PET Scanning

Spec mapping: OCR H556 Module 6.5 — Medical Imaging (positron emission tomography: positron-emitting radiotracer (e.g. $^{18}$ F-FDG) administered to the patient; $\beta^{+}$ -decay produces a positron; annihilation with an electron yields two 511 keV gamma photons travelling back-to-back; coincidence detection by a ring of scintillator detectors defines the line of response; tomographic reconstruction of the 3-D tracer distribution; functional imaging of metabolism, typically combined with CT for anatomical reference). Refer to the official OCR H556 specification document for exact wording.

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 sophisticated shadow of the body. Ultrasound (with its Doppler extension) uses non-ionising sound waves to give a different kind of anatomical image, real-time and free of radiation hazard. Both modalities 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 — for that you need CT or MRI — 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 closing topic of the OCR A-Level Physics A specification (H556), Module 6.5 — Medical Imaging.

This lesson is also the synoptic finale of the course. It draws on almost every idea from the nuclear and particle modules: beta-plus decay, annihilation, coincidence detection, conservation of momentum and energy, and tomographic reconstruction. Medical imaging is the part of the OCR A-Level physics syllabus that sets it apart from AQA and Edexcel — and PET is its centrepiece.

The Principle

A PET scan works as follows:

A small dose of a positron-emitting radiotracer is injected into the patient.
The tracer circulates through the body, preferentially accumulating in tissue where it is metabolically useful.
Each tracer nucleus decays by beta-plus emission, $p \to n + e^{+} + \nu_e$ , emitting a positron.
The positron travels a short distance through the tissue (typically $\lesssim 1$ mm), losing kinetic energy by ionisation, and then meets a (free or atomic) electron.
The electron and positron annihilate: $e^{-} + e^{+} \to 2\gamma$ . Two gamma photons, each carrying an energy of $E_\gamma = m_e c^{2} \approx 511$ keV, are produced. By conservation of momentum (the pair was essentially at rest at the moment of annihilation), the two photons travel in opposite directions — they are emitted back-to-back, 180° apart.
A ring of gamma detectors surrounding the patient records the arrival of each photon, together with its time of arrival to a precision of a few hundred picoseconds.
When two detectors on opposite sides of the ring both register a 511 keV photon within a short time window (typically a few nanoseconds), the scanner declares a coincidence event. The line joining the two detectors is recorded as the line of response (LoR) — the annihilation must have occurred somewhere along this line.
By collecting millions of such lines from the same slice and many adjacent slices, the computer reconstructs a 3-D image of tracer concentration throughout the body.

The reconstruction mathematics is essentially the same as in CT: the set of lines is inverted (via filtered back-projection or, in modern scanners, iterative algorithms) to produce a 3-D map of tracer uptake.

The Radiotracer

The chemistry of the radiotracer is what makes PET clinically useful. An ideal PET tracer has four properties:

A positron-emitting isotope.
A short enough half-life to minimise dose to the patient.
A long enough half-life to allow synthesis at a cyclotron, transport to the patient, injection, and scanning — in practice tens of minutes to a few hours.
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 (the body's basic metabolic fuel) with a single $^{18}$ F atom substituted for one of the hydroxyl groups on the sugar ring. Biologically, FDG behaves almost like glucose: it is taken up into cells by glucose transporters and phosphorylated by the enzyme hexokinase. But because of the fluorine substitution it cannot proceed down the glycolysis pathway, and it gets trapped inside the cell as 6-phosphate-FDG. Tissues that consume a lot of glucose — brain, heart muscle and (most importantly for oncology) most cancers — therefore accumulate FDG strongly, lighting up brightly on the PET image.

Fluorine-18 decays by $\beta^{+}$ emission:

$\,^{18}_{9}\text{F} \to \,^{18}_{8}\text{O} + e^{+} + \nu_e,$

with a half-life $t_{1/2} \approx 110$ minutes. This is the engineering sweet spot: long enough to synthesise FDG at a cyclotron, transport it to a nearby hospital (typically within an hour), inject and scan the patient (about 30 minutes), and short enough that the residual activity has decayed away within a day. After about 10 half-lives ( $\sim 18$ hours), the radioactivity in the patient is essentially indistinguishable from natural background.

