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Diffusion Tensor Processing and Visualization

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).

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Diffusion Tensor Processing and Visualization

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  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. Matrices Square Nonsquare Skew symmetric Symmetric Positive 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 • Lambda’s > 0 Quadratic equation – implicit equation for ellipse (ellipsoid in 3D)

  8. v3 l3 l1 l2 v1 v2 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 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 – (l1 + l2 + l3)/3 • (l12 + l22+ l32)1/2 • Area – (l1 l2 + l1 l3 + l2 l3) • Volume – (l1 l2 l3) Sometimes used. Also called: “Root sum of squares” “Diffusion norm” “Frobenius norm” Generally used. Also called: “Mean diffusivity” “Trace”

  14. 3 Barycentric shape space 2 1 l1 >= l2 >= l3 (CS,CL,CP) Westin, 1997 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. Problem: Visual ambiguity Glyphs: ellipsoids

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

  25. Problem: missing symmetry 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. 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

  37. DTI Tractography Seed point(s) Move marker in discrete steps and find next direction Direction of principle eigen value

  38. Tractography J. Fallon

  39. Whole-Brian White Matter ArchitectureL. O’Donnell 2006 Atlas Generation Analysis Saved structure information High-Dimensional Atlas Automatic Segmentation

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

  41. 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

  42. 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

  43. Averaging Tensors • What should be the average of these two tensors? Linear Average Componentwise

  44. 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

  45. 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

  46. Riemannian Arithmetic Example Interpolation Interpolation Linear Riemannian

  47. 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…

  48. Questions

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