Towards large nuclear imaging system optical simulations with optiGAN, a generative adversarial network

Abstract

Optical Monte Carlo (MC) simulations are essential for modeling light transport in radiation detectors used in nuclear imaging and high-energy physics. However, full-system simulations remain computationally prohibitive due to the need to track optical photons across large detector arrays. To address this challenge, optiGAN, a conditional Wasserstein generative adversarial network (GAN) was developed to accelerate detailed optical simulations while maintaining high fidelity. Our approach trains optiGAN on high-dimensional optical photon distributions generated using GATE 10, the new Python-based version of the well-established MC simulation toolkit. Two datasets were constructed from 511 keV interactions in bismuth germanate crystals: one included multidimensional features (spatial coordinates, kinetic energy, and time), and another focused solely on time distributions. OptiGAN employs a combination of conditional GAN and Wasserstein GAN with gradient penalty (WGAN-GP) to enhance training stability and accuracy. Model performance was evaluated using the Jensen–Shannon distance, achieving similarity scores exceeding 90% for most photon properties, with further improvements when focusing exclusively on timing distributions. To validate optiGAN ability to reproduce system-level detector performance, its output was used to generate silicon photomultiplier signals using a validated SiPM simulation toolkit. The resulting energy and timing resolutions closely matched those obtained from full MC simulations, demonstrating that optiGAN preserves key detector characteristics while improving computational efficiency by up to two orders of magnitude. These findings establish optiGAN as a promising tool for large-scale detector simulations, enabling rapid evaluation of new detector technologies, also because it has been integrated into the new version of GATE. Future work will focus on further optimizing model performance and extending its applicability to system-level nuclear imaging simulations.

1. Introduction

Monte Carlo (MC) simulations for optical systems are widely used to analyze, enhance, and develop crystal-based gamma-ray detectors, which are crucial in many physics applications like nuclear medicine and high-energy physics (Staelens et al 2003, Roncali et al 2017, Ghabrial et al 2018). For example, the goal of engineering gamma detectors with 10 ps time resolution (Lecoq 2017) is driving the development of detector instrumentation for time-of-flight positron emission tomography (TOF-PET), and MC simulation is an indispensable tool to access information not captured experimentally. Ultra-fast detectors necessitate precise modeling of light propagation within the crystal and collection by photodetectors for a more detailed understanding of the light transport and to improve their spatial, timing, and energy performance (Bauer et al 2009, Ariño-Estrada et al 2020, He et al 2022). However, performing optical simulations is time-intensive because each individual optical photon generated after gamma-ray interaction in the scintillator must be tracked from emission to detection (Geant4 Collaboration 2021) to simulate realistic waveforms.

Single detector simulations remain feasible and useful for testing photon nature, exploring different crystal shape and properties, evaluating various photodetectors, and testing emerging technologies in optical photon transport. However, large system simulations are needed to account for advances in radiation detector technology and assess their impact on system performance (e.g. energy, spatial and timing resolution, and ultimately image contrast and signal-to-noise ratio). MC methods are already frequently used to model full clinical and preclinical imaging systems (Ricci et al 2019, Ahmed et al 2020, Sarrut et al 2021, Sanaat et al 2022, Zaidi and Andreo 2022), but analyses are typically restricted to the first gamma-ray interaction point to evaluate system performance due to computational limitations. In addition, the photodetector response is never included (Gu et al 2015, Abbaszadeh et al 2018, Borghi et al 2018, Surti and Karp 2018). Indeed, simulating the optical photons travelling inside hundreds or thousands of individual scintillator elements and hundreds of photodetectors is computationally prohibitive. As a result, current system simulations lack precision and do not include the possibility of using optical photon behavior for detector optimization (e.g. prompt photons to be used for timing performances) or for testing new detector technology. Furthermore, these simulations often generate massive output datasets, reaching several gigabytes per detector.

Recently, multiple research groups have explored the use of machine learning techniques to accelerate MC simulations and reduce their computational burden (de Oliveira et al 2017, Paganini et al 2018, Erdmann et al 2019, Sarrut et al 2019, Fanelli and Pomponi 2020, Sarrut et al 2021b, Hashemi et al 2023). These efforts mainly utilize generative adversarial networks (GANs), a type of generative model introduced by Goodfellow et al (2014). GANs consist of two neural networks capable of producing multi-dimensional data distributions that replicate the complex, high-dimensional patterns found in the training datasets.

