Role of very high order and degree B₀ shimming for spectroscopic imaging of the human brain at 7 Tesla

Abstract

With the advent of ultrahigh field systems (7T), significant improvements in spectroscopic imaging (SI) studies of the human brain have been anticipated. These gains are dependent upon the achievable B₀ homogeneity, both globally (σB₀^Global, over the entire ROI or slice) and locally (σB₀^Local, influencing the linewidth of individual SI voxels within the ROI). Typically the B₀ homogeneity is adjusted using shim coils with spatial distributions modeled on spherical harmonics which can be characterized by a degree (radial dependence) and order (azimuthal symmetry). However, the role of very high order and degree shimming (e.g. 3^rd and 4^th degree) in MRSI studies has been controversial. Measurements of σB₀^Global and σB₀^Local were determined from B₀ field maps of 64×64 resolution. In a 10mm thick slice taken through the region of the subcortical nuclei, we find that in comparison to 1^st−2^nd degree shims, use of 1^st−3^rd and 1^st−4^th degree shims reduces σB₀^Global by 29% and 55% respectively. Using a spectroscopic imaging voxel size of ~1cc with an estimate of σB₀^Local from 3×3×3 B₀ map pixels in this subcortical region, the number of pixels with σB₀^Local of less than 5Hz increased from 24% to 59% with 1^st−3^rd and 1^st−4^th over 1^st−2^nd degree shims respectively.

INTRODUCTION

Anticipated increases in SNR and sensitivity to B₀ dependent parameters for susceptibility weighted imaging and functional imaging have formed much of the driving force for the development of 3T and ultra-high field MR systems. Although at least linear improvements in SNR have been demonstrated at 3–7T (1–4), limitations due to increasing static field (B₀) inhomogeneity have also been noted (5–11). Smith and colleagues simulated and mapped the B₀ field inhomogeneity created by the presence of air-tissue interfaces in the human brain (12,13), and reported large shifts of several parts per million (ppm) over centimeter distances near the sinuses and auditory canals. Smaller but significant shifts can also be seen at the periphery of the brain adjacent to the skull (14). These susceptibility interfaces that effectively act as magnetic sources imply that the spherical harmonic solution to the resulting magnetic field will be non-orthogonal, i.e., that weighting coefficients will change depending on the number of shim terms used (15). As a result, the field shifts caused by tissues of differing magnetic susceptibility and their interfaces can commonly result in signal dropout and distortions in gradient echo, echo-planar and susceptibility weighted imaging. In spectroscopic imaging studies poor B₀ homogeneity can result in large numbers of unusable voxels (6,7,11). Nonetheless, with implementation of accurate field maps, our group has shown that even over large regions of interest, excellent shimming performance can be achieved (16–18). From these field maps, it has become clear that, although the required strengths for all shim terms decay as the distance away from the origin of the susceptibility shift increases, higher order and degree terms are seen throughout the brain. Unfortunately, most human 3T, 7T MR systems have typically been equipped with only 2^nd degree shims, such that corrections for susceptibility differences can be difficult with large residual inhomogeneity.

At 7T, the necessary hardware requirements in terms of shim strength, degree and order have been controversial. The arguments against the efficacy of higher shim terms (>2^nd degree) include: 1) the required degree/order and strengths for higher degree shim terms (≥3^rd degree) to achieve significant improvement in homogeneity over the large regions of interest (e.g. those selected for MRSI studies) are not achievable with resistive shim technology (19,20) and 2) even if sufficient strength were achievable, the effects of higher dependence terms across individual voxels will be minimal with small voxel sizes (e.g. ~1cc in MRSI studies, 22).

In this paper, we describe the 1) the strength requirements and improvements for 3^rd degree shimming of the human brain at 7T; 2) the shim strength requirements for 4^th degree shimming and their predicted improvements; 3) an implementation of a 4^th degree shim insert for the human brain and its performance; 4) the impact of 3^rd and 4^th degree shimming on spectroscopic imaging studies of the human brain at 7T and 5) the limitations and potential role of even higher degree shimming. The extent to which the achievable shim strength improves the overall homogeneity for imaging and spectroscopic imaging studies can be characterized by the global homogeneity, $\sigma {\text{B}}_0^{\text{Global}}$, the standard deviation of the B₀ field over the entire ROI. The scalar value of $\sigma {\text{B}}_0^{\text{Global}}$ is a significant factor in determining the performance of narrow band frequency selective pulses applied over the entire ROI, such as those generally used for water suppression or spectral editing in MRSI studies. However the extent to which higher order and degree shims improve the homogeneity of individual spectroscopic voxels is characterized by the local homogeneity, $\sigma {\text{B}}_0^{\text{Local}}$ (standard deviation of the B₀ field measured over an individual MRSI voxel). The value of $\sigma {\text{B}}_0^{\text{Local}}$ affects spectral resolution for any given voxel, and over the entire ROI, can significantly impact both the number of pixels and specific brain locations which provide usable spectra (6,7,11,21). We assess how $\sigma {\text{B}}_0^{\text{Global}}$ and $\sigma {\text{B}}_0^{\text{Local}}$ depend on higher order and degree shims in a variety of brain regions and demonstrate their relationship with spectral resolution in MRSI data at 7T. Finally, it is worth noting that the shim terminology “order” and “degree” have been used interchangeably in the literature. To clarify this issue we have included an appendix on the formal use of these terms. Briefly, consistent with the NIST Digital Library of Mathematical Functions, (http://dlmf.nist.gov/ Chapter 14), “degree” characterizes the rate at which the strength of the magnetic field changes with respect to distance from the origin along a polar radius changes and “order” characterizes the periodicity of the strength of magnetic field in the azimuthal direction.

