The effect of and correction for through‐slice dephasing on 2D gradient‐echo double angle B1+ mapping

Abstract Purpose To show that B0 variations through slice and slice profile effects are two major confounders affecting 2D dual angle B1+ maps using gradient‐echo signals and thus need to be corrected to obtain accurate B1+ maps. Methods The 2D gradient‐echo transverse complex signal was Bloch‐simulated and integrated across the slice dimension including nonlinear variations in B0 inhomogeneities through slice. A nonlinear least squares fit was used to find the B1+ factor corresponding to the best match between the two gradient‐echo signals experimental ratio and the Bloch‐simulated ratio. The correction was validated in phantom and in vivo at 3T. Results For our RF excitation pulse, the error in the B1+ factor scales by approximately 3.8% for every 10 Hz/cm variation in B0 along the slice direction. Higher accuracy phantom B1+ maps were obtained after applying the proposed correction; the root mean square B1+ error relative to the gold standard B1+ decreased from 6.4% to 2.6%. In vivo whole‐liver T1 maps using the corrected B1+ map registered a significant decrease in T1 gradient through slice. Conclusion B0 inhomogeneities varying through slice were seen to have an impact on the accuracy of 2D double angle B1+ maps using gradient‐echo sequences. Consideration of this confounder is crucial for research relying on accurate knowledge of the true excitation flip angles, as is the case of T1 mapping using a spoiled gradient recalled echo sequence.


INTRODUCTION
Several applications rely on the accurate knowledge of the true flip angles (FAs) exciting the spins, 1-6 including T 1 mapping using the variable flip angle (VFA) spoiled gradient recalled echo (SPGR) sequence.The double-angle method (DAM) for B + 1 mapping 7 can be implemented using widely available and scanner-agnostic pulse sequences such as gradient recalled echo (GRE) with an echo-planar imaging (EPI) readout.
][10][11] It has been assumed that B 0 inhomogeneity should not be relevant to GRE-based DAM B + 1 maps 7 because the two signals, acquired at excitation FAs of 2α and α, have the same echo time (TE).However, the large FAs typically required in the DAM 1,12 mean that through-slice B 0 variation can affect the measured ratio.
On-resonance in the small angle approximation, the phase roll accrued due to the slice select gradient is set to zero by the rephasing gradient.However, for FAs above 30 • , where the small angle approximation breaks down, even in the absence of B 0 field inhomogeneities, the phase accrued during the slice-select gradient is not completely refocused by the refocusing gradient for the usual convention of using a refocusing gradient with half the moment of the slice select gradient. 13This results in a residual phase roll across the slice.The degree of signal loss that results depends on the shape of the slice profile, being greater for slice profiles with more energy farther from the slice center.Consequently, the ratio of the integrated complex signal in the DAM B + 1 factor calculation will be altered compared to the case of equal phase roll for the two FAs.
Using a 2D multi-slice implementation of the GRE-EPI for DAM B + 1 mapping suffers from slice profile effects. 12,14or a constant B 0 offset, integrating the Bloch-simulated complex signal through slice, 15 before taking the ratio of the signals in the DAM, will account for the phase roll differences between 2α and α as well as the different slice profiles.
However, B 0 inhomogeneities along the slice direction impose an additional dephasing of the spins on top of the dephasing from the slice-select gradient.Organs located at interfaces with large susceptibility differences suffer from large B 0 inhomogeneities, reaching values at the liver dome of 200 Hz at 3T.These inhomogeneities in the liver decrease with distance from the lung, creating a B 0 gradient through slice (∇ z B 0 ).We hypothesized that the existence of this ∇ z B 0 would alter the phase roll profile for each FA and lead to a dependence of the calculated B + 1 factor on the ∇ z B 0 .
In this work, the effect of a ∇ z B 0 on the B + 1 factor was studied.Simulations using linear ∇ z B 0 were used to estimate the error introduced in the B + 1 map when neglecting this effect.A novel correction for the ∇ z B 0 effects on the GRE-EPI B + 1 map is proposed, resulting in accurate B + 1 maps using widely available pulse sequences.Validation of the proposed corrections to the B + 1 map was conducted in a phantom.The correction method was also applied to the nominal FAs in VFA SPGR T 1 maps of the liver in vivo.

