Gamma-Ray Position-of-Interaction Estimation in a Thick Monolithic LaBr3 Detector Using Artificial Neural Networks

Monolithic gamma-ray detectors can be used in single photon emission computed tomography systems for monitoring the delivered dose during boron neutron capture therapy treatments. Gamma-ray hit localization in thick monolithic scintillator crystals is a challenging task due to internal reflections and Compton scattering. Existing methods like the center of gravity (CoG) are susceptible to reconstruction uncertainties at the crystal edges, while approaches, including nonlinear analytical and statistical models, such as the maximum-likelihood, require significant computational resources. Artificial neural networks (ANNs) offer significant improvements in terms of accuracy and computational speed. In this study, we develop a supervised ANN regression algorithm for real-time position reconstruction in a thick square lanthanum bromide crystal [LaBr $_{3}(\text {Ce}+\text {Sr})$ ] with $5\, \text {cm}\times 5\,\text {cm}\times 2\,\text {cm}$ dimensions, coupled with an $8\times 8$ matrix of silicon photomultipliers. The implemented neural network was trained and tested using calibration data acquired irradiating the crystal with a collimated 137Cs source (pencil-beam irradiation). The detector in combination with the ANN model achieves a positioning accuracy for single-gamma-ray events of approximately 2.6 mm in the central region, evaluated as the full width at half maximum (FWHM) of the prediction error distribution, slightly worsening toward the edges. The imaging capabilities of the detector in combination with a channel-edge pinhole collimator were then evaluated by acquiring images of a movable uncollimated 137Cs point source. The source was shifted in nine different positions at 3 mm distance from each other and the resolution of the system was evaluated fitting the images with a Gaussian curve. An image spatial resolution of around 8 mm FWHM was obtained, dominated as expected by the collimator geometry, with an accuracy of 0.7 mm in estimating the position of the point source.

SECTION I. Introduction

The reconstruction of the gamma-ray interaction position in thick, monolithic scintillator crystals represents a fundamental challenge in several medical imaging applications. In particular, our interest is for detection systems for dose monitoring in boron neutron capture therapy (BNCT) [1], [2]. In this technique, patients are administered specific pharmaceuticals containing ¹⁰B, a stable isotope of boron, which selectively accumulates in tumors. Patients are then irradiated with a beam of thermal neutrons, which interact with ¹⁰B through the neutron capture reaction ¹⁰B(n,$\alpha $ )⁷Li. This reaction has a large cross section and in 94% of the cases produces a ⁷Li in excited state, which decays by emitting gamma rays at 478keV that may eventually leave the patient’s body and be detected for dose monitoring and localization. To date, detection systems proposed for this dose monitoring are primarily based on a pixelated matrix of semiconductor detectors, such as cadmium zinc telluride (CZT) [3] and monolithic scintillator crystals, such as lanthanum bromide (LaBr₃) [4]. CZT detectors are renowned for their outstanding energy resolution, but they are typically limited by their maximum physical thickness, which is usually only a few millimetres, limiting the overall detection efficiency. On the other hand, scintillator-based detectors like LaBr₃, provide good energy resolution and have the advantage of being manufactured in thicker crystals, enhancing their efficiency for capturing high-energy gamma rays. High detection efficiency is a fundamental aspect of BNCT, because of the need to detect low concentrations of boron within a limited time span with sufficient statistical accuracy. For this application, a monolithic crystal has been chosen over a pixelated one. Detectors based on monolithic crystals can simultaneously achieve excellent energy resolution, good spatial resolution, and the possibility to estimate the depth of interaction (DOI), if necessary, whereas in pixelated crystals usually it is necessary to compromise between these different characteristics. Moreover, LaBr₃ is highly hygroscopic and therefore difficult to be machined, so monolithic crystal offer also reduced costs [5], [6], [7], [8], [9], [10].

Despite the favorable characteristics of monolithic detectors, the complexity of position reconstruction methods, the presence of edge effects, and the necessity for time-consuming calibration procedures represent a potential limitation to their use [10]. The reconstruction of 2-D gamma-ray position of interaction in monolithic scintillators can be obtained by estimating the centroid of the light distribution (LD) read out by all photodetector pixels [11], [12]. This is achieved by using methods, such as the center of gravity (CoG). However, a considerable proportion of the scintillation photons may be reflected or absorbed by the crystal lateral sides before detection, which results in a discrepancy between the signal pattern observed by the photodetectors and the light pattern due to direct photons [13]. In the case of thick monolithic crystals, the issue is further complicated by the fact that the width of the LD is influenced by the gamma-ray DOI in the crystal. Another limitation of monolithic crystals is the inaccurate assessment of the interaction positions near the edges [11]. As the crystal thickness increases, the positioning inaccuracy becomes more pronounced, resulting in a spatial resolution degradation closer to the margins of the crystal [14]. Some algorithms, based on nonlinear analytical models [15] and statistical models, such as the maximum-likelihood [16], [17], attempt to implement a pattern-shape analysis of the detector response for all gamma-ray interaction positions in the detector, with a high level of detail. The former require an exact reproduction of the detector’s geometry and they need significant computational time, which may be a limiting factor in real-time applications. The latter, despite their speed, require time-consuming estimations of prior lookup tables to function properly.

