1

Diffusion Tensor Imaging

2011. 10. 4.KAIST 바이오및뇌공학과

이정원

2

Diffusion MRI

• Tissue cellularity • ConnectivityDW-MRI DTI

3

Outline

• Diffusion Anisotropy

• Diffusion Tensor

• Quantitative parameters from Diffusion Tensor

• Visualization

• Fiber Tracking

• Probabilistic Fiber Tracking

• High b-value q-space imaging

What is diffusion?

• Einstein recognized that Brownian motion was associated with diffusion– No macroscopic concentration gradient is needed.– Self-diffusion arising from local concentration fluctuations

• Einstein derived the self-diffusion coefficient of the Brownian particle– Einstein expressed the energy change as the total work done by the particles contained

within the volume

– Diffusion coefficient

• Einstein rewrote Fick’s laws for the diffusion– in terms of diffusion under probability gradients

𝐷=𝑘𝐵𝑇6𝜋𝜂 𝑅

: Sutherland-Einstein relation (1905) viscosityT absolute temperatureR radius of the spherical particle

𝐾𝑛𝜁−𝐷

𝛿𝑛𝛿𝑥

=0 the number of Brownian particles per unit volume frictionD diffusion coefficientX position

𝑃 (𝑟|𝑟 ′ , 𝑡 ) :𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑝𝑟𝑜𝑏𝑎𝑏𝑙𝑖𝑡𝑦 h𝑡 𝑎𝑡 𝑎𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔𝑎𝑡 𝑟 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑧𝑒𝑟𝑜𝑤𝑖𝑙𝑙𝑚𝑜𝑣𝑒𝑡𝑜𝑟 ′ 𝑎𝑓𝑡𝑒𝑟 𝑎 𝑡𝑖𝑚𝑒𝑡

: Einstein equation for diffusion

4

5

Equation of diffusion attenuation, and b-value

𝐴 (𝑇𝐸 ): ln( 𝑆𝑆𝑜 )=−𝛾 2𝐺2 𝛿2 (∆− 𝛿

3 )𝑫=−𝑏𝑫

ln ( SSo )=−b𝐀𝐃𝐂ln ( SSo )=− ∑

i= x , y , z. ∑j= x , y, z

bij𝐃𝐢𝐣=−(bxx𝐃𝐱𝐱+2bxy𝐃𝐱𝐲+2bxz𝐃𝐱𝐳+b yy𝐃𝐲𝐲+2b yz𝐃𝐲𝐳+bzz𝐃𝐳𝐳)

6

Diffusion Anisotropy

• Diffusion anisotropy– Diffusivity is not the same in all directions

• Diffusion Tensor (Jost, 1960)

D=[D xx D xy D xz

D xy D yy D yz

D xz D yz D zz]

Diffusion Ellipsoid

7

How to measure the components of Diffusion Tensor

• DTI needs more than 7 DW-MRIs

D=[D xx D xy D xz

D xy D yy D yz

D xz D yz D zz]

zzz

yyy

xxx

zyx

zyx

zyx

zzyzxz

yzyyxy

xzxyxx

VVV

VVV

VVV

VVV

VVV

VVV

DDD

DDD

DDD

321

321

321

3

2

1

333

222

111

D

321

Diffusion Tensor

• Major Eigen-Value :

• Major Eigen-Vector:

1

Tzyx VVV 111

• Mean Diffusivity: 1 2 3

3D

Diffusion Ellipsoid

• Eigen-system

9

Diffusion Ellipsoid

Quantitative parameters provided by DTI

• Size of diffusion ellipsoid– Trace(D) is intrinsic to the tissue– It is independent of fiber orientation, gradient directions– Corresponding to tissue’s structure or physiological state

• Shape of diffusion ellipsoid– Diffusion Deviation Tensor

• The anisotropic part of D

– Relative Anisotropy (RA): • coefficient of variation of the eigenvalues

– Fractional Anisotropy (FA): • the fraction of the diffusion tensor’s total magnitude that is anisotropic

– Westin• The degree to which D is line-like; plane-like; sphere-like

Trace (D )=Dxx+D yy+D zz

𝑅𝐴=√𝑉𝑎𝑟 ¿ ¿¿

𝐷=𝑫− ⟨ 𝐷 ⟩ 𝐈

)

10

11

The two main challenges with DTI

Visualization Fiber Tracking

12

Visualization (1) 2D Maps

• Scalar diffusion measures– Fractional Anisotropy (FA)– Mean/Longitudinal/Transverse Diffusivity– Relative Anisotropy (RA)– Westin Measures

• Direction-encoded color (DEC) maps– R: x (left-right)– G: y (front-back)– B: z (bottom-top)

Fractional Anisotropy (FA) Directionally-Encoded Color Map

FA DEC

13

Visualization (1) 2D Maps: scalar parameters

Westin

14

Visualization (1) 2D Maps: Direction-encoded color (DEC) maps

Diffusion Ellipsoid

15

Visualization (2) Glyph Representations

16

Visualization (3) 3D Display

• Volume rendering• Fiber Tractography

– Streamlines and streamtubes– Hyperstreamlines

HyperstreamlinesVolume Rendering Streamlines

17

Visualization of orientation

Eigenvector color map

Ellipsoidal glyphs

Diffusion tensor field Tractography

18

Fiber Tractography

• Fiber Tracking– The reconstruction of long neuronal fibers from the information about anisotropic dif-

fusion– In vivo and noninvasively

• Methods– Streamlines

• FACT (Fiber Assignment by Continuous Tracking)

