numerical simulation and macroscopic model formulation for...

French-Vietnam Master in Applied Mathematics, Sept 23, 2017 1

Modeling and simulation of brain diffusion MRI

Numerical simulation and macroscopic model formulation

for diffusion magnetic resonance imaging in the brain

Jing-Rebecca Li Equipe DEFI, CMAP, Ecole Polytechnique

Institut national de recherche en informatique et en automatique (INRIA) Saclay, France

L' HABILITATION À DIRIGL' HABILITATION À DIRIGER DES RECHEHERCHES



DeFI

Denis Le Bihan Cyril Poupon

Luisa Ciobanu Khieu Van Nguyen (PhD) Hang Tuan Nguyen (PhD)

Houssem Haddar Simona Schiavi (PhD)

Gabrielle Fournet (PhD) Dang Van Nguyen (PhD)

Julien Coatleven (Post-doc) Fabien Caubet (Post-doc)

Master 2 interns

Vu Duc Thach Son (April-August 2017), Hoang An Tran (April-August 2016), Hoang

Trong An Tran (April-August 2015), Thi Minh Phuong Nguyen (April-August 2014),

Khieu Van Nguyen (April 2013 - Aug 2013), Tri Tinh Nguyen (March 2011 - July

2011), Thong Nguyen (March 2011 - July 2011), Cesar Retamal Bravo (March 2007

- July 2007), Sonia Fliss (Sept. 2004 - Mar. 2005).



Outline

1. Brain micro-structure is complex

2. MRI using “diffusion encoding” to “see” micro-structure

3. DMRI signal due to tissue (neurons+other cells)

a) Full-scale numerical simulation

b) Macroscopic model formulation via homogenization

c) Experimental validation in simpler organism



MRI useful for studying

Gray: cortical surface.

Teal: fMRI activations

Red: arteries in red

Bright green: tumor

Yellow: white matter fiber

Diffusion Tensor and Functional

MRI Fusion with Anatomical MRI

for Image-Guided Neurosurgery.

Sixth International Conference on

Medical Image Computing and

Computer-Assisted Intervention -

MICCAI'03.



Large-scale electron

microscopy of serial thin

sections, mouse primary

visual cortex.

Pink: blood vessels

Yellow: nucleoli,

oligodendrocyte nuclei,

and myelin

Aqua: cell bodies

and dendrites.

Scale bars: a, b, 100 mm;

c–e, 10mm; f, 1 mm. Bock et al. Nature 471, 177-182 (2011)



Magnetic resonance imaging (MRI)

Non-invasive, in-vivo MRI signal: water proton magnetization over a volume called a voxel. To give image contrast, magnetization is weighted by some quantity of the local tissue environment.

Contrast: (tissue structure)

1. Spin (water) density

2. Relaxation (T1,T2,T2*)

3. Water displacement (diffusion)

in each voxel

MRI

Spatial resolution:

One voxel = O(1 mm)

Much bigger than micro-structure



???

o Standard MRI: T2 relaxation (T2 contrast)

at different spatial positions of brain

o In diffusion MRI (recently developed)

magnetization is weighted by water

displacement due to Brownian motion over

10s of ms (called measured diffusion time).

o Water displacement depends on local cell

environment, hindered by cell membranes.

o Right: T2 contrast does not show dendrite

beading hours after stroke, diffusion weighted

image (DWI) does.



DMRI measures incoherent water motion during “diffusion

time” between 10-40ms.

Root mean squared displacement: 6-13 mm

Voxel : 2mm x 2mm x 2 mm (human, clinical scanner)

100mm x 100mm x 100mm (small animal, high field scanner)



This problem difficult because:

1. Dendrites (trees) and extra-cellular (EC) space (complement of densely

packed dendrites) are anisotropic, numerically lower dimensional (dendrites

1 dim, EC 2 dim).