METHODS

All data were acquired with an Agilent Direct Drive system and a head only (68cm ID) actively shielded 7T magnet. The gradient system (Agilent, ID/OD 42/68cm) included a full set of 2^nd and 3^rd degree shims, with each shim channel driven by 20A shim power supplies (Resonance Research Inc., Billerica MA). Based on the results of 1^st−3^rd degree shimming (a total of 15 shim channels), and the projected improvements for 4^th degree shimming (total 24 shim channels), a very high degree shim insert was designed, constructed and evaluated (Resonance Research Inc., Billerica MA). The insert consisted of eight 4^th degree terms (all but Z4), two 5^th degree (4^th order) terms ZC4 and ZS4, and four 5^th order shims C5, S5, ZC5 and ZS5. The inductances of this static shim insert vary between 120 to 350nH for each of the individual channels. Physically, the insert weighed 27.2kg with ID/OD of 36/41.5cm, length 44.5cm. All data were acquired using an 8 element transceiver array (16,17). Due to limitations in the maximum aggregate output of this shim insert’s power supply (18A), the four 5^th order shims were not used; otherwise each shim insert channel was driven by 5A.

We evaluated the role of shimming for human brain spectroscopic imaging studies, assessing slice selective (10mm thick) 2D MRSI studies in three different regions: the frontal lobes at the level of the supplementary motor area (SMA); the frontal-parietal lobes at the level of the subcortical nuclei (SCN); and the mid-temporal lobes at the level of the hippocampi (MTL). To evaluate the effects of 3^rd degree shimming and determine the required strengths for 4^th degree shims, N=20 volunteer studies from each region were assessed to provide group statistics. The projected improvements for 4^th degree shimming were then used to design a very high degree shim insert. The performance of this shim insert was then validated for slice selective MRSI studies (N=10 subjects) in the SMA and SCN regions. We also assessed the performance of the higher degree shimming over a 44mm thick slab centered at the top of the ventricles spanning the superior portion of the temporal lobe to superior frontal lobe locations including much of the motor cortex in N=10 subjects.

A non-iterative multi-slice B₀ mapping method was used to acquire B₀ maps, calculate necessary shim currents and set all currents (18). The B₀ maps were generated using 5 evolution periods, corresponding to additional delays of 0 (for reference), 1, 2, 4 and 8ms. The B₀ maps had 64×64 resolution, 192×192mm FOV, 11 slices with a thickness/gap of 2mm/2mm, such that the 10mm SI slice thickness was spanned by three B₀ map slices. The ROI for the shim optimization was determined using scout images using the three central B₀ map slices. The shim currents were optimized using a least squares minimization (18) calculated based on the defined degree and order of shim terms. For 3D studies, all 11 slices were used. With each B₀ map acquisition, the measured and predicted (i.e., after shim correction) $\sigma {\text{B}}_0^{\text{Global}}$ was calculated over the target ROI. For verification of the predicted map, a second B₀ map was acquired after shimming. The measured and predicted $\sigma {\text{B}}_0^{\text{Global}}$ typically varied by less than 1Hz for a single iteration of shimming. To examine the limits of higher degree and order shimming, the associated Legendre polynomials in spherical polar coordinates (up to 15^th degree, or 256 shim terms) were used. In this theoretical analysis, each B₀ slice was analyzed independently with no limitation on applied shim strength to allow the greatest flexibility in shim terms.

To determine the effect of shimming over a given SI pixel, the ROI used for shimming was divided into a moving boxcar array of ~1cc volumes consisting of 3×3×3 B₀ map pixels (9mm × 9mm × 10mm). The standard deviation of the B₀ field across each ~1cc sub-volume ( $\sigma {\text{B}}_0^{\text{Local}}$) was calculated (scaled by 2.35 in order to generate the full width half maximum linewidth equivalent). $\sigma {\text{B}}_0^{\text{Local}}$ histograms and maps were generated as a function of brain region, shim degree and analyzed in correlation with spectroscopic imaging data.

Spectroscopic imaging data were acquired using short (15ms) or moderate (40ms) TE sequences (Fig. 1a). Short TE acquisitions were used to visualize the effect of $\sigma {\text{B}}_0^{\text{Local}}$ on spectral resolution of different metabolite resonances, while moderate TE acquisitions were used to minimize spectral overlap and provide greater accuracy for quantitative measurements of metabolite linewidth. As previously described, localization was performed using an 8 element elliptical transceiver with two RF distributions (“homogeneous”, “ring”, both determined by RF shimming, ref. 17) and gradient based slice-selective pulses. Rectangular phase encoding (24×24) was performed over a FOV of 192×192mm² achieving a nominal voxel of 0.64cc. The TR was 1.5s resulting in an acquisition time of 14.4min. To provide phase and amplitude scaling for the reconstruction of the metabolite SI data, an unsuppressed water SI was acquired using the same sequence but without water suppression and TR=0.5s as previously described (16). For the purposes of visualization, metabolite data evaluating spectral quality were processed with a Gaussian filter in the time domain (4Hz FWHM equivalent in the spectral domain) and a spatial cosine filter (±π/3) in the spatial domain. Integration of the point spread function of this filter gave a spatial FWHM ±0.53cm with a 1.1cc sampling volume.

