Timing, Energy, and Spatial Characterization of Highly Sampled Monolithic PET Detectors with Different Thicknesses

Abstract

The HyPET project proposes a hybrid dedicated TOF-PET for prostate imaging, with pixelated detector blocks in the front layer and monolithic blocks in the back layer. In this work, four detector configurations have been experimentally evaluated for the rear detector layer. The detector configuration consists of LYSO monolithic blocks with the same size (25.4 mm × 25.4 mm) but different thicknesses (5, 7.5, 10, and 15 mm) coupled to the same SiPM array.

Each detector configuration has been experimentally characterized in terms of time, energy and spatial resolution by scanning the crystal surface using a fan beam in steps of 0.25 mm. Regarding spatial resolution, the interaction position was estimated using a Neural Network technique.

All resolutions except energy, which remains nearly constant at 17% for all cases, show better values for the 5 mm detector thickness. We have achieved spatial resolution values of FWHM of 1.02 ± 0.10, 1.19 ± 0.13, 1.53 ± 0.17, 2.33 ± 0.55 mm, for the 5, 7.5, 10, and 15 mm blocks, respectively. The detector time resolution obtained was 275 ± 26, 291 ± 21, 344 ± 48, and 433 ± 45 ps respectively, using the energy weighted average method for the time stamps.

I. INTRODUCTION

Positron Emission Tomography (PET) is a medical imaging technique widely used in the field of nuclear medicine to detect, identify, and assess oncological diseases, among other pathologies. Nowadays, PET detectors typically consist of scintillation crystals optically coupled to an array of silicon photomultipliers (SiPMs). The two gamma rays originated from the positron-electron pair annihilation are stopped in the scintillator. These two gamma rays, also named annihilation photons, generate optical photons that are detected by the SiPMs. A line of response (LOR) is generated by connecting the two impact points detected in coincidence on opposite detector blocks. To accurately decode the 3D gamma ray interaction positioning and correct for the parallax error associated with photons entering the detectors at oblique angles, detectors with good spatial resolution and depth of interaction (DOI) capabilities are necessary [1], [2], [3]. Moreover, including time of flight (TOF) information about the different arrival times of the two gamma rays generates a probability distribution along the LOR, improving the signal to noise ratio (SNR) of the reconstructed image [4].

The most common PET detector configuration in commercially available scanners consists of an array of pixelated scintillation crystals coupled one-to-one or one-to-many to the photosensor array [2],[4]. Accurate timing performance is obtained when a single SiPM photosensor collects all or most of the scintillation light from a single event, as the rise time of the signal decreases. This enhances the signal-to-noise ratio (SNR) and results in a more precise determination of the timestamp. However, as this detector configuration just provides information from one sensor, it hinders decoding the DOI position [5],[6].

An alternative to pixelated scintillators is the monolithic crystal approach. Because of the depth dependent light distribution associated with monolithic blocks, their light distribution inherently contains 3D positioning information [5],[6],[7],[8]. This inherent positioning capability is one of the major advantages of monolithic detectors over pixelated detectors, although sophisticated positioning algorithms are usually required to reduce the so-called edge effect or compression effect [1],[9]. Moreover, since light spreads along the whole crystal volume, the amount of scintillation light detected at each SiPM decreases, reducing the SNR and, therefore, challenging the timing performance [10].

In this work, we evaluate the time, energy, and spatial resolution performance of monolithic LYSO crystal blocks of 25.4 mm × 25.4 mm with different thicknesses ranging from 5 to 15 mm and wrapped with enhanced specular reflector (ESR). These detectors are being evaluated in the context of a hybrid configuration detector concept, originated from the HyPET project [11]. Therein, it is proposed a first detector layer based on an array of pixelated scintillation crystals optimized for coincidence time resolution (CTR), and the second layer based on a monolithic crystal for 3D spatial resolution and high effective detection efficiency. Our HyPET detector is designed to provide accurate performance for human prostate imaging. In the following, we present the results obtained with four different monolithic crystals read out by analog SiPMs and state-of-the-art commercial readout electronics.

