Analytic ODF Reconstruction and Validation in Q-Ball Imaging Maxime Descoteaux 1 Work done with E. Angelino 2, S. Fitzgibbons 2, R. Deriche 1 1. Projet Odyssée, INRIA Sophia-Antipolis, France 2. Physics and Applied Mathematics, Harvard University, USA McGill University, Jan 18th 2006

Plan of the talk Introduction Background Analytic ODF reconstruction Results Discussion

Introduction Cerebral anatomy Basics of diffusion MRI

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

Background High Angular Resolution Diffusion Imaging Q-Space Imaging Q-Ball Imaging …

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 points642 points

High Order Reconstruction We seek a spherical function that has maxima that agree with underlying fibers Diffusion profile Fiber distribution Diffusion Orientation Distribution Function (ODF)

Diffusion Orientation Distribution Function (ODF) Method to reconstruct the ODF Diffusion spectrum imaging (DSI) Sample signal for many q-ball and many directions Measured signal = FourierTransform[P] Compute 3D inverse fourier transform -> P Integrate the radial component of P -> ODF

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]

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

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]

My Contributions The Funk-Radon can be solved ANALITICALLY Spherical harmonics description of the signal One step matrix multiplication Validation against ground truth evidence Rat phantom Knowledge of brain anatomy Validation and Comparison against Tuch reconstruction [collaboration with McGill]

Analytic ODF Reconstruction Spherical harmonic description Funk-Hecke Theorem

Sketch of the approach S in Q-space Spherical harmonic description of S ODF Physically meaningful spherical harmonic basis Analytic solution using Funk-Hecke formula For l = 6, C = [c 1, c 2, …, c 28 ]

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. SPIE-MI 06]

Spherical Harmonics (SH) coefficients In matrix form, S = C*B S : discrete HARDI data 1 x N C : SH coefficients 1 x m = (1/2)(order + 1)(order + 2) B : discrete SH, Y j  m x N (N diffusion gradients and m 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

For l = 6, C = [c 1, c 2, …, c 28 ] S = [d 1, d 2, …, d N ] SH description of the signal For any (  )

Funk-Hecke Theorem Solve the Funk-Radon integral Delta sequence

Funk-Hecke Theorem [Funk 1916, Hecke 1918]

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

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 (if time, proof)

Funk-Hecke formula Solving the FR integral => Delta sequence

Final Analytic ODF expression (if time bigO analysis with Tuch’s ODF reconstruction)

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

Validation and Results Synthetic data Biological rat spinal chords phantom Human brain

Synthetic Data Experiment

Multi-Gaussian model for input signal Exact ODF

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

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

Real Data Experiment Biological phantom Human Brain

Biological phantom T1-weigthedDiffusion tensors [Campbell et al. NeuroImage 05]

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]

Genu of the corpus callosum - frontal gyrus fibers FA map + diffusion tensors ODFs

Corpus callosum - corona radiata - superior longitudinal FA map + diffusion tensorsODFs

Corona radiata diverging fibers - longitudinal fasciculus FA map + diffusion tensorsODFs

Discussion & Conclusion

Summary S in Q-space Spherical harmonic description of S ODF Physically meaningful spherical harmonic basis Analytic solution using Funk-Hecke formula Fiber directions

Advantages of our approach Analytic ODF reconstruction Discrete interpolation/integration is eliminated Solution for all directions is obtained in a single step Faster than Tuch’s numerical approach Output is a spherical harmonic description which has powerful properties

Spherical harmonics properties Can use funk-hecke formula to obtain analytic integrals of inner products Funk-radon transform, deconvolution Laplacian is very simple Application to smoothing, regularization, sharpening Inner product Comparison between spherical functions

What’s next? Tracking fibers! Can it be done properly from the diffusion ODF? Can we obtain a transformation between the input signal and the fiber ODF using spherical harmonics

Thank you! Key references: Maxime.Descoteaux/index.en.html Tuch D. Q-Ball Imaging, MRM 52, 2004 Thanks to: P. Savadjiev, J. Campbell, B. Pike, K. Siddiqi

 n is a delta sequence => 1) 2)

3) Nice trick! =>

Spherical Harmonics SH SH PDE Real Modified basis

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]

Limitations of classical DTI Classical DTI rank-2 tensor HARDI ODF reconstruction