B1 mapping using an EPI‐based double angle approach: A practical guide for correcting slice profile and B0 distortion effects

Aim of this study was to develop a reliable B1 mapping method for brain imaging based on vendor MR sequences available on clinical scanners. Correction procedures for B0 distortions and slice profile imperfections are proposed, together with a phantom experiment for deriving the approximate time‐bandwidth‐product (TBP) of the excitation pulse, which is usually not known for vendor sequences.


INTRODUCTION
Over the last years, applications of quantitative MRI (qMRI) have increased. However, the reliable quantification of tissue parameters like T 1 and T 2 requires careful corrections for inhomogeneities of the RF transmit field B 1 . Without B 1 correction, T 1 mapping based on the variable flip angle method, 1,2 which derives T 1 from two gradient echo (GE) data sets acquired with different flip angles (FA), yields an apparent T 1 scaling with the square of B 2 1 . 3 Therefore, for a 10% underestimation of B 1 , the real T 1 is 24% higher than the apparent value. 4 Similarly, B 1 mapping is required for T 2 mapping based on the acquisition of several turbo spin echo data sets with different TE, because corrections for stimulated and secondary echoes are B 1 -dependent, 5,6 for mapping the magnetization transfer ratio 7,8 and to improve results of MR fingerprinting. 9 Several B 1 mapping methods have been described, using the Bloch-Siegert phase shift, 10 acquisitions with different excitation angles, [11][12][13][14][15][16] different preparation angles, 8,17 actual FA imaging, 18 combined acquisitions of a stimulated echo with a spin echo 19 or an FID, 20 and STEAM-based methods. 21 However, most methods are not available as release sequences, requiring pulse sequence programming expertise and time consuming implementations. 4 This poses particular problems for qMRI-based multicenter studies, as B 1 mapping methodology and accuracy may differ across sites. In a recent study, inter-site differences between measured qMRI parameters were partly attributed to the use of different B 1 mapping methods. 22 To overcome this problem, several publications have proposed techniques based on release sequences, such as the acquisition of a series of 3D GE data sets with excitation angles close to 180 • . 23,24 A frequently used technique dubbed "double angle method" (DAM) acquires several (mostly two) data sets with different excitation angles, deriving B 1 from the signal quotient. [4][5]12,[15][16][25][26] Because of its relative simplicity and the broad availability of the underlying acquisition techniques, DAM served as a reference method for assessing the accuracy of other B 1 mapping techniques. 17,[27][28] Most publications on DAM propose excitation angles differing by a factor of 2, especially 60 • -120 • for optimum B 1 sensitivity, 5,12,[15][16][17] and signal readout via spin echo EPI 12,15,25 or GE-EPI. 12,26 For full spin relaxation before each acquisition, DAM requires a pulse TR of at least five times the maximum expected T 1 value. 12,16,25,27,29 To speed up acquisition, 2D multi-slice implementations are frequently used, covering all slices within a single TR. 29 However, deviations from an ideal rectangular slice profile yield a non-linear relationship between the FA and the signal integral across the slice, 4,12,[26][27]30 resulting in erroneous B 1 values. A correction procedure was proposed for 2D-DAM with spin echo EPI readout 30 and successfully used for T 2 mapping, 5 requiring, however, knowledge of the exact RF pulse shape, 4,[29][30] which may be problematic for release sequences. To avoid this problem, 3D-DAM approaches with non-selective or slab-selective pulses have been used, 23 proposing techniques for a faster acquisition, (e.g., by introducing RF pulses for compensation, saturation, and catalyzation purposes.) 12,16,29 Alternatively, the DAM concept has been applied to preparation rather than excitation pulses. 27 Again, these solutions usually require customized instead of release sequences.
There is contradictory evidence about the impact of slice profiles on the B 1 mapping accuracy in 2D-DAM. One study 15 found that a certain 2D-DAM implementation yielded the same B 1 maps as 3D-DAM with non-selective pulses, so the authors concluded that the slice profile of the release sequence was excellent. In a subsequent study, the same group reported a crucial slice profile dependence of B 1 in 2D-DAM with different excitation pulses, yielding deviations of up to 15% from reference values. 26 Concerning the effect of B 0 distortions on the accuracy of DAM-based B 1 maps, it has been claimed that they should be considered in 3D-DAM with non-selective pulses as they reduce the effective B 1 , 23,26 but can be neglected in 2D-DAM with strong slice gradients where they only yield a minor shift or deformation of the slice. 12,17 Because they cause distortions of the underlying EPI images, 2D-DAM should be accompanied by B 0 mapping and standard image distortion correction, but the B 1 accuracy should not be impaired as images acquired with different FA are affected in an identical fashion, unless the resulting SNR loss prevents a reliable derivation of B 1 from the signal quotient. 26 However, a far more serious effect, which to the best of our knowledge has not been considered so far is the impact of B 0 distortions on the slice refocusing gradient: because of the non-linearity of the Bloch equations, this effect depends on the excitation angle, therefore, affecting the two acquisitions in 2D-DAM differently, giving rise to systematic B 1 errors.
The goal of this study was to establish a 2D-DAM protocol based on multi-slice GE-EPI as available on commercial MR systems for routine fMRI experiments, and to propose a correction procedure for slice profile imperfections and B 0 distortions. As the excitation angle may be restricted to a maximum value of 90 • in release sequences, optimizations were performed for the combination 45 • -90 • , which also reduces the risk of RF amplifier nonlinearities. 4 In summary, the goals were: 1. To propose a phantom experiment for deducing the approximate RF excitation pulse shape of the GE-EPI sequence; 2. To show that B 0 distortions can seriously affect the accuracy of 2D-DAM; and 3. To propose a correction procedure for imperfect slice profiles and B 0 distortions.

