Illustrated Physics
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The depth an ultrasound image pays for its sharpness

An ultrasound scanner sees finer detail at higher frequencies, because shorter waves make shorter pulses, and sees less deep, because tissue absorbs higher frequencies more strongly. Both effects scale as one over the frequency, so they cancel in the one number that matters: how many separate depths fit along a line of the image. For soft tissue, which absorbs nearly in proportion to frequency, that number is about four hundred whatever the frequency — set by the tissue and by the scanner's electronics and by nothing else. A scan of the abdomen at three megahertz and a scan of the skin at fifty are the same picture at different magnifications.
14 min read 5 figures What stays the sameMade into an instrument

Assumes: The distance that takes the treble out · The answer that cannot come first

The distance that takes the treble out found that sound loses its high frequencies with distance because absorption grows with frequency: as its square in a simple fluid, more slowly in real air, where molecular relaxations take their share. The answer that cannot come first found that absorption and the bending of waves are one fact, tied by causality, and where the loudness goes followed the absorbed energy into heat. Later essays carried absorption into X-rays, semiconductors and pigments.

This essay takes absorption into the body, where it decides what an ultrasound scanner can see. The trade is familiar to anyone who has watched a scan: a probe for the abdomen sees deep and blurry, a probe for a tendon sees sharp and shallow. What is less familiar is how exact the trade is. In soft tissue, sharpness and depth are exchanged one for one, and the number of distinct depths a scanner can resolve along a line of its image comes out independent of the frequency it uses. The reason is a peculiarity of how tissue absorbs sound — and that peculiarity is itself a fact about tissue’s structure.

From submarines to the womb

Ultrasound imaging began as a way to find submarines. Paul Langevin, in Paris during the First World War, used the piezoelectric effect of quartz — a crystal that changes shape when a voltage is applied and makes a voltage when squeezed — to send pulses of ultrasound into the sea and hear their echoes, the beginning of sonar. Industry used the same pulse-echo principle from the 1930s to find flaws inside metal castings. The step into medicine was taken in the 1940s and 1950s, and its turning point is usually dated to 1958, when Ian Donald, an obstetrician in Glasgow, and the engineer Tom Brown published images of abdominal masses and, soon after, of the foetus in the womb, made with an industrial flaw detector adapted for the body.

What made it medicine rather than engineering was that soft tissue, nearly transparent to the method’s predecessors, turned out to return echoes from every boundary between tissues and from the fine structure within them, faint but plentiful. The images were coarse, because the frequencies used were low enough to see through a whole abdomen, and the trade this essay is about was there from the first scan: to see further in, the operators had to accept seeing less sharply.

How tissue absorbs sound

How tissue and water absorb sound, against frequency. The attenuation of ultrasound, in decibels per centimetre, against frequency in megahertz, on logarithmic scales, for typical values in fat, soft tissue such as liver and muscle, blood, and water. Water absorbs as the square of the frequency, by the viscous mechanism of the air's treble loss, and is nearly transparent: 0.055 dB/cm at 5 MHz. Tissues absorb fifty to two hundred times more, and nearly in proportion to frequency: about 2.5 dB/cm at 5 MHz in soft tissue, 7.5 at 15. The near-proportional law is the signature of many relaxation processes with a broad spread of times, each absorbing best near its own frequency — the proteins, membranes and structures of tissue — summed into a nearly straight line.
Fig. 1 Attenuation in decibels per centimetre against frequency for fat, soft tissue, blood and water. Water absorbs as f2f^2, 0.055 dB/cm at 5 MHz; soft tissue nearly as f, about 2.5 dB/cm at 5 MHz and 7.5 at 15.

Water absorbs ultrasound weakly and as the square of the frequency, by the viscous and thermal mechanism that also removes the air’s treble: at five megahertz a beam loses only five hundredths of a decibel per centimetre. Soft tissue, which is mostly water, absorbs fifty to two hundred times more, and with a different law: almost exactly in proportion to frequency, at about half a decibel per centimetre for every megahertz, with variations between tissues — more in muscle across its fibres, less in blood, more in fat.

