Stochastic‐offset‐enhanced restricted slice excitation and 180° refocusing designs with spatially non‐linear ΔB0 shim array fields

Developing a general framework with a novel stochastic offset strategy for the design of optimized RF pulses and time‐varying spatially non‐linear ΔB0 shim array fields for restricted slice excitation and refocusing with refined magnetization profiles within the intervals of the fixed voxels.


INTRODUCTION
Fetal MRI suffers from unpredictable and substantial fetal motion that also precludes many standard diagnostic contrasts used in adult brain imaging. 1 Clinical fetal scans use fast MRI sequences, like half-Fourier single-shot rapid acquisition with relaxation enhancement, 2,3 to encode a single-slice in ∼500 ms, significantly decreasing sensitivity to fetal motion artifacts.Despite the rapid acquisition, inter-and intra-slice motion during encoding still routinely degrades image quality, requiring repeated scans for reacquisition. 4To further improve motion robustness, reduced FOV (rFOV) fetal brain imaging could substantially reduce acquisition time because the region of interest (ROI), the fetal brain slice, occupies only a small fraction of the gravid abdomen and requires many fewer phase-encoding to fully sample.With fast fetal brain tracking and pose estimation techniques, 5,6 2D spatially selective pulses could excite a slice restricted to the fetal brain in real time.
For small-tip-angle excitation designs of a restricted slice or arbitrarily shaped magnetization patterns, Pauly et al. 7 linearized the Bloch equation and exploited the Fourier relationship between RF pulse in excitation k-space and the spatial transverse magnetization profile.][10][11] To address the lengthy durations of the resultant RF pulses, UNFOLD 12 is combined with 2D RF pulse designs to control the aliasing from shorter RF pulses. 13][19] To address the nonlinearity of the Bloch equations for large-tip-angle excitations, the Shinnar-Le Roux (SLR) algorithm proposed a mapping of RF pulses to pairs of polynomials that represent Cayley-Klein parameters with the hard pulse assumption. 20,21Later, Setsompop et al. 22 proposed a two-stage design with an initial (small-tip) approximation refined through iterative Bloch optimization.Other methods include dividing the large-tip angle design into a number of additive small-tip-angle RF pulse designs, 23 optimal control-based techniques combined with the small-perturbation approximation by Grissom et al. 24,25 and 3D tailored excitation with auto-differentiation and time-discrete Bloch simulator by Luo et al. 26 Auto-differentiation tools break down computations into stages and generate Jacobian operations for each stage.These Jacobians for each stage are then merged using the chain rule.
Recent developments in independently driven multi-coil shim arrays 27,28 provide additional design flexibility for selective excitations, which has been demonstrated in mouse brains 29 and in the visual cortex and peripheral cerebrum. 30In particular, Zhang et al. 31 incorporated the time-varying spatially non-linear ΔB 0 shim fields into the voxelwise auto-differentiation design of 2D and 3D spatially selective excitations with short RF duration in the imaging application of pregnancy.Compared with the Cayley-Klein polynomial framework, which assumes a static and linear gradient field, the time-discrete Bloch simulator offers an approach to incorporate a more general set of ΔB 0 fields into the excitation and refocusing design, including spatially varying non-linear, and time-varying multi-coil shim fields.However, even with the non-linear shim array fields, spatially selective excitation for a 2D restricted slice with short RF duration requires large coil currents and still suffers from leakage near the transition regions.For the 2D spin echo imaging of a restricted slice of the fetal brain, a spatially selective 180 • refocusing pulse would help mitigate aliasing by eliminating accidentally excited magnetizations in the suppressed regions (stop-bands) while refocusing the magnetizations in the desired ROI (fetal brain slice).
The previous designs 31 suffer from two key issues.First, the design is formulated as a voxelwise optimization problem with fixed spatial points for evaluating the loss function.However, this approach fails to adequately excite magnetization profiles within the intervals of the fixed locations, resulting in suboptimal designs.Second, the previous method requires specification of exact initial magnetization that is not compatible with refocusing pulse design.In this work, we introduce a novel design framework that optimizes non-linear, spatially time-varying ΔB0 fields and RF pulses for both spatially selective excitation and 180 • refocusing.
Our approach extends prior work on selective fetal brain excitation 31 by incorporating the decomposition property of the time-discrete Bloch simulator into auto-differentiation optimization to enable restricted slice excitation and refocusing with arbitrary initial magnetizations.To capture the effect of idealized crusher gradients in refocusing designs, we use magnetization requirements for the optimization derived from the extended phase graph (EPG) approach, 32 which avoids costly sub-voxel isochromatic simulations.We also propose a stochastic offset strategy that randomly shifts spatial points that are used to evaluate the loss function during optimization to refine resultant slice profiles.This differs from prior optimization methods that relied on fixed spatial points for evaluating the cost function, which led to undesired magnetizations in the intervals between fixed points.
Simulation experiments use a single-channel, quadrature RF excitation transmit model with assumed uniform B1+ field to model conventional body coils.We first applied our proposed framework to design standard slice-selective excitation and refocusing pulses that closely match pulses designed with the traditional SLR algorithm.Next, we demonstrated the use of our technique in a simulated set of shim-array coils for MRI in pregnancy by jointly designing RF-pulses and non-linear shim array fields for restricted excitation and refocusing of the fetal brain in the slice and phase-encoding directions.Both the selective excitation and refocusing designs achieved high performance under a variety of metrics.

