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

Abstract

Abstract Objective. To derive an exact analytical expression for the diffusion MRI signal arising from diffusion confined to a cylindrical surface under finite rectangular pulsed-gradient spin-echo (PGSE) gradients, and to develop computationally efficient strategies for repeated model evaluations. Approach. The Bloch–Torrey equation was solved using a spectral matrix formalism of the Laplace operator in the eigenbasis of the cylindrical surface. The resulting signal is expressed as a product of non-commuting matrix exponentials and is valid for arbitrary rectangular gradient durations and separations, without approximations to the diffusion propagator or spin phase distribution. A reduced real spectral basis was introduced to decrease the dimensionality of the problem. Accelerated implementations based on Strang splitting and Gauss–Legendre quadrature were developed for repeated signal evaluations and computation of the spherical mean. The analytical signal was validated against Monte Carlo diffusion simulations. Main results. The exact formulation showed excellent agreement with Monte Carlo simulations over a wide range of cylinder radii and acquisition parameters. The spectral expansion converged rapidly with mode order, and the reduced basis substantially decreased computational cost without loss of accuracy. The accelerated implementations provided large additional speed-ups while retaining controllable errors relative to the exact solution. Significance. The proposed framework extends exact finite-pulse solutions of the Bloch–Torrey equation to diffusion confined to cylindrical surfaces and provides practical implementations for directional and orientationally averaged diffusion MRI signals. It therefore offers an approximation-free reference for assessing faster signal approximations and for applications requiring large numbers of model evaluations.