Figure 1.

(A) Pulse sequence used for MRSI acquisitions and (B,C) B1 maps of the two RF distributions used. RF pulses for slice selection, broad band semi-selective refocusing and the frequency selective inversion pulse for water suppression are delivered with the homogeneous RF distribution (B). Outer volume suppression uses the ring RF distribution (C) and a double inversion recovery method.

A quantitative assessment of the relationship between fitted linewidth σ_Fit and σB₀^Local was made using TE=40ms acquisitions (N=10 volunteers) from the MTL slice selecting a rectangular ROI spanning both hippocampi and shimmed using 1^st−3^rd degree shims. The MTL slice was chosen based on the substantial range in $\sigma {\text{B}}_0^{\text{Local}}$ seen over the relatively small ROI and its well characterized anatomy. Due to inter-individual anatomical differences, the ROI varied in size and shape yielding on average 27.2±4.4 pixels per subject. To minimize issues associated with spectral processing and overlap, these spectra were processed without any line broadening and the creatine resonance analyzed using a Voigt lineshape, i.e., a fixed Lorentz linewidth (2.92Hz, according to a T2 of 109msec, ref. 23,24) convolved with a variable Gaussian lineshape. The values for the fitted Gaussian lineshape, σ_Fit, was plotted against $\sigma {\text{B}}_0^{\text{Local}}$ to determine the correlation.

RESULTS

$\sigma {\text{B}}_0^{\text{Global}}$ and Higher Degree Shimming

Based on goals of achieving spectroscopic imaging over extended brain regions and the known sources of susceptibility in the brain, we selected ROIs for shimming based on the objective of maximizing homogeneity over large extents of single axial slices. Figure 2 shows data from three such slices, where the two superior slices (SMA, SCN) have ROIs that include virtually all of the brain within the slice. For the MTL slice, the ROI was chosen similar to that previously studied (18), excluding regions directly over the ear canals (extreme lateral temporal lobe) and the frontal sinus. This allows the B₀ optimization to target the hippocampi, which are critical sites in the pathophysiology of epilepsy and dementia.

Figure 2.

Left column: Scout images from three regions (top-I supplementary motor area SMA, middle-II subcortical nuclei SCN, bottom -III medial temporal lobe MTL) showing the ROIs (in red) used for shimming. Middle column: B₀ maps after 1^st+2^nd degree shimming. Right column: B₀ maps after 1^st − 3^rd degree shimming. For each map, the $\sigma {\text{B}}_0^{\text{Global}}$ is shown. For each region, the dynamic range of the color bar is shown on the right.

3^rd degree shims

Fig. 2 shows B₀ maps acquired from the three ROIs after adjustment of all 1^st − 2^nd degree and 1^st − 3^rd degree shims. As expected, the SMA (most superior slice) shows the best overall homogeneity. Although $\sigma {\text{B}}_0^{\text{Global}}$ for the MTL slice is smaller than the SCN slice, this is due to the smaller ROI used for the MTL slice. Addition of 3^rd degree shims to 1^st and 2^nd degree shims reduced $\sigma {\text{B}}_0^{\text{Global}}$ by 26, 29 and 37% in all three slices (Fig. 3). Fig. 2 also shows that after correction with 1^st − 3^rd degree terms, there is a residual strong 4^th order inhomogeneity (C4) in both the SMA and SCN slices. When including the 4^th degree terms in the analysis (without constraints on 4^th degree currents), we found that $\sigma {\text{B}}_0^{\text{Global}}$ significantly decreased further from 8.5, 16.7 and 10.5 Hz to 5.3, 10.5, and 8.1 Hz for the three slices (SMA 38%, SCN 37% and MTL 23% respectively). The more modest fractional improvement in the MTL slice is most likely due to the presence of very high B₀ gradients from the local anatomy (cavernous sinus, ear canals) which cannot be completely corrected for by use of 4^th degree terms. The Table lists our maximum shim strength, the mean, standard deviation and maximum absolute value (for any subject) of the shim strengths used for 2^nd, 3^rd and predicted for 4^th degree shims for each region. Notably none of the 2^nd and 3^rd degree values used exceeded more than 39% of the available shim strength in any subject in any of these regions, indicating that sufficient strength was achievable with the existing hardware configuration to support 3^rd degree shimming in these locations. As expected however, the 4^th degree values were significantly higher in the MTL in comparison to more superior locations, reflecting its locally high B₀ gradients.

Figure 3.

Bar plot showing the mean and standard deviation of $\sigma {\text{B}}_0^{\text{Global}}$ measured for 1^st, 1^st and 2^nd, 1^st − 3^rd, predicted for 1^st − 4^th degree shimming from the SMA, SCN and MTL for N=20 subjects for each region. Also shown is the measured performance with the 1^st − 4^th degree shim insert for N=10 subjects in the SMA and SCN slices (MTL region not included).

