Jason Ostenson^{1,2}, Bruce M. Damon^{1,2,3,4,5}, and E. Brian Welch^{1,3,5}

Unbalanced steady-state free precession MR fingerprinting (uSSFP-MRF) may be used to quantify fat signal as well as water T1 and T2, but is subject to off-resonance (B0) and blurring. We propose a variable echo time uSSFP and post-processing method that estimates fat signal fraction, water T1 and T2, and B0 within a single sequence. The method is tested in phantoms and in vivo. Preliminary results indicate the proposed method provides sharp fat fraction maps that account for chemical-shift and B0 effects that normally lead to blurring, as well as generate water-only T1 and T2 maps.

Using a modification of the method of Brodsky et al.^{5} combined with singular value decomposition^{}^{6}, the on-resonance fat-water MRF signal vector in k-space $$$\mathbf{s}_{\mathbf{o}}\left( \mathbf{k} \right)$$$ can be modeled as $$\mathbf{s}_{\mathbf{o}}\left( \mathbf{k} \right)\mathbf{= \ }\left\lbrack \mathbf{U}_{\mathbf{o}}\ \mathbf{F} \right\rbrack\mathbf{\beta}\left( \mathbf{k} \right) \qquad (1),$$ where $$$\mathbf{U}_{\mathbf{o}}$$$ is the matrix of left-singular dictionary vectors of on-resonance water signal, $$$\mathbf{F}$$$ is the fat signal model, and $$$\mathbf{\beta}$$$ is a vector of weights. The off-resonance fat-water signal, $$$\mathbf{s}\left( \mathbf{k} \right)$$$, is $$\mathbf{s}\left( \mathbf{k} \right)\mathbf{= \ }\mathbf{Uc(k)} \qquad (2),$$ where $$$\mathbf{U}$$$ is the matrix of left-singular dictionary vectors that includes T1, T2, and off-resonant/chemical shift effects and $$$\mathbf{c(k)}$$$ is the coefficient vector. If a fixed repetition time and linear ramp of TEs are used over the MRF acquisition, then when using multi-frequency interpolation (MFI),^{7-8} the signal corrected for off-resonance effects can be given as $$\mathbf{s}_{\mathbf{o}}\left( \mathbf{k} \right)\mathbf{\approx \ }\sum_{l\mathbf{=}1}^{L}a_{l}\mathbf{J}_{l}\mathbf{s(k)} \qquad (3);$$ $$$\left\{ \mathbf{J}_{l} \right\}$$$ are diagonal matrices that modify the phase of $$$\mathbf{s(k)}$$$ according to specific frequencies, and $$$\left\{ a_{l} \right\}$$$ are the MFI weights. A set of fat-water coefficients is obtained by fitting the phase modulated measurements $$$\left\{ \mathbf{J}_{l}\mathbf{s(k)} \right\}$$$ in Eq. (3) to Eq. (1), projecting along the columns of $$$\mathbf{U}$$$ using Eq. (2), and added to projected residuals to give total measurement coefficients $$$\left\{ \mathbf{c}_{l\mathbf{,}\text{tot}}\mathbf{(k)} \right\}$$$. By inverse Fourier transform of the coefficients, the on-resonance solution at position $$$\mathbf{r}$$$ can be expressed as $$\mathbf{s}_{\mathbf{o}}\left( \mathbf{r} \right)\mathbf{\approx \ }\sum_{l\mathbf{=}1}^{L}a_{l}\mathbf{(}\Delta B_{o})\mathbf{U}\mathbf{c}_{l\mathbf{,}\text{tot}}\mathbf{(r)} \qquad (4).$$ $$$\left\{ a_{l}\mathbf{}(\Delta B_{o}) \right\}$$$ depend on a single variable, $$$\mathbf{}\Delta B_{o}(\mathbf{r})$$$. Eq. (1) in the spatial domain also holds. Using variable projection, the residuals $$$\mathbf{\sigma}$$$ to the solution $$$\mathbf{s}_{\mathbf{o}}\left( \mathbf{r} \right)$$$ can be given as $$\mathbf{\sigma = \ }\left\lbrack \mathbf{I -}\left\lbrack \mathbf{U}_{\mathbf{o}}\ \mathbf{F} \right\rbrack\left\lbrack \mathbf{U}_{\mathbf{o}}\ \mathbf{F} \right\rbrack^{\mathbf{\dagger}} \right\rbrack\mathbf{\ }\sum_{l\mathbf{=}1}^{L}a_{l}\mathbf{}(\Delta B_{o})\mathbf{U}\mathbf{c}_{l\mathbf{,}\text{tot}}\mathbf{(r)} \qquad (5),$$ with $$$\mathbf{\dagger}$$$ designating the pseudo-inverse. By exhaustive search of discretized frequencies using pre-calculated $$$\left\{ a_{l}\mathbf{}(\Delta B_{o}) \right\}$$$, the off-resonance frequency that minimizes error can be found for each position, giving an estimated $$$\mathbf{}\Delta B_{o}$$$ and on-resonance fat and water coefficients. The water signals can then be reconstructed and fitted to a water dictionary to estimate T1 and T2.

