Diffusion Tensor Imaging - Computer Science - Lecture Slides | CS T101, Study notes of Computer Science

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Medical Image Analysis

CS 593 / 791

Computer Science and Electrical Engineering Dept. West Virginia University

31st March 2006

Outline

1 Diffusion Process

(^2) Diffusion Equation

(^3) Diffusion Weighted MRI

(^4) Diffusion Tensor Imaging

(^5) Visualization Techniques

Diffusion in Neuronal Tissue

White matter structure. Tissue forms a barrier to diffusing molecules, changing the rate and direction of diffusion. Anisotropic material can induce anisotropic diffusion in surrounding water. Diffusion near fiberous structures, such as white matter, can be very anisotropic.

Diffusion in Neuronal Tissue

We can infer properties of tissue by observing the diffusion of water surrounding the tissue.

3 main tissue classes: White matter: fiberous connective structure. Grey matter: less coherent functional units. Cerebro-spinal fluid (CSF): free diffusing water.

Diffusivity

Diffusivity : the rate of mass transfer through a unit area of material under a unit concentration gradient.

For isotropic materials the diffusivity, d, is a constant (independent of direction).

For anisotropic materials the diffusivity may be modelled using the diffusion tensor:

d ( g ) = gT^ Dg

Diffusion-Weighted MRI

The MRI signal intensity is attenuated when water molecules undergo diffusion. Stejskal-Tanner Equation (1965):

S = S 0 exp(− bd )

b is a known property of the gradient pulse sequence called ”the diffusion weighting factor.” S 0 is the ideal image formed with no diffusion weighting b = 0. Every MRI pulse sequence has some nonzero diffusion weighting, so S 0 cannot be measured, but b can be made small. By acquiring two (or more) images, one with a low b-value and one with a high b-value we can estimate S 0 and d.

Diffusion-Weighted MRI

d may be computed from as few as 2 DWIs. We only get a scalar measure of diffusivity. The DWI changes as the direction, g, of the diffusion encoding gradient changes.

Diffusion Tensor MRI

Peter Basser (1994): Substituted the tensor diffusivity

d ( g ) = gT^ Dg

into the Stejskal-Tanner equation

S = S 0 exp(− bd )

to obtain S = S 0 exp(− bgT^ Dg )

b is the diffusion weighting factor g is the diffusion encoding gradient direction

Diffusion Tensor MRI

If we acquire m images we can estimate the symmetric tensor, D, and S 0 by finding the least squares solution to the linear system:

2 (^66) 4

ln S^1 .. . ln Sm

3 (^77) 5 =

2 (^66) (^64)

1 − b^1 xx − b^1 yy − b^1 zz − 2 b^1 xy − 2 b^1 yz − 2 b^1 xz .. .

.. .

.. .

.. .

.. .

.. .

.. . 1 − bmxx − bmyy − bmzz − 2 bmxy − 2 bmyz − 2 bmxz

3 (^77) (^75)

2 (^66) (^66) (^66) (^64)

ln S 0 Dxx Dyy Dzz Dxy Dyz Dxz

3 (^77) (^77) (^77) (^75)

Solve for 7 unknowns at each voxel: 6 tensor components (since D is symmetric) and S 0. We must acquire at least 7 images with noncoplanar gradient directions.

The Diffusion Tensor

We can characterize anisotropic diffusion by a 3 × 3 matrix, D. D describes diffusivity rates over all directions. D is symmetric and positive-definite It is useful to examine the eigenvalue decomposition D = E T^ Λ E.

D =

e 1 T e 2 T e 3 T

λ 1 0 0 0 λ 2 0 0 0 λ 3

[

e 1 e 2 e 3

]

Orthonormal Basis e 1 , e 2 , e 3. Real positive eigenvalues λ 1 , λ 2 , λ 3.

Diffusion Ellipsoid




 



Sort λ so that λ 1 = λ 2 = λ 3 Principal diffusion direction is e 1

Probabalistic Interpretation

The diffusion tensor is an isoprobability surface: molecules starting from the center have equal probability of diffusing through points on the surface of the ellipsoid.

The probability that a molecule undergoes displacement r over time t is pt ( r ) = N ( r | 0 , 2 tD )

Zero-mean Gaussian distribution. Covariance is a scaled diffusion tensor. The scale is related to the diffusion time. We can consider the tensor field to also be a field of probability distributions.

Scalar Field Visualization

S 0 and fractional anisotropy.

Scalar Field Visualization

Tensor Components