The use of artificial intelligence (AI) methods offers a promising solution to address the aforementioned challenges [6], [7], [8], [9]. These methods enable more accurate and faster estimation of gamma-ray interactions, taking into account the system nonidealities. Furthermore, AI techniques can adapt and improve over time by learning from new data. In recent years, numerous studies have been conducted on the application of machine learning (ML) and deep learning (DL) techniques for gamma-ray localization in monolithic scintillators, working as a reference for this work. One of the most frequently employed ML techniques is the k-nearest neighbors (kNNs) algorithm [18], [19], [20]. The kNN model, implemented by Seifert et al. [20] using a 10-mm thick LaBr$_{3}(\text {Ce}+\text {Sr})$ monolithic crystal coupled to a $4\times 4$ silicon photomultiplier (SiPM) matrix, achieved an average full width at half maximum (FWHM) of 1.64 mm. Some research has focused on developing artificial neural network (ANN) models for event-position reconstruction [21], [22], [23], [24]. Among the most notable contributions, Ulyanov et al. [24] implemented three ANN models to reconstruct the x–z coordinates of gamma-ray interactions. They achieved an 8 mm FWHM at 356keV by using a gamma-ray detector consisting of a 20 mm thick LaBr$_{3}(\text {Ce}+\text {Sr})$ monolithic crystal coupled to an $8\times 8$ SiPM matrix. Kawula et al. research [25] has resulted in the development of a convolutional neural network (CNN) with a spatial resolution of 0.96 mm FWHM at 662 keV. They used a 30 mm thick LaBr$_{3}(\text {Ce}+\text {Sr})$ monolithic crystal coupled to a multianode photomultiplier tube (PMT).

In this article, we propose the development of an ANN model for real-time position reconstruction of gamma-ray interactions with a $50\,\text {mm} \times 50\,\text {mm}$ large and 20 mm thick LaBr$_{3}(\text {Ce}+\text {Sr})$ monolithic crystal coupled to an $8\times 8$ SiPM matrix. This detector has been developed for BNCT applications, where the detection of 478keV gammas resulting from the capture of neutrons by ¹⁰B could be used to monitor the treatment by means of single photon emission computed tomography (SPECT). Section II provides a detailed description of the detection module and the various stages involved in the ANN model’s implementation and validation. Moreover, the experimental measurements conducted to evaluate the capacity of the model to reconstruct the interaction position are presented. Section III presents and discusses the results obtained regarding the model’s metrics, including the FWHM achieved, in comparison with the state of the art. Furthermore, the model’s predictive capabilities are evaluated in the context of experimental data. Finally, Section IV presents the conclusions of the work and highlights potential future developments.

SECTION II. Materials and Methods

A. Detection Module

The detection module used to acquire gamma-ray events is a commercial LaBr$_{3}(\text {Ce}+\text {Sr})$ monolithic scintillator crystal by Saint-Gobain, measuring $5\,\text {cm}\times 5\,\text {cm}\times 2\,\text {cm}$ . All sides have a diffusive surface treatment with a standard reflector, except for the one coupled with the light guide which presents a polished surface treatment. The crystal is coupled with a matrix of $8\times 8$ near ultraviolet high-density (NUV-HD) SiPMs from FBK, Trento, Italy [26], [27]. It features excellent energy resolution (${\approx }3\%$ at 662 keV) and good detection efficiency (60% at 478 keV). Four 16-channel custom gain amplitude modulation multichannel ASICs (GAMMA) are used as front-end electronics for the SiPMs readout, enabling charge-to-voltage conversion and programmable gain [28]. Finally, a field programmable gate array (FPGA) manages data acquisition and communication with a PC to transfer, save and display the data acquired. The detector is combined with a channel-edge pinhole lead collimator, with dimensions of $9.4\,\text {cm} \times 9.4\,\text {cm}\times 10\,\text {cm}$ , which is used specifically to deal with high energy particles, effectively directing incoming radiations toward the detection module [29]. The collimator used for dose monitoring in BNCT applications, has been designed ad hoc through the aid of analytical formulas and Monte Carlo simulations, with the twofold focus on enhancing spatial resolution (achieving < 1 cm) and maximizing geometric efficiency (attaining $\gt 10{^{-}6 }$ ). In Fig. 1, the structure of the designed collimator is shown. It is characterized by a round channel with a diameter of 5 mm and a depth of 48 mm. The aperture angle is 9.25°, to fit a field of view (FOV) of $5 \times 5 \, \text {cm}^{2}$ on the detector, at 30 cm distance. The collimator features a spatial resolution of 8.2-mm FWHM and a geometric efficiency of $3.84 \times 10^{-6}$ at 478keV [29].

Fig. 1.

Channel-edge pinhole collimator designed for BNCT applications.

