Processing HARDI Data to Recover Crossing Fibers

1. Processing HARDI Data to Recover Crossing Fibers Maxime Descoteaux PhD student Advisor: Rachid Deriche Odyssée Laboratory, INRIA/ENPC/ENS, INRIA Sophia-Antipolis, France

2. Plan of the talk Introduction of HARDI data Spherical Harmonics Estimation of HARDI Q-Ball Imaging and ODF Estimation Multi-Modal Fiber Tracking

3. Brain white matter connections Short and long association fibers in the right hemisphere ([Williams-etal97])

4. Cerebral Anatomy Radiations of the corpus callosum ([Williams-etal97])

5. Diffusion MRI: recalling the basics • Brownian motion or average PDF of water molecules is along white matter fibers • Signal attenuation proportional to average diffusion in a voxel [Poupon, PhD thesis]

6. Classical DTI model DTI --> • Brownian motion P of water molecules can be described by a Gaussian diffusion process characterized by rank-2 tensor D (3x3 symmetric positive definite) Diffusion MRI signal : S(q) Diffusion profile : qTDq

7. Principal direction of DTI

8. Limitation of classical DTI • DTI fails in the presence of many principal directions of different fiber bundles within the same voxel • Non-Gaussian diffusion process True diffusion profile DTI diffusion profile [Poupon, PhD thesis]

9. High Angular Resolution Diffusion Imaging (HARDI) 162 points 252 points • N gradient directions • We want to recover fiber crossings Solution: Process all discrete noisy samplings on the sphere using high order formulations [Wedeen, Tuch et al 1999]

10. Our Contributions • New regularized spherical harmonic estimation of the HARDI signal • New approach for fast and analytical ODF reconstruction in Q-Ball Imaging • New multi-modal fiber tracking algorithm

11. Sketch of the approach Data on the sphere For l = 6, C = [c1, c2 , …, c28] Spherical harmonic description of data ODF ODF

12. Spherical Harmonic Estimation of the Signal Description of discrete data on the sphere Physically meaningful spherical harmonic basis Regularization of the coefficients

13. Spherical harmonicsformulation • Orthonormal basis for complex functions on the sphere • Symmetric when order l is even • We define a real and symmetric modified basis Yj such that the signal [Descoteaux et al. MRM 56:2006]

14. Spherical Harmonics (SH) coefficients • In matrix form, S = C*B S : discrete HARDI data 1 x N C : SH coefficients 1 x R = (1/2)(order + 1)(order + 2) B : discrete SH, Yj(R x N (N diffusion gradients and R SH basis elements) • Solve with least-square C = (BTB)-1BTS [Brechbuhel-Gerig et al. 94]

15. Regularization with the Laplace-Beltrami ∆b • Squared error between spherical function F and its smooth version on the sphere ∆bF • SH obey the PDE • We have,

16. Minimization & -regularization • Minimize (CB - S)T(CB - S) + CTLC => C = (BTB + L)-1BTS • Find best  with L-curve method • Intuitively,  is a penalty for having higher order terms in the modified SH series => higher order terms only included when needed

17. Effect of regularization [Descoteaux et al., MRM 06]  = 0 With Laplace-Beltrami regularization

18. Fast Analytical ODF Estimation Q-Ball Imaging Funk-Hecke Theorem Fiber detection

19. ODF can be computed directly from the HARDI signal over a single ball Integral over the perpendicular equator = Funk-Radon Transform Q-Ball Imaging (QBI) [Tuch; MRM04] ODF -> [Tuch; MRM04] ~= ODF

20. FRT -> ODF Illustration of the Funk-Radon Transform (FRT) Diffusion Signal

21. Funk-Hecke Theorem [Funk 1916, Hecke 1918]

22. Funk-Hecke ! Problem: Delta function is discontinuous at 0 ! Recalling Funk-Radon integral

23. Funk-Hecke formula Delta sequence => Solving the FR integral:Trick using a delta sequence

24. Final Analytical ODF expression in SH coefficients • Fast: speed-up factor of 15 with classical QBI • Validated against ground truth and classical QBI  [Descoteaux et al. ISBI 06 & HBM 06]