THEORY
For two multi-slice GE-EPI data sets acquired with FA nom of 45 • and 90 • , the signal quotient Q (calibrated to Q = 1 for B 1 = 1 and ideal rectangular slice profiles) is calculated: In the absence of saturation effects, the signal S is proportional to the sine of FA real , yielding: Therefore, a theoretical B1 theo can be derived according to: Three main effects yield deviations of B1 theo from the real B 1 .

Incomplete spin relaxation before EPI acquisition
For full relaxation, TR should be at least five times the maximum T 1 (4.5 s for CSF at 3 T). Because frequency adjustment procedures before data acquisition may also cause saturation effects, an initial dummy scan should be inserted. To avoid cross talk between adjacent slices, interleaved slice coverage and an interslice gap are recommended.

2.2
Deviations of the excitation slice profile from a rectangular shape This effect can be corrected via simulations, solving the Bloch equations for the excitation pulse used. Here, the concept presented previously for T 2 mapping 6 and MR fingerprinting 9 will be used, using as "trial pulses" Hanning-filtered sinc pulses with different values of the time-bandwidth-product (TBP): P is the duration and BW the bandwidth of the pulse with the time profile: In the simulation below, Q and therefore, B1 theo will be determined for different values of TBP and B 1 , showing that the quotient B 1 /B1 theo is largely independent of B 1 , displaying only a TBP dependence, so it can be used as a correction factor C 0 (TBP) for deriving correct B 1 values: An equation for C 0 (TBP) will be derived, allowing to correct for slice profile effects, provided the TBP of the excitation pulse is known. Because this is usually not the case for vendor sequences, a phantom calibration experiment is here suggested: because the scanner performs an average B 1 calibration across the whole phantom, it can be assumed that B 1 averaged across the phantom is <B 1 > = 1, which yields: In summary, a single phantom calibration experiment yields C 0 from which the approximate excitation pulse TBP can be derived. The relationship C 0 (TBP) can then be used for the evaluation of any in vitro or in vivo experiment.

