Purpose: To achieve free-breathing quantitative fat and $R_2^*$ mapping of the liver using model-based iterative reconstruction, dubbed as MERLOT. Methods: For acquisition, we use a multi-echo radial FLASH (fast low-angle shot) sequence that acquires multiple echoes with different complementary radial spoke encodings. We investigate real-time single-slice and volumetric multi-echo radial acquisition. Model-based reconstruction based on generalized nonlinear inversion is used to jointly estimate water, fat, $R_2^*$, $B_0$ field inhomogeneity, and coil sensitivity maps from the multi-coil multi-echo radial spokes. Spatial smoothness regularization is applied onto the $B_0$ field and coil sensitivity maps, whereas joint sparsity regularization is employed for the other parameter maps. The method integrates calibration-less parallel imaging and compressed sensing and was implemented in Berkeley Advanced Reconstruction Toolbox (BART). For the volumetric acquisition, the respiratory motion is resolved with self-gating using Adapted Singular Spectrum Analysis (SSA-FARY). The quantitative accuracy of the proposed method was validated via numerical simulation, the NIST phantom, a water/fat phantom, and in in-vivo liver studies. Results: For real-time acquisition, the proposed model-based reconstruction allowed acquisition of dynamic liver fat fraction and $R_2^*$ maps at a temporal resolution of 0.3 seconds per frame. For the volumetric acquisition, whole liver coverage could be achieved in under 2 minutes using the self-gated motion-resolved reconstruction. Conclusion: The proposed multi-echo radial sampling sequence achieves fast k-space coverage and is robust to motion. The proposed model-based reconstruction yields spatially and temporally resolved liver fat fraction, $R_2^*$ and $B_0$ field maps at high undersampling factor and with volume coverage.
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