Low Peak Power Multiband Spokes Pulses for $B_1^{+}$ Inhomogeneity-Compensated Simultaneous Multislice Excitation in High Field MRI

Abstract

Purpose

To design low peak and integrated power simultaneous multislice excitation radiofrequency pulses with transmit field inhomogeneity compensation in high field MRI.

Theory and Methods

The ‘interleaved greedy and local optimization’ algorithm for small-tip-angle spokes pulses is extended to design multiband spokes pulses that simultaneously excite multiple slices, with independent spokes weight optimization for each slice. The peak power of the pulses is controlled using a slice phase optimization technique. Simulations were performed at 7T to compare the peak power of optimized multiband spokes pulses to unoptimized pulses, and to compare the proposed slice-independent spokes weight optimization to a joint approach. In vivo experiments were performed at 7T to validate the pulse’s function and compare them to conventional multiband pulses.

Results

Simulations showed that the peak power-minimized pulses had lower peak power than unregularized and integrated power-regularized pulses, and that the slice-independent spokes weight optimization consistently produced lower flip angle inhomogeneity and lower peak and integrated power pulses. In the brain imaging experiments, the multiband spokes pulses showed significant improvement in excitation flip angle and subsequently signal homogeneity compared to conventional multiband pulses.

Conclusion

The proposed multiband spokes pulses improve flip angle homogeneity in simultaneous multislice acquisitions at ultra-high field, with minimal increase in integrated and peak RF power.

Introduction

In recent years advancements in high field and parallel imaging techniques have provided significant momentum to simultaneous multislice (SMS) imaging [1, 2]. To realize SMS imaging, multiband (MB) pulses are used to simultaneously excite multiple slices using a single RF excitation pulse. This is of high interest for scan time reductions, especially in functional MRI and diffusion-weighted imaging at high main field strength [1–3] which brings intrinsic improvements in signal-to-noise ratio (SNR), blood oxygenation level dependent contrast and spatial resolution. However, at ultra-high field strengths (7 T and higher), nonuniform and subject-dependent transmit RF ($B_1^{+}$) fields lead to large spatial variations in image contrast [4]. Imaging at high field also poses potential safety risks due to elevated specific absorption rate (SAR) caused by increased RF power deposition which increases with transmit RF frequency [5, 6]. In SMS imaging, the effects of high field are compounded by the fact that the peak power of a MB pulse increases with the square of the number of simultaneously excited slices [7, 8]. This is a concern for spokes pulses, which inherently demand a high peak power due to the necessary short slice-selective subpulse durations, so it can be difficult to simultaneously achieve low peak power and homogeneous flip angle. Thus, to accomplish widespread application of MB pulses at high fields, there is a need for a solution to overcome the challenge of nonuniform $B_1^{+}$ fields while controlling peak RF power.

Previously, Wu et al [9] have described a method for RF shimming of MB pulses in both single and parallel transmit settings. They introduced two approaches to design the RF amplitude and phase weights (collectively called shim weights) called ‘MB B1 Shim’ and ‘Full pTx MB’. In the simpler ‘MB B1 Shim’ approach, channel-specific RF shim weights are jointly calculated for all the excited slices. In the ‘Full pTx MB’ approach, the shim weights are independently calculated for each transmit channel and each excited slice. It was shown that independently designing the shim weights provides a more homogeneous flip angle pattern compared to the joint design. They also demonstrated a two-spoke extension of this concept, with fixed target phase and excitation k-space trajectory and a hard peak-power constraint built into the pulse design problem [10]. Here, we report an analogous approach to design multidimensional multiband pulses by extending the single-slice spokes [11] pulse. The proposed pulse is henceforth denoted as ‘MB spokes’. In addition to the shim weights, we optimize the spokes excitation k–space trajectory and the target in-plane phase pattern for each slice. To design MB spokes pulses, the ‘interleaved greedy and local’ optimization method [12] for small-tip-angle spokes pulse design was extended to the multiband case. This method was chosen because it is computationally efficient and allows the incorporation of off-resonance effects due to main field inhomogeneity, which are amplified at high fields. It also jointly optimizes the RF pulses, the target excitation phase patterns, and the excitation k-space locations visited by the spokes trajectory. For the computation of shim weights the ‘Full pTx MB’ approach proposed by Wu et al [9] was adapted for MB spokes. Wu et al also demonstrated the use of a regularization term in the pulse design cost function to control the integrated power of the shimmed MB pulse. However, this technique does not regulate peak RF power. It has been shown that peak RF power can be reduced by optimizing the relative phases of the slices excited by a multiband pulse prior to summing the pulses [7, 8]. We generalize this method to the MB spokes pulses. For a given slice, the spokes subpulses have phase relationships that are optimized to achieve a uniform excitation pattern. Therefore, directly changing the phase of the slice-select sub-pulses will destroy these inter-spoke phase relationships leading to non-uniform excitation. To address this problem we propose a straightforward extension of the optimal excitation phase method to achieve reduced peak RF power of MB spokes pulses.

