Figure 4: a) Sagittal slice of the 27 3D B₁⁺ predictions using the default shim setting (left) and the 4kT-UP (right). Library: B₁⁺ maps 1-22, test-cases: B₁⁺ maps 23-27. The 4kT-UP results in a homogeneous FA in the heart ROI of all 5 test cases. Moreover, the 4kT-UP achieves a homogeneous FA also in surrounding tissues such as the aorta. b-c) Boxplot of the FA spread in the 3D heart ROI of all 27 subjects for default (b) and 4kT-UP (c) demonstrating the FA homogeneity across all B₁⁺ predictions using the 4kT-UP pulse.

Figure 5: B₁⁺ predictions and reconstructed, respiration corrected 3D GRE images for two unseen test cases with 4kT-UP. These two volunteers were not part of the optimization. The 3D images are free of breathing artefacts and demonstrate, despite some differences close to the coil elements, the feasibility to achieve a homogeneous calibration-free FA of the whole heart. The remaining signal changes in the AP direction of the acquired images are a result of receive (B₁^-) variations.

Fig 4. Designed flip angle distribution (top row) and acquired EPI images (bottom row) from one representative slice in the human brain at 7T with system default quadrature mode, dynamic multislice RF shimming with conventional MLS, and with proposed FD-MLS. The RF pulse design ROI is labeled in red. With fixed system gain, the FD-MLS improved the signal homogeneity and SNR over default quadrature, while not forming the null as the conventional MLS.

Fig 5. Evaluation of FD-MLS with knee data from an 8Tx array with two different postures. The FD-MLS not only improved over null solutions (in slices marked by the green stars), but also improved over local minimum solutions (marked by green circles).

FIGURE 4. In vivo imaging results in the cervical spinal cord. (a) The standard and sheared 2DRF pulses had 16.0 ms and 9.5 ms durations, respectively. (b) The localizer showing the reduced-FOV region corresponding to 200 mm X 50 mm FOV. T2-weighted EPI images acquired using (c) 1DRF pulse with additional fat suppression (cropped full FOV of 200 mm X 162.5 mm), and (d) standard and (e) sheared 2DRF pulses. Reduced-FOV imaging achieves higher resolution with reduced in-plane distortion. The sheared 2DRF pulse further improves SNR in regions with high off-resonance effects (red arrows).

FIGURE 2. Simulated 2DRF excitation profiles for (a) the standard and (b) the sheared 2DRF designs. For the standard 2DRF pulse, the profile is periodic in the SS direction, limiting the slice coverage due to partial saturation effects. In contrast, the sheared 2DRF pulse places the replicas along the sheared axis, effectively pushing them outside the potential slice locations. (c) 1D cross-sectional excitation profiles along SS and PE directions demonstrate identical sharpness, with a slab thickness of 40 mm and a slice thickness of 4 mm.

Figure 3. (A-C) Phantom results illustrating localization performance of the RF pulses compared, selective for a 7 cm slab, and (D) illustrates traces along the X direction for the cases with a zoomed cut out illustrating the transition profile obtained for the three pulses. Excellent signal suppression throughout the slice (M_z/M₀ < 0.02) and minimal perturbation outside the inversion band is seen for all pulses.

Figure 5. An in vivo MRSI data acquired (TE/TR = 30 ms/2000 ms, 8.2 min acquisition time, 10 x 10 mm² grid from a 5 mm axial slab) utilizing ECLIPSE localization. The anatomical image illustrates six-voxel positions on the left, and 9 voxel locations on the right, corresponding to illustrated spectra. The ellipse in red color illustrates the water ROI selected with ECLIPSE, and the blue ellipse corresponds to the lipid ROI due to chemical shift ~1.6 mm smaller in radius along the x-direction.

Figure 1: A comparison of different water excitation schemes. Top: RECT kT-point excitation as used in [5]. Middle: “Classical” binomial 1-1 excitation applied to kT-points pulses. The second pulse starts at an odd number of half phase cycles of fat. Bottom: Proposed interleaved version of the 1-1 binomial kT-points pulse.

Figure 3: In vivo MP-RAGE acquisition employing a 7sh interleaved binomial kT-points pulse (Table 1) for excitation (bottom) compared to a non-selective CP-mode MP-RAGE acquisition (top). Each with (right) and without (left) fat suppression. Yellow arrows point at better homogenization with kT-points. Green arrows point at missing fat artifacts with water excitation (WE).

