Standardized universal pulse: A fast RF calibration approach to improve flip angle accuracy in parallel transmission

In parallel transmission (pTX), subject‐tailored RF pulses allow achieving excellent flip angle (FA) accuracy but often require computationally extensive online optimizations, precise characterization of the static field ( ΔB0 ), and the transmit RF field ( B1+ ) distributions. This costs time and requires expertise from the MR user. Universal pulses (UPs) have been proposed to reduce this burden, yet, with a penalty in FA accuracy. This study introduces the concept of standardized universal pulses (SUPs), where pulses are designed offline and adjusted to the subject through a fast online calibration scan.


| INTRODUCTION
In MRI at ultra-high field (UHF), radiofrequency (RF) excitation profiles produced by the RF transmit coil suffer from spatial inhomogeneities due to destructive interferences occurring within the body. They show reproducible patterns (typically low signal in the temporal lobe and cerebellum for head imaging at 7T), but present to a certain degree an intersubject and intersession variability that depends on the position in the coil, morphology, and composition. 1 To mitigate this effect, the use of parallel transmission (pTX) with an array of local transmit coils has been shown to be a very efficient approach. [2][3][4][5] In pTX, the amplitudes and phases of the pulses transmitted by the channels of the array can be modulated independently to homogenize the excitation profile over the volume of interest, provided the actual static field offset (ΔB 0 ) and the transmit RF field (B + 1 ) distributions are known. For nonselective excitations, various approaches have been proposed to design optimized RF and magnetic field gradient (MFG) waveforms, in particular with k T -points, 6 SPINS 7 , and recently, GRAPE. 8 The design of such subject-tailored pulses results in very homogeneous RF excitation profiles, at the cost of the scan time required to map ΔB 0 and B + 1 fields and to design the pulses, along with a certain expertise from the MR user. Strategies have been proposed to speed up and ease the acquisition workflow using either fast B + 1 mapping, 9 fast pulse computation 10 , or universal approaches. [11][12][13][14] Within the universal pulse (UP) framework, the pulse design step is performed offline so that no time or expertise is required during the scanning session. UPs make use of the reproducibility of the ΔB 0 and B + 1 distributions across subjects to design a pulse that homogenizes the excitation profiles simultaneously across all subjects of a so-called design database. UPs can then be applied to new subjects without further calibration and with good excitation performance, albeit with some performance penalty compared to the subject-tailored pulse design approach. For many imaging applications, UP performance for brain imaging at 7T appears to be sufficient. 12 It has also shown robustness across subjects, though some variability remains due to differences in head morphology, placement in the RF coil and composition. As a result, an UP designed to realize a given uniform FA value across the entire brain can occasionally produce a FA distribution whose average value across the brain can deviate significantly from the expected value. For instance, in Ref. [15], the average deviation from the nominal FA could reach up to 10 % over the brain in some volunteers.
In this study, a new UP design approach is introduced to improve the stability of the excitation pattern across subjects through a standardization of the pulse design database and a session-specific fast calibration scan. It relies on the normalization of the N c -dimensional B + 1 distribution of the subject, where N c is the number of transmit channels (8 in this work) to map the B + 1 amplitudes and phases of the subject to a referenceB + 1 distribution. This step is performed through a linear transformation with an N c × N c complex adjustment matrix, which is meant to account for variations in coil loading and coupling between transmitters. This transformation is applied on every subject of the database to produce a so-called standardized design database; a UP designed on the standardized design database is then called a standardized UP (SUP). This approach imposes a calibration step during the MRI exam, where the adjustment matrix of the subject is estimated from a fast B + 1 scan (< 10 s). [16][17][18] However, as for UPs, SUPs do not require online pulse design, so that only a fast calibration procedure is performed online, resulting in a simplified workflow and shorter processing times compared to a subject-tailored approach.
The aim of the current study was to assess the performance of adjusted SUPs by comparison to UP and subject-tailored pulses in the small tip angle regime, with nonselective pulses as used for spin excitation in the magnetization prepared two rapid acquisition gradient echoes (MP2RAGE) sequence. 19,20 A GRAPE algorithm was chosen for the pulse design to allow for short pulse duration and improved robustness to off-resonance artifacts. 8 UP, SUP, and subject-tailored pulse performances were compared in terms of the normalized root mean square error (FA-NRMSE) and coefficient of variation of the simulated FA profile, along with in vivo acquisitions at 7T.
In this work, the same inversion pulse-a GRAPEoptimized pTX pulse-was used throughout to ease the comparison between the three pulse design strategies. The two inversion contrasts of the MP2RAGE were further used to perform T 1 quantification so that T 1 maps obtained with adjusted SUP were compared to the ones obtained with UP with a B + 1 correction. 14,21 Following this approach, any confounding effect such as receive profile variation, T * 2 contrast can reasonably be ignored.