Measurements were performed using 3 T Philips Achieva
and Ingenia scanners (Philips Healthcare, The Netherlands) with either a
transmit-receive head coil or multi-channel anterior and integrated posterior coils. The MRF sequence used
the flip angle pattern reported by Jiang et al.,^{1} a fixed TR of 18 ms, and a linearly ramped TE
from 3.25 to 7.25 ms. Variable density spirals with an acquisition time of 7 ms
and independently measured k-space trajectories^{9} with gridded reconstruction were used for
spatial encoding and reconstruction. Fitting for FSF and B0 with the proposed
method used 31 off-resonance MFI basis frequencies and the SVD basis rank preserved
99.99% of the dictionary energy. The fat model consisted of six peaks with
previously reported T1 and T2 values for adipose tissue.^{10} Reference fat fraction measurements used
multi-echo spoiled gradient echo scans and post-processing by a graph cut algorithm^{11} or scanner software. Phantom experiments
consisted of seven 50 mL tubes of varying fat fraction
subject to off-resonance conditions and a water-only MRI system phantom^{12} subject to off-resonances. One
subject’s liver was scanned after providing written informed consent.

1. Jiang Y, Ma D, Seiberlich N, Gulani V, Griswold MA. MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout. Magnetic Resonance in Medicine. 2015;74(6):1621–1631.

2. Han D, Kim M-O, Lee H, Hong T, Kim D-H. A Free-breathing water/fat separation and T1, T2 quantification method using dual TR FISP in abdomen. In: Proc Intl Soc Mag Reson Med. 2016. p. 575.

3. Ostenson J, Welch EB. Fat Signal Fraction Determination Using MR Fingerprinting. Proc Intl Soc Mag Reson Med. 2017:134.

4. Cencici M, Biagi L, Kaggie J, Schulte RF, Rosetti MT, Buonincontri G. MR Fingerprinting in the knee with dictionary-based fat and water separation. In: ISMRM Workshop on MRF. 2017.

5. Brodsky EK, Holmes JH, Yu H, Reeder SB. Generalized k-space decomposition with chemical shift correction for non-Cartesian water-fat imaging. Magnetic Resonance in Medicine. 2008;59(5):1151–64.

6. McGivney DF, Pierre E, Ma D, Jiang Y, Saybasili H, Gulani V, Griswold MA. SVD compression for magnetic resonance fingerprinting in the time domain. IEEE Transactions on Medical Imaging. 2014;33(12):2311–2322.

7. Man LC, Pauly JM, Macovski A. Multifrequency interpolation for fast off-resonance correction. Magnetic resonance in medicine. 1997;37(5):785–92.

8. Ostenson J, Robison RK, Zwart NR, Brian Welch E. Multi-frequency interpolation in spiral magnetic resonance fingerprinting for correction of off-resonance blurring. Magnetic Resonance Imaging. 2017:1–10.

9. Welch EB, Robison RK, Harkins KD. Robust k-space trajectory mapping with data readout concatenation and automated phase unwrapping reference point identi cation. In: Proc Intl Soc Mag Reson Med. 2017. p. 1387.

10. Hamilton G, Smith DL, Bydder M, Nayak KS, Hu HH. MR properties of brown and white adipose tissues. Journal of Magnetic Resonance Imaging. 2011 [accessed 2011 Jul 20];34(2):468–473. http://doi.wiley.com/10.1002/jmri.22623

11. Hernando D, Kellman P, Haldar JP, Liang ZP. Robust water/fat separation in the presence of large field inhomogeneities using a graph cut algorithm. Magnetic Resonance in Medicine. 2010;63(1):79–90.

12. Keenan KE, Stupic KF, Boss MA, Russek SE, Chenevert TL, Prasad P V, Reddick WE, Cecil KM, Zheng J, Hu P, et al. Multi-site, multi-vendor comparison of T1 measurement using ISMRM/NIST system phantom. In: Proc Intl Soc Mag Reson Med. 2016. p. 3290.

Figure 1. The fat signal fraction (FSF) maps from a
Cartesian acquired multi-echo Dixon measurement (a) are shown in comparison
with MR fingerprinting FSF maps generated using a direct dictionary match in
the image domain (b); fitting in k-space without considering B0 (c); and
k-space fitting with B0 (d). Chemical shift and B0 cause blurring in the image
domain, which confound the FSF measurement.

Figure 2. MR fingerprinting (MRF) T1 (a) and T2 (b) maps of an MRI system phantom using the variable echo time MRF sequence in this study are shown under homogeneous B0 conditions ("well shimmed"), as measured
by a Cartesian acquired B0 map (c). The
T1 (d) and T2 (e) maps exhibit blurring and large bias when subjected to off-resonance ("poorly shimmed") as seen in the
Cartesian B0 map (f). The proposed post-processing method improves the T1 and T2 fitting and
simultaneously yields a B0 map (i) under the same off-resonance condition
as (d-f).

Figure 3. The fat signal fraction (FSF) maps from a
Cartesian acquired multi-echo Dixon map (a) and the proposed MR fingerprinting (MRF)
method (b) are displayed. MRF T1 (c) and T2 (d) maps are simultaneously
estimated using the proposed method. B1+ mapping for corrections to the MRF
fitting were not applied.