Confounder-Corrected T1 Mapping in the Liver through Simultaneous Estimation of T1, PDFF, R2* and B1+ in a Single Breath-hold Acquisition

Abstract

Purpose:

Quantitative volumetric T1 Mapping in the liver has the potential to aid in the detection, diagnosis, and quantification of liver fibrosis, inflammation and spatially resolved liver function. However, accurate measurement of hepatic T1 is confounded by the presence of fat and inhomogeneous B1+ excitation. Furthermore, scan time constraints related to respiratory motion require tradeoffs of reduced volumetric coverage and/or increased acquisition time. This work presents a novel 3D acquisition and estimation method for confounder-corrected T1 measurement over the entire liver within a single breath-hold through simultaneous estimation of T1, fat and B1+.

Theory and Method:

The proposed method combines chemical shift encoded MRI and variable flip angle MRI with a B1+ mapping technique to enable confounder-corrected T1 mapping. The method was evaluated theoretically and demonstrated in both phantom and in vivo acquisitions and 1.5T and 3.0T. At 1.5T, the method was evaluated both pre- and post- contrast enhancement in healthy volunteers.

Results:

The proposed method demonstrated excellent linear agreement with reference inversion-recovery spin-echo based T1 in phantom acquisitions at both 1.5T and 3.0T, with minimal bias (5.2ms and 45ms, respectively) over T1 ranging from 200-1200ms. In vivo results were in general agreement with reference saturation-recovery based 2D T1 maps (SMART₁Map, GE Healthcare).

Conclusion:

The proposed 3D T1 mapping method accounts for fat and B1+ confounders through simultaneous estimation of T1, B1+, PDFF and R2*. It demonstrates strong linear agreement with reference T1 measurements, with low bias and high precision, and can achieve full liver coverage in a single breath-hold.

1. Introduction

Fibrosis is the histological change most predictive of disease progression to clinically relevant outcomes (1). Untreated, fibrosis can progress to cirrhosis, a precancerous condition increasing the risk of hepatocellular carcinoma (HCC). Together, cirrhosis and HCC account for roughly 2 million deaths globally each year, making them the fifth leading cause of death worldwide (1). Earlier detection, diagnosis, and quantification of liver fibrosis, before the disease progresses to cirrhosis, would facilitate targeted interventions and treatment monitoring.

While MR elastography is the most accurate imaging technique for non-invasive measurement of clinical liver fibrosis, T1 is emerging as a new quantitative biomarker of liver fibrosis (2-5). Specifically, recent studies have linked increased levels of fibrosis and inflammation to the lengthening of native T1 in the liver (4,6). Furthermore, there is emerging evidence that contrast-enhanced T1 mapping may be able to provide a quantitative measure of hepatobiliary function (7-9). As a result, MRI based T1 mapping may be positioned to provide an additional non-invasive means for detection, staging, and treatment monitoring of chronic liver disease.

Relaxometry using single point inversion recovery spin echo (IR-SE) is accepted as the definition of T1. However, IR-SE generally requires TRs to be four or five times the duration of the T1 times to be measured (10,11) making the acquisition time required for IR-SE impractical for clinical liver imaging. Beginning with Look and Locker (12), many IR-based T1 measurement strategies have been proposed that greatly reduce imaging time and enable abdominal T1 mapping. However, the duration of these T1 mapping methods typically limits the acquisition to just one or a few slices per breath-hold, making volumetric T1 mapping of the liver challenging. More recently, variable flip angle (VFA) strategies employing spoiled gradient echo (SGRE) acquisitions, favored in the liver for their short scan times and robustness to motion (13,14), have been proposed for rapid volumetric T1 mapping (14). Unfortunately, VFA T1 mapping methods are confounded by fat (5,15) and B1+ heterogeneities (i.e., flip angle errors) (16,17).

Chemical shift-encoded (CSE)-MRI is routinely used to measure several quantitative biomarkers of liver disease, including proton density fat fraction (PDFF), an established biomarker of tissue triglyceride concentration (18), and R2*, an emerging biomarker of liver iron content (19-21). Accordingly, methods that combine multi-echo CSE-MRI with multi-pass VFA methods can remove T1 bias from CSE-MRI-based PDFF estimation (22,23). However, while combined CSE-MRI and VFA methods account for fat, they do not inherently account for flip angle errors, resulting in T1 measurements biased by B1+ heterogeneity (23).

Conventional B1+ correction requires a separate calibration acquisition to measure and correct for B1+ field variations (17). This increases overall scan time and risks inter-acquisition misregistration. Therefore, there is an unmet need for quantitative B1+- and fat-corrected T1 mapping amenable to abdominal liver imaging within a breath-hold. The purpose of this work is to develop and present a novel approach to simultaneous estimation of confounder-corrected T1, PDFF, R2* and B1+ in a single breath-hold across the entire liver.

2. Theory

Our approach incorporates a phase-based B1+ encoding method, based on a previously reported B1+ mapping technique (24), with multi-echo CSE-MRI and multi-pass VFA (14) to jointly estimate PDFF, R2*, B1+, and T1 without the need for a separate B1+ calibration. The orthogonal-α B1+ mapping method (24) was selected for incorporation because it is amenable to short TRs and rapid scan times. The pulse sequence diagram for the proposed 3D SGRE-based acquisition strategy is shown in Figure 1. In the proposed method, data are acquired in three passes. As with conventional CSE-MRI, all passes acquire multi-echo SGRE data enabling estimation of PDFF and R2*. The first pass is a multi-echo SGRE acquisition that acquires data after a single slab-selective excitation. The remaining passes are modified SGRE acquisitions that employ a composite excitation to facilitate phase-based B1+ estimation. Importantly, a different flip angle is used in pass 1 than in the following two passes to enable variable T1 weighting, necessary for T1 estimation. The following sections explain the theory behind both the acquisition and reconstruction, including calculations and simulations used to determine optimal acquisition parameters.

Figure 1.

The proposed method acquires data in three passes. The first pass is a standard multi-echo SGRE acquisition that acquires data after a single slab-selective excitation. Passes 2 and 3 are modified SGRE acquisitions that employ a composite excitation to allow phase-based B1+ estimation. The schematic displays the readout, phase encode, slab select, RF, and excitation phase (theta) boards of the proposed pulse sequence. Cramér-Rao lower bound analysis (detailed in Figure 4) showed that a 120° phase difference between consecutive RF pulses improved noise performance.

2.1. Signal Model and Estimation

The signal model for pass 1 of the proposed acquisition is a T1 weighted CSE-MRI SGRE expression with terms added to reflect the spatial variation of the RF excitation (B1+) field. Equation [1] shows the total pass 1 signal model as the sum of the water and fat signal contributions:

$$\begin{gathered} S_1(TE) \\ =M_0((1-\eta )e^{i{\varphi }_1}\frac{\sin ({\beta }_W{\alpha }_1)\left(1-e^{-\frac{T{R}_1}{T{1}_W}}\right)}{1-e^{-\frac{T{R}_1}{T{1}_W}}\cos ({\beta }_W{\alpha }_1)}{e}^{-R_{2,W}^{\ast }TE}\phantom{)} \\ \phantom{(}+\eta e^{i{\varphi }_1}\frac{\sin ({\beta }_F{\alpha }_1)(1-e^{-\frac{T{R}_1}{T{1}_F}})}{1-e^{-\frac{T{R}_1}{T{1}_F}}\cos ({\beta }_F{\alpha }_1)}{e}^{-R_{2,F}^{\ast }TE}\sum \begin{array}{c}6\hfill \\ k=1\hfill \end{array}{\rho }_k{e}^{i2\pi f_{k}TE})e^{i2\pi \psi TE} \end{gathered}$$ (1)

