Application of an Integrated Radio-Frequency/Shim Coil Technology for Signal Recovery in fMRI

Abstract

Purpose:

Gradient-echo EPI, which is typically used for BOLD fMRI, suffers from distortions and signal loss caused by localized B₀ inhomogeneities. Such artifacts cannot be effectively corrected for with the low-order spherical harmonic (SH) shim coils available on most scanners. The integrated parallel reception, excitation, and shimming (iPRES) coil technology allows RF and direct currents to flow on each coil element, enabling imaging and localized B₀ shimming with one coil array. iPRES was previously used to correct for distortions in spin-echo EPI and is further developed here to also recover signal loss in gradient-echo EPI.

Methods:

The cost function in the shim optimization, which typically uses a single term representing the B₀ inhomogeneity, was modified to include a second term representing the signal loss, with an adjustable weight to optimize the trade-off between distortion correction and signal recovery. Simulations and experiments were performed to investigate the shimming performance.

Results:

Slice-optimized shimming with iPRES and the proposed cost function substantially reduced the signal loss in the inferior frontal and temporal brain regions compared to shimming with iPRES and the original cost function or 2^nd-order SH shimming with either cost function. In breath-holding fMRI experiments, the ΔB₀ and signal loss root-mean-square errors decreased by −34.3% and −56.2%, whereas the EPI signal intensity and number of activated voxels increased by 60.3% and 174.0% in the inferior frontal brain region.

Conclusion:

iPRES can recover signal loss in gradient-echo EPI, which is expected to improve BOLD fMRI studies in brain regions suffering from signal loss.

1. Introduction

BOLD fMRI is one of the most widely used noninvasive neuroimaging methods to study the function of the human brain. It typically uses gradient-echo single-shot EPI to achieve a high temporal resolution, with a fairly long TE (e.g., 30 ms at 3T) to maximize the BOLD sensitivity. However, magnetic susceptibility differences at air/tissue interfaces cause localized B₀ inhomogeneities that result in both geometric distortions (due to the B₀ offset (ΔB₀) in each voxel) and signal loss (due to the B₀ variation over each voxel, which is more severe for larger voxels). These B₀ inhomogeneities and image artifacts are particularly severe for ultra-high field (e.g., 7T) applications and are most prominent in the inferior frontal and temporal brain regions, which makes BOLD fMRI studies of cognitive functions associated with these regions very difficult, if not impossible.

Various methods have been proposed to correct for these image artifacts, but each has some limitations. For example, post-processing techniques can correct for distortions but not signal loss; z-shimming techniques^1–4 may have some trade-off between shimming effectiveness and scan time and can only compensate first-order spatial field variations; and early localized shimming methods using small shim coils⁵ or passive shims^6,7 placed in the subject’s mouth have limited subject comfort and flexibility. In addition, the 2^nd-order or sometimes 3^rd-order spherical harmonic (SH) shim coils available on most MRI scanners cannot effectively shim localized B₀ inhomogeneities, while higher-order SH shim coils are limited by available space, coil efficiency and cooling, and number of available current amplifiers⁸. Multi-coil shimming with a localized shim coil array was proposed to address some of these limitations and was shown to be more effective than low-order SH shimming^9–12. However, as shim coil array elements are placed closer to the subject and increase in number, they begin to compete with the RF coil array elements for space, potentially resulting in a decrease in receive SNR, transmit RF efficiency, and/or shimming performance.

Integrated parallel reception, excitation, and shimming (iPRES) is a recent coil technology that has been proposed to address these limitations by allowing an RF current and a direct current (DC) to flow on each coil element, thereby enabling imaging and localized B₀ shimming with a single coil array¹³. Combining both the RF and shim coil arrays into one integrated RF/shim coil array placed close to the subject can save valuable space in the scanner bore without reducing the SNR^14–18. The iPRES technology has already been implemented in various coil arrays to effectively shim localized B₀ inhomogeneities in the human brain^14,18–22, abdomen^15,23, breast¹⁷, and spinal cord^24–26, as well as the monkey brain²⁷.

However, it has so far mostly been used to correct for distortions in spin-echo EPI, which is used for diffusion-weighted imaging, but not to recover signal loss in gradient-echo EPI, which is used for BOLD fMRI. The purpose of this work is therefore to further develop the iPRES technology to recover signal loss as well, thereby enabling the use of BOLD fMRI to investigate brain regions suffering from signal loss, which have so far proven difficult to study. Preliminary results from our group have been presented in abstract form²⁸.

2. Methods

2.1. Theory

In our previous studies using iPRES coil arrays^{14–18,20,23,25}, the optimal DC currents for shimming were determined by minimizing a cost function defined as the root-mean-square error (RMSE) of the ΔB₀ map obtained after shimming with iPRES (ΔB_0,shimmed):

$$C_1=\sqrt{\text{mean}\left(\text{Δ}{B}_{0,\text{shimmed}}^2\right)}$$ [1]

where ΔB_0,shimmed is the sum of: 1) a baseline ΔB₀ map acquired on a subject before shimming with iPRES (ΔB_{0,to shim}) and 2) a weighted combination of basis ΔB₀ maps (ΔB_0,basis) representing the magnetic field generated by a DC current of 1 A separately applied in each iPRES coil element:

$$\text{Δ}{B}_{0,\text{shimmed}}=\text{Δ}{B}_{0,\text{to\hspace{0.17em}shim}}+{\sum }_{k=1}^n{I}_k\text{Δ}{B}_{0,\text{basis,}k}$$ [2]

and where I_k is the DC current applied in the k^th iPRES coil element and n is the number of iPRES coil elements.