2. Multiple scales (5 orders of magnitude difference).

3. Cell membranes are permeable to water. Cells must be coupled together.

Extra-cellular

space thickness

10-30nm

Soma diameter

1-10mm

Dendrite radius

0.5-0.9 mm

DMRI voxel

2mm

Goal: quantify dMRI contrast in terms of tissue micro-structure



Simple (original) model of dMRI

Brain: 70 percent water

Brownian motion of water molecules

2

4

2

0

)4(

)|,( 0,d

Dt

Dt

xx

extxu

Mean-squared displacement Can be obtained by dMRI

dDtdxxxxtxuMSD 2)|,( 200,

D = 2 x 10-3 mm²/s if no cell membranes

Experimentally measured = 1x10-3 mm²/s

../otherstalks/nbt929-S2.mov



Pulsed gradient spin echo (PGSE) sequence (Stejskal-Tanner-1965)

D

RF180

d d g g

Echo

f(t)

TE

Diffusion time

Gradient duration

𝐵 𝐱, 𝑡 = 𝑓 𝑡 𝐠 ⋅ 𝐱

How diffusion MRI assigns contrast to displacement

Water 1H (hydrogen nuclei), spin ½

Precession Larmor frequency:

𝛾𝐵 𝐱, 𝑡 𝑑𝑡𝑡

Proton: g/2 = 42.57 MHz / Tesla



𝑡 = 0: 𝑀𝛿 = 𝑀0𝑒−𝑖𝛾𝛿𝐠⋅ 𝐱0

𝑡 = Δ + 𝛿,𝑀Δ+𝛿 = 𝑀0𝑒𝑖𝛾𝛿𝐠⋅ 𝐱𝚫+𝜹−𝐱0



𝑢 𝐱, 𝑡|𝐱0 =𝑒−| 𝐱−𝐱0|

2

4𝐷𝑡

4𝜋𝐷𝑡32

MSD = ADC*(2D)

Brain gray matter: ADC around10-3 mm²/s

Root MSD: 6-13 mm

𝑆 𝑏 = 𝑢 𝐱, Δ + 𝛿|𝐱0 𝑒𝑖𝛾𝛿𝐠⋅ 𝐱 𝛥+𝛿 −𝐱 0 𝑑𝐱𝑑𝐱𝟎

𝐱0∈𝑉𝐱∈𝑉

= 𝑒−𝐷 𝛾2𝛿2 𝐠 2 Δ−

𝛿3

𝑏 𝐠, Δ, 𝛿 ≡ 𝛾2𝛿2 𝐠 2 Δ −𝛿

3,

Experimental

parameters

g D, d can be varied

𝐴𝐷𝐶 ≡ −d

dblog (𝑆 𝑏 ):

“apparent diffusion

coefficient”

Fitted at every voxel



2.5

3

3.5

4

4.5

5

0 1000 2000 3000 4000

b value

ln(s

ign

al)

Human visual cortex (Le Bihan et al. PNAS 2006).

bADCeS

S )(

0

Log plot not a straight line.

Simple model is “wrong”

Physicists try a different

simple model

.0

bDbD slowslow

fastfast efef

S

S

Diffusion is not Gaussian in biological tissues

(In each voxel)

Free diffusion:

ln(S/S0) = -bD

ffast= 65.9%, fslow= 34.1%

Dfast = 1.39 10-3 mm²/s,

Dslow = 3.25 10-4 mm²/s



DMRI signal due to tissue (neurons+other cells)

A. Direct simulation of reference model (Bloch-Torrey PDE)

oFinite elements + RKC time stepping (PhD Dang Van Nguyen)

B. Asymptotic models using homogenization

C. Experimental validation in simpler organism: Aplysia



Ω𝑖 , 𝐷 Ω𝑒 , 𝐷 𝜅𝑖𝑒

Reference model: Bloch-Torrey PDE

PDE with interface condition between cells and the extra-cellular space

𝜕𝑀𝑗 𝐱, 𝑡 𝐠

𝜕𝑡= 𝑖 𝛾𝑓 𝑡 𝐠 ⋅ 𝐱 𝑀𝑗 𝐱, 𝑡 𝐠 + 𝛻 ⋅ 𝐷 𝛻𝑀𝑗 𝐱, 𝑡 𝐠 , 𝐱 ∈ Ω𝑗 .

