Nathan Tibbitts Roberts^{1,2}, Diego Hernando^{1,3}, Timothy J Colgan^{1}, Xiaoke Wang^{1,4}, and Scott B Reeder^{1,3,4,5,6}

Spatially varying B1 inhomogeneities and tissue fat are known to be confounders of quantitative T1 mapping methods that use multiple flip-angle techniques. Separately acquired B1 calibration maps can be used to correct flip angle errors caused by B1 inhomogeneities, but this requires an additional acquisition. In this work we propose a comprehensive approach that combines concepts from actual flip-angle imaging with variable flip-angle imaging to simultaneously estimate B1 inhomogeneity, T1, proton density fat-fraction and R2*. The feasibility and noise performance of this joint acquisition and fitting approach are evaluated using Cramer-Rao Lower Bound analysis, simulations, phantom experiments, and preliminary in vivo examples.

**Introduction**

T1 mapping using the variable flip angle (VFA) method has multiple applications, particularly when imaging speed and motion robustness are important (1,2). However, VFA is confounded by the presence of tissue fat (due to its short T1) and B1 inhomogeneities (which cause spatially varying flip angle errors) (3,4).

The presence of fat can be addressed through fat-water separated chemical shift encoded MRI (CSE-MRI) techniques (5,6); however, standard B1-corrected VFA methods require a separate calibration acquisition (4,7).

**Therefore, the purpose of this work** is to present a novel B1- and fat-corrected CSE-MRI T1 mapping method. Our approach combines an alternating TR SGRE
“actual flip angle” (AFI) acquisition strategy (7)
with a traditional VFA strategy (2)
to jointly estimate B1- and fat-corrected T1, as well as provide proton density
fat fraction (PDFF) and R2* maps without an additional calibration scan.

AFI allows fast and accurate 3D B1 measurements through repeated SGRE excitations with identical flip angles and alternating TRs (7). Similar to work done by Hurley et al (8), we propose a joint estimation, including a term for B1 inhomogeneity ($$$\beta$$$), from one AFI dataset and one VFA SGRE dataset; however, distinct from this previous work, we utilize a multi-echo, fat/water separated approach (9-11). Given complete spoiling of transverse magnetization after each TR, the respective steady-state signals are as follows:

$$S_{1}(TR_{1},TR_{2},\alpha_{1},\alpha_{2},TE;\theta)=(\rho_{W}\cdot{K(TR_{1},TR_{2},\alpha_{1},\alpha_{2},T1_{W})}+\rho_{F}\cdot{K(TR_{1},TR_{2},\alpha_{1},\alpha_{2},T1_{F})}\cdot{C_{n}})\cdot{e^{-R2^{*}\cdot{TE}}}\cdot{e^{i(2\pi\psi\cdot{TE}+\phi)}}\\\\S_{2}(TR_{1},TR_{2},\alpha_{1},\alpha_{2},TE;\theta)=(\rho_{W}\cdot{K(TR_{2},TR_{1},\alpha_{2},\alpha_{1},T1_{W})}+\rho_{F}\cdot{K(TR_{2},TR_{1},\alpha_{2},\alpha_{1},T1_{F})}\cdot{C_{n}})\cdot{e^{-R2^{*}\cdot{TE}}}\cdot{e^{i(2\pi\psi\cdot{TE}+\phi)}}\\\\S_{3}(TR_{3},\alpha_{3},TE;\theta,\beta)=(\rho_{W}\frac{sin(\beta\cdot{\alpha_{3}})\cdot{(1-e^{-\frac{TR_{3}}{T1_{W}}}})}{1-e^{-\frac{TR_{3}}{T1_{W}}}\cdot{cos(\beta\cdot{\alpha_{3}})}}+\rho_{F}\frac{sin(\beta\cdot{\alpha_{3}})\cdot{(1-e^{-\frac{TR_{3}}{T1_{F}}}})}{1-e^{-\frac{TR_{3}}{T1_{F}}}\cdot{cos(\beta\cdot{\alpha_{3}})}}\cdot{C_{n}})\cdot{e^{-R2^{*}\cdot{TE}}}\cdot{e^{i(2\pi\psi\cdot{TE}+\phi)}}$$