In this context, a conditional GAN model named optiGAN was developed (Trigila et al 2023) (Srikanth et al 2024). This model is designed to generate high-fidelity optical photon distributions at the scintillator-photodetector interface, conditioned on their emission position (gamma-ray interaction point within the crystal), with the goal of skipping the optical photon tracking process (figure 1, top row) and was trained using a large multidimensional dataset generated through accurate optical simulation with the toolkit GATE (Sarrut et al 2021) inside a single crystal. The comparison of the simulation and GAN-generated distributions showed excellent agreement, with similarity values higher than 93.5% for all distributions (energy, photon position, photon momentum). Moreover, the GAN accelerated the optical photon distribution generation by up to two orders of magnitude. These single-crystal results demonstrated the feasibility of using GANs to speed up the MC simulation, which has the potential to drastically change the use of nuclear imaging system optical simulations by enabling high-fidelity system-level simulations in reasonable computation times.

Figure 1.

Schematic of the methodology presented to estimate the capability of optiGAN to generate real system features. Detailed optical MC simulations were used to build the high-fidelity training dataset. Once trained, the model was used to generate distributions. Simulated and generated distributions were then compared to assess the fidelity of the training. These distributions were then used to generate SiPM signals, which in turn served to estimate both simulated and generated detector performance.

Here, the capability of using optiGAN-generated distributions to extract the performance of the detector in terms of timing and energy resolution, including the response of the photodetector, which was not included in our previous studies, is presented. Timing distributions are complex due to the combined effect of the light emission, transport in the crystal with multiple reflections, and detection and are thus challenging to reproduce accurately with the GAN. Bismuth germanate (BGO) was selected as the crystal, as it is a well-established scintillator for PET applications with high density, excellent stopping power, high photo-fraction, appropriate refractive index, and low production costs (Brunner and Schaart 2017, Cates and Levin 2019, Roncali et al 2019). It also emits Cherenkov photons, which are of great interest for TOF-PET application (Kwon et al 2016, Gundacker et al 2020, Kratochwil et al 2020).

Using both simulated and GAN-generated optical photon distributions, realistic silicon photomultipliers (SiPM) signals were generated using a well-validated SiPM code (SiPMSiM) (Mehadji et al 2024). They were used to extract the detector energy and timing resolution (figure 1, bottom row). This is a crucial step in evaluating the capability and the limitation of the GAN to generate accurate energy and timing data before scaling up to system simulations.

2. Materials and methods

2.1. High-fidelity optical Monte Carlo simulations to build the training dataset

All training datasets were prepared using optical simulations using the open-source software GATE, a specialized toolkit for numerical simulations in medical imaging and radiotherapy built upon the Geant4 platform (Geant4 v. 11.02). In particular, the new version of GATE was used, GATE v10 (Krah et al 2024), which is entirely based on Python. Simulations were conducted in a 3 × 3 × 10 mm³ in BGO crystal (light yield of ~8500 ph MeV⁻¹). The simulation integrated the bulk and optical properties of BGO, including density, composition, emission spectra, decay time, absorption length, and refractive index. To control the source emission position, electrons were used as the source. The tortuous path of the electrons was simulated with a realistic inelastic mean free path on the scale of fractions of a micron (Trigila et al 2022). More details can be found in our previous work (Trigila et al 2023).

The optical photons generated through scintillation or Cherenkov emission were modelled and tracked using the Livermore Model in GATE. Optical photon transport was implemented with the LUT Davis model at the crystal edges. This model relies on a crystal reflectance look-up table derived from three-dimensional crystal surface measurements (Roncali and Cherry 2013, Roncali et al 2017b) to determine photon behavior at each crystal surface interaction.

Surfaces were modelled as polished and wrapped in Teflon using customized LUT (Trigila et al 2021), except for the side coupled with the photodetector, which was simulated with optical grease (index of refraction 1.56). The photodetector was considered ideal with a photodetection efficiency of 1, which considers both geometrical and quantum efficiencies.

To create the training dataset, simulations were conducted at 30 distinct emission positions within the crystal, each defined by specific X-Y-Z coordinates (figure 2). At each position, electrons were emitted, generating at least 30k optical photons that reached the photodetector interface. The characteristics of these photons, measured at the coupling-photodetector boundary, were saved in a phase-space GATE root file. The data included the X and Y coordinates at the interface (with the Z coordinate fixed to match the photodetector location), the kinetic energy (EKine), and the time elapsed since the event beginning (Time). The Time distribution was highly skewed, rapidly decreasing with a sharp peak at zero, making it challenging for the GAN to learn effectively (not shown). To address this, a logarithmic transformation was applied to reshape the distribution, mitigating the sharp peak at zero and facilitating the learning process.

