Diffusion Tensor Processing and Visualization

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Diffusion Tensor Processing and Visualization

• Ross Whitaker
• University of Utah
• National Alliance for Medical Image Computing

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Acknowledgments

• Contributors
• A. Alexander
• G. Kindlmann
• L. ODonnell
• 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 space
Diffusion 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)

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D Is A Special Kind of Matrix

• The universe of matrices

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
• Lambdas gt 0

Quadratic equation  implicit equation for
ellipse (ellipsoid in 3D)
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Eigen Directions and Values(Principle Directions)
v1
l1
l2
v2
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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

Physical constants Strength of gradient Duration
of gradient pulse Read-out time
kth DW Image
Base image
Gradient direction
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Tensors From Diffusion-Weighted Images

• Stejskal-Tanner equation
• Relationship between the DW images and D

Physical constants Strength of gradient Duration
of gradient pulse Read-out time
kth DW Image
Base image
Gradient direction
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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 lambdagt0)

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 (l1 l2 l3)/3
• (l12 l22 l32)1/2
• Area (l1 l2 l1 l3 l2 l3)
• Volume (l1 l2 l3)

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
G. Kindlmann
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Reducing Shape to One NumberFractional Anisotropy
Properties Normalized variance of eigenvalues
Difference from sphere
FA (not quite)
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FA As An Indicator for White Matter

• 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?

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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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Coloring by Principal Diffusion Direction

• Principal eigenvector, linear anisotropy
  determine color

e1
Coronal
Axial
R e1.x G e1.y B e1.z
Sagittal
Pierpaoli, 1997
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
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Worst case scenario ellipsoids
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Glyphs cuboids
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SuperquadricsBarr 1981
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Superquadric Glyphs for Visualizing DTIKindlmann
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
direction
Direction of principle eigen value
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Tractography
J. Fallon
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Whole-Brian White Matter ArchitectureL.
ODonnell 2006
Atlas Generation
Analysis
Saved structure information
High-Dimensional Atlas
Automatic Segmentation
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Path of InterestD. Tuch and Others
A
Find the path(s) between A and B that is most
consistent with the data
B
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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 Average Componentwise
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Arithmetic Operations On Tensor

• Dont preserve size
• Length, area, volume
• Reduce anisotropy
• Extrapolation gt 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