General design framework for selective excitation and refocusing
The previous auto-differentiation framework 26,31 required specification of the exact initial and final magnetizations for optimization.However, refocusing pulse design requires consideration of undetermined initial magnetization because of factors such as B 0 inhomogeneity, etc.In this work, we propose a general framework that uses the decomposition property of the time-discrete Bloch-simulator to enable voxelwise auto-differentiation optimization with arbitrary initial magnetizations.

Decomposition property
For RF excitation and refocusing process, the T 1 and T 2 effects are often assumed to be negligible for design regimes where the pulse duration  ≪ T1, T2.Under this assumption, in a given voxel, the time-discrete Bloch simulator can be written in the form of sequential rotations that is a linear operator that represents arbitrary initial magnetizations with a combination of basis vectors where is the total time-discrete Bloch operator and R is the rotation matrix determined by the total z-directed magnetization field, B(t), and the transverse RF-pulse field at time t.Note that B(t) = B 0 (t) + ΔB 0 (I(t)) with the presence of the current-driven spatially non-linear shim array coil field ΔB 0 (I(t)).For notational simplicity, we omit reference to the spatial position of the voxel here.M 0 and M T are the initial and resultant magnetization before and after the forward Bloch simulation, respectively.M 0x , M 0y , M 0z are the three spatial components of the magnetization vector.

Modeling for excitation
For excitation designs, the initial magnetization is assumed to be ] for all voxels and the corresponding final target magnetization is ] in the ROI (pass-band) and in the suppressed regions (stop-band), assuming excited to x-axis.

Modeling for refocusing with crusher gradient
To address the issues of undetermined initial magnetizations for refocusing design, the three independent basis vectors mentioned above are used as the exact input for the voxelwise auto-differentiation optimization, where is the resultant magnetization after applying RF and B(t) to the corresponding initial magnetization.
The requirements could be derived either from EPG with a mapping between Mx, My, and Mz basis and Mxy, Mxy*, and Mz basis, or mathematics in the Mx, My, and Mz basis.Details are given in the Appendix A.

Stochastic offset strategy
The previous auto-differentiation pulse designs 26,31 used fixed and equally spaced points, { r i,,k } , based on the resolution and FOV, for the computation of the voxelwise loss function during optimization.However, even when the resultant slice profiles achieve high performance on the selected, fixed points of evaluation, there are no performance guarantees on locations in-between these fixed points, which can yield undesired features of the design.Alternatively, here, we propose a stochastic offset strategy to better impose the design constraints on the magnetization across the entire spatial domain.This approach is more memory efficient than simply increasing the resolution of the fixed-point approach but requires more optimization iterations.
To implement this stochastic offset method, for each optimization iteration, a new offset Δr i,,k is applied to each fixed point.As a result, the cost function is calculated on the point set { r i,,k + Δr i,,k (iter) } .Inspired by the stochastic gradient descent method, the offset Δr i,,k is chosen independently and randomly from a uniform distribution in the range of for each optimization iteration.To help improve the convergence of this optimization, an adaptive learning rate is used.
For the slice profile performance evaluation, 80 equally spaced points within a voxel (distributed along one diagonal line) are evaluated and the magnetizations of these 80 points are summed up to represent the net magnetization of the voxel.We note that this is a good approximation of dense Cartesian-sampled points (512 sampled points), whereas the simulation time is less (fewer sampled points), as shown in Figure S7.

