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Free-Breathing 3D T1 Mapping of the Whole-Heart Using Low-Rank Tensor Modeling
Paul Kyu Han1, Debra E. Horng1, Yoann Petibon1, Jinsong Ouyang1, Nathaniel Alpert1, Georges El Fakhri1, and Chao Ma1

1Gordon Center for Medical Imaging, Massachusetts General Hospital and Harvard Medical School, Boston, MA, United States

### Synopsis

T1 of the myocardium is an emerging quantitative biomarker for a variety of heart diseases. However, T1 mapping is challenging in the heart due to the cardiac and respiratory motion. To overcome this issue, T1 mapping methods utilizing ECG triggering with breath-hold or respiratory control using bellow or navigators have been developed in the past. Recently, low-rank tensor-based approach has been proposed for MR to enable image reconstruction at extremely high acceleration factors, and has demonstrated promising results. This work presents a low-rank tensor-based approach for free-breathing 3D T1 mapping of the whole-heart.

### Introduction

Myocardium T1 is an emerging quantitative biomarker for a variety of heart diseases, including infiltration, acute myocardial injury, and fibrosis1. Quantification of T1 over the entire myocardium is desirable for complete assessment and diagnosis of the heart, however, T1 mapping is challenging in the heart due to cardiac and respiratory motion. To overcome this issue, methods have been developed combining inversion2-11, saturation13-20, variable-flip-angle21 or signal-model22 based methods with electrocardiography (ECG) trigger-controlled single/multi-slice 2D2-4,11-16 or single-shot/segmented 3D5,9,10,17-22 acquisitions over breath-hold2-5,9-16 or with respiratory control using bellow/navigators for free-breathing acquisition17-22. Recently, low-rank tensor-based approaches have been proposed for MR to exploit correlations in high-dimensional data and enable image reconstruction at high acceleration factors23-26. Promising results have been demonstrated in various applications27-33 including quantitative cardiovascular imaging32,33. This work presents a low-rank tensor-based data acquisition and processing method to enable free-breathing 3D T1 mapping of the whole-heart.

### Methods

Low-Rank Tensor Modeling

We propose to model the image function $\rho(x,t_1,t_2)$ as high-order partially-separable (PS) functions (Figure 1)23,24,33:$$\rho(x,t_1,t_2)=\sum_{l=1}^L\sum_{m=1}^M\sum_{n=1}^Nc_{l,m,n}\theta_l(x)\phi_m(t_1)\psi_n(t_2)\space\space\space\space\space\space\space\space\space\space(1)$$ where $x$ denotes the spatial dimension, $t_1$ denotes the respiratory motion phase, $t_2$ denotes the inversion-recovery (IR) time, $\left\{\theta_l(x)\right\}_{l=1}^L$, $\left\{\phi_m(t_1)\right\}_{m=1}^M$ and $\left\{\psi_n(t_2)\right\}_{n=1}^N$ denotes the spatial, respiratory, and T1 recovery basis functions, respectively, $L$, $M$, and $N$ denotes the tensor rank, and $c_{l,m,n}$ denotes the core-tensor coefficients. The proposed model indicates that the high-dimensional imaging function resides in a low-dimensional subspace and thus can be recovered even from a highly under-sampled data. To ensure sufficient number of measurements are sampled for image reconstruction, we propose to fully-sample a limited number of locations at the k-space center and sparsely-sample all the other locations to provide extensive k-space coverage over the entire $(k,t_1,t_2)$-space (Figure 2). Note that the proposed model allows to reconstruct the image function (i) at different respiratory motion phases without involving any binning or image registration, and (ii) at each different IR times resulting from the natural variations in the heart-rate over time.

Reconstruction

With known respiratory and T1 recovery basis functions (denoted as $\left\{\hat{\phi}_m(t_1)\right\}_{m=1}^M$ and $\left\{\hat{\psi}_n(t_2)\right\}_{n=1}^N$), $\left\{c_{l,m,n}\right\}_{l,m,n=1}^{L,M,N}$ and $\left\{\theta_l(x)\right\}_{l=1}^L$ can be estimated by solving the following optimization problem:$${arg\min_{c_{l,m,n},\theta_l(x)}}\parallel{d(k,t_1,t_2)-\Omega F_s\left\{\Sigma_{l=1}^L\Sigma_{m=1}^M\Sigma_{n=1}^Nc_{l,m,n}\theta_l(x)\hat{\phi}_m(t_1)\hat{\psi}_n(t_2)\right\}}\parallel_2^2+R_1(vec\left\{c_{l,m,n}\right\})+R_2\left\{\theta_l(x)\right\}\space\space\space\space\space\space\space\space\space\space(2)$$where $d(k,t_1,t_2)$ denotes the sparsely-sampled k-space measurement, $\Omega$ denotes the sampling matrix, and $F_s$ denotes the spatial Fourier transform matrix. The optimization problem in Eq.(2) can be solved using alternating minimization algorithms with verified convergence property24.

