Diffusion Tensor Imaging (DTI): An overview of key concepts

(Supplemental material for presentation)

Prepared by: Nadia Barakat BMB 601

Chris Conklin Thursday, April 8th 2010

Diffusion Concept [1,2]:

Brownian Motion:

Refers to constant, thermal and random motion of molecules Phenomenon observed by Robert Brown

Fick’s Law [2]:

Physical law governing molecular diffusion Differences in solutes concentration in a medium leads to net flux of solute particles from

high concentration to low concentration regions

𝐽 = −𝐷∇𝐶 → 𝐽𝑥𝐽𝑦𝐽𝑧 = − 𝐷𝑥𝑥 𝐷𝑥𝑦 𝐷𝑥𝑧𝐷𝑦𝑥 𝐷𝑦𝑦 𝐷𝑦𝑧𝐷𝑧𝑥 𝐷𝑧𝑦 𝐷𝑧𝑧 𝜕𝐶𝜕𝑥𝜕𝐶𝜕𝑦𝜕𝐶𝜕𝑧

where J is the net particle flux, D is the diffusion coefficient (tensor in terms of DTI), and C is the particle concentration. The minus sign indicates that the particle is moving in direction of decreasing concentration.

Einstein’s Contribution [2]:

Displacement distribution of a group of particles undergoing diffusion In free diffusion with a large number of molecules, it was predicted that after time, t, a

molecule will end up somewhere within a sphere of radius R Mean square displacement of particles during diffusion time, t, and diffusion coefficient,

D

𝑅2 = 2𝐷𝑡 1D- Gaussian function 2D-circular cross section of Gaussian distribution (for isotropic medium) 3D-spherical distribution function

Diffusion Tensor Imaging [1-3]:

An MRI technique used to quantify diffusion of water molecules in each voxel of an image

By using linear magnetic field gradients inhomogeneities are introduced whereby diffusion can be detected as signal attenuation

What is being Measured? [1-4]

H.C. Torrey modified the Bloch equations to account for diffusion in 1956 [4]. In essence, the classical magnetization equations incorporate an additional term that

reflects the echo signal attenuation

𝑆 = 𝑆0𝑒(−𝑏𝐷) where S is the signal intensity at a given voxel, S0 is the signal intensity for a non-diffusion weighted sequence, b is the diffusion weighting factor that depends upon the shape and duration of sensitizing gradients, and D is the effective diffusion tensor quantifying motion of water molecules in desired non-collinear directions.

DTI Parameters [1-3]:

To extract information from the collected DTI data, a few parameters (scalar values and

parametrical maps) are derived from the estimated diffusion tensor.

Trace is the sum of the three diagonal elements of the tensor, which is equal to the sum of the three eigenvalues. Dividing trace by three yields the averaged mean diffusivity. It is found using equation 3-3

Trace = 𝐷𝑥𝑥+ 𝐷𝑦𝑦+ 𝐷𝑧𝑧 or Trace = λ1 + λ2 + λ3

Mean diffusivity (MD), in 𝑚𝑚2/s, describes the overall mean-squared displacements of water

molecules and the existence of any obstacles to diffusion. It does not account for anisotropic

diffusion. Mean diffusivity is given by

MD =Dxx+Dyy +Dzz3 or MD =λ1+λ2+λ33

In the case of anisotropic diffusion, several indices are calculated.

Fractional anisotropy (FA) is the most commonly used anisotropy index. It is a unit-less

parameter which characterizes the degree (or magnitude) of anisotropy, or how the molecules

displacements vary in space. In the same manner the magnitude of a vector is calculated (i.e.

summing the squares of its components), the magnitude of a tensor is calculated by summing the

squares of its eigenvalues. In the ideal case of diffusion occurring along one direction, a

maximum FA value of one is expected. In gray matter for example, a low FA value is measured

because the medium is isotropic (not organized in fibers as in white matter). FA is calculated by

the following equation:

𝐹𝐴 = 3[(λ1−λ)2+(λ2−λ)2+(λ3−λ)2] 2(𝜆12+𝜆22+𝜆32) where λ =λ1+λ2+λ33

Volume ratio (VR), in 𝑚𝑚2/s, is the ratio of the ellipsoid volume to the volume of a sphere of

radius λ. VR ranges from one (isotropic diffusion) to zero and calculated by

𝑉𝑅 =λ1λ2λ3λ3

Relative anisotropy (RA), in 𝑚𝑚2/s, represents the ratio of the anisotropic part of diffusivity to its

isotropic part. RA varies between zero (isotropic medium) to 2 (anisotropic medium).

𝑅𝐴 = (λ1 − λ)2 + (λ2 − λ)2 + (λ3 − λ)2 3𝜆

Finally it is worth noting that the DTI parameters mentioned above represent the overall diffusion

(isotropic and anisotropic) in an ellipsoid-shaped tensor where λ1 > λ2 > λ3 . They do not

provide any information on the ellipsoid shape where λ1 > λ2 = λ3 or λ1 = λ2 > λ3. In order to

characterize the tensor’s ellipsoid shape , three other indices are used:

inear anisotropy coefficient (Cl) describes the linearity if the ellipsoid, and is defined by

𝐶𝑙 =(λ1 − λ2)3𝜆

Planar anisotropy coefficient (Cp) is used to find how planar the tensor ellipsoid is

𝐶𝑝 =2(λ2 − λ3)3𝜆

Spherical anisotropy coefficients (Cs) shows the ellispsoid’s spherical characteristics, and is

determined by the following equation

𝐶𝑠 =λ3𝜆

Below are a series of pictures showing several DTI parameter maps that can reveal useful clinical information.

(A) – T2 weighted, (B) – FA map, (C) - Trace map

(A) – color FA map (red-left/right; blue – superior/inferior; green – anterior/posterior), (B)– normal FA map

(A) (B) (C) (A) (B) (C) (A) (B)

(A) (B)

Sequences:

Spin-Echo (SE)

Developed by Hahn [5] Most Basic Sequence 900-1800 pulses Trilobed slice selection to eliminate transverse magnetization from imperfect 1800

Pulsed Gradient Spin Echo (PGSE)

Image: Pulsed gradient spin echo sequence [6]

Carr and Purcell SE sequence [7] modified in 1965 by Stejskal-Tanner [8] Constant field gradient replaced by pulsed scheme Accurate measure of diffusion as distinction between encode time(δ) & diffusion time(Δ)

is pronounced As δ-> Δ sequence reverts to typical SE

Echo Planar Imaging (EPI)

Peter Mansfield initially showed the concept for EPI and multi-planar image formation [9]

Fill k-space in 1 TR Single shot vs Multishot readout

o Single shot – fast but phase errors propogate o Multi shot – less stress on gradients, reduction in phase errors, but slow

Diffusion Weighted Echo Planar Imaging (DW-EPI)

Shaded portions of pulse diagram are diffusion gradients which can be applied in any of the principal directions [10]

Mathematics:

Tensor

Mathematical formation conceived by Bernhard Riemann and Elwin Bruno Christoffel Representation that expresses how vectors relate

o Invariant to choice of coordinate system Created to solve stress problems in mechanics

Tensor that describes the force on each face of the cube Tensors hold great utility in a wide variety of problems

Diffusion Tensor, D

)3()2()1( eee TTT

333231

232221

131211

[10] [11]

Diffusion of molecules along applied gradients make up elements of D Can represent the diffusion pattern at each point in the brain using an ellipsoid

b-Matrix

To accurately estimate the diffusion tensor it is necessary to quantify the contributions of all gradient pulses

o The net effect of the gradient on each echo for each desired direction make up the elements of the b matrix

b-matrix for pulsed gradient sequence is related to the area of the gradient pulses