Design and modeling
This work designs the RF pulse b(t) and current in the shim coils I(t), which generate the time-varying non-linear fields ΔB 0 (x, y, z, t) = ∑ n coil i=1 I i (t)ΔB i 0,coil (x, y, z), for selective excitation and refocusing.Similar to the previous work, 30 we numerically designed a 64-channel body shim array with a simple heuristic to densely distribute the 16-cm diameter coils at a distance of 5 cm away from the mother's body, and with approximately equal surface density as shown in Figure 1A. Figure 1B-D show the spatially non-linear field patterns for the chosen coils.
Cartesian sampling is assumed for the selective spin echo imaging and the aliasing de-selection method 33 is adopted where only regions that get aliased into the ROI are suppressed, denoted as ROI interferers.Other regions will be aliased into non-ROI regions in the FOV.The reduction of suppressed regions (stop-band) will improve restricted-slice performance.
The framework is based on the time-discrete Bloch simulator and optimized with auto-differentiation.The linear gradient is used but fixed during the RF process.The cost function is modeled as below, argmin where f is a L2 norm cost function; M D is the desired excitation pattern; M 0 is the initial magnetization; M Txy is the actual transverse excitation profile generated by the RF pulse and the time-varying spatially nonlinear ΔB 0 field under the time-discrete Bloch simulator; The sets of {M 0 , M D } are different for the excitation and the refocusing design.r is the random spatially stochastic offset to every fixed point at each iteration.W is a voxelwise hyper-parameter to implement spatially varying weights for the target voxels in ROI and background; ℛ 1 is the infinity norm regularizer of coil currents with weight λ 1 to encourage minimization of the maximum current in the coil array; ℛ 2 is the L2 norm regularizer, with weight λ 2 , of the initial and final values of coil currents at the beginning and end of the RF waveform to minimize unwanted accumulated phase because of any non-zero coil current at the start and end of the RF; ℛ 3 is the L2 norm regularizer that controls RF power with weight λ 3 ; I is the vector of coil currents; b is the RF pulse; n t is the number of time points; n coil is the number of shim coils in the array; b max is the maximum RF power constraint; I max is the maximum absolute current constraint for each coil in the array, and s max is the bandwidth constraint for the coil current.
There are two optimization stages for the spatially selective excitation: selectively excite the ROI regardless of the phase distribution followed by a rewinding segment to maximize phase coherence of spins within the ROI through the application of gradient and shim-array fields.For the excitation stage, only the magnitude of M Txy and M D is needed for the computation of the objective function f with an initial magnetization ; for the rewinding stage, only the coil current I is optimized to provide time-varying rewinding ΔB 0 field starting with the end of current at the excitation stage and the phase of M Txy and M D is used for the computation of the objective function f with the initial magnetization as the resultant magnetization from the excitation stage.RF is set to zero during rewinding, and for both stages, the same set of regularizations are used.This two-stage excitation design is compared to a self-rewinding excitation design that excites phase-aligned magnetization in the ROI without the need for a dedicated rewinding stage after the application of the RF waveform.
For the selective refocusing design, the two different initial magnetizations ered for the optimizations as described above.We assume the magnetizations rotate around the x-axis in the rotating frame at the Larmor frequency.The complex value of M Txy and M D is used for the computation of the objective function f .

Simulated model parameters
In this work, numerical simulations are performed to demonstrate the performance of our proposed design framework for both selective excitation and refocusing enabled by the time-varying, spatially non-linear shim array fields ΔB 0 (x, y, z, t).
A model of pregnancy at gestational age of 30 weeks is derived from 3D fetal MRI obtained at the Boston Children's Hospital, from which we obtain the outer contour of the mother's abdomen and the outline and position of the fetal brain.Simulation experiments are conducted on a resolution of 4 × 4 × 4 mm 3 with a voxel grid of 90 × 90 × 60 (resulting in a mother's FOV of 36 × 36 × 24 cm 3 ) as depicted in Figures 1 and 2. The fetal brain's extent is approximately one-quarter of the mother's body dimension along the superior-inferior (S-I) axis and one third along the right-left (R-L) and anterior-posterior (A-P) axes.

Slice selective excitation and refocusing designs (R-L direction)
The proposed decomposition-based design as well as the modeled effects of idealized crusher gradients in our framework is first demonstrated and compared against conventional SLR designs 20 for slice-selective excitation and 180 • refocusing pulses.The stochastic offset strategy is used and compared with optimized RF pulse with fixed-point optimization.A slice in the center of the fetal brain with the slice thickness of 12 mm (3 layers) is selected as the ROI.The width of the transition band is 4 mm straddling the ROI.To comply with constraints of the SLR algorithm, these designs assume a constant linear gradient and zero currents in the shim array coil.
The tradeoff between the larger memory usage with finer resolution and the longer convergence time with coarser resolution is studied with different resolutions and different ranges of the stochastic offset.As the optimization is a highly non-convex problem, the effect of different the RF waveform initializations are also explored.Related results and figures are shown in the Supporting Information because of figure number limits.

