Alexandre Triay Bagur^{1}, Chloe Hutton^{1}, Benjamin Irving^{1}, Michael L. Gyngell^{1}, Matthew D. Robson^{1}, and Michael Brady^{1}

Complex-based MRI chemical-shift encoded water-fat separation depends on
accurate field map convergence, which is often mitigated with spatial regularization.
This is prone to error propagation and over-smoothing of fat-fraction maps. Magnitude-based
separation circumvents field mapping but is reportedly limited in fat-fraction range
(0-50%). We have recently presented *MAGO*,
a magnitude-based method that resolves this water-fat ambiguity. In this study,
we compare *MAGO* to state-of-the-art fat-fraction
quantification on N=150 volunteers, and we expand the method for field map calculation
using previously estimated water and fat images. *MAGO* is comparable to regularized hybrid-based decomposition and
shows promise in higher field inhomogeneity regimes.

The *MAGO* algorithm uses the phase-constrained
signal model where $$$\phi_W(x)=\phi_F(x)=\phi(x)$$$^{6}:

$$|s[t_i]|=\left|\left(\rho_W+\rho_F\cdot\sum_{p=1}^{P}\alpha_p\cdot e^{j2\pi f_pt_i}\right)\cdot e^{j(2\pi \psi t_i+\phi)}\cdot e^{-R_2^*t_i} \right|=\left|\rho_W+\rho_F\cdot\sum_{p=1}^{P}\alpha_p\cdot e^{j2\pi f_pt_i}\right|\cdot e^{-R_2^*t_i}$$

Water $$$w(x)$$$, fat $$$f(x)$$$ and $$$R_2^*(x)$$$ are estimated directly at each pixel
$$$x$$$ from magnitude images using multi-peak fat modelling and multipoint search
coupled to non-linear optimization (ITK LevenbergMarquardtOptimizer). Two separate
runs of the algorithm at each voxel suffice to ensure correct convergence
(initial conditions $$$\left\{\rho_W,\rho_F,R_2^*\right\}_1=\left\{1000,0,50\right\}$$$ and $$$\left\{\rho_W,\rho_F,R_2^*\right\}_2=\left\{0,1000,50\right\}$$$). At each pixel independently, the converged parameters with lower
associated residual sum of squares are chosen. Field map $$$\psi(x)$$$ and phase offset $$$\phi(x)$$$ can then be estimated given $$$w(x)$$$, $$$f(x)$$$ and $$$R_2^*(x)$$$ using the full complex-valued data (Matlab *lsqcurvefit*, $$$\psi_0=\phi_0=0$$$ for all pixels) and the reduced expression

$$FW_i\equiv\left(\rho_W+\rho_F\cdot\sum_{p=1}^{P}\alpha_p\cdot e^{j2\pi f_pt_i}\right),\,\,\,R_i\equiv e^{-R_2^*t_i},\,\,\,s[t_i]/(FW_iR_i)=e^{j(2\pi \psi t_i+\phi)}$$

PDFF and field maps were calculated on N=150 UK Biobank^{7} volunteers (Siemens 1.5T, single-slice
six-echo 2D spoiled gradient echo protocol, $$$\text{TE}_1=1.2$$$ ms, $$$\Delta\text{TE}\approx2$$$ ms, 5° flip angle) with *MAGO* and
with two implementations of the *Hybrid IDEAL* algorithm^{5}, one
pixel-independent (“*IDEAL*”, $$$\psi_0=0$$$ for all pixels), the other with spatial regularization (“*RG-IDEAL*”, includes an initial region
growing step from Yu et al., 2005^{2}). One case was discarded due to poor positioning. Median hepatic PDFF and
field map values from all three methods were extracted using automatic liver segmentation
masks^{8} and compared for absolute agreement
with Bland-Altman analyses (mean ± 95% CI).

Pixel-independent *MAGO* PDFF
and field map calculation showed comparable performance to a regularized state-of-the-art
method on UK Biobank data, which generally consists of well-shimmed single-slice
acquisitions. Figure 4 includes a comparison on a peripheral slice from another
study (Siemens 1.5T) where *Hybrid IDEAL*
suffered from a full-liver swap; this shows the potential of the *MAGO* approach in more challenging cases
with higher field variation. Future work will aim to validate this approach in 3T acquisitions, notably under bipolar readouts, and explore field map regularization, e.g.

$$\widehat{\psi(x)}=\text{arg min}\sum_x|s(x)-\hat{s}(x)|^2+\lambda\cdot\text{reg}(\psi)$$

where $$$\hat{s}(x)$$$ is the estimated signal, $$$\lambda$$$ a Lagrange multiplier and $$$\text{reg}(\cdot)$$$ a suitable regularizer. We may initially follow a region growing approach for comparison, but regularizers that respect tissue boundaries are of particular interest, including anisotropic diffusion (total variation) or Markov measure fields.