Figure 2.

Schematic of the training dataset. Accurate optical simulations were carried out at various emission positions to generate the dataset for training the conditional generative adversarial network optiGAN. (left) A total of 30 emission points located within 1/8 of the crystal volume were used to build the training dataset. Images are not to scale. (b) Optical photon distributions (position X and Y, energy EKine, and Time) were recorded for all training points (30 k entries per point) in a multidimensional matrix that included the emission positions as a condition (Dataset1, shown in green). A second dataset was also generated (Dataset2), which included only the Time distribution and the 3D class label.

These distributions were stored as tabular data, with the 3D emission source position used as the class label to tag the distributions. The tabular data for the 30 training points was concatenated to create a single training matrix (figure 2).

This formed a first multidimensional dataset (Dataset1) used to train our conditional GAN, comprising a total of four distributions (X, Y, Ekine, log(Time)), and a 3D class label defined by the optical photon emission position. To evaluate the GAN’s ability to accurately reproduce the detector’s timing characteristics, an additional training dataset containing only the logarithm of the Time distribution and the 3D class label (Dataset2) was created. Both datasets are available online (Trigila and Roncali 2025). Since this dataset included a single distribution, it was anticipated that the GAN would achieve better performance on this specific distribution than when trained on all four distributions.

These two training datasets were used to train optiGAN.

2.2. GAN architecture, parameters, and training

OptiGAN belongs to the class of generative networks. GANs (Goodfellow et al 2014) are a class of deep learning architectures designed for generating high-quality synthetic data that closely resemble real samples from the training dataset. They consist of two neural networks, a generator G and a discriminator D, which compete in an adversarial training process. The generator maps a noise vector z ~ p_z(z) from a prior distribution (e.g. Gaussian or uniform) to a synthetic data sample G(z), while the discriminator attempts to distinguish real samples × ~ p_data (x) and generated samples G(z).

For optiGAN (Trigila et al 2023), a conditional GAN (Mirza 2014) and a Wasserstein GAN with Gradient Penalty (WGAN-GP) (Gulrajani et al 2017) were combined to enhance the stability and accuracy of photon transport simulation. WGAN-GP improves adversarial training by using the Wasserstein-1 distance, which provides meaningful gradients even when real and generated distributions are disjoint, reducing mode collapse. By incorporating the conditional GAN framework, optiGAN can generate photon properties conditioned on specific characteristics. In our case, the model is conditioned on the point of emission of optical photons: it takes as input a noise vector z and a class label representing the 3D source position within the crystal. It then outputs a sampled multidimensional distribution targeting a real distribution in the training dataset, enabling the generation of physically meaningful photon transport distributions.

The architecture of the latest optiGAN version was designed to efficiently process multidimensional matrix inputs. Both the generator G and discriminator D were implemented as fully connected neural networks with four hidden layers. The generator follows a layer configuration of H, 2H, 2H, 4H, while the discriminator follows a structure of 4H, 2H, 2H, H. Here, H was set to 128, chosen empirically to balance model complexity and computational efficiency. ReLU activation was applied to all hidden layers, except for the final layer of the generator, where a linear activation was used, following recommendations from (Arjovsky et al 2017). This architecture builds upon our previous work (Trigila et al 2023), with the key modification of an additional hidden layer in both G and D. Our experiments showed that this adjustment improves training accuracy (not shown here).

The model was implemented and trained using PyTorch and optimized on GPU (Srikanth et al 2024), following an update ratio of 3 and 1 for the discriminator and generator, respectively. G was responsible for generating a multidimensional distribution indistinguishable from the real one in the dataset. G took as input z, sampled from a simple multidimensional normal distribution, and a class label L. It output a multidimensional distribution targeting a real data distribution. D took as input a multidimensional distribution with class labels and was responsible for classifying it as either real (drawn from the dataset) or fake (generated). Both the generator and the discriminator were optimized using the Adam optimizer with its hyperparameter β₁ and β₂ equal to 0.5 and 0.9, and a learning rate of 5 × 10⁻⁵.

Training was conducted for 30 k and 20 k epochs for Dataset1 and Dataset2, respectively, with each epoch consisting of 10 iterations. At each iteration, the model processed a stochastic batch of 90k samples. The dataset was normalized before training to facilitate convergence, and the mean and standard deviation of each feature were stored for denormalizing optiGAN’s generated outputs. The training was performed on an Apple M1 Max GPU (5 s/epoch), and generator checkpoints were saved every 5 epochs for evaluation.

