Quantitative magnetization transfer MRI unbiased by on-resonance saturation and dipolar order contributions

Purpose: To demonstrate the bias in quantitative MT (qMT) measures introduced by the presence of dipolar order and on-resonance saturation (ONRS) effects using magnetization transfer (MT) spoiled gradient-recalled (SPGR) acquisitions, and propose changes to the acquisition and analysis strategies to remove these biases


Purpose:
To demonstrate the bias in quantitative MT (qMT) measures introduced by the presence of dipolar order and on-resonance saturation (ONRS) effects using magnetization transfer (MT) spoiled gradient-recalled (SPGR) acquisitions, and propose changes to the acquisition and analysis strategies to remove these biases.

Methods:
The proposed framework consists of SPGR sequences prepared with simultaneous dual-offset frequency-saturation pulses to cancel out dipolar order and associated relaxation (T 1D ) effects in Z-spectrum acquisitions, and a matched quantitative MT (qMT) mathematical model that includes ONRS effects of readout pulses. Variable flip angle and MT data were fitted jointly to simultaneously estimate qMT parameters (macromolecular proton fraction [MPF], T 2,f , T 2,b , R, and free pool T 1 ). This framework is compared with standard qMT and investigated in terms of reproducibility, and then further developed to follow a joint single-point qMT methodology for combined estimation of MPF and T 1 . Results: Bland-Altman analyses demonstrated a systematic underestimation of MPF (−2.5% and −1.3%, on average, in white and gray matter, respectively) and overestimation of T 1 (47.1 ms and 38.6 ms, on average, in white and gray matter, respectively) if both ONRS and dipolar order effects are ignored.

INTRODUCTION
Quantitative magnetization transfer (qMT) imaging aims at quantitatively assessing the magnetization exchange processes occurring between motion-restricted macromolecules and surrounding water protons. For brain imaging, the classical qMT methodology uses a two-pool model [1][2][3] applied to MT-prepared spoiled gradient-recalled (SPGR) data. 4 The most relevant quantitative parameter derived from this approach is the macromolecular proton fraction (MPF), which presents a fair sensitivity to brain tissue changes. 5,6 In the qMT framework, an independent measurement of T 1 relaxation is required to disentangle relaxation and MT effects, and provide access to an estimation of MPF. Variable flip angle (VFA)-SPGR and MT-SPGR data are usually combined and either used separately [7][8][9] or jointly 10 to estimate the apparent free pool T 1 and other qMT parameters. The joint estimation is advantageous, as it explicitly considers magnetization exchanges between the macromolecular and the free pools in the two-pool model for a more accurate T 1 and MPF estimation. 11 However, two important features are typically not considered in common SPGR-based qMT methodology: • First, on-resonance saturation effects induced by the readout pulses on the macromolecular lineshape are usually ignored. Of interest, Mossahebi et al. 10 have argued that on-resonance saturation effects of the SPGR readout pulse at the macromolecular pool level may have a nonnegligible effect on the qMT parameters' estimation. This was further demonstrated by ensuing work in the context of magnetic resonance fingerprinting 12 and VFA-T 1 mapping. 13,14 It is therefore worth investigating whether these saturation effects (referred hereafter to as on-resonance saturation [ONRS]) may bias the estimated values of qMT parameters if not accounted for in the joint VFA-SPGR and MT-SPGR data modeling.
• Second, the off-resonance RF saturation of the semisolid pool should be treated using the Provotorov theory, 15,16 which demonstrates a coupling between the macromolecular Zeeman and dipolar orders when applying single-offset RF irradiation. Early studies concluded a negligible effect from dipolar order in the estimation of the qMT parameter values 17,18 from single-offset frequency saturation (SOFS; RF irradiation applied at a single side with respect to the resonance frequency) Z-spectrum data. Hence, it was argued that the classical binary spin-bath model-which neglects dipolar order contributions-was appropriate for the modeling of SOFS-based qMT data. However, these early experiments were performed either on agar gel phantoms, which are not representative of the brain tissue and have weak dipolar order effects, 18 or on ex vivo white matter samples at room temperature, 17 which strongly limits the influence of dipolar order on the macromolecular magnetization. 19-21 Recent investigations have shown that myelinated central nervous system tissues are associated with relatively strong dipolar order effects as denoted by a long dipolar order relaxation time (T 1D ; T 1D ≈ 6-10 ms) and as identified by inhomogeneous magnetization transfer (ihMT) imaging in studies with ex vivo (at physiological temperature) and in vivo specimens from animals, such as in mouse brains 22-24 and rat spinal cord, 25 and in humans brains. 26-28 Hence, and given the typical irradiation powers used in clinical MR scanners for pulsed SOFS qMT experiments, the conditions for dipolar order to contribute to the MT signal in in vivo central nervous system tissues can be satisfied. 29 Because dipolar order effects manifest as a reduced RF saturation rate of the macromolecular pool, 28 which varies with the preparation pulse's power and frequency, 21,25,30,31 biases can be expected in the estimated qMT parameters from SOFS-based Z-spectra analyzed with the two-pool model. Conversely, an alternative approach involving symmetric dual-offset frequency saturation (DOFS; RF irradiation applied on both sides regarding the resonance frequency) pulses, which effectively decouples the dipolar orders from their associated macromolecular Zeeman orders, 32,33 would make the two-pool model more accurate for Z-spectrum data analyses.
In this work, qMT simulations and experiments were performed to confirm the existence of biases in quantitative MPF and longitudinal relaxation derived from SOFS-based Z-spectra analyzed with the classical two-pool model. Additionally, we propose an alternative framework based on DOFS pulses to cancel out dipolar order effects and a two-pool model that accounts for ONRS effects when deriving quantitative MT parameters from combined VFA-SPGR and MT-SPGR data in the human brain. The proposed framework was repeated for reproducibility assessment and further extended in an adapted joint single-point qMT methodology 7 for simultaneous estimation of MPF and apparent free pool T 1 .

