The Influence of Radio-Frequency Transmit Field Inhomogeneities on the Accuracy of G-ratio Weighted Imaging

G-ratio weighted imaging is a non-invasive, in-vivo MRI-based technique that aims at estimating an aggregated measure of relative myelination of axons across the entire brain white matter. The MR g-ratio and its constituents (axonal and myelin volume fraction) are more specific to the tissue microstructure than conventional MRI metrics targeting either the myelin or axonal compartment. To calculate the MR g-ratio, an MRI-based myelin-mapping technique is combined with an axon-sensitive MR technique (such as diffusion MRI). Correction for radio-frequency transmit (B1+) field inhomogeneities is crucial for myelin mapping techniques such as magnetization transfer saturation. Here we assessed the effect of B1+ correction on g-ratio weighted imaging. To this end, the B1+ field was measured and the B1+ corrected MR g-ratio was used as the reference in a Bland-Altman analysis. We found a substantial bias (≈-89%) and error (≈37%) relative to the dynamic range of g-ratio values in the white matter if the B1+ correction was not applied. Moreover, we tested the efficiency of a data-driven B1+ correction approach that was applied retrospectively without additional reference measurements. We found that it reduced the bias and error in the MR g-ratio by a factor of three. The data-driven correction is readily available in the open-source hMRI toolbox (www.hmri.info) which is embedded in the statistical parameter mapping (SPM) framework.


INTRODUCTION
The g-ratio [i.e., the ratio between the inner (r) and outer (R) radius of an axon with myelin sheath (g-ratio = r/R)] of a given axon quantifies the degree of relative myelination, ranging between 0 (no axon) and 1 (no myelin). The g-ratio captures both axonal and myelin damage by incorporating axonal and myelin volumes in one metric, making it potentially more specific to tissue integrity than focusing on one of these aspects only. For example, in multiple sclerosis, the g-ratio increases if the underlying disease mechanism is solely driven by demyelination (Yu et al., 2019), but is expected to remain unaffected if demyelination is accompanied by axonal degeneration. To differentiate such processes and understand their functional implications, neuroscience and clinical research would greatly benefit from in-vivo whole-brain measurements of MR g-ratio. Until recently, the g-ratio was measurable only by means of histology (Hildebrand and Hahn, 1978), which restricted the analyses to a small number of axons and a limited number of small brain regions or pathways. Stikov et al. (2011Stikov et al. ( , 2015 introduced a methodology for an MRI-based whole-brain "aggregate" g-ratio mapping, to which we refer as "MR g-ratio" or "g-ratio weighted imaging." In g-ratio weighted imaging, the MR g-ratio is computed on a voxel-by-voxel basis from the axonal (AVF) and myelin volume fraction (MVF) maps and reflects a weighted mean of g-ratio values within the voxel (West et al., 2016). Therefore, g-ratio weighted imaging requires the acquisition of separate sets of images that are sensitive to AVF and MVF, respectively (Campbell et al., 2018;Mohammadi and Callaghan, 2020). To generate MVF and AVF from the measured MR parameters, a calibration step is required that converts the measured MR-visible water signals into the respective volume fractions (Mohammadi and Callaghan, 2020).
Magnetization transfer saturation (MT sat ) has often been used as proxy for MVF (Mohammadi et al., 2015) as it is minimally affected by the longitudinal relaxation time (Helms et al., 2008) and is expected to show high correlation with macromolecular content (Sereno et al., 2013;Callaghan et al., 2015a;Campbell et al., 2018), making it a sensitive metric of MVF. One common approach to estimate AVF complements the parameters from neurite orientation and dispersion density imaging (NODDI Zhang et al., 2012) with a MVF-proxy, e.g., MT sat (Ellerbrock and Mohammadi, 2018;Kamagata et al., 2019), to correct for the missing myelin water signal in diffusion MRI measurements (Stikov et al., 2015). Maps of MT sat can be obtained, among others, from the multi-parameter mapping (MPM) protocol  in combination with the hMRI toolbox 1 Tabelow et al., 2019).
Although the MT sat measure is largely insensitive to transmit field (B 1 +) inhomogeneities (Helms et al., 2008), it still shows a residual dependence which introduces a bias and/or error in the MT sat maps that can propagate into the MR g-ratio and lead to systematic bias. Such B 1 + inhomogeneities can be corrected based on an independently acquired B 1 + field map measurement (Helms, 2015;Helms et al., 2021). Residual B 1 + inhomogeneity effects on MT sat have been shown to be not negligible when the B 1 + correction was omitted (Helms, 2015;Helms et al., 2021). However, the impact of B 1 + correction on MR g-ratio estimates is unknown. Additionally, it is unclear whether these residual B 1 + inhomogeneity in MT sat and the MR g-ratio can retrospectively be corrected using a data-driven B 1 + field inhomogeneities estimation approach such as the "unified segmentation based correction of R1 maps for B 1 + inhomogeneities" (UNICORT, (Weiskopf et al., 2011)). 1 www.hMRI.info In this study, we investigate the effect of B 1 + inhomogeneities on MR g-ratio maps when omitting the B 1 + correction. As a reference, we use the B 1 + corrected MR g-ratio from a dataset of healthy controls. We compare the reference MR g-ratio values against (i) values obtained without B 1 + correction and (ii) values obtained with B 1 + correction using the data-driven UNICORT approach.

