explain that, in PET scanning, positrons emitted by the decay of the tracer annihilate when they interact with electrons in the tissue, producing a pair of gamma-ray photons travelling in opposite directions

Medical Physics – X‑rays and Positron Emission Tomography (PET)

1. Production of X‑rays

• Bremsstrahlung (continuous) spectrum – Electrons are accelerated through a potential difference \(V\) (typically 30–150 kV) and strike a high‑Z target (usually tungsten). Their rapid deceleration produces a broad, continuous distribution of photon energies up to a maximum.
• Maximum photon energy (minimum wavelength) occurs when the whole kinetic energy of an electron is converted into a single photon:

  \$\lambda_{\min}= \frac{hc}{eV}\$

  where \(h\) is Planck’s constant, \(c\) the speed of light and \(e\) the elementary charge.

• Characteristic X‑rays (discrete lines) – When an inner‑shell (K‑shell) electron is ejected, an outer‑shell electron falls into the vacancy. The energy difference is emitted as a photon with a well‑defined wavelength (e.g. K‑α, K‑β lines). These lines appear as sharp peaks superimposed on the bremsstrahlung background.

2. Interaction of X‑rays with Matter

The intensity of a mono‑energetic beam after passing through a material of thickness \(x\) follows the exponential law

\$I = I_{0}\,e^{-\mu x}\$

• \(I_{0}\) – incident intensity.
• \(I\) – transmitted intensity.
• \(\mu\) – linear attenuation coefficient (depends on photon energy and atomic number).

Typical attenuation coefficients for 511 keV photons (relevant to PET) are:

3. Clinical Uses of X‑rays

• Plain radiography – A single projection recorded on film or a digital detector; contrast arises from differential attenuation.
• Computed Tomography (CT) – The X‑ray tube and detector rotate, acquiring many angular projections. Reconstruction algorithms (e.g. filtered back‑projection) produce cross‑sectional images.
• Fluoroscopy – A continuously (or pulsed) emitted X‑ray beam is visualised in real time on a monitor, allowing dynamic studies such as cardiac catheterisation.
• Mammography – Uses a low‑energy (≈20–30 kV) X‑ray beam and a high‑resolution detector to exploit the greater attenuation of soft tissue versus adipose tissue, giving excellent contrast for breast imaging.

4. Radiation Protection and Dose

• Shielding – Lead (or equivalent high‑Z material) is used to attenuate scattered X‑rays; thickness is chosen using the same exponential law.
• Absorbed dose – Energy deposited per unit mass, measured in grays (Gy). 1 Gy = 1 J kg\(^{-1}\).
• Effective dose – Weighted sum of organ doses that reflects the differing radiosensitivities of tissues; expressed in sieverts (Sv).
• ALARA principle – “As Low As Reasonably Achievable”. Dose should be minimised by optimisation of technique, shielding and exposure time.

5. Positron Emission Tomography (PET)

PET is a functional imaging technique that visualises the distribution of a positron‑emitting radiotracer inside the body.

5.1 Physics of Positron Annihilation

A positron (\(e^{+}\)) emitted by β\(^+\) decay loses kinetic energy through collisions (thermalisation) and then annihilates with an electron (\(e^{-}\)):

\$e^{+}+e^{-}\;\longrightarrow\;\gamma+\gamma\$

• Each photon carries the electron rest‑mass energy: \(m_{e}c^{2}=511\ \text{keV}\).
• Conservation of momentum forces the two photons to travel in (approximately) opposite directions – ideally \(180^{\circ}\) apart.

5.2 Sequence of Events in a PET Scan

1. Tracer administration – A biologically active molecule labelled with a positron‑emitting radionuclide (e.g. \(^{18}\)F‑FDG) is injected.
2. β\(^+\) decay – The radionuclide emits a positron with kinetic energy up to a few MeV.
3. Thermalisation – The positron travels a few millimetres, losing energy through collisions, until it is essentially at rest.
4. Annihilation – The thermalised positron meets an electron, producing two 511 keV γ‑photons emitted in opposite directions.
5. Coincidence detection – A ring of scintillation detectors records the two photons simultaneously (within a few nanoseconds). The line joining the two detectors is called the line‑of‑response (LOR).
6. Image reconstruction – Tomographic algorithms (e.g. filtered back‑projection, iterative reconstruction) combine many LORs to locate the annihilation sites and generate a three‑dimensional map of tracer concentration.

5.3 Key Quantities

5.4 Worked Example – Decay and Attenuation

Problem: A patient receives an injection of \(^{18}\)F‑FDG containing \(5.0\times10^{9}\) nuclei. Calculate:

1. The activity (Bq) 30 min after injection.
2. The fraction of the 511 keV photons that emerge from 10 cm of soft tissue (use \(\mu =0.095\ \text{cm}^{-1}\)).

Solution:

1. Decay constant: \(\displaystyle\lambda =\frac{\ln2}{t_{1/2}}=\frac{0.693}{109.8\times60\ \text{s}}=1.05\times10^{-4}\ \text{s}^{-1}\).
2. Number of nuclei after \(t=30\ \text{min}=1800\ \text{s}\):

   \$N(t)=N_{0}e^{-\lambda t}=5.0\times10^{9}\,e^{-1.05\times10^{-4}\times1800}=5.0\times10^{9}\,e^{-0.189}=4.14\times10^{9}.\$

3. Activity \(A=\lambda N = (1.05\times10^{-4}\ \text{s}^{-1})(4.14\times10^{9})\approx4.3\times10^{5}\ \text{Bq}.\)
4. Attenuation through 10 cm of soft tissue:

   \$I = I{0}e^{-\mu x}=I{0}e^{-0.095\times10}=I{0}e^{-0.95}\approx0.387\,I{0}.\$

5. Approximately 38 % of the annihilation photons survive the 10 cm path; the remaining 62 % are lost to absorption or scattering and must be corrected for during reconstruction.

6. Comparison – X‑ray/CT vs PET

7. Why PET Uses Gamma Photons Instead of Conventional X‑rays

• Gamma photons have a precisely known energy (511 keV), allowing efficient energy discrimination and high detector efficiency.
• Simultaneous detection of two photons travelling in opposite directions provides intrinsic localisation (coincidence detection) without the need for external collimators.
• Because the photons originate from within the body, PET directly maps the radiotracer’s distribution, revealing physiological processes rather than merely static anatomy.