B. Position Estimation Method Based on ANN

To achieve good spatial resolution, despite the thickness of the scintillator, and real-time reconstruction of the interaction positions, we employed a supervised approach and developed a simple and low-computational-demanding algorithm. The algorithm’s high computational speed allows for the real-time reconstruction of event interaction positions in practical applications, where the detection module is subjected to a high detection rate. Furthermore, the reduced number of model parameters allows for further embedding the reconstruction on the FPGA of the detection system, which is responsible for sending the data packages to a PC where they are processed. In the preliminary development of the module, different methods were explored, such as k-NN, decision tree, random forest, AdaBoost, and ANNs. Among them, the regression ANN models demonstrated to best match our requirements in terms of the number of trainable parameters and processing time. DL models were excluded from consideration due to their significant computational complexity. In the context of this SPECT application, the spatial resolution is predominantly determined by the geometric characteristics of the collimator, which provides a contribution to the final image spatial resolution in the order of 8 mm. Consequently, enhancing the complexity of the reconstruction algorithm would not yield a corresponding improvement in overall performance. A supervised learning approach was employed to develop the predictive model, which was trained on input data comprising $\gamma $ -ray events whose 2-D interaction positions on the crystal surface were known. The algorithm was trained to identify the relationship between the SiPM signal patterns and the true interaction positions. Subsequently, the model was subjected to a validation process, whereby its ability to predict the x and y coordinates of new, previously unseen input events was evaluated. The fully connected ANN architecture that we propose is composed of four layers. The input layer comprises 64 neurons, corresponding to the 64 SiPM signals after being processed. The two hidden layers have 25 neurons each and feature a hyperbolic tangent activation function. This choice is justified by the fact that the estimation of gamma-ray interaction positions based on the SiPM signal pattern provided as input to the ANN is a nonlinear separable problem. Therefore, a single hidden layer was insufficient for the ANN to learn the information about the input data properly. Accordingly, two hidden layers were selected as the optimal number to effectively interpret the input information while avoiding the addition of undue complexity for real-time purpose. Finally, the output layer, comprising two neurons, employs a linear activation function to provide the prediction of the x and y interaction coordinates, respectively.

1) Data Acquisition:

In order to acquire the experimental data required for training and testing the ANN model, the detector surface was scanned in 481 positions using a narrow pencil beam of 662keV gamma rays, obtained using a ¹³⁷Cs point source of eight mCi positioned inside a cylindrical tungsten collimator with an aperture diameter of 1 mm. The irradiation points were uniformly distributed on the crystal surface in order to acquire an optimal dataset for the event position reconstruction task. The experimental setup is shown in Fig. 2.

Fig. 2.

Experimental setup used to acquire the data to train the network. The detection module is attached to a system made with two stages, used to move the detector in the x and y directions. The system is moving, while the ¹³⁷Cs source inside the tungsten-alloy collimator with 1 mm aperture, is fixed.

This approach was employed to guarantee a successful learning process and to reduce the risk of biasing the model toward particular patterns. Before starting the acquisitions, the crystal was meticulously aligned with the collimated beam by measuring count rate profiles across the detector edges in both the vertical and horizontal directions. After the alignment procedure between the source and the detector module, the acquisition protocol was implemented. This entailed a uniform scan of the entire crystal surface at 481 positions, intending to produce the most homogeneously distributed dataset possible (Fig. 3). The dual linear guide system, which moved the detector along the x and y axis with respect to the collimated ¹³⁷Cs source, allowed for accurate scanning of all interaction positions, with 2.21 mm step. The acquisition times were adjusted to guarantee the collection of a minimum number of events per position.

Fig. 3.

This figure shows the 481 scanning positions with their respective measurement time. The green points have a duration of 30 s while the red one 20 s. The x and y labels at each point are assigned according to the coordinate reference system.

2) Data Preparation:

The processing of the data started with the selection of the events of interest. First, the unprocessed spectrum for each position was smoothed in order to highlight the photopeak of ¹³⁷Cs. Subsequently, an adaptive window was defined to perform a Gaussian fitting. Once the 662keV peak had been suitably fitted, the distribution center was identified and only those events falling within the interval [mean $\pm ~2\sigma $ ] were retained, in order to keep a total of 95% of the events under the peak. This approach permitted the selection of events situated close to the centroid of the Cs peak, thereby reducing the probability of background noise events. A total of 1400 events were then randomly selected for each scanning position in order to have a dataset with a homogeneous number of events for each point.

Subsequently, a calibration procedure was implemented to render the 64 values associated with each event independent of the baseline and gain of the respective ASIC channels [28]. Initially, the baseline value of the corresponding channel was subtracted from each component of the event vector. Then, the resulting value was normalized by the gain of the channel. Finally, in order to ensure comparable magnitude values, the derived value was multiplied by the gain of a reference channel, which in our case was the gain of channel 1 of ASIC 1. The procedure employed ensured that the neural network received input values from each event that accurately reflected the SiPMs’ signal response to the energy of the incident gamma ray. This correction is necessary to maintain the same calibration in the event that the ASICs need to be replaced, thus ensuring that the neural networks remain unaffected.

Then, the dataset underwent normalization. Each of the 64 values describing an event is divided by the sum of the 64 values of the event itself.

Finally, the dataset was partitioned into training, validation, and test sets. The 10% of the acquired data were used as testing set, while the remaining 90% was further divided as 75% for training and 25% for validation.

3) Model Implementation:

Fig. 4 shows the architecture of the designed ANN. Hidden neurons have hyperbolic activation functions, while linear activation functions characterize output neurons. Due to its low sensitivity to outliers, the mean absolute error (MAE) was chosen as the cost function to be optimized during the training phase. In our case, the majority of the outliers are reasonably due to events that undergo Compton scattering within the crystal, which results in their absorption at a point that is far from the expected interaction position. The MAE metric quantifies prediction error without the amplification that occurs with root mean-squared error (RMSE). Therefore, during the training phase, the network deals with prediction errors of approximately the same magnitude. This aspect is of particular importance because, when faced with prediction errors of different magnitudes, the neural network tends to prioritize the instances with the highest error, thereby focusing on trying to accurately predict the interaction position of the Compton photons, which have LDs not determined only by their first point of interaction and therefore do not improve the network accuracy.