Subjects
This study included 25 healthy control subjects (12 females, age (mean ± standard deviation) of 25.4 ± 2.4 years). They were recruited at the University Medical Centre Hamburg-Eppendorf and screened for neurological or psychiatric illness. The study was in agreement with the Declaration of Helsinki and was approved by the local ethics committee (Ärztekammer Hamburg #PV5141).

Data Acquisition
Each subject was scanned twice within 1 week in a whole-body 3T Tim TRIO MR scanner (Siemens Healthcare, Erlangen, Germany) using the body RF-coil for transmission and a 32channel radiofrequency (RF) head coil for signal reception, respectively. The MR acquisition on both scan days included a multi-parameter mapping (MPM) Callaghan et al., 2015b) and a diffusion-weighted imaging (DWI) protocol. The MPM protocol consists of three differently weighted 3D-multi-echo spoiled gradient echo sequences (Siemens FLASH). The echo train length and flip angle for the proton density (PD) weighted, T1-weighted, and magnetization transfer (MT) weighted sequences were 8/6, 8/21, and 6/6 • , respectively. The MT-weighted sequence had a Gaussian RF pulse (2 kHz off resonance with 4 ms duration and a nominal flip angle of 220 • ). All other sequence parameters were the same for the three sequences: repetition time (TR) 25 ms, echo spacing, resolution 0.8 mm isotropic; field of view (FoV) 166 × 224 × 256 mm 3 , readout bandwidth 488 Hz/pixel, partially parallel imaging using the GRAPPA algorithm was employed in each phase-encoded direction (anterior-posterior and right-left) with 40 reference lines and a speed up factor of two, total acquisition time: ∼25 min. The B 1 + field reference map was acquired using the three-dimensional echo-planar imaging (3D EPI) method, including field maps for distortion correction (Lutti et al., 2010).
The DWI sequence was a twice-refocused single-shot spinecho EPI scheme (Reese et al., 2003), consisting of 12 nondiffusion-weighted images (b 0 images), equidistantly distributed across the diffusion weighted images. The diffusion-weighted images were acquired at two b-values (1000 s mm 2 and 2000 s mm 2 ), sampled along 60 unique diffusion-gradient directions within each shell. The entire protocol was repeated with identical parameters but with reversed phase encoding direction (anteriorposterior) to correct for susceptibility-related image distortions (blip-up, blip-down correction). In total, 264 images were acquired per subject (120 diffusion-weighted images, 12 b 0 images, each acquired twice). Other acquisition parameters were: 86 slices with no gap, TR = 7.1 s, TE = 122 ms, an isotropic voxel size of (1.6 mm) 3 , FoV = 224 × 224 × 138 mm 3 , 7/8 partial Fourier imaging in phase encoding direction, readout bandwidth. To accelerate the data acquisition, GRAPPA (inplane acceleration with factor two) and simultaneous multi-slice acquisitions ("multiband, " slice acceleration factor two) (Feinberg et al., 2010;Moeller et al., 2010;Xu et al., 2013) were used as described in Setsompop et al. (2012). The image reconstruction algorithm was provided by the University of Minnesota Centre for Magnetic Resonance Research. The total acquisition time was ∼37 min.