Where $M_0$ (arbitrary units) is the steady state signal amplitude proportional to the equilibrium longitudinal magnetization, and PDFF is modeled directly as $\eta$ (percent fat fraction expressed as a decimal) (25). ${\varphi }_1$ (radians) is the initial phase of pass 1. $\beta$ is the RF/B1+ transmission efficiency (the unitless ratio of actual flip angle over the expected flip angle, typically ranging between 0 and 2)(26). ${\alpha }_1$ is the prescribed flip angle (degrees) for pass 1. $\psi$ is B0 off-resonance (Hz). ${\rho }_k$ and $f_k$ are the relative signal amplitudes (${\rho }_k$, arbitrary units) and frequency offsets ($f_k$, Hz) of the 6-peak spectral model of fat, such that ${\sum }_{k=1}^6{\rho }_k=1$ (27,28). Note that $\beta$, T1 (s), and R2* (s⁻¹) are each modeled independently for water and fat, as denoted by the subscripts W and F, respectively. Modeling $\beta$ independently for water and fat reflects the effect of off-resonance during RF excitation and is explained in greater detail in the following section. Additionally, note that for pass 1 the initial phase is common to both water and fat (29); this constrained phase configuration accurately reflects the physics of SGRE acquisitions so long as ${\alpha }_1\le \sim 5^{{}_{°}}$ (23).

The signal model for passes 2 and 3 of the proposed acquisition were derived using the Bloch equations assuming instantaneous RF excitation and perfect spoiling of transverse magnetization at the end of each TR using the MATLAB symbolic toolbox. For brevity, the derivation and resulting signal models, $S_2(TE)$ and $S_3(TE)$, are shown in Appendix A. As in pass 1, terms for B1+ transmission efficiency ($\beta$), T1, and R2* are all modeled independently for fat and water. Unlike pass 1, the initial phase terms in passes 2 and 3 are modeled independently for fat and water as ${\alpha }_2$ is greater than the threshold (~5°) where the constant phase for fat and water are known to be different and vary with flip angle and T1 (23). Note that TR, flip angle, and echo times are identical between passes 2 and 3, but do not necessarily match those of pass 1; however, all passes share a common echo spacing to avoid phase-related estimation errors from concomitant gradients (30-32).

Estimation of the unknown parameters was accomplished by fully complex-based nonlinear least squares (NLLS) (33,34) according to the following formulation:

$$\widehat{\theta }=\text{arg}\underset{\theta }{\text{min}}\left[\sum _{p=1}^3\sum _{n=1}^{N_p}\mid S_p(t_{p,n};\theta )-s_{meas,p,n}{\mid }^2\right]$$ (2)

In this notation, $p$ is the pass index, $n$ the echo index, and N_p the number of echoes acquired in pass $p$. Accordingly, $S_p(t_{p,n};\theta )$ is the calculated signal for the n^th echo of pass $p$ as a function of unknown parameters $\theta =\{{\beta }_W,{\beta }_F,T{1}_W,T{1}_F,M_0,{\varphi }_1,{\varphi }_{2,W},{\varphi }_{2,F},\eta ,R_{2,W}^{\ast },R_{2F}^{\ast },\psi \}$ and $s_{meas,p,n}$ is the corresponding measured (i.e., acquired) signal.

Estimation was performed as follows. First, an initial estimate of $M_0$, ${\varphi }_1$, $\eta$, $R_2^{\ast }$, and $\psi$ was made from the data acquired in pass 1. This was performed using an implementation of the graph-cut algorithm developed by Hernando et al (35) followed by a NLLS fitting using the following simplified signal model that ignores T1 weighting and assumes a common R2* of fat and water (25,36):

$$s(TE)=M_0{e}^{i{\varphi }_1}\left((1-\eta )+\eta \sum \begin{array}{c}6\hfill \\ k=1\hfill \end{array}{\rho }_p{e}^{i2\pi f_{k}TE}\right)e^{-R_2^{\ast }TE}{e}^{i2\pi \psi TE}$$ (3)

This was followed by three iterations of the full NLLS fitting, using data from all three passes (as shown in Eq. [2]), with each iteration used as the initialization for the following fitting. The first iteration fixed ${\beta }_W={\beta }_F=1.0$ and estimated all other unknown parameters. The second iteration unfixed ${\beta }_W$ and ${\beta }_F$ and jointly estimated all unknown parameters. Finally, as B1+ maps are generally smooth (37), estimated ${\beta }_W$ and ${\beta }_F$ maps were low-pass filtered (using a Kaiser-Bessel filter to 75% and 13% of the original resolution for phantom and in vivo estimation, respectively) and fixed at smoothed values to improve noise performance of the T1 estimation during a final joint fitting of the remaining parameters. Finally, in vivo estimation was determined to be more robust to B0 off-resonance when $\psi$ was determined during the simplified NLLS fitting (Eq. [3]) and held constant during the subsequent joint estimations.

2.1.1. Separate Estimation of Fat and Water B1+

Under the assumption of instantaneous RF excitation, B1+ transmission efficiency is independent of B0 off-resonance during excitation. However, in practice, RF pulses have a finite duration during which off-resonance affects B1+ transmission. Figure 2 shows the effective B1+ transmission as a function of off-resonance for a single simulated water voxel. Signals were simulated using extended phase graphs (38) for pass 2 of the proposed pulse sequence (Figure 1) at 1.5T with parameters common to the liver. These parameters included T1_W ranging from 200-1000ms, T1_F=280ms, R2*_W ranging from 20-60s⁻¹, and R2*_F=16s⁻¹ (23,39,40). RF excitation was simulated using 1048μs rectangular (hard) pulses in place of the truncated and shifted sinc RF pulses (of the same duration) used in the actual sequence. Results from the 3.0T simulation are in Supporting Information Figure S1. In pure water, off-resonance would simply contribute to the apparent B1+ heterogeneity. However, for a voxel containing both water and fat, the off-resonance of fat confounds the measurement of the effective B1+ transmission in water. To account for this effect, our proposed signal model has separate B1+ terms for water and fat.

Figure 2.

Noiseless simulations showing effective B1+ transmission as a function of off-resonance for non-instantaneous RF excitation suggest that B1+ of fat and water should be modeled separately. Although instantaneous excitation is assumed when deriving the SGRE signal model, in practice off-resonance during RF excitation affects B1+ transmission. Simulations were performed twice: once assuming instantaneous RF excitation and again assuming hard pulses with finite duration. RF excitations with finite duration were simulated by dividing the α₂₋pulse into N=150 instantaneous rotations of flip angle α₂/N separated by a 4μs time interval during which precession and off-resonance were modeled. Effective B1+ transmission was calculated as the ratio of the signal magnitude after non-instantaneous RF over the signal magnitude after instantaneous RF.

2.2. B1+ Estimation

The orthogonal-α B1+ mapping method (24) uses two consecutive non-selective RF pulses of equal magnitude, but orthogonal phase, to encode the transmitted flip angle into the phase of the MR signal. This strategy exploits the non-commutative relationship of rotations about orthogonal axes (using two passes to remove background phase) to estimate local B1+. Our method incorporates the orthogonal-α B1+ mapping method into passes 2 and 3 of the proposed acquisition with the following major adaptations: 1) the composite excitation is now slab selective, 2) the gradient echo acquisition is modified to include both gradient and RF spoiling, and 3) the phase difference between consecutive RF pulses is optimized for improved noise performance. These adaptations are discussed below. Note that the simulation results shown in Figures 2, 3, and 4 are for 1.5T. See Supporting Information Figure S1 for the corresponding 3.0T figures.