Minimizing this cost function effectively minimizes the overall B₀ inhomogeneity, as shown in previous studies. However, when applied to a single slice (i.e., for slice-optimized rather than whole-brain shimming), this method only reduces the susceptibility-induced in-plane B₀ gradients, which cause distortions, but does not take into account the susceptibility-induced through-plane B₀ gradients, which cause signal loss. Even when performing the shim optimization in a thin slab including the slice of interest and the two adjacent slices, which is typically done by our group^{14–18,23,25} and others^9,11,19, this method tends to reduce the in-plane B₀ gradients at the expense of the through-plane B₀ gradients.

To more effectively minimize the signal loss, we propose a second cost function defined as the RMSE of the signal loss map obtained after shimming with iPRES (L_shimmed):

$$C_2=\sqrt{\text{mean}\left(L_{\text{shimmed}}^2\right)}$$ [3]

Assuming an ideal rectangular slice profile, a uniform spin density across the slice, and a linear through-plane B₀ gradient across the slice, L_shimmed can be computed as follows⁴:

$$L_{\text{shimmed}}=1-\left|\mathrm{sinc}\left(\frac{\varphi }{2}\right)\right|$$ [4]

from the phase dispersion ϕ induced by the through-plane B₀ gradient, which can in turn be computed as follows:

$$\varphi =\text{γ}\frac{\partial B_{0,\text{shimmed}}}{\partial z}\text{Δ}z\text{\hspace{0.17em}TE}$$ [5]

from ΔB_0,shimmed, where γ is the gyromagnetic ratio and Δz is the slice thickness. Signal loss maps can thus be derived from ΔB₀ maps by using Eqs. [4–5] in each voxel. This signal loss, which is a unitless quantity between 0 (no signal loss) and 1 (complete signal loss), represents the signal loss due to the macroscopic B₀ inhomogeneities induced by susceptibility differences at air/tissue interfaces and does not depend on the intrinsic T₂* of the tissue.

We hypothesize that minimizing the cost function C₂ can reduce the signal loss more effectively than minimizing the cost function C₁, but that it may not reduce the in-plane B₀ gradients and the resulting distortions as effectively. To address this issue, we propose to minimize a third cost function:

$$C_3=C_1+w{C}_2$$ [6]

where w is a weight that controls the contribution of C₂ relative to C₁ and that can be adjusted to achieve an optimal trade-off between a reduction in B₀ inhomogeneity and a reduction in signal loss.

2.2. Simulations

Simulations were performed to determine the optimal weight and to compare the shimming performance of the proposed method with that of other shimming methods. First, baseline ΔB₀ maps were acquired in the brain of 7 healthy adult volunteers (4 males, 3 females) with various head shapes and sizes. All volunteers provided written informed consent to participate in this study, which was carried out in accordance with a protocol approved by the Duke University Health System Institutional Review Board. All scans were performed on a 3T MR750 MRI scanner (GE Healthcare, Milwaukee, WI) with a 32-channel iPRES head coil array previously described in Ref. 14.

In our previous work, iPRES was only implemented into the 16 anterior RF coil elements and eight adjustable DC currents were applied to the pairs of right/left coil elements. In this work, iPRES was implemented into all 32 RF coil elements and an individually adjustable DC current was applied to each of them. Briefly, iPRES was implemented by adding: 1) a 5-A, 16-V, 32-channel DC power supply (W-IE-NE-R, Plein & Baus Corp., Springfield, OH), 2) a shielded twisted-pair DC cable with a pair of chokes between the power supply and each coil element to provide RF-isolation between them, and 3) inductors to bypass the capacitors in the RF coil elements and to allow a DC current to flow on each of them, as described in more detail in Ref. 14.

Axial ΔB₀ maps were acquired with an 8-echo gradient-echo sequence (TR = 1.2 s, TEs = 2.0, …, 14.2 ms, echo spacing = 1.7 ms, flip angle = 50°, voxel size = 2.5 mm isotropic, number of slices = 65) after applying 1^st-order shimming from the scanner. There was still a sufficient signal intensity at all TEs to ensure a linear phase evolution (after phase unwrapping along the TE dimension with the unwrap Matlab function) and an accurate ΔB₀ mapping, even in the presence of large B₀ inhomogeneities in the inferior frontal brain region (Supporting Information Fig. S1). Axial T₁-weighted anatomical images were also acquired with a 3D inversion-prepared fast spoiled gradient-echo sequence (TR = 7.4 ms, TE = 2.8 ms, TI = 450 ms, flip angle = 12°, voxel size = 1x1x2 mm, acceleration factor = 2).

To determine the optimal weight in the proposed cost function C₃ and to compare the shimming performance of different methods, a shim optimization was performed for each of the 7 subjects and each of 12 shimming methods consisting of all possible combinations of:

• two different cost functions: C₁ or C₃ (with 15 empirically chosen weights ranging from 0.25e-5 to 1.5e-5 with a 0.125e-5 step size and from 1.5e-5 to 2.5e-5 with a 0.25e-5 step size);

• two different shim volumes: slice-optimized or whole-brain shimming;

• three different sets of basis ΔB₀ maps: 2^nd-order SH, iPRES, or iPRES + 2^nd-order SH.

