47.13. Identifying Scattered Coincidences

Table of Contents

Understanding Scattered Coincidences in PET

Scattered coincidences are PET coincidence events in which at least one of the two detected photons has undergone one or more interactions that changed its direction or energy before detection. In practice, this almost always means Compton scattering in the patient or in surrounding materials.

For an absolute beginner, it is useful to think of scattered coincidences as events that no longer point back correctly to the original annihilation position. Even though the detector still records two photons within the coincidence timing window, the line of response that you build from their detector positions is wrong, which degrades image contrast and quantitative accuracy.

In a GATE PET simulation you can exploit the detailed Monte Carlo information to identify these scattered coincidences very precisely, at the level of hits, singles, and finally coincidence records.

Where Scatter Occurs in the PET Workflow

Within the broader “Practical Example: PET Scanner” project, your simulation already generates:

True physics histories at the Geant4 level, including all gamma interactions.
Hits in detector volumes.
Singles, which are digitized detector events.
Coincidences, which are time-correlated pairs of singles.

Scatter can appear in different places in this chain. Two important categories are:

Patient or phantom scatter. A 511 keV photon scatters in the patient or phantom before it ever reaches the detector. Its direction changes, its energy decreases, and if it is finally detected, the corresponding coincidence often appears at an incorrect location and/or outside the photopeak energy window.

Detector or environmental scatter. The photon may scatter in detector housing, shielding, or other components before depositing its final detectable energy in the scintillation crystal. This can also distort the true event geometry.

GATE gives you access to enough information to distinguish these situations. At the coincidence-analysis level, you usually classify scattered coincidences according to photon history flags and the total energy of each detected photon, rather than by re-tracing every interaction.

Information Needed to Classify Scatter in GATE

To tag scattered coincidences, you need extra information carried from the physics level to the digitizer and then written into the coincidence output. This is typically done by:

Adding user-level flags in hits or singles that describe whether a photon has scattered.

Preserving the annihilation event ID or some parent ID so that you can relate each detected photon back to its origin.

Two types of information are especially important:

Scatter-related flags. GATE can record flags such as “photon scattered in phantom,” “photon scattered in detector,” or similar process tags, depending on how your PET example is configured. These are often stored as integer or boolean fields in hits, singles, or coincidences, and can be used directly for classification.

Energy-related quantities. The total energy recorded in a single should correspond to about 511 keV for an unscattered photon that deposits all its energy in the detector. In practice the energy is blurred and partial-energy deposition can occur, but significantly reduced energy can be a strong hint of scatter.

In the PET practical example, your digitizer configuration and coincidence sorter will already define output branches such as single energy, coincidence energies, and possibly scatter flags. When you configure the PET digitizer and coincidences, it is good practice to ensure that these scatter-related attributes are included in the output tree.

Logical Rules for Identifying Scattered Coincidences

Once the required information is present in your coincidence data, you can define clear logical rules to classify coincidences. It is helpful to separate two layers:

A “truth-based” classification that uses Monte Carlo history information such as scatter flags.

A “measurement-like” classification that uses only energies and detector information, similar to what you can do experimentally.

For the purposes of this PET example, you will primarily use truth-based information, because it is unambiguous and directly available in the simulation.

The core idea is simple: if at least one of the two detected photons was scattered at any point before its final detection, then the coincidence is classified as a scattered coincidence.

You can summarize this as a rule:

A coincidence is a scattered coincidence if and only if at least one of its two photons has undergone one or more scattering interactions before the final detection.
Equivalently:
If $s_1$ and $s_2$ are boolean scatter flags for the two photons, then
$$\text{is\_scatter} = s_1 \lor s_2$$
A coincidence is a “purely true” coincidence only if
$$\text{is\_true} = \neg s_1 \land \neg s_2$$

In a GATE coincidence tree, $s_1$ and $s_2$ may correspond to integer count fields such as number of Compton interactions in the phantom or detector for each photon. In that case, you apply the same logic with comparisons such as “number of Compton > 0.”

For example, if your coincidence output has branches:

nCompton_phantom_1, nCompton_phantom_2

nCompton_detector_1, nCompton_detector_2

then one convenient scatter definition is:

The coincidence is classified as scattered if
$$n_{\text{Compton,phantom,1}} + n_{\text{Compton,phantom,2}} > 0$$
to focus on patient or phantom scatter, which is clinically more relevant.

