Studies and Rejection of Intercrystal Crosstalk on FPGA in a High-Energy Photon-Counting System

Abstract

Intercrystal scatter reduces system sensitivity and spatial resolution, a phenomenon that has been extensively studied in positron emission tomography (PET) systems. However, the issue is even more significant in high-energy systems. The purpose of this study is to propose a practical crosstalk rejection technique and demonstrate its applicability in high-energy photon-counting systems. The effect of inter-crystal scattering interactions between ⁶⁰Co γ photons and lutetium yttrium oxyorthosilicate (LYSO) scintillator crystals is investigated through Monte Carlo simulations conducted using the Geant4 toolkit. To suppress the crosstalk phenomenon, a field-programmable gate array (FPGA)-based algorithm is proposed to suppress inter-crystal scattering events, characterized by a time window of 5 nanoseconds and detector window sizes of one or two. The 250 mm Fe steel penetration model is used to evaluate the proposed algorithm, showing improved radiation image quality, particularly with a detector window size of two, which performs better under low-count-rate conditions. Laboratory testing indicates that the proposed algorithm can enhance steel penetration (SP) by 60–70 mm of Fe when compared to the existing current integration system under the same settings. The suggested method has been proven effective in producing higher-quality images and demonstrates good adaptability by adapting the detector window width according to different system count rates.

1. Introduction

Advances in photon-counting detectors have made it possible to utilize the time, position, and energy information of the detected photon signal, which is useful to distinguish intercrystal scatter due to Compton scattering [1,2,3]. Intercrystal scatter has a bad effect on system sensitivity and can lead to a degradation of spatial resolution [4]. Unfortunately, system sensitivity and spatial resolution are two key factors in radiation detection systems, especially for positron emission computed tomography (PET) systems that require accurate positioning. It has been suggested that intercrystal scattering is the primary cause of event mispositioning in high-resolution PET systems [5]. Moreover, the tendency to utilize smaller crystal sizes leads to a higher fraction of intercrystal scatter [6].

Many experiments have been performed to look at the impact of intercrystal crosstalk on positioning error and the implications for the spatial resolution of PET detectors [7,8,9,10,11]. Simulations also suggest that these effects will ultimately lead to image blurring and a loss in contrast at the system level [12]. The conventional method used to reduce the effect of intercrystal scattering is setting a proper energy threshold mostly on the photopeak. However, this method is somewhat limited when the energy resolution is poor, as is the case for the high-energy system presented in this study, and inevitably, object scatter will be partly lost. Other attempts appeared to improve the structure of the detector. Multi-layer detectors with depth-of-interaction (DOI) detection capability can extract the position information of photons. Intercrystal scattering events can be discriminated by reading out each layer independently and utilizing the pulse information obtained [13,14]. Also, utilizing a thin intercrystal septa has been proven to be effective [6]. But it will greatly increase the cost and complexity in large-scale systems with hundreds of detectors, and the septa, like lead strips, may introduce additional interference, especially in high-energy systems. Many researchers have proposed different filters concerning the light distribution of intercrystal crosstalk, which have the capability to improve image quality significantly [15,16,17]. However, some complicated filters, for instance, the one based on maximum likelihood algorithms [18], require high computational effort and are therefore very challenging to implement directly in the field-programmable gate array (FPGA) data processing module. Moreover, with an object to be detected in our system, the energies of photons will spread after penetrating the object, while PET systems detect coincidence events of 511 keV photons, which is a further difference [19].

As can be seen above, although many publications dealing with intercrystal scatter interactions in PET detectors already exist, there are few relevant studies in other radiation detection systems. Here, we investigate the intercrystal scatter phenomenon in a high-energy large container inspection γ-ray photon-counting system, and the details of the studied system can be found in reference [20]. The system uses a ⁶⁰Co radioisotope source, which emits photons of 1.17 MeV and 1.33 MeV, with an average energy of 1.25 MeV [21,22]. Theoretically, photons with higher energy are more likely to scatter with the lutetium yttrium oxyorthosilicate (LYSO) [23] scintillation crystal (see

Table 1. The attenuation coefficient ratio for LYSO crystal.

