Introduction to Diffusion Tensor Imaging

Why DTI? Diffusion – what it is, how it affects MR

signal Tensor – how we represent diffusion Imaging – how we measure it in MRI

Stroke/ischemia Alzheimer’s Disease Multiple Sclerosis Brain maturation studies Ischemia and stroke Neoplasm Preoperative planning Traumatic brain injury Congenital anomalies and

diseases of white matter Encephalopathies Neurodegenerative diseases Spinal Cord Injury Epilepsy Dementia, schizophrenia,

depression Developmental disorders Autism Aging

Why diffusion?

http://www.vh.org/Providers/Textbooks/BrainAnatomy/Ch5Text/Section18.html

http://eclipse.nichd.nih.gov/nichd/DTMRI/mri/

Conceptually: in vivo histology

Why diffusion? Diffusion is EXTREMELY SENSITIVE to

differences and changes in tissue microstructure Myelination/Demyelination Axon damage/loss Inflammation/Edema Necrosis

It is NOT a biomarker of white matter integrity

It is NOT just about white matter Gray matter Cardiac tissue

Example DTI image

“Fractional Anisotropy” map “map” is a computed

parameter, unlike an “image” which is acquired signal

Also called a “tractogram” since it clearly shows major white matter fiber tracts

What is Diffusion?

stochastic movement of particles in a solvent, driven by the thermal molecular motion of the solvent…

… and also applies to motion of the solvent itself (Einstein, 1905)

time NOTE: In the limit N→∞, use the Central Limit Theorem to assume “step size” is fixed and equal to the average of individual displacements i.

1D Fick’s Law - what the flux?

t = t0 +

x

t = t0

0 2x 0x 0x 0x0 2x

t = t0 +

x

t = t0

0 2x 0x 0x 0x0 2x

What is the flux (J) through x0 after one time interval ?

C1(x)C2(x)

dxdCJ

2

21

Adolf Fick, 1855: Flux is proportional to the particle concentration gradient(conservation of mass)

The Diffusion Coefficient

3D Fick’s Law Note the minus sign: flux

goes from high to low concentration

del operator replaces partial derivative

factor of 6, not 2 (why?) D is the diffusion

coefficient This is the expression for

isotropic diffusion

62D

CDJ

CJ

6

2

Isotropic Diffusion (water)

water

ink

Dtr 6

r1

t1

r2

t2

62D

Diffusion in Tissue (Anisotropic)

t

ink

r2

r3

r1

diffusionellipsoid

tDr 11 2

tDr 22 2

tDr 33 2x

y

z

laboratoryframe

DON’T try this

at lab!!!!!

The Diffusion Tensor

zzyzxz

yzyyxy

xzxyxx

DDDDDDDDD

x

y

z

r2

r3

r1

3

2

1

000000

DD

Ddiagonalization

Lab frame Intrinsic frame

Tensor Invariants

Eigenvalues: diagonalization (iterative QR factorization)

Eigenvectors

xx xy xz

xy yy yz

xz yz zz

D D DD D DD D D

1 2 3D D D

321 eee

Tensor Invariants

Shape invariants: analytical calculation directly from tensor coeffs

xx xy xz

xy yy yz

xz yz zz

D D DD D DD D D

13av xx yy zzD D D D

12

2 2 2

13

xx yy xx zz yy zz

surfxy xz yz

D D D D D DD

D D D

12 3

2 22xx yy zz xx yz

volxy xz yz zz xy yy xz

D D D D DD

D D D D D D D

Scalar Anisotropy Indices

avND

D

DADC

DDDADC

32131

2

2

23

22

21

23

22

21

1

23

mag

surfND

D

DD

FA

DDD

DDDDDDFA

FA vs. ADC

FAx 10 -4 )

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Prob

abili

ty

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

WMGMCSF

MD x 10 -3 mm2/s)

0 500 1000 1500 2000 2500 3000 3500 4000

Prob

abili

ty

0.000

0.001

0.002

0.003

0.004

0.005

WMGMCSF

FA and ADC are very useful clinically, but are very different.

Tensor has a LOT of information!

