47.9. Applying Energy Blurring

Table of Contents

Understanding Energy Blurring in PET

Energy blurring models the limited energy resolution of real PET detectors. In your PET simulation, perfect energy measurements would produce unrealistically sharp photopeaks at 511 keV. By applying energy blurring to singles, you make the detector response more realistic and obtain energy spectra that resemble experimental data.

In GATE digitizers, energy blurring is typically applied to singles after energy summation in a crystal, before energy windowing and coincidence sorting. You will attach an energy blurring step to the PET digitizer chain and configure it with a realistic resolution, usually specified at 511 keV.

Energy resolution is almost always specified as a full width at half maximum (FWHM) at 511 keV, and energy blurring in GATE is applied using a Gaussian distribution. Forgetting to convert between FWHM and $\sigma$ or misinterpreting the percentage can seriously distort your simulated spectra.

From Ideal to Blurred Energy

In a PET detector crystal, many physical effects cause the measured energy to deviate from the true deposited energy. Even if a photon deposits exactly $E_{\text{true}}$ in the crystal, the measured energy $E_{\text{meas}}$ will vary event by event.

In your PET simulation, before blurring, each single has an attribute that represents the total energy deposit in the crystal, often called something like energy or edep. Without blurring, this value is perfectly sharp and directly equals the simulated true deposit. After blurring, this value is replaced by a random sample that follows a Gaussian distribution around the true energy.

Mathematically, for each event you draw
$$
E_{\text{meas}} \sim \mathcal{N}
\left(
E_{\text{true}},
\sigma_E^2
\right)
$$
where $\sigma_E$ controls the width of the response.

In PET simulations you must always apply energy blurring to singles, not to raw hits. Hits represent microscopic interactions. The detector measures the summed energy in a crystal, so blurring must be applied after summation at the crystal level.

Defining Energy Resolution

Energy resolution is commonly given as a relative FWHM at a reference energy, usually 511 keV. For example, a typical LYSO PET detector might be described as having a resolution of 12 percent at 511 keV. This means that the FWHM of the 511 keV peak is

$$
\text{FWHM}_{511} = 0.12 \times 511\ \text{keV} \approx 61\ \text{keV}.
$$

To use this in a Gaussian model, you must convert FWHM to the standard deviation $\sigma$.

The relation between FWHM and $\sigma$ is

$$
\text{FWHM} = 2 \sqrt{2 \ln 2} \ \sigma \approx 2.355 \ \sigma.
$$

So for a given relative FWHM resolution $R$ (for example 0.12 for 12 percent) at energy $E$, you have

$$
\text{FWHM}(E) = R \, E,
$$
$$
\sigma_E = \frac{\text{FWHM}(E)}{2.355} = \frac{R \, E}{2.355}.
$$

Energy blurring in PET simulations often assumes that the relative resolution scales as $1 / \sqrt{E}$, which reflects the fact that more scintillation photons are produced at higher energies. If you know the resolution $R_{511}$ at 511 keV, and if your digitizer supports energy dependent resolution, you can model the energy dependence as

$$
R(E) = R_{511} \sqrt{\frac{511\ \text{keV}}{E}}.
$$

Then

$$
\sigma_E(E) = \frac{R(E) \, E}{2.355}.
$$

In practice you usually only need to specify a resolution at 511 keV. The digitizer will use a built-in model for the energy dependence if it supports that option.

Important formula:
$$
\sigma_E = \frac{R \, E}{2.355}
$$
with $R$ the relative FWHM (for example 0.12 for 12 percent). Never use the percentage directly as $\sigma$. Always convert through FWHM.

Configuring Energy Blurring in the PET Digitizer

In your PET example, you already have a digitizer chain that processes hits into singles and then coincidences. Energy blurring is an additional step in this chain, often implemented as a dedicated module, for example an "energy blurring" or "energy resolution" digitizer.

The typical sequence for the PET digitizer chain is:

Hit collection from detector crystals.
Energy summation per crystal to create singles.
Energy blurring on the singles energy.
Application of an energy window on blurred energy.
Coincidence sorting based on time and energy.

Your task in this chapter is to add step 3 to your existing digitizer configuration.

In a Python based GATE configuration, you typically do something conceptually similar to the following:

You create or access your digitizer, for example pet_digi, then add an energy blurring module to it and set its parameters. While the exact function and parameter names depend on the specific OpenGATE version, you will usually specify at least:

The input collection, for example the singles.
The output collection, which might reuse the same name or create a new branch.
The type of resolution model, for example Gaussian.
The resolution value at 511 keV, given as a relative FWHM.