As in our previous work, slice-optimized shimming with the cost function C₁ was performed in a thin slab including each slice of interest and the two adjacent slices. On the other hand, slice-optimized shimming with the cost function C₃ was performed in each slice of interest alone to achieve the best shimming performance. All shim optimizations were performed within a brain mask generated from the anatomical images with the Brain Extraction Tool²⁹ from FSL.

For the 2^nd-order SH simulations, the nine basis ΔB₀ maps corresponding to the 0^th- to 2^nd-order SH terms were analytically calculated and the DC currents were not constrained. For the iPRES simulations, the basis ΔB₀ maps were acquired on a 21-cm diameter spherical water phantom filling the entire head coil array (and thus including all voxels within the brain of all subjects) and the DC currents were constrained to be within ±2.5 A, which is the maximum current rating for the inductors used in the iPRES head coil array. For the iPRES + 2^nd-order SH simulations, the two sets of basis ΔB₀ maps above were combined and the DC currents were constrained for iPRES and unconstrained for SH.

All shim optimizations were performed using the fmincon Matlab function with the interior-point method. The output of fmincon showed that the Hessian of the cost function C₃ is positive semi-definite, meaning that the optimization problem is convex and that a global minimum can be achieved.

Additional simulations were performed to assess the shimming performance of the proposed method at a higher resolution and field strength. The baseline ΔB₀ maps were interpolated from a 2.5-mm to a 1-mm isotropic voxel size and scaled from 3T to 7T, and simulations were performed with a TE of 22 ms for slice-optimized shimming with iPRES or iPRES + 2^nd-order SH and the cost function C₃.

2.3. Experiments

A first experiment was performed on a healthy volunteer to compare the shimming performance of different methods. After applying 1^st-order shimming from the scanner, axial ΔB₀ maps and gradient-echo single-shot EPI images were acquired through the inferior frontal brain region before and after slice-optimized shimming with iPRES and either the cost function C₁ or the cost function C₃ with different weights. The ΔB₀ maps were acquired as described above and the EPI images were acquired with TR = 2 s, TE = 30 ms, flip angle = 60°, voxel size = 2.5 mm isotropic, number of slices = 10, echo spacing = 532 μs, SENSE acceleration factor = 3, phase-encoding direction = anterior-posterior, so that voxels with a positive ΔB₀ offset are shifted in the anterior direction. T₁-weighted anatomical images were also acquired as described above. Dynamic shimming was not implemented, so only one slice was shimmed with the cost function C₃ and only one slab of three slices including the slice of interest and the two adjacent slices was shimmed with the cost function C₁.

A second experiment was performed on two additional healthy volunteers to demonstrate the effectiveness of the proposed method for signal recovery in a BOLD fMRI application. Axial ΔB₀ maps and gradient-echo EPI time series (with the same parameters as above and with 4 dummy scans and 72 time points) were acquired before and after slice-optimized shimming with iPRES, the cost function C₃, and the optimal weight. The activation paradigm was a block design consisting of nine alternating normal breathing and end-expiration breath-holding periods, each 16 s in duration. A breath-holding task was chosen to induce a global BOLD response over the whole brain independent of cognitive processes. Two runs were performed without iPRES and two runs with iPRES in an interleaved order.

The acquired fMRI data were averaged across both runs of each type, detrended with a 3^rd-order polynomial, temporally realigned to account for the interleaved slice acquisition, spatially smoothed with a Gaussian filter (FWHM = 1.5 voxels), cross-correlated with a hemodynamic response function convolved with the experimental paradigm, converted to Z-score maps, restricted within a gray matter mask generated from the anatomical images with the FMRIB’s Automated Segmentation Tool³⁰ from FSL, and overlaid on the anatomical images. All simulations and data analyses were performed in Matlab (The MathWorks, Natick, MA).

3. Results

3.1. Experiment and Simulations with Different Weights

Figure 1 shows the experimental results after slice-optimized shimming with iPRES and different cost functions and weights for a 2.5-mm isotropic voxel size at 3T. Shimming with the original cost function C₁ in a slab of three slices can achieve a very effective reduction in B₀ inhomogeneity (Fig. 1B), but at the expense of an increase in signal loss (Fig. 1E). Shimming in a single slice rather than a slab of three slices results in a larger reduction in B₀ inhomogeneity (first map in Fig. 1C), but also a larger increase in signal loss (first map in Fig. 1F), as expected. In contrast, shimming with the proposed cost function C₃ and increasing weights results in a progressively less effective reduction in B₀ inhomogeneity (last three maps in Fig. 1C), but a progressively more effective reduction in signal loss (last three maps in Fig. 1F). The gradient-echo EPI images shimmed with increasing weights (Figs. 1H,I) show slightly worse distortions near the anterior edge of the brain, but a substantially higher signal in the ventromedial prefrontal cortex (pink circle), which is consistent with the ΔB₀ and signal loss maps, respectively. These results are expected, since the ΔB₀ fields generated by the iPRES coil elements need to extend through the orbitofrontal cortex to reach the ventromedial prefrontal cortex and since shim coil elements located outside the head cannot fully shim B₀ inhomogeneities induced by air/tissue interfaces inside the head, as Laplace’s equation no longer holds⁸.

Fig. 1.

