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. O’Donnell 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 Matrices NonsquareSquare Symmetric Positive Skew symmetric 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 –Lambda’s > 0 Quadratic equation – implicit equation for ellipse (ellipsoid in 3D)

8. Eigen Directions and Values (Principle Directions) 1 2 3 v3v3 v2v2 v1v1 v1v1 v2v2 2 1

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 k th DW Image Base image Gradient direction Physical constants Strength of gradient Duration of gradient pulse Read-out time

10. Tensors From Diffusion-Weighted Images Stejskal-Tanner equation –Relationship between the DW images and D k th DW Image Base image Gradient direction Physical constants Strength of gradient Duration of gradient pulse Read-out time

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 lambda>0) 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 – ( 1 + 2 + 3 )/3 –( 1 2 + 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”

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

15. Reducing Shape to One Number Fractional Anisotropy FA (not quite) Properties: Normalized variance of eigenvalues Difference from sphere

16. 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

17. Various Measures of Anisotropy A1A1 VFRAFA A. Alexander

18. Visualizing Tensors: Direction and Shape Color mapping Glyphs

19. Principal eigenvector, linear anisotropy determine color e1e1 R = | e 1. x | G = | e 1. y | B = | e 1. z | Coloring by Principal Diffusion Direction Pierpaoli, 1997 Axial Sagittal Coronal 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(e 1 ) G. Kindlmann

23. Glyphs: ellipsoids Problem: Visual ambiguity

24. Worst case scenario: ellipsoids one viewpoint: another viewpoint:

25. Glyphs: cuboids Problem: missing symmetry

26. Superquadrics Barr 1981

27. Superquadric Glyphs for Visualizing DTI Kindlmann 2004

28. Worst case scenario, revisited

29. Backdrop: FA Color: RGB(e 1 )

30. Backdrop: FA Color: RGB(e 1 )

31. Backdrop: FA Color: RGB(e 1 )

32. Backdrop: FA Color: RGB(e 1 )

33. Backdrop: FA Color: RGB(e 1 )

34. Backdrop: FA Color: RGB(e 1 )

35. Backdrop: FA Color: RGB(e 1 )

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 Architecture L. O’Donnell 2006 Saved structure information Analysis High-Dimensional Atlas Atlas Generation Automatic Segmentation

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

43. 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

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 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

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 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