Introduction

In a positron emission tomography (PET) scanner, a method of improving the spatial resolution without reducing the sensitivity is to use a depth-of-interaction (DOI) detector that is able to measure the depth position of the interaction of incident gamma rays. A project aimed at the development of a DOI-PET scanner named “jPET-D4” has been undertaken at the National Institute of Radiological Sciences of Japan [1–8]. In this project, various structures for the DOI detector have been proposed for the realization of high sensitivity and high spatial resolution. These DOI detectors consist of four layers of a Gd2SiO5:Ce (GSO) scintillation crystal array and a 256-channel flat-panel position-sensitive photomultiplier tube (PSPMT).

In relation to the development of jPET-D4, a detector simulator was also needed. Analyzing the behavior of the gamma rays and optical photons inside the DOI detector with a simulator leads to an improved understanding of the detector’s performance and therefore improvement of the design. More practically, the detection characteristics determined with the detector simulator can be used in the development of an algorithm for detector element identification and estimation of the identification accuracy.

Several studies have been carried out on the DOI detector by use of the Monte Carlo simulation [9, 10]. DETECT2000 is a typical detector simulator [9]. In the jPET-D4, various detector configurations that have different surface conditions of crystal and positions for inserted reflection material have been examined. For rapid simulation of each of these configurations, a simulator with a high degree of freedom was required. Thus, we first developed a Monte Carlo simulator [11, 12].

This simulator traces all optical photons generated by the interaction of gamma rays with a crystal element and predicts the output signals of photomultiplier tubes (PMTs). It has been used mainly for investigating the relationship between certain properties of the detector such as crystal element size, reflectance of the reflector, and the detector identification performance. This simulator is precise, but slow. Thus, we developed a much faster simulator by introducing some approximation. This paper describes the newly developed simulator and its validation.

DOI Detector in jPET-D4

Figure 1 schematically shows the structure of the finally adopted detector in the jPET-D4 project. A DOI detector block consists of 1024 crystal elements, and each crystal element has dimensions of 2.9 mm × 2.9 mm × 7.5 mm. The block consists of four layers and each layer has 16 × 16 elements. In Fig. 1 right, a vertical cross section of the detector is shown where only 8 elements  × 4 layers appear for illustration purpose. The detector block has an array of fundamental structures. The fundamental structure is composed of 2 × 2 × 4 crystal elements as shown in Fig. 1 left. We call the first layer to the forth layer from bottom to top as shown in this figure. While four crystal elements are separated by an optical reflector in the first and second layer, there is no reflector between crystal elements in the third and forth layer. Instead, in the third and forth layer, the fundamental structure is covered by a reflector. Eight by eight fundamental structures are connected hard and the whole detector block is wrapped in a reflector except for the bottom surface. The bottom is optically coupled to a flat-panel PSPMT with an area of 52 mm × 52 mm.

Fig. 1
figure 1

Structure of four-layer DOI detector treated in this study. Left a fundamental structure. Right a vertical cross section of the detector

Full size image

When a gamma ray enters the detector, it causes an interaction with the scintillation crystal, and this produces optical photons which correspond to the deposited energy of the gamma rays. The produced photons reach the anodes of the PSPMT with certain probabilities. The distribution pattern of the photons depends on the position of the crystal element in which the interaction takes place. The PSPMT generates 256-channel anode signals, and these signals are summarized into four signals by a resistor chain circuit. Coordinates, X and Y, are calculated from these four output signals on the basis of Anger logic. Due to the unique characteristics of the DOI detector, the position are localized at specific areas in two-dimensional (2D) space depending on the crystal element of interaction, and therefore identification of the crystal element of interaction is easily performed [5, 6].

In addition, this four-layer DOI detector is a phoswich detector with two kinds of GSO crystals which differ in terms of their doped Ce concentration. It discriminates between the upper parts (2nd and 4th layers) and lower parts (1st and 3rd layers) on the basis of the difference in the scintillation decay time caused by different Ce concentrations. Namely the pulse shape of the dynode signal is used for the layer discrimination.

Simulator

The simplified simulator uses the statistical parameters that can be obtained with the previously developed Monte Carlo simulator. Therefore, the Monte Carlo simulator is first reviewed, and then the simplified simulator will be described. In the simulation, the following parameters were assumed.

  • Scintillator

    • Material: Gd2SiO5Ce(GSO)

    • Light yield: 10,000 photons/MeV [13]

    • Refractive index: n = 1.85

    • Surface: chemical etching

  • Optical reflector

    • Material: multiple polymer layers

    • Reflectance: 96%

  • Interlayer material

    • Material: silicon oil

    • Refractive index: n = 1.45

Monte Carlo simulator

In this study, the Monte Carlo simulator uses the Monte Carlo technique to trace both the gamma rays in the crystal and the optical photons emitted due to the interaction of the gamma rays with the crystal. For each gamma ray irradiation, the number of optical photons detected is counted at each anode. Finally X, Y coordinates of the interaction position is estimated from the outputs of a resistor chain circuit.

