Precise positioning of gamma ray interactions in multiplexed pixelated scintillators using artificial neural networks

Introduction. The positioning of γ ray interactions in positron emission tomography (PET) detectors is commonly made through the evaluation of the Anger logic flood histograms. machine learning techniques, leveraging features extracted from signal waveform, have demonstrated successful applications in addressing various challenges in PET instrumentation. Aim. This paper evaluates the use of artificial neural networks (NN) for γ ray interaction positioning in pixelated scintillators coupled to a multiplexed array of silicon photomultipliers (SiPM). Methods. An array of 16 Cerium doped Lutetium-based (LYSO) crystal pixels (cross-section 2 × 2 mm2) coupled to 16 SiPM (S13360-1350) were used for the experimental setup. Data from each of the 16 LYSO pixels was recorded, a total of 160000 events. The detectors were irradiated by 511 keV annihilation γ rays from a Sodium-22 (22Na) source. Another LYSO crystal was used for electronic collimation. Features extracted from the signal waveform were used to train the model. Two models were tested: i) single multiple-class neural network (mcNN), with 16 possible outputs followed by a softmax and ii) 16 binary classification neural networks (bNN), each one specialized in identifying events occurred in each position. Results. Both NN models showed a mean positioning accuracy above 85% on the evaluation dataset, although the mcNN is faster to train. Discussion The method’s accuracy is affected by the introduction of misclassified events that interacted in the neighbour’s crystals and were misclassified during the dataset acquisition. Electronic collimation reduces this effect, however results could be improved using a more complex acquisition setup, such as a light-sharing configuration. Conclusions The methods comparison showed that mcNN and bNN can surpass the Anger logic, showing the feasibility of using these models in positioning procedures of future multiplexed detector systems in a linear configuration.