METHODS
Experiments were performed on a 3T clinical scanner (MAGNETOM Vida, software version XA20A; Siemens Healthineers, Erlangen, Germany) with body coil transmission and a 32-channel receive head coil on 3 healthy

F I G U R E 1
Magnetization transfer (MT)-prepared spoiled gradient-recalled (SPGR) sequences with a single-offset (+Δ) saturation pulse (SOFS; RF irradiation applied at a single side with regard to the resonance frequency) (A), and with a simultaneous symmetric dual-offset (±Δ) saturation pulse (DOFS; RF irradiation applied at both sides regarding the resonance frequency) (B). The use of a simultaneous DOFS pulse effectively decouples the macromolecular Zeeman order from its associated dipolar order. Conversely, SOFS couples dipolar order and Zeeman order. The classical and commonly used two-pool quantitative MT (qMT) model, which does not include a dipolar order reservoir, is more accurately adapted to the DOFS-SPGR conditions. volunteers (2:1 men:women; mean age: 30.2 ± 2.9 years). Experiments were approved by the institutional ethics committee on clinical investigations (CRMBM, Marseille), and written informed consent was obtained from each participant before the study.

2.1
Sequence description Figure 1 shows the MT-prepared SPGR pulse sequences used in the experiments. The MT preparation includes either a SOFS ( Figure 1A) or a DOFS ( Figure 1B) pulse, both of identical duration τ SAT , followed by a spoiling gradient applied in the readout direction within a resting delay interval (RD 1 ). The 3D-SPGR acquisition module includes a nonselective Hann-shaped readout pulse of duration τ RO , followed by a delay to TR (RD 2 ), encompassing a gradient-echo readout and a spoiling gradient in the same direction as in the preparation. SOFS pulses consisted of Hann-shaped pulses applied with off-resonance: Where Δ is the pulse offset frequency from free water resonance, and A(t) its normalized amplitude, defined as follows: Simultaneous symmetric dual-offset saturation was achieved by modulating the SOFS pulse amplitude with a symmetric sine function, thereby guaranteeing that the on-resonance component (i.e., water) remains unaffected: The saturation pulse power B 1,RMS SAT is given by B 1,RMS SAT = B 1,peak SAT √ p 2 SAT , where B 1,peak SAT denotes the pulse peak B 1 and p 2 SAT denotes the pulse power integral. 34 To apply equivalent saturation power, the B 1,peak SAT of the DOFS pulses was increased by a √ 2 factor to compensate for its halved power integral in comparison to that of the SOFS pulse.
In the following, and as classically modeled in qMT frameworks, we assume that RF irradiations are applied at an exact frequency offset Δ despite originating from finite pulses characterized by a spectral response.