Experimental ΔB₀ (or voxel shift) maps (A–C), signal loss (Eq. [4]) maps (D–F), and gradient-echo EPI images (G–I) through the inferior frontal brain region of subject 1, before and after slice-optimized shimming with iPRES and different cost functions and weights for a 2.5-mm isotropic voxel size at 3T. Shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. A weight of 0 is equivalent to the original cost function C₁. The ΔB₀ RMSE (Eq. [1]) and signal loss RMSE (Eq. [3]) in the brain mask of this slice (red outline) and their changes relative to the baseline are listed in white below each map. The ΔB₀ and signal loss RMSEs and the average EPI signal intensity in the inferior frontal brain region (pink circle) and their changes relative to the baseline are listed in pink above each map or image. The ΔB₀ maps also represent voxel shift maps when scaled by the EPI readout duration.

Figure 2 shows the simulated ΔB₀ and signal loss RMSEs averaged over all 7 subjects after shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Whole-brain shimming with 2^nd-order SH, iPRES, or iPRES + 2^nd-order SH (Fig. 2A,B) and slice-optimized shimming with 2^nd-order SH (Fig. 2C,D, circles) all show little dependence on the weight and are unable to provide a significant reduction in signal loss. In contrast, slice-optimized shimming with iPRES or iPRES + 2^nd-order SH (Fig. 2C,D, diamonds and squares) shows the same trade-off seen in Figure 1 between a better reduction in B₀ inhomogeneity (for lower weights) and a better reduction in signal loss (for higher weights). On average, a weight of 1.75e-5 achieves a significant reduction in both B₀ inhomogeneity (−34.0% and −51.3%) and signal loss (−39.1% and −46.9%), with iPRES + 2^nd-order SH performing better than iPRES, as expected. For simplicity, 1.75e-5 was therefore chosen to be the optimal weight used with the cost function C₃ for all methods, subjects, and slices in the comparison of different shimming methods shown below. However, other similar weights may provide a better trade-off depending on the shimming method, subject, and/or slice used in a particular experiment.

Fig. 2.

Simulated ΔB₀ RMSE (Eq. [1]; A,C) and signal loss RMSE (Eq. [3]; B,D) changes relative to the baseline (1^st-order shimming) after shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Slice-optimized shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. A weight of 0 is equivalent to the original cost function C₁. The RMSEs are calculated in the brain mask of each slice (red outline in Figs. 3–4 and Supporting Information Figs. S2–S3). The symbols and shaded regions (or error bars) represent the average and standard deviation, respectively, calculated over 3 slices through the inferior frontal brain region of each of the 7 subjects (i.e., 21 different slices, including those shown in Figs. 3–4 and Supporting Information Figs. S2–S3). The weight can be adjusted to achieve an optimal trade-off between a reduction in ΔB₀ RMSE and a reduction in signal loss RMSE (Eq. [6]). The optimal weight of 1.75e-5 is indicated by the black vertical line.

3.2. Simulations of Different Shimming Methods

Figures 3–5 and Supporting Information Figures S2–S3 show the simulated ΔB₀ and signal loss maps and RMSEs after shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Whole-brain shimming with the cost function C₁ or C₃ and with 2^nd-order SH, iPRES, or iPRES + 2^nd-order SH achieves a reduction in B₀ inhomogeneity in the inferior frontal and temporal brain regions ranging from −26.7% to −36.7% when averaged over all 7 subjects (Figs. 3D–I; Supporting Information Figs. S2D–I; Fig. 5A, first six bars). However, all six methods (including the currently most widely used shimming method, which is whole-brain shimming with the cost function C₁ and 2^nd-order SH) result in an increase in signal loss in the inferior frontal and temporal brain regions ranging from +12.1% to +25.0% (Figs. 4D–I; Supporting Information S3D–I; Fig. 5B, first six bars).

Fig. 3.

ΔB₀ (or voxel shift) maps through the inferior frontal brain region of two representative subjects, before (A, experimental) and after (D–O, simulated) shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Slice-optimized shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. Anatomical images with the brain mask (B, red outline) and sagittal localizers with the slice position (C, red line) are included for anatomical reference. For the cost function C₃, an optimal weight of 1.75e-5 was used. The ΔB₀ RMSE (Eq. [1]) in the brain mask of each slice and its change relative to the baseline are listed below each map. Results for all subjects are shown in Supporting Information Figure S2.

Fig. 5.

Simulated ΔB₀ RMSE (Eq. [1]; A) and signal loss RMSE (Eq. [3]; B) changes relative to the baseline (1^st-order shimming) after shimming with different methods for a 2.5-mm isotopic voxel size at 3T. Slice-optimized shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. The RMSEs are calculated in the brain mask of each slice (red outline in Figs. 3–4 and Supporting Information Figs. S2–S3). The colored bars and error bars represent the average and standard deviation, respectively, calculated over 3 slices through the inferior frontal brain region of each of the 7 subjects (i.e., 21 different slices). For the cost function C₃, an optimal weight of 1.75e-5 was used. For a fair comparison, the RMSEs were averaged in the same 3 slices for all shimming methods. These slices are also the same as those used in Figure 2.

Fig. 4.

Signal loss (Eq. [4]) maps in the same slices as those shown in Figure 3, before (A, experimental) and after (D–O, simulated) shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Slice-optimized shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. Anatomical images with the brain mask (B, red outline) and sagittal localizers with the slice position (C, red line) are included for anatomical reference. For the cost function C₃, an optimal weight of 1.75e-5 was used. The signal loss RMSE (Eq. [3]) in the brain mask of each slice and its change relative to the baseline are listed below each map. Results for all subjects are shown in Supporting Information Figure S3.