Introduction
The use of machine learning in nuclear instrumentation is becoming common, especially for the estimation of photon interaction position in the detectors in applications where high spatial resolution is crucial (up to a few millimeters), such as in PET preclinical imaging [1][2][3].Conventional methods were usually based on Anger logic, which is vulnerable to the signalto-noise ratio of the data recorded, which results in a decline in performance along the edge pixels readout.
Therefore, the spatial resolution of these systems is significantly impacted by this phenomena.
To overcome these limitations, there is an increased interest in integrating machine learning techniques directly into the acquisition hardware front-end (using microcontrollers or FPGAs-field programmable gate arrays).This is of particular interest in high-energy and medical physics applications that involve highly irradiated detectors and where the acquisition dead-time needs to be minimized while maintaining a higher data throughput [4][5][6][7][8].This approach not only saves time and computing resources but also has the potential to improve data acquisition and processing by capturing more accurate and usable data.FPGAs are highly attractive due to their cost-effectiveness and flexibility compared to application-specific integrated circuits (ASICs), when used for the same tasks.FPGAs can be reprogrammed with new circuits even while they are in operation, unlike ASICs, which are hardware-specific and cannot be modified after production.Both FPGAs and ASICs find extensive use in PET instrumentation, and they can be used simultaneously to leverage each other capabilities [9][10][11][12].
Pulse shape discrimination (PSD) combined with classification methods has been demonstrated to be a successful approach for depth-of-interaction (DOI) determination in scintillator detectors, which allows for reducing the parallax effect [13,14].More recently, it was shown the estimation feasibility of the photon's time-of-flight in PET detectors by using their waveform combined with convolutional neural networks, with a time-of-flight timing resolution improvement of up to 23% compared with typical methods, with timing resolution up to 185 ps [5].Several machine learning algorithms have successfully been applied to similar classification problems using a supervised approach [15][16][17].
Pixelated scintillators are common in the field of medical imaging due to their ability to provide exceptional spatial resolution, which is closely linked to the size of individual pixels.They also offer simplified signal extraction compared to their monolithic counterparts [18].However, this advantage comes at the cost of potential signal loss due to the increased number of reflections and absorptions [19], at higher price.
The ideal scenario for achieving optimal performance involves a one-to-one pixel-to-photodetector coupling.Nevertheless, this approach comes with increased costs in front-end electronics.Additionally, when dealing with very small pixel sizes (e.g., below 1 mm 2 ), the sensitive areas of current photodetectors are expensive and have reduced effective area, making it challenging to meet the demands of extremely small pixel dimensions.The fabrication of such tiny pixels also presents significant mechanical and financial challenges.As a result, in the quest for a balance between spatial resolution, scintillator pixel size, and photodetector area, compromises are often necessary [20,21].
When the photodetectors inter-connection reduces the required number of channels for data acquisition (DAQ) to a value below the total number of photodetectors, this arrangement is referred to as a multiplexed configuration.The most commonly used method for multiplexing is based on charge modulation, which is chosen for its simplicity in implementation.In this approach, the charge generated by each photodetector is shared through a multiplexing network.When constructing such networks using resistors, the topological approach includes the use of a 1D linear resistor array or 2D matrices, resulting in 2 or 4 channels, respectively [22,23], from which the energy and localization information is recovered by applying Anger logic algorithms [23][24][25][26].More complex schemes, such as employ weighting resistor circuits [27], use capacitive multiplexing networks [28] or hybrid charge division methods combining a multiplexing network of capacitors and resistors [29], which can further reduce the number of required channels while improving the positioning algorithms.
Anger logic rely on the pulse height measured on each electronic channel, resulting in successful events classification if undesired effect are absent from the data, which is often not the case.In particular, for low energy events (smaller pulse heights), the electronic noise, events pileups and baseline shifts, among other effects, affects the data quality and can result in misclassified event positioning [30].
There is a research line dedicated to innovative methods to reduce the number of readout channels while maintaining optimal performance in terms of interaction localization [22,26].These methods can include strictly analytical analysis and signal filtering [31,32].However, machine learning approaches are also being employed for DOI estimation.These methods use convolutional neural networks [33,34] with pixel light sharing information, both simulated and experimental data, or employ Random Forest algorithms [35] to extract signal features in a single-ended readout without light sharing.
Artificial neural networks (NN) can also be used as an alternative classification method, given that the multiple pulse waveform features extracted (in addition to the pulse height used by Anger logic) can be used as input features to train the NN.
This works aims to evaluate the feasibility of using NNs to identify the γ ray interactions in pixelated scintillators coupled to photodetectors and multiplexed using a resistive chain.Several NNs architectures were tested and compared against Anger logic classification.To our knowledge, this is the first investigation into multiplexed pixelated scintillators, where light sharing between scintillators is not employed.The study establishes the feasibility of utilizing neural networks for classifying events based on their interaction positions.

Materials and methods
The problem we want to address here is interpreted as a classification problem (predicting 1 out of 16 possible positions).Feedforward, fully connected NN were used as the classifier, considering both binary and multiclass classification.

