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# A photon source model for alpha-emitter radionuclides

D Sarrut, A Etxebeste, Jean Michel Létang

► To cite this version:

D Sarrut, A Etxebeste, Jean Michel Létang. A photon source model for alpha-emitter radionuclides. Physics in Medicine and Biology, 2024, 69 (9), pp.095009. 10.1088/1361-6560/ad3881. hal-04564153

HAL Id: hal-04564153

https://hal.science/hal-04564153v1

Submitted on 17 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
1 A photon source model for alpha-emitter radionuclides

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D. Sarrut¹, A. Etxebeste¹, JM. Létang¹

¹ Université de Lyon, CREATIS; CNRS UMR5220; Inserm U1044; INSA-Lyon; Université Lyon 1; Centre Léon Bérard, France.

E-mail: david.sarrut@creatis.insa-lyon.fr

**Abstract.**

*Objective.* A Monte Carlo virtual source model named PHID (Photon from Ion Decay) that generates photons emitted in the complex decay chain process of alpha-emitter radionuclides is proposed, typically for use during the simulation of SPECT image acquisition.

*Approach.* Given an alpha-emitter radionuclide, the PHID model extracts from Geant4 databases the photon emission lines from all decaying daughters for both isometric transition and atomic relaxation processes. According to a given time range, abundances and activities in the decay chain are considered thanks to the Bateman equations, taking into account the decay rates and the initial abundances.

*Main results.* PHID is evaluated by comparison with analog Monte Carlo simulation. It generates photons with the correct energy and temporal distribution, avoiding the costly simulation of the complete decay chain thus decreasing the computation time. The exact time gain depends on the simulation setup. As an example, it is 30× faster for simulating 1 MBq of ²²⁵Ac in water for 1 sec. Moreover, for ²²⁵Ac, PHID was also compared to a simplified source model with the two main photon emission lines (218 and 440 keV). PHID shows that 2 times more particles are simulated and 60% more counts are detected in the images.

*Significance.* PHID can simulate any alpha-emitter radionuclide available in the Geant4 database. As a limitation, photons emitted from Bremsstrahlung are ignored, but they represent only 0.7% of the photons above 30 keV and are not significant for SPECT imaging. PHID is open-source, available in GATE 10, and eases the investigation of imaging photon emission from alpha emitters.

29 1. Introduction

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Radiopharmaceutical therapy (RPT) with alpha emitters or Targeted Alpha Therapy (TAT), exploits the short range and high linear energy transfer (LET) of the emitted alpha particle (⁴He) to increase dose to targeted tissues [1]. Examples of radionuclide used for TAT are: ²²³Ra (radium, Xofigo™) for castration-resistant prostate cancer (CRPC) with metastatic bone disease [2], or, more recently ²²⁵Ac (actinium, Ac-225-PSMA-617) combined with PSMA for metastatic prostate cancer [3]. Other alpha emitters are under investigation, such as ¹⁴⁹Tb (terbium), ²¹¹At (astatine), ²¹²Pb (lead), ²¹³Bi (bismuth), ²²³Ra (radium) or ²²⁷Th (thorium). The LET of alpha is around 50-200 keV/µm which is about 3 orders of magnitude higher deposited energy per unit path length than with
conventional $\beta^-$ emitters such as $^{177}$Lu or $^{90}$Y ($\approx$ 0.2 keV/µm) [4]. The range of emitted alpha is around 50 to 100 µm roughly corresponding to the few cells diameters [5]. Such high impact comes with associated toxicity risks as the absorbed dose is also large when the radiopharmaceutical is not located near/in cancer cells.

It is hence crucial to evaluate the spatial biodistribution, and, ideally, the temporal biokinetics of the injected compound in the patient body. For some alpha emitters, in-vivo imaging is feasible thanks to emitted energetic gamma rays and X-ray photons that may be detected by SPECT systems [6]. For example, feasibility of imaging with $^{212}$Pb [7] (phantom) or with $^{225}$Ac [8] (patient) have been reported. Alternatively, companion surrogates, such as $^{226}$Ac, are under investigation [9] but can only be useful for pre-treatment and not per-treatment. Note also that $^{149}$Tb leads to 7% of $\beta^+$ that makes it suitable for PET imaging [10].

