Optimized interferometric encoding of presaturated TurboFLASH B1 mapping for parallel transmission MRI at 7 T: Preliminary application for quantitative T1 mapping in the spinal cord

The acquisition of accurate B1 maps is critical for parallel transmit techniques (pTx). The presaturated turboFLASH (satTFL) method has been widely used in combination with interferometric encoding to provide robust and fast B1 maps. However, typical encodings, mostly evaluated on brain, do not necessarily fit all coils and organs. In this work, we evaluated and improved the accuracy of the satTFL for cervical spine at 7 T, proposing a novel interferometric encoding optimization. The benefits of such improvements were investigated in an exploratory study of quantitative T1 mapping with pTx‐MP2RAGE.


INTRODUCTION
MRI at 7 T has shown great promises, demonstrating the potential of high-resolution and quantitative techniques to study pathologies. 1 However, higher static magnetic field B 0 leads to several challenges, including B 1 inhomogeneity and high specific absorption rate (SAR). 2 Parallel transmit (pTx) methods, which independently control the different transmit channels of a given multi-element RF coil, have been introduced to mitigate these SAR and B 1 effects. 3,4 Different levels of complexity can be used for pTx MRI, 5 but all require a precise prior knowledge of the B 1 generated by each RF channel to provide customized B 1 optimization. Several B 1 mapping sequences have been proposed in the literature, each offering different characteristics, [6][7][8] with the presaturated TurboFLASH (satTFL) sequence 9 offering a great trade-off between accuracy and speed. This sequence, which uses the ratio of two images with and without preconditioning pulse to calculate the saturation flip angle (FA), is particularly SAR-efficient 8 and has shown robustness to flow and motion. 10 Although it has been developed for 2D B 1 mapping, 9 recent works have implemented it in 3D. 7,10 It is possible to measure individual B 1 maps with the satTFL by transmitting with only one channel at a time. However, this was shown to be particularly inefficient because low SNR leads to poor accuracy with increasing distance from the driven channel. 11 Alternatively, it was proposed that B 1 maps from linear combinations of the coil elements, B lc 1 , may be acquired with several RF modes, hence offering a better dynamic range while limiting the potential to have low SNR regions. 12 Individual B 1 maps may be derived from this so-called interferometric encoding according to: and where i is the i-th RF mode, k is the k-th RF channel, and is the so-called interferometry encoding matrix. 12 The matrix A must be invertible, limit noise amplification, and provide sufficient information from all regions across the different RF modes to limit low SNR and inaccuracies. Different methods have been introduced to improve the satTFL B 1 mapping. For instance, although all RF modes may be acquired with and without preconditioning pulse, a hybrid method was designed to obtain individual B 1 maps based on a single reference absolute satTFL B 1 -map scaled with each TFL RF mode. 13 This method was improved by acquiring two complementary RF shims to limit the loss of accuracy of the reference satTFL in low FA regions, as well as interleaved RF coil cycling for faster acquisitions (B 1 TIAMO). 14 Other studies have investigated the impact of the interferometry on the performance of the satTFL, and a few encoding matrices have been proposed, such as "One-inv" (all channels have identical amplitudes, one has opposite phase) and Fourier encoding, which have been used in vivo in particular for brain applications. 11,15 However, it was also shown that adjustment of the amplitude and phase of the diagonal elements of the interferometric matrix A led to improved B 1 mapping quality because it depends on the coil configuration and imaging target. 16 Indeed, although certain coils may provide similar B 1 mapping performance with a specific encoding matrix, less standard RF coil configurations (e.g., posterior or anterior only, [17][18][19] distributed over rows 20 ) may suffer from destructive interference across all RF modes in some regions. The choice of encoding matrix is therefore critical for such applications.
As a result of using unreliable B 1 maps, pTx pulse optimization may not provide adequate level of precision for excitation. Although some level of B 1 bias may be acceptable for anatomical MRI, 21 it may be particularly problematic for some quantitative measurements. 22 For instance, in the case of quantitative T 1 mapping (T 1 q) from MP2RAGE, 23 homogeneous B 1 is required for both the inversion and excitation pulses. Used in single channel mode, such T 1 q was shown to be inaccurate in the presence of B 1 inhomogeneities. 24,25 To mitigate those effects, MP2RAGE typically relies on long adiabatic inversion pulses, usually at the expense of more elevated SAR. Offline B 1 bias correction was also proposed, 19 but it adds complexity and may not be reliable in regions suffering from high B 1 -inhomogeneity.
Parallel transmission also bears great potential for other organs such as the spinal cord. Despite a small cross-sectional area in the transverse plane (cord diameter ∼ 1 cm), its elongated shape and surrounding structures may result in B 1 inhomogeneities along the z-axis, in the order of 30% between the C3 and C7 cervical levels, up to 70% in some cases. 26 This may lead to unreliable results in some subjects, whose data may be partly discarded. 26 Because T 1 q was shown to be useful in the identification and characterization of pathologies such as multiple sclerosis, 27 pTx is a great candidate to provide an improved quantification from MP2RAGE. However, no studies of B 1 mapping with interferometric encoding have been performed so far for this organ or when using cervical spinal cord RF coils.
This work presents a novel RF coil and organ specific optimization of interferometric encoding, which modifies the amplitudes and phases of all the elements of the matrix A (Eq. 1 and 2), and of the reference mode when using the hybrid approach, to reach better B 1 accuracy while including sensitivity to noise. This general method is applied to cervical spinal cord 7 T MRI, both on phantom and in vivo. The performance of the optimized satTFL is evaluated by comparing it with a typical encoding matrix (One-inv) 12 using the actual flip angle (AFI) technique 28 as standard due to its accuracy. 8 A practical benefit of having improved B 1 accuracy is finally evaluated in an exploratory study by acquiring T 1 q from MP2RAGE with the classic single-channel implementation, and with two sets of optimized pTx pulses: the first obtained from standard satTFL data, the second one from optimized satTFL data.