A loss proportional to frequency is not what any single mechanism gives. A single relaxation process, a molecular arrangement that takes a definite time to respond to the compression, absorbs as f2f^2 well below its relaxation frequency and levels off above it, the shape that the earlier essay found for the oxygen and nitrogen in air. Tissue contains a great many such processes — proteins changing shape, water moving between bound and free states, membranes and fibres responding at their own rates — with relaxation times spread over many decades. Each contributes a hump of absorption near its own frequency, and the sum of a great many humps spread evenly in logarithm of frequency is a nearly straight line, the power law with exponent near one. The same superposition of relaxations gives many polymers and biological materials their nearly frequency-independent loss per cycle.

Causality adds a consequence that can be measured. Absorption rising in proportion to frequency forces, through the Kramers–Kronig relations, a speed of sound that rises slowly with frequency, by about the logarithm of the frequency, a fraction of a per cent across the diagnostic band. Ultrasound pulses in tissue are therefore slightly dispersive, and the dispersion has been measured and found to match the absorption, as the answer that cannot come first says it must.

An echo that fades with depth

The echo that fades with depth. The strength of an echo from a reflector at a given depth in soft tissue, relative to one at the skin, against depth, for scanners at 2, 5, 10 and 20 MHz: a loss of 2α₀fd decibels, out and back, with α₀ = 0.5 dB per cm per MHz. The dashed line is a usable dynamic range of 60 dB, below which echoes are lost in the electronic noise. The deepest echo each frequency can bring back is 30.0 cm at 2 MHz, 12.0 cm at 5 MHz, 6.0 cm at 10 MHz, 3.0 cm at 20 MHz. That is why abdominal and obstetric scans use 2 to 5 MHz, to see fifteen or twenty centimetres in, and scans of the thyroid, tendons or breast use 10 to 15 MHz, which see only a few centimetres but see them more sharply.
Fig. 2 Echo strength from a reflector against depth in soft tissue, out and back, at 2, 5, 10 and 20 MHz, with a 60 dB usable range dashed. The deepest echo comes from 30, 12, 6 and 3 cm.

A scanner sends a short pulse into the body and listens for echoes from boundaries between tissues, and the time each echo takes says how deep the boundary is, at a speed of sound close to 1,540 metres a second in soft tissue. An echo from a depth d has travelled 2d, and lost 2α0fd2\alpha_0 f d decibels on the way. The scanner can amplify echoes from deeper down more than those from near the skin, compensating for the expected loss, but it cannot recover an echo weaker than its own electronic noise. The ratio between the strongest echo and the noise, the usable dynamic range, is typically sixty to eighty decibels, and it sets the deepest echo the scanner can use: dmax⁡=D/2α0fd_{\max} = D/2\alpha_0 f.

At two megahertz that is thirty centimetres, deeper than the abdomen; at five, twelve; at ten, six; at twenty, three. The choice of probe follows: two to five megahertz for the liver, the kidneys and a pregnancy, ten to fifteen for the thyroid, a tendon or a breast, twenty to fifty for the skin and the front of the eye, and higher still for probes threaded inside blood vessels, which need to see only a few millimetres into the vessel wall.

Sharpness as a pulse length

The sharpness along the beam is set by the pulse. Two reflectors closer together in depth than half the pulse’s length return echoes that overlap and cannot be told apart, the limit that the layer too thin to see found for seismic pulses in rock. A transducer rings for a cycle or two when struck by its driving voltage, so the pulse is a couple of wavelengths long, and the axial resolution is about one wavelength: half a millimetre at three megahertz, a tenth of a millimetre at fifteen. Making the pulse shorter means using more bandwidth, the price sharpness always pays, and a transducer’s bandwidth is a fraction of its centre frequency, so the pulse’s length in cycles is roughly fixed and its length in millimetres goes as one over the frequency.

Two lines that run parallel

Depth and sharpness, both falling as 1/f. Against frequency on logarithmic scales: the deepest echo a 60 dB scanner can recover in soft tissue, D/2α₀f, and its axial resolution, half the length of a two-cycle pulse, Nc/2f, both in centimetres. Both fall as one over the frequency, so the two lines are parallel: at 3 MHz the scanner sees 20 cm deep with 0.51 mm resolution; at 15 MHz, 4 cm with 0.10 mm; at 50 MHz, 1.2 cm with 31 μm. Choosing a frequency slides the image's window up and down the scale of sizes without changing how many details fit into it.
Fig. 3 The deepest echo for a 60 dB scanner, D/2α₀f, and the axial resolution, Nc/2f, against frequency on logarithmic scales: parallel lines. At 3 MHz, 20 cm and 0.51 mm; at 15 MHz, 4 cm and 0.10 mm; at 50 MHz, 1.2 cm and 31 μm.