Figure 3.

116° is an optimal RF spoiling phase increment for passes 2 and 3. The figure shows expected T1_W estimation bias determined from noiseless non-linear least squares simulations for a range of T1_W and PDFF values as a function of RF spoiling phase increment. Simulated parameters were chosen to match the subsequent phantom acquisition (Protocol 1, Table 1).

Figure 4.

CRLB analysis shows the theoretical noise performance (as standard deviation of the estimated parameters) of the proposed acquisition and estimation strategy as a function of selected flip angles and RF phase difference. Standard deviation is given in the units specified at the top of each column (B1+_W: unitless, T1_W: ms, and PDFF: % PDFF). The top row shows that using a 120° RF phase difference between the consecutive pulses of passes 2 and 3, instead of the original 90°, gives improved noise performance over a range of T1_W values. The bottom row shows noise performance as a function of selected flip angles, α₁ and α₂. Acquisitions in this work used α₁=3° and α₂=60°, as indicated by the star. Note that α₂ was limited to 60° by system constraints and a desire to limit flow and motion artifacts that can accompany higher flip angles. CRLB analysis used parameters common to the liver (i.e., 20% PDFF, 30s⁻¹ R₂*_W, 16s⁻¹ R₂*_F, 0Hz $\psi$) and an assumed SNR of 100 (on the order of that achieved in the subsequent in vivo experiments at 1.5T). SNR was calculated as the magnitude signal averaged over all acquired echoes and divided by the standard deviation of the additive zero-mean Gaussian noise. Note that CRLB results have a direct inverse linear relationship to SNR (e.g., an acquisition with SNR reduced by a factor of two should expect the standard deviation of estimated T1_W to double, etc.) and thus these results can be generalized to acquisitions with higher or lower SNR performance.

2.2.1. Slab Selective B1+ Mapping

In the orthogonal-α B1+ mapping method, as in our proposed method, the time interval separating the two RF pulses of the composite excitation is defined as $\tau$. A key assumption of the orthogonal-α method is that $\tau$ is short compared to T2*. When true, the phase of the two acquired passes can be approximated as symmetric about $\frac{\pi }{4}$ and the transmitted flip angle can be calculated according to a simple expression relating the two phases. In the original method, $\tau$ is made short by using short non-selective RF pulses. However, slab-selective excitation is necessary for pragmatic in vivo acquisitions. Playing consecutive $\alpha$-pulses, with their accompanying slab-selective and refocusing gradients, significantly lengthens $\tau$, violating the assumption of the original method and increasing the overall scan time. As shown in Figure 1, our proposed solution uses bipolar gradients and a time-reversed RF waveform to maintain short $\tau$ while permitting slab-selective excitation. Simulated slab profiles for water and fat are shown in Supporting Information Figure S2.

2.2.2. Gradient and RF Spoiling

In addition to crusher gradients, a well-chosen RF spoiling phase-increment must be selected to eliminate transverse coherences (41). Signals simulated with extended phase graphs (38) were given as noiseless input to NLLS estimation using the derived signal model to determine the optimal RF phase increment. Figure 3 shows the results for simulated 1.5T data using common liver tissue parameters: T1_W ranging from 200-1000ms, T1_F=280ms, R2*_W=30s⁻¹, R2*_F=16s⁻¹, and PDFF ranging from 0-20% (23,39,40). With T1_W estimation bias relative to the simulated T1_W values, simulations show that optimal RF spoiling is achieved with a phase increment between 115° and 116°. Similar results are observed for 3.0T, shown in Supporting Information Figure S1. Repeated phantom acquisitions fine-tuned the RF phase increment for minimal T1_W estimation bias with respect to a reference spin echo inversion recovery T1 measurement. These phantom experiments paralleled the procedure and conditions of the phantom experiments to be described in the Methods section. Accordingly, the proposed acquisition uses an RF phase increment of 116° for passes 2 and 3.

2.2.3. Optimization of RF Pulse Phase Difference and Selection of Flip Angles

Consecutive RF pulses in the orthogonal-α B1+ mapping method are, as the name implies, 90° out of phase. In our proposed joint estimation strategy, Cramér-Rao Lower Bound (CRLB) analysis (42) was used to determine the optimal RF phase difference between consecutive pulses, the results of which can be seen in Figure 4. Based on these results, an RF phase difference of 120° is more optimal in terms of noise performance than the 90° orthogonal configuration. As shown in Figure 4, the increased RF phase difference improves noise performance for higher T1 values.

Figure 4 also shows the flip angles selected for this work at 1.5T, ${\alpha }_1=3^{{}_{°}}$ and ${\alpha }_2={60}^{{}_{°}}$, determined from the same CRLB analysis. Flip angles were selected based on CRLB calculations assuming a typical 1.5T liver T1_W value of 576ms (43). While the selected flip angles give near optimal performance for T1_W estimation over a range of clinically relevant T1 values, we comment that ${\alpha }_1$ was selected to be just slightly lower (3°) than the optimal flip angle (5°) to provide an independent PDFF estimate free of T1 bias (22). Additionally, ${\alpha }_2$ was limited to a maximum of 60° by system constraints (e.g., specific absorption rate – SAR – heating limitations) and a desire to limit the flow and motion artifacts that can occur with higher flip angles due to inflow effects (23). At 3.0T, SAR heating constraints limited ${\alpha }_2$ to a maximum of 45°. The corresponding CRLB analysis for 3.0T calculations is shown in Supporting Information Figure S1.

3. Methods

We evaluated the proposed method at 1.5T (Signa Artist, GE Healthcare) and 3.0T (Signa Premier, GE Healthcare). Four protocols were used in this study (Table 1). Protocols 1 and 2 were designed for phantom experiments performed without parallel imaging acceleration at 1.5T and 3.0T, respectively. Protocols 3 and 4 were designed for in vivo liver imaging, with total scan time reduced to a 19-second breath-hold at the expense of SNR and resolution by using parallel imaging acceleration (reconstructed with ARC, GE Healthcare, Waukesha WI) and partial Fourier acquisition (reconstructed by zero-filling interpolation).

Table 1.

Acquisition parameters for the protocols employed in this manuscript. Protocols 1 and 2 were designed for phantom experiments performed without parallel imaging acceleration. Protocols 3 and 4 were designed for in vivo liver imaging, with total scan time reduced to a 19-second breath-hold using parallel imaging acceleration and partial Fourier acquisition. Aside from minor differences in sequence timings (i.e., TEs, TRs, $\tau$), the main difference between 1.5T and 3.0T acquisitions is α₂ which had to be reduced at 3.0T due to SAR induced heating constraints.

