Restricted diffusion in annular geometrical pores

Nuclear magnetic resonance (NMR) diffusion (including diffusion MRI) experiments are only as powerful as the models used to analyse the NMR diffusion data. A major problem, especially with measurements on biological systems, is that the existing models are only very poor approximations of cellular shape. Here, diffusion propagators and pulsed gradient spin-echo attenuation equations are derived in the short gradient pulse limit for diffusion within the annular region of a concentric cylinder of finite length and, similarly, within the annular region of a concentric sphere. The models include the possibility of relaxation at the boundaries and, in the case of the concentric cylinder, having the cylinder arbitrarily oriented with respect to the direction of the applied field gradient. The two models are also of interest due to their direct analogy to optical double slit diffraction. Also expressions for the mean square displacements, which are very useful information for determining the diffusion coefficient within these complex geometries, are obtained as well as for the limiting cases of diffusion on cylindrical and spherical shells and in a ring.