Data-driven Retrospective Correction of B₁ Field Inhomogeneity in Fast Macromolecular Proton Fraction and R₁ Mapping

Abstract

Correction of B₁ field non-uniformity is critical for many quantitative MRI methods including variable flip angle (VFA) T₁ mapping and single-point macromolecular proton fraction (MPF) mapping. The latter method showed promising results as a fast and robust quantitative myelin imaging approach and involves VFA-based R₁ = 1/T₁ map reconstruction as an intermediate processing step. The need for B₁ correction restricts applications of the above methods, since B₁ mapping sequences increase the examination time and are not commonly available in clinics. A new algorithm was developed to enable retrospective data-driven simultaneous B₁ correction in VFA R₁ and single-point MPF mapping. The principle of the algorithm is based on different mathematical dependences of B₁-related errors in R₁ and MPF allowing extraction of a surrogate B₁ field map from uncorrected R₁ and MPF maps. To validate the method, whole-brain R₁ and MPF maps with isotropic 1.25 mm³ resolution were obtained on a 3 T MRI scanner from 11 volunteers. Mean parameter values in segmented brain tissues were compared between three reconstruction options including the absence of correction, actual B₁ correction, and surrogate B₁ correction. Surrogate B₁ maps closely reproduced actual patterns of B₁ inhomogeneity. Without correction, B₁ non-uniformity caused highly significant biases in R₁ and MPF (P < 0.001). Surrogate B₁ field correction reduced the biases in both R₁ and MPF to a non-significant level (0.1 ≤ P ≤ 0.8). The described algorithm obviates the use of dedicated B₁ mapping sequences in fast single-point MPF mapping and provides an alternative solution for correction of B₁ non-uniformities in VFA R₁ mapping.

I. Introduction

NON-UNIFORMITY of transmitter radiofrequency (RF) magnetic field (B₁) is commonly acknowledged as a major source of errors in various quantitative MRI methods [1]. B₁ inhomogeneity is dramatically amplified with an increase in magnetic field strength, thus rendering mandatory B₁ field mapping for error correction in certain high-field quantitative imaging applications [1]. One of such applications is the variable flip angle (VFA) T₁ mapping method [2]-[4], which gained broad popularity due to its acquisition speed and the capability to generate high-resolution 3D parametric maps with a clinically acceptable scan time. In the most time-efficient design [3], the VFA method utilizes two spoiled gradient-echo (GRE) images with T₁ and proton density (PD) contrast weightings to compute T₁ and PD parametric maps. However, this technique is extremely sensitive to B₁ errors that may cause up to several hundred percent bias in T₁ estimates obtained using 3 T MRI equipment [4]. Another quantitative MRI method that heavily relies on B₁ correction is single-point macromolecular fraction (MPF) mapping [5], [6], which attracted significant interest over the past years as a fast and robust technique for quantitative imaging of myelin in the brain [7]-[16]. Particularly, MPF demonstrated strong correlations with histologically measured myelin density in animal models [7]-[10] and showed promise as a biomarker of myelin in human studies of multiple sclerosis [11], [12], mild traumatic brain injury [13], and brain development [14]-[16]. The single-point method in its recent synthetic-reference modification [6] reconstructs an MPF map from a single spoiled GRE image with magnetization transfer (MT) contrast and T₁ and PD maps obtained by the two-point VFA technique, thus utilizing only three source images (T₁-, PD-, and MT-weighted) as input data. Although to a lesser degree than T₁, MPF estimates obtained by this method are also subjected to B₁-related errors, which may approach 30-40% in the relative scale for a typical range of B₁ non-uniformities occurring at 3 T [17]. The need for B₁ correction substantially restricts clinical applications of both VFA T₁ and MPF mapping techniques due to examination time penalties associated with a supplementary B₁ mapping scan and limited availability of specialized B₁ mapping sequences on clinical MRI systems.

As substantiated by the detailed analysis provided below, propagation of B₁ errors into MPF and longitudinal relaxation rate R₁ = 1/T₁ is described by principally different mathematical relations. This study aimed to exploit distinctions between the mathematical descriptions of B₁-related errors in R₁ and MPF in order to develop a data-driven algorithm for simultaneous retrospective correction of B₁ inhomogeneity in corresponding parametric maps without additional B₁ field mapping sequences.

II. Theory

A. Theory of Errors in MPF Due to B₁ Inhomogeneity

As a starting point, we use the matrix formulation of a signal equation derived for a spoiled gradient-echo sequence with MT preparative pulse (MT-GRE) based on the two-pool pulsed MT theory [18], [19]. This model considers periodic pulsed saturation applied to the two-pool model with cross-relaxation, where tissue is presented as a system containing free water protons (free pool, identified below by superscript index “F”) and macromolecular protons (bound pool, identified below by superscript index “B”). Evolution of magnetization is analyzed separately during four time intervals comprising the sequence cycle: off-resonance saturation pulse (t_m), delay for spoiling gradient (t_s), readout pulse (t_p), and delay for signal readout and relaxation (t_r). Assuming that the pulsed steady state is established, the resulting equation for the longitudinal magnetization can be written in matrix form as

$$\begin{gathered} {\mathbf{M}}_z=(\mathbf{I}-{\mathbf{E}}_{\mathrm{s}}{\mathbf{E}}_{\mathrm{m}}{\mathbf{E}}_{\mathrm{r}}\mathbf{C}{)}^{-1}\{[{\mathbf{E}}_{\mathrm{s}}{\mathbf{E}}_{\mathrm{m}}(\mathbf{I}-{\mathbf{E}}_{\mathrm{r}})+(\mathbf{I}-{\mathbf{E}}_{\mathrm{s}})]{\mathbf{M}}_{\text{eq}} \\ +{\mathbf{E}}_{\mathrm{s}}(\mathbf{I}-{\mathbf{E}}_{\mathrm{m}}){\mathbf{M}}_{\text{ss}}\}, \end{gathered}$$ (1)

where M_eq is the vector of equilibrium magnetization with elements M₀ (1 – f) and M₀ f, where f is MPF; M_ss is the vector of steady-state longitudinal magnetization with explicit notation given elsewhere [19]; M_z is the vector with components $M_z^{\mathrm{F}}$ and $M_z^{\mathrm{B}}$ corresponding to the longitudinal magnetization immediately before the excitation pulse; I is the unit matrix; the matrix term E_m = exp((R + W)t_m) describes off-resonance saturation by an RF pulse with duration t_m; the terms E_s = exp(Rt_s) and E_r = exp(Rt_r) describe relaxation during delays before (t_s) and after (t_r) an excitation RF pulse; and the diagonal matrix C corresponds to instant rotation of the magnetization $M_z^{\mathrm{F}}$ by an excitation pulse. The relaxation matrix R is defined as:

$$\mathbf{R}=\left[\begin{array}{cc}\hfill -R_1^{\mathrm{F}}-Rf∕(1-f)\hfill & \hfill R\hfill \\ \hfill Rf∕(1-f)\hfill & \hfill -R_1^{\mathrm{B}}-R\hfill \end{array}\right],$$ (2)

where $R_1^{\mathrm{F}}$ and $R_1^{\mathrm{B}}$ are the longitudinal relaxation rates of the pools, and R is the exchange rate constant for MT from the bound to the free pool.

