Analytical phase kurtosis of the constant gradient spin echo

The Gaussian phase approximation (GPA) underlies many standard diffusion magnetic resonance (MR) signal models, yet its validity is rarely scrutinized. Here, we assess the validity of the GPA by analytically deriving the excess phase kurtosis $\kappa_4/\kappa_2^2$, where $\kappa_n$ is the $n^{\text{th}}$ 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 $\Delta x$ and mean inter-hop time $\tau_{\text{hop}}$; (2) a trapped-release model in which spin isochromats are initially immobilized and then released with diffusivity $D$ following an exponentially-distributed release time, $\tau_{\text{rel}}$; 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, $\kappa_4/\kappa^2_2 = (9/5)\tau_{\text{hop}}/T$, inversely proportional to the mean hop number, $T/\tau_{\text{hop}}$. In the trapped-release system, $\kappa_4/\kappa_2^2$ is positive and decreases roughly log-linearly with $T/\langle\tau_{\text{rel}}\rangle$, where $\langle\tau_{\text{rel}}\rangle$ is the average release time. For restriction, $\kappa_4/\kappa_2^2$ vanishes at small and large $L/\sqrt{DT}$, but has complicated intermediate behavior. There is a negative peak at $L/\sqrt{DT}\approx 1.2$ and a positive peak at $L/\sqrt{DT}\approx 4.4$. 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., $T=10\;\mathrm{ms}$, $g\approx 0.2-0.6\;\mathrm{T/m}$.