Figure 3. Simulated comparison of three lipid suppression scenarios in the brain. For each case, we simulate the off-resonance histogram (c) and z-magnetization (d) for water and lipids. (e) chemical shift artifacts in brain slices due to residual lipid are also simulated (5x fat for a better contrast). Using the 32-ch AC/DC shim array, LARDS performs better than simple global homogeneity shimming because the lipid mask used for optimization is limited only to voxels within the passband of the water excitation pulse, reducing constraints on the solver.

Figure 4. Three simulated cases of lipid suppression in the abdomen. The simulated off-resonance histogram, z-magnetization and chemical shift artifacts (5x fat for a better contrast) clearly show the benefit of LARDS compared to conventional shimming.

On the left hand side, the desired pattern is shown which is a 3D volume. On the right hand, the excited results under the four different experiments settings are shown. The error is based on the pixel-wise L2 norm. We could see that proposed joint designed time-varying field and RF pulse achieves best among all experiments.

The optimization results for different initializations for setting 3 and 4. We could see from the figure that the proposed method is more robust to the initialization with similar error ~6.2, while setting 3 is sensitive to the initialization as the error for initialization 2 is 13.85. Another thing to notice is that for the proposed method, different initializations may require different current and RF power levels.

Linear phase slice selection pulses and their transverse magnetization profiles after refocusing. The improved SLR (iSLR) pulse has a much flatter phase response than the original one. Pulse energy is reduced by 18.6% and peak is reduced by 9.1%. Note that the improved pulse is asymmetric, which shows that the proposed design compensates for the phase of α to generate a linear phase profile.

Zero phase spin-echo refocusing pulses and their longitudinal magnetization profiles. Pulse energy is reduced by 18.6% and peak is reduced by 13.4%.

Figure 2: FA maps and corresponding B₁⁺/B₀ maps. First row: the DeepControl result. Second row: the corresponding OC result. The printed numbers in white are normalized root-mean-square errors. Row three to five, between the two dashed line, are intended to recap what missing field information results in with respect to FA maps. Using the previous version of DeepControl¹⁴, which did not account for field inhomogeneities, distortions like these should be expected. In all cases both B₁⁺ map/B₀ maps were included in the Bloch simulations (raster grid: 128x128).

Figure 3: DeepControl vs Optimal Control (row 1 and 2 in Fig. 2) 3D and 2D histograms of voxel FA values inside the target region. The target FA was 30^o. Numbers printed below the plots are mean and standard deviation FAs.

Figure 2 - Magnetization Transfer Ratio (MTR) maps from one volunteer obtained with saturation prep-pulse applied using CP (top row) and DSatC (bottom row) modes using Target=0.6μT² and TR=23ms. For both modes the saturation prep-pulse was applied with different number of sub-pulses: 1 (left column), 2 (middle column) and 3 (right column). The observed MTR correlates well with the predicted <B₁²> (Figure 3). The solution for 1 sub-pulse is not uniform, and this is also predicted from <B₁²>.

Figure 5 – Results from the gradient demonstration using the non-blipped (top row) and blipped (bottom row) saturation pulses in Figure 4. (a,d) MTR maps, (b,e) <B₁²> of the saturation pulse and (c,f) flip angle from the RF and gradients applied.

Figure 4: Evaluation of four different pulses optimized for the heart ROI on exhale, intermediate, inhale and optimized on all respiration states (respiration robust) for deep breathing of all 10 subjects. Depicted are boxplots containing the CVs of all respiration states. The respiration robust pulse performs best across all subjects and respiration states with the lowest median and spread.

Figure 5: Side-by-side comparison of B₁⁺ predictions (a) and reconstructed 3D GRE images (b) with the 4 kT-points pTx pulses optimized on inhale (left) and respiration robust (right) of subject 7. Depicted are the views of the acquired 3D volume close to the position of the B₁⁺ prediction. The arrows point to signal drop-out regions in using the inhale pulse which are corrected by the respiration robust pulse.

Figure 1: Pulse design methodology in the volunteer for the MPRAGE excitation 8° GRAPE pulse. a) A calibration matrix (L) is computed to adjust the standardized universal pulse (SUP) designed over the standardized pulse design database to the subject B1⁺ map. Its magnitude is close to the identity matrix while diagonal phase terms are close to 0 radian. b) RF and gradient amplitudes of the resulting adjusted SUP. c and d) Flip angle maps and histograms retrospectively simulated on the subject for the UP and adjusted SUP with the subject’s full B1⁺ map. NRMSEs are 9.5% and 7.5%, respectively.

