Analytic ODF Reconstruction and Validation in Q-Ball Imaging

1. Analytic ODF Reconstruction and Validation in Q-Ball Imaging Maxime Descoteaux1 Work done with E. Angelino2, S. Fitzgibbons2, R. Deriche1 1. Projet Odyssée, INRIA Sophia-Antipolis, France 2. Physics and Applied Mathematics, Harvard University, USA McGill University, Jan 18th 2006

2. Plan of the talk Introduction Background Analytic ODF reconstruction Results Discussion

3. Introduction Cerebral anatomy Basics of diffusion MRI

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

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

6. 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]

7. 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

8. Principal direction of DTI

9. 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]

10. Background High Angular Resolution Diffusion Imaging Q-Space Imaging Q-Ball Imaging …

11. High Angular Resolution Diffusion Imaging (HARDI) 162 points 642 points • N gradient directions • We want to recover fiber crossings Solution: Process all discrete noisy samplings on the sphere using high order formulations

12. High Order Reconstruction • We seek a spherical function that has maxima that agree with underlying fibers Diffusion profile Diffusion Orientation Distribution Function (ODF) Fiber distribution

13. Diffusion Orientation Distribution Function (ODF) • Method to reconstruct the ODF • Diffusion spectrum imaging (DSI) • Sample signal for many q-ball and many directions • Measured signal = FourierTransform[P] • Compute 3D inverse fourier transform -> P • Integrate the radial component of P -> ODF

14. 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] [Tuch; MRM04]

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

16. z = 1 z = 1000 J0(2z) [Tuch; MRM04] (WLOG, assume u is on the z-axis) Funk-Radon ~= ODF • Funk-Radon Transform • True ODF

17. My Contributions • The Funk-Radon can be solved ANALITICALLY • Spherical harmonics description of the signal • One step matrix multiplication • Validation against ground truth evidence • Rat phantom • Knowledge of brain anatomy • Validation and Comparison against Tuch reconstruction [collaboration with McGill]

18. Analytic ODF Reconstruction Spherical harmonic description Funk-Hecke Theorem

19. Sketch of the approach S in Q-space Physically meaningful spherical harmonic basis For l = 6, C = [c1, c2 , …, c28] Spherical harmonic description of S Analytic solution using Funk-Hecke formula ODF

20. 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. SPIE-MI 06]

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

22. 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,

23. 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

24. SH description of the signal • For any () S = [d1, d2, …, dN] For l = 6, C = [c1, c2 , …, c28]

25. Funk-Hecke Theorem Solve the Funk-Radon integral Delta sequence

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

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

28. 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 (if time, proof)

29. Funk-Hecke formula Delta sequence => Solving the FR integral

30. Final Analytic ODF expression (if time bigO analysis with Tuch’s ODF reconstruction)

31. Time Complexity • Input HARDI data |x|,|y|,|z|,N • Tuch ODF reconstruction: O(|x||y||z| N k) (8N) : interpolation point k := (8N) • Analytic ODF reconstruction O(|x||y||z| N R) R := SH elements in basis

32. Time Complexity Comparison • Tuch ODF reconstruction: • N = 90, k = 48 -> rat data set = 100 , k = 51 -> human brain = 321, k = 90 -> cat data set • Our ODF reconstruction: • Order = 4, 6, 8 -> m = 15, 28, 45 => Speed up factor of ~3

33. Validation and Results Synthetic dataBiological rat spinal chords phantom Human brain

34. Synthetic Data Experiment

35. Synthetic Data Experiment • Multi-Gaussian model for input signal • Exact ODF

36. Strong Agreement Multi-Gaussian model with SNR 35 Average difference between exact ODF and estimated ODF b-value

37. 55 crossing b = 3000 Field of Synthetic Data b = 1500 SNR 15 order 6 90 crossing

38. Real Data Experiment Biological phantom Human Brain

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

40. Tuch reconstruction vsAnalytic reconstruction Analytic ODFs Tuch ODFs Difference: 0.0356 +- 0.0145 Percentage difference: 3.60% +- 1.44% [INRIA-McGill]

41. Human Brain Analytic ODFs Tuch ODFs Difference: 0.0319 +- 0.0104 Percentage difference: 3.19% +- 1.04% [INRIA-McGill]

42. Genu of the corpus callosum - frontal gyrus fibers FA map + diffusion tensors ODFs

43. Corpus callosum - corona radiata - superior longitudinal FA map + diffusion tensors ODFs

44. Corona radiata diverging fibers - longitudinal fasciculus FA map + diffusion tensors ODFs

45. Discussion & Conclusion

46. Summary S in Q-space Physically meaningful spherical harmonic basis Spherical harmonic description of S Analytic solution using Funk-Hecke formula ODF Fiber directions

47. Advantages of our approach • Analytic ODF reconstruction • Discrete interpolation/integration is eliminated • Solution for all directions is obtained in a single step • Faster than Tuch’s numerical approach • Output is a spherical harmonic description which has powerful properties

48. Spherical harmonics properties • Can use funk-hecke formula to obtain analytic integrals of inner products • Funk-radon transform, deconvolution • Laplacian is very simple • Application to smoothing, regularization, sharpening • Inner product • Comparison between spherical functions

49. What’s next? • Tracking fibers! • Can it be done properly from the diffusionODF? • Can we obtain a transformation between the input signal and the fiber ODF using spherical harmonics

50. Thank you! Key references: • http://www-sop.inria.fr/odyssee/team/ Maxime.Descoteaux/index.en.html • Tuch D. Q-Ball Imaging, MRM 52, 2004 Thanks to: P. Savadjiev, J. Campbell, B. Pike, K. Siddiqi