25. Biological phantom [Campbell et al. NeuroImage 05] T1-weigthed Diffusion tensors

26. Corpus callosum - corona radiata - superior longitudinal fasciculus FA map + diffusion tensors ODF + maxima

27. Multi-Modal Fiber Tracking Extract ODF maxima Extension to streamline FACT

28. Streamline Tracking • FACT: Fiber Assignment by Continuous Tracking • Follow principal eigenvector of diffusion tensor • Stop if FA < thresh and if curving angle >  typically thresh = 0.15 and = 45 degrees) • Limited and incorrect in regions of fiber crossing • Used in many clinical applications [Mori et al, 1999 Conturo et al, 1999, Basser et al 2000]

29. Classical DTI Principal tensor direction HARDI ODF maxima Limitations of DTI-FACT

30. DTI-FACT Tracking start->

31. DTI-FACT + ODF maxima start->

32. Principal ODF FACT Tracking start->

33. Multi-Modal ODF FACT start->

34. DTI-FACT Tracking start->

35. Principal ODF FACT Tracking start->

36. Multi-Modal ODF FACT start->

37. DTI-FACT Tracking start-> start->

38. DTI-FACT Tracking start-> start-> Very low FA threshold <-start start-> Lower FA thresh

39. Principal ODF FACT Tracking start-> start->

40. Multi-modal FACT Tracking start-> start->

41. Summary Signal S on the sphere Spherical harmonic description of S Multi-Modal tracking ODF Fiber directions

42. Contributions & advantages • Regularized spherical harmonic (SH) description of the signal • Analytical ODF reconstruction • Solution for all directions in a single step • Faster than classical QBI by a factor 15 • SH description has powerful properties • Easy solution to : Laplace-Beltrami smoothing, inner products, integrals on the sphere • Application for sharpening, deconvolution, etc…

43. Contributions & advantages 4) Tracking using ODF maxima = Generalized FACT algorithm => Overcomes limitations of FACT from DTI • Principal ODF direction • Does not follow wrong directions in regions of crossing • Multi-modal ODF FACT • Can deal with fanning, branching and crossing fibers

44. Perspectives • Multi-modal tracking in the human brain • Tracking with geometrical information from locally supporting neighborhoods • Local curvature and torsion information • Better label sub-voxel configurations like bottleneck, fanning, merging, branching, crossing • Consider the full diffusion ODF in the tracking and segmentation • Probabilistic tracking from full ODF [Savadjiev & Siddiqi et al. MedIA 06], [Campbell & Siddiqi et al. ISBI 06]

45. BrainVISA/Anatomist • Odyssée Tools Available • ODF Estimation, GFA Estimation • Odyssée Visualization • + more ODF and DTI applications… • http://brainvisa.info/ • Used by: • CMRR, University of Minnesota, USA • Hopital Pitié-Salpétrière, Paris

46. Thank you! Thanks to collaborators: C. Lenglet, M. La Gorce, E. Angelino, S. Fitzgibbons, P. Savadjiev, J. Campbell, B. Pike, K. Siddiqi, A. Andanwer Key references: -Descoteaux et al, ADC Estimation and Applications, MRM 56, 2006.-Descoteaux et al, A Fast and Robust ODF Estimation Algorithm in Q-Ball Imaging, ISBI 2006. -http://www-sop.inria.fr/odyssee/team/Maxime.Descoteaux/index.en.html -Ozarslan et al., Generalized tensor imaging and analytical relationships between diffusion tensor and HARDI, MRM 2003. -Tuch, Q-Ball Imaging, MRM 52, 2004

47. Principal direction of DTI

49. Spherical Harmonics • SH • SH PDE • Real • Modified basis

50. Trick to solve the FR integral • Use a delta sequence n approximation of the delta function  in the integral • Many candidates: Gaussian of decreasing variance • Important property