METHODS
All data were acquired using a 7 T Magnetom TERRA (Siemens Healthcare, Erlangen, Germany) equipped with an 8-channel transceive cervical spine coil (Rapid Biomedical GmbH, Rimpar, Germany). 19 The study was performed on the SAM phantom (SPEAG, Zurich, Switzerland) and on four healthy volunteers with informed consent and approval of the local ethics committee. The 10 g-averaged SAR (SAR 10g ) was calculated using electromagnetic simulations of different human models (Duke and Ella, from the Virtual Family 29 ) in several positions inside the coil (with three different shifts along the z-axis to account for positioning variations) 30 with Sim4Life (ZMT, Zurich, Switzerland), and monitored online using virtual observation points (VOP). 31 Identical VOPs were also used to take SAR 10g into account for the design of the matrix A and of the pTx pulses.

Acquisition of the B 1 maps
The 2D-satTFL were acquired using a custom sequence with a Shinnar-Le Roux saturation pulse to mitigate slice cross-interference 32 and enable the import of external encoding matrices. Two different satTFL methods were included in this study: • Interferometry 12 : Images were acquired with eight RF modes i, with and without presaturation pulse (sig sat i and sig nosat i , respectively). The effective saturation FA with linear combination of channels (β lc i ), was calculated as: • Hybrid: The "hybrid" method only uses one reference satTFL and eight TFL RF modes, reducing the number of acquisitions from 16 to 10. 13  The signals corresponding to different pTx pre-saturation (if any) and excitation pulses were acquired as repetitions of turbo-FLASH acquisition using the following sequence parameters: resolution: 5 × 5 × 2.5 mm 3 , matrix size = 48 × 36, 48 slices acquired in an interleaved manner, TE = 1.81 ms, FA readout = 5 • (Sinc pulse with duration = 0.8 ms), FA presaturation = 60 • (minimum phase Shinnar-Le Roux pulse with duration = 5 ms, time bandwidth product = 9, and pass-band ripple 1%). The time TR between two consecutive turbo-FLASH acquisitions was 15 s, allowing for the magnetization to fully recover. In addition, B 0 maps were acquired as they were used during the calculation of FA distribution when optimizing the encoding matrix and the pTx pulses (sequence parameters: 3D multi-echo gradient echo, resolution: 2.5 × 2.5 × 2.5 mm 3 , TE 1 /TE 2 /TE 3 /TR = 5/6.5/8/300 ms, FA = 20 • , TA = 2 min 52 s).
An initial satTFL was acquired using "One-inv" encoding (B acq 1 ) to calculate each channel B 1 , with a reference voltage calculated based on the average FA inside a region of interest (ROI) approximating the cervical cord (C1-C7) in the phantom and including C1-T1 in vivo. Although not shown, a comparison between the Fourier and One-inv encoding 11 showed that the latter produced higher quality B 1 maps.
Data from four healthy volunteers (two males and two females, with age and weight range = [28,33] years and [55, 90] kg, respectively) were acquired. The sat-TFL B 1 maps acquired with "One-inv" were denoised using block matching with 4D filtering (BM4D) 33 separately on the real and imaginary components, and were considered as ground truth during the optimization of the matrix A. The satTFL was optimized for all four volunteers simultaneously, aiming to find a preliminary proof-of-concept "generic" optimized satTFL, as an alternative to subject-specific optimization.