2.3. Generation of synthetic distributions

By saving the generator model, we enable the generation of optical photon distributions for a specific interaction point at a given training epoch. Indeed, the generator takes as input: (i) a random latent vector z, whose dimensionality defines the size of the output distribution, and (ii) a class label corresponding to a specific 3D emission point within the crystal. It outputs a synthetic sample—(X, Y, EKine, log(Time)) or only (log(Time))—drawn from the learned multidimensional distribution associated with that position. An exponential was applied to the log(Time) to obtain the Time distribution. Both simulated and generated distributions were histogrammed with 100 bins.

The simulated and generated distributions were compared using the Jensen–Shannon distance (JSD), a similarity measure of probability distributions (Menéndez et al 1997), which computes the statistical distance between two distributions. The JSD outcome is confined to the range [0,1], where 0 indicates that the two distributions are identical and 1 represents complete dissimilarity. The similarity S was computed as a percentage from (1–JSD). It was estimated for all emission points, distributions, and training iterations for the two trainings (Dataset1 and Dataset2).

The similarity results were analyzed to determine the optimal training epoch to be used for the generation of SiPM signals. This selection was made by determining the best compromise between the mean and the standard deviation of the S values estimated from all points for each distribution across all epochs. The mean S across all points was used to evaluate the overall similarity between training and synthetic distributions. The S standard deviation provided a measure of consistency in the synthetic data quality across points. The scoring metric used to select the best epoch applied an equal weight (0.5) for both the mean and the standard deviation. The best epoch was defined as one that achieved a high mean S (indicating high similarity) and a small standard deviation (indicating consistency) for a given distribution across all labels.

2.4. SiPMs waveform generation

To simulate the behavior of SiPMs, the SiPMSiM toolkit, an advanced MC simulation framework (Mehadji et al 2024), was employed. This standalone tool operates at the micro-cell level, enabling the generation of realistic SiPM signals including noise by simulating individual photon interactions and the SiPM optical and electronic response. The toolkit has been validated experimentally and is openly accessible for research and development purposes (Mehadji et al 2023).

The SiPMSiM toolkit requires two inputs: a configuration file (in XML format) that stores the specifications and parameters necessary for simulating the behavior of specific silicon photomultipliers, and a npy (binary NumPy) file containing the 3D spatial coordinates, time, and, optionally, the energy of optical photons striking the SiPM surface. Using these inputs, it generates a signal for each photon, considering parameters such as trigger probability and recovery time. It also incorporates realistic noise sources such as crosstalk, afterpulse, dark count rate, delayed crosstalk, and electronic noise (Piemonte and Gola 2019). Output signals are generated in a npy format and are suitable for energy and timing resolution characterization. The toolkit is implemented in C++ following Geant4 standards but does not necessitate a Geant4 installation. The XML-formatted input files and NumPy binary compatibility facilitate integration with existing workflows. This implementation makes SiPMSiM versatile for a wide range of applications, including detector optimization and signal analysis. For further details on the methodology and parameterization, readers are directed to Mehadji et al (2024) and the SiPMSiM GitHub repository (Mehadji 2024).

To test the full dataset training accuracy (Dataset1), for each emission point, the simulated and generated multidimensional distributions of the optical photons at the photodetector face (X, Y, Ekine, Time) coming from each electron were saved as npy and used as input to the SiPMSiM standalone. The generated signals were used to calculate the MC-simulated and optiGAN-generated energy and timing resolution (figure 1).

To test optiGAN ability to reproduce the Time distribution (Dataset2) and how it affects the SiPM signals, for each emission point, the GAN-generated Time distribution was concatenated with the simulated (X, Y, Ekine) and saved as npy, which was then given as input to SiPMSiM. These signals were then used to estimate only the timing resolution.

2.5. Energy resolution

The energy resolution was estimated using signals generated by simulating the response of a Hamamatsu S13360-3050 MPPC, optimized for energy measurement (Mehadji et al 2023). The specific SiPM.xml input file used is available on GitHub (Mehadji 2024). For each gamma event, the signals produced by individual electrons were integrated to determine the total deposited energy of the event. The histogram of the energy from all the events emitted at the point was used to estimate the energy resolution, assuming ideal energy linearity.

2.6. Timing resolution

Timing signals were obtained simulating S13360-3050 MPPC, using SiPMSiM optimized for time measurements (Mehadji et al 2023). The specific SiPM.xml input file used is available on GitHub (Mehadji 2024). For both the simulated and GAN-generated signals, the time resolution was determined by calculating the standard deviation of the histogram of time differences between event time stamps. Each time stamp was obtained from the leading-edge time pick-off of its respective time signal, similar to traditional experimental detector timing measurements.