𝐷 𝛻𝑀𝑗 𝐱, 𝑡 𝐠 ⋅ 𝐧j 𝐱 = −𝐷 𝛻𝑀𝑘 𝐱, 𝑡 𝐠 ⋅ 𝐧k(𝐱), 𝐱 ∈ Γ𝑗𝑘 ,

𝐷 𝛻𝑀𝑗 𝐱, 𝑡 𝐠 ⋅ 𝐧j 𝐱 = 𝜿 𝑀𝑗 𝐱, 𝑡, 𝐠 − 𝑀𝑘 𝐱, 𝑡 𝐠 , 𝐱 ∈ Γ𝑗𝑘 ,

M: magnetization

g: magnetic field gradient

Tend: diffusion time

From signal, want to quantify

cell geometry and membrane permeability.

𝑀 𝑗 𝐠, 𝑡 = 𝑀𝑗 𝐱, 𝑡 𝐠 𝑑𝐱 .

𝐱∈Ω𝑗

𝑆 𝐠, 𝑇𝑒𝑛𝑑 = 𝑀 𝑗(𝑔, 𝑡)

𝑗

≈ exp (−𝐴𝐷𝐶 𝑏𝑒𝑥𝑝𝑒𝑟𝑖)



M 𝐱, 𝑡 𝐠

1. Numerical simulation of diffusion MRI signals using an adaptive time-

stepping method, J.-R. Li, D. Calhoun, C. Poupon, D. Le Bihan. Physics in

Medicine and Biology, 2013.

2. A finite elements method to solve the Bloch-Torrey equation applied to

diffusion magnetic resonance imaging, D.V. Nguyen, J.R. Li, D.

Grebenkov, D. Le Bihan, Journal of Computational Physics, 2014.



To do:

Define/segment -- mesh –

solve Bloch-Torrey PDE on

realistic brain tissue geometry



Asymptotic models of dMRI using periodic homogenization

1. Coupled ODE model

Works for large diffusion displacement and wide range of gradient amplitudes.

(Post-doc Coatleven, PhD Hang Tuan Nguyen)

2. Simpler PDE model for wide range of diffusion displacement and low gradient amplitudes (ADC).

(Ph.D. Schiavi)

Both are limited to low membrane permeability



Choose space and time scales

Spatial scale

membrane’s permeability

coefficient

Valid when diffusion displacement long

compared to cell features. low membrane permeability.

We fix a

non-dimensional

parameter ε

which goes to 0

intrinsic diffusion coefficient 𝜎 =

𝜎𝑒 𝑖𝑛 𝑌𝑒𝜎𝑐 𝑖𝑛 𝑌𝑐

𝐿 = 𝜀𝐿0

𝜅 = 𝜀𝜅0



𝑀 =

𝜖𝑖∞

𝑖

𝑀𝑖𝑒 𝐱, 𝐲, 𝑡 𝑖𝑛 Ω𝑒

𝜀𝑖∞

𝑖=1

𝑀𝑖𝑐(𝐱, 𝐲, 𝑡) 𝑖𝑛 Ω𝑐

Define two space scales: coarse and fine scales

And expand magnetization in terms of

𝐲 =𝐱

ε

Match powers (0,1,2) of using the PDE, get

relationships between coefficient functions



𝜕𝑀 𝑂𝐷𝐸𝑒(𝐠, 𝑡)

𝜕𝑡= −𝑐(𝑡)𝛾2𝐷 𝑒 ∥ 𝐠 ∥2 𝑀 𝑂𝐷𝐸

𝑒(𝐠, 𝑡) −

1

𝜏𝑒𝑀 𝑂𝐷𝐸

𝑒(𝐠, 𝑡) +

1

𝜏𝑖𝑀 𝑂𝐷𝐸

𝑖(𝐠, 𝑡)

𝜕𝑀 𝑂𝐷𝐸𝑖(𝐠, 𝑡)

