A2 · Paper 4 practice · Medical physics

Imaging the body, with physics.

Ten original structured questions in the style of Paper 4, covering the whole of Medical physics: ultrasound and acoustic impedance, X-ray production and attenuation, CT, and PET scanning. The last questions compare the methods and link the tracer back to radioactive decay. Each is tagged with its lessons; attempt them all, then reveal the worked solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level Paper 4 style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Data: c = 3.00 × 10⁸ m s⁻¹ · speed of sound in soft tissue ≈ 1540 m s⁻¹ · h = 6.63 × 10⁻³⁴ J s · e = 1.60 × 10⁻¹⁹ C · m₋ = 9.11 × 10⁻³¹ kg
Keep these straight

Echo, shadow, annihilation.

01
Paper 4
[7 marks]

Soft tissue has density 1060 kg m⁻³ and the speed of sound in it is 1540 m s⁻¹.

(a) State what is meant by the specific acoustic impedance of a medium. [1]

(b) Calculate the acoustic impedance of soft tissue. [2]

(c) Explain how a piezoelectric transducer both generates and detects ultrasound. [3]

(d) State why ultrasound, rather than audible sound, is used for imaging. [1]

  • (a)Z = ρc, the product of density and speed of sound in the medium ✓
  • (b)Z = 1060 × 1540 ✓ = 1.6 × 10⁶ kg m⁻² s⁻¹ ✓
  • (c)An alternating p.d. makes the piezoelectric crystal vibrate and emit ultrasound ✓; returning ultrasound deforms the crystal ✓, generating an e.m.f. that is detected ✓
  • (d)Its short wavelength gives better resolution (smaller detail can be imaged) ✓
02
Paper 4
[8 marks]

At a boundary, the intensity reflection coefficient is Iᵣ/I₀ = (Z₁ − Z₂)²/(Z₁ + Z₂)². Impedances (×10⁶ kg m⁻² s⁻¹): fat 1.38, muscle 1.70, air 0.0004.

(a) Calculate the fraction of intensity reflected at a fat-muscle boundary. [2]

(b) Calculate the fraction reflected at a tissue (Z = 1.63)-air boundary. [2]

(c) Use your answers to explain why a coupling gel is used. [2]

(d) State why little of the ultrasound is reflected at a fat-muscle boundary. [2]

  • (a)(1.38 − 1.70)²/(1.38 + 1.70)² = 0.1024/9.486 ✓ = 0.011 (1.1%) ✓
  • (b)(1.63 − 0.0004)²/(1.63 + 0.0004)² ✓ ≈ 0.999 (99.9%) ✓
  • (c)An air gap reflects almost all the ultrasound ✓; the gel (matched to tissue) excludes the air so the pulse can enter and echoes return ✓
  • (d)Fat and muscle have nearly equal impedances ✓, so Z₁ − Z₂ is small and little is reflected (most is transmitted) ✓
03
Paper 4
[6 marks]

In an A-scan, an echo from a boundary returns 40 µs after the pulse is sent. The speed of sound in the tissue is 1540 m s⁻¹.

(a) Calculate the depth of the boundary below the probe. [3]

(b) Explain why the factor of one half appears in your calculation. [2]

(c) State what the size of an echo indicates. [1]

  • (a)d = ½ct = ½ × 1540 × 40 × 10⁻⁶ ✓✓ = 0.031 m = 3.1 cm ✓
  • (b)The pulse travels to the boundary and back ✓, so the measured time covers twice the depth ✓
  • (c)The size of the impedance mismatch at that boundary (a bigger mismatch gives a larger echo) ✓
04
Paper 4
[8 marks]

In an X-ray tube, electrons are accelerated through a p.d. of 80 kV before striking the target.

(a) Calculate the maximum kinetic energy of an electron, in joules. [2]

(b) Hence state the maximum energy of an X-ray photon produced. [1]

(c) Calculate the minimum wavelength of the X-rays. [3]

(d) State what happens to most of the electrons' energy at the target. [2]

  • (a)E = eV = 1.60 × 10⁻¹⁹ × 80000 ✓ = 1.28 × 10⁻¹⁴ J (= 80 keV) ✓
  • (b)hfₘₐₓ = eV = 1.28 × 10⁻¹⁴ J (80 keV) ✓
  • (c)λₘₐₘ = hc/E = (6.63 × 10⁻³⁴ × 3.00 × 10⁸)/1.28 × 10⁻¹⁴ ✓✓ = 1.6 × 10⁻¹¹ m ✓
  • (d)Most becomes heat in the target ✓, so it must be cooled (e.g. a rotating anode) ✓
05
Paper 4
[8 marks]

An X-ray beam passes through soft tissue of linear attenuation coefficient 0.20 cm⁻¹.