II. Materials and Methods

A. Materials

Four different detector configurations were evaluated under the same conditions in terms of time, energy, and spatial resolution. For the four cases under study, all the detector components were the same except for the scintillation block. Four LYSO monolithic crystals (EPIC Crystals, China) with the same size (25.4 mm × 25.4 mm) but different thicknesses (5, 7.5, 10, 15 mm) were studied, see Fig. 1. All crystal surfaces were polished and wrapped with ESR, except one of the 25.4 mm × 25.4 mm side, which was coupled to the photosensor array using optical grease (Bluesil Paste 7, Silicon Solutions, USA).

Fig. 1.

LYSO blocks evaluated with 25.4×25.4 mm² base. From left to right: 15, 10, 7.5, and 5 mm thickness.

The scintillation light was collected using a SiPM photosensor array of 8×8 channels with 3×3 mm² active area each (model S13361–3050AE-08 from Hamamatsu Photonics, Japan). As reference detector, we placed a single LYSO pixel of 3 mm × 3 mm × 5 mm, ESR wrapped, attached to one channel of an identical SiPM array via one-to-one coupling (see Fig. 2 (a)). The readout system used to decode the SiPM signals into energy and timing information was the TOFPET2 BGA ASIC-based system from PETsys Electronics (Portugal) [12].

Fig. 2.

(a) Sketch of the coincidence measurement. (b) Upper view of a sketch of the experimental set-up using the linear actuator and the beam collimator in the collimated set measurement. (c) Sketch of the position scan performed in the collimated set measurement.

In this study, we have acquired three types of data sets for each crystal thickness (see Table I). First, several uniform radiation flood maps were obtained to optimize the readout parameters in terms of spatial, energy, and timing performance, named parameters set. Then, to apply different filters and processing techniques, a second uniform flood irradiation with higher statistics was acquired, so-called uniform. Thereafter, to evaluate the detector performance the acquisition with the fan beam was carried out, named collimated set. In this measurement, a high-resolution mechanical position set-up was implemented using a linear actuator connected to a stepper motor and controlled by an Arduino module (see Fig. 2 (b)). This actuator allows one to scan a collimated fan beam along one dimension of the crystal surface in steps of 0.25 mm (see Fig. 2 (c)). The collimator is made of tungsten blocks generating a slit of 420 μm width and 56 mm height. A ²²Na source of 0.25 mm in diameter and 770 kBq activity was used for all experimental data collection. The temperature was maintained in 25 ± 2 °C using a vortex and compressed air inside a light-tight box.

TABLE I.

EXPERIMENTS PERFORMED, NUMBER OF MEASUREMENTS AND ACQUISITION TIME.

Data set name	# Measurements per crystal size	Acquisition time per measurement (s)
Parameters measurements	52(x3)	30
Uniform measurements	1	1000
Collimated measurements	100	2000
Calibration measurement	1 for all sizes	1200

Note that before the previous measurements, the studied detector was coupled to a crystal array of 8×8 elements of 3 mm × 3 mm × 5 mm each (one-to-one coupling) in order to apply skew and energy corrections for the studied sensor array. In this measurement, named calibration measurement, the source was placed close to the reference pixel at a distance of 84 mm from the array of crystals to ensure homogeneous irradiation. This measurement lasted 20 minutes yielding 2000 coincidence events per channel.

B. Methods

a. Reference characterization

As the two coincidence detectors are not identical, the CTR obtained is the combination of the detector time resolution (DTR) of the reference detector $\left(DT{R}_{Ref}\right)$ and the DTR of the monolithic detector $\left(DT{R}_{Monolothic}\right)$ following the formula:

$$CTR=\sqrt{DT{R}_{Ref}^2+DT{R}_{Monolithic}^2}$$ (1)

To characterize the timing contribution of the reference detector, two identical single crystal pixels were measured in coincidence. With the radioactive source placed at the middle distance, the $CT{R}_{Ref}$ was obtained, and thus the $DT{R}_{Ref}$ deduced to be 142 ± 6 ps FWHM.

