Sen Ma^{1,2}, Christopher Nguyen^{1}, Anthony Christodoulou^{1,3}, Daniel Luthringer^{4}, Jon Kobashigawa^{3}, and Debiao Li^{1,2}

We present an image reconstruction technique to accelerate cardiac diffusion tensor imaging by jointly applying a low-rank and spatial sparsity constraint. We evaluated four acquisition schemes at different undersampling levels on 9 ex vivo diseased human heart, evaluating the reconstruction quality based on the resulting helix angle (HA) maps and helix angle transmurality (HAT) values. A Wilcoxon signed rank test was performed to statistically evaluate changes in HAT to determine the highest achievable acceleration factor for each acquisition scheme. Our framework shows promise in greatly reducing scan time while preserving the fiber architecture features of heart failure.

The low-rank structure in cardiac diffusion-weighted
images lies in the strongly correlated behaviors of diffusion-weighted signals
at different voxels^{3}. This correlation induces low-rankness of the
matrix $$$\mathbf{S}=[s_1,s_1,\dots,s_N]$$$, where the diffusion-weighted image for direction $$$n$$$ is reshaped as a vector $$$s_n\in\mathbb{C}^{M}$$$, such that it can be factorized as $$$\mathbf{S}=\mathbf{U}_\mathrm{s}\mathbf{V}_\mathrm{d}$$$, where $$$\mathbf{V}_\mathrm{d}\in\mathbb{C}^{L\times{N}}$$$ contains “diffusion basis functions” that span the
diffusion subspace, and $$$\mathbf{U}_\mathrm{s}\in\mathbb{C}^{M\times{L}}$$$ contains spatial coefficients. $$$\mathbf{S}$$$ is a low-rank matrix when $$$L\lt\min\{M,N\}$$$. Here we enforce the low-rank property by predefining $$$\mathbf{V}_\mathrm{d}$$$ from center k-space data via singular value
decomposition^{3}; we further include a sparsity constraint to also leverage
the compressed sensing framework^{6}. Our reconstruction model is thus:$$\mathbf{S}=\hat{\mathbf{U}}_\mathrm{s}\mathbf{V}_\mathrm{d}\textrm{, where }\hat{\mathbf{U}}_\mathrm{s}=\arg\min_{\mathbf{U}_\mathrm{s}}\|\mathbf{d}-\Omega(\mathbf{F}_\mathrm{s}\mathbf{U}_\mathrm{s}\mathbf{V}_\mathrm{d})\|_2^2+\lambda\|\mathbf{W}\mathbf{U}_\mathrm{s}\mathbf{V}_\mathrm{d}\|_1\qquad(1)$$where $$$\Omega$$$ is the k-space sampling operator, $$$\mathbf{F}_\mathrm{s}$$$ applies the spatial Fourier transform, $$$\mathbf{W}$$$ applies a spatial sparsifying transform, and $$$\lambda$$$ is the regularization parameter. Equation
(1) can be efficiently
solved by an ADMM algorithm.

Our study was performed on 9 ex vivo hearts extracted from heart failure patients (Cedars-Sinai Medical Center Heartome database). Data was acquired on a 3T mMR PET/MRI Siemens scanner using a diffusion-weighted spin echo sequence. Table 1 shows the acquisition and reconstruction parameters. HAT was calculated by radially sampling the HA along 25 transmural directions and fitting the slope between HA and transmural depth using linear regression. A Wilcoxon signed rank test was performed to evaluate the difference between fully sampled-derived HAT (reference) and undersampled-derived HATs at varying acceleration factors.

1. Mekkaoui C, Reese T G, Jackowski M P, et al. Diffusion MRI in the heart. NMR in Biomedicine, 2015.

2. Nguyen C, Fan Z, Sharif B, et al. In vivo three-dimensional high resolution cardiac diffusion-weighted MRI: A motion compensated diffusion-prepared balanced steady-state free precession approach. Magnetic resonance in medicine, 2014, 72(5): 1257-1267.

3. Gao H, Li L, Zhang K, et al. PCLR: Phase-constrained low-rank model for compressive diffusion-weighted MRI. Magnetic resonance in medicine, 2014, 72(5): 1330-1341.

4. Liang Z P. Spatiotemporal imaging with partially separable functions. IEEE International Symposium on Biomedical Imaging, 2007: 988-991.

5. Pedersen H, Kozerke S, Ringgaard S, et al. k-t PCA: Temporally constrained k-t BLAST reconstruction using principal component analysis. Magnetic resonance in medicine, 2009, 62(3): 706-716.

6. Zhao B, Haldar J P, Christodoulou A G, et al. Image reconstruction from highly undersampled-space data with joint partial separability and sparsity constraints. IEEE transactions on medical imaging, 2012, 31(9): 1809-1820.

7. Feng L, Axel L, Chandarana H, et al. XD-GRASP: Golden-angle radial MRI with reconstruction of extra motion-state dimensions using compressed sensing. Magnetic resonance in medicine, 2016, 75(2): 775-788.

8. Lustig M, Donoho D, Pauly J M. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magnetic resonance in medicine, 2007, 58(6): 1182-1195.