Data Processing
MT sat maps were generated in the SPM-based hMRI toolbox (Tabelow et al., 2019). Note that the hMRI toolbox also generates additional maps of longitudinal (R 1 ) and effective transverse relaxation rates (R 2 ) and PD. Three MT sat maps were generated: (i) MT NO sat maps, without B 1 + correction; (ii) MT B1 sat map, using the reference B 1 + field map for correction (Lutti et al., 2010); and (iii) MT UN sat maps, using the datadriven UNICORT approach for B 1 + estimation (Weiskopf et al., 2011; see Supplementary Figure 2). UNICORT is a probabilistic framework for unified-segmentation based correction of R 1 maps for B 1 + inhomogeneities. The framework incorporates a physically informed generative model of smooth B 1 + inhomogeneities and their multiplicative effect on R 1 estimates (Weiskopf et al., 2011). Parameters used in UNICORT such as the smoothness and regularization were optimized for R 1 B 1 + correction in a 3T scanner (i.e., Tim Trio scanner- Weiskopf et al., 2011).
For B 1 + correction, we used the following heuristic correction factor as detailed in Helms (2015), and Helms et al. (2021): where C has been calibrated to be 0.4 for the MT pulse used in this paper. B 1 + can be either measured (MT Corr sat = MT B1 sat ) or estimated with the UNICORT approach (MT Corr sat = MT UN sat ). The DWI data were processed based on the pipeline described in Ellerbrock and Mohammadi (2018) using the SPM-based ACID toolbox 2 . It included several artifact corrections such as Rician signal bias correction (i.e., denoising) (André et al., 2014), correction for eddy current and motion artifacts (Mohammadi et al., 2010(Mohammadi et al., , 2014, and correction for image distortions due to susceptibility artifact using reversed phase encoding (Ruthotto et al., 2012(Ruthotto et al., , 2013Macdonald and Ruthotto, 2018). The corrected images were fitted with the NODDI signal model (Zhang et al., 2012) to estimate the intra-cellular volume fraction (ν icvf ), the isotropic volume fraction (ν iso ), and the orientation dispersion index (ODI) in each voxel.

Co-registration
The voxel-wise arithmetic between the MT sat and ν icvf maps, necessary for MR g-ratio computation, requires an accurate spatial alignment between the two maps (Mohammadi et al., 2015). To this end, we created two white matter (WM) tissue probability maps (TPMs) based on the ODI and MT B1 sat maps, respectively (Figure 1). To reduce the influence of contrast-specific artifacts (e.g., due to subject motion) on the registration quality, the WM TPM of the ODI map was coregistered to the WM TPM of the MT B1 sat map using rigid-body registration (spm_coreg algorithm, SPM toolbox). The estimated transformation parameters were applied to all other NODDI maps as well. Note that the segmentation quality of the second session was unsatisfactory for two subjects, and the R B 1 1 map (R 1 with B 1 + inhomogeneities bias correction using the B 1 + reference measurements) was used to generate the WM TPM instead. In another subject, the ν iso was segmented instead of the ODI to achieve satisfactory WM segments.

Normalization
Spatial normalization was performed in four steps. First, a rough alignment of the MT B1 sat maps with the T1-weighted MNI template image was achieved using the Auto-Reorient function (hMRI toolbox) and this was applied on the NODDI maps as well. Second, both MT B1 sat maps of each subject (corresponding to two sessions) were registered to the mid-point average using the Pairwise Longitudinal Registration (SPM12). Hereby, values below zero and above 10 were excluded to improve the registration. Third, the resulting mid-point average image was normalized to the MNI space using the DARTEL-based (Ashburner, 2007) Spatial Processing module (hMRI toolbox). Fourth, a combined deformation field was generated per subject and session, combining the deformation fields from steps 2 and 3.

