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We review the mathematical background of diffusion nuclear magnetic resonance (NMR), also known as NMR diffusometry, diffusion magnetic resonance imaging (dMRI) or diffusion weighted imaging (DWI). This non-invasive experimental technique aims at infering geometric and structural information on a medium at micrometer length scales (surface-to-volume ratio, pore size distribution, membrane permeability, connectivity, anisotropy, etc.) through understanding how the diffusive motion of nuclei in the medium is affected by its microstructure. We start from the Bloch–Torrey equation that provides an accurate microscopic description of the transverse magnetization evolution and then overview various theoretical and phenomenological approaches developed to relate the microstructure to the macroscopic signal. In particular, we discuss the classical narrow pulse approximation and Gaussian phase approximation, as well as more recent advances on diffusion in compartmental tissues and localization at high gradients.

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