KAHIBARO
28.8. PET Performance Metrics
Table of Contents
Sensitivity
In PET, sensitivity tells you how efficiently the scanner detects annihilation events and turns them into usable coincidence data. For beginners, you can think of sensitivity as “how many coincidence counts you get per unit of source activity and per unit time.”
Sensitivity is usually reported in units of counts per second per kilobecquerel, often written as cps/kBq. Conceptually, if you place a standard source in a standard phantom and run a PET simulation, you can measure how many true (or prompt) coincidences you detect and normalize this by the activity and the acquisition time.
If $N_{\text{coinc}}$ is the number of detected coincidences in a time interval $\Delta t$, and the average source activity during that time is $A$, then the total sensitivity $S$ is defined as
$$
S = \frac{N_{\text{coinc}}}{A \, \Delta t}
$$
where $A$ is often expressed in kBq and $S$ then has units cps/kBq.
In a GATE simulation you usually obtain $N_{\text{coinc}}$ from the coincidence output. If your simulation includes true, scatter, and random coincidences, you must decide what you want to call “sensitivity.” In system specifications, sensitivity usually refers to prompt coincidences that pass the standard PET energy and timing windows. However, in performance studies you often compute:
True sensitivity:
$$
S_{\text{true}} = \frac{N_{\text{true}}}{A \, \Delta t}
$$
Prompt sensitivity:
$$
S_{\text{prompt}} = \frac{N_{\text{true}} + N_{\text{scatter}} + N_{\text{random}}}{A \, \Delta t}
$$
Here $N_{\text{true}}$ is the number of true coincidences, $N_{\text{scatter}}$ the number of scattered coincidences, and $N_{\text{random}}$ the number of random coincidences. The PET Coincidence Analysis chapter explains how to classify coincidences, so in this section the focus is only on how they feed into the sensitivity metric.
In GATE, you typically:
- Simulate a known activity $A$ placed in a standard phantom or point source setup.
- Run for a known acquisition duration $\Delta t$ or a known number of decays.
- Count how many coincidences pass your standard energy and timing windows.
- Apply the formula above to compute $S$, $S_{\text{true}}$ or $S_{\text{prompt}}$.
Because Monte Carlo simulations generate counts with statistical fluctuations, you should also estimate the uncertainty on sensitivity. If $N_{\text{coinc}}$ is large and governed by Poisson statistics, then the standard deviation on $N_{\text{coinc}}$ is approximately $\sqrt{N_{\text{coinc}}}$, so the relative uncertainty on $S$ is
$$
\frac{\sigma_S}{S} \approx \frac{1}{\sqrt{N_{\text{coinc}}}}
$$
This shows that to get precise sensitivity values in GATE, you must simulate enough coincidences, which usually means a large number of decays or a long simulated acquisition time.
Sensitivity strongly depends on scanner geometry, detector material, energy window, and coincidence timing window. In GATE you can change these parameters and recompute sensitivity to study how each design choice affects scanner performance.
Scatter fraction
Scatter fraction quantifies how much of your detected coincidence data is contaminated by scattered events that do not follow the original line of response. It is a key measure of image quality, because a high scatter fraction usually leads to more background and less contrast.
If you can separate coincidence events into true, scatter, and random categories, then the scatter fraction $SF$ is defined as the fraction of scattered coincidences among all “nonrandom” coincidences, that is all true plus scatter events. Formally,
$$
SF = \frac{N_{\text{scatter}}}{N_{\text{true}} + N_{\text{scatter}}}
$$
Often the scatter fraction is expressed as a percentage, so you simply multiply by 100.
In a real PET system, identifying scatter and true events is very difficult, but in GATE you have full access to the Monte Carlo truth information. You can therefore classify coincidences according to whether one or both photons underwent scattering before detection. The PET Coincidence Analysis chapter describes how to label coincidences as true or scatter in practice. Once you have $N_{\text{true}}$ and $N_{\text{scatter}}$, computing $SF$ is straightforward using the formula above.