| Universal pulse design
Let us consider a design database consisting of ΔB 0 and B + 1 brain field maps acquired over N s subjects with N c transmit coils (ΔB 0,S:1 , B + 1,S:1 , ⋯, ΔB 0,S:N s , B +

1,S:N s
). For each subject, ΔB 0 and B + 1 brain field maps are defined over a set of positions ℛ S:n which correspond to the brain volume. For simplicity, we represent ΔB 0 and B + 1 as M n × 1 and M n × N c matrices (ΔB 0 being real and B + 1 complex), where each row of the matrix corresponds to one voxel in ℛ S:n and where M n = |ℛ S:n | represents the cardinality of ℛ S:n . Let P UP be a UP defined by its RF and MFG waveforms: U UP (t) ∈ ℂ 1×N c in volt and G UP (t) ∈ ℝ 3 in T/m, respectively. The universal pulse P UP , denoted by P UP = U UP ⊙ G UP , is designed offline on the design database with a minimization of the average FA-NRMSE across subjects: where: and where t is the targeted FA and n (r) is the actual FA simulated at position r for subject n. The latter is given in the STA approximation by Ref. [22]: where |z| corresponds to the magnitude of z, T to the pulse duration in s, = 42.57 MHz/T to the gyromagnetic ratio of the proton, and k(t) to the transmit k-space trajectory in m −1 : We note in Equation (3) that the product U (t) ⋅ B + 1,S:n (r) is a matrix product where the individual TX channel contributions are summed together.

| Standardized universal pulse design
In a normalization process of the design database called standardization, every B + 1 map of the database is expressed as the linear transform of a reference, while the ΔB 0 maps are left unchanged. The first step of this process is thus to define a reference B + 1,ref over the region of interest ℛ ref . Let ℐ(r) represent the set of subjects where ℛ S:n is defined (i.e., ℐ(r) = 1 ≤ n ≤ N s , r ∈ ℛ S:n ). B + 1,ref is taken here as the average B + 1 over ℛ ref , chosen as the set of voxels where ℛ S:n is defined in at least half of the subjects: where |ℐ(r)| is the cardinal of ℐ(r). The second step is to compute the dimensionless N c × N c adjustment matrix L n that fits B + 1,S:n to B + 1,Ref as the solution to the least-squares problem: where ‖. ‖ Fro represents the Fröbenius norm and, for any set of positions A ∈ ℛ S:n , B + 1,S:n (A) denotes the submatrix of B + 1,S:n containing the rows that correspond to A. The standardized RF field of this subject, B + 1,S:n , is then defined as: This operation is repeated for every subject of the design database. A standardized UP, P SUP = U SUP ⊙ G SUP , is an UP designed on the above-defined standardized database. As the databases used to design the UP and the SUP are different (i.e., "raw" vs. standardized), the resulting RF and MFG waveforms of these pulses differ in general.