All studies have been carried out keeping the acquisition parameters constant for the reference pixel, regardless of the optimization carried out for the acquisition parameters in the detector under characterization.

b. Timing calibration

When more than one SiPM channel is considered to provide the timing information of an impinging annihilation photon, both time skew and time walk corrections are needed [13]. Concerning time skew, the calibration measurement was used and the CTR centroids of the 64 coincidence pairs between the reference and the studied detector (pixelated array) were used to measure and correct for the time skew [14].

The time walk of each detector configuration has also been corrected using the so-called uniform measurement, from which the relationship between the time stamp assigned to each event and the energy associated with it is compensated [15].

c. Energy calibration and resolution analysis

Using the calibration measurement acquired with the 3 mm × 3 mm × 5 mm crystal array, the energy of each event (E) was calculated as the sum of the detected signals. The energy photopeak position of all 64 channels of the SiPM array was obtained and a Look-Up-Table (LUT) was generated to normalize the SiPM gain, named as LUT1. To calibrate the energy from arbitrary ADC units into keV, we have fitted a polynomial using the 511 keV and 1274 keV peaks of the ²²Na spectrum for each SiPM and assuming the channel zero corresponds to 0 keV. We noticed saturation effects since the second photopeak of the ²²Na was observed at an ADC channel that was approximately 2 times higher than the first photopeak instead of about 2.5. Due to this, we used a second order polynomial for the fitting [16].

The energy resolution will be measured by the FWHM of the Gaussian fitted curve to the energy spectrum around the photopeak.

d. Acquisition parameters

The methodology followed to evaluate the DTR performance of the four configurations was to first acquire scans to optimize the acquisition parameters of the PETsys data acquisition electronics using the parameters set measurement [15],[17]. The V_bias is defined as the sum of the breakdown voltage and overvoltage (OV). When using PETsys electronics, it is required to define the voltage threshold parameters related to the timing, the integration window, and the energy, namely vth_t1, vth_t2, and vth_te, respectively. A breakdown voltage of 53 V was predefined based on the datasheet of the photodetector and, then, a OV scan was carried out before the thresholds were studied. Values from 1 to 5 V in steps of 0.5 V were tested while constant thresholds values of (vth_t1, vth_t2, vth_te) = (14, 14, 12) were selected [10],[17]. Once the OV value that optimizes the DTR has been determined, the thresholds scans were performed. vth_te was kept stable at 12, as it has no impact on the time resolution, and vth_t1 and vth_t2 were scanned. For vth_t1 values from 4 to 10 in steps of 1 and vth_t2 from 6 to 16 in steps of 2. The obtained DTR was corrected only for time skew.

To evaluate the detector performance with these parameters, the DTR, the energy resolution (in ADC units) and the flood map compression was calculated for each studied case. The flood map compression will be evaluated as the difference between the maximum and minimum x-coordinate obtained using the center of gravity (CoG) method [18]. Notice that three different measurements were taken at each acquisition value. Each measurement was independently processed and the mean value of the DTR, energy resolution and flood map compression is obtained. The error is obtained as the standard deviation of the centroid of each of the three acquired datasets that comprise the parameters set.

e. Filters and processing algorithms

The uniform measurement was used to determine the best filters for timing calculation. After the application of the time skew correction, the photopeak position of each SiPM channel was normalized to equalize possible detector gain drifts. Herein, we applied the energy correction factor previously obtained and stored for LUT1. The time walk correction was also applied. Then, a data filter was implemented to avoid multiple random hits of a given event. Thus, for each gamma impact, if the first and third arriving time stamps in the studied detector differ more than 2 ns, this event was discarded [10].

