Diffusion tensor tractography

In neuroscience, tractography is a 3D modeling technique used to visually represent neural tracts using data collected by Difusion Weighted images (DWI). It uses special techniques of magnetic resonance imaging (MRI), and computer-based image analysis. The results are presented in two- and three-dimensional images.

In addition to the long tracts that connect the brain to the rest of the body, there are complicated neural networks formed by short connections among different cortical and subcortical regions. The existence of these bundles has been revealed by and biological techniques on post-mortem specimens. Brain tracts are not identifiable by direct exam, CT, or MRI scans. This difficulty explains the paucity of their description in neuroanatomy atlases and the poor understanding of their functions.

Using diffusion tensor MRI, one can measure the apparent diffusion coefficient at each voxel in the image, and after multilinear regression across multiple images, the whole diffusion tensor can be reconstructed.

Suppose there is a fiber tract of interest in the sample. Following the Frenet–Serret formulas, we can formulate the space-path of the fiber tract as a parametrized curve:

where ${\displaystyle {\mathbf {T} }(s)}$ is the tangent vector of the curve. The reconstructed diffusion tensor ${\displaystyle D}$ can be treated as a matrix, and we can easily compute its eigenvalues ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}$ and eigenvectors ${\displaystyle {\mathbf {u} }_{1},{\mathbf {u} }_{2},{\mathbf {u} }_{3}}$. By equating the eigenvector corresponding to the largest eigenvalue with the direction of the curve:

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