written informed consent before being scanned and the study was conducted in accordance with the Massachusetts General Hospital Institutional Review Board. Scans were performed by using the scanner body coil for RF transmission and the “32-channel orient 1” array for signal reception. The shimming performance was evaluated by calculating the standard deviation of the ΔB₀ field map (σB₀). To measure the ΔB₀ field map, we used a dual-echo gradient-echo sequence after shimming the magnetic field by scanner’s 2^nd-order SH shimming. The magnitude images and field maps were then preprocessed by Brain Extraction (30) and Prelude in FSL to create a 3D brain mask and phase-unwrapped off-resonance field maps, respectively. The shim current was calculated by the ΔB₀ field map and 39 basis field maps (see the paragraph above) with the constraint that the shim current on each shim coil did not exceed 2 A and that the total shim current across all shim coils did not exceed 35 A in both global shimming and slice-optimized shimming. After shimming, a residual field was measured again using the dual-echo gradient-echo sequence.

The performance of the “7-channel orthogonal” array and “39-channel orthogonal” array was evaluated using a single-shot EPI sequence (TE = 35ms, TR = 4500 ms, FA = 90°, matrix: 110×110×40, FOV: 220×220 mm, slice thickness = 2 mm, BW =1820 Hz/pixel, 7/8 phase partial Fourier, echo spacing: 0.76ms). This sequence had a long readout trains that were sensitive to B₀ inhomogeneity. EPI had both “blip-up” (from negative k_y to positive k_y with a negative k_y pre-phasing) and “blip-down” (from positive k_y to negative k_y with a positive k_y pre-phasing) phase encoding directions. Ideally without any off-resonance, images should completely overlap between these two versions of EPI. We compared the overlap between the two differently distorted images with and without shimming. The slice-optimized shimming scenario was calculated to reduce the off-resonance in an 8-mm slab of interest.

Result

We used simulations to compare shimming across seven different multi-coil shim arrays and SH shimming up to the sixth order. Overall, we found that slice-optimized shimming was better than global shimming. And the residual off-resonance field decreased as the number of shim coils or the order of SH shimming increased, due to a higher degree of freedom in the shim current design.

Figure 3 show averages and standard deviations of σB₀ across 7 participants in various shimming simulations. Compared to the scanner’s 2^nd-order SH shimming over the whole head (σB₀ = 27.9 Hz), restricting the volume-of-interest of the 2^nd-order SH shimming within the brain reduced σB₀ to 19.2 Hz. Higher order SH shimming improved σB0: with the 6^th-order SH shimming, σB₀ was 11.5 Hz. Compared with “32-channel orient 1” array (σB₀ = 15.8Hz), “32-channel orient 2” array slightly improved the field homogeneity (σB₀ = 15.2 Hz) in global shimming. This σB₀ improvement in comparison to “32-channel orient 1” array was statistically significant (p = 0.01). The “64-channel orient 1” array had significantly smaller σB₀ than “64-channel orient 1+2” array (p = 0.01). The σB₀ for 7-channel orthogonal coil (14.9 Hz) was found similar to the performance of 4^th-order (16.2 Hz) SH shimming.

Figure 3.

Average and standard deviation of σB₀ across 7 participants after various shimming approaches. For multi-coil shimming scenarios, the comparison between with and without SH shim basis have shown aside. The array geometries inside dot-frames were further tested experimentally.

The “32-channel orient 2” array in slice-optimized shimming (σB₀ = 9.3 Hz) was worse than the “32-channel orient 1” array (σB₀ = 7.3 Hz). Slice-optimized shimming using either “64-channel orient 1” array (σB₀ = 6.8 Hz) or “64-channel orient 1+2” (σB₀ = 6.8 Hz) array was comparable. In slice-optimized shimming, σB₀ in the “7-channel orthogonal” array was 12.2Hz, which was comparable to the 3^rd-order SH shimming (11.5 Hz).

Multi-coil shimming using 32-channel orient 1 and orient 2 achieved similar shimming performance as the 4^th-order SH shimming in both global and slice-optimized shimming. Adding 7 channels of orthogonal coils (“39-channel orthogonal” array) had the shimming performance comparable with 5^th-order SH shimming in both global and slice-optimized shimming.

