Fast online‐customized (FOCUS) parallel transmission pulses: A combination of universal pulses and individual optimization

To mitigate spatial flip angle (FA) variations under strict specific absorption rate (SAR) constraints for ultra‐high field MRI using a combination of universal parallel transmit (pTx) pulses and fast subject‐specific optimization.


| INTRODUCTION
Ultra-high field (UHF) MRI (B 0 ≥7T) yields improved SNR 1 and, often, also improved image contrast. Therefore, it provides unprecedented anatomic detail in neuroimaging. 2 However, there are also substantial challenges that need to be overcome to translate the full potential of UHF MRI in the clinic. 3 In particular, the spatial variation in the transmit magnetic field (B + 1 ) results in spatially varying flip angles (FA) leading to undesirable variations in signal intensity and image contrast. In addition, spatial variation of the local specific absorption rate (SAR) is becoming more dominant at higher field strengths.
To overcome these limitations, the concept of parallel transmission (pTx) has been applied successfully. 4 PTx makes use of multi-channel transmit (Tx) RF coils that are driven by individual, independent RF pulses and provide an effective means of optimizing the FA distribution while minimizing local and global SAR. Static pTx, 4 often denoted as B + 1 shimming, uses a single RF shape for all Tx channels with individual channel amplitudes and phases designed to optimize B + 1 . This technique has made a strong impact on mitigating local and global B + 1 field inhomogeneities in the human head, 5,6 aorta, 7 prostate, 8 and whole body, 9 yet, has limited effect in highly inhomogeneous areas of the B + 1 field. Dynamic pTx typically consists of both a "transmit k-space" trajectory and corresponding individual RF shapes for each Tx channel. During the transmission of a dynamic pTx RF pulse, the resulting B + 1 field is usually spatially inhomogeneous at a given time point and temporally changing, but the aim is to achieve a homogeneous FA distribution at the end of the pulse. 4,10-12 Therefore, dynamic pTx pulses provide more degrees of freedom and, thus, higher potential to produce uniform FA patterns than static pTx pulses. Several techniques to design dynamic pTx pulses have been proposed including slice/slab selective, 13,14 tailored 3D volume excitation, 15,16 and nonselective 17,18 pulses.
Applying a subject-specific pTx pulse first requires the mapping of B + 1 and B 0 within the region of interest and, then, the calculation of RF pulse shapes and potentially gradient trajectories. Presently, the mapping and calculation step hinder a wider application of such pulses, because both steps may require ~15 min total. 19,20 Therefore, they are often regarded as not suitable for clinical routine. A recent major step toward clinically applicable pTx pulses is the concept of "universal pulses" (UPs) by Gras et al, 21 which do not require any online calibration. Using B + 1 and B 0 maps of multiple training subjects, pTx pulses have been designed that produced much more homogeneous FA distributions than circularly polarized (CP) pulses, which are considered to be equivalent to single Tx, in subjects not included in the training set. Non-selective UPs have been presented for both T 1 -and T 2 -weighted imaging [21][22][23] and slice-selective UPs have been created for T 1 weighted imaging. 24 Furthermore, the technique has also been shown to be applicable for various pTx head coils. 24 Nevertheless, although this technique allows substantially improved excitation homogeneity without the need for online pulse calculation, Gras et al also demonstrated that subject-specific pTx pulses (i.e., pulses that are not designed based on the training set but on B + 1 and B 0 maps of the actual subject) achieve even higher spatial signal homogeneity. 21 In this work, we present a method that combines both UPs and a fast subject-specific online calibration requiring <1 min additional sequence preparation time. This short preparation time paves the way for the application of individually optimized pulses (IOPs) in a clinical routine. We call these IOP pulses as fast online-customized (FOCUS) pTx pulses. In the first step, parameters to define the k-space trajectories, energy regularization weights, universal pulse shapes, and fitting curves for maximum local specific energy dose (SED) with respect to further adjustable regularization weights were optimized offline using 12 training data sets. Based on these parameters, subject-specific RF pulse calculation can be performed online during the sequence preparation phase after rapid B 0 and B + 1 mapping. Stable improvements of the pulses' performance regarding FA homogeneity and maximum local SED exposure (often considered to be the most restrictive constraint) 25 are shown in Bloch simulations of an additional 60 subjects (36 healthy Europeans, 12 Asians, and 12 Europeans with pathological or incidental findings). Unlike the original UPs, 21 the basic trajectory used here is based on a "SPINS" (spiral-nonselective) trajectory 18 with several additional parameters to create more degrees of freedom. These FOCUS pTx pulses were designed for a low FA excitation in a 3D MPRAGE sequence. We demonstrate improved excitation behavior of IOPs as compared to UPs and a reliable performance within simulation across all study subjects. MPRAGE images using CP pulses, UPs, and IOPs were acquired from 7 additional subjects to prove consistency.