Other tracers are used for specific clinical questions: $^{11}$ C-methionine for amino-acid metabolism (half-life $\sim 20$ min); $^{15}$ O-water for blood flow (half-life $\sim 2$ min — must be made on-site immediately before injection); $^{13}$ N-ammonia for cardiac perfusion (half-life $\sim 10$ min). All are made at a cyclotron — most often a nearby on-site one for the short-lived tracers.

The Physics of the Coincidence Measurement

Recall from the lesson on annihilation and pair production that when a positron annihilates with an electron at rest, conservation of momentum requires the two resulting photons to fly in exactly opposite directions, and conservation of energy fixes their energy: each photon carries $m_e c^{2} \approx 511$ keV. This 180°-apart photon pair, with its precisely fixed energy, is the key to PET imaging.

The scanner consists of a ring (or in modern scanners, multiple parallel rings forming a short cylinder) of many thousands of small scintillating crystal detectors arranged around the patient. The crystals are typically bismuth germanate (BGO), lutetium oxyorthosilicate (LSO) or lutetium yttrium oxyorthosilicate (LYSO), chosen for high stopping power at 511 keV and fast scintillation decay. When a 511 keV photon strikes a crystal, it produces a brief flash of visible light, which is recorded by an attached photomultiplier tube or silicon photomultiplier. The electronics record the time of each detected event to a precision of a few hundred picoseconds.

A coincidence is declared when two detectors on opposite sides of the ring both register photons within a short coincidence time window — typically a few nanoseconds. (The window is set just wider than the maximum expected time-of-flight across the patient; a 30 cm patient gives a maximum time difference of $0.3 / 3 \times 10^{8} \approx 1$ ns.) Such a pair of events is taken to have come from a single annihilation, and the scanner records the line connecting the two hit detectors — the line of response.

The annihilation event must have occurred somewhere along this line. From one event alone you learn only that there is some tracer activity along the LoR — but you do not know where along the line. With many millions of LoRs collected from different directions through the body, the points where many lines intersect emerge as concentrations of tracer activity. This is a tomographic reconstruction problem, mathematically identical to CT (although with annihilation-line data rather than X-ray-attenuation data), and is solved by filtered back-projection or iterative methods.

Why Coincidence Detection? — Three Important Advantages

Coincidence detection of the pair of annihilation photons offers several crucial advantages over single-photon detection (as used, for example, in a conventional gamma camera or SPECT scanner):

Spatial localisation without mechanical collimation. A conventional gamma camera needs heavy lead collimators to define the direction from which each photon arrives — because there is only one photon per event. In PET the pair of photons defines the line of response by pure geometry, with no collimator needed. This dramatically increases sensitivity (no signal lost to the collimator) and improves spatial resolution.
Background rejection. Cosmic-ray muons, scattered photons, random single events from natural background — none of these produce back-to-back photons on opposite sides of the ring within a few nanoseconds, so they are automatically rejected. The 511 keV energy gating further filters out scattered photons that have lost energy in transit.
Attenuation correction. Because the photon pair travels through the entire body along a single straight line, the total attenuation along that line is the same regardless of where the annihilation occurred. This makes attenuation correction straightforward — typically done using a co-acquired low-dose CT image as the attenuation map.

Worked Example 1: Photon energy from FDG decay

Fluorine-18 decays by $\beta^{+}$ emission, and the emitted positron annihilates with an electron in the surrounding tissue. What is the energy of each of the two photons produced? What frequency corresponds to that energy?