Optimization of the encoding matrix
Starting from B acq 1 , signal images were simulated in MatLab (R2018a, Mathworks, Natick, MA) to study the effect of modifying the encoding matrix on the accuracy of the interferometric encoding in the presence of noise. 16 The signal images were calculated as: and with n 1 and n 2 complex Gaussian noise scaled to obtain a SNR of 60 for the phantom study, and a SNR of 20 for the in vivo study (based on estimations from acquired data) inside the ROI when using the coil manufacturer default RF shim (Shim def ). The FA distributions of the i-th RF mode i and i were calculated from B acq 1 by numerical integration of the Bloch equation 34 and by considering B 0 inhomogeneity effects from the measured B 0 maps. Following this calculation, simplified signal equations were used (Eq. 6 and 7) as the effects of T 1 , T 2 , and proton density are identical for sig nosat i and sig sat i . Those terms were ignored as they would have no effect on the ratio of the images used in Eq. 3-5.
The calculated images were used to simulate the reconstruction of the magnitude and phase of the individual B 1 maps, B sim 1 , using the interferometry and hybrid methods for any given interferometric matrix A. The interferometric matrix was optimized with the aim to minimize the following loss function, only including data inside the ROI: where NRMSE stands for normalized RMS error. For the case of the hybrid method, the RF shim used to acquire the reference FA map was included in the optimization as well. The MatLab built-in "pattern search" global optimization was used, with One-inv and Shim def as starting points of the interferometry matrix and reference mode, respectively. A maximum number of iterations of 3000 was set as trade-off between convergence and computation time, but the optimization could stop earlier if the convergence target was reached. At each iteration of the optimization algorithm, two constraints were implemented. First, the maximum FA inside the ROI was limited to 140 • to prevent inaccurate measurements at large saturation FA. 35 Secondly, the SAR 10g generated by each mode was calculated, and peak SAR 10g was limited to the recommended value of 10 W/kg averaged over 6 min, and 20 W/kg averaged over 10 s. 36 Those two constraints were automatically checked sequentially during the optimization and could lead to a linear scaling of the magnitudes of some RF modes to stay within those values. As a preliminary step to isolate the effect of matrix amplitude on the accuracy of the encoding, the optimization only allowed linear scaling of the One-inv modes, giving an optimized matrix A opt,linear . A full optimization of all elements of the encoding matrix A was then performed to generate A opt,full , using One-inv as starting point.
Flow chart of the different steps of the optimization of the encoding matrix A, which was done for the interferometry and hybrid methods, with linear scaling of the modes of One-inv and with full optimization of the matrix A.
In addition to the encoding matrix, the optimization calculated the required reference voltage, which was then used during the acquisitions. A summary of the method is shown in Figure 1. In addition, the condition (Cond) of the different optimized matrices was calculated because this metric represents the tendency of a matrix to amplify noise.

2.3
Validation of the optimization

Phantom study
To demonstrate the benefit of a full encoding matrix optimization as compared to a simple optimization of the reference voltage, individual-channel B 1 maps were acquired for the interferometry and hybrid satTFL using A opt,linear and A opt,full matrices as well as different scaling of the One-inv matrix. Acquisitions using One-inv were performed with a reference voltage calculated based on the mean FA inside the ROI, and with a reference voltage achieving the maximum SAR 10g according to the optimization constraints (corresponding to the normal mode of operation). The RF modes of the One-inv matrix were also individually scaled to reach the maximum constraints considered during the optimization (FA and SAR 10g ), regardless of the NRMSE. The performance of the interferometric encoding was evaluated in a ROI representing the approximate location of the C1-C7 levels of the spinal cord in the SAM phantom.
In order to evaluate and compare the performance of each encoding matrix, the acquired satTFL were combined in MatLab after denoising to predict the B 1 distribution from different static or dynamic RF shims inside the ROI: (i) Shim def with a 300 μs rectangular pulse; (ii) static RF shim obtained by using the scanner built-in tool (Shim scanner ); (iii) two nonselective 500 μs pTx pulses using the B acq 1 gradient ascent pulse engineering (GRAPE) method, 37 designed to cover a range of higher FA values (≈ 40-70 • , Shim pTx_high , the upper FA was limited by the SAR 10g constraint of the AFI), and lower FA values (≈ 20-40 • , Shim pTx_low ). Details about the calculation and constraints used for the GRAPE pulses are provided in section 4.
The simulated combined B 1 were used to calculate corresponding FA maps (from Bloch simulations) and were compared with denoised FA maps derived from AFI acquired using identical RF shims and pulses, with sequence parameters: 3D acquisition, resolution 5 × 5 × 2.5 mm 3 , TE 1 /TE 2 /TE 3 /TR = 3.1/4.9/7/130 ms, FA = 60 • , TA = 5 min 49 s. The NMRSE and the mean percentage error (MPE, the ratio of their absolute difference and the FA from AFI) were evaluated for each method.

2.3.2
In vivo study A similar comparison, used for in vivo validation, was performed on two of the four volunteers (volunteer 1: male, aged 33 years, 89 kg; volunteer 2: female, aged 32 years, 85 kg), but only included a comparison between hybrid sat-TFL and AFI. The B 1 maps were combined, calculated as FA and compared with AFI using two shim configurations: Shim def (volunteers 1 and 2) and a 500 μs GRAPE pTx pulse (volunteer 1) (Shim pTx ).

Linear correction of satTFL
As previously shown, 38 a linear bias may exist between the satTFL and AFI. The AFI data were therefore used to estimate this bias: for the different methods introduced earlier, linear regressions were calculated between the FA calculated from the satTFL and acquired from the AFI. The regression coefficients were then used to correct for this bias.