B₁ non-uniformity can be described by the scaling factor c defined as the ratio of actual to nominal B₁ field in a voxel:

$$c=B_{1\mathrm{a}}∕B_{1\text{nom}}.$$ (3)

In the presence of B₁ inhomogeneity, the matrix C describing the effect of the excitation pulse with the nominal flip angle α is expressed as

$$\mathbf{C}=\text{diag}(\cos (c\alpha ),1).$$ (4)

The saturation matrix W is scaled with the squared c:

$$\mathbf{W}=-c^2\text{diag}(W^{\mathrm{F}},W^{\mathrm{B}}),$$ (5)

where W^F,B are the saturation rates for the pools averaged over the saturation pulse duration, which are expressed as

$$W^{\text{F,B}}=\pi {\omega }_{1\text{rms}}^2{g}^{\text{F,B}}(\Delta ,T_2^{\text{F,B}}),$$ (6)

where g^F,B(Δ, $T_2^{\text{F,B}}$) are the spectral line shapes of the pools dependent on their T₂ and offset frequency Δ, and ω_1rms is the root-mean-square nominal saturation power:

$${\omega }_{1\text{rms}}=\frac{{\text{FA}}_{\text{MT}}}{t_{\mathrm{m}}}\frac{\pi }{180}\frac{{\left({\int }_0^{t_m}{b}_1^2(t)\mathrm{d}t\right)}^{1∕2}}{{\int }_0^{t_m}{b}_1(t)\mathrm{d}t},$$ (7)

which is calculated from the flip angle (FA_MT), duration (t_m), and envelope function (b₁(t)) of the saturation pulse [5]. The line shapes g^F,B(Δ, $T_2^{\text{F,B}}$) are commonly approximated by the Lorentzian and SuperLorentzian functions for the free and bound pool, respectively [18], [20]. Note that according to equations (4) - (7), all RF-related variables are defined by the nominal values calculated from the pulse sequence parameters, while their spatial dependence on B₁ field is solely described by the scaling factor c.

Applying the first-order approximation to the exponential terms and using the identity RM_eq = (R + W)M_ss, (1) can be rewritten as:

$${\mathbf{M}}_z\approx (\mathbf{R}+\tau \mathbf{W}+{\text{TR}}^{-1}\text{ln}\mathbf{C}{)}^{-1}{\mathbf{RM}}_{\text{eq}},$$ (8)

where τ = t_m/TR is the duty cycle of the saturation pulse. If the sequence is completely spoiled, the equation for the MT-weighted signal can be obtained from (8):

$$S_{\text{mt}}=A{M}_z^{\mathrm{F}}\sin (c\alpha )\text{exp}(-\text{TE}∕T_2^{\ast }),$$ (9)

where A is the coil reception profile and $M_z^{\mathrm{F}}$ is given by

$$\begin{gathered} M_z^{\mathrm{F}} \\ \approx \frac{M_0(1-f)(R_1^{\mathrm{F}}{R}_1^{\mathrm{B}}+R_1^{\mathrm{F}}R+R_1^{\mathrm{B}}Rf(1-f{)}^{-1}+R_1^{\mathrm{F}}{c}^2\tau W^{\mathrm{B}})}{R_1^{\mathrm{F}}{R}_1^{\mathrm{B}}+R_1^{\mathrm{F}}R+R_1^{\mathrm{B}}Rf(1-f{)}^{-1}} \\ +(R_1^{\mathrm{F}}+Rf(1-f{)}^{-1})c^2\tau W^{\mathrm{B}} \\ +(R_1^{\mathrm{B}}+R+c^2\tau W^{\mathrm{B}})(c^2\tau W^{\mathrm{F}}-{\text{TR}}^{-1}\text{ln}\cos (c\alpha )) \end{gathered}$$ (10)

To obtain explicit equation describing the relationship between the measured MPF and B₁ non-uniformity, we further assume that the relaxation rates of the pools are equal to the observed R₁, $R_1^{\mathrm{F}}=R_1^{\mathrm{B}}=R_1$ that is the standard assumption in the single-point MPF mapping method [5], [6]. After rearrangements, an approximated first-order equation for the observed MT-weighted signal can be expressed as

$$\begin{gathered} S_{\text{mt}}\approx \text{PD}{R}_1\sin (c\alpha )\{R_1-{\text{TR}}^{-1}\text{ln}\cos (c\alpha )+c^2\tau W^{\mathrm{F}} \\ +\frac{fR({\text{TR}}^{-1}\text{ln}\cos (c\alpha )+c^2\tau W^{\mathrm{B}}-c^2\tau W^{\mathrm{F}})}{R+(1-f)(R_1+c^2\tau W^{\mathrm{B}})}{\}}^{-1}, \end{gathered}$$ (11)

where the effective proton density (PD) absorbs the factors of the coil reception profile A, actual proton density M₀(1 – f), and $T_2^{\ast }$ decay:

$$\text{PD}=A{M}_0(1-f)\text{exp}(-\text{TE}∕T_2^{\ast }).$$ (12)

The PD value is identical for all three scans (MT-, PD, and T₁-weighted) acquired in the single-point synthetic reference MPF mapping method [6].

Computation of MPF in the single-point synthetic reference method with unknown B₁ non-uniformity can be approximated as the single-parameter fit of the signal model $S_{\text{mt}}^{\mathrm{m}}$ calculated with the nominal values of RF-related variables (c = 1) and measured by the VFA method values of PD and R₁ containing B₁-related errors (PD_m and R_1m, respectively) to the experimental signal S_mt given by (11):

$$f_{\mathrm{m}}=\text{arg}\text{min}[S_{\text{mt}}(f,\text{PD},R_1,c)-S_{\text{mt}}^{\mathrm{m}}(f_{\mathrm{m}},{\text{PD}}_{\mathrm{m}},R_{1\mathrm{m}},c=1){]}^2,$$ (13)

where f_m is the experimentally measured MPF value containing an error caused by B₁ non-uniformity. To incorporate the effect of B₁ non-uniformity on PD and R₁ measurements, we adopt the earlier results by Helms at al. [4] for the corresponding errors in the two-point VFA method based on a small flip angle approximation:

$${\text{PD}}_{\mathrm{m}}=c\text{PD}$$ (14)

and

$$R_{1\mathrm{m}}=c^{-2}{R}_1.$$ (15)