Several filters have been applied to the collimated measurement. The x- and y- annihilation photons coordinates were pre-estimated using the Raise to the Power (RTP) positioning algorithm [19]. For each beam position, an energy filter of 30% (358 keV – 664 keV) around the energy photopeak position was applied. In addition, a contour filter removing the lower 10% of the data from the RTP distribution profile of the transversal beam axis was also applied to avoid scattered events.

f. Timing resolution analysis

After sorting the time stamps by the arrival times, the energy weighted average method ($\mathrm{E}\mathrm{W}\mathrm{A}\mathrm{M}$) was used to determine the time stamp of the event [15][10].

$$EWAM=\frac{\sum _{i=1}^n{Energy}_i\times {Timestamp}_i}{\sum _{i=1}^n{Energy}_i}$$ (2)

Where, $\mathrm{n}$ is the total number of hits (SiPM channel signals) in the event, $Energ{y}_i$ is the energy of the hit, and ${Timestamp}_i$ is the time stamp of the hit.

As we are using a monolithic based detector, light spreads in such a way that most of the SiPMs in the array are fired for a given event. To discard dark counts and improve timing resolution, just the time stamps of the 10 most energetic hits will be considered in the EWAM. Using the uniform measurement, we have evaluated the implementation of this method averaging over the 1^st to the 6^th earliest time stamps (over the 10 selected). Finally, using the collimated set, the number of time stamps studied will be reduced to provide optimum performance.

We have studied the DTR performance as a function of the DOI for each detector configuration. We have also studied the relationship between the centroid of the CTR with DOI. For this, the x- and y- annihilation photon coordinates were pre-estimated using the RTP method, and the DOI or z-coordinate was obtained by the ratio between the total energy (E) and the maximum intensity (I) of the light distribution profile using row and columns signals (E/I estimator) [5],[20]. For each crystal geometry, we have selected 3 regions of interest (ROI) in the flood maps (see Fig. 3 (a)), corresponding to the central, lateral, and corner areas with approximately 13% of the total flood map surface. Each DOI distribution has also been divided into 3 equal segments considering the attenuation coefficient of the LYSO for 511 keV, namely low (near the photosensors), medium, and high (entrance of the crystal), see Fig. 3 (b).

Fig. 3.

Examples of (a) ROIs selected to study the DTR performance as a function of the DOI for the 10 mm block and (b) DOI (E/I) distribution for the 10 mm block in the center area and the segmentation applied.

g. Spatial resolution analysis

In this work, the x- annihilation photon interaction position was calculated using a supervised Neural Network (NN) technique based on a multilayer perceptron (MLP) with 2 layers and 64 nodes each. Each MLP contains 16 inputs, corresponding with the summed signal of the 8 rows and 8 columns of the SiPM array. Since the NN inputs are the summed energy signals of the 8 rows and 8 columns of the sensor array, a pre-processing stage was needed to reduce the 64 signals into 16. The MLP was trained using an Adagrad optimizer, rectified linear activation function (RELU) and the root mean squared error (RMSE) loss function [9]. The filtered collimated measurement data was divided into 3 datasets named: train (50%), evaluation (45%), and test (5%). The MLP was trained with the train dataset and after training, the MLPs were evaluated using the evaluation dataset to avoid overfitting. Three parameters were calculated to evaluate the spatial resolution performance of the detector block using our described NN. The parameters calculated were bias, mean absolute error (MAE) and full width at half of the maximum (FWHM).

1. The bias is obtained as the difference of the estimated and real position:

   $$Bias=\frac{1}{n}\times {\sum }_{i=1}^{i=n}\left(x_{predicted}\left(i\right)-x_{real}\left(i\right)\right)$$ (3)

   where, $\mathrm{n}$ is the total number of row and columns, $x_{predicted}$ is the estimated x- position and $x_{real}$ is the mechanical x- position of the slit.
2. The MAE is calculated as the probability-weighted mean of the absolute value of the positioning error:

   $$MAE=\frac{1}{n}\times \sum _{i=1}^{i=n}\left|x_{predicted}\left(i\right)-x_{real}\left(i\right)\right|$$ (4).

   where, $\mathrm{n}$ is the total number of row and columns, $x_{predicted}$ is the estimated x- position and $x_{real}$ is the mechanical x- position of the slit.

