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 in density Diffusion – matrix, tensor (2x2 or 3x3) Derivatives (gradients) in space

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\ 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 Symmetric D is a “square, symmetric, positive definite matrix” (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

l 2

v

1 Eigen Directions and Values (Principle Directions)

v

3 l 1 l 1 l 3 l 2

v

1

v

2

v

2

Tensors From Diffusion-Weighted • Big assumption Images – 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

k th – Relationship between the DW images and Physical constants D Strength of gradient Duration of gradient pulse Read-out time DW Image Base image Gradient direction

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 k th DW Image Base image Gradient direction

Tensors From Diffusion-Weighted Images 2D • 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) 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 – ( l 1 – ( l 1 2 + l 2 2 + l 2 + l 3 2 ) 1/2 + l 3 )/3 • Area – ( l 1 l 2 + l 1 l 3 + l 2 l 3 ) • Volume – ( l 1 l 2 l 3 ) Sometimes used.

Generally used.

Also called: “Root sum of squares” “Diffusion norm” “Frobenius norm” Also called: “Mean diffusivity” “Trace”

l 1 l 3 Shape Other Than Size l 1 >= l 2 >= l 3 l 2 Barycentric shape space (C S ,C L ,C P ) Westin, 1997 G. Kindlmann

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 A 1 VF RA FA A. Alexander

Visualizing Tensors: Direction and Shape • Color mapping • Glyphs

Coloring by Principal Diffusion Direction • Principal eigenvector, linear anisotropy determine color e 1 Coronal R = |

e

1

.

x | G = |

e

1

.

y | B = |

e

1

.

z | Axial Pierpaoli, 1997 Sagittal G. Kindlmann

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”?

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

Backdrop: FA Color: RGB(

e

1 ) G. Kindlmann

Glyphs: ellipsoids

Problem:

Visual ambiguity

Worst case scenario: one viewpoint: ellipsoids 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(

e

1 )

Backdrop: FA Color: RGB(

e

1 )

Backdrop: FA Color: RGB(

e

1 )

Backdrop: FA Color: RGB(

e

1 )

Backdrop: FA Color: RGB(

e

1 )

Backdrop: FA Color: RGB(

e

1 )

Backdrop: FA Color: RGB(

e

1 )

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 Automatic Segmentation

Find the path(s) between A and B that is most consistent with the data Path of Interest D. Tuch and Others A

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 Interpolation Example 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