From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 214

Deterministic tractography is a well established tool that has seen considerable
success in researching neurological disorders [ 20 ]. Deterministic tractography
begins from a seed point and traces along the dominant fiber direction by locally
connecting the “fiber” vectors or mathematically becoming tangent to these. Classi-
cally the major eigenvector of the diffusion tensor in DTI represented these “fiber”
vectors [ 7 , 38 , 47 ]. However, since DTI is ambiguous and cannot accurately describe
the fiber directions in regions with complex fiber configurations, DTI tractography,
in spite of its successful usage, is known to be prone to errors. Hence the trend
in recent years to extend tractography to complex shaped SDFs that describe the
underlying fiber directions more accurately [ 26 , 67 , 72 ].
Probabilistic tractography was proposed to address the reliability of deterministic
tractography which remains sensitive to a number of parameters. The concept and
output of probabilistic tractography is, however, subtly different from determinist
tractography. While the latter attempts to find the connectivity between two regions,
the former measures the likelihood that two regions are connected, or it provides a
connectivity confidence. Given the capabilities and ambiguities of dMRI acquisition
and reconstruction schemes of today, due to partial voluming, noise, etc., probabilis-
tic tractography provides a more complete statement. However, probabilistic trac-
tography is also computationally more expensive than deterministic tractography.

6.6.1 Deterministic Tractography

Of the many deterministic tractography algorithms, the continuous streamline
tractography is a widely used scheme. The continuous version of streamline
tractography [ 7 ] defined for DTI, considers a fiber tract as a 3D space curve
parametrized by its arc-length, r

( s ) , and describes it by its Frenet equation:

d r ( s )

t s ε 1 ( r s ,

= ( )= ( )) (6.33)

ds

where t ( s ) the tangent vector to r ( s ) at s is equal to the unit major eigenvector

ε 1 ( . This implies that fiber tracts are locally

tangent to the dominant eigenvector of the diffusion tensor at every spatial position.
The differential equation Eq. ( 6.33 ) along with the initial condition r

r ( s )) of the diffusion tensor at r ( s )

r 0 means
that starting from r 0 , a fiber can be traced by continuously integrating Eq. ( 6.33 )
along the direction indicated locally by the major eigenvector of the diffusion tensor
at that point.
However, integrating Eq. ( 6.33 ) requires two things—first, a spatially continuous
tensor (or SDF) field, and second, a numerical integration scheme. In [ 7 ], the
authors proposed two approaches for estimating a spatially continuous tensor field
from a discrete DTI tensor field, namely approximation and interpolation. They
also proposed the Euler's method, the second order Runge-Kutta method, and
the adaptive fourth order Runge Kutta method as numerical integration schemes.

(0) =