Magnetic resonance fingerprinting (MRF) ^[1] provides a new paradigm to simultaneously acquire multiple tissue parameter maps (e.g., T1, T2, and spin density maps). The conventional MRF reconstruction is a matched filter based, non-iterative approach, which estimates tissue parameter maps from artifact-corrupted gridded reconstructions of a highly-undersampled spiral data. This approach often requires relatively long acquisition time to obtain accurate parameter maps. To alleviate such difficulty, a maximum likelihood (ML) approach ^[2] has recently been proposed to enable parameter estimation directly from undersampled k-space data. Nonetheless, the ML reconstruction only makes use of the Bloch equation based data model and noise characteristics. Further improvement can be achieved by incorporating a proper image prior model into the reconstruction process. In this work, we present a model-based approach using joint low-rank and sparse structure to improve the spatial reconstructions of the MRF time series. Tissue parameter maps are then derived from improved time series by the dictionary matching, which are shown to substantially benefit from high-quality time series reconstruction. Representative results from an in vivo experiment are shown to demonstrate the effectiveness of the proposed method.

Let $$$\mathbf{C} \in \mathbb{C}^{N \times M}$$$ denote the Casorati matrix ^[3] representing a contrast-weighted image sequence associated with an MRF experiment. Due to the strong spatiotemporal correlation of these images, a low-rank model is introduced to represent $$$\mathbf{C} $$$, i.e., $$$\mathbf{C} = \mathbf{U}\mathbf{V}$$$, where $$$\mathbf{U} \in \mathbb{C}^{N \times L}$$$ and $$$\mathbf{V} \in \mathbb{C}^{L \times M}$$$ respectively denote the spatial and temporal subspaces of $$$\mathbf{C}$$$, and $$$L$$$ denotes the rank. We further pre-estimate the temporal subspace for the low-rank model, denoted as $$$\hat{\mathbf{V}}$$$, from the ensemble of magnetization dynamics using the principled component analysis ^[3-6]. Additionally, we incorporate a joint sparsity constraint to capture correlated edge structure of co-registered contrast-weighted images, regularizing the ill-conditioned low-rank reconstruction ^{[7, 8]}. Putting together the above constraints, the reconstruction problem can be formulated as

$$\hat{\mathbf{U}}=\text{arg min}_{\mathbf{U}}\sum_{c=1}^{N_c}||\mathbf{d}_c-\mathbf{F}_u\mathbf{S}_c\mathbf{U}\hat{\mathbf{V}}||_2^2+\lambda ||\mathbf{D}\mathbf{U}\hat{\mathbf{V}}||_{1, 2},$$

where $$$\mathbf{d}_c$$$ denotes the acquired $$$\mathbf{k}$$$-space data from the $$$c$$$th coil, $$$\mathbf{F}_u$$$ the undersampled Fourier encoding matrix, $$$\mathbf{S}_c$$$ the coil sensitivities associated with the $$$c$$$th coil, $$$\mathbf{D}$$$ the spatial finite difference matrix, and $$$\lambda$$$ the regularization parameter. The above formulation results in a convex optimization problem, for which we apply the alternating direction method of multipliers algorithm ^[9]. After obtaining $$$\hat{\mathbf{U}}$$$, we form $$$\hat{\mathbf{C}} = \hat{\mathbf{U}}\hat{\mathbf{V}}$$$, from which we apply the dictionary matching as the conventional approach.

A 2D in vivo MRF experiment was performed on a 3T Siemens scanner with a 32 channel head coil using an identical IR FISP sequence, spiral trajectory, and flip angles and TRs as in ^[10]. The acquisition parameters include: FOV = 300×300 mm², matrix size = 256×256, and slice thickness = 5 mm. For each acquisition parameter, a single spiral interleave was used (the fully sampled data requires 48 interleaves). We acquired data with 1000 TRs (i.e., corresponding to the duration of 13.1s) to obtain a set of sparsely-sampled data for image reconstruction. To obtain a fully sampled “gold standard” for comparison, the above experiment was repeated 48 times, rotating the spiral trajectory. For the sparsely-sampled data, we applied the conventional MRF reconstruction, ML reconstruction, and the proposed method. For the proposed method, we empirically chose the model order and regularization parameter for the optimized performance. Fig. 1 shows the MRF time series reconstructions from the gridding reconstruction and proposed method. As opposed to the conventional approach, which estimates parameter maps from the artifact-corrupted gridding reconstruction, the proposed method provides much higher quality contrast-weighted images for subsequent parameter estimation. Quantitatively, Figs. 2-4 show the reconstructed T1, T2, and spin density maps from different methods. Clearly, the proposed method significantly outperforms the conventional MRF reconstruction, and results in more than a factor of two reduction in the reconstruction errors for all three parameter maps. Compared to the ML reconstruction, it also shows noticeable improvement, especially for the T2 map. This illustrates the power of incorporating a prior image model to improve the reconstruction performance.

To enhance the computational efficiency of the conventional approach, a low-rank/subspace model ^[11] was recently proposed to compress the dictionary (or k-space data) before the gridding reconstruction. Note that this is fundamentally different from the proposed method in which the low-rank model is used to enable high-quality reconstruction of contrast-weighted images from sub-Nyquist data.This work presented a new model-based approach using low-rank and sparsity constraints for accelerated MRF. It enables much more accurate reconstructions of tissue parameter maps over the conventional approach and recently proposed ML approach with limited data.