With (14) and (15), the equation for $S_{\text{mt}}^{\mathrm{m}}$ takes the form:

$$\begin{gathered} S_{\text{mt}}^{\mathrm{m}}\approx c\text{PD}{R}_1\sin \alpha \{R_1-c^2{\text{TR}}^{-1}\text{ln}\cos \alpha +c^2\tau W^{\mathrm{F}} \\ +\frac{c^2{f}_{\mathrm{m}}R({\text{TR}}^{-1}\text{ln}\cos \alpha +\tau W^{\mathrm{B}}-\tau W^{\mathrm{F}})}{R+(1-f_{\mathrm{m}})(c^{-2}{R}_1+\tau W^{\mathrm{B}})}{\}}^{-1}. \end{gathered}$$ (16)

It should be pointed out that the expression for $S_{\text{mt}}^{\mathrm{m}}$ (16) now explicitly depends on c as a result of substitution of (14) and (15) for PD_m and R_1m. Equation (13) can be exactly solved with respect to f_m by equating the analytical signal expressions $S_{\text{mt}}^{\mathrm{m}}=S_{\text{mt}}$ given by (11) and (16). For a small flip angle α, the solution can be greatly simplified by applying the approximations c sin α ≈ sin(cα) and c² ln cos α ≈ ln cos(cα), which follow from the Maclaurin series expansion. Under the above approximations,

$$f_{\mathrm{m}}\approx f\frac{c^2(\tau W^{\mathrm{B}}+R)+R_1}{c^2(c^2\tau W^{\mathrm{B}}+R+R_1)+f(1-c^2)(c^2\tau W^{\mathrm{B}}+R_1)}.$$ (17)

Equation (17) can be further simplified assuming that R₁ ≪ R. This assumption is supported by the experimental data suggesting that R is about 20-40-fold larger than R₁ for most tissues [21]. Additionally, the second term in the denominator of (17) is much smaller than the first term and can be neglected. Accordingly, (17) is reduced to

$$f_{\mathrm{m}}\approx f\frac{\tau W^{\mathrm{B}}+R}{c^2\tau W^{\mathrm{B}}+R}.$$ (18)

Analysis of formulas (17) and (18) results in several important conclusions:

1. The main source of B₁-related errors in single-point MPF mapping [5], [6] is the propagation of B₁ non-uniformity into the spatial variations of the saturation rate W^B.

2. According to (18), the relative error in MPF is almost independent of tissue properties in the brain, since the parameters R and $T_2^{\mathrm{B}}$ are nearly identical for brain tissues and kept constant in the single-point algorithm [5]. The residual tissue dependence associated with the small terms including R₁ and f in (17) is negligible due to the dominant effect of W^B.

3. The errors caused by the effect of B₁ non-uniformity on the excitation flip angle and direct saturation of the free pool (W^F) are negligibly small, since the corresponding terms are eliminated during the derivation of (17).

B. Estimation of B₁ Inhomogeneity From the Mismatch Between MPF and R₁ Measurement Errors

The key idea of the proposed data-driven algorithm is the derivation of the B₁ field non-uniformity factor c from the measured R₁ and MPF values based on the fact that B₁-related errors in these parameters have different functional forms given by (15) and (17). To apply this approach, one needs to find a function defining a relationship between R₁ and MPF in the absence of B₁ errors. For this purpose, we adopt a general linear relaxometry model [22], which describes the longitudinal relaxation rate in brain tissues as a weighted sum of macromolecular and paramagnetic contributions:

$$R_1=R_{1\mathrm{w}}+r_{\mathrm{m}}[\mathrm{M}]+{\sum }_i{r}_{\mathrm{p}i}[\mathrm{P}{]}_i,$$ (19)

where R_1w is the longitudinal relaxation rate of pure water, r_m and [M] are the relaxivity and molar concentration of the macromolecular protons modulating R₁ via cross-relaxation and chemical exchange, and r_pi and [P]_i are the relaxivities and molar concentrations of paramagnetic ions (primarily iron). Equation (19) can be rewritten with inclusion of MPF:

$$R_1=R_{1\mathrm{w}}+r_{\mathrm{m}}[\mathrm{W}]\frac{f}{1-f}+\frac{1}{V(1-f)}{\sum }_i{r}_{\mathrm{p}i}{n}_i,$$ (20)

where [W] is the molar concentration of water protons, V is the tissue volume, and n_i are the molar amounts of paramagnetic substances. Note that the concentration of macromolecular protons in (20) is expressed from the definition of MPF, f = [M]/([M] + [W]), and concentrations [P]_i are defined as the molar amounts n_i per water volume in tissue, which can be approximated as V(1 – f). After rearrangements, (20) can be presented as a linear function in coordinates R₁ and f/(1 – f):

$$R_1=r_0+r_{f}f∕(1-f),$$ (21)

where the coefficients r₀ and r_f absorb all unrelated to MPF contributions. We further assume that the constants r₀ and r_f for the brain can be determined as population-average values from experimental B₁-corrrected measurements, and their tissue-dependent variability is negligible for the purpose of B₁ field estimation. The validity of these assumptions is investigated later.

Measured R_1m and f_m values in the presence of B₁ errors can be substituted into (21) after transforming into the actual values according to the error models given by (15) and (17). For MPF, the explicit recalculation formula is obtained from (17) by solving for f:

$$f\approx f_{\mathrm{m}}\frac{c^2+R(\tau W^{\mathrm{B}}+R_{1\mathrm{m}}{)}^{-1}}{1+R(\tau W^{\mathrm{B}}+R_{1\mathrm{m}}{)}^{-1}-f_{\mathrm{m}}(1-c^2)}.$$ (22)

After substitution of (15) and (22) into (21), the resulting equation can be solved relative to the B₁ field scaling factor c. To emphasize the distinction between the approximated and actual B₁ estimates, we further denote this solution as surrogate B₁ field (B_1s) with the scaling factor c_s, which should be treated as an approximation of the actual field B_1a:

$$c\approx c_{\mathrm{s}}=\sqrt{\frac{r_0(1-f_{\mathrm{m}})+r_{f}P{f}_{\mathrm{m}}}{R_{1\mathrm{m}}(1-f_{\mathrm{m}})-r_f(1-P)f_{\mathrm{m}}}},$$ (23)

where P = R/(R + τ W^B + R_1m). As such, a surrogate B_1s field map can be reconstructed according to (23) from uncorrected R₁ and MPF maps under the assumption of constant regression coefficients r₀ and r_f across the tissues of interest and further used to compute corrected R₁ and MPF maps, similar to an actual B_1a map.

