The image made of frequencies

The angular momentum that is not a rotation found that a spin carries angular momentum without anything going round, and with it a magnetic moment. The turn that has to be made twice rotated one and found it needed seven hundred and twenty degrees to come back. The experiment that defines spin followed Stern and Gerlach’s magnet, which sorts spins by the force a field gradient puts on their moments. And the push that comes out sideways found that a torque on a spinning thing makes it precess rather than topple.

Put those together for the nucleus of a hydrogen atom, a single proton, in a magnetic field. Its moment feels a torque, its angular momentum turns the torque into precession, and the rate of precession is proportional to the field. That is the whole of nuclear magnetic resonance, and it became one of the most important instruments of medicine through a single further step: make the field different in different places. Then the precession frequency says where a proton is, and a picture can be made from a signal that never passes through a lens. This essay follows that step, and why the picture it makes is so much sharper than the radio waves that carry it.

A clock in every proton #

A proton’s magnetic moment in a field B precesses about the field at the Larmor frequency $f = \bar\gamma B$, where for a proton $\bar\gamma$ is 42.58 megahertz per tesla. In the 1.5 tesla field of a common clinical scanner every hydrogen nucleus in the body is a clock running at 63.87 megahertz, in the band used for FM radio. The body is mostly water and fat, each molecule carrying two or more hydrogen nuclei, about seventy million million million million of them per litre.

Left alone the nuclei point every which way, and their moments cancel. In the field a slight excess lines up with it — about five in a million at body temperature, a balance struck between the tiny energy difference of the two orientations and the thermal agitation that the exponential that decides everything weighs — and that excess is a net magnetisation, small but made of an enormous number of nuclei. A pulse of radio waves at exactly the Larmor frequency tips it away from the field, by resonance, as the frequency that gets an answer found a driven oscillator answering only at its own frequency. Tipped, the magnetisation precesses, and a precessing magnet is a rotating magnet: it induces a voltage in a coil of wire placed near it, at the precession frequency, as any spinning magnet in a dynamo does.

That voltage is the signal, and it is worth being clear about what kind of signal it is. It is not radiation picked up by an antenna. The coil sits close to the body, inside the near field of the precessing magnetisation, and picks up its changing flux directly, the inductive coupling that the coupling that is the same both ways showed is equal in both directions — which is why the coil’s sensitivity to a patch of tissue is the field the same coil would make there if it carried a current.

From a beam of molecules to a body #

The resonance was first seen not in a body or a beaker but in a beam. In 1938 Isidor Rabi sent a beam of molecules through a pair of magnets of the kind Stern and Gerlach had used, which deflected each molecule according to its nuclear magnetic moment’s orientation, with a uniform field between them carrying a weak oscillating field. When the oscillation’s frequency matched the nuclei’s precession, the oscillating field flipped their orientations, the second magnet no longer refocused them onto the detector, and the beam’s current dropped. The Stern–Gerlach apparatus, which cannot measure a free electron’s spin, became a detector of nuclear flips, and the frequency of the dip measured the nuclear moment to a part in a thousand.

Eight years later Felix Bloch and Edward Purcell, independently, detected the same resonance in ordinary solid and liquid matter — water and paraffin wax — by the signal the precessing nuclei induced in a coil, which is the method every scanner uses. The step from there to imaging took twenty-seven more years, because a uniform field, which is what a spectroscopist works hard to make, is exactly the thing that hides where each nucleus is.

Position written as frequency #

In a perfectly uniform field every proton precesses at the same frequency and the signal says nothing about where any of them are. Paul Lauterbur’s idea, published in 1973, was to add to the strong uniform field a weak field that rises steadily across the body: a gradient. A gradient of ten millitesla per metre, less than a hundredth of the main field, makes the field at one side of a head differ from the field at the other by a couple of millitesla, and the precession frequency then rises across the head by 425.8 hertz for every millimetre.

Now each position has its own frequency, and the signal from the whole body is a chord, a sum of all those frequencies, each as loud as the number of protons at the corresponding position. Separating a chord into its notes is what a spectrum does, and the spectrum is a picture: for the four tubes in the figure, four bands, each as wide as its tube and as tall as its density of water. The axis of position and the axis of frequency are the same axis.

A signal that is a transform #

What the coil records looks nothing like the tubes. At any instant it is a single number, the sum over the whole row of every proton’s contribution, each with the phase its precession has accumulated since the gradient was switched on. At the centre of the readout the phases have been arranged to agree and the signal is large; away from it the protons at different positions have drifted out of step by amounts proportional to their positions, and the sum oscillates and shrinks.