For multi-coil shimming scenarios, the global shimming was initially shimmed by the 2^nd-order SH basis sets, while slice-optimized shimming was initially shimmed by the 1^st-order SH basis sets. Compared with the field shimmed without SH shimming basis sets, adding additional SH Shimming basis set further improve the global and slice-optimized shimming performance for “32-channel orient 1” array, “32-channel orient 2” array and “7-channel orthogonal” array,

Figure 4 shows the residual off-resonance field maps at three representative slices showing prominent B₀ inhomogeneity in the frontal or temporal lobes from a representative participant. The spatial distributions of the residual off-resonance field shimmed by “32-channel orient 1” array and by “32-channel orient 2” array were similar, suggesting that the shim coil orientation had little effect in shimming. This result was also supported by the comparison between “64-channel orient 1” array and “64-channel orient 1+2” array. Adding 7 orthogonal shim coils to the “32-channel orient 1” array led to significant off-resonance correction at the frontal lobe: σB₀ was reduced from 15.1 Hz to 11.3 Hz.

Figure 4.

Simulated off-resonance magnetic field maps at two representative slices using two SH shim arrays and seven multi-coil shim arrays in global and slice-optimized shimming. σB₀ for different shim arrays were also reported in yellow numbers. The array geometries marked with yellow were further tested experimentally.

The residual off-resonance field maps after slice-optimized shimming was more homogeneous than that after global shimming using either MC arrays. Comparing to the scanner’s 2^nd order SH shimming, slice-optimized shimming using the 7-channel orthogonal array with 1^st-order SH shimming improved the B₀ homogeneity, especially in the middle and upper slices covering the frontal cortex.

Figure 5 shows single coil SNR measurements before and after converting the RF coil into a RF-shim coil. The orthogonal shim coil design shows a marginal interaction with the receiver coil. The SNR profiles of 1 turn and 2 turns orthogonal shim coil design show marginal SNR loss compared with RF only scenarios, 5% and 7% SNR loss alone the SNR profiles for 1 turn and 2 turns orthogonal shim coil design respectively.

Figure 5.

(a) SNR experiment coil placement. (b) SNR profiles through the dotted white lines of three SNR maps in (c).

Figures 6a and 6b show the implemented “32-channel orient 1 array” and “7-channel orthogonal” array, respectively. “7-channel orthogonal” array had marginal interaction with the receiver coil array: There was minor change in the unloaded/loaded Q-ratios (5.7 vs. 5.6) and tuning frequencies (123.25 MHz vs. 123.26 MHz) at the RF coil closest to the “7-channel orthogonal” shim array. The SNR maps of “32-channel orient 1” array were similar to those of “39-channel orthogonal” array, suggesting that adding 7 orthogonal shim coils caused negligible SNR reduction. Quantitatively, we found 3.6%, 2.0%, and 0.8% SNR loss in an axial, mid-sagittal, and a coronal slice, after adding the 7 orthogonal shim coils, respectively.

Figure 6.

(a) Converting “32-channel orient 1” array to “39-channel orthogonal” array by adding “7-channel orthogonal” array, which was placed around the forehead as the brim of a baseball cap. (b) “7-channel orthogonal“ array. (c) SNR maps at three representative slices of the “32-channel orient 1“ array (top) and “39-channel orthogonal” array (bottom).

Figure 7 shows the S11 reflection of loaded eye loops on “32-channel orient 1” array, which are located right under the “7-channel orthogonal” array. Adding “7-channel orthogonal ” array slightly changed the impedance of the closest eye loops.

Figure 7.

S11 reflection of loaded eye loops on “32-channel orient 1” array before and after adding “7-channel orthogonal” shim array.

Figure 8 shows empirical residual magnetic field maps in slice-optimized shimming. Three representative slices from two participants were shown to evaluate the shimming performance. Prominent B₀ inhomogeneity in the frontal or temporal lobes persisted after scanner’s 2^nd-order SH shimming in global shimming. Comparing to scanner’s 2^nd order SH shimming, shimming using “7-channel orthogonal” array provided 35% improvement in σB₀ (from 22.5 Hz to 14.5 Hz). The residual off-resonance field maps showed clear improvement especially in the frontal cortex in middle and upper slices. Compared with the “32-channel orient 1” array, “39-channel orthogonal” array reduced σB₀ from 8.6 Hz to 7.2 Hz and from 8.9 Hz to 8.5 Hz for participant 1 and 2, respectively. The limited improvement might be because “32-channel orient 1” array already corrected much off-resonance in slice-optimized shimming.