Fig. 4.

Model architecture is a fully connected regression ANN, comprising 64 input neurons, two hidden layers with 25 neurons each, and two output neurons for the x- and y-coordinate predictions, respectively.

In order to minimize the cost function and thereby tune the parameters of the ANN, we initiated the learning rate (LR) at 0.01 and employed the “Adam” optimizer, which adapts the LR for each parameter based on the gradient history, thus accelerating convergence and reducing memory allocation. The model features a total of 2327 trainable parameters and was trained for 1500 epochs. The batch size was set to 606060, with each training epoch consisting of a single batch. The ANN model was implemented in Python using the TensorFlow library.

4) Model Evaluation:

The RMSE and MAE, were computed on the training, validation, and test sets to assess the model’s performance and to detect potential overfitting [30].

1. RMSE is defined as follows:\begin{equation*} \text {RMSE} = \sqrt {\frac {1}{n} \sum _{{i}=1}^{n} \left ({{{y}_{\text {true}, {i}} - {y}_{\text {pred}, {i}}}}\right )^{2}} \tag {1}\end{equation*} where n is the number of observations in the set, ${y}_{\text {true}}$ is the true irradiation position, and ${y}_{\text {pred}}$ is the interaction positions predicted by the model. Large errors are magnified as a result of the squaring. Since the RMSE metric penalizes models more severely for larger errors, it shows greater sensitivity to outliers.

1. MAE is defined as follows:\begin{equation*} \text {MAE} = \frac {1}{n} \sum _{{i}=1}^{n} |{y}_{\text {true}, {i}} - {y}_{\text {pred}, {i}}|. \tag {2}\end{equation*}

The MAE metric is less sensitive to outliers than the RMSE. Large errors are not further amplified since the difference between the true and the predicted values is not squared.

In order to evaluate more accurately the spatial resolution of the model, we divided the crystal’s surface into six regions, as illustrated in Fig. 5. Given that the spatial resolution of monolithic scintillator detectors varies based on the interaction position and frequently deteriorates toward the crystal’s edges [14], [18], we examined the prediction error distribution across the different regions. The test set was divided into smaller sets according to the irradiation coordinates (x and y). For each area, two 1-D histograms were generated, one for the x-coordinate and one for the y-coordinate. The histograms plot the discrepancies between test events’ actual and estimated interaction positions within the specified region. Given the non-Gaussian error distributions, several metrics were considered in order to achieve a more comprehensive evaluation of the model’s x and y spatial resolution. First, the FWHM and full width at tenth maximum (FWTM) values of the prediction error distributions over the entire crystal surface and in the different regions were calculated. A Gaussian fitting was deemed inappropriate to calculate these values due to the non-Gaussian nature of the error curves. Therefore, linear interpolation was employed to determine the FWHM and FWTM values from the experimental error distributions of the total and individual regions. Without applying any smoothing to linearly interpolated the error distribution, we found the bin corresponding to the maximum height of the interpolated curve, then we calculated half of the maximum height and measured the width of the curve in this point to evaluate the FWHM. In the same way, by taking one-tenth of the maximum height, we measured the width to obtain the FWTM. Furthermore, the MAE and bias values were calculated for the x and y error distributions of each region. In particular, the bias was defined as the difference between the mean error value and the zero value, which denotes an unbiased prediction by the model. The objective was to identify whether the model tended to overestimate or underestimate the predictions with respect to the true values of the x and y interaction coordinates, in the presence of a positive or negative bias, respectively.

Fig. 5.

Six different regions were defined to evaluate the spatial resolution and the model’s bias across the crystal surface. The dimensions of each region are reported in relation to the $50\,\text {mm}\times 50\,\text {mm}$ crystal surface.

C. Experimental Measurements

In order to assess the performance of the new ANN model in a more realistic setting with the collimator, imaging measurements were also acquired using a ¹³⁷Cs point source with an active diameter of 1 mm and an activity of 7.8$\mu $ Ci. The source was mounted on a horizontal linear guide, which permitted precise control over the source’s position in relation to the detection module (Fig. 6). The source was positioned 30 cm from the center of the pinhole collimator and 60 cm from the detection module, both horizontally and vertically aligned with the FOV center. Starting from this position, nine different images of the point source were acquired, moving it in four different positions on the left and four on the right, in discrete steps of 3 mm, spanning an interval of 2.4 cm within the 5 cm FOV. Due to the nonuniform geometric efficiency of the collimator across the FOV, the measurement time varied from 45 min in the central position up to 1 h in the most external positions. This was done in order to acquire the same statistics for all positions. Finally, a 10-min background measurement was also acquired.

Fig. 6.

Experimental setup used to acquire the measurements with $^{137}{\mathrm { Cs}}$ source and evaluate the positioning reconstruction performance of the model. The source is placed on a linear guide, at a distance of 30 cm from the collimator center and of 60 cm from the detection module.