Preliminary application for pTx quantitative T 1 mapping
Following the comparison with the AFI, the potential of using the optimized satTFL was evaluated by measuring quantitative T 1 maps (T 1 q) from MP2RAGE, 21 using the sequence parameters described in Massire et al. 19 (coronal acquisition, FA 1 /FA 2 = 4/5, TE/TR = 2.4/5000 ms, TI 1 /TI 2 = 700/2400 ms, FOV = 260 mm, number of slices = 192). The reference T 1 q (T 1 q ref ) was obtained after standard MP2RAGE acquisition (using Shim def and an hyperbolic secant 4 adiabatic inversion pulse) and included correction for B 1 -induced bias of the T 1 . Its calculation assumed an inversion efficiency of 1, as previously done. 19 Different pTx-MP2RAGE 21 were then acquired with pulses calculated from hybrid satTFL with One-inv (the default protocol on our MR scanner) and optimized encoding. Importantly, only the latter included a correction for the bias between satTFL and AFI, showing the combined effects of the matrix optimization and the AFI-based bias correction. T 1 q were calculated without correcting the T 1 for B 1 -induced bias, leading to T 1 q One-inv and T 1 q opt,corr , respectively. Denoising using BM4D was applied to all T 1 q.
The GRAPE method 37 was used to calculate the nonselective excitation (duration = 500 μs) and non-adiabatic inversion (duration = 7500 ms) pTx pulses. This method uses parametrization-free RF waveforms and 3D gradient trajectories to lower the NMRSE between the calculated and target FAs inside the ROI used in the satTFL optimization. Optimization of the excitation and inversion pulses were performed in parallel (calculation time of about 2 min on a DELL Precision 7560).

2.4.1
Phantom study MP2RAGE was acquired with a resolution of 0.7 × 0.7 × 1 mm 3 . The different T 1 q were calculated and compared inside the entire ROI used for the satTFL and pTx pulse optimizations. Because it was expected that different types of RF pulses (in particular, adiabatic and non-adiabatic inversion pulses) may lead to differences in T 1 q, the purpose of this phantom study was to correct for this bias with a calibration factor κ, used in the T 1 q calculation to better match T 1 q ref .

2.4.2
In vivo study MP2RAGE images were acquired at 0.7 mm isotropic resolution. A single volunteer (volunteer 1) was included in this preliminary study, with a ROI including the C1-C7 cervical levels. T 1 q was calculated with = 1 and 0.89 for the reference and pTx sequences, respectively, based on the phantom calibration presented in the previous section. Automatic segmentation of the cord was performed using the Spinal Cord Toolbox (v5.6). 39,40 Mean cord (gray matter and white matter) T 1 q at different cervical levels were then calculated using ITK-SNAP v3.8 (www.itksnap.org). 41 Figure 2 shows the magnitude and phase of different A-matrices used for the interferometry and hybrid sat-TFL in the SAM phantom and in vivo. All optimizations increased the magnitude of A, with maximum values of 2.04 for the linear optimization of One-inv, and 2.98 for the full optimization of hybrid satTFL. Substantially larger amplitudes were reached when only considering the SAR 10g and/or FA constraints compared with the linear optimization of One-inv. However, mean amplitudes of the encoding matrices with maximum constraints (interferometry: 1.35, hybrid: 2) were similar F I G U R E 2 (A) Magnitudes of One-inv used on phantom with the (left) interferometry and (right) hybrid satTFL when scaled to reach (top) the maximum SAR 10g , (center) the maximum constraints used in the optimization, (bottom) after linear optimization of the RF modes. (B) Encoding matrices for phantom after full optimization for the (left) interferometry and (right) hybrid satTFL. (C) Encoding matrix for hybrid satTFL based on data from four volunteers. The standard One-inv encoding had a uniform magnitude = 1. Rows and columns of the matrices correspond to the RF modes and channels, respectively. A matrix with a Cond = 1 is perfectly conditioned, whereas large condition numbers tend toward ill conditioning and may result in greater impact of noise on measured data. Cond, condition of the matrix; SAR, specific absorption rate; satTFL, presaturated turboFLASH.

