A feasibility study on 3D interaction position estimation using deep neural network in Cherenkov-based detector: a Monte Carlo simulation study

Positron emission tomography (PET) relies on the detection of positron-annihilation gamma rays using a ring-shaped detector constructed by opposing multiple detectors (Phelps 2004). To accurately measure a couple of gamma rays along the line of response between a pair of PET detectors, the interaction positions of the gamma rays in the crystals should be determined. Using a monolithic scintillator coupled to a position-sensitive photodetector is a promising method for 3D position estimation (Yoshida et al 2011, Borghi et al 2016, Etxebeste et al 2016, Marcinkowski et al 2016) and for time-of-flight measurement (Seifert et al 2012, Borghi et al 2016, Borghi et al 2018). Moreovermonolithic crystals can increase the packing fraction of the detector as no reflecting materials are required, unlike finely pixelated scintillator arrays. Monolithic crystals are also cheaper and easier to fabricate than scintillator arrays (Marcinkowski et al 2016). In addition, some position estimation methods using machine learning have been developed to extract accurate features of the detector response. Bruyndonckx et al 2004 and Wang et al 2013 have developed an imping/interaction position estimation method using artificial neural networks. Pedemonte et al 2017 have developed a DOI estimation method using nonlinear dimensionality reduction.

Recently, the timing and spatial performance of monolithic Cherenkov-radiator-based PET detectors have been determined on time-of-flight PET (Somlai-Schweiger and Ziegler 2015, Ota et al 2018). These approaches may improve the PET time resolution compared to scintillation-based detectors because Cherenkov photons are emitted in an extremely short time within 1 ps. In fact, an excellent coincidence time resolution better than 100 ps full width at half maximum (FWHM) has been experimentally obtained using the Cherenkov radiator and microchannel-plate photomultiplier tube (Korpar et al 2011, Ota et al 2019). However, only up to 10 Cherenkov photons can be detected when a 511 keV gamma ray interacts with a Cherenkov radiator such as lead tungstate and lead fluoride (PbF₂) (Brunner et al 2014, Ota et al 2018). Therefore, it is challenging to accurately estimate the interaction position using the monolithic radiator. Ota et al (2018) proposed a 3D Cherenkov-based detector using the monolithic PbF₂ radiator coupled to a photodetector array. Cherenkov photons are promptly emitted with a fixed angle along the trajectory of a photoelectron, and the Cherenkov cone is consequently drawn on the detector plane. However, they found that Cherenkov photons induced by a 511 keV gamma ray do not draw the Cherenkov cone due to multiple scattering of the photoelectron and estimated the interaction position using the center of gravity (CoG) and principal component analysis (PCA) defined in section 2.4. However, the detector performance can be further improved, especially in terms of the position resolution, because better approaches than the CoG–PCA method may exist (Ota et al 2018).

In this paper, we present a method for the 3D estimation of the interaction position in monolithic Cherenkov-radiator-based detectors using a deep neural network. Many kinds of nonlinear phenomena occur in the Cherenkov radiator, e.g., the photoelectric effect, multiple scattering, and Cherenkov radiation could be defined as nonlinear functions. Therefore, we used neural networks because of their generalization ability to model highly nonlinear functions (Gardner and Dorling 1998). In addition, we focused on the 3D estimation of the interaction position and not on improving the time-of-flight performance. Therefore, we used a simplified detector to exclusively assess the feasibility of using a deep neural network for the estimation. We confirmed that the proposed approach is capable of 3D position estimation through a Monte Carlo simulation. Moreover, we verified the feasibility for position reconstruction retrieved by the proposed method compared to the method based on CoG and PCA.

2.1. Cherenkov-based detector

Figure 1 shows the Cherenkov-based detector proposed by Ota et al (2018). This detector provides a high temporal and spatial performance. We used a Cherenkov radiator with monolithic PbF₂ crystal of 40 × 40 × 10 mm³. In addition, the readout pitch of the photodetector ranged from 0 to 3 mm in intervals of 0.1 mm. Moreover, we varied the single photon time resolution (SPTR) of the photodetector as Gaussian distribution of σ = 0, 5, 10, 25, 50, 75, and 100 ps; an SPTR of 50–100 ps is a practical condition according to Nemallapudi et al (2016). A photodetection efficiency (PDE) of 100% was considered for simplicity. The PDE does not affect the performance in terms of position resolution although it affects efficiency. Each photodetector was assumed to be topped with a 300 μm thick protection resin (n = 1.5), and the photodetector array and radiator were connected by a 100 μm thickness grease (n = 1.5). A monolithic PbF₂ crystal was employed as the Cherenkov radiator, classified in category 2, owing to its excellent properties listed in table 1.

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Table 1.  Lists of the optical properties of PbF₂ crystal.