However, each radionuclide has its own decay chain leading to a specific amount of exiting photons with different energy lines, see figure 1. For example, $^{225}$Ac decays to $^{221}$Fr that emits 218.12 keV (11.4%) and $^{213}$Bi 440.45 keV (25.94%), $^{223}$Ra emits gamma rays of 154.21 keV (5.7%), 269.46 keV (13.9%) and lower energy X-rays 81.07 keV (15%), 83.79 keV (24.7%), etc. According to the energy ranges and quantities of those photons, SPECT acquisition must be tuned by selecting adapted energy windowing and appropriate scatter correction and reconstruction methods. For high energy photons, larger than 400-450 keV, that are difficult or impossible to detect with current SPECT technology, other imaging systems such as Compton Camera may be envisioned [11].

In this situation, Monte Carlo simulation is a crucial method [12] for investigating SPECT image acquisition parameters and advancing the development of new imaging systems [13]. GATE-Geant4 offers the capability to simulate various ion sources using the Monte Carlo engine, considering the complete decay chain and all relevant emissions. However, this process is time-consuming as it requires individual creation and tracking of all by-product elements such as alpha and beta particles, photons, and daughter ions. Additionally, the timestamps of particles may not be directly compatible with the digitizer modules of SPECT systems within GATE.

In this study, we propose a source model that efficiently generates gammas and X-rays emissions from an alpha-emitter source, including the associated energy spectra, intensity, and timing.

## 2. Material and methods

In this section, we first recall the photon emission processes involved during alpha emitters decay (subsections 2.1), followed by the description of GATE/Geant4 main simulation mechanisms for those processes in 2.2. Then, we describe the proposed PHID source model in 2.3.
<figure><img src="image_3.png" /><figcaption>Figure 1: Example of decay chains for $^{211}$At, $^{212}$Pb, $^{225}$Ac and $^{227}$Th</figcaption></figure>

## 2.1. Photon emission process of alpha emitters

Photon emissions in isomeric transitions. Isomeric transitions can occur when an alpha particle is emitted from a nucleus, resulting in the formation of a daughter nucleus in an excited state. They refer to the decay of an excited nuclear state to a lower energy state without changing the number of protons and neutrons. Photon emission processes during isomeric transitions from the nucleus with an excess of energy can occur through either gamma emission or internal conversion mechanism. The latter leads to X-ray emissions by atomic relaxation.

Isomeric transitions often involve gamma ray emission with specific energy levels
characteristic of the particular nucleus involved. The daughter nucleus subsequently undergoes a transition to a lower energy state by emitting gamma rays, the emitted gamma rays carry away the excess energy and stabilize the nucleus.

The exited nucleus instead of emitting a gamma ray can interact with an orbital electron (internal conversion process). The excited atom undergoes *atomic relaxation* to return to its ground state, releasing excess energy in the form of characteristic X-rays. Atomic relaxation involves the rearrangement of electrons within the atom to achieve a stable configuration. As electrons transition between different energy levels, X-rays can be emitted to balance the energy changes. The emitted X-rays carry information about the atomic structure and energy levels of the daughter atom.

*Decay equilibrium.* In addition to the photon energy spectra, the relative intensity of all emissions from the different daughters must be known. Indeed, the *decay equilibrium* refers to the balance between the decay rates of the parent radionuclide and its daughter products. Equilibrium can be categorized into two types: secular equilibrium and transient equilibrium. *Secular equilibrium* occurs when the half-life of the parent radionuclide is significantly longer than the half-lives of its daughter products (i.e., more than about 100 times longer). In this case, the decay rate of the parent is much slower compared to the decay rates of its daughters. As a result, the activities of the parent and daughter radionuclides reach a relatively constant ratio over time. The daughter products are continuously generated through the decay of the parent, but their decay rates are approximately equal to their production rates. Thus, the overall activity of the system remains stable.

