Spatial mapping of translational diffusion coefficients using diffusion tensor imaging: A mathematical description

Anil N. Shetty, Sharon Chiang, Mirjana Maletic-Savatic, Gregor Kasprian, Marina Vannucci, Wesley Lee
2014 Concepts in magnetic resonance. Part A, Bridging education and research  
In this article, we discuss the theoretical background for diffusion weighted imaging and diffusion tensor imaging. Molecular diffusion is a random process involving thermal Brownian motion. In biological tissues, the underlying microstructures restrict the diffusion of water molecules, making diffusion directionally dependent. Water diffusion in tissue is mathematically characterized by the diffusion tensor, the elements of which contain information about the magnitude and direction of
more » ... n and is a function of the coordinate system. Thus, it is possible to generate contrast in tissue based primarily on diffusion effects. Expressing diffusion in terms of the measured diffusion coefficient (eigenvalue) in any one direction can lead to errors. Nowhere is this more evident than in white matter, due to the preferential orientation of myelin fibers. The directional dependency is removed by diagonalization of the diffusion tensor, which then yields a set of three eigenvalues and eigenvectors, representing the magnitude and direction of the three orthogonal axes of the diffusion ellipsoid, respectively. For example, the eigenvalue corresponding to the eigenvector along the long axis of the fiber corresponds qualitatively to diffusion with least restriction. Determination of the principal values of the diffusion tensor and various anisotropic indices provides structural information. We review the use of diffusion measurements using the modified Stejskal-Tanner diffusion equation. The anisotropy is analyzed by decomposing the diffusion tensor based on symmetrical properties describing the geometry of diffusion tensor. We further describe diffusion tensor properties in visualizing fiber tract organization of the human brain. The symbol < > in Eq.(3a) describes an average value. The above relation is referred to as Einstein's equation of diffusion, where diffusion, D, relates the random spatial variation (displacement: ξ) of spins to the density (density: ρ) on a MRI time scale as (22) : SHETTY et al.
doi:10.1002/cmr.a.21288 pmid:27441031 pmcid:PMC4948124 fatcat:o7vvzbfawvfa3aazysx46swpsu