FR35 · MRI

MRI Diffusion Encoding: How Far Water Wanders

Two gradient pulses cancel for still water. How much signal do wandering molecules lose, and what changes when walls hem them in? Two equal gradient pulses around a refocusing pulse wind and exactly unwind the phase of still water. Water that wanders between them keeps a leftover phase, so the echo shrinks. In free water the echo falls exponentially with the b-value, which grows with the gradient area squared and the pulse spacing. Membranes slow the fall, so the measured diffusion coefficient is lower and timing-dependent.

Water molecules random-walk while two gradient pulses wind and unwind their phases. Free water loses signal exponentially as the weighting grows; water hemmed in by membranes loses less.

Diffusion weighting b $534\,\mathrm{s/mm^2}$

Echo with gradients / echo without $0.356$

Apparent diffusion coefficient from the fit $1.94\,\mathrm{\mu m^2/ms}$

Typical distance moved along the gradient between pulses $14.1\,\mathrm{\mu m}$

Set the gradient pair

Gradient strength

40

G\;(\mathrm{mT/m})

Pulse duration

10

\delta\;(\mathrm{ms})

Pulse separation

50

\Delta\;(\mathrm{ms})

Where the water is

Drag across the stage to scrub time; colour shows phase.

Diffusion coefficient, walkers and seed

Free diffusion coefficient, assumed (square micrometres per ms)

2

Simulated walkers

6000

Seed of the random walk

3541

Model, units and sources

$x_{n+1}=x_n+\sqrt{2D\,\Delta t}\,\xi_n,\qquad \varphi\mathrel{+}=\gamma G_{\mathrm{eff}}\Big(\tfrac{x_n+x_{n+1}}{2}\Delta t+\sqrt{\tfrac{2D\,\Delta t^3}{12}}\,\eta_n\Big)$

Time step 0.1 ms. The second term in the phase is the spread of the time integral of a Brownian path between its two end points, so for free water the simulated phase has exactly the variance the formula predicts. Between membranes, steps that cross a wall are mirrored back. The echo ratio is computed once per gradient of the sweep from the same walk, because the phase is proportional to the gradient strength.

Hrabe, Kaur and Guilfoyle (2007), Eqs. 1–3 and 12–14 · Szczepankiewicz and Hutter, ISMRM 2019 problem set

Why a spin echo can measure how far water moves

Model assumptions

One-dimensional random walk along the gradient, in the rotating frame, for a pulsed-gradient spin echo with rectangular pulses. The 180-degree pulse is ideal and instantaneous, so it is folded into an effective gradient whose second pulse counts negative. Each echo ratio is the magnitude of the mean phase factor over 6000 seeded walkers; relaxation is the same with and without gradients and cancels in the ratio. Membranes are impermeable parallel planes; each walker reflects inside its own slab. The free diffusion coefficient 2.0 square micrometres per millisecond is an assumed round value for water near body temperature, not taken from a cited measurement. Motion across the gradient is drawn only to make the stage readable and changes no number.

Pulse sequence (white line: now)

Orange: gradient; dashed: effective (second pulse flipped). Purple: RF. Yellow dashes: echo.

Echo ratio across gradient strengths

Yellow: simulated (bars: two standard errors). Pink: held out. Grey: not fitted. Green: fit. Blue dashes: free-diffusion reference.

Walker phases at the echo

More numbers and how they are checked

Diffusion weighting in microscopic units: $0$
Monte Carlo standard error of the echo ratio: $0$
Free-diffusion reference at this weighting: $0$
Held-out point against the one-exponential fit: $0$
Spread of phases at the echo (simulated): $0$
Spread expected for free diffusion: $0$
Gradient area times gamma: $0$
Echo time: $0$
Effective diffusion time: $0$
Free-diffusion distance over the pulse separation: $0$
Simulated walkers: $0$

The yellow points come only from the simulated walkers. The dashed blue curve is the closed-form free-diffusion result and never uses the simulation. The fit uses only points well above the Monte Carlo noise; the largest of them is held out and predicted, so a large miss means one exponential does not describe the data.

Physics tutorial

A spin echo that measures motion

BackgroundA pulsed-gradient spin echo plays the same gradient pulse twice, with a 180-degree pulse in between. The 180-degree pulse reverses every phase, so for a molecule that stays put the second pulse undoes exactly what the first one did.

Why it mattersWater molecules do not stay put. A molecule that has moved between the pulses keeps a leftover phase, and when thousands of leftover phases are added, the echo shrinks. The shrinkage tells how far water wandered, which is what diffusion-weighted imaging maps.

Start with the essentials

Focus question: Two gradient pulses cancel for still water. How much signal do wandering molecules lose, and what changes when walls hem them in?
One-sentence intuition: For free water the leftover phases are Gaussian with variance $2bD$ , so the echo falls as $e^{-bD}$ . The weighting $b$ grows with the square of the gradient area and with the time between pulses. Walls cap how far a molecule can go, so the echo falls more slowly than the free curve and the fitted coefficient depends on the timing.

Core mathematical model

Echo ratio for free diffusion

\frac{S}{S_0}=e^{-bD},\qquad b=\gamma^2G^2\delta^2\left(\Delta-\frac{\delta}{3}\right)

Rectangular pulses of strength $G$ and length $\delta$ , starting $\Delta$ apart (Hrabe, Kaur and Guilfoyle 2007, Eqs. 1 and 2). The ISMRM 2019 problem set works the default case: 40 mT/m, 10 ms and 50 ms give about $0.53\,\mathrm{ms/\mu m^2}$ , which is $530\,\mathrm{s/mm^2}$ .

Where the leftover phase comes from

\varphi=\gamma\int_0^{T_E}G_{\mathrm{eff}}(t)\,x(t)\,\mathrm{d}t,\qquad \frac{S}{S_0}=\left|\left\langle e^{i\varphi}\right\rangle\right|

The effective gradient counts the second pulse negative because of the 180-degree pulse. A molecule that does not move ends with $\varphi=0$ .

Common difficulties

The gradient does not push water

Typical misconceptionThe diffusion gradient makes water flow, and the signal drops because water leaves the voxel.

Better mental modelThe water wanders the same way with the gradient on or off. The gradient only labels each molecule with a phase that depends on where it was; the signal drops because the labels no longer agree at the echo.

Run the experiment

01
Watch phases wind and unwind
Start with Free water, a clinical b-value and watch one cycle.
What to observe: The first pulse paints a rainbow across the tissue; after the second pulse most colours return to one hue, but not all. The echo is about 0.36 of the echo without gradients, close to the free-diffusion curve.
02
Double the gradient
Choose Double the gradient: four times b.
What to observe: The weighting rises fourfold and the echo nearly vanishes; the phase histogram now spreads over more than a full turn.
03
Add membranes
Choose Water between membranes, then Wait longer: walls matter more.
What to observe: The echo stays high, the fitted coefficient drops well below the free value, and it drops further when the pulses are farther apart. The held-out point no longer sits on a single exponential.

MRI Diffusion Encoding: How Far Water Wanders

A spin echo that measures motion

Core mathematical model

Echo ratio for free diffusion

Where the leftover phase comes from

Common difficulties

The gradient does not push water

Run the experiment

Watch phases wind and unwind

Double the gradient

Add membranes

Related experiments and articles

From precessing water to k-space, undersampling and diffusion