X-Ray CT & Therapy

When an X-ray beam of initial intensity I₀ passes through a material of thickness x, its intensity decreases exponentially: I = I₀ e^(−μx). The quantity μ (mu) is the linear attenuation coefficient, measured in m⁻¹, and it depends on both the photon energy and the material's atomic number and density. Bone has a much higher μ than soft tissue, which is why bones appear bright on X-ray images. In CT, the scanner measures the transmitted intensity along thousands of paths at many angles, and from these measurements it infers how μ varies throughout the slice — effectively mapping tissue type across the entire cross-section.

For each rotation angle θ, the detector records a projection — a set of line integrals of μ along parallel (or fan-shaped) X-ray beams. Mathematically, each projection value p(r, θ) equals the integral of μ(x, y) along a line at distance r from the centre at angle θ. This mapping from the 2D function μ(x,y) to its line integrals is called the Radon transform. Reconstruction means inverting this transform. The most widely used algorithm is filtered back-projection (FBP): each projection is sharpened using a high-pass filter (to correct for blurring), then 'smeared back' across the image plane at its acquisition angle. Summing all back-projections over 180° or more converges to the true μ distribution.

CT images are displayed using the Hounsfield scale, which normalises attenuation relative to water: HU = 1000 × (μ_tissue − μ_water) / μ_water. Air registers at −1000 HU, water at 0 HU, and dense cortical bone at around +1000 HU. Radiologists use 'windowing' — adjusting the display range — to highlight structures of interest: a narrow window centred near 0 HU reveals soft tissue contrast, while a wide window centred higher shows bone detail. This quantitative scale makes CT uniquely useful for measuring tissue density and monitoring changes over time.

A medical linac accelerates electrons using a series of microwave-powered copper cavities. Microwaves (typically at 3 GHz, S-band) create oscillating electric fields that push electrons forward in synchrony with the wave, gaining ~6–18 MeV of kinetic energy over about 1 metre. At these energies electrons travel at >99.9% of the speed of light, so relativistic effects are significant — their kinetic energy far exceeds their rest-mass energy (0.511 MeV). The accelerated electron beam hits a water-cooled tungsten target, producing bremsstrahlung (braking radiation) X-rays in the megavoltage range. These high-energy photons penetrate deeply into the body with a dose profile that peaks a few centimetres below the skin — the 'skin-sparing' effect — delivering maximum dose to deep tumours.

A multi-leaf collimator (MLC) sits in the linac head: it consists of up to 160 individually motorised tungsten leaves that shape the beam cross-section to match the tumour's outline from any angle. During IMRT, the gantry rotates around the patient while the MLC continuously reshapes the beam, modulating the fluence (photon flow per unit area) so that the dose conforms tightly to the tumour volume in 3D. A computerised treatment planning system uses CT data to calculate the dose distribution, optimising leaf positions using inverse-planning algorithms. The goal is to exceed the tumour lethal dose (typically 60–80 Gy delivered in 30 fractions) while keeping nearby organs below their dose tolerance thresholds.

• I — Transmitted intensity (W m⁻²): Intensity of the X-ray beam after passing through the material
• I_0 — Incident intensity (W m⁻²): Original intensity of the X-ray beam before entering the material
• \mu — Linear attenuation coefficient (m⁻¹): Material-specific constant describing how strongly the material absorbs and scatters X-rays; depends on photon energy and atomic number
• x — Thickness of material (m): Path length of the X-ray beam through the absorbing material

Problem. A CT scanner sends X-rays through 24 cm of soft tissue (μ = 0.18 cm⁻¹) followed by 2.0 cm of bone (μ = 0.48 cm⁻¹). If the initial beam intensity is I₀ = 1.20 × 10⁵ W m⁻², calculate the transmitted intensity and express the result in Hounsfield units assuming μ_water = 0.18 cm⁻¹ for the bone region (as a simplified check).

Answer. Transmitted intensity ≈ 607 W m⁻²; the bone's Hounsfield value ≈ +1667 HU.

• X-Ray Production and Attenuation — CT scanning relies directly on X-ray attenuation through tissue; understanding bremsstrahlung and the photoelectric effect underpins both CT image contrast and linac beam production.
• Ultrasound Imaging — Ultrasound is a complementary non-ionising imaging modality; comparing it with CT illustrates different trade-offs in resolution, tissue contrast, and radiation dose.
• Radiation Dose and Safety — Both CT (diagnostic) and linac radiotherapy (therapeutic) involve ionising radiation, making absorbed dose, effective dose, and dose limits essential companion knowledge.
• Relativistic Kinetic Energy — Electrons in a medical linac reach >99.9% of the speed of light; relativistic mechanics is needed to correctly describe their energy gain and momentum.