Introduction

Multi-echo chemical shift-encoded (CSE)-MRI is an established method for measuring proton density fat fraction (PDFF) and R2* as biomarkers of tissue fat and iron content, respectively^1–4. At 3.0T, CSE-MRI is routinely acquired in two or more time-interleaved passes (i.e., multi-pass). Multi-pass acquisitions permit high spatial resolution while maintaining optimal echo spacing^5–7; however, they also exacerbate concomitant gradient (CG) phase errors^8,9. Away from isocenter, CG phase errors cause spatially-varying bias when estimating B0 off-resonance, R2*, and PDFF^7,8.

While methods have been proposed to correct CSE-MRI multi-pass acquisitions for CG phase errors^6–8, they generally require knowledge of the gradient waveforms. In this work, we present a fitting method that addresses concomitant gradient phase errors in PDFF and R2* estimation without prior knowledge of gradient waveforms.

Theory

CGs are magnetic field variations induced as the by-product of the imaging gradients⁹. For a position relative to isocenter (x=0, y=0, z=0), CGs are approximated by: \begin{equation}\begin{aligned}B_{CG}(x,y,z,t)&=\frac{1}{2B_0}[G_x^2(t)z^2+G_y^2(t)z^2+G_z^2(t)\frac{x^2+y^2}{4}\\&-G_x(t)G_z(t)xz-G_y(t)G_z(t)yz]\end{aligned}\tag{1}\end{equation}where $$$G_x(t)$$$, $$$G_y(t)$$$, and $$$G_z(t)$$$ are the imaging gradients^8,9.

We begin with the following phase-constrained CSE-MRI spoiled gradient echo (SGRE) signal model^10,11:\begin{equation}s_n(\eta,R_2^*,\psi,M_0,\phi)=M_0e^{i\phi}(1-\eta+\eta\,C_F(t_n))e^{-R_2^*t_n}e^{i2\pi\psi{}t_n}\tag{2}\end{equation}for N echoes acquired at TEs $$$t_1,t_2,...,t_N$$$. $$$M_0e^{i\phi}$$$ is the complex signal amplitude ($$$M_0$$$) and constant phase ($$$\phi$$$), $$$\eta$$$ is PDFF, $$$\psi$$$ is the B0 field map (Hz), and $$$C_F$$$ is the sum of complex exponentials reflecting a multi-peak spectral model of fat, typically 6 peaks¹².

CG phase errors can then be included as:\begin{equation}\begin{aligned}s_{n,CG}(x,y,z,\pmb{\theta})=s_n(\pmb{\theta})e^{-i\phi_{CG}(x,y,z,t_n)}\end{aligned}\tag{3}\end{equation}for $$$\pmb{\theta}=\{\eta,R_2^*,\psi,M_0,\phi\},~$$$where$$$~\phi_{CG}(x,y,z,TE)$$$ is the spatially-dependent phase error attributed to CGs given by\begin{equation}\begin{aligned}\phi_{CG}(x,y,z,TE)=\gamma\int_{0}^{TE}B_{CG}(x,y,z,\tau)\,d\tau\end{aligned}\tag{4}\end{equation}and $$$\gamma$$$ is the gyromagnetic ratio.

Multi-pass acquisitions introduce a delay between echo trains of different passes (Figure 1). Off-isocenter, this results in a constant phase offset between interleaved echo trains and a non-linear phase error across sequential echoes that confounds estimation of PDFF and R2*. As a solution for CG phase error correction without requiring knowledge of the imaging gradients, we propose the following SGRE signal model:\begin{equation}s_{n,p}(\eta,R_2^*,\psi,M_0,\phi_1,…,\phi_P)=M_0e^{i\phi_p}(1-\eta+\eta\,C_F(t_n))e^{-R_2^*t_n}e^{i2\pi\psi^*{}t_n}\tag{5}\end{equation}where $$$\phi_p$$$ is a new pass-specific phase intercept (constant phase term) for the p^th pass. As shown in Figure 1, this solution recognizes that passes accrue CG-induced phase errors in parallel: at the same rate but separated by a constant phase offset. We propose to estimate that constant phase offset on a voxel-by-voxel basis. As in single-pass acquisitions, CGs will still bias the estimated B0 off-resonance (now denoted $$$\psi^*$$$).

