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Analytical phase kurtosis of the constant gradient spin echo

Teddy X. Cai1,*, Nathan H. Williamson1,2,3,4, and Peter J. Basser1,2,†

  • *Contact author: teddy.cai@nih.gov
  • †Contact author: basserp@mail.nih.gov
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Phys. Rev. E 114, 044113 – Published 8 October, 2026

DOI: https://doi.org/10.1103/tl57-pmyp

Abstract

The Gaussian phase approximation (GPA) underlies many standard diffusion magnetic resonance signal models, yet its validity is rarely scrutinized. Here, we assess the validity of the GPA by analytically deriving the excess phase kurtosis κ4/κ22, where κn is the nth cumulant of the accumulated phase distribution due to motion. We consider the signal behavior of the spin echo with constant gradient amplitude g and echo time T in several one-dimensional model systems: (1) a stationary Poisson pore-hopping model with uniform pore spacing Δx and mean interhop time τhop; (2) a trapped-release model in which spin isochromats are initially immobilized and then released with diffusivity D following an exponentially distributed release time, τrel; and (3) restricted diffusion in a domain of length L. To our knowledge, this is among the first systematic analytical treatments of spin-echo phase kurtosis without assuming Gaussian compartments or infinitesimally short gradient pulses. In the pore-hopping system, κ4/κ22=(9/5)τhop/T, inversely proportional to the mean hop number, T/τhop. In the trapped-release system, κ4/κ22 is positive and decreases roughly log-linearly with T/〈τrel〉, where 〈τrel〉 is the average release time. For restriction, κ4/κ22 vanishes at small and large L/DT, but has complicated intermediate behavior. There is a negative peak at L/DT≈1.2 and a positive peak at L/DT≈4.4. Monte Carlo simulations are included to validate the analytical findings. Overall, we find that the GPA does not generally hold for these systems under moderate experimental conditions, i.e., T=10ms, g≈0.2–0.6T/m. These results suggest that signal models reliant on the GPA must be carefully examined, particularly for strong, extended gradients and in media with complex kinetics, morphology, and/or microstructure.

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Phys. Rev. E 114, 044113– Published 8 October, 2026
  • Received 27 January 2026
  • Accepted 8 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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