Computation of MVF MR , AVF MR and g MR
In this section, our approach to estimating MVF and AVF from the measured MR parameters is introduced. The MR-based MVF (MVF MR ) was assumed to be proportional to MT sat without intercept, following (Mohammadi and Callaghan, 2020): The proportionality constant α was estimated from Equation (2) in a region where the histological MVF (MVF hist ) was known. Due to the lack of own histological data, we used published histological data which contain the frequency distribution of inner-axon radius (r) and myelin sheath thickness (m) of 2,400 myelinated fibers in the medullary pyramids of a 71 years old human (see Table 1 in Graf von Keyserlingk and Schramm, 1984). The total volume (TV) of the sample is the sum of the total volume of myelinated axons (TAV m ), unmyelinated axons (TAV u ), myelin volume (TMV), and extracellular volume (TEV). TAV m was calculated as N m i=1 πr 2 i with i indexing the N m myelinated axons only, and TMV was computed as − TAV m . TAV u , while not reported in Graf von Keyserlingk and Schramm (1984), was found to be Frontiers in Neuroscience | www.frontiersin.org FIGURE 1 | Illustration of the spatial alignment pipeline of the MT sat and NODDI maps. The pipeline consists of (i) co-registration between MT sat and NODDI maps (driven by ODI map), (ii) normalization into MNI space, and (iii) back-projection of ROIs into the native space. Note that each subject consists of two sets of images acquired in separate sessions. In the co-registration step (section "Co-registration"), the white matter (WM) tissue probability map (TPM) of the ODI was co-registered to the WM TMP of the MT sat in each subject and session using rigid-body registration (spm_coreg algorithm, SPM12). The resulting transformation was applied to all other NODDI maps as well. In the normalization step (section "Normalization"), MT sat maps were roughly aligned with the T1-weighted MNI template in each subject and session using the Auto-Reorient function. The realigned MT sat maps from both sessions were then registered to their mid-point average using the Pairwise Longitudinal Registration (SPM12). In each subject, the mid-point average MT sat map was normalized to the MNI space using the DARTEL-based (Ashburner, 2007) Spatial Processing module. Finally, all deformation fields were converted to a single deformation field and applied on the NODDI maps. In the last step (section "Region of Interest Selection"), the ROIs and the WM masks were back-projected into the native space using the inverse of the combined deformation field.
approximately 43% of TAV m for multiple mammals (Swadlow et al., 1980;LaMantia and Rakic, 1990;Olivares et al., 2001;Wang et al., 2008;Liewald et al., 2014). Note that the aforementioned papers typically reported the unmyelinated axons as 30% of the total volume of axons, which corresponds to 43% ( = 0.3 1−0.3 · 100) of TAV m . EVF was estimated to be 25%, according to Lehmenkühler et al. (1993), Nicholson and Hrabìtová (2017), Tønnesen et al. (2018). Finally, MVF was calculated as with j indexing all N fibers, yielding MVF hist ≈ 0.3623. Plugging this value into Equation (2) (assuming that MVF MR ≈ MVF hist ) along with the group-average MT sat within the medullary pyramids (see Figure 2 for ROI definition) yielded an α of 0.2496 for MT B1 sat , 0.2414 for MT UN sat , and 0.2884 for MT NO sat . The MR-based AVF (AVF MR = (1−MVF MR ) AWF MR ) was calculated as where AWF = (1−ν iso ) ν icvf is the axonal water fraction estimated from the NODDI parameters (Stikov et al., 2015) and MVF MR = αMT sat . The MR g-ratio was then computed according to Stikov et al. (2011Stikov et al. ( , 2015 g Note that three versions of MT sat , AVF MR , and g MR were generated according to notation in section "Data Processing": (i) FIGURE 2 | Location of the pyramidal tracts in the medulla oblongata ROI, overlaid on the group-averaged MT B1 sat map, that was used to determine the calibration constant, converting MT sat into MVF MR (section "Computation of MVF MR , AVF MR , and g MR ). To create this ROI, the corticospinal tract ROI of the JHU-ICBM-DTI-81 atlas, which extends across the pons and medulla pyramids, was modified to cover only the medulla pyramids. Left-right position: X = 82; anterior-posterior position: Y = 77; superior-inferior position, Z = 30.

Definition of White Matter Masks
As g MR and its constituents (MVF MR , AVF MR ) are defined only in the WM, we restricted the analysis to the WM by creating binary WM masks (Mohammadi and Callaghan, 2020). WM tissue probability maps (WM-TPM) were created for each subject by segmenting AWF and MT B1 sat using the hMRI toolbox, and taking their intersection according to Mohammadi and Callaghan (2020). In two subjects, the MT B1 sat segmentation was of insufficient quality for segmentation and was replaced by the R B 1 1 map. A groupspecific binary WM mask (WM group ) was generated by averaging all individual WM-TPMs in the MNI space and thresholding it at 0.95.
A so-called high-SNR WM group was also defined by taking the intersection of the WM group and a binary signal-to-noise ratio (SNR) map. Hereby, the latter was used to reduce the number of voxels with unrealistically high values of ν icvf (ν icvf ≥ 0.999). In 6 of 25 subjects, an SNR map was created by dividing the mean b 0 image by a single noise estimate in the native space and multiplied by the square root of the number of b 0 images per DWI dataset (n = 12). The noise was estimated within a noise ROI outside the brain in 72 images (6 subjects, both timepoints and 6 b0 images FIGURE 3 | Relationship between signal-to-noise ratio (SNR) and unrealistically high ν icvf values-here defined as ν icvf ≥ 0.999. (A) Sagittal, coronal, and axial view of the whole-brain SNR map (i), with a zoom-in view of the brainstem (ii). The brainstem is characterized by low SNR due to the spatial characteristics of the receive coil array (ii) and high occurrence of unrealistically high ν icvf (iii), also shown as a binary mask (iv). (B) Given the co-occurrence of low SNR and unrealistically high ν icvf , a binary SNR mask was created to exclude low-SNR voxels. To determine the optimal threshold for the SNR mask, the ratio between the number of voxels with unrealistically high ν icvf and the total number of voxels within the mask were plotted against the SNR threshold. The solid dots and error bars represent the group mean and group standard deviation of the ratio, respectively. The SNR value that yielded the minimum of this ratio was considered optimal (SNR = 39, shown in red).  Table 1) are part of the JHU-ICBM-DTI-81 WM atlas (Hua et al., 2008) and are displayed here on the group-averaged normalized MT B1 sat image. Note that for ROI analysis, the ROIs were projected into the native space using the inverse of the combined deformation field. each) using the ACID toolbox, with the values averaged to obtain a single noise estimate. The threshold for SNR maps to create binary SNR map was chosen such that it minimizes the ratio between the number of artifactual voxels where ν icvf ≥ 0.999 and the total number of voxels in the SNR mask (Figure 3B), yielding a value of 39. This was motivated by the observation that unrealistically high ν icvf values typically occur in low-SNR areas (Figures 3Aii,iii). This threshold selection represents a tradeoff between removing unrealistic voxels while retaining as many voxels as possible. This table lists the dynamic range ( DR ), lowest (min i∈ROI ) and highest (max i∈ROI ) ROI average value, mean value of the 21 analyzed ROI's (mean) with its corresponding standard deviation (SD).

