1. ADC and ODF estimation from HARDI
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
Presente-toi
Parle des groupes
Collaboration et connait bien les gens du MNI et BIC, collaboration avec les vaisseaux sanguins
Overview: Specific and technical talk, it can give a good overview of diffuion MRI and challenges around fiber tractography.
More importantly, all those working with discrete data on spheres, this talk can good mathetimatical tools and ideas to treat those data.
Last time I had focused on the SH formulation, this time I’ll will focus more on the signal itself.
Max Planck Institute, March 28th 2006

2. Plan of the talk Introduction Spherical Harmonics Formulation
Applications:
1) ADC Estimation 2) ODF Estimation
Discussion
Intro -> Basics of dMRI
Background -> HARDI, DSI, QBI
Method -> SH formulation, Funk-Hecke theorem, delta sequence, nice mathematics
Results -> Synthetic data, phantom, humn brain
conlusion

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

4. 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
Tres important pour la tractographie….

5. High Order Descriptions
Seek to characterize multiple fiber diffusion
Apparent Diffusion Coefficient (ADC)
Orientation Distribution Function (ODF)
-Diffusion profile is in the wrong space. Still in signal space where the maxima of the diffusion profile do not agree with true fibers
-Need a transformation from Signal to ODF q-space to real space.
Input: 3D dataset where at each voxel you have 3D data on the unit sphere.
Output: spherical function, ODF in real space -> extract fiber directions
Fiber distribution
ADC profile
Diffusion ODF

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

7. Spherical harmonics Description of discrete data on the sphere
Regularization of the coefficients

8. Spherical harmonics formulation
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
Analogy to the Fourier series
[Descoteaux et al. SPIE-MI 06]

9. 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, 

10. Minimization & -regularization
Minimize
(CB - S)T(CB - S) + CTLC
=>
C = (BTB + L)-1 BTS
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

11. Characterize multi-fiber diffusion High order anisotropy measures
Estimation of the ADC
Characterize multi-fiber diffusion
High order anisotropy measures

12. Apparent diffusion coefficient
Diffusion MRI signal : S(g)
Standard narrow-pulse assumption ->
Blue failble, rouge fort
2 facon de visualizer, sur la sphere ou plus d’info visuellement de deformer la sphere proportionelement aux valeurs
Gauche, signal d’attenuatation, faible dans la direction de la fibre
Droite: coefficient apparent de diffusion -> fort le long de la fibre
Example for 1 fiber. Maxima of ADC profile does not agree with fiber orientation in general
Explain color map
ADC profile : D(g) = gTDg

13. In the HARDI literature…
2 class of high order ADC fitting algorithms:
Spherical harmonic (SH) series
[Frank 2002, Alexander et al 2002, Chen et al 2004]
High order diffusion tensor (HODT)
[Ozarslan et al 2003, Liu et al 2005]

14. from linear-regression
Summary of algorithm
Spherical
Harmonic (SH)
Series
High Order
Diffusion Tensor
(HODT)
HODT D
from linear-regression
Modified SH basis Yj
Least-squares with
-regularization
Fast since B and M matrices need only be computed once
eliminates high order terms modeling noise while leaving those necessary to describe the underlying diffusion
Validate on synthetic, phantom and real data
Add reference in the diagram M
transformation
C = (BTB + L)-1BTX
D = M-1C
[Descoteaux et al. SPIE-MI 06]

15. 3 synthetic fiber crossing

16. Splenium, x-ssing between cc, slf, cst

17. Limitations of the ADC Maxima do not agree with underlying fibers
ADC is in signal space (q-space)
HARDI
ADC profiles
[Campbell et al.,
McGill University, Canada]
Need a function that is in real space with maxima that agree with fibers => ODF

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

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

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

21. Funk-Hecke Theorem [Funk 1916, Hecke 1918]
Funk-Hecke thm relates the inner product of any spherical harmonic and the projection onto the unit sphere of any function f conitnuous on [-1,1]
[Funk 1916, Hecke 1918]

22. Recalling Funk-Radon integral
Funk-Hecke ! Problem: Delta function is discontinuous at 0 !
FR in direction x,
-express signal has a SH series
-get the coefficients out

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

24. Final Analytical ODF expression in SH coefficients
[Descoteaux et al. ISBI 06]

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

26. Corpus callosum - corona radiata - superior longitudinal
Coronal slice
FA map + diffusion tensors
ODFs

27. Corona radiata diverging fibers - longitudinal fasciculus
Sagittal
Diverging fibers
FA map + diffusion tensors
ODFs

28. Contributions & advantages
SH and HODT description of the ADC
Application to anisotropy measures
Analytical ODF reconstruction
Discrete interpolation/integration is eliminated
Solution for all directions in a single step
Faster than Tuch’s numerical approach by a factor 15
Spherical harmonic description has powerful properties
Smoothing with Laplace-Beltrami, inner products, integrals on the sphere solved with Funk-Hecke

29. Perspectives
Tracking and segmentation of fibers using multiple maxima at every voxel
Consider the full diffusion ODF in the tracking and segmentation

