Water self-diffusion in tissue follows the Stejskal–Tanner relation between the diffusion-weighted MRI signal and a symmetric 3×3 diffusion tensor D:
S(b,g) = S₀ · exp(−b · gᵀDg)
⟨x²⟩ = 2·D·t (Einstein relation, per axis)
Inside myelinated axon bundles water diffuses freely along the fiber but is restricted across it, so D is anisotropic. Diagonalizing D (via Jacobi eigen-decomposition, computed live here for every voxel) gives eigenvalues λ₁≥λ₂≥λ₃ and eigenvectors e₁,e₂,e₃. The tensor ellipsoid glyphs you see are literally these eigenvectors as axes, scaled by the eigenvalues — a squashed cigar shape means one dominant fiber direction, a sphere means isotropic diffusion (grey matter or CSF).
Fractional Anisotropy:
FA = √(3/2) · √Σ(λᵢ−λ̄)² / √Σλᵢ² , λ̄ = mean(λ₁,λ₂,λ₃)
This scene mixes two synthetic bundles — a left–right sheet and an anterior–posterior sheet — that overlap near the center, a classic "crossing-fiber" phantom used to stress-test DTI algorithms. Each voxel's tensor is the weighted sum of both bundles' single-fiber tensors, which is exactly how partial-volume averaging works in a real multi-fiber voxel.
Streamline tractography (FACT algorithm) seeds a point, follows the principal eigenvector e₁ as a direction field, and takes discrete steps:
x_{n+1} = x_n + step · e₁(x_n)
- Anisotropy — sets how restricted diffusion is across a fiber (λ⊥) relative to along it (λ∥); low values wash the phantom back toward isotropic grey-matter-like diffusion.
- FA stop threshold — a streamline terminates once local anisotropy drops below this, the standard criterion for leaving white matter.
- Max turning angle — a streamline terminates if the direction would bend more than this per step, preventing tracks from jumping across unrelated bundles at the crossing.
- Step size — the Euler integration step; smaller steps track curvature more faithfully at the cost of more steps to reach the boundary.
Real-world relevance: this is the same principle behind clinical diffusion-MRI tractography, used to map white-matter connectivity and plan around eloquent tracts before neurosurgery.