| Measurement system and data acquisition
All measurements were conducted on a 7T whole-body MR system (MAGNETOM Terra, Siemens Healthcare GmbH, Erlangen, Germany) with an 8Tx/32Rx head coil (Nova Medical, Wilmington, MA). B 0 mapping was performed with a sagittally oriented gradient echo sequence (TE1 = 2.39 ms, TE2 = 4.59 ms, TE3 = 7.09 ms; resolution = 4 × 4 × 6 mm 3 ; 28 slices; no slice gaps; T Aq = 12 s). B + 1 mapping was performed using an interferometric transversely oriented magnetization prepared saturation recovery turbo flash sequence (TE = 1.63 ms, TR = 3.76 s, resolution = 4 × 4 × 5 mm 3 , 30 slices, 2.5 mm slice gapping, T Aq = 40 s), 26 and involved 8 GRE images, where always 1 transmit channel was left out. Absolute single channel B + 1 maps were calculated using these 8 GRE images and a reference map (all channels combined) derived from those. A mask was generated based on the B 1 map of the CP mode, excluding voxels with intensities in the lower 10th percentile. Additionally, voxels, whose phases in prepared and unprepared GRE images differed by more than 80° were regarded as erroneous measurement points and consequently also excluded. No brain extraction was performed, and lower and peripheral regions (upper cervical spine, skull, nose, and jaw) were kept because those regions might also be relevant for diagnostics. These B + 1 /B 0 mapping protocols each had the same FOV as the MPRAGE sequence (FOV = 250 × 218.75 × 167 mm 3 in sagittal orientation, see below). These acquired B + 1 /B 0 maps are generally available on the 7T system and were used for both offline and online optimizations described below.
The pTx excitation pulses were calculated for use in a sagittally oriented 3D-MPRAGE prototype sequence. The following timing parameters were applied: TR = 3 s, TI = 1.1 s, acquisition time = 7 min 11 s, minimum TE (either TE = 2.92 ms using CP rectangle pulses of 100 µs duration or TE = 3.37 ms using UPs or FOCUS pulses of 1 ms duration), matrix size: 384 × 336 × 256, FOV = 250 × 218.75 × 167 mm 3 , 0.65 mm isotropic resolution, BW = 250 Hz/Px, GRAPPA acceleration factor 3, and echo spacing (ES) = 6.9 ms using CP rectangle pulses and 7.8 ms using pTx pulses. The transmitter voltage was adjusted by a vendor provided routine consisting of a fast saturation recovery turbo FLASH B + 1 mapping sequence that measures B + 1 in 3 transversal slices (1 located in the isocenter and 2 located ±4 cm off-center). The transmitter voltage is then set such that the upper 20th percentile of F reaches at least the nominal value. All B + 1 and B 0 maps were interpolated into a 4 × 4 × 6 mm 3 matrix in sagittal orientation covering the FOV of the sequence.
In addition to Bloch simulations that were performed on each subject and for all pulses, 7 additional European subjects were examined experimentally using the MPRAGE sequence in combination with FOCUS pulses. Six of these patients were healthy, and one was a brain surgery patient, who was considered to have the most apparent anomalies among all. The study was approved by the local Ethical Review Board, and all subjects provided informed consent before the scan.

