Download - On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function
![Page 1: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/1.jpg)
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
![Page 2: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/2.jpg)
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.
![Page 3: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/3.jpg)
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.
![Page 4: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/4.jpg)
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
![Page 5: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/5.jpg)
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.
![Page 6: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/6.jpg)
Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian Tensors
Real Symmetric Spherical Functions
Real Spherical Harmonics
Real Symmetric Cartesian Tensors
![Page 7: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/7.jpg)
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
![Page 8: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/8.jpg)
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.
![Page 9: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/9.jpg)
Classification of Stationary PointsUse the principle curvatures (k1, k2) to
classify each stationary point:
Minima Inflection Points
Principle Directions Secondary Maxima
![Page 10: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/10.jpg)
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
![Page 11: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/11.jpg)
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 )
![Page 12: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/12.jpg)
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
![Page 13: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/13.jpg)
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…
![Page 14: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/14.jpg)
Computing Principle Curvatures
![Page 15: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/15.jpg)
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
![Page 16: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/16.jpg)
Real Spherical Harmonics of Even Order
Images of the first few?
![Page 17: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/17.jpg)
Symmetric Cartesian Tensors
Ref Max.
![Page 18: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/18.jpg)
Relationship between Anisotropy and Mean Curvature
Single tensor model
mean diffusivity of 700 mm2/sec
SNR = 35
Red line = absence of noise
![Page 19: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/19.jpg)
False Positives Rates
SNR # of PDS
10 65%
15 92%
>25 100%
![Page 20: On Computing the Underlying Fiber Directions from the Diffusion Orientation Distribution Function](https://reader036.vdocuments.net/reader036/viewer/2022062807/5681516e550346895dbf9e86/html5/thumbnails/20.jpg)
Equivalence of Real Spherical Harmonic Expansion and Symmetric Cartesian TensorsReal Symmetric Spherical
Functions Real Symmetric Cartesian TensorsReal Spherical
Harmonics