Region of Interest Selection
For the region of interest (ROI) analysis, the JHU-ICBM-DTI-81 WM atlas (Hua et al., 2008) was transformed into the native space using the inverse of the combined deformation field. Two sets of ROIs were defined: (i) whole-WM ROIs and (ii) high-SNR ROIs, used for the main analysis. The whole-WM ROIs included those of the JHU-ICBM-DTI-81 WM atlas that were completely in WM group defined in 2.6, yielding 43 ROIs (out of 48, leaving out the column and body of the fornix, the left and right cingulum part in the vicinity to the hippocampus, and the left and right uncinate fasciculus). The high-SNR ROIs included only those whole-WM ROIs that overlapped with the high-SNR WM group to at least 95%, yielding 21 ROIs (Figure 4 and Table 2).
For the analyses, group-averaged g MR , AVF MR , and MVF MR were calculated within the WM group . Note that averaging included both sessions of each subject for all analyses except for the analysis in section "Test-Retest Analysis of the Group-Averaged MR G-ratio, Axon, and Myelin Volume Fraction." Test-Retest Analysis of the Group-Averaged MR G-ratio, Axon, and Myelin Volume Fraction The group-averaged g B1 MR of the first and second session were compared within the previously mentioned 21 high-SNR ROIs using Bland-Altman plots (Bland and Altman, 1986). In the Bland-Altmann plots, the differences in g B1 MR between the first (g B1 MR 1 ) and second (g B1 MR 2 ) session (δ retest yielding the relative error (δ retest DR% = retest DR · 100) and relative bias (δ retest DR% = δ retest DR · 100). The same procedure was also applied to AVF B1 MR and MVF B1 MR . The distinction between bias and error is important, because while a potential bias can be retrospectively corrected, the error in the MR g-ratio method defines its sensitivity to detect differences between individuals, groups, or time points. To reliably capture these differences, the error must be significantly lower than the expected effect size.

Influence of B 1 + Correction in the Group-Averaged MR G-ratio, Axon, and Myelin Volume Fraction
Bland-Altman analysis was used to compare g MR with and without B 1 + correction. In particular, the difference δ B1 i in g MR between (g B1 MR ) i , when using the reference method B 1 + correction, and (g k MR ) i , when using no (k = NO) or UNICORT (k = UN) B 1 + correction: , with i being the index of the 21 high-SNR ROIs. The bias and error associated with the lack of (or UNICORT) B 1 + correction are defined respectively.
The computed B1 and δ B1 were normalized by the dynamic range of g B1 MR within the high-SNR ROIs, yielding the relative error ( B1 DR% = B1 DR · 100) and relative bias (δ B1 DR% = δ B1 DR · 100). The same procedure was also applied to AVF MR and MVF MR , comparing them to their respective reference method and dynamic range. For MVF MR , the Bland-Altman analysis was additionally done using the whole-WM ROIs instead of the high-SNR ROIs (see section "Region of Interest Selection") to assess the influence of including low-SNR voxels in the analysis.

Group Variability in MR G-ratio, Axon, and Myelin Volume Fraction
To assess group variability for each correction method, the coefficient-of-variation (CoV) across subjects and sessions was calculated for MVF MR , AVF MR , and g MR in the MNI space after applying tissue-weighted smoothing (Tabelow et al., 2019), yielding: CoV B1 MR , CoV UN MR , and CoV NO MR , where MR ∈ {g MR , AVF MR , and MVF MR }. For tissue-weighted smoothing, a full width at half maximum Gaussian smoothing kernel of 6 mm was used. Bland-Altman analysis (see section "Test-Retest Analysis of the Group-Averaged MR G-ratio, Axon, and Myelin Volume Fraction") was used to compare CoV UN MR and CoV NO MR against CoV B1 MR based on the reference method, yielding bias (δ CoV ) and error ( CoV ) values. A higher variability across the brain is expected to increase δ CoV whereas a higher local variability is expected to increase CoV .