You may also set whether to discard events with negative blurred energy, but in most cases this is handled automatically.

After this configuration, when you run the simulation again, the energy in your singles will no longer be equal to the summed energy deposits, but will instead be drawn from the appropriate Gaussian distribution.

Make sure that:

Energy blurring uses the correct units for energy, usually keV in GATE output.
You configure resolution at 511 keV, not at 1 keV or another arbitrary value.
The blurring step is placed before the energy window digitizer and before coincidence sorting.

Observing the Effect on PET Singles

Once you have energy blurring configured, rerun your PET simulation and inspect the singles energy spectrum. The changes compared to the unblurred case are important for understanding real detector behavior.

Before blurring, you would see a very narrow peak at 511 keV and sharply separated Compton edges. This is not realistic.

After blurring, the 511 keV peak becomes broader and has a Gaussian like shape, and the Compton edge becomes less sharp. The spectrum more closely matches what you would see with a real PET scanner.

A simple way to verify that blurring is working is to plot histograms of the singles energy before and after you activate the blurring, using the same number of events. You should see:

A wider FWHM around 511 keV.
A smoother, less structured spectrum at lower energies.

If you choose an unrealistically good resolution, for example 1 percent at 511 keV, the peak will still be extremely narrow. If you choose a very poor resolution, for example 30 percent, the peak will be so wide that it overlaps strongly with scattered events.

For many PET systems made of LYSO, typical values are in the range 10 to 15 percent at 511 keV. For NaI based gamma cameras, values are often poorer, for example around 9 to 11 percent at 140 keV, but PET is almost always based on faster crystals.

Choosing Realistic Resolution Parameters

The choice of energy resolution has a direct impact on PET image quality, scatter fraction, and quantitative accuracy. You should select a value that matches the detector you want to model.

The table below gives rough guidelines at 511 keV:

Crystal material	Typical resolution at 511 keV (FWHM)
LYSO / LSO	10 to 15 percent
BGO	15 to 20 percent
NaI(Tl)	12 to 15 percent
Plastic scint.	20 percent or more

For a beginner PET example using LYSO, a value around 12 percent at 511 keV is a reasonable default if you do not have specific experimental data.

When you change the energy resolution in your simulation:
Always adjust your PET energy window settings accordingly. A wider energy resolution usually requires a wider window to keep a similar true coincidence efficiency, but this will also let in more scattered events.

Interaction with the PET Energy Window

In the previous step of your PET example, you defined an energy window, for example something like 350 keV to 650 keV, to accept events near the 511 keV photopeak and reject low energy scatter and noise.

With ideal, unblurred energy, this window sharply separates events. After blurring, some events that were just outside the window might be blurred inside, and some that were originally inside might be blurred outside. This effect is essential for a realistic PET performance study.

The energy window should always operate on the blurred energy, because the detector makes decisions based on what it actually measures, not on the true physical deposit.

This has several consequences:

The true coincidence sensitivity depends not only on the window limits but also on the resolution.
The scatter fraction is affected because scattered events can sometimes be blurred into the window.
The optimization of the PET energy window must consider the chosen resolution.

In your practical PET example, once you have both energy blurring and the window configured, you can change the resolution parameter and observe how the number of accepted singles and coincidences changes. This is a realistic way to study the trade off between energy resolution and scatter rejection.

Practical Checks and Common Pitfalls

When you first add energy blurring to your PET digitizer, it is helpful to perform a few quick checks on the simulation output.

First, verify that the units are consistent. GATE usually stores energy in keV in ROOT output. Confirm that your histograms are labeled accordingly, so that the 511 keV peak appears at 511 on the axis.

Second, confirm that the number of singles does not change drastically when you enable blurring, before applying the energy window. Blurring alone should mostly change the distribution of energy values, not the number of entries. A large reduction of entries often indicates a filtering or a threshold applied automatically with the blurring step.

Third, check that your energy window acts on the blurred energy. One way is to plot the singles spectrum with the window labeled and verify that the accepted events correspond to the correct energy region.

Finally, test a few artificial settings: for example, temporarily set the resolution to a very large value such as 30 percent at 511 keV and inspect the spectrum. The peak should become obviously wider. Then set it to an unrealistically small value, for example 1 percent, and verify that the peak narrows. These tests help you confirm that energy blurring is indeed active and controlled by your chosen parameter.

By correctly applying energy blurring at this stage, you ensure that the subsequent PET steps such as applying the PET energy window and sorting coincidences rely on realistic detector measurements, which is essential for meaningful PET performance studies and for comparing your simulation with experimental data.