Acquisition Parameters
	Phantom Protocols		In Vivo Protocols
	Protocol 1	Protocol 2	Protocol 3	Protocol 4
Field Strength	1.5T	3.0T	1.5T	3.0T
Coil	19 Channel Head/Neck	21 Channel Head/Neck	30 Channel Body	30 Channel Body
Field of View	40.0 x 20 x 25.2 cm3	40.0 x 20 x 25.2 cm3	40.0 x 32 x 25.2 cm3	40.0 x 32 x 25.2 cm3
Matrix Size	128 x 128 x 28	128 x 128 x 28	128 x 128 x 28	128 x 128 x 28
In-Plane Resolution	3.1 x 3.1 mm2	3.1 x 3.1 mm2	3.1 x 3.1 mm2	3.1 x 3.1 mm2
Slice Thickness	9 mm	9 mm	9 mm	9 mm
Bandwidth	±83.33 kHz	±83.33 kHz	±83.33 kHz	±83.33 kHz
Flip Angle
Pass 1 (α1)	3°	3°	3°	3°
Passes 2 and 3 (α2)	60°	45°	60°	45°
Number of Echoes
Pass 1	4	4	4	4
Passes 2 and 3	4	4	4	4
First Echo
Pass 1	1.1 ms	1.0 ms	1.1 ms	1.0 ms
Passes 2 and 3	1.6 ms	1.4 ms	1.6 ms	1.4 ms
Echo Spacing	1.5 ms	1.4 ms	1.5 ms	1.4 ms
TR
Pass 1	7.7 ms	7 ms	7.7 ms	6.9 ms
Passes 2 and 3	8.8 ms	8 ms	8.8 ms	7.8 ms
$\tau$ (Passes 2 and 3)	576 μs	512 μs	576 μs	480 μs
Acceleration
Phase	1	1	2	2
Slice	1	1	2	2
Partial kz	1	1	0.75	0.75
Corner Cutting	No	No	Yes	Yes
RF Phase Increment
Pass 1	117°	117°	117°	117°
Passes 2 and 3	116°	116°	116°	116°
Scan Duration	45 s	43 s	19 s	19 s

3.1. Phantom Experiments

3.1.1. Phantom Design

A 2% agar gel phantom was fabricated using varying concentrations of NiCl₂₋ and peanut oil to modulate T1 and PDFF. The phantom consisted of 3 sets of 5 vials for 15 total vials. Each set varied T1 from 200ms to 1000ms (for 3.0T) over the five vials using NiCl₂₋ (6.2mM NiCl₂₋ Concentration: 200ms Target at 3.0T T1_W, 2.8mM:400ms, 1.6mM:600ms, 1.1mM:800ms, 0.7mM:1000ms). PDFF was fixed at 0%, 10%, and 20% across the 3 sets using peanut oil. Single-slice, single-echo, 2D IR-SE with TR 4000ms and inversion times 50ms, 103ms, 212ms, 436ms, 897ms, 1846ms, and 3800ms, was used to determine the ground truth T1 for the fabricated vials at both 1.5T (Signa Artist, GE Healthcare) and 3.0T (Signa Premier, GE Healthcare) independently.

3.1.2. Acquisition

Acquisitions in the fabricated agar gel phantom were conducted at 1.5T (Signa Artist, GE Healthcare) and 3.0T (Signa Premier, GE Healthcare) using the parameters given in Table 1. The first set of experiments were designed to test the repeatability of the proposed acquisition by acquiring multiple independent datasets in rapid succession to maintain identical experimental conditions. Before each acquisition, product prescan updated shim and center frequency values to account for any frequency shift over time. In the first experiment, Protocol 1 was acquired 40 times in succession at 1.5T (Signa Artist, GE Healthcare) using a 19-channel head/neck coil (GE Healthcare). In the second experiment, Protocol 2 was acquired 40 times in succession at 3.0T (Signa Premier, GE Healthcare) using a 21-channel head/neck coil (GE Healthcare).

Reference B1+ maps were acquired using a commercially available version of the Bloch-Siegert method (GE Healthcare) (44) with the following parameters: 40 × 40 cm² FOV, 8 mm slice thickness, 15° flip angle, 64 × 64 matrix size, ±15.63 kHz receiver bandwidth, using an 8ms Fermi excitation pulse.

Due to the lack of a reference standard T1 mapping application suitable to in vivo breath-hold acquisitions, the following sections use a product single-point saturation recovery bSSFP 2D single-slice T1 mapping application (SMART₁Map, GE Healthcare) as an in vivo reference (45). It should be noted that while SMART₁Map serves as a reasonable reference for comparison due to its availability in vivo, it measures a mixed T1 (i.e., combining the contributions from water and fat). In a final phantom experiment, SMART₁Map is evaluated against the measured IR-SE T1 values and compared against the proposed in vivo protocol (Protocol 3, Table 1). Using the in vivo protocols in phantom data provides a reference for their subsequent use in volunteer in vivo liver acquisitions. Both protocols were acquired 10 times in succession at 1.5T (Signa Artist, GE Healthcare) using the 30-channel body coil (GE Healthcare). These experiments will be discussed in the results section, but figures will be left as supporting information.

3.1.3. Reconstruction

Data acquired using the proposed acquisition were reconstructed offline using the NLLS fitting method described in the Theory section (Eq. [2]). Product T1 maps (SMART₁Map, GE Healthcare) were reconstructed online using the vendor provided software.

3.1.4. Analysis

All phantom experiments underwent identical analysis. After reconstruction and estimation, regions of interest (ROI) were placed in each vial of a single central slice in each estimated T1_W map. Bias and precision were then determined with linear regression and Bland-Altman analysis using the average values of the ROIs from the independent repeated measurements.

3.2. In Vivo Evaluations

3.2.1. Acquisition

All in vivo data were collected after obtaining approval from the local institutional review board (IRB) and informed written consent in a Health Insurance Portability and Accountability Act–compliant manner.

Feasibility of the proposed method to measure native (i.e., non-contrast enhanced) T1_W of the liver was evaluated at 1.5T (Signa Artist, GE Healthcare) and 3.0T (Signa Premier, GE Healthcare) in healthy volunteers. At 1.5T, the livers of 5 volunteers were imaged using Protocol 3 (Table 1). At 3.0T, the livers of 3 volunteers were imaged using Protocol 4 (Table 1). All data were acquired during end-expiration breath-holding, with scan times noted in Table 1.

The proposed method was also tested in contrast-enhanced acquisitions at 1.5T (Signa Artist, GE Healthcare) in 5 additional healthy volunteers, after the intravenous bolus injection of 0.025 mmol/kg of gadoxetic acid (Eovist, Bayer Healthcare, Berlin, Germany). Using Protocol 3 (Table 1), data were acquired prior to contrast injection and 5-, 10-, 15-, and 20-minutes after injection. All data were acquired during end-expiration breath-holding, with scan times noted in Table 1.

In both non-contrast and contrast-enhanced studies, a product 2D single-slice T1 mapping application (SMART₁Map, GE Healthcare) was used to acquired T1 maps in each volunteer as a reference for T1. Key parameters for SMART₁Map included 40 cm x 32 cm FOV, 8mm slice thickness, 1.4ms TE, 3.0ms TR, 131ms saturation recovery time, SSFP readout, 65° flip angle, 160 x148 matrix size, ±83.33 kHz bandwidth, with a phase acceleration factor of 2. Reference B1+ maps were acquired using the Bloch-Siegert method with the following parameters: 40 × 40 cm² FOV, 15 mm slice thickness, 10° flip angle, 64 × 64 matrix size, ±31.25 kHz receiver bandwidth, using a 2ms adiabatic excitation pulse. Multi-TE-TR MR Spectroscopy (STEAM) data were acquired as a reference for PDFF (46). Both the product SMART₁Map (10 second duration) and the STEAM acquisition (20 second duration) were acquired during end-expiration breath-holding.

3.2.2. Reconstruction

Data acquired using the proposed acquisition were reconstructed offline using the NLLS fitting method described in the Theory section (Eq. [2]). Product T1 maps (SMART₁Map, GE Healthcare) were reconstructed online using the vendor provided software. STEAM data were reconstructed offline using a joint fitting of the acquired spectra accounting for T2 decay and T1 recovery. Each spectrum was fitted assuming a single water peak and 6 fat peaks using Voigt line shapes (47).