Experimental setup
The experimental setup is shown in figure 1.An array of 16 polished scintillator crystals made of lutetium-yttrium oxyorthosilicate cerium doped (LYSO:Ce 0.5%) with dimensions of 2 × 2 × 30 mm 3 (from EPIC-Crystal4 ) was used, totally wrapped in a reflector made of barium sulfate (BaSO 4 , 100 μm thickness) in all but one face (2 × 2 mm 2 ), for light collection using a photodetector.Optical coupling was performed with silicone grease (BC-630 from Saint-Gobain).These scintillators have excellent properties for radiation detectors and instrumentation for medical physics such as the high quantum light yield conversion (over 26000 photons MeV −1 ), fast scintillation decay time (40 ns-60 ns), high density (7.2 g cm −3 ) and radiation stopping power [20,36].
An array of 16 silicon photomultipliers (SiPM) (S13360-1350 series from Hamamatsu Photonics, Japan5 ) was used.Those were mounted in a custommade Printed Circuit Board (PCB).The SiPM cathodes share the voltage bias, while their anodes are connected using a simple resistor chain made of 17 surface-mounted device (SMD) resistors of 5 Ω and 0.5% tolerance.This readout acts as a 16:2 multiplexer by resistive charge division and allows the use of only two electronic channels for signal amplification and event reconstruction using Anger logic methods, as depicted in figure 1(a).In this linear configuration, two pulses will appear at the front-end every time one SiPM is activated.Depending on the position of the activated SiPM in the resistor chain, the time features Another LYSO crystal is used to perform electronic collimation and filter events by position.The SiPM signal is preamplified and shaped by a front-end board, whose output is sent to the oscilloscope for the waveform digital conversion.The process is repeated for each LYSO pixel, moving the source and collimator jointly.The distances between the collimator and source, D CS , and between the source and LYSO array, D SA , were defined to minimize the source penumbra in neighboring array pixels.
of each of the two pulses will be different but somehow correlated.Figure 1(b) depicts the use of the front-end electronics connected to an oscilloscope for signal digitization.The time features of each signal are extracted and the activated SiPM position is decoded from signals measured relations.
In this setup, the scintillators are irradiated by the γ rays from a 22 Na source.The source is a cylinder of 0.5 mm diameter and 0.5 mm height, embedded in a polymethyl methacrylate (PMMA) rectangular tank of 3 mm×3 mm×8 mm with approximately 56 kBq .On the other side, a single LYSO scintillator crystal, with the same dimensions as the ones on the array, is used to perform electronic collimation for the annihilation photons from the source.The distance between the collimator LYSO entrance face and the 22 Na source center, D CS , was settled to 61 mm and to reduce the coincidence irradiation penumbra, as depicted in figure 1(b), the distance between the 22 Na source center and the scintillator array entrance face, was defined to D SA , 1.5 mm which is the physical minimum distance possible due to the PMMA tank dimensions.
The setup was manually moved between positions using a micrometer screw connected to a calibrated linear guide rail to guarantee minimum oscillations.
For data acquisition, the electronics front-end used was a dedicated printed circuit board (from RI-TE, LDA, Portugal6 ) that allows up to 4 electronic channels in single or coincidence readout mode.Each channel, based on the OPA656 (Texas Instruments, USA 7 ) amplifiers, has three stages: a preamplifier, a fast discriminator for timing and a slow integration stage for real-time pulse height measurement.

Computer configuration
The computer setup used consists of a laptop with an Apple®M1 Pro 8 Processor, a 10-core CPU with a maximum frequency of 3.22 GHz and a L2 cache size of 24 MB, equipped with 16 GB LPDDR5 RAM and an integrated GPU with 16 cores and 2048 arithmetic logic units.Whenever possible, code parallelization was performed at the processor level, with up to 10 cores processing data simultaneously to accelerate the data analysis.

Data acquisition and features extraction
Two channels of the front-end board were used to read the two signals from the resistor chain.The output of each channel is readout by an oscilloscope (HDO series 6104, 2.5 GSs −1 , 12-bits vertical scale from Teledyne LeCroy, USA 9 ) to record the pulse waveform.For each pulse, data is converted during 5 μs at a frequency of 100 MHz (10 ns per sample).Pulse signals are smoothed using the wavelet denoising function wdenoise (from Matlab®), with the parameters: sym4 wavelet, level integer 6, Bayes denoising method and median threshold rule.After this, several features are extracted for both channels (left and right, respectively): pulse heights (V L and V R ), areas (A L and A R ), rise times (t r,L and t r,R ) from 10% to 90% of the signal height, fall times (t f,L and t f,R ) from 90% to 10% of the pulse height, signal time trigger (at 10% signal crossing) t t,L and t t,R , time instant of the signal maximum, t m,L and t m,R , and time-over-threshold (above 10% of the signal) t ToT,L and t ToT,R as depicted in figure 2.
The data acquisition process was repeated for each SiPM in the resistor array.For each position, a dataset of 5000 events was recorded, which gives a total of 80000 events.For each event, several features were measured as described in section 2.1.2:pulse heights, areas, rise times and fall times, trigger times, times at signal maximum and time-over-threshold.
Oscilloscope time trigger fluctuations can have a significant impact on the absolute values of the signal trigger features.To address this issue, the left and right signal triggers are not directly used in the machine learning algorithm but combined, using their difference in a variable called Difference Start Time, d t,start .This also applies to the time at signals maximum, which are combined into Difference Time at Maximum, d t,max , as shown in equations (2.1) and (2.2).
In addition, is calculated and stored the ratio R and energy E of each event, according to equations (2.3) and (2.4).When the pulse height on each channel is measured (V R and V L for channel 1 (right) and 2 (left), respectively) the energy E deposited by the particle and the spatial position of the activated detector can be estimated using equations (2.3) and (2.4).The energy is given by the sum of the two pulse heights while the position is determined using the ratio R between the two measured pulse heights difference and their energy.The time-over-threshold of the summed signal is also extracted and stored in a feature called Time Over Threshold Sum, t ToT,Sum .
While E has the physical dimension in which the pulse height is measured (commonly V, ADC channel or keV if energy calibration is performed), R is dimensionless.
Finally, the scintillator relative position (classification label) in the SiPM array (in this case ranging from 1 to 16) is also stored.Data is stored in a data-frame of events (rows) and features (columns) with dimensions of [ ] 80000 15 .