Simulations
Bloch equations with 201 points covering the slice profile were used to simulate 16 transverse signals at FAs of 130 • and 65 •12 using the vendor's Hamming-windowed sinc GRE-EPI excitation pulse (FWHM of 0.55 ms for a 3.2 ms long simulation window).Signals were simulated on-resonance, with a constant off-resonance and with a ∇ z B 0 varying between −45 Hz/cm and 45 Hz/cm, in steps of 5 Hz/cm.Note that these ∇ z B 0 are small relative to the slice-select gradient (2.54 kHz/cm); thus, there is no detectable slice distortion.The code is available here: https://github.com/gabrielaBelsley/ThroughSliceDephasing&uscore;2DGRE (SHA-1 hash c694 101).
The effect of the ∇ z B 0 on the B + 1 factor was quantified through the deviation of the estimated B + 1 factor from true B + 1 biases of 0.59, 1, and 1.14.These correspond to the range of liver B + 1 factors observed in vivo at 3T. 17

Image acquisition
Imaging data were acquired on a phantom and 10 healthy volunteers, five male and five female, on a 3T Prisma (Siemens Healthineers, Germany) scanner.Volunteers were scanned according to our institution's ethical practices and gave informed consent.A 2D multi-slice GRE single-shot EPI was used for the B + 1 mapping with fat saturation and nominal FAs of 65 • and 130 • . 18Acquisition parameters were FOV = 450 × 366 mm 2 , matrix = 64 × 52, 15 slices interleaved, slice thickness/spacing 8/2 mm, TE/TR = 11/10000 ms, linear phase encoding, no acceleration or phase partial Fourier, bandwidth (BW) 3906 Hertz/pixel, and acquisition time 10 s breath-hold.
A 2D multi-slice double-echo spoiled GRE acquisition was acquired to compute a B 0 map.The B 0 map was used for distortion correction of the GRE-EPI images through fsl fugue 19,20 and modeling of ∇ z B 0 in the B + 1 map calculation.Acquisition parameters were TR/TE1/TE2 = 20/4.78/7.17ms, FA= 15  18 each during a 15 s breath-hold.Caipirinha 23 with 3× acceleration along the slice direction with 24 separate GRE reference lines was used.Spatial saturation was turned off because it perturbs the steady-state signal.
Gold standard (GS) B + 1 and T 1 maps were performed on the phantom.The GS B + 1 map consisted of a 3D nonselective DAM GRE acquisition with a long TR of 10 s for full relaxation of longitudinal magnetization and without an EPI readout.For the GS T 1 map, a slice-selective inversion recovery (IR) spin echo (SE) was used with inversion times logarithmically increasing between 25 ms and 5000 ms.Acquisition details, maps, and linear fits between the SPGR T 1 and GS T 1 are in the Supporting Information.

DAM B + 1 map corrected for slice profile effects
The DAM B + 1 mapping 7 takes the ratio between two signals from fully relaxed spins, S 2 and S 1 , acquired respectively at nominal FAs 2α and α to estimate the true FA exciting the spins: where r is the voxel slice center coordinate position.
A nonuniform slice profile invalidates Equation 1.To correct for slice profile effects, the Bloch-simulated complex transverse signal was integrated across the slice for each FA from 1 • to 360 • .The ratio was taken between the absolute value of the integrated complex signals at 2α and α (Equation 2) to generate a B + 1 lookup table (LUT) of the ratio (R) as a function of the FA (Figure S1).
At each pixel, the ratio of the signal acquired at FAs 2 and  was linearly interpolated from the LUT to obtain the true FA exciting the spins.The B + 1 correction factor is the ratio between the true FA and the nominal FA prescribed at the scanner.

B 0 gradient through slice correction to the slice profile-corrected B + 1 map
The correction for the ∇ z B 0 effect was based on Bloch simulations 16 of the signal, using the RF excitation pulse, slice-select gradient (G SS ), and slice refocusing gradient (G Ref ) information, and including varying off-resonances in the slice direction.To determine the through-slice B 0 at a 1 mm spatial resolution for each pixel in the B + 1 map, the B 0 values across the liver along the slice direction were fit to a cubic spline. 24The Bloch-simulated complex signals, for FAs 2 and , were calculated at a 1 mm spatial resolution over a spatial extension of ±1 cm from slice center for each voxel.The transverse signal immediately after the RF excitation pulse, calculated with the interpolated off-resonance, was propagated until TE, including free precession at the corresponding off-resonance and T 1 ∕T 2 relaxation with values of 900/30 ms, respectively.Lastly, the complex signal at TE for each voxel was integrated across the slice dimension, and the ratio was taken between the absolute value of the two signals (Equation 2).A nonlinear least squares (NLLS) fit was used to find the B + 1 factor corresponding to the Bloch-simulated ratio (Ratio Sim ) that best matched the ratio between the distortion-corrected GRE-EPI images acquired at nominal FAs of 130 • and 65 • (Ratio acq ) for each pixel (Equation 3). ( To speed up the NLLS fit convergence, the starting point for the B + 1 correction factor was the B + 1 map factor corrected only for slice profile effects, using the LUT described in section C.Moreover, the B + 1 was only calculated for voxels within the liver by applying a liver mask.The mask was manually delineated for each slice on the high spatial-resolution SPGR FA 15 • and down-sampled to the B + 1 and B 0 resolutions.