F I G U R E 3
Comparison of the acquired linear combinations of B 1 + , (B lc 1 using Eq. 4 and 5) and shown as sagittal FA maps on a volunteer using One-inv and optimized hybrid satTFL with the encoding matrix shown in Figure 2C Figure 2B). The red contour shows the ROI in which the B 1 + accuracy was optimized. Channel 8 was used a reference for the phase calculation. The color scale was chosen to better show signal variations inside the ROI. The black dotted line corresponds to the C6/C7 level, below which great improvement of the satTFL was observed after optimization. Black arrows show low accuracy regions outside the ROI, which were corrected by the optimization. matrices) compared with the interferometry approach.
Although not constrained, the conditions after optimization were close to Cond(One-inv) = 3, with the exception of the full optimization using the interferometry approach (Cond = 5.52). The stopping criteria was reached in about 1 h when using data from a single subject. An intermediate step of the B 1 -map reconstruction is shown in Figure 3, as the B lc 1 (shown as sagittal FA maps) with the One-inv and optimized hybrid satTFL on a volunteer, using the matrix shown in Figure 2C. The red contour shows the ROI in which the accuracy and robustness of the B 1 was optimized, illustrating that the One-inv encoding included almost no signal below the black dotted line, which represents the region under the C6 cervical level. After optimization, some of the RF modes generated signal in this region, providing information that can be used in the reconstruction of the individualchannel B 1 .
This observation was confirmed in Figure 4, which displays the effect of the optimization on the individual hybrid satTFL B 1 maps on volunteer 1. B 1 maps are shown before denoising to better appreciate the impact of the optimization of the encoding. Although One-inv and optimized-satTFL provide similar results above the C6 cervical level (black dotted line), a qualitative comparison shows improvements in the magnitude for lower cord levels, displaying as a reduction of noise. In particular, a substantial difference between One-inv and optimized satTFL was observed for channel 4 in those regions, where very poor signal was partially recovered using the optimized satTFL acquisition. Although not included in the optimization, improvements were also noticed in some regions outside the ROI (black arrows). Similar phase distributions were obtained with the different methods. Results on phantom are shown in Figure S1. Figure 5 shows, for every channel, the difference between the BM4D-denoised B 1 maps (available in Figure S2) and the "raw" B 1 maps (Figure 4). The denoising operation notably improves the measured B 1 + of every channel while preserving the distribution. As indicated by the average noise level inside the ROI, denoising has a more pronounced effect on One-inv due to lower noise robustness of this strategy. An average noise reduction factor of 27.5% (up to 44% for channel 4) was obtained with optimized encoding. Figure 6 shows scatter plots comparing acquired AFI and calculated FA maps from the hybrid satTFL, with different encoding methods for the SAM phantom ( Figure 6A) and for volunteers 1 and 2 ( Figure 6B). The comparisons combine results from different RF shims, as described in the Methods section, and includes FA inside the ROI which were higher than 10 • . Results show that the One-inv encoding leads to relatively high deviation from the AFI, for a wide range of FA, for all calibrations and scaling, even when similar amplitudes were used in the encoding matrices (see Figure 2). This deviation is somewhat reduced using "One-inv optimized" (phantom only); yet, a full interferometric matrix optimization further reduces this deviation (SD) for both the interferometry and hybrid methods. Scaling of the One-inv to reach maximum SAR 10g constraints is shown in Figure S3. As previously reported, a linear bias was measured between the AFI and the satTFL, 38 justifying the need for a

F I G U R E 6
Scatter plots comparing the FA calculated from different satTFL and encoding matrices, with the FA from acquired AFI, including FA > 10. (A) Interferometry and hybrid satTFL on the SAM phantom, with different scalings of the One-inv matrix, as shown in Figure 2. Results combine data from Shim def , Shim scanner , Shim ptx_high , and Shim ptx_low . (B) In vivo comparison with volunteers 1 and 2, combining data from Shim def and Shim ptx . The linear regression model fit of their combined data is plotted in orange. Regression coefficients and quality of the adjustment are indicated on top of each graph. The blue line represents the y = x plot. These regression coefficients were then used to correct each satTFL to correct for the bias with AFI. pTx, parallel transmit. linear correction of the satTFL. A linear regression model fit was performed for all methods, showing superior R 2 after optimization (from R 2 = 0.90 to 0.98, and R 2 = 0.84 to 0.89 for the hybrid satTFL on phantom and in vivo, respectively, excluding results from Hybrid: One-inv maximum SAR 10g ). Although only combined data is shown, the differences between the linear slopes of volunteers 1 and 2 with Shim def were 10% and less than 1%, based on One-inv and optimized-satTFL, respectively, indicating that a unique linear correction could be used for different subjects after optimization. Because AFI may be inaccurate in the low FA regime, 6,8 Figure S4 also shows results when considering FA greater than 20 • . However, excluding FA between 10 • and 20 • only led to variations lower than 1% in the slope of the linear correction. Table 1 complements those results with a comparison between the different methods based on the NRMSE and MPE metrics, for FA > 20 • . Without linear correction, similar NRMSE and MPE were observed with different encoding methods, although increasing the reference voltage or optimizing the encoding matrix reduced the SD of the MPE by up to 63%. With linear correction of the fully optimized encoding matrices, the MPE was reduced to less than 3.7% (phantom) and to around 12% (in vivo).
Finally, different scaling of One-inv led to MPE greater than 5% and up to 57% larger SD compared with A opt,full . The NRMSE from One-inv to corrected optimized-satTFL was reduced by approximately 60% and 40% in phantom and in vivo, respectively. Because the hybrid encoding method achieved better results than interferometry encoding with a lower acquisition time, it was exclusively used for the following sections of this study.
Finally, Figure 7 illustrates the benefit of optimizing and using linear correction for the hybrid satTFL. FA maps were calculated with different RF shims and were compared with acquired AFI. The main benefit of the optimization can again be observed in the lower section of the ROI, where the One-inv encoding sometimes leads to poor prediction of the FA due to insufficient information from the B lc 1 .

