Synopsis

A new navigator concept is presented which enables the measurement of accurate rigid-body motion information from spherical navigators at extremely small k-space radii. This is achieved by using a 3D Lissajous navigator trajectory and a rotation estimation algorithm known from computer vision. The new method provides motion information with sufficient accuracy and precision for many motion correction techniques and significantly outperforms current spherical navigator methods in terms of acquisition and computation time.

Introduction

Motion is a frequent problem in MRI causing image artefacts which limit clinical and scientific utility. Motion correction techniques have been developed most of which require intra-scan motion information¹. Spherical navigators are an efficient way to measure complete rigid-body subject motion using only the available scanner hardware without any external devices. However, current spherical navigator methods, which use helical spiral trajectories^2,3, have significant drawbacks limiting their further adoption: Rotation estimation is typically carried out via pattern matching algorithms (PMAs) which are computationally expensive². Since PMAs require a high amount of magnitude features on the sphere, the navigators need to be acquired at large k-space radii k_r≥0.4/cm ². Unfortunately, however, at large k_r, the SNR is low, making the optimal k_r a tradeoff between feature richness and SNR³. In addition, current navigator designs exhibit problematic gradient characteristics and require two excitations leading to acquisition times >25ms ^2,3.

In this work we present a new spherical navigator concept which overcomes the aforementioned drawbacks. It comprises a spherical Lissajous trajectory which can be acquired in a single shot in less than 10ms with full sphere coverage and gentle gradient requirements throughout⁴. A spherical harmonics-based rotation estimation known from computer vision^5,6 is used which enables accurate estimations from small-k_r Lissajous navigators with minimal computational overhead.

Methods

The spherical navigator trajectory used in this work was a 3D Lissajous curve given by

$$k_x=k_r\sin(\omega_{\theta}t)\cos(\omega_{\phi}t)\\k_y=k_r\sin(\omega_{\theta}t)\sin(\omega_{\phi}t)\\k_z=k_r\cos(\omega_{\theta}t)$$

where t is the time, $$$\omega_{\theta}=42\pi/\delta_{nav}$$$, $$$\omega_{\phi}=40\pi/\delta_{nav}$$$ and $$$\delta_{nav}=8ms$$$ is the net navigator duration. Two thousand sampling points were acquired equidistantly in time. The trajectory and the corresponding gradient waveforms are shown in Fig. 1. A non-selective 200μs RF pulse with 15° flip angle was used for excitation. To assess the accuracy of our technique, two series of 121 spherical Lissajous navigators (see Fig. 2) were acquired on a commercial Siemens 3T Trio scanner (Siemens Healthineers, Erlangen, Germany) from a brain-shaped water phantom in a 32-channel head coil. Phantom motion was induced by sinusoidally rotating and translating the FOV by up to 10° about and 10mm along each gradient axis, respectively. In another experiment 450 Lissajous navigators were acquired without moving the FOV, allowing a quantification of the precision, i.e. reproducibility, of the results. Shimming was performed once at the beginning of the measurements using the standard scanner routines.

The Euler angles of the rotations were estimated using an algorithm which was developed for robotic cameras and exploits the rotation theorem of spherical harmonics as described by Makadia et al.^5,6. Translations were estimated conventionally by solving an equation system as described by Welch et al.². The k-space radius of all spherical navigators was k_r=0.1/cm and the first navigator of every series was used as the reference. All computations were implemented in Python and performed on a standard laptop with 16GB RAM and a 2.5GHz CPU (Core i7-4710MQ, Intel, USA).

Results

Fig. 3 shows the results for the rotation and translation estimations. The applied FOV motion was accurately extracted from the Lissajous navigators. The maximum deviations from the applied values were below 0.4° and 0.2mm, respectively. Fig. 4 shows the reproducibility of the motion estimations which are very stable and deviate no more than 0.1° and 0.04mm from zero, respectively. The computation times of the complete rigid-body motion estimations from the experiments in Fig. 3 are depicted in Fig. 5. The calculations took on average approx. 65ms and did not exceed 100ms.

Discussion and Conclusions

As shown in this proof of concept, the Lissajous navigator design in combination with the proposed rotation estimation algorithm is, in principle, suited for rigid-body motion measurements from spherical navigators sampled at unprecedentedly small k_r. The accuracy and the precision are clearly within the range of what is required for many motion correction techniques. The navigator is acquired more than 50% quicker than in previous techniques^2,3 with only a single excitation. Due to the simplicity of the rotation estimation algorithm^5,6, the computation times are on the order of tens of milliseconds making this technique significantly faster than previously reported PMA-based approaches² and applicable to prospective motion corrections.

Since the applied FOV motion was taken as a ground truth although the scanner certainly introduces errors as well, the stated accuracies are worst-case estimates. The investigated motion is still somewhat “artificial” because the phantom itself remained motionless while only the FOV was moved. Our current efforts are towards extending the investigation to “real” motion and switching to a registration-based method as the ground truth. The proposed method has the potential to significantly increase the utility of spherical navigator techniques for motion correction in MRI.

Acknowledgements

References

¹Godenschweger F, Kägebein U, Stucht D, Yarach U, Sciarra A, Yakupov R, Lüsebrink F, Schulze P, Speck O. Motion correction in MRI of the brain. Physics in medicine and biology 2016;61:R32

²Welch EB, Manduca A, Grimm RC, Ward HA, Jack Jr CR. Spherical navigator echoes for full 3D rigid body motion measurement in MRI. Magnetic Resonance in Medicine 2002;47:32–41

³Petrie DW, Costa AF, Takahashi A, Yen Y-F, Drangova M. Optimizing spherical navigator echoes for three-dimensional rigid-body motion detection. Magnetic Resonance in Medicine 2005;53:1080–1087

⁴Ullisch M, Stöcker T, Vahedipour K, Pracht E, Tellmann L, Herzog H, Shah N. Towards Lissajous navigator-based motion correction for MR-PET. Proceedings of the 18th Annual Meeting of ISMRM, Stockholm, Sweden. 2010. p. 3060

⁵Makadia A, Daniilidis K. Direct 3d-rotation estimation from spherical images via a generalized shift theorem. In: Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. 2003

⁶Makadia A, Sorgi L, Daniilidis K. Rotation estimation from spherical images. In: Proceedings of the 17th International Conference on Pattern Recognition, 2004