The echo that gives back everything but the wandering

The image made of frequencies took a proton’s precession in a magnetic field and made it into a map: add a gradient to the field, and every position precesses at its own rate, so the signal’s spectrum is a picture. That essay listened to the signal for a few milliseconds and stopped, and it noted in passing that a readout cannot usefully go on much longer, because the protons’ precession falls out of step and the signal fades.

How it fades turns out to be two different stories that look identical on an oscilloscope. In one the spins really are losing their magnetisation, exchanging energy and phase with their surroundings in ways nothing can undo. In the other they have simply drifted apart, each keeping its magnetisation perfectly and pointing in a slightly different direction from the rest, so that the sum is small. Erwin Hahn found in 1950 that the second kind of fading can be reversed by a single pulse, and that the signal then reappears, unprompted, a while later. The echo he found divides apparent loss from real loss as cleanly as any experiment in physics. It also turned out to have a residue — something an echo cannot give back — and that residue is now one of the most useful numbers in medicine.

A signal that dies with nothing lost #

A tipped proton is a small magnet turning about the field at the Larmor rate, $\bar\gamma B$, 42.58 megahertz per tesla, the precession that the push that comes out sideways found in every spinning top under a torque. A sample holds an enormous number of them, and the signal a coil picks up is their sum. Immediately after a pulse has tipped them all together they point the same way and the sum is as large as it will ever be.

No magnet is perfectly uniform, and no sample is either: a few parts per million of variation across a centimetre is good engineering, and the sample’s own magnetism distorts the field near every boundary between materials. Each proton therefore precesses at a rate set by the field where it happens to sit, and in the frame that turns at the average rate each one drifts steadily forward or back. After a while they point in every direction round the circle and their sum is nearly nothing. The time this takes is written $T_2^*$, with the star marking that it belongs to the apparatus and not to the protons.

The first figure has a spread of 20 hertz, a modest inhomogeneity, and the free signal falls to a third of its first value within ten milliseconds and to three per cent by twenty. Meanwhile the true relaxation — the loss each individual spin suffers from the jostling of its neighbours, written $T_2$ — has been set at 80 milliseconds. At ten milliseconds each proton still carries nearly nine-tenths of its magnetisation. The signal is gone and the magnetisation is not; it has only stopped adding up.

Turning the fan over #

Hahn’s remedy is a pulse twice as long as the one that tipped the spins: a π pulse, which turns every spin through half a revolution about an axis lying in the plane of precession. Seen from above, as in the second figure, it mirrors the fan. A spin that had run ahead of the average by an angle $\delta\tau$ is now behind it by the same angle, and a spin that had lagged is now ahead.

Nothing about the rates has changed. The fast spin is still fast, because it still sits in the same patch of stronger field, and it goes on gaining on the average at the same rate as before. So after a further time τ it has gained back exactly the $\delta\tau$ it was set behind, and so has every other spin, each by its own amount. At $2\tau$ they all point together again and the signal rises out of nothing into a peak. The fastest spin in the figure gained 162 degrees in thirty milliseconds; the pulse put it 162 degrees behind; thirty milliseconds later it is level.

The usual picture is a race. Runners of different speeds leave the starting line together and spread out along the track; at a signal every one of them turns round and runs back at the same speed; they all reach the line together, the fastest having run furthest out and furthest back. The picture is exact as long as every runner’s speed stays the same throughout — and that condition is the whole content of what follows.

What the echo measures #

The echo’s height in the first figure is 0.472, and the dashed curve passing through it is $e^{-t/T_2}$ with $T_2$ = 80 milliseconds. The agreement is exact because the calculation behind the figure averages over the whole spread of rates and every one of them cancels. What is left is the loss that was never a matter of pointing in different directions: the slow destruction of each spin’s own phase by the fluctuating fields of its moving neighbours.

So a single echo separates two numbers that the free signal mixes. $T_2^*$ is a fact about the magnet and the sample’s shape, and changes when either is moved. $T_2$ is a fact about the molecules — how quickly they tumble, what they are bound to — and that is the number chemists and physicians want. The tissue contrast that the image made of frequencies described, fluid bright and fat dark or the reverse, comes from imaging at echoes timed to weigh $T_2$, which no free signal could show through a scanner’s own inhomogeneity.

The echo is also an answer to a question about time’s arrow that the ring that forgets and then remembers put on a toy model. Josef Loschmidt objected to Boltzmann that every motion of a mechanical system can be run backwards, so any decay towards disorder can be undone by reversing every velocity; Boltzmann is said to have replied, “Then go and reverse them.” The π pulse does exactly that to the phases of some 10²² protons, with a pulse lasting a few microseconds, and a disorder that looked final — a fan of directions with nothing left in the sum — returns to order. What the experiment shows is that the decay of the free signal was never an increase of entropy in the spins. The information about where each spin pointed was kept in its rate, untouched, and the pulse made use of it. Real irreversibility is what is left over when every such reversal has been made.

