Eunju Cha^{1}, Eung Yeop Kim^{2}, and Jong Chul Ye^{1}

Time-resolved angiography with interleaved stochastic trajectories(TWIST) has been widely used for dynamic contrast enhanced (DCE) MRI. To achieve highly accelerated acquisitions for improved temporal and spatial resolution, the high frequency region is randomly sub-sampled at each time frame. Therefore, the periphery of the k-space data from multiple time frames are combined to obtain the uniformly sub-sampled k-space data so that the temporal resolution of TWIST is limited. The purpose of this research is to improve the temporal resolution of TWIST by reducing the view-sharing. Furthermore, we proposed the algorithm that can reconstruct the imagesat various number of view sharing using k-space deep learning.

It was shown that the missing k-space data for parallel MRI can be interpolated by exploiting the redundancies along the coils and images. Moreover, a recent mathematical discovery reveals that the Hankel matrix decomposition is closely related to a convolutional neural network (CNN). By synergistically combining these findings, we propose a multi-coil k-space deep learning approach that immediately interpolates the missing k-space data using a deep convolutional neural network.

As shown in Fig. 1(b), the reduced view sharing results in irregular sampling pattern which cannot be reconstructed using the existing GRAPPA algorithm. We demonstrated that the missing elements in k-space data from two frames can be interpolated using annihilating filter based low-rank Hankel matrix approach (ALOHA)^{2}, but its computational cost is too expensive. It was shown that the missing k-space data for parallel MRI can be interpolated by exploiting the redundancies along the coils and images. Moreover, a recent mathematical discovery reveals that the Hankel matrix decomposition is closely related to a convolutional neural network (CNN)^{3}. By synergistically combining these findings, we propose a multi-coil k-space deep learning approach that immediately interpolates the missing k-space data using a deep convolutional neural network. The neural network is constructed in the k-space domain by stacking multi-coil k-space data along the channel direction of the network as shown in Fig. 1(c). To address the lack of ground-truth data and to provide the flexible reconstruction for various number of view sharing, our neural network learned the k-space interpolation relationship between the minimum number of k-space samples and the fully sampled k-space data from GRAPPA reconstruction. The proposed network was implemented in Python using TensorFlow library.

We used four sets of in vivo 3D DCE data for carotid vessel scan obtained with TWIST sequence using Siemens 3T Verio scanners. The acquisition parameters for two sets were as following: 159×640×80 matrix, 16 coils, and 30 temporal frames. For other two sets, the acquisition parameters were same as above, except for 37 temporal frames.The down sampling rate was 3 and 2 along k_{x} and k_{y} direction, respectively. Due to partial Fourier, only 63% of data was acquired. Among four patients, two patients data were used for training, and one patient data was used for validation. We used the remaining patient data to test the trained network.

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2. Kyoung Hwan J, Dong-wook L, and Jong Chul Y. A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix. IEEE Trans. on Computational Imaging.2016;2(4):480-495.

3. Jong Chul Y, Yoseob H, and Eunju C. Deep convolutional framelets: A general deep learning framework for inverse problems. SIAM Journal on Imaging Sciences 11.2 (2018): 991-1048.

(a) Conventional view sharing scheme for 2D GRAPPA reconstruction, and (b) an example of view sharing. The center and periphery of k-space are denoted as A and B, respectively. (c) An overall scheme of k-space deep learning for parallel MRI.

Temporal resolution comparison of the subtracted MIP results of GRAPPA,raw data and the proposed methods for various number of view sharing. VS stands for the number of view sharing.