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Math ématiques de la diffusion restreinte dans des milieux poreux

Math ématiques 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.

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Math ématiques de la diffusion restreinte dans des milieux poreux

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  1. 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

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

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

  4. B0 B0 B0 B0 Two physical states Different populations m Local magnetization Schematic principle of NMR Nuclei of spin ½ (e.g., protons) Application of a magnetic field

  5. Time-dependent linear magnetic field gradient Static magnetic field B0 z z y y x x Phase at time T Schematic principle of NMR

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

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

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

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

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

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

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

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

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

  15. Moments of the dephasing

  16. Multiple correlation functions

  17. Multiple correlation functions

  18. Multiple correlation functions

  19. Reflecting boundaries

  20. First moment

  21. Second moment For weak magnetic fields, one has

  22. f(t) 1 T t -1 Slow diffusion regime (small p)

  23. Slow diffusion regime (small p)

  24. Slow diffusion regime (small p) Grebenkov, RMP (submitted)

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

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

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

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

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

  30. Open problems and questions • Efficient numerical implementation, in particular, for model structures (sphere packs, fractals, …) • Computation of the high moments, transition to the localization regime • 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

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