G-ratio, Myelin, and Axonal Volume Fraction Across the White Matter
Voxel-wise maps of group-averaged g B1 MR , AVF B1 MR , and MVF B1 MR in WM are shown in Figure 5. The group-averaged mean and standard deviation of g B1 MR , MVF B1 MR , and AVF B1 MR in 21 high-SNR ROIs are reported in Table 1 and Figure 6. The dynamic range ( DR ), minimum and maximum values, and mean and standard deviation of g B1 MR , AVF B1 MR , and MVF B1 MR across ROIs are listed in Table 2. The largest g B1 MR and AVF B1 MR were found in the right anterior limb of the internal capsule (0.688 and 0.384, respectively), while the largest MVF B1 MR was in the genu of corpus callosum (0.445), where also the lowest g B1 MR (0.642) can be found. The lowest AVF B1 MR , and MVF B1 MR were found in the right posterior thalamic radiation (AVF B1 MR = 0.308) and in the left external capsule (MVF B1 MR = 0.408), respectively. The DR was the smallest for MVF B1 MR (0.037), followed by g B1 MR (0.046) and AVF B1 MR (0.076).
Test-Retest Analysis of the Group-Averaged MR G-ratio, Axon, and Myelin Volume Fraction The relative error ( retest DR% ) and bias (δ retest DR% ) values of the test-retest analysis are summarized in Table 3 and shown as Bland-Altmann plots in Figure 7. The test-retest analysis    Table 1. The mean and standard deviation of the distribution are indicated by solid dot and whiskers, respectively.
Frontiers in Neuroscience | www.frontiersin.org FIGURE 7 | Depicted are scatter and Bland-Altman plots of g B1 MR (first row), AVF B1 MR (second row), and MVF B1 MR (third row) from two session across 21 WM regions (denoted high-SNR ROIs, see Figure 4). The Bland-Altman plot illustrates the differences between values obtained from the two sessions (e.g., g B1   (Figure 7 and Table 3).
The retest DR% was below 22.2% for each metric, where the AVF B1 MR showed the lowest retest DR% with 20.5% (Figure 7 and Table 3). List of the bias (δ B1 ) and error ( B1 ) values as defined in Figure 9, along with their relative value with respect to the dynamic range DR : B1 · 100. Note that the error and bias in the last two rows were obtained when using the whole-WM ROIs instead of the high-SNR ROIs (see Supplementary  Figure 1).
Group Variability in MR G-ratio, Axon, and Myelin Volume Fraction g MR showed on average smaller CoV than AVF MR and MVF MR (Figure 10). In all maps, the CoV was the highest in the deep brain areas. The relative error ( CoV CoV B1 · 100) and bias ( δ CoV CoV B1 · 100) values of CoV with respect to the B 1 + reference measurement FIGURE 8 | Scatter plots of g MR , AVF MR , and MVF MR , plotting values obtained without B 1 + correction (superscript: NO, top row) and with UNICORT B 1 + correction (superscript: UN, bottom row) against values obtained with the reference method, i.e., B 1 + field map correction (superscript: B1). A dashed unit slope line is plotted for reference. Each point in the scatter plot represents the group-averaged value in a single ROI (see Figure 4 for the locations of the 21 high-SNR ROIs).
FIGURE 9 | Bland-Altman plots of g MR , AVF MR , and MVF MR , comparing values obtained without B 1 + correction (NO, top row) and with UNICORT B 1 + correction (UN, bottom row) against values obtained by B 1 + field map correction (superscript: B1). The Bland-Altman plot illustrates the differences between values obtained by two different methods (reference vs. tested method); e.g., , with k = UN, NO and i indexing the i th ROI). Each point in the scatter plot represents the group-averaged value in a single ROI (see Figure 4 for the locations of the 21 high-SNR ROIs). The bold black line represents the bias (δ , while the dashed line shows error ( B1 = 1.96 · SD(δ B1 i )) between the reference and the tested method. Error and bias values averaged across all ROIs and subjects are listed in Table 5. are summarized in Table 5 and the error and bias are also displayed as Bland-Altman density plot in Figure 11. For g MR , compared to the no correction case, UNICORT showed similar CoV (UNICORT vs. no correction: 0.6% vs. 0.6%) but lower δ CoV (−0.1% vs. −0.4%). UNICORT yielded higher CoV (UNICORT vs. no correction; 1.0% vs. 0.8%) and lower δ CoV (−0.2% vs. −0.4%) for AVF MR , and higher CoV (1.2% vs. 0.4%) and higher δ CoV (−0.5% vs. −0.1%) for MVF MR . The lower δ CoV of g MR and AVF MR associated with UNICORT reveals itself as a slight shift of the points toward the unit slope line in the scatter density plot (Figure 12).

