Medical physics

Piezo-electric transducers: generating and detecting ultrasound

A piezo-electric crystal, such as quartz or a lead zirconate titanate ceramic, changes shape when a potential difference is applied across it, and it generates an e.m.f. across it when its shape is changed. The effect works in both directions, and that is exactly why one crystal can both send and receive a pulse of ultrasound. A transducer is that crystal mounted in a housing with electrodes on its opposite faces, backed with a damping material so that it rings for only a few cycles.

To generate ultrasound you apply an alternating p.d. at the resonant frequency of the crystal across the electrodes. The crystal is cut so that its thickness is half a wavelength of sound in the material, so it vibrates strongly at that frequency and radiates a beam of ultrasound into whatever it is pressed against. To detect ultrasound the process is reversed: an arriving wave forces the crystal to vibrate, the vibration changes its shape, and it produces an alternating e.m.f. of the same frequency across its faces. That tiny signal is amplified and displayed. Most scanners use the same crystal alternately as transmitter and receiver, switching between the two thousands of times a second.

The audible limit of hearing is about 20 kHz, so ultrasound is anything above that, and medical scanners work between about 1 MHz and 20 MHz. Frequency decides the trade-off that shapes every scan: a shorter wavelength gives better resolution, because two boundaries closer together than about one wavelength cannot be told apart, but the attenuation coefficient also rises with frequency, so a higher frequency is absorbed sooner and penetrates less deeply. Abdominal and obstetric scanning therefore uses about 2 MHz to 5 MHz, while an eye, which is only a few centimetres across, is scanned at 10 MHz to 20 MHz. Ultrasound is used in preference to X-rays wherever it can do the job because it is not ionising and so carries no radiation risk.

Because the pulse is short, the scanner can time it. The crystal is excited for a few microseconds, then listens for echoes. The time between transmitting the pulse and receiving an echo gives the depth of the boundary, and because pulses are repeated many times a second, the image can be updated fast enough to show a moving fetus or a beating heart.

• Piezo-electric effect: a p.d. across the crystal makes it change shape, and changing its shape makes it produce a p.d.
• An alternating p.d. at the resonant frequency (set by the thickness of the crystal) generates the ultrasound.
• An incoming ultrasound wave vibrates the crystal, which generates an e.m.f. that is amplified and displayed.
• Medical ultrasound is 1 MHz to 20 MHz: higher frequency gives better resolution but is attenuated more.
• Ultrasound is non-ionising, which is why it is preferred for scanning a fetus.

Echoes from boundaries and specific acoustic impedance

A pulse of ultrasound travels through tissue until it reaches a boundary between two media. Part of the pulse is reflected there and part continues, and the reflected pulse - the echo - returns to the transducer. The echo is detected and timed, and because the pulse has travelled down to the boundary and back again, the depth is depth = c t / 2, where c is the speed of sound in the tissue (about 1540 m s-1 for soft tissue) and t is the time between transmitting and receiving. A boundary that reflects strongly gives a large echo; one that reflects weakly gives a small echo, and that difference is what builds up the picture.

How much is reflected depends on the specific acoustic impedance of each medium, defined as Z = rho c, where rho is the density of the medium in kg m-3 and c is the speed of sound in it in m s-1. The unit of Z is therefore kg m-2 s-1. Physically, Z measures how hard it is to make the medium move: a dense, stiff medium such as bone has a large Z and resists being vibrated, while air has a tiny Z because it is light and springy.

The fraction of the incident intensity reflected at a boundary is the intensity reflection coefficient, IR / I0 = (Z1 - Z2)^2 / (Z1 + Z2)^2. For example, at a fat-muscle boundary Z1 = 1.4 x 10^6 kg m-2 s-1 and Z2 = 1.7 x 10^6 kg m-2 s-1, so the difference is 3.0 x 10^5 and the sum is 3.1 x 10^6. Then IR / I0 = (3.0 x 10^5)^2 / (3.1 x 10^6)^2 = 9.0 x 10^10 / 9.6 x 10^12 = 9.4 x 10-3, so only about 1% of the intensity is reflected and 99% carries on. That is why echoes from soft tissue are so weak that the receiver has to be very sensitive.

