Sen Ma^{1,2}, Christopher T Nguyen^{2,3}, Anthony G Christodoulou^{2}, Sang-Eun Lee^{2,4,5,6}, Hyuk-Jae Chang^{4,5,6}, and Debiao Li^{1,2}

We propose to sparsely sample in vivo cardiac diffusion tensor imaging (CDTI) by combining a phase-corrected low-rank model and sparsity constraint. The proposed method was evaluated on 7 hypertrophic cardiomyopathy patients. Helix angle and mean diffusivity maps were compared against employing single constraint, and changes in helix angle transmurality and mean diffusivity were evaluated using Wilcoxon signed rank test to statistically determine the highest achievable acceleration factors preserving CDTI measurements with no significant difference. Our framework shows promise in accelerating acquisition window while preserving myofiber architecture features, and may allow higher spatial resolution or shorter temporal footprint in the future.

Due to the strong correlation between free-breathing,
diffusion-weighted images along different diffusion directions, we leverage the
low-rank structure of
diffusion-weighted
images $$$\textbf{X}\in\mathbb{C}^{M\times N}$$$ using a partially separable model^{8}. This model is particularly useful when a phase map $$$\textbf{P}\in\mathbb{C}^{M\times N}$$$
is applied to
compensate for the drastic phase inconsistency across diffusion directions^{7}, i.e., $$$\textbf{X}=\textbf{P}\circ(\textbf{U}\textbf{V})$$$, where $$$M$$$ represents the
number of voxels,
$$$\textbf{V}\in\mathbb{C}^{L\times N}$$$ contains “temporal”
basis functions (containing contributions from diffusion and respiratory
motion), and $$$\textbf{U}\in\mathbb{C}^{M\times L}$$$
contains spatial
coefficients with $$$L\lt \min\{M,N\}$$$. We also include a group sparsity constraint to
leverage the CS framework. We propose to reconstruct $$$\textbf{X}$$$ in three steps:

*1) Estimate phase map*: We propose to estimate $$$\textbf{X}$$$ from an intermediate solution of employing only a sparsity
constraint:

$$\widetilde{\textbf{X}}=\arg\min_{\textbf{X}}||\textbf{d}-E(\textbf{X})||_2^{2}+\lambda \text{R}_{s}(\textbf{X})\quad\quad\quad\quad\quad\quad\quad (1)$$

with $$$P_{jk}=\exp{(i\angle\widetilde{X}_{jk})}$$$, where $$$\textbf{d}$$$ denotes undersampled k-space data, $$$E(\cdot)$$$ performs spatial encoding and sparse sampling, $$$\lambda$$$ represents the regularization factor, and $$$\text{R}_{s}(\cdot)$$$ is the regularization penalty promoting group sparsity.

*2) Estimate "temporal" subspace*: We propose
to estimate $$$\textbf{V}$$$ from the SVD of the magnitude image $$$|\widetilde{\textbf{X}}|$$$ by collecting $$$L$$$ most significant right singular vectors.

*3) Recover spatial coefficients*: Lastly, we recover the spatial coefficient matrix by

$$\textbf{U}=\arg\min_{\textbf{U}}||\textbf{d}-E(\textbf{P}\circ(\textbf{U}\textbf{V}))||_2^{2}+\lambda \text{R}_{s}(\textbf{UV}).\quad\quad\quad\quad\quad\quad\quad (2)$$

Data
were acquired from $$$n$$$=7 hypertrophic
cardiomyopathy (HCM) patients. Diffusion MRI was performed on a 3T Siemens
Prisma scanner. A second-order motion-compensated diffusion tensor sequence^{2} was used replacing
the bSSFP readout with a single-shot EPI readout. Imaging protocol and
reconstruction details are displayed in Table 1. The diffusion tensor, along
with helix angle (HA) and mean diffusivity (MD), was log linearly-fitted after standard
mutual information affine registration. Helix angle transmurality (HAT) was
calculated by radially sampling the HA along 60 transmural directions and
linearly fitting between HA and transmural depth. Both global (7 samples) and regional
(16 AHA segments/subject
× 7 subjects = 112 samples) HAT and MD of the entire group were
compared between fully-sampled (reference) and reconstructed data at varying
acceleration factors (*R*)
using a Wilcoxon signed rank test with Bonferroni correction.

1. Mekkaoui C, Reese TG, Jackowski MP, Bhat H, Sosnovik DE. Diffusion MRI in the heart. NMR Biomed 2017;30(3).

2. Nguyen C, Fan Z, Xie Y, Pang J, Speier P, Bi X, Kobashigawa J, Li D. In vivo diffusion-tensor MRI of the human heart on a 3 tesla clinical scanner: An optimized second order (M2) motion compensated diffusion-preparation approach. Magn Reson Med 2016;76(5):1354-1363.

3. Wu Y, Zhu YJ, Tang QY, Zou C, Liu W, Dai RB, Liu X, Wu EX, Ying L, Liang D. Accelerated MR diffusion tensor imaging using distributed compressed sensing. Magn Reson Med 2014;71(2):763-772.

4. McClymont D, Teh I, Whittington HJ, Grau V, Schneider JE. Prospective acceleration of diffusion tensor imaging with compressed sensing using adaptive dictionaries. Magn Reson Med 2016;76(1):248-258.

5. Ma S, Nguyen C, Christodoulou A, Luthringer D, Kobashigawa J, Li D. Accelerated Cardiac Diffusion Tensor Imaging Using Joint Low-Rank and Sparsity Constraint. In Proceedings of the 25th Annual Meeting of ISMRM, Honolulu, Hawaii, USA, 2017. Abstract 6603.

6. Huang J, Wang L, Chu C, Zhang Y, Liu W, Zhu Y. Cardiac diffusion tensor imaging based on compressed sensing using joint sparsity and low-rank approximation. Technology and Health Care 2016;24(s2):S593-S599.

7. Gao H, Li L, Zhang K, Zhou W, Hu X. PCLR: phase-constrained low-rank model for compressive diffusion-weighted MRI. Magn Reson Med 2014;72(5):1330-1341.

8. Liang Z-P. Spatiotemporal imaging with partially separable functions. IEEE ISBI, 2007: 988-991.

Table 1. Imaging protocol and specific reconstruction details

Figure 1. Reconstructed helix angle (HA) maps, mean diffusivity (MD) maps with respective voxel-wise error maps using LR Only, CS Only and LR/CS at acceleration factors of *R*=2, 3, and 4.

Figure 2. Left: p-value analysis between reference and reconstructed global HAT (7 samples for the entire group) using LR Only, CS Only and LR/CS at acceleration factors *R=*2, 3, and 4. Significance level (*p*=0.05) is shown in red horizontal line. Right: p-value analysis between reference and reconstructed global MD using LR Only, CS Only and LR/CS at acceleration factors *R*=2, 3, 4, and 5. Significance level (p=0.05) is shown in red horizontal line.

Table 2. p-value analysis between reference and reconstructed regional HAT measurements (112 samples for the entire group) using LR Only, CS Only and LR/CS at acceleration factors R=2, 3, and 4. Star represents significance (*p*<0.0031 after Bonferroni correction).

Table 3. p-value analysis between reference and reconstructed regional MD measurements (112 samples for the entire group) using LR Only, CS Only and LR/CS at acceleration factors R=2, 3, and 4. Star represents significance (p<0.0031 after Bonferroni correction).