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Exact, time-dependent analytical equations for spiral trajectories and matching gradient and density-correction

Guruprasad Krishnamoorthy¹, James G Pipe¹

¹Department of Radiology, University of Wisconsin, Madison, Wisconsin, USA.

Magnetic Resonance in Medicine

|September 13, 2025

Abstract

Purpose:

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

Theory And Methods:

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.

Results:

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.

Conclusion:

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.

Keywords:

spiral gradient, k‐space trajectory design

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