Figure 4: a) UNI contrast computed from the two inversion volumes of the MP2RAGE sequence acquired with the adjusted SUP. b to d) T1 map computed by injecting UNI signal intensities and sequence parameters in Bloch equation simulations for the MP2RAGE sequence acquired in the CP mode, CP mode with B1⁺ postprocessing correction and adjusted SUP, respectively. e) T1 distributions extracted from the brain for figures 4b to 4d, showing an improvement on the distributions.

Fig. 2: 90°, 2ms “universal” excitation pulses and their performance. For all pulses, the voltage is limited to a maximum of 400V. A: “Universal” kT-point pulse. B: “Universal” RF + gradient optimization pulse (no shim currents). C: “universal” RF + gradient + shim currents pulse (INSTANT). D: RMSE performance of the different pulse strategies and typical sagittal flip-angle map (first of the three subjects included in the universal design). The black arrow points to typical under-flip in the frontal lobe, which is largely corrected by the optimized pulses.

Fig. 3: Flip-angle maps acquired on the last four subjects of the cohort using a standard rectangular pulse (RECT) and the “universal” kT-point and RF + gradient optimized pulses shown in Fig. 2. The target flip-angle is a uniform 90° distribution in the brain (skull has been stripped from the optimization mask). The red and black arrows point to typical under-flip in the temporal and frontal lobes, respectively.

Figure 3 MPRAGE images for a fixed adiabatic inversion pulse and various excitation pulses. Experimental images and corresponding FA simulations are shown for coil-specific CP pulses and all generated pTx excitation pulses.

Figure 4 MPRAGE images using the UP of the respective coil for excitation and various inversion pulses. Experimental images and corresponding FA simulations are shown for coil-specific CP pulses and all generated pTx excitation pulses.

Figure 2. Magnitude and Phase profiles for a -5° roll rotation about the centre of the head. Magnitude nRMSE was reduced from 14% (reference pulse) to 5.4% (MRP). Maximum magnitude error was reduced from 64% to 20%. Phase RMSE was reduced from 17° to 3.6°, and maximum phase error from 68° to 15°. The MRP was successful at areas of high motion-induced error in magnitude and phase.

Figure 5. (a) Magnitude nRMSE, (b) phase RMSE, (c) maximum magnitude error, (d) maximum phase error of the MRP vs the reference pulse. The green shaded area represents where the MRP outperformed the reference pulse. Cases where MRP performs worse than the reference still report very similar errors.

Absolute value and angle of the oil-water phantom image measured with a B₁-scaling of 100% (a-b). The red line shows the position of the line plots depicted in (c) and (d). The signal intensity profile without and with inversion pulse $$$\textbf{optim}$$$ with different B₁-scalings are shown in (c). In (d), the measured normalized signal intensity after application of the $$$\textbf{optim}$$$ inversion pulse by a measurement without inversion is plotted. Thus, the influence of the coil sensitivity distribution and the varying excitation flip angle is eliminated.

Absolute value and angle of the cylindrical MR phantom image measured with a B₁-scaling of 100% (a-b) and the nominal B₁-map (c). The red line shows the position of the line plots depicted in (d) and (e). The signal intensity profile without and with inversion pulse $$$\textbf{optim}$$$ with different B₁-scalings are shown in (d). In (e), the measured normalized signal intensity after application of the inversion pulse by a measurement without inversion is plotted. Thus, the influence of the coil sensitivity distribution and the varying excitation flip angle is eliminated.

Fig 2: Normalized excitation patterns (top row) and error maps (bottom row) in central slices for k-space domain (left) and spatial domain designs (right). The calculated RMSEs were 2.13% (spatial) and 2.31% (k-space), indicating uniform inner volume excitation while maintaining the outer volume intact. The parallelized k-space domain design required 6.7 s computation versus 30.3 s for the spatial domain method, a 78% decrease. The size of the system matrix (W) of the k-space domain design was 0.8 Gb, while the system matrix of the spatial domain design had a size of 79 Gb, a 99% decrease.