C. Reparameterization of the Direct Saturation Effect

In the single-point MPF mapping algorithm [5], the two-pool model parameters with small tissue-dependent variations are constrained by using the experimentally determined standard whole brain values. Particularly, constant values of R = 19 s⁻¹, $T_2^{\mathrm{B}}=10$ μs, and the product of the observed R₁ and T₂ of the free pool, $R_{1\mathrm{m}}{T}_2^{\mathrm{F}}=0.022$, were used for MPF mapping at 3 T [5]. The latter quantity is field-dependent [5], [23], [24] and needs to be specified for a particular field strength. The product $R_{1\mathrm{m}}{T}_2^{\mathrm{F}}$ is used to estimate $T_2^{\mathrm{F}}$, which determines the saturation rate of the free pool W^F (6). If off-resonance saturation is applied at relatively high offset frequencies (4-7 kHz) as suggested for the optimal fast MPF mapping protocol design [5], the effect of W^F is negligibly small. However, careful consideration of direct saturation is required for applications of the proposed method with clinical MT-weighted sequences, where offset frequencies are typically in a range 1-1.5 kHz [14]. If measured in the presence of large B₁ non-uniformity, R_1m may artificially inflate the errors associated with the direct saturation term. As a more robust approach to correct for the direct saturation effect, we propose an estimate based on the global correlation given by (21), such that the constraint for $T_2^{\mathrm{F}}$ takes the form

$$T_2^{\mathrm{F}}=0.022(1-f_{\mathrm{m}})(r_0(1-f_{\mathrm{m}})+r_f{f}_{\mathrm{m}}{)}^{-1}$$ (24)

for 3 T magnetic field. The main advantage of this parameterization is that an estimate of $T_2^{\mathrm{F}}$ and, therefore, calculated W^F, becomes much less dependent on B₁ inhomogeneity. More detailed comparison between both parameterization approaches is given in the Results section.

III. Methods

A. Simulations

To investigate R₁ and MPF measurement errors caused by B₁ inhomogeneity and the effect of their correction by the proposed algorithm, PD-, T₁-, and MT-weighted signal intensities were simulated using the matrix two-pool pulsed steady-state model (1)-(7) for a series of B₁ field scaling factors c. Then R₁, PD, and MPF values were computed using the single-point synthetic reference algorithm [6] at the nominal flip angles of RF pulses in all sequences. The resulting R₁ and MPF estimates (R_1m and f_m) containing B₁-related errors were used to compute the surrogate B₁ field scaling factor c_s according to (23). To simulate the effect of surrogate B₁ field correction, R₁, PD, and MPF values were refitted using the same procedure after scaling all flip angles by the factor c_s. The models with both the previous ($R_{1\mathrm{m}}{T}_2^{\mathrm{F}}=0.022$) and new (24) parameterization of the direct saturation effect were tested. Additionally, to assess the validity of approximations applied in the algorithm, R_1m and f_m values were generated using the analytical expressions (15) and (17).

Simulations were carried out for the two sets of the two-pool model parameters corresponding to the average values in the brain white matter (WM) and gray matter (GM). The following parameters were used: MPF = 12.5%, R₁ = 0.95 s⁻¹, R = 19 s⁻¹, $T_2^{\mathrm{F}}=23$ ms, and $T_2^{\mathrm{B}}=10$ μs for WM and MPF = 6.5%, R₁ = 0.63 s⁻¹, R = 19 s⁻¹, $T_2^{\mathrm{F}}=35$ ms, and $T_2^{\mathrm{B}}=10$ μs for GM. Pulse sequence parameters were the same as those detailed in the experimental protocols outlined below. The coefficients r₀ and r_f were set according to their experimentally measured values as detailed below. The effect of uncertainties in the constants r₀ and r_f was assessed by varying their values in a range of two standard deviations (SD) of their experimental measurements. Simulations were performed using custom-written C-language software.

B. Study Participants

The study was approved by the Institutional Review Board and all participants provided written informed consent. 3D R₁ and MPF maps were obtained from 11 healthy volunteers (7 females; mean age ± SD = 44.8 ± 13.8 years; age range 26-65 years) using an optimized high-resolution protocol [6], [17] on a research MRI scanner. Additionally, one participant (male, 53-year-old) was scanned with a lower-resolution protocol on a clinical MRI unit using standard manufacturer’s sequences.

C. MRI Acquisition

Images from 11 participants were acquired on a 3 T research MRI scanner (Achieva; Philips Medical Systems, Best, Netherlands) with an eight-channel head coil and modified software for optimal performance of imaging sequences. Software modifications included optimal settings of the saturation pulse [5] in the MT-weighted sequence and optimal RF and gradient spoiling [25]. The 3D MPF mapping protocol was implemented according to the single-point synthetic reference method [6] and included the PD-weighted (TR = 21 ms, α = 4°, scan time 4 min 21 s), T₁-weighted (TR = 21 ms, α = 25°, scan time 4 min 21 s), and MT-weighted (TR = 28 ms, α = 10°, scan time 5 min 48 s) imaging sequences. Off-resonance saturation in the MT-weighted sequence was achieved by the single-lobe-sinc pulse with Gaussian apodization, Δ = 4 kHz, FA_MT = 560°, and t = 12 ms. These settings resulted in ω_1rms = 940 rad/s, τ = 0.42, and W^B = 18.1 s⁻¹. All images were acquired in the sagittal plane with dual-echo readout (TE₁/TE₂ = 2.3 ms/6.9 ms), FOV = 240 × 240 × 180 mm³, and actual voxel size of 1.25 × 1.25 × 1.25 mm³ interpolated to 0.63 × 0.63 × 0.63 mm³ after zero-padded Fourier transform. Actual flip-angle imaging (AFI) B₁ maps [26] were obtained with TR₁/TR₂/TE = 40/160/2.3 ms, α = 60°, scan time of 3 min 26 s, and voxel size of 2.5 × 2.75 × 5.0 mm³ interpolated to 0.63 × 0.63 × 0.63 mm³. Parallel imaging (SENSE) [27] was used for all scans in two phase encoding directions with acceleration factors 1.5 and 1.2.

A prototype clinical protocol was tested on a single study participant. Data were obtained using a 3 T clinical scanner (Ingenia; Philips Medical Systems, Best, Netherlands) with a 16-channel head coil. The fast MPF mapping technique was built on unmodified manufacturer’s sequences. The PD-weighted (TR = 20 ms, α = 3°, scan time 2 min 38 s), T₁-weighted (TR = 20 ms, α = 20°, scan time 2 min 38 s), and MT-weighted (TR = 70 ms, α = 10°, scan time 9 min 12 s) images were acquired. The preset manufacturer’s three-lobe-sinc pulse with Gaussian apodization, Δ = 1.1 kHz, FA_MT = 520°, and t_m = 34 ms was used for off-resonance saturation. These parameters corresponded to ω_1rms = 612 rad/s, τ = 0.49, and W^B = 14.9 s⁻¹. Images were acquired in the axial plane with TE = 2.3 ms, FOV = 240 × 200 × 240 mm³, and actual voxel size of 1.25 × 1.5 × 4.0 mm³ interpolated to 1.0 × 1.0 × 2.0 mm³ after zero-padded Fourier transform. Additionally, an AFI B₁ map (scan time 4 min 22 s) was obtained with the sequence parameters same as above and voxel size of 2.5 × 2.75 × 5.0 mm³ interpolated to 1.0 × 1.0 × 2.0 mm³.