B 0 distortions
B 0 distortions are critical for DAM-based B 1 mapping if the resulting susceptibility gradient (G susc ) in slice encoding direction modifies the slice rephasing gradient (G R ). Figure 1A shows a normal slice rewinding process where the shaded areas are equal, so the rewinding factor R is: This assumes that the isodelay of the RF pulse, that is, the critical time constant for calculating G R , is the time difference between the maximum RF pulse amplitude and the end of the RF pulse, corresponding to P/2. It has been shown that this is a very good approximation for small FA 31 and can therefore, be expected to be integrated in release sequences. Figure 1B shows the effect of a susceptibility gradient in slice encoding direction. Assuming TE > > P and G susc < < G S , R is given by: For a pulse with BW, G S for achieving a slice thickness Δz is given by: where γ is the gyromagnetic ratio (42.57 MHz/T). Combining Eqs. (3a), (5b), and (6) yields: This effect is critical, since especially for large FA values, the isodelay can deviate from P/2 because of the non-linearity of the Bloch equations, 31 so full slice refocusing and therefore, maximum signal is achieved for modified R values deviating from 1. Because this effect is FA-dependent, the numerator and denominator in Eq. (1a) are affected differently, giving rise to further discrepancies between B1 theo and B 1 .
To account for this effect, the simulations proposed above will be extended, simulating Q values for different TBP, B 1 , and x. In each case, B1 theo will be calculated and compared to the assumed B 1 . The dependence of the quotient C = B 1 /B1 theo on TBP, B1, and x will be analyzed. If the B 1 dependence of C is insignificant, C can be written as a function C (TBP, x) and fitted as a 2D polynomial of order N: The correction factor C 0 corresponds to C (TBP, x = 0) and is given by: The polynomial coefficients P kl will be derived from the simulations and listed below in a table. In summary, the following procedure is proposed: Step 1: In a phantom calibration experiment, Q and B1 theo are calculated using Eqs. (1a) and (2), respectively. B1 theo is averaged across the whole phantom and C 0 is set to 1/<B1 theo >, so TBP can be derived via Eq. (10). This calibration experiment has to be performed only once for a given DAM protocol.
Step 2: For any in vitro or in vivo experiment performed with this protocol, Q and B1 theo are calculated in the same fashion. G susc in slice encoding direction is derived from a B 0 map and converted into x values via Eq. (8), allowing to derive the correction factor C (TBP, x) from Eq. (9). Corrected B 1 values are then obtained via: The polarity of x as calculated from Eq. (8) depends on the relative polarities of G susc and G S , which are unknown without access to the sequence code. Therefore, it is recommended to test for both polarities in the calibration part: for the correct polarity, B 0 induced distortions in the corrected B 1 map are reduced, in the other case exacerbated. This polarity is maintained for analyzing all further data.

General
All experiments were performed on a 3 T whole body MR scanner (Magnetom Prisma, Siemens, Erlangen, Germany) using the body transmit and a 20-channel head/neck receive coil. All simulations and data analyses were based on custom-built programs written in MATLAB (The MathWorks, Natick, MA). Standard procedures such as data coregistration and EPI distortion correction were performed with the FMRIB Software Library (version 4.1.9, http://www.fmrib.ox.ac.uk/fsl). The phantom for the in vitro experiments has physiological relaxation times (T 1 ≈ 1 s, T 2 ≈ 65 ms) with details described previously. 32 Throughout this manuscript, B 1 is given in relative units with B 1 = 1.0 corresponding to equality of nominal (FA nom ) and actual (FA real ) excitation angles. B 1 and B 0 maps were acquired either with a reference protocol (using established in-house sequences) or a DAM protocol (using vendor sequences).

3.1.1
Reference protocol Reference B 1 values were obtained as described previously, 8 acquiring two GE data sets with centric phase encoding, one after full spin relaxation and one after magnetization preparation via a saturation RF pulse with the nominal saturation angle (SA nom ) = 45 • . Therefore, the quotient of both data sets yields the cosine of the effective saturation angle and comparison with SA nom yields B 1  Please note that data were acquired as part of a more comprehensive study, comparing qMRI protocols based on custom-built versus vendor sequences. Therefore, separate B 0 maps were acquired, using sagittal slices in the reference protocol (matching a variable flip angle scan for T 1 mapping), and axial slices matching the EPI sequences of the vendor-based protocol. Otherwise, the B 0 mapping concept and post-processing were identical (acquisition of two GE data sets with a TE difference of 2.46 ms, yielding constant phase differences between water and lipid spins). For the data presented, reference B 0 mapping was only used for phantom experiment 1 (with sagittal EPI data acquisition) but not for the other phantom and in vivo experiments.

Simulation 1
The signals after 45 • and 90 • excitations were calculated for different combinations (TBP, B 1 , x), deriving for each pair Q and B1 theo according to Eqs. (1a) and (2), respectively. The goals were: 1. To calculate the correction factor C = B1/B1 theo as a function of TBP, B 1 , and x; 2. To show that C has a very weak dependence on B 1 , so it can be approximated as a function C (TBP, x); and 3. To fit C (TBP, x) as a 2D polynomial according to Eq. (9) and to supply the required matrix elements P kl .
Parameter ranges are listed in Table 1, exceeding the ranges used in the literature for similar simulations. 9 The B 1 range is fully sufficient to cover typical B 1 values on a T A B L E 1 Range of values for B 1 , TBP, and x as used in the simulations.