Two-pool model for SPGR-based sequences
A two-pool model was adapted into a general matrix formalism accounting for magnetization saturation, tilting, relaxation, and exchange events throughout the MT-SPGR and VFA-SPGR sequences. The objective was to isolate the longitudinal magnetization of the free pool (M z,f ) from the magnetization vector The general cross-relaxation matrix was subsequently defined as follows: where M 0,i , R 1,i (=1/T 1,i ) and T 2,i are the thermal equilibrium, longitudinal relaxation rate and transverse relaxation time of the free pool (i = f) and macromolecular pool (i = b), and R is the exchange rate between the two pools. The longitudinal relaxation times of both pools are assumed equal (T 1,b = T 1,f = T 1 ), and we denote the resulting apparent relaxation time as T 1 in the following. In addition, the convention M 0,f + M 0,b = 1 was adopted, and MPF-defined as the ratio of the macromolecular magnetization over the total proton magnetization-then reduces to MPF = M 0,b .
The saturation matrix associated with the MT preparation pulse was defined as follows: where W b SAT = πg b (Δ, T 2,b ) × (γB 1,RMS SAT ) 2 is the macromolecular saturation rate, using the super-Lorentzian absorption lineshape. This lineshape includes a residual broadening term as proposed by Pampel et al., 35 an alternative to the extrapolation method for on-resonance saturation rate calculation. 12,36 The value of W f SAT = πg f (Δ, T 2,f ) × (γB 1,RMS SAT ) 2 is the free pool saturation rate using a Lorentzian lineshape. Regarding the VFA-SPGR sequences, the terms W b SAT and W f SAT are set to 0. In contrast, the rotation-saturation matrix corresponding to the on-resonance readout pulses was defined as follows, assuming a rotation about the x-axis, a constant pulse of duration τ RO , and no B 0 -related deviation as the pulse duration is sufficiently short: where W b RO = πg b (Δ = 0, T 2,b ) × (γB 1,RMS RO ) 2 is the macromolecular saturation rate induced by the readout pulse, and FA (expressed in radians) and τ RO are the readout pulse flip angle and duration, respectively, and B 1,RMS RO is the readout pulse B 1,RMS , as B 1,RMS RO = B 1,peak RO √ p 2 RO . The matrix formalism proposed by Malik et al. 37 was used to compute M z,f 's amplitude in steady-state just before the readout pulse, assuming a perfect spoiling condition and piecewise constant events. Each event-related matrix A i was rewritten as follows: where C = [0 0 R 1,f M 0,f R 1,b M 0,b ] T and "0" is a row vector of five "0" elements to match the matrix dimensionality. The value of M z,f was then calculated as the third eigenvector element associated with the eigenvalue equal to 1 of theX matrix, defined as follows: (8) where ϕ = diag([0,0,1,1,1]) ensures the transverse magnetization spoiling condition, and RD 1 and RD 2 refer to the resting delays after the saturation pulse and after the readout pulse, respectively ( Figure 1). Finally, the transverse magnetization was computed as M xy,f = M z,f × sin(FA).
To eliminate the contribution of scaling factors including T 2 * decay, proton density, coil sensitivity profile and receiver gain, all signals were classically normalized by a reference obtained without saturation (referred hereafter to as MT 0 ) for both simulations and experiments.
This two-pool model was used in simulations and experiments to estimate quantitative MT parameters according to different methodologies: (i) without considering ONRS effects (W b RO = 0; model corresponds to the classical qMT model, hereafter referred to as qMT-noONRS) and (ii) considering ONRS effects (W b RO >0; hereafter referred to as qMT-ONRS).

