Paul Kyu Han^{1}, Debra E. Horng^{1}, Yoann Petibon^{1}, Jinsong Ouyang^{1}, Nathaniel Alpert^{1}, Georges El Fakhri^{1}, and Chao Ma^{1}

T_{1} of the myocardium is an emerging quantitative biomarker for a variety of heart diseases. However, _{1} mapping is challenging in the heart due to the cardiac and respiratory motion. To overcome this issue, T_{1} mapping methods utilizing ECG triggering with breath-hold or respiratory control using bellow or navigators have been developed in the past. Recently, _{1} mapping of the

*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 T_{1} 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 T_{1} 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 property^{24}.

In this work, the T_{1} recovery basis functions were estimated by performing singular value decomposition (SVD) on a dictionary of magnetization signal generated using Bloch equation simulation^{37} over the clock-time with a range of T_{1} (100 to 2000ms) and flip angle values (5 to 15$$$^\circ$$$) to account for the B_{1} 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 reconstruction^{35}. 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 model^{23}. Lastly, regularizations exploiting sparsity and spatial-temporal frequency sparsity was used for $$$R_1(\cdot)$$$ and for $$$R_2(\cdot)$$$, respectively.

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$$$\times$$$144mm^{3}, 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, T_{1} map of a single 2D slice was acquired using MOLLI^{2,3}. T_{1} was estimated for each voxel via least-square fitting using variable-projection-algorithm^{36} and Look-Locker equation for MOLLI^{37} 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 T_{1} over the whole-heart was also feasible using the proposed method, with mean myocardium T_{1} values similar to those from 2D MOLLI (Figure 4).

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**Figure 2. Illustration of the acquisition scheme. A: **Schematic diagram of acquisition. A non-selective 180° inversion pulse is inserted for T_{1} 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,t_{1},t_{2}*)-space and to ensure a sufficient number of measurements are sampled for reconstruction.