| k-Space trajectory parametrization and individual RF pulse optimization
The transmit k-space trajectory used was based on a 3D SPINS. 18 SPINS pulses provide more pulse samples, and therefore, more degrees of freedom for RF pulse optimization as compared to the k T point trajectory. 17 To increase the degrees of freedom for the trajectory, several new parameters were added. In spherical coordinates (k r , k θ , and k φ denoting the radial, polar, and azimuthal components), the modified trajectory is defined by: k max describes the maximum distance from the k-space center, and α and β (0 < β < 1) are parameters describing the evolution of k r based on the original SPINS trajectory, and a second order term with parameters γ and δ (0 < δ < 1) was added. θ 0 , φ 0 , ω θ , ω φ , υ θ , and υ φ indicate the initial angles, angular velocities and acceleration in polar and azimuthal direction, respectively. θ 0 and φ 0 were added to introduce more degrees of freedom and to avoid high slew rates at the beginning of the trajectory for the x (k x = k r ⋅ sin k θ cos k φ ) and z (k z = k r ⋅ cos k θ ) gradient coils. In addition, the acceleration terms υ θ and υ φ , as well as γ and δ, were added to provide second order terms. After transforming the spherical coordinates into Cartesian coordinates, the z coordinate of the trajectory was scaled with a constant factor k z (t) = c z k � z (t). This was done to introduce yet 1 more additional degree of freedom for the trajectory in z direction (head-feet), because all transmit elements are located at the same z-position (head-foot direction).
To optimize the individual RF pulse profile, the following problem was solved, which aimed to minimize the squared deviation of the FA across the head volume from a target FA using the variable exchange method 27,28 : where α t = 7 • denotes the target FA, and A ∈ ℂ N V × (NC ⋅ N S) denotes the spins' dynamics matrix for all (N V ) head voxels of a single subject with N C = 8 Tx channels, each having N S pulse shape samples (detailed description in Hoyos-Idrobo et al 29 ). The matrix A is defined by the B + 1 /B 0 maps with individually generated masks described above (see Measurement system and data acquisition), as well as the k-space trajectory. b ∈ ℂ N C ⋅ N S describes the concatenated complex waveforms of all Tx channels and λ is the energy regularization weight. A fixed pulse duration of T = 1 ms at a sampling rate of 100 kHz was used, resulting in N S = 100 pulse shape samples. To enable fast online-customization, we used a single regularization weight λ to meet both local and global SAR constraints even though a better trade-off between local and global SAR might be achieved using a combination of k T points and simultaneously constraining local SAR and power as shown by Guerin et al. 30

| Universal pulses and energy regularization weights
For the following optimizations, B + 1 and B 0 maps from N p = 12 healthy European training subjects (6 female, 6 male, age = 25 ± 2.8 years, ranging from 22-29 years), as well as virtual observation point (VOP) data 31 were used as input data. Offline calculations described below (see Equations 6-8) were performed on a desktop computer (Intel Core i7-67008 CPU; 2.7 GHz; 4 Kernels; 64 GB of RAM) using MATLAB 2017b (The MathWorks, Natick, MA) with the parallel computing toolbox.
In a first step, the SPINS trajectory parameters and the (initial) energy regularization weight λ 0 (see Equation 2) were optimized simultaneously. Consequently, their values were defined as combined optimization values (COV): During the optimization of the COV (Equation 6), for each set of COV, individual RF pulse calculation (Equation 2) was performed. For evaluation, the normalized root mean square error (NRMSE) of the FA and the pulse's maximum local SED were minimized for each individual pulse. For a single subject p, the NRMSE was calculated as using a Bloch simulation where α (v) denotes the simulated FA in voxel v among N V voxels in total, and the overall mean FA is denoted by α m . The SED was chosen as a measure of energy absorbed per mass of tissue in [J/kg], which is proportional to the pulse's contribution to the maximum local SAR in [W/kg]. To specify the individual maximum local SED values, the N VOP = 8 VOPs derived from 3 different models (Duke, Ella, and Hugo) 32,33 simulating SAR in each 10-g volume in the subject's head were used. The VOPs and head (including shoulders) models were provided by the manufacturer of the MR system (Siemens Healthcare GmbH, Erlangen, Germany) and are generally available on the new type of 7T system. The maximum local SED value for a single subject p from these different VOPs was calculated via the following formula: The matrix B ∈ ℂ N S × N C describes the complex waveforms of all channels, containing the same information as the vector b, and the matrix VOP j ∈ ℂ N C × N C denotes the SAR estimation matrix of VOP j, G p ∈ ℝ 1 × N C denotes the subject-specific gains on each transmit channel (difference between measured and nominal voltage supplied by the RF power amplifier, measured automatically during each examination), ℜ(…) denotes the real part of a number, ⊙ denotes the Hadamard product, and ‖…‖ 1 denotes the sum norm of a matrix. All other VOP types (e.g., global SAR and hardware protection) were neglected because they were found to be much less constrictive.