A 32-by-32-pixel image was reconstructed for each source position using the x- and y-coordinate of the ¹³⁷Cs events estimated by the NN. In each image, the image of the background was subtracted on a pixel-by-pixel basis, after rescaling it to have a statistic equivalent to the duration of the other measurements. The 1-D (1D) histograms used to calculate the point-source position were extracted from each reconstructed image by selecting the region having the y coordinate comprised in the interval $[{-}4.5; +9.5]$ mm, where the source was localized, and summing along the columns. Subsequently, each obtained profile was fitted with a Gaussian distribution in order to evaluate the measured values of the FWHM and the centroids. Lastly, the theoretical values, whose computations are detailed in the Appendix, and the measured values were compared. The MAE between the measured and expected FWHM values was calculated to quantify the average absolute deviation of the measurements from the theoretical values. A similar analysis was conducted for the measured and theoretical centroid values.

SECTION III. Results and Discussion

A. Model Performance Analysis

Table I reports the regression metrics values obtained for the training, validation, and test sets. For each metric, the average value resulting from the x- and y-coordinate reconstruction is shown.

TABLE I Model Performance Evaluation: Regression Metric Values Obtained for Training, Validation, and Test Sets

After optimizing the model based on MAE during training to reduce sensitivity to outliers, this metric was considered the most relevant for evaluating performance. RMSE, however, was also regarded as a reliable metric. The model exhibits consistent performance across all the datasets, as evidenced by the uniformity of metric values. Consequently, there is no evidence of overfitting.

Concerning the model’s spatial resolution, the FWHM and FWTM values of the prediction error distributions over the entire crystal surface and across the different regions are presented in Table II. The corresponding MAE and bias values are provided in Table III.

TABLE II FWHM and FWTM of the Model Evaluated on the Different Crystal Regions Along Both the x and y Axis

TABLE III MAE and Bias of the Model Evaluated on the Different Crystal Regions Along Both the x and y Axis

The model exhibits an FWHM of 2.9 mm and an FWTM of 9.2 mm across the entire crystal surface. Fig. 7 illustrates the 1-D histograms of the prediction error distributions in the x and y direction for the total test set, which have been linearly interpolated. The MAE associated with events occurring over the entire crystal surface is found to be consistent between the x and y-coordinate reconstructions. Furthermore, the zero-biased error distributions indicate that the model-predicted values are, on average, equally distributed around the true values.

Fig. 7.

1-D histograms of the prediction error distribution for the total test set along the x and y axis, calculated as the difference between the true and estimated interaction coordinate of all test events. The average FWHM values, calculated following the application of linear interpolation, are presented for both the (a) x and (b) y coordinates.

The relatively low FWHM values of approximately 2.6 mm in the central and intermediate regions indicate a narrow spread of the prediction error distributions. The bias values are found to be approximately zero, which suggests that predictions of the events interacting in both the central and intermediate regions are not biased.

In the upper and lower regions, we observed a degradation in spatial resolution along the y-axis. In fact, the FWHM value for the x-coordinate remains relatively consistent with the previous results obtained for the other regions, indicating that the model’s spatial resolution is not significantly affected along this axis. Conversely, the model demonstrates suboptimal performance in reconstructing the interaction points along the y-axis, as evidenced by the elevated FWHM values. A similar pattern is observed in the MAE associated with the reconstruction of the y-coordinate, which is greater than that for the x-coordinate. This can be attributed to the truncation of the LD along the y-axis for interacting events in the lower and upper regions, as well as to internal reflection phenomena occurring at the edges of the crystal. In both the upper and lower regions, the FWHM deteriorates by 50% or more, whereas the MAE increases by a maximum of 30%. This can be explained by the fact that the FWHM is highly sensitive to the shape of the distribution, whose peak significantly decreases in the lateral regions. Consequently, the MAE can be considered a more representative value of the entire distribution. Furthermore, in the lower region, the negative bias indicates that the model tends to overestimate the value of the y-coordinate, with ${y}_{\text {pred}}\gt ~{y}_{\text {true}}$ . Conversely, the positive bias of the upper region indicates that the model tends to underestimate the value of the y-coordinate, with ${y}_{\text {pred}}\lt {y}_{\text {true}}$ . Consequently, the y-coordinate of interacting events in the lower and upper regions is reconstructed with a bias toward the crystal center.

Similarly to the upper and lower regions, the left and right ones exhibit a deterioration in spatial resolution along the x-axis, as indicated by the larger FWHM values. This can be attributed to the truncation of the LD along the x-axis, which involves events occurring at the lateral areas of the crystal. The MAE associated with the reconstruction of the x-coordinate is higher than that for the y-coordinate. The same variation in the FWHM with respect to the MAE that is observed in the upper and lower regions is also evident in both the left and right regions. In the left region, the model tends to overestimate the value of the x-coordinate, as evidenced by the negative bias. Conversely, the positive bias observed in the right region indicates that the model has a tendency to underestimate the value of the x-coordinate. Therefore, the x-coordinate of interacting events in the left and right regions is also reconstructed closer to the crystal center. Fig. 8 shows the linearity diagram for the 31 scanned positions along the crystal diagonal (Fig. 3). For each scanned point, the average of the reconstructed x and y coordinate for all events is compared with the true position. The linearity deteriorates near the edges, resulting in an effective FOV around 4 cm, defined as the region of the crystal where the linearity remains monotonically increasing [23], [31].

Fig. 8.

Linearity diagram along the crystal diagonal, obtained by comparing the true scanning position with the average reconstructed x and y coordinates.