The equation also explains the coupling gel. The specific acoustic impedance of air is about 4.3 x 10^2 kg m-2 s-1 and that of soft tissue about 1.6 x 10^6 kg m-2 s-1, so (Z1 - Z2)^2 / (Z1 + Z2)^2 is almost 1: a thin layer of air between the transducer and the skin would reflect about 99.9% of the ultrasound straight back and almost nothing would enter the body. A gel whose impedance is close to that of soft tissue removes that boundary and lets the pulse through. The same idea explains why an air-filled lung, a bone surface or a gallstone produces such a strong echo, and why a crack in a metal casting - a metal-air boundary - is so easy to detect.

Attenuation, and turning echoes into a picture: A-scans and B-scans

Ultrasound does not travel through tissue for free: it is attenuated by absorption, as energy is turned into internal energy of the tissue, and by scattering at small irregularities. The intensity that survives a path length x is given by I = I0 e^(-mu x), where mu is the linear attenuation coefficient of the tissue in m-1 and x is in metres. Because mu x must be dimensionless, convert every length to metres before you substitute. Take ultrasound of intensity 5.0 x 10-3 W m-2 entering a tissue with mu = 12 m-1 and travelling 8.0 cm: mu x = 12 x 0.080 = 0.96, so I = 5.0 x 10-3 x e^(-0.96) = 5.0 x 10-3 x 0.38 = 1.9 x 10-3 W m-2. Note how the intensity falls by an ever-decreasing amount and never reaches zero, because the attenuation is exponential rather than a fixed loss per centimetre.

An A-scan (amplitude scan) displays the received signal as the height of a voltage spike against time on an oscilloscope-like display. The transmitter pulse starts the time base, and each boundary along the line of the beam produces a spike at a time that gives its depth. An A-scan therefore gives distances along one line - the length of an eye, the position of the midline of the brain, or the depth of a flaw inside a casting - but it is not a picture of a cross-section.

A B-scan (brightness scan) moves the transducer across the body, and instead of plotting the echo height it sets the brightness of a spot from the echo amplitude, using the position of the transducer for the horizontal coordinate and the echo time for the depth. Electronics combines thousands of these lines and builds up a two-dimensional cross-section of the body, which is what most people picture when they think of an ultrasound scan. Scanning fast enough produces a moving image, so the movement of a heart valve or of a fetus can be watched directly.

Attenuation and depth of penetration are inseparable. mu increases roughly in proportion to frequency, so doubling the frequency roughly doubles the attenuation in dB. A 2 MHz beam may penetrate 20 cm of soft tissue, whereas a 15 MHz beam used to measure the thickness of a cornea is useful over only a centimetre or so. Choosing the frequency is a compromise between the resolution you want and the depth you need to reach.

Producing X-rays: electron bombardment and the minimum wavelength

X-rays are produced when fast-moving electrons are stopped by a metal target. In an X-ray tube, a tungsten filament is heated so that electrons are emitted from it by thermionic emission, and a large potential difference - typically 30 kV to 150 kV - accelerates them across an evacuated tube towards a heavy metal target, usually tungsten. When an electron strikes the target it is decelerated violently, and most of its kinetic energy becomes internal energy in the target while a small fraction, well under 1%, is radiated as X-ray photons. Because so much energy goes into heating, the target is a rotating anode so that the hot spot moves, and it is cooled by oil or water.

The spectrum produced has two parts. The continuous (bremsstrahlung) spectrum comes from electrons that lose different amounts of energy in different numbers of deflections, so every photon energy up to a maximum is possible. Superimposed on it are sharp characteristic peaks at wavelengths fixed by the energy levels of the target atoms. The continuous spectrum has a sharp short-wavelength cut-off, lambda_min, produced when a single electron gives all of its kinetic energy to a single photon: eV = hc / lambda_min, so lambda_min = hc / (eV) - or, if you prefer the frequency form, fmax = eV / h.

A worked value. With an accelerating p.d. of 80 kV, eV = 1.60 x 10-19 x 8.0 x 10^4 = 1.28 x 10-14 J and hc = 6.63 x 10-34 x 3.00 x 10^8 = 1.99 x 10-25 J m, so lambda_min = 1.99 x 10-25 / 1.28 x 10-14 = 1.55 x 10-11 m. Every photon in the beam has a wavelength longer than this, and the cut-off is a hard edge on the spectrum.