D. Image Processing and Analysis

Eleven datasets obtained with the optimal high-resolution protocol were processed to determine the mean algorithm constants r₀ and r_f and compare B₁ field correction options including the absence of correction, correction with an actual B₁ map (B_1a), and surrogate B₁ field (B_1s) correction. A single dataset acquired using the prototype clinical protocol was processed with the above B₁ correction options to demonstrate the compatibility of the proposed algorithm with unmodified manufacturer’s MRI sequences.

Prior to map reconstruction, individual echo images in the high-resolution datasets were averaged to increase SNR [28]. Extracranial tissues were removed from source images by applying a brain mask created from the PD-weighted image using the brain extraction tool [29] available in FSL software (FMRIB Software Library; http://www.fmrib.ox.ac.uk/fsl). R₁ maps were computed using the analytical solution for the two-point VFA method [3]. MPF maps were reconstructed using the single-point synthetic reference algorithm [6] with $T_2^{\mathrm{F}}$ calculated according to (24) and the fixed values of R = 19 s⁻¹ and $T_2^{\mathrm{B}}=10$ μs. MPF fit was performed for the voxels with the non-zero saturation effect defined as (S_ref-S_mt)/S_ref > 0.01, where S_ref is the synthetic reference image intensity calculated from uncorrected PD_m and R_1m maps. Surrogate B_1s maps were calculated from uncorrected R_1m and MPF (f_m) maps according to (23) with the same coefficients r₀ and r_f for all datasets. Their values were determined from the regression analysis of R₁ and MPF measurements in segmented brain tissues as detailed below. Noise and residual tissue-dependent variations in both actual and surrogate B₁ maps were minimized by applying a 3D trimmed mean filter with 12-voxel radius and field non-uniformity thresholds 0.3 < c < 2.0. All reconstruction procedures were carried out using custom-written C-language software.

High-resolution MPF maps reconstructed with actual B₁ correction were segmented into three tissue classes (WM, GM, and a mixed class containing voxels with partial volume of cerebrospinal fluid (PVCSF)) similar to the procedure described earlier [6], [17]. Segmentation was performed using the automated tool FAST [30] in FSL software with the Markov random field weighting parameter 0.25 and tissue-specific priors of 12% for WM, 6% for GM, and 1% for PVCSF. Examples of segmented tissue masks for similar datasets can be found elsewhere [6], [17]. Mean R₁ and MPF values calculated within each tissue mask for each reconstruction option were used in subsequent analyses.

To examine agreement between actual and surrogate B₁ corrections on the voxel level, voxel-based Bland-Altman plots were constructed for each individual dataset from the B₁ scaling factor (c or c_s) maps and corrected R₁ and MPF maps. The voxel-based Bland-Altman plots were computed as two-dimensional histograms of the mean value and difference for each pair of voxel intensities with the bin sizes of 0.004, 0.02%, and 0.004 s⁻¹ for B₁, R₁, and MPF, respectively. Additionally, relative error maps for B₁, R₁, and MPF were reconstructed as the difference between actual and surrogate B₁ corrections divided by the actual map.

E. Statistical Analysis

The coefficients r₀ and r_f were determined from linear regression of R₁ on MPF/(1-MPF) values in segmented brain tissues measured from the datasets reconstructed with actual B₁ correction. The effect of B₁ field corrections on R₁ and MPF in each tissue class was examined using repeated-measures ANOVA with three levels of the within-subject factor (no correction, actual B₁ correction, and surrogate B₁ correction). Greenhouse-Geisser correction for non-sphericity was applied to the degrees of freedom. Significance of the biases between B₁ correction options was assessed using post-hoc Tukey honest significant difference tests. Bland-Altman plots were used to investigate an agreement between corrected and uncorrected R₁ and MPF values across the brain tissues. The limits of agreement were calculated as the mean difference ±1.96SD of the mean difference. Agreement between different reconstructions was also characterized by the within-subject coefficient of variation (CoV). Two-tailed tests were used with the significance level of P < 0.05. Statistical analysis was carried out in Statistica (StatSoft Inc, Tulsa, OK, USA) software.

IV. Results

A. Estimation of the Algorithm Constants

Linear regression across the mean parameter measurements based on B_1a-corrected maps in the brain tissues (WM, GM, and PVCSF) for all subjects (Fig. 1) showed strong linear relationship between R₁ and the MPF-derived quantity f/(1 – f), which was described by the empirical equation R₁ = 0.31 + 4.47 f/(1 – f) (r = 0.998, P < 0.001). Regression analysis of individual datasets resulted in the following mean values ± SD of the regression coefficients: r₀ = 0.31 ± 0.01 (range 0.29-0.32) and r_f = 4.47 ± 0.13 (range 4.29-4.76). Based on these results, the rounded values of the constants r₀ = 0.3 and r_f = 4.5 were uniformly used in subsequent reconstruction of surrogate B₁ field maps and simulations.

Fig. 1.

Linear regression of R₁ on MPF/(1-MPF) measurements in segmented brain tissues based on 11 high-resolution datasets reconstructed with actual B₁ correction. Black, red, and blue points correspond to the PVCSF, GM, and WM tissue classes, respectively.

B. Simulations

Simulated dependences of R₁ and MPF in WM and GM on the B₁ scaling factor c in the absence of B₁ correction and after surrogate B₁ field correction for the optimized protocol are presented in Fig. 2. Relative errors before and after surrogate B₁ correction and the effect of variations in the algorithm coefficients r₀ and r_f are illustrated in Fig. 3. In agreement with the literature [4], B₁-related errors result in an inverse quadratic dependence of R₁ with a four-fold overestimation (300% relative error) at c = 0.5 (Fig. 2(a), 3(a), and 3(b)). The B₁ dependence of MPF is nearly linear with the errors being in a range from −25% to +35% for c varying from 1.5 to 0.5 (Fig. 2(b), 3(c), and 3(d)). Analytical formulas for B₁-related errors (15) and (17) provide accurate approximations of the numerical simulations (Fig. 2). It is also important to emphasize that the relative errors for WM and GM appear very similar (Fig. 3), thus confirming their negligible dependence on the tissue parameters in accordance with the analytical approximations (15) and (18).

Fig. 2.

Simulated dependences of R₁ (a) and MPF (b) measurements on the B₁ scaling factor before (solid and dot lines) and after (dash lines) correction using the surrogate B₁ field estimation algorithm for the sequence parameters from the high-resolution research protocol and tissue parameters corresponding to WM (black lines) and GM (gray lines). Uncorrected R₁ (a) and MPF (b) values were simulated either numerically (solid lines) or analytically using (15) and (17) (dot lines). Thin solid lines correspond to the R₁ (a) and MPF (b) values in the absence of B₁ inhomogeneity (c = 1).

Fig. 3.

Simulated relative errors of surrogate B₁ field correction in R₁ (a, b) and MPF (c, d) caused by uncertainties in the algorithm coefficients r₀ (a, c) and r_f (b, d). Relative errors in each parameter are plotted for WM (solid lines) and GM (dash lines) as functions of the B₁ scaling factor for the following settings: the absence of correction (black lines), correction with exact r₀ and r_f values (red lines), and correction with either r₀ or r_f deviating by ±2SD from its exact value (blue lines). Simulations are presented for the sequence parameters from the high-resolution research protocol.