On the other hand, the *transient equilibrium* occurs when the half-lives of the parent radionuclide and its daughters are comparable. In this case, the activities of the parent and daughter radionuclides initially change at different rates, but eventually, they reach a state of equilibrium. However, unlike secular equilibrium, this equilibrium is not permanent, and it is a temporary state that occurs during the decay process.

For example, $^{225}$Ac can be considered to be in secular equilibrium within its decay series [1]. The half-life of $^{225}$Ac (10 days) is larger than its daughter products $^{221}$Fr (4.9 min), $^{217}$At (32.3ms), $^{213}$Bi (45.59 min). This leads to a relatively constant ratio of activities between the parent and daughter isotopes, as the production and decay rates of the daughters are approximately equal, see figure 2. On the other hand, the half-life of $^{217}$Th (18.7 days) is close to one of its direct daughter $^{223}$Ra (11.43 days) and this leads to a transient equilibrium. Figure 2 has been generated by computing Bateman equation using the *radioactivedecay* python package [14]: https://radioactivedecay.github.io.

*Importance of decay and recoil energy.* During gamma ray emission via isomeric transition, the recoil energy imparted to the daughter is much higher than the binding energy of the chemical bond, resulting in the release of the daughter [15]. Redistribution of recoil progeny depends on the recoil distance, diffusion, active transport, and organ affinity. The half-life of the daughter affects the time to target and toxicity. Redistribution
<figure><img src="image_4.png" /><figcaption>Figure 2: Time activity curves for $^{225}$Ac and $^{227}$Th and their daughters. For $^{225}$Ac secular equilibrium is reached after about 24 hours. For $^{227}$Th we observe a transient equilibrium.</figcaption></figure>

is difficult to measure and is mostly done post-mortem.

## 2.2. Geant4 and GATE simulation.

There is a difference between Geant4 and GATE in how they handle source activity management. In Geant4, when simulating an ion source, users only define the number of ions, and the engine will model *all* the subsequent decay products that will occur to the ions, regardless of *when* they are emitted [16]. Products include alpha particles, beta particles, photons, and daughter ions. This means that if we are only interested in a specific time range, such as in the case of a SPECT acquisition, we must filter the emitted particles based on their timestamps. Consequently, a potentially substantial number of particles will be generated and tracked, even if they are not of prime interest. To illustrate this, consider simulating an activity of $A_0 = 10$ Bq of $^{225}$Ac. The total number of atoms $N$ that must be simulated can be calculated with $N = A_0/\lambda$, using $\lambda = \ln 2/T^{1/2}$, where $\lambda$ is the decay constant in $s^{-1}$ (the inverse of the lifetime), and $T^{1/2}$ is the half-life. The half-life of $^{225}$Ac is approximately 10 days (864,000 seconds),
which yields $N = 12.5 \times 10^6$ events that need to be simulated (as an input parameter of Geant4). Each of these events will produce decay products, all with their corresponding timestamps, which will be created and tracked in the simulation.

In contrast, GATE employs a more efficient approach to manage the decay process, producing only the ions that are undergoing decay. Users define the simulation time range by specifying start and end times, the half-life, and the source's initial activity. The GATE engine then generates ion events that decay within this defined time frame, along with their corresponding timestamps. This process unfolds on an event-by-event basis, where the decay time is stochastically computed using Equation 1. Here, $t$ represents the current GATE simulation time, RND$_{[0:1]}$ is a randomly generated number uniformly distributed between 0 and 1, and $A(t)$ denotes the current activity at time $t$. After each event, the activity is updated to account for the half-life, following Equation 2, beginning with the initial activity $A_0$ provided by the user.

$$t_{\text{decay}} = t - \frac{\log(\text{RND}_{[0:1]})}{A(t)} \qquad (1)$$

$$A(t) = A_0 \exp^{-\lambda(t-t_0)} \qquad (2)$$

In this approach, the radionuclide's associated lifetime in Geant4 is set to zero, allowing GATE to set the timestamps of the first daughters. This method has been utilized for years in GATE, such as in simulating fast $^{177}$Lu sources. In this example, instead of modelling a decaying ion source, it is replaced by the corresponding photon emission distribution. This principle can also be applied to ion sources. However, in this scenario, the daughter ion products of decay will still retain their intrinsic lifetime properties, leading to tracking all decays produced regardless of time.