4^th degree shims

Based on the projected improvements for 4^th degree shimming and their required strengths, a 4^th degree shim insert was designed with eight 4^th degree shims (all except Z4), two 5^th degree (4^th order) shims ZC4 and ZS4 and four 5th order terms (C5, S5, ZC5, ZS5). As indicated in the Table, the shim insert achieves its target performance criteria for the majority of 4^th degree shims with a 5A power supply for each channel. We then evaluated the performance of the shim insert in N=10 subjects from the SMA and SCN regions. Due to limitations arising from the larger size of the transceiver array commonly needed for hippocampal studies, hippocampal studies were not performed with the shim insert. Fig. 3 displays the achieved homogeneity $\sigma {\text{B}}_0^{\text{Global}}$, which are consistent with the predicted values, of 31% and 38.5% in the SMA and SCN respectively. In this group, the average aggregate current applied for the eight 4^th degree and two 5^th degree 4^th order shims for the SMA and SCN was 3.9 and 7.1A respectively. We also evaluated the improvement in $\sigma {\text{B}}_0^{\text{Global}}$ achievable using 1^st − 4^th degree shimming for a 4.4cm slab. Displayed in Fig 4 are ROIs, and B₀ maps from the 4.4cm slab. From n=10 subjects, addition of 4^th degree shims provided a 29% improvement in $\sigma {\text{B}}_0^{\text{Global}}$ (12.16±1.68Hz) in comparison to 1^st − 3^rd degree shimming (17.13±2.22Hz) at an average aggregate current of 3.6A.

Figure 4.

Scout images (each slice shown with ROI) and B₀ maps (11 slices) from a 4.4cm slab showing the stepwise improvements of 1^st−2^nd, 1^st − 3^rd and 1^st − 4^th degree shimming for $\sigma {\text{B}}_0^{\text{Global}}$ and on each slice. All B₀ maps are plotted on a scale of ±50Hz.

$\sigma {\text{B}}_0^{\text{Local}}$ and Spectral Resolution

The data in the three single slice regions show that significant improvements in $\sigma {\text{B}}_0^{\text{Global}}$ can be achieved using successively higher degree shims. However for spectroscopic performance, a key performance measure for shimming is $\sigma {\text{B}}_0^{\text{Local}}$, i.e., the field inhomogeneity in any given spectroscopic voxel. We examined the relationship between $\sigma {\text{B}}_0^{\text{Local}}$ and spectral resolution in these three regions (Fig. 5). The MTL is commonly a difficult region for B₀ shimming due to the high gradients of B₀ in the vicinity of the cavernous sinus and ear canals. However with 3^rd degree shimming, the degradation of $\sigma {\text{B}}_0^{\text{Local}}$ along the hippocampal formation is improved such that even the most anterior pixels yield sufficient resolution to fully resolve major singlets (Fig. 5 top panel, spectra A and F). With 3^rd and 4^th degree shimming, similar behavior can be seen in the SCN (Fig. 5 middle panel) where in well shimmed regions $\sigma {\text{B}}_0^{\text{Local}}$ <4Hz (thalami, posterior brain, insula), excellent spectral resolution is achieved. In this SCN plane, the high B₀ gradients are over the frontal sinus, where the $\sigma {\text{B}}_0^{\text{Local}}$ ranges from 3.5 to 10.8 Hz over the distance of 3.2cm or 4 contiguous pixels (Fig. 5 middle panel, spectra E through H). The degradation in spectral resolution is apparent as the resonances between 3.4 and 4.0 ppm (inositol, creatine, α-amino acids) merge into a single non-descript broad plateau when $\sigma {\text{B}}_0^{\text{Local}}\ge 7\text{Hz}$, and the creatine and choline resonances begin to overlap when $\sigma {\text{B}}_0^{\text{Local}}>10\text{Hz}$. Again, however, major singlets are maintained throughout. Finally, it is not surprising that the SMA (Fig. 5 bottom panel) shows the greatest consistency, with $\sigma {\text{B}}_0^{\text{Local}}$ generally less than 3Hz and all spectra showing excellent resolution. In this example, the increased glutamate associated with gray matter in comparison to white matter (25–27) is clearly visible (left vs. right column of spectra in Figure 5 bottom panel).

Figure 5.

Spectroscopic and shimming data from the MTL (top), SCN (middle) and SMA (bottom panel). For each location, the scout (I), B_o (II) and $\sigma {\text{B}}_0^{\text{Local}}$ (III) maps are shown with spectra from indicated locations and their corresponding $\sigma {\text{B}}_0^{\text{Local}}$ values. All $\sigma {\text{B}}_0^{\text{Local}}$ maps are shown with color bar 0–12Hz.