(a) Calculate the fraction of the beam transmitted through 10 cm of tissue. [2]

(b) Calculate the half-value thickness of the tissue. [2]

(c) Bone has a much larger attenuation coefficient. State and explain what this does to the image. [2]

(d) State one advantage and one disadvantage of a CT scan over a single X-ray image. [2]

  • (a)I/I₀ = e⁻μᵛ = e⁻⁽⁰.²⁰ × ¹₀⁽ = e⁻² ✓ = 0.14 (14%) ✓
  • (b)x½ = ln2/μ = 0.693/0.20 ✓ = 3.5 cm ✓
  • (c)Bone transmits far less, so it casts a brighter shadow ✓; the difference in transmitted intensity gives contrast, making the bone visible ✓
  • (d)Advantage: a 3D image with much better soft-tissue contrast ✓. Disadvantage: a much higher radiation dose ✓
06
Paper 4
[6 marks]

A CT scanner builds an image from many X-ray exposures.

(a) Outline how a CT scan produces a three-dimensional image. [3]

(b) State why image contrast for soft tissue is better in CT than in a single X-ray. [2]

(c) State why the radiation dose is a concern. [1]

  • (a)X-ray images are taken from many angles around the body ✓; a computer combines these slices ✓ to reconstruct a 3D image that can be viewed from any direction ✓
  • (b)Combining many views distinguishes small differences in attenuation ✓ that a single overlapping projection would hide ✓
  • (c)X-rays are ionising, so a higher dose raises the risk of cell damage ✓
07
Paper 4
[8 marks]

In PET, a positron from the tracer annihilates with an electron.

(a) Show that the energy of each gamma photon produced is about 0.51 MeV. [3]

(b) State the directions in which the two photons travel, and why. [2]

(c) Explain how detecting the pair locates a line of response. [2]

(d) State what a PET scan shows that a CT scan does not. [1]

  • (a)Each photon carries the electron rest energy: E = m₋c² = 9.11 × 10⁻³¹ × (3.0 × 10⁸)² ✓ = 8.2 × 10⁻¹⁴ J ✓ = 0.51 MeV ✓
  • (b)In opposite directions (180° apart) ✓, so that momentum is conserved ✓
  • (c)Two detectors firing at the same instant (in coincidence) ✓ fix the straight line the photons travelled, which passes through the annihilation point ✓
  • (d)Physiological function (e.g. metabolic activity), not just structure ✓
08
Paper 4
[7 marks]

Describe how a PET scan is carried out and how the image is formed.

(a) State the type of tracer used and how it reaches the region of interest. [2]

(b) Describe what happens to the emitted positrons. [2]

(c) Explain how many lines of response build up an image. [2]

(d) State why the tracer must have a short half-life. [1]

  • (a)A positron-emitting (β⁺) isotope attached to a molecule such as glucose ✓; it is taken up by the tissue that uses that molecule ✓
  • (b)Each positron travels a short distance, then annihilates with an electron ✓, producing two 0.511 MeV gamma photons in opposite directions ✓
  • (c)Each coincidence gives one line of response through the tracer ✓; many lines cross where the tracer concentrated, reconstructing the image ✓
  • (d)To limit the radiation dose to the patient ✓
09
Paper 4
[8 marks]

A common PET tracer is fluorine-18, which has a half-life of 110 minutes. (ln2 = 0.693)

(a) Calculate the decay constant of fluorine-18, in min⁻¹. [2]

(b) A sample is prepared with activity 8.0 × 10⁸ Bq. State its activity after 220 minutes. [2]

(c) Calculate the activity after 60 minutes. [3]

(d) State why the short half-life is both useful and inconvenient. [1]

  • (a)λ = 0.693/110 ✓ = 6.3 × 10⁻³ min⁻¹ ✓
  • (b)220 min = 2 half-lives, so A = 8.0 × 10⁸/4 ✓ = 2.0 × 10⁸ Bq ✓
  • (c)A = A₀e⁻λᵗ = 8.0 × 10⁸ × e⁻⁶.³⁸₋⡊⠿ × ⁶₀ ✓ = 8.0 × 10⁸ × e⁻⁰.³⁸ ✓ = 5.5 × 10⁸ Bq ✓
  • (d)Useful: the dose falls quickly; inconvenient: it must be made near the scanner and used at once ✓
10
Paper 4
[9 marks]

A clinic must choose an imaging method for three cases.

(a) State which of the three techniques use ionising radiation, and which does not. [2]

(b) Suggest the best method to monitor a developing fetus, with a reason. [2]

(c) Suggest the best method to check for a bone fracture, with a reason. [2]

(d) Suggest the best method to find an area of unusually high metabolic activity, with a reason. [3]

  • (a)X-rays/CT and PET use ionising radiation ✓; ultrasound does not ✓
  • (b)Ultrasound ✓; it is non-ionising and so safe for the fetus, and gives real-time soft-tissue images ✓
  • (c)X-ray ✓; bone has a high attenuation coefficient, giving strong contrast against soft tissue ✓
  • (d)PET ✓; a tracer that follows metabolism concentrates in active tissue ✓, and the annihilation gamma pairs map where it gathers ✓

Mark this once you have attempted all ten questions and checked your working against the solutions. Revealing the solutions alone does not count.