Let's consider simulating 10 Bq $^{225}$Ac source over a 1-hour acquisition, about 35,984 events (decaying $^{225}$Ac ions) will be simulated. These events will subsequently decay into $^{221}$Fr, $^{217}$At, $^{213}$Bi, etc., and all their respective decay products will be simulated and tracked. In this case, the half-lives of the daughters are shorter than the one of $^{225}$Ac, which allows significant time savings during the simulation even if all decay products of daughters are still tracked.

In both cases (Geant4 source or GATE source), when simulating a SPECT imaging system, it is theoretically possible that several hits occurring in the same crystal may be combined to form a single while they did not correspond to the same timing period, leading to incorrect single. In practice, the probability of such incorrect hits combination is very low because it only occurs when two photons coming from two different daughters of the same event are hitting the same crystal. Alternatively, to define an ion source, the user can define a source of photons by directly providing the energy spectra and time activity such as in [17], but, in general only very few photon lines are simulated, ignoring a lot of other photon emissions that may impair detection. Moreover, the equilibrium and variable time activity are not taken into account.
In both cases (Geant4 source or GATE source) also, the daughters that are already present when the simulation starts are not handled. However, in the proposed PHID source model described below, they are handled thanks to the Time Activity Curves automatically associated with all daughters (section 2.3.4). In the following, we describe a new PHID source model that generates only the required photons, with the correct timestamps.

## 2.3. PHID: a photon source model for alpha emitters

The proposed photon source model is named PHID (Photon from Ion Decay) and is based on two key elements.

* First, it involves identifying and compiling all photon lines, including their energies and intensities, from all decay daughters and processes into a comprehensive list of energy-intensity pairs. The origin of the photons comes from isomeric transition and atomic relaxation. Bremsstrahlung radiation due to the emitted charged particles is not included in the model. The next sections 2.3.1 and 2.3.2 describe this step.

* Second, it automatically computes and utilizes Time Activity Curves to express the relative intensities of decaying daughters as a function of time. During simulation runtime, the Geant4 engine will subsequently generate primary gamma events sequentially, considering their relative intensities, energy levels, and associated timestamps. This is described in the next sections 2.3.3 and 2.3.4.

### 2.3.1. Modeling gamma rays from isomeric transition
The information regarding the gamma ray emissions from isomeric transitions is obtained from the Geant4 database. The relative emission intensities are computed using a recursive process that takes into account all decaying channels and excitation energy levels. First, all decaying channels for a specific radionuclide are considered by examining the decay table of the G4RadioactiveDecay process object. This information is obtained through the functions GetDecayTable and GetDecayChannel. The relevant data are stored in a file, named for example z89a225 in the case of $^{225}$Ac, within a folder indicated by the G4LEVELGAMMADATA environment variable. This file contains information about all levels, starting from the ground state (level 0), including the excitation energy of each level in keV and the number of possible deexcitation channels for each level. For each deexcitation channel, the file provides the energy of the gamma transition, the relative gamma ray emission intensity, and the total internal conversion coefficient, denoted as $\alpha = Ic/Ig$. Here, Ic represents the rate of internal-conversion electrons, and Ig represents the rate of emitted gamma rays. The final gamma emission rate is then recursively computed for each channel using the formula $p = BR \times Pg \times It \times pc$, where BR is the branching ratio, Pg is the gamma ray emission probability, It = Ic + Ig is the total transition probability, and pc represents the recursively accumulated channel probability. As an example, consider Figure 3, where level 2 (440.439 keV) can deexcite to the ground level 0 with a probability of 26.1%, or to level 1 (292.796 keV) with a probability of 0.0128%,
and then further deexcite to the ground state with a probability of 0.421%. Similar deexcitation lines of the same levels are merged.

This process is performed for all channels and stored in a file specific to each radionuclide.