30. Thank you! Key references:
Maxime.Descoteaux/index.en.html
-Ozarslan et al. Generalized tensor imaging
and analytical relationships between diffusion tensor
and HARDI, MRM 50, 2003
-Tuch D. Q-Ball Imaging, MRM 52, 2004

32. Classical DTI model Diffusion MRI signal : S(q)
-->
Brownian motion P of water molecules can be described by a Gaussian diffusion
process characterized by rank-2 tensor D (3x3 symmetric positive definite)
Narrow pulse approx. Diffusion is neglible during the gradient pulse duration. Brownian motion ~ probability averaged over a voxel that a spin starting point in the voxel will have displaced by some amount during the time of the gradient pulse
Diffusion MRI signal : S(q)
Diffusion profile : qTDq

33. Spherical Harmonics (SH) coefficients
In matrix form, S = C*B
S : discrete HARDI data 1 x N
C : SH coefficients 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]
Brechbuhel: 3D shape description of closed surfaces

34. Introduction Limitations of Diffusion Tensor Imaging (DTI)
High Angular Resolution Diffusion Imaging (HARDI)

35. Our Contributions Spherical harmonics (SH) description of the data
Impose a regularization criterion on the solution
Application to ADC estimation
Direct relation between SH coefficients and independent elements of the high order tensor (HOT)
Application to ODF reconstrution
ODF can be reconstructed ANALITYCALLY
Fast: One step matrix multiplication
Validation on synthetic data, rat biological phantom, knowledge of brain anatomy

36. High order diffusion tensor (HODT) generalization
Rank l = x3
D = [ Dxx Dyy Dzz Dxy Dxz Dyz ]
Rank l = x3x3x3
D = [ Dxxxx Dyyyy Dzzzz Dxxxy Dxxxz Dyzzz Dyyyz Dzzzx Dzzzy Dxyyy Dxzzz Dzyyy Dxxyy Dxxzz Dyyzz ]

37. Tensor generalization of ADC
Generalization of the ADC,
rank-2 D(g) = gTDg
rank-l
General tensor
Pour ordre superieur, on peut simplement exprimer l’ADC comme un produit matriciel
3x3 add ind.
Independent elements
Dk of the tensor
[Ozarslan et al., mrm 2003]

38. Left: corpus callosum in the splenium
Right: corticospinal tract crossing with the cc
Center for Magnetic Resonance Research, University of Minnesota, on a 3 Tesla Siemens MagnetomTrio whole-body scanner. We acquired 3 repetitions per direction, each with b = 1000 s/mm2 , TR = 5100 s and TE = 109 ms. The three measurements are averaged by default by the scanner to produce 81 individual measurements to process. The voxel size was 3 mm3 cube and there were 24, 64x64 slices.

39. 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

40. n is a delta sequence 1) 2) =>
Using definte integral formula in Schaum’s outline 2)
=>

41. Nice trick! 3)
=>

42. Spherical Harmonics SH
SH PDE
Real
Modified basis

43. Funk-Hecke Theorem Key Observation:
Any continuous function f on [-1,1] can be extended to a continous function on the unit sphere g(x,u) = f(xTu), where x, u are unit vectors
Funk-Hecke thm relates the inner product of any spherical harmonic and the projection onto the unit sphere of any function f conitnuous on [-1,1]
<f,g> = Int_sphere f*g dsphere

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

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

46. 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
Surface area: 4pir^2
Interpolation angle every sqrt(4pi / N) rad
sqrt(8piN) point on the great circle
->Getting SH coefficients transform O(Nm^2), inversion O(m^3)
Multiplication N directions by mxm matrix

47. 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
In practice, we find that the order 6 approximation gives best result. Hence, at least, a factor of 2 speed up. Does not depend on input N.
In our experiments, on my lap top, the speed up is more important. At least a factor of 10.
=> Speed up factor of ~3

48. Time complexity experiment
Tuch -> O(XYZNk)
Our analytic QBI -> O(XYZNR)
Factor ~15 speed up

49. ODF evaluation

50. Tuch reconstruction vs Analytic reconstruction
B = 3000
90 directions
2.8 cubic resolution
Analytic ODFs
Tuch ODFs
Difference:
Percentage difference: 3.60% %
[INRIA-McGill]

51. Human Brain Analytic ODFs Tuch ODFs Difference: 0.0319 +- 0.0104
crossings between the cortical spinal tract (cst) and corpus callosum (in the plane) and the cst and longitudinal superior fibers
(coming out of the plane).
Data was acquired: 99 diffusion encoding directions, 2 mm isotropic voxel size, and b = 3000 s/mm 2 (q = 0.35 um −1 ).
Tuch ODFs
Analytic ODFs
Difference:
Percentage difference: 3.19% %
[INRIA-McGill]

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

53. Limitations of classical DTI
rank-2 tensor
HARDI
ODF reconstruction
DTI classique trouve la direction au milieu des deux fibres (Gaussienne est la moyenne des 2). Montre les limites du DTI. Tandis que avec un TOS et les données on peut décrire et repérer plusieurs directions

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

55. High order anisotropy measures
Summary
Signal S on the sphere
Physically meaningful
spherical harmonic
basis
Spherical harmonic
description of S
High order anisotropy measures
ADC
Analytic solution using
Funk-Hecke formula
ODF
Fiber directions