Mathématiques de la diffusion restreinte dans des milieux poreux Denis S. Grebenkov Laboratoire de Physique de la Matière Condensée CNRS – Ecole Polytechnique, Palaiseau, France Séminaire du groupe « Milieux poreux », 12 Janvier 2007, Paris, France

Outline of the talk Studying porous structures… Studying porous structures… Basic principles of NMR diffusion imaging Basic principles of NMR diffusion imaging Pulsed-gradient spin-echo (PGSE) experiments Pulsed-gradient spin-echo (PGSE) experiments General description via matrix formalisms General description via matrix formalisms Different diffusion regimes Different diffusion regimes Conclusions and perspectives Conclusions and perspectives Grebenkov, Rev. Mod. Phys. (submitted)

Studying porous structures… Material sciences: rocks, sols, colloids, tissues,... Petrol search: sedimentary rocks Medicine: brain, lung, bone, kidney, etc. Length scales: μm - mm Time scales: ms - s

Schematic principle of NMR Nuclei of spin ½ (e.g., protons) Application of a magnetic field B0B0 Two physical states B0B0 Different populations B0B0 Local magnetization B0B0 m

Schematic principle of NMR Phase at time T Static magnetic field B 0 x z y Time-dependent linear magnetic field gradient x z y

Schematic principle of NMR is the projection of a 3D Brownian motion of a nucleus onto a given gradient direction Local magnetization:Total transverse magnetization:

Example: free diffusion can be seen as 1D Brownian motion Isotropy of 3D Brownian motion is a Gaussian variable, therefore t f(t) T 1 with the rephasing condition to cancel the imaginary part

Apparent diffusion coefficient Free diffusion: D is a measure of how fast the nuclei diffuse in space

Apparent diffusion coefficient Effective « slow down » of the diffusive motion Restricting geometry Smaller ADC Smaller length scale

Apparent diffusion coefficient Normal volunteer Healthy smoker Patient with severe emphysema van Beek et al. JMRI 20, 540 (2004) Can one make a reliable diagnosis at earlier stage?

Pulsed-gradient spin-echo (PGSE) t f(t) T 1 δ Tanner & Stejskal, JCP 49, 1768 (1968)

PGSE: diffusive diffraction For T long enough, one “measures’’ a form-factor Diffusion in a slab of width L: Coy and Callaghan, JCP 101, 4599 (1994).

PGSE: pro & contro Direct access to the propagator Direct access to the propagator Easy experimental implementation Easy experimental implementation Characteristic length scales of the geometry via diffusive diffraction Characteristic length scales of the geometry via diffusive diffraction Assumption of very narrow pulses is not always valid, especially for gas diffusion Assumption of very narrow pulses is not always valid, especially for gas diffusion Material inhomogeneity may destroy diffraction peaks Material inhomogeneity may destroy diffraction peaks Lost information about the motion between 0 and T. Lost information about the motion between 0 and T. Pro Contro

Axelrod & Sen, JCP 114, 6878 (2001); Grebenkov, RMP (submitted) General description Total dephasing of a diffusing spin: echo time gyromagnetic ratiotemporal profile spatial profile spin trajectory (Brownian motion) field intensity Averaging individual magnetizations:

Moments of the dephasing

Multiple correlation functions

Reflecting boundaries

First moment

Second moment For weak magnetic fields, one has

Slow diffusion regime (small p) t f(t) T 1

Slow diffusion regime (small p)

Grebenkov, RMP (submitted)

Fast diffusion regime (large p) Robertson, PR 151, 273 (1966)

Example: cylinder Hayden et al. JMR 169, 313 (2004); Grebenkov, RMP (submitted)

Localization regime (large q) Stoller et al., PRA 44, 7459 (1991); de Swiet & Sen, JCP 100, 5597 (1994) Hurlimann et al. JMR 113, 260 (1995) Water proton NMR

Diagram of diffusion regimes Grebenkov, Rev. Mod. Phys. (submitted)

Summary Geometry and field inhomogeneity: Geometry and field inhomogeneity: Temporal dependence : Temporal dependence : Physical parameters: Physical parameters: A general theoretical description of restricted diffusion in inhomogeneous magnetic fields Slow diffusion regime (small p): S/V Slow diffusion regime (small p): S/V Fast diffusion regime (large p): sensitivity to L Fast diffusion regime (large p): sensitivity to L Localization regime: non-Gaussian behavior Localization regime: non-Gaussian behavior

Open problems and questions Efficient numerical implementation, in particular, for model structures (sphere packs, fractals, …) Efficient numerical implementation, in particular, for model structures (sphere packs, fractals, …) Computation of the high moments, transition to the localization regime Computation of the high moments, transition to the localization regime Inverse problem: what can one say about the geometry from experimental measurements? Inverse problem: what can one say about the geometry from experimental measurements? Development and optimization of the temporal and spatial profiles to probe porous structures Development and optimization of the temporal and spatial profiles to probe porous structures