𝜕𝑡= −𝑐 𝑡 𝛾2𝐷 𝑖 ∥ 𝐠 ∥2 𝑀 𝑂𝐷𝐸

𝑖(𝐠, 𝑡) −

1

𝜏𝑖𝑀 𝑂𝐷𝐸

𝑖(𝐠, 𝑡) +

1

𝜏𝑒𝑀 𝑂𝐷𝐸

𝑒(𝐠, 𝑡)

𝑐(𝑡) = 𝑓 𝑠 𝑑𝑠𝑡

0

21

𝜏𝑖 =𝜅|𝜕Ω𝑖|

|Ω𝑖| , 𝜏𝑒 =

|Ω𝑖|

|Ω𝑒| 𝜏𝑖

𝑆𝑂𝐷𝐸 𝑏 = 𝑀 𝑂𝐷𝐸 𝑗𝐠, 𝑡 ,

𝑗

Formulated macroscopic model for the compartment

magnetizations for arbitrary pulse shape: is an ODE model

D

RF180

d d g g Echo

f(t)

TE Diffusion time

Gradient duration



𝛿 = 40𝑚𝑠, Δ = 40𝑚𝑠

Simulation box 7.5 mm x 14 mm x 7.5 mm

Membrane permeability k= 1e-5 m/s



Spatial scale

membrane’s permeability

coefficient

Time scale

gradient strength

intrinsic diffusion coefficient 𝜎 =

𝜎𝑒 𝑖𝑛 𝑌𝑒𝜎𝑐 𝑖𝑛 𝑌𝑐

𝐿 = 𝜀𝐿0

𝜅 = 𝜀𝜅0

𝑡 = 𝜀2𝑡0

q =𝑞0

𝜀2

Diffusion displacement not long w.r.t. cell features



Homogenized model for Deff

𝐹 𝑡 = 𝑓 𝑠 𝑑𝑠𝑡

0

𝐷𝑒𝑓𝑓𝑖𝑙

𝜎≔ 𝑒𝑖 ⋅ 𝑒𝑙 −

1

𝛿2 Δ −𝛿3

𝐹 𝑡 𝑝𝑖𝑙(𝑡) Δ+𝛿

0

𝑝𝑖𝑙 𝑡 ≔ 1

𝑌 𝜕

𝜕𝑥𝑖

𝑌

𝜔𝑙(𝐱, 𝑡)

D

RF180

d d

Echo

F(t)

TE

Diffusion time

Gradient duration

d

𝐴𝐷𝐶𝑛𝑒𝑤 ≔ 𝐃𝐞𝐟𝐟𝐮𝑔 ⋅ 𝐮𝑔



Homogeneous diffusion equation

with constant coefficient and

Neumann boundary condition on the

interface

Need to analyze

𝑝𝑖𝑙 𝑡 ≔ 1

𝑌 𝜕

𝜕𝑥𝑖

𝑌

𝜔𝑙(⋅, 𝑡)