Where:

$$$K(TR_{A},TR_{B},\alpha_{A},\alpha_{B},T1)=\frac{sin(\beta\cdot{\alpha_{A}})(1-e^{-\frac{TR_{B}}{T1}}+(1-e^{-\frac{TR_{A}}{T1}})\cdot{e^{-\frac{TR_{B}}{T1}}cos(\beta\cdot{\alpha_{B}})})}{1-e^{-\frac{TR_{A}}{T1}}e^{-\frac{TR_{B}}{T1}}cos(\beta\cdot{\alpha_{A}})cos(\beta\cdot{\alpha_{B}})}\\\theta=(\beta,~T1_{W},~T1_{F},~\rho_{W},~\rho_{F},~R2*,~\phi,~\psi)$$$

$$$\rho_{W}~\text{and}~\rho_{F}~=~\text{real-valued}~\text{signals}~\text{from}~\text{water}~\text{and}~\text{fat}$$$

$$$T1_{W}~\text{and}~T1_{F}~\text{=}~\text{T1}~\text{of}~\text{water}~\text{and}~\text{fat}$$$

$$$\phi~=~\text{initial}~\text{phase}~\text{(11)}$$$

$$$\psi~=~\text{fieldmap}$$$

$$$\theta$$$ is the set of unknown parameters; β relates the transmitted flip angle (α_{T}) to prescribed flip angle (α_{P}) by the equation α_{T}=βα_{P}; Fat is corrected by the inclusion of a 6-peak spectral model, denoted $$$C_{n}$$$(9,12).

A diagram and description of the proposed pulse sequence are shown in Figure 1.

Optimization of Acquisition Parameters

A Cramer-Rao Lower Bound (CRLB) (13) was calculated for the acquisition strategy
described above (ETL_{1}=6, ETL_{2}=1, ETL_{3}=2) and used to determine the set of SNR-optimal acquisition
parameters (SNR defined as: average |S_{n}|/σ,
where σ=standard deviation of the noise). Optimization was constrained to realizable TRs,
TEs, and flip angles.

Numerical Simulations

The proposed acquisition and a nonlinear least squares joint
reconstruction strategy was tested in simulated data using common 1.5T liver tissue
parameters, i.e. β=0.95,T1_{W}=576ms,T1_{F}=288ms,PDFF=20%,R2*=30 (14,15)
and acquisition parameters determined by the CRLB and scan time constraints (α_{1}=60^{o}/α_{2}=60^{o}/α_{3}=3^{o},
TR_{1}=29.4ms/TR_{2}=4.9ms/TR_{3}=6.8ms, TE_{1}=1.3ms,ΔTE=2ms, ETL_{1}=6,ETL_{2}=1,ETL_{3}=2). For each SNR value (ranging 5-50), 10000
realizations of CSE-MRI were simulated with added zero-mean complex Gaussian
noise. For each realization, a joint nonlinear least squares fitting algorithm
was used to estimate the set of unknown parameters.

Phantom Acquisitions

The proposed acquisition and reconstruction strategy was
tested in a multi-vial agar gel phantom including varying T1_{W} and
PDFF values.
Imaging was performed with a 32-channel body
coil on a 1.5T system (GE Healthcare Optima
MR450W, Waukesha, WI) with acquisition parameters identical to the numerical
simulation. Multi-TE multi-TR spectroscopy (STEAM) (16)
was acquired once in each vial to provide reference T1 and PDFF measurements.
Parameters were estimated
using nonlinear least squares and a region of interest (ROI) analysis compared
our proposed method to STEAM-based measurements in each vial.