Energy	μph:μc 1
500 keV	1:2.14
1 MeV	1:6.97
1.25 MeV	1:9.59

¹ μ_ph is the photoelectric linear attenuation coefficient; μ_c is the Compton linear attenuation coefficient. The reaction material here is the LYSO crystal with 10% Yttrium (Y) participation.

). Experimentally, according to the Geant4 [24] simulation results of our system, as the photon energy increases, crosstalk is more likely to occur and to be observed in detectors that are far from the incident one (see Figure 1).

In this paper, we study the effect and properties of the measured intercrystal scatter interactions and reject them using a time window of 5 ns and a detector window of one or two accordingly.

2. Methodology

2.1. Intercrystal Scattering Simulation Model

In Section 3, a simplified steel plate model is employed as an example to analyze the scattering problem within the system using the developed Geant4 5.2 simulation platform. Figure 2 shows the setup of the studied system, in which a line array of scintillation detector modules moves relative to the detected object to obtain the scanned image. Each detector module consists of a 10 × 10 × 30 mm³ LYSO crystal and a silicon photomultiplier (SiPM). The distance between the ⁶⁰Co source and the detector array is 6 m, which is necessary for the passage of large containers, and the detected object is placed in the middle. To investigate the effect of crosstalk at different settings, two different collimator widths (5 mm and 10 mm) were simulated with 10, 20, and 30 cm thick iron plates, respectively.

2.2. Steel Penetration (SP) Simulation Model

There are three commonly used indicators for large container inspection system to assess the performance: Contrast Index (CI) [25,26], Image Quality Indicator (IQI) [27], and SP [20]. The SP indicator is tested in Section 5, which is defined as the maximum thickness of the iron plate (Fe) when an absorber behind the plate can still be recognized. We built a SP simulation model based on Geant4 and verified the rejection performance of the proposed method for intercrystal scattering in Section 5.1. A lead block (Pb, 100 × 100 × 200 mm³) is usually taken as the absorber in the SP model. Simulation is conducted in Geant4 as shown in Figure 3.

2.3. SP Test Platform

In Section 5.2, the SP ability of the ⁶⁰Co large-object photon-counting imaging system is evaluated using the constructed detection system. This evaluation aims to verify the advantages of the photon-counting imaging approach and assess the effectiveness of the temporal discrimination method in enhancing the system’s imaging performance. In the laboratory setup depicted in Figure 4, the intensity of ⁶⁰Co is 6.82 Ci (100 Ci; in November 2002), the distance between the radioactive source and the detector is about 3 m, and the sample time is 92 ms. According to the no-load count rate calculated by Equation (2), the system tested is equivalent to a typical system with a 100 Ci intensity, 6 m distance, and a scan speed of 24 m/min (i.e., 25 ms sample time).

3. Studies of Intercrystal Scatter

3.1. Proportion and Effect

Events with different incident and deposition detectors are defined as intercrystal crosstalk. The typical crosstalk interactions occur in multi-Compton scattering, in which the annihilation photon scatters from one scintillation crystal element to another, depositing a portion of its energy in each [28]. Crosstalk is very likely observed in the second and subsequent interactions (see the dotted lines of case 3 in Figure 5). Here, we define the energy deposition event in the incident crystal as the incident event, and the second and subsequent interactions that do not occur in the incident place are defined as crosstalk events. Another instance is when a photon passes through its incident crystal without interacting with it and then completely or partially deposits its energy in neighboring crystals (see the dotted lines of cases 1 and 2 in Figure 5). Simulation findings of more than 40,000 measured signals reveal that 16.7% of incident photons have experienced intercrystal interactions in the studied high-energy photon-counting system, in which crosstalk occupies 98.0% of all interference events.

The percentages of various types of events are denoted with different colored areas in Figure 6. When an incident photon undergoes the photoelectric effect within a material, the resulting signal is classified as a non-interfering event if the photon’s energy is deposited entirely or partially in the crystal without altering its path. In contrast, signals involving any deviation from the original path are considered interfering events. As can be seen in Figure 6, the proportion of crosstalk (marked in dark blue) in the total signals is fairly stable between 15% and 16%, while its proportion in interference signals (marked in light blue) varies in different models. The proportion of crosstalk in interference tends to be higher with thicker iron plates and narrower collimator widths. For example, the proportion can reach 90.1% in the model of a 10 cm thick iron plate combined with a 5 mm width front collimator (marked as 10 cm Fe_5mm), indicating that crosstalk at this time is the main factor that degrades the image quality. Meanwhile, in the model of 30 cm Fe_10mm, the proportion drops to 18.6%, in which crosstalk slightly worsens the image quality.