We show in 7 T in vivo experiments and single-channel and two-channel parallel transmit simulations that MB spokes pulses improve flip angle homogeneity in multiple simultaneously excited slices compared to conventional multiband excitations. It is demonstrated that spokes pulses designed using a ‘Slice-Independent’ design approach analogous to Wu’s ‘Full pTx MB’ approach gives a more uniform excitation pattern than a ‘Slice-Joint’ design approach analogous to Wu’s ‘MB B1 Shim’. Simulation results show that the maximum power of peak power-minimized pulses increases more gradually with the number of excited slices than unregularized and integrated power-regularized pulses. A preliminary report of this work was presented in [13].

Theory

A spokes pulse comprises of a train of N_s slice-selective RF subpulses that are played concurrently with slice-select gradient trapezoids, and are separated in time by small transverse-plane gradient blips that move the excitation k-space trajectory to different locations in the in-slice spatial frequency dimensions. Such a pulse can be extended to simultaneously excite multiple slices by replacing the single-slice-selective subpulses with multiband waveforms. Here we present an extension of a single-slice spokes pulse design method [12] to multiband pulses. Though Ref. [12] describes a pulse design method for multiple off-resonance frequencies, for simplicity we will discuss the MB spokes pulse design for a single isochromat at each spatial location.

Slice-Independent Design of MB Spokes Pulses

Here we introduce the ‘Slice-Independent’ design approach to MB spokes pulse design that optimizes shim weights for each slice independent of the others, allowing each slice’s excitation pattern to be tailored to its local $B_1^{+}$ and B₀ fields. Figure 1 illustrates this approach in comparison to a ‘Slice-Joint’ design approach wherein the spokes subpulse weights are solved for jointly across slices.

Figure 1.

Illustration of ‘Slice-Joint’ and ‘Slice-Independent’ design approaches to multiband spokes pulse design. The ‘Slice-Independent’ approach designs separate RF amplitude and phase weights for each single-slice subpulse, whereas in the ‘Slice-Joint’ approach a single amplitude and phase are determined for all slices simultaneously. Both independent and joint designs start with a single slice-selective pulse which is modulated to produce pulses for the N_sl number of desired slices. In the ‘Slice-Joint’ design approach the single-slice pulses are summed to obtain a multiband subpulse which is copied for the desired N_s number of spokes. The optimization routine optimizes the shim weight and excitation k-space trajectory of each multiband spoke to produce the final MB spokes pulse. In the ‘Slice-Independent’ approach the slice-select pulses are copied over to N_s spokes to produce N_sN_sl number of subpulses. The design algorithm independently optimizes the shim weights for each of the N_sN_sl subpulses and jointly optimizes the k-space trajectory. The final MB spokes pulse is obtained by summing the optimized slice-select spokes subpulses.

