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Diffusion Tensor Processing and Visualization

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Title: Diffusion Tensor Processing and Visualization


1
Diffusion Tensor Processing and Visualization
  • Ross Whitaker
  • University of Utah
  • National Alliance for Medical Image Computing

2
Acknowledgments
  • Contributors
  • A. Alexander
  • G. Kindlmann
  • L. ODonnell
  • J. Fallon
  • National Alliance for Medical Image Computing
    (NIH U54EB005149)

3
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
4
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)
5
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)

6
D Is A Special Kind of Matrix
  • The universe of matrices

D is a square, symmetric, positive-definite
matrix (SPD)
7
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)
8
Eigen Directions and Values(Principle Directions)
v1
l1
l2
v2
9
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
10
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
11
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
12
Shape Measures on Tensors
  • Represent or visualization shape
  • Quanitfy meaningful aspect of shape
  • Shape vs size

Different sizes/orientations
Different shapes
13
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
14
Shape Other Than Size
G. Kindlmann
15
Reducing Shape to One NumberFractional Anisotropy
Properties Normalized variance of eigenvalues
Difference from sphere
FA (not quite)
16
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?

17
Various Measures of Anisotropy
A1
VF
RA
FA
A. Alexander
18
Visualizing Tensors Direction and Shape
  • Color mapping
  • Glyphs

19
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
20
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?

21
Visualization with Glyphs
  • Density and placement based on FA or detected
    features
  • Place ellipsoids at regular intervals

22
Backdrop FA
Color RGB(e1)
G. Kindlmann
23
Glyphs ellipsoids
24
Worst case scenario ellipsoids
25
Glyphs cuboids
26
SuperquadricsBarr 1981
27
Superquadric Glyphs for Visualizing DTIKindlmann
2004
28
Worst case scenario, revisited
29
Backdrop FA
Color RGB(e1)
30
Backdrop FA
Color RGB(e1)
31
Backdrop FA
Color RGB(e1)
32
Backdrop FA
Color RGB(e1)
33
Backdrop FA
Color RGB(e1)
34
Backdrop FA
Color RGB(e1)
35
Backdrop FA
Color RGB(e1)
36
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37
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38
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

39
DTI Tractography
Seed point(s)
Move marker in discrete steps and find next
direction
Direction of principle eigen value
40
Tractography
J. Fallon
41
Whole-Brian White Matter ArchitectureL.
ODonnell 2006
Atlas Generation
Analysis
Saved structure information
High-Dimensional Atlas
Automatic Segmentation
42
Path of InterestD. Tuch and Others
A
Find the path(s) between A and B that is most
consistent with the data
B
43
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
44
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

45
Averaging Tensors
  • What should be the average of these two tensors?

Linear Average Componentwise
46
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

47
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

48
Riemannian Arithmetic Example
Interpolation
Interpolation
Linear
Riemannian
49
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

50
Questions
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