3.2.3. Analysis

With placement overseen by a senior radiologist with 22 years of experience, ROIs were drawn in a central slice of the liver in reconstructed T1 maps, avoiding major vessels, bile ducts, and the liver edge. In the contrast-enhanced study, the ROI was propagated to each reconstructed map (including SMART₁Map T1 and R2* maps), with slight manual adjustments to account for variability in breath-hold positions between different acquisitions. After contrast injection, measured T1_W values were expected to show a statistically significant decrease (48). To verify this, paired Student’s t-tests with significance level of 0.01, and Bonferroni correction, were used to compare pre-contrast T1_W values of the liver with the post-contrast (5, 10, 15, and 20 min) values. Statistical analysis was performed using MATLAB (MathWorks).

4. Results

4.1. Phantom Experiments

IR-SE is considered the reference-standard for T1 mapping (49,50); however, IR-SE is confounded by the presence of fat. Assuming that the true T1_W is dependent on NiCl₂₋ but independent of fat concentration, the IR-SE T1 measured in the vials with 0% PDFF (i.e., without added peanut oil) can be used as a surrogate for T1_W in the remaining vials. Accordingly, the ground truth T1_W values in the fabricated agar gel phantom were established as: 245ms, 479ms, 705ms, 927ms, and 1155ms for 1.5T and 225ms, 444ms, 664ms, 886ms, and 1131ms for 3.0T.

Figure 5 shows linear regression and Bland Altman analysis of 40 repeated acquisitions collected to determine the repeatability of the proposed method using Protocol 1 at 1.5T. The proposed T1_W shows excellent agreement with the IR-SE T1 reference measurements (slope=1.01, R²=1.00, RMSE=22ms) with an average bias of −5.2±23ms. Figure 5 also shows the proposed method compared against IR-SE T1 as measured independently in each vial, highlighting the confounding effect of fat on traditional T1 mapping. Supporting Information Figure S3 shows maps of all estimated parameters, averaged across repetitions, to supplement the non-averaged T1_W and PDFF maps shown in Figure 5.

Figure 5.

The proposed B1+- and fat- corrected T1_W measurements showed excellent linear agreement with Inversion Recovery Spin-Echo (IR-SE) T1 measurements (B) in phantom data at 1.5T. IR-SE T1 is biased by the presence of fat (A). However, by assuming true T1_W is dependent on NiCl₂₋ concentration and independent of fat, then IR-SE T1 can be measured in vials without added peanut oil (0% PDFF) and used as a reference in all vials (B). A sample T1_W and PDFF map are shown on the left.

Phantom experiments at 3.0T demonstrated the proposed method’s ability to jointly estimate B1+ nonuniformity to return unbiased estimates of T1_W. Across the 40 repeated acquisitions at 3.0T, estimated B1+_W ranged from 0.64-1.25 (1.1 median with 0.88-1.18 5^th-95^th percentiles). In the liver, B1+ inhomogeneities can range from approximately 0.4 to 1.3 at 3.0T (though most commonly observed between 0.6 and 1.2) and 0.7 to 1.2 at 1.5 T (though most commonly observed between 0.8 and 1.1) (26). The range of observed B1+ inhomogeneities in this study is consistent with these published ranges from the literature. A reconstructed B1+_W map is shown with a reference Bloch-Siegert B1+ map for comparison in Figure 6. Linear regression and Bland Altman analysis of the 40 repeated acquisitions of the proposed method using Protocol 2 at 3.0T are shown in Figure 6. The proposed T1_W shows good agreement with the IR-SE T1 reference measurements (slope=1.00, R²=0.99, RMSE=24ms) with an average bias of −45±24ms. Figure 6 also shows an analysis of the proposed method when errors from B1+ inhomogeneities are ignored, highlighting the confounding effect of B1+ on T1 estimation. Supporting Information Figure S4 shows maps of all estimated parameters, averaged across repetitions, to supplement the non-averaged B1+_W, T1_W and PDFF maps shown in Figure 6.

Figure 6.

Ignored B1+ inhomogeneity can lead to substantial variability and spatially-dependent bias (A). However, the proposed B1+- and fat- corrected T1_W measurements showed strong linear agreement with Inversion Recovery Spin-Echo (IR-SE) T1 measurements in phantom data at 3.0T in the presence of B1+ inhomogeneity (B). The Bloch-Siegert B1+ map (i) demonstrates a clear B1+ artifact (RF nonuniformity) that will result in T1_W errors if ignored (iii). In the proposed joint estimation, RF non-uniformity is successfully estimated (ii) to return B1+-corrected estimates of T1_W (iv). Note: The estimated B1+ map shown above is B1+_W.

Supporting Information Figure S5 shows linear regression and Bland Altman analysis of both the 1.5T in vivo protocol (Protocol 3) and the product SMART₁Map acquired in the fabricated agar gel phantom. The proposed T1_W shows better agreement with the IR-SE T1 reference (slope=1.04, R²=0.99, RMSE=26ms) than the SMART₁Map T1 map (slope=0.8, R²=0.89, RMSE=89ms).

4.2. In Vivo Evaluations

Feasibility of the proposed method was demonstrated in vivo in 5 healthy volunteers at 1.5T and 3 healthy volunteers at 3.0T. Figure 7 shows estimated T1_W maps along with estimated B1+ inhomogeneities and PDFF maps from a subset of volunteers (see Supporting Information Figure S6 for estimated maps, including R2*_W, from all volunteers). Estimated T1_W showed fair agreement with the vendor provided SMART₁Map (slope=0.7, intercept=245ms, R²=0.74) and are consistent with liver T1 values reported in the literature (43,48). Estimated PDFF was correlated with MRS-STEAM results (slope=0.83, intercept=0.6%, R²=0.95), although with only 5 data points it was difficult to demonstrate agreement definitively. Supporting Information Figure S7 shows that B1+_W field maps estimated by the proposed method were in general agreement with the reference Bloch-Siegert B1+ field maps and also shows side-by-side comparisons of the estimated T1_W maps with and without B1+ correction.

Figure 7.

Proposed T1_W values (row 2) are generally in agreement with the product T1 mapping sequence (SMART₁Map, row 1) in reconstructed maps from healthy volunteers. Multi-TE-TR MR spectroscopy (STEAM) PDFF values are also shown. For in vivo estimation, B0 off-resonance was estimated from the data acquired in pass 1 and fixed in the subsequent joint estimation. After joint estimation, the estimated B1+ map was smoothed (as shown) and held constant in a final joint estimation. See Supporting Information Figure S6 for results from the full cohort of healthy volunteers.

The proposed method was evaluated in a contrast-enhanced study at 1.5T in 5 healthy volunteers. Figure 8 shows several slices of a T1_W map from a single volunteer acquired using the proposed method before and after (5, 10, 15, and 20 min) contrast injection. Figure 8 also shows the single-slice SMART₁Map acquired immediately after each of the proposed method acquisitions.

Figure 8.

Representative T1_W maps acquired with the proposed 19 second breath-hold acquisition before and after (5, 10, 15, and 20 minutes) administration of gadoxetic acid contrast agent. While most T1 mapping acquisitions are limited to a small number of 2D slices per breath-hold, the proposed acquisition permits 3D T1_W mapping with full liver coverage, corrected for B1+ inhomogeneities, fat content, R2* decay, and B0 off-resonance. Shown are three (non-consecutive) slices of the proposed T1_W map and a reference product T1 map (SMART₁Map, GE Healthcare) in a single healthy volunteer.

