PET Scanning

So far in the medical imaging module we have met two classes of technique. X-ray radiography and CT project ionising electromagnetic waves through the patient and record the attenuation — an image of anatomy, essentially a shadow of the body. Ultrasound uses high-frequency sound waves and gives a different kind of anatomical image, non-invasive and in real time. Both show structure: where things are and what shape they have.

Positron emission tomography (PET) is fundamentally different. A PET scan shows function rather than structure. It maps the distribution of a radioactive tracer that has been designed to concentrate in metabolically active tissue. The image reveals not where the organs are — you need CT or MRI for that — but what those organs are doing: where glucose is being burned, where cell division is happening, where the brain is active. PET gives doctors a uniquely direct view of biochemistry in vivo. It is a remarkable synthesis of nuclear physics, chemistry, biology and engineering, and it is the final topic of the OCR A-Level Physics A specification (H556), Module 6.5.

This lesson brings together almost every idea from the nuclear and particle physics modules: beta-plus decay, annihilation, coincidence detection, conservation of momentum, and computed tomography. It is both the culmination of the course and — unique to OCR among the A-Level physics boards — the area that sets this specification apart from AQA and Edexcel.

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The Principle

A PET scan works as follows:

1. A small dose of a positron-emitting radiotracer is injected into the patient.
2. The tracer circulates through the body, preferentially accumulating in tissue where it is metabolically useful.
3. Each tracer atom decays by beta-plus emission: p → n + e^+ + \nu_e. A positron is emitted.
4. The positron travels a short distance (typically <1 mm in tissue), losing kinetic energy, and then meets an electron.
5. The electron and positron annihilate: e^- + e^+ → 2\gamma. Two 511 keV gamma photons are produced, travelling in opposite directions (180° apart).
6. A ring of gamma detectors surrounding the patient records the arrival of each photon.
7. When two photons are detected simultaneously on opposite sides of the ring — coincidence detection — the scanner records a line between the two detectors: the line of response.
8. The annihilation event must have occurred somewhere along this line. By collecting millions of such lines from the same slice, the computer reconstructs a 3-D image of tracer concentration throughout the body.

The reconstruction mathematics is similar to CT: the set of lines is inverted (via filtered back-projection or iterative methods) to produce a 3-D map of tracer uptake.

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The Radiotracer

The key piece of chemistry in PET is the choice of radiotracer. The ideal tracer has:

• A positron-emitting isotope.
• A short half-life (to minimise dose to the patient).
• A long enough half-life to allow synthesis, delivery and imaging (tens of minutes).
• A chemical form that concentrates in the tissue of interest.

The overwhelmingly dominant tracer is fluorine-18 fluorodeoxyglucose (abbreviated ^{18}F-FDG, or just FDG). It is glucose with a single ^{18}F atom substituted for one of the hydroxyl groups. Biologically, FDG behaves almost like glucose: it is taken up into cells by glucose transporters and phosphorylated by hexokinase. But because of the fluorine substitution, it cannot proceed down the glycolysis pathway and gets trapped inside the cell. Tissues that consume a lot of glucose (brain, heart muscle, and most importantly many cancers) therefore accumulate FDG strongly.

Fluorine-18 decays by beta-plus emission:

¹⁸₉F → ¹⁸₈O + ⁰₊₁e + ν_e

with a half-life of about 110 minutes. This is conveniently short: long enough to synthesise FDG at a cyclotron, transport it to a nearby hospital (~1 hour), inject it, and run the scan (~30 minutes); but short enough that the residual activity decays away within a day or so. After about 10 half-lives (~18 hours), the radioactivity is negligible.

Other tracers are used for specific applications: ^{11}C-methionine for amino acid metabolism, ^{15}O-water for blood flow, ^{13}N-ammonia for cardiac perfusion. All are made at a nearby cyclotron (fluorine and carbon tracers) or on demand (oxygen-15, half-life just 2 minutes).

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The Physics of the Coincidence Measurement

Recall from Lesson 8 that when a positron annihilates with an electron at rest, the two resulting photons fly in exactly opposite directions (to conserve momentum) and each carries an energy of m_e c^2 = 511 keV. This 180°-apart pair is the key to PET imaging.