𝑏= 𝒌 𝑡 − 2𝐻 𝑡 − 𝜏 𝒌 𝜏 (𝒌 𝑡 − 2𝐻 𝑡 − 𝜏 𝒌 𝜏 )𝑇𝑑𝑡2𝜏0

where 2τ = TE , H(t) is the unit-step Heaviside function, 𝒌 𝑡 = 𝛾 𝑮 𝑡′ 𝑑𝑡′𝑡0 and

𝑮 𝑡 = [𝐺𝑥 𝑡 , 𝐺𝑦 𝑡 , 𝐺𝑧 𝑡 ]𝑇 [12]

Estimating the Diffusion Tensor

ln 𝐴 𝑏 𝐴 0 = − 𝑏𝑖𝑗𝐷𝑖𝑗3𝑗 =13𝑖=1

Since the b matrix can be calculated, the diffusion tensor, D, can be found. For isotropic medium the b matrix degenerates to a scalar and the above equation reduces to

𝐷 =ln 𝐴 𝑏 𝐴 0 /𝑏

A(b) is the signal intensity for a diffusion weighted image, with a known b value, and A(0) is a non-diffusion weighted image.

Applications:

Diffusion MRI emerging as a routine clinical protocol

Ischemia Stroke MS Schizophrenia

Tractography

Allows for the visualization of networks in the body. This is particularly useful for neuroimaging and tracking of fibers in the CNS. When used in conjunction with fMRI this modality could serve to establish connectivity pathways for a variety of tasks.

Limitations:

[13]

[14]

Motion

Diffusion weighted sequences optimized to be sensitive to diffusion processes. This can cause ghosting and aliasing artifacts from several sources

Subject Motion Pulsation of blood vessels CSF Flow Cardiac/Respiratory motion

Gradient Inhomogeneities/ Non-Linearity

Warping

Eddy Currents

[15]

[2]

Gradient coils that create magnetic gradient also induce electric field Can lead to warping due to eddy currents induced from diffusion weighted gradients

References

[2]

[1] LeBihan, D., et al. 2001. Diffusion Tensor Imaging: Concepts and Applications. Journal of Magnetic Resonance Imaging 13, 534-546.

[2] Johansen-Berg, H., and Behrens, T. 2009. Diffusion MRI: From Quantitative Measurement to in Vivo Neuroanatomy. 1st Edition. Elsevier Inc.

[3] Susumu, M. 2007. Introduction to Diffusion Tensor Imaging. 1st Edition. Elsevier.

[4] H.C. Torrey. “Bloch equations with diffusion terms.” Phys. Rev. 104, 563 (1956)

[5] E.L. Hahn. “Spin Echoes.” Phys. Rev. 80, 4 (1950)

[6] - Hagmann P, Jonasson L, Maeder P, et al. Understanding diffusion MR imaging techniques: from scalar diffusion-weighted imaging to diffusion tensor imaging and beyond. Radiographics 2006;26:S205e23.

[7] H.Y. Carr and E.M. Purcell. “Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments.” Phys. Rev. 94,3 (1954)

[8] E.O. Stejskal and J.E. Tanner. “Spin diffusion measurements: spin echos in the presence of a time dependent field gradient.” J. Chem. Phys. 42, 10 (1964)

[9] P. Mansfield. “Multi-planar image formation using NMR spin echoes.” J.Phys. C: Solid State Phys., 10 (1977)

[10] Bernstein, Matt A., King, Kevin F., and Zhou, Xiaohong J. 2004. Handbook of MRI. Pulse Sequences. Elsevier Academic Press.

[11] Webpage http://en.wikipedia.org/wiki/Tensor

[12] Basser, P., J. Mattiello, and D. LeBihan. 1994. Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance Imaging 103, 247-254.

[13] Webpage http://www.scielo.br/img/revistas/rbp/v25n3/a13fig01.gif

[14] Webpage http://csdl.computer.org/comp/proceedings/vis/2004/8788/00/87880028p.pdf

[15] Webpage http://www.mr-tip.com/serv1.php?type=art&sub=Motion%20Artifact