Ultrasound imaging

A crystal that emits and detects

Ultrasound is sound above about 20 kHz, and medical scanners run in megahertz. One component both makes it and detects it, the piezoelectric transducer. That is a crystal which deforms when a pd is applied across it, and which generates a pd when something deforms it. Drive it with an alternating pd at its resonant frequency and it vibrates, launching a sound pulse. Let a returning echo squeeze it and it produces a measurable pd.

So the transducer alternates between two roles. It transmits for a microsecond, then detects the echoes that return. That switching is what lets the whole technique work with one probe held against the skin.

Pulse and echo

Everything else is timing. A pulse leaves the transducer, part of it reflects off each boundary between tissues, and the echoes arrive back after a delay. Sound of speed c that took time t to go there and back has found a boundary at depth

That factor of two is what most exam questions turn on, because the pulse covers the distance twice. An A-scan plots echo strength against time along one line, which suits a single depth measurement such as the length of an eye. A B-scan sweeps the beam and turns each echo into a bright dot on a two-dimensional map, and that is the moving picture of an antenatal scan. Nobody is photographing anything. You are looking at a map of boundaries, assembled from arrival times.

How much a pulse can tell you

Timing an echo gives a position. It does not give unlimited detail, and the two things that cap the detail are worth knowing because they are the same two on every pulse-echo instrument ever built, from a bat to a ship's sonar to a radar set.

Start with the duration of the pulse. A pulse is not an instant; it lasts some time Δt and therefore occupies a length cΔt of tissue while it travels, and its echo is just as long. Two boundaries a distance Δd apart return echoes separated in time by 2Δd/c, that factor of two arriving for the same reason it arrived in d = ct/2. If that separation is shorter than the pulse itself, the second echo begins before the first has finished, the two run into one blur, and the scanner reports one boundary where there are two. The boundaries are just separable when 2Δd/c equals Δt, so the finest detail the pulse can resolve is about cΔt/2.

So shorter pulses see finer detail, and that pushes you straight into the second limit. A pulse cannot be shorter than about one cycle of the wave carrying it, so the shortest pulse a beam of wavelength λ$\lambda$λ can make is roughly λ$\lambda$λ long. Detail finer than about a wavelength is out of reach whatever the electronics do, and medical scanners are built around that fact: at 1540 m s⁻¹ a 3 MHz beam has a wavelength near half a millimetre, and half a millimetre is about the resolution such a scan achieves.

Both limits work against penetration depth. Raising the frequency shortens both the wavelength and the pulse, which is the gain, but attenuation climbs steeply with frequency, so the finer beam is attenuated sooner. Shallow structures are scanned at high megahertz and deep organs at low, and the sonographer's choice of probe is that trade-off in practice.

Acoustic impedance, and the gel

How much of a pulse reflects at a boundary is set by the acoustic impedance of the media either side,

the product of density and sound speed. At every interface between two media the arriving wave splits in two: part of it is reflected straight back and the remainder is transmitted onwards into the second medium, and for the head-on, normal-incidence arrival every exam question assumes, with no absorption at the boundary itself, the two impedances alone set the share. Similar impedances let most of the pulse through. Badly mismatched ones reflect most of it. The fraction of the intensity reflected is

That equation sends the sonographer reaching for the coupling gel. Air's impedance is thousands of times smaller than tissue's, so an air gap between probe and skin reflects nearly all the ultrasound before it ever gets in. The gel displaces the air with a material whose impedance sits close to tissue, letting the pulse through. This is called impedance matching, and the term is worth using. Inside the body, the useful echoes come from mild mismatches between organs, and the strong ones from tissue meeting bone.

Ultrasound also fades as it travels. Absorption and scattering attenuate the pulse exponentially with depth, just as matter attenuates X-rays, and the attenuation rises steeply with frequency. That sets the sonographer's trade. High frequency resolves finer detail but is attenuated sooner, so shallow work uses high megahertz and deep organs use low. One board asks you to put a number on that fade, and the closing sections take it up.

The other non-ionising imagers

The option puts two more non-ionising instruments alongside ultrasound. The endoscope is total internal reflection put to work. A coherent bundle of optical fibres carries an image out of the body, with each fibre holding its place in the bundle so the picture arrives intact, while a second, incoherent bundle carries light in.

The third instrument is the MR scanner, and it gets a section to itself, because it images without ionising rays and without sound, and gives the best soft-tissue contrast of the methods compared here.