T 1 Mapping calculation and analysis
The corrected B + 1 map was interpolated to the SPGR spatial resolution and multiplied by the nominal SPGR FAs to obtain the true excitation FAs.A correction for incomplete spoiling was applied to the SPGR signal, through extended phase graphs simulations, 25 which was then fit to the steady-state SPGR function through a NLLS regression. 26 1 maps were analyzed by defining three 8-pixel diameter circular regions of interest (ROIs) per slice.ROIs were placed in vessel and bile-free areas, avoiding the liver edges.A linear fit between the mean T 1 values, extracted from each ROI, and the slice dimension before and after the ∇ z B 0 correction was used to quantify the effect of the B 0 gradient on liver T 1 .The resulting slope multiplied by the number of slices gives the T 1 variation (ΔT 1 ) in ms across the liver.
A linear mixed model was used to assess the ∇ z B 0 effect on the T 1 values before and after the ∇ z B 0 correction.The linear mixed model was applied at the slice level-that is, each mean T 1 across the slice is one data point-over all the 10 volunteers.In Equation 4, β 0 and β 1 are the regressor coefficients for the intercept and the ∇ z B 0 , respectively; (1|volunteer) represents the random effects term in the intercept and  is the error term.

Simulations: On-resonance and constant off-resonance phase roll at M xy signals 𝛂 and 2𝛂
Figure 1B shows the residual phase on-resonance post-excitation pulse, which is larger for 2α.A constant off-resonance for all the z positions introduces an overall constant phase offset (purple lines), which is equal for both FAs and will not affect the B + 1 ratio.Figure 1C shows the breakdown in the small angle approximation, confirming that phase roll increases with FA when the rephasing gradient moment is half the slice-select gradient moment. 13

3.2
Simulations: Effect of a  z B 0 on the 2D GRE-EPI DAM B + 1 On-resonance, the phase roll for 2α is larger than α (dashed vs. solid green line in Figure 2B).In the presence of a negative off-resonance gradient (orange lines in Figure 2B), the preexisting phase roll is reinforced.The phase difference for the two peaks of the absolute transverse magnetization at 2 is 0.92, close to the condition for complete destructive interference.This destructive interference reduces the overall integrated signal for the 2 pulse.In contrast, for the  pulse the largest contribution comes from a region around the slice center where the phase roll remains small.Consequently, the ratio is less than that on-resonance.When the off-resonance gradient is of opposite sign to the slice-select gradient, the phase roll for the 2 pulse is smaller near the slice center compared to on-resonance (green dashed line vs. blue dashed line in Figure 2B), resulting in a larger integrated signal for 2.The overall effect of a positive off-resonance gradient is an increased ratio compared to on-resonance.Failure to address variations in the ratio due to the ∇ z B 0 will result in over-or underestimation of the B + 1 factor.The B + 1 factor error scales by approximately 3.8% per every 10 Hz/cm variation in B 0 along the slice direction, at a B + 1 factor of 1.

Phantom
Figure 3A shows the B + 1 factor error for the phantom, measured against a B + 1 GS 3D DAM, before (blue) and after (green), applying the ∇ z B 0 correction for four different slices.The median error decreased from −1.8% to 0.01%, and the root mean square error relative to the GS B + 1 decreased from 6.4% to 2.6%.The interquartile range using only slice profile correction was (−4.5 −0.5)%, becoming approximately three times tighter, (−0.8 0.8)% after including corrections for ∇ z B 0 .

F I G U R E 1
(A) Absolute value of the transverse magnetization in arbitrary units (a.u.) for RF excitation FAs of α equal to 65 • (solid line) and 2α equal to 130 • (dashed line) on-resonance (green) and with a constant off-resonance of 50 Hz (purple).(B) Phase of the transverse magnetization on-resonance for 2α (dashed green line) and α (solid green line).A constant B 0 off-resonance shifts the phase by a constant offset (purple lines).The signals correspond to a time moment immediately after the rephasing gradient.(C) Phase roll extent quantified as the difference between the maximum and minimum phase accrual across slice positions varying between −0.5 cm and 0.5 cm for FAs varying between 5 • and 160 • on-resonance.FA, flip angle.