To obtain the most suitable COV, the following cost function was minimized: To evaluate a single cost value on a set of COV, individual RF pulse shape (b) optimization (Equation 2) was performed for all N p training data sets based on the COV and starting with CP mode and rectangular pulse shapes. NRMSE p and SED p were derived from the individually optimized pulse for subject p and applied on subject p. For this optimization, we used MATLABs global search algorithm ("GlobalSearch", see Ugray et al 34 for more details). For the input variables (COV), fixed lower and upper bounds (Supporting Information Table S1) and a boundary condition to guarantee feasibility with respect to a maximum slew rate of 200 mT/m/ms were established. A single function evaluation took 234 s and a time constraint of 1 week for the optimization was set. This approach enables a tradeoff, which is adjustable with the ratio of 2 different relative weights w Hom and w SED to achieve both good B + 1 -homogeneity and low SED, respectively. The exponential behavior of each single data set's contribution to the cost was chosen to strongly penalize single patients' high NRMSE and SED values aiming to generate pTx pulses, which work universally. In a second step, when an optimized set COVopt has been found they were used to generate corresponding universal pulse shapes b 0 . This was done by solving the following problem using the same interior-point-based global search method with starting points generated in the same way as described above: Here, a single cost value was calculated via all training subjects' NRMSE and SED for a specific universal pulse shape b 0,COVopt . A single function evaluation took 168 s on the same desktop computer and a time constraint of 1 week for the optimization was set. The magnitude of each RF sample in b 0,COVopt was limited to 150 V (option "nonlcon" in MATLAB's function "createOptimProblem").

| Individually optimized pulses
Based on the so created UPs (trajectory derived from COVopt and corresponding universal RF shape b 0,COVopt ), individual RF shape optimization was performed (Equation 2) for each training subject using varying energy regularization weights λ. For each regularization weight, each training subject's individual maximum SED value of the subject's respective pulse was calculated. Thereby N p different SED data points were generated for each value of λ. The following function was used to fit these points: The parameters a, b, and c are specified by minimizing the RMSD of the data points from the fitting curve. This exponential fitting curve was found empirically and outperformed polynomial, linear, sum of sine, and Fourier fitting curves with respect to RMSE. Based on the 10-s limit for local SAR, because it was found to be the most restrictive constraint (20 W/kg; according to IEC guideline 60601-2-33), the given pulse sequence parameters (FA, TR, number of pTx pulses per TR) and the SED of the fixed adiabatic hyperbolic secant inversion pulse (nominal voltage 350 V, pulse duration 12.8 ms, SED = 34.03 ± 1.25 J/kg), a limit for the pTx excitation pulse's SED was also calculated (SED ub : 122 ± 5.9 mJ/kg). This SED (λ) curve enables quick adjustment of λ online to reach SED values below the pulse's SED limit (see section below).
Based on a set of COV (Equation 6), b 0 (Equation 7) and SED(λ) (Equation 8) data that were calculated offline, individual pulse optimization was performed online during the scan on the scanner control computer (Intel Xeon E5-1620 CPU; 3.5 GHz; 4 Kernels; 32 GB of RAM) using MATLAB Runtime 8.0. RF pulse optimization could be performed within 15 s by solving Equation 2. In the case that SAR exceeded the allowed limit, was adjusted using an online SED regularization algorithm described in Supporting Information Pseudocode S1. All online optimization algorithms were embedded into the scanner's software system to enable short online computation times and to facilitate future clinical applicability.
