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

Overview Registration nTensor orientation pattern matching – Non-rigid registration nDiffusion profile matching – Affine transformations Visualization nAnatomically labeled fiber tracts – Partitioning the corpus callosum

Orientation Pattern Matching Duda et al, SPIE Med Imag 2003

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 1  1 2  2  x y

Non-Rigid Registration of 2D Tensor Data Want to find a smooth continuous mapping, u(s), from I 2 to I 1 This is accomplished by minimizing a function consisting of three separate terms

Objective Function Eigenvalue (ellipsoid shape) difference Smoothness constraint

Objective Function Local neighborhood orientation-pattern difference

Local Neighborhood Orientation-Pattern Difference r1r1 s1s1 r1r1 s1s1 r2r2 s2s2 r2r2 s2s2

Gradient of Objective Function The following function is minimized:

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

Preliminary Registration Example A B B  A A  B

Diffusion Profile Matching Zhang, Yushkevich, Gee, ISBI 2004

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

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

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

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

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

Objective function TemplateSubject Affine Registration Specialized to diffusion tensors where

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

Affine Transform Parametrization Affine transformation with Parametrization of 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 Affine Parameters Polar Decomposition-based: with and

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

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 derivativeswithout derivatives Relative Error Distribution

Piecewise Affine, Non-rigid Extension 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 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)

Anatomically Labeled Fiber Tracts

Cortex-based Partitioning of the Corpus Callosum

Callosal Morphometry Witelson 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

“Modern” Callosal Morphometry Template Deformations Correspondences between callosa are obtained by registering the curve geometry at different scales 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 Boundary correspondence is interpolated in a geometrically correct way throughout the interior (or even exterior) of the curve Related work 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 , 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 , 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 , 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 , 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 , 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 , 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 , 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 , 2003.

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

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

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

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

Atlas-based Segmentation Variational Diffeomorphic Matching Chimp to Human Atlas Image Subject Anatomic Labels Individualized Atlas Atlas-based Localization Spatial Transformation REGISTER OVERLAYWARP Gee et al, JCAT, 1993; Avants and Gee, MedIA, NeuroImage, in press

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

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

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

Project Aims To Develop a Brain Atlas To Develop a Brain Atlas An MRI template for registration An MRI template for registration A segmentation into anatomical structures A segmentation into anatomical structures Desired Properties of the Atlas: Desired Properties of the Atlas: High spatial resolution High spatial resolution Large number of structures Large number of structures Brodmann’s areas Brodmann’s areas Subcortical organs Subcortical organs Atlas of Brodmann’s areas

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

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

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

Cortical Segmentation Expert marks a dense set of landmarks on the surface of the cerebral cortex Expert marks a dense set of landmarks on the surface of the cerebral cortex Landmarks are projected onto the white matter surface and connected, forming ribbons Landmarks are projected onto the white matter surface and connected, forming ribbons Ribbons are used as boundaries for level-set segmentation Ribbons are used as boundaries for level-set segmentation Grey matter with curves Curves projected on the white matter

Initial Approach: Limitations Curves projected on the white matter surface have discontinuities Curves projected on the white matter surface have discontinuities Currently breaks in the projection are connected by tracing the shortest path on the surface Currently breaks in the projection are connected by tracing the shortest path on the surface This results in ribbons that are unnatural and in suboptimal segmentation This results in ribbons that are unnatural and in suboptimal segmentation White matter projection discontinuity Ribbons formed by projectionExample of segmentation error caused by a inadequate ribbon (the ribbon is pink)

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

Current Approach: Fine Step Expert draws curves on the surface of a cortical region using Livewire-style user interface 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 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 Patch segmentation will be propagated onto the GM volume using level set/flow techniques BrainTracer tool used to mark the cortical surface

Summary Diffusion profile metric Diffusion profile reorientation Analytical derivative-based affine registration Piecewise-affine registration Thank You!

Towards Diffusion Profile Image Registration Hui Zhang, Paul A. Yushkevich and James C. Gee University of Pennsylvania ISBI’04 Funded in part by NIH grants MH62100, NS044189, DA015886, NS045839

Extra Slides

Motivation Diffusion Weighted Imaging Diffusion Profile Images Spatial Normalization among Populations New Insight into White Matter Structure

Reorientation: Rigid Case

Affine Registration Closed-form Objective function TemplateSubject Applied to diffusion tensors

Analytical Derivatives Some notations: Derivatives of rotation matrix Derivatives of deformation matrix Derivatives of translations

TemplateSubject Non-rigid Extension Piecewise affine registration Multi-resolution hierarchy Number of regions are Displacement fields recovered at each level are smoothed with B-Spline

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Our Contributions Methodology for diffusion profile registration metric reorientation affine parametrization scheme affine registration objective function with analytical derivatives Results applied to diffusion tensors affine registration piecewise affine registration

Diffusion Profile Reorientation The reorientation effect of rigid transformation Reorientation Operator A function of transformation’s Jacobian matrix Rigid transformation Applied to diffusion tensors

Affine Reorientation One possible solution like “Preservation of principle directions” Non-linear and not shape preserving Affine transformation is not an isometry and

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

Atlas Image Subject Anatomic Labels Individualized Atlas Atlas-based Localization Spatial Transformation REGISTER OVERLAYWARP Anatomic Atlas

Witelson partitioning approach to morphometry Witelson partitioning approach to morphometry Deformation based morphometry Deformation based morphometry Cite studies Cite studies New, anatomy-based partitioning approach New, anatomy-based partitioning approach How How Atlas Atlas Matching Matching Propagating labels---see Behrens? Propagating labels---see Behrens?