For N_s subpulses that simultaneously excite N_sl slices using N_c transmit channels, with $B_1^{+}$ fields ${\{s_c^m(x,y)\}}_{m=1}^{N_{sl}}$ where m is the slice index, the goal of ‘Slice-Independent’ multiband spokes pulse design is to determine N_sN_cN_sl subpulse weights and N_s transverse-plane (k^x-k^y) excitation k-space locations to be visited by the trajectory, which together produce a uniform flip angle pattern in each slice. The flip angle pattern excited in the center of slice m by a small-tip-angle single-slice spokes pulse is represented by θ_m and is given by equations [1] and [2] in Ref. [12]. The system matrix for slice m is represented by A_m (𝕂) and is given by equation [3] in Ref. [12]. For N_c transmit channels, a combined system matrix for slice m, denoted Ã_m (𝕂), can be constructed as:

$${\tilde{\mathit{A}}}_m(𝕂)=[{\mathit{S}}_{1,m}{\mathit{A}}_m(𝕂)\dots {\mathit{S}}_{N_c,m}{\mathit{A}}_m(𝕂)],$$ [1]

where ${\mathit{S}}_{c,m}=\text{diag}\{s_c^m(x^m,y^m)\}$. And a single length-N_sN_c column vector of shim weights for slice m is similarly constructed by concatenating the shim weight for all the coils:

$${\tilde{\mathit{b}}}_m={[{\mathit{b}}_{1,m}^T\dots {\mathit{b}}_{N_c,m}^T]}^T.$$ [2]

With these definitions θ_m can be equivalently written as:

$${\mathit{\theta }}_m={\tilde{\mathit{A}}}_m(𝕂){\tilde{\mathit{b}}}_m.$$ [3]

With the discretized system model defined, the MB spokes pulse design problem is cast as a penalized least squares problem:

$$\underset{\mathit{k},\tilde{\mathit{b}},\varphi }{\text{min}}\sum _{m=1}^{N_{sl}}[\frac{1}{2}{\Vert {\tilde{\mathit{A}}}_m(𝕂){\tilde{\mathit{b}}}_m-{\mathit{d}}_m({\varphi }_m)\Vert }^2+\frac{\lambda }{2}{\Vert {\tilde{\mathit{b}}}_m\Vert }^2]$$ [4]

where d_m is the length- $N_x^m$ desired flip angle pattern vector for each slice, with target phase pattern ϕ_m. We note that $N_x^m$ counts the number of voxels containing tissue in slice m, and can be different for each slice. For $B_1^{+}$ inhomogeneity-compensating pulse design the d_m patterns have uniform magnitude inside the slice’s brain masks, and the ϕ_m are optimized independently for each slice. The second term in the objective function is a Tikhonov regularizer that penalizes the ℓ₂ norm of each slice’s shim weights, where λ is a tunable factor that determines the relative weighting given to the regularization term. In the supplemental materials we show that $\sum _{m=1}^{N_{sl}}{\Vert {\tilde{\mathit{b}}}_m\Vert }^2$ is proportional to the integrated power of the coil-combined time-domain pulses.

The design problem in Eq. 4 is solved using the ‘interleaved greedy and local optimization’ method for single-slice spokes design [12]. In each iteration, the shim weights b̃_m are calculated slice-by-slice by calculating the pseudo-inverse of each slice’s contribution to the objective function. The target phase profiles ϕ_m are updated slice-by-slice using Eq. 6 from Ref. [12]. The excitation k-space locations 𝕂 of the spokes are updated using Eq. 9 from Ref. [12], where all the slices contribute to the gradient calculations. Once the algorithm converges to the optimal shim weights ${\{{\tilde{\mathit{b}}}_m\}}_{m=1}^{N_{sl}}$, the MB spokes pulses for each transmit channel are computed by summing the individual spokes pulses across slices.

