ADC and ODF estimation from HARDI

Maxime Descoteaux1

Work done with E. Angelino2, S. Fitzgibbons2, R. Deriche1

1. Projet Odyssée, INRIA Sophia-Antipolis, France2. Physics and Applied Mathematics, Harvard University, USA

Max Planck Institute, March 28th 2006

Plan of the talk

Introduction

Spherical Harmonics Formulation

Applications:

1) ADC Estimation 2) ODF Estimation

Discussion

• 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 642 points

High Order Descriptions

Seek to characterize multiple fiber diffusion• AApparent DDiffusion CCoefficient (ADCADC)• OOrientation DDistribution FFunction (ODFODF)

ADC profileFiber distribution Diffusion ODF

Sketch of the approachData on the

sphere

Spherical harmonic

description of data

ODF

For l = 6,

C = [c1, c2 , …, c28]

ADC

ADC ODF

Spherical harmonics

Description of discrete data on the sphere

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. SPIE-MI 06]

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

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]

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. SPIE-MI 06]

3 synthetic fiber crossing

Limitations of the ADC

• 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]

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

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

Final Analytical ODF expression in SH coefficients

[Descoteaux et al. ISBI 06]

Biological phantom

T1-weigthed Diffusion tensors

[Campbell et al.NeuroImage 05]

Corpus callosum - corona radiata - superior longitudinal

FA map + diffusion tensors ODFs

Corona radiata diverging fibers - longitudinal fasciculus

FA map + diffusion tensors ODFs

Contributions & advantages

• SH and HODT description of the ADC• Application to anisotropy measures

• Analytical ODF reconstruction• Discrete interpolation/integration is eliminated

• Solution for all directions in a single step• Faster than Tuch’s numerical approach by a factor 15

• Spherical harmonic description has powerful properties• Smoothing with Laplace-Beltrami, inner products,

integrals on the sphere solved with Funk-Hecke

Perspectives

• Tracking and segmentation of fibers using multiple maxima at every voxel

• Consider the full diffusion ODF in the tracking and segmentation

Thank you!

Key references:• http://www-sop.inria.fr/odyssee/team/

Maxime.Descoteaux/index.en.html

-Ozarslan et al. Generalized tensor imagingand analytical relationships between diffusion tensorand HARDI, MRM 50, 2003

-Tuch D. Q-Ball Imaging, MRM 52, 2004

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-->

Spherical Harmonics (SH) coefficients

• In matrix form, S = C*BS : discrete HARDI data 1 x NC : SH coefficients 1 x m = (1/2)(order + 1)(order + 2)

B : discrete SH, Yjm x N(N diffusion gradients and m SH basis elements)

• Solve with least-square C = (BTB)-1BTS

[Brechbuhel-Gerig et al. 94]

Introduction

Limitations of Diffusion Tensor Imaging (DTI)

High Angular Resolution Diffusion Imaging (HARDI)

Our Contributions

• Spherical harmonics (SH) description of the data• Impose a regularization criterion on the solution

• Application to ADC estimation• Direct relation between SH coefficients and

independent elements of the high order tensor (HOT)• Application to ODF reconstrution

• ODF can be reconstructed ANALITYCALLY• Fast: One step matrix multiplication

• Validation on synthetic data, rat biological phantom, knowledge of brain anatomy

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]

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!

=>

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(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]

Field of Synthetic Data

90 crossing

b = 1500SNR 15order 6

55 crossingb = 3000

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

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]

Synthetic Data Experiment

• Multi-Gaussian model for input signal

• Exact ODF

Limitations of classical DTI

Classical DTIrank-2 tensor

HARDIODF reconstruction

Strong Agreement

b-value

Average differencebetween exact ODFand estimated ODF

Multi-Gaussian model with SNR 35

SummarySignal S on the

sphere

Spherical harmonic

description of S

ODF

Physically meaningfulspherical harmonicbasis

Analytic solution usingFunk-Hecke formula

Fiber directions

ADCHigh order anisotropy measures