So both the depth a scanner reaches and the finest detail it resolves fall as one over the frequency, and on logarithmic axes they are parallel lines. Raising the frequency moves the image’s window down the scale of sizes, making both its depth and its pixels smaller, without changing their ratio. A scanner at three megahertz sees twenty centimetres with half-millimetre detail; at fifteen, four centimetres with tenth-millimetre detail; at fifty, a centimetre with thirty-micrometre detail. They are the same image at three magnifications.

A count of details that frequency cannot move

A number of details that frequency cannot change. The number of separately resolvable depths along one line of an ultrasound image, the deepest echo divided by the axial resolution, against frequency, for dynamic ranges of 50, 60 and 80 dB in soft tissue; dashed, the same for a medium that absorbed like water, as the square of the frequency, with 60 dB. In tissue the count is D/(α₀Nc), the same at every frequency: 325 for 50 dB, 390 for 60 dB, 519 for 80 dB. It is a property of the tissue, through α₀, and of the electronics, through D, and changing the frequency only trades depth for sharpness one for one. In a medium with an f² law the count would fall as 1/f, from 88,548 at 1 MHz to 1,771 at 50 MHz. It is the near-linear absorption of tissue that makes the trade exact.
Fig. 4 Resolvable depths along a line — deepest echo over axial resolution — against frequency, for 50, 60 and 80 dB in soft tissue: 325, 390 and 519 at every frequency. Dashed: an absorber with an f2f^2 law, where the count falls as 1/f.

Dividing one line by the other gives the number of separately resolvable depths along a line of the image,

dmax⁡Δz=Dα0Nc,\frac{d_{\max}}{\Delta z} = \frac{D}{\alpha_0 N c},

where N is the pulse’s length in cycles. The frequency has cancelled. For sixty decibels of dynamic range, two-cycle pulses and tissue’s half a decibel per centimetre per megahertz, it is about three hundred and ninety, at any frequency. Only two things can raise it: a scanner with a larger dynamic range, more decibels between the strongest echo and the noise, or tissue that absorbs less per cycle, which is not available.

The cancellation depends on the absorption law. A medium that absorbed as f2f^2, like water, would lose depth as the square of the frequency and resolution only as its first power, and the count of resolvable depths would fall as one over the frequency, from tens of thousands at one megahertz to under two thousand at fifty. It is the near-linear absorption of tissue, the sum of its many relaxations, that makes the trade exactly one for one. The structure of tissue at the scale of its molecules and membranes sets the information content of every ultrasound image of it.

Ultrasound shares this kind of invariant with other imaging methods. An optical microscope’s field of view and resolution trade through its numerical aperture and magnification, keeping a roughly fixed number of resolvable spots across an image; a radar’s range and range resolution trade through its bandwidth and power. In each, choosing a scale does not change how many details fit, and the number that does is set by something the user cannot choose.

Across the beam as well as along it

The same cancellation applies, more loosely, to the other direction in the image. A modern probe is an array of a hundred or more small transducer elements, each driven with its own delay, so that the pulses from all of them arrive at a chosen point together and the beam is focused there; on reception the echoes are delayed and summed in the same way, focusing again. The width of the focused beam is about the wavelength times the focal depth divided by the width of the array, the same diffraction limit as any lens, and so it too shrinks as one over the frequency.

The field of view across the image is set by the width of the array and the angle it sweeps, which do not change with frequency, so across the beam a higher frequency does buy more resolvable spots. But a high-frequency probe for shallow work is made small, to be held against a neck or a wrist, and its focal depth is short, and in practice the images from probes of every frequency come out a few hundred resolvable spots on a side. A scanner’s display can show many more pixels than that; it does not show more detail.