3.2. Characters

Intercrystal scatter interactions mostly occur in multi-Compton scattering, thus generating more than two events. We can study the characters of each event pair and find the relationship between them using Geant4 5.2 software, which helps to capture and process multiple kinds of information of transporting particles. Meanwhile, in cases 1 and 2 of Figure 5, crosstalk is produced independently, so such instances without a comparable reference are out of the scope of our discussion.

3.2.1. Energy

The energies of one incident event and its corresponding first crosstalk event are dotted in Figure 7, in which the second and subsequent crosstalk events with a very low likelihood of arising are disregarded. In accordance with the Compton scattering law stating that the combined energies are equal to the energies of photons decayed by ⁶⁰Co, a pair of intercrystal events have a combined energy of 1.17 or 1.33 MeV (see Figure 7). The density is greater below the dotted line, which means incident events tend to have higher energies than crosstalk events. However, there is no definite relationship between the energies of the two events, and the energies are distributed in a wide range from 0 to the maximum value of 1.17 or 1.33 MeV.

3.2.2. Position

Figure 8 presents the deposition position distribution of events that make up intercrystal scatter interaction pairs. All incident events are assumed to be detected by crystal i. The crosstalk events are distributed in a concentrated area with a small interval from the incident crystal. Statistics show that 89.93% of crosstalk events are deposited in crystals i + 1 or i − 1, and this scenario is denoted as ΔDet = 1. Meanwhile, the percentages are 8.47%, 1.25%, and 0.35% for the cases in which ΔDet equals two, three, and four, respectively. Thus, ΔDet ≤ 2 accounts for the majority of cases with a probability of 98.40%, while cases in which ΔDet > 2 can be neglected, having only a small share of less than 2%.

3.2.3. Time

We first take the instant at which 0 V is exceeded as the beginning time of each event. Under this condition without a trigger, the timing difference has a narrow dispersion with a full width at half maximum (FWHM) of just 0.66 ns. However, the start time cannot be acquired directly. The leading-edge timing method is usually employed in the electronic readout to discriminate times of response signals. The response signals are created using a PSpice electronics model of the readout circuits [20]. The FWHM of the distribution expands to 4.41 ns after using a leading-edge timing circuit with a 5 mV trigger voltage, as shown in Figure 9. As the trigger voltage rises, the dispersion widens even further.

There are always negative distributions in the three curves of Figure 9, indicating the possibility of incident events being detected following crosstalk events. To explain this phenomenon, we should first clarify factors that influence the timing of events. There are three main factors that can have an impact: (1) the response times of detectors, (2) the time discrimination method employed, and (3) the flight times of the detected photons. SiPMs utilized in the system have the same specifications, guaranteeing a consistent performance, including the time response characteristics. In addition, the leading-edge timing method with the same trigger voltage is applied to each signal. Therefore, the leading factor impacting the timing of events is the flight time.

The flight trajectories of the photons that make up an intercrystal interaction pair are shown in Figure 10 to better illustrate the time difference between them. L1 and L3 are the minimum flight distances for the recoil electron and scattered photon to reach the SiPM device, respectively. L2 is the flight distance of the scattered photon before its deposition. Therefore, their time difference can be expressed as ΔT = (L2 + L3 − L1)/c, where c is the speed of light. As can be seen in Figure 10, the difference in the flight time between the incident event and its accompanying crosstalk event is mainly related to the scatter angle. L2 + L3 tends to be shorter than L1 with a smaller scatter angle θ, which results in a negative distribution of ΔT. Meanwhile, the flight distance of the scattered photon will very easily exceed that of the recoil electron with larger scatter angles, particularly in back-Compton scattering.

The correlation between the time difference and the scatter angle of over 1500 pairs of intercrystal interaction events is shown in Figure 11a, aligning with the previous discussion regarding negative time differences, which tend to appear in interactions with small scatter angles. Conversely, there is only a minimal likelihood of a negative distribution when the scatter angle is large. Further illustrating this trend, the distribution in Figure 11a is transformed into Figure 11b.