Simulations
Theoretical synthetic brain signals for VFA-and MT-SPGR sequences were generated with the same parameters used in the experiments (see Section 2.4) and using the Provotorov theory of RF saturation ( Figure 1A), which accounts for the coupling between the Zeeman and dipolar orders of the macromolecular pool, as already described in the literature. 25,32 In practice, Eqs. (4)-(7) were modified to include the dipolar order contribution to the RF saturation (T 1D and RF-induced exchange rate with the Zeeman pool; see Appendix A) and applied for the generation of MT signals as a function of T 1D spanning from 10 μs to 20 ms (hence mimicking an increasing contribution of dipolar order).
Representative qMT parameters of white matter (WM) and gray matter (GM) were evaluated, with reference parameters (p ref ; p = {T 2,f , T 1 , T 2,b , MPF, R}) set to T 2,f = 20/30 ms, T 2,b = 11/10 μs, T 1 = 1100/1600 ms, MPF = 15.0/9.0%, and R = 20/20 s −1 , respectively. 38 Simulations were performed for both SOFS and DOFS conditions, and the synthetic signals were then fitted using the qMT-noONRS and qMT-ONRS mathematical models to estimate the qMT parameters (p est ). The effects of dipolar order and ONRS on the estimation of qMT parameters were evaluated by computing the relative variation (RV p ; expressed in %) between the estimated and reference parameters as well as the deviation of parameters (Δp = p est − p ref ; expressed in absolute units).

MRI experiments
The imaging protocol consisted of 3D sagittal full Z-spectra and VFA acquisitions using the described MT-SPGR sequences ( = 420 Hz/voxel, matrix size = 128 × 128 × 72 and voxel size = 2-mm isotropic, total gradient spoiling moment in the readout direction = 40 mT⋅ms/m after the MT pulse and post-echo readout, respectively, 5-s dummy scans, 6/8 partial Fourier in the phase direction, and 2 × 2 CAIPIRINHA acceleration (24 × 24 integrated autocalibration lines); and acquisition time per volume = 1 min 23 s. SOFS and DOFS sequences paired with regard to Δ and B 1,RMS SAT values were run consecutively, and each pair was randomly distributed throughout the protocol. VFA acquisitions were randomly interleaved. An identical order was conserved for all subjects.

Image processing
For each saturation protocol (i.e., SOFS and DOFS), the 51 SPGR images acquired per subject ( were stacked together and denoised jointly using the Marchenko-Pastur principal component analysis routine from the MRtrix3 package (v. RC301). [39][40][41] Each multi-echo SPGR image was then combined with a sum-of-squares operation to further enhance the SNR. Finally, VFA-weighted and MT-weighted (MTw) images were rigidly registered onto the respective VFA-SPGR image at FA = 25 • to compensate for motion. The latter was rigidly registered onto the MPRAGE image, and the transformation and resampling operations were applied to all other SPGR images.
The qMT parameter values derived from qMT analyses were evaluated from regions of interest (ROIs) selected in WM and deep gray matter (DGM). For WM, ROIs were retrieved from the JHU probabilistic atlas 42 through label propagation of the MNI template (symmetric ICBM 2009c) onto the anatomical MPRAGE volume using a multistage, rigid, affine, and diffeomorphic registration (SyN 43 ; antsRegistrationSyN.sh) implemented in the Advanced Normalization Tools (ANTs; v. 2.0.1 44 ). DGM ROIs were automatically segmented using the FreeSurfer (v. 6.0.0) default recon-all pipeline on the MPRAGE image. 45