Slice-optimized shimming with the cost function C₁ or C₃ and with 2^nd-order SH achieves a reduction in B₀ inhomogeneity of −33.6% and −33.9% (Figs. 3J,M; Supporting Information Figs. S2J,M; Fig. 5A, 7^th and 10^th bars), but results either in a minimal increase in signal loss of +0.5% or only in a small reduction in signal loss of −13.6%, respectively (Figs. 4J,M; Supporting Information Figs. S3J,M; Fig. 5B, 7^th and 10^th bars).

Slice-optimized shimming with the cost function C₁ and with iPRES or iPRES + 2^nd-order SH achieves the largest reduction in B₀ inhomogeneity among all methods of −57.3% and −60.3% (Figs. 3K,L; Supporting Information Figs. S2K,L; Fig. 5A, 8^th and 9^th bars), but at the expense of a much larger increase in signal loss of +70.4% and +45.7% (Figs. 4K,L; Supporting Information Figs. S3K,L; Fig. 5B, 8^th and 9^th bars), as expected.

Finally, slice-optimized shimming with the cost function C₃ (and the optimal weight) and with iPRES or iPRES + 2^nd-order SH still achieves a significant reduction in B₀ inhomogeneity of −34.0% and −51.3% (Figs. 3N,O; Supporting Information Figs. S2N,O; Fig. 5A, last two bars), while resulting in the largest reduction in signal loss among all methods of −39.1% and −46.9% (Figs. 4N,O; Supporting Information Figs. S3N,O; Fig. 5B, last two bars). The reduction in signal loss is most notable in the inferior frontal brain region of all subjects and in the inferior temporal brain regions of subject 4.

Similar results were obtained in even more inferior slices (Fig. 6), where slice-optimized shimming with the cost function C₃ (and the optimal weight) and with iPRES or iPRES + 2^nd-order SH achieves a substantial reduction in B₀ inhomogeneity of −42.0% and −49.7% (Figs. 6D,E) and a substantial reduction in signal loss of −53.4% and −55.9% (Figs. 6G,H) in both the inferior frontal and temporal brain regions.

Fig. 6.

ΔB₀ (or voxel shift) maps (C–E) and signal loss (Eq. [4]) maps (F–H) in more inferior slices than those shown in Figures 3–4 and Supporting Information Figures S2–S3, before (C,F; experimental) and after slice-optimized shimming with iPRES (D,G; simulated) or iPRES + 2^nd-order SH (E,H; simulated), the cost function C₃, and the optimal weight (1.75e-5) for a 2.5-mm isotropic voxel size at 3T. Anatomical images with the brain mask (A, red outline) and sagittal localizers with the slice position (B, red line) are included for anatomical reference. The ΔB₀ RMSE (Eq. [1]) and signal loss RMSE (Eq. [3]) in the brain mask of each slice and their changes relative to the baseline are listed below each map.

Figure 7 and Supporting Information Figure S4 show the simulated ΔB₀ and signal loss maps and RMSEs after slice-optimized shimming with the cost function C₃ (and the optimal weight) and with iPRES or iPRES + 2^nd-order SH for a 1-mm isotropic voxel size at 7T. Both methods achieve a significant reduction in B₀ inhomogeneity of −31.9% and −48.7% (Figs. 7B,C; Supporting Information Figs. S4B,C) and a significant reduction in signal loss of −34.7% and −55.9% (Figs. 7E,F; Supporting Information Figs. S4E,F). These results are comparable to those obtained with a 2.5-mm isotropic resolution at 3T and predict that the proposed method can still be beneficial at a much higher resolution and field strength.

Fig. 7.

Simulated ΔB₀ maps (A–C) and signal loss (Eq. [4]) maps (D–F) in similar slices as those shown in Figures 3–4, before (A,D) and after slice-optimized shimming with iPRES (B,E) or iPRES + 2^nd-order SH (C,F), the cost function C₃, and the optimal weight (6e-5) for a 1-mm isotropic voxel size at 7T. The ΔB₀ RMSE (Eq. [1]) and signal loss RMSE (Eq. [3]) in the brain mask of each slice (red outline in Figs. 3–4) and their changes relative to the baseline are listed below each map. Results for all subjects are shown in Supporting Information Figure S4.

3.3. BOLD fMRI Experiments

Figure 8 shows the fMRI experimental results after slice-optimized shimming with the cost function C₃ (and the optimal weight) and with iPRES for a 2.5-mm isotropic voxel size at 3T. The ΔB₀ maps (Figs. 8A,E) and signal loss maps (Figs. 8B,F) show a substantial reduction in B₀ inhomogeneity and signal loss, resulting in a drastic increase in the EPI signal intensity (Figs. 8C,G) and number of activated voxels (Figs. 8D,H) in the inferior frontal brain region of both subjects. On average, the ΔB₀ and signal loss RMSEs are reduced by −35.7% and −44.4% in the brain mask of each slice and by −34.3% and −56.2% in the inferior frontal brain region (pink oval), respectively, whereas the average signal intensity in the EPI images and the number of activated voxels are increased by 60.3% and 174.0% in the inferior frontal brain region, respectively.

Fig. 8.