4. Intercrystal Scattering Rejection Algorithm

4.1. Determination of Time Window

A wide time window is helpful to retain the majority of intercrystal scatter events, but meanwhile, the events coming from various photons (different decays) are more likely to be misjudged. Therefore, determining an appropriate coincident time window width is necessary. According to Figure 9, we can derive a time window with a width of around 5 ns.

Equation (1) is utilized to estimate the probability of n decays occurring in time t. λ is the decay constant of the radioactive source, n₀ is the number of nuclei in the radioactive source, and n₀λ is the number of decays per unit time. Here the probability of more than two decays (P(n ≥ 2)) occurring within three adjacent detector units is calculated, so n₀λ is three times the no-load count rate per detector unit (N), which can be calculated by (2). In Equation (2), N₀ is the initial intensity of ⁶⁰Co (unit: Ci, i.e., 3.7 × 10¹⁰/s), typically 30~300 Ci, d² is the entrance window area of the detector unit (1 × 1 cm²), r stands for the distance between the radioactive source ⁶⁰Co and the detector array (6 m), and η is the detection efficiency (about 70%). N₀ is multiplied by two, for the reason that ⁶⁰Co emits both 1.17 and 1.33 MeV γ photons [20].

$$P\left(n\right)={\displaystyle \frac{(n_0\lambda t{)}^n}{n!}}{e^{-}}^{n_0\lambda t},$$

(1)

$$n_0\lambda =3\cdot N=3\times 2N_0\cdot {\displaystyle \frac{d^2}{4\pi r^2}}\cdot \eta$$

(2)

The calculated results are listed in

Table 2. The probability of more than two decays occurring within three adjacent detector units for different time gaps.

Source Intensity N0	No-Load Count Rate/Unit N	P (n ≥ 2)
		t = 2 ns	t = 5 ns	t = 7 ns	t = 10 ns
30 Ci	3.44 × 105/s	0%	0%	0%	0.01%
50 Ci	5.73 × 105/s	0%	0%	0.01%	0.01%
100 Ci	1.15 × 106/s	0%	0.01%	0.03%	0.06%
300 Ci	3.44 × 106/s	0.02%	0.13%	0.25%	0.50%

. For the 300 Ci source intensity, the probability of more than two decays happening within 5 ns in three adjacent detectors is only 0.13%. And this probability decreases further under conditions of a lower source intensity and non-empty load. If the time gap extends to 10 ns, the probability reaches 0.50%. Therefore, the probability of more than two decay events occurring within the 2~10 ns time window is quite low. Combined with the results of Figure 9, a time window of 5~10 ns can include more intercrystal scatter events.

As evidenced by the simulation results, when the criterion of ΔDet = 1 is applied with a 5 ns time window, the RMSE under 30 Ci no-load conditions increases from 5.44 to 26.06. Meanwhile, when ΔDet = 1 with a 10 ns time window, the RMSE rises to 26.10. When the no-load condition is adjusted from 30 Ci to 300 Ci, the RMSE increases from 5.44 to 26.89 and 26.49 under the 5 ns and 10 ns time windows, respectively. Therefore, the 5 ns and 10 ns time windows exhibit nearly equivalent levels of effectiveness. With the aim of retaining the majority of intercrystal scatter events, decreasing the potential for misjudging events from different photon decays, as well as improving the counting accuracy, the smaller 5 ns time window is selected as the optimal criterion.

4.2. Selection of Detector Window

When considering the detector window width, cases in which ΔDet > 2 can be disregarded due to their relatively low occurrence rate. To determine the optimal detector window width, a simulation is conducted using several models with different count rates. The findings demonstrate that the criterion of ΔDet = 1 is preferred for systems with high count rates (more than 10³/s), for example, when the detected object is an empty container or is made of materials with low atomic numbers, whereas ΔDet ≤ 2 should be selected for systems with low count rates. The justification for this recommendation is explained below.

Judgment errors mainly come from the misjudgment of events from different photon decays and omission of true coincidence events, which are illustrated in Figure 12. The events marked as No.24446 and No.24440 come from different photon decays and would be incorrectly judged under the ΔDet ≤ 2 criterion. However, the true coincidence events from the same photon, No.26578, might be omitted under the ΔDet = 1 criterion. High count rates typically result in an increased frequency of events from different photon decays appearing in adjacent detector windows, in which the impact of a misjudgment is greater than that of an omission. Hence, the ΔDet = 1 criterion proves effective in reducing misjudgments. Meanwhile, for conditions with low count rates, misjudgments rarely occur; therefore, the ΔDet ≤ 2 criterion is utilized to mitigate the chance of overlooking true coincidence events.