2.5.1
Joint fitting of VFA and Z-spectrum data Three frameworks of data analysis were used to derive the quantitative maps of MT parameters (T 2,f , T 2,b , T 1 , MPF, and the exchange rate R), combining VFA data with (i) SOFS Z-spectra and fitted by the qMT-noONRS model (SOFS-noONRS; "A"), (ii) SOFS Z-spectra and fitted by the qMT-ONRS model (SOFS-ONRS; "B"), and (iii) DOFS Z-spectra and fitted by the qMT-ONRS model (DOFS-ONRS; "C"). Hence, only the DOFS-ONRS framework is conceptually consistent, because it combines a model that is adapted to the input data (no dipolar order). All reconstructions included transmit-field inhomogeneity correction of all pulse-related quantities in Eqs. (5) and (6), as provided by the estimated and resampled relative B 1 + map.
Comparison of the three frameworks was achieved by performing linear regression and Bland-Altman analyses on quantitative parameter values derived in WM and DGM regions.
Finally, the reproducibility of the DOFS-ONRS framework was assessed in a test-retest experiment. Bland-Altman and linear regression analyses of the estimated qMT parameters were performed between the two experiments. For each metric, the coefficients of variation (CoV = SD/mean; between both sessions) averaged over all selected regions in WM and DGM regions were reported.
In Bland-Altman analyses, biases and limits of agreement (LOAs; defined as mean difference ± 1.96 SD of the mean difference) were reported. For each framework, the average values of parameters in ROIs and associated RMS error (RMSE) of the fits were calculated.

2.5.2
Joint fitting of VFA and single-point MT data The DOFS-ONRS framework was further extended following the single-point qMT approach 7 : VFA data and a single optimized and MT 0 -normalized DOFS-MTw image were used jointly (JSP) for the simultaneous estimation of T 1 and MPF while constraining other qMT parameters. In practice, R, T 2,b , and the product R 1 ⋅T 2,f were fixed to the median values of the estimated parameter distributions from all voxels in the brain parenchyma over the 3 subjects, and as computed from the previous full Z-spectra DOFS-ONRS framework. Then, MPF and T 1 maps were estimated for each possible experimental Δ/B 1,RMS SAT pair. The optimized DOFS-MTw image for JSP-qMT was selected as that yielding the minimum deviation (δ p ) in MPF and T 1 values estimated in the whole-brain parenchyma relative to those obtained from the full spectra analysis.
where p Z i and p JSP i are the estimated voxel values for p = {T 1 ;MPF} from full Z-spectrum and JSP-qMT, respectively. Linear regressions and Bland-Altman analyses were performed between both reconstructions over MPF and T 1 .

Simulations
The theoretically expected biases on the estimation of qMT parameters caused by the dipolar order and ONRS effects are highlighted in the simulation results ( Figure 2). Focusing on MPF and T 1 , the relative variations between the reference and estimated values highlight systematic overestimation of T 1

3.2.1
Comparison of frameworks Figure 3A shows representative Z-spectra of SOFS-MT and DOFS-MT average signals in the anterior corona radiata. The dual-offset frequency pulses (black curves) yield a higher apparent saturation effect compared with single-offset frequency pulses (red curves). For both saturation strategies, signal attenuation increases with B 1,RMS SAT . The difference between SOFS-MTw and DOFS-MTw normalized data is shown in Figure 3B and values in WM can be observed between the SOFS-noONRS framework ( Figure 4D) and the two other frameworks ( Figure 4E,F). Mean absolute values of qMT parameters averaged over the 3 subjects and the RMSE of the fits are reported in Table 1. Of interest, RMSE of the fits were significantly different (two-sampled t-test; p < 0.05) only for average WM between frameworks DOFS-ONRS (mean RMSE = 8.06) and SOFS-noONRS (mean RMSE = 8.65) and SOFS-ONRS (mean RMSE = 8.63), but not in DGM (mean RMSE of 9.47, 9.43, and 9.35 for frameworks SOFS-noONRS, SOFS-ONRS, and DOFS-ONRS, respectively).
Comparisons between frameworks are provided with the linear regressions and Bland-Altman plots for MPF ( Figure 5) and T 1 (Figure 6). Other qMT parameters (R, T 2,b , and T 2,f ) are reported and commented in Figures S6-S8.