DISCUSSION
In this study, we showed that omitting the correction of the magnetization transfer saturation map (MT sat ) for residual B 1 + effects introduces large error and bias in the MR g-ratio and the constituents (myelin and axon volume fractions, or in short MVF MR and AVF MR ). We also demonstrated that this error and bias can be reduced by roughly a factor of three using the data-driven UNICORT B 1 + correction (implemented in the hMRI toolbox, see text footnote 1) when a B 1 + field measurement is unavailable.
The Effect of Omitting the B 1 + Field Measurement MT sat have been often used as a proxy for the MVF MR in g-ratio weighted imaging (Mohammadi et al., 2015;Campbell et al., 2018;Ellerbrock and Mohammadi, 2018;Hori et al., 2018;Kamagata et al., 2019), because they are directly linked to the macromolecular pool with an intrinsic correction for underlying longitudinal relaxation time and B 1 + field inhomogeneities effects (Helms et al., 2008). Despite the latter intrinsic correction for B 1 + field inhomogeneities, we found that the residual B 1 + effects on MT sat map were still observable. In particular, the bias and error of the MR g-ratio (g MR ) was about −89 and 37% higher, respectively, when omitting the B 1 + correction. We found the same trend for MVF MR and AVF MR ; while the error and bias were even larger for MVF MR when B 1 + correction was omitted, it was smaller but still substantial for the AVF MR . We found that omitting B 1 + leads to a substantially higher (more than 10-fold) bias in the MR g-ratio and its constituents when compared to a test-retest analysis of our data (Figure 7 and Table 3). Also, the error due to omitting the B1+ correction was twice as large as the error observed in the test retest analysis for the MR g-ratio and the MVF, whereas for AVF the errors were similar. We expect that the high error will be of particular relevance for group studies because it can be regarded as an error that evolves when replacing the reference method with the alternative method. For comparison, age-related changes assessed by g-ratio weighted imaging (Cercignani et al., 2017;Berman et al., 2018) have been reported to vary between 30 and 100% (in absolute values: g MR 0.02-0.04 (Figure 5 in Cercignani et al., 2017). Consequently, the reported effect size of age-related changes would have become potentially undetectable if the B 1 + field correction has been omitted in the study of Cercignani et al. (2017). The B 1 + effect is particularly relevant for the MR g-ratio method by Cercignani et al. (2017) that combined quantitative MT (Gloor et al., 2008) with NODDI, because the qMT method does not possess an intrinsic correction for B 1 + field inhomogeneities as opposed to the MT sat methods used here. Note that we reported, for better intuition, the bias and error relative to the dynamic range of the parameters across the investigated white matter (WM) ROIs (the dynamic range of g MR is DR = 0.046; the absolute bias and error can be found in Table 4).
To reduce this source of bias and error, we propose a datadriven approach to correct for B 1 + field inhomogeneities when no B 1 + field measurement is available. To this end, we used UNICORT to estimate the B 1 + field (Weiskopf et al., 2011). We found that using the UNICORT-estimated B 1 + field to correct residual B 1 + field inhomogeneities in MT sat reduces at the group level the bias and error in the MR g-ratio and its constituents by roughly a factor of three. However, the UNICORT estimated B 1 + inhomogeneity can be erroneous with the error varying across subjects. To assess this variability, we estimated coefficientof-variance (CoV) maps of g MR , AVF MR , and MVF MR for all List of the bias (δ CoV ) and error ( CoV ) values as defined in Figure 11, along with their relative value with respect to the group-average CoV across the MR g-ratios using the reference B 1 + field correction method:   Table 5. three methods. In general, an increased CoV can be found at tissue boundaries (e.g., cerebral spinal fluid to WM) due to slight misregistration between the maps of axonal and myelin markers and/or imperfect normalization (Figure 10). Additionally, we found a strong increase in the bias and error of the CoV of MVF maps (increase in bias: 11% and in error: 18%) when UNICORT B 1 + correction was used as compared to no correction. The CoV of g MR and AVF MR did not show a consistent trend: while the bias decreased, the error increased for both parameters. In other words, the UNICORT B 1 + correction leads to higher accuracy in the g-ratio and its constituents but comes at the cost of a lower precision in MVF.

G-ratio, Myelin, and Axonal Volume Fraction Across the White Matter
Our g B1 MR and AVF B1 MR across the white matter were within the range of the reported values of previous studies (g MR : 0.64-0.76; AVF MR : 0.26-0.43 in (Cercignani et al., 2017;Berman et al., 2018). The range of MVF B1 MR was in the upper half of previously reported values (0.17-0.42 in Cercignani et al., 2017). Our slightly higher MVF MR values might be due to differences in the calibration approach: while we calculated the reference MVF REF from previously published ex-vivo histology data (Graf von Keyserlingk and Schramm, 1984), Cercignani et al. (2017), used a reference from previously published ex-vivo histology g-ratio data in the corpus callosum and Berman et al. (2018), did not perform any calibration assuming that macromolecular tissue volume and MVF MR are equal. An error in the calibration constant can lead to a bias in the MVF estimates which in turn leads to an error and bias in the MR g-ratio (Campbell et al., 2018).