3. Results

3.1. GAN synthetic distribution generation capability

After 2000 training epochs, preliminary similarity values greater than 90% were obtained between the optiGAN-generated and training dataset distributions for all distributions (not shown) and both training datasets.

When using the full training dataset (Dataset1), the selected epoch for future analysis was identified from the Time distribution, where the best compromise between mean S and its standard deviation was obtained at 27505 epochs, with a mean S of (92.8±0.3)% among all points. It matched the best epoch obtained with the kinetic energy, with a mean S of (92.7±0.2)%. Although the best S for X and Y occurred at a different epoch, the difference in S from the value at the best epoch was less than 2%.

When only training using the Time distribution, the best epoch (7600 epochs) yielded a mean S among all points of (93.2±0.2)%.

The GAN successfully learned the distributions corresponding to various emission point positions in the crystal, capturing differences in shape and levels of complexity (figure 3). For example, the distributions of optical photons emitted at the center of the crystal near the photodetector and those of photons emitted near the crystal edges away from the photodetector were generated with a mean similarity S higher than 90% across all distributions. However, the GAN achieved better performance (S higher than 94%) when generating the Ekine and Time distributions compared to the X and Y distributions, where S dropped as low as 88% (figure 3). This outcome aligns with the selection of the best epoch from the Time distribution and is particularly evident for highly asymmetric X and Y distributions, such as those corresponding to a point near the edge of the crystal and the photodetector (figure 3).

Figure 3.

Example of simulated and generated distribution for two points in the training dataset, one at the center far from the photodetector and one at a crystal edge close to the photodetector.

Moreover, when trained on the complete Dataset1, the GAN achieved high overall S values but failed to capture low-level features of the Time distribution, as shown for both representative points in the inset of figure 4(a). Accurately reproducing these features is crucial, as they correspond to the time contribution from fast optical photons (e.g. Cherenkov photons), which significantly impact the signal rise time used for time pick-off and timing resolution estimation.

Figure 4.

Log(Time) distribution for two points: (a) top row: comparison of simulated (red) and GAN-generated (black) distributions using the complete training dataset. These distributions match those in figure 3 but are shown again for clarity. OptiGAN successfully generates distributions that closely fit the simulated ones at high S values but does not fully capture fine details at small Time values, as highlighted in the inset. (b) Bottom row: identical simulated distributions, but with GAN-generated results obtained from training on the Dataset2, containing only the Time distribution.

When using the generated distribution from the model trained exclusively on Time, the GAN’s ability to reproduce fine-scale details significantly improved for both representative points, as qualitatively observed in figure 4(b) insets.

However, the overall S values remain nearly unchanged despite the improved reproduction of fine details (e.g. 95.39% vs 95.92%, figure 4 first column), highlighting the need for further refinement of the metric used to evaluate our distributions similarity.

The distributions served as input to SiPMSiM to generate signals useful for energy and time estimation.

3.2. Energy resolution

Figure 5(a) shows a representative example of SiPM signals used for energy resolution estimation, obtained using either the simulated or generated matrix from the complete training as input. The high fidelity of the generated signals is evident when comparing signals from a single event at two different positions within the crystal.

Figure 5.

Example of SiPM signals generated for energy resolution estimation. (Top) A single signal coming from one single event in two different point positions ([X, Y, Z] in mm) inside the crystal, using the simulated or the generated matrix as input. (Bottom) Mean signal among all events in the same position, together with a dispersion halo defined by the mean standard deviation.

The plots in figure 5(b) present the mean signal across all events emitted at the same position, together with a dispersion halo defined by the mean standard deviation. Qualitatively, the signals exhibit strong agreement, and similarity values exceeding 95% were attained. This demonstrates the model’s ability to robustly reproduce signals from events involving a large range of energy depositions.

This can also be seen from the histogram of the integral of each signal, representative of the energy, which yielded the spectra shown in figure 6. The presence of the shoulder in the GAN-generated signals histogram indicates that all types of events are accurately reproduced and represented (figure 6(b)). Furthermore, the energy resolution of the simulated and generated signals exhibits strong agreement within one standard deviation and aligns well with values reported in the literature (Brunner and Schaart 2017).

Figure 6.

Energy spectra obtained as the histogram of the simulated and generated signals integral for two emission points inside the crystal ([X, Y, Z] in mm). Good agreement was found between the simulated and GAN-generated energy resolution.