• Euler (EUL)• Runge-Kutta (RK)

– Tensor Deflection• TEND (Tensor Deflection)

• Tensorlines

– Other variations• G-TRACT• Gibbs tracking algorithm

19

Fiber Tractography: Streamlines

• Using only major eigenvector

• Linear step-wise integration methods

– FACT method• Changes the direction at the interfaces• Same e1 over the entire voxel

– Euler (EUL) method• Fixed step size

• High-order integration methods– Runge-Katta (RK) method

• Nonlinear and more accurate

)

ssds

sdrt

r1

20

Fiber Tractography: Seeding and Stopping Cri-teria

• Seed location– Regional seed methods

• To extract a specific pathway or mapping tracts from a specific region

– Whole-brain seed methods• To generate nearly al possible pathways• Higher redundancy in the pathways

• Stopping Criteria– FA threshold– Curvature Criteria

21

Limitations of DTI

• Tract Dispersion– The variance or uncertainty in tractography results– Tract dispersion increases as the SNR is decreased

• Tract Deviation– Systematic errors in the tract position– Tract deviation is influenced by the step size

• Tract Divergence and Convergence– Local heterogeneity of the diffusion tensor field

• Crossing Fibers

22

Strong diffusion weighting with Crossing Fibers

Diffusion Circular Spectrum Imaging

• Case 1: Single Fiber

W.Zhan et. al. MRM 2003, 49:1077

Diffusion Circular Spectrum Imaging

• Case 2: Fiber-Crossing

W.Zhan et. al. MRM 2003, 49:1077

25

HARDI (high angular resolution DW imaging) acquisition

26

Q-space Imaging

• High b-value q-space analysis – Signal decay is not monoexponential at high b-value (>10000 s/mm2) DW-MRI – Reflect a smaller structure information in stronger gradient or longer diffusion time– To emphasizes the diffusion characteristics of the slow-diffusing component– Extremely useful for detection restricted diffusion (Assaf et al., MRM, 2002)

• Q-space analysis

27

Displacement in PGSE with short/long gradi-ent pulses

28

Displacement and probability maps from DWI

29

Q-space imaging

30

31

Probabilistic Fiber Tracking

• A limitation of deterministic tractography– The lack of information regarding the error in the tracking procedure in any given

experiment• Probabilistic tractography methods

– To explicit characterization of the confidence with which connections may be estab-lished through the diffusion MRI

– Uses probability density functions (PDFs) of fiber orientation• To capture the expected distribution of possible fiber orientations

– Is effective at identifying multiple possible routes for connections from the chosen single start point

• Advantages– Better accounts for experimental errors– More robust tracking results– Better deals with crossing fibers, low SNR

• Disadvantages– Computationally intense– Probabilities will be modified by crossing fibers

32

Probabilistic Fiber Tracking

• Methods– Monte Carlo Streamlines

• Simulation the rage of possible outputs of a deterministic tracking process• Random sampling of the PDF• At least 1000 samples for each voxel’s PDF

– Simulated Diffusion Random Walk• To invoke the diffusion process itself as a propagator for the tractography process• generate PDFs on the basis of measures derived from the diffusion orientation distribution

function• Each jump during the random walk is typically independent of any of the previous jumps

– Front evolution methods• The evolution of the front is deterministic rather than statistical• Front evolution methods generate maps of a distributed ‘degree of connection’ index (fast

marching tractography)

33

Deterministic- vs. Probabilistic Tractography

Tournier, J.-D., Mori, S., and Leemans, A. (2011). “Diffusion tensor imaging and beyond.” Magnetic resonance in medicine : official journal of the Society of Magnetic Resonance in Medicine / Society of Magnetic Resonance in Medicine, 65(6), pp. 1532-56.

34

Applications of the DTI

• The visualization the organization of specific WM pathways noninvasively• The segmentation of WM• The visualization specific WM patterns relative to pathology

35

Limitations of the DTI

1. It is not possible to differentiate afferent from efferent pathways, antero-grade and retrograde pathways, inhibitory and excitatory connections, and direct versus indirect route in diffusion data

2. Tractography picks up mainly large-fiber pathways; smaller pathways, or those through regions of fiber crossing of interrupted by synapses may not be detected

3. The probability of tracing a pathway between two points will be influenced by factors other than the true existence of an anatomical connection – for example, longer or more tortuous paths are less likely to be traced

36

References

[1] Tournier, J.-D., Mori, S., and Leemans, A. (2011). “Diffusion tensor imaging and beyond.” Magnetic resonance in medicine : official journal of the Society of Magnetic Resonance in Medicine / Society of Magnetic Resonance in Medicine, 65(6), pp. 1532-56.

[2] Callaghan, P., Codd, S., and Seymour, J. (1999). “Spatial coherence phenomena arising from translational spin motion in gradient spin echo experiments.” Concepts in Magnetic Reso-nance, 11(4), pp. 181–202.

[3] Cohen, Y. and Assaf, Y. (2002). “High b-value q-space analyzed diffusion-weighted MRS and MRI in neuronal tissues - a technical review.” NMR in biomedicine, 15(7-8), pp. 516-42.