In Vivo Acquisition

The proposed acquisition and reconstruction
strategy was tested with a 16-channel Flex coil on a 1.5T system (GE Healthcare
Optima MR450W, Waukesha, WI) in the knee of a healthy volunteer. The following
acquisition parameters were used: α_{1}=60^{o}/α_{2}=60^{o}/α_{3}=3^{o},
TR_{1}=80ms/TR_{2}=5.5ms/TR_{3}=10ms, TE_{1}=1.4ms,ΔTE=2.2ms, ETL_{1}=6,ETL_{2}=1,ETL_{3}=2.

Optimization of Acquisition Parameters

CRLB analysis showed T1_{W} estimation to be
largely orthogonal to PDFF and R2* estimation which dictated optimal
echo time choices.
Plots showing the CRLB optimization are
shown in Figure 2.

Numerical Simulations

Monte-Carlo simulations demonstrated the proposed strategy converges to unbiased estimators for all parameters of interest. Figure 3 shows mean and standard deviations of estimated β, T1W, R2*, and PDFF.

Phantom Acquisitions

Phantom experiment results showed linear agreement with STEAM, with variability of estimated T1 increasing with T1. Results are plotted in Figure 4.

In Vivo Acquisition

Reconstructed images are shown in Figure 5.

**Discussion **

In this work we have successfully presented a novel B1- and fat-corrected CSE-MRI T1 mapping method. This strategy combines alternating TR SGRE acquisitions with traditional VFA strategies to estimate B1- and fat-corrected T1. Theoretical analysis determined optimal acquisition parameters and performance was confirmed in Monte-Carlo simulations and phantom experiments.

Continued development will focus on improving estimation performance over a wider range of T1 values to address the variability of T1 estimates in the phantom experiments. Additionally, the STEAM MRS pulse sequence used to make reference measurements in the phantom experiments is susceptible to B1 inhomogeneities. Future work will validate performance of the proposed strategy against flip-angle corrected T1 mapping.

Overall, the proposed strategy demonstrates initial feasibility for a B1- and fat-corrected T1-mapping technique without needing a separate B1 calibration scan.

1. Brookes JA, Redpath TW, Gilbert FJ, Murray AD, Staff RT. Accuracy of T1 measurement in dynamic contrast-enhanced breast MRI using two- and three-dimensional variable flip angle fast low-angle shot. Journal of Magnetic Resonance Imaging 1999;9(2):163-171.

2. Deoni SC, Rutt BK, Peters TM. Rapid combined T1 and T2 mapping using gradient recalled acquisition in the steady state. Magn Reson Med 2003;49(3):515-526.

3. Deoni SC. Correction of main and transmit magnetic field (B0 and B1) inhomogeneity effects in multicomponent-driven equilibrium single-pulse observation of T1 and T2. Magn Reson Med 2011;65(4):1021-1035.

4. Cheng H-LM, Wright GA. Rapid high-resolution T1 mapping by variable flip angles: Accurate and precise measurements in the presence of radiofrequency field inhomogeneity. Magnetic Resonance in Medicine 2006;55(3):566-574.

5. Liu CY, McKenzie CA, Yu H, Brittain JH, Reeder SB. Fat quantification with IDEAL gradient echo imaging: correction of bias from T(1) and noise. Magn Reson Med 2007;58(2):354-364.

6. Wang X, Hernando D, Wiens C, S. R. Fast T1 Correction for Fat Quantification Using a Dual-TR Chemical Shift Encoded MRI Acquisition. . 2017.

7. Yarnykh VL. Actual flip-angle imaging in the pulsed steady state: a method for rapid three-dimensional mapping of the transmitted radiofrequency field. Magn Reson Med 2007;57(1):192-200.

8. Hurley SA, Yarnykh VL, Johnson KM, Field AS, Alexander AL, Samsonov AA. Simultaneous Variable Flip Angle – Actual Flip Angle Imaging (VAFI) Method for Improved Accuracy and Precision of Three-dimensional T1 and B1 Measurements. Magnetic Resonance in Medicine 2012;68(1):54-64.

9. 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(5):1122-1134.

