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Processing HARDI Data to Recover Crossing FibersPowerPoint Presentation

Processing HARDI Data to Recover Crossing Fibers

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### Processing HARDI Data to Recover Crossing Fibers

### Plan of the talk

### Spherical Harmonic Estimation of the Signal

Maxime Descoteaux

PhD student

Advisor: Rachid Deriche

Odyssée Laboratory, INRIA/ENPC/ENS,

INRIA Sophia-Antipolis, France

Introduction of HARDI data

Spherical Harmonics Estimation of HARDI

Q-Ball Imaging and ODF Estimation

Multi-Modal Fiber Tracking

Brain white matter connections

Short and long association fibers in the right hemisphere

([Williams-etal97])

Cerebral Anatomy

Radiations of the corpus callosum ([Williams-etal97])

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]

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

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]

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]

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

Sketch of the approach

Data on the sphere

For l = 6,

C = [c1, c2 , …, c28]

Spherical harmonic

description of data

ODF

ODF

Description of discrete data on the sphere

Physically meaningful spherical harmonic basis

Regularization of the coefficients

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]

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]

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,

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

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

Funk-Hecke Theorem single ball

[Funk 1916, Hecke 1918]

Funk-Hecke ! single ball Problem: Delta function is discontinuous at 0 !

Recalling Funk-Radon integralFunk-Hecke formula single ball

Delta sequence

=>

Solving the FR integral:Trick using a delta sequenceFinal Analytical ODF expression in SH coefficients single ball

- Fast: speed-up factor of 15 with classical QBI
- Validated against ground truth and classical QBI

[Descoteaux et al. ISBI 06 & HBM 06]

Corpus callosum - corona radiata - superior longitudinal fasciculus

FA map + diffusion tensors

ODF + maxima

Streamline Tracking fasciculus

- 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]

DTI-FACT Tracking fasciculus

start->

DTI-FACT + ODF maxima fasciculus

start->

Principal ODF FACT Tracking fasciculus

start->

Multi-Modal ODF FACT fasciculus

start->

DTI-FACT Tracking fasciculus

start->

Principal ODF FACT Tracking fasciculus

start->

Multi-Modal ODF FACT fasciculus

start->

Summary fasciculus

Signal S on the sphere

Spherical harmonic

description of S

Multi-Modal tracking

ODF

Fiber directions

Contributions & advantages fasciculus

- 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…

Contributions & advantages fasciculus

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

Perspectives fasciculus

- 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]

BrainVISA/Anatomist fasciculus

- 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

Thank you! fasciculus

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

Principal direction of DTI fasciculus

Spherical Harmonics fasciculus

- SH
- SH PDE
- Real
- Modified basis

Trick to solve the FR integral fasciculus

- Use a delta sequence n approximation of the delta function in the integral
- Many candidates: Gaussian of decreasing variance

- Important property

Funk-Hecke Theorem fasciculus

- 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]

z = 1 fasciculus

z = 1000

J0(2z)

[Tuch; MRM04]

(WLOG, assume u is on the z-axis)

Funk-Radon ~= ODF- Funk-Radon Transform
- True ODF

Synthetic Data Experiment fasciculus

- Multi-Gaussian model for input signal
- Exact ODF

ODF evaluation fasciculus

Tuch reconstruction vs fasciculusAnalytic reconstruction

Analytic ODFs

Tuch ODFs

Difference: 0.0356 +- 0.0145

Percentage difference: 3.60% +- 1.44%

[INRIA-McGill]

Human Brain fasciculus

Analytic ODFs

Tuch ODFs

Difference: 0.0319 +- 0.0104

Percentage difference: 3.19% +- 1.04%

[INRIA-McGill]

Time Complexity fasciculus

- Input HARDI data |x|,|y|,|z|,N
- Tuch ODF reconstruction:
O(|x||y||z| N k)

(8N) : interpolation point

k := (8N)

- Analytic ODF reconstruction
O(|x||y||z| N R)

R := SH elements in basis

Time Complexity Comparison fasciculus

- Tuch ODF reconstruction:
- N = 90, k = 48 -> rat data set
= 100 , k = 51 -> human brain

= 321, k = 90 -> cat data set

- N = 90, k = 48 -> rat data set
- Our ODF reconstruction:
- Order = 4, 6, 8 -> m = 15, 28, 45

=> Speed up factor of ~3

Time complexity experiment fasciculus

- Tuch -> O(XYZNk)
- Our analytic QBI -> O(XYZNR)
- Factor ~15 speed up

In the HARDI literature… fasciculus

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]

High order diffusion tensor (HODT) generalization fasciculus

Rank l = 2 3x3

D = [ Dxx Dyy Dzz Dxy Dxz Dyz ]

Rank l = 4 3x3x3x3

D = [ Dxxxx Dyyyy Dzzzz Dxxxy Dxxxz Dyzzz Dyyyz Dzzzx Dzzzy Dxyyy Dxzzz Dzyyy Dxxyy Dxxzz Dyyzz ]

Tensor generalization of ADC fasciculus

- Generalization of the ADC,
rank-2 D(g) = gTDg

rank-l

General tensor

Independent elements

Dk of the tensor

[Ozarslan et al., mrm 2003]

Summary of algorithm fasciculus

Spherical

Harmonic (SH)

Series

High Order

Diffusion Tensor

(HODT)

HODT D

from linear-regression

Modified SH basis Yj

Least-squares with

-regularization

M

transformation

C = (BTB + L)-1BTX

D = M-1C

[Descoteaux et al. MRM 56:2006]

3 synthetic fiber crossing fasciculus

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