The two controls on the tube do different jobs. Raising the accelerating p.d. increases the energy of each electron, which moves the cut-off to shorter wavelength and makes the beam more penetrating, so the image improves; the cut-off depends only on the p.d. and not on the target material. Raising the filament current boils off more electrons per second, which increases the intensity of the beam without changing the cut-off, because the energy each individual electron can hand over is fixed by the p.d. alone. The target material fixes the wavelengths of the characteristic lines and helps determine how efficiently X-rays are produced.

• Electrons are released by thermionic emission from a heated tungsten filament.
• They are accelerated through a high p.d. and stopped by a heavy metal target such as tungsten.
• Almost all of the kinetic energy becomes heat, so the anode rotates and is cooled.
• The continuous spectrum ends at lambda_min = hc / (eV), where one electron gives all of its energy to one photon.
• Increasing the p.d. shortens lambda_min and makes the beam more penetrating; increasing the filament current increases the intensity.

X-ray imaging, attenuation and contrast in tissue

A single X-ray image is a shadow picture. X-rays leave a small source, pass through the patient and reach a detector or film on the far side. Where the beam passes through material that absorbs strongly, less intensity reaches the detector, so that part of the film is less exposed and appears white. Bone, which absorbs well, shows white; fat and muscle show grey; air-filled lung shows nearly black because it absorbs almost nothing.

X-rays are removed from the beam by the same exponential law as ultrasound, I = I0 e^(-mu x), but the physics of the removal is quite different. Being ionising electromagnetic radiation, an X-ray photon interacts with the atoms of the material in one of two main ways. In photoelectric absorption the photon is completely absorbed and its energy is used to eject a tightly bound electron from an inner shell; this is a strong interaction with high atomic number atoms and low-energy photons, and it grows roughly as the cube of the atomic number. In Compton scattering the photon collides with a loosely bound outer electron, transfers part of its energy and is deflected out of the beam in a new direction, so it never reaches the detector. Compton scattering depends mainly on the number of electrons present, and therefore on the density of the tissue rather than on the atomic number.

The attenuation of X-rays in a material is described by its linear attenuation coefficient mu in m-1, which depends on the photon energy, the density of the material and the atomic number of its atoms. Because bone contains calcium and phosphorus (higher Z and denser) and soft tissue is mostly water, bone has a much larger mu. Contrast in an X-ray image is the difference in the transmitted intensity between two parts of the image, and it is largest where the two tissues differ most in mu x. Bone against soft tissue gives good contrast; soft tissue against soft tissue, such as a tumour inside a muscle, gives almost none, because mu x is nearly the same. That is why contrast media of high atomic number are used: a suspension of barium sulphate is swallowed to outline the digestive tract, and iodine compounds are injected into blood vessels, because they absorb X-rays much more strongly than the tissue around them.

X-rays are ionising, so every X-ray examination delivers a dose to living tissue and that is the reason for the safety rules. Absorbed radiation can ionise atoms and molecules in cells, breaking chemical bonds and producing free radicals, which can kill cells outright or damage DNA so that a cell mutates or starts to divide out of control and becomes a cancer. The effect of the radiation on matter is therefore ionisation, and its consequences in tissue depend on the dose and on which tissue is exposed. Radiographers use the smallest exposure that gives a usable image, collimate the beam to the region of interest, use lead shielding over the gonads and a lead apron for anyone else in the room, stand behind a lead screen, and keep a record of the dose received by staff. The dose from a diagnostic X-ray must always be justified by the benefit of the diagnosis.

Computed tomography: many projections, one slice

A single X-ray image has a fundamental weakness: the body is three-dimensional but the film is flat, so everything along a beam is superimposed. A tumour lying in front of a rib may be completely hidden by the brighter image of the bone, and no amount of staring at the film will separate them. Computed tomography (CT) solves this by measuring the beam from many directions and letting a computer do the separating.