It is carried out for a set of common radionuclides ($^{149}$Tb, $^{211}$At, $^{212}$Pb, $^{213}$Bi, $^{223}$Ra, $^{227}$Th, etc.), including all their decay daughters. Consequently, the final source model for a given radionuclide considers all of its daughters and retrieves the list of gamma ray emission energies and associated probabilities from the corresponding files.

<figure><img src="image_5.png" /><figcaption>Figure 3: Example of $^{213}$Bi deexcitation schema extracted from http://www.lnhb.fr/Laraweb.</figcaption></figure>

### 2.3.2. Modeling photons from atomic relaxation
The photons emitted from atomic relaxation can be extracted from the IAEA web site, thanks to the provided web API‡. The web IAEA API only allows to automatically retrieve a simplified version of the X-rays data [18]. Here, we use a more complete database, kindly provided by the Department of Nuclear Sciences and Applications from IAEA that extracted data from Livechart, based on the ENSDF snapshot of April 2022 (ENSDF stands for Evaluated Nuclear Structure Data Files). Similar to the previously described isomeric transition process, the retrieved data are then stored in a file that can then be read to list photon emission energies and associated probabilities. Note that those data only include transitions to K and L shells. Moreover, Geant4 manages several fluorescence databases. The one

<footnote>‡ The API is described here: https://nds.iaea.org/relnsd/vcharthtml/api_v0_guide.html</footnote>
that corresponds to the IEAE website is called Bearden [19] and this is enabled using `/process/em/fluoBearden true`. As the decay time of atomic relaxation is of the order of a nanosecond, the characteristic X-ray emission is considered to be prompt (no delayed fluorescence) [20].

### 2.3.3. Bremsstrahlung and annihilation photons
Bremsstrahlung can also be emitted from alpha emitter radionuclides. Bremsstrahlung occurs when the alpha particle interacts with the surrounding matter and undergoes acceleration or deceleration. As the alpha particle slows down when deflected by another charged particle, electromagnetic radiation is emitted. Because this process depends on the material, it can not be directly included in the source model. However, in water, the Bremsstrahlung energy distribution is a continuous spectrum mostly of low energy and with very low intensity. This is illustrated in figure 7 where Bremsstrahlung photons represent around 3% of the total number of photons, with only 0.7% above 30 keV. We thus decided to neglect this process. Moreover, some radionuclides such as $^{149}$Tb also emit $\beta^+$ that will annihilate into pairs of back-to-back 511 keV photons. This process is not included in PHID. If needed, it can be considered as an additional source of $\beta^+$ in GATE.

### 2.3.4. Time Activity Curves of relative daughters activities
Previous sections described how we retrieved the isomeric transition and atomic relaxation photon emissions. However, the relative abundance of each daughter radionuclide varies with time and equilibrium. The user controls this by providing the initial and final time points of his/her simulation. The Bateman equations are then used to compute the relative abundances of all daughters, describing the time-dependent decay of radioactive nuclides and their daughter products. These equations consider changes in population over time, accounting for the decay of parent nuclides and subsequent decay of daughters. By utilizing decay constants and initial abundances, these equations model and predict radioactive decay processes. Based on these equations and the specified time points, Time Activity Curves (TAC) are automatically computed for all daughters.
In conclusion, the PHID source model automatically retrieves all information (daughters, half-life, TAC, etc) of the given radionuclide. The overall description is illustrated in the schema figure 4. It incorporates the relative activity from the computed TACs, along with the emitted photons from IT and AR, to generate only the required primary photons with the correct energy, intensity, and timing in the desired time frame.