Peak Power Minimization

The peak power of a conventional multiband pulse can be minimized by applying optimized ex-citation phases to each band [7, 8]. This strategy can be extended to MB spokes by optimizing the relative phases of the individual slices’ spokes pulses. After the ‘interleaved greedy and local optimization’ method has converged to obtain the shim weights ${\{{\tilde{\mathit{b}}}_m\}}_{m=1}^{N_{sl}}$, excitation phases that reduce the peak power can be obtained by solving the optimization problem:

$$\underset{\mathit{\alpha }}{\text{min}}\underset{n,c,i}{\text{max}}\left|\sum _{m=1}^{N_{sl}}{p}_i^m{b}_{c,n,m}{e}^{ı{\alpha }_m}\right|,$$ [5]

where $p_i^m$ is sample i of the slice-select pulse for slice m and the ‘max’ operation returns the maximum RF amplitude across spokes (n), coils (c), and time points (i), after summing the individual N_sl slice-selective spokes pulses. This problem can be solved to obtain the N_sl optimal excitation phases ${\{{\alpha }_m\}}_{m=1}^{N_{sl}}$ using the MATLAB (MathWorks Inc., Natick, MA, USA) fminsearch function (Nelder-Mead simplex [14]). A feature of this method is that it does not require hand-tuning of regularization.

Overall Algorithm

The overall pulse design approach described here has two steps. In the first, the extended interleaved greedy and local spokes pulse design method is used to solve for 𝕂, ${\{{\tilde{\mathit{b}}}_m\}}_{m=1}^{N_{sl}}$, and ${\{{\varphi }_m\}}_{m=1}^{N_{sl}}$. The output of this step is a set of spokes pulses for each slice that have been regularized for low integrated RF power. These spokes pulses are input to the second step which solves for the optimal phases ${\{{\alpha }_m\}}_{m=1}^{N_{sl}}$ using the method described above to reduce the peak power of the slice-combined pulses.

Methods

Pulse Design Parameters

The following simulations and experiments were carried out with conventional MB and MB spokes pulses designed in MATLAB running on a MacBook Air computer (Apple Inc., Cupertino, CA, USA) with a 1.7 GHz Intel Core i7 processor and 8 GB RAM. All slice-selective subpulses were designed using the Shinnar-Le Roux [15] algorithm and had a time-bandwidth product of 2. Fly-back spokes trajectories were used, and RF and gradient waveforms were sampled with a dwell time of 6.4 μs. All gradient trapezoids were designed subject to a maximum amplitude of 15 mT/m and maximum slew rate of 120 T/m/s. For ‘Slice-Independent’ MB spokes designs, peak power minimization was implemented by initializing the α_m to zero, and applying MATLAB’s fminsearch to optimize them using 30 restarts, each with its own uniformly-distributed random initialization. Unless otherwise noted, MB spokes pulses comprised five spokes subpulses of duration 1.8 ms each, and had a total duration of 8.8 ms. Figures 3(c,d) show that the knees of the flip angle standard deviation and power versus number of spokes curves occur around five spokes, indicating that this number of spokes gives a good balance between flip angle uniformity, power, and pulse duration.

Figure 3.

Parallel transmit simulations using measured $B_1^{+}$ maps of a 2-channel head volume coil, shown in (a). (b) The 5 spokes ‘Slice-Independent’ approach yields more uniform patterns compared to ‘Slice-Joint’ design. (c) Plot of predicted flip angle standard deviation versus number of spokes for 2-channel ‘Slice-Joint’- and ‘Slice-Independent’-designed pulses. For any given number of spokes, the ‘Slice-Independent’ approach has the lowest flip angle standard deviation. Pulse design times are also shown. The design times for the 5-subpulse MB spokes pulse with ‘Slice-Joint’ and ‘Slice-Independent’ design were 3.4 s and 6.2 s, respectively. (d) Plot of normalized integrated and peak RF powers versus number of spokes for the ‘Slice-Joint’- and ‘Slice-Independent’-designed pulses. For any given number of spokes, the ‘Slice-Independent’ method produces pulses with lower integrated and peak power than the ‘Slice-Joint’ approach.

