Advantages of radial k-space sampling schemes over Cartesian sampling are a regular acquisition of the k-space center, giving increased information on changes to contrast and insensitivity to motion artifacts. Furthermore, under-sampling artifacts appear as incoherent streaking artifacts, which are in many applications more tolerable than Cartesian artifacts [1] and are more suited to a compressed sensing reconstruction [2]. However, since projections are acquired at varying angles, radial acquisitions are more sensitive to gradient delays and eddy currents, leading to image distortions. Additionally, phase errors introduced by B₀ inhomogeneities may cause artifacts due to signal cancellations during gridding. Gradient delays can be measured and corrected for by adapting the gradient timing or by retrospectively shifting the trajectory [3], [4]. Phase corrections have also been shown to improve image distortions [5]. Under ideal conditions, the phase of intersection points of two projections is identical. Without gradient delays, all profiles meet in k-space center and can be corrected for by measuring the center phase φ and multiplying the projection by e^-iφ. However, if data are affected by gradient delays, profiles will not intersect in k-space center anymore. In the 3D case, they might not intersect at all. In this work, a novel method is presented, in which gradient delays are retrospectively corrected by shifting the trajectory during gridding and these shifts are incorporated into a 3D phase correction.

Phantom data were acquired using a 3D golden angle radial sequence on a 3T MRI system (Magnetom Skyra, Siemens Healthcare, Erlangen, Germany). Gradient delays were measured and corrected for according to [3]. For additional phase correction, the following steps were performed:

1) A random reference profile p₀ is chosen and its corresponding complex data are stored in the vector p_ref.
2) All projections {p_j} are found for which the closest absolute distance d to p₀ is smaller than a given threshold t.
3) The projections {p_j} are phase corrected as described below with respect to p₀ and also recorded in p_ref.
4) This procedure is repeated for all projections in p_ref on an individual basis, where each corrected profile serves as new additional reference projection.
5) The algorithm stops when all projections are corrected for.

The phase correction of projection p₁ with respect to projection p₂ is performed as follows (fig. 1). Both profiles are linearly interpolated and the points r₁ on p₁ and r₂ on p₂ with the closest distance d between both projections are determined. The phases φ₁ and φ₂ at the points r₁ and r₂ and their difference ∆φ = φ₁-φ₂ are calculated. For a small threshold t (here t=0.001 pixel), the phase in both points should be approximately the same, when no phase errors are present. Under this assumption, the complex signal of p₂ is multiplied by a factor e^-iΔφ making both phases consistent.

For a spherical phantom, the gradient delays were measured to be ∆t_x=1.638 μs, ∆t_y=0.792 μs and ∆t_z=0.335 μs. The uncorrected, only gradient delay corrected, and additionally phase corrected images along with their difference images are shown in figure 2. The red arrows indicate regions of artifact reduction. The ratio s of the mean background and the phantom signal is calculated for the 3D volume. A reduction of 62 % for gradient delay correction and 12 % for the applied phase correction is found.A novel method for phase correction incorporating gradient delays has been shown to reduce artifacts. In comparison with other works such as [6], no additional measurements are needed. The algorithm is simple and straightforward to apply. Phase corrections have been performed in a similar study [5], however, there it was not taken into account that profiles do not intersect in the center in the presence of gradient delays. Since the gradient delays are assumed to be exactly known, the accuracy of gradient delay measurements affects the quality of the phase correction. The presented algorithm is furthermore based on the assumption that two points in close proximity have approximately the same phase, which may become invalid in the case of fast changing phases. In the experiment shown, phase errors are relatively small compared to gradient delay errors, however still clearly visible. Especially in the case of field strengths greater than 3T, B₀ effects increase and phase errors are likely to become more relevant.