Methods

Fitting Algorithms
All fittings were solved by nonlinear least-squares¹³, as follows, after initialization with a graph cut algorithm¹⁴.
Complex Fitting - Conventional:\begin{equation}\pmb{\widehat{\theta}}=\text{arg}\min\limits_{\pmb{\theta}}\,[\sum_{n=1}^{N}|s_n(\pmb{\theta})-s_{n,meas}|^2],\qquad\,for\,\pmb{\theta}=\{\eta,R_2^*,\psi,M_0,\phi\}\tag{6}\end{equation}
Complex Fitting - Proposed:\begin{equation}\pmb{\widehat{\theta}}=\text{arg}\min\limits_{\pmb{\theta}}\,[\sum_{p=1}^{P}\sum_{n\in{N_p}}^{}|s_{n,p}(\pmb{\theta})-s_{n,p,meas}|^2],\qquad\,for\,\pmb{\theta}=\{\eta,R_2^*,\psi,M_0,\phi_1,...,\phi_P\}\tag{7}\end{equation}
where $$$N_p$$$ denotes the set of echo indices acquired in the p^th pass.
Magnitude Fitting - Reference:\begin{equation}\pmb{\widehat{\theta}}=\text{arg}\min\limits_{\pmb{\theta}}\,[\sum_{n=1}^{N}(|s_n(\pmb{\theta})|-|s_{n,meas}|)^2],\qquad\,for\,\pmb{\theta}=\{\eta,R_2^*,M_0\}\tag{8}\end{equation}
Cramér-Rao Lower Bound Analysis
Cramér-Rao Lower Bound (CRLB) analysis¹⁵ was used to predict and compare the noise performance of the proposed (Eq. [5]), conventional (Eq. [2]), and magnitude (magnitude of Eq. [2]) fittings.

Phantom Acquisitions
Multi-pass phantom acquisitions were performed on a 3.0T clinical MRI system (Discovery MR750, GE Healthcare, Waukesha, WI) with a 32-channel phased-array torso coil (NeoCoil, Pewaukee, WI). Estimates from the conventional (with and without CG correction), magnitude, and proposed fitting methods were compared.

In-Vivo Acquisitions
Data were acquired at and 12cm off isocenter in the liver of a healthy volunteer. PDFF estimates from the conventional, magnitude, and proposed fittings were compared.

Data from 29 clinical patients, acquired in a previous study¹⁶, were reprocessed. Acquisitions were performed using a clinical protocol with the imaging volume positioned at isocenter. Regions-of-interest (ROI) were placed in each Couinaud segment of the liver¹⁷ and Bland-Altman analysis was used to evaluate the proposed fitting.

Results

Cramér-Rao Lower Bound Analysis
Figure 2 shows the CRLB results. Compared to magnitude fitting, the proposed fitting method demonstrates improved noise performance within the range of common echo time combinations.

Phantom Acquisitions
Figure 3 shows PDFF and R2* estimated along a profile in the z-direction through the middle of the water phantom. The proposed fitting method removed PDFF and R2* estimation errors up to 12% and 10s^-1, respectively, at ±12cm off isocenter.

In-Vivo Acquisitions
Figure 4 shows the proposed method reduced PDFF bias by ~10% in the liver 12cm off isocenter. Using magnitude fitting as a reference, analysis of the 29 clinical liver datasets showed the proposed fitting reduced absolute PDFF bias by 0.6% on average and reduce the coefficient of variation from 16% to 7.6% compared to the conventional fitting (Figure 5).

Discussion

We have presented a novel fitting method that provides PDFF and R2* measurements corrected for CG-induced phase errors in multi-pass acquisitions with uniformly spaced echoes. This is particularly relevant at 3.0T where multi-pass acquisitions are routinely used to achieve high spatial resolution while maintaining optimal echo spacing for water-fat separation.

CG-induced phase errors in multi-pass CSE-MRI acquisitions can result in significant estimation biases that increase quadratically away from isocenter. The severity is particularly notable for PDFF, with clinically relevant bias across the liver, even when the imaging volume is centered at isocenter. The proposed fitting enables PDFF and R2* quantification corrected for CG phase errors without prior knowledge of imaging gradient waveforms, reducing both bias and apparent variability in PDFF and R2* estimates between different locations across the liver.

Acknowledgements

This work was supported by the NIH (K24 DK102595, R01 DK083380, R01 DK088925, R01 DK100651, R01 DK117354, R41 EB025729, T32CA009206, UL1TR00427). The authors also acknowledge GE Healthcare who provides research support to the University of Wisconsin-Madison. Finally, Dr. Reeder is a Romnes Faculty Fellow, and has received an award provided by the University of Wisconsin-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation.

