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

Spatially varying B_{1} inhomogeneities and fat
content are well known to be confounders of quantitative T_{1} mapping that
use multi flip angle techniques. Separate B_{1} calibration maps can be
acquired to correct flip angle errors cause by B_{1} inhomogeneities, but this
requires an additional acquisition. In this work we consider an alternative
approach that acquires variable flip angles and variable repetition times, and
simultaneously estimates B_{1} inhomogeneity, T_{1}, fat-fraction and R_{2}*. The
feasibility and noise performance of this approach are evaluated using
theoretical Cramer-Rao Lower Bound analysis and simulations. Our results
demonstrate that this approach is feasible, but suffers from relatively poor
noise performance at typical acquisition parameters.

T_{1} mapping using the variable flip angle (VFA)
method has multiple applications, particularly for body imaging where imaging
speed and motion robustness are important (1,2).
However, VFA is affected by two confounding factors: the presence of 1) fat and
2) B_{1} inhomogeneities.

The presence of fat results confounds T_{1} mapping of tissue
parenchyma due to the short T_{1} of fat signals. The presence of fat can be
addressed through fat-water separated chemical shift encoded (CSE) techniques (3,4).

B_{1} inhomogeneities result in spatially varying flip angle
errors which, if uncorrected, lead to significant errors in VFA-based T_{1}
mapping (5,6).
Previous B_{1}-corrected VFA methods use an external B_{1} calibration acquisition to
correct for spatially varying B_{1} errors (6,7).
In this work, we consider extending the CSE VFA method to jointly estimate fat-fraction
and R_{2}* maps, as well as B_{1}, T_{1W}, and T_{1F} maps
without an additional B_{1} calibration scan. **The
purpose of this work is to propose and analyze a novel B _{1}- and fat-corrected
CSE-MRI T_{1} mapping method.**

Transmitted
flip angle (α_{T}) is related to prescribed flip angle (α_{P})
by the equation:

α_{T}=βα_{P},

where
β is the B_{1}
calibration
coefficient. The
spoiled gradient echo (SGRE) signal model (8-10) can
be rewritten taking advantage of the relationship between transmitted flip
angle, prescribed flip angle, and B_{1} inhomogeneity as:

$$\pmb{S_n(\beta,\theta,\text{TR}_n,\alpha_{P_n},\text{TE}_n)=(\rho_W\frac{\text{Sin}\left(\beta\alpha_P\right)\left(1-e^{-\text{TR}\left/T_{1w}\right.}\right)}{1-\cos\left(\beta\alpha_P\right)e^{-\text{TR}\left/T_{1w}\right.}}+\rho_F\frac{\text{Sin}\left(\beta\alpha_P\right)\left(1-e^{-\text{TR}\left/T_{1F}\right.}\right)}{1-\cos\left(\beta\alpha_P\right)e^{-\text{TR}\left/T_{1F}\right.}}(\sum_{m=1}^Ma_me^{\text{j2$\pi\mathit{f}$}_m\text{TE}}))e^{j(\phi+\text{$\psi$TE})-R_2^*\text{TE}}}$$

Where θ=(ρ_{W}, ρ_{F}, T_{1W}, T_{1F}, Φ, ψ, R_{2}*) is the set of unknown parameters where ρ_{W} and ρ_{F} are the real-valued signals from water and
fat, respectively, and T_{1F} and T_{1W} are their respective T_{1}
values; Φ is the initial phase for the fat and water
signals (10); and ψ is the magnetic field
inhomogeneity map. Additionally fat is jointly estimated, and therefore inherently
corrected, by the inclusion of a 6-peak spectral model (11) of fat, where a_{m} and f_{m} are the
corresponding relative amplitudes and frequencies of fat signals (8).

The subsequent sections examine the theoretical limitations of simultaneously estimating β as an unknown parameter from SGRE CSE-MRI data using variable flip angles and multiple TRs.

Analysis was designed for data acquired with 3 flip angles (α_{P}), 3 TRs, and
6 echo times per TR/α_{P}
combination, resulting in 18 total acquisitions per reconstruction.

