Editing Tractography (section)

== Mathematics ==

Using [[Diffusion MRI|diffusion tensor MRI]], one can measure the [[Diffusion MRI#ADC|apparent diffusion coefficient]] at each [[voxel]] in the image, and after [[Multi-linear regression|multilinear regression]] across multiple images, the whole diffusion tensor can be reconstructed.<ref name="Basser_2000">{{cite journal | vauthors = Basser PJ, Pajevic S, Pierpaoli C, Duda J, Aldroubi A | title = In vivo fiber tractography using DT-MRI data | journal = Magnetic Resonance in Medicine | volume = 44 | issue = 4 | pages = 625–632 | date = October 2000 | pmid = 11025519 | doi = 10.1002/1522-2594(200010)44:4<625::AID-MRM17>3.0.CO;2-O | doi-access =  }}</ref>

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 parameterized curve:

:<math> \frac{d\mathbf{r}(s)}{ds} = \mathbf{T}(s), </math>

where <math>\mathbf{T}(s)</math> is the tangent vector of the curve.  The reconstructed diffusion tensor <math> D </math> can be treated as a matrix, and we can compute its [[eigenvalues]] <math> \lambda_1, \lambda_2, \lambda_3 </math> and [[eigenvectors]] <math> \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3 </math>.  By equating the eigenvector corresponding to the largest eigenvalue with the direction of the curve:

:<math> \frac{d\mathbf{r}(s)}{ds} = \mathbf{u}_1(\mathbf{r}(s)) </math>

we can solve for <math> \mathbf{r}(s) </math> given the data for <math> \mathbf{u}_1(s) </math>. This can be done using numerical integration, e.g., using [[Runge–Kutta]], and by interpolating the principal [[eigenvector]]s.