Lesson 19: Medical Physics (Option)

Medical physics applies waves, fields, and quantum physics to the human body — imaging the invisible and treating the untreatable. This A-level option covers X-ray imaging (attenuation and contrast), CT scanning (reconstruction from projections), ultrasound (reflection and Doppler blood-flow measurement), MRI (nuclear magnetic resonance), and the physics of vision and hearing. Every technique is core physics pointed at the body: exponential attenuation, wave reflection, resonance, and the Doppler effect saving lives daily.

X-rays — Seeing Through the Body

X-rays (0.01–10 nm, keV–MeV photons) attenuate exponentially: I = I₀e^{−μx} — μ the attenuation coefficient (depends strongly on Z — photoelectric absorption ∝ Z³ roughly — bone (Ca, Z=20) absorbs far more than soft tissue). Half-thickness x½ = ln2/μ. Contrast: bone vs tissue differ in μ — the shadow image. Worked: 50 keV X-rays, μ_bone = 0.8 cm⁻¹, μ_tissue = 0.2 cm⁻¹. Through 10 cm tissue + 2 cm bone: I/I₀ = e^{−(0.2×10 + 0.8×2)} = e^{−3.6} ≈ 0.027; through 12 cm tissue: e^{−2.4} ≈ 0.091 — bone shadows are 3× darker. Contrast agents: barium (Z=56) meals coat the gut; iodine (Z=53) injected for blood vessels — high-Z, high-μ, temporary.

X-ray production: bremsstrahlung (electrons decelerated in tungsten target — continuous spectrum) + characteristic lines (K-shell knocked out, L→K transition emits Kα — tungsten 59 keV). Tube voltage sets max photon energy (E_max = eV — 100 kV → 100 keV photons); current sets intensity (more electrons). Filters (aluminium) remove soft X-rays (absorbed by skin, contributing dose without image) — beam hardening as a safety feature. Rotating anodes spread the heat (kW of electron beam on mm²).

Dose and risk: absorbed dose (gray = J/kg), equivalent dose (sievert = gray × radiation weighting factor — alpha 20, gamma 1), effective dose (tissue weighting). Chest X-ray ~0.02 mSv; CT abdomen ~8 mSv; annual background ~2.7 mSv (UK — radon-heavy Cornwall higher). ALARP governs medical exposures: benefit must outweigh risk (a CT finding a tumour outweighs 8 mSv). Lead aprons, collimation (narrow beams), and digital detectors (less dose than film) minimise exposure.

Worked — attenuation thickness: a beam must be reduced to 10% of its incident intensity passing through tissue with μ = 0.35 cm⁻¹. From I/I₀ = e^{−μx} = 0.10: x = ln(10)/0.35 = 2.303/0.35 ≈ 6.6 cm. In half-thicknesses: x½ = ln2/0.35 ≈ 1.98 cm, and 10% ≈ (½)^3.32, so 3.32 × 1.98 ≈ 6.6 cm ✓. Fluency both ways (exponent and half-thickness counting) is what exam speed looks like.

CT Scanning — Reconstruction

CT: an X-ray tube rotates around the patient, detectors measuring attenuation along hundreds of angles; computation reconstructs the 2D slice (Radon transform — the mathematics of projections). Each voxel's μ mapped to Hounsfield units: HU = 1000(μ − μ_water)/μ_water — water 0, air −1000, bone +1000+, blood ~+40. Windowing displays tissues: brain windows, bone windows — the radiologist's contrast controls. A chest CT is ~500 slices — millimetre anatomy in 3D.

Worked — Hounsfield units: a tissue has μ = 0.21 cm⁻¹ where water has μ_water = 0.20 cm⁻¹. HU = 1000(0.21 − 0.20)/0.20 = +50 — soft-tissue range (muscle and many tumours live at +20 to +80; fat is negative, around −50 to −100). The HU scale converts a raw attenuation measurement into a diagnostic number with water as the universal reference — calibrate once, compare everywhere.

