Physical foundations, models, and methods of diffusion magnetic resonance imaging of the brain: A review

The foundations and characteristics of models and methods used in diffusion magnetic resonance imaging, with particular reference to in vivo brain imaging, are reviewed. The first section introduces Pick's laws, propagators, and the relationship between tissue microstructure and the statistical properties of diffusion of water molecules. The second section introduces the diffusion-weighted signal in terms of diffusion of magnetization (Bloch-Torrey equation) and of spin-bearing particles (cumulant expansion). The third section is dedicated to the rank-2 tensor model, the b-matrix, and the derivation of indexes of anisotropy and shape. The fourth section introduces diffusion in multiple compartments: Gaussian mixture models, relationship between fiber layout, displacement probability and diffusivity, and effect of the 6-value. The fifth section is devoted to higher-order generalizations of the tensor model: singular value decompositions (SVD), representation of angular diffusivity patterns and derivation of generalized anisotropy (GA) and scaled entropy (SE), and modeling of non-Gaussian diffusion by means of series expansion of Pick's laws. The sixth section covers spherical harmonic decomposition (SHD) and determination of fiber orientation by means of spherical deconvolution. The seventh section presents the Fourier relationship between signal and displacement probability (Q-space imaging, QSI, or diffusion-spectrum imaging, DSI), and reconstruction of orientation-distribution functions (ODF) by means of the Funk-Radon transform (Q-ball imaging, QBI).