3.3. Time resolution

Figure 7(a) presents examples of single event signals generated for time estimation when using the complete training dataset. The left plot of figure 7(a) shows a case where the GAN-generated signals deviate from the simulated ones, as evidenced by a shifted rising edge, leading to a different timestamp at pick-off. This discrepancy is due to the GAN’s suboptimal ability to reproduce the Time distribution, especially at very low time values, as previously discussed and shown in figure 4. In contrast, the right plot of figure 7(a) shows a case where the generated signal closely matches the simulated one, with almost identical timestamps. These are representative examples; across all points and events, some signals were well reproduced, while others deviated, as reflected in the mean signal and the dispersion halo given by the standard deviation. As a consequence, a discrepancy is observed between the mean generated and simulated signals at low time values (as highlighted by the blue circle in the left plot of figure 7(b); shown only for one representative point), while similar signals are visible at higher time values (more than 20 ns). This directly affects the time delay distribution, where a significant difference in the distribution’s shape can be observed, for instance, the generated time delay curve exhibits extended tails that are absent in the simulated one. Moreover, differences exceeding 40% in the standard deviation of the distributions after Gaussian fitting can be seen at certain points (right plot of figure 7(b)).

Figure 7.

(a) Representative examples of single-event signals generated for time estimation using the complete training dataset using [1.45, 1.45, 9] mm as a representative emission point. The left plot shows a case where the GAN-generated signal deviates from the simulated one, with a shifted rising edge leading to a different timestamp. The right plot shows a case where the generated signal closely matches the simulated signal with nearly identical timestamps. (b) On the left, comparison of the correspondent mean generated and simulated signals, highlighting discrepancies at low time values (blue circle) and better agreement at higher time values (greater than 20 ns), which directly affects the time delay distributions shapes and their standard deviations (on the right), with differences of up to 40%.

However, when analyzing the signals from the generated data trained solely on the Time distribution (Dataset2), the simulated signals are more accurately reproduced. This is evident from the improved overlap of the mean signals and their standard deviation halo (left plot of figure 8). This improvement is reflected in the time resolution (right plot of figure 8), where the sigma values of the time difference distribution are now in closer agreement, with significantly reduced sigma differences (less than 10% difference) than the one shown before (up to 40%, figure 7(b)).

Figure 8.

On the left, comparison of the mean generated and simulated signals generated for time estimation at the emission point [1.45, 1.45, 9] mm using the training dataset with only the Time distribution. Very good agreement can be found even at small time values (blue circle). On the right, time delay distributions and their standard deviations show differences of less than 10%.

Similar results are observed across all points in the training dataset (figure 9). When considering the time resolution obtained from signals generated by the GAN model trained on the complete training dataset (figure 8(a)), some points show low agreement with the simulated results (e.g. point [0,0,1] mm), while others exhibit full agreement (e.g. point [1,1,1] mm, figure 8(a)).

Figure 9.

Timing resolution obtained from the simulated or generated signals, for all points in the training dataset. (a) Results were obtained using the model trained across four distributions (X, Y, Ekine, Time). (b) Results obtained when training the GAN only with the Time distributions.

In GANs models, these results could be due to several reasons.

First, if some labels (conditions) have more training samples than others, a GAN might prioritize learning those distributions. Indeed, underrepresented labels may not be well-learned because the discriminator sees them less frequently. In optiGAN, the training dataset contained a homogeneous number of entries (30 k) per point. However, if specific labels were overrepresented in batches during the training, the generator may have disproportionately focused on those.

Then, in GANs the generator might focus on generating a subset of the distributions that are easier to learn, neglecting others, or some classes might be simpler for the generator to approximate, leading to faster learning. Some distributions might have sharper or more complex structures, making them harder to learn. Consequently, a GAN might first learn easier distributions before refining more complex ones, resulting in uneven learning across labels. However, this does not seem to be the case with optiGAN. Indeed, for example, the distributions at point [1,1,1] mm are qualitatively very similar to the one at point [0,0,1] mm (not shown), yet they produced significantly better results (figure 9).

At the same time, optiGAN model trained exclusively on the Time distribution successfully learned these distributions, as evidenced by the results in figure 9(b), where the generated time resolution closely matches the simulated one across all points.

This suggests that optiGAN is capable of learning the distributions; however, when handling a large number of distributions and labels, the training process may slow down or require further refinement and optimization to ensure stable learning, especially when distribution with low-level features that have to be captured are included in the training dataset. When having a large and complex training dataset, optimizing and extending the training duration (more epochs) could, therefore, be a potential solution to improve performance.

