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Diffusion Tensor Processing and Visualization
Ross WhitakerUniversity of Utah
National Alliance for Medical Image Computing
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Acknowledgments
Contributors:• A. Alexander • G. Kindlmann • L. O’Donnell • J. Fallon
National Alliance for Medical Image Computing (NIH U54EB005149)
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Diffusion in Biological Tissue
• Motion of water through tissue• Sometimes faster in some directions than others
Kleenex newspaper• Anisotropy: diffusion rate depends on direction
isotropic anisotropic
G. Kindlmann
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The Physics of Diffusion
• Density of substance changes (evolves) over time according to a differential equation (PDE)
Change in
density
Derivatives (gradients) in spaceDiffusion –
matrix, tensor(2x2 or 3x3)
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Solutions of the Diffusion Equation
• Simple assumptions– Small dot of a substance (point)– D constant everywhere in space
• Solution is a multivariate Gaussian– Normal distribution– D plays the role of the covariance matrix\
• This relationship is not a coincidence – Probabilistic models of diffusion (random walk)
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
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D Is A Special Kind of Matrix
• The universe of matrices
Matrices
NonsquareSquare
Symmetric
Positive
Skew symmetric
D is a “square, symmetric, positive-definite matrix” (SPD)
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Properties of SPD• Bilinear forms and quadratics
• Eigen Decomposition
– Lambda – shape information, independent of orientation
– R – orientation, independent of shape– Lambda’s > 0
Quadratic equation – implicit equation for ellipse (ellipsoid in 3D)
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Eigen Directions and Values
(Principle Directions)
12
3
v3
v2
v1
v1
v2
21
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Tensors From Diffusion-Weighted Images
• Big assumption– At the scale of DW-MRI measurements– Diffusion of water in tissue is approximated by Gaussian•Solution to heat equation with constant diffusion tensor
• Stejskal-Tanner equation– Relationship between the DW images and D
kth DW Image Base image Gradient direction
Physical constantsStrength of gradientDuration of gradient pulseRead-out time
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Tensors From Diffusion-Weighted Images
• Stejskal-Tanner equation– Relationship between the DW images and D
kth DW Image Base image Gradient direction
Physical constantsStrength of gradientDuration of gradient pulseRead-out time
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Tensors From Diffusion-Weighted Images
• Solving S-T for D– Take log of both sides– Linear system for elements of D– Six gradient directions (3 in 2D) uniquely specify D
– More gradient directions overconstrain D•Solve least-squares
» (constrain lambda>0)2D
S-T Equation
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Shape Measures on Tensors
• Represent or visualization shape
• Quanitfy meaningful aspect of shape
• Shape vs size
Different sizes/orientations Different shapes
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Measuring the Size of A Tensor
• Length – (1 + 2 + 3)/3
– (12 + 2
2 + 3
2)1/2
• Area – (1 2 + 1 3 + 2 3)
• Volume – (1 2 3)
Generally used.Also called:
“Mean diffusivity”“Trace”
Sometimes used.Also called:
“Root sum of squares”“Diffusion norm”“Frobenius norm”
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Shape Other Than Size
1
2
3
l1 >= l2 >= l3
Barycentricshape space
(CS,CL,CP)Westin, 1997
G. Kindlmann
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Reducing Shape to One Number
Fractional Anisotropy
FA (not quite)Properties:Normalized variance of eigenvalues Difference from sphere
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• Visualization – ignore tissue that is not WM
• Registration – Align WM bundles• Tractography – terminate tracts as they exit WM
• Analysis– Axon density/degeneration– Myelin
• Big question– What physiological/anatomical property does FA measure?
FA As An Indicator for White Matter
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Various Measures of Anisotropy
A1 VF RA FA
A. Alexander
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Visualizing Tensors: Direction and Shape
• Color mapping• Glyphs
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• Principal eigenvector, linear anisotropy determine color
e1
R=| e1.x
|G=| e1
.y |B= | e1
.z |
Coloring by Principal Diffusion Direction
Pierpaoli, 1997
Axial
Sagittal
Coronal
G. Kindlmann
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Issues With Coloring by Direction
• Set transparency according to FA (highlight-tracts)
• Coordinate system dependent• Primary colors dominate
– Perception: saturated colors tend to look more intense
– Which direction is “cyan”?
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Visualization with Glyphs
• Density and placement based on FA or detected features
• Place ellipsoids at regular intervals
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Backdrop: FA
Color: RGB(e1)
G. Kindlmann
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Glyphs: ellipsoids
Problem:Visual
ambiguity
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Worst case scenario: ellipsoidsone viewpoint:
another viewpoint:
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Glyphs: cuboids
Problem:missing
symmetry
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SuperquadricsBarr 1981
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Superquadric Glyphs for Visualizing DTI
Kindlmann 2004
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Worst case scenario, revisited
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Backdrop: FA
Color: RGB(e1)
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Backdrop: FA
Color: RGB(e1)
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Backdrop: FA
Color: RGB(e1)
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Backdrop: FA
Color: RGB(e1)
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Backdrop: FA
Color: RGB(e1)
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Backdrop: FA
Color: RGB(e1)
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Backdrop: FA
Color: RGB(e1)
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Going Beyond Voxels: Tractography
• Method for visualization/analysis• Integrate vector field associated with grid of principle directions
• Requires– Seed point(s)– Stopping criteria
•FA too low•Directions not aligned (curvature too high)
•Leave region of interest/volume
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DTI Tractography
Seed point(s)
Move marker in discrete steps and find next directionDirection of principle eigen value
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Tractography
J. Fallon
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Whole-Brian White Matter ArchitectureL. O’Donnell 2006
Saved structureinformation
Analysis
High-DimensionalAtlas
Atlas Generation
Automatic Segmentation
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Path of InterestD. Tuch and Others
A
B
Find the path(s) between A and B that is most consistent with the data
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The Problem with TractographyHow Can It Work?
• Integrals of uncertain quantities are prone to error– Problem can be aggravated by nonlinearities
• Related problems– Open loop in controls (tracking)– Dead reckoning in robotics
Wrong turn
Nonlinear: bad information about where to go
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Mathematics and Tensors
• Certain basic operations we need to do on tensors– Interpolation– Filtering– Differences– Averaging– Statistics
• Danger– Tensor operations done element by element• Mathematically unsound• Nonintuitive
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Averaging Tensors
• What should be the average of these two tensors?
Linear AverageComponentwise
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Arithmetic Operations On Tensor
• Don’t preserve size– Length, area, volume
• Reduce anisotropy• Extrapolation –> nonpositive, nonsymmetric
• Why do we care?– Registration/normalization of tensor images
– Smoothing/denoising– Statistics mean/variance
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What Can We Do?(Open Problem)
• Arithmetic directly on the DW images– How to do statics?– Rotational invariance
• Operate on logarithms of tensors (Arsigny)– Exponent always positive
• Riemannian geometry (Fletcher, Pennec)– Tensors live in a curved space
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Riemannian Arithmetic Example
Interpolation Interpolation
Linear
Riemannian
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Low-Level Processing DTI Status
• Set of tools in ITK– Linear and nonlinear filtering with Riemannian geometry
– Interpolation with Riemannian geometry
– Set of tools for processing/interpolation of tensors from DW images
• More to come…
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Questions