MathMathématiques de la ématiques de la diffusion restreinte dans diffusion restreinte dans

des milieux poreuxdes milieux poreux

Denis S. GrebenkovDenis S. Grebenkov

Laboratoire de Physique de la Matière Laboratoire de Physique de la Matière CondenséeCondensée

CNRS – Ecole Polytechnique,CNRS – Ecole Polytechnique, Palaiseau, Palaiseau, FranceFrance

Séminaire du groupe « Milieux poreux », 12 Janvier 2007, Paris, France

Outline of the talkOutline of the talk

Studying porous structures…Studying porous structures… Basic principles of NMR diffusion Basic principles of NMR diffusion

imagingimaging Pulsed-gradient spin-echo (PGSE) Pulsed-gradient spin-echo (PGSE)

experimentsexperiments General description via matrix General description via matrix

formalismsformalisms Different diffusion regimes Different diffusion regimes Conclusions and perspectivesConclusions and perspectives

Grebenkov, Rev. Mod. Phys. (submitted)

Studying porous Studying porous structures…structures…

• Material sciences: rocks, sols, colloids, tissues, ...• Petrol search: sedimentary rocks

• Medicine: brain, lung, bone, kidney, etc.

Length scales: μm - mmTime scales: ms - s

Schematic principle of Schematic principle of NMRNMR

Nuclei of spin ½ (e.g., protons)

Application of a magnetic fieldB0

Two physical states

B0

Different populations

B0

Local magnetization

B0

m

Schematic principle of Schematic principle of NMRNMR

Phase at time T

Static magnetic field B0

x

z

y

Time-dependent linear magnetic field gradient

x

z

y

Schematic principle of Schematic principle of NMRNMR

is the projection of a 3D Brownian motion of a nucleus onto a given gradient direction

Local magnetization:

Total transverse magnetization:

Example: free diffusionExample: free diffusion

can be seen as 1D Brownian motion

Isotropy of 3D Brownian motion

is a Gaussian variable, therefore

t

f(t)

T1

-1

with the rephasing conditionto cancel the imaginary part

Apparent diffusion Apparent diffusion coefficientcoefficient

Free diffusion:

D is a measure of how fast the nuclei diffuse in space

Apparent diffusion Apparent diffusion coefficientcoefficient

Effective « slow down »of the diffusive motion

Restricting geometrySmaller ADC

Smaller length scale

Apparent diffusion Apparent diffusion coefficientcoefficient

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-Pulsed-gradient spin-echo (PGSE)echo (PGSE)

t

f(t)

T1

-1

δ

Tanner & Stejskal, JCP 49, 1768 (1968)

PGSE: diffusive PGSE: diffusive diffractiondiffraction

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 & controPGSE: pro & contro

Direct access to the propagatorDirect access to the propagator Easy experimental implementationEasy experimental implementation Characteristic length scales of the Characteristic length scales of the

geometry via diffusive diffractiongeometry via diffusive diffraction

Assumption of very narrow pulses is not Assumption of very narrow pulses is not always valid, especially for gas diffusionalways valid, especially for gas diffusion

Material inhomogeneity may destroy Material inhomogeneity may destroy diffraction peaksdiffraction peaks

Lost information about the motion Lost information about the motion between 0 and T.between 0 and T.

Pro

Contro

Axelrod & Sen, JCP 114, 6878 (2001); Grebenkov, RMP (submitted)

General descriptionGeneral description

Total dephasing of a diffusing spin:echo

time

gyromagnetic ratio

temporal profile

spatial profile

spin trajectory(Brownian

motion)field intensity

Averaging individual magnetizations:

Moments of the Moments of the dephasingdephasing

Multiple correlation Multiple correlation functionsfunctions

Multiple correlation Multiple correlation functionsfunctions

Multiple correlation Multiple correlation functionsfunctions

Reflecting boundariesReflecting boundaries

First momentFirst moment

Second momentSecond moment

For weak magnetic fields, one has

Slow diffusion regime Slow diffusion regime (small p)(small p)

t

f(t)

T1

-1

Slow diffusion regime Slow diffusion regime (small p)(small p)

Slow diffusion regime Slow diffusion regime (small p)(small p)

10-2

10-1

100

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

p

Grebenkov, RMP (submitted)

Fast diffusion regime Fast diffusion regime (large p)(large p)

Robertson, PR 151, 273 (1966)

Example: cylinderExample: cylinder

10-2

100

102

10-2

100

102

104

p

Dapp

(cm2/s)

experimentinterpolation (165)slow diffusion (160)

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

Localization regime Localization regime (large q)(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 Diagram of diffusion regimesregimes

Grebenkov, Rev. Mod. Phys. (submitted)

100

101

102

103

10-2

10-1

100

101

102

q

p

inaccessible experimentally

localization

motional narrowing

slow diffusion

SummarySummary

Geometry and field inhomogeneity: Geometry and field inhomogeneity: Temporal dependence :Temporal dependence : Physical parameters:Physical parameters:

A general theoretical description A general theoretical description of restricted diffusion in of restricted diffusion in

inhomogeneous magnetic fieldsinhomogeneous magnetic fields

Slow diffusion regime (small p): S/VSlow diffusion regime (small p): S/V Fast diffusion regime (large p): Fast diffusion regime (large p):

sensitivity to Lsensitivity to L Localization regime: non-Gaussian Localization regime: non-Gaussian

behaviorbehavior

Open problems and Open problems and questionsquestions

Efficient numerical implementation, in Efficient numerical implementation, in particular, for model structures (sphere particular, for model structures (sphere packs, fractals, …)packs, fractals, …)

Computation of the high moments, Computation of the high moments, transition to the localization regimetransition to the localization regime

Inverse problem: what can one say about Inverse problem: what can one say about the geometry from experimental the geometry from experimental measurements?measurements?

Development and optimization of the Development and optimization of the temporal and spatial profiles to probe temporal and spatial profiles to probe porous structuresporous structures