title: Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance.
abstract: In order to bridge microscopic molecular motion with macroscopic diffusion MR signal in complex structures, we propose a general stochastic model for molecular motion in a magnetic field. The Fokker-Planck equation of this model governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad hoc selection of a propagator. In particular, the boundary conditions that implicitly incorporate the microstructure into the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal.
authors.affiliation: Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. Electronic address: inbarse@post.tau.ac.il.
authors.forename: Inbar
authors.initials: I
authors.lastname: Seroussi
authors.affiliation: Laboratoire de Physique de la Matière Condensée, CNRS - Ecole Polytechnique, University Paris-Saclay, F-91128 Palaiseau, France.
authors.forename: Denis S
authors.initials: DS
authors.lastname: Grebenkov
authors.affiliation: Departments of Psychiatry and Radiology, Brigham and Women's Hospital, Harvard Medical School, Boston, MA 02215, USA.
authors.forename: Ofer
authors.initials: O
authors.lastname: Pasternak
authors.affiliation: Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel.
authors.forename: Nir
authors.initials: N
authors.lastname: Sochen
mesh_terms: D000465:Algorithms; D004058:Diffusion; D009682:Magnetic Resonance Spectroscopy / Q000379:methods* / Q000706:statistics & numerical data*; D008280:Magnetics; D008853:Microscopy
keywords: Bloch-Torrey equation; Diffusion MR; Inhomogeneous diffusion; Random phase approach; Stochastic model