From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 216

Euclidean interpolation in the SH basis is equivalent to the L2-norm interpolation
of the SDFs, since the SHs form an orthonormal basis [ 24 ].

6.6.2 Probabilistic Tractography

General SDF-based (ODF etc.) probabilistic tractography have recently been pub-
lished in the literature [ 9 , 17 , 33 , 35 , 48 , 52 , 57 , 60 ] to generalize several existing
DT-based methods. First, in [ 35 ] parametric spherical deconvolution is used as the
SDF [ 65 ]andin[ 9 ] a mixture of Gaussian model is used to extend the probabilistic
Bayesian DT-based tracking [ 10 ]. Related to these techniques, [ 33 ] uses a Bayesian
framework to do global tractography instead of tracking through local orientations.
In [ 52 ], Monte Carlo particles move inside the continuous field of q-ball diffusion
ODF and are subject to a trajectory regularization scheme. In [ 48 ], an extension to
their DT-based approach [ 49 ] is also proposed using a Monte Carlo estimation of
the white matter geometry and recently, a Bingham distribution is used to model
the peak anisotropy in the fiber distributions [ 60 ]. Finally, in [ 17 ], large number of
M-FACT QBI streamlines are reconstructed and all pathways are reversed-traced
from their end points to generate of map of connection probability. In this chapter,
a new probabilistic algorithm is presented based on the ODF using a Monte Carlo
random walk algorithm.
The new algorithm is an extension of the random walk method proposed in [ 36 ]
to use the distribution profile of the fiber ODF. It starts off a large number of particles
from the same seed point and lets the particles move randomly according to the local
ODF estimate, F , and counts the number of times a voxel is reached by the path of a
particle. This yields higher transitional probabilities along the main fiber directions.
The random walk is stopped when the particle leaves the white matter mask.
For each elementary transition of the particle, the probability for a movement
from the seed point x to the target point y in direction u xy

is computed as the

product of the local ODFs in direction u xy , i.e.,

P ( x → y )= F ( u xy ) x · F ( u xy ) y , (6.35)

where P u xy ) x

is the ODF at point x in direction xy (by symmetry, direction xy and yx are the
same).
The transition directions in the local model are limited to 120 discrete directions
corresponding to the angular sampling resolution of the acquired brain data and
the step size of the particle step was fixed to 0.5 times the voxel size. A trilinear
interpolation of the ODF was used for the subvoxel position and a white matter
mask computed from a minimum FA value of 0.1 and a maximum ADC value of
0.0015 was used. A total of 100,000 particles were tested for each seed voxel. The
connectivity of any given voxel with the seed voxel is estimated by the number of
particles that reach the respective voxel, called a tractogram .

( x → y ) is the probability for a transition from point x to point y , F (