The scanner consists of a ring (or multiple rings) of many thousands of scintillating crystal detectors arranged around the patient. When a photon strikes a detector, it produces a flash of light that is recorded by an attached photomultiplier tube. The electronics record the time of each event very precisely (to within a few hundred picoseconds).

A coincidence is declared when two detectors on opposite sides of the ring both register photons within a short time window (typically a few nanoseconds). Such a pair of events is assumed to have come from a single annihilation event, and the scanner records the line connecting the two hit detectors — the line of response.

The annihilation event must have occurred somewhere on this line. From one event alone you learn only that there is some activity along the line — you do not know where along the line. But with millions of lines from different directions across the body, the points of highest concentration become those where many lines intersect. Just as in CT, this is a tomographic reconstruction problem, solved by filtered back-projection or iterative methods.

flowchart TB
    T["Positron-emitting<br/>tracer in tissue"]
    P["Positron travels<br/><1 mm then annihilates"]
    Ann["Annihilation<br/>e⁻ + e⁺ → 2γ"]
    G1["γ (511 keV)<br/>→ detector A"]
    G2["γ (511 keV)<br/>→ detector B"]
    Coinc["Coincidence:<br/>both detected<br/>within ~ns"]
    LoR["Line of response<br/>A-B"]
    Recon["Computer reconstructs<br/>3-D tracer map"]
    T --> P
    P --> Ann
    Ann --> G1
    Ann --> G2
    G1 --> Coinc
    G2 --> Coinc
    Coinc --> LoR
    LoR --> Recon

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Why Coincidence Detection?

Coincidence detection offers several crucial advantages over single-photon detection:

• Spatial localisation without mechanical collimation. In a conventional gamma camera, you need heavy lead collimators to define the direction of each incoming photon. In PET, the line of response is defined purely by the geometry of the two detectors — no lead collimator is needed. This dramatically improves sensitivity.
• Background rejection. Cosmic rays and random single events are automatically rejected because they do not produce coincident photons on opposite sides of the ring.
• Attenuation correction. Because both photons travel through the body, the total attenuation along the line of response is the same regardless of where the annihilation occurred. This makes attenuation correction straightforward (usually done using a low-dose CT scan acquired at the same time).

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Worked Example 1: Photon energy from FDG decay

Fluorine-18 decays by beta-plus emission and the emitted positron annihilates with an electron. What is the energy of each of the two photons produced? What frequency corresponds to this energy?

Solution. The positron annihilates at rest (after losing its kinetic energy to surrounding tissue within ~1 mm). Each photon carries m_e c^2 = 0.511 MeV = 8.18 × 10^{-14} J.

Frequency:

f = E/h = 8.18 × 10⁻¹⁴ / 6.63 × 10⁻³⁴ ≈ 1.23 × 10²⁰ Hz

This is in the gamma-ray region of the electromagnetic spectrum — well beyond any medical X-ray and a sign that PET detectors need quite different physics from ordinary X-ray detectors.

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Worked Example 2: Dose from FDG

A typical PET scan delivers about 350 MBq of ^{18}F-FDG to the patient, with a half-life of 110 minutes. How many positrons are emitted during the first half-life? How much of the initial activity remains after 6 hours?

Solution.

Number of decays (= number of positrons emitted) during the first half-life:

Initial activity A₀ = 350 × 10⁶ s⁻¹
Average activity over first t₁/₂ ≈ (3/4) × A₀ (rough average of A₀ and A₀/2)
t₁/₂ = 110 × 60 = 6600 s
N ≈ (3/4)(350 × 10⁶)(6600) ≈ 1.7 × 10¹² decays

More precisely, the number of nuclei decaying between t = 0 and t = t_{1/2} is N_0/2 = A_0 \tau (1 - 1/2) = A_0 t_{1/2} / (2 \ln 2). Let's use the exact form:

N₀ = A₀/λ = 350 × 10⁶ × 6600 / ln 2 ≈ 3.33 × 10¹² initially
N decayed in first half-life = N₀/2 ≈ 1.67 × 10¹²

Activity after 6 hours:

n = 6/1.833 ≈ 3.27 half-lives
A = A₀ × 2^{-3.27} ≈ 350 × 10⁶ × 0.104 ≈ 3.6 × 10⁷ Bq = 36 MBq