Processing HARDI Data to Recover Crossing Fibers

Maxime DescoteauxPhD student

Advisor: Rachid Deriche

Odyssée Laboratory, INRIA/ENPC/ENS,INRIA Sophia-Antipolis, France

Plan of the talk

Introduction of HARDI data

Spherical Harmonics Estimation of HARDI

Q-Ball Imaging and ODF Estimation

Multi-Modal Fiber Tracking

Short and long association fibers in the right hemisphere

([Williams-etal97])

Brain white matter connections

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

Cerebral Anatomy

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

Diffusion profile : qTDqDiffusion 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)

DTI-->

Principal direction of DTI

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

• DTI fails in the presence of many principal directions of different fiber bundles within the same voxel

• Non-Gaussian diffusion process

Limitation of classical DTI

[Poupon, PhD thesis]True diffusion

profileDTI diffusion

profile

High Angular Resolution Diffusion Imaging (HARDI)

• N gradient directions • We want to recover fiber crossings

SolutionSolution: Process all discrete noisy samplings on the sphere using high order formulations

162 points 252 points

[Wedeen, Tuch et al 1999]

Our Contributions

1. New regularized spherical harmonic estimation of the HARDI signal

2. New approach for fast and analytical ODF reconstruction in Q-Ball Imaging

3. New multi-modal fiber tracking algorithm

Sketch of the approachData on the

sphere

Spherical harmonic

description of data

ODF

For l = 6,

C = [c1, c2 , …, c28]

ODF

Spherical Harmonic Estimation of the Signal

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*BS : discrete HARDI data 1 x NC : SH coefficients 1 x R = (1/2)(order + 1)(order + 2)

B : discrete SH, YjR 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)-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

Effect of regularization

= 0

With Laplace-Beltrami regularization

[Descoteaux et al., MRM 06]

Fast Analytical ODF Estimation

Q-Ball Imaging

Funk-Hecke Theorem

Fiber detection

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

ODF ->

Illustration of the Funk-Radon Transform (FRT)

Diffusion Signal

FRT->

ODF

Funk-Hecke Theorem

[Funk 1916, Hecke 1918]

Recalling Funk-Radon integral

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

Funk-Hecke formula

Solving the FR integral:Trick using a delta sequence

=>

Delta sequence

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

[Descoteaux et al. ISBI 06 & HBM 06]

Final Analytical ODF expression in SH coefficients

Biological phantom

T1-weigthed Diffusion tensors

[Campbell et al.NeuroImage 05]

Corpus callosum - corona radiata - superior longitudinal fasciculus

FA map + diffusion tensors ODF + maxima

Multi-Modal Fiber Tracking

Extract ODF maxima

Extension to streamline FACT

Streamline Tracking

• FACT: FFiber AAssignment by CContinuous TTracking• 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, 1999Conturo et al, 1999,Basser et al 2000]

Limitations of DTI-FACT

Classical DTIPrincipal tensor direction

HARDIODF maxima

DTI-FACT Tracking

start->

DTI-FACT + ODF maxima

start->

Principal ODF FACT Tracking

start->

Multi-Modal ODF FACT

start->

DTI-FACT Tracking

start->

Principal ODF FACT Tracking

start->

Multi-Modal ODF FACT

start->

DTI-FACT Tracking

start->

start->

DTI-FACT Tracking

start->

<-start

Lower FA thresh

start->

start->

Very low FAthreshold

Principal ODF FACT Tracking

start->

start->

Multi-modal FACT Tracking

start->

start->

SummarySignal S on the

sphere

Spherical harmonic

description of S

ODF Fiber directionsMulti-Modal

tracking

Contributions & advantages

1) Regularized spherical harmonic (SH) description of the signal

2) Analytical ODF reconstruction• Solution for all directions in a single step• Faster than classical QBI by a factor 15

3) SH description has powerful properties• Easy solution to : Laplace-Beltrami smoothing, inner

products, integrals on the sphere• Application for sharpening, deconvolution, etc…

Contributions & advantages

4) Tracking using ODF maxima = Generalized FACT algorithm

=> Overcomes limitations of FACT from DTI1. Principal ODF direction

• Does not follow wrong directions in regions of crossing

2. Multi-modal ODF FACT• Can deal with fanning, branching and crossing

fibers

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]

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

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

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

Principal direction of DTI

Spherical Harmonics

• SH

• SH PDE

• Real• Modified basis

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

n is a delta sequence

=>

1)

2)

3)

Nice trick!

=>

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]

• Funk-Radon Transform

• True ODF

Funk-Radon ~= ODF

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

J0(2z)

z = 1 z = 1000

[Tuch; MRM04]

Synthetic Data Experiment

• Multi-Gaussian model for input signal

• Exact ODF

Field of Synthetic Data

90 crossing

b = 1500SNR 15order 6

55 crossingb = 3000

ODF evaluation

Tuch reconstruction vsAnalytic reconstruction

Tuch ODFs Analytic ODFs

Difference: 0.0356 +- 0.0145Percentage difference: 3.60% +- 1.44%

[INRIA-McGill]

Human Brain

Tuch ODFs Analytic ODFs

Difference: 0.0319 +- 0.0104Percentage difference: 3.19% +- 1.04%

[INRIA-McGill]

Time Complexity

• 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

• 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

Time complexity experiment

• Tuch -> O(XYZNk)• Our analytic QBI -> O(XYZNR)

• Factor ~15 speed up

Estimation of the ADC

Characterize multi-fiber diffusion

High order anisotropy measuresanisotropy measures

Apparent Diffusion Coefficient

ADC profile : D(g) = gTDgDiffusion MRI signal : S(g)

In the HARDI literature…

2 class of high order ADC fitting algorithms:

1) Spherical harmonic (SH) series[Frank 2002, Alexander et al 2002, Chen et al 2004]

2) High order diffusion tensor (HODT)[Ozarslan et al 2003, Liu et al 2005]

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 ]

Tensor generalization of ADC

• Generalization of the ADC,

rank-2 D(g) = gTDg

rank-l

Independent elementsDk of the tensor

General tensor

[Ozarslan et al., mrm 2003]

Summary of algorithm

High OrderDiffusion Tensor

(HODT)

SphericalHarmonic (SH)

Series

Modified SH basis Yj

C = (BTB + L)-1BTX

Least-squares with-regularization

D = M-1C

M transformation

HODT Dfrom linear-regression

[Descoteaux et al. MRM 56:2006]

3 synthetic fiber crossing