Aurelien Bustin^{1}, Gastao Cruz^{1}, Olivier Jaubert^{1}, Karina Lopez^{1}, RenĂ© Botnar^{1}, and Claudia Prieto^{1}

In magnetic resonance imaging, multiple contrasts are exploited to extract clinically relevant tissue parameters and pathological tissue changes. Multi-contrast acquisitions find important applications in parameter mapping (e.g. T_{1} and T_{2} mapping) and magnetic resonance fingerprinting. However, these acquisitions lead to long scan times since multiple images with different contrasts need to be acquired. In this study, we present a new reconstruction technique, termed as High-Dimensionality undersampled Patch-based RecOnSTruction (HD-PROST), for highly accelerated 2D and 3D multi-channel multi-contrast MRI.

**Reconstruction:** HD-PROST combines the promising performance of patch-based reconstructions [1,2] and the potential of low-rank image reconstruction through higher-order tensor decomposition [3]. HD-PROST jointly reconstructs multi-contrast 2D or 3D MR images by exploiting the highly redundant information, on a local and non-local scale, and the strong correlation shared between the multiple contrast images (Fig.1). The joint multi-contrast undersampled reconstruction can be combined with parallel imaging and can be formulated as the following joint optimization problem:

$$\mathcal{L}\left(X,\mathcal{T} \right) := \underset{X,\mathcal{T}}{\operatorname{argmin}} \frac{1}{2}\Vert EX-Y \Vert_F^2 + \sum_p \lambda_p \Vert \mathcal{T}_p \Vert_\ast \quad s.t. \quad \mathcal{T}_p = R_p \left( X \right)$$

Where $$$Y$$$ denotes the acquired undersampled data, $$$E$$$ is the encoding operator (including coil sensitivities, Fourier operator, and sampling), and $$$X$$$ denotes the multi-contrast MR images to reconstruct. The operator $$$R_p \left( . \right)$$$ constructs a third order tensor of similar 2D/3D patches from the patch $$$p$$$ centered at pixel $$$p$$$ (see optimization 2). The nuclear norm is used to enforce multi-dimensional low-rank on a multi-contrast patch scale and $$$\lambda_p>0$$$ controls the strength of sparsity. This optimization problem can be solved via alternating direction method of multipliers (ADMM) by iteratively solving the two following sub-problems:

**Optimization 1:** A regularized multi-contrast MR reconstruction (optimization on $$$X$$$) is performed using the 3D denoised volume ($$$\mathcal{T}$$$) obtained from optimization 2 as prior knowledge. This optimization is solved using the conjugate gradient (CG) algorithm.

**Optimization 2:** A high-order denoising on a patch level (optimization on $$$\mathcal{T}_p$$$) is performed by building a 3D tensor from similar 2D/3D+L patches (L being the number of contrasts) and applying a high-order singular value thresholding for each block $$$p$$$. The denoised images are then obtained by patch aggregation and used as prior in optimization 1.

**2D-MRF Experiments:** 2D radial MRF phantom [4] and in vivo brain acquisitions were performed on a 1.5T MR system (Ingenia, Philips) to evaluate the performance of HD-PROST for highly-accelerated simultaneous T_{1} and T_{2} mapping. Relevant scan parameters included: radial bSSFP, TE/TR=2/4.4ms, FOV=160x160mm^{2}, resolution=1x1mm^{2}, slice thickness=8mm, bandwidth=723Hz/pixel. One radial spoke was acquired at each time-point (total of 2000 time-points, scan time 8.8seconds) with a flip angle pattern similar to [5]. Five healthy subjects (four men, range 28-37 years) were scanned with the same parameters. The MRF dictionary was generated using the EPG formalism [6] for a T_{1} in the range of ([50:10:1400,1430:30:1600,1700:100,2200,2400:200:3000]ms) and T_{2} in the range of ([5:2:80,85:5:150,160:10:300,330:30:600]ms). HD-PROST reconstructions were compared to the low-rank inversion reconstruction (LRI) [7,8].

**3D-MTC Experiments:** In vivo experiments for reconstructing multiple undersampled 3D Magnetization Transfer (MT)-weighted images were conducted on a 1.5T MR scanner (Siemens Magnetom Aera) to illustrate the impact of HD-PROST for high-resolution multi-contrast 3D imaging. Scan parameters included: 3D GRE sequence, axial orientation, FOV=230x230x160mm^{3}, resolution=1x1x2mm^{3}, TE/TR/FA=1.78ms/4.06ms/15°, bandwidth=925Hz/pixel. Six contrasts were acquired with different MT pulse flip angles ($$$\alpha_{MT}=\left[ 0^{\circ}, 160^{\circ}, 320^{\circ}, 480^{\circ}, 640^{\circ}, 800^{\circ} \right]$$$) on two healthy subjects with an accelerated variable density Cartesian sampling with spiral profile order [2,9] (acceleration 6.5x, total scan time for 6 contrasts 13min18s). For comparison purposes, an additional fully-sampled acquisition was performed for the reference image only ($$$\alpha_{MT}=0^{\circ}$$$) (total scan time for a single contrast 12min57s).

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