That phase is the key. After a time t in the gradient, a proton at position x has gained a phase $2\pi\bar\gamma G x t$ relative to one at the centre, so the signal is

$$S(t) = \int \rho(x)\,e^{-2\pi i\,\bar\gamma G x t}\,dx,$$

which is the Fourier transform of the density ρ(x), evaluated at the spatial frequency $k = \bar\gamma G t$. Every sample is one number in that transform, and as the readout goes on the scanner walks through spatial frequencies from low to high. The image is not recorded; its transform is, one point at a time, and the image is computed afterwards.

This is the mathematics the fan of plane waves inside every beam used to describe a beam as a sum of waves travelling at every angle, and that sharpness has to be paid for used to show that a short pulse needs a wide band of frequencies. Here the same pair of variables, position and spatial frequency, is laid out in time, because a gradient turns elapsed time into spatial frequency.

Sharpness bought with time #

The inverse transform of the 256 samples returns the four tubes exactly. Keeping fewer of them — only the samples near the centre of the readout, which is the same as listening for a shorter time — returns a blurrier picture: with sixteen samples the narrow tube is a single bump and the edges ring, with sixty-four they are recognisable. The finest detail an image can show is set by the highest spatial frequency recorded, which is $\bar\gamma G T$ for a readout of length T, so the pixel is

$$\Delta x = \frac{1}{\bar\gamma\,G\,T}.$$

The gradient and the time trade against each other. A stronger gradient spreads the frequencies further apart per millimetre, so the same readout reaches higher spatial frequencies; a longer readout lets the phases from a given gradient wind further. Neither depends on the wavelength of anything.

Resolution a thousand times finer than the wave #

The radio waves that tip the protons and the signal they give back are at sixty-four megahertz, with a wavelength of 4.7 metres in air and about half a metre inside the body, where the water slows them. Any imaging method that focused those waves would be limited to resolving detail of about that size. Magnetic resonance imaging resolves a millimetre routinely and a few tenths of a millimetre in research scanners, five hundred to five thousand times finer.

There is no conflict with the diffraction limit, because no wave is focused. The wiggle faster than any wave in it found one way past that limit, by catching the evanescent near field before it decays. MRI uses another: it never asks the wave where anything is. The position is written into the frequency by the gradient, inside the body, and the wave only carries the frequency out. The resolution is a property of the gradient coils and the receiver’s clock.

What does limit it is physiology and physics of a different kind. A readout cannot usefully last longer than the time the protons’ precession stays in step — tens of milliseconds in tissue, the relaxation time the width that is a lifetime identifies with a spectral width — because after that the signal is gone. And gradients cannot be made arbitrarily strong or switched arbitrarily fast inside a person: a rapidly changing field induces currents in the body, and above a certain rate of change those currents stimulate nerves and make muscles twitch. Clinical gradients are a few tens of millitesla per metre, switched in a fraction of a millisecond, close to that limit.

What makes tissues look different #

The density of protons varies little between soft tissues — most are seventy to eighty per cent water — so a picture of density alone would be nearly featureless. What makes magnetic resonance images so useful in medicine is that the signal also depends on how quickly the protons’ magnetisation recovers after being tipped and how quickly their precession falls out of step, two relaxation times that depend strongly on the water’s molecular surroundings. Water bound loosely to large molecules, as in fat or in the fibres of white matter, relaxes quickly; free water, as in fluid or swollen tissue, slowly. In 1971 Raymond Damadian reported that tumour tissue relaxed more slowly than normal tissue, one of the observations that pointed towards medical use.

The scanner chooses how much each relaxation time weighs in an image by when it sends its pulses and when it listens: repeating the excitation before the magnetisation has fully recovered makes slowly recovering tissue dark, and waiting a long time before reading out makes quickly dephasing tissue dark. The same anatomy can be shown with fluid bright or dark, fat bright or suppressed, by changing timings rather than hardware. None of that changes where things appear, which the gradients decide; it changes how bright they are, and that is where most of the clinical information lies.

Three dimensions from one axis #

The figures use one gradient and one axis. A real image needs three, and the same idea supplies them in three different forms. A gradient applied while the tipping pulse is sent, with the pulse’s frequency band chosen narrow, tips only the protons in a thin slice where the field matches — slice selection, the frequency picking a plane. Within the slice, one direction is read out with a gradient on, as in the figures — frequency encoding. The other direction is encoded by applying a gradient briefly before the readout, so each row of protons acquires a phase in proportion to its position, and repeating the whole measurement with different strengths of that brief gradient walks through the second spatial frequency — phase encoding.