Minimum Maximum
Step width Abbreviation: TBP, time-bandwidth-product.
3 T system. 9,27,33 The TBP range was based on the following considerations: For TBP values below 2, slice profiles would be poor, making the use of such pulses unlikely in release sequences. For TBP ≥ 12, strong slice gradients would hamper the choice of thin slices: for TBP = 12 and P = 5120 μs, Eq. (3a) yields BW = 2344 Hz, requiring G S = 28 mT/m for exciting a 2 mm slice (see Eq. (6)), which is close to the maximum nominal gradient for most clinical systems.
Equation (8) shows that x is Δ / where Δ is the dephasing of spins across the excited slice, induced by G susc . Therefore, values of ±2 correspond to complete dephasing and total signal loss, rendering further data evaluation impossible. Even for values of ±1 the signal loss would be considerable, so simulating Q for −1 ≤ x ≤ +1 was deemed sufficient. For TE = 20 ms and slices of 2 mm thickness, x values between −1 and +1 correspond to a G susc between −300 and +300 μT/m covering the majority of B 0 distortions across the brain at 3 T. At 2 T, susceptibility gradients between 100 and 200 μT/m were reported for the orbitofrontal cortex. 34 Further simulation parameters were P = 2560 μs and Δz = 2 mm (arbitrary values that do not affect the results). For each combination (TBP, B 1 , x), the effect of RF pulses with the respective TBP, the excitation angles B 1 × 45 • and B 1 × 90 • and the rewinding factors R = 1 − x/TBP on a completely relaxed spin system was calculated numerically by solving the Bloch equations using a Runge-Kutta algorithm. 6,35 The respective signals were determined by integrating the transverse magnetization across the excited slice and two adjacent slices on each side (to account for slice profile imperfections). From the signal quotients, the parameters Q, B1 theo and C were calculated and analyzed as described above.

Simulation 2
To assess the accuracy of the method for a different EPI excitation pulse type, signals after 45 • and 90 • excitations with a Gaussian pulse (TBP = 3) were simulated as described above, using identical parameter ranges (B 1 , x). Q and B1 theo where obtained for each case, C 0 followed from 1/B1 theo for (B 1 = 1, x = 0) and a best matching TBP was fitted via Eq. (10) with the P kl obtained above. C (TBP, x) was derived from Eq. (9), the fully corrected B 1 from Eq. (11), and B 1 without B 0 correction from C 0 × B1 theo .

Phantom experiment 1: RF pulses with known TBP
Purpose of this experiment was the experimental evaluation of the simulation results, particularly the dependence of Q on TBP and B 1 for x ≈ 0.
Reference B 1 and B 0 data were acquired as described above. For DAM, a home written GE-EPI sequence with a sinc-shaped excitation pulse was used, allowing for free TBP choice. EPI acquisition parameters were: geometrical parameters as for reference B 1 mapping, but using 2 mm slice thickness with 2 mm gap, TR = 10 s (allowing for full spin relaxation for the phantom's T 1 of approximately 1 s), TE = 21 ms, receiver bandwidth = 2604 Hz/Px, no acceleration techniques, echo spacing = 450 μs, pulse duration 2500 μs, 1 dummy scan plus 2 acquisitions, duration: 30 s per FA. For each TBP value (2, 4, 5.5, 10), data for FA = 45 • and FA = 90 • were acquired. To increase the B 1 range, each measurement was repeated twice, either reducing or increasing the RF pulse voltage amplitude by 15%, therefore, yielding three ranges of B 1 values centered around 0.85, 1.00, and 1.15.