Experimental ΔB₀ (or voxel shift) maps (A,E), signal loss (Eq. [4]) maps (B,F), gradient-echo EPI images (C,G), and BOLD fMRI activation maps (D,H) through the inferior frontal brain region, before (A–D) and after (E–H) slice-optimized shimming with iPRES, the cost function C₃, and the optimal weight for a 2.5-mm isotropic voxel size at 3T. The ΔB₀ RMSE (Eq. [1]) and signal loss RMSE (Eq. [3]) in the brain mask of each slice (red outline) and their changes relative to the baseline are displayed in white below each map. The ΔB₀ and signal loss RMSEs, average EPI signal intensity, and number of activated voxels (i.e., Z-score > 1.5) in the inferior frontal brain region (pink oval) and their changes relative to the baseline are displayed in pink above each map or image.

4. Discussion

4.1. Comparison of Different Shimming Methods

In this work, we further developed the iPRES technology by introducing a new cost function C₃ for the shim optimization that can reduce both the susceptibility-induced B₀ inhomogeneity and signal loss in gradient-echo EPI. The simulations and experiments performed with a 2.5-mm isotropic voxel size at 3T show that slice-optimized shimming with iPRES is much more effective at reducing signal loss with the proposed cost function C₃ and the optimal weight than with the original cost function C₁. When 2^nd-order SH shim coils are available on the scanner, slice-optimized shimming with iPRES + 2^nd-order SH and the cost function C₃ can achieve even better results, providing the best signal recovery among all shimming methods. However, dynamic 2^nd-order SH shimming is typically not implemented on clinical scanners, which limits the applicability of the latter method. In addition, both methods can achieve a larger reduction in B₀ inhomogeneity as well as a much more effective signal recovery than whole-brain shimming with 2^nd-order SH and the cost function C₁, which is currently the most widely used shimming method (Figs. 5A,B, first bar vs. last two bars).

Whole-brain shimming with 2^nd-order SH and the cost function C₁ reduced the B₀ inhomogeneity (Fig. 3D, Supporting Information Fig. S2D), but increased the signal loss (Fig. 4D, Supporting Information Fig. S3D) in the inferior frontal and temporal brain regions relative to the 1^st-order shimming baseline for all subjects. On average, the signal loss RMSE (Eq. [3]) in the brain mask of these slices increased by 15.7% (Fig. 5B, first bar). To verify this result experimentally, we acquired ΔB₀ and signal loss maps and gradient-echo EPI images in the inferior frontal brain region of two subjects after 1^st-order shimming or whole-brain 2^nd-order SH shimming (Supporting Information Fig. S5). On average, the signal loss RMSE in the brain mask of each slice increased by 13.1% when using whole-brain 2^nd-order SH rather than 1^st-order shimming, whereas the signal loss RMSE and average EPI signal intensity in the inferior frontal and temporal brain regions (pink circles) increased by 19.7% and decreased by 11.1%, respectively. In contrast, the signal loss RMSE in the whole brain decreased on average by 2.4% and 8.3% when using whole-brain 2^nd-order SH rather than 1^st-order shimming in the simulations and experiment described above, respectively. Because this work focuses on signal recovery in the inferior frontal and temporal brain regions and because whole-brain 2^nd-order SH shimming increased the signal loss in those regions, all experiments were performed with a 1^st-order rather than whole-brain 2^nd-order SH shimming baseline to start with less signal loss and achieve better results, which may initially seem counter-intuitive. In other applications, it may be more advantageous to perform the experiments with a whole-brain 2^nd-order SH shimming baseline.

When performing slice-optimized shimming with iPRES and the original cost function C₁ in a slab of three slices rather than a single slice, the signal loss was reduced, but still remained much larger than in the baseline, both in the experiment (Figs. 1E,H vs. 1F,I, C₁ vs. C₃ with weight = 0) and in the simulations for nearly all subjects (Fig. 2D, diamonds, C₁ vs. C₃ with weight = 0; Fig. 4K; Supporting Information Fig. S3K), showing that this approach is not sufficient to recover the signal loss. The change in signal loss RMSE (in the brain mask of each slice) relative to the baseline was reduced from +240.2% in the experiment and +163.3% on average in the simulations to +83.1% and +70.4%, respectively, by including the two adjacent slices in the shim optimization, which still represent an increase relative to the baseline, while it was reduced to −47.5% and −39.1%, respectively, by using the proposed cost function C₃, which represent a substantial decrease relative to the baseline.

Interestingly, changing the cost function (or the weight) only has a minimal impact on the ΔB₀ and signal loss RMSEs obtained with 2^nd-order SH, but has a much greater impact on those obtained with iPRES or iPRES + 2^nd-order SH (Fig. 2), which is likely because the ΔB₀ fields generated by the 32 iPRES coil elements are much more localized than those generated by the nine 0^th- to 2^nd-order SH terms, providing many more degrees of freedom and more flexibility to adjust the ΔB₀ field for shimming. Compared to shimming with iPRES, shimming with higher-order (e.g., 3^rd- to 6^th-order) SH and the cost function C₃ may provide similar or better results, but such setups could require more hardware, space, and cost.

In this work, subject motion was effectively minimized by inserting foam paddings between the head and the coil array, and no degradation in shimming performance was observed over the course of the fMRI experiments. In general, however, subject motion can potentially affect the shimming performance, but a quantitative assessment of this effect would require the acquisition and shimming of multiple ΔB₀ maps with different head positions.

