Zhengguo Tan^{1}, Dirk Voit^{1}, Jost M Kollmeier^{1}, Martin Uecker^{2,3}, and Jens Frahm^{1,3}

To achieve dynamic water/fat separation even in the presence of rapid physiological motions and large magnetic field inhomogeneities, this work presents a multi-echo multi-spoke radial FLASH sequence and a model-based non-linear inverse reconstruction. Asymmetric echoes are integrated into the sequence to shorten echo times. A spatial-smoothness constraint on field inhomogeneity maps is developed to counteract local minima in the non-convex inverse problem.

Figure
1 illustrates the multi-echo multi-spoke radial FLASH sequence and its corresponding
k-space trajectory, where an echo asymmetry 75% is integrated to shorten TE and TR^{7}. To achieve optimal coverage of k-space, radial
spokes with the same TE are uniformly distributed in k-space, and the
incremental angle between frames is empirically chosen as Golden angle (68.75^{o}).

The signal model is:

$$F_{j,l} (x) = P_l \mathcal{F} \{ (W + F \cdot z_l) \cdot e^{i 2\pi f_{B0} \text{TE}_l \cdot c_j} \} \; \text{with} \; x = (W, F, f_{B0}, c_1, ... c_N)^T$$

Here, $$$P_l$$$ and $$$\mathcal{F}$$$ are the sampling pattern for the $$$l$$$th echo and 2D FFT, respectively. The unknown $$$x$$$ consists of water ($$$W$$$), fat ($$$F$$$), B0 field inhomogeneity (off-resonance) frequency ($$$f_{B0}$$$), and coil sensitivity maps ($$$c_j$$$). The fat modulation^{8} follows $$$z_l = \sum_{p=1}^{6} a_p \cdot e^{i 2\pi f_p \text{TE}_l}$$$. To jointly estimate all unknowns, the cost function is

$$\Phi(\hat{x}) = \text{argmin}_{\hat{x}} \left \| y - F(T\hat{x}) \right \|_2^2 + \alpha \left \| \hat{x} \right \|_2^2 \; \text{with} \; x = T\hat{x} $$

$$$y$$$ is the gridded k-space data without roll-off corrections. The weighting matrix is $$$T = \mathcal{F}^{-1} \Big(1 + w \cdot \left \| \vec{k} \right \| \Big)^{-h}$$$, with $$$w=11$$$ and $$$h=18$$$ for the B0 field map, $$$w=880$$$ and $$$h=16$$$ for coil sensitivity maps, while an identity matrix ($$$T=I$$$) is applied onto water and fat maps. $$$\vec{k}$$$ is a 2D Cartesian grid matrix. Although weaker than coil sensitivity maps, the weighting on the B0
field map enforces spatial smoothness and assures
accuracy. This cost function is minimized by the iteratively regularized
Gauss-Newton method^{9} with automatic scaling^{10} for water and fat, while the
scaling for the B0 field map is kept as 1.5 to warrant convergence. The reconstruction starts with $$$W=F=1$$$, $$$f_{B0}=0$$$, and $$$c_j=0$$$, while the initialization for the following frames is set as the estimate
from the preceding frame damped by 0.9 to enforce temporal continuity.

Data acquisitions were performed on a 3T scanner (Magnetom Prisma,
Siemens Healthineers, Erlangen, Germany) with an 18-channel body matrix coil. Acquisition
parameters were: 8^{o} FA, standard shimming, 1560 Hz/Px bandwidth, 320 x 320 mm^{2} FoV, 200 x 200 matrix size, 1.6 x 1.6 x 6 mm^{3} spatial resolution, and 40 ms temporal resolution with 9 RF excitations per frame and TR = 4.43 ms, TE = 1.26/2.66/3.69 ms.

In addition, a whole-body scan with the
built-in 2-channel body receiver coil was conducted. The volunteer was pulled
through the isocenter from the lower leg to the head. Acquisition parameters
were: 16^{o} FA, tune-up shimming, 1200 Hz/Px bandwidth, 448 x 448 mm^{2} FoV, 320 x 320 matrix size, 1.4 x 1.4 x 10 mm^{3} spatial resolution, and 50 ms temporal resolution with 9 RF excitations per frame and TR = 5.54 ms, TE = 1.54/3.12/4.70 ms. This
scan covers the whole body within only 40 s. To accompany with rapid change of anatomies from slice to
slice, a real non-negative constraint is applied on water and fat maps during
image reconstructions.

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