In this work, the T1 recovery basis functions were estimated by performing singular value decomposition (SVD) on a dictionary of magnetization signal generated using Bloch equation simulation37 over the clock-time with a range of T1 (100 to 2000ms) and flip angle values (5 to 15$^\circ$) to account for the B1 inhomogeneity effect. The respiratory basis functions were estimated via a multi-step procedure. The proposed model in Eq.(1) was first reduced to a PS model ($\rho(x,t)$, where $t$ denotes clock-time)23,24,34,35 and underwent a preliminary low-rank-based PS reconstruction35. Afterwards, the clock-time ($t$) was correlated with the respiratory motion phase ($t_1$) based on the liver motion in the reconstructed image. The respiratory basis functions were then estimated by performing SVD on the spatial-respiratory motion phase dataset $\rho(x,t_1)$ at a fixed IR time. Note that the proposed model is not limited to this approach of basis estimation. For example, the respiratory basis functions can also be estimated via alternating minimization with known $\left\{\hat{\psi}_n(t_2)\right\}_{n=1}^N$ and estimated temporal basis functions from the fully-sampled data acquired over the k-space center using the PS model23. Lastly, regularizations exploiting sparsity and spatial-temporal frequency sparsity was used for $R_1(\cdot)$ and for $R_2(\cdot)$, respectively.

### Results

All experiments were performed on a whole-body MR scanner. One healthy volunteer was imaged under a study protocol approved by the Institutional Review Board (IRB). Acquisitions were performed using an IR-FLASH sequence at the end-diastole with ECG-triggering. Non-selective inversion pulses were inserted every 4-cardiac-cycles with alternating inversion-time delays of 100 and 200ms. The imaging parameters were: FOV=360$\times$304$\times$144mm3, matrix size=192$\times$162$\times$24, TR/TE=4/2ms, flip-angle=10$^\circ$, PE lines sampled per cardiac-cycle=54 (in which 10 lines were fully-sampled at the center of k-space), total acquisition time=10min. for heart-rate of 80bpm. For comparison, T1 map of a single 2D slice was acquired using MOLLI2,3. T1 was estimated for each voxel via least-square fitting using variable-projection-algorithm36 and Look-Locker equation for MOLLI37 and Bloch equation simulation for our proposed method.

Overall, the proposed approach successfully reconstructed the 3D image including the whole-heart over different respiratory motion phases and IR times (Figures 3 and 5). Estimation of T1 over the whole-heart was also feasible using the proposed method, with mean myocardium T1 values similar to those from 2D MOLLI (Figure 4).

### Conclusion

Free-breathing 3D T1 mapping of the whole-heart can be achieved by low-rank tensor modeling. The proposed method can enable new clinical applications of T1 mapping in cardiac MR imaging.

### Acknowledgements

This work was partially supported by the National Institutes of Health (P41EB022544, R01CA165221, R01HL137230, R01HL118261, and T32EB013180).

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### Figures

Figure 1. Illustration of the proposed partially-separable low-rank tensor model. A: Schematic diagram of the proposed model. The imaging function is modeled as a multi-dimensional partially-separable function. B: Singular value analysis of the basis functions. The basis functions display low-rank property, indicating that the high-dimensional imaging function resides in a low-dimensional subspace.

Figure 2. Illustration of the acquisition scheme. A: Schematic diagram of acquisition. A non-selective 180° inversion pulse is inserted for T1 relaxation and data acquisition is performed at end-diastole with ECG triggering. B: Schematic diagram of data-sampling. A limited number of k-space locations at the k-space center (blue) are fully-sampled over the entire cardiac-cycle, whereas all the other k-space locations (green) are sparsely-sampled to provide extensive k-space coverage over the entire (k,t1,t2)-space and to ensure a sufficient number of measurements are sampled for reconstruction.

Figure 3. Reconstruction result of the proposed method. A: Result of 3D image at respiratory motion phase = 8 (blue box) and inversion-time (TI) = 2 (green box). B: Result of respiratory motion of liver at TI = 2 (green box). C: Result of T1 relaxation at slice = 13 (red box) and respiratory motion phase = 8 (blue box). Note that only representative images are shown in this figure since there are more than 8 TIs resulting from variation in subject heart-rate over time.

Figure 4. Whole-heart 3D T1 mapping result. A: 3D T1 map estimated using the proposed method. B: Slice view of the 3D T1 map along the blue dotted-line. C: Comparison with 2D MOLLI at a single-slice. The estimated myocardium T1 was 1238.5 ± 120 ms and 1214.9 ± 90.3 ms using the proposed method and 2D MOLLI, respectively.

Figure 5. Animation of clock-time reconstruction. A: Reconstruction result of the proposed method. B: Result of sliding-window reconstruction combining k-space lines acquired over 40 cardiac-cycles in a single frame.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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