4. Discussion

In this project, a conditional Wasserstein Generative Adversarial Network capable of generating multidimensional optical photon distributions on the photodetector of a radiation detector was developed. This approach not only demonstrates the feasibility of using generative models for simulating complex photon distributions but also lays the foundations for accurate simulations of large systems. We demonstrated that the GAN can accurately reproduce optical photon distributions and thus can enable high-fidelity extraction of detector properties, such as energy and timing, directly from the generated data. This indicates that detector or system properties can be reliably inferred using the GAN-generated distributions.

Previously, the high performance of optiGAN model was shown by generating multidimensional distributions in a 3 × 3 × 3 mm³ BGO crystal, achieving similarity values above 94%, even when testing it on emission points that were not part of the training set (Trigila et al 2023). These results highlighted not only the excellent distribution generation abilities of the trained optiGAN but also its potential to replace detailed MC optical simulations of interaction within the entire crystal volume without the need to perform their detailed and accurate MC optical simulations. Moreover, our approach offers computation time reductions of up to two orders of magnitude, underscoring its promise for efficient and scalable radiation detection simulations.

In this study, optiGAN’s capability to accurately reproduce both the energy and time resolution of a 3 × 3 × 10 mm³ BGO detector was demonstrated, achieving results that closely align with simulated performances for a realistic scintillator size, which are, in turn, consistent with values reported in the literature.

Although high mean similarity values are obtained (higher than 90% among all points and distributions), they remain lower than the performance previously reported for a 3 × 3 × 3 mm³ crystal (higher than 94%) (Trigila et al 2023). This reduction in S can be explained by the greater complexity and variability of the optical photon distribution in a thicker material, in contrast to the more symmetric and smaller crystal examined in the previous study. The complexity of distributions such as (X, Y, EKine) is rooted in the variable photon transport paths within the scintillation crystal, which differ depending on the emission point’s proximity to the photodetector or the crystal’s edges. Each of these emission points, described by multiple distinct distributions, results in a highly diverse and challenging dataset, which makes the training more complex and requires more iterations to converge. Nevertheless, with a carefully chosen set of hyperparameter and sufficient training time, the GAN has consistently demonstrated strong generative performance. While the optimal hyperparameters are not known a priori, the configuration adopted in this study, specifically, the number of layers, the dimensionality of the latent space, and the learning rate, proved effective across different scenarios. These parameters can be adapted based on dataset size and typically yield reliable results even without exhaustive optimization.

In addition, the results highlighted the necessity of capturing fine details in the distributions to enable reliable performance estimations, as illustrated in the case of the Time distribution. When considering a crystal like BGO, which produces both scintillation and Cherenkov photons, the time distribution is influenced by contributions from both types of light, making it more challenging for the GAN to learn. Indeed, excellent results were obtained when training the GAN with a simplified dataset consisting only of the Time distribution and 3D labels, where the GAN was able to capture low-level details in the distributions. This demonstrates the model’s ability to lead to accurate performance estimations when generating multimodal distributions with high fidelity.

However, when training with a more complex dataset, the GAN exhibited lower capacity in learning fine-scale features in the training dataset distributions. These observations suggest that training on individual distribution types might improve performance further. Thus, the main challenge lies not only in the shape complexity of individual distributions but also in the number of distributions the model must learn simultaneously.

This opens the question of the most effective training strategy: whether to train a single model on a large, diverse set of distributions or to perform multiple, independent training runs on smaller, more focused datasets. Investigations are underway to understand the effect on the simulation time when using a GAN-integrated approach when having one model (e.g. capable of generating lots of distributions) or more models (e.g. two models, one trained with X, Y and one on Time). The latter makes the presented results even more promising.

However, while the results remain promising, they also show that there is the potential and the necessity of enhancing the model generation capabilities to minimize the deviation between the optiGAN-generated and the training data.

Also, when using a complex training dataset, the GANs showed uneven learning across labels, and similar distributions yielded different results. This suggests that the model can learn complex distributions but may require optimization when handling a large number of distributions and labels. Extending training duration (more epochs) could help improve performance when dealing with complex datasets containing fine-scale features.

Potential contributing factors include the model architecture (e.g. number of layers), hyperparameter selection, the choice of gradient penalty function, and the overall dimension of the training dataset.

Further investigations concerning hyperparameter optimization, using Bayesian optimization with Radial Basis Surrogate functions (Ilievski et al 2017) (Dumont et al 2022), and gradient-based dataset reduction (Killamsetty et al 2021) are currently underway to address these limitations.

