The Physics of Diffusion MRI: Interactive Phantom
Synthetic Brain Diffusion MRI Phantom
Explore how diffusion weighting and directional fiber architecture influence image contrast, then take the signal apart with spherical harmonics.
Diffusion-weighted signal
Voxel signal
Diffusion tensor · O(b)
b = 1,000 fit
fODF · CSD
b = 3,000 · ℓ ≤ 6
Diffusion signal
S(g) / S₀ · selected shell
Drag any shape to rotate all three · R–L / A–P / S–I
Faster diffusion → lower signal. The tensor’s long axis and fODF peaks usually align with fibre directions, while diffusion-weighted signal is attenuated most strongly along them. The signal therefore has an opposite directional profile—not a mathematical reciprocal. A single tensor cannot resolve crossing fibres.
Signal across the three shells
Fixed S/S₀ axis: 0–1. Nonphysical values outside this range are clipped in this plot; voxel readouts retain them.
Build the angular signal, one order at a time.
The isotropic term sets the mean. Higher even orders describe finer angular structure. The fODF stays fixed; this control truncates the signal reconstruction.
500 equal-area Fibonacci directions over the full sphere. Drag to rotate · arrow keys also rotate. Both surfaces share one radial scale. Radius = |S/S₀|; negative values appear in red.
What is being simulated? Data, model & limitations
One axial slice (z = 18 mm) from Santiago Coelho’s HCP-derived BrainDiffusionPhantom, generated at 1.25 mm isotropic resolution. The browser loads the tissue labels, measured b₀ contrast, normalized MRtrix fODF coefficients through ℓ = 6, and response coefficients for b = 1,000, 2,000 and 3,000 s/mm². These are simulated shells, not uploaded raw HCP measurements.
White matter uses the phantom’s voxel-specific two-compartment Standard Model, including finite intra-axonal transverse diffusivity. Gray matter uses its anisotropic Kärger / NEXI forward kernel at Δ = 30 ms; CSF uses D = 3 µm²/ms. The same ARMNEXI-phantom response routines used by the source phantom generate all three shells. This forward path does not include the phantom’s optional soma maps.
fODF provenance: the original MRtrix image header records dwi2fod csd with response_b3000.txt and -lmax 6. The website reuses these coefficients; it does not rerun deconvolution. Tensor: a log-linear least-squares fit of −ln(S/S₀) = b gᵀDg to 500 simulated b = 1,000 directions, with S₀ fixed. This is a finite-b estimate of the O(b) log-signal model, not the exact b→0 limit. Samples outside 0 < S/S₀ ≤ 1 are excluded. Non-positive-definite fits are flagged instead of displayed. The ellipsoid maps a unit sphere through D (axis lengths proportional to its eigenvalues), independently scaled for visibility.
The normalized fODF P has integral 1. For orthonormal real SH, sℓm = 4π kℓ pℓm, where kℓ = ∫₀¹ K(b, ξ) Pℓ(ξ) dξ. SH conventions follow MRtrix3. Coefficients are in world RAS axes; the slice uses radiological display (right on the left).
The slice toggle switches between S/S₀ and S/S₀ multiplied by measured b₀ contrast (stored in fixed display units, scaled by its 99th percentile). Both use the same user-controlled display gain; it does not auto-normalize each shell. Spherical mean is the ℓ = 0 term. Glyphs are independently scaled to emphasize shape, and negative lobes are clipped only for glyph/image display. SH truncation or negative source fODF lobes can produce small nonphysical signal values; SH surfaces use absolute radius and mark negative values red; readouts retain their signs. No noise, motion, partial-volume model, or gradient nonlinearity is added. This is an educational prototype.