## 3. Experiments

The first experiment compared the gamma ray emission from the reference ion source model from Geant4 with the PHID source model. The reference energy spectrum was obtained by using an ion point source of $37.4 \times 10^6$ ions of $^{225}$Ac (30 Bq), in a water box. The output of the reference simulation was a phase-space that contains all particles that have been created and tracked during the whole ion life-time, together with the
<figure><img src="image_6.png" /><figcaption>Figure 4: Schematic description of the process to create the PHID source model for a given radionuclide.</figcaption></figure>

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kinetic energy, their Geant4 creator process, and their associated timestamps. The PHID model considered an acquisition time frame of 30 minutes starting 24 hours after injection of the exact same number of ions. At the end of the simulation, the phase space was processed to extract gamma rays whose creator process is `model_RDM_IT` (isometric transition) and whose timestamps are between 24 hours and 24 hours plus 30 minutes. The resulting reference gamma energy spectrum was compared to the one obtained from the second simulation using the proposed source model of isometric transition only. The second experiment was dedicated to atomic relaxation and illustrated here for $^{212}$Pb, with a starting time at 5 hours after injection and 2000 seconds acquisition time ions (also 30 Bq). The third experiment combined photons from IT and AR using the same $^{225}$Ac point source and analysed the complete gamma spectra including bremsstrahlung radiation. Additional comparisons between Geant4 ion sources and PHID sources were also performed for three other radionuclides $^{212}$Pb, $^{223}$Ra and $^{149}$Tb with respectively 2000, 1000 and 100 seconds acquisition duration, 18000, 50000 and 10000 seconds start times.

In the fourth experiment, a SPECT projection acquisition was simulated. The modeled SPECT imaging system was a Siemens Intevo Bold device with HE collimator (High Energy, 8000 holes of 4 mm, 2mm septal thickness, 9.5 mm NaI crystal thickness). The acquisition was one single-angle orientation. We simulated the NEMA IEC body phantom with six spheres filled with 100 kBq/mL $^{225}$Ac (around 0.5 MBq total) acquired with a time frame of 210 seconds starting at t=24h like in [8]. Three source types were compared: (1) the pristine reference $^{225}$Ac ion source, (2) a simplified dual gamma source
of 218.12 keV (11.4%) and 440.45 keV (25.9%), like the one in [21] for example, and (3) the proposed PHID source that includes all isomeric transition and atomic relaxation photons. The SPECT energy windows were: two windows centered on the main peaks having 20% width (196-240 keV and 396-484 keV), and 4 scatter energy windows below and above the peaks with 10% energy width. For the reference ion source only, we simulated the number of decays of $^{225}$Ac occurring during the acquisition period from t=24h to t=24h+210 seconds, which amounted to approximately 10$^9$ decays. During this process, the Geant4 engine tracked the photons emitted throughout the entire lifetime of the decays and GATE used them to generate the final image. This approach is an approximation, as in reality, we should have only considered and filtered photons whose timestamps fell within the specified time range (from 24h to 24h+210 seconds) originated from the radioactive daughters of $^{225}$Ac decays occurred in the previous 24 hours, totalling around $4 \times 10^{11}$ ions. Given the computational constraints, processing at a rate of approximately 9000 ion events per second (on a single thread) would have required more than a year of computation time. Consequently, we chose to ignore the photon timestamps and considered all emitted photons, irrespective of their timestamps. This approximation hence considers all ion's daughters whatever their emission time. Because the secular equilibrium of $^{225}$Ac is already reached (figure 2), the number of considered photons is correct, but their timestamps, which are disregarded, are not. For the PHID source, however, the starting and end times for the acquisition period do not impact the computation time as the corresponding TAC of all daughters is computed. Finally, note that the SPECT digitizer chain simulated within GATE does not utilize the particle timestamp. Instead, it groups all crystal hits into singles for each event.

## 4. Results

The results of experiment 1 are shown in figure 5 where the energy spectrum of isomeric transition of the reference simulation is compared to the one obtained with the PHID source model. The overall number of gamma ray differences was 0.3%. For example, the two main peaks at 218.10 keV and 440.40 keV show respectively -1.8% and 1.3% differences.