Position classification using Anger logic
The number of unequivocal solutions for equation (2.4) depends only on the number of nodes in the resistor chain-in a perfectly behaved system without external sources of noise, for an array with N nodes (being N the total number of SiPM, one per node), only N solutions are possible, independently on the deposited energy E. As a result, for each detected particle, measuring R is sufficient to determine in which node the particle was detected.
The position detection method based on Anger logic relies on the discrimination of the values of R by its relative frequency in a flood map: in our case scenario events can be classified according to the ratio vector R in equation (2.5) since for a resistor array with 16 SiPM, equation (2.4)only 16 ratio values are possible. 5 For each event i, depending on the measured ratio r i , that event is sorted to the position n with the closest R n value, i.e. finding n that minimizes d n,i , the difference between R n and r i , as depicted in equations (2.6), (2.7).

Position classification using artificial neural networks (NN)
To detect the position of the first interaction in the crystal array, a set of events is acquired, and their positions are classified using a supervised approach [13,17].The electronic collimation configuration is employed to avoid events with a first interaction position in a crystal different from the one being collimated.Then, the pulse features measured at each position are used to train the model and predict the position of the first interaction of the incoming photons.

NN architecture
In this work fully connected feedforward NNs were used, in which the first layer is connected to the input data and the last layer to the classification output label.
The NNs architecture can be customized to achieve an upgraded NN for our classification problem.This can be done by changing the shape and number of layers, the activation functions, the function to initialize fully connected layer weights, the type of initial fully connected layer biases, the regularization term strength and the loss function.Matlab® Statistics and Machine Learning Toolbox (Matlab 2021b toolbox) was used to implement the described architecture.From this toolbox, the function fitcnet was used to create, train and test the NNs.By default, this function creates a NN whose architecture is specialized for classification problems.The rectified linear unit (ReLU) activation function was used for the inner layers connections, while the softmax activation function was used at the output of the NN to give the classification scores and the Pulses waveform from the resistor chain.The features pulse heights, V L and V R , areas, A L and A R , rise times, t r,L and t r,R , fall times,t f,L and t f,R , signal time trigger, t t,L and t t,R , time of signal maximum, t m,L and t m,R , and time-over-threshold t ToT,L and t ToT,R are extracted from the two resultant signals produced by resistor chain when a single γ ray is detected.Some of these features are not directly used in the NN models but originate subsequent features in the model.The left and right plots highlight the feature extraction for the left L and right R sides of the resistor chain, respectively.Pulses are normalized to the maximum pulse height for illustrative purposes.
correspondent labels.The cross-entropy loss function was used, and the ridge (L2) regularization strength hyperparameter used was λ = 1 × 10 −7 .The layers were initialized using the 'He' initializer [37] for weights and with a bias of ones.No parameter optimization was performed except in the experiment with different sizes of the inner layers of the NNs.
NNs with up to three layers and different neuron numbers in each layer were tested in two different approaches: binary classification (bNN ) and multipleclass classification (mcNN ).The difference between the binary and the multiple-class approaches, depicted in figure 3, relies on the training procedure and the network output: • mcNN , a single NN is trained to classify the events into one out of 16 possible original positions.
• bNN , 16 NN are trained, each one specialized in identifying events of every single position, being the prediction 1 or 0 depending on whether the event is classified as belonging or not to that position.Then, for each event, the final labeling is given by the NN with the highest belonging classification score (probability).
The size of the mcNN was optimized selected after testing a different number of hidden layers and number of neurons.The optimum mcNN size found (among the tested ones) was also used for bNN experiments, adapting the output layer for the binary result.

