Synopsis

We previously presented a framework to reconstruct 3D non-rigid motion-fields from highly undersampled k-space data by exploiting the lower-dimensional nature of motion-fields. Here, a quantitative comparison with a well-established image-based motion estimation is performed on 3D abdomen images. To also exploit temporal compressibility the framework is extended to reconstructions of spatio-temporal non-rigid motion-fields at high frame-rate, and validation is performed on 2D+t abdomen cine-acquisitions. Additionally, a dedicated non-Cartesian 3D trajectory was employed to prospectively acquire highly undersampled k-space data, and reconstruct 3D+t head motion at 16Hz. High quality respiratory/head motion-fields are obtained and the framework outperforms the fully-sampled image-based motion estimation even for undersampling up to 8x.

Introduction

3D time-resolved motion information is essential for MR-guided interventions/radiotherapy and cardiac imaging, but remains a challenge for state-of-the-art dynamic MRI methods. Image-based methods are limited by long acquisition times required for image reconstruction, despite possible accelerations through PI/CS^1,2, and are therefore sub-optimal for motion estimation at high frame-rate (see Figure 1). Our previous work³ demonstrates a framework that avoids image reconstruction, and reconstructs 3D non-rigid motion-fields directly from highly undersampled$$$\hspace{1.5mm}k$$$-space data by inverting our dynamic MR-signal model. Significantly higher accelerations were possible compared to image-based methods by exploiting the spatial compressibility of motion-fields.

We first validate the framework by a quantitative comparison with well-established image-based motion estimation (optical-flow). To also exploit temporal compressibility, the framework is extended from spatial to spatio-temporal motion-field reconstructions. The spatio-temporal framework is validated on retrospectively highly undersampled$$$\hspace{1.5mm}k$$$-space data generated from 2D+t cine-acquisitions. Finally, a dedicated non-Cartesian 3D trajectory was designed and implemented to prospectively acquire highly undersampled$$$\hspace{1.5mm}k$$$-space data and reconstruct 3D+t head motion at 16Hz.

Theory

Methods

We assume a reference image$$$\hspace{1.5mm}q_\text{ref}\hspace{1.5mm}$$$is available, and acquire highly undersampled $$$k$$$-space snapshot signals $$$\mathbf{s}=[\mathbf{s}_1,\dots,\mathbf{s}_M]^T\hspace{1.5mm}$$$of$$$\hspace{1.5mm}N\hspace{1.5mm}$$$interleaves/snapshot in$$$\hspace{1.5mm}N\cdot\text{TR}\hspace{1.5mm}$$$seconds/snapshot. $$$\mathbf{T}\hspace{1.5mm}$$$is spatially parametrized with non-rigid affine and cubic B-spline motion-models⁴ for in-vivo brain and abdomen motion, respectively, and temporally with cubic B-splines. Motion-fields are reconstructed at $$$1/(N\cdot\hspace{1.5mm}\text{TR})$$$ Hz by solving a minimization problem with respect to motion-model parameters $$$\mathbf{\theta}$$$ (see Figure 2):$$\min_\mathbf{\theta}\quad\lVert\mathbf{F}(\mathbf{\theta})-\mathbf{s}\rVert_2^2+\mathcal{R}(\mathbf{T}).\quad\quad(3)$$Here the regularizer $$$\mathcal{R}$$$ is chosen as the curvature⁵ to penalize non-smooth deformations in space/time. (3) is solved using Matlab's interior-point algorithm and the reconstructed $$$\mathbf{T}$$$ are inverted to$$$\hspace{1.5mm}\mathbf{U}\hspace{1.5mm}$$$using Picard-iterations⁶. Finally, the reference image is warped using (1) to asses reconstruction quality. See Table 3a for sequence details regarding the following experiments.

First validation: comparison with well-established image-based motion estimation.

A quantitative comparison with optical-flow^7,8,9 was performed using two fully-sampled, breath-hold, Cartesian 3D abdomen images. A snapshot $$$k$$$-space was generated from one image and was undersampled with several factors (see Table 3b) to cubes around the $$$k$$$-space origin, while the other was used as reference. The relative differences with the fully sampled acquisition were computed for the warped reference images.

Second validation: 2D+t reconstructions.

Snapshot $$$k$$$-space data was synthetically generated from free-breathing abdomen 2D+t cine MR-images and retrospectively undersampled to $$$30\times30\hspace{1.5mm}k$$$-space cubes around the origin (=125x undersampling). 2D+t motion-fields were reconstructed from this $$$k$$$-space data according to (3) for all dynamics at once. Dynamic images were obtained from (1) and compared with cine MR-images.

Reconstruction from prospectively acquired snapshots.

To reconstruct motion at high frame-rate, a dedicated fast non-Cartesian 3D trajectory was designed to prospectively acquire highly undersampled $$$k$$$-space snapshots at 16Hz, and implemented on a 1.5T scanner (Philips,Ingenia) as a repetition of 6 interleaves of a 3D cone trajectory¹⁰ (see top-right of Figure 2 for illustration). This corresponds to 480x undersampling in the number of read-outs.

Results

Table 3a shows that our method outperforms optical-flow even for undersampling up to 8x, and stays within 2% relative error for undersampling up to 64x. Higher undersampling requires more regularization, and quality degrades rapidly. This is likely caused by our sub-optimal curvature regularization for breathing motion.

Figure 4 shows good agreement with the ground-truth despite possible through-plane motion and intensity variations over time.

Figure 5 shows a dynamic 3D image sequence that was reconstructed using motion-fields acquired at 16Hz. The affine model recovered rigid motion and is consistent with the 8-shaped motion performed by the volunteer.

Discussion & Conclusion

Our framework³ bypasses the requirement of image reconstruction for motion estimation by focusing on the lower-dimensional problem of directly estimating motion-fields from $$$k$$$-space data. The framework is compared with state-of-the-art image-based motion estimation and extended to reconstruction of spatio-temporal deformation-fields. Reconstruction quality was evaluated on data generated from a 2D+t cine acquisition. Experimental feasibility was demonstrated by reconstructing 3D+t motion-fields at 16Hz from prospectively acquired highly undersampled $$$k$$$-space data using a dedicated 3D non-Cartesian trajectory. In all cases the proposed framework showed high quality reconstructions and higher accuracy than fully-sampled image-based motion estimation even for up to 8x undersampling of the input data.

Our method reconstructs 3D motion-fields at sub-second frame-rate, and has therefore great potential for MR-guided radiation therapy, which relies on similar dynamic information. Future research will address other spatio-temporal regularization and optimal trajectory design for even faster trajectories required by applications like cardiac imaging.

Acknowledgements

References

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