Application of the surrogate field correction algorithm with the accurate values of the coefficients r₀ and r_f results in nearly complete elimination of B₁-related errors from R₁ and MPF (Fig. 2 and 3). Potential errors of the algorithm associated with inaccuracy of the constants r₀ and r_f were simulated under assumption of their variations within the range of ±2SD of the experimental measurements given above. The corresponding residual errors in R₁ and MPF appeared rather small (<10% for R₁ and <5% for MPF, Fig. 3), being substantially less than B₁-related errors without correction for a wide range of B₁ non-uniformities. It should be noted that if either r₀ or r_f deviates from its exact value, the corresponding error in the surrogate B₁ field estimate c_s (23) becomes dependent on the actual field scaling factor c. These dependences are nearly linear and rather weak (data are not shown). For the simulations presented in Fig. 3, the absolute error in c_s varies from ±0.01 at c ≈ 0.5 to ±0.05 at c ≈ 1.5. The errors in c_s translate differently into the corresponding errors in R₁ and MPF. For R₁, the relative errors are nearly constant across the entire range of c_s values (Fig. 3(a) and 3(b)), whereas the MPF errors replicate the linear trend in c_s (Fig. 3(c) and 3(d)). Such a behavior of errors can be easily proven by the analysis of approximated equations (15) and (22). Uncertainties in the coefficients r₀ or r_f may introduce additional errors in corrected maps as compared to the uncorrected case, if B₁ field is uniform. As seen in Fig. 3, the errors without surrogate B₁ correction in the vicinity of c = 1 (c in a range of 0.95-1.05) can be smaller than the errors caused by variations in r₀ or r_f. Accordingly, if uncertainties in the coefficients r₀ and r_f are assumed to vary within ±2SD, the simulations suggest that the algorithm is expected to improve accuracy of both R₁ and MPF measurements for B₁ inhomogeneities exceeding ±5%. Conversely, the utility of the proposed method may be limited in the applications with a smaller range of B₁ variations.

Fig. 4 illustrates the rationale for reparameterization of the direct saturation effect described above. Simulations were performed with the listed above parameters of the clinical sequence, which is characterized by a relatively small offset frequency and, therefore, more prone to the errors associated with direct saturation of the free pool. The model based on the constant product $R_{1\mathrm{m}}{T}_2^{\mathrm{F}}$ results in a non-monotonic behavior of B₁-related errors in MPF (Fig. 4(b)), which translates in inaccurate estimation of surrogate B₁ field (Fig. 4(c)). Correspondingly, correction of both R₁ and MPF with this parameterization appears substantially inaccurate for large B₁ non-uniformities (Fig. 4(a) and 4(b)). In contrast, new parameterization (24) eliminates this problem and results in both close agreement between the actual and surrogate B₁ non-uniformity factors (Fig. 4(c)) and accurate correction of R₁ and MPF values in a wide range of B₁ inhomogeneities (Fig. 4(a) and 4(b)).

Fig. 4.

Simulated dependences of the uncorrected (solid and short-dash lines) and corrected with surrogate B₁ field (dash and dot lines) R₁ (a) and MPF (b) measurements and the surrogate B₁ field scaling factor c_s (c) on the actual B₁ scaling factor for the sequence parameters from the prototype clinical protocol and tissue parameters corresponding to WM (black lines) and GM (gray lines). Simulations are presented for the two parameterizations of the direct saturation effect in the single-point MPF fitting algorithm based on either MPF-related formula (24) or the constant product $R_1{T}_2^{\mathrm{F}}$. The parameterization with (24) results in the uncorrected MPF (b) and c_s (c) dependences plotted by solid lines and the corrected R₁ (a) and MPF (b) dependences plotted by dash lines. The parameterization with the constant product $R_1{T}_2^{\mathrm{F}}$ results in the uncorrected MPF (b) and c_s (c) dependences plotted by short-dash lines and the corrected R₁ (a) and MPF (b) dependences plotted by dot lines. Thin solid lines correspond to the R₁ (a) and MPF (b) values in the absence of B₁ inhomogeneity (c = 1).

C. Experimental Comparison Between Actual and Surrogate B₁ Corrections

An example of the application of the surrogate B₁ field correction algorithm to a dataset obtained with the high-resolution 3D R₁ and MPF mapping protocol is presented in Fig. 5. Uncorrected R₁ and MPF (Fig. 5(a) and 5(b)) demonstrate large B₁-dependent variations, as seen in the difference images (Fig. 5(i) and 5(j)) obtained by their subtraction from the maps reconstructed with actual B_1a correction (Fig. 5(c) and 5(d)). The surrogate B_1s map (Fig. 5(h)) reconstructed from uncorrected R₁ and MPF maps shows a very similar pattern of B₁ non-uniformity to that observed in the actual B_1a map (Fig. 5(g)). The R₁ and MPF maps corrected by applying surrogate B_1s field (Fig. 5(c) and 5(d)) demonstrate close agreement with the maps reconstructed using actual B_1a correction, as evidenced by their subtraction images (Fig. 5(k) and 5(l)), which show minor residual variability.

Fig. 5.

Example application of the surrogate B₁ field correction algorithm to high-resolution 3D R₁ and MPF maps. 3D cross-sections show uncorrected R₁ (a) and MPF (b) maps; corrected with actual B₁ (B_1a) field R₁ (c) and MPF (d) maps; corrected with surrogate B₁ (B_1s) field R₁ (e) and MPF (f) maps; B_1a (g) and B_1s (h) field maps; subtraction images between the corrected with actual B_1a field and uncorrected R₁ (i) and MPF (j) maps; subtraction images between the corrected with actual B_1a and surrogate B_1s fields R₁ (k) and MPF (l) maps; difference between actual and surrogate B₁ maps (m); maps of residual percentage errors in R₁ (n) and MPF (o) after surrogate B₁ correction; and a percentage error map of surrogate B₁ field estimation (p).

More detailed examination of the residual errors of surrogate B₁ correction using percentage difference maps (Fig. 5(n), 5(o), and 5(p)) suggests that local discrepancies in R₁ and MPF (Fig. 5(n) and 5(o)) closely reproduce the patterns of the difference between the actual and surrogate B₁ scaling factors (Fig. 5(m)) but have different quantitative scales. Local percentage errors across the brain parenchyma varied in ranges of about ±10-12% for R₁ (Fig. 5(n)), ±2-3% for MPF (Fig. 5(o)), and ±5-6% for B₁ (Fig. 5(p)). These errors appeared in good agreement with the simulation results presented in Fig. 3. Larger negative errors (up to −20% for R₁, −6% for MPF, and −10% for B₁) were observed at certain locations on the brain boundary and around the frontal pole. The patterns of local errors exemplified in Fig. 5 were consistent across all datasets. Anatomic locations of the most discrepant regions suggest that the largest errors of surrogate B₁ correction may be associated with substantial non-uniformities of B₀ field, proximity of large blood vessels, or sub-voxel misregistration between high-resolution source images and a low-resolution actual B₁ map.