Training and testing the NN
After extracting the features from each event and organizing the dataset, the training/test and evaluation datasets were divided into 80%/20% partitions, respectively.This represents 64000 events (4000 per position) for the training/test dataset and 16000 (1000 per position) for the evaluation dataset.The data was standardized before training, using Matlab's function normalize with the zscore option.
The partitions were obtained using random stratification, which means that each class will be equally represented in both the training and test datasets.This ensures that the mcNN partitions are data-balanced, given the fact that each one of the possible outcomes (classes) has the same number of events per position (4000 for training and 1000 for testing).
However, the same approach doesn't ensure wellbalanced data for the case of the individual 16 bNN.
For this case, when each bNN will indicate whether an event belongs or does not belong to a given position (one position out of the 16 possible ones), the initial dataset to be considered for each NN is highly unbalanced: only 1/16 of the events will be true (1), all the others will be false (0).
To ensure data balancing in this dataset, for each bNN, the events classified as true were oversampled, so new events were synthesised as a copy of the less representative class [38][39][40].

Accuracy
The classification success rate, denoted by accuracythe percentage of correctly classified events-can be calculated using equation (2.8).
Wrong Events Total Events 2.8 3. Results

Anger logic classification
Histograms of R and the energy spectrum from 80000 events, shown in figure 5, are drawn using the recorded pulse height and the outputs from equations (2.3) and (2.4).
The visible 16 peaks in the histogram of R (figure 5(a)) are the resistive charge division possible values in the resistor array.In the energy spectrum, the peak of total absorption of the 511 keV γ rays from the source is also visible.This is the total spectrum, which means that it results from the accumulation of the signals acquired at different positions of the resistor array.

NN classification
Initial attempts were performed changing the size and shape of the hidden layers of the mcNN.Results are shown in figure 6.Among the tested configurations, the longest mcNN took less than 3 hours to train all the folds, while the time needed for the events classification on the test and evaluation datasets is almost negligible.
The fastest mcNNs to train were the ones with only 1 hidden layer, the number of considered neurons is negligible for the training time.On the other hand, the mcNN that took more time to train was the one with 3 hidden layers, followed by the mcNN with 2 hidden layers of (64,64) neurons, as would be expected.While the (10,32) model showed less stability in the evaluation dataset (higher accuracy fluctuation), it also revealed the highest average accuracy (above 85.5%) and the maximum accuracy result among all the models, with a performance closer to 86%.For these reasons, the mcNN with hidden layers shape (10,32) was the selected configuration for the mcNN and bNN experiments in the next sections.The confusion matrix for the best-case scenario of each NN is shown in figure 7. Compared with the mcNN, the bNN training time is the same per neural network, but since 16 bNN are needed for the experiment, the total training time is multiplied by 16. Figure 8 shows the bivariate histogram E vs R for the testing data and misclassified events for each classification algorithm, giving visual information about event classification quality.
The separation of data into 16 groups in the R coordinate is evident.Although it is more noticeable for higher energies than for low-energy events, it shows that the classification by Anger logic is mostly energy-dependent and error-prone at low energy.For those events, electronic noise and non-linear effects have a higher impact on the pulse height measurements, affecting the Anger algorithm performance.
A visual comparison of the misclassified events for the three methods is also shown, making it noticeable that all the methods are giving wrong classifications at all the energy ranges.The last layer has a softmax activation function, so the output of the mcNN is the label corresponding to the position with the highest probability.(b) Representation of 2 (out of 16) binary NN trained to identify events in each position.Each event is assigned a label of '1' or '0' by the bNN, depending on the probability of that event occurring in that bNN.The position of each event is determined by identifying the bNN that outputs '1' with the highest probability.
NN methods were also able to correctly classify more events even when E vs R is not linear for a given position, especially for high energies when one of the signals is deformed by the circuit saturation limit, as can be visually observed in figure 8. Anger logic misclassifications are also more spread along the flood map, while the NNs produce wrong classifications mostly for fixed values of R. A summary of the results is shown in table 1.