The total preparation time of the pulse sequence was 90 s. During this period, standard preparation (transmitter adjustment, frequency adjustment, adjustments of the directional couplers [DICO]), B 0 mapping, B + 1 mapping, and pTx pulse design were performed automatically. Default third order B 0 shim currents using the entire brain as target ROI were applied to all measurements. An overview of the whole process to design FOCUS pulses is shown in Figure 1.
For this study, various sets of COV, b 0 , and SED(λ) data were calculated with different relative weightings. The most relevant combinations were evaluated and further examined in this paper, being [w Hom , w SED ] = [1,1] and [5,1]. These 2 combinations have been chosen among others, because they reflect 2 different tradeoffs: 1 combination that puts weight on low NRMSE values [5,1] whereas still providing not excessively high SED values and 1 set [1,1] that puts weight on low SED while still providing acceptable homogeneity. Consequently, the resulting UPs or IOPs are denoted by UP 11 , UP 51 , IOP 11 , and IOP 51 . Their respective trajectories are shown in Figure 2 and the optimized COV of both pulses are shown in Supporting Information Table S1. Evaluations of other combinations of [w Hom , w SED ] can be found in Supporting Information Figure S1.

| RESULTS
The simulated NRMSE and SED values of all 72 subjects for all pTx pulses and a fixed CP rectangular shaped pulse are reported in Figure 3. The NRMSE values of 28.2 ± 2.4% (mean ± SD) by the fixed CP rectangle pulse could be reduced to 18.3 ± 2.2% by UP 11 and 16.2% ± 1.9% by UP 51 . Individual optimization could substantially further decrease NRMSE to 13.4 ± 0.8% (IOP 11 ) and 10.8 ± 0.7% (IOP 51 ). Yet, pTx pulses with [w Hom , w SED ] = [1,1] also slightly increased the maximum local SED exposure from 27.49 ± 1.0 mJ/kg (CP) to 30.0 ± 0.96 mJ/kg (UP 11 ) and 35.9 ± 5.2 mJ/kg (IOP 11 ). Higher weighting of homogeneity leading to lower NRMSE values, also causes higher amounts of SED, such as 83.5 ± 2.7 mJ/kg (UP 51 ) and 111.5 ± 19.1 mJ/kg (IOP 51 ). It has to be noted that the CP rectangle pulse has a shorter duration of 0.1 ms than the 1 ms pTx pulses and a lower mean FA (CP = 4.0°, UP 11 = 6.5°, UP 51 = 6.6°, IOP 11 = 6.8°, and IOP 51 = 6.9°). IOP 11 outperformed UP 51 in both NRMSE (mean decrease by 17%) and SED (mean decrease by 57%). Although all pTx pulses were only trained with healthy European subjects, differences regarding homogeneity between study groups were marginal for all pulses. For each pTx pulse, mean NRMSE values were highest for the "European, anomalies" F I G U R E 1 Schematic description of the pTx pulse design workflow. The offline optimization process uses previously measured data sets and predefines the transmit k-space trajectory, initial pulse shapes, SED dependence of the energy regularization parameter, and its initial value. Subject-specific pulse shapes were calculated online using the B + 1 , and B 0 maps were acquired during preparation time. Therefore, the amount of additional time required for online optimization is 67 s F I G U R E 2 k-Space trajectories of both optimized UP and corresponding FOCUS pulses UP 11 /IOP 11 and UP 51 /IOP 51 . The individual sampling points are denoted as black points, with the center part of k-space sampled more densely than the peripheral regions group, and lowest for the "European, training" group, yet, those differences were generally minor ( Figure 3).
Pulse energies for all subject groups and for all pulse types are displayed in Supporting Information Figure S2. In addition, different IOPs were designed based on various training groups and evaluated on all data sets (Supporting Information Figure S3). Compared to the choice of [w Hom , w SED ], the choice of the training group had only minor influences on the results regarding pulse performance. We also optimized pulses using various numbers of training subjects (Supporting Information Figure S4), yet, both increasing the number of subjects to 24 (2 weeks optimization time) and using only single subjects (2 days optimization time) did not improve the results. Furthermore, IOPs trained with single subjects did not show lower cost values for the respective subject it was trained on.
