Daniel Stäb^{1}, Tobias Wech^{2,3}, and Markus Barth^{1}

3D planes-on-a-paddlewheel (POP) echo-planar imaging (EPI) is an effective non-Cartesian readout scheme realized by rotating conventional EPI readout planes about the phase encoding axis. Navigator based phase correction schemes are typically employed to account for gradient timing errors, associated trajectory errors and artifacts. In this work, we propose to use “Self Consistency for an Iterative Trajectory Adjustment” SCITA for an improved and purely data-driven removal of trajectory misalignment artifacts. As the actual k-space trajectory is derived from the imaging data, navigator acquisitions can be omitted and echo, repetition and acquisition times may be considerably shortened.

3D planes-on-a-paddlewheel (POP) echo-planar
imaging (EPI) is a non-Cartesian readout scheme realized by rotating
conventional EPI readout planes about the phase encoding axis (Fig. 1). The technique
is effective for distortion minimization and was recently employed for
whole-brain quantitative susceptibility mapping (QSM) at ultra-high field^{1}.

Gradient timing errors are known to cause
severe artifacts in EPI. Phase correction algorithms^{2}, based on additional
navigator echoes, typically have to be employed to remove the Nyquist ghosting,
which arises from the shift of odd and even echoes with respect to each other
along the phase encoding axis in k-space. In 3D POP EPI, an additional
misalignment of the projections within the radial readout planes is introduced,
which is typically neglected by navigator-based phase correction algorithms.

In this work, we propose to exploit “Self
Consistency for an Iterative Trajectory Adjustment” (SCITA)^{3} for a purely
data-driven removal of trajectory misalignment artifacts in 3D POP EPI. As the
additional acquisition of navigator echoes can be omitted, the approach may
facilitate shorter echo, repetition and acquisition times.

SCITA exploits
the inherent oversampling of central k-space in radial imaging and obtains an
estimate of the actual k-space trajectory that can be finally used for
gridding. To that end, the trajectory correction technique iteratively maximizes
the data consistency across the individual projections in the oversampled part
of k-space by shifting individual projections with respect to the rest of the
acquired k-space data by means of the GRAPPA operator^{4}.

Measurements
were performed on a non-commercial 7 T whole-body research MRI scanner (Siemens
Healthcare, Erlangen, Germany) under institutional review board permission. The system was equipped with a Tx/32
channel Rx head array (Nova Medical, USA) and 3^{rd} order shimming was
employed for all measurements.
One healthy
34-year-old volunteer was examined, after obtaining written informed consent and
approval of the local ethics committee, and 3D POP EPI images were obtained at
1.5 mm isotropic spatial resolution using following acquisition parameters: FOV
= 170 x 170 x 37 mm3, 172 projections with linearly increasing
azimuthal angle within the interval [0;π[, TR = 36 ms,
TE = 19 ms, ES = 1.03 ms, flip angle = 12°, ramp sampling = 6.4%. At the
beginning of each readout train, 3 additional navigator echoes were acquired. All
images were reconstructed offline using Matlab (The Mathworks, Natick, USA). The non-uniform fast Fourier transform (NUFFT)
framework^{5} was used for gridding, followed by
a Fourier transformation along the phase encoding axis and sum-of-squares coil
combination.

To test the performance of SCITA, images were reconstructed

- without any correction,
- using
a navigator based phase correction
^{2}for each projection before the gridding step, - using a SCITA trajectory estimate obtained within the radial plane containing the echo (phase encoding step 13) for the reconstruction of all radial planes, and
- using a separate SCITA trajectory estimate for odd and even readout planes/phase encoding steps.

For the latter reconstruction, the trajectory estimates were obtained from the radial plane containing the echo and the adjacent radial plane (phase encoding step 12). To achieve a proper alignment of those two trajectories in the separate odd/even reconstruction, each trajectory was centered by iteratively maximizing the sum-of-squares k-space DC signal at k = 0 upon the application of multiple linear phase ramps in image space.

In all cases, SCITA was employed iteratively as
described elsewhere^{3} with continuously decreasing shift lengths of 0.33,
0.15, 0.1, 0.07, 0.03 and 0.01 Δk with 20 iterations per shift length.

- Stäb D, Bollmann S, Langkammer C, et al. Accelerated mapping of magnetic susceptibility using 3D planes-on-a-paddlewheel (POP) EPI at ultra-high field strength. NMR Biomed. 2017;30:e3620.
- Heid O. Method for the phase correction of nuclear magnetic resonance signals. United States Patent Application Publication US6043651 (2000).
- Wech T, Tran-Gia J, Bley TA, Köstler H. Using self-consistency for an iterative trajectory adjustment (SCITA). Magn. Reson. Med. 2015;73:1151–1157.
- Seiberlich N, Breuer FA, Blaimer M, Barkauskas K, Jakob PM, Griswold MA. Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG). Magn Reson Med 2007;58:1257–1265.
- Fessler JA, Sutton BP. Nonuniform fast Fourier transforms using min–max interpolation. IEEE Trans. Signal Process. 2003; 51: 560–574.

Fig 2: Trajectory estimates (central
k-space) for the 12^{th} and 13^{th} phase encoding step. The
trajectories of odd and even echoes differ considerably.