Discussion
Despite the drawback of requiring additional feature extraction from the signal waveform, increased data storage, and the necessity of a pre-trained model, promising results are attained with NN for the given dataset when real-time classification is required.Both the mcNN and bNN exhibit an accuracy exceeding 85%, as depicted in figure 6.The confusion matrix obtained for the best case (highest accuracy) trained mcNN and bNN using the evaluation data is shown in figure 7. The bNN showed similar performance, compared to the mcNN, but with 16×more time to training.Although is not obvious that the size optimization obtained for the mcNN and then used for the bNN is optimal for the bNN, it was decided to use the same size to keep consistency during the experiment and minimize the number of parameters to be optimized.Both NN perform better than the Anger logic method.
The dataset was acquired using the LYSO collimator moving along the LYSO/SiPM array.Although the experimental setup was designed to reduce the number of coincidence events in LYSO pixels, different than the one collimated at each position (denoted by neighbours), it is not possible to disentangle these events from the ones that occurred in the LYSO pixel in front of the collimator, without filter them by the ratio R.However, doing so would bias the results since the method would evaluate the positioning accuracy of the NNs and compare them against Anger logic in a dataset already filtered using the Anger logic.This means that a percentage of misclassified events will be introduced in the study when considering the LYSO collimator position as the labeling feature.On top of that, with the current experimental setup, it is not possible to disentangle scattered events in the LYSO array, which originated signals in the neighbour pixels from the ones that, due to the sourcecollimator solid-angle aperture, interacted directly in the neighbour pixels and not in the collimated one.In both cases, this justifies why neither one of the classification methods is able to surpass classification accuracy's above 85%, as depicted in figure 6.
The confusion matrices also show that overall higher accuracy is obtained for the two extreme positions (1 and 16) of the array, which can be explained by the reduced number of neighbours for the LYSO pixels in the extreme positions-only 1-and therefore the number of photons scattered coincidences is reduced.The differences in accuracy for the inner pixels can also be justified by the mechanical alignment of the experiment at each position-although a precision micrometer screw (0.1 mm resolution) was used to move the source and the collimator LYSO, the LYSO array manufacturer only guarantees the pixel positions   with 0.1 mm resolution, which on top of the distance between the collimator and the LYSO array and the difficulties to ensure a perfect alignment between the collimator, source and LYSO array, can justify slightly different coincidence acceptance angles for each position on the LYSO array and the accuracy differences.Under ordinary acquisition conditions, e.g. when the detectors are irradiated at a higher radiation rate, with higher counts of false and scattered coincidences, higher electronic noise reducing the signal-to-noise ratio and the information quality from digital converter, the overall accuracy of classification methods is expected to decrease.
Contrary to what would be expected, the incorrect classifications occur at all the energy ranges and not mostly at lower energy, as one might expect due to the lower signals.However, this is likely related to the initial misclassification of some events in the data acquisition phase-these misclassified events are not energy-dependent.Therefore, they will show all over the energy range.Despite that, in PET instrumentation, the use of low energy events is avoided, since they are due to scattered photons and ultimately degrade the image quality.
The performance of the bNNs indicates that given their simplicity, the increased number of NNs increases the training time without an evaluation accuracy improvement.This can also result from the original unbalanced dataset: the bNNs probably lack sufficient information to better distinguish one class from all the others in the current format.

Conclusions
Machine learning algorithms, such as neural networks, have been used for event classification PET detectors, showing great results in the position identification of the γ rays interaction in radiation detectors.
In this work, NN were used to identify the positions of interaction of 511 keV γ rays within a LYSO crystal array, coupled one-to-one to a multiplexed (16:2) linear resistor array of SiPMs readout in both ends by charge preamplifiers.The electric pulse resultant from the γ rays interactions in the scintillators, recorded using the oscilloscope, was used to determine the position of the LYSO in the SiPM array using features extracted from the pulse waveform.
Traditional Anger logic classification (using the flood map histograms) was compared with artificial neural network classification, using two approaches: multi-class (mcNN) and binary-class (bNN) neural networks.While the mcNN consists of a single NN trained to identify in which position the event occurred, among the 16 possible locations, the bNN approach uses 16 NN with a binary output, each one trained to identity if a given event occurred in its position or not, being then chosen the position with higher belonging probability among the 16 bNN.
For the considered dataset, results indicate that over 84.5% of events are well resolved using either the Anger logic or the NN approaches.Both NN best-case configurations showed a performance with accuracy closer to 86%.The approach with the bNN, although simpler than the mcNN, required a higher number of NN to be trained (1 vs 16), therefore more time to train, which is also a current drawback of this approach.
Future optimization of the NNs structure (shape and layers size) and its implementation in the hardware front-end (FPGAs) will be tested, aiming a lowlatency classification with high accuracy directly at the electronics.Both approaches will also be tested using more noisy signals, commonly found detectors with a higher number of channels.Results will be tested in real preclinical PET scanners that use a readout multiplexed configuration, comparing PET image quality using events classified using Anger logic and the developed NN approaches.