The loss that wandering costs #

The race picture assumed that each runner keeps the same speed. A proton’s rate is set by the field where it is, and in a liquid nothing stays where it is. A water molecule at room temperature wanders, by the jiggle that proved atoms, with a diffusion coefficient D of 2.3 × 10⁻⁹ square metres per second, so in a tenth of a second it moves on average about twenty micrometres. Twenty micrometres is a long way in a gradient. If the field changes across the sample by 5 millitesla per metre, a molecule that moves twenty micrometres changes its precession rate by four and a half hertz, and over a tenth of a second that is nearly half a turn.

The π pulse reverses the phase a molecule has gained. It cannot reverse its wandering, and after the pulse the molecule is no longer where it was before, so the rate it now runs back at is not the rate it ran out at. Each molecule returns to the line a little early or a little late, by an amount that depends on its private walk, and the echo is smaller by the spread of those errors.

The size of the spread follows from how a random walk grows. A molecule that has wandered for a time $t$ is typically $\sqrt{2Dt}$ from its start, so the extra phase it has gained is that distance times $\gamma G$ times the time it has spent there, a quantity growing as $t^{3/2}$. The echo loses in proportion to the square of the phase errors, which grows as $t^3$. Carried through exactly for a pulse at τ and an echo at $TE = 2\tau$, the result is

$$\frac{S(TE)}{S(0)} = \exp\!\left(-\frac{\gamma^2 G^2 D\, TE^3}{12}\right),$$

where γ is the proton’s ratio of magnetic moment to angular momentum, 2.675 × 10⁸ per second per tesla. The third figure tests this against a simulation that knows nothing of the formula: three thousand molecules, each taking a random step every half millisecond, each accumulating phase at a rate set by its position, each having its phase reversed at the halfway point. The averages land on the curve — 0.713 against 0.710 at a hundred milliseconds, 0.067 against 0.064 at two hundred — and the shape is unmistakably not exponential. The loss is hardly noticeable for fifty milliseconds and then catastrophic, so that doubling the echo time from 100 to 200 milliseconds takes the echo from seven-tenths to six-hundredths.

This is the same diffusion the walk that comes home studied as a pure random walk, read here through a phase that records every step. And it is the reason the echo of the first figure cannot be pushed to arbitrarily long times in a real magnet, whatever the true $T_2$: in a gradient the wandering, not the relaxation, sets the limit.

A train of pulses instead of one #

The cube is also the remedy. If the damage done by wandering grows as the cube of the time between pulses, then many short intervals do far less damage than one long one. Herman Carr and Edward Purcell showed in 1954 that a train of π pulses — one at τ, then every $2\tau$ thereafter — produces an echo midway between each pair, each echo a little smaller than the last, and that the decay of the train is governed by the spacing rather than by the total time.

Over a train of echoes spaced $t_s$ apart the wandering costs $\exp(-\gamma^2 G^2 D\, t_s^2\, t/12)$ by the time $t$, a loss that now grows only in proportion to time and is smaller by the square of how short the spacing is. In the fourth figure, a sample of water in a magnet whose residual gradient is 5 millitesla per metre, the free signal is gone in well under a millisecond, the 2,129 hertz of spread across a centimetre of sample having fanned the spins out almost at once. A single echo does far better and is still halved by 123 milliseconds. A train with pulses every 10 milliseconds keeps 0.586 of the signal after a full second, within four per cent of the 0.607 the water’s own relaxation allows; with pulses every 40 milliseconds it keeps 0.350.

The practical consequences were immediate. Measuring the true relaxation time of a pure liquid, which may be seconds, became possible in magnets far from perfect, and the train, with a refinement by Saul Meiboom and David Gill in 1958 that keeps small errors in the pulses from accumulating, is still the standard way of measuring $T_2$. It also became the basis of fast imaging, which reads a different line of the spatial-frequency grid at each echo of one train, and of a branch of quantum technology: a qubit dephasing in a noisy environment is protected by exactly such a train, under the name of dynamical decoupling, and the spacing needed to protect it measures how fast the noise changes. The wait that sets the precision noted that an atomic clock with inhomogeneous atoms can be echoed in the same way, and pays the same kind of price for anything that changes during the wait.

The loss turned into a measurement #

The wandering that limits the echo can be put to work. If the loss depends on D, then measuring the loss measures D — the rate at which molecules move, read without following any of them. Edward Stejskal and John Tanner arranged this in 1965 by switching the gradient on in two short, strong pulses, one on each side of the π pulse, rather than leaving a weak gradient on throughout. The first pulse labels each molecule with a phase proportional to its position; the second, after the π pulse, unwinds exactly that label for any molecule that has not moved. A molecule that has moved keeps a residue proportional to its displacement over the time between the pulses, and the echo is multiplied by $e^{-bD}$, where $b = \gamma^2 G^2 \delta^2 (\Delta - \delta/3)$ gathers the gradient’s strength G, the pulses’ length δ and their separation Δ into a single number.

