Some Developments in DT-MRI Registration and Visualization

1. Some Developments in DT-MRI Registration and Visualization James Gee, Hui Zhang, Jeffrey Duda, Paul Yushkevich, Brian Avants, Abraham DubbUniversity of Pennsylvania Dagstuhl-Seminar, April 18-23, 2004

2. Overview • Registration • Tensor orientation pattern matching • Non-rigid registration • Diffusion profile matching • Affine transformations • Visualization • Anatomically labeled fiber tracts • Partitioning the corpus callosum

3. Orientation Pattern Matching Duda et al, SPIE Med Imag 2003

4. y 11  x 22 Definitions • In this case the tensor may be represented by a symmetric 2x2 matrix, I • For this formulation it is useful to represent the tensor by a column vector of the eigenvalues and an orientation angle

5. Non-Rigid Registration of 2D Tensor Data • Want to find a smooth continuous mapping, u(s), from I2 to I1 • This is accomplished by minimizing a function consisting of three separate terms

6. Objective Function • Eigenvalue (ellipsoid shape) difference • Smoothness constraint

7. Objective Function • Local neighborhood orientation-pattern difference

8. s1 s1 r1 r2 s2 r1 s2 r2 Local Neighborhood Orientation-Pattern Difference

9. 0 0 0 0 0 0 0 0 0 0 0 45 0 0 45 0 0 45 Local Neighborhood Orientation-Pattern Difference 0 0 45 0 0 45 0 0 45

10. Gradient of Objective Function • The following function is minimized:

11. Gradient of Objective Function • U’3(u) give a non-linear term and is approximated with the following term: • Where upis the value of u at the previous iteration and:

12. Preliminary Registration Example A B  A B A  B

13. Diffusion Profile Matching Zhang, Yushkevich, Gee, ISBI 2004

14. Diffusion Profiles Diffusion Tensors A diffusion profile is a function that describes the rate of diffusion in any given direction at a point in space A diffusion tensor is a Cartesian tensor (s.p.d. matrix) that represents a Gaussian diffusion profile

15. Generality Why Profiles? Well-defined physical interpretation Alexander DC. UCL Research Notes 2000 Inclusion of non-Gaussian profiles Alexander AL et al. MRM 2001, Frank LR. MRM 2001, Tuch DS et al. MRM 2002

16. with Diffusion Profile Metric Given two diffusion profiles and : the distance metric is a function that measures their similarity Since diffusion profiles form a subset of Hilbert space, the functional distance metric is used:

17. Express diffusion profiles in terms of truncated spherical harmonic series: Alexander DC et al. MRM 2002 Computing the Metric inner product computed as: where and are spherical harmonic coefficients of and

18. Inner product and distance metric Specializing to Diffusion Tensors Coefficients from spherical harmonic expansion where and are the Cartesian tensor inner product and distance metric respectively

19. Objective function where Template Subject Specialized to diffusion tensors Affine Registration

20. Finite Strain reorientation Finite Strain-based Reorientation Best orthogonal approximation to Polar Decomposition Theorem is an orthogonal matrix (pure rotation) is symmetric and positive-definite (pure deformation) Ignore the reorientation effect of Alexander DC et al. TMI 2001

21. with with and Affine Parameters Affine Transform Parametrization Affine transformation Parametrization of Polar Decomposition-based: Compared to the standard parametrization, this scheme allows us to explicitly express the finite strain-based reorientation operator in terms of and thus differentiate analytically

22. Experimental setup Data: 2D slices of human brain DT A collection of 288 synthetic transformations Each transformation generates a synthetic subject image Relative error between the original and the recovered transformations is computed as the sum of the relative errors in each parameter Preliminary Affine Registration Results The original image is registered to the synthetic subject to recover the transformation

23. Relative error statistics registration with analytical derivatives using conjugate gradient optimizer average relative error = 3% registration without derivatives using direction set optimizer average relative error = 207% with derivatives without derivatives Relative Error Distribution

24. 1 2 3 Piecewise Affine, Non-rigid Extension a. The trace and the anisotropy image of the original image b. The trace and the anisotropy image of the image deformed by the displacement field (1) 1. The original B-Spline based displacement field 2. The piecewise affine displacement field recovered with the Cartesian distance metric 3. The piecewise affine displacement field recovered with the diffusion profile metric