Simulations

Peak Power Minimization

The peak power minimization strategy was tested in simulations of single-channel transmit and two-channel parallel transmit by measuring the increase in peak RF power of MB spokes pulses with the number of excited slices. Pulses were designed with the ‘Slice-Independent’ approach to excite a flip angle of 30 degrees in three to nine axial brain slices of 3 mm thickness. The $B_1^{+}$ maps for pulse design were measured in a healthy volunteer in 9 axial brain slices with a slice gap of 10 mm and slice thickness of 2 mm on a 7 T Philips Achieva scanner (Philips Healthcare, Cleveland, OH, USA) with a 32 receive channel head coil and a two-channel birdcage transmit coil (Nova Medical Inc., Wilmington, MA, USA), with the approval of the Institutional Review Board at Vanderbilt University. The ‘STE first’ DREAM $B_1^{+}$ mapping sequence [16] was used to map each transmit channel independently, where the nominal flip angle of the stimulated echo preparation pulse was 40 degrees and the flip angle of the imaging pulse was 15 degrees. The maps were acquired over a 20 × 20 cm in-plane field-of-view (FOV) and a 64 × 64 matrix size. The increase in peak RF amplitude versus number of excited slices was compared among pulses designed with no power regularization, integrated power regularization only, and integrated power regularization plus peak power minimization. The pulses with no regularization were computed by setting λ = 0 in Eq. 4. The integrated power-regularized-only pulses were computed by skipping the peak power minimization step.

‘Slice-Joint’ versus ‘Slice-Independent’ Design

To investigate the advantages of the ‘Slice-Independent’ design approach, pulses designed by that method were compared with pulses designed using the ‘Slice-Joint’ approach. Pulses with 1 to 50 spokes subpulses were designed to excite three slices at 20 mm slice gap, and the predicted flip angle standard deviation and integrated and peak RF powers were recorded for the comparison.

In Vivo Experiments

In vivo experiments were performed in 2 subjects at 7 T, using the same two-channel Nova birdcage coil driven in its single-channel circularly-polarized mode. Experiments were performed in axial and coronal orientations. $B_1^{+}$ and off-resonance maps were measured for 3 slices spaced 2 cm apart in the axial orientation and 3 cm apart in the coronal orientation with 64 × 64 matrix size. The in-plane FOV’s were 20×20 cm for axial slices and 21×21 cm for coronal slices. The Bloch-Siegert [17] method with optimized encoding pulses [18] was used for $B_1^{+}$ mapping, and off-resonance maps were measured using the double echo-time (TE) method [19] with a TE difference of 1 ms. A tissue mask was formed for pulse design for the transverse slices by manually choosing a threshold level. Since brain masks are more difficult to form manually in coronal slices, the Otsu’s image segmentation algorithm [20] was used to automatically calculate thresholds. Conventional MB pulses and MB spokes pulses with the ‘Slice-Independent’ approach were designed to excite a flip angle of 75 degrees in the three slices with thickness 3 mm each.

Images were collected using the pulses for excitation, followed by a 3D gradient echo readout with 144 × 144 matrix size and TR/TE of 150/6.8 ms. The flip angle excited by the pulses was measured using a 3D gradient echo Actual Flip Angle (AFI) [21] mappping sequence with 6×6×3 mm resolution, 114 × 114 matrix size and TE/TR₁/TR₂ of 7.3/100/400 ms. Due to slice profile effects [22] the measured AFI maps were biased down, so the maps were normalized by their median value for comparison to the predicted patterns. Aliased slice images were also acquired by exciting with multiband pulses followed by a 2D Cartesian readout. The aliases were separated using the slice-GRAPPA reconstruction algorithm [23]. Calibration data for the GRAPPA kernel were calculated from 3D images acquired using the conventional multiband and MB spokes pulse, respectively. The separated images were normalized by the coil receive sensitivity maps which were estimated by dividing out the $B_1^{+}$ maps from low-resolution small-tip single-spoke images.

Results

Simulations

Peak Power Minimization

Figure 2(a) shows a plot of the increase in peak RF amplitude versus number of excited slices for the unregularized, the integrated power-regularized and the peak power-minimized single-channel MB spokes pulses. The peak amplitudes of the unregularized and integrated power-regularized pulses rise almost linearly with the number of slices. In comparison, the peak power-minimized pulses show a more gradual increase, and a 9-slice pulse had only 1.64× higher peak amplitude than a 3-slice pulse. As expected, the peak power minimization did not increase the integrated power of the MB spokes pulse.

