Introduction

Cardiac T₁ mapping is a powerful magnetic resonance (MR) imaging technique that allows quantitative assessment of tissue characteristics and underlying pathology of the myocardium¹. Up to date, various methods have been developed to overcome the limitations due to cardiac and respiratory motion and enable cardiac T₁ mapping with free-breathing acquisitions^2-5. However, existing methods are still limited in terms of imaging speed, spatial resolution and coverage (i.e., in the slice selection direction). In this work, we propose a new method to enable free-running (i.e., free-breathing and no ECG gating), simultaneous 3D cardiac T₁ mapping and cine imaging using a linear tangent space alignment (LTSA) model. The performance of the proposed method is validated using in vivo experimental data obtained from healthy volunteers at 3T.

Method

Image Reconstruction

The data is reconstructed using the Linear Tangent Space Alignment (LTSA) model⁶. Consider the Casorati matrix of the dynamic image series $$$X=[x_1,...,x_N]\in{\mathbb{C}^{M\times{N}}}$$$ (M is the number of voxels and N the number of frames) sampled from a low-dimensional manifold $$$\mathcal{M}$$$. The images $$${x_i}$$$ can be grouped into $$$C$$$ neighborhoods based on a similarity metric, forming $$$X_c=[x_{i^{(c)}_{1}},...,x_{i^{(c)}_{K_c}}]\in{\mathbb{C}^{M\times{K_c}}}$$$ where $$${i^{(c)}_{k}}$$$ corresponds to the frame index of the $$$k$$$-th image at the $$$c$$$-th neighborhood and $$$K_c$$$ to the total number of frames in that neighborhood. The LTSA model for the neighborhood $$$c$$$ is then:
$$X_c=T{L}_c^{-1}{\Phi}_c^T$$

Where $$$T\in{\mathbb{C}^{M\times{d}}}$$$ is composed of the global coordinates of the underlying manifold $$$\mathcal{M}$$$, $$${L}_c^{-1}\in{\mathbb{C}^{d\times{d}}}$$$ is a linear transformation matrix for the neighborhood $$$c$$$ and $$${\Phi}_c\in{\mathbb{C}^{{K_c}\times{d}}}$$$ is a matrix forming an orthonormal basis of the subspace of the neighborhood $$$c$$$. Note that $$${\Phi}_c$$$ can be learned from training data. The LTSA model is a nonlinear generalization of the conventional subspace models (i.e., the low-rank and low-rank tensor models). It leverages the intrinsic low-dimensional structure of the dynamic MRI images for accelerated real-time imaging with sparse sampling.

In image reconstruction, we minimize the representation error of $$$T$$$ and $$$\left\{L_c^{-1}\right\}_{c=1}^C$$$ with a sparsity constraint on $$$T$$$:

$$arg\min_{T,\left\{{{L}_c}^{-1}\right\}_{c=1}^C}\Big\{\frac{1}{2}\sum_{c=1}^C{||A_c\{T{L}_c^{-1}{\Phi}_c^T\}-s_c||_2^2}+{\lambda}_1||DT||_1\Big\}$$

where, $$$s_c$$$ is the undersampled k-space data of the neighborhood c, $$$A_c$$$ is the forward imaging matrix, $$$D$$$ is the finite difference operator along (x,y,z) directions and $$$\lambda_1$$$ is a scaling parameter for the sparsity constraint. This optimization problem is solved using the alternating direction methods of multipliers (ADMM) algorithm⁷.

We seek to form neighborhoods such that the matrices $$$X_c,c=1,\ldots,{C}$$$ have each a low modelling rank. As shown in [4], forming neighborhoods based on respiratory motion is a good candidate for that purpose. Therefore in this work, we form $$$X_c$$$ with images belonging to the same respiratory phase.

In Vivo Study

Experiments were performed on a 3T MR scanner. In vivo experiments were performed on a healthy volunteer (approved by our local IRB) using a free-running continuous acquisition sequence with spoiled gradient-recalled echo (GRE) readout and adiabatic inversion pulse (Fig. 1b). A special data acquisition scheme was employed to sample data along a stack-of-stars trajectory (Figs. 1b and c) and enable image reconstruction using the LTSA model⁸. One study was performed using the following imaging parameters: FOV=300×300×128 mm³, matrix-size=160×160×32, TR/TE=5.5/2.0 ms, flip-angle=5°, number-of-TRs-per-frame=9, time-between-IR-pulses=1528 ms, and acquisition time=~12.4 min. Another study was performed to achieve higher spatial resolution and coverage with all other imaging parameters the same: FOV=300×300×160 mm³ and matrix-size=160×160×64. For comparison, 2D T₁ maps were acquired using MOLLI⁹ in three slices in the short-axis view from the apical, mid-cavity, basal regions of the heart and for one slice in the long-axis view.

Estimation of $$$T_1$$$ was performed by first estimating the $$$T_1^{*}$$$ for each voxel via least-squares fitting of the inversion recovery signal to the following signal equation:
$$S[n]=A\cdot{}exp\Big(-\frac{n\cdot{}T_{frame}}{T_1^{*}}\Big)+B,\quad{n=0,\ldots,N-1}$$

where $$$S$$$ denotes the signal, $$$n$$$ denotes the inversion-time index, $$$T_{frame}$$$ denotes the temporal resolution of each frame, and $$$A$$$ and $$$B$$$ denote linear coefficients. Once $$$T_1^{*}$$$ was estimated, $$$T_1$$$ was estimated using the following equation:
$$T_1=\frac{1}{\frac{1}{T_1^{*}}+\frac{log(cos(\alpha\cdot{}B_1))}{TR}}$$

where $$$\alpha$$$ denotes the flip angle and $$$B_1$$$ denotes the transmit B₁. In this work, uniform transmit B₁ was assumed which was determined by calibrating the $$$T_1$$$ of the myocardium to literature value.

Results

Images reconstructed using LTSA showed fewer artifacts compared to those from low-rank plus sparsity-based image reconstruction as shown in Fig. 2. Figure 3 shows the LTSA-reconstructed images along different axes, demonstrating the capability of the proposed method in spatial coverage and in revealing respiratory motion, cardiac motion (with synthesized bright- and dark-blood contrasts) and contrast changes during inversion recovery. The 3D T₁ maps (12 out of 32 slices) from the proposed method were comparable to those acquired from MOLLI (Fig. 4). The myocardium T1 was 1208.7 ± 60.0 ms for MOLLI and 1189.8 ± 168.9 ms for proposed method. Figure 5 shows our preliminary results when pushing the proposed method to achieve a spatial resolution of 1.9x1.9x2.5 mm³ (64 slices) in a 12-min acquisition.

Conclusion

A new method is proposed to achieve simultaneous 3D T1 mapping and cine imaging of the heart using a linear tangent space alignment-based image reconstruction.

Acknowledgements

This work was supported in part by the National Institutes of Health (P41EB022544, R01CA165221, R01HL137230, and K01EB030045).

References

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