The scanner thus fills a grid of spatial frequencies, a plane called k-space, line by line, and a two-dimensional inverse transform turns it into a picture. A typical image of 256 by 256 pixels needs 256 lines of k-space, each a separate excitation of the slice, which is why a scan takes minutes and why a patient must keep still: motion during the minutes of acquisition mixes lines of k-space recorded with the body in different places. Peter Mansfield’s echo-planar method of 1977 reads a whole plane of k-space after a single excitation by switching the gradients rapidly back and forth, which makes a picture in a tenth of a second and is the basis of functional imaging of the brain.

Chemistry mistaken for place #

The method reads every frequency as a position, and not every difference in frequency is a difference in position. The electrons round a nucleus shield it slightly from the applied field, by an amount that depends on the molecule it is in, so the same field gives slightly different frequencies to protons in different chemical surroundings. Protons in the fat chains of lipids precess about three and a half parts per million slower than those in water. Reconstructed as though every proton were water, fat appears shifted along the frequency-encoding axis, by half a millimetre at the gradient in the figure and by more at gentler ones, leaving a bright band on one side of a fat–water boundary and a dark one on the other.

That artefact is the image’s version of the measurement nuclear magnetic resonance was first used for. Felix Bloch and Edward Purcell observed the resonance in bulk matter in 1946, and within a few years chemists found that the few-parts-per-million shifts identify the chemical surroundings of each nucleus, which made NMR spectroscopy the most powerful method for working out the structures of molecules. In imaging the shift is something to correct for, or to exploit: scanners can separate fat and water deliberately, by exploiting exactly the frequency difference that displaces them, and spectroscopic imaging maps chemistry instead of water.

The price of a smaller pixel #

Resolution has a cost the formula does not show. Each pixel’s signal comes from the protons inside it, so halving the pixel’s width in all three directions leaves an eighth as many protons to supply it, while the electrical noise from the body and the coil stays the same. Recovering the lost signal-to-noise ratio by averaging repeated measurements needs sixty-four times as long, since noise falls only as the square root of the time spent. That is why a research scanner resolving a tenth of a millimetre in a small animal uses a much stronger magnet, which raises the five-in-a-million polarisation and with it the signal, and why the millimetre of clinical imaging is set as much by the minutes a patient can lie still as by the gradients.

Where the model stops #

The figures are one-dimensional and idealised. The tubes are sampled on a grid that matches the reconstruction’s exactly, which is why the full transform returns them perfectly; real objects are not confined to a grid and their transforms are truncated, giving the ringing visible in the short readouts at every edge. The protons are taken to keep precessing in step for the whole readout, with no relaxation and no noise; in tissue the signal decays during the readout and noise is often what sets the useful resolution, since a smaller pixel holds fewer protons. The field is taken to be perfectly uniform apart from the gradient, where real magnets and real bodies distort it by parts per million, which shifts positions just as the chemical shift does. And the receiving coil is assumed equally sensitive everywhere, which a real coil is not.

What the figures establish is the principle: that a gradient makes position and frequency the same variable, and that the resulting image is the Fourier transform of the signal, with a resolution set by gradient and time.

Still open: how few samples an image needs #

A full image needs a full grid of k-space only if nothing is known about the picture in advance, and medical images are not arbitrary: they are mostly smooth regions with edges, which can be described with far fewer numbers than pixels. Since the mid-2000s, compressed-sensing methods have reconstructed images from a fraction of k-space, sampled irregularly, by finding the simplest image consistent with the samples taken, and machine-learning methods now do the same by learning what images of a particular organ look like. Both shorten scans several-fold. What remains open is how far such reconstructions can be trusted — whether an image built partly from prior expectation can hide a small abnormality that was not in the training set, or invent a plausible structure that was never there — and how to certify a reconstruction that a radiologist cannot check against the signal by eye.

The habit worth carrying away is to ask whether an instrument measures a thing or its transform. A gradient G makes a proton’s precession frequency say where it is, 425.8 Hz per mm at 10 mT/m in a 1.5 T scanner, so the signal is the Fourier transform of the body and the image is computed, with a pixel of 1/(γ̄GT) — 0.62 mm from a 3.76 ms readout, while the wave carrying it is 4.7 m long. Nothing is focused, and nothing about the wavelength enters.