Figure 1 .
Figure 1.(a) Schematics of the data acquisition process for an array of 16 SiPM in a linear arrangement.The SiPM anodes are connected to different points of the resistor chain, and the charge is collected according to the principle of resistive charge division at both ends of the resistive chain.The amplitude ratio shift can be evaluated as a function of the radiation interaction position.This effect is exemplified for 3 different crystal positions, showing the signals measured on the left and right amplifiers, V L and V R , respectively.(b) Schematics of the experimental setup used to measure the response for each collimation position (not to scale).The LYSO array is coupled to the SiPM array (1-to-one coupling, no light sharing).Another LYSO crystal is used to perform electronic collimation and filter events by position.The SiPM signal is preamplified and shaped by a front-end board, whose output is sent to the oscilloscope for the waveform digital conversion.The process is repeated for each LYSO pixel, moving the source and collimator jointly.The distances between the collimator and source, D CS , and between the source and LYSO array, D SA , were defined to minimize the source penumbra in neighboring array pixels.

Figure 2 .
Figure2.Pulses waveform from the resistor chain.The features pulse heights, V L and V R , areas, A L and A R , rise times, t r,L and t r,R , fall times,t f,L and t f,R , signal time trigger, t t,L and t t,R , time of signal maximum, t m,L and t m,R , and time-over-threshold t ToT,L and t ToT,R are extracted from the two resultant signals produced by resistor chain when a single γ ray is detected.Some of these features are not directly used in the NN models but originate subsequent features in the model.The left and right plots highlight the feature extraction for the left L and right R sides of the resistor chain, respectively.Pulses are normalized to the maximum pulse height for illustrative purposes.

Figure 3 .
Figure 3. (a) Schematic of the multiclassification NN.The last layer has a softmax activation function, so the output of the mcNN is the label corresponding to the position with the highest probability.(b) Representation of 2 (out of 16) binary NN trained to identify events in each position.Each event is assigned a label of '1' or '0' by the bNN, depending on the probability of that event occurring in that bNN.The position of each event is determined by identifying the bNN that outputs '1' with the highest probability.

Figure 4 .
Figure 4. Procedure used to train and evaluate the NN approaches (Multiclass NN and Binaries NN) and compare with Anger logic (previous methods).Implemented using fitcnet framework from Matlab.

Figure 5 .
Figure 5. (a) Acquired histogram of R and correspondent (b) E spectrum.R and E Calculated using signals pulse height.(c) is the correspondent bivariate histogram.Visual organization of data into 16 distinguishable peaks in (a) and (c) indicates that the majority of events can be correctly associated with their original position in the resistor array only by measuring R, although there is still some overlap between adjacent positions to a certain extent.

Figure 6 .
Figure 6.Box plots of the training duration time (a) and the measured accuracy in the evaluation dataset (b) of the different mcNN models tested after 10-folds repetitions.The maximum accuracy and higher average accuracy were obtained by the mcNN with hidden layers shape of (10,32) neurons.All the mcNN models show higher classification accuracy compared to the Anger logic method.

Figure 7 .Figure 8 .
Figure 7. Confusion matrices examples of the mcNN (a) and bNN (b) application, both with a similar accuracy of 85.9%.Most of the events are correctly classified to their true classification position.Results are shown in the percentage of correctly classified events.

Table 1 .
Summary of the results comparison between the Anger logic algorithm and the two NN approaches explored for the best and worst case scenario using k-fold with 10-folds.