Two recent studies used a 16-channel iPRES coil array (also referred to as an integrated ΔB₀/Rx array or AC/DC array) and a 16-channel multicoil array for BOLD fMRI applications in the monkey brain²⁷ and human brain¹², respectively. However, these studies are not directly comparable, because they used a visual task and a finger-tapping task involving brain regions such as the visual cortex, motor cortex, and cerebellum, which are not affected by severe signal loss, and neither study used a cost function specifically designed to recover the signal loss.

4.2. Extensions to Other Applications

Slice-optimized shimming is designed to work with a 2D multi-slice acquisition, which has been the most widely used method for BOLD fMRI, and dynamic shimming, which has previously been implemented with iPRES¹⁹. However, more advanced techniques based on 3D or multi-band acquisitions have recently become more popular and dynamic shimming may not always be available. In such cases, slab-optimized shimming with iPRES or iPRES + 2^nd-order SH and the cost function C₃ can be used instead to shim a slab in the region of interest. Alternatively, the shim optimization can be performed simultaneously in two or more non-adjacent slices for multi-band acquisitions, which has recently been shown to be almost as effective as slice-optimized shimming for both iPRES²¹ and high-order SH³¹ and which could also be done with the cost function C₃ in future work.

In this work, the cost function C₂ (Eqs. [3–5]) only accounted for the signal loss caused by the through-plane B₀ gradients, which is typically more severe than the signal loss caused by the in-plane B₀ gradients in axial slices acquired in the human brain (Supporting Information Fig. S6), while the in-plane B₀ gradients were indirectly penalized in the cost function C₁ (Eq. [1]), which is included in the cost function C₃ (Eq. [6]). However, the signal loss caused by the in-plane B₀ gradients could also be included in the cost function C₂, and hence in the cost function C₃, to more directly penalize the in-plane B₀ gradients and obtain a more general case.

In addition, we assumed that the through-plane B₀ gradients were linear across the slice, which was estimated to be the case in the vast majority of voxels in the inferior frontal brain region (Supporting Information Fig. S7). While nonlinear B₀ gradients could also be modeled and their impact on the shim optimization investigated in future work, it would require the ΔB₀ maps to be acquired at a higher resolution (at least in the slice direction), which would be more time-consuming and may not be practical.

In this work, the optimal weight in the cost function C₃ was chosen to achieve a reduction in signal loss (in the brain mask of each slice) on the order of −40%, which was experimentally shown to result in a sufficient signal recovery in the gradient-echo EPI images (Fig. 8) and which still provided a reduction in B₀ inhomogeneity on the order of −35%. Since the reduction in signal loss depends on the voxel size (Eqs. [4–5]), but the reduction in B₀ inhomogeneity does not, the optimal weight for the 1-mm isotropic voxel size at 7T was adjusted to achieve similar results as for the 2.5-mm isotropic voxel size at 3T. Different choices could also be made for different applications, depending on whether it is more important to reduce the distortions or the signal loss. For example, a higher weight could be chosen to further improve the signal recovery if post-processing is used to correct for the distortions, which is typically done in fMRI studies, or for applications such as T₂* mapping (Supporting Information Fig. S8) that are based on conventional gradient-echo imaging rather than gradient-echo EPI and are thus mostly affected by signal loss rather than both distortions and signal loss.

Finally, the proposed method was demonstrated in a breath-holding experiment as a proof-of-concept, but it is expected to enable the application of BOLD fMRI to investigate the function of anatomical regions such as the inferior frontal and temporal brain regions, which are affected by signal loss and are currently difficult to study. In addition, it could potentially benefit other applications based on gradient-echo imaging (e.g., T₂* mapping, susceptibility-weighted imaging, or quantitative susceptibility mapping), for both human and animal imaging, including applications outside the brain (e.g., spinal cord) and at ultra-high field strength (as predicted by the simulations performed with a 1-mm isotropic voxel size at 7T), which can be affected by severe distortions and signal loss.

5. Conclusion

The simulation results show that slice-optimized shimming with iPRES and the proposed cost function C₃ can recover the susceptibility-induced signal loss in the inferior frontal and temporal brain regions much more effectively than shimming with iPRES and the original cost function C₁ or than 2^nd-order SH shimming with either cost function. The BOLD fMRI experimental results obtained with a 2.5-mm isotropic voxel size at 3T further show that it can substantially reduce the ΔB₀ and signal loss RMSEs (by −34.3% and −56.2%) and increase the EPI signal intensity and number of activated voxels (by 60.3% and 174.0%) in the inferior frontal brain region with a single integrated RF/shim coil array.

Supplementary Material

fS1-8

Supporting Information Fig. S1. Signal intensity (A) and unwrapped phase (B) from a representative voxel in the inferior frontal brain region (white square) of the baseline (1^st-order shimming) 8-echo gradient-echo images of subject 1, and corresponding ΔB₀ map (C) for a 2.5-mm isotropic voxel size at 3T. These results show that there is a sufficient signal intensity at all TEs to ensure a linear phase evolution and an accurate ΔB₀ mapping.

Supporting Information Fig. S2. ΔB₀ (or voxel shift) maps through the inferior frontal brain region of all subjects, before (A, experimental) and after (D–O, simulated) shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Slice-optimized shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. Anatomical images with the brain mask (B, red outline) and sagittal localizers with the slice position (C, red line) are included for anatomical reference. For the cost function C₃, an optimal weight of 1.75e-5 was used. The ΔB₀ RMSE (Eq. [1]) in the brain mask of each slice and its change relative to the baseline are listed below each map.