Figure 2 shows a flow of the Monte Carlo simulation in which (a) represents the gamma ray tracing part and (b) represents the optical photon tracing part. The following describes details of the simulation.

Fig. 2
figure 2

Flow of the Monte Carlo simulation. a Gamma ray tracing part b optical photon tracing part

Full size image

Gamma ray tracing part

In this simulation, we only treat the photoelectric (PE) absorption and the Compton scattering and neglect the other interaction phenomena such as coherent scattering (Thomson scattering) and pair production because those occurrence probability is very low for a gamma ray with the energy range of this simulation. For each gamma ray that enters the detector, an interaction with a crystal takes place with a probability that depends on the depth of the crystal. In the simulation, the distance is determined by

$$ z = \frac{{- \ln \xi}}{{\sigma_{{\rm PE}} + \sigma_{{\rm Compton}}}} $$
(1)

Here, ξ is a computer-generated random number ranging from 0 to 1. σPE and σCompton is the photoelectric absorption factor and the Compton scattering factor of linear attenuation coefficient, respectively, and determined by the kind of crystal and the energy of the gamma ray. Figure 3 shows the relationship between the linear attenuation coefficients of GSO used in this study and the gamma ray energy. If the interaction position expected from the incident angle and the distance given by Eq. (1) is out of the detector block, the trace of the gamma ray is terminated and a next gamma ray is tested. If inside the detector block, one of two kinds of interactions, i.e., photoelectric absorption or Compton scattering, is selected according to the following probabilities

$$ P_{{\rm PE}} = \frac{{\sigma_{{\rm PE}}}}{{\sigma_{{\rm PE}} + \sigma _{{\rm Compton}}}}, P_{{\rm Compton}} = \frac{{\sigma_{{\rm Compton}}}}{{\sigma _{{\rm PE}} + \sigma_{{\rm Compton}}}} .$$
(2)
Fig. 3
figure 3

Relationship between the linear attenuation coefficients of GSO and gamma ray energy

Full size image

The energy deposited by the interaction is then calculated. For photoelectric absorption, the energy of the gamma ray is all deposited. For Compton scattering, a scattering angle and deposited energy are calculated on the basis of the Nishina–Klein formula describing the differential cross section of Compton scattering.

Depending on the energy deposited by the interaction, many optical photons are produced. Each photon acts upon optical phenomena such as reflection, absorption, and transmission. Its details are described in the optical photon tracing part below. The number of optical photon finally detected by the PSPMT is counted and summed with the number of optical photon produced and counted by the other interactions.

If the interaction is a Compton scattering, some energy remains in the scattered gamma ray. If it is greater than a predetermined threshold level, the gamma ray moves to the direction calculated above. Then the same processing as written in this part is repeated.

Optical photon tracing part

The mean number of optical photon produced, \(\bar{N},\) is given by

$$ \bar{N} = {\rm Lightyield} \times \Updelta E $$
(3)

Here Lightyield is the number of optical photons produced by a gamma ray with 1 MeV and was referred from [13]. ΔE denotes the deposited energy and is described in MeV. Actual number of photon produced is fluctuated according to the Poisson distribution. As the number is large, however, such fluctuation is very small in fact.

Due to the arrangement of crystal elements and the optical reflectors, an optical photon can reach not only the anode right below the interaction position but the different anodes. An optical photon produced at the interaction position moves into random direction. A wavelength is given to each photon. The wavelength is determined so that the total emission spectra correspond to the spectral characteristics of scintillation light of the crystal used. Each optical photon with a certain wavelength is subject to the spectral characteristics in the subsequent phenomena such as absorption, transmission and reflection.

When a moving direction is given to a photon, the first surface in the forward direction is found and the moving distance to the surface is calculated. The photon reaches at the surface with the probability given by the transmittance of the moving distance.

There are three kinds of interface hit by a moving photon. Those are an adjacent crystal, a reflector, and the glass of PMT. At each interface, one phenomenon among reflection, transmission and absorption is selected with proper probabilities. In the case of reflector, reflection is selected subject to the inherent reflectance of the material. Otherwise the photon is absorbed. In the case of glass of PMT, the reflectance is first calculated according to the Fresnel’s law. The direction of reflection is given by the Cook–Torrance model. The detail should be referred to [14]. If a photon transmits the glass of PMT, it is detected with 100% of probability.

Simplified simulator

The gamma-ray tracing part of the simplified simulator is the same as that in the Monte Carlo simulator described above. On the other hand, an approximation is used in the photon tracing part.