F I G U R E 7
Combined sagittal B 1 maps (shown as FA maps) of (A) the SAM phantom (with Shim def , Shim scanner , Shim ptx_high , and Shim ptx_low ); (B) of volunteer 1 (with Shim def and Shim ptx ) from acquired AFI and calculated from channel-wise B 1 maps. Different encoding matrices are compared when using the hybrid satTFL and additional correction from the linear regression fit (Figure 4). The red contour shows the ROI in which the B 1 + accuracy was optimized. The black arrows show regions where the One-inv encoding failed, which were corrected by the optimization unintentionally. The black dotted line corresponds to the C6/C7 level, under which Optimized satTFL shows improved match with the AFI. All shown data was denoised. One-inv and optimized satTFL. The latter included a linear correction based on the comparison with the AFI. The T 1 was averaged over the ROI (phantom) or cord levels, including GM and WM (in vivo). The in vivo reference and pTx T 1 q were calculated with = 1 and 0.89, respectively. Although the reference MP2RAGE was run at the maximum level of SAR, a more than 50% reduction was achieved for the pTx sequences. Abbreviations: C, cervical level, GM, gray matter; GRAPE, gradient ascent pulse engineering; opt, corr , optimized and corrected; ROI, region of interest; T 1 q, quantitative T 1 maps; WM, white matter.

F I G U R E 8
(Top) Sagittal view of the T 1 maps obtained from (left) the reference MP2RAGE, and with pTx pulses from (center) One-inv and (right) optimized satTFL with linear correction, optimizing the FA inside a ROI from C1-C7 (blue contour, the signal outside the ROI was ignored). (Bottom) Axial views of the corresponding T 1 q in two slices, at the C4 and C7 cervical levels. To maintain a total SAR 10g lower than 10 W/kg, the maximum SAR 10g of the excitation and inversion pulses were designed to be lower than 6 and 4 W/kg, respectively. The higher SAR requirement of the excitation pulse was due to the high turbo factor of 192. To reduce the number of degrees of freedom and accelerate the GRAPE optimization, values of the gradients and RF channels were discretized with 20 and 200 μs steps for the excitation and inversion pulses, respectively. Constraints were set to limit the gradient slew rate, the SAR 10g with VOPs, global SAR, total power, and channel-by-channel power. As shown in Massire et al., 19 T 1 in the CSF is outside of the bijective range of the T 1 -map estimation curve, which may result in failed calculations. Corresponding voxels were arbitrarily assigned a T 1 of 3 s and were ignored in the analysis. GRAPE, gradient ascent pulse engineering; T 1 q, quantitative T 1 map; VOP, virtual observation points.
(T 1 q One-inv ), and optimized encoding followed by linear correction (T 1 q opt,corr ) in the phantom. FA maps calculated during the GRAPE optimization are shown in Figure S5. An important bias was measured when using = 1 between the reference and pTx T 1 q calculations, with mean T 1 q ref = 1219 ms and mean T 1 q opt,corr, = 1 = 1102 ms (this difference of almost 10% is greater than intersubject or intersession variability of T 1 q 19 ). For this exploratory study, it was chosen to empirically calibrate relative to T 1 q ref to reduce this bias. This calibration was done by matching the mean T 1 q over the whole ROI in the phantom for the two methods and was estimated to = 0.89. However, even lower T 1 were measured from the MP2RAGE based on One-inv, with a remaining global error of 12.6% after calibration. Furthermore, greater SD inside the ROI was measured from the pTx-MP2RAGE. Figure 8 shows in vivo T 1 q (volunteer 1), acquired and reconstructed in the same conditions as for the phantom experiments. The mean T 1 over the whole cord, using the vendor pulses and Shim def values with B 1 + correction and = 1, was found equal to 1205 ± 87 ms (it is worth noting that, for volunteer 1, FA deviations from the target FA value were up to 50%; no correction from these heterogeneities would have led to T 1 q errors up to 12%).
The mean T 1 over the whole cord using the optimized pTx-MP2RAGE with linear correction and = 0.89 was found equal to 1173 ± 112 ms. Table 2 (bottom) shows a quantitative comparison of the measured T 1 values averaged over the gray matter and white matter in the different cervical levels. Whereas T 1 q One-inv substantially underestimated the T 1 , with errors of about 8% in C2-C7, and almost 19% in C1, T 1 q opt,corr, = 0.89 presented good agreement with T 1 q ref , with deviations of about 4% in the C1-C3 levels and lower than 2% for C4-C7 levels. Finally, it is worth noting that the reference MP2RAGE was acquired using 100% SAR availability (corresponding to the maximum SAR authorized in first-level mode of operation), whereas pTx-MP2RAGE only required 49% (corresponding to nearly the maximum SAR in normal mode of operation).