Framework SOFS-noONRS Framework SOFS-ONRS
The comparison between the SOFS-ONRS and DOFS-ONRS frameworks highlights only the effect of dipolar order contributions, as ONRS effects are accounted for. A systematic, albeit more reduced, underestimation of MPF was also observed (MPF C = 1.06 × MPF B + 0.31%; R 2 adj = 0.97; Figure 5D). The corresponding biases were

Reproducibility
The reproducibility of the DOFS-ONRS framework is synthetized in the Bland-Altman and linear regression analyses performed between the test and retest experiments reported in Table 2 Figure S2. Details on the reproducibility of T 2,f , T 2,b , and R are provided in Table S2.

DISCUSSION
In this study, we demonstrated both theoretically and experimentally that dipolar order effects and MT effects induced by readout pulses in SPGR-based acquisitions can lead to biases in the MPF and T 1 values estimated with the standard two-pool qMT model. As a potential solution, we proposed a framework that uses DOFS pulses (canceling out dipolar order effects) combined with a two-pool qMT model that considers the ONRS of the readout pulses. This framework shows good performance in terms of reproducibility and is suitable for fast single-point qMT imaging.

T A B L E 2
Summary of qMT parameters of the reproducibility experiment estimated by the DOFS-ONRS framework and evaluated by Bland-Altman and linear regression analyses. The coefficient of variation between test and retest experiments are provided.

Understanding and reducing biases in standard qMT-SPGR frameworks
The combined effects of dipolar order and ONRS lead to an underestimation of MPF and an overestimation of T 1 when the standard qMT model and SOFS-based acquisitions are used. The amplitude of the theoretically expected biases estimated from simulated data were in line with experiments, with larger effects in WM in comparison to GM (Figures 2, 5 and 6, and Table 1). The overestimation of T 1 was predominantly a consequence of ONRS effects, given the low difference in bias values between the SOFS-ONRS and DOFS-ONRS frameworks, which differ by the effect of dipolar order. Overall, this is consistent with already-established observations in ihMT studies, in which higher dipolar order effects are known to occur preferentially in WM compared with GM, 22,23,32,46-48 and further supported by the resulting mean RMSE significantly decreasing in WM from the SOFS-noONRS and SOFS-ONRS frameworks to the DOFS-ONRS framework (Table 1). Overall, accounting for ONRS effects almost cancels T 1 -related biases, whereas dipolar order effects need to be removed to minimize MPF-related biases, hence making the binary spin-bath model appropriate to process the data.
In most qMT-SPGR frameworks, MT effects occurring in VFA experiments are not modeled (i.e., ONRS are not considered) and dipolar order effects are neglected, despite the use of SOFS pulses. 5,7,[49][50][51][52][53][54][55] As a result, the estimated qMT parameter values remain dependent on the experimental conditions (Δ, B 1,RMS SAT , readout flip angles, and sequence delays); hence, different protocols prevent consistent reporting and cross-platform reproducibility. More specifically, variations related to Δ and B 1,RMS SAT parameters-and intrinsically to the saturation pulse shape-strongly influence the resulting T 1D -weighting effects when relying on SOFS data, 29 and hence the apparent MT weighting.
The proposed DOFS-ONRS framework was designed to remove dipolar order effects and therefore to make the two-pool model suitable. It has demonstrated a good intrasession reproducibility, thereby representing a promising solution for cross-scanner investigations. Studies on animals with preclinical scanners are likely to be affected by similar biases, hence calling for similar sequence implementations and analyses. This is even more important for translational purposes, as the MPF or its equivalents (e.g., pool size ratio) can be used to synthetize biomarkers such as the g-ratio. [56][57][58]