Supporting Information Fig. S3. Signal loss (Eq. [4]) maps in the same slices as those shown in Supporting Information Figure S2, before (A, experimental) and after (D–O, simulated) shimming with different methods for a 2.5-mm isotropic voxel size at 3T. Slice-optimized shimming with the cost function C₁ or C₃ was performed in a slab of three slices or a single slice, respectively. Anatomical images with the brain mask (B, red outlne) and sagittal localizers with the slice position (C, red line) are included for anatomical reference. For the cost function C₃, an optimal weight of 1.75e-5 was used. The signal loss RMSE (Eq. [3]) in the brain mask of each slice and its change relative to the baseline are listed below each map.

Supporting Information Fig. S4. Simulated ΔB₀ maps (A–C) and signal loss (Eq. [4]) maps (D–F) in similar slices as those shown in Supporting Information Figures S2–S3, before (A,D) and after slice-optimized shimming with iPRES (B,E) or iPRES + 2^nd-order SH (C,F), the cost function C₃, and the optimal weight (6e-5) for a 1-mm isotropic voxel size at 7T. The ΔB₀ RMSE (Eq. [1]) and signal loss RMSE (Eq. [3]) in the brain mask of each slice (red outline in Supporting Information Figs. S2–S3) and their changes relative to the baseline are listed below each map.

Supporting Information Fig. S5. Experimental ΔB₀ (or voxel shift) maps (A,D), signal loss (Eq. [4]) maps (B,E), and gradient-echo EPI images (C,F) through the inferior frontal brain region, acquired with 1^st-order shimming (A–C) or whole-brain 2^nd-order SH shimming (D–F) with a 2.5-mm isotropic voxel size at 3T. The ΔB₀ RMSE (Eq. [1]) and signal loss RMSE (Eq. [3]) in the brain mask of each slice (red outline) and their changes relative to the 1^st-order shimming baseline are displayed in white below each map. The ΔB₀ and signal loss RMSEs and the average EPI signal intensity in the inferior frontal and temporal brain regions (pink circles) and their changes relative to the 1^st-order shimming baseline are displayed in pink above each map or image. These results show that whole-brain 2^nd-order SH shimming results in more signal loss in the inferior frontal and temporal brain regions than 1^st-order shimming.

Supporting Information Fig. S6. Maps of the signal loss (Eq. [4]) caused by the in-plane B₀ gradients in the right/left (A,B) and anterior/posterior (C,D) directions and by the through-plane B₀ gradients (E,F), calculated using Eqs. [5–6] and similar equations with ∂B₀/∂x and ∂B₀/∂y, in the same slices as those shown in Supporting Information Figures S2–S4, for a 2.5-mm isotropic voxel size at 3T (A,C,E; experimental) and a 1-mm isotropic voxel size at 7T (B,D,F; simulated). The signal loss RMSE in the brain mask of each slice (red outline in Supporting Information Figs. S2–S3) is listed below each map. These results show that the signal loss caused by the through-plane B₀ gradients is typically more severe than the signal loss caused by the in-plane B₀ gradients for both voxel sizes.

Supporting Information Fig. S7. Experimental baseline (1^st-order shimming) ΔB₀ maps (A) and signal loss (Eq. [4]) maps (B) through the inferior frontal brain region of three representative subjects for a 2.5-mm isotropic voxel size at 3T. The ΔB₀ and signal loss RMSEs in the brain mask of each slice (red outline in Supporting Information Figs. S2–S3) are listed below each map. (D) Intravoxel ΔB₀ field estimation in each of the three voxels boxed in (A–C) and the two adjacent voxels in the slice direction (purple circles: measured ΔB₀ field; green crosses: intravoxel ΔB₀ field variation estimated with a cubic interpolation (interp1 Matlab function with makima option); blue, red, and yellow lines: linear fit within each voxel, with corresponding R² values shown in the legend). (C) R² maps, with the average R² and percentage of voxels with R² > 0.8 in each slice listed below each map. (E) R² histograms over each slab of three slices, with the percentage of voxels with R² > 0.8 shown in each histogram. These results show that voxels with a large (e.g., voxel 1) or moderate (e.g., voxel 2) signal loss have a highly linear intravoxel ΔB₀ field variation and that there are only few voxels with a highly nonlinear intravoxel ΔB₀ field variation (e.g., voxel 3).

Supporting Information Fig. S8. Experimental T₂* maps in the same slice as that shown in Figure 8 for subject 10, before (A) and after (B) slice-optimized shimming with iPRES, the cost function C₃, and the optimal weight for a 2.5-mm isotropic voxel size at 3T. The average T₂* values in the inferior frontal brain region (pink oval) are listed above each map. These T₂* maps were calculated by fitting an exponential decay to the magnitude of the 8-echo gradient-echo images that were already acquired to calculate the ΔB₀ maps. As such, the low range of TEs may not have been optimal for T₂* mapping and voxels with cerebrospinal fluid (which has a longer T₂* than gray matter and white matter) were excluded. In addition, unlike the gradient-echo EPI images used for BOLD fMRI, which are affected by both distortions and signal loss, these conventional gradient-echo images are mostly affected by signal loss, so the weight in the cost function C₃, which was optimized for BOLD fMRI, may also not have been optimal for T₂* mapping. Nevertheless, these results show that the proposed method enables a more accurate T₂* mapping in the inferior frontal brain region.