Solution. By the time the positron annihilates it has lost essentially all of its kinetic energy to ionisation in the tissue (typically within $\lesssim 1$ mm of the emission point). Conservation of energy and momentum then fixes each photon's energy:

$E_\gamma = m_e c^{2} = 0.511\,\text{MeV} = 8.18 \times 10^{-14}\,\text{J}.$

The corresponding frequency is

$f = \dfrac{E}{h} = \dfrac{8.18 \times 10^{-14}}{6.63 \times 10^{-34}} \approx 1.23 \times 10^{20}\,\text{Hz}.$

This is in the gamma-ray region of the electromagnetic spectrum — well above any medical X-ray (typically a few times $10^{19}$ Hz at most) and a sign that PET detectors need different physics from ordinary X-ray detectors: scintillator + photomultiplier rather than scintillator + photodiode.

Worked Example 2: Dose-budget from FDG

A typical clinical PET scan delivers about $A_0 = 350$ MBq of $^{18}$ F-FDG to the patient, with $t_{1/2} \approx 110$ min. Take $\lambda = \ln 2 / t_{1/2}$ . (a) How many positrons are emitted during the first half-life? (b) What fraction of the initial activity remains after 6 hours?

Solution.

(a) The total number of $^{18}$ F nuclei present at the start is $N_0 = A_0/\lambda = A_0 t_{1/2} / \ln 2$ . The number of decays during the first half-life is exactly $N_0/2$ (by definition of half-life). With $t_{1/2} = 110 \times 60 = 6600$ s:

\begin{aligned} N_0 &= \dfrac{(350 \times 10^{6})(6600)}{0.693} \approx 3.33 \times 10^{12}, \\ N_\text{decayed in first }t_{1/2} &= \dfrac{N_0}{2} \approx 1.67 \times 10^{12}. \end{aligned}

About $1.7 \times 10^{12}$ positrons are emitted during the first 110 minutes of the scan. Each annihilation produces a back-to-back photon pair, so a similar number of coincidence-detectable events are generated within the patient over that period — of which the scanner records of order $10^{8}$ (a small fraction; most photons either escape the ring without being detected, or are absorbed in tissue before reaching it).

(b) After 6 hours $= 360$ minutes, the number of half-lives elapsed is $n = 360 / 110 \approx 3.27$ . The remaining activity is

$A = A_0 \times 2^{-n} = (350 \times 10^{6}) \times 2^{-3.27} \approx (350 \times 10^{6})(0.104) \approx 3.6 \times 10^{7}\,\text{Bq} = 36\,\text{MBq}.$

About 10% of the initial activity remains after 6 hours. After another 6 hours it will be about $1\%$ , and after 24 hours about $0.01\%$ — essentially background. This is why the 110-minute half-life is the engineering sweet spot: short enough to limit total dose, long enough to allow synthesis, transport and a full clinical scan.

Worked Example 3: The intrinsic PET resolution limit

An $^{18}$ F positron is emitted with maximum kinetic energy $\sim 0.63$ MeV. Estimate its initial speed (note that this is mildly relativistic) and hence the approximate distance it travels in soft tissue before annihilating, given that the practical range of a 0.63 MeV positron in water is about 2 mm.

Solution. Kinetic energy $K = 0.63$ MeV; rest energy $m_e c^{2} = 0.511$ MeV. The total energy is $E = K + m_e c^{2} = 1.14$ MeV, so $E/(m_e c^{2}) \approx 2.23$ . Using the relativistic relation $E^{2} = (pc)^{2} + (m_e c^{2})^{2}$ :

\begin{aligned} (pc)^{2} &= E^{2} - (m_e c^{2})^{2} = 1.14^{2} - 0.511^{2} \approx 1.04 \,\text{MeV}^{2}, \\ pc &\approx 1.02\,\text{MeV}, \\ v/c &= pc/E \approx 1.02/1.14 \approx 0.90, \end{aligned}

so $v \approx 0.90 c \approx 2.7 \times 10^{8}$ m s $^{-1}$ . The positron is comfortably relativistic.

The range in tissue is set by ionisation losses, not by initial kinematics. For a 0.63 MeV positron in water the range is about 2 mm — but most positrons are emitted with energies below the maximum and therefore travel less. The mean range is more like 0.5 mm.