As preparation, the probability that an optical photon emitted from each crystal element reaches each anode in the PSPMT is calculated in advance by use of the Monte Carlo method. We call this probability the photon transfer efficiency (PTE) and denotes the PTE from ith crystal to jth anode by p ij . For an arbitrary crystal element of a fundamental structure, we assume that the interaction position is the center of the crystal element. A number of optical photons are emitted and the finally detected photons at each anode are counted. The PTE is given by the ratio of the number of detected photons to the number of emitted ones. Those obtained probabilities are saved as a look-up-table and used in the simplified simulator. For more exact simulation, p ij should be calculated for interaction at not only the center but also many positions in the crystal element. However, it makes the look-up table size very huge. In this paper, we chose the above-mentioned simplification for practical use.

When an interaction takes place at ith crystal element, a number of optical photons produced, N i is determined in a way described in the previous section. Mean number of photons that reaches at jth anode is calculated by multiplying the number of generated photons with the PTE, i.e., N i p ij . Then the final number of photons captured is randomized subject to a Poisson distribution. The rationale using the Poisson distribution is described in Appendix [15].

Validation of simulator

Comparison between two simulators

The simplified simulator was compared with the Monte Carlo simulator. It was assumed in the simulation that 200,000 gamma rays of 662 keV uniformly irradiated the 4 × 4 × 4 crystal array composed of 2 × 2 fundamental structures. We also carried out the validation of simulator under the condition of gamma rays of 511 keV used in PET. The results of 511 keV were similar to those of 662 keV photons described below. In this paper, comparisons with gamma rays of 622 keV are presented as a canonical case. Of course, in the practical use of this simulator for PET, the simulation should be done with gamma rays of 511 keV.

Figure 4 shows the two-dimensional position histogram (2D position histogram), and Figure 5 shows the histogram of the sum of the light output obtained with the two simulators, which is corresponding to the energy spectrum. The results for the simplified simulator show very good agreement with those of the Monte Carlo simulator in both the 2D position histogram and the energy spectrum. To process 200,000 histories of gamma rays, the Monte Carlo simulator took approximately 5,397 min with use of a Pentium-4 2.4 GHz PC. On the other hand, the simplified simulator took 43 min, including the time taken for calculating the PTE by use of the Monte Carlo simulator. Once the PTE is calculated, it can be used repeatedly under the same detector condition. It is evident that the simplified simulator makes a much faster simulation possible.

Fig. 4
figure 4

2D position histogram for 4 × 4 × 4 crystal array. a and b Monte Carlo simulator, c and d simplified simulator. Left column 1st and 3rd layers (1.5 mol%), right column 2nd and 4th layers (0.5 mol%)

Full size image
Fig. 5
figure 5

Energy spectrum obtained by the two simulators

Full size image

Comparison with experiment

2D position histograms

The simulation result was also compared with the experimental result. An experiment was conducted under the conditions described above. Figure 6 shows the 2D position histograms for the experimental and simulation results. Both similar and dissimilar points are observed. In this subsection, the similar points are first described and the similarity is quantitatively evaluated. After that the dissimilar points are discussed.

Fig. 6
figure 6

Comparison between experiment and simulation with respect to 2D position histogram

Full size image

The basic patterns observed in these 2D histograms, such as the spatial relationships among clusters and the frequency of occurrence of each cluster are similar. The degree of similarity between the experimental and simulator-generated histograms was evaluated quantitatively as follows. In the 2D histogram, at first, the position of the maximum count of each cluster is found. As shown in Fig. 6, eight clusters that consist of four inner and four outer ones form a group. We denote the maximum count position of the jth inner cluster of the ith group by A ij , and that of the jth outer cluster of the ith group by B ij . We then let the point at which the centroid of four centers of inner clusters be O i . We further let a side of the square 2D histogram be L. Then, the following ratios are calculated.

$$ D_{ij}=\frac{O_{i}A_{ij}}{L}, \quad i=1,\ldots,4,\ j=1,\ldots, 4$$
(4)
$$ R_{ij}=\frac{O_{i}B_{ij}}{O_{i}A_{ij}}, \quad i=1,\ldots,4,\ j=1,\ldots,4 $$
(5)

Here O i X ij (X is A or B) denotes the distance between O i and X ij . D ij means the degree of mutual separation of inner clusters normalized by the side of the square of the 2D histogram. On the other hand, R ij provides a measure of the degree of separation between the inner and outer clusters.

Because the ratios D ij and R ij vary depending on i and j, each ratio was calculated for all (i, j) combinations. Then the minimum, maximum, mean and median of those values were compared for the histograms of the experiment and the simulation. Table 1 shows the result. Top and bottom of this table shows the statistics with respect to the ratios D ij and R ij , respectively. For mean and median values, the differences between the experimental values and the simulation values are also calculated and presented. The difference here is defined by the absolute difference between two values divided by the simulation value.