Relying solely on the JSD similarity metric may be insufficient for evaluating training quality. This is evident from the fact that similar S values were observed whether the optiGAN-generated distributions successfully captured (figure 4(b)) or failed to capture (figure 4(a)) essential distribution features. Consequently, JSD seems not to adequately assess the model generation capacity. Indeed, while S values remained consistently high, they were not precise enough to accurately reflect detector performances. Future works include investigating other metrics beyond JSD, such as Hellinger Distance, Wasserstein Critic, Kolmogorov-Smirnov, and the Intrinsic Multiscale distance, which are known alternative metrics for quantifying differences in probability data distributions (Panaretos and Zemel 2019, Massey Jr 1951, Lee 2006).

Several tests also showed that the binning of the distributions impacted the final similarity values, causing up to 2% differences (not shown). Experiments are being conducted to improve histograms using alternative static binning methods such as Knuth (2019) and Scott (1979), as well as an adaptive method, the Bayesian Blocks (Pollack et al 2019).

Due to the continuous nature of GANs output, future work also includes testing various probability density function estimation methods. This includes the use of kernel density estimators with a variety of kernels such as Gaussian and Epanechnikov (Silverman 1986), but also static and variable width kernels. Additionally, Gaussian Mixture Models estimation optimized with Bayes Information Criterion (Fraley and Raftery 2002), Bayesian Gaussian Mixture Models (Attias 1999), Manifold Parzen Windows (Bengio 2022), Orthogonal Series Estimators (Efromovich 2010), and Wavelet Estimation (Donoho et al 1996) will be tested.

One limitation of optiGAN is its restricted generalizability. At present, each distinct simulation setup, defined by factors such as crystal geometry, material composition, and photon energy, requires a separate training process. As a result, models trained on a specific configuration (e.g. 3 × 3 × 10 mm³ BGO crystals) may not perform effectively when applied to different scenarios.

However, generating the training dataset and training the GAN model to achieve good performance would take less than two weeks at this time of development and indicates that optiGAN will achieve greater portability. This implies that within two weeks, users could develop a custom GAN capable of producing high-fidelity simulations, something that would otherwise be prohibitively expensive or practically impossible with full-scale MC approaches.

Despite recent advances in GAN architecture, generalizing the model capable of handling various crystal types and geometries remains challenging due to the high dimensionality and complexity of the associated data and models. Aside from the work presented here, we are investigating transfer learning and modular model architectures to enable the adaptation of pretrained networks to new configurations with minimal additional training. Furthermore, refining the composition of the training dataset through methods such as active sampling, data augmentation, or simulation-informed priors could help reduce the data requirements for new configurations.

5. Conclusions

In this work, the proof-of-concept of using optiGAN in estimating detector performance is presented, including generating realistic waveforms that could be used for timing pickoff for TOF PET. This approach represents a major step forward in integrating GANs into system-level simulations and provides a practical tool for physics applications that necessitate detailed modeling of optical photon behavior in large-scale systems, where conventional simulations are computationally prohibitive. The GAN-generated distributions will be used to extract system properties, which will be equivalent to those obtained from full MC simulations. We believe that this approach will be highly beneficial to the community, given the extensive ongoing efforts to develop novel detector technologies that will require evaluation through large-scale system simulations.

OptiGAN has a broad range of applications. First, once a high-fidelity training dataset comprising multidimensional distributions from accurate optical MC simulations is available, researchers can readily adopt this method. By tailoring these datasets to specific applications, crystals, geometries, surface treatments, and technologies, as well as different sources and interaction processes, the approach can be broadly extended. The next step involves demonstrating optiGAN’s ability to model a nine-crystal array, thereby accounting for the effects of Compton scattering and inter-crystal optical crosstalk. Future work will focus on enhance the adaptabilityx of configuration-specific training datasets, ensuring broader applicability across different detector setups.

Users can also access OptiGAN, which has been recently integrated into the newly developed GATE 10 and can be used to bypass low-level detail of optical photon tracking. However, its full validation within the new GATE version is currently in progress and is the focus of a separate paper.

Further developments will build on these results with the ultimate goal of demonstrating optiGAN capability for system-level simulations. All the features will be ultimately combined, providing a comprehensive AI-accelerated framework for simulations. Efforts are underway to make the training code publicly available on GitHub, enabling the community to take full advantage of this new simulation approach. The datasets used in this work are already available (Trigila and Roncali 2025).

Data availability statement

The data that support the findings of this study are openly available at the following URL/DOI: https://ieeedataport.org/documents/optigan-training-datasets.