Worked concept: two projections of a 2×2 grid suffice to illustrate — horizontal beams give row sums, vertical give column sums; solving the equations yields the four pixels (with ambiguities resolved by more angles). Real CT uses filtered back-projection over 1000+ angles — the algorithm is applied mathematics (Fourier methods). Spiral CT: continuous rotation + table movement — whole-body scans in seconds; cardiac CT gates to the heartbeat (ECG-triggered acquisition — freezing motion). Contrast with plain X-ray: CT separates overlapping structures (slices vs shadows) and quantifies μ (Hounsfield numbers diagnose: fresh blood vs old, tumour vs cyst). Cost: higher dose (more projections) and computation. PET-CT fuses metabolic (Lesson 16's positrons) with anatomical imaging — cancer staging's gold standard. The option's techniques combine: physics is modular, medicine uses the modules together.

Ultrasound — Sound as a Probe

Ultrasound (> 20 kHz; medical 1–15 MHz): generated by piezoelectric crystals (AC voltage → mechanical vibration — the reverse of the microphone effect). Wavelength in tissue (v ≈ 1540 m/s): at 5 MHz, λ = 0.31 mm — millimetre resolution (resolution ~λ). Reflection at boundaries: fraction reflected depends on acoustic-impedance mismatch Z = ρv — soft-tissue/bone reflects strongly (bright), tissue/tissue weakly. Time of flight gives depth: d = vt/2 (there and back) — the A-scan principle; B-scan sweeps to build 2D images.

Worked: echo returns 0.13 ms after the pulse: d = 1540(0.13×10⁻³)/2 ≈ 0.10 m = 10 cm deep. Gel coupling: air's Z mismatches skin hugely (99.9% reflected) — gel matches impedances, letting sound in. Frequency trade-off: higher f → better resolution but more attenuation (α ∝ f — high frequencies absorbed) — obstetrics uses 3–5 MHz (deep), eyes/thyroid 10–15 MHz (shallow detail). No ionising radiation — safe in pregnancy (the decisive advantage over X-ray/CT).

Doppler ultrasound: moving blood shifts the reflected frequency — Δf = 2f₀v cosθ/c (factor 2: the blood acts as moving observer AND moving source). Worked: 5 MHz probe, blood at 0.5 m/s toward probe (θ = 0): Δf = 2(5×10⁶)(0.5)/1540 ≈ 3.25 kHz — audible! (Doppler fetal monitors make it audible deliberately.) Worked — angled probe: a 7.5 MHz probe views a vessel at θ = 60° with blood speed 0.35 m/s: Δf = 2(7.5×10⁶)(0.35)(cos60°)/1540 = 2.63×10⁶/1540 ≈ 1,705 Hz ≈ 1.7 kHz. The cosθ factor is the clinical trap: near θ = 90° the shift vanishes — sonographers keep θ ≤ 60° so the correction stays reliable. Colour Doppler maps velocity (red toward, blue away — conventions) — cardiologists watch valves leak in real time. The Doppler effect (Lesson 6) as a blood-flow meter.

MRI — Resonance Imaging

Protons (hydrogen nuclei) are magnetic dipoles — in a strong B₀ field (1.5–3 T, superconducting magnets — Lesson 11) they align parallel/antiparallel (tiny excess parallel — Boltzmann). Larmor precession: spins precess at f = γB₀/2π (γ/2π = 42.6 MHz/T for protons) — at 1.5 T: 63.9 MHz (radiofrequency). Worked: at 3.0 T, f = 42.6 × 3.0 ≈ 128 MHz — FM-radio frequencies, which is why MRI suites are Faraday-caged against radio interference. RF pulse at the Larmor frequency tips spins (resonance — like pushing a swing); they re-emit RF as they relax — the signal. Different tissues relax at different rates (T1: realignment; T2: dephasing) — the contrast mechanism (fat vs water vs tumour distinct).

Spatial encoding: gradient coils make B vary with position — Larmor frequency becomes position-dependent (frequency encodes x, phase encodes y) — Fourier transform reconstructs the image (mathematics again!). No ionising radiation (RF is non-ionising) — safe for repeated scans; but: projectile risk (ferromagnetic objects fly into the bore — screening essential), claustrophobia, loud gradients, and cost (£1M+ scanners). Functional MRI: blood-oxygen changes alter T2* (BOLD signal) — mapping brain activity (neuroscience's workhorse).