To quantify the relationship between $\sigma {\text{B}}_0^{\text{Local}}$ and the observed linewidth we analyzed N=10 moderate TE studies from the MTL selecting a rectangular ROI spanning both hippocampi and shimmed with 1^st − 3^rd degree shims. Fig. 6 shows a scatter plot of $\sigma {\text{B}}_0^{\text{Local}}$ versus σ_Fit as determined from the spectral fitting. Not surprisingly, in this range of $\sigma {\text{B}}_0^{\text{Local}}$ between 1.9 and 10.6 Hz, the data show excellent correlation, with R=0.87, p<0.00001, and a slope of 0.91±0.03. As a measure of the residual inhomogeneity, the mean difference $\overline{({\mathit{\sigma }}_{\mathbf{Fit}}-\mathit{\sigma }{\mathbf{B}}_{\mathbf{0}}^{\mathbf{Local}})}$ is 1.18±0.69Hz, close to the intercept at 1.51±0.24Hz. Fig. 7 shows SCN data acquired with 1^st−4^th vs. 1^st−2^nd degree shims in the same subject during a single imaging session (i.e., the subject was not moved). There is marked improvement in both $\sigma {\text{B}}_0^{\text{Global}}$ (Fig. 7E versus 7C) and $\sigma {\text{B}}_0^{\text{Local}}$ (Fig. 7F versus 7D) with 4^th degree shims. Spectra from a 3×6 pixel ROI (Fig. 7B) overlaying the thalamus are displayed in Figs. 7G and 7H. As predicted from the maps of $\sigma {\text{B}}_0^{\text{Local}}$, spectral quality from the anterior medial portions of the thalamus are markedly improved, consistent with reductions in $\sigma {\text{B}}_0^{\text{Local}}$ from 8–10Hz to 4–6Hz.

Figure 6.

A scatter plot of $\sigma {\text{B}}_0^{\text{Local}}$ versus σ_Fit for pixels spanning both hippocampi from N=10 subjects (268 pixels total).

Figure 7.

Data from the SCN displaying maps of B₀, $\sigma {\text{B}}_0^{\text{Local}}$ and spectra with 1^st − 2^nd and 1^st − 4^th degree shimming. (A, B) Scout image and the thalamic region selected for display. (C, D and G) acquired with 1^st − 2^nd degree shims. (E, F and H) acquired with 1^st − 4^th degree shims. Spectra from the thalamus (G & H), show marked improvement of spectral resolution with 4^th degree shims. For purposes of comparison the spectra are cropped to show the region between NAA and choline.

It is useful to quantify the impact of higher degree shimming on $\sigma {\text{B}}_0^{\text{Local}}$ with histograms that show the fraction of SI pixels at a given $\sigma {\text{B}}_0^{\text{Local}}$ (Figs. 8A–C). Depending upon the $\sigma {\text{B}}_0^{\text{Local}}$ required for accurate interpretation of the data, the fraction of usable pixels in the ROI for a given slice and shim degree can be estimated. For each successive shim degree, the curve shifts to the left, indicating a larger number of pixels at a lower value of $\sigma {\text{B}}_0^{\text{Local}}$. For example, if σB₀^Local <5Hz is used as a threshold for sufficient homogeneity, integrating the area under the curves shows that use of 1^st − 3^rd or 1^st − 4^th in comparison to only 1^st − 2^nd degree shimming increases the number of pixels achieving sufficient homogeneity for the SCN slice by 24% and 59% respectively. Fig. 8D–F shows the spatial distribution of $\sigma {\text{B}}_0^{\text{Local}}$ for increasing shim degree from the three locations. As expected the greatest improvements in $\sigma {\text{B}}_0^{\text{Local}}$ are seen for cortical brain locations (SMA), regions affected by the frontal sinuses i.e. the thalamus, basal ganglia, insula and frontal cortex, (SCN slice) and the anterior hippocampus (MTL slice).

Figure 8.

(A–C) shows histograms of $\sigma {\text{B}}_0^{\text{Local}}$ for three regions studied SMA (A), SCN (B) and MTL (C), where the $\sigma {\text{B}}_0^{\text{Local}}$ values are binned by 0.25Hz increments and normalized as fraction of total pixels. In each plot, four shim configurations are displayed, 1^st (green) 1^st and 2^nd (blue), 1^st − 3^rd (red) and 1^st − 4^th(black) degree terms. (D–F) shows scout images and maps of σB₀^Local for these regions, calculated for 1^st − 2^nd, 1^st − 3^rd and predicted for 1^st − 4^th degree shimming.

Discussion

$\sigma {\text{B}}_0^{\text{Global}}$

Our data indicate that for the three 10mm thick single slice target regions (SMA, SCN, MTL), substantial gains in $\sigma {\text{B}}_0^{\text{Global}}$ are achieved with higher degree shimming at 7T. With 3^rd degree shims for the three regions evaluated, decreases in $\sigma {\text{B}}_0^{\text{Global}}$ of 26–37% were achieved and were well within the capability of the existing hardware. Notably even superior locations (SMA, commonly thought not to have substantial higher degree inhomogeneity terms) also showed significant improvement. For all three locations, the maximum required strength for any 3^rd degree shim did not exceed 39% of its full capacity. However, it is important to note that our head only system uses a gradient set with an ID/OD of 42/68cm and the shims are driven by 20A power supplies. Incorporation of 3^rd degree shims on a body gradient set (larger diameter) would most likely substantially reduce the achieved strength per amp of applied current.

The B₀ maps after 1^st − 3^rd degree shimming showed clear 4^th order asymmetries (C4) in both the SMA and SCN slices. As predicted by our 4^th degree calculations and verified using our 4^th degree shim insert, additional improvements of 31% for the SMA and 38% for the SCN were achieved. A more modest improvement is predicted for the MTL, 23%.

The improvement in $\sigma {\text{B}}_0^{\text{Global}}$ with higher degree shims is also seen in slab selective shimming. For a 4.4cm thick slab center at the top of the ventricles, addition of 4^th degree shims, provides a 29% improvement over 1^st − 3^rd degree shims. Thus despite the substantial brain volume involved, 4^th degree shimming provides significant gains.