Figure 9 shows the averages of ROIs placed in each liver T1_W map acquired using the proposed method and the reference SMART₁Map before and after contrast injection. Paired Student’s t-tests (with Bonferroni correction) determined that the proposed T1_W values decreased significantly (P<0.01) after contrast injection with an average decrease of 572ms after 5 minutes, 603ms after 10 minutes, 612ms after 15 minutes, and 622ms after 20 minutes. In agreement with previously reported work (51), measured R2*_W values were found to increase significantly (P<0.01) after contrast administration. Figure 9 also reports the pre-contrast MRS-STEAM PDFF measurements for each volunteer.

Figure 9.

T1_W values, measured using the proposed 19 second breath-hold acquisition, were significantly shorter (P<0.01) after contrast administration while $R_{2,W}^{\ast }$ were significantly (P<0.01) increased. Graphs show the average of a single ROI drawn in each liver map, avoiding major veins and arteries, which was co-localized between the three maps (proposed T1_W, vendor-provided SMART₁Map, and proposed $R_{2,W}^{\ast }$) and across the 5 timepoints (Pre, 5, 10, 15, and 20 minutes). Results from the pre-contrast MRS STEAM measure of liver PDFF are shown for each volunteer.

5. Discussion

In this work, we have proposed and demonstrated the feasibility of a novel acquisition and reconstruction method that enables confounder-corrected 3D T1_W estimation of the entire liver within a single 20s breath hold. This is achieved through simultaneous estimation of fat content in addition to fat and water specific T1, R2* and B1+ (without the need for a separate B1+ calibration acquisition). The method was demonstrated at both 1.5T and 3.0T, and validated in simulations, phantoms and human volunteers with and without contrast. In vivo pre- and post- contrast liver T1_W values were comparable to previously published results (4,6,48,52).

Confounder corrected T1 mapping has the potential to provide noninvasive biomarkers of chronic liver disease. In addition to correlating with liver fibrosis (4), T1 measurements made with gadoxetic acid, a contrast agent with a T1-shortening effect, can be used to provide a quantitative measure of liver function (7-9). Differences in measured liver T1 before and after enhancement correlate with the indocyanine green (ICG) clearance test (an established reference standard for measuring hepatobiliary function) especially when paired with measures of liver volume (53). Unlike the ICG clearance test, which is limited to a single global measure of liver function, T1 mapping can assess the spatial heterogeneity of liver function typical of chronic liver disease and advanced fibrosis. We observed that our proposed T1 mapping method has improved noise performance for lower T1 values which may make it particularly well suited for contrast enhanced T1 mapping. We also observed that in four out of the five volunteers given contrast at 1.5T, the SMART₁Map reference estimated higher post-contrast injection T1 than our proposed method. Importantly, it is well known from the cardiac relaxometry literature that saturation recovery T1 mapping methods may be discordant with other T1 mapping methods (54). While future work should seek to understand this difference more fully, we note that the average T1_W estimated with MRS-STEAM (145ms) 20 minutes after injection was closer to the average of the proposed T1W (111ms) than the four SMART₁Maps (296ms).

This work also demonstrated how the previously non-selective orthogonal-α B1+ mapping method (24) could be adapted for slab selective volumetric imaging through the use of both bipolar slab select gradients and time reversed RF pulses. Experiments outside the present scope suggest the orthogonal-α B1+ mapping method could be made robust to the presence of fat by acquiring echoes when fat and water are in-phase. However, further work would be required to confirm these observations independent of the joint acquisition and estimation method proposed in this work.

Additionally, this work has demonstrated that careful selection of an RF spoiling phase-increment allows accurate T1_W estimation using a closed-form expression of the derived signal model. However, we note that both simulations and preliminary phantom experiments revealed that the accuracy our proposed method is sensitive to small changes in RF spoiling phase-increment. To this end, future work is being planned to account for RF spoiling directly during parameter estimation (e.g., inclusion of Bloch equation simulation-based lookup tables or direct modeling of RF spoiling (55)). We note, however, that lookup table strategies are computationally intensive, especially for tables with a large number of degrees of freedom such as the signal model used in this work. Future strategies will need to include computational optimization strategies to address this expected challenge.

This work used a signal model with independent water and fat specific R2*. Although modeling a single common R2* of water and fat has more favorable noise performance (25), previous work in CSE-MRI has demonstrated that independent estimation of R2* values for water and fat more accurately reflects the underlying physics and can yield more accurate quantitation (56). During development of the proposed method, we found that independent fat and water R2* modeling yielded more accurate water-specific T1 estimates without any appreciable effect on noise-performance in phantoms. However, the optimal method of modeling R2* in the proposed method is still uncertain and the effect of this additional degree of freedom will be evaluated in future studies.

This work exists in the context of the ongoing exploration for rapid confounder-corrected T1 mapping. Repeated single point IR-SE is the traditional reference standard that defines T1; however, the acquisition time required for IR-SE makes it impractical for clinical imaging, especially for breath-hold applications such as liver MRI. Look and Locker proposed an inversion-recovery-based T1 measurement strategy (12) that has been subsequently modified (Modified Look-Locker—MOLLI) (11) and shortened (Shortened MOLLI—ShMOLLI) (57) for breath-hold durations. These methods, and others like them (e.g., saturation recovery single-shot acquisition—SASHA), enable T1 mapping to be used clinically and are used widely today (5). More recently, Thompson et al proposed a saturation recovery based T1 mapping method in the liver similar to the cardiac SMART₁Map application (58). However, the duration of these T1 mapping methods typically limit the acquisition to just a few slices per breath-hold, making volumetric T1 mapping of the liver infeasible.

An alternative 3D VFA T1 mapping method (DESPOT1), based on the variable nutation angle method (59), was proposed by Deoni et al in which T1 is measured volumetrically with repeated SGRE acquisitions (14). VFA methods removed the obstacle of scan time constraints for volumetric liver T1 mapping; however, the methods themselves were still susceptible to both B1+ nonuniformities (16,17) and the presence of fat. The novelty of our contribution is that by combining CSE- and VFA-MRI with a simultaneous B1+ mapping technique, the proposed method can estimate T1 corrected for both fat and RF non-uniformity.

It should also be noted that recent advances in 3D T1 mapping have addressed the limitation of scan time constraint with motion-robust free breathing acquisitions. Some examples include magnetization-prepared golden-angle radial sparse parallel MRI (MP-GRASP)(60) and multi-echo MR multi-tasking (61), both of which can be used to generate 3D water-specific T1 maps in 5 minutes or less.

Our present work has limitations. First, this work treats PDFF and R2* primarily as confounders to accurate T1 quantification. While a clinically relevant range of PDFF was explored in phantoms, it was done so primarily to validate the main technical focus of the paper: confounder corrected T1 mapping. Indeed, some acquisitions parameters, including flip angles, were selected to optimize T1 estimation performance at the expense of PDFF and R2* estimation performance. While preliminary comparisons against reference standards (including MRS STEAM) suggest PDFF and R2* estimated by the proposed method remain unbiased, rigorous quantitative validation of the jointly-estimated confounders is beyond the scope of this introductory work and will be the focus of future investigation.

On a related note, a limited number of healthy volunteers were used to evaluate the proposed method in vivo. Previous studies have demonstrated that in addition to the range of liver PDFF and R2* expected in a population, both B0 and B1+ vary from person to person and depend on both field strength and system geometry (26). Our limited sample size has not adequately assessed the comprehensive range and combinations of T1, PDFF and R2* values that can be experience in a variety of liver diseases with varying levels of fibrosis, fat and iron. However, the present study demonstrates feasibility while future studies are currently being planned to assess the bias and precision of the method in a wide variety of patients at both 1.5T and 3.0T.