25. Anatomically Labeled Fiber Tracts

26. Cortex-based Partitioning of the Corpus Callosum

35. Callosal MorphometryWitelson Partition • Anterior (rostrum, genu) ↔ motor cortex • Anterior half ↔ somatosensory cortex • Posterior two-thirds, dorsal splenium ↔ auditory regions, limbic cortex • Isthmus ↔ posterior parietal and superior temporal cortex, cortical regions related to functional asymmetry • Splenium, body and anterior portion ↔ visual cortex

36. “Modern” Callosal MorphometryTemplate Deformations • Correspondences between callosa are obtained by registering the curve geometry at different scales • Boundary correspondence is interpolated in a geometrically correct way throughout the interior (or even exterior) of the curve • Related work • Pettey, D.J., Gee, J.C. Using a linear diagnostic function and non-rigid registration to search for morphological differences between populations: An example involving the male and female corpus callosum. Information Processing in Medical Imaging, Insana, M., Leahy, R., eds., Heidelberg:Springer-Verlag, LNCS 2082, pp. 372-379, 2001. • Dubb, A., Avants, B., Gur, R., Gee, J.C. Shape characterization of the corpus callosum in Schizophrenia using template deformation. Medical Image Computing and Computer-Assisted Intervention, Kikinis, R., ed., Heidelberg:Springer-Verlag, LNCS 2489, pp. 381-388, 2002. • Pettey, D.J., Gee, J.C. Sexual dimorphism in the corpus callosum: A characterization of local size variations and a classification driven approach to morphometry. NeuroImage, 17 (3), pp. 1504-1511, 2002. • Dubb, A., Gur, R., Avants, B., Gee, J.C. Characterization of sexual dimorphism in the human corpus callosum using template deformation. NeuroImage, 20(1), pp. 512-519, 2003.

37. DTI-based Partitioning of Corpus Callosum • Validate Witelson partition • Enable more anatomically rigorous and detailed segmentation of corpus callosum and associated morphological studies • Behrens et al, Nature Neuroscience, 2003

38. Combined GM and WM volume is divided into three regions using graph partitioning software METIS Regions: Left cortex Right cortex Cerebellum & brain stem Cortex Labeling: Coarse Step Coarse level GM+WM partition

39. Cortex Labeling: Fine Step • Expert draws curves on the surface of a cortical region using Livewire-style user interface • These curves will be used to partition the surface into patches • Patch segmentation will be propagated onto the GM volume using level set/flow techniques BrainTracer tool used to mark the cortical surface

40. Brain Atlas • Cortical structures: • Segmented using surface-painting procedure • Sub-cortical structures • Automatic, level set based segmentation for caudate and ventricles • Manual SNAP segmentation Level set segmentation of the ventricles 2D and 3D views of Left Cortical Structures

41. Atlas-based Localization Atlas Image Subject REGISTER Spatial Transformation Anatomic Labels WARP OVERLAY Individualized Atlas Atlas-based SegmentationVariational Diffeomorphic Matching Chimp to Human Gee et al, JCAT, 1993; Avants and Gee, MedIA, NeuroImage, in press

42. Deterministic Tractography in the Presence of Noise • DT-MRI signal noisy near gray-white interface • Early termination of reconstructed pathwaysBehrens et al, 2003 • Cortex segmentation is propagated medially using a Voronoi partitioning scheme

43. Acknowledgements • This work was supported by the USPHS under NIH grants MH62100, NS044189, DA015886, NS045839, and a Whitaker graduate fellowship

46. Building a Brain Atlas Using Graph Theory P. Yushkevich, A. Dubb and J. Gee University of Pennsylvania April, 2004

47. Project Aims • To Develop a Brain Atlas • An MRI template for registration • A segmentation into anatomical structures • Desired Properties of the Atlas: • High spatial resolution • Large number of structures • Brodmann’s areas • Subcortical organs Atlas of Brodmann’s areas http://ist-socrates.berkeley.edu/~jmp/LO2/6.html

48. Talairach Atlas • Frequently used to map SPM locations to anatomical regions • Contains many structures, including Brodmann’s areas • Based on a post-mortem study of an elderly subject so there is no matching MRI • Low spatial resolution Talairach Brain Atlas Courtesy of www.brainvoyager.com

49. Our Approach • Build atlas for the widely used SPM ’96 template • Constructed by scanning a volunteer multiple times to reduce noise • Freely available from the BrainWeb Project • Segmentation into GM, WM and CSF is also freely available SPM’96 Template from BrainWeb

50. Initial Approach • Cortical structures: • Segmented using surface-painting procedure • Sub-cortical structures • Automatic, level set based segmentation for caudate and ventricles • Manual SNAP segmentation Level set segmentation of the ventricles 2D and 3D views of Left Cortical Structures