A spectral framework for NMR signal with restricted diffusion

By the multiple correlation function (MCF) formalism, the nuclear magnetic resonance magnetization of diffusing spins can be represented for simple pore geometries. It may be used to infer geometric structure at the scale of microns. This is to be compared to the diffusion tensor imaging, which provides geometric information at the scale of millimeters. The MCF formulation was derived to special cases in which the gradients of the magnetic field are oriented in specific directions. A generalized approach allowing an arbitrary magnetic field direction was introduced by Özarslan. In this article, we present a complete account of the generalized MCF mathematical derivation starting from Bloch-Torrey equations. It is aimed to the experts and novices alike. We present two approaches-the indirect derivation is based on Özarslan's work, where the standard MCF equations are adopted to arbitrary gradient directions. Our alternative approach is based on direct calculations of the MCF matrices for specified gradient directions. We prove that the two approaches lead to the same equations. Finally, we revise Mitra's microscopic approach and show the relation to the macroscopic MCF approach. We prove that in some limit conditions the two signal equations coincide.