Data analysis
Reference B 1 maps were obtained as described above and in the literature 8 and post-processed as suggested, 4 restricting B 1 to values between 0.5 and 1.5 and smoothing resulting maps by convolution with a 3 × 3 × 3 kernel. B 0 maps in rad/s were derived by dividing the phase difference between both B 0 mapping data sets by their TE difference (2.46 ms). Phase unwrapping and smoothing was performed with the FMRIB Software Library functions "prelude" and "fugue." Before further analysis, all EPI data were distortion corrected, using the B 0 map and the FMRIB Software Library function "fugue." For each scan (45 • /90 • ), Q was derived from the signal quotient according to Eq. (1a). Pairs of values (B 1 , Q) were collected across a phantom mask derived via signal thresholding. To avoid spurious outliers, pixels comprising the 5% lowest and highest B 1 values were excluded from the analysis. Q values were collected separately for the three experiments acquired with the normal and the changed pulse amplitudes. The corresponding B 1 values were the reference B 1 values, multiplied with the factors 1.0, 0.85, and 1.15.
For analyzing the dependence Q (B 1 ), data were sorted into bins according to their B 1 values, using a bin size of 0.025. Empty bins or bins with a very low number of data points (<10% of the expected average number of data points per bin) were excluded. The resulting Q (B 1 ) was then compared to the simulation results.
For each TBP, data acquired with unchanged RF pulse amplitudes were used for B 1 mapping, following the procedure proposed at the end of the section "Theory". Furthermore, a B 1 map corrected for slice profiles only (but not for B 0 distortions) was calculated by multiplying B1 theo with C 0 . Both B 1 maps were post-processed in the same way as the reference B 1 map and compared to the latter.

Phantom experiment 2: RF pulses with unknown TBP
Purpose of this experiment was to apply the presented theory to B 1 mapping using the manufacturer's release GE-EPI sequence with unknown RF excitation pulse and to derive its TBP value.
Data were acquired both with the reference protocol (B 1 mapping part only) and the DAM protocol. Data analysis was identical to the previous experiment. For the DAM-based method, B 1 maps both with and without B 0 correction were derived. For comparison of the results, all sagittal reference B 1 maps were coregistered to the axial data space.

In vivo experiment
The study was approved by the local ethics committee of the university hospital and all subjects gave written informed consent before participation. For three healthy subjects (2 male, age 33/33/25 years), the same acquisitions as for phantom experiment 2 were performed (B 1 mapping part from reference protocol and DAM protocol). Data were evaluated as described above, assuming a TBP of 5.8 (see Section 4.4). For post-processing of reference and DAM-based B 1 maps, B 1 was restricted to values between 0.7 and 1.3, spurious outliers were corrected and maps were smoothed by convolution with a 3 × 3 × 3 kernel. DAM-based B 1 maps were again calculated with and without B 0 correction and compared to the reference B 1 map, which was coregistered into axial space. Figure 2 shows the dependence C 0 (TBP) for three B 1 values: 0.7 (blue), 1.0 (green), and 1.3 (red). The deviation between the curves is minor. A more quantitative investigation showed that relative deviations between C 0 (B 1 = 1) and C 0 (B 1 ) in dependence on TBP are largest for small TBP and large B 1 deviations. However, even for TBP = 2 and a B 1 of 0.7 or 1.3, deviations do not exceed 0.5%. For TBP values of approximately 6 (most likely matching the TBP of RF pulses in release sequences) and B 1 values between 0.8 and 1.2, errors do not exceed 0.2%. Therefore, C 0 is largely independent of B 1 , warranting the introduction of the function C 0 (TBP), which is derived from the simulation results for B 1 = 1. To investigate the B 1 -dependence of C (TBP, x, B 1 ), relative deviations between C (TBP, x, B 1 ) and C (TBP, x, B 1 = 1) were assessed. For common ranges of B 1 (0.8 to 1.2) and x (−0.79 to 0.79) at 3 T, all deviations were <1%. Therefore, it is reasonably accurate to introduce the function C (TBP, x) derived from the simulation results for B 1 = 1. This function was fitted as a 2D polynomial of seventh order according to Eq. (9). Table 2 lists the elements P kl as a matrix with k referring to the row and l to the column number, both starting from 0. The relative deviation between original and fitted versions of C (TBP, x) was <0.25% for all combinations of TBP and x. For large TBP, C 0 attains values below 1.0, although slice profiles should approach a rectangular shape. The reason is the use of Hanning-filters and the non-linearity of the Bloch equations, yielding modified slice profiles for large angles. Further investigation of the 90 • excitation profile showed that it is wider for small TBP, which increases S (90 • ), therefore, reducing Q and therefore, B1 theo (see Eqs.

T A B L E 2
Elements P kl for the polynomial fit according to Eq.(9), given as a matrix.