The MR scanner

The patient lies inside a strong magnetic field, a few teslas from a superconducting magnet. The body is mostly water, every water molecule carries two hydrogen nuclei, and each of those protons behaves as a tiny magnet. In the field they do not simply snap into line. They precess about the field direction, the way a leaning spinning top wheels around the vertical, at a frequency set by the strength of the field, and for the fields an MR magnet supplies that frequency sits in the radio range.

The scanner then supplies energy at exactly that frequency. A radio-frequency pulse at the precession frequency is absorbed by the precessing protons and tips them away from the field direction. When the pulse ends they relax back into line, giving the energy up again as a radio signal that coils around the patient pick up.

The relaxation time, the time the protons take to settle back, depends on the tissue their water sits in. Protons in fat relax at a different rate from protons in fluid or in a tumour, so the timing of the returning signal labels the tissue that sent it. That is the contrast mechanism, and it is why MR tells apart soft tissues that X-rays cross almost identically. One board note: AQA states that de-excitation relaxation times will not be examined, so for AQA this paragraph is the why behind the scan, not a required point.

Locating the signal is done with the field itself. The precession frequency depends on the field strength, so gradient coils make the field vary slightly from place to place across the patient, and position along a gradient then maps to frequency. One gradient alone cannot pin down a point, so real scanners switch gradients to encode the remaining directions in the signal's frequency and phase, and a computer reconstructs cross-sectional images in any plane from that encoded signal, with nothing rotating around the patient.

Weigh it as the exam does, both ways at once. In its favour: excellent soft-tissue contrast, the finest of the modalities compared here, sections in any plane, and no ionising radiation, so repeat scans add no radiation dose, though every scan still passes the safety checks on implants, heating and any contrast agent first.

Against it: the superconducting magnet makes it expensive to buy and to run; a scan is slow and loud, with the patient holding still for tens of minutes inside a narrow bore; and the strong field rules out patients with MR-unsafe pacemakers or ferromagnetic implants, which the field would disturb or drag. Many modern implants are MR Conditional and scannable under controlled protocols; the safety label, rather than the presence of a pacemaker, determines whether a scan can proceed.

Attenuation with depth, a CIE equation

CIE alone sets numerical attenuation questions on ultrasound, so read this section only if CIE is your specification. Absorption and scattering strip a fixed fraction of the surviving intensity out of every further millimetre of tissue, which is the recipe for an exponential and gives the law the X-rays lesson already uses:

with μ the attenuation coefficient of the tissue, in per metre, and x the distance travelled through it. None of the mathematics is new. What is new is a feature of the board's own formula sheet: it prints the radioactive-decay exponential, x = x₀e^−λt, where x stands for an activity or a number of nuclei, and it does not print this one. An exponential on the page is not the exponential you need, so this equation has to be recalled, for ultrasound and for X-rays alike.

The coefficient is where the sonographer's trade-off appears numerically. μ climbs steeply with frequency, so the higher frequency that resolves finer detail is also attenuated faster. μ depends on the tissue too, and bone attenuates so much more strongly than soft tissue that ultrasound cannot usefully see through the skull or into air-filled lungs.

Doppler ultrasound and the speed of blood, an OCR equation

OCR adds one measurement the other boards leave out, so read this section only if OCR A is your specification. Bounce ultrasound off something that is moving and the echo returns at a shifted frequency, the same Doppler effect that drops the pitch of a siren as it passes. Red blood cells are the moving reflectors, and the shift in the returning echo measures how fast they are travelling.

Two features of the equation are worth understanding rather than memorising. First, the shift is doubled. The cells are moving relative to the transmitter, so they meet an already shifted frequency; they then re-radiate that wave while still moving relative to the receiver, so it shifts a second time. Two shifts, one factor of two. Second, only the part of the velocity along the beam shifts anything, and with θ the angle between the beam and the flow that part is v cos θ. Put the two together:

Here c is the speed of ultrasound in tissue, about 1540 m s⁻¹, and not the speed of light. The equation is printed on the OCR data sheet, so the work is in choosing the right angle and the right c rather than in recalling it.

The cosine explains how the probe is held. Aim it square across the vessel, θ = 90°, and cos θ is zero: the blood is moving neither towards the probe nor away from it, and the machine reports no flow whatever, however fast the blood is going. Tilt the probe along the flow and the shift grows, which is why scanning angles are chosen deliberately, typically somewhere around 45° to 60°.