Fig 1: (a) Central slices of the target excitation pattern used for all pulse designs, which comprised an ellipse centered on the ventricles with AP/HF/LR semi-axes of 4.8/3.2/3.2 cm. (b) The SPINS trajectory used in the designs. (c) The 24-channel loop Tx array that was simulated in a human head model to obtain B₁⁺ maps. The array has diameter 32 cm, height 28 cm, and 16×11 cm rectangular loops in 3 rows of 8. (d) 10 ms minimum time gradient waveforms that produce the SPINS trajectory, subject to the scanner's gradient amplitude and slew rate constraints of 200 mT/m and 700 T/m/s, respectively.

Figure 1. Concomitant field effects approximated as a Bloch-Siegert shift. (A) Reference frame visualization of (a,b) the concomitant $$$B_c(\vec{r},t)$$$, $$$B_0$$$, and gradient field $$$\vec{G}(t)\cdot\vec{r}$$$, and Rotating frame visualization (c) showing that the $$$\tilde{B_c}(\vec{r},t)$$$ approximation can be treated as additional off-resonance in the rotating frame. (B) Detailed derivations.

Figure 2. Concomitant field estimation accuracy. (A) (left) Original 2D RF pulse excitation profile (magnitude: top & phase: bottom). (right) Magnitude (row a, b) and phase (row c, d) of the reference profiles (a, c) and estimated profiles (b, d) using Bloch-Siegert approximation at 0.2T, 0.55T, 1.5T, 3T and 7T. NRMSE results are 4.30%, 1.60%, 0.64%, 0.33% and 0.16% from 0.2T to 7T, respectively. Reference method simulates the concomitant field in the reference frame. (B) NRMSE as a function of field strength. Note that NRMSE is < 5% for B₀ ≥ 0.2 Tesla.

Figure 5: Zoomed in SE-EPI images acquired with excitation FA of 60° and refocusing FA of 120°. Significant signal dropouts can be observed in the case of CP pulses (green arrows). Excellent signal homogeneity is observed when using any of the 3 pTx optimisation methods. Signal losses at the edges (red arrows) are further recovered with matched excitation and joint optimised pulses.

Figure 1: Pulse optimisation work flow for (2) separately optimised pTx pulses, (3) matched excitation pTx pulses and (4) jointly optimised pTx pulses.

Figure 3. FA and error maps of the human head calculated by the AFI method. Pixels with erroneous, aliased FAs were found in the nominal FAs of ≥ 60°.

Figure 2. FA, phase difference, and error maps of the phantom calculated by the AFI method. FA maps determined by AFI are considerably different from those by the VFA method for nominal FAs of 90 and 130°, and the map is slightly different only at the central area in that of 50°. Pixels with erroneous FA values due to the aliasing phenomenon are readily detected by using the phase difference images by applying the threshold of 1/6 rad.

Figure 1. Summary of DeepRF-Grad for a slice-selective inversion pulse. Compared to the SLR results, the DeepRF-Grad designed RF pulse and z-gradient produced a substantial reduction in SAR (62% reduction) while sustaining a similar slice profile (SLR: blue line, DeepRF-Grad: red dashed line) and satisfying hardware constraints (slew rate and maximum gradient). This idea of designing both RF and gradient to reduce SAR is similar to VERSE and comparisons including VERSE design continue in Figs 3 and 4.

Figure 2. Comparison of a) DeepRF and b) DeepRF-Grad. Both methods use deep reinforcement learning (DRL) and gradient descent. a) DeepRF designs only RF magnitude and phase, while z-gradient is fixed as uniform. The loss function of DeepRF consists of the slice profile and SAR terms. b) In DeepRF-Grad, z-gradient is also designed simultaneously with an RF pulse. A slew rate term is added to the loss function.

Comparison of designs on outer volume excitation objective (OV90). Our proposed method is much faster than the method in [2] (3 min vs 8.5 min). The optimized pulse waveforms and the k-space trajectory resemble that of method [2].

Comparison of simulated excitation profiles for the pulses in Fig. 2. The goal is to excite only the spins in the outer volume. In terms of NRMSE, our proposed method performance is slightly better than that of method [2].