Figure 2.

Increase in peak RF amplitude of 5 MB spokes pulses using ‘Slice-Independent’ design versus number of excited slices. Peak RF power of the unregularized, integrated power-regularized and peak power-minimized MB spokes pulses are compared. Single channel and parallel transmit $B_1^{+}$ maps and simulation results are shown for a target mean flip angle of 30 degrees. (a) Single transmit channel results: The peak power-minimized pulses show a drastic reduction of peak power compared to the unregularized and integrated power regularized pulses. (b) 2-channel parallel transmit results: Again, the peak power-minimized parallel transmit MB spokes pulses have the lowest peak RF amplitude.

Figure 2(b) shows the same plot for the two-channel parallel transmit MB spokes pulses. The peak power-minimized pulses showed the most gradual increase in peak power with the number of slices, and a 9-slice pulse had only 1.96× higher peak amplitude than a 3-slice pulse. As expected, the peak power minimization did not increase the integrated power of the parallel transmit MB spokes pulse.

‘Slice-Joint’ versus ‘Slice-Independent’ Design

Figure 3(b) shows that among conventional multiband, ‘Slice-Joint’ and ‘Slice-Independent’ MB spokes pulses, the ‘Slice-Independent’ MB spokes pulses produced qualitatively the most homogeneous excitation patterns. The plot of flip angle standard deviation versus number of spokes in Fig. 3(c) shows that for any given number of spokes the ‘Slice-Independent’-designed two-channel MB spokes pulse produced a more uniform excitation compared to the ‘Slice-Joint’-designed pulse. As more spokes are added the flip angle standard deviation is reduced for both the methods. The pulse design times are also shown. The design times for the 5-spoke MB spokes pulses with ‘Slice-Joint’ and ‘Slice-Independent’ design were 3.4 s and 6.2 s, respectively. The mean compute time for peak-power optimization was 0.17 s. Figure 3(d) shows that for any given number of spokes the ‘Slice-Independent’-designed pulses have lower integrated and lower peak power than the ‘Slice-Joint’-designed pulses.

Experiments

Figures 4 and 5 show results from axial and coronal head scans conducted in two human subjects. In both subjects the MB spokes images show a significant reduction in the center brightening artifact which obscures the anatomical information near the center of the brain when conventional MB pulses are used. Both axial and coronal MB spokes patterns show a significant improvement over the conventional MB patterns which are marred by the center-brightening artifact caused by the non-uniform $B_1^{+}$ field. The flip angle standard deviation is smaller in all the slices with MB spokes pulses. Compared to conventional pulses, in the MB spokes patterns the flip angle standard deviation was reduced by 31.2% averaged across all the slices in both the subjects. Also, the measured patterns match well with the predictions.

Figure 4.

$B_1^{+}$ maps, images and predicted and measured flip angle maps from in vivo axial slices. Excitations using conventional MB and MB spokes pulses with 5 subpulses and ‘Slice-Independent’ design are compared. Arrows point to regions with center brightening artifact in conventional MB images which are reduced when the MB spokes pulse is used for excitation. The measured AFI maps match well with the predictions and MB spokes show a more uniform excitation pattern than conventional MB pulse. The flip angle standard deviation was lower for all the slices acquired using MB spokes pulse compared to conventional MB.

Figure 5.

$B_1^{+}$ maps, images and predicted and measured flip angle maps from in vivo coronal slices. Excitations using conventional MB and MB spokes pulses with 5 subpulses and ‘Slice-Independent’ design are compared. Acquired aliased images, 3D images and slice images separated using slice-GRAPPA [23] reconstruction are shown. Arrows point to center brightening in conventional multiband images that are reduced when the MB spokes pulse is used for excitation. The separated images match well with the acquired 3D images. The measured AFI maps match well with the predictions and MB spokes show a more uniform excitation than conventional MB pulses. The flip angle standard deviation was lower for all the slices acquired using MB spokes pulse compared to conventional MB.