UPs based on SPINS trajectories markedly improved NRMSE compared to CP-mode and they also proved useful as starting point (b 0 ) for fast online optimization of subject-specific pulse shapes (Equation 1), as Figure 4 shows. RF pulse optimization was stopped when the cost function decreased by <0.1% of its previous value. When initialized with universal RF shapes, IOP 11 reached up to 5% better cost values (average = 1%) in 0.3 s less time on average, and IOP 51 reached up to 41% better cost values (average = 8%) in 2.3 s less time on average on our PC. Figure 5 shows the performance of IOPs with online SED regularization for different limits. Lower NRMSE values are reached by those pulses whose initial SED distributions are closer to the given SED limits (low mean_ΔSED values). Doubling the upper SED limit from 100 mJ/kg to 200 mJ/kg resulted in a relatively low improvement of NRMSE (IOP 11 = 0.26%, IOP 51 = 0.49%), whereas changing from 30 mJ/kg to 100 mJ/kg resulted in a higher decrease of NRMSE (IOP 11 = 2.58%, IOP 51 = 9.48%). Because both IOPs generate more SED than their corresponding UPs, Supporting Information Figure S5 shows both IOPs regularized with SED-bounds equal to the maximum and minimum SED values of their corresponding UPs to evaluate the benefit of individual RF shape optimization. When generating the SED values in the same bounds, both IOPs still show lower NRMSE values than the corresponding UPs. Figure 6 shows MPRAGE images and corresponding simulated FA maps of a single subject (m26) using either CP pulses, UP 11 , UP 51 , IOP 11 , or IOP 51 without SED regularization as excitation pulses. Both UPs gained higher signal homogeneity and intensity especially in the cerebellum and in the lower head regions compared to the CP mode. IOPs further improved signal homogeneity throughout the entire F I G U R E 3 NRMSE and maximum SED values of different pulse types for all 72 subjects grouped in Europeans used for optimization (12), other Europeans (36), Asians (12), and Europeans with anomalies (12). The asterisk denotes the mean value, the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively and the red crosses denote outliers. UP 11 and IOP 11 were able to decrease the NRMSE compared to the CP pulse, UP 11 requires a 9% higher, IOP 11 requires a 31% higher amount of SED. Using the same pairs of w Hom and w SED , IOPs further improved the homogeneity at the expense of higher SED exposure. Yet, IOP 11 reaches both lower NRMSE and SED than UP 51 . Differences between study groups were generally lower than differences between pulse types | 3147 HERRLER Et aL.

F I G U R E 4
Online RF shaping cost function decrease for both IOPs as a function of the iteration i using either UP shapes or CP mode as starting points (b 0 ), indicated with "UPinit" (green) and "CPinit" (blue), respectively. The 2 upper plots show the mean curves among the validation data (n = 60). The lower plots show the corresponding individual curves. The optimization stopped, when the cost improves <0.1% of its own value in 1 iteration. When initialized with universal RF shapes, IOP 11  head. Yet, differences between UPs with different weights, as well as between IOPs with different weights are smaller than between UPs and IOPs. For better comparison of CP and FOCUS pulses, MPRAGE images and corresponding simulated FA maps of another single subject (m29) including a CP pulse having higher voltage to achieve a mean FA of 7° next to UP 11 and IOP 11 are shown in Figure 7. Here, both UP 11 and IOP 11 show a more homogeneous FA distribution than the CP pulses. The differences between both UPs and their corresponding IOPs are shown in Figure 8 on 4 exemplary subjects.