In the limit of small regions (on the order of a few cubic centimeters) such as that used in single voxel spectroscopy, it is clear that good homogeneity can be achieved in many brain regions using only 2^nd degree shims (22,25). In this case, the parameters $\sigma {\text{B}}_0^{\text{Global}}$ and $\sigma {\text{B}}_0^{\text{Local}}$ become identical. However, the definition of a “small region” depends on where it is in the brain, as small voxels taken in regions of high B₀ inhomogeneity can still suffer from large $\sigma {\text{B}}_0^{\text{Local}}$. As the target ROI is increased, the spatial dependence from inhomogeneities which have 3^rd and 4^th degree and order components becomes substantial, and cannot be completely compensated for by lower degree shims.

The limits of $\sigma {\text{B}}_0^{\text{Global}}$

Given the complexity of the susceptibility maps in the human head (12,13), we anticipate that higher degree shims should always be able to decrease $\sigma {\text{B}}_0^{\text{Global}}$ and $\sigma {\text{B}}_0^{\text{Local}}$ over large regions of brain. It is clear however that how much of an improvement can be achieved will depend on the algorithm used, ROI selected and experimental goals of the shimming process. To address the boundaries of this question, it is useful to consider the extent to which very high degree shims terms could further improve homogeneity. For this analysis we defined our ROI to be an entire slice of brain as it is a common target for spectroscopy and imaging. The starting point for this analysis are high resolution B₀ maps (thick/gap 2mm/2mm) acquired after 4^th degree shimming over a 10mm thick slice centered between the two slice positions (superior, inferior) shown in Fig. 9.

Figure 9.

Data from ultra-high degree shim corrections from two slices (A,B superior, inferior) showing scouts and B₀ maps that are corrected to 3^rd, 7^th and 15^th degree shims, shown at ±50Hz and ±10Hz scaling. The ±10Hz maps are shown in color and grayscale for clearer identification of details. (C) Bar plot showing the asymptotic behavior of $\sigma {\text{B}}_0^{\text{Global}}$ with succeeding degrees of shim corrections (up to 15^th degree). In the superior slice, $\sigma {\text{B}}_0^{\text{Global}}$ approaches an asymptote of ~2.3Hz at n ~7^th and higher degree shims.

The consistency of our actual shimming performance is shown by the relatively small differences between the measured (shimmed 1^st−4^th degree) and 1^st, 2^nd and 3^rd degree mathematically corrected field maps. With the lower degree terms, the remaining difference most likely arises from the fact that the head was actually shimmed over a 10mm slab while the mathematical correction was applied to the individual B₀ slices. With increasing shim degree correction n, an asymptotic behavior for $\sigma {\text{B}}_0^{\text{Global}}$ is seen, with the majority of improvement achieved by 7^th degree. At this level of shim correction in the superior slice, the $\sigma {\text{B}}_0^{\text{Global}}$ approaches an asymptote of ~2.3Hz. This represents a FWHM frequency spread of 5.3Hz, approximately equal to the shift due to known susceptibility differences between gray and white matter (28,29), suggesting that the residual inhomogeneity is likely dominated by the intrinsic differences between gray and white matter. The calculated maps indicate that the very high spatial complexity of susceptibility that distinguishes vessels, structures such as the striatum and pallidum, white and gray matter tissues remains. As described by Hillenbrand (15) structures of biological interest that are characterized by discrete susceptibility changes are effectively “islands” of B₀ inhomogeneity, such that they cannot be wholly corrected using spherical harmonics or any other externally generated field profiles. Overall, Fig. 9 suggests that for such slice based regions, shim degrees above 7^th may yield only minor improvements in $\sigma {\text{B}}_0^{\text{Global}}$. Finally, this simulation does not address the strength required to achieve the corrections which may substantially exceed that which is technically feasible at this time.

$\sigma {\text{B}}_0^{\text{Local}}$

As shown above, $\sigma {\text{B}}_0^{\text{Local}}$ is an important influence on spectral quality. However the extent to which the residual inhomogeneity affects small voxel SI studies and σB₀^Local has been controversial (22,30,31). For spectroscopy, the actual FWHM linewidth (LW) of any given pixel is determined by several factors, including the intrinsic T2 (1/π*T2) and the (potentially multiple) distribution(s) of local susceptibility. The local susceptibility variation (or LW − 1/π*T2) has also been separated into two components, “macroscopic” σB₀^macro and “microscopic” σB₀^macro, i.e., the susceptibility variations that can and cannot be eliminated by shimming respectively (22). While functionally these two components are useful, the parameter σB₀^Local is a specific measure of susceptibility variation independent of shimming, and is a potentially good definition of the amplitude and distribution of susceptibility. As a result, this parameter can provide guidance on how best to optimize or analyze such data. For example, high resolution B₀ maps acquired in well shimmed regions (e.g., Fig. 5 and 9) show that shifts in B₀ appear dependent on white-gray matter boundaries. (Notably, these variations in field values contributing to σB₀^Local cannot be generally wholly eliminated by shimming, and yet are clearly “macroscopic”.) As has been discussed (32,33), knowledge of the distribution of the susceptibility variation is highly useful for spectral analysis, e.g., distortions in lineshape would be anticipated from pixels near strong susceptibility shifts. Caveats for this type of analysis however should be stated, for example, while gray-white matter shifts in the water resonance intrinsically determine the measured B₀ value and $\sigma {\text{B}}_0^{\text{Local}}$, metabolites may not shift to the same amount (34), in which case the effect of $\sigma {\text{B}}_0^{\text{Local}}$ on linewidth would be overestimated.