Q. Which metric would you use to detect brain cancer?

Vector anisotropy measures

We can use eigenvector information from the tensor as well Represent direction of

primary eigenvector as color on a scalar map

Or render the primary eigenvectors as “fibers” for astonishing* 3D visualizations

Red = R/LGreen = A/P Blue = S/I

*but how “real” is it? Many PhD theses have asked….

Diffusion tensor coefficients Diffusion tensor invariants

Scalar anisotropy indices

Vector anisotropy indices

Effect of Diffusion on MRI signal

Signal attenuation!

Diffusion term

Diffusion weighted MRI

G G

echo

2

0

δexp δ 3M G DM

2 δδ 3b G

(boxcar gradients)“b-value”

Consider simplified diffusion experiment…

MR Measurement of Diffusion Tensor

jTj j

j

pG G q

r

0

xx xy xz j

j j j xy yy yz j

xz yz zz j

D D D pp q r D D D q

D D D rj

b

S S e

22 2γ δ Δ δ 3b G

1

6j NN

jth diffusion-weighted

image

Diffusion magnitude

Diffusion direction

Gz

Gy

Gx

...

...

...

Solving for D

20

0

xx xy xz j

j j j xy yy yz j

xz yz zz j

D D D pp q r D D D q

D D D rj

b

S S e

1. Acquire T2W image (b = 0 s/mm2)

3. Choose a diffusion gradient orientation2. Choose a b-value

4. Acquire image (Sj)5. Repeat steps 1 – 3, j = 1 … N times

6. Solve for D…. How?

Let’s do some linear algebra…

zj

yj

xj

zzzyzx

yzyyyx

xzxyxx

zjyjxjj

DDDDDDDDD

bSS

exp0

yz

xz

xy

xx

yy

xx

T

jxy

jxy

jxy

jzz

jyy

jxx

j

DDDDDD

bSS

222

ln2

2

2

01661 xNxNx xAY

B-matrix formalism

22

yzyzxzxzxyxyzzzzyyyyxxxx DDDDDDb 222222

yzyzxzxzxyxyzzzzyyyyxxxx DbDbDbDbDbDb 222

3

1

3

1i jijij Db

The “b-matrix”

The b-matrix formalism summarizes total attenuating effect of all gradient waveforms in all directions (including imaging gradients)

T2W(b = 0 s/mm2)

Y, -ZY, Z-X, Y

X, Y-X, Z+X, Z

24

SVD

DIAG

T2W(b = 0 s/mm2)

…DWI

(j = 1, 2, 3 … N)

Dij

N=27 N=55

N=13N=6

N NEX # DWI6 8 5613 4 5627 2 5655 1 56

#DWI = (N + 1) x NEX

If TR = 4 sec, then acq time = 56*4sec = 3.7 minutes

Tradeoff: N vs NEX

Rotational invariance

Hasan et al, JMRI 2001Jones MRM 2004

27

Empirical Image Qualityincreasing N, decreasing NEX

incr

eas i

ng b

-va l

ue

How low can you go?

High b-values mean more attenuation, lower SNR

Lower b-values mean higher SNR, room for more N

At very low b-values, imaging gradients’ diffusion effects are no longer negligible

Lower b-values also do not probe same diffusion scale, less clinically interesting

b=100 s/mm2 b=500 s/mm2

(N = 6, 8 NEX)

Echo-Planar Imaging (EPI)

Advantages Minimal motion

artifacts NEX N

Disadvantages Eddy current artifacts T2* limits spatial

resolution Geometric distortion

(susceptibility)

29

DT-MRI Alexander

SLFSLF CRCR CCCC CINGCING

Partial Volume Effects on Anisotropy

DT-MRI Alexander

Mapping Complex Diffusion

Based Upon Q-Space Theory – Model Independent ODF – orientation density function (Tuch et al., Neuron 2003)

Diffusion Spectrum Imaging (DSI) (Tuch et al. Neuron 2003, Wedeen et al. 2005)

High Angular Diffusion Imaging (HARDI), Q-Ball (Frank 2002; Tuch et al. Neuron 2003)