F I G U R E 3
Result of phantom experiment 1, using radiofrequency pulses of known time-bandwidth-product (TBP). For each TBP value chosen in the experiment (a: 2, b: 4, c: 5.5, d: 10), the respective panel shows the measured signal quotients (dotted, blue), the respective simulation result, assuming the TBP that was chosen in the experiment (dashed, red), and the respective simulation result, assuming a fitted TBP (solid, green). Fitted TBP values were 3.25 (a), 4.25 (b), 6.0 (c), and 12 (d).

Simulation 2
C 0 was 1.186 and the best matching TBP was 2.3. For a B 1 range of 0.8 to 1.2 and an x-range of −0.7 to 0.7, deviations of fully corrected from real B 1 values were <2.7%. However, deviations attained 25% (for large positive x) for B 1 without B 0 correction and 30% (for large negative x) for B1 theo . Figure 3A-D shows a comparison of the measured and the simulated dependence Q (B 1 ) for TBP RF values of 2 (a), 4 (b), 5.5 (c), and 10 (d). In each case, the measured data (blue, dotted), the simulated curve best matching the data (green, solid), and the curve simulated for TBP sim = TBP RF (red, dashed) are shown. There is a good correspondence between measured and simulated data, especially for central TBP values: for TBP RF of 4 and 5.5 the best match is achieved for TBP sim of 4.25 and 6.0, respectively. For TBP RF = 10, there is a mismatch and data are best described for TBP sim = 12. However, the simulated curves for TBP sim = 12 (green) and 10 (red) show only minor differences. For TBP RF = 2, data are best described for TBP sim = 3.25. This discrepancy may be because of the poor slice profile of RF pulses with a low TBP. Importantly, for central TBP values, which are most likely used for excitation pulses in release sequences, the differences are minor. B 1 maps were derived from each of the four data sets acquired with different TBP RF as proposed, yielding fitted TBP values of 3.4, 4.4, 6.1, and 12.0, corresponding approximately to the TBP sim obtained above. In each case, there was a good agreement between the fully corrected and the reference B 1 maps, with median deviations of 0.8%, 0.5%, 0.2%, and 0.1% for TBP RF values of 2, 4, 5.5, and 10, respectively. As an example, Figure 4 shows for TBP RF = 4 the resulting B 1 map with all corrections (top left), the B 1 map with corrections for slice profiles only (top right), the reference B 1 map (bottom left), and a histogram of the quotient of the fully corrected and the reference B 1 map (bottom right). Comparison of the B 1 maps with and without B 0 distortion correction show marked errors in the latter in areas where B 0 is distorted (top right, arrows).

Phantom experiment 2
The TBP as derived for the release EPI sequence was 5.8. Figure 5 shows

F I G U R E 6
Reference B 1 map (left), fully corrected double angle method (DAM)-based B 1 map (center) and DAM-based B 1 map without B 0 correction (right) as a transparent overlay on T 1 -weighted data. Differences (white arrows) are visible in the orbitofrontal and temporal cortex, areas typically affected by B 0 distortions.

4.5
In vivo experiment  either without (top) or with (bottom) B 0 correction. The x-axes show the mean and the y-axes the difference of B 1 values. Without B 0 correction, the bias (blue line) always exceeds 1% (DAM-based method yielding larger B 1 values than reference method) and shows a certain dependence on the mean B 1 . With B 0 correction, the bias, the B 1 dependence and the agreement limits (red dashed lines) are reduced.

