On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution FunctionLuke Bloy1, Ragini Verma2

The Section of Biomedical Image AnalysisUniversity of PennsylvaniaDepartment of Bioengineering1

Department of Radiology2

Diffusion Tensor Imaging

DTI model is incapable of representing multiple orientations

Diffusion imaging rests on the assumption that the diffusion process correlates with the underlying tissue structure.

Diffusion Orientation Distribution Function

ODF: Approximates the radial projection of the diffusion propagator.

It essentially describes the probability that a water molecule will diffuse in a certain direction.

Its maxima have been shown to correspond with principle directions of the underlying diffusion process.

How to find the maxima of Orientation Distribution Function? Existing Methods:

Optimization Methods Spherical Newton’s method Powell’s method Need to ensure convergence Need to avoid small local maxima

Finite Difference Method Accuracy is limited by Mesh Size (accuracy of 4 degrees

requires 1280 mesh points

NEEDS REFS

Computing Maxima of the Diffusion Orientation Distribution Function

Our method:

•Represent ODF as symmetric Cartesian tensor

•Compute the stationary points of the ODF from a system of polynomial equations

• Classify the stationary points using the local curvature of the ODF graph into principle directions, secondary maxima, minima and inflection points.

Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian Tensors

Real Symmetric Spherical Functions

Real Spherical Harmonics

Real Symmetric Cartesian Tensors

Orientation Distribution Function as a Cartesian Tensor

In spherical coordinates the from of Funk-Radon transform allows a the computation of the ODF RSH expansion in terms of the RSH expansion of the MRI signal.

Since M is a change of basis matrix it is invertible and the ODF tensor can be computed

Stationary points

Stationary points are points on the sphere ( ) where the derivative of the ODF vanishes. Using the tensor representation of the ODF, they are solutions to the following system of equations.

t is a solution to an lth order polynomial

Use the method of resultants to solve for v and u.

Classification of Stationary PointsUse the principle curvatures (k1, k2) to

classify each stationary point:

Minima Inflection Points

Principle Directions Secondary Maxima

Stationary points of the Orientation Distribution Function

Diffusion ODF reconstructions from simulated fiber populations performed with a rank 4 tensor. Red lines indicate principle directions, Blue minima, Black lines saddle points and green lines indicate secondary maxima.

One Fiber Two Fibers Three Fibers

Affect of Signal to Noise on Principle Direction Calculation

Single Tensor Model

b = 3000 sec /cm2

64 gradient direction

50 DWI signals, each with randomly chosen principle direction, at each SNR

SNR: 5,10,15,25,35,45

Angular Error = true calcPD ,PD )

Application to Clinically Feasible Data

3Tesla scanner

64 Gradient Directions

Single average

Scan time ~ 8 min

B = 1000 sec /cm2

CC : Corpus CollosumSCR : Superior Corona RadiataALIC : Anterior Limb of the Internal Capsule

Future Work

Implementation within fiber tracking framework

Investigation of geometric features (mean curvature/Gauss curvature) of the ODF surface as measures of diffusion anisotropy

Thanks…

Computing Principle Curvatures

Limitations of Diffusion Tensor ImagingSingle Single

FibersFibersMultiple Multiple

FibersFibers

Behrens et al, Neuroimage, 34 (1) Behrens et al, Neuroimage, 34 (1) 20072007

As many as 1/3 of white Matter voxels may be effected .

DTI model is incapable of representing multiple orientations

Real Spherical Harmonics of Even Order

Images of the first few?

Symmetric Cartesian Tensors

Ref Max.

Relationship between Anisotropy and Mean Curvature

Single tensor model

mean diffusivity of 700 mm2/sec

SNR = 35

Red line = absence of noise

False Positives Rates

SNR # of PDS

10 65%

15 92%

>25 100%

Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian TensorsReal Symmetric Spherical

Functions Real Symmetric Cartesian TensorsReal Spherical

Harmonics