3. The spatial resolution calculated as the FWHM of the Gaussian fit of the $x_{predicted}$ accumulative distribution.

III. Results

A. Time skew correction

Fig. 4 shows the 64 relative values of the CTR centroids obtained between timestamps from each SiPM channel and the reference channel. This information is used to correct the time skew for each SiPM signal in the acquired data with the monolithic blocks. Two main different timing areas are easily observed, as already reported elsewhere [15]. The maximum relative timing difference between SiPMs of the same array is about 1.3 ns. In the following, this information is considered when we plot the time difference between two coincidence events.

Fig. 4.

Map of the relative centroids (ps) of the time skew offset per channel in the SiPM sensor.

B. Energy calibration

Fig. 5 shows the gain differences observed in the SiPM array of the studied detector over the mean value measured in arbitrary ADC units. Deviations up to 15% were observed, and no trend has been identified.

Fig. 5.

Map of the values of the energy gain (ADC units) per channel in the SiPM sensor.

As an example, Fig. 6 shows the energy spectrum of the 15 mm block before normalization and calibration, after normalization, and after calibration into keV. The energy resolution measured was 14.2 ± 0.3 % in arbitrary ADC units, after normalization it improves to 13.5 ± 0.3% and finally after the gain calibration into keV it increases up to 17.8 ± 0.4% as the energy spectrum was compressed due to the SiPM saturation. The same procedure was applied to all crystal blocks.

Fig. 6.

(a) Energy spectrum of the collimated measurement considering all beam positions for the 15 mm block. (a) Energy resolution before gain normalization, (b) after gain normalization (c) after energy calibration.

C. Acquisition parameters

Fig. 7 shows the DTR, energy resolution, and flood map compression values for each of the OV value tested and crystal thickness. All crystal geometries follow the same trend for the three indicators studied. The energy resolution (before calibration) improves as we increase the OV value. However, the uniform irradiation flood map compression worsens as the OV increases. The DTR deteriorates as we increase the crystal thickness. The scans showed that the optimum OV setting was 4 V for all detector blocks, which corresponds to a bias voltage of 57 V.

Fig. 7.

Performance of all crystal thickness for each OV value scanned. (a) Detector time resolution; (b) Energy resolution before energy calibration; (c) Flood map compression.

Fig. 8 shows the DTR values for each of the OV value tested and crystal thickness. It can be appreciated that very low vth_t1 values are required to achieve the best DTR performance. We observe that DTR got worse when raising the threshold value.

Fig. 8.

DTR values obtained per each threshold value and per each crystal thickness.

D. Filters and processing algorithms

Finally, time walk correction, time stamp averaging, and a gain equalization procedure were applied. Table II shows: first line, uniform measurement when just time skew is applied; second, uniform measurement applying all corrections and filters; and third collimated measurement applying all corrections and filters. After optimizing the acquisition parameters and calibration settings, an improvement of more than 150 ps in the DTR was observed in all cases when going from raw to optimized configurations.

TABLE II.

DTR OBTAINED PER EXPERIMENT PERFORMED.

	DTR (ps, mean values)
Data type	5mm	7.5mm	10mm	15mm
Uniform measurement just time skew applied	381±3	518±3	610±4	716±4
Uniform measurement with all corrections and filters applied	268±2	307±1	343±2	439±3
Collimated measurement with all corrections and filters applied	275±26	291±21	344±48	433±45

Fig. 9 shows the evaluation of the DTR using different filters and number of time stamps for the uniform measurement. In these graphs, we can observe that the 2 ns time window introduces an improvement of approximately 100 ps in all crystal geometries. We can also observe that using more than 4 time stamps worsens the timing resolution for all crystal geometries [10].

Fig. 9.

DTR values applying EWAM timing estimation with different number of time stamps for each timing correction and for each crystal thickness.