Comparing modalities: X-ray (cheap, fast, ionising, 2D shadows); CT (3D, quantitative, higher dose); ultrasound (safe, real-time, operator-dependent, poor through bone/air); MRI (superb soft-tissue contrast, safe, slow, expensive, metal restrictions); PET (metabolic, radioactive tracer, fused with CT). 'Choose the modality' exam questions: match the physics to the clinical need (pregnancy → ultrasound; brain tumour detail → MRI; trauma bleed → CT speed; fracture → X-ray).

Vision, Hearing, and Exam Technique

The eye: cornea + lens focus (converging system, ~60 dioptres total) onto the retina; accommodation (ciliary muscles change lens shape); defects — myopia (eyeball too long, corrected with diverging lenses), hyperopia (too short, converging), astigmatism (cylindrical). The ear: pinna → canal (resonance ~3 kHz) → eardrum → ossicles (lever + area ratio amplifying pressure ~20×) → cochlea (frequency-separated hair cells — place theory). Hearing loss: conductive (mechanical) vs sensorineural (nerve) — diagnosed by comparing air/bone conduction (Rinne/Weber tests — physics in the clinic).

Exam technique: (1) attenuation questions — write I = I₀e^{−μx}, identify μ for each material, sum exponents for layers; (2) ultrasound — d = vt/2 (the half is the mark), Doppler with the factor 2 derived; (3) MRI — Larmor frequency from f = (γ/2π)B₀ = 42.6 MHz/T × B₀, resonance condition stated, T1/T2 as the contrast source; (4) 'compare techniques' — table format: physical principle, resolution, safety, cost, best use — examiners reward structured comparison over prose; (5) dose questions — gray vs sievert distinguished, ALARP invoked, background compared (mSv scale familiarity: background 2.7, chest X-ray 0.02, CT 8).

Medical physics is applied core physics — attenuation (Lesson 15's gamma), waves (Lessons 6–7), resonance (Lesson 6), magnetism (Lesson 11), quantum (Lesson 8's photons). The option tests whether you can recognise old physics in new clothes — the ultimate synoptic skill.

Common mistakes

• Forgetting the factor 2 in ultrasound ranging (pulse goes there AND back) and in Doppler (double shift). Both factors are the commonest lost marks in this option.
• Gray vs sievert: gray is absorbed energy per kg; sievert folds in radiation type and tissue sensitivity. A dose in 'gray' cannot be compared with background (quoted in mSv) until weighted.
• μ units and layer sums: exponents add layer by layer with each material's own μ and thickness — a single average μ for mixed tissue/bone is wrong.
• MRI 'radiation' confusion: RF photons are non-ionising (MHz, ~10⁻⁷ eV each); MRI's hazards are projectiles, implants and heating — not dose. Stating MRI is 'radiation-free' without qualification also loses the nuance.
• Larmor slip: f = (γ/2π)B₀ with γ/2π = 42.6 MHz/T — don't divide by 2π twice.

Exam Experience

(a) A beam passes through tissue with μ = 0.35 cm⁻¹. Calculate the thickness that reduces the intensity to 10% of its incident value. [3] (b) A tissue has μ = 0.21 cm⁻¹; for water μ = 0.20 cm⁻¹. Calculate its Hounsfield unit value and comment on the tissue type. [3] (c) A 7.5 MHz ultrasound probe views blood flowing at 0.35 m/s at 60° to the beam (speed of sound in tissue 1540 m/s). Calculate the Doppler shift. [3] (d) Calculate the proton Larmor frequency in a 3.0 T MRI scanner (γ/2π = 42.6 MHz/T). [2] (e) State, with a physics justification, the most suitable imaging method for routine imaging of a fetus. [2]

Mark scheme (a) 0.10 = e^{−μx} → x = ln10/0.35 = 2.303/0.35 ≈ 6.6 cm. [3] (b) HU = 1000(0.21 − 0.20)/0.20 = +50 [2]; soft-tissue range (e.g. muscle) — far below bone (+1000), above fat (negative). [1] (c) Δf = 2f₀v cosθ/c = 2 × 7.5×10⁶ × 0.35 × cos60°/1540 ≈ 1.7×10³ Hz. [3 — factor 2 AND cosθ both required] (d) f = 42.6 × 3.0 ≈ 128 MHz. [2] (e) Ultrasound [1]; it uses non-ionising sound (no dose to fetus/mother), with real-time imaging — unlike X-ray/CT (ionising) it is safe for repeated pregnancy scans. [1]