This sets the fundamental spatial-resolution limit of PET: the scanner records where the positron annihilated, not where it was emitted, and the two may be up to about $0.5$ to $1$ mm apart. Together with detector-pixel size, this gives clinical PET resolution of about $4$ – $6$ mm — worse than CT ( $\sim 1$ mm) or MRI ( $\lesssim 1$ mm), but adequate for distinguishing organ-scale functional features.

Combined PET-CT

Modern PET scanners are almost always combined with a CT scanner in a single instrument — a PET-CT scanner. The patient is imaged first with low-dose CT (a few mSv, providing both anatomical reference and the attenuation map needed for PET correction), then — without moving — with PET (typically about 7 mSv). The two images are spatially co-registered, and the radiologist views them as either separate panels or as a transparent overlay: the PET activity map ("hot spots") on top of the CT anatomical map.

This combination is enormously more useful than either modality alone. CT gives the structural map; PET gives the metabolic activity. A tumour appears as a "hot spot" on the PET image (high FDG uptake) whose precise location is given by the co-registered CT image (behind the rib, in the upper lobe of the right lung, etc.). PET-CT is the standard technique for staging most common cancers, for assessing treatment response, and for many neurological and cardiac investigations.

flowchart TB
    Q["Clinical question:<br/>structure or function?"]
    A["Anatomy / structure<br/>(where is it?)"]
    F["Function / metabolism<br/>(what is it doing?)"]
    XR["Plain X-ray"]
    CT["CT scanning"]
    US["Ultrasound"]
    PET["PET<br/>(positron emission)"]
    Combo["PET-CT<br/>(anatomy + function)"]
    Q --> A
    Q --> F
    A --> XR
    A --> CT
    A --> US
    F --> PET
    A -.-> Combo
    F -.-> Combo

PET answers the functional question (what is the tissue doing?). CT answers the anatomical question (where is the tissue?). PET-CT answers both at once — which is why it has become the standard oncology imaging modality.

Synoptic Links

Beta-plus decay and lepton-number conservation (Lesson 1, this module): Every PET image starts with a $\beta^{+}$ decay $p \to n + e^{+} + \nu_e$ inside the tracer nucleus. The neutrino carries away part of the energy invisibly; the positron is what we detect (indirectly, via its annihilation photons). The whole modality is a downstream consequence of the weak interaction described in Lessons 1 and 8.
Mass-energy equivalence and annihilation (Lessons 3 + 8, this module): The 511 keV photon energy is $m_e c^{2}$ — pure mass-energy conversion of an electron-positron pair. Conservation of momentum forces the two photons to fly back-to-back in the centre-of-mass frame, which is essentially the rest frame because the positron has lost almost all its kinetic energy by the time it annihilates.
CT and tomographic reconstruction (Lesson 11, this module): PET uses exactly the same tomographic reconstruction algorithms (filtered back-projection, iterative methods) as CT — only the input data differ. CT inputs are line-integrals of X-ray attenuation; PET inputs are coincidence-detection lines of response. The shared mathematical machinery is why every modern PET-CT scanner can run both reconstructions on the same hardware and present them as a single fused image.

Specimen question modelled on the OCR H556 paper format

Question (9 marks): Fluorine-18 fluorodeoxyglucose ( $^{18}$ F-FDG) is the most widely used radiotracer for PET imaging. Fluorine-18 decays by $\beta^{+}$ emission with a half-life of $t_{1/2} \approx 110$ min, with an emitted positron of maximum kinetic energy about $0.63$ MeV. The electron rest mass is $m_e = 9.11 \times 10^{-31}$ kg; $c = 3.00 \times 10^{8}$ m s $^{-1}$ ; $h = 6.63 \times 10^{-34}$ J s.

(a) Write a balanced nuclear equation for the $\beta^{+}$ decay of $^{18}$ F. [2]

PET Scanning

PET Scanning

The Principle

The Radiotracer

The Physics of the Coincidence Measurement

Why Coincidence Detection? — Three Important Advantages

Worked Example 1: Photon energy from FDG decay

Worked Example 2: Dose-budget from FDG

Worked Example 3: The intrinsic PET resolution limit

Combined PET-CT

Synoptic Links

Specimen question modelled on the OCR H556 paper format

More in Physics