If you want to include detector scatter as well, you extend the condition to the detector counters.

Using Energy-based Criteria as Additional Checks

Although Monte Carlo truth is the most accurate way to tag scattered coincidences, it is useful to understand energy-based criteria, because they approximate what experimental PET systems can do. You will need this intuition later for performance metrics and for designing realistic energy windows.

A scattered photon has lower energy than 511 keV. If your singles digitizer records energy including blurring, and your coincidence sorter saves the energies of both singles as $E_1$ and $E_2$, then typical patterns are:

True coincidence. Both $E_1$ and $E_2$ lie around the photopeak (for example between 400 and 650 keV, depending on your PET energy window).

Scattered coincidence. At least one of the energies is significantly lower. In many scattered events only one of the photons has scattered, so you see one energy near the peak and one in the lower tail of the spectrum.

Random coincidence. Energies can have any combination permitted by the singles spectra and may or may not lie in the window.

Even though energy criteria alone cannot perfectly distinguish scatter from randoms or true events, they provide a test that often correlates with the Monte Carlo based scatter flags.

For analysis, you can think of an auxiliary criterion:

Energy-based scatter indicator:
If either $E_1$ or $E_2$ lies below the lower bound of the main PET photopeak window, the probability that the coincidence is scattered is high.

In your PET example, you will not reconfigure the energy window here, but when you later analyze scattered coincidences, you may overlay energy distributions for truth-classified true and scattered events to see how strongly they differ.

Practical Steps to Tag Scattered Coincidences in the PET Example

Once your PET simulation is configured to record coincidences with scatter-related information, the identification step is mostly an analysis task.

In ROOT or Python you will:

Open the coincidence file produced by GATE. This is often a ROOT file with a tree such as “Coincidences”.

Inspect the available branches to locate scatter-related fields. Common branches include flags for Compton interactions in various volumes and sometimes a summary scatter flag written directly by GATE.

Define a selection expression that classifies coincidences as “scattered.” With ROOT you can do this as a branch expression or cut string. With Python and uproot you can implement the logical conditions with NumPy arrays or Pandas filters.

For instance, if your tree has branches:

comptInPhantom1, comptInPhantom2

you can define a scattered coincidence in analysis code as:

coincidence is scattered if comptInPhantom1 > 0 or comptInPhantom2 > 0.

In conceptual form, this rule is exactly the logical formula:

$$\text{is\_scatter} = (n_{\text{Compton,1}} > 0) \lor (n_{\text{Compton,2}} > 0)$$

You can similarly define other categories, such as coincidences with scatter only in the detector materials, by using detector-specific scatter counters.

The result is a tagged set of coincidence events, each labeled “true” or “scattered” according to their Monte Carlo history. This classification is central for the later step where you compute PET performance metrics like scatter fraction and where you assess how scatter affects energy spectra, lines of response, and reconstructed images.

Interpreting Scattered Coincidences in PET Performance

After you have identified scattered coincidences, you can study how they differ from true coincidences within your PET example. Important patterns include:

Spatial distortion. The reconstructed line of response uses the final detector positions of the scattered photons. Since scattering changes the travel direction, these LORs often do not pass through the true annihilation point. This produces a background that fills regions between true activity and lowers image contrast.

Energy distribution. Scattered photons populate the lower-energy tail of the singles and coincidence spectra. By comparing the energy histograms of true and scattered coincidences, you can see how well an energy window can suppress scatter.

Angular distribution. Scatter often changes the emission angle so that the two detected photons are no longer back-to-back. If your output keeps angular information or line-of-response geometry, you can examine deviations from the ideal 180 degree relation.

A clear classification of scattered coincidences in your GATE simulation therefore provides you with a powerful “ground truth” reference. It tells you exactly how much of your coincidence data is degraded by scatter, and lets you test how different detector designs, energy windows, and phantoms influence that scattered fraction.

In the context of this practical PET scanner example, identifying scattered coincidences is not the final goal, but an essential step toward computing PET performance metrics, validating your PET system design, and understanding how realistic imaging conditions affect image quality.