B. Experimental Measurement Analysis

Fig. 9 shows the images obtained with the collimator moving the ¹³⁷Cs point source in nine different positions. The images are the 2-D histograms of the x and y coordinates of the ¹³⁷Cs events estimated by the ANN. In Fig. 9, for each image is also reported the corresponding 1-D histogram fitted with a Gaussian distribution used to compute the FWHM of the image. The negative values observed in the histograms result from the background subtraction and represent statistical noise fluctuations. The average FWHM of the nine reconstructed positions was approximately 8 mm. This result is consistent with the dominant contribution of the 8 mm geometric resolution of the collimator. The reconstructed images and the measured centroids of the corresponding distributions are consistent with the displacement of the source with a 3 mm step. This represents a significant achievement, as it illustrates the model’s ability to track the positional changes of the source. The comparison between the expected and measured values of FWHM and centroids is shown in Fig. 10. The theoretical and measured centroid values demonstrate an inverse relationship with the source position, which is due to the pinhole geometry of the collimator. The results reported indicate that the model can accurately reconstruct the ¹³⁷Cs radiation source position, achieving a mean error with respect to the theoretical expected value of FWHM of about 0.48 mm. Additionally, the model demonstrates an accuracy of 0.7 mm in tracking the source position based on the MAE between the measured and theoretical centroid values.

Fig. 9.

$32\times 32$ -pixel reconstructed images of the ¹³⁷Cs source shifted in nine positions along the x-axis and the corresponding 1-D histograms. The FWHM values are obtained by a Gaussian fit.

Fig. 10.

Comparison between the measured and theoretical FWHM (a) and centroid (b) values for the nine reconstructed source positions. The theoretical values are obtained with the formulas reported in the Appendix, while the measured values are determined by a Gaussian fit of the 1-D histograms (Fig. 9).

C. Comparison With Literature Benchmark

Following the comprehensive evaluation of the model using the metrics outlined in Section II-B4, we now focus on the FWHM parameter, which is commonly used in the literature to estimate the spatial resolution of the implemented models. The FWHM values considered, are the one reported in Table II, and were obtained by fitting the network prediction error distribution for the test dataset. All data in the test set were acquired with ¹³⁷Cs source inside a tungsten collimator with 1 mm aperture diameter. This allows us to compare the performance of the model implemented in the present study with that of the state-of-the-art. Results from different studies are summarized in Table IV.

TABLE IV Comparison With the State of the Art

The positioning algorithm we developed is based on a supervised ANN regression model. With a system based on a LaBr₃ square monolithic scintillator crystal, measuring $5~\text {cm}\times 5~\text {cm}\times 2~\text {cm}$ , and coupled with a matrix of $8\times 8$ SiPMs, the model achieves a spatial resolution of 2.9 mm FWHM at 662keV over the majority of the crystal surface. In particular, it reaches a 2.5 mm FWHM in the crystal center, exhibiting a slight decrease toward the edges.

The obtained results are encouraging when considering the ANN models developed by Ulyanov et al. [24]. Their system consists of a LaBr₃ scintillator with a reflector housing, with dimensions of $2.8\,\text {cm}\times 2.8\,\text {cm}\times 2\,\text {cm}$ and an $8\times 8$ SiPM matrix. In addition to employing a separate ANN for each coordinate to be reconstructed, trained by scanning the crystal with a ¹³³Ba source with 2 mm beam diameter, the resulting spatial resolution, with an FWHM value of 8 mm at 356keV, is considerably inferior to that achieved in our study, although the difference in energy also contributes to the inferior position resolution of this system. In the case of ANNs employed as 2-D positioning algorithms, our result (2.9 mm FWHM) demonstrates a notable improvement over the resolutions (at 356keV) of 5.4 mm with a 2 cm thick and 2.9 mm with a 1 cm thick crystal reported in [22]. In this other study, Ulyanov et al. employed CeBr₃ crystals with a diffusive reflector and unpolished surface, coupled with a $4\times 4$ SiPM matrix. The network training was performed by scanning the crystal surface with a ¹³³Ba source inside 5 cm thick lead brick with 1.4 mm aperture. Despite the larger thickness of the crystal, which is generally associated with lower positional accuracy, our model has been shown to achieve the same spatial resolution as the ANNs used with an 1 cm thick CeBr₃ crystal. The spatial resolution obtained here for the 20 mm thick crystal is comparable to the 3.01 mm FWHM value at 511keV reported by Babiano et al. [23] using LaCl₃ crystal with a diffusive reflector and polished surface, of $5~\text {cm}\times 5~\text {cm}\times 2\,\text {cm}$ dimensions coupled to $8\times 8$ SiPM matrix. The scanning of crystal surface is made with a ²²Na source inside a tungsten box with a central hole diameter of 1 mm. Regarding KNN algorithms, the model developed by Seifert et al. [20] using a 10 mm thick LaBr$_{3}(\text {Ce}+\text {Sr})$ monolithic crystal with a reflector housing and polished surface, coupled to a $4\times 4$ SiPM matrix, demonstrates significantly better performance, with an average FWHM of 1.64 mm at 511keV, in comparison to the ANN developed by this study. The network training was performed by scanning the LaBr₃ surface with a ²²Na source collimated at 0.5 mm. The discrepancy in resolution can be attributed to the higher thickness of the crystal employed in our system and a narrower pencil beam to scan the detector. It has been demonstrated that spatial resolution is superior in the case of CNN-based position algorithms, as in the work proposed by Kawula et al. [25]. This work shows the difference between the categorical averaged pattern (CAP) algorithm, an optimized version of the KNN method, and a CNN for a system made by LaBr₃ scintillator with reflective housing, and dimensions of $5\,\text {cm}\times 5\,\text {cm}\times 3\,\text {cm}$ , coupled to a 64-fold segmented multianode PMT. Both networks were trained with the same scanned positions on the crystal surface, obtained using a ¹³⁷Cs source inside a collimator with a 1 mm diameter channel. The CAP method achieves an FWHM of 2.9 mm at 1.33MeV, in comparison to the 0.9 mm obtained with the CNN. Furthermore, the CNN exhibits the same FWHM value at 662keV. Another work by Balibrea-Correa at al. [31] proposed different models for gamma rays position reconstruction for a LaCl₃(Ce) monolithic crystal sealed in a diffusive reflector with $5\,\text {cm}\times 5\,\text {cm}$ surface and 20 mm thicknesses, coupled with $8\times 8$ SiPM matrix. Using a CNN implemented in Keras they obtained a resolution of 2.4 mm FWHM at 511keV. The crystal was scanned with a ²²Na source inside a tungsten brick with 1 mm aperture diameter. Notwithstanding the favorable results, the computational demands of DL models render them rather unsuitable for real-time implementation.