| SUP adjustment
In contrast to P UP which can be applied to new subjects without any calibration, a calibration step is required to adjust P SUP to the subject (see Figure 1). This calibration consists in estimating the session-specific adjustment matrix L n from the subject B + 1 field (rewritten as a matrix) with Equation (6), and then applying the linear transform to the SUP RF waveforms, while its MFG waveforms are left unchanged: The adjusted standardized pulse P adjSUP = U adjSUP ⊙ G SUP thus obtained is defined as: To understand this, we insert U adjSUP and G adjSUP in Equation (3) and examine the product U adjSUP ⋅ B + 1 . From Equations (7) and (8), we have: that is, the action of P adjSUP on B + 1 is equivalent to the action of P SUP on B + 1 . Since the least-squares problem for computing L is overdetermined (N c × N c ≪ M n complex numbers to fit), it can be estimated from a subsampled B + 1 map, for example, from a partial coverage of the brain with three slices. This allows achieving a significant reduction of the scan time as compared to a full-brain coverage B + 1 mapping protocol. A verification is then performed to ensure that the adjusted SUP fits scanner and SAR limits (maximal energy per channel E ch,max , average energy summed over channels E tot, max and peak power limit V max ). Let be the pulse scaling parameter defined as follows: If > 1 (i.e., any of the above limits is exceeded), the pulse length is scaled according to the equation: The pulse is then reinterpolated on a 10 s raster time to comply with the pulse definition imposed by the scanner's hardware.

| MP2RAGE UP and SUP excitation pulse
A small tip angle (max 8 • ) UP and SUP were designed with the GRAPE algorithm. 8 The designs of these pulses were performed offline on a design database comprising the brain field maps of 20 subjects that were collected in a previous study. 11 The UP was designed on this "raw" database, while the SUP was designed on the standardized database according to the processing steps described in the Theory section.
The UP and the SUP were computed using the activeset algorithm available in Matlab (R2020a release, the Mathworks, Natick, MA, USA). The optimization algorithm was initialized with G(t) = 0 and U(t) obtained using the variable exchange method. 23,24 These pulses were 240 s long and the design constraints to account for scanner and SAR limits were (at the coil plug): energy per channel E ch,max = 15 mW, average energy E tot,max = 90 mW, RF peak power limit V max = 165 V, maximum slew rate = Methodology of the adjusted standardized universal pulse (SUP) approach. A, The design database that has been normalized to a reference (standardized database) is used to design a SUP with the same formalism as for UP. This pulse design step is performed offline. b-d) Calibration steps performed during the scanning session to adjust the SUP to the subject. B, During the scan, a subsampled B + 1 map is acquired with, for example, three-axial slices equally spaced over the brain, as represented here with dark lines. C, An N c × N c matrix is then estimated by fitting the B + 1 map to the reference. The magnitude of the adjustment matrix is close to the identity matrix, while diagonal phase terms are close to 0 rad. D, The RF waveforms of the SUP are multiplied by the adjustment matrix, while its gradient waveforms are initially left unchanged. MFG ramps are added and, if needed, the RF and gradient waveforms are scaled and re-interpolated to match scanner and SAR limits. The adjusted SUP can then be run on the MR system 180 T/m/s. The number of iterations was limited to 2 × 10 3 and the discretization time was set to 10 s. As the GRAPE optimization can lead to a gradient waveform with nonvanishing initial and final values, MFG ramps of 50 s were added at the beginning and the end of the UP and the SUP to match slew rate constraints.

| MP2RAGE UP inversion pulse
A 7-ms-long 180 • GRAPE UP was design offline on the "raw" database using the active-set algorithm, and taking a low-amplitude hyperbolic-secant pulse for the initialization of the RF waveform. Here, a discretization time of 40 s was used, the RF energy was limited to 900 mW per channel and 6 W total, and the maximum RF voltage was constrained to not exceed V max = 165 V. For the design of the inversion pulse, an additional ΔB 0 weighting term in the objective function to increase inversion accuracy in voxels exhibiting strong ΔB 0 excursions. 8