3.4
Restricted-slice excitation and refocusing designs Next, our proposed framework is used to demonstrate a restricted slice excitation, for instance, a slice-selective RF pulse with a truncated in-plane ROI in the phase-encoding direction targeted at applications for rapid fetal brain MRI with reduced encoding burden.
As shown in Figure 2, the 2D target for the spatially selective excitation of a restricted slice is a sagittal slice with a thickness of 4 mm in the R-L orientation and a FOV in the A−P phase-encoding direction of 12 cm.This target region of excitation includes the fetal brain and covers approximately one-third of the A-P FOV necessary to image the full extent of the mother's abdomen.The RF pulse excites the full FOV in the S-I readout encoding direction, but S-I aliasing is avoided by adequate sampling rate along the readout direction.The desired excitation target is the intersection of the fetal brain and the excited slice.In the through-slice direction, a pair of 4 mm transition-bands are stacked on either side of the target slice and serve as a buffer between the excited target and the suppressed regions.We adopt the aliasing de-selection technique in the phase encoding direction that exploits "do not care" regions within the FOV. 33

RF pulse, linear gradients, and shim current initializations
The durations for both slice and restricted slice excitation designs are 3 ms in total, 2 ms for the excitation, and 1 ms for the rewinding, with a time resolution of 4 μs.The amplitudes of the fixed linear gradient in the slice direction are 4 mT/m and −2 mT/m, for the excitation and rewinding, respectively.The initialized RF pulse is a Gaussian envelope with a 20 μT peak amplitude.
For slice refocusing, the pulse duration is 4 ms with a time resolution of 8 μs.The amplitudes of the fixed linear gradient in the slice direction are 2 mT/m.The initialized RF pulse is a Gaussian envelope with a 25 μT peak amplitude for the slice selective design.The initial pulse for the restricted slice design is a conventional SLR design that refocuses the fetal brain slice without FOV restriction.

Optimization settings
The optimization is performed on an Nvidia TITAN V graphics processing unit (GPU) card with a 12 GB memory.The L-BFGS algorithm is used with a learning rate of 0.1/null/0.01for the update of the RF pulse and 0.05/0.2/0.005 for the update of the coil current, excitation, rewinding, and refocusing pulses, respectively.We set the number of iterations equal to 300, λ 1 = 1, λ 2 = 1000, λ 3 = 1, b max = 0.025 mT, I max = 60 A. As the number of voxels in the target and the background are not balanced, a judicious choice of the loss weight W improves overall performance in comparison to a uniform loss weighting.The loss weight for the target in the ROI is 12/4/4, for the region in the background that gets excited by the initialization is 2/null/1.1,and 1/null/0.1 for all other voxels.For the slice design experiments, the learning rate is 0.005 for the update of the RF pulse.The loss weight for the ROI is 12 and for the suppressed regions is 0.1.In this case, the aliasing de-selection technique is not needed because it is a 1D design problem.
The overall optimization process takes ∼2 min for the slice design and ∼3 h for the restricted brain designs with Pytorch.

RESULTS
Figures of results present the magnitude and the phase of our optimized RF pulses, time-varying currents in the shim array coils, the corresponding excitation and refocusing profiles of the fetal brain slice, and several performance statistics (phase range, violin plot of the magnetization, maximum and minimum magnitude, and root mean square error [RMSE]).Note, that the violin plots show the magnetizations of all the ROI and ROI interferers in the total 3D volume.

Results of the whole slice design: excitation and refocusing
For the slice excitation design, we use the SLR forward simulation to obtain the slice profiles for both our optimized RFs and the comparison SLR designs.We note that although the SLR transform assumes a hard pulse approximation, whereas our framework uses time-discrete Bloch simulator, the difference in the resultant slice profiles is negligible.Note that for the slice excitation, RF pulse that achieves a linear through-slice phase is preferred as the phase could be easily rewinded by a linear gradient.In this work, we only demonstrated and imposed amplitude constraints without any constraints on the through-slice phase.To achieve a linear phase, relative phase constraints could readily be incorporated into our loss function.
Under the 4-mm resolution constraint, only three fixed points are optimized in the ROI/pass-band.
Figure 3 shows the slice profiles of different methods.Our design framework achieved similar excitation profiles as SLR designs with the minimum-phase filter The results of the experiments of the slice excitation design at 4-mm resolution.The first row (A) shows the optimized pulse in real value (imagine value is almost zero) with min-phase Shinnar-Le Roux algorithm and its corresponding slice profile in magnitude and phase.Note that we use a thinner line for the phase in the stop-band for a better visualization.The second row (B) shows the optimized pulse in magnitude with our method and its corresponding slice profile in magnitude and phase.The phase of the optimized pulse is not zero, but negligible.The red dashed line is the initial Gaussian RF pulse for the optimization.The third row (C) shows the results of the fixed-points method.The fixed points for the optimization locate at 0 ± 700n Hz where n are integers.It could be clearly seen that the profiles at the fixed points are well excited but fail in the interval.
setting.Additionally, the substantially improved slice profile obtained with the stochastic offset method versus the fixed-points design shows the use of this strategy.
For the 180 • slice refocusing design, we optimized the RF pulse with a Gaussian pulse (with a slightly reduced bandwidth compared to the initial condition for the excitation design) as the initial RF pulse for the optimization.The optimized RF pulses are also fed into the SLR forward simulator and the 180 • refocusing slice profiles are calculated assuming the presence of idealized crusher gradients.
Table 1 shows the convergence time and steps used by the stochastic offset strategy and fixed-point method under different resolution.Coarser resolution tends to result in shorter convergence time, but this is not always true.
As Figure 4 shows, the results for the 180 • slice refocusing design demonstrate similar performance characteristics between the SLR and the decomposition framework pulses, both of which incorporate and model idealized crusher gradients.Much like for the excitation design, the stochastic offset method significantly improved the slice profile compared with the fixed-points strategy.