Voxel-wise agreement between actual and surrogate B₁ correction is illustrated by the voxel-based individual Bland-Altman plots for the B₁ field scaling factor, MPF, and R₁ (Fig. 6 and Supplementary Material), and the summary of their analysis given in Table I. In all datasets, Bland-Altman analysis showed no significant bias and relatively narrow limits of agreement (Table I). Both limits of agreement and voxel-based coefficients of variations (Table I) suggest that MPF is characterized by the smallest residual correction errors, followed by B₁ and R₁. In the relative scale, the voxel-based limits of agreement indicate that residual deviations for 95% of voxels do not exceed ±11% for R₁, ±3% for MPF, and ±6% for the B₁ scaling factor. These estimates are in close correspondence with the observations from percentage error maps (Fig. 5) and simulations (Fig. 3).

Fig. 6.

Example voxel-based Bland-Altman plots for a single study participant illustrating an agreement between the actual and surrogate B₁ field scaling factors (a), the MPF maps corrected with actual and surrogate B₁ field (b), and the R₁ maps corrected with actual and surrogate B₁ field (c). Solid and dashed lines correspond to the mean difference and limits of agreement calculated from all non-zero voxels in the 3D maps. Grayscale (from white to black) represents voxel density ranges 0–0.15% for B₁ maps, 0–0.02% for MPF maps, and 0–0.04% for R₁ maps.

TABLE I.

Mean Statistical Metrics for Voxel-Based Bland-Altman Plots Comparing Actual and Surrogate B₁ Field Correction

Metric	B1 (c or cs)	MPF (%)	R1 (s−1)
B1a mean	0.897±0.017	7.90±0.46	0.712±0.014
B1s mean	0.896±0.018	7.90±0.45	0.710±0.013
Bias	0.001±0.006	0.00±0.02	0.002±0.009
SD of Bias	0.028±0.003	0.11±0.01	0.040±0.004
Limits of	−0.055±0.009	−0.22±0.04	−0.077±0.012
Agreement	0.056±0.006	0.23±0.03	0.081±0.011
Voxel-based CoV (%)	2.23±0.26	1.14±0.10	4.33±0.42

Note: All data are presented as the mean ± SD across 12 study participants.

Mean R₁ and MPF values in segmented brain tissues across 11 study participants and statistical comparisons between the parameter measurements obtained with actual and surrogate B₁ correction and without correction are presented in Table II. Bland-Altman plots exemplifying the effect of surrogate B₁ field correction on R₁ and MPF measurements in brain tissues are shown in Fig. 7. In the absence of B₁ correction, the mean R₁ and MPF values in WM, GM, and PVCSF demonstrate highly-significant biases relative to the measurements corrected using the actual B₁ maps (Table II, Fig. 7(a) and 7(b)). Application of the surrogate B₁ field correction algorithm reduced the biases in both R₁ and MPF to a non-significant level for all tissues (Table II, Fig. 7(c) and 7(d)). Surrogate B₁ correction also dramatically reduced the coefficients of variation (Table II).

TABLE II.

Comparisons Between Corrected With Actual and Surrogate B₁ Fields and Uncorrected R₁ and MPF in Segmented Brain Tissues

	R1(s−1)			MPF (%)
Value	WM	GM	PVCSF	WM	GM	PVCSF
Uncorrected mean ± SD	1.187±0.051	0.809±0.022	0.526±0.027	13.04±0.39	6.71±0.13	2.08±0.10
B1a corrected mean ± SD	0.930±0.021	0.626±0.009	0.384±0.012	12.34±0.34	6.33±0.11	1.92±0.09
B1s corrected mean ± SD	0.918±0.019	0.622±0.008	0.390±0.011	12.32±0.35	6.32±0.12	1.93±0.09
Bias B1a–Uncorrected ± SD	−0.257±0.042	−0.183±0.023	−0.143±0.018	−0.70±0.10	−0.38±0.04	−0.16±0.01
(P)	(<0.001)	(<0.001)	(<0.001)	(<0.001)	(<0.001)	(<0.001)
Bias B1a–B1s ± SD	0.012±0.009	0.003±0.008	−0.007±0.007	0.02±0.03	0.01±0.02	−0.01±0.01
(P)	(0.49)	(0.83)	(0.35)	(0.57)	(0.57)	(0.09)
CoV B1a–Uncorrected (%)	17.4	18.2	22.4	3.9	4.2	5.7
CoV B1a–B1s (%)	1.2	1.0	1.8	0.2	0.2	0.4
ANOVA F (P)	424 (<0.001)	664 (<0.001)	569 (<0.001)	551 (<0.001)	956 (<0.001)	1140 (<0.001)

Fig. 7.

Bland-Altman plots comparing R₁ (a, b) and MPF (c, d) measurements in segmented brain tissues between reconstructions with actual B_1a field correction and without correction (a, c) and between reconstructions with actual B_1a and surrogate B_1s field corrections (b, d). Black, red, and blue colors correspond to the PVCSF, GM, and WM tissue classes, respectively. Solid and dash lines indicate mean differences (MD) and limits of agreement (LA).

D. Compatibility With Clinical MRI Sequences

Application of the surrogate B₁ correction algorithm to the R₁ and MPF mapping dataset obtained using unmodified manufacturer’s pulse sequences is demonstrated in Fig. 8. The surrogate B_1s field map (Fig 8(a)) shows substantial similarity to the actual B_1a distribution, as seen in the subtraction B_1a - B_1s image (Fig. 8(f)). The only notable distinction is some B₁ underestimation by the surrogate field map in the proximity to paranasal sinuses (Fig. 8(a) and 8(f)), which may be attributed to high B₀ inhomogeneity in this area. Despite this discrepancy, large B₁-dependent deviations in uncorrected MPF (Fig. 8(b) and 8(g)) and R₁ (Fig. 8(d) and 8(i)) maps are effectively eliminated by surrogate B_1s correction (Fig. 8(c), 8(e), 8(h), and 8(j)). Corresponding subtraction images calculated as the difference between the maps reconstructed with actual B_1a and surrogate B_1s corrections (Fig. 8(h) and 8(j)) demonstrate remarkable uniformity with minor propagation of the above-mentioned artefact into the R₁ map (Fig. 8(j)).

Fig. 8.

Application of the surrogate B₁ field correction algorithm to the R₁ and MPF maps obtained using clinical imaging sequences. A cross-section at the midbrain level shows the surrogate B_1s map (a); uncorrected (b) and corrected with B_1s field (c) MPF maps; uncorrected (d) and corrected with B_1s field (e) R₁ maps; subtraction image between the actual B_1a and surrogate B_1s maps (f); subtraction image between the corrected with actual B_1a field and uncorrected MPF maps (g); subtraction image between the MPF maps corrected with actual B_1a and surrogate B_1s fields (h); subtraction image between the corrected with actual B_1a field and uncorrected R₁ maps (i); and subtraction image between the R₁ maps corrected with actual B_1a and surrogate B_1s fields (j).

