Matrix Formalism for Restricted Diffusion in Porous Media

by D. Grebenkov

(Laboratory of Condensed Matter Physics, 
CNRS - Ecole Polytechnique, F-91128 Palaiseau France)

A geometrical confinement considerably affects the diffusive motion of
the nuclei and the consequent signal attenuation under inhomogeneous
magnetic fields.  We focus on theoretical and numerical aspects of 
resticted diffusion in NMR.  Starting from the classical Bloch-Torrey 
equation, we derive the free induction decay (FID) and the spin-echo 
or gradient-echo signal in a compact matrix form.  Each attenuation 
mechanism (restricted diffusion, gradient dephasing, surface or bulk 
relaxation) is represented by a matrix which is constructed from the 
Laplace operator eigenbasis and thus depending only on the geometry 
of the confinement.  In turn, the physical parameters (free diffusion 
coefficient, gradient intensity, surface or bulk relaxivity) characterize 
the "strengths" of the underlying attenuation mechanisms and naturally 
appear as coefficients in front of these matrices.  Once the Laplacian 
eigenfunctions for a given confinement are found (analytically or 
numerically), further computation of the macroscopic signal is more 
accurate and much faster than by using conventional simulation methods.  
The matrix technique is actually a simple numerical tool to deal with 
arbitrary gradient waveforms, including simple or stimulated, single 
or multiple spin echoes.  We present recent extensions of the matrix
formalism to multilayered structures and its use for fast Monte Carlo
simulations in multiscale porous media.