Results of the restricted slice design: excitation and rewinding
Figure 5A-C show the optimized RF pulse and five representative time-varying coil currents.Note that we optimized both the excitation and the rewinding process.Figure 5D shows the distribution of magnetization magnitude in the ROI and the ROI interferers as well as related statistics.Figure 5E shows the rewinding phase across the ROI. Figure 5F shows the corresponding excitation profile within the slice of interest.Figure 5G shows the profile along the center red line 1 in Figure 5F across the ROI in the direction of phase encoding.Figure 5H shows the profile along the center red line two across the ROI in the direction of frequency encoding.Figure 5I shows the through-slice excitation profile crossing the center point (R-L).
The self-rewinding excitation design is also compared and shown in Figure 6.The magnetizations in the ROI interferers in the fetal brain slice are not well suppressed, and the phase across the ROI is poorly rewinding in The convergence time, convergence steps, optimization loss when converged, and evaluation loss at a finer resolution (fixed-point method and stochastic offset method with different range of offset at different resolutions).Note: The optimization loss shown in this table is evaluated at a set of fixed point at the top-left-front corner for the check of convergence.Otherwise the loss would keep changing as the magnetization might not be uniform at different offset for coarser resolution.We consider convergence as the optimization loss remains unchanged for five iterations.Different weights are assigned at pass-bands and stop-bands as discussed in the experiment section.For the stochastic offset methods, we ran the experiments five times and calculate the mean and SD of the statistics.For the evaluation loss, we show the RMSE loss at a resolution of 0.03 mm.RF optimized at 3 mm resolution with 2 mm range offset could achieve similarly uniform excitation profiles as that at 1.5 mm with the fixed-point method, but the stochastic offset method takes shorter time to converge.For the convergence time and steps, the number before the brackets is the mean and within the brackets is the SD.Abbreviations: FP, fixed-point method; RMSE, root mean square error; SO, stochastic offset method.

Methods
comparison to the alternatively proposed two-stage rewinding method.
To show the effect of the stochastic offset strategy, we designed the two-step excitation process with fixed spatial optimization points and evaluate the slice profile summing the magnetizations of the same 80 equally spaced points within the voxel while only optimizing one fixed point (center point) within the voxel.The results are shown in Figure 7.The slice profile and the RMSE are worse than those of our proposed method because of non-uniform magnetization in the interval between the fixed optimized points.
To demonstrate the issue of sub-voxel dephasing, mostly introduced by the S-I direction fixed linear gradient, we show the magnetization along line 1 (phase encoding direction).Instead of summing over different spatial points within the voxel to obtain the net magnetization, the magnetizations along a diagonal line at a finer resolution (×16, 0.25 mm) are evaluated.All 16 magnetizations for each voxel are plotted in order.It could be seen that our method obtains a more uniform slice profile within all the voxels in both ROI and ROI interferers as shown in Figure 8.

Results of restricted slice design 4: refocusing
For the evaluation, without loss of generality, we assume the initial magnetization of the refocusing process is The results of the experiments of the slice refocusing design at 4-mm resolution.Similar as Figure 3A,B show that our method achieves similar slice profile compared with Shinnar-Le Roux algorithm, which proves the correctness of our design framework and the equivalent crusher gradient effect.(C) Shows the results of the fixed points method and the fixed points for the optimization locate at 0 ± 350n Hz where n are integers.Because of the presence of the crusher gradient, the issue of non-uniform profile is more severe. is , the resultant magnetizations are M Tx = M Tx,2 + M Ty,1 and M Ty = M Ty,2 − M Tx,1 .As a result, this evaluation is sufficient for the demonstration of our proposed restricted 180 • refocusing design.Figure 9A-C show the optimized RF pulse and five representative time-varying coil current.Overall, the suppression performance metrics of the 180 • refocusing pulse is superior to the excitation pulse design, and when applied together in a spin-echo pair, the suppression is further enhanced.