Confounding Factors
As this study calculates the in-vivo MR g-ratio, there is no histological data available from the participants of this study, which could be used for calibration or as a gold standard reference. For calibration of MT sat to MVF MR , we estimated the histological MVF (MVF hist ) from published ex-vivo data within the human medulla oblongata (Graf von Keyserlingk and Schramm, 1984). Since the reference MVF hist and the calibrated MT sat map were taken from different subjects, this might introduce a systematic bias in the MR g-ratio. However, since we found a relatively good agreement between our g MR , FIGURE 12 | Scatter density plots of CoV g (left column), CoV AVF (middle column), and CoV MVF (right column), plotting values obtained with no correction (superscript: NO, top row) and with UNICORT B 1 + correction (UN, bottom row) against values obtained by B 1 + field map correction (superscript: B1). The unit slope line is plotted for orientation (dotted line). The dots in the scatter plots represent the WM voxels in the CoV maps in Figure 10 (yellow indicates high voxel density).
AVF MR , and MVF MR values with previously reported values obtained by a different calibration approach (Cercignani et al., 2017;Berman et al., 2018), we expect that it had a small effect on the results. Moreover, we focused on the effect of omitting B 1 + correction, which will lead to additional inaccuracies in g-ratio weighted imaging, independent of the quality of the calibration.
Although, not reported in previous NODDI-based g-ratio mapping studies (Stikov et al., 2015;Cercignani et al., 2017;Jung et al., 2017;Mancini et al., 2017;Ellerbrock and Mohammadi, 2018;Hori et al., 2018), we found that the intra-cellular volume fraction (ν icvf ) determined with NODDI tends to be biased at small signal-to-noise ratios (SNR < 39), resulting in a ceiling effect, i.e., ν icvf ≈ 1. To avoid a corresponding bias in g MR (and AVF MR ), we restricted the analysis to regions with sufficiently high SNR (Figure 3). To investigate whether our findings generalize to low-SNR regions as well, we performed an additional Bland-Altman analysis of MVF MR in whole-WM ROIs. To this end, a larger set of ROIs was used covering the entire white matter. Although the bias was smaller for the whole-WM as compared to the high-SNR ROI analysis, we found the same trend: the error and bias were reduced when using UNICORT B 1 + correction relative to no correction. Note that the smaller bias for the whole-WM analysis is most probably an artifact of the calibration procedure. Since the ROI used for calibration was not part of the high-SNR ROIs but was part of the whole-WM ROIs, we think it could have reduced the bias in the whole-WM ROI analysis as compared to the high-SNR analysis.
We note that the presented results were based on a customized B 1 + mapping method (Lutti et al., 2010). Using vendor specific protocols for B 1 + and MT sat mapping may influence the results (Leutritz et al., 2020). Moreover, the calibration factor in Equation (1) may have to be recalibrated for different MT-pulses. Future studies should investigate the effect of B 1 + correction on MR g-ratio mapping when using alternative biomarkers to estimate AVF MR and MVF MR (e.g., Ellerbrock and Mohammadi, 2018). Moreover, there are alternative B 1 + mapping approaches available which might vary in precision (Lutti et al., 2010) and therefore can affect the MR g-ratio values. However, the differences in the precision of these methods are in the order of few percentage and thus much smaller than the effect of omitting the B 1 + field or using the data-driven UNICORT B 1 + estimate (Weiskopf et al., 2011).

CONCLUSION
In this study, we assessed the effect of B 1 + correction on the accuracy of MR g-ratio as well as axonal and myelin volume fraction based on MT sat and NODDI. Our results demonstrate that B 1 + correction via a measured B 1 + field map is the method of choice. If the B 1 + field map cannot be acquired, we propose the retrospective, data-driven UNICORT B 1 + correction to estimate and correct for B 1 + field inhomogeneities, which reduces the error and bias by a factor of three. UNICORT is implemented in the free and open-source hMRI toolbox (see text footnote 1).

DATA AVAILABILITY STATEMENT
The datasets presented in this article are not readily available because the data that support the findings of this study are available on request from the corresponding author. The data have not been made freely available on the internet due to privacy or ethical restrictions. Requests to access the datasets should be directed to corresponding author.

ETHICS STATEMENT
The studies involving human participants were reviewed and approved by the Ärztekammer Hamburg. The patients/participants provided their written informed consent to participate in this study.

AUTHOR CONTRIBUTIONS
SM and TE contributed to the conception and design of the study, performed statistical analysis and MRI processing, and wrote the first draft of the manuscript. All authors contributed substantially to revising the manuscript critically for intellectual content and have approved the submitted version.