It is clear that on a microscopic level, determinants of linewidth are complex and beyond the scope of this paper. Nonetheless, Fig. 6 clearly shows a strong correlation (R=0.87, slope=0.91±0.03; p<0.00001) between $\sigma {\text{B}}_0^{\text{Local}}$ and σ_Fit, supporting the view that over a wide range of shimming performance, $\sigma {\text{B}}_0^{\text{Local}}$ provides a good estimate of the linewidth. This information can be very useful, e.g., during the shimming process prior to MRSI acquisition, maps of $\sigma {\text{B}}_0^{\text{Local}}$ can be used to refine/modify target ROIs depending on experimental goals.

Overall, given the evidence on the reduction of $\sigma {\text{B}}_0^{\text{Local}}$ by 3rd and 4th degree shimming, for CSI based studies where large 2D or 3D regions of the brain are sampled with ~1cc voxel resolution, the improvements in decreased $\sigma {\text{B}}_0^{\text{Local}}$ clearly translate into improvements in linewidth and spectral resolution. At values of $\sigma {\text{B}}_0^{\text{Local}}>10\text{Hz}$, choline and creatine show substantial spectral overlap and resolution of amino acid resonances is significantly degraded. As $\sigma {\text{B}}_0^{\text{Local}}$ approaches 7Hz, there is improvement in resolution for the resonances between 3.4 and 4ppm. Finally, at and below $\sigma {\text{B}}_0^{\text{Local}}$ of 2–3Hz, spectral resolution is generally excellent. The largest gains for higher degree shimming are seen at the cortical periphery, frontal regions above the sinuses and anterior regions along the hippocampal formations. All of these regions represent important brain regions based on their functional roles, including motor and visual control, visual/auditory perception and memory.

Conclusions

In conclusion, for 2D and 3D spectroscopic imaging studies of the human brain at 7T, higher degree shimming including 3^rd and 4^th degree shims can provide significant gains in both $\sigma {\text{B}}_0^{\text{Global}}$ and $\sigma {\text{B}}_0^{\text{Local}}$. Improvements in $\sigma {\text{B}}_0^{\text{Local}}$ in the SMA, SCN and MTL correlate strongly, qualitatively and quantitatively with the measured linewidth and the spectral resolution achieved. Additionally, we find a very significant correlation (R=0.87) between spectral linewidth and $\sigma {\text{B}}_0^{\text{Local}}$. While smaller values for $\sigma {\text{B}}_0^{\text{Global}}$ may be thought to be always desirable, it should be realized that for different experimental goals, the spatial distribution of $\sigma {\text{B}}_0^{\text{Local}}$ may be more important. Mathematical correction of higher degree shimming to 15^th degree shows that in these slice based regions, the large majority of the potential improvement in $\sigma {\text{B}}_0^{\text{Global}}$ can be achieved with 7^th degree shims. In high brain regions, the remaining inhomogeneity appears to be dominated by the ~5Hz susceptibility shift of gray and white matter. Finally, in our system, realization of 4^th degree shimming was achieved with a shim insert located within the gradient coils thereby allowing minimization of overall size while maintaining shim efficiency.

Table.

Shim strengths and maximum values by degree and region of interest

Second Degree Terms (n=20)
		Z2	ZX	ZY	C2	S2
Max. Strength		5.70E−01	2.12E+00	2.09E+00	1.30E+00	1.32E+00
SMA	AVG	−2.54E−02	−3.65E−03	5.60E−03	3.80E−03	−7.20E−04
	STD	1.04E−02	7.82E−03	1.40E−02	5.01E−03	2.05E−03
	MAX	4.18E−02	2.05E−02	3.13E−02	1.47E−02	4.32E−03
	% of MAX	7.32%	0.96%	1.50%	1.12%	0.33%
SCN	AVG	−2.56E−02	−1.70E−03	−7.50E−03	−1.06E−02	−2.43E−03
	STD	1.08E−02	1.06E−02	5.36E−02	9.51E−03	4.43E−03
	MAX	4.71E−02	2.61E−02	9.99E−02	2.94E−02	1.21E−02
	% of MAX	8.26%	1.23%	4.78%	2.26%	0.92%
MTL	AVG	−9.83E−02	−2.17E−02	1.26E−01	−2.33E−02	−2.85E−03
	STD	2.99E−02	2.31E−02	4.82E−02	3.84E−02	1.07E−02
	MAX	1.71E−01	7.94E−02	2.02E−01	1.26E−01	2.39E−02
	% of MAX	29.91%	3.74%	9.65%	9.68%	1.81%

