Researchers interested in designing nonlinear gradient trajectories.Previous work has shown that nonlinear gradients can provide more efficient encoding to allow reduced scan time and undersampling artifacts.^1,2 A metric is required to fully take advantage of the flexibility in nonlinear trajectory design and to find efficient encoding strategies. Such a metric should reflect image quality and be computationally efficient. Recent works have proposed metrics for trajectory design but have only shown modest acceleration factors.^3,4,5 Recently, it has been proposed that analyzing the point spread functions (PSFs) that can be reconstructed from the k-space encoding matrix gives insights into how nonlinear gradient encoding affects the ability to reconstruct features in the image.⁶ In this work, we propose using a k-space PSF metric to find a repeating nonlinear gradient trajectory that best complements the information of an undersampled linear gradient trajectory.

The signal equation in k-space can be written in the form , where S_i is a vector of samples from the ith readout, x is a vector of frequency components from the object’s magnetization, and E(Ɵ_i) is the k-space encoding matrix with trajectory parameters, Ɵ_i. The encoding matrix is the set of all k-space sampling functions resulting from the applied encoding. When each frequency can be represented by a delta function, all the features in the image are well represented. The PSF can be calculated by applying the encoding matrix to point objects at all locations in k-space (Eq. 1). By selecting trajectories with parameters Ɵ_i that minimize the widths of the PSF at each location, high encoding efficiency is enforced (Eq. 2).

\[ PSF(i ̂)=P(null^⊥ (E)) δ_i ̂ \ \ \ \ \ (1) \]

\[ G=\sum(width(PSF(i ̂ ) ))\ \ \ \ \ \ (2) \]

Trajectories are designed using 2 linear gradients (x, y), 3 nonlinear gradients (x²+y², 2xy, x²-y²), and 8 radiofrequency (RF) receive coil profiles. A small subset of 16 successive encoding times for an 8 fold undersampled Cartesian linear gradient trajectory is selected (Fig. 1b). During the optimization nonlinear gradients, with amplitudes Ɵ_i=[w₁, w₂, w₃], are added at each encoding time via PSF metric calculation over a small 8 x 8 subregion of k-space containing the encoding functions (Fig. 1c). The gradient slew rate is constrained to 45 T/s. The optimized nonlinear gradients (Fig 1e) are then repeated over the full linear trajectory that covers the desired extent in k-space (Fig. 1f,g). The images are reconstructed from data simulated with intra-voxel dephasing effects and noise added. Undersampling artifacts are most severe in the 128 x128 image reconstructed from the undersampled Cartesian trajectory (Fig. 2a). When a x²+y² field is added the undersampling artifacts become less coherent and noisy (Fig. 2b). Addition of the optimized repeating nonlinear gradient trajectory results in the least undersampling artifact in reconstructed images (Fig. 2c). Fig. 3 shows a 64 x64 image reconstruction of the same optimized nonlinear gradients. A k-space PSF metric for repeating nonlinear gradient trajectory design is introduced. It is shown that undersampling artifacts can be reduced by using optimization with this metric to find efficient nonlinear encoding trajectories to add to an undersampled linear trajectory. The method is computationally efficient since the metric must only be calculated for a subregion of k-space and the solution can be repeated to obtain any desired image resolution.