FA variations produced by UP 11 can be minor (subject 1), but also apparent (subject 2), whereby IOP 11 reliably improves homogeneity, especially in the latter case. UP 51 also shows inhomogeneities (subjects 3 and 4), which are improved by switching to IOP 51 . Figure 9 shows several MPRAGE images and FA maps of the subject considered to have the most apparent anomalies. UP 11 still yields acceptable signal and FA homogeneity in most brain regions, yet, in some regions, certain anatomical structures (especially parts of the cerebellum) are not visible. IOP 11 shows improvement in the FA homogeneity, F I G U R E 6 MPRAGE images of a single subject either CP or pTx pulses for excitation and the pulses' corresponding simulated FA maps. UPs substantially improve homogeneity, yet, tend to overexcite with high FAs in the center of the head. IOPs do improve homogeneity and reach exactly the chosen 7° excitation FA. For CP/UP 11 /UP 51 /IOP 11  having no such regions with severe signal loss. In none of the acquired MPRAGE images artifacts have been identified, which could be attributed to the adiabatic pulse.

| DISCUSSION
In this work, we combined the concept of UPs 21 and individual pulse optimization to achieve a fast online calculation for subject-specific or FOCUS pTx pulses. The individual pulse optimization under strict SAR constraints required only ~67 s of additional sequence preparation time (B + 1 mapping and pulse calculation). Therefore, the applicability of these pulses is clinically feasible. UPs reliably reduced FA-NRMSE compared to CP mode requiring no additional preparation time, and the proposed FOCUS pulses further improved pulse performances by further decreasing NRMSE. Because only marginal differences in image quality and contrast between MPRAGE images obtained with IOP 11 and IOP 51 were observed, we consider IOP 11 better suited for clinical routine because of its decreased SED exposure. In addition, an online SEDregularization method was implemented that allowed us to explore the full set of available SEDs tailored to subject and sequence parameters. By setting a lower bound for SED, we can enforce lower NRMSE values for subjects, whose SED(λ)-function is strongly differing from the SED fitting curve. By using FOCUS excitation pulses with controlled SED exposure, it was possible to apply a standard adiabatic inversion pulse, which worked sufficiently well. In contrast to the original UPs, as proposed by Gras et al, 21 which are based on k T points, 17 we used a SPINS trajectory 18 that allows distribution the RF energy along an excitation k-space trajectory that includes a larger number of k-space points as than the k T points trajectory. This trajectory was originally designed for 3T MRI, and in this work, we introduced several additional parameters that characterize the trajectory to increase the degrees of freedom. Furthermore, the concept of FOCUS pulses can also be used for different excitation k-space trajectories, for example, k T points 17 for nonselective, spiral/spherical shaped ones for both nonselective and 3D volume selective 15,16 or spokes for slice selective 13,14 pTx pulses. For trajectories like spokes or k T points, which use only a few pulse shape samples compared to a SPINS trajectory, online RF pulse calculation needs even less time. To further save online computation and measurement time during the sequence preparation phase, pulse shapes could be calculated with lower temporal resolution. Additionally, the applied B + 1 mapping sequence could be compared to different turbo FLASH protocols 35 or various B + 1 mapping sequences (e.g., DREAM or actual flip-angle imaging [AFI]), 36,37 as also investigated in Chung et al 35 and Pohmann and Scheffler. 38 Furthermore, we included extracranial regions into the optimization process, which most likely increased our NRMSE values. Additionally, the fact that IOP 10,1 does not improve NRMSE ( Figure S1), and entirely individually optimized FOCUS pulses do not reach lower NRMSE values for their respective subject than other pulses ( Figure S4), indicates a general limitation of this approach (limited offline optimization time and number and range of the COV, limitations of the "GlobalSearch" algorithm itself, as well as using a predefined SPINS trajectory and pulse duration). Yet, when using a mask that ignores non-brain tissues in the pTx pulse design, we reached lower mean NRMSE values among all data sets of ~7% (see Supporting Information Figure S6). These values are in good agreement to the NRMSE values F I G U R E 8 MPRAGE images and corresponding FA maps with both types of UPs and their respective IOPs. The green/red numbers in each row denote the NRMSE/SED of the respective UP and IOP. The red arrows indicate the most obvious differences in the MPRAGE images. In the FA maps, the improvements of both IOPs compared to their respective UPs are clearly visible. Yet, in the MPRAGE images of subject 1, almost no difference between UP 11 and IOP 11 can be seen. However, in subject 2, UP 11 shows regions with signal loss, especially in the cerebellum (red arrow). Especially in these regions, IOP 11 improves signal intensity and homogeneity. UP 51 (applied to subject 3 and 4) also produces FA inhomogeneities (red arrows), which also can be further mitigated by switching to IOP 51 observed by Gras et al 21 for "subject tailored" pulses (individual k-space trajectory and RF shapes). In general, online RF pulse calculation can also be done with several predefined trajectories with corresponding UPs, whereas the most suitable FOCUS pulse for the specific sequence and patient is chosen online. Furthermore, to reduce SAR, the adiabatic inversion pulse could be replaced by a FOCUS pulse as well (using ~78% of this adiabatic pulse's SAR, data not shown). On the contrary, this would increase online optimization time (from ~15 s to ~70 s online pTx calculation time). In principle, this approach can also be used with a k T point trajectory, although this would limit the available degrees of freedom for online optimization. On the contrary, in this case, the online optimization could use other fast and clinically applicable methods as presented in Guérin et al 30 and Hoyos-Idrobo et al 39 that more directly use local, global SAR, and power constraints.
