Diffusion Tensor Processing
and Visualization
Ross Whitaker
University of Utah
National Alliance for Medical Image
Computing
Acknowledgments
Contributors:
• A. Alexander
• G. Kindlmann
• L. O’Donnell
• J. Fallon
National Alliance for Medical Image Computing
(NIH U54EB005149)
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
The Physics of Diffusion
• Density of substance changes (evolves)
over time according to a differential
equation (PDE)
Change
Derivatives
in
(gradients) in
density Diffusion –
space
matrix, tensor
(2x2 or 3x3)
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\
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
• This relationship is not a coincidence
– Probabilistic models of diffusion (random
walk)
D Is A Special Kind of Matrix
• The universe of matrices
Matrices
Square
Skew symmetric
Nonsquare D is a “square,
symmetric, positivedefinite matrix”
Symmetric
(SPD)
Positive
Properties of SPD
• Bilinear forms and quadratics
Quadratic equation – implicit equation for ellipse
(ellipsoid in 3D)
• Eigen Decomposition
– Lambda – shape information, independent of
orientation
– R – orientation, independent of shape
– Lambda’s > 0
Eigen Directions and Values
(Principle Directions)
v3
v1
l2
l1
l1
l3
l2
v1
v2
v2
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
Physical constants
Strength of gradient
D
kth DW Image
Base image
Gradient direction
Duration of gradient
pulse
Read-out time
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
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
2D
» (constrain lambda>0)
S-T Equation
Shape Measures on Tensors
• Represent or visualization shape
• Quanitfy meaningful aspect of shape
• Shape vs size
Different sizes/orientations
Different shapes
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)
Sometimes used.
Generally used.
Also called:
Also called:
“Root sum of squares”
“Mean diffusivity”
“Diffusion norm”
“Trace”
“Frobenius norm”
l3
Shape Other Than Size
Barycentric
shape space
l1
l2
l1 >= l2 >= l3
(CS,CL,CP)
Westin, 1997
G.
Reducing Shape to One Number
Fractional Anisotropy
Properties:
Normalized variance of
eigenvalues
Difference from sphere
FA (not quite)
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?
Various Measures of Anisotropy
A1
VF
RA
FA
A. Alexander
Visualizing Tensors: Direction
and Shape
• Color mapping
• Glyphs
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.
Issues With Coloring by
Direction
• Set transparency according to FA (highlighttracts)
• Coordinate system dependent
• Primary colors dominate
– Perception: saturated colors tend to look more
intense
– Which direction is “cyan”?
Visualization with Glyphs
• Density and placement based on FA or
detected features
• Place ellipsoids at regular intervals
Backdrop: FA
Color: RGB(e1)
G.
Glyphs: ellipsoids
Problem:
Visual
ambiguity
Worst case scenario:
ellipsoids
one viewpoint:
another viewpoint:
Glyphs: cuboids
Problem:
missing
symmetry
Superquadrics
Barr 1981
Superquadric Glyphs for Visualizing
DTI
Kindlmann 2004
Worst case scenario, revisited
Backdrop: FA
Color: RGB(e1)
Backdrop: FA
Color: RGB(e1)
Backdrop: FA
Color: RGB(e1)
Backdrop: FA
Color: RGB(e1)
Backdrop: FA
Color: RGB(e1)
Backdrop: FA
Color: RGB(e1)
Backdrop: FA
Color: RGB(e1)
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
DTI Tractography
Seed point(s)
Move marker in
discrete steps
and find next
direction
Direction of
principle eigen
value
Tractography
J. Fallon
Whole-Brian White Matter
Architecture
L. O’Donnell 2006
Atlas Generation
Analysis
High-Dimensional
Saved structure
information
Atlas
Automatic Segmentation
Path of Interest
D. Tuch and Others
A
Find the
path(s)
between A
and B that is
most
consistent
with the data
B
The Problem with
Tractography
How 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
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
Averaging Tensors
• What should be the average of these
two tensors?
Linear Average
Componentwise
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
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
Riemannian Arithmetic
Example
Riemannian
Linear
Interpolation
Interpolation
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…
Questions