SECTION IV. Conclusion

A positioning algorithm for the reconstruction of $\gamma $ -ray interactions in thick monolithic scintillators was developed. This algorithm is based on a supervised ANN regression model, and its performance has been validated through experimental measurements. With a system based on a 20 mm thick LaBr₃ monolithic scintillator crystal, coupled with a matrix of $8\times 8$ SiPMs, the model reaches a spatial resolution of approximately 2.6 mm in the central region of the crystal, slightly worsening toward the edges. While the current study focused on estimating the x and y coordinates of the interactions, the estimation of DOI was not investigated. This choice is motivated by the fact that the collimator features a maximum acceptance angle of 9.52°, which limits the maximum incidence angle of gamma rays impinging on the crystal and therefore also the maximum parallax error due to DOI variation. Since the total system resolution, around 8 mm, is dominated by the collimator contribution, we estimated that DOI calculation, along with the use of more complex network architectures, would determine an almost negligible improvement in the image spatial resolution adding complexity to the calibration procedures, therefore it was not implemented. The network is currently running on a personal computer utilizing C++ code within a graphical user interface designed for the display of data during acquisitions. The PC features an AMD Ryzen 7 5700G with 8 cores and 16 logic processors, and 32GB of RAM, and processes a single event on average in 1.8$\mu $ s. The subsequent stage of development will be to embed the ANN into the system FPGA, guaranteeing a more compact gamma module that fulfills the applicability of real-time reconstruction. Furthermore, in the long term, such ANN could be implemented in an analogue manner using a circuit, monolithically integrated within the front-end ASIC that reads the SiPM signals [33].

Appendix Analythical Determination of The Collimator Resolution

We conduct analytical calculations to assess the theoretical geometric resolution of our collimator, aiming to compare it with the results obtained from experimental measurements. This method revolves around evaluating the area that a point $V(x,y,z)$ projects onto the detection plane as it passes through the collimator channel (Fig. 11). The dimensions of the channel are fixed as follows.

1. CHX = 5 mm, is the width of the hole in the x direction.

2. CHY = 5 mm, is the dimension of the hole in the y direction.

3. CHZ = 48 mm, represents the depth of the channel along the z-axis.

Fig. 11.

This figure reports the schematic for the first case, where the point x coordinate is between $-\textrm {CHX}/2$ and $\textrm {CHX}/2$ .

For simplicity, in accordance with the experimental measurement setup, we will consider the straightforward case of a point moving solely in the x direction with its y coordinate set to zero. To perform these calculations, we need to divide the problem into three parts depending on the point position. We report in Table V with the obtained results of the theoretical resolution and centroids calculated with the formulas reported below, considering a point source moving along the x axis, with fixed y = 0. To take into account penetration effects we used ANTS2 simulations from a previous work [29], [32], to simulate the geometric resolution $R_{\mathrm { sim}}$ for a point source in the FOV center, evaluated to be 8.2 mm, and with that evaluating the effective diameter $d_{\mathrm { eff}} = 6$ mm\begin{equation*} d_{\mathrm { eff}} = \frac {R_{\mathrm { sim}}}{\sqrt {\frac {x+f}{x+{\mathrm { CHZ}}/2}}}. \tag {3}\end{equation*}

TABLE V Theoretical Values of Geometrical Resolution and Centroid of the Projected Points Onto the Detection Plane for a Given Position of a Point Source Along the x-Axis

The resolution at the object was then evaluated with the following equation:\begin{equation*} R_{o} = \sqrt {R_{\mathrm { thx}}^{2} + R_{d}^{2}} \tag {4}\end{equation*} where $R_{d}$ is the resolution at the detector reported in Table II.

