Processing HARDI Data to Recover Crossing Fibers

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Processing HARDI Data to Recover Crossing Fibers
Maxime Descoteaux PhD student Advisor Rachid
Deriche Odyssée Laboratory, INRIA/ENPC/ENS, INRI
A Sophia-Antipolis, France
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Plan of the talk

• Introduction of HARDI data
• Spherical Harmonics Estimation of HARDI
• Q-Ball Imaging and ODF Estimation
• Multi-Modal Fiber Tracking

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Brain white matter connections
Short and long association fibers in the right
hemisphere (Williams-etal97)
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Cerebral Anatomy
Radiations of the corpus callosum
(Williams-etal97)
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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
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Classical DTI model
DTI --gt

• 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 profile qTDq
Diffusion MRI signal S(q)
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Principal direction of DTI
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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

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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 2002
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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

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Sketch of the approach
Data on the sphere
For l 6, C c1, c2 , , c28
Spherical harmonic description of data
ODF
ODF
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Spherical Harmonic Estimation of the Signal

• Description of discrete data on the sphere
• Physically meaningful spherical harmonic basis
• Regularization of the coefficients

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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 562006
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Spherical Harmonics (SH) coefficients

• In matrix form, S CB
• 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

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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,

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Minimization ?-regularization

• Minimize
• (CB - S)T(CB - S) ?CTLC
• gt
• 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
• gt higher order terms only included when needed

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Effect of regularization
Descoteaux et al., MRM 06
? 0
With Laplace-Beltrami regularization
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Fast Analytical ODF Estimation

• Q-Ball Imaging
• Funk-Hecke Theorem
• Fiber detection

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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

ODF -gt
Tuch MRM04
ODF
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Illustration of the Funk-Radon Transform (FRT)
Diffusion Signal
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Funk-Hecke Theorem
Funk 1916, Hecke 1918
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Recalling Funk-Radon integral
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Solving the FR integralTrick using a delta
sequence
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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
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Biological phantom
Campbell et al. NeuroImage 05
T1-weigthed
Diffusion tensors
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Corpus callosum - corona radiata - superior
longitudinal fasciculus
FA map diffusion tensors
ODF maxima
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Corona radiata diverging fibers - superior
longitudinal fasciculus
FA map diffusion tensors
ODF maxima
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Multi-Modal Fiber Tracking

• Extract ODF maxima
• Extension to streamline FACT

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Streamline Tracking

• FACT Fiber Assignment by Continuous Tracking
• Follow principal eigenvector of diffusion tensor
• Stop if FA lt thresh and if curving angle gt ?
• ?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
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Limitations of DTI-FACT
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DTI-FACT Tracking
start-gt
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DTI-FACT ODF maxima
start-gt
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Principal ODF FACT Tracking
start-gt
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Multi-Modal ODF FACT
start-gt
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DTI-FACT Tracking
start-gt
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Principal ODF FACT Tracking
start-gt
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Multi-Modal ODF FACT
start-gt
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DTI-FACT Tracking
start-gt
start-gt
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DTI-FACT Tracking
start-gt
start-gt
Very low FA threshold
lt-start
start-gt
Lower FA thresh
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Principal ODF FACT Tracking
start-gt
start-gt
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Multi-modal FACT Tracking
start-gt
start-gt
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Summary
Signal S on the sphere
Spherical harmonic description of S
Multi-Modal tracking
ODF
Fiber directions
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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

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Contributions advantages

• 4) Tracking using ODF maxima Generalized FACT
  algorithm
• gt 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

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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
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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

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Thank you!

• Thanks to collaborators
• C. Lenglet, P. Savadjiev, J. Campbell, B. Pike,
  K. Siddiqi, E. Angelino,
• S. Fitzgibbons, 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.Desco
  teaux/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

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Principal direction of DTI
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Spherical Harmonics

• SH
• SH PDE
• Real
• Modified basis

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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

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?n is a delta sequence
1)
2)
gt
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Nice trick!
3)
gt
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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

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Funk-Radon ODF

• Funk-Radon Transform
• True ODF

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Synthetic Data Experiment

• Multi-Gaussian model for input signal
• Exact ODF

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Field of Synthetic Data
b 1500 SNR 15 order 6
90? crossing
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ODF evaluation
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Tuch reconstruction vsAnalytic reconstruction
Analytic ODFs
Tuch ODFs
Difference 0.0356 - 0.0145 Percentage
difference 3.60 - 1.44
INRIA-McGill
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Human Brain
Analytic ODFs
Tuch ODFs
Difference 0.0319 - 0.0104 Percentage
difference 3.19 - 1.04
INRIA-McGill
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Time Complexity

• Input HARDI data x,y,z,N
• Tuch ODF reconstruction
• O(xyz N k)
• ?(8?N) interpolation point
• k ?(8?N)
• Analytic ODF reconstruction
• O(xyz N R)
• R SH elements in basis

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Time Complexity Comparison

• Tuch ODF reconstruction
• N 90, k 48 -gt rat data set
• 100 , k 51 -gt human brain
• 321, k 90 -gt cat data set
• Our ODF reconstruction
• Order 4, 6, 8 -gt m 15, 28, 45

gt Speed up factor of 3
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Time complexity experiment

• Tuch -gt O(XYZNk)
• Our analytic QBI -gt O(XYZNR)
• Factor 15 speed up

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Estimation of the ADC

• Characterize multi-fiber diffusion
• High order anisotropy measures

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Apparent Diffusion Coefficient
ADC profile D(g) gTDg
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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

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High order diffusion tensor (HODT) generalization

• 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
  

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Tensor generalization of ADC

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

General tensor
Independent elements Dk of the tensor
Ozarslan et al., mrm 2003
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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
M transformation
C (BTB ?L)-1BTX
D M-1C
Descoteaux et al. MRM 562006
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3 synthetic fiber crossing
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ADC limitations for tracking

• 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 gt ODF
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High Order Descriptions

• Seek to characterize multiple fiber diffusion
• Apparent Diffusion Coefficient (ADC)
• Orientation Distribution Function (ODF)

ADC profile
Diffusion ODF
Fiber distribution