E. Timing resolution

Fig. 10 shows the results obtained as a function of the number of time stamps considered in the EWAM calculation and for each slit position of the collimated measurement across the surface of each crystal. To reduce the density of the graphs, we have represented the mean value of each group of 4 positions. The DTR has been studied by applying the EWAM from 1 to 4 time stamps. As the previous steps demonstrated, when using more than 4 time stamps the DTR worsens (see Fig. 9). We observe that the DTR at the edges of the crystal is always better than the DTR at the center, which becomes more pronounced with increasing the crystal thickness. Optimum performance was achieved using 3 time stamps for all cases except for the 15 mm thick crystal, where using only 2 time stamps provided the best timing performance. The optimum DTR mean value for each detector configuration is shown in Table II, third row. The timing performance using the optimum number of timestamps in the uniform and the collimated datasets show approximately the same mean results for all crystal surfaces, demonstrating consistency in the results.

Fig. 10.

Mean DTR obtained per each group of 4 slit positions and number of time stamps considered in the EWAM for all crystal thickness. The mean value over all slit positions per each number of time stamps is shown next to the legend.

DTR values and DTR centroids obtained for each crystal ROI and DOI regions (see Fig. 3) for each crystal thickness are plotted in Fig. 11 (a) and (b), respectively. We observed the same behavior in all crystal blocks and all areas regarding DTR.

Fig. 11.

Results obtained selecting 3 different ROIs (central, lateral and corner) and 3 different DOI regions (low, medium, and high) per each ROI for all crystal thicknesses. (a) DTR (FWHM, ps) (b) CTR centroid (ps).

As expected, better timing was found for low DOIs (closer to the photosensor). Also, we can see better timing performance at the corners. On the other hand, focusing on the CTR centroid, we observed that there is a relationship between the CTR centroid and the DOI (see Fig. 11 (b)). We observe that all central areas show lower centroids values for lower DOIs; however, this performance is not observed at the corner areas.

F. Energy resolution

Concerning the energy resolution, it was calculated at the same ROI and applying the same filters described for the timing resolution study. As an example, Fig. 6 shows the accumulated spectrum of all beam positions obtained before (a) and after (b) gain normalization and energy calibration in keV (c) for the 15 mm crystal block.

Fig. 12 (a) shows the local energy resolution for each crystal thickness and beam position. Notice that we have represented the mean value of each group of 4 positions. We can observe that the results obtained are very similar, with an average energy resolution in the range of 17% for all crystal geometries.

Fig. 12.

Comparison of the optimum resolution performance per each detector configuration and position sampling (representing the mean value of each four positions). (a) Energy resolution (FWHM, %), (b) detector time resolution (FWHM, ps), and (c) spatial resolution (FWHM, mm).

G. Spatial resolution

Fig. 13 shows the bias, MAE and FWHM for each slit position of the filtered collimated measurement and for each crystal thickness. We can see that the spatial resolution degrades with crystal thickness. Furthermore, the spatial resolution worsens near the edges of the crystal and degrades more as the thickness increase.

Fig. 13.

Spatial parameters (bias, MAE, and FWHM) for each beam position and for each crystal thickness. The mean value over all beam positions is shown next to the legend.

An average spatial resolution FWHM of 1.02 ± 0.10, 1.19 ± 0.13, 1.53 ± 0.17, 2.33 ± 0.55 mm was found for the 5, 7.5, 10, and 15 mm thick crystals, respectively. The mean values and standard deviation of the studied parameters considering all the beam positions are shown next to the legends of Fig. 13.

IV. Discussion

We have evaluated a monolithic crystal PET detector design studying four different crystal thicknesses. Each of the detectors are suitable for high resolution PET systems and offer DTR to support TOF PET. The study has been carried out in terms of energy, timing, and spatial resolution. The same treatment configuration based on polished walls and ESR film was applied to the four blocks, with size of 25.4 mm × 25.4 mm and thicknesses of 5, 7.5, 10 and 15 mm, see Fig. 1. Each detector configuration has been optimized to obtain best performance. The procedure was to first determine the acquisition parameters that provided the best DTR and energy resolution while minimizing compression (i.e., bias) observed in the flood maps, see Fig. 7. Then using the uniform measurement, the optimum filters and processing algorithms were determined. Finally, the detector performance was evaluated by scanning the entrance crystal surface using a slit collimator of approximately 420 μm in width. A comparative study with the energy, time and spatial resolution as a function of the thickness of the crystal block has been presented, Fig. 12.

