Introduction

Adjacent spokes are often spaced closely in zero echo time (ZTE) 3D-radial imaging to limit eddy-current effects, gradient-ramping time, and noise. Conventional approaches have arranged spoke endpoints along carefully spaced spherical spirals [1]. With appropriate interleaving [2], spherical spirals can achieve uniform k-space coverage over a full scan extent, but shorter time windows leave large gaps in the polar direction -- less suitable for dynamic imaging. A recent work adapts spherical spirals to better control polar versus azimuthal angular velocity [3], but increasing polar velocity in spherical spirals leads either to oversampling near the poles or greater slew variation.

We consider slew-constrained 3D-radial trajectory design as an optimal path-finding problem on a sphere. Our solution resembles the seam lines of a tennis ball for a certain path length, so we call the general class of solutions "tennisball" trajectories. Here we characterize tennisball trajectory properties and report on initial results in static and motion-resolved imaging.

Theory

Consider the space of all length-$$$T$$$ continuous paths on a unit sphere $$$\mathbb{S}_2$$$. We would like to construct a path in this space that explores the sphere as efficiently as possible. We define a candidate path $$$\vec{\mathbf{k}}$$$'s inefficiency as the maximum geodesic distance between the path and any point on the sphere. We seek a solution $$$\vec{\mathbf{k}}^\star$$$ that minimizes this maximal gap:
$$\vec{\mathbf{k}}^\star\in\,\operatorname{arg}\,\operatorname{min}_{||\vec{\mathbf{k}}||_2=T}\,\operatorname{max}_{\substack{\mathbf{p}\in\mathbb{S}_2}}\,d_{\mathbb{S}_2}(\vec{\mathbf{k}},\mathbf{p}).$$
This problem might be difficult to solve in general, but remarkably simple solutions exist for many values of $$$T$$$. These solutions and their constructions (Fig. 1) are inspired by recent related results in geometry [4] but are otherwise new to our knowledge. Suppose $$$T$$$ can be expressed as the sum of the circumferences of $$$L$$$ evenly spaced latitudes on the sphere, where $$$L$$$ is any positive integer. This set of latitudes achieves the minimum worst-case gap by construction (Fig. 1A). Rotating either hemisphere about any plane perpendicular to the poles does not change the maximal gap, but a continuous path emerges for certain rotation angles. Multiple rotation modes exist for $$$L>2$$$ (Fig. 1B) and offer a natural mechanism to tradeoff polar versus azimuthal velocity. We explore this property later for dynamic imaging.

Methods and Results

Tennisball implementation:
We modified GE's product 3D-radial sequence to calculate and prescribe tennisball trajectories. The sequence chooses $$$L$$$ based on FOV, resolution, and slew constraint settings. Under mild slew limits, $$$L$$$ can be adjusted to balance spacing between latitudes versus spacing along the path, thereby yielding uniform coverage. Under more stringent slew limits, the sequence designs the longest path possible and operates close to the slew limit for the duration of the scan. All calculations occur at prescription time. All experiments were performed on a GE 3T MR750w scanner.

Static phantom experiment:
We first compared tennisball to a variation [5] of the conventional spherical spiral trajectory. All scans used 1mm isotropic resolution, 24cm FOV, 2$$$^\circ$$$ flip -- for which spherical spiral acquired 27648 spokes in 72s with 29 T/m/s peak slew. We tested two tennisball trajectories: one designed not to exceed this peak slew, and a second with a lower peak slew constraint. A rotation mode suitable for isotropic imaging (Fig. 2) was selected manually. All scans were reconstructed with BART [6]. Fig. 3 demonstrates that tennisball achieves superior image quality for the same peak slew and comparable quality at 31% lower peak slew.

Brain experiment:
We next compared tennisball to AZTEK [3], a state-of-the-art spherical spiral trajectory that has been demonstrated for static and motion-compensated imaging. We found the publicly accessible AZTEK implementation to be largely compatible with our system, except that peak slew rates briefly and periodically exceed 400 T/m/s using recommended AZTEK parameter and product gradient amplitude settings, well in excess of hardware limits. We implemented a minimally invasive workaround by inserting intermediate spokes between large jumps as needed. The added overhead was small -- <1% when constraining to 120 T/m/s. Separate high-resolution phantom experiments showed negligible difference when constraining AZTEK slew further, so here we reduced AZTEK peak slew down to 50 T/m/s, leading to ~6% overhead in extra spokes. To compensate, we allowed AZTEK to set scan duration and adjusted tennisball parameters to meet this time limit. Tennisball and AZTEK were acquired in a healthy volunteer with a 22ch head coil. Both scans used 2mm isotropic resolution, 32cm FOV, 2$$$^\circ$$$ flip, 80896 spokes, 1m47s scan time. Figure 4 shows that tennisball achieves comparable image quality as AZTEK but with 42% lower peak slew.

Motion phantom experiment:
We explored setting tennisball rotation mode $$$M$$$ such that $$$L/M$$$ approximates the golden ratio, and found that that this results in small maximal gaps for short temporal windows. We adjusted $$$M$$$ accordingly and repeated the prior experiment in a motion phantom with 16ch anterior array. Fig. 5 again shows comparable performance with AZTEK but with 42% lower peak slew.

Discussion

Our results thus far demonstrate that tennisball trajectories are competitive for slew-limited 3D-radial static imaging and have potential for slew-limited imaging in the presence of motion. We suspect that golden-ratio tennisball modes and/or multi-component tennisballs (Fig. 1c) could be useful for full-scale dynamic volumetric imaging [7]. Tennisball trajectories could have relevance for other path design problems, e.g. tailoring RF pulses.