References

1. Caussy C, Reeder SB, Sirlin CB, Loomba R. Noninvasive, Quantitative Assessment of Liver Fat by MRI-PDFF as an Endpoint in NASH Trials. Hepatol. Baltim. Md 2018;68:763–772 doi: 10.1002/hep.29797.
2. Hernando D, Levin YS, Sirlin CB, Reeder SB. Quantification of liver iron with MRI: state of the art and remaining challenges. J Magn Reson Imaging 2014;40:1003–21 doi: 10.1002/jmri.24584.
3. Wood JC, Enriquez C, Ghugre N, et al. MRI R2 and R2* mapping accurately estimates hepatic iron concentration in transfusion-dependent thalassemia and sickle cell disease patients. Blood 2005;106:1460–1465 doi: 10.1182/blood-2004-10-3982.
4. Hernando D, Cook RJ, Qazi N, Longhurst CA, Diamond CA, Reeder SB. Complex confounder-corrected R2* mapping for liver iron quantification with MRI. Eur. Radiol. 2020 doi: 10.1007/s00330-020-07123-x.
5. Hernando D, Hines CD, Yu H, Reeder SB. Addressing phase errors in fat-water imaging using a mixed magnitude/complex fitting method. Magn Reson Med 2012;67:638–44 doi: 10.1002/mrm.23044.
6. Ruschke S, Eggers H, Kooijman H, et al. Addressing phase errors in quantitative water-fat imagin at 3T using a time-interleaved multi-echo gradient-echo acquisition. In: Proceedings of the 23rd Annual Meeting of ISMRM. Toronto, Ontario, Canada; p. 3657.
7. Ruschke S, Eggers H, Kooijman H, et al. Correction of phase errors in quantitative water–fat imaging using a monopolar time-interleaved multi-echo gradient echo sequence. Magn. Reson. Med. 2017;78:984–996 doi: https://doi.org/10.1002/mrm.26485.
8. Colgan TJ, Hernando D, Sharma SD, Reeder SB. The effects of concomitant gradients on chemical shift encoded MRI. Magn. Reson. Med. 2017;78:730–738 doi: https://doi.org/10.1002/mrm.26461.
9. Bernstein MA, Zhou XJ, Polzin JA, et al. Concomitant gradient terms in phase contrast MR: Analysis and correction. Magn. Reson. Med. 1998;39:300–308 doi: https://doi.org/10.1002/mrm.1910390218.
10. Bydder M, Yokoo T, Yu H, Carl M, Reeder SB, Sirlin CB. Constraining the initial phase in water–fat separation. Magn. Reson. Imaging 2011;29:216–221 doi: https://doi.org/10.1016/j.mri.2010.08.011.
11. Horng DE, Hernando D, Reeder SB. Quantification of Liver Fat in the Presence of Iron Overload. J Magn Reson Imaging 2017;45:428–39 doi: 10.1002/jmri.25382.
12. Yu H, Shimakawa A, McKenzie CA, Brodsky E, Brittain JH, Reeder SB. Multiecho water-fat separation and simultaneous R2* estimation with multifrequency fat spectrum modeling. Magn Reson Med 2008;60:1122–34 doi: 10.1002/mrm.21737.
13. Galassi M, Davies J, Theiler J, et al. GNU Scientific Library Reference Manual (3rd Ed.).
14. Hernando D, Kellman P, Haldar JP, Liang Z-P. Robust water/fat separation in the presence of large field inhomogeneities using a graph cut algorithm. Magn. Reson. Med. 2010;63:79–90 doi: https://doi.org/10.1002/mrm.22177.
15. Scharf LL, Mcwhorter LT. Geometry of the Cramer-Rao Bound. Signal Process. 1993;31:301–311 doi: Doi 10.1016/0165-1684(93)90088-R.
16. Hernando D, Zhao R, Taviani V, et al. Repeatability and reproducibility of confounder-corrected R2* as a biomarker of liver iron concentration: interim results from a multi-center, multi-vendor study at 1.5T and 3T. In: Montreal, Canada; 2019.
17. Campo CA, Hernando D, Schubert T, Bookwalter CA, Pay AJV, Reeder SB. Standardized Approach for ROI-Based Measurements of Proton Density Fat Fraction and R2* in the Liver. Am. J. Roentgenol. 2017;209:592–603 doi: 10.2214/AJR.17.17812.