Figure 8.

Empirical ΔB₀ field maps in slice-optimized shimming using “7-channel orthogonal” array, “32-channel orient 1” array, and “39-channel orthogonal” array. Three representative axial slices were shown from 2 participants. The σB₀ within the representative slice or across the whole volume were reported in white and yellow numbers, respectively.

Figure 9 shows the reduction in EPI geometric distortion after slice-optimized shimming. Single-shot EPI sequences exhibited strong geometric distortions in the middle and lower brain slices, where B₀ inhomogeneity was severe. Compared with the scanner’s 2^nd order SH shimming, using “7-channel orthogonal” array itself reduced the distortion at the middle and upper slices of the brain. Adding “7-channel orthogonal” array to “32-channel orient 1” array further reduced the distortion in middle and upper slices and recovered slightly more signal at the frontal lobe.

Figure 9.

Geometric distortions in EPI in two axial slices using different shimming. EPI had less distortion after shimming. Red contours are estimated brain parenchyma boundaries from gradient-echo images. Red arrows indicated improvement of distortions in the frontal cortex region.

Discussion

In this study, we proposed to place the local shim coil such that its plane is perpendicular to the plane of neighboring RF receiver coils in order to minimize the coupling and SNR degradation. The empirical measurements show that the 7-channel orthogonal array has around 35% improvement for the B₀ homogeneity in a slice-optimized shimming basis (Figure 8; σB₀ 22.5 Hz → 14.5 Hz). The 39-channel orthogonal array improved the field homogeneity by 68% (Figure 8; σB₀ 22.5 Hz → 7.2Hz). Comparing field maps before (“32-channel orient 1” array) and after adding 7 rectangular shim coils (“39-channel orthogonal” array), the field homogeneity can be improved by 15% (Figure 8; σB₀ 8.6 Hz → 7.2 Hz), in slice-optimized shimming. Compared to “32-channel orient 1” array, “39-channel orthogonal” array had marginal SNR degradation (between 0.3% and 3.5%; Figure 9). Taken together, our results suggested that the 7-channel orthogonal array can provide efficient shimming on a 3T MRI system with marginal SNR compromise. We acknowledge that the shimming performance gains are not as compelling as those provided by dedicated shim coils (6,10,25). However, the present work is meant only as a proof of concept for the orthogonal shim coil approach. We believe different, optimized coil geometries could provide larger gains, but we reserve such optimization for future work.

The difference between field homogeneity improvement in slice-optimized shimming experiment (Figure 8) and simulations (Figure 3) may be due to 1) the discrepancy between the designed and implemented geometry of rectangular shim coils, 2) the difference in the off-resonance maps (from an arbitrarily chosen participant in simulations), and 3) the relative positions between the head and the shim array.

In our simulation, the global shimming was initially shimmed by the 2^nd-order SH basis sets, while slice-optimized shimming was initially shimmed by the 1^st-order SH basis sets (Figure 3). The different baseline conditions in global and slice-optimized shimming was motivated by the feasibility study (26). We did not constraint the maximal strength of the shim current was imposed in the simulation, because the purpose of spherical harmonic shimming in this study was meant to provide an ideal scenario to compare the performance of our orthogonal shim coil.

In our work, we demonstrated that shim coils in orientation 2 had minimal RF interference while achieving similar shimming performance like that provided by shim coils in orientation 1. Shim coils in orientation 2 can be advantageous, because they will not be constrained to be at RF coil locations. This corresponds to the degree of freedom in optimizing shim coil shapes and locations in off resonance reduction. Note that shim coils in orientation 1 typically share current paths with RF coils (13,14). This current path sharing limits the degree of freedom in shim current design. With minimal interactions with RF coils, separating shim current paths from RF current paths allows more versatile shim current and thus field distributions. Simulations suggested (Figure 3) that a combination of shim coils in both orientation 1 and 2 achieve better shimming and minimal interaction to RF coils. The “38-channel face loop orient 1” array had similar shimming performance to the “39-channel orthogonal” array (Figure 3). But the orthogonal coils can be more comfortable to patients, because coils are not tightly covering the face.