Fast MPF and T 1 mapping
We demonstrated the feasibility of fast joint MPF and T 1 mapping using the derived JSP-qMT method, and within the optimized and accurate DOFS-ONRS framework (Figures 7 and 8) with a good agreement between full Z-spectrum and JSP-qMT outputs, resulting in a total acquisition time of 6 min 25 s for a 2-mm isotropic resolution. The method, as proposed, requires a minimum of five images: a pair of MTw and MT 0 SPGR images, unsaturated proton density-weighted and T 1 -weighted SPGR-images, and a B 1 + map. Of interest, the constrained values for R

F I G U R E 8
Bland-Altman and linear regression plots of MPF (A,B) and T 1 (C,D) estimated from full DOFS Z-spectra analysis (Z) and joint single-point (JSP) DOFS data analysis. Bias and LOAs are indicated in dashed and solid lines, respectively, in Bland-Altman plots. Regression and unity lines are depicted as dashed and solid lines, respectively, in regression plots. Biases and LOAs in WM and DGM are reported independently, whereas the reported slopes and intercepts of the linear regression were calculated from the joint distributions. Each subject is depicted by a different symbol (circle, diamond, and square). and T 2,b are rather close to that estimated by Yarnkykh 7 at 3 T (R = 21.1 s −1 vs. 19.0 s −1 and T 2,b = 10.0 μs vs. 9.7 μs), while the product R 1 ⋅T 2,f is reduced (R 1⋅ T 2,f = 0.016 vs. 0.022). Applicability of this methodology to pathologies (e.g., multiple sclerosis) and animal models (e.g., cuprizone 59 and experimental autoimmune encephalomyelitis 60 ) nonetheless requires a dedicated validation on respective subjects, although previous works revealed very close values of the constrained qMT parameters between healthy and nonhealthy groups. 7,55