DISCUSSION
This study demonstrates that interferometric encoding of satTFL B 1 mapping such as One-inv, which is largely used in the brain, is not fully adapted to the used spinal cord RF coil. Due to coil-dependent constructive and destructive interference, linear combinations of channels used with these methods may not provide sufficient signal in some regions to accurately reconstruct the individual channel B 1 maps. In the case of a cervical spinal cord RF coil with posterior only elements, as used in this study, suboptimal encoding resulted in poor SNR and inaccuracies, in particular near the lower cervical levels.
To overcome these weaknesses and in the perspective of full deployment of pTx, a novel full optimization of the interferometric encoding matrix was proposed. The optimized encoding matrices had different phases from One-inv, which indicates that the interference pattern was changed to better suit the coil configuration. Scaling of the RF modes of One-inv was also shown to be insufficient because it did not correct the distribution of the B lc 1 , which provided little signal across all RF modes in some regions below the C6 cervical level, even with substantial increase of the reference voltage. A full optimization of the encoding was shown to be necessary to provide interferometric encoding adapted to this RF coil, based on its channels and B 1 distributions, and using constraints for the maximum FA and local SAR. Because the number of RF channels is lower near the lower part of the cervical cord with the coil used in this study, the B 1 accuracy in this region was improved with the proposed method, admittedly with remaining noise for some channels (e.g., channel 4). Furthermore, the similar conditions of the One-inv and optimized matrices, in particular with hybrid encoding, indicate that well-conditioned matrices were found, with low sensitivity to noise variations. The lower values measured inside the ROI from the difference between acquired B 1 + before and after denoising implies that the optimization of the encoding matrix led to a SNR increase in the measured B 1 + -map, in particular near the lower cervical levels. This indicates that iteratively optimizing the encoding matrix, based on optimized satTFL, may further improve the performance of the proposed approach, although this remains to be evaluated.
In addition to the optimization of the interferometric encoding, it was observed that there existed a linear bias between the AFI and satTFL. This had previously been reported, 38,42,43 and was shown to be influenced by the echo train length and at least partly due to the transient state of the longitudinal magnetization after the saturation pulse. A thorough study of this effect was beyond the scope of this preliminary work; however, it would be of great interest to prevent it. Although different linear slopes were required to correct the phantom and in vivo satTFL, the two volunteers used in this comparison had similar linear regression results. The stability among more subjects will be investigated in future studies. Because hybrid satTFL had slightly better accuracy than interferometry satTFL (MPE and NRMSE better by 4% and 8%, respectively), with 60% faster acquisition time, this method was chosen as the default B 1 mapping sequence for pTx studies at our institution.
Direct benefit from accurate B 1 mapping was demonstrated with preliminary investigation of T 1 mapping from pTx-MP2RAGE. The standard MP2RAGE sequence often relies on the correction of the B 1 bias to provide T 1 maps in the presence of B 1 inhomogeneities up to a certain range, and the current study only included the C1-C7 spine levels to accommodate this limitation. 19 However, pTx-MP2RAGE and optimized-satTFL may enable increasing the coverage of the spinal cord (including pons and medulla oblongata and/or upper thoracis levels, for instance). In addition, the inversion pulse of the default product sequence, which was primarily optimized for brain imaging, relies on high SAR adiabatic pulses and may not be adapted to other applications due to SAR constraints. As a consequence, in the spinal cord, the maximum SAR level is always reached at our institution when running the standard sequence. In this work, it was shown that similar performance could be achieved in vivo without the need for B 1 correction, and with a 50% reduction of predicted local SAR. This allowed running the sequence in normal SAR operating mode, which is currently a requirement for pTx sequences at our institution. Differences were nonetheless observed between T 1 q ref and T 1 q from the pTx-MP2RAGE, with larger SD for the latter, and different mean values when the pulse calculation was based on uncorrected One-inv B 1 maps (T 1 q One-inv ) and on corrected optimized B 1 maps (T 1 q opt,corr ). This variation, which may arise from the difference in pulses (adiabatic and non-adiabatic, and different magnetization transfer bias 44 ) was already observed in the brain, for instance, when using universal pTx pulses. 45 The calibration factor κ used in this exploratory study was shown to partially compensate for this effect. However, this approach is limited as it uses the calibration of a single parameter to correct the effect of, potentially, different causes. More accurate results may be obtained by properly characterizing those variations in the future, as well as investigating phantom and in vivo bias differences.
In this work, although it was tested on a limited number of volunteers to show preliminary results, the potential of using a "generic" encoding matrix based on the combination of four datasets was also investigated. Indeed, the optimization of the satTFL interferometric encoding could not realistically be performed for each subject because of the large number of degrees of freedom (128 and 144 for the interferometry and hybrid methods, respectively). Further work is required to investigate the robustness of the calculated encoding matrix and linear correction with more subject anatomies. In particular, the current implementation of the optimization provides a unique generic encoding matrix and reference voltage for all subjects. However, although not specific to the proposed approach, it can be expected that variations of the FA distributions of the different RF modes without subject-specific calibration may reduce the performance of the optimized satTFL in some cases. A potential solution may be to normalize the mean FA of all RF modes relative to Shim def and use a single calibration of this RF shim for all modes.
Current limitations of the proposed method include the potential uncertainties of the AFI data in low FA regions, as biases around 7%, 27%, and 55% were measured for FA = 20 • , 15 • , and 10 • . 6 For this reason, FA < 20 • were excluded from quantitative comparisons. Other B 1 mapping sequences, such as the double angle method 43 or phase-based techniques, 8 have shown better accuracy in the low FA regime but are less practical for in vivo applications due to scan time, SAR, or sensitivity to magnetic susceptibility variations (particularly present in the spinal cord). 8 In addition, denoising of the B 1 + maps was performed using the spatial domain filter BM4D due to its low computational complexity, 46 providing a satisfactory reduction of noise (average of 28%) to improve the quality of data used for the optimization of the encoding. However, applications such as MR electrical properties tomography, which rely on low-noise B 1 maps because some implementations may amplify image noise, have shown that geometric nonlinear diffusion filters may be preferred. 47 This work focused on optimization for a given ROI centered on the cervical spinal cord because surrounding tissues were not of interest. However, whole FOV optimizations may be required when pTx is applied for techniques such as reduction of FOV. 48 In such cases, when pTx is used to cancel signal excitation outside a smaller FOV to prevent fold-over artifacts, the inaccuracy of satTFL B 1 maps may result in ineffective pTx-based cancelation outside the FOV. In particular, high FA regions generated by the encoding RF modes should be avoided to remain within the satTFL accurate FA range, 35 as observed in this study in some regions outside the ROI near the coil elements. In this study, optimization of the satTFL was applied to cervical spinal cord MRI, but the same methodology could in principle be applied to other RF coil architectures and other organs. It can be expected that certain coil configurations may strongly benefit from a coil-specific optimization of the interferometric encoding. The presented method only requires B 1 and B 0 maps, as well as accurate SAR 10g prediction (with VOPs for instance) to ensure the feasibility of the optimized encoding. It could therefore be combined with other variations of satTFL B 1 mapping, such as 3D acquisitions 7,10 or interleaved RF cycling and complementary reference modes (B 1 TIAMO), 14 aiming to provide robust and accurate inputs for better pTx techniques. The method and code will be shared on demand by directly contacting the authors.