A. First Case:$-\textrm {CHX}/\textit {2}\,\lt \,Vx\,\lt \textrm {CHX}/\textit {2}$

First we evaluate the distance VH refers to the distance from the point V to the point H, Considering Vy=0 and x and f are the distances from the source plane and the detection plane to the hole center h and are fixed to 30 cm\begin{align*} \textrm {VH}=& \sqrt {(Vx-Hx)^{2}+(Vy-Hy)^{2}+(Vz-Hz)^{2}} \tag {5}\\ \textrm {VH}=& \sqrt {(Vx)^{2}+(x+\textrm {CHZ}/2)^{2}}. \tag {6}\end{align*} ${\mathrm { CDP}}x$ is the coordinate of the point V projected to the detection plane in the x axis\begin{equation*} {\mathrm { CDP}}x = -Vx*\frac {x-\textrm {CHZ}/2}{x+\textrm {CHZ}/2}. \tag {7}\end{equation*} VD is the distance from the point V to the detection plane passing through H\begin{equation*} \textrm {VD} = \sqrt {(Vx-{\mathrm { CDP}}x)^{2}+(2x)^{2}}. \tag {8}\end{equation*}

For a knife-edge pinhole collimator [34], the point h is coincident with the point H, so if the point V is at the FOV center ($Vx\,=\,0$ , $Vy\,=\,0$ ), the distances VH and VD are respectively equal to x and x+f. Considering the hole diameter d and the shadows of the hole in the detection plane that have the same length for both x and y direction equal to $R_{\mathrm { th}}$ , we can derive the equation of the resolution as\begin{align*} R_{\mathrm { th}}^{2}=& d^{2} * \frac {x+f}{x} \tag {9}\\ R_{\mathrm { th}}=& d * \sqrt {\frac {x+f}{x}}. \tag {10}\end{align*}

For a channel-edge collimator, the projections onto the detection plane of the hole, CHDx and CHDy, are not the same. Therefore, the resolution has two components\begin{align*} R_{\mathrm { thx}}=& {\mathrm { CHD}}x = \textrm {CHX} * \sqrt {\frac {\textrm {VD}}{\textrm {VH}}} \tag {11}\\ R_{thy}=& {\mathrm { CHD}}y = \textrm {CHY} * \sqrt {\frac {\textrm {VD}}{\textrm {VH}}}. \tag {12}\end{align*}

For this work, considering all the measurements points with Vy=0, the resolution on the y direction is calculated as\begin{equation*} R_{thy} = \textrm {CHY} * \sqrt {\frac {x+f}{x+\textrm {CHZ}/2}}. \tag {13}\end{equation*}

B. Second Case: $Vx\,\lt \,-\textrm {CHX}/\textit {2}$ (Ref. To Fig. 12)

\begin{align*} dx=& \frac {|Vx+\textrm {CHX}/2|*\textrm {CHZ}}{x-\textrm {CHZ}/2} \tag {14}\\ Hx=& \frac {dx}{2} \tag {15}\\ \textrm {VH}=& \sqrt {(Vx-Hx)^{2}+(Vy-Hy)^{2}+(Vz-Hz)^{2}} \tag {16}\\ \textrm {VH}=& \sqrt {(Vx-Hx)^{2}+(x+\textrm {CHZ}/2)^{2}} \tag {17}\\ {\mathrm { CDP}}x=& \frac {-(Vx-Hx)*(x-\textrm {CHZ}/2)}{x+\textrm {CHZ}/2} + Hx \tag {18}\\ \textrm {VD}=& \sqrt {(Vx-{\mathrm { CDP}}x)^{2}+(Vz-{\mathrm { CDP}}z)^{2}} \tag {19}\\ \textrm {VD}=& \sqrt {(Vx-{\mathrm { CDP}}x)^{2}+(2x)^{2}} \tag {20}\\ R_{\mathrm { thx}}=& {\mathrm { CHD}}x = (\textrm {CHX}-dx) * \sqrt {\frac {\textrm {VD}}{\textrm {VH}}}. \tag {21}\end{align*}

Fig. 12.

This figure reports the schematic for the second case, where the point x coordinate is minor than $-\textrm {CHX}/2$ .

C. Third Case: $Vx\,\gt \textrm {CHX}/\textit {2}$ (Ref. To Fig. 13)

\begin{align*} dx=& \frac {|Vx-\textrm {CHX}/2|*\textrm {CHZ}}{x-\textrm {CHZ}/2} \tag {22}\\ Hx=& -\frac {dx}{2} \tag {23}\\ \textrm {VH}=& \sqrt {(Vx-Hx)^{2}+(Vy-Hy)^{2}+(Vz-Hz)^{2}} \tag {24}\\ \textrm {VH}=& \sqrt {(Vx-Hx)^{2}+(x+\textrm {CHZ}/2)^{2}} \tag {25}\\ {\mathrm { CDP}}x=& \frac {-(Vx-Hx)*(x-\textrm {CHZ}/2)}{x+\textrm {CHZ}/2} + Hx \tag {26}\\ \textrm {VD}=& \sqrt {(Vx-{\mathrm { CDP}}x)^{2}+(Vz-{\mathrm { CDP}}z)^{2}} \tag {27}\\ \textrm {VD}=& \sqrt {(Vx-{\mathrm { CDP}}x)^{2}+(2x)^{2}} \tag {28}\\ R_{\mathrm { thx}}=& {\mathrm { CHD}}x = (\textrm {CHX}-dx) * \sqrt {\frac {\textrm {VD}}{\textrm {VH}}}. \tag {29}\end{align*}

Fig. 13.

This figure reports the schematic for the Third case, where the point x coordinate is greater than $\textrm {CHX}/2$ .

ACKNOWLEDGMENT

All authors declare that they have no known conflicts of interest in terms of competing financial interests or personal relationships that could have an influence or are relevant to the work reported in this article.

NOTE

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