Another limitation of this method is its dependence on high flip angles for B1+ mapping. Although noise performance increases with increasing flip angle in passes 2 and 3, imaging above the Ernst angle leads to unmodeled signal amplification in the drop off of the slab profile (62). The practical implication of this limitation is that a greater number of slices near the edges of the acquired slab may need to be discarded. In our experience, this limitation did not preclude volumetric liver imaging, although it did require extending the FOV above and below the liver in the slice encoding direction (See Supporting Information Figures S2 and S8). Extending the volumetric coverage or improving the spatial resolution of the proposed method may require more aggressive parallel acceleration, and/or increased use of partial Fourier sampling.

Conclusion

B1+ inhomogeneity and fat are known confounders of T1 estimation in the liver. Our proposed method accounts for both confounders intrinsically through simultaneous estimation of T1, B1+, PDFF and R2*. The proposed method demonstrates good linear agreement with reference T1 measurements, with low bias and high precision, and can achieve full-volume liver coverage in a single breath-hold at both 1.5T and 3.0T.

Supplementary Material

Supporting Information

Supporting Information Figure S1. This figure shows the complementary 3.0T analyses and results in (A), (B), and (C) for Figures 2, 3, and 4, respectively. Please refer to the in-manuscript captions for Figures 2, 3, and 4 (which give the 1.5T analyses and results) for a detailed explanation of each element of this supplementary figure.

Supporting Information Figure S2. The proposed method uses bipolar gradients and a time-reversed RF waveform to maintain short inter-RF separation ($\tau$) while permitting slab-selective excitation. Shown are the slab profiles for both water (T1=576ms, T2=46ms) and fat (3.4 ppm peak, T1=300ms, T2=56ms) simulated for α₂=60° at 1.5T. Note the signal peaks near the edge of the slab profile for water caused by using a flip angle above the Ernst angle. As addressed in the discussion, this unmodeled signal amplification may confound accurate T1 estimation near the edges of the acquired slab.

Supporting Information Figure S3. The proposed method jointly estimates 12 parameters: B1+ of water and fat (${\beta }_W,{\beta }_F$), T1 of water and fat ($T{1}_W,T{1}_F$), signal amplitude ($M_0$), signal phases (${\varphi }_1,{\varphi }_{2,W},{\varphi }_{2,F}$), PDFF ($\eta$), R2* of fat and water ($R_{2,W}^{\ast },R_{2,F}^{\ast }$), and B0 field map ($\psi$). Shown above are all estimated parameter maps, averaged across repetitions of the 1.5T phantom experiments, to supplement the non-averaged maps shown in Figure 5. Note that only PDFF and water-specific T1 were controlled during phantom construction.

Supporting Information Figure S4. The proposed method jointly estimates 12 parameters: B1+ of water and fat (${\beta }_W,{\beta }_F$), T1 of water and fat ($T{1}_W,T{1}_F$), signal amplitude ($M_0$), signal phases (${\varphi }_1,{\varphi }_{2,W},{\varphi }_{2,F}$), PDFF ($\eta$), R2* of fat and water ($R_{2,W}^{\ast },R_{2,F}^{\ast }$), and B0 field map ($\psi$). Shown above are all estimated parameter maps, averaged across repetitions of the 3.0T phantom experiments, to supplement the non-averaged maps shown in Figure 5. Note that, consistent with the physical properties of fat, T1_F was bound to an upper limit of 400ms. Simulations suggested this would have a negligible effect on the estimated T1_W. Note that only PDFF and water-specific T1 were controlled during phantom construction.

Supporting Information Figure S5. The proposed method (3D full liver acquisition, accelerated for a 19 second acquisition) shows both reduced bias and improved precision compared to the vendor provided T1 mapping application (2D single-slice, 10 second acquisition). Plots show both linear regression and Bland-Altman analysis for both the vendor provided SMART₁Map (GE Healthcare) (A) and proposed (B) T1 mapping methods with IR-SE T1 as a reference (as measured in the phantom vials without fat). Data were acquired in the constructed agar gel phantom at 1.5T using the accelerated in vivo protocol (Protocol 3).

Supporting Information Figure S6. This figure is supplemental to Figure 7. Proposed T1_W values (row 2) are generally in agreement with the product T1 mapping sequence (SMART₁Map, row 1) in reconstructed maps from healthy volunteers at both 1.5T (5 volunteers) and 3.0T (3 volunteers). Multi-TE-TR MR spectroscopy (STEAM) PDFF values are also shown.

Supporting Information Figure S7. Ignoring spatially varying B1+ inhomogeneities in the liver results in significant T1 estimation errors (row 1 compared to row 2). The proposed method accounts for B1+ errors in a simultaneous joint estimation. In 5 volunteers, the estimated B1+_W maps (row3) in were in general agreement with the reference Bloch-Siegert B1+ reference maps (row 4). Shown are estimated T1_W and B1+ maps acquired in 5 healthy volunteers at 1.5T.

Supporting Information Figure S8. The unmodeled signal amplification near the edges of the slab profile, caused by imaging above the Ernst angle (see Supporting Information Figure S2), is demonstrated in this coronal reformat (and accompanying axial slices) of an estimated T1_W map in a healthy volunteer at 1.5T. As noted in the discussion, this issue may be avoided by extending the FOV in the superior/inferior direction.

Appendix A: Derivation of Steady State Signal after Composite Dual RF Excitation

The signal model for passes two and three of the proposed acquisition were derived assuming instantaneous RF excitation and perfect spoiling of transverse magnetization at the end of each TR. This appendix provides a derivation of the steady-state signal for passes 2 and 3 of the proposed acquisition. Note that simulations showed that the number of TRs required to reach steady-state was similar to that required for a standard SGRE acquisition (~30-40). However, by acquiring a 3D volume with standard phase encoding, the actual number of discarded acquisitions can be reduced in recognition that the signal will reach steady state long before acquiring data near the center of k-space. As a result, our pulse sequence implementation used the vendor default of 4 discarded TRs. The steady-state magnetization following the dual RF excitation (abbreviated DRF) can be calculated as follows, beginning with the initial magnetization, $m_{Z0}$, along the $\widehat{z}$ direction:

$$\overrightarrow{\mathit{M}}{}_{{}_1}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill m_{Z0}\hfill \end{array}\right]$$ [4]

$$\text{After first excitation:}\overrightarrow{\mathit{M}}{}_{{}_2}=R(\beta {\alpha }_2,{\varphi }_2+{\theta }_1)\overrightarrow{\mathit{M}}{}_{{}_1}$$ [5]

$$\text{Inter}-\text{RF precession:}\overrightarrow{\mathit{M}}{}_{{}_3}=A_{\tau }{R}_Z(2\pi \psi \tau )\overrightarrow{\mathit{M}}{}_{{}_2}+M_0\overrightarrow{\mathit{B}}{}_{{}_{\tau }}$$ [6]

$$\text{After second excitation:}\overrightarrow{\mathit{M}}{}_{{}_4}=R(\beta {\alpha }_2,{\varphi }_2+{\theta }_2)\overrightarrow{\mathit{M}}{}_{{}_3}$$ [7]

$$\text{Precession to end of TR:}\overrightarrow{\mathit{M}}{}_{{}_5}=A_{T{R}_2}{R}_Z(2\pi \psi TR)\overrightarrow{\mathit{M}}{}_{{}_4}+M_0\overrightarrow{\mathit{B}}{}_{T{R}_2}$$ [8]

