Processing HARDI Data to Recover Crossing Fibers Maxime Descoteaux PhD 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 : q T DqDiffusion 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

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 profile DTI diffusion profile

High Angular Resolution Diffusion Imaging (HARDI) N gradient directions We want to recover fiber crossings Solution Solution: Process all discrete noisy samplings on the sphere using high order formulations 162 points252 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 approach Data on the sphere Spherical harmonic description of data ODF For l = 6, C = [c 1, c 2, …, c 28 ] ODF

Spherical Harmonic Estimation of the Signal Description of discrete data on the sphere Physically meaningful spherical harmonic basis Regularization of the coefficients

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 Y j 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, Y j  R x N (N diffusion gradients and R SH basis elements) Solve with least-square C = (B T B) -1 B T S [Brechbuhel-Gerig et al. 94]

Regularization with the Laplace-Beltrami ∆ b Squared error between spherical function F and its smooth version on the sphere ∆ b F SH obey the PDE We have,

Minimization & -regularization Minimize (CB - S) T (CB - S) + C T LC => C = (B T B + L) -1 B T S 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-weigthedDiffusion tensors [Campbell et al. NeuroImage 05]

Corpus callosum - corona radiata - superior longitudinal fasciculus FA map + diffusion tensorsODF + maxima

Corona radiata diverging fibers - superior longitudinal fasciculus FA map + diffusion tensorsODF + maxima

Multi-Modal Fiber Tracking Extract ODF maxima Extension to streamline FACT

Streamline Tracking FAC T FACT: Fiber Assignment by Continuous Tracking Follow principal eigenvector of diffusion tensor Stop if FA   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]

Limitations of DTI-FACT Classical DTI Principal tensor direction HARDI ODF 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->

DTI-FACT Tracking start-> <-start Lower FA thresh start-> Very low FA threshold

Principal ODF FACT Tracking start->

Multi-modal FACT Tracking start->

Summary Signal S on the sphere Spherical harmonic description of S ODFFiber directions Multi-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 DTI 1. 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… Used by: CMRR, University of Minnesota, USA Hopital Pitié-Salpétrière, Paris

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, Descoteaux et al, A Fast and Robust ODF Estimation Algorithm in Q-Ball Imaging, ISBI Ozarslan et al., Generalized tensor imaging and analytical relationships between diffusion tensor and HARDI, MRM 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(x T u), 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) J 0 (2  z) z = 1z = 1000 [Tuch; MRM04]

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

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

ODF evaluation

Tuch reconstruction vs Analytic reconstruction Tuch ODFs Analytic ODFs Difference: Percentage difference:3.60% % [INRIA-McGill]

Human Brain Tuch ODFs Analytic ODFs Difference: Percentage difference:3.19% % [INRIA-McGill]

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

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 anisotropy measures High order anisotropy measures

Apparent Diffusion Coefficient ADC profile : D(g) = g T Dg Diffusion 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 = [ D xx D yy D zz D xy D xz D yz ] Rank l = 4 3x3x3x3 D = [ D xxxx D yyyy D zzzz D xxxy D xxxz D yzzz D yyyz D zzzx D zzzy D xyyy D xzzz D zyyy D xxyy D xxzz D yyzz ]

Tensor generalization of ADC Generalization of the ADC, rank-2D(g) = g T Dg rank- l Independent elements D k of the tensor General tensor [Ozarslan et al., mrm 2003]

Summary of algorithm High Order Diffusion Tensor (HODT) Spherical Harmonic (SH) Series Modified SH basis Y j C = (B T B + L) -1 B T X Least-squares with -regularization D = M -1 C M transformation HODT D from linear-regression [Descoteaux et al. MRM 56:2006]

3 synthetic fiber crossing

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

High Order Descriptions Seek to characterize multiple fiber diffusion 1. ADCADC 1. Apparent Diffusion Coefficient (ADC) 2. ODFODF 2. Orientation Distribution Function (ODF) ADC profile Fiber distribution Diffusion ODF