F I G U R E 2
(A) Absolute transverse signal for α equal to 65 • (solid line) and 2α (dashed line) on-resonance (green) and for a B 0 gradient in the slice direction of −45 Hz/cm (orange) and +45 Hz/cm (blue).(B) Phase in radians for the transverse signal on-resonance (green) at TE, with a negative B 0 gradient of −45 Hz/cm (orange) and a positive B 0 gradient of +45 Hz/cm (blue).The phase accrual for 2α (dashed green line) is larger than for α (solid green line) on-resonance.A negative B 0 gradient across the slice position reinforces the dephasing of the negative slice-select gradient, whereas a positive B 0 gradient results in a phase accrual in the opposite direction.(C) B + 1 factor error (|B + 1,sliceprofile − B + 1,true |) as a function of B 0 gradient through slice for three different B + 1 factors of 0.59 (orange), 1.0 (green), and 1.14 (blue).The error increases for larger absolute B 0 gradients as well as increasing B + 1 factors.A linear fit (pink dashed line) to the blue curve shows that at B + 1 =1, the error in B + 1 factor scales by approximately 3.8% per every 10 Hz/cm variation in B 0 along the slice direction.

F I G U R E 3 (A) B +
1 factor error in the phantom B + 1 map, normalized by the B + 1 factor gold standard value, without any corrections (orange), with slice profile correction (blue), and with the B 0 gradient through slice correction added to the slice profile correction (green) for four GRE-EPI slices.Without any corrections, the median B + 1 factor error was −5.2% with an IQR of (−7.4 −3.7)% over the four slices.Using only slice profile correction, the median B + 1 factor error was −1.8% with an IQR of (−4.5 −0.5)% over the four slices.Including the correction for off-resonance variations through slice, the B + 1 factor error decreased to 0.01% (−0.8 0.8)% over the four slices.(B) Percentage T 1 error, normalized by the IR SE, in the phantom T 1 map for four VFA-SPGR slices matching the GRE-EPI slices in (A).The colors are the same as described in (A).Without any corrections, the median and IQR T 1 error was 10.5% (6.7 17.5)% over the four slices.Using only slice profile correction, the median and IQR T 1 error was 2.6% (−0.4 10.2)% over the four slices.Including the correction for off-resonance variations through slice, the T 1 error decreased to −0.3% (−2.5 2.1)% over the four slices.GRE, gradient recalled echo; EPI, echo-planar imaging; IQR, interquartile range; IR, inversion recovery; SE, spin echo; SPGR, spoiled gradient recalled echo; VFA, variable flip angle.
Figure 3B shows the error in T 1 values for the phantom, measured against the GS T 1 , before and after applying the ∇ z B 0 corrections.The median T 1 error decreased from 2.6% to −0.3%.The interquartile range using only slice profile correction was (−0.5 10.2)% and decreased by more than a half, (−2.5 2.1)%, after including corrections B 0 gradient through slice effect on in vivo T 1 maps for four volunteers.Coronal T 1 maps without the B 0 gradient through slice correction (first column) show a gradient in T 1 as a function of slice number (vertical direction), with lower T 1 values for slices close to the liver dome and increasing T 1 values as the slice number increased.Applying the developed B 0 gradient through slice correction (second column), the T 1 values at the dome of the liver increased, resulting in a more homogeneous T 1 map.Coronal B 0 maps showing a large variation in B 0 values across the liver (third column).Data for all volunteers is in Figure S7.
for ∇ z B 0 .The slice with the largest median B + 1 factor error of −6.6% (giving a median T 1 error of 16.3%) before correction had the largest ∇ z B 0 of 18.7 Hz/cm.After applying the ∇ z B 0 correction, this slice had the largest T 1 error reduction.Phantom B + 1 and T 1 maps are shown in the Supporting Information (Figure S6).

In vivo
Figure 4 shows coronal cuts of the in vivo T 1 maps before and after the ∇ z B 0 correction for four volunteers.T 1 maps for all 10 volunteers are in the Supporting Information (Figure S7).The T 1 is expected to be homogenous throughout the whole liver for a healthy population.After applying the developed correction, the T 1 gradient disappeared and the T 1 maps were visually more homogeneous.Without a ∇ z B 0 correction, an average difference in T 1 of 90 ms was observed between the superior and inferior parts of the liver and reached a maximum of 221 ms.After applying the ∇ z B 0 correction, the variation in T 1 through slice reduced to between −44 ms and 26 ms (Table 1).For eight out of 10 subjects, the 95% confidence interval in the ΔT 1 included zero.The linear mixed model gave a p-value <10 −6 for the regressor coefficient of the gradient in B 0 (β 1 in Equation 4), confirming that the ∇ z B 0 significantly affects the uncorrected T 1 .After applying the proposed correction, the p-value is 0.14 (>0.05).