Table: Coincidence categories for scatter fraction
| Category | Description | Used in $SF$ calculation? |
|---|---|---|
| True coincidences | Both photons without significant scatter in object | Included in denominator |
| Scattered coincidences | At least one photon scattered before detection | Included in numerator and denominator |
| Random coincidences | Unrelated decays paired within timing window | Excluded from $SF$ in standard definitions |
Random coincidences are excluded when computing scatter fraction, because $SF$ is meant to describe the balance between true and scatter only. If your coincidence dataset contains randoms, you should either remove them based on timing information or at least not count them in the $SF$ formula.
In GATE simulations of standard PET performance tests, such as those inspired by NEMA protocols, you usually:
- Simulate a standard phantom with realistic activity distribution.
- Apply the clinical energy and time windows to your coincidence data.
- Use Monte Carlo truth to classify each coincidence as true or scatter.
- Compute $SF$ from $N_{\text{true}}$ and $N_{\text{scatter}}$ using the formula above.
Uncertainty on $SF$ comes from the finite number of events, and you can estimate it by standard propagation of Poisson errors in $N_{\text{true}}$ and $N_{\text{scatter}}$. In practice, you reduce this uncertainty by simulating enough coincidences.
Scatter fraction depends on patient size, phantom geometry, energy window, and scanner design. In GATE, you can systematically vary these parameters to see how $SF$ changes, which is very useful when comparing different PET geometries or energy window choices.
Count rate
Count rate performance describes how many events per second your PET system processes, and how this changes with activity. It determines how well the system can handle high activities without being dominated by randoms and dead time.
You can define several types of count rates. The three most important are:
- True count rate $T$.
- Scatter count rate $S$.
- Random count rate $R$.
If $N_{\text{true}}$, $N_{\text{scatter}}$, and $N_{\text{random}}$ are counts detected in an acquisition time $\Delta t$, then
$$
T = \frac{N_{\text{true}}}{\Delta t}, \quad
S = \frac{N_{\text{scatter}}}{\Delta t}, \quad
R = \frac{N_{\text{random}}}{\Delta t}.
$$
The prompt count rate $P$ includes all coincidences that pass your selection criteria, so
$$
P = T + S + R.
$$
In a GATE simulation you can compute these quantities by counting how many coincidences of each type occur in a chosen time interval. If your simulation is configured in terms of a specified number of decays rather than real acquisition time, you can convert decays to time using the activity, or configure your simulation directly with a known duration.
Count rate behavior is not only about raw numbers. The key question is how the relationship between $T$, $S$, and $R$ changes when you increase activity. At low activity the true count rate typically rises almost linearly with activity, while scatter and randoms are still relatively small. At higher activity, randoms increase roughly with the square of activity, and detector dead time may reduce the effective true count rate. You can explore these regimes in GATE by simulating the same scanner and phantom with different source activities and recording $T$, $S$, and $R$.
For assessing overall PET performance, a common figure of merit is noise equivalent count rate (NECR). It compresses the combined effects of true, scatter, and randoms into a single metric that relates to image noise. NECR is defined as
$$
\text{NECR} = \frac{T^2}{T + S + k R}.
$$
Here $k$ is a factor that describes how randoms are corrected in the reconstruction, and is often taken as 1 or 2 depending on the assumed correction method. A higher NECR usually corresponds to better image signal to noise for a given scan duration.
In GATE simulations, you can compute NECR at each activity level using the true, scatter, and random count rates you derived. The resulting NECR versus activity curve often has a maximum at some intermediate activity, beyond which additional activity does not improve image quality because randoms and dead time dominate. This is important for optimizing injected activity in PET protocols.
For a basic performance study in GATE you typically:
- Choose a scanner geometry and a realistic phantom.
- Simulate several activity levels.
- For each activity, compute $T$, $S$, $R$, and $P$ from the coincidence categories.
- Compute NECR using the formula above.
- Plot $T$, $S$, $R$, $P$, and NECR as a function of activity.
This set of metrics, combined with sensitivity and scatter fraction, gives a concise quantitative description of how well a PET system performs, and it is one of the main reasons to use Monte Carlo simulation when designing or optimizing PET scanners.
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