Similarly, the results of experiment 2 for atomic relaxation are shown in figure 6 for $^{212}$Pb. In that case, the PHID source model is composed of 25 photon lines for $^{212}$Pb (≈80% photon per decay), 49 for $^{212}$Bi (≈14%) and 25 $^{208}$Tl (≈16%). The overall difference is larger, with about 15% less photons observed with PHID compared to the reference ion simulation. This is mainly due to the low energy photons with -33% differences below 20 keV, but only 4% above. Indeed, the used database only includes photon line transitions to K and L shells and not to the others ones (below 10 keV). The two main peaks at 74.81 keV and 77.11 keV are well reproduced (6.3% and 5.2% difference). Moreover, between 86.8 and 89.8 keV, the gamma transitions are provided on the IAEA website as a range of energy, not with a list of individual energy values and we can observe slight energy differences with the photons retrieved from the reference simulation.
Similar discrepancies were observed with other radionuclides (larger differences at lower energy). Comparable gamma production cuts have been employed. These discrepancies are inconsequential for SPECT imaging as it solely influences low-energy X-rays. It is worth noting that disagreements between ENSDF/EADL and Geant4 datasets have already been reported [22].

<figure><img src="image_7.png" /><figcaption>Figure 5: Energy spectra of isomeric transition generated gamma-rays from $^{225}$Ac (including all daughters) obtained from reference and PHID source model simulations.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Energy spectra of atomic relaxation generated photons from $^{212}$Pb (including all daughters) obtained from reference and PHID source model simulations.</figcaption></figure>

The figures 7 display first the three different types of photons (Isomeric Transition, Atomic Relaxation, Bremsstrahlung) obtained from the reference Monte Carlo simulation with an $^{225}$Ac ion source, and, in the second plot, the comparison with all photons obtained from the reference and the PHID source model simulations.

The figure 8 displays comparisons between photons emitted from the Geant4 ion source vs photons produced by PHID for three other radionuclides with various start and end acquisition times: $^{212}$Pb, $^{223}$Ra and $^{149}$Tb. In those examples again, the ion source requires emission and tracking of all secondary particles followed by a filter on the timing range, while PHID only generates the photons with the right timestamps. Discrepancies occur in the low energy regions, below 20 keV. Overall number of photons differences above 20 keV were : 0.5%, 1.3% and 1.8% respectively for $^{212}$Pb, $^{223}$Ra and $^{149}$Tb.
<figure><img src="image_9.png" /><figcaption>Figure 7: Top: the three types of photons (Isomeric Transition, Atomic Relaxation, and Bremsstrahlung) obtained from a reference $^{225}$Ac ion source simulation. Bottom: comparison of photon spectra obtained from the reference ion source and the proposed PHID source model.</figcaption></figure>

Figure 9 shows projections obtained from simulating the IEC 6 spheres phantom with the reference ion source, the simplified source model (two emission lines at 218 and 440 keV), and the proposed PHID source model. The images show the 218 keV energy window only (the 440 keV energy window looks similar). The bottom plots show integrated profiles along the horizontal and vertical directions respectively, with the associated $2\sigma$ uncertainty around the reference curve. This uncertainty was computed with the batch method, running 10 times the reference simulation and considering the standard deviation.

Comparing the computation time between the ion source and the model source is not a straightforward task. Using reference ion source involved the tracking of a lot of different particles and required to start from $t=0$, and filter the desired photons. If we consider the simulation of $1.2 \times 10^6$ $^{225}$Ac ions source, about $32 \times 10^6$ particles will be tracked for a total of $1.1 \times 10^5$ photons emitted during the whole lifetime (1.3 minutes of computation time). With the PHID source, the same number of simulated photons lasts only 10 seconds. Indicative computation speeds for the SPECT experiments were reported in table 1. The speed per tracked particle is not exactly equivalent for the two-photon sources because the PHID source tracks also low-energy photons that may require fewer steps and so slightly fewer computation times. Almost 2 times more photons were simulated with the PHID model compared to the two-peaks model, requiring
<figure><img src="image_10.png" /><figcaption>Figure 8: Photons emitted from the reference ion sources and from the PHID sources, for $^{212}$Pb, $^{223}$Ra and $^{149}$Tb.</figcaption></figure>

366 correspondingly more computation time. About 60% more counts were detected in the 367 two photopeaks energy windows images and about three times more counts in the scatter 368 windows.