DISCUSSION
In this work, we demonstrated RF excitation and refocusing designs that are enabled by a multi-coil shim array when applied together with conventional single-channel RF and gradient waveforms.Specifically, we showed in a numerical example a 4-mm thick slice with in-plane truncation of the target ROI in the phase-encoding direction limited to a 12-cm width that cuts down the FOV approximately three-fold, thereby reducing the number of phase encodes required to encode an alias-free slice.Furthermore, the proposed designs achieved their specified targets with relatively short RF pulses, here, 2 ms and 4 ms, for excitation and refocusing, respectively.The shim array that was simulated for these results was populated with 64 loops of current, approximately equally distributed in a geometry that fit the abdomen of a body model of a late-stage pregnant woman.The maximum shim current reached ∼50 A, which is on the order of previously achieved shim coils array currents by Juchem et al. 34 with multi-turn coil implementations.We performed comparisons of our proposed design framework, including the modeling of idealized crusher gradients effects, against the established SLR method for a conventional slice-selective RF pulse.For both the excitation and refocusing designs, the proposed framework and the SLR algorithm yielded similar pulse envelopes and magnetization profiles (Figures 2 and 3).This comparison is not expected to result in identical RF waveforms, as our design framework uses voxelwise optimization based on the time-discrete Bloch simulator with some differences in objective functions of the two designs, such as equi-ripple SLR design versus L2 norm least squares.

