Youngwook Kee^{1}, Junghun Cho^{2}, Thanh Nguyen^{1}, Pascal Spincemaille^{1}, and Yi Wang^{1}

We propose a second-order directional total generalized variation (dTGV) that makes use of directional edge information in T1w to reconstruct highly undersampled T2w and T2FLAIR data. This allows a further doubling of the acquisition speed over the standard four fold accelerated protocol. The proposed dTGV regularizer promotes structural similarity between contrasts.

In clinical MRI, multiple contrasts such as T1w, T2w, and T2 FLAIR are sequentially acquired one at a time. The lengthy scan time increases patient discomfort and decreases patient compliance. Although there has been abundant research in parallel imaging and compressed sensing to shorten scanning time, many techniques focus on acceleration a single scan. Anatomical images in different contrasts have structural information in common, which is exploited here to further accelerate multicontrast MRI data acquisition. To this end, we propose second-order directional total generalized variation (dTGV) that makes use of such structural information for a variational image reconstruction of highly undersampled MRI data.

1) Rank-2 Orthogonal Projector

In this work, the structural information that coexists between contrasts is encoded using the orientation of edges (up to sign) of T1w. We extract this information using the following rank-2 orthogonal projector:

$$ \mathbb{M} = \left( \begin{array}{ccc} \mathbb{M}_{11} & \mathbb{M}_{12} \\ \mathbb{M}_{21} & \mathbb{M}_{22} \\ \end{array} \right) = I - \frac{(\nabla w)(\nabla w)^\top}{(\nabla w)^\top(\nabla w)},$$

where $$$w$$$ is the T1w image. That is, the operator $$$\mathbb{M}$$$ acts on a vector and extracts perpendicular component of the vector with respect to $$$\nabla w$$$. Fig. 1 shows this projector computed from a T1w image.

2) Direction Total Generalized Variation (dTGV) based Multicoil/Multicontrast MRI Reconstruction

The dTGV MRI reconstruction solves

$$ \min_{u,v} \frac{1}{2}\sum_{j=1}^{P}\|\Omega{\cal F}S_ju - g_j \|_2^2 + \alpha_1 \|\mathbb{M}\nabla u - v \|_1 + \alpha_0\|Ev\|_1,$$

where $$$g_1, \dots, g_P$$$ is the undersampled multichannel k-space raw data, P the number of channels, $$$S_1, \dots, S_P$$$ coil sensitivity maps, and $$$\Omega$$$ the k-space sampling mask. The last two terms are a variation of second-order TGV [2]. Observe that the rank-2 orthogonal projector is involved in the first L1 regularization term, which promotes parallel orientation of edges that coexist in T1w and T2w/T2 FLAIR. The operator is the symmetrized gradient. This optimization is then solved by the Chambolle—Pock algorithm with further dualization on the first term.

Four healthy subjects were scanned at a 3T GE scanner with 32ch coils to acquire SAG 3D T1 BRAVO, SAG CUBE T2, and SAG CUBE T2 FLAIR with 1mm isotropic resolution and 1 healthy subject was scanned at the same system to acquire SAG 3D T1 BRAVO and SAG CUBE T2 with 1mm isotropic resolution. For T2w and T2 FLAIR, we acquired 1) fully sampled data, 2) standard data (ACC slice 2 x phase 2), 3) undersampled data (ACC slice 2 x phase 4). Matrix size is 256x256x192 and scan time was: 5:11 min for T1w, 34:46 min for fully sampled T2 FLAIR, 8:56 min for standard T2 FLAIR, 5:23 min for the proposed sampling scheme, 18:59 min for fully sampled T2w, 3:58 min for standard T2w, and 2:10 min for the proposed sampling scheme.

The regularization parameters $$$\alpha_1$$$ and $$$\alpha_0=2*\alpha_1$$$ and an optimized sampling scheme were selected by repeating the reconstruction of a single slice using 30 different regularization values and 10 different patterns (Fig. 2), respectively. Two radiologists scored (on a scale of 1 (bad) to 5 (excellent)) these images and we chose the sampling pattern by consensus. Coil sensitivity maps were estimated using ESPIRiT [1].

Reconstructed images were quantitatively evaluated with respect to fully sampled data using RMSE (root mean squared error), HFEN (high frequency error norm), SSIM (structure similarity index).

[1] Uecker et al. ESPIRiT--an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA. MRM 2014.

[2] Bredies, Kunisch, and Pock. Total Generalized Variation. SIIMS 2010.

10 sampling patterns with the numbers of samples
(%) with respect to the entire k-space samples. (1) Acceleration factor (ACC) of 4 along the
slice encoding direction (SL) and ACC of 2 along the phase encoding direction
(PH); (2) ACC of 2 along the SL and ACC of 4 along the PH; (3) ACC of 3 along
both SL and PH; (4), (5), and (6) random sampling; (7), (8), and (9) random
sampling including the corners; (10) ACC of 4 along both SL and PH including
the corners. Based on the scores by two radiologists, we chose

Reconstructed T2/T2 FLAIR images from
retrospectively sampled data.