𝜕

𝜕𝑡𝜔𝑙 − 𝑑𝑖𝑣 𝜎𝛻𝜔𝑙 = 0 𝑖𝑛 𝑌 × 0, 𝑇

𝜎𝛻𝜔𝑙(⋅, 𝑡) ∙ 𝜈 = 𝐹(𝑡)𝜎𝑒𝑙 ∙ 𝜈 𝑜𝑛 Γ × 0, 𝑇

𝜔𝑙 ∙, 0 = 0 𝑖𝑛 𝑌

𝜔𝑙 𝑖𝑠 𝑌 𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐



𝜕

𝜕𝑡𝜔1 − 𝑑𝑖𝑣 𝜎𝛻𝜔1 = 0 𝑖𝑛 𝑌 × 0, 𝛿

𝜎𝛻𝜔1 ∙ 𝜈 = 𝑡𝜎𝑒1 ∙ 𝜈 ≔ 𝑔 ⋅, 𝑡 𝑜𝑛 Γ × 0, 𝛿

𝜔1 ∙, 0 = 0 𝑖𝑛 𝑌

𝜔1 ⋅, 𝑡 = 𝑆𝐿 ⋅, 𝑡 = 𝐺 𝐱 − 𝐲, 𝑡 − 𝜏 𝜇 𝐲, 𝜏 𝑑𝑠𝐲𝑑𝜏

Γ

𝑡

0

𝐹𝑢𝑛𝑑𝑎𝑚𝑒𝑛𝑡𝑎𝑙 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

𝐺 𝐱, 𝑡 = (4𝜋𝜎𝑡)−𝑑2𝑒−

𝐱 2

4𝜎𝑡

𝜎

2𝜇 𝐱𝟎, 𝑡 + 𝜎𝐾∗ 𝜇 𝐱𝟎, 𝑡 = 𝑔(𝐱𝟎, 𝑡)

𝐾∗ 𝜇 (𝐱𝟎, 𝑡) ≔ 𝜕

𝜕𝑛𝐱𝟎𝐺 𝐱𝟎 − 𝐲, 𝑡 − 𝜏 𝜇 𝐲, 𝜏 𝑑𝑠𝐲𝑑𝜏

Γ

𝑡

0

𝑝11 𝑡 ≔ 1

𝑌 𝜕

𝜕𝑥

𝑌

𝜔𝑙(⋅, 𝑡) ≈4

3 𝜋𝜎𝑆𝐴𝑥𝑉𝑂𝐿

𝑡32

First Pulse: 𝑡 ∈ 0, 𝛿 , single layer potential

𝑆𝐴𝑥𝑉𝑂𝐿

≔1

|𝑌| 𝑛𝑥

2 Γ



𝜕

𝜕𝑡𝜔1 − 𝑑𝑖𝑣 𝜎𝑐𝛻𝜔1 = 0 𝑖𝑛 𝑌𝑐 × 𝛿, Δ

𝜎𝛻𝜔1 ∙ 𝜈 = 𝛿𝜎𝑒1 ∙ 𝜈 𝑜𝑛 Γ × 𝛿, Δ

𝜔1 ∙, 𝛿 = 𝑆𝐿 ⋅, 𝛿 𝑖𝑛 𝑌

𝜕

𝜕𝑡𝜔1 − 𝑑𝑖𝑣 𝜎𝛻𝜔1 = 0 𝑖𝑛 𝑌 × 𝛿, Δ

𝜎𝛻𝜔1 ∙ 𝜈 = 0 𝑜𝑛 Γ × 𝛿, Δ

𝜔1 ∙, 𝛿 = 𝑆𝐿 ⋅, 𝛿 − 𝛿𝑥 𝑖𝑛 𝑌

𝜔1 ⋅, 𝑡 ≔ 𝜔1 +𝛿𝑥

𝜔1 ⋅, 𝑡 = 𝛿𝑥 + 𝑐𝑛𝜙𝑛 ⋅ 𝑒−𝜆𝑛𝜎(𝑡−𝛿)

∞

𝑛=1

Between pulses: 𝑡 ∈ [𝛿, Δ], eigenfunction expansion



𝜔1 ⋅, 𝑡 = 𝛿𝑥 + (𝑏𝑛 + 𝛿𝑎𝑛)𝜙𝑛 ⋅ 𝑒−𝜆𝑛𝜎(𝑡−𝛿)

∞

𝑛=1

Between pulses: 𝑡 ∈ [𝛿, Δ] eigenfunction expansion

𝑝11 𝑡 =1

𝑌 𝜕

𝜕𝑥

𝑌

𝜔1(⋅, 𝑡) ≔ 𝛿 + 𝛿𝑎𝑛 + 𝑏𝑛 ℎ𝑖𝑛𝑒−𝜆𝑛𝜎(𝑡−𝛿)

∞

𝑛=1

ℎ𝑖𝑛 ≔1

|𝑌| 𝜕𝜙𝑛 ⋅

𝜕𝑥

𝑏𝑛 ≔ 𝑆𝐿 ⋅, 𝛿 𝜙𝑛 ⋅

𝑌

𝑎𝑛 ≔ 𝑥 𝜙𝑛(⋅)

𝑌

𝑎𝑛 are the first moment

of the eigenfunctions.