F I G U R E 5
Optimization results of our method for the restricted slice excitation design.Our method uses two-stage optimization where first excites the region of interest (ROI) with loss functions on the magnitude and second uses the magnetization results of stage one as the initial magnetizations, then rewinds the phase in the ROI with loss functions on the phase.Stochastic offset strategy is used for refinement.(A), (B) Optimized RF pulse in magnitude and phase, respectively.(C) Five representative optimized time-variant shim coil current.Current for all coils are shown in the Supporting Information.The maximum current in the 64 coils is 53A.(D) The distribution of excitation magnitude within the ROI and the suppressed regions.Within the ROI, the maximum magnitude is 0.971 and the minimum magnitude is 0.930.The mean magnitude is 0.962 and the median magnitude is 0.965.Within the suppressed regions, the maximum magnitude is 0.279 and the minimum magnitude is 0. The mean magnitude is 0.001 and the median magnitude is 0.000.The root mean square error is 0.0076.).Note that the plot in the through-slice direction is of a finer resolution (0.5 mm) compared with the optimization resolution (4 mm) to show better details.As a result, the intra-voxel dephasing effect is less under this finer resolution compared with figure (G) and (H).All the following through slice profile will use the same resolution for this plot.
The optimization method as proposed applies a stochastic offset of spatial coordinates of the cost function evaluation at each iteration.Under a relatively coarser resolution, in our case 4-mm for the chosen application, this approach was found to significantly improve the uniformity of the slice profile compared to a fixed-point method (i.e., where the cost function is evaluated on the same set of spatial points in each iteration).As the spatial resolution of the fixed-point method is increased, the resulting slice profile becomes smoother as demonstrated Optimization results of the self-rewinding method for the restricted slice excitation design.The design is optimized in one stage where the magnetizations are supposed to be both excited and rewinding with loss functions on magnitude and phase.Stochastic offset strategy is used.(A), (B) Optimized RF pulse in magnitude and phase, respectively.(C) Five same representative optimized time-variant shim coil current as Figure 5.The maximum current in the 64 coil is 41A.(D) The distribution of excitation magnitude within the ROI and the suppressed regions.Within the region of interest (ROI), the maximum magnitude is 0.874 and the minimum magnitude is 0.844.The mean magnitude is 0.869 and the median magnitude is 0.871.Within the suppressed regions, the maximum magnitude is 0.713 and the minimum magnitude is 0. The mean magnitude is 0.024 and the median magnitude is 0.000.The root mean square error is 0.0625.(E) The phase in ROI after the excitation.The phase across the ROI ranges in ∼0.113π.
in Figure S1, which shows optimized RF pulse and slice profile of the fixed-point method under different resolutions.Meanwhile, the stochastic offset strategy could achieve nearly uniform excitation magnetization with different resolution as shown in Figures S2-S4.However, with increased resolution in the fixed-point method, the computation burdens increases and especially so for 3D designs, which was limited to a 4-mm resolution for our GPU and chosen dimensions of the FOV.Moreover, the stochastic offset approach considers all points within the interval between the fixed points coordinates given a sufficient number of optimization steps, whereas the fixed-point method does not apply explicit magnetization constraints except at the fixed-point coordinates.Note that fully random coordinate selection strategies did not converge.
As Table 1 shows, generally, the speed of convergence for the stochastic-offset approach is slower than for the fixed-points designs, which is a tradeoff compared with GPU memory.However, the stochastic-offset approach with coarse resolution converges faster than the fixed-point method at finer resolution while achieving Optimization results of the fixed-point method for the restricted slice excitation design.The design has the same two-stage optimization as our proposed method, except the stochastic offset strategy.(A), (B) Optimized RF pulse in magnitude and phase, respectively.(C) Five same representative optimized time-variant shim coil current as Figure 5.The maximum current in the coil is 6A.(D) The distribution of excitation magnitude within the region of interest (ROI) and the suppressed regions.Within the ROI, the maximum magnitude is 0.716 and the minimum magnitude is 0.704.The mean magnitude is 0.710 and the median magnitude is 0.710.Within the suppressed regions, the maximum magnitude is 0.758 and the minimum magnitude is 0. The mean magnitude is 0.023 and the median magnitude is 0.007.The root mean square error is 0.0884.(E) The phase in ROI after the excitation.The phase across the ROI ranges in almost 0.03π.similar slice profile quality with less GPU memory usage (stochastic offset with 3 mm vs. fixed-point with 7 mm).
As shown in Table 1, smaller offset values in the proposed stochastic strategy do not necessarily guarantee faster convergence times.In particular, selecting the offset to be half the resolution yields the longest convergence times.The fixed-point artifacts are reduced with the increasing of the covered offset within the voxel.The minimum offset range to achieve uniform excitation magnetization is dependent on the resolution as shown in supporting information from Figures S2-S4.For example, the minimum offset to achieve a desired excitation profile is 3 mm for 4-mm resolution whereas it is 1.5 mm for 3-mm resolution.There is always a tradeoff between the memory usage and the convergence time.However, although the stochastic offset approach is proposed to address the issues of the fixed point method at coarser resolution, the slice profile may not be guaranteed to be good at extremely coarser resolution.As shown in Figure S2, the slice profile (6-mm resolution) even under full offset range is less desired compared with finer resolution.As Figure S5 shows, for the slice selective excitation, the ripples (fixed-point artifacts) do not come from the rapid shape change in RF waveform, which is addressed by stochastic offset approach with adequate coverage of the size of the voxel.Note that in our optimization, using The dispersion within voxels along line 1 (anterior-posterior [A-P] direction).Within each voxel along line 1, the magnitude of magnetizations of 16 diagonally equally spaced points (as illustrated in the bottom right panel, from the top-front-left corner to the bottom-back-right corner) are calculated and plotted to show the dispersion within the voxels.The blue line represents the dispersion of the magnetizations along line 1 of our method and the red line represents the fixed-point method.Our method refines the profile, whereas the fixed-point method only optimizes the magnetization at the fixed points, but fails in the interval.smoothness constraints on the RF waveform without the stochastic offset approach in our optimization framework is not able to get the desired RF waveform.The study of better regularizations of the RF waveform remains as a rich topic for future steps.
The choice of initial conditions has a significant impact on the output of the design.For instance, we showed the effect of the choice of an initial RF pulse for 1D designs where we adjusted the degree of asymmetry of a Gaussian envelope.Figure S6 shows the results of optimized RF pulses and the corresponding slice profiles of our proposed method with such different Gaussian initial RF pulses.Our design framework could achieve similar maximum, minimum, linear phase and other RF pulse phase profiles by varying the shape of the initial RF envelope.
For the restricted slice excitation design, we exploited the superiority of our two-stage excitation and rewinding design over the self-rewinding design.In terms of the degrees of freedom of the optimization, the solution of the self-rewinding excitation method is a superset of our excitation-rewinding method.However, our two-stage method achieves better slice profile and, more importantly, less phase range, whereas the self-rewinding method fails to rewind the phase.Compared with conventional rewinding with a linear gradient of a symmetric RF pulse where approximately half the gradient area yields optimal rewinding, the optimized rewinding shim fields do not comply with this heuristic as the "imbalance" of the refocusing part of the coil driving current could be related to the non-orthogonality of the used fields of the target ROI.This shows that our problem is highly non-convex and non-linear, which is sensitive to the initializations, loss function design, and regularizations.However, the current needed for the selective excitation is relatively high and may have more signal leakage near the transition band if the maximum current is more limited.
We also showed the function of the stochastic offset strategy that improves the uniformity of the magnetization profiles in restricted slice designs.Shown in Figure 8, the magnetizations at the fixed points are well optimized, whereas those in the intervals are not.Therefore, the fixed-point optimization has severe signal loss because of intra-voxel dephasing as there are no constraints in the interval between the fixed points.Stronger current is needed to eliminate the signals in the suppressed regions and excite the signals between the intervals.
Although the profile of the restricted slice with both excitation and 180 • refocusing demonstrates excellent performance, we note that this problem, as posed, is non-convex and the global optimality of our Optimization results of our method for the restricted slice 180 refocusing design.The design optimized two different initial magnetizations at the same iteration based on the loss function with modeled idealized crusher gradient effect.(A), (B) Optimized RF pulse in magnitude and phase, respectively.(C) Five same representative optimized time-variant shim coil current as Figure 5.The maximum current in the coil is 25.6A.(D) The distribution of refocusing magnitude within the region of interest (ROI) and the suppressed regions.Within the ROI, the maximum magnitude is 0.957 and the minimum magnitude is 0.952.The mean magnitude is 0.955 and the median magnitude is 0.956.Within the suppressed regions, the maximum magnitude is 0.077 and the minimum magnitude is 0. The mean magnitude is 0.003 and the median magnitude is 0.002.The root mean square error is 0.0087.(E) The refocusing phase in ROI.The magnetizations in the ROI are well refocused with the range of 0.008π.
Bloch-optimized results is not guaranteed.A more general study of the optimization toward global optimality could be studied in future work.
On current hardware, the overall optimization takes ∼3 h, which is clearly not suitable for on-the-fly designs at the scanner.In future work, strategies for faster convergence (e.g., via multi-scale approaches) 35 to speed up the optimization while maintaining the target excitation profile performance could be exploited.Alternate strategies for improved convergence could explore a more sophisticated means of RF initialization or other priors.