- Reeder SB, Wen Z, Yu H, Pineda AR, Gold GE, Markl M, Pelc NJ. Multicoil Dixon chemical species separation with an iterative least-squares estimation method. Magn Reson Med. 2004 Jan;51(1):35-45. DOI: 10.1002/mrm.10675.
- Yu H, Reeder SB, Shimakawa A, Brittain JH, Pelc NJ. Field map estimation with a region growing scheme for iterative 3-point water-fat decomposition. Magn Reson Med. 2005 Oct;54(4):1032-9. DOI: 10.1002/mrm.20654.
- Bydder M, Yokoo T, Hamilton G, Middleton MS, Chavez AD, Schwimmer JB, Lavine JE, Sirlin CB. Relaxation effects in the quantification of fat using gradient echo imaging. Magn Reson Imaging. 2008 Apr;26(3):347-59. DOI: 10.1016/j.mri.2007.08.012.
- Bagur A, Hutton C, Irving B, Gyngell ML, Robson MD, Brady M. Magnitude-Intrinsic Water-Fat Ambiguity can be Resolved with Multi-Peak Fat Modelling and a Multipoint Search Method. [Submitted manuscript, 2018].
- Yu H, Shimakawa A, Hines CD, McKenzie CA, Hamilton G, Sirlin CB, Brittain JH, Reeder SB. Combination of complex-based and magnitude-based multiecho water-fat separation for accurate quantification of fat-fraction. Magn Reson Med. 2011. DOI: 10.1002/mrm.22840.
- Yu H, Reeder SB, McKenzie CA, Brau AC, Shimakawa A, Brittain JH, Pelc NJ. Single acquisition water-fat separation: feasibility study for dynamic imaging. Magn Reson Med. 2006 Feb;55(2):413-22. DOI: 10.1002/mrm.20771.
- Sudlow C, Gallacher J, Allen N, Beral V, Burton P, Danesh J, Downey P, Elliott P, Green J, Landray M, Liu B, Matthews P, Ong G, Pell J, Silman A, Young A, Sprosen T, Peakman T, Collins R. UK biobank: an open access resource for identifying the causes of a wide range of complex diseases of middle and old age. PLoS Med. 2015 Mar 31;12(3):e1001779. DOI: 10.1371/journal.pmed.1001779.
- Irving B, Hutton C, Dennis A, Vikal S, Mavar M, Kelly M, Brady, JM. Deep Quantitative Liver Segmentation and Vessel Exclusion to Assist in Liver Assessment. MIUA 2017. DOI: 10.1007/978-3-319-60964-5_58.

Intermediate images of the *MAGO* algorithm. The first set of initial conditions will lead to the PDFF 1 map; this
map is similar to magnitude-based PDFF maps previously reported in the
literature, where liver PDFF values are reported in the expected range, but
subcutaneous and visceral PDFF values are aliased to values below 50%. PDFF 2
has subcutaneous and visceral PDFF values in the expected range, but liver PDFF
is infeasibly high. The Final PDFF map is constructed by choosing the converged
set with lower residual sum of squares at each pixel.

Bland-Altman
comparisons show median PDFF and median field map values within the liver
segmentation masks of the following implementations: pixel-independent *Hybrid IDEAL* (i.e. without the prior
region growing step; *IDEAL* in the
plots), regularized *Hybrid IDEAL* (i.e.
with the prior region growing step; *RG-IDEAL*
in the plots), and pixel-independent *MAGO*
(*MAGO* in the plots).
Pixel-independent *MAGO* shows
excellent agreement with the regularized *Hybrid
IDEAL* algorithm. The case shown in Figure 3 had poor field map agreement
when comparing pixel-independent *MAGO*
to pixel-independent *Hybrid IDEAL*,
but excellent agreement with *RG-IDEAL*.

Convergence analysis
of pixel-independent methods. Field map values from pixel-independent *Hybrid IDEAL* (*IDEAL* in the plots) and pixel-independent *MAGO* show disagreement in the posterior liver region (automatic liver
segmentation included for reference), where a pixel was chosen for further
investigation (red square in the images). The cost function of field map
convergence shows pixel-independent *Hybrid
IDEAL* falling into a local field map minimum for regions of higher
inhomogeneity, and therefore mis-calculating PDFF. Pixel-independent *MAGO* estimates PDFF correctly, and therefore
converges to the correct field map solution.

Field map regularization
in *Hybrid IDEAL* will propagate wrong field
map and PDFF values to the entire liver if initial estimations on down-sampled
data have converged to incorrect minima. This may be the case in out-of-center
slices of multi-slice acquisitions, where field inhomogeneity is high. Pixel-independent
*MAGO PDFF* is correct over the entire
range and correct field map solutions are achieved with the previously calculated
water and fat. This suggests that the *MAGO*
approach is more robust than complex- or hybrid-based methods, where correct
PDFF estimation critically depends on correct field map convergence.