Third Degree Terms (n=20)
		Z3	Z2X	Z2Y	ZC2	ZS2	C3	S3
Max. Strength		4.15E−03	7.76E−03	7.71E−03	1.24E−02	1.28E−02	4.47E−03	4.60E−03
SMA	AVG	3.49E−06	5.14E−05	2.79E−04	−5.79E−05	−3.47E−04	−7.21E−05	7.46E−06
	STD	1.23E−04	1.84E−04	2.57E−04	9.33E−05	2.57E−04	2.88E−05	4.74E−05
	MAX	2.90E−04	5.39E−04	7.93E−04	2.41E−04	8.58E−04	1.37E−04	1.11E−04
	% of MAX	6.97%	6.94%	10.29%	1.94%	6.70%	3.07%	2.41%
SCN	AVG	−6.17E−05	2.79E−05	1.24E−03	−1.67E−05	−4.62E−04	−6.27E−05	−8.35E−05
	STD	1.90E−04	1.34E−04	5.63E−04	3.75E−04	4.10E−04	4.96E−05	5.63E−05
	MAX	4.02E−04	2.92E−04	2.67E−03	1.30E−03	1.66E−03	1.53E−04	2.39E−04
	% of MAX	9.67%	3.76%	34.63%	10.51%	12.98%	3.41%	5.21%
MTL	AVG	−3.72E−05	1.05E−04	−1.62E−03	7.55E−07	1.58E−03	−1.41E−05	−1.28E−04
	STD	2.91E−04	3.41E−04	7.80E−04	4.65E−04	7.06E−04	1.81E−04	3.04E−04
	MAX	5.97E−04	7.48E−04	2.97E−03	9.13E−04	3.18E−03	5.49E−04	6.72E−04
	% of MAX	14.37%	9.64%	38.51%	7.36%	24.81%	12.27%	14.61%

Fourth Degree Terms (n=10)
		Z4	Z3X	Z3Y	Z2C2	Z2S2	ZC3	ZS3	C4	S4
Max. Strength			5.05E−05	5.41E−05	3.38E−05	3.41E−05	4.36E−05	3.96E−05	1.41E−05	1.38E−05
SMA	AVG	5.47E−07	−6.90E−07	−5.32E−06	−8.80E−06	4.95E−07	−3.19E−07	5.34E−07	1.45E−06	−2.52E−07
	STD	2.61E−06	6.51E−06	6.13E−06	9.68E−06	2.57E−06	2.29E−06	3.31E−06	7.87E−07	4.25E−07
	MAX	6.94E−06	1.93E−05	1.65E−05	2.33E−05	4.45E−06	7.08E−06	7.29E−06	2.94E−06	1.01E−06
	% of MAX		38.22%	30.50%	68.93%	13.05%	16.24%	18.41%	20.85%	7.32%
SCN	AVG	−3.20E−06	8.89E−07	−2.25E−05	−2.31E−05	−9.57E−07	−6.20E−07	8.49E−06	2.13E−06	−1.55E−07
	STD	3.67E−06	7.71E−06	1.59E−05	1.36E−05	6.01E−06	2.47E−06	4.97E−06	6.85E−07	6.07E−07
	MAX	1.03E−05	2.77E−05	5.12E−05	5.84E−05	1.22E−05	5.88E−06	2.62E−05	3.63E−06	1.47E−06
	% of MAX		54.85%	94.64%	172.78%	35.78%	13.49%	66.16%	25.74%	10.65%
MTL	AVG	7.43E−06	6.42E−06	−1.69E−05	4.70E−06	1.57E−06	−4.16E−06	−3.61E−05	1.02E−05	−3.73E−08
	STD	7.41E−06	1.45E−05	2.14E−05	3.07E−05	1.16E−05	1.23E−05	1.10E−05	4.93E−06	3.22E−06
	MAX	2.30E−05	3.37E−05	5.75E−05	5.82E−05	3.23E−05	3.27E−05	6.35E−05	1.93E−05	1.04E−05
	% of MAX		66.73%	106.28%	172.19%	94.72%	75.00%	160.35%	136.88%	75.36%

The maximum shim strength in Hz/mmⁿ deliverable by the shims and power supply is listed under Max Strength. The average (AVG), standard deviation (STD) for each group (number of volunteers shown for each) and maximum (absolute value) used in any subject (MAX) are listed for each location (SMA, SCN and MTL) and shim term. Values for the all terms (except 4th degree) are measured and are within power supply range. Values for the 4th degree terms are predicted and therefore some for the MTL exceeded 100% of the developed insert and 5A power supplies used.

APPENDIX

Definitions and Notations of degree and order

Because in a region that is free of magnetic sources the solution of the magnetic scalar potential (as provided by Laplace’s equation) is readily given by the spherical harmonics, the spatial dependencies of the magnetic fields used in shimming are commonly described by the associated Legendre functions in spherical polar coordinates. As a result, when describing a particular shim coil associated with a given spherical harmonic, the terms degree and order are used. Unfortunately, these terms are used inconsistently and interchangeably in the literature. Our text adopts the notation of Abramowitz and Stegun (35) and The NIST Digital Library of Mathematical Functions, (http://dlmf.nist.gov/), whereby the term n is called the degree and the term m is called the order for the associated Legendre function $P_n^m(u)$ (u = cos(θ) where θ is the polar angle). The term degree (n) is associated with the polar radius r and characterizes the rate at which the strength of the magnetic field changes with respect to distance from the origin along a polar radius. The term order (m) is associated with the azimuthal angle ϕ and characterizes the periodicity of the strength of magnetic field in the azimuthal direction. The present text defines all harmonics which have a common order, m, as belonging to the same azimuthal symmetry family. Each order m comprises an infinite number of spherical harmonic members, each of which has a different degree, n. Within a given order m, the lowest degree, n, is numerically equal to that order m.