DISCUSSION AND CONCLUSION
The method presented here facilitates the correction of 2D-DAM-based B 1 maps acquired with a GE-EPI readout for the deleterious effects of slice profile imperfections and B 0 distortions.
The algorithm requires knowledge of the excitation pulse TBP, which in general is not known for vendor sequences, but can be derived from a single phantom calibration experiment: if correct B 1 values are available (e.g., via a reference B 1 mapping method), the quotient B 1 /B1 theo yields the correction factor C 0 , so TBP can be fitted via Eq. (10). This approach of using reference B 1 values was tested in phantom experiment 2, yielding a TBP of 5.7. This agrees with the value of 5.8, as obtained with the method proposed here, which does not require reference B 1 mapping: C 0 is derived from 1/<B1 theo >, assuming an average <B 1 > = 1 across the whole phantom, provided the initial adjustment procedure calibrates B 1 using the average signal received. For the phantom experiments described here, this condition was met with 1% accuracy, <B 1 > being 1.0081 and 1.0078 in experiments 1 and 2, respectively. There are prerequisites for this assumption. First, the phantom should neither exceed the FOV nor the sensitive areas of the transmitting and receiving coils, rendering TBP derivation from an in vivo experiment inadvisable. Second, for strong non-uniformities of the receive coil sensitivity, signal contributions from different phantom areas will be weighted differently during adjustment, so the assumption of an average <B 1 > = 1 may not be met. In this study, a 20-channel head receive coil with relatively small sensitivity variations was used. If such a coil is not available, the body coil can be used both for RF transmission and reception. The concomitant SNR loss would not be problematic as the procedure is only required for the phantom calibration experiment. Still, the use of this method is critical for scanners that perform a calibration on a single-slice basis or may be prone to misadjustments during the calibration experiment, yielding deviations of <B 1 > from 1. These critical cases would require reference B 1 mapping, but a slow and motion sensitive method would be acceptable, as it has to be performed only once on a phantom for calibration. Furthermore, reduced volume coverage (e.g., single-slice) would suffice as B 1 /B1 theo is a constant. A simple method based exclusively on release sequences would be to acquire two single-slice EPI data sets, one after full spin relaxation and one after volume saturation of an area extending considerably beyond the image slice. If the voltage of the saturation pulse is halved manually, this corresponds to a 45 • preparation pulse, so the quotient of both data sets yields the cosine of the actual preparation angle, and a reference B 1 map for the whole slice follows from comparison with the nominal value of 45 • . If it is not possible to influence the saturation pulse voltage, a calibration is still feasible by identifying areas of minimum residual signal in the saturated EPI where the actual saturation angle is 90 • , so B 1 = 1. C 0 can then be derived by averaging 1/B1 theo across these areas. The results of a single calibration experiment, namely the best match for the excitation pulse TBP, can subsequently be used for any in vitro or in vivo scan performed on any scanner with an identical hard and software platform.
The plot of the correction factor C 0 (TBP) (Figure 2) shows that for a TBP close to 6, C 0 is approximately 1.0, so correction of B 1 maps for slice profile effects is not required. This explains findings, where a release implementation of 2D-DAM yielded accurate B 1 values. 15 For the release sequence used in the study presented here, an approximate TBP of 5.8 was found, corresponding to studies where a TBP of 6 was assumed. 17 For a Gaussian excitation pulse, simulations yielded TBP = 2.3 and C 0 = 1.186. This explains the finding of B 1 underestimations by 10% to 15% for Gaussian excitation pulses or sinc-shaped pulses with a low TBP. 26 The B 0 corrected B 1 maps of the in vivo data still show deviations from the reference B 1 maps in areas with very strong B 0 distortions ( Figure 6). This seems to be because of insufficient distortion correction of the underlying EPI data in these areas. A potential improvement would be to use parallel imaging techniques to reduce the echo train length of the EPI readout. This was avoided in the experiments described here because the explicit goal was to propose a B 1 mapping method based on versions of standard MRI release sequences available on any MR scanner. However, because the correction factor C (TBP, x) depends only on the shape of the excitation pulse, findings should also be valid for EPI sequences with in-plane parallel acquisition. Furthermore, the TOPUP method 39 could improve the distortion correction.
Potential limitations of the proposed method are: (1) 2D sequences are likely to suffer from inflow effects. 29 (2) At 7 T, B 1 variations between 0.4 and 1.5 have been reported across the brain. 28 As this range was not covered in the simulation performed here, a revised simulation and a different P matrix are required for higher fields. (3) The method was developed and tested for brain imaging only. In theory, it can also be used for B 1 mapping of other body parts. However, it crucially depends on an acceptable quality of the underlying EPI images, which may be impaired in body areas where B 0 is strongly distorted. (4) The method assumes that different excitation angles are achieved merely by altering the pulse amplitude. It is not applicable to cases where the pulse shape and/or the slice rephasing gradient are angle-dependent.
In conclusion, the method presented helps to set up qMRI studies on clinical scanners using release sequences only, as it does not require knowledge of the exact RF-pulse profiles or the use of in-house sequences. This can be especially useful for multi-center studies.

ACKNOWLEDGEMENT
Open Access funding enabled and organized by Projekt DEAL.

DATA AVAILABILITY STATEMENT
The codes for deriving B1 maps from EPI data, including all corrections described, are available upon request from the authors.