Table 1 Quantitative evaluation of similarity with respect to 2D position histograms of experimental and simulation results
Full size table

As the clusters are distributed asymmetrically in the results of both experiment and simulation, each ratio varies widely. Particularly in the experiment, a stronger asymmetry can be observed, possibly due to the non-uniform sensitivity characteristics of the PSPMT channels. The range of each value in the simulation is, in most cases, within the range for corresponding value in the experiment. Mean values and median values have the difference of about 15% at maximum between the simulation and the experiment. This result would be acceptable considering that the experiment includes many factors producing the variation of data.

In the experiment, each cluster has blurred distribution more than the simulation. There are several reasons described below. As seen in the energy histogram later, the experimental data shows higher frequency characteristics in the low energy region compared to the simulation. It is supposed that this is mainly due to the scattering from peripheral parts such as the supporting bench in the experimental setup. Dark light with greater photon noise makes anode signals more fluctuated and consequently brings wider clusters in the 2D position histogram. It is obvious that the electric circuit itself generates noise more or less and causes the fluctuated anode signals. In the simulation, any noise originated from the electric circuit is not assumed.

Another different point between the experiment and the simulation is the bias appeared in the 2D position histogram. It seems that the experiment has a larger bias in the central area in the histogram. If in a certain crystal element in a detector a Compton scattering takes place as the first interaction, the second interaction can take place in a different crystal element in the same detector. Since the detector can not discriminate this multiple interactions, summed signals due to the multiple interactions is used for position calculation. As a result, the position is estimated at around an intermediate point between the multiple interactions. This causes a bias-like pattern in the 2D histogram. Particularly, the discrepancy between the experiment and the simulation in terms of the bias-like patter is caused by multiple interactions among crystals with different Ce concentrations. In the simulation, we do not use the pulse shape for the discrimination of two kinds of layers. The discrimination is performed based on the first interaction layer. On the other hand, in the experiment, larger error take place in the pulse shape discrimination in the case of the multiple interactions. It is supposed that this difference makes the histogram in the experiment have a larger bias.

Energy spectrum

The similarity between the simulation and experiment was also evaluated in the energy spectrum. Figure 7 shows the energy spectra for the experimental and simulation results. In this figure, the scales of both horizontal and vertical axes are different between the simulation and the experiment because we did not intend to match them. In this case, the similarity of the relative shape should be evaluated. In each figure, four spectra corresponding to the first to the forth layer are plotted and the photopeak of each spectrum is indicated by the arrow. Since in the higher layer more interactions take place, the energy spectrum of the higher layer has higher level. The energy spectrum is very similar between the simulation and experiment, particularly around the photopeaks. A slightly higher level of frequency in the low energy region in the experimental data is presumably due to scattering from the peripheral parts as mentioned above.

Fig. 7
figure 7

Comparison between experiment and simulation with respect to energy spectrum

Full size image

The photopeak position of the first layer is highest. This would be because the optical photons emitted in the first layer nearest to the PMT can reach at the PMT with the higher probability than the other layers. However, the order of the position of photopeaks is not necessarily the layer order. This phenomenon can not be explained exactly but the statistical distribution of the path length of photon in the unique configuration of this detector would cause this result.

A quantitative evaluation of the similarity was again performed for the energy spectra. In this case, the relative position of the photopeaks was focused because this measures if the above mentioned unique characteristics are simulated well. With the position of the photopeak of the first layer used as a standard, the photopeak positions of the three other layers were calculated. Namely, the position of the photopeak of i-th layer is calculated by the channel number at the photopeak of i-th layer divided by the channel number at the photopeak of the first layer. Table 2 shows the results. The values for each layer are very close, and the similarity in terms of the energy spectra between the experiment and the simulation is clearly evident.

Table 2 Quantitative evaluation of similarity with respect to energy spectra of experimental and simulation results
Full size table

Conclusion

We have developed a simplified simulator of a four-layer DOI detector on the basis of the previously developed Monte Carlo simulator. We compared the performance of the simplified simulator with the Mote Carlo one and found that very similar results can be obtained within about 1/125 computational time. The simulator was also validated through the comparison with the experiment. The 2D position histogram and the energy spectra of the simulation were similar to those obtained by the experiment. However, there were some dissimilar points although those are not serious so much. Finer tuning of the simulator would be a next work.

It should be noted that the simulator constructed in this study is used in practice in the following two areas. When a method for discriminating in which detector element the first interaction with a gamma ray takes place is developed and evaluated, a computer simulation for the detection of each event is required. This simulator was used in such an application [16, 17]. The other area is in the use of the detector response characteristics in a reconstruction algorithm. It is known that the exact representation of an imaging system in an iterative reconstruction algorithm is effective for obtaining a high image quality. The representation of the detector response characteristics in such a reconstruction algorithm should contribute to a high image quality. The simulator we developed has also been used in such application [18, 19].