Seeing with the harmonic

A harmonic made inside the body. The amplitude, in decibels, of a 3 MHz ultrasound beam going into soft tissue and of the 6 MHz second harmonic that the tissue's own nonlinearity generates from it, against depth, one way, with the harmonic's scale set 20 dB below the fundamental at its peak. The fundamental fades steadily, 15.0 dB by 10 cm. The harmonic starts at nothing, grows as the wave distorts, and fades at twice the rate, peaking at 1/2α₁ = 2.9 cm. Imaging with the harmonic gains a sharper beam, made only where the fundamental is strongest, and loses the haze of echoes from near the skin, where the harmonic has not yet formed — paid for by working at twice the frequency over the return path.
Fig. 5 A 3 MHz beam’s amplitude against depth in soft tissue, one way, and the 6 MHz harmonic the tissue’s nonlinearity makes from it, which grows from nothing and peaks at 1/2α₁ = 2.9 cm.

There is one way round the trade that scanners now use routinely. Tissue is slightly nonlinear: the compressions of a strong ultrasound wave travel a little faster than its rarefactions, so the wave steepens as it goes, the process the front that steepens until it cannot followed in air, and in steepening it generates harmonics, energy at twice, three times the frequency it was sent at. The second harmonic grows from nothing near the transducer, strongest where the fundamental is strongest, and is itself absorbed at twice the rate; it peaks a few centimetres in and fades.

A scanner that transmits at three megahertz and listens at six — tissue harmonic imaging — gets the sharper resolution of six megahertz on the way back without paying six megahertz’s absorption on the way in. The harmonic is generated mostly in the strongest part of the beam, its centre, so the effective beam is narrower and the image’s lateral resolution better, and it is not generated in the first centimetre or two below the skin, so the haze of reverberations between skin, fat and muscle that blurs ordinary images is largely missing from it. Most modern diagnostic images are harmonic images.

Why the transmitter cannot simply shout

The obvious way to raise the dynamic range, and with it the count of resolvable depths, is to send stronger pulses. Two safety limits stop it. Absorbed sound heats tissue, at a rate proportional to the absorption coefficient and the intensity, and regulators cap the time-averaged intensity a scanner may deliver, reported on its screen as a thermal index, so that no tissue warms by more than a degree or so. And a strong pulse’s rarefactions pull on the liquid in tissue, and below a threshold of negative pressure small gas nuclei can grow into bubbles and collapse violently — cavitation, which can damage cells. The tendency to cavitate falls with frequency, since a short rarefaction gives a bubble less time to grow, and scanners display a mechanical index, the peak negative pressure divided by the square root of the frequency, kept below a fixed ceiling.

Both limits bound the transmitted pressure, so the dynamic range is set mainly at the other end, by how quiet the receiving electronics can be made, and that has improved by tens of decibels since the first scanners. Every ten decibels of extra range adds about sixty-five resolvable depths at any frequency, by the formula above — one of the few ways the fixed count has actually been raised.

Ultrasound’s other limits

Depth and axial resolution are only part of an image. Resolution across the beam is set by the transducer’s width and focusing, as for any lens, and is usually several times coarser than the axial resolution. Every image is covered in speckle, the granular texture that the grain that is in the light found in laser light, made by the interference of echoes from many scatterers too small to resolve; it looks like tissue texture and is not. And sound must get into the body at all: the impedance of skin is so close to water’s and so far from air’s that a probe in air sends almost nothing in, the mismatch the lever that lets air speak to water found, which is why every scan begins with a layer of gel.

The same physics that limits imaging is used deliberately elsewhere. Absorbed ultrasound heats tissue, and focused beams of a few megahertz at high power raise the temperature at their focus enough to destroy a small volume of tissue deep inside the body without a cut, the basis of focused-ultrasound treatments of tumours and of some movement disorders, where the beam is aimed through the skull. Doppler shifts of the echoes from moving blood, which the shift a mirror gives twice explained, measure its speed.

Where the model stops

The figures use a single attenuation coefficient for soft tissue, half a decibel per centimetre per megahertz, a single exponent of one and a single speed of sound, where real tissues vary by tens of per cent in each and real paths cross several tissues, fat absorbing more than liver and blood less. The dynamic range is a single number standing for the whole receiving chain, and the axial resolution is idealised as half a two-cycle pulse. The harmonic figure uses the simplest model of second-harmonic growth in a medium with linear absorption, without diffraction of the beam, and gives the shape of the depth dependence rather than absolute levels. And the count of resolvable depths is along one line; an image is a fan of lines, and its lateral resolution follows other rules.