<table><caption>Table 1: Indicative computation speeds for each source (Apple M1 pro 2021, one thread) and associated number of events (ion or photons). PPS means "Particles Per Seconds".</caption><thead><tr><th>Source types</th><th>PPS</th><th>Events</th><th>Total time</th></tr></thead><tbody><tr><td>Ion source</td><td>10 000</td><td>10<sup>9</sup> ions</td><td>28 hours</td></tr><tr><td>Two peaks source</td><td>65 000</td><td>3.8 × 10<sup>8</sup> photons</td><td>1.6 hours</td></tr><tr><td>PHID source</td><td>75 000</td><td>7.6 × 10<sup>8</sup> photons</td><td>3.7 hours</td></tr></tbody></table>
<figure><img src="image_11.png" /><figcaption>Figure 9: Top: $^{225}$Ac SPECT projections of the IEC phantom within the 218 keV energy window, with the reference ion source (left), the simplified two-peaks model (middle), and the PHID (right). Bottom: integrated horizontal and vertical profiles for both models, with uncertainty ($2\sigma$) for the reference curve.</figcaption></figure>

## 5. Discussion and conclusion

We present a comprehensive Monte Carlo source model tailored for the decay of alpha emitters, focused on the precise generation of gamma rays within the desired time frame. This approach offers advantages in terms of computational efficiency when compared to conventional ion sources, which necessitate the generation and tracking of all decay products. For instance, consider the case of 1 MBq of $^{225}$Ac over a 1-second interval (in water). The conventional ion source computation takes approximately 1 minute (Apple M1 Pro, one single core), involving the generation and tracking of 25.8 million particles. In contrast, the PHID model accomplishes the same task in just 2 seconds, only generating about 408,000 photons. Also, the PHID source considers 2 times more photons than a simple two photo-peaks source, enhancing its precision significantly as illustrated in figure 9.

In this study, we considered Geant4 physics as the ground truth. The accuracy of Geant4 with respect to experimental data is beyond the scope of this study and interested readers can find useful information for example in [23] and the Geant4 Validation Portal https://geant-val.cern.ch.

Discrepancies were observed at low energy photons (below 20 keV) and the PHID model must not be used if interested in photons at those energies as it does not reproduce well Geant4 data. Moreover, differences around 80-90 keV photons have also been observed
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because IAEA data provide energy ranges rather than individual lines, but with very
low energy difference (2 keV). For studying SPECT imaging, such discrepancies are
negligible.

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The PHID source is not a variance reduction technique and is not intended to be used to compute absorbed dose as only the photons are considered while the deposited energy of alpha emitters comes first from alpha and electrons. It is an optimization that allows only generating photons of interest for some particular types of applications (SPECT imaging) providing some assumptions (the user is not interested in low-energy photons, below 20 keV). The PHID source applies to a wide range of alpha emitters, as long as they are available in Geant4. PHID is open-source and has been seamlessly integrated into the open-source simulation software GATE [24], as part of the version 10.beta. This version is a major evolution of the GATE software (with Python interface, multithreading) [25]. It is already openly available but still in beta, with the first non-beta release planned in 2024. We also include additional tools in GATE to gather information about alpha emitters. The tools started with "phid": phid_info <ion> prints basic information about the ion (decay characteristics, half-life, daughters products), phid_gammas <ion> plots all emitted photons with their corresponding yields for the given ion. Examples of obtained plots are shown in figure 10.

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The proposed PHID source model could be useful for several applications, particularly in the exploration of the imaging capabilities of alpha emitters through the simulation of diverse imaging systems like current SPECT systems and Compton cameras. It serves as a valuable tool for optimizing acquisition protocols, fine-tuning scatter correction parameters, and investigating advanced reconstruction methods.

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Acknowledgments

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This work was performed within the framework of the MOCAMED project (ANR-20-CE45-0025), the SIRIC LYriCAN+ Grant INCa-DGOS-INSERM-ITMO cancer18003. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research. This work was granted access to the HPC resources of IDRIS under the allocation 2019-101203 made by GENCI (Jean Zay computing center).

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