| Validation in simulation
The performance of the MP2RAGE excitation pulse was assessed in terms of fidelity to the targeted FA for the UP, the adjusted SUP and subject-tailored pulses using a control database of ΔB 0 and B + 1 maps acquired over 15 subjects different from those used in the design database. These maps were acquired in a previous study. 8,11 For FA simulations, the UP was applied to the subjects of the database without further adjustments. For the SUP, B + 1,S:n and B + 1,ref were first subsampled to match the B + 1 sampling pattern chosen for in vivo experiments (3 axial slices equally spaced over the brain and distant by 3 cm, see Figure 1B). The adjustment matrix L n defined by Equation (6) was then computed by pseudo-inverse ( Figure 1C). The SUP was finally adjusted to the subjects and scaled using Equations (8) and (12). For comparison with the subjecttailored approach, a subject-specific pulse was designed for every subject of the control database using the same constraints as for UP and SUP, and using the UP for the initialization of the RF and MFG waveforms. 10 The simulation results were analyzed in terms of (i) the average FA across the subjects and (ii) the coefficient of variation (CV) of the FA, defined for every subject n and for every voxel in the domain ℛ S:n as: In order to evaluate the importance of the off-diagonal elements of the adjustment matrix on SUP performance, an alternative definition of the adjustment matrix was investigated wherein the off-diagonal coefficients of the adjustment matrix were forced to zero: where where diag( ) denotes the N c × N c matrix whose diagonal coefficients are 1 , …, N c . These two adjustment strategies, called full (Equation 6) and diagonal (Equation 14) adjustments, were compared in terms of FA-NRMSE and residual error of the fit, computed as the average ||B + 1,S:n − L n B + 1,ref || Fro over the control database. Finally, a measure of the distance of the adjustment matrix to the identity matrix was also performed with two different metrics: the L ∞ norm and the maximum coefficient over the diagonal, denoted in this work by L ∞,D .

| Experimental validation
Acquisitions were performed on three healthy volunteers (21 ± 1 years, 1 female/2 males) on a whole-body investigative 7T MR system (Magnetom 7T, Siemens Healthcare, Erlangen, Germany) equipped with a Nova 8Tx-32Rx head coil (Nova Medical, Wilmington, MA). This study was approved by the local Institutional Review Board (approval number 2018-A01761-54) and the volunteers provided informed written consent prior to examinations. Sequences were run within the Siemens protected mode, that is, with a peak power limit per channel of 540 W, average power limits of 1.5 W per channel and 8 W total at the coil plug. For each volunteer, a second order shim was performed, then a ΔB 0 scan, consisting in a 3D-GRE sequence, and two B + 1 scans, consisting in interferometric turbo-FLASH sequences, [16][17][18] were acquired: (i) a fast B + 1 scan that was used for the computation of the L matrix, and (ii) a full-brain B + 1 scan that was used for the correction of B + 1 inhomogeneities in T 1 quantification based on the MP2RAGE image pairs. Anatomical scans consisted in three MP2RAGE sequences 19,20 : the first run used the UP version of the excitation pulse, the second run the adjusted SUP, and the third run the subject-tailored one. Given the low FA-NRMSE achieved with the UP inversion, all MP2RAGE acquisitions used this inversion pulse for magnetization preparation. Hence, any difference amongst these acquisitions is expected to originate mostly from variations in the FA profile of the excitation pulse. Sequence parameters are listed in Table 1.
Uniform (UNI) contrasts were processed from the two inversion volumes of the MP2RAGE sequences using an in-house Matlab script. 19,20 T 1 maps were computed both without and with B + 1 inhomogeneity correction using the full-brain B + 1 maps according to the procedure introduced in Refs. [14,21]. UNI images were segmented into five tissue classes (gray matter, white matter, CSF, bone and soft tissues) using spm12 (R7219, http://www.fil.ion.ucl.ac.uk/ spm). 25 The resulting tissue probability maps were thresholded to 5% to obtain gray matter and white matter masks.

| Standardized design database
Using a statistical representation of the interindividual B + 1 variation, 15 maps of the second order statistics (i.e., mean and covariance matrix) of the transmit RF field can be computed from a B + 1 database as: where A H denotes this Hermitian conjugate of A. In this study, this has been done from the 35 B + 1 obtained by pooling the design (20 subjects) and control (15 subjects) databases. For the estimation of the variance matrix, a Ledoit-Wolf correction 26 was applied as the number of maps remained too small given the high dimensionality (8 complex variables) of the measured B + 1 maps (the observable quantity). Not using this correction could lead to rank-deficient correlation matrices which are physically not realistic. The second order statistics estimated over the "raw" design database and the standardized design database are displayed in the Supporting Information Figure S1. While the mean transmit RF fields ̂ B + 1 and ̂ B + 1 (Supporting Information Figure  S1A,B) remain very similar, it can be observed that the covariances of the standardized RF field (ĈB+ 1 ) are significantly smaller than the covariances of the "raw" RF field (Ĉ B + 1 ) (Supporting Information Figure S1C,D). In other words, a significant part of the variability of the "raw" RF field is contained in the adjustment matrix L.