4.3. Realization on FPGA Chip

To realize the stated function of time and position judgment, the proposed algorithm has been implemented on a cyclone IV FPGA chip with a 200 MHz clock frequency, enabling the identification of a 5 ns time difference. The steps of the algorithm are outlined in Figure 13. Additionally, the algorithm features two distinct criteria with varying detector window widths, which can be interchanged according to the count rate or material of the detected object. This flexibility allows for a more tailored approach that is adaptable to various scenarios.

5. Imaging Performance Validation Results

5.1. Simulation Validation Results

In order to visualize the effect of the two criteria, the simulation results for the SP model with a 250 mm thick iron plate are shown in Figure 14. The image quality is quantified by the root mean square error (RMSE) compared with the ideal image (the image only counts non-interference signals).

Table 3. Quantification results of the proposed algorithm applied to the SP (250 mm Fe) model.

	Ideal	Origin	ΔDet = 1	ΔDet ≤ 2
RMSE	/	4.328	3.322	3.273
Interference Removed %	100%	/	24.09%	25.46%
Non-interference Misremoved %	0	/	14.38%	15.70%

includes various quantitative indicators that reflect these results. The proposed algorithm is proven to be capable of improving the radiation image quality, as evidenced by the decrease in RMSE from 4.328 to 3.273. For the tested model with a low count rate of 200/s, the criterion of ΔDet ≤ 2 has better results compared with the algorithm of ΔDet = 1. The rejection ratio of interference (compared with the original image) is raised from 24.09% to 25.46%, leading to an overall improvement in image quality despite a few extra non-interferences being mistakenly deleted in the ΔDet ≤ 2 judgement. Meanwhile, under no-load conditions with a count rate of 3.44 × 10⁶/s, the criterion of ΔDet = 1 functions better, increasing the RMSE from 5.44 to 26.89.

5.2. Actual Test Results

In this paper, we verify the performance of the crosstalk rejection algorithm on the experimental platform described in Section 2.3 and obtain the corresponding imaging results. The results obtained are displayed in Figure 15. After applying the algorithm of ΔDet ≤ 2 to reject intercrystal scatter, the SP ability improves from 260 mm to 280 mm Fe. Meanwhile, the SP ability of the current integration systems in use now can only reach around 210~220 mm Fe under the same settings. The results reflect the effect of the improvement of the photon-counting measurement mode and the application of the time screening method on the performance enhancement of the large-scale guest radiation imaging system, which verifies the effectiveness of the technical scheme proposed in this paper.

6. Conclusions

The intercrystal scatter phenomenon has a bad effect on image quality and is significant in our designed high-energy large-container-inspection ⁶⁰Co γ-ray photon-counting system, and this has been verified both theoretically and experimentally. To solve this problem, we analyzed the characters of intercrystal scatter events, taking into account their energy, position, and time relationships, which can be determined with photon-counting detectors. The analysis shows that intercrystal scatter events mainly distribute in the adjacent detector units, with time gaps of around 5 ns. Then, the proper time and detector window widths are selected and applied to reject the intercrystal scatter events. The algorithm is implemented on the FPGA and can switch the detector window width according to the applied scenarios.

The simulation on an SP 250 mm Fe model validates the effectiveness of the proposed approach, with the criterion of ΔDet ≤ 2 functioning better in this low-count-rate scenario, reducing the RMSE from 4.328 to 3.273. Finally, the laboratory test shows that with the application of the algorithm of ΔDet ≤ 2, the photon-counting system under an 100 Ci intensity and 24 m/min scan speed can reach 280 mm Fe SP, which is 60~70 mm Fe more than that of the current integration system, proving the advantages of the algorithm we proposed. Further, FPGAs enable the real-time timestamping of detected signals with a sub-nanosecond resolution, thus having the potential to be applied to more coincidence event discrimination tasks (e.g., PET imaging), positioning them as a pivotal enabler for next-generation scientific and industrial systems.