TEACHING / AN INTERACTIVE EXPERIMENT
Physics of Diffusion MRI
Water moves. Gradients label its position. When spins move between the two gradient pulses, their phases no longer fully cancel—and the signal falls.
The PGSE sequence
The physical gradient lobes have the same sign. The 180° RF pulse reverses phase evolution, so the effective second lobe has the opposite sign.
Preparing the visual experiment…
Same sequence. Different environments.
● Anisotropic Gaussian diffusion
Free motion, faster along x than y at the default settings.
● Restricted diffusion in a circle
Isotropic motion inside an impermeable reflecting wall.
A circular boundary has no preferred direction.
80 representative spins are shown. The two views use different spatial scales, labeled in µm. The dashed circle on the left marks starting positions, not a boundary.
Aligned phases add up. Scattered phases cancel.
Rotate the gradient to compare diffusion along and across the fast axis.
The two ideas behind the experiment
b = γ²G²δ²(Δ − δ/3)
Longer or stronger pulses increase sensitivity to motion.
S/S₀ = |〈eiφ〉| · φ = γ ∫ Geff(t) · r(t) dt
Signal is the length of the average phase vector. For Gaussian diffusion at the echo: S/S₀ = exp(−b Dg).
About this teaching prototype & further exploration
This is a fast visual explanation, not a research sequence simulator. Ideal rectangular PGSE gradients and instantaneous RF pulses are assumed. The displayed echo time is 10 + Δ + δ ms: a schematic with 5 ms margins, not a scanner prescription. Relaxation, imaging gradients, ramps, exchange and measurement noise are omitted.
The Gaussian echo uses the analytic expression with Dg = Dₓ cos²θ + Dᵧ sin²θ. Illustrative trajectories use 1,200 seeded walkers per environment, 0.1 ms integration steps, and midpoint phase accumulation. The restricted environment is a 2D disk with reflecting steps at the wall, not a 3D sphere. The restricted echo is a Monte Carlo magnitude estimate, with sampling variability of order 1/√1,200 ≈ 0.03; tiny signals should not be interpreted quantitatively. It is intentionally a small ensemble for responsive interaction. The same seed keeps comparisons stable.
Both populations start uniformly inside a disk. Gaussian spins then move freely, without wrapping; spins leaving the displayed window still contribute to the signal. Phases are shown in the toggling frame (RF sign absorbed into the effective gradient). The 180° pulse therefore does not create a visual phase jump. Playback replays the calculated paths at a teaching-friendly speed.
Useful related teaching and simulation examples: ISMRM Interactive Diffusion Course, SimDRIFT theory, dmipy simulation examples, and the DW-MRI Random Walk Simulator. This page is an independent implementation.