| Simulation results
The FA simulation results over the control database are reported in Figures 2 and 3 for the UP, the adjusted SUP and for the subject-tailored pulses. An example of set of pulses obtained in one subject is reported in Supporting Information Figure S2. One can observe that the RF and MFG waveforms of the UP and the SUP differ significantly. The value of the pulse scaling parameter was generally defined by the energy per channel exceeding the 15 mW limit. The distance of the adjustment matrix to the identity matrix, as well as the pulse energy and maximal amplitude are represented in the Supporting Information Figure S3. From this figure, no clear correlation was found between the energy or the peak amplitude of the adjusted SUP and ‖ L n − I‖ ∞ . Moreover, comparing ‖ L n − I‖ ∞ and ‖ L n − I‖ ∞,D , we found that the maximum difference between L n and I was located on the diagonal only for half of the subjects. Finally, we observe from this figure that the scaling procedure applied on the adjusted SUP to satisfied the imposed energy constraints led to an average increase of the pulse length of 25% (ranging from 0% to 50%).
The FA distributions averaged over the control database (see Equation 13) illustrated in Figure 2A,B, showed

F I G U R E 3
Boxplots of the subject-wise FA normalized root mean square error (FA-NRMSE) measured on the control database of 15 subjects for a) the hard pulse in CP mode, and b) the UP, adjSUP Not scaled , adjSUP Scaled and the subject-tailored pulses. CP results were displayed in a separate boxplot as they required a different scale. Two boxplots are displayed for the adjusted SUPs: the dark-colored ones represent the results obtained in with the full adjustment, that is, from an adjustment matrix with off-diagonal coefficients, while the lightcolored boxplots were obtained from a diagonal adjustment. Adjusted SUPs showed improved performance compared to UP in terms of FA-NRMSE, outliers and reduced intersubject variability. The pulse scaling to satisfy scanner and SAR limits only slightly decreased pulse performance. The adjustment with a diagonal matrix lead to lower FA-NRMSE compared to the approach with off-diagonal terms an improved accuracy and spatial homogeneity with adjusted SUPs compared to UP. Interestingly, the coefficient of variation computed from Equation (13) ( Figure 2C) demonstrates more reproducible FA patterns across subject using the SUP approach. This was also quantified by the histograms of the CV ( Figure 2D) where values are centered around 7% for the UP and 4% for the adjusted SUPs, both of them outperforming the CP mode. The effect of pulse rescaling (see Equation 12) can be observed in Figure 2 by comparing the performance of the the scaled (AdjSUP Scaled ) and non-scaled (AdjSUP Not scaled ) adjusted SUPs, the latter being obtained by forcing to 1 and thereby presenting the risk of exceeding either of the energy or RF amplitude thresholds imposed for the excitation pulse. From the FA profiles and the histograms, we can conclude that the effect of scaling the pulse had only a minimal effect on pulse performance. Finally, the subjecttailored pulses showed excellent performances, both in terms of spatial homogeneity and intersubject robustness.
In terms of FA accuracy, as shown in Figure 3, the FA-NRMSE values were 8.8% for the UP, 7.1% for the adjusted SUP and 1.3% for the subject-tailored pulses, on average across the 15 subjects of the control database. As a result, the adjusted SUP and the tailored approaches allow reducing the intersubject variability of the FA pattern as compared to UP.
A diagonal adjustment led to an increased residual error of the fit (residual error of 0.22 ± 0.04) compared to a full adjustment (0.13 ± 0.03). In addition, the FA-NRMSE was systematically higher for the diagonal adjustment as compared to the full adjustment. A paired t-test with a significance threshold of 5% returned a p < 0.01. Finally, the difference between the FA-NRMSE of the adjusted SUP and the UP appeared to be statistically significant for the full adjustment (p-value of 0.01), but not for the diagonal adjustment (p-value of 0.09).

