There are many ways to choose the spoke directions in 3D radial imaging. The problem can be viewed as how to place N points “evenly” on the the surface of a sphere, however there is no universally agreed upon notion of "even”. Nevertheless several simple methods exist for generating sets of points that appear to be evenly spaced. Three such methods are considered here: Saff spiral¹ Rusin disco ball² and Carlson golden angle³.

It is helpful to see simple examples of the points generated by these methods (Figure 1 with N = 106 points). The Saff method places points regularly along a smooth spiral curve. The Rusin method places equally spaced points in discrete, equally spaced bands (note: only certain numbers of points are supported in this configuration). The Carlson method places points equally along the z-direction with golden angle rotations in the xy-plane, which results in a much more irregular pattern (note: the points have been rearranged to produce a smaller angular increment).

The purpose of the present study was to implement and compare strategies for choosing the radial spoke directions in a 3D center-out radial sequence.

The points from Figure 1 were projected onto a map view in Figure 2. To serve as a metric for comparison, the color scale represents the distance to the nearest point (blue = close to a point and red = far from a point). The Saff method leaves the largest gaps between points (near the poles) and the Rusin method leaves the smallest gaps. The Carlson method is in-between. According to this metric, the Rusin method can be said to produce the most evenly spaced points of these 3 methods.

In addition to looking at the point distributions (in k-space), an important aspect for imaging is the point-spread-function in image space. These are shown in Figure 3 using a larger number of points (N = 7942) to give a more realistic example. The more regular point distributions in k-space (Saff and Rusin) give rise to structured aliasing artefacts whereas the less regular distribution (Carlson) has incoherent aliasing artefacts.

To verify the simulation results using actual data, the different radial spoke directions were implemented in a center-out radial 3D spoiled gradient echo sequence. Data were acquired on a Siemens Magnetom 7T using 7942 spokes, 68 readout points per spoke, TR 10 ms, TE 0.03 ms, flip angle 2 deg, BW 444 Hz/pixel and reconstructed in Matlab onto a 128×128×128 matrix (Kaiser-Bessel regridding, width 4, oversampling factor 2). Images are shown in Figure 4. It can be seen that the streaking artefacts are higher with the Saff and Rusin methods than with the Carlson method, which follows the simulation results.

In choosing a set of spoke directions for 3D radial imaging, it is important to consider the distribution of the points in k-space - such as distance to nearest neighbor, Voronoi polygon area, furthest distance to any point (as used in Figure 2) - but also the point-spread-function in image-space.

This study has observed that some sets of evenly spaced points (e.g. the Rusin method, which produces points that have an almost equal distance to nearest neighbor and also results in the smallest furthest distance to any point) can produce relatively high levels of aliasing in the image. Conversely, a less evenly spaced set of points (e.g. the Carslon method, which produces points that do not excel in any particular metric) produce relatively low levels of aliasing.

Of the methods presented in this study, the Carlson golden angle appears to produce the least aliasing in the image. This is attributed to the irregular undersampling pattern, which tends to cause incoherent aliasing⁴. Methods not presented (but of considerable interest going forward), are the electrostatic repulsion method discussed in Ref 1 and a uniform spiral method⁵, which are computationally more intensive but represent good examples of irregular and regular point distributions.