Introduction

The free-running simultaneous acquisition of 3D myocardial $$$T_1$$$ and $$$T_2$$$ maps with isotropic spatial resolution and cardiac cine images has recently been achieved with a ~10-minute scan using a 3D golden-angle radial trajectory and rigid-respiratory-motion correction^1,2. In that study, k-space spokes were ECG-gated to cardiac phases, and hence only a fraction of the acquired data was used in the reconstruction of each phase. Here, we propose to incorporate non-rigid cardiac motion correction³ into the reconstruction framework, allowing all acquired data to be used at each phase and thus enabling the simultaneous acquisition of 3D whole-heart $$$T_1$$$ and $$$T_2$$$ maps and cine images in ~3 minutes.

Methods

Data were acquired using a low flip angle spoiled gradient echo readout with a 3D golden-angle radial trajectory². Sufficient $$$T_1$$$ and $$$T_2$$$ encoding was achieved by repeating the three-shot cycle in Figure 1(a); each shot was preceded by an IR pulse alongside (for two of the shots) a $$$T_2$$$-prep pulse with an echo time of 30 ms or 60 ms.
Respiratory binning was performed using a respiratory signal extracted from the centre of k-space¹, with auxiliary bin images used to estimate and correct the k-space data for 3D rigid respiratory motion.
The respiratory-motion-corrected k-space spokes were sorted into $$$Q$$$ cardiac phases ($$$Q=10$$$ for $$$T_1$$$/$$$T_2$$$ maps, $$$Q=16$$$ for cine images) and for each phase a soft-weighted⁴ GROG⁵ reconstruction was implemented. Non-rigid 3D motion fields were estimated between each phase and a reference phase ($$$T_1$$$/$$$T_2$$$ maps) or between every pair of phases (cine) using a free-form deformation algorithm⁶.
A dictionary of signal evolutions over the three-shot cycle for different $$$(T_1,T_2)$$$ combinations was simulated and then compressed by retaining only the highest singular values from an SVD decomposition of the dictionary matrix⁷.
The reconstruction of multi-singular-contrast 3D images followed the low-rank motion-corrected (LRMC) approach of Cruz et al.³ by integrating non-rigid cardiac motion correction with the previous dictionary-based low-rank inversion (LRI)⁷ reconstruction and low-rank patch-based HD-PROST⁸ denoising, and thus proceeded by minimising
$$\mathcal{L}\left(\boldsymbol{\rho},\mathcal{T},\mathbf{Y}\right)=\|E\boldsymbol{\rho}-W\mathbf{k}\|_2^2+λ\sum_p\|\mathcal{T}_p\|_*+\frac{\mu}{2}\|\mathcal{T}_p-P_p(\boldsymbol{\rho})-P_p\left(\mathbf{Y}\right)\|_F^2$$
via an ADMM iteration scheme⁹. Here, for $$$N$$$ voxels, $$$N_c$$$ coils, $$$r$$$ singular contrasts, $$$K$$$ total k-space samples and $$$K_q$$$ k-space samples at the $$$q$$$th cardiac phase, $$$\boldsymbol{\rho}\in\mathbb{C}^{Nr\times\,1}$$$ is the multi-singular-contrast 3D image vector we are solving for; $$$\mathbf{k}\in\mathbb{C}^{KN_c\times\,1}$$$ is the vector of rigid-motion-corrected multi-coil k-space data; $$$W\in\mathbb{R}^{KN_c\times\,KN_c}$$$ contains radial-sampling-density-compensation weights; $$$\mathcal{T}_p$$$ is the HD-PROST tensor of patches similar to the patch centred at the $$$p$$$th voxel; $$$P_p$$$ is a patch-selection operator; $$$\mathbf{Y}$$$ is the augmented Lagrangian; $$$\lambda$$$ and $$$\mu$$$ are penalty weightings; $$$\ast$$$ and $$$F$$$ denote the nuclear and Frobenius norms, respectively; and $$$E=\sum_{q}WU_rA_qF_qSM_q$$$ is the encoding operator, which consists of $$$M_q\in\mathbb{R}^{Nr\times\,Nr}$$$, which applies the motion distortion for the $$$q$$$th cardiac phase, $$$S\in\mathbb{C}^{NN_cr\times\,Nr}$$$, the estimated coil sensitivity maps, $$$F_q\in\mathbb{C}^{K_qN_cr\times\,NN_cr}$$$, the non-uniform Fourier operator that transforms the 3D multi-coil multi-singular-contrast image to a corresponding 3D radial k-space with only those spokes acquired at the $$$q$$$th cardiac phase, $$$A_q\in\mathbb{N}^{KN_cr\times\,K_qN_cr}$$$, which positions $$$q$$$th-phase spokes into the full radial trajectory, and $$$U_r\in\mathbb{R}^{KN_c\times\,KN_cr}$$$, which contains the dictionary compression weights that relate the singular contrasts to the $$$T_1$$$- and $$$T_2$$$-weighting of the contrast that each spoke was acquired with.
To obtain cine images, the process was repeated with each cardiac phase set as the reference. To obtain 3D $$$T_1$$$ and $$$T_2$$$ maps, the multi-singular-contrast images were compared voxel-wise with the entries of the compressed dictionary. The $$$T_1$$$ and $$$T_2$$$ values used in the generation of the dictionary entry with the closest fit (determined by a dot product of normalised vectors) were assigned to that voxel in the maps.
For comparison, LRI (motion-resolved) reconstructions² ($$$M=I$$$) were also implemented. In this case, $$$\mathbf{k}\in\mathbb{C}^{K_qN_c\times\,1}$$$ contained only the rigid-motion-corrected k-space data acquired at the $$$q$$$th cardiac phase.

Results

Singular-contrast images and parameter maps for a healthy subject, obtained using LRMC and LRI reconstruction with varying amounts of k-space data, are presented in Figure 2 without regularisation ($$$\lambda=\mu=0$$$) to best demonstrate the improvement from non-rigid motion correction. $$$T_1$$$ and $$$T_2$$$ maps obtained with the proposed approach (30% of data, LRMC, HD-PROST) are shown in Figure 3 alongside conventional 2D mapping techniques (MOLLI and GraSE) for two subjects. AHA bullseye plots¹⁰ are presented in Figure 4 for one subject, alongside boxplots of the means and standard deviations of the mean recorded in each segment across 11 subjects for LRMC and LRI reconstructions and 100% or 30% of the acquired data. A cine of the 2nd singular-contrast image using the proposed approach, shown for nine short-axis slices, is presented in Figure 5.

Discussion

Results suggest that similar levels of accuracy and precision can be attained using only 30% of the acquired data when cardiac motion correction is incorporated. The proposed approach is seen to produce 3D maps of comparable quality to the conventional 2D mapping sequences and prevents the large increase in the standard deviation of myocardial $$$T_1$$$ and $$$T_2$$$ values seen in the motion-resolved reconstruction when using 30% of data.

Conclusion

By incorporating motion correction into the previous free-running simultaneous 3D myocardial $$$T_1$$$ and $$$T_2$$$ mapping and cine imaging framework, similar image quality was achieved with only 30% of the acquired k-space data, corresponding to a scan time reduction from ~10 minutes to ~3 minutes.

Acknowledgements

This work was supported by EPSRC (EP/P001009, EP/P032311/1, EP/P007619/1) and Wellcome EPSRC Centre for Medical Engineering (NS/ A000049/1).

References

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