PET Scans (OCR A Level Physics)

• Positron Emission Tomography (PET) is defined as:

A type of nuclear medical procedure that images tissues and organs by measuring the metabolic activity of the cells of body tissues

• In PET scanning, a beta-plus emitting radioactive tracer is used in order to stimulate positron-electron annihilation to produce gamma photons
  • These are then detected using a ring of gamma cameras

Principles of PET Scanning

Before the scan

• The patient is injected with a beta-plus emitting isotope, usually fluorine-18 (F-18)

During the scan

• The part of the body being studied is surrounded by a ring of gamma cameras
• The positrons from the F-18 nuclei annihilate with electrons in the patient
• The annihilation of a positron and an electron produces two identical gamma photons travelling in opposite directions
• The delay time between these two gamma ray photons is used to determine the location of the annihilation due to the F-18 tracer
  • Photons that do not arrive within a nanosecond of each other are ignored, since they cannot have come from the same point

After the scan

• Computer connected to the gamma cameras detect the signal and an image is formed by the computer

Detecting gamma rays with a PET scanner

Annihilation

• When a positron is emitted from a tracer in the body, it travels less than a millimetre before it collides with an electron
• The positron and the electron will annihilate, and their mass becomes pure energy in the form of two gamma rays which move apart in opposite directions
• Annihilation doesn’t just happen with electrons and positrons, annihilation is defined as:

When a particle meets its equivalent antiparticle they are both destroyed and their mass is converted into energy

• As with all collisions, the mass, energy and momentum are conserved

Annihilation of a positron and electron to form two gamma-ray photons

• The gamma-ray photons produced have an energy and frequency that is determined solely by the mass-energy of the positron-electron pair
• The energy E of the photon is given by

E = hf = m_ec²

• The momentum p of the photon is given by

• Where:
  • m_e = mass of the electron or positron (kg)
  • h = Planck's constant (J s)
  • f = frequency of the photon (Hz)
  • c = the speed of light in a vacuum (m s^–1)

Part (a)

Step 1: Write down the known quantities

• • Mass of an electron = mass of a positron, m_e = 9.11 × 10^–31 kg
  • Total mass is equal to the mass of the electron and positron = 2m_e

Step 2: Write out the equation for mass-energy equivalence

E = m_ec²

Step 3: Substitute in values and calculate energy E

E = 2 × (9.11 × 10^-31) × (3.0 × 10⁸)² = 1.6 × 10^–13 J

Part (b)

Step 1: Determine the energy of one photon

• • Planck's constant, h = 6.63 × 10⁻³⁴ J s
  • Two photons are produced, so, the energy of one photon is equal to half of the total energy from part (a):

Step 2: Write out the equation for the energy of a photon

E = hf

Step 3: Rearrange for frequency f, and calculate

Part (c)

Step 1: Write out the equation for the momentum of a photon

Step 2: Substitute in values and calculate momentum, p

• Once the tracer is introduced to the body it has a short half-life, so, it begins emitting positrons (β⁺) immediately
  • This allows for a short exposure time to the radiation
  • A short half-life does mean the patient needs to be scanned quickly and not all hospitals have access to expensive PET scanners

• In PET scanning:
  • Positrons are emitted by the decay of the tracer
  • They travel a small distance and annihilate when they interact with electrons in the tissue
  • This annihilation produces a pair of gamma-ray photons which travel in opposite directions

Annihilation of a positron and an electron is the basis of PET Scanning

Image Formation on a Computer

• The signals produced by the photomultiplier tubes are used to produce an image
• The γ rays travel in straight lines in opposite directions when formed from a positron-electron annihilation
  • This happens in order to conserve momentum
• They hit the detectors in a line – known as the line of response
• The tracers will emit lots of γ rays simultaneously, and the computers will use this information to create an image
• The more photons from a particular point, the more tracer that is present in the tissue being studied, and this will appear as a bright point on the image
• An image of the tracer concentration in the tissue can be created by processing the arrival times of the gamma-ray photons