V. Discussion

This study introduces a new data-driven retrospective algorithm for correction of B₁-related errors in quantitative imaging. The described algorithm uses actually measured VFA R₁ and single-point MPF maps to derive an approximated B₁ field map based on the rigorous analytical theory. The presented results indicate that the application of surrogate B₁ field enables accurate correction of R₁ and MPF maps across a wide range of B₁ non-uniformities and native R₁ and MPF values in tissues.

The primary area of use of the described algorithm is B₁ correction in the single-point MPF mapping method [5], [6]. This method becomes increasingly popular over the past years as a fast and reliable tool for quantitative myelin mapping in pre-clinical and clinical neuroscience [7]-[17], [23], [24]. The need for correction of B₁-related errors constitutes the major obstacle in routine applications of fast MPF mapping, especially in high magnetic fields. While not as large as errors in R₁, these errors require correction even at the level of global brain tissue analysis due to a statistically significant bias they introduce [17]. The common practice in current fast MPF mapping applications in magnetic fields of 3 T and higher is to map B₁ field using specialized sequences, which typically add 2-5 minutes to the examination time, if a B₁ map is required to provide continuous 3D whole-brain coverage and sufficient resolution for correction of quantitative MPF and R₁ maps. Also, the majority of advanced B₁ mapping techniques are based on customized pulse sequences and unavailable to users of routine clinical MRI equipment. Retrospective correction of B₁ non-uniformity during post-processing provides an attractive alternative solution of the above problem. While no post-processing method has been proposed to specifically correct errors caused by B₁, more general non-uniformity correction algorithms based on biasfield reconstruction can be applied to quantitative parametric maps [31]-[33]. If B₁ distribution is the main source of low-frequency intensity variations, such as in R₁ maps, bias field can be viewed as a proxy of B₁ field [31], [32]. However, a general limitation of all bias field correction approaches is their non-quantitative nature. Particularly, the earlier bias field correction technique for R₁ mapping [31] relies on the assumption of unbiased global B₁ calibration, which is not always correct, as indicated by the systematic deviation of whole-brain B₁ values from unity (c ≈ 0.9, Table I) found in the present study. Another approach [32] is based on the assumption of a constant T₁ in a reference tissue, which is used to extract global flip angle calibration. This assumption may not hold due to tissue changes caused by age or disease and variability in the definition of reference anatomic regions. In contrast to alternative approaches, the proposed surrogate B₁ correction algorithm eliminates the need for acquisition of separate B₁ maps and provides unbiased quantitative R₁ and MPF measurements based on a simple post-processing procedure.

Besides MPF mapping, the described algorithm demonstrated reasonable performance in correction of R₁ maps. Accordingly, it potentially can be used to correct B₁-related errors in R₁ maps alone, if R₁ or T₁ is a primary parameter of interest. In such a prospective application, an MT-weighted image can provide supplementary data for reconstruction of a surrogate B₁ field map instead of any B₁ mapping sequence. An attractive feature of this technique is the broad availability of MT-weighted sequences on clinical MRI systems. Additionally, the described algorithm potentially can be combined with multi-parameter quantitative MT techniques utilizing VFA data for R₁ estimation [18], [19], [34] and an empirical approach to quantify MT saturation, R₁, and PD using three GRE source images [35].

Accuracy of the described algorithm depends on the correct values of the regression coefficients r₀ and r_f, which should match particular experimental settings and an MPF reconstruction model. Since R₁ strongly depends on magnetic field strength, whereas MPF is independent of magnetic field [23], [24], the values of r₀ and r_f are expected to have a similar to R₁ field dependence. To a lesser extent, the algorithm constants may be influenced by certain instrumental factors not accounted by the VFA and MPF reconstruction models, such as imperfect spoiling [25] or finite duration of excitation pulses [36]. Due to variations in the sequence design between MRI systems of different manufacturers, the optimal values of r₀ and r_f may be platform-dependent. The coefficients r₀ and r_f may also be affected by some modifications of the MT signal model. Examples of such modifications include different assumptions about R₁ of macromolecular protons [37], [38], correction of bi-exponential biases in R₁ and MPF [39], inclusion of a non-exchangeable water compartment [40], and different parameterizations to constrain $T_2^{\mathrm{F}}$ discussed above. Until more experience is gained in applications of the surrogate B₁ correction method, it would be advisable to measure the coefficients r₀ and r_f for a particular MRI system, protocol, and reconstruction software in a pilot series of experiments with actual B₁ mapping as a gold standard.

Certain limitations of the surrogate B₁ field mapping algorithm should be noted. First, an object under investigation must exhibit a significant MT effect enabling a non-zero estimate of MPF. Consequently, the method cannot be used for tissues with a negligible MT effect, such as cerebrospinal fluid or fat [41], as well as for fluid-filled phantoms. Second, the algorithm may result in incorrect surrogate B₁ field estimation in certain tissues, for which the relationship between R₁ and MPF strongly deviates from the linear form (21). This situation may occur due to substantial T₁ shortening caused by the paramagnetic effect, particularly in enhancing lesions after administration of a contrast agent, hemorrhages, or tissues with excessive iron deposition. For such special cases, additional post-processing approaches may be needed, for example, lesion masking followed by interpolation of surrogate B₁ field. Third, for clinical MT sequences with low saturation offset frequency, some influence of B₀ inhomogeneities may be expected. While the effect of B₀ on MPF measurements was demonstrated to be negligibly small for optimal protocol settings [17], it may be more pronounced for clinical MT sequences in the anatomic regions with large B₀ non-uniformity, particularly, in the brain areas proximal to paranasal sinuses or in the presence of metallic implants. More research is needed to evaluate practical importance of this potential limitation. Fourth, the MT model used in this study does not explicitly include the dependence of R₁ on cross-relaxation parameters [39], [42], which may affect both R₁ [39], [42] and MPF [39] estimates. Theoretically, this effect can be incorporated into the proposed algorithm in future studies with the use of recalculation formulas described elsewhere [39]. Finally, the application of the described method to non-brain tissues remains a topic of future studies. Such applications, while generally possible, would require knowledge of the two-pool MT model parameters and coefficients r₀ and r_f for particular organs and tissues of interest.

In conclusion, this study demonstrated and validated a new data-driven approach for accurate correction of errors caused by B₁ field non-uniformity in VFA R₁ and single-point MPF mapping of the human brain. The described surrogate B₁ field correction algorithm obviates the use of dedicated B₁ mapping sequences in fast MPF mapping, thus enhancing time-efficiency and availability of this increasingly popular quantitative myelin imaging technology. This algorithm may also provide an alternative solution for correction of B₁ non-uniformities in VFA R₁ mapping applications by replacing a B₁ mapping sequence with an MT-weighted scan. Future research is needed to expand the applicability of the surrogate B₁ field correction method to non-brain tissues.