In a CT scanner, an X-ray tube and a ring of detectors rotate around the patient in one plane. At each of several hundred positions a narrow fan of X-rays crosses the same thin section of the body and the detectors record how much intensity has been transmitted, which is a measure of the total mu x along each of hundreds of different lines. Because each line passes through many different tissues, any single measurement is a sum of contributions, but the whole set of sums, taken at enough different angles, can be solved by computer to give the attenuation of each small volume element (voxel) in the section. The result is a two-dimensional image of that section in which each pixel is labelled with a number related to mu, on the Hounsfield scale: air is about -1000, water is 0, and dense bone is about +1000. Nothing is hidden, because the overlapping has been undone mathematically.

To build a three-dimensional image, the patient is moved a little along an axis of the body and the whole process is repeated for the next section. Stacking the reconstructed 2D sections one behind another gives a volume that can be displayed as any chosen slice or viewed from any angle, which is what makes CT so useful for planning surgery and for locating a tumour precisely. A modern scanner reconstructs slices about 1 mm thick, and a whole scan takes only a few seconds.

The drawback of gathering all this information is the radiation dose. A CT scan uses many exposures rather than one, so the dose to the patient is typically two or three orders of magnitude larger than that from a single plain radiograph of the same region. CT is therefore reserved for the cases where the extra information changes what is done - a suspected bleed in the head, a tumour that must be measured, a complex fracture - and the benefit of the diagnosis has to be weighed against the risk from the ionising radiation.

PET scanning: tracers, positron annihilation and the gamma camera

A tracer is a substance containing radioactive nuclei that is introduced into the body and absorbed by the tissue being studied, so that the tissue can be imaged from outside. In positron emission tomography the tracer is one that decays by beta-plus decay. The most common is a glucose analogue labelled with fluorine-18, which has a half-life of 110 minutes and is made in a cyclotron. Cells that use a lot of glucose - cancer cells, and the brain - take up the tracer most, so the concentration of the tracer is a map of metabolic activity rather than of anatomy.

In beta-plus decay a proton-rich nucleus converts a proton into a neutron and emits a positron, the antiparticle of the electron: proton -> neutron + positron. The positron is the positively charged partner of the electron, with the same mass, and in tissue it travels only about a millimetre before it meets an electron. When a particle meets its antiparticle the two annihilate: both masses disappear and their total mass-energy is released as radiation. Two particles of rest mass 9.11 x 10-31 kg each therefore release 2 m c^2 = 2 x 9.11 x 10-31 x (3.00 x 10^8)^2 = 1.6 x 10-13 J, and because two photons are produced, each carries half of it: 8.2 x 10-14 J, which is 5.1 x 10^5 eV (about 511 keV).

Both conservation laws decide what the photons do. Mass-energy is conserved, which fixes the energy of each photon at m e c^2 for an electron at rest. Momentum is also conserved, and the electron and positron are almost at rest when they annihilate, so their total momentum is close to zero; two photons of equal energy therefore have to fly off in exactly opposite directions so that their momenta cancel. That single fact is the basis of the whole technique. It also means the process is impossible with one photon, because a single photon cannot carry away both the energy and the momentum.

The two gamma-ray photons are penetrating enough to leave the body, unlike the positron, which is stopped within about a millimetre. A ring of detectors surrounds the patient, and when two detectors on opposite sides record a photon at the same instant - a coincidence - the annihilation event must have happened somewhere on the straight line joining them. Each coincidence therefore defines a line in space, and after many thousands of events the computer adds the lines together (back-projection) to produce an image of the tracer concentration in each part of the tissue. Recording the arrival times precisely is what makes the coincidence possible, and because the two photons are detected together, absorption in the surrounding tissue affects both of them and largely cancels out, so a PET scan gives a quantitative measure of activity rather than just a picture.

• The tracer decays by beta-plus decay, emitting a positron: proton -> neutron + positron.
• The positron travels about 1 mm in tissue and annihilates with an electron.
• Mass-energy is conserved, so the pair's rest energy 2 m c^2 appears as two photons of 5.1 x 10^5 eV each.
• Momentum is conserved, so the two photons travel in exactly opposite directions.
• Detectors in coincidence define a line of response; processing many lines gives an image of tracer concentration.