Optimization of Acquisition Parameters

A Cramer-Rao Lower Bound (CRLB) (12)
was calculated for the 18-acquisition strategy described above. Results were used
to determine the set of optimal acquisition parameters that would result in the
smallest standard deviation of T_{1W} when β ≈1.
Optimization was constrained to “reasonable” TR, TE, and flip angle (α_{P}) limits,
such that TR<20ms, 1ms<TE_{1}<3ms, ΔTE >0.25ms, and 1^{o}< α_{P}<60^{o}.
Additionally, SNR, defined here as mean(|S_{n}|)/σ, where σ=standard deviation of
the noise), was set to a constant value of 60, which is an achievable
value, eg: for 3D CSE-MRI of the liver (13).

Numerical Simulations

For a given set of unknown parameters including a small
intentional B_{1} inhomogeneity (β=0.9), T_{1W}=822ms, and PDFF=5% and
acquisition parameters selected with CRLB analysis. 10,000 realizations of CSE-MRI
were simulated with added zero-mean complex Gaussian noise. For each
realization, a nonlinear least squares algorithm was used to estimate the set
of unknown parameters, including
β. This was repeated for SNR values over a range of 10-100.

A second set of identical simulations was performed, except β was not estimated but was instead erroneously assumed to be 1.

A final set of identical simulations was repeated, expect SNR was fixed at 60 and β was varied (0.5-1.5) for sets of 1,000 realizations.

Optimization of Acquisition Parameters

CRLB analysis determined optimal α_{P}/TR pairs of 20^{o}/20ms, 5^{o}/20ms,
and 60^{o}/20ms, with an initial echo time of TE_{1}=1.1ms
and echo spacing ΔTE=0.3ms.
Plots showing the CRLB optimization space are shown in Figure 1.

Numerical Simulations

Numerical simulations demonstrated convergence to an
unbiased estimator for both T_{1W} and β when average SNR is approximately 80 when
estimating β
simultaneously. Results also demonstrated that relatively small flip
angle errors cause large bias in T_{1} estimation (Figures 2,3).
Results also show β
dependent bias in T_{1} estimation for both simultaneously estimating
and ignoring β methods (Figure
4).

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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. 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.

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

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8. 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.

9. 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.

10. 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.

11. 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.

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13. Motosugi U, Hernando D, Wiens C, Bannas P, Reeder SB. High SNR Acquisitions Improve the Repeatability of Liver Fat Quantification Using Confounder-corrected Chemical Shift-encoded MR Imaging. Magn Reson Imaging 2017.

Figure 1. Optimal acquisition parameters using Cramer-Rao
Bound analysis show a preference for longer TRs, and specific flip angle pairs,
but is fairly robust to choice of echo times. Shown above are regions
surrounding the parameters that result theoretically in the lowest variability
achievable with an unbiased estimator of T_{1W}. Any parameter not
specified in each plot is held constant at the optimal parameters (Flip Angle/TR: 20^{o}/20ms,
5^{o}/20ms, and 60^{o}/20ms, TE_{1}=1.1ms, ΔTE=0.3ms). The stars indicate
the optimal acquisition parameters. SNR has been fixed at 60.

Figure 2. Simultaneous estimation of – and correction for – B_{1} inhomogeneities
(β) is feasible when average SNR is sufficiently high. Monte Carlo simulations
demonstrate large variability, but low bias when β is estimated. The plot on
the right has non-zero variability because β is estimated as an unknown
parameter, where the simulated truth is β=0.9.

Figure
3. Uncorrected B_{1} inhomogeneities can lead to substantial T_{1} estimation bias.
Monte Carlo simulations demonstrate low variability, but large bias when β is
erroneously assumed to be 1. The plot on the right has zero variability because
β is not estimated, but instead assumed to be 1, although the simulated truth
is β=0.9.

Figure 4. Even at a fixed SNR, estimated T_{1} is strongly
dependent on B_{1} inhomogeneity. Further, when uncorrected (assuming β=1), B_{1}
inhomogeneity can result
in large T_{1} bias. Of note is the difficulty of simultaneously estimating β when
β<0.25; this is possibly a result of using acquisition parameters optimized for
β ≈1 and may
improve with a different choice of parameters.