DISCUSSION
The B + 1 DAM approach using a GRE-EPI sequence was chosen due to its wide availability and ability to provide whole liver coverage within two 10 s breath-holds.We have shown that the 2D multi-slice nature of the acquisition and the use of a GRE sequence make it sensitive to slice profile effects and B 0 variations across the slice direction that affect the B + 1 factor accuracy.An alternative to the proposed B 0 postprocessing correction would be to use a SE sequence, although slice profile effects would still be relevant.However, not all manufacturers offer SE-EPI sequences with control over the FAs.Moreover, GRE-EPI is available even at 7 T, where 180 • refocusing pulses may be inaccessible.Rapid B + 1 DAM has also been proposed for abdominal and cardiac applications.

T A B L E 1
Average T 1 for each subject before and after applying the B 0 gradient through slice correction to the B + 1 map, which is subsequently used to correct the SPGR excitation FAs.Note: The mean T 1 increases on average 104 ms over all subjects after applying the correction.Columns 4 and 5 correspond to the variation in T 1 between the superior and inferior parts of the liver for the 10 volunteers.After the B 0 gradient through slice correction, the variation of T 1 with slice number decreased significantly.Abbreviations: CI, confidence interval; FA, flip angle; SPGR, spoiled gradient recalled echo; ∇ z B 0 , B 0 gradient through slice.

Subject
Nonselective saturation pulses 27 or a catalyzation RF pulse chain 28 reset the longitudinal magnetization after each TR and thus permit short TRs without a T 1 bias.][31] After applying the developed slice profile and ∇ z B 0 corrections to the B + 1 map, the median error relative to the GS dropped to 0.01%.Applying the corrected B + 1 map to the SPGR FAs resulted in accurate T 1 values with a median error of −0.3%.The ∇ z B 0 correction greatly improved the homogeneity of in vivo liver T 1 maps.The mean T 1 difference across the liver volume, along the slice direction, for the 10 volunteers decreased from 90 ms to −3 ms.
Malik et al. 15 corrected slice profile effects in 2D actual FA imaging 32 B + 1 maps by Bloch-simulating the signal for the slice-selective RF excitation pulse and generating an offline LUT.This approach is identical to that adopted in this work to correct for slice profiles alone.Wang et al. 14 studied both the effect of the RF pulse shape and off-resonance on estimating the B + 1 factor using the GRE DAM.The authors reported differences in the actual FA measured across a phantom for three different RF pulse profiles and attributed these to slice profile effects.The authors calculated the average FA across the slice using the DAM formula without slice profile correction (Equation 1) and compared it to the nominal FA to calibrate for slice profiles.This scheme assumes that the average B + 1 factor in a slice is 1, which is not guaranteed either in vivo or in a phantom.The authors only studied B 0 offsets and found variations of the FA map of less than 4% using an 800 Hz off-resonance RF pulse, and of less than 1% for B 0 inhomogeneities of ±50 Hz.Thus, the authors concluded, as do we, that off-resonance effects have no impact on the B + 1 map.More recently, Nöth et al. 33 also showed that B 0 distortions affect the accuracy of the 2D-DAM GRE-EPI.The B + 1 correction consisted of a seventh-order 2D polynomial that depended on the time BW product of the RF pulse and the B 0 gradient value.Their correction method does not account for the effect of the B 0 gradient during RF excitation.As shown by our Bloch simulations (Figure 2), the differences in dephasing between the α and 2α during excitation are crucial to explain the effect of B 0 variations through slice on the DAM ratio and consequently the error in the B + 1 map.The dephasing will proceed during the slice refocusing gradient until TE but should already be considered during excitation.
Several limitations are present in the Nöth et al. 33 method.The polynomial coefficients were calculated based on a FA pair of 45 • /90 • .As shown in our previous work, this is not the optimal FA pair for DAM GRE-EPI acquisition. 18Presumably, the polynomial coefficients need to be recalculated when using different FA pair values, whereas our algorithm is valid for any combination of FAs.Moreover, the authors mention their method is not valid when the RF pulse shape is angle-dependent.This is not the case for our algorithm.For example, our algorithm would still work if the pulse duration is increased instead of scaling the pulse amplitude.Related • , 2 • , 15 • , and 15 • ,