Where matrices $A_t$ and $B_t$ are defined as

$$\begin{gathered} A_t=\left[\begin{array}{ccc}\hfill E_{2,t}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill E_{2,t}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill E_{1,t}\hfill \end{array}\right] \\ {A}_t=\left[\begin{array}{ccc}\hfill E_{2,t}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill E_{2,t}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill E_{1,t}\hfill \end{array}\right] \\ \overrightarrow{\mathit{B}}{}_{{}_t}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1-E_{1,t}\hfill \end{array}\right], \end{gathered}$$ [9]

rotational matrices, $R$ and $R_Z$, are defined as

$$\begin{gathered} R(\alpha ,\theta )=R_Z(\theta )R_Y(\alpha )R_Z(-\theta ) \\ {R}_Z(\alpha )=\left[\begin{array}{ccc}\hfill cos(\alpha )\hfill & \hfill -sin(\alpha )\hfill & \hfill 0\hfill \\ \hfill sin(\alpha )\hfill & \hfill cos(\alpha )\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right] \\ {R}_Y(\alpha )=\left[\begin{array}{ccc}\hfill cos(\alpha )\hfill & \hfill 0\hfill & \hfill sin(\alpha )\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -sin(\alpha )\hfill & \hfill 0\hfill & \hfill cos(\alpha )\hfill \end{array}\right], \end{gathered}$$ [10]

and $E_{1,t}$ and $E_{2,t}$ are defined as

$$E_{1,t}=e^{-t∕T1}\text{and}{E}_{2,t}=e^{-R_2^{\ast }t}.$$ [11]

The steady-state solution for the magnetization of a single species after the dual RF excitation at echo time TE can then be determined as:

$$\overrightarrow{\mathit{M}}{}_{{}_{TE}}=A_{TE}{R}_Z(2\pi \psi TE)\overrightarrow{\mathit{M}}{}_{{}_4}+M_0\overrightarrow{\mathit{B}}{}_{{}_{TE}}$$ [12]

The corresponding general (i.e., arbitrary single species) signal is the complex sum of the magnetization along $\widehat{x}$ and $\widehat{y}$ of $\overrightarrow{\mathit{M}}{}_{{}_{TE}}$:

$$\begin{gathered} S_{DRF,General}(TE)=\overrightarrow{\mathit{M}}{}_{TE}^{(1)}+i\overrightarrow{\mathit{M}}{}_{TE}^{(2)} \\ =E_{2,TE}sin({\alpha }_2\beta )e^{i({\varphi }_2+{\theta }_2+2\pi TE\psi )}(M_0(1-E_{1,\tau }) \\ +E_{1,\tau }{m}_{Z0}cos({\alpha }_2\beta )+i{E}_{2,\tau }{m}_{Z0}sin({\theta }_1-{\theta }_2+2\pi \psi \tau ) \\ +\frac{E_{2,\tau }{m}_{Z0}cos({\theta }_1-{\theta }_2+{\alpha }_2\beta +2\pi \psi \tau )}{2} \\ +\frac{E_{2,\tau }{m}_{Z0}cos({\theta }_1-{\theta }_2-{\alpha }_2\beta +2\pi \psi \tau )}{2}) \end{gathered}$$ [13]

The steady-state signal can be determined by solving for $m_{Z0}$, which can be achieved by solving $m_{Z0}=\overrightarrow{\mathit{M}}{}_5^{(3)}$ (i.e., equating the initial magnetization along $\widehat{z}$ in $\overrightarrow{\mathit{M}}{}_{{}_1}$ to that in $\overrightarrow{\mathit{M}}{}_{{}_5}$, which should be equivalent in steady-state) resulting in the following solution:

$$\begin{gathered} m_{Z0} \\ =\frac{4M_0(E_{1,T{R}_2}-E_{1,T{R}_2}\cos (\beta {\alpha }_2)+E_{1,T{R}_2}{E}_{1,\tau }\cos (\beta {\alpha }_2)-1)}{\left(\begin{array}{c}\hfill 2E_{1,T{R}_2}{E}_{1,\tau }+2E_{1,T{R}_2}{E}_{1,\tau }\cos (2\beta {\alpha }_2)-2E_{1,T{R}_2}{E}_{2,\tau }\cos ({\theta }_1-{\theta }_2+2\pi \psi \tau )\hfill \\ \hfill +E_{1,T{R}_2}{E}_{2,\tau }cos({\theta }_1-{\theta }_2-2\beta {\alpha }_2+2\pi \psi \tau )\hfill \\ \hfill +E_{1,T{R}_2}{E}_{2,\tau }\cos ({\theta }_1-{\theta }_2+2\beta {\alpha }_2+2\pi \psi \tau )-4\hfill \end{array}\right)} \end{gathered}$$ [14]

The general expression given in Eq. [13] is adapted to fat/water separated CSE-MRI through the following substitutions:

$${\phantom{\mid }{S}_{DRF,Water}(TE)=S_{DRF,General}(TE)\mid }_{\begin{array}{c}\hfill M_0=M_0(1-\eta ),{\varphi }_2={\varphi }_{2,W}\hfill \\ \hfill \beta ={\beta }_W,T1=T{1}_W,R_2^{\ast }=R_{2,W}^{\ast }\hfill \end{array}}$$ [15]

$$\phantom{(}{S}_{DRF,Fat}(TE)=\sum \begin{array}{c}6\hfill \\ p=1\hfill \end{array}{(S_{DRF,General}(TE)\mid }_{\begin{array}{c}\hfill M_0=M_0\eta {\rho }_k,{\varphi }_2={\varphi }_{2,F}\hfill \\ \hfill \beta ={\beta }_F,T1=T{1}_F,R_2^{\ast }=R_{2,F}^{\ast }\hfill \\ \hfill \psi =\psi +f_k\hfill \end{array}})$$ [16]

$$S_{DRF}(TE)=S_{DRF,Water}(TE)+S_{DRF,Fat}(TE)$$ [17]

Finally, the signal models for passes two and three are distinguished by opposite substitutions for ${\theta }_1$ and ${\theta }_2$, (the phases, respectively, of the consecutive dual RF pulses):

$${\phantom{\mid }{S}_2(TE)=S_{DRF}(TE)\mid }_{{}_{{\theta }_1=0,{\theta }_2={\Theta }_{DRF}}}$$ [18]

$${\phantom{\mid }{S}_3(TE)=S_{DRF}(TE)\mid }_{{}_{{\theta }_1={\Theta }_{DRF},{\theta }_2=0}}$$ [19]

where ${\Theta }_{DRF}$ is the phase difference between the RF pulses.

In all expressions above, $M_0$ is the steady state signal amplitude at TE=0; PDFF is modeled directly as $\eta$ (25); ${\varphi }_{2,W}$ and ${\varphi }_{2,F}$ are the common initial phases of passes two and three, modeled independently for fat and water because ${\alpha }_2$ is above the threshold (~5°) where a constrained phase would be valid (23); ${\beta }_W$ and ${\beta }_F$ reflect B1+ transmission efficiency of the prescribed flip angle, ${\alpha }_2$, for water and fat independently; similarly, both T1 and R2* for water and fat are modeled separately, as denoted by the subscripts W and F; $\psi$ is B0 off-resonance (Hz); ${\rho }_k$ and $f_k$ are the relative signal amplitudes and frequency offsets of the 6-peak spectral model of fat (27,28). Note that TR₂ is the TR of both passes 2 and 3, while TR₁ is the TR of pass 1.