Future biophysical and mathematical modeling
In this work, we expanded on the classical two-pool model methodology 3,61 that has raised interest over the last two decades because of its simplicity, practicability, and reproducibility. 52 More advanced models may help provide better understanding of the MT signal behavior in brain tissues, the most common application of qMT. 53 This is highlighted here by the discrepancies in the exchange rate when fitting SOFS and DOFS data ( Figure S6), which complements the difference in values between classical qMT frameworks (R ≈ 20 s −1 ) 7,50,54,55 and that found in quantitative ihMT frameworks (R spanning from 20 to 70 s −1 , depending on the compartmental model). 32 As a perspective, more detailed models could lead to a better description of myelinated brain tissues. For instance, the four-pool model-recently documented quite comprehensively by Manning et al. 62 -considers two exchanging myelin and nonmyelin water pools, respectively exchanging with the myelin and nonmyelin macromolecular matrices. In addition, evidence was found that multiple dipolar order reservoirs exist in myelinated tissues 25 and in synthetized lipid samples representative of myelin, 63 and their intrinsic association to macromolecular Zeeman reservoirs (with respective and distinct MT parameters) has yet to be investigated. Of note, the proposed DOFS-MTw acquisitions effectively remove the need to consider the dipolar order magnetization reservoirs, and the qMT-ONRS model remains valid as long as the respective exchange rate and macromolecular T 2,b parameters are considered equal, contrary to the fitting of SOFS-based acquisitions for which the impact of dipolar order reservoir on the saturation attenuation is less predictable. Note also that equality between the two pools' relaxation times (T 1,f = T 1,b ) was assumed. Although this is a common hypothesis in standard qMT analyses, 7,10,36,64,65 some studies have reported shorter values for the estimated T 1,b in healthy human brain WM. 66,67 Upgrading the current models to allow for the discrimination of multiple free and bound Zeeman reservoirs would necessitate accumulation of a high number of images with varying sequence parameters, and for which classical Z-spectra acquisitions (i.e., varying Δ and B 1,RMS SAT ) may not be sufficient to yield an adequate sensitivity to the parameters. Further investigation is therefore warranted.
Perfect transverse-magnetization spoiling (i.e., complete extinction of M xy f just before the readout pulse in this case) is a pervasive assumption made in VFA and qMT mathematical models-here numerically achieved with the "ϕ" matrix in Eq. (8). However, slight deviations of the experimental signal from Eq. (8) may occur and depend primarily on the apparent tissue parameters (e.g., T 1 , T 2 , diffusion coefficients, and MT parameters) and local B 1 + . 12,68,69 Gradient spoiling-induced transverse magnetization attenuation through diffusion effects, in addition to the RF spoiling, helps in mitigating such effects but remains mildly efficient with standard clinical systems, and depends on hardware capacities and resolutions. [69][70][71] This calls for deepened modeling of the signal behavior using the configuration states formalism, to account for these effects, 12,72 at the cost of a substantial computational burden with regard to the voxelwise inverse problem solving. Further investigation is therefore warranted on the contributions of imperfect spoiling effects about the estimated qMT parameters.
The longitudinal magnetization attenuation of the free pool during the application of an off-resonance RF pulse is classically calculated using a Lorentzian lineshape for the calculation of the absorption rate (W f SAT ), assuming a constant RF saturation pulse applied at an exact frequency Δ. Recent work showed the effect of the off-resonance pulse shape and duration, which yielded complex effects straightforwardly related to an on-resonance spectral leakage, affecting the free pool magnetization 34 (also referred to as direct saturation effects). These results emphasize the modeling limitation of W f SAT when using finite RF pulse whose spectral response depends on the pulse shape, duration and power, and remain important for the estimation of T 2,f , as the parameter's sensitivity is localized at low offset frequencies. 7,55 In this work, we used a super-Lorentzian absorption lineshape, including a residual broadening 35 term (mostly affecting the on-resonance part of the line), for the calculation of the macromolecular absorption rates (W b SAT and W b RO ), as it represents a more legitimate alternative to the extrapolation method for on-resonance saturation. 12,36 However, bandwidth effects of the finite readout pulse are not considered in the calculation of W b RO , and its subsequent effect on the parameters' estimation may still depend on the readout pulse power, shape and duration, as the absorption lineshape varies rapidly around the resonance. As such, we chose a rather spectrally concentrated and spatially nonselective Hann-shaped pulse for readout with a reasonable duration (τ RO = 1 ms) instead of seemingly often used short rectangular pulses that may exhibit a large spectral response with nonnegligible sinc ripples.
In the proposed DOFS-ONRS framework, VFA data provided the necessary sensitivity to longitudinal relaxation. Although SPGR sequences are attractive for their simplicity, acquiring high-resolution images requires an important acquisition time increase, for which advanced acceleration methods such as compressed sensing 73 should be beneficial. Alternatively, as an efficient high-resolution solution, the widely used MP2RAGE sequence 74 -which also has been shown to be sensitive to MT effects 75 -may substitute the VFA-SPGR protocol to provide the desired T 1 sensitization, and by adapting the DOFS-ONRS framework to encompass an advanced MP2RAGE signal model.

CONCLUSIONS
SOFS-prepared SPGR acquisitions are inconsistent with the classical binary spin-bath model, which does not account for dipolar order and associated relaxation (T 1D ) effects. In this study, we have documented how this model inconsistency biases the estimated qMT parameters. In addition, we also showed that MT effects arising from the on-resonance saturation and exchanges during a VFA experiment can lead to biases on apparent T 1 and MPF estimations. The present work demonstrated the importance of sequence design and signal modeling in quantitative MT imaging. For the sake of reproducibility and accuracy, we therefore encourage investigators to systematically consider all MT effects in models. The use of DOFS pulses in MT-prepared SPGR experiments eliminates dipolar order-related errors in qMT and is therefore recommended for future studies.

SUPPORTING INFORMATION
Additional supporting information may be found in the online version of the article at the publisher's website.
The saturation matrix (A ′ SAT ) associated with the MT preparation pulse is now defined as