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

Ludovico Minati, Władysław P. Weglarz

Research output: Contribution to journalArticlepeer-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).

Original languageEnglish
Pages (from-to)278-307
Number of pages30
JournalConcepts in Magnetic Resonance Part A: Bridging Education and Research
Issue number5
Publication statusPublished - Sep 2007


  • Diffusion physics
  • Diffusion spectrum imaging (DSI)
  • Diffusion-weighted imaging (DWI)
  • Generalised diffusion-tensor imaging (GDTI)
  • Q-ball imaging (QBI)
  • Spherical harmonic decomposition (SHD) and deconvolution

ASJC Scopus subject areas

  • Chemistry(all)
  • Physical and Theoretical Chemistry
  • Spectroscopy


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