Figure 5 also shows the slice images separated in reconstruction along with the acquired 3D and aliased images. The reconstructed slice images show good correspondence with the 3D images and in both cases the MB spokes pulse gives a reduced center brightening artifact compared to images excited with the conventional pulse.

Discussion

We have introduced an approach to design multi-dimensional multiband excitation pulses that mitigate flip angle variations due to inhomogeneous $B_1^{+}$ fields in multiple simultaneously-excited slices. MB spokes have all the advantages of the pulses proposed by Wu et al in Ref. [9] when designing small-tip pulses and also inherit the benefits of the interleaved and greedy method from Grissom et al [12]. The pulses can also be implemented with a single transmit channel.

Though bipolar gradients enable shorter pulse durations, flyback gradients were used in the presented results since it was found that the excitation patterns produced by pulses with bipolar gradients were degraded in comparison. We believe those errors were primarily due to zeroth- and first-order eddy current effects, which could be compensated using gradient trajectory measurement [24] or predistortion [25, 26]. The proposed algorithm is fully compatible with bipolar gradient trajectories.

The results from our single-channel experiments show that ‘Slice-Independent’-designed MB spokes pulses yield reduced center brightening compared to conventional MB pulses. The pulses can be implemented with a single transmit channel. The simulation results presented in the supplemental materials demonstrate that for the same number of subpulses, MB spokes gains from the use of multiple transmitters. In the slice-GRAPPA [23] reconstructions of 3× slice-accelerated acquisitions using MB spokes excitations, the separated slices show a reduction in the center-brightening compared to conventional multiband pulse excitations. The supplemental material also shows MB spokes slice profiles measured in a human head. The MB spokes pulse excited the intended slices at the desired slice gaps.

The integrated RF power was minimized using a regularization scheme and we proposed a straightforward method to minimize the peak power of the pulses. This approach was taken because it is simple to implement and worked consistently in all simulations and experiments. An alternative approach would be to extend the pulse design problem (Eq. 4) to incorporate a peak-power regularization term in the cost function directly, which may enable still lower peak power via simultaneous adjustment of k-space locations, target excitation phase, and subpulse weights, at the cost of increased algorithm complexity [10]. The additional regularization term would be:

$$\frac{\beta }{2}\underset{n,c,i}{\text{max}}\left|\sum _{m=1}^{N_{sl}}{p}_i^m{b}_{c,n,m}\right|.$$ [6]

A problem formulation with a hard peak power constraint could also be used. Another approach to reducing the peak power of multiband RF pulses is to shift the pulses in time with respect to each other, so that their main lobes do not overlap [27]. This approach could be incorporated into the proposed MB spokes design algorithm by replacing the slice-selective subpulses for each slice with time-shifted pulses. We note that constraining the peak power of the multiband pulse does not directly limit the local and global SAR values, which can be explicitly controlled by incorporating SAR matrices in the pulse design problem [28].

Conclusions

We have presented a fast method to design low peak power patient-tailored multiband spokes pulses to mitigate $B_1^{+}$ inhomogeneity at high field. In vivo experiments showed that the MB spokes pulses improve the flip angle homogeneity in all excited slices. Compared to conventional pulses, with a 5-subpulse MB spokes pulse the flip angle standard deviation was reduced by 31.2% averaged across all the slices in all the subjects. Simulations with in vivo field maps demonstrated that the algorithm was able to mitigate increased peak RF power as the number of excited slices increased. For a peak-power minimized two-channel pulse that excites 9 slices, the peak amplitude increased by 1.96× of the peak amplitude of a 3-slice pulse, compared to 3.27× for an unregularized MB spokes pulses. The use of multiple transmit channels further improved the performance of MB spokes pulses. Simulation results show that a MB spokes pulse with 5 subpulses gave a flip angle standard deviation of 1.1%. MB spokes pulses are expected to be most useful for applications such as high field BOLD fMRI and diffusion weighted imaging (DWI) where multiple slices need to be imaged within a short acquisition time window.