# Exact, time‐dependent analytical equations for spiral trajectories and matching gradient and density‐correction waveforms

## TL;DR

This paper introduces a new analytical method for spiral MRI that simplifies implementation while maintaining performance comparable to numerical methods.

## Contribution

The novel contribution is the derivation of exact analytical equations for spiral trajectories and gradient waveforms using an involute circle approach.

## Key findings

- The proposed analytical method performs similarly to numerical solutions in spiral MRI.
- The method is easier to implement and analyze compared to numerical approaches.

## Abstract

To analytically define a spiral waveform and trajectory that match the constraints of gradient frequency, slew rate, and amplitude.

Piecewise analytical solutions for gradient waveforms under the desired constraints are derived using the circle of an involute rather than an Archimedean spiral. Also given are the analytical equations for the time‐dependent k‐space trajectory and sampling density compensation weights, and analytical expressions for the time dependence of data acquisition in k‐space. Open‐source software implementing all these equations is shared. Performance is measured against numerically derived solutions to an Archimedean spiral. Scanner implementation is illustrated.

The performance of the proposed equations is very similar to that of numerically derived solutions, but this method is much easier to implement and analyze.

The proposed method, WHIRLED PEAS (Winding Hybrid Interleaved Radial Lines Encoding Described by Piecewise Exact Analytical Solution), is an easy‐to‐implement solution for spiral MRI that performs comparable to optimal numerical designs.