PET scanning
Section: Medical Physics | Syllabus: Cambridge AS Level Physics 9702
PET Scanning Positron Emission Tomography (PET) is a medical imaging technique that uses radioactive tracers to produce images showing the metabolic activity of tissues, rather than just their structure.
Radioactive Tracers Tracer A substance containing radioactive nuclei that can be introduced into the body and is then absorbed by the tissue being studied. For PET scanning, the tracer must decay by β^+ (positron) emission .
Common tracers include: Fluorine-18 (F-18): Half-life of 110 minutes, attached to glucose to form FDG (fluorodeoxyglucose) Carbon-11, Nitrogen-13, Oxygen-15: Shorter half-lives, used for specific applications The tracer is typically injected into the bloodstream or inhaled.
It accumulates in tissues with high metabolic activity (e.g., tumours, active brain regions). Positron Emission and Annihilation β^+ Decay In β^+ decay, a proton in the nucleus converts to a neutron, emitting a positron (e^+) and a neutrino (ν): p n + e^+ + ν Annihilation The process that occurs when a particle interacts with its antiparticle, resulting in the conversion of their combined mass into energy.
Mass-energy and momentum are conserved. The Annihilation Process in PET A positron (e^+) is emitted from the tracer nucleus during β^+ decay The positron travels a short distance (typically < 1 mm) through tissue, losing kinetic energy When it meets an electron (e^-), annihilation occurs The mass of both particles is converted into energy in the form of two gamma-ray photons The photons travel in opposite directions (180° apart) to conserve momentum Figure: Electron-Positron Annihilation Diagram showing the annihilation process: Before annihilation: A positron (e^+) and electron (e^-) approach each other.
Both particles have rest mass m_e = 9.11 × 10^-31 kg At annihilation: The two particles meet and their combined mass is converted entirely into energy (mass is destroyed) After annihilation: Two gamma-ray photons (γ) are produced, travelling in exactly opposite directions (180° apart) Energy of each photon: E = m_e c^2 = 0.511 MeV (shown as arrows of equal length) Momentum conservation: Total momentum before ≈ 0 (particles nearly at rest), so photon momenta must be equal and opposite The diagram emphasises that this is not a collision but a complete conversion of mass to energy according to E = mc^2.
Energy of Gamma-Ray Photons The energy released in annihilation equals the total mass-energy of the electron and positron. Since both particles have the same rest mass: Total energy released: E = 2m_e c^2 Energy of each photon: E_γ = m_e c^2 = 0.511 MeV where m_e = 9.11 × 10^-31 kg (electron/positron rest mass) Worked Examples Worked Example: Annihilation Photon Energy Question: Calculate the energy of each gamma-ray photon produced when an electron and positron annihilate at rest.
Solution Total mass converted to energy: 2m_e = 2 × 9.11 × 10^-31 kg Total energy: E = 2m_e c^2 = 2 × 9.11 × 10^-31 × (3.00 × 10^8)^2 E = 1.64 × 10^-13 J Energy per photon: E_γ = 1.64 × 10^-132 = 8.2 × 10^-14 J Converting to MeV: E_γ = 8.2 × 10^-141.60 × 10^-13 = 0.511 MeV Why Two Photons at 180°?
Why Two Photons at 180°? Conservation of momentum requires the two photons to travel in opposite directions . If the electron and positron are approximately at rest before annihilation, the total initial momentum is approximately zero.
For the final momentum to also be zero, the two identical photons must have equal and opposite momenta. Detection and Image Formation The gamma-ray photons produced by annihilation have high penetrating power and can travel outside the body where they are detected.
How PET Images are Created A ring of gamma-ray detectors surrounds the patient When two detectors on opposite sides of the ring detect photons simultaneously (within a few nanoseconds), this is called a coincidence event The annihilation must have occurred somewhere along the line joining the two detectors By recording many coincidence events and processing the arrival times of the photons, a computer can determine where annihilations occurred This builds up an image showing the concentration of the tracer in different tissues Figure: PET Scanner Detection (Coincidence Detection) Cross-section of a PET scanner showing the detection principle: Detector ring: A complete ring of gamma-ray detectors (typically hundreds) surrounds the patient Annihilation event: Occurs at point P inside the patient where positron meets electron Gamma photon paths: Two photons travel in exactly opposite directions (180°) from point P Coincidence detection: Detectors A and B (on opposite sides) detect photons within a few nanoseconds of each other Line of response (LOR): Dashed line connecting detectors A and B; the annihilation occurred somewhere along this line Multiple coincidence events from different angles are recorded.
The computer uses these lines of response to determine the precise location of annihilation events, building up an image of tracer concentration. Region…
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