Medical imaging: X-rays, CT, nuclear medicine and ultrasound

Core ideas

X-rays: production, attenuation and contrast

Electrons from a heated filament are accelerated through 20–150 kV onto a tungsten anode. Only about 1% of their energy becomes X-rays; the rest becomes heat, so the anode rotates and is cooled. The continuous spectrum runs up to a maximum photon energy $eV = hc/\lambda_{\min}$, with characteristic lines on top. The tube current sets the intensity and the tube voltage sets the photon energy. An aluminium filter removes low-energy photons that the skin would absorb anyway, cutting the dose without spoiling the image.

$\mu$ is the linear attenuation coefficient and $x_{1/2}$ is the half-value thickness. AQA also uses the mass attenuation coefficient $\mu_m = \mu/\rho$.

At diagnostic energies the photoelectric effect dominates and rises steeply with proton number (roughly $\propto Z^3$). Bone, which contains calcium, absorbs far more than soft tissue, so it shows up with good contrast. Soft tissues have similar values of $\mu$, so high-$Z$ contrast media are used: a barium sulfate meal for the gut, or iodine injected into blood vessels. Intensifying screens and flat-panel detectors need fewer photons, which lowers the dose.

CT: an X-ray tube rotates around the patient, sending a thin fan beam through one slice to a ring of detectors. A computer combines the readings from many angles to find the attenuation of each small volume (voxel), building a 2D image of the slice. Stacking the slices gives a 3D image. CT can distinguish soft tissues with similar $\mu$, but its dose is hundreds of times that of a chest X-ray.

Tracers and effective half-life

A tracer is a radioactive substance, attached to a compound the target organ absorbs, whose distribution is imaged from outside the body.

A good diagnostic tracer emits gamma only (alpha or beta would just add dose), has a half-life of a few hours, and can be attached to a suitable compound. The body also excretes it, with a biological half-life $T_B$, so the effective half-life is shorter than both $T_B$ and $T_P$:

The gamma camera

1. Collimator: lead with long parallel holes. It absorbs every photon that is not travelling along the holes, so each point on the crystal matches one point in the patient.
2. Scintillator: a sodium iodide crystal. Each gamma photon it absorbs produces a flash of thousands of visible photons.
3. Photomultiplier tubes turn each flash into an electrical pulse and amplify it.
4. Position logic and pulse-height analysis: the relative signals from neighbouring tubes locate the flash, and pulses that are too small (from scattered photons) are rejected. A computer builds the image.

The resolution is only about 5 mm, but the image shows how an organ is working.

PET scanning

A $\beta^+$ tracer (usually F-18 FDG) collects in metabolically active tissue such as tumours. Each positron travels about 1 mm, then annihilates with an electron. Their total momentum is about zero, so two photons fly off in opposite directions, each with energy

A ring of detectors records coincident pairs of photons. Each annihilation lies on the line joining the two detectors, a distance $c\Delta t/2$ from its midpoint, where $\Delta t$ is the difference in arrival times. Millions of events build up a 3D map of the tracer.

Ultrasound

Medical ultrasound uses frequencies of about 1–15 MHz. A piezoelectric crystal changes shape when a p.d. is applied, so an alternating p.d. makes it vibrate and emit ultrasound. Returning echoes deform it and generate a p.d. The transducer sends short pulses and listens between them.

The acoustic impedance is $Z = \rho c$ ($\text{kg m}^{-2}\,\text{s}^{-1}$). At a boundary, the fraction of the intensity reflected is

At an air–skin boundary almost all of the ultrasound is reflected, so a coupling gel with $Z$ close to that of skin is used to exclude the air.

• A-scan: echo amplitude against time along one line. Depth $= ct/2$, because the pulse travels there and back.
• B-scan: many lines, each echo shown as a bright dot, building a 2D image.
• Doppler (OCR A, Eduqas, WJEC): $\Delta f = 2fv\cos\theta/c$ for blood moving at speed $v$. The factor 2 appears because the cells both receive and re-emit a shifted wave.

MRI and dose (AQA, Eduqas, WJEC)

MRI: a strong field of 1–3 T makes hydrogen nuclei precess at the Larmor frequency, $f = 42.6 \times 10^{6}B$. A radio-frequency pulse at that frequency excites them, and as they relax they emit radio signals. Different tissues relax at different rates, which gives excellent soft-tissue contrast, and gradient coils locate each signal. MRI is non-ionising, but it is expensive, slow and noisy, and it is unsuitable for some patients with metal implants.

Dose (Eduqas, WJEC): absorbed dose $D = E/m$ in grays; equivalent dose $H = DW_R$ in sieverts ($W_R = 1$ for X-rays, gamma and beta, 20 for alpha); effective dose . A chest X-ray is about 0.02 mSv, a chest CT about 6 mSv, and UK background radiation about 2.7 mSv per year.

Worked examples

Exam technique

• Layers: add the $\mu x$ terms for each layer, then take one exponential. An intensity of $1/16$ means 4 half-value thicknesses.
• A-scans: halve the echo time. For a layer’s thickness, use the time between its two echoes.
• Describe questions (gamma camera, PET): name each part and say what it does, in order along the photon’s path.
• Comparisons: for each technique, give the physics, what the image shows, its resolution, and its risk and cost.

Common mistakes

Quick recap

• $eV = hc/\lambda_{\min}$. The current sets intensity and the voltage sets photon energy.
• $I = I_0e^{-\mu x}$, , and contrast media are high- absorbers.