Prospective motion correction is typically performed by re-adjusting the slice position and orientation data once per TR prior to the RF excitation. However, for sequences with an extended signal preparation as diffusion-weighted imaging^1,2 or with long trains of refocusing pulses as 3D-T2-weighted imaging (aka SPACE)³, a single correction per TR may become insufficient. In such cases it is essential to compensate for the additional phase accumulated due to motion in presence of signal weighting or imaging gradients. Especially for multi-pulse experiments it is crucial that these predictions remain accurate and quantitative for arbitrary combinations of translations and rotations. In this work we present a compact and elegant approach to perform such calculations, based on the rigid body kinematics.

According to Mozzi-Chasle's theorem, any rigid-body displacement can be represented as a combination of a translation along a line combined with a rotation around that same line, termed the screw axis. This screw motion between two rigid body poses describes uniquely a trajectory with a constant translational velocity along and a constant angular velocity about the screw axis. The displacement of all spins inside this rigid body during motion can be described by two vectors: parallel and tangential to the screw axis. Generally a combination of the two motions has to be considered, which can be decoupled because of the orthogonality of the translational and tangential motions of the spins.

For any complex MR pulse sequence, an arbitrary combination of gradients still results in a single gradient of an appropriate direction and a magnitude. To describe its effect it is convenient to decompose this gradient into two components: parallel and orthogonal to the screw axis. Due to the orthogonality of the corresponding displacements the parallel component will only interfere with the linear velocity, whereas the orthogonal with the angular, respectively.

The orthogonal movement of very voxel in can be described as: $$P_{orth(x,y)}(t)=r\cdot{}\left(\begin{array}{c}cos(t\omega+\beta)\\sin(t\omega+\beta)\end{array}\right)+ \left(\begin{array}{c}x_{rotaxis}\\y_{rotaxis}\end{array}\right)$$ and parallel: $$P_{par(z)}(t)=z_0+t\cdot{}v_z$$ with: $$$r$$$ as radius of the rotation, $$$\omega$$$ as rotation speed, $$$\beta$$$ an angle-offset to define the start position on the circle, $$$x_{rotaxis},~y_{rotaxis}$$$ the rotation-point in xy-plane. The variables that are varying between voxel are only: $$$r$$$, $$$\beta$$$ and $$$z_0$$$. Using these two equations it is trivial to calculate the phase, as it is the integration over the circular motion and the translation along the z-axis. $$\phi_{orth}=\int_{t_{0}}^{t_{1}}\!P_{orth}(t)\cdot{}G_{orth}\,dx$$ can then be integrated and gives the phase produced by this part. $$\phi_{par}=\int_{t_{0}}^{t_{1}}\!P_{par}(t)\cdot{}G_{par}\,dx$$ the z-component, afterwards: $$\phi_{total}=\phi_{orth}+\phi_{par}$$.

Our first experiment was conducted to verify that the screw interpolation appropriately describes true motion of a subject’s head. This was performed by measuring the head position every 12.5ms using a Moiré phase tracking (MPT) system. The screw interpolation was verified by interpolating between every 10th position and comparing the interpolated positions to the true positions recorded.

The second experiment was performed with a rotation phantom (shown in Figure 1), using a velocity-compensated GRE sequence with an additional bipolar pulse in phase encoding or readout direction, introduced between the excitation and readout. The MPT system was used to record the phantom pose at specific time points within the sequence. These data were used to predict the phase using a simulation in Matlab (The Mathworks, USA based on the formulas above.

The quality of the screw interpolation from the first experiment was measured by calculating the errors of two trivial motion patterns: nodding and shaking of the head. The error of the interpolated data is shown in Table 1.

The observed errors are comparable to the noise of the MPT system. The screw interpolation is therefore appropriate, especially when even shorter time periods are considered, e.g. between the optical frames.

The data of the second experiment were acquired with the rotating phantom inside an MRI scanner, with the slice positioned as in Figure 1. Figure 2 shows the measured phase with different bipolar gradients and the results of the simulation. Eddy current induced artifacts were subtracted by imaging a stationary phantom.

The measured phase and the simulated phase have the same phase-wraps and as shown in Table 2, similar phase gradients. The new method to calculate the phase with an analytical integration over the screw motion is in good agreement with the experimental measurement.

With the accurate and fast prediction of the phase changes, methods to compensate for arbitrary rigid body motion in sequences with extended signal preparation periods can be developed. The proposed algorithm is fast enough to be used for real-time motion correction in sequences such as T2-SPACE or SPACE-FLAIR, sequences of high clinical importance.