An overview of DeepRF. (a) In the RF generation module, a series of RF values (i.e., actions) are generated from the RNN agent to shape an RF envelope (N^th RF), and the virtual MRI simulates a slice profile. Then, a value of the objective function (e.g., a difference between simulated and desired profiles) is calculated from which the agent changes its behavior to generate a next RF pulse ((N+1)^th RF). (b) In the RF refinement module, an RF pulse is refined (Mth RF to (M+1)^th RF) with respect to the objective function using RF value changes (∆RF) calculated from the Bloch graph.

The results of the DeepRF and SLR pulse designs with different TBWs (4.3, 5.6, 6.8, and 8.1). For all the designs, the pulse shapes of the DeepRF pulses are clearly different from those of the SLR pulses (1^st and 2^nd columns). The slice profiles from the DeepRF and SLR pulses are almost identical (3^rd and 4^th columns). The SARs of the DeepRF pulses are smaller than those of the SLR pulses in all TBWs (1^st column).

Fig.1. The deep reinforcement learning (DRL) pulse design framework. In a forward pass, the neural network takes the input target slice profile and outputs the predicted RF pulse. The Bloch simulator embedded in the system gives the produced slice profile, which in turn is used (along with the target slice profile) to evaluate the loss. In a backward pass, the gradient of the loss with respect to the weights of the neural network are calculated using backpropagation to update the neural network. The procedures described above are iterated during the training.

Fig.5. Comparison of the SLR algorithm vs our DRL method when used to design (A) 90◦ excitation, (B) refocusing, (C) inversion and (D) 90◦ excitation multiband (MB=2, gap=50 mm) pulses. For each scenario a separate neural network was trained. Note that despite larger fluctuation observed in the predicted RF pulse, our DRL method led to visually comparable slice profiles for refocusing and inversion, while improving slice profiles for single band and multiband excitation pulses with noticeable suppression of stopband ripples.

Figure 2. Pulse Performance Tradeoff. (a) Velocity profile sharpness vs. pulse duration (left) and signal loss (right) with (b) detailed parameters of plotted designs. T₁ and T₂ values were used as 1121ms, 263ms at 0.55T; and 1650ms, 150ms at 3T, respectively. Normalized signal loss is defined as (M₀-M_z)/M₀. Note that 0.55T system (blue) shows higher fidelities of tradeoffs. The RF and gradient waveforms of two representative designs (marked “A” and “B” asterisks) are plotted in Figure 1.

Figure 4. Impact of gradient distortions and pre-compensation. (a) GIRF predicted excitation profiles as a function of ∆f and B₁⁺ scale. (b) GIRF pre-compensated profiles. Notice that stripe artifacts appear as spatial modulations (red arrows) and as sidelobe peaks (green arrows) in (a). These are completely resolved in (b) by using GIRF-based pre-compensation.

Figure 4 MPRAGE images using FOCUS pTx pulses, based on B₀ maps, which were acquired with either a CP pulse or a UP. The purple arrows indicate B₀ related artifacts, which were weakened when using the UP-improved B₀ mapping for designing the pTx inversion pulse. Different behavior of the excitation pulses could not be observed.

Figure 3 FA simulations of FOCUS excitation (Exc) and inversion (Inv) pulses using either the B₀ map acquired with a CP pulse (B0CP) or a UP (B0UP), as well as the difference maps (‘UP case – CP case’). All Bloch simulations were performed using the B₀ map acquired with the UP. B₀ maps acquired with CP pulses may lead to a worse performance of the FOCUS inversion pulse, especially in the sinus region (purple arrow).

Figure 4: 9.4T T2^*-weighted GRE acquisitions (voxel size: 0.8x0.8x0.8 mm³, matrix size: 224x280, TR = 18 ms, TE = 8ms) from the three non-database heads. Once, the respective TPs (FA7) were applied, once the UP (FA7) was applied.

Figure 3: Bloch simulated FA profiles of the TPs for FA90 (first line of profiles), UP2D for FA90 (second line), the TPs for FA7 (third line) and UP2D for FA7 (fourth line). The eight columns on the left present the database heads. The three columns on the right present the non-database heads. The bar plot below the profiles illustrates the NRMSEs for each pulse and head.

EPI images acquired on an oil phantom with a pulse voltage of 180V and different numbers of repetitions of the saturation module. The left, middle, and right columns show 1, 2, and 3 repetitions of the saturation module, respectively. The bottom, middle, top rows show images acquired at the bottom, in the middle, and at the top of the inversion slab, respectively.

Schematic of the labelling strategies for the proposed method. A cylindrical ROI is chosen.