1. On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function
Luke Bloy1, Ragini Verma2
The Section of Biomedical Image Analysis
University of Pennsylvania
Department of Bioengineering1
Department of Radiology2

2. Diffusion Tensor Imaging
Diffusion imaging rests on the assumption that the diffusion process correlates with the underlying tissue structure.
DTI model is incapable of representing multiple orientations

3. 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.

4. 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

5. 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.

6. Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian Tensors
Real Spherical Harmonics
Real Symmetric Spherical Functions
Real Symmetric Cartesian Tensors

7. 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

8. 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.

9. Classification of Stationary Points
Use the principle curvatures (k1, k2) to classify each stationary point:
Inflection Points
Minima
Principle Directions
Secondary Maxima

10. Stationary points of the Orientation Distribution Function
One Fiber
Two Fibers
Three Fibers
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.

11. 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 =

12. Application to Clinically Feasible Data
3Tesla scanner
64 Gradient Directions
Single average
Scan time ~ 8 min
B = 1000 sec /cm2
CC : Corpus Collosum
SCR : Superior Corona Radiata
ALIC : Anterior Limb of the Internal Capsule

13. 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…

14. Computing Principle Curvatures

15. Limitations of Diffusion Tensor Imaging
Single Fibers
Multiple Fibers
DTI model is incapable of representing multiple orientations
As many as 1/3 of white Matter voxels may be effected .
Behrens et al, Neuroimage, 34 (1) 2007

16. Real Spherical Harmonics of Even Order
Images of the first few?

17. Symmetric Cartesian Tensors
Ref Max.

18. Relationship between Anisotropy and Mean Curvature
Single tensor model
mean diffusivity of mm2/sec
SNR = 35
Red line = absence of noise

19. False Positives Rates
SNR
# of PDS 10
65% 15
92%
>25
100%

20. Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian Tensors
Real Symmetric Spherical Functions
Real Symmetric Cartesian Tensors
Real Spherical Harmonics