| In vivo results
An axial view of the MP2RAGE UNI images acquired over the three volunteers of the study is displayed in Figure 4. UP images showed small signal inhomogeneities along the right-left and antero-posterior axis, while the signal in adjusted SUP images looked more homogeneous, and seemed occasionally to offer a better delineation between white matter and gray matter. Surprisingly, the signal obtained using subject-tailored pTX pulses for subjects 1 and 2 also showed spatial inhomogeneities in the frontal lobe. This was possibly caused by motion-induced local B 0 and B + 1 changes between the time of acquisition of the static F I G U R E 4 UNI contrasts measured for the three subjects of the study from the two inversion volumes of the MP2RAGE sequence acquired with a universal inversion pulse and an excitation A, universal pulse, B, adjusted SUP, C, subject-tailored pulse. A zoom in the frontal cortex is provided for each scan to emphasize the poorer symmetry of the white matter signal in UP-and subject tailored pulse-based as compared to SUP-based acquisitions. Yellow arrows on the splenium of the corpus callosum indicate hypersignals, symptomatic of a FA overshoot for the case of the UP-based acquisitions field offset and RF maps and the MP2RAGE sequence (approximately 30-min delay). In subject 3, where the subjecttailored MP2RAGE sequence was acquired straight after pulse design, this artifact was not present. The T 1 maps measured in subject 3 without and with B + 1 correction are presented in Figure 5. From the difference images (third row) and the associated boxplots, it can be seen that without B + 1 correction, T 1 error can locally largely exceed 100 ms for the UP-based acquisition. That bias was reduced using the adjusted SUP, the subject-tailored pulse remaining the most accurate solution though.
The T 1 Histograms measured in the joint masks of white and gray matter (i.e., cerebrospinal fluid excluded) for each subject are presented in Figure 6. As expected from the simulations, and noting that the three participants have the same age, T 1 values were quite reproducible across subjects when measured from adjusted SUP and subject-tailored images, while UP images showed a greater intersubject variability.