Each straight line on the logarithmic scale of the last figure is one tissue, and its slope is its diffusion coefficient. Water free to move at body temperature has a coefficient of 3.0 × 10⁻³ square millimetres per second and keeps only five per cent of its signal at a b-value of 1000. In brain tissue the water is hindered by cell membranes and fibres and the apparent coefficient is about a quarter of that, so it keeps nearly half.

The surprising use came in the 1990s. In the first minutes after the blood supply to part of the brain is cut off, its cells lose the energy to pump ions across their membranes, take in water and swell; the water that was outside the cells, where it moved most freely, is squeezed into narrow gaps, and the apparent diffusion coefficient of the region falls by a third to a half. Nothing else visible has happened yet — the tissue’s water content, its relaxation times and its appearance on every other kind of scan are almost unchanged for hours. On a diffusion-weighted image the affected region keeps 0.670 of its signal where its surroundings keep 0.449, and it stands out, bright, within minutes of the stroke. Because treatments that dissolve clots must be given within hours, this has become one of the most consequential applications of magnetic resonance: a measurement of how far water molecules wander in forty milliseconds, read from the one part of an echo that the echo could not restore.

Imaging the same loss along different directions gives more. Water in the brain’s white matter moves easily along the bundles of nerve fibres and poorly across them, so the loss depends on which way the gradient points, and following the direction of easiest motion from one point to the next traces the fibre bundles through the living brain — the technique called tractography.

How far the echo reaches #

The arguments here assume a proton population with a smooth spread of fixed rates, and real spreads are often lumpy or skewed; that changes the shape of the free decay but not the echo, which cancels any static spread whatever its shape. They assume the π pulse is perfect. A real pulse turns some spins by a little more or less than half a revolution, depending on where they sit in the coil, and the errors accumulate along a long train unless the train is designed to cancel them, which is the point of Meiboom and Gill’s refinement. And the wandering is treated as free diffusion in an unbounded liquid. In tissue it is not: molecules meet membranes, and over long times their displacement grows more slowly than free diffusion predicts, the slower spread that the diffusion that slows the longer it is watched found in other systems. The coefficient a scanner reports is therefore an apparent diffusion coefficient, which depends on the time between the gradient pulses, and treating it as a property of the tissue alone is itself an approximation.

What the curves cannot show #

The figures show the size of a signal, never its source. An echo that comes back smaller than $e^{-2\tau/T_2}$ says that something changed during the wait, and it does not say what: diffusion through a gradient, a chemical exchange that moves a proton from one molecule to another with a different shielding, a flowing liquid carrying spins to a different field, a field that itself drifts. Each leaves a characteristic dependence on the echo time — the cube of the time for diffusion, a different signature for exchange — and it is by varying the timing that an experiment tells them apart. A single echo cannot.

Nor do the figures show that a stroke has occurred. A region of reduced diffusion is bright on a diffusion-weighted image whatever the cause, and abscesses, some tumours and densely packed cells of other kinds restrict water in the same way. The image is a measurement of motion; the diagnosis is a reading of it with everything else that is known.

Still open: what the wandering says about the walls #

The single number D treats tissue as if it were a uniform liquid with a smaller coefficient. It is not, and the way the apparent coefficient changes with the time between gradient pulses carries information about what the molecules are bumping into: the sizes of cells, the spacing of fibres, the permeability of membranes. Methods that fit a model of the tissue’s microstructure to measurements at many b-values, directions and timings now claim to estimate the diameters of nerve fibres a few micrometres across, far below the resolution of the image itself, from the way the wandering is restricted. Whether those estimates are measurements or the output of models that happen to fit — whether a fitted fibre diameter would survive a different, equally plausible model of the same tissue — is argued, and validating them against tissue examined under a microscope is slow and incomplete. The question is how much of the structure of a material can be read from statistics of motion alone, and the answer is still being worked out tissue by tissue.

The habit worth carrying away is to ask which part of a decay is a loss and which is only a disagreement. A π pulse reverses every static difference among the spins, so an echo at 2τ returns the signal to e^(−2τ/T₂) — 0.472 at 60 ms in the first figure, from a signal that had vanished at 10 — and what it cannot return is the phase gained while a molecule wandered, a loss of exp(−γ²G²D·TE³/12) that grows as the cube of the time and, read as e^(−bD), shows a stroke as tissue keeping 0.670 of its signal where its surroundings keep 0.449. The echo gives back everything the spins remembered, and what they forgot is the record of how far they walked.