Zhengguo Tan^{1,2}, Sebastian Rosenzweig^{1,2}, Xiaoqing Wang^{1,2}, Nick Scholand^{1,2}, H Christian M Holme^{1,2}, Moritz Blumenthal^{1}, and Martin Uecker^{1,2,3,4}

^{1}Institute for Diagnostic and Interventional Radiology, University Medical Center Göttingen, Göttingen, Germany, ^{2}German Center for Cardiovascular Research (DZHK), Göttingen, Germany, ^{3}Cluster of Excellence "Multiscale Bioimaging: from Molecular Machines to Networks of Excitable Cells" (MBExC), University of Göttingen, Göttingen, Germany, ^{4}Campus Institute Data Science, University of Göttingen, Göttingen, Germany

To achieve free-breathing liver fat and $$$R_2^\star$$$ mapping, this work combines multi-echo radial FLASH with stack-of-stars volumetric acquisition and SSA-FARY to resolve respiratory motion. Moreover, regularized model-based reconstruction is implemented in BART to directly estimate quantitative parameter maps from acquired k-space data. Joint spatial and temporal regularization is used in this work. The proposed method is validated with NIST and water/fat phantoms. Furthermore, free-breathing liver studies show repeatability and good agreement between single-slice real-time and volumetric acquisition.

Figure 1 illustrates the multi-echo radial FLASH sequence and its corresponding $$$k$$$-space trajectory. With the blip gradients, multiple echoes (7 echoes in this work) with different radial spoke encoding are sampled per RF excitation.

The independent water/fat signal sampled by multi-gradient-echo sequences is modeled as [3],

$$$B : x \mapsto \rho_m = \bigg( \text{W} \cdot e^{-{R_2^\star}_\text{W} \text{TE}_m} + \text{F} \cdot z_m \cdot e^{-{R_2^\star}_\text{F} \text{TE}_m} \bigg) \cdot e^{i2\pi f_{B_0} \text{TE}_m}$$$ with $$$x = (\text{W}, {R_2^\star}_\text{W}, \text{F}, {R_2^\star}_\text{F}, f_{B_0})^T$$$.

$$$\rho_m$$$ denotes the $$$m$$$th echo signal. $$$\text{W}$$$ and $$$\text{F}$$$ are the water and fat proton density, respectively. Plus, water and fat have their independent $$$R_2^\star$$$ relaxation rate. The fat chemical-shift-induced phase modulation is $$$z_m = \sum_p a_p \cdot e^{i2\pi f_p \text{TE}_m}$$$ with $$$f_p$$$ being the 6-peak fat spectrum with their corresponding amplitude $$$a_p$$$ [10]. The last term is the $$$B_0$$$ field inhomogeneity ($$$f_{B_0}$$$) induced phase modulation. Noteworthy, this model can be reduced to the well-known single $$$R_2^\star$$$ model when assuming water and fat share the same $$$R_2^\star$$$ [1,2] or in the presence of only water. The nonlinear model is then combined with the parallel imaging model [11] and thus $$$F_{j,m} (x) = P_m \mathcal{F} S B_m$$$, where $$$x$$$ also includes coil sensitivity maps and $$$P_m \mathcal{F}$$$ denotes the non-uniform FFT operator of the $$$m$$$th echo. To jointly estimate all unknowns, the objective function is

$$$\arg\!\min_x \left\lVert y - F(x) \right\rVert_2^2 + \alpha R(x)$$$

This regularized nonlinear inverse problem is solved by IRGNM [11] with ADMM [12], allowing generalized regularization terms. This work utilized (1) joint $$$\ell^1$$$-Wavelet spatial regularization onto the parameter maps (including $$$\text{W}$$$, $$${R_2^\star}_\text{W}$$$, $$$\text{F}$$$, $$${R_2^\star}_\text{F}$$$), (2) spatial smoothness constraint on $$$B_0$$$ field inhomogeneity and coil sensitivity maps, (3) $$$\ell^2$$$-Tikhonov regularization onto all unknowns, (4) non-negativity constraints on $$$R_2^\star$$$ maps, and (5) temporal TV regularization for dynamic acquisition [13]. This nonlinear inverse problem is initialized with the estimate from model-based 3-point water/fat separation [7], while $$$R_2^\star$$$ and coil sensitivity maps are initialized as 0. For 3D data, initialization uses an initial reconstruction that uses sequential reconstruction along the partition dimension with regularization relative to the previous partition.

Both the NIST and a simple water/fat phantom were used for validation. The water/fat phantom consists of four tubes filled with Rama (7% fat), Kochsahne (15% fat), Schlagsahne (at least 30% fat), and peanut oil (92g fat per 100mL). The tube in the center is filled with distilled water. MERLOT was compared with reference $$$R_2^\star$$$ and fat fraction maps obtained via multi-gradient-echo Cartesian acquisition with pixel-wise fitting.

Free-breathing liver studies were conducted with both 2D single-slice and 3D stack-of-stars protocols. Detailed acquisition parameters were flip angle $$$5^o$$$, FOV $$$320$$$ mm, voxel size $$$1.6 \times 1.6 \times 5.0$$$ mm$$$^3$$$, base resolution $$$200$$$, bandwidth $$$1090$$$ Hz/pixel, and 7 echoes with TEs $$$1.31, 2.54, 3.77, 5.00, 6.23, 7.46, 8.69$$$ ms and TR $$$9.89$$$ ms. For single-slice acquisition, each frame consisted of $$$33$$$ RF excitation, leading to a temporal resolution of $$$326$$$ ms per frame. For stack-of-stars acquisition [14], a total of $$$36$$$ partitions and $$$330$$$ excitation were used (i.e. total scan time of $$$1.95$$$ min).

As shown in Figure 2, the spatial smoothness constraint on $$$f_{B_0}$$$ avoids artifacts which appear in the pixel-wise fitting reconstruction. Quantitative analysis of the reconstructed $$$R_2^\star$$$ and fat fraction values of the selected ROIs shows good match between these two methods.

Figure 3 shows reconstruction results with only spatial sparsity and with joint spatial and temporal sparsity regularization. The latter reduces noise and streaking artifacts in the reconstructed parameter maps.

Figure 4 shows motion-resolved reconstruction results via SSA-FARY and joint spatial and temporal regularization. Quantitative analysis of both single-slice and stack-of-stars acquisition in Table 1 shows overall agreement of quantitative $$$R_2^\star$$$ and fat fraction values between different acquisitions. In addition, the result from the 2nd scan shows close similarity to the 1st scan, which demonstrates the repeatability of the proposed method.

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