| DISCUSSION
In the current study, a new pTX UP design methodology called SUP was proposed where a set of RF and MFG waveforms is designed offline on a so-called standardized database, and the RF waveforms adjusted to the subject during the MRI session thanks to a fast calibration scan (< 10 s). For neuroimaging at 7T, UPs (and thus SUPs) provide already better homogeneity than a CP mode pulse at 3T 27 so that emphasis was put here on quantitative methods (MP2RAGE). Scans at higher fields or targeting pediatric imaging, where B + 1 variability is expected to increase, could also benefit from this method. Although the theory has been presented in the light of the small tip angle approximation, the proposed technique, potentially, could be applied to any type of RF pulses, that is, including slice selective, large FA and refocusing pulses. For the inversion pulse, however, we have observed that the gain offered by the adjusted SUPs was not significant, probably due to the already excellent FA homogeneity obtained with UPs in this nonlinear regime (FA-NRMSE < 2 %). 8 In addition, we observed that the high peak power demand of inversion pulse (the universal inversion pulse used in this study reached to maximum RF amplitude allowed by the system) can lead to an increase in pulse duration after scaling the pulse, possibly leading to a performance deterioration in brain regions exposed to strong static field offsets.
In simulation, adjusted SUPs improved signal homogeneity in the brain as compared to the UP approach, which translated in a decreased FA-NRMSE along with a better F I G U R E 5 A, T 1 maps computed in subject 3 by injecting UNI signal intensities and sequence parameters in the Bloch simulator (numerical integration) for the MP2RAGE sequence acquired with a universal inversion pulse and from the left to the right: an universal excitation pulse, an adjusted SUP excitation pulse and a subject-tailored excitation pulse. ). The quality of the MP2RAGE UNI images was very satisfactory for all RF pulse, although for some subjects a small signal asymmetry remained along the left-right and antero-posterior axis. We note here that this asymmetry was absent in the UNI images obtained from the SUP-based MP2RAGE acquisitions. The subject-tailored seemed to show a greater sensitivity to head motion, perhaps due to a decreased immunity to ΔB 0 variations. 12 This weakness however could be alleviated by employing, for example, a multifrequency band pulse optimization, 7 yet at the cost of a higher computational burden. The adjusted SUPs furthermore led to a robust T 1 quantification similar to that obtained with subject-tailored pulses. Compared to the subject-tailored approach, the main advantage of this method thus remains the absence any pulse optimization during the MRI session (Equation 6 being in essence a simple quadratic problem that is solved by pseudo-inverse).
In this study, we observed that the adjustment matrix can deviate significantly from the identity matrix, on the diagonal as well as off-diagonal coefficients. Furthermore, we observed that a full adjustment was superior to a diagonal adjustment, where the off-diagonal coefficients are forced to zero. This observation suggests that the variations in head positions and anatomies can lead to RF coupling variations that can be accounted for by using a full adjustment, but not a diagonal adjustment. The observed effect might be correlated with the variations of the scattering matrix of the transmitter array, which can be measured by the MR system itself, and which can be exploited for motion tracking purposes. 28 The adjustment procedure proposed in the current study can increase the pulse amplitude as well as its energy. To mitigate this problem, the pulse was scaled by a factor that exactly divides the pulse amplitude and energy by . This factor never exceeded 1.5, leading to a pulse duration increase of less than 120 s, of minimal impact on the echo spacing of the MP2RAGE sequence. Other strategies, consisting in adding enforcing RF energy constraints in Equation (6) to avoid scaling the pulse, were envisaged but lead to a significant performance degradation.
The least-squares problem to compute the adjustment matrix is overdetermined, which allows adjusting a SUP from a subsampled B + 1 scan that included only 3 axial slices. As such, the scan time spent to map B + 1 was reduced compared to the subject-tailored approach, where typically a minimum of 10 to 20 slices are required. 10 The putative link between the adjustment matrix and the scattering matrix of the transmitter array could be explored to eventually be able to further speed-up this pulse adjustment step. In future developments, the computation of the adjustment matrix could be implemented directly on the scanner as part of an automated calibration procedure, providing in pTX a generalization Although not shown in this work, we have tested the possibility of using any B + 1 map of a particular subject of the design database as the reference B + 1,ref . All these definitions were acceptable in the sense that the mean and covariance of the FA (see Equation 13) improved as compared to the UP. Taking the average resonant RF field seemed a better choice as it tends to eliminate the influence of a particular B + 1 in the database. However, we do not exclude that other (potentially better) definitions for B + 1,ref exist. Finally, an interesting field of application of SUPs would be MRI of other body parts at UHF using pTX. When imaging the abdomen or the thorax, one has to face greater B + 1 variations due to an increased variability of the morphology across subjects. In this potential field of application, it is foreseeable that the definition of the reference B + 1 can significantly vary as compared to what has been proposed in this work. Possibly a multiplicity of references, and therefore a large amount of data, would be necessary to make the approach generalizable to other body parts. 13 In conclusion, the use of adjusted SUP has shown potential to further improve excitation profiles and intersubject robustness as compared to the UP approach without session-specific pulse design, which simplifies the workflow compared to a subject-tailored pulse design.

SUPPORTING INFORMATION
Additional supporting information may be found in the online version of the article at the publisher's website.

FIGURE S1
Axial view of the second-order statistics of B + 1 computed on (A, B) the "raw" design database and (C, D) on the database normalised to a reference (i.e. standardised design database). On the left panels (A, C), images 1 to 8 correspond to the mean B + 1 measured on each transmission channel, while the ninth image correspond to the CP mode. On the right panels (B, D), diagonal images correspond to the variance of the transmission channels and off-diagonal images correspond to the covariance between the transmission channels. The reduction of the inter-subject variability through the design database standardisation can be appreciated here FIGURE S2 RF (top) and MFG waveforms (bottom) generated for the Universal Pulse (UP), and, for one example subject of the control database, for the adjusted SUP (adjSUP Not scaled ), the adjusted and scaled SUP (adjSUP Scaled ), and tailored pulse (Tailored pulse) FIGURE S3 A, Distance of the adjustment matrix to the identity matrix computed as the L ∞ norm (maximum difference betwen coefficients) versus the maximum coefficient over the diagonal. B, Scatter plot of the L ∞ -distance of the adjustment matrix to the identity matrix as a function of the adjusted SUP normalised total energy (blue dots), energy per channel (orange dots) and peak voltage (gray dots) for the 15 subjects of the control database