MRI measures how water molecules move. In biological tissue, that movement is confined — along axons, within cell membranes, through tubular structures. The mathematical models that interpret these measurements typically assume simple geometries (spheres, cylinders) and simple approximations (narrow gradient pulses, Gaussian phase distributions). These approximations lose accuracy precisely when the measurement is most informative: at high gradient strengths that probe fine-scale structure.
Canales-Rodríguez, Tax, Górriz, Jones, Thiran, and Rafael-Patiño (arXiv:2603.23421) derive an exact analytical solution for the PGSE (pulsed gradient spin echo) signal of molecules diffusing on a cylindrical surface. No narrow-pulse approximation. No Gaussian phase assumption. The solution uses the spectral decomposition of the Laplacian on the cylinder — the eigenfunctions of diffusion on that geometry — and expresses the signal as a matrix exponential in this basis. The computation is exact because the eigenbasis is exact.
Symmetry reduction cuts the computational cost. The cylindrical geometry means many matrix elements vanish, and Strang splitting further accelerates the matrix exponential evaluation. The result is fast enough for fitting — where you need to evaluate the model thousands of times at different parameter values.
The validation against Monte Carlo simulation confirms the formula works across parameter regimes where the standard approximations fail. The practical value is for fitting diffusion MRI data to biophysical models of axons or other tubular structures, where the curvature of the confining surface matters. Approximating a cylinder as a flat slab discards the curvature. The exact solution keeps it. The geometry is the signal, not noise to be averaged away.