10. Horng DE, Hernando D, Hines CD, Reeder SB. Comparison of R2* correction methods for accurate fat quantification in fatty liver. J Magn Reson Imaging 2013;37(2):414-422.

11. Bydder M, Yokoo T, Yu H, Carl M, Reeder SB, Sirlin CB. Constraining the initial phase in water–fat separation. Magnetic Resonance Imaging 2011;29(2):216-221.

12. Hamilton G, Yokoo T, Bydder M, Cruite I, Schroeder ME, Sirlin CB, Middleton MS. In vivo characterization of the liver fat (1)H MR spectrum. NMR Biomed 2011;24(7):784-790.

13. Scharf LL, Mcwhorter LT. Geometry of the Cramer-Rao Bound. Signal Process 1993;31(3):301-311.

14. Stanisz GJ, Odrobina EE, Pun J, Escaravage M, Graham SJ, Bronskill MJ, Henkelman RM. T1, T2 relaxation and magnetization transfer in tissue at 3T. Magn Reson Med 2005;54(3):507-512.

15. Gold GE, Han E, Stainsby J, Wright G, Brittain J, Beaulieu C. Musculoskeletal MRI at 3.0 T: relaxation times and image contrast. AJR American journal of roentgenology 2004;183(2):343-351.

16. Hamilton G, Middleton MS, Hooker JC, Haufe WM, Forbang NI, Allison MA, Loomba R, Sirlin CB. In vivo breath-hold (1) H MRS simultaneous estimation of liver proton density fat fraction, and T1 and T2 of water and fat, with a multi-TR, multi-TE sequence. J Magn Reson Imaging 2015;42(6):1538-1543.

Figure 1. We propose a
novel CSE-MRI acquisition strategy that combines concepts from AFI and VFA to jointly
estimate B1, T1, PDFF and R2*. Two passes are acquired in the same overall
acquisition. In the first pass, ETL_{1} echoes are acquired in an
excitation with flip angle α_{1} and delay TR_{1} followed by an
excitation with flip angle α_{2} and delay TR_{2} in
which ETL_{2} echoes are acquired. In the second pass, ETL_{3}
echoes are acquired in an excitation with flip angle α_{3} and delay TR_{3}. Acquired data are used to
jointly estimate B1, T1, PDFF, and R2*.

Figure 2. Cramer-Rao Lower Bound
analysis revealed optimal flip angle values of 60^{o}, 60^{o},
and 3^{o} (white “X”) for
achieving minimum variance of T1_{W} estimated by the proposed
acquisition strategy.

Figure 3.
The proposed acquisition strategy converges to an unbiased estimator of T1_{W},
B1 inhomogeneity, R2*, and PDFF in simulation as SNR increases. Data were simulated with: β=0.95, T1W=576ms, T1F=288ms,
PDFF=20%, R2*=30
and with acquisition parameters: α_{1}=60^{o},
α_{2}=60^{o},
α_{3}=3^{o};
TR1=29.4ms, TR2=4.9ms, TR3=6.8ms; TE_{1}=1.3ms,
ΔTE=2ms. The average (solid line) and standard
deviation (fill) of 10000 simulated estimates per SNR is plotted for β,
T1W, R2* and PDFF.

Figure 4. The proposed
acquisition strategy demonstrated good linear agreement with reference measurements
from STEAM-MRS in a gel agarose phantom of 32 vials of varying T1 and PDFF
values. T1 and PDFF
were jointly estimated using nonlinear least squares and seven ROIs were drawn in
each vial (1 ROI/vial/slice) in the estimated maps. The mean of
each ROI was compared to the single STEAM measurement taken in each vial. Each
point and error-bar represents the mean and standard deviation within a single
vial. Linear regressions of STEAM-MRS
versus the proposed method are shown above for T1_{W} (A) and PDFF (B).

Figure 5. Examples
reconstructed T1_{W}, T1_{F}, PDFF, and β maps taken in two axial
slices of a knee of a healthy volunteer. Measurements in several ROIs are
shown.