Within those simplifications the central claim is robust, because it rests only on two proportionalities that hold well across the diagnostic band: absorption in tissue proportional to frequency, and pulse length proportional to wavelength.

Still open: how much an image can be pushed past its count

Several methods now aim to beat the fixed count. Coded excitation sends long pulses with a known pattern of phases and compresses them on reception, as radar does, raising the energy sent without raising the peak power that tissue safety limits, and so buying dynamic range. Ultrafast imaging sends plane waves instead of focused beams and forms images thousands of times a second, trading resolution per image for averaging over many. Super-resolution methods localise injected microbubbles, each smaller than the resolution, and build images of blood vessels at a tenth of the wavelength by plotting where thousands of them pass, as fluorescence microscopy beat its diffraction limit. How far each can go, given the tissue’s absorption and the limits on heating and on the mechanical effects of the pressure, and whether together they can raise the number of resolvable details by an order of magnitude, are being tested in clinics now.

The habit worth carrying away is to divide one limit by another before trying to improve either. In soft tissue the deepest echo a scanner recovers, D/2α₀f, and its axial resolution, Nc/2f, both fall as 1/f, so the number of resolvable depths along a line is D/(α₀Nc) — about 390 for 60 dB — at three megahertz and at fifty alike. Frequency chooses the magnification; the tissue and the electronics choose how much there is to see.

Part 8 of 8

This essay is one argument about Attenuation. The others:

The same absorber, spread evenly and gathered into patches. The fraction of light transmitted by a layer holding a fixed amount of absorber, on a logarithmic axis, against the optical depth it would have if spread evenly, for the absorber spread evenly (solid) and gathered into patches covering 50 per cent and 10 per cent of the area, the rest left clear. At an average optical depth of 2 the even layer transmits 13.5 per cent; with the absorber in patches over half the area, 50.9 per cent; over a tenth, 90.0 per cent — almost all of it through the gaps. The average of e^(−τ) over a layer is never less than e^(−average τ): Jensen's inequality, which holds for any convex function. Beer's law describes a uniform absorber and overstates the attenuation of every other kind. The pigment that absorbs less for being packed Part 7 — Dissolve a dye evenly in water and it absorbs a certain fraction of the light passing through. Gather the same dye into grains — the same molecules, the same amount, in the same volume — and it absorbs less. The light has not changed and neither have the molecules. The grains shadow their own insides, and the light that would have been caught by the shaded molecules slips through the gaps between grains instead. The effect is called packaging, and it is why the chlorophyll inside large algae is worth less per molecule than the chlorophyll in small ones, and why an ocean's colour does not say directly how much chlorophyll is in it.

Where this goes next

An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound. The distance that takes the treble out Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles. Absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else. The answer that cannot come first A medium absorbs at some frequencies and bends light at all of them, and those look like two separate facts to be measured separately. They are one fact. The requirement that a material respond after it is asked rather than before ties the absorption at every frequency to the refraction at every other, by an integral — and a material given the two halves independently answers before the question arrives. Echoes from a layer that thins to nothing. Synthetic seismic echoes from a layer between two identical rocks — the top reflecting +R, the base −R — for thicknesses of 0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 75, 100 m, sounded with a 30 Hz pulse in rock of 3000 m/s, whose wavelength in the layer is 100 m. Time runs downward; each trace is one thickness. Thick, the layer gives two separate echoes, one from each face, and its thickness can be read from the time between them. As it thins the two echoes overlap and merge. Near 20 m, about a fifth of a wavelength, they interfere constructively and the combined echo is strongest, 1.45 times one face's echo. Thinner still, the combined echo keeps a fixed shape — the pulse's derivative — and simply weakens in proportion to the thickness; nothing in its timing changes. The layer too thin to see A seismic survey sends a pulse of sound into the ground and listens to the echoes from every boundary where the rock changes. A thick layer returns two echoes, one from its top and one from its base, and the time between them gives its thickness. A thin one returns one echo, and below about a quarter of a wavelength the timing says nothing more: every thin layer looks the same, just weaker as it thins. It has not become invisible. It has become unmeasurable in the one way that was being used, and its thickness has moved into the echo's strength and its spectrum.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Acoustic attenuationAxial resolutionDynamic rangeHarmonic imagingPower law absorptionPulse echo imagingRelaxationUltrasound