Design of 3D Non-Cartesian Trajectories for Fast Volumetric MRI via Analytic Coordinate Discretization

3D non-Cartesian trajectories offer several advantages over rectilinear trajectories for rapid volumetric imaging, including improved sampling efficiency and greater robustness to motion, flow, and aliasing artifacts. In this paper, we present a unified framework for designing three widely used non-Cartesian trajectories: 3D Radial, 3D Cones, and Stack-of-Spirals. Our approach is based on the idea that a non-Cartesian trajectory can be interpreted as a discretized version of an analytic coordinate defined by a set of template trajectories. Equivalently, the analytic coordinate is conceptualized as a non-Cartesian trajectory composed of an infinite number of copies of a set of template trajectories. The discretization is accomplished by constructing a continuous spiral path on a surface and sampling points along this path at unit intervals, leaving only the essential spokes/interleaves, thereby yielding the practical non-Cartesian trajectory from the analytic coordinate. One of the advantages of our approach is that the analytic density compensation factor can be readily derived using Jacobian determinants, which quantify changes in unit areas due to the transformation from the analytic coordinate to the Cartesian grid. Additionally, the proposed approach derives analytic formulae to compute the number of readouts based on prescribed parameters, allowing us to specify the trajectory's acceleration factor for a given total scan time. Furthermore, variable-density sampling can be easily incorporated, and spokes/interleaves are smoothly distributed in k-space along the derived spiral path, even for a small number of readouts. In a preliminary phantom study, the proposed method demonstrated improved sampling efficiency and image quality compared to the conventional approach.