The time resolution was studied by measuring the DTR using the EWAM approach, see Table II. The results obtained demonstrate that the thinner the crystal block, the better the time resolution. Similar values were obtained for the 5 and 7.5 mm blocks, but a significant worsening for the 15 mm case with respect to the other crystals was observed. Only the results for the 5 mm case are comparable with state-of-the-art PET detectors based on pixelated crystals. Performance worsens for thicker crystals due to the direct light flux being focused on more SiPM sensors leading to lower signal-to-noise ratio. The improvement of the DTR towards the edges of the crystals is indeed related to the increase of signal and a fewer number of SiPMs. Novel timing determination methods based on machine learning algorithms are envisaged in the future [21],[22].

As one of the main advantages of detectors based on monolithic scintillators is the intrinsic DOI information, we have also studied the time resolution as a function of DOI [6]. The results show that the lower the DOI (i.e., closer to the photosensor plane), the better the CTR. This might be due to the higher amount of light collected in that area due to optical effects and also to the reduced number of SiPMs fired, improving the SNR, similar to the case of the improvement at the edges[10],[23]. This demonstrates the possibility of correcting the timing according to the DOI to obtain better timing performance.

Regarding the energy resolution, all the crystal blocks are made of the same material (LYSO) and purchased at the same time from the same supplier and, thus, the expected light-yield is approximately identical in all cases. Since the crystals are treated with the same reflector type (ESR) and the thresholds were set low, most of the direct and bouncing scintillation light was collected by the photosensors, almost independently of the crystal thickness. This resulted in comparable energy performance for all detectors, see Fig. 12 (a).

Concerning the spatial resolution, the x-coordinate for the different monolithic crystals was studied using a neural network positioning technique. The spatial performance study has been done as a function of the impact position within the scintillator. We calculated the bias, MAE, and FWHM at each slit position and for all crystal thicknesses. Better spatial resolution was found for thinner crystals. Similar resolution results were obtained for the 5 and 7.5 mm block, whereas results significantly worsen for the 15 mm block. The two bumps at the edges in the FWHM plots increase for higher crystal thickness (see Fig. 13). This is most likely generated because the Light Distribution (LD) shape is not preserved due to the high amount of scintillation light reflections at the crystal edges, challenging the prediction of the impact position determination in these regions. Notice that these results are obtained for the x-coordinate. For the y-coordinate, another slit measurement would have to be carried out rotating the studied crystal. However similar performance is expected to be obtained along the y-direction of the crystal.

V. Conclusions

In this paper, we have presented a study of the influence of the crystal thickness in a PET detector based on LYSO monolithic scintillators using commercially available electronics. We have studied four different thicknesses: 5, 7.5, 10, and 15 mm. All of them have been evaluated using the same methodology to obtain the optimum performance for each of the detectors in terms of energy, time, and spatial resolution. The energy, timing, and spatial resolution obtained is represented in the Table III.

TABLE III.

Mean resolution obtained per crystal thickness.

	5mm	7.5mm	10mm	15mm
DTR (ps)	275 ± 26	291 ± 21	344 ± 48	433 ± 45
Energy Resolution (%)	17.97 ± 1.15	16.56 ± 0.60	16.5 ± 0.63	17.25 ± 0.58
Spatial Resolution (mm)	1.02 ± 0.10	1.19 ± 0.13	1.53 ± 0.17	2.34 ± 0.55

The results obtained demonstrate that the best option to perform the HyPET absorber detector should be a crystal block no higher than 7.5 mm, as up to this size, all features are preserved: DTR, energy, and timing resolutions.