Exact analytical PGSE signal for diffusion confined to a cylindrical surface using a spectral Laplacian formalism

TL;DR

This paper derives an exact analytical solution for the PGSE MRI signal of diffusion confined to cylindrical surfaces, improving accuracy over existing models and enabling efficient computations for high diffusion weightings.

Contribution

It introduces a spectral Laplacian formalism to obtain an exact solution for cylindrical surface diffusion MRI signals with finite gradient durations, surpassing previous approximate models.

Findings

Validated the analytical model against Monte Carlo simulations.

Developed efficient numerical strategies for repeated signal evaluations.

Benchmarking shows the method's accuracy and computational efficiency.

Abstract

Pulsed-gradient spin-echo (PGSE) MRI experiments probe molecular self-diffusion through spin phase accumulation under time-dependent magnetic field gradients. For diffusion confined to cylindrical surfaces, existing analytical signal models typically rely on the narrow-pulse limit, approximate treatments of finite gradient durations, or the Gaussian phase approximation, which become increasingly inaccurate at high diffusion weightings. Here, we derive an exact analytical solution of the Bloch-Torrey equation for diffusion confined to a cylindrical surface under finite PGSE gradients and obtain the corresponding diffusion MRI signal expression valid for arbitrary gradient durations and separations. The derivation is based on a spectral matrix formalism of the Laplace operator in the eigenbasis of the confining geometry. The signal is expressed as a product of non-commuting matrix exponentials, without approximations to the diffusion propagator or the spin phase distribution. We further introduce a reduced real spectral basis exploiting the symmetry of the cylindrical surface, substantially improving computational efficiency. Building on this exact formulation, we develop efficient numerical strategies for repeated signal evaluations, including a Strang splitting approximation of the matrix exponentials and an efficient computation of the spherical mean signal using Gauss-Legendre quadrature. The analytical signal is validated against Monte Carlo simulations over a wide range of cylinder radii and experimental parameters. The accelerated implementations are benchmarked against the exact formulation to quantify accuracy-runtime trade-offs. These results establish a computationally efficient framework for evaluating directional and orientationally averaged diffusion MRI signals in applications requiring large numbers of model evaluations.