Synopsis

Keywords: Data Acquisition, Artifacts, Optimized 3D Trajectory

The commonly used golden angle trajectory for 3D radial dynamic MRI suffers from spiral-shaped sample clustering around the z-axis, often resulting in image streaking artifact. We develop a method that corrects for this for center-out 3D radial trajectories. The electric potential energy of the trajectory is minimized through the use of repulsive forces, producing a spherically uniform distribution while maintaining the quasirandom quality of the golden angle trajectory. ELECTRic potential energy Optimized (ELECTRO) trajectories are relatively inexpensive to obtain and can produce images with lower MSE compared to the golden angle counterpart.

Introduction

In MRI, "golden angle" 3D radial trajectories are used because of their versatility in the reconstruction of dynamic images. Under this quasirandom acquisition scheme¹, data in a temporal window of arbitrary size will have relatively adequate and non-redundant directional coverage of $$$k$$$-space, enabling 1) post-acquisition choice of the reconstructed temporal resolution and 2) retrospective physiological sorting of the data (e.g. cardiac, respiratory, bulk motion). However, the spokes in this trajectory do not always fill one of the largest gaps in $$$k$$$-space. In fact, the trajectory forms spiral clusters around the z-axis (Figure 1), which can lead to significant loss in image quality.

We develop a method for optimizing 3D center-out radial imaging that uses electric/repulsive forces to produce ELECTRic potential energy Optimized (ELECTRO) trajectories. ELECTRO trajectories are quasirandom and do not have any sample clustering.

Proposed Method

The direction of spokes in a 3D center-out radial trajectory can be represented as point charges on the surface of the unit sphere. For a set of $$$N$$$ points, let the set of their positions be $$$S=\left\{r_1,...,r_N\right\}$$$, $$$r_i\in R^3$$$. Let the total electric potential energy (EPE) of $$$S$$$ be
$$U(S)=U(r_1,...,r_N)=\sum_{1\leq i<j\leq N}\frac{1}{|r_i-r_j|}.$$

The objective is to find a set of positions that locally minimize $$$U(S)$$$, subject to the constraints $$$|r_i|=1$$$ for $$$i=1,...,N$$$. We carry out the minimization by using gradient descent on each point $$$r_i$$$ simultaneously, and then projecting each point back onto the unit sphere. Thus the steps in the $$$t$$$-th iteration are
$$u=r_i^{(t)}-\gamma^{(t)}\frac{\partial U(S^{(t)})}{\partial r_i^{(t)}}$$
$$r_i^{(t+1)}=u/|u|$$
for $$$i=1,...,N$$$. Evaluating the gradient results in a sum of repulsive forces (dropping the $$$t$$$-th iteration superscript):
$$u=r_i+\gamma\sum_{j=1,j\neq i}^N\frac{r_i-r_j}{|r_i-r_j|^3}.$$
This procedure produces sets of points that tend towards uniformity on the sphere.

We initialize the optimization using the points from the golden angle trajectory¹. The step size, $$$\gamma^{(t)}$$$, is allowed to change at every iteration. Since the points at the north/south poles are initially in close proximity to each other, a smaller step size is used for the first few iterations to prevent large displacements, and is then gradually increased as the points spread out. This helps preserve the relative position of the points.

The sum in the update step is expensive to calculate for large $$$N$$$. Instead, it is approximated by only considering a small spherical neighborhood of points around $$$r_i$$$. The animation in Figure 1 shows how the 10000 spoke golden angle trajectory changes throughout the optimization process.

Experiments

A static 160x160x160 digital phantom (MRXCAT) was used in a simulated imaging experiment². The phantom was filtered using a spherical mask so that its frequency representation could be entirely sampled by a radial trajectory. A set of 24 analytically calculated birdcage coil sensitivity maps were used for $$$k$$$-space simulation and image reconstruction. Seven ELECTRO trajectories were created, with acceleration factors ranging from 1 to 64. Multicoil $$$k$$$-space data was calculated for each ELECTRO trajectory, as well as for the golden angle trajectory. A center-out radial readout without gradient ramping was used, and no noise was added to $$$k$$$-space. Reconstruction was performed using CG-SENSE³. Density compensation factors (DCF) were calculated for each trajectory using an iterative method⁴.

Results and Discussion

Point spread function (PSF) analysis shows that the golden angle PSF has a disk-like structure near the mid xy-plane (Figure 2), likely caused by the sample clustering around the z-axis. The ELECTRO PSFs are improved and do not have this feature. The iterative DCFs used in this work help to compensate for regions with high trajectory overlap, but do not adequately compensate for the spiral clustering, which has large sub-Nyquist spacings between samples.

The ELECTRO trajectories have an EPE much closer to that of a uniform radial trajectory (Fibonacci sphere), compared to the golden angle trajectory (Figure 3a). This correlates with a lower MSE in the images produced by the ELECTRO trajectories (Figure 3b). Improvements in the MSE come from both the correction of clustering and the more evenly distributed points around the sphere. We expect that the former reduces coherent streaking artifacts while the latter reduces noise-like artifact.

The orthogonal center planes of the reconstructed images for the 20000 spoke trajectories are shown in Figure 4, along with the difference images. Streaking artifact can be seen in the image using the golden angle trajectory, which is unnoticeable in the image using the ELECTRO trajectory.

To evaluate the quasirandom quality of the ELECTRO trajectory, the EPE of a sliding window of spokes was calculated. Each section of spokes has a different EPE, so the mean of EPE was calculated over all sections. Figure 5 shows this result as a function of different sliding window sizes. The curve corresponding to the ELECTRO trajectory is nearly identical to that of the golden angle trajectory. The 2D bit-reversed trajectory⁵, another commonly used quasirandom trajectory, is plotted as a reference for comparison.

The optimization objective behind ELECTRO trajectories is relatively general, compared to methods that optimize the sampling pattern based on specific patient anatomies, contrasts, reconstruction algorithms/parameters, and coil geometries⁶. Such methods would improve image quality, but may require a new optimization whenever a new imaging setting is encountered. ELECTRO trajectories are relatively inexpensive to obtain and remain uncommitted to any of these imaging settings.

Acknowledgements

Funding provided by NSERC Canada - Discovery Grant (RGPIN-2019-06483)