𝜕

𝜕𝑡𝜔1 − 𝑑𝑖𝑣 𝜎𝛻𝜔1 = 0 𝑖𝑛 𝑌 × Δ, Δ + 𝛿

𝜎𝛻𝜔1 ∙ 𝜈 = Δ + 𝛿 − 𝑡 𝜎𝑒1 ∙ 𝜈 𝑜𝑛 Γ × Δ, Δ + 𝛿

𝜔1 ∙, Δ = 𝜔1 ⋅, Δ 𝑖𝑛 𝑌

𝜕

𝜕𝑡𝜔1 − 𝑑𝑖𝑣 𝜎𝛻𝜔1 = 0 𝑖𝑛 𝑌 × 0, 𝛿

𝜎𝑐𝛻𝜔1𝑐 ∙ 𝜈 = −𝑡𝜎𝑐𝑒1 ∙ 𝜈 𝑜𝑛 Γ × 0, 𝛿

𝜔1𝑐 ∙, 0 = 0 𝑖𝑛 𝑌𝑐

𝜔1 ⋅, 𝑡 = 𝜔1(⋅, Δ) − 𝑆𝐿 ⋅, 𝑡 − Δ

Second Pulse: 𝑡 ∈ [Δ, Δ + 𝛿] single layer potential

𝑝11 𝑡 = 𝛿 + 𝛿𝑎𝑛 + 𝑏𝑛 ℎ𝑖𝑛𝑒−𝜆𝑛𝜎(Δ−𝛿)

∞

𝑛=1

−4


𝑡32



𝐷𝑒𝑓𝑓11

𝜎≔ 1 −

1

𝛿2 Δ −𝛿3

𝐼 + 𝐼𝐼 + 𝐼𝐼𝐼

𝐼𝐼𝐼 =𝛿3

2+𝛿2

2 𝛿𝑎𝑛 + 𝑏𝑛 ℎ𝑖𝑛 (𝑒

−𝜆𝑛𝜎 Δ−𝛿 )

∞

𝑛=1

+ 4


−2

5𝛿72 +2

7𝛿72

𝐼 ≔ 𝐹 𝑡 𝑝𝑖𝑙 𝑡 𝛿

0

= 8


𝛿72

𝐼𝐼 ≔ 𝐹 𝑡 𝑝𝑖𝑙 𝑡 Δ

𝛿

= 𝛿2 Δ − 𝛿 + 𝛿 (𝛿𝑎𝑛 + 𝑏𝑛)ℎ𝑖𝑛1

𝜆𝑛𝜎(1 − 𝑒−𝜆𝑛𝜎(Δ−𝛿))

∞

𝑛=1

𝑎𝑛 are the first moment

of the eigenfunctions.

ℎ𝑖𝑛 ≔1

|𝑌| 𝜕𝜙𝑛 ⋅

𝜕𝑥

𝑏𝑛 ≔ 𝑆𝐿 ⋅, 𝛿 𝜙𝑛 ⋅

𝑌

𝑆𝐴𝑥𝑉𝑂𝐿

≔1

|𝑌| 𝑛𝑥

2 Γ




Numerical results



To do: Model is not so simple like ODE, must work to simplify it to

make it numerically efficient for parameter estimation.

Use mix of layer potentials and eigenfunctions

Account for multiple scales and anisotropic shape of neurons.



Estimating cell size and membrane permeability using reference PDE model in neuron network of the Aplysia (sea slug).

(Ph.D. Khieu Van Nguyen)

Cells are really big.



o Segment high resolution T2 image of buccal ganglia.

o Use literature values of feathers that cannot be seen in T2 image

o Simulate Bloch-Torrey PDE

o Obtain diffusion images

Carlo Musio et al. Zoomorphology. 1990;110:17-26



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numerical simulation and macroscopic model formulation for...

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