Potential disadvantages for shim coils in orientation 2 are: 1) shim coils in orientation 2 occupy larger space in the bore and are not suitable when the room in the bore is limited. 2) Higher shim current is required to generate the desired shim field. (Supplementary Figure S1) However, this issue may be addressed by building shim coils in orientation 2 coil with multiple turns and a closer placement to the RF array helmet. 3) Shim coils in orientation 2 need more complex mechanical housing to endure the torques on shim coils inside MRI. The Lorentz force induced by the rapidly switching of the gradient system caused strong vibration on coils during EPI acquisitions. Hence, a well-design mechanical housing is necessary for orthogonal shim coils.

In this study, one challenge is the lack of a gold standard to evaluate the shimming performance. Therefore, we can only report relative performance between different shimming approaches. In Supplementary Table 1, we compiled the performance of different orders of SH shimming and multi-coil shimming from different studies. The 2nd-order static SH shimming at 3T (19.2 Hz; 44.8 Hz as converted to 7T) reported in this study is higher than the values reported in some of previous studies (ranging between 29.6 Hz and 43.8 Hz; Supplementary Table 1)(6,10,17,23). However, the simulated 6th order static SH shimming (11.5 Hz, 26.8 Hz as converted to 7T) was comparable to reported measurement for 4th-order at 7T (ranging between 22.4 Hz and 32.6 Hz)(6,10,17,23). We hypothesize that discrepancies in reported values may be related to errors or noise in the field map estimation, differences in shim volume and brain masking, and inter-subject variability. In these comparisons, there were different factors may bias the results, including brain masking and accuracy of off-resonance field estimates. Specifically, reducing the size of the mask from the whole head to the brain-only can reduce the σB₀ in the 2^nd-order shim by as much as 15-20%. The other potential source of error is the estimation of the field map. We note that the scanner’s approach and our approach were different in the calibration procedure: We used a point-by-point B₀ field map for calibration, while the scanner appears to use a different procedure. Potentially the same shim volume should be used for shim performance comparisons, and attention to bias from scanner eddy currents should also be studied. Consensus on these factors can establish a gold standard needed for B₀ shim simulations and experimental acquisitions across sites and different scanner vendors.

The power supply in this study has a total current limitation at 60 A, so we could safely use 50 A without getting too close to the limit. For the experiment we built in an extra safety factor and used lower current limits because raising the total current limit from 35 A to 50 A showed very little change in shim performance (likewise for raising the channel-wise limit from 2 A to 3 A). The simulated global shimming performance for seven in vivo B₀ field maps (shimmed up to the 2nd order) for 7ch orthogonal, 32ch ortient 1, and 39ch orthogonal arrays were 14.9 Hz, 15. 8 Hz and 11.6 Hz under the 3 A per coil (50 A total) limit and 15.3 Hz, 15.9 Hz, and 11.7 Hz under the 2.5 A per coil (35 A total) limit, respectively. The simulated slice-optimized shimming for the same three arrays were 12.2 Hz, 7.3 Hz, and 6.7 Hz under the 3 A per coil (50 A total) limit and 12.7 Hz, 7.7 Hz, and 7.0 Hz under the 2.5 A per coil (35 A total) limit, respectively. The performance change due to the current constraint is less than 4% for global shimming and 3% for slice-optimized shimming. The two current limits have only very small differences in shim performance at 3T. For purposes of simplicity and robustness, this motivated us to chose the more conservative current limits for our initial proof-of-concept.

In this study, we evaluated the shimming performance of our orthogonal coil in global and slice-optimized shimming. However, multi-band imaging requires several slices to be shimmed at the same time, posing a different shim challenge from the two cases considered here. There have been several explorations of multi-coil and SH coil performance for multi-band acquisitions (17,31,32). We plan to explore how our shim coil array performs in multi-band imaging in the near future.

In summary, we presented the orthogonal shim coil design to reduce off-resonance. We demonstrated that the “7-channel orthogonal” array was able to reduced the B₀ inhomogeneity in the frontal cortex. In slice-optimized shimming experiment, the shim coil reduced image geometric distortions. It is anticipated variety of research and clinical MR applications, such as functional MRI, spectroscopy, and spectroscopic imaging, that sensitive to B₀ inhomogeneity can be benefited by orthogonal shim coils.

Figure 1.

(a) ΔB₀ field maps from 7 participants. Participant 5 (highlighted) was chosen as the target participant (as shown in Figure 4). (b) Left: the definition of coil orientations, with orientation 2 perpendicular to orientation 1. Right: the volume and slice definitions in global and slice-optimized shimming methods. The white arrow indicates the orientations of the field maps.