CONCLUSIONS
In this study, a novel full optimization of interferometric encoding of satTFL was introduced for 7 T MRI of the spinal cord. The method was evaluated on phantom and in vivo, indicating a need to adjust encoding matrices depending on the coil configuration. The optimization was shown to reduce the deviation between the satTFL and AFI, in particular in low SNR regions such as below the C6 cervical levels. A linear correction of the satTFL was additionally shown to be necessary to better match AFI results, leading to substantial improvement of the accuracy of the optimized satTFL. This improved B 1 mapping was used for preliminary investigation of quantitative T 1 mapping from standard and pTx-MP2RAGE. Reference and pTx-based T 1 q using the corrected and optimized satTFL were in good agreement, with no need to correct T 1 values for the effect of B 1 -inhomogeneity, and substantially lower SAR 10g , for the latter. The combination of fast and accurate B 1 mapping, and rapid pTx pulse calculation, paves the way for improved use of pTx sequences outside the brain in future works.

SUPPORTING INFORMATION
Additional supporting information may be found in the online version of the article at the publisher's website.  Figure 2B). The red contour shows the ROI in which the B 1 + accuracy was optimized. Channel 8 was used a reference for the phase calculation.  Figure 2B). The red contour shows the ROI in which the B 1 + accuracy was optimized. Channel 8 was used a reference for the phase calculation. The color scale was chosen to better show signal variations inside the ROI. The black dotted line corresponds to the C6/C7 level, below which great improvement of the satTFL was observed after optimization. Black arrows show low accuracy regions outside the ROI which were corrected by the optimization. FIGURE S3. Scatter plots comparing the FA calculated from different satTFL and encoding matrices, with the FA from acquired AFI, including FA > 10: (A) Interferometry and hybrid satTFL on the SAM phantom, with scaling the One-inv matrix to the maximum allowed SAR 10g . Results combine data from Shim def , Shim scanner , Shim ptx_high and Shim ptx_low . Scaling of the One-inv to reach maximum SAR 10g constraints with the hybrid satTFL resulted in inaccurate B 1 mapping as the scaling was applied to RF modes which do not require saturation pulses, leading to inaccurate calibration of the FA. represents the y = x plot. These regression coefficients were then used to correct each satTFL to correct for the bias with AFI. FIGURE S5. Sagittal slices of the FA maps calculated during the GRAPE optimization of the inversion pulses with two ROIs: C1 to C7, as used to calculate the pulses for the pTx-MP2RAGE; Pons to T1, to evaluate the potential of using a larger ROI in the future.
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