In previous works, pTx pulses were applied to smaller study groups of up to 20 subjects to demonstrate the feasibility. 8,17,21,24 In this work, we evaluated the design of the pTx pulses on 72 subjects in total and included both patients and different healthy ethnic groups (Europeans and Asians). Compared to CP-mode, the pTx pulses-both UPs and IOPsshowed substantial improvements in FA homogeneity for all subjects despite varying age, head shapes and sizes, as well as anatomical anomalies, which were present in a subgroup of the subjects. Therefore, the designed pTx pulses prove to be robust and thereby clinically applicable. The choice of the regularization parameters such as [w Hom , w SED ] or the choice of the pulse type (e.g., IOPs instead of UPs) had a larger influence on NRMSE than the differences between study groups. Pulses derived from different training groups showed similar performances ( Figure S3), therefore, dividing the data sets into "healthy Europeans", "Asians", and "Europeans with anomalies" might not be required, and other features remain to be explored that allow creating certain clusters having their own optimized COV. As Figure 5 suggests, there is a need for adapting the trajectory and regularization weight to each other, which was done in this work by optimizing the respective parameters simultaneously. For abdominal MRI, Tomi-Tricot et al 40 found a larger dependence on the training data. This might be caused by a larger anatomical variability in the abdominal region and also the use of different solvers, pulse calculation techniques and k-space trajectories (SPINS trajectory with individual RF optimization vs. entirely precalibrated UP-based k T -point trajectories).
FOCUS pulses have shown great performance on a large number of subjects. However, there is still potential to further improve it by finding better subject-specific trajectories online in a clinically feasible amount of time. Additionally, the offline optimization process might be improved by using different global solvers, input value ranges, and function and input tolerances, which could potentially lead to COV or UPs that are more tailored to single subjects or groups. Using our solver, we did not observe improved homogeneity or lower SAR values for entirely individually optimized pulses for their respective subjects (see Supporting Information Figure S4). Because the applied solver was used F I G U R E 9 MPRAGE images and corresponding FA maps using UP 11 and IOP 11 of a patient who had brain surgery and is considered to be the subject with the most apparent anomalies in brain tissue. The NRMSE values of UP 11  with a fixed time constraint, it might be possible that our COV and UPs are derived from local minima. Furthermore, using subject-tailored trajectories might be essential for other body parts such as the abdomen where inter-subject variability is higher. 40

| CONCLUSION
We presented a method to generate FOCUS pTx pulses based on UPs in only ~67 s additional sequence preparation time. Our simulations performed on 72 subjects show, that FOCUS pulses generally reach lower FA-NRMSE values and can also reach a better tradeoff between FA-NRMSE and SAR values than the designed UPs. On 7 additional subjects, we have shown experimentally that, first, UPs and FOCUS pulses improve FA homogeneity compared to CP pulses. Second, the presented FOCUS pulses provide better FA homogeneity than UPs, which is particularly evident in a patient having apparent anomalies.