CONCLUSION
We proposed and demonstrated a novel and general framework for slice-selective RF designs with in-plane restriction of the excited FOV.The magnetizations are significantly refined by the stochastic offset strategy.With the optimized, short-duration RF waveforms, a constant slice-selective gradient, and time-varying, spatially non-linear ΔB 0 fields, a slice profile of 4-mm thickness, and reduction of in-plane FOV by a factor three was achieved.These designs are expected benefit rapid fetal MRI, as well as other applications in body and brain where such restricted-slice excitation could be used to reduce encoding burden and speed up imaging.

ACKNOWLEDGMENTS
, for all magnetizations (de-phase or in-phase) within the voxel.Note that the basis vectors Mx, My, Mz instead of Mxy, Mxy * , Mz are used as a result of the input requirements of our auto-differentiation framework.A-I could be calculated from the formulas above.We will show that only D is needed to be computed for the 180 refocusing design along x-axis.

A.1. Mxy, Mxy
For a given voxel, the magnetizations within the voxel are in-phase with phase angle  and angle between z direction  and denoted as Mxy = sine α , Mz = cos.As EPG suggested, the crusher gradient effect will be analyzed with Fourier transformation which also follows the equation above.
For the initial magnetization, F 0 = sine α , Z 0 = cos.After applying the first crusher gradient causing 2 dephasing, it becomes F +1 = sin with phase  and Z 0 = cos.Based on the EPG theory, only F −1 would be rephased to echo signal, where F ′ −1 = DF +1 .Note that the phase for F ′ −1 becomes −.We could get the resultant magnetization from The first crusher gradient applies 2 dephasing within a voxel along z direction by rotating the magnetization with different angles (z) where (z) is in the range of 2.The magnetization at location z can be expressed as After the first crusher gradient, RF and B(t) are applied to the voxel.We assume that within the voxel, RF and B(t) are not strong such that they are all the same within the voxel.
Based on the decomposition property, the magnetiza-

E 1
The model of the pregnancy, fetal brain and simulated body shim coils in our demonstrations.(A) The distribution of 64 shim coils near the surface of the abdomen of the pregnancy.The green bubble in the center represents the fetal brain and the light gray bubble represents the numerical model of the pregnancy.The gray circles represent the shim coil.(B-D) Non-linear shim array ΔB 0 field patterns at the sagittal slice of interest, shown in (A) of the three selected coils (blue circles).

2
The desired design target (region of interest[ROI]) and suppressed regions (ROI interferers) of our restricted slice experiments.(A) The ROI, which is the intersection of the fetal brain and the sagittal slice of interest.The anterior-posterior (A-P) direction is the phase encoding direction where the aliasing de-selection technique is used and the blue region is the ROI interferers (suppressed regions) at this slice.Note that the ROI interferers through slice is not shown here.(B) The 2D plot of the desired pattern of the restricted slice design by expanding all right-left (R-L) slices into the same 2D plane.The blue represents the ROI interferers, the red represents the ROI and the red represents the do not care region.The white box is the FOV.

0 ].
Note that with the presence of the crusher gradients, the resultant magnetizations are M Tx = M Tx,1 − M Ty,2 and M Ty = M Tx,2 + M Ty,1 .If the initial magnetization (E) The phase in ROI after the excitation.The white contour represents the outline of the ROI.The phase across the ROI ranges in almost 0.092π.(F) The full excitation slice profile.The yellow contour represents the outlines of the ROI and ROI interferers.The region between two black dashed lines is the FOV.(G) The excitation profile along the red line 1 in (F) in the direction of phase encoding (anterior-posterior [A-P]).(H) The excitation profile along the red line 2 in (F) in the direction of frequency encoding (superior-inferior [S-I]).(I) The through-slice excitation profile crossing the center point (right-left [R-L] the magnetization after the RF process, given the response (M T{x,y,z},{1,2,3} ) of the RF process to the chosen three basis magnetizations.Given the transformation between Mx, My, Mz and Mxy, Mxy * , Mz,

Mz basis vectors with EPG
* ,