Density (g cm−3)	Refractive index	Cutoff (nm)
7.7	1.82	245–280

2.2. Simulation dataset

A Monte Carlo simulation was performed using the Geant4 particle simulator (Geant4 Collaboration) (Agostinelli et al 2003, Allison et al 2006, Allison et al 2016) to generate the measurement data considering the physical parameters from Ota et al 2018, except for the detector dimensions. Geant4 Livermore libraries were used for low-energy electromagnetic processes according to Brunner et al 2014. Bremsstrahlung, ionization, and multiple-scattering were implemented against the electrons throughout simulations as physics lists. In Geant4, a function that defines the degree of accuracy of particle tracking can be optimized for a particular study by setting the parameters of the function, particularly the maximum step length (MSL), which is important when implementing simulations with Cherenkov effects because the MSL highly depends on both the electron tracking and the amount of Cherenkov photons in the low-energy region. Therefore, we used the MSL of 1 μm based on Ota et al (2018). The optical properties of PbF₂, e.g., the transmittance, refractive index, and density, were determined based on the definitions provided by Anderson et al 1990 and the Refractive Index Database (available online at https://refractiveindex.info/?shelf=main&book=PbF2&page=Malitson). The center of the radiator in xy plane and z axis was defined as the origin of the coordinate system, i.e., (x, y) = (0, 0) and z = 0, as shown in figure 1. Here, z is equivalent to the depth of interaction (DOI).

Figure 2 shows the flowchart of Monte Carlo simulation performed in this study. First, a 511 keV gamma ray was generated 10 mm away from the entrance surface and the center of the radiator on the xy-plane. Then, its momentum direction was randomly determined in polar and azimuth angles, which varied from −π/4 to +π/4 and 0 to 2π, respectively. After fixing the momentum direction, the gamma ray was emitted and its interaction with the radiator was observed. If an electron is generated by the interaction, its position is recorded. After recording, we track the electron and see if it emits Cherenkov photons. If one or more Cherenkov photons are emitted from the electron, they are tracked until they are absorbed or reach the photodetector plane. After tracking all the particles, the next event is executed.

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We performed the following sequential recordings to construct an event:

• (x_true, y_true, z_true): true position of the interaction with radiator.
• N: number of photons reaching the xy (photodetector) plane.
• (x_i, y_i, t_i), 1 ≤ i ≤ N: xy position and time each photon reaching the photodetector plane, where t₁ = 0 s.

The photon detection threshold was assumed to be five, and if more than seven photons were detected per event, the first seven were considered as effective photons, because events where more than eight photons were detected were very few for training the neural network. We obtained 2.3 × 10⁶, 1.0 × 10⁶, and 6.6 × 10⁵ events for five, six, and seven detected photons, respectively, and considered 90% of them as training dataset and the remaining 10% as the validation dataset. Actually, approximately only 10% of all the detected events have more than five photons.

2.3. Neural network architecture

We propose the multilayer perceptron (MLP) architecture illustrated in figure 3 for 3D estimation of gamma-ray interaction position on the radiator, because we consider that the full connection of the MLP is the most suitable for the proposed detector. Although one of the difficulties of MLP is the large number of fully connected layers, the proposed detector provides a low amount of data related to the small number of detected Cherenkov photons. Three MLPs, one per number of detected photons, are used for estimation, and their input data are consistent with the corresponding number of detected photons. Each MLP consists of input, hidden, and output layers. We evaluated several configurations of hidden layers and hidden units. Considering the position resolution, we selected 3 hidden layers and 256 hidden units.

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The input layer receives the position of the detected photons (x, y) and records time t. Then, the vector-valued input data are processed with an affine transformation into several hidden units, normalized by batch normalization, and activated by rectified linear unit. A similar process is applied throughout the hidden layers. Finally, the final hidden layer is affine transformed into 3D position ($\hat{x},$ $\hat{y},$ $\hat{z}$). We used a mean square error as a loss function in this study.

Training was performed on a computer running Ubuntu 16.04 and containing a graphics processing unit (NVIDIA GeForce GTX TITAN X with 12 GB of memory) using Chainer 3.3.0 (Tokui et al 2015). The input data were standardized by removing the mean and scaling to unit variance using a standardscaler from the scikit-learn library (Pedregosa et al 2011). We considered 200 epochs and used the Adam method (Kingma and Ba 2014) with weight decay of 10^–4 for optimization and regularization.

2.4. Evaluation

To evaluate effectiveness of the proposed approach, we compared it to the method based on CoG and PCA (Ota et al 2018). The interaction position of gamma rays with respect to the point on the xy plane (x_CoG, y_CoG) was estimated using the CoG method and given by

$$\begin{eqnarray}&&{\hat{x}}_{{\rm{CoG}}}=\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}{x}_{i},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\hat{y}}_{{\rm{CoG}}}=\displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}{y}_{i}.\end{eqnarray} \tag{ 2 }$$

Anomalous photons scattered on the radiator were rejected using

$$\begin{eqnarray}&&\begin{array}{l}{d}_{ij}=\,\sqrt{{\left({x}_{j}-{x}_{i}\right)}^{2}+{\left({y}_{j}-{y}_{i}\right)}^{2}}\\ \,\,-\varepsilon \displaystyle \frac{\displaystyle {\sum }_{i=1}^{n}\sqrt{{\left({x}_{{\rm{CoG}}}-{x}_{i}\right)}^{2}+{\left({y}_{{\rm{CoG}}}-{y}_{i}\right)}^{2}}}{n},\end{array}\end{eqnarray} \tag{ 3 }$$

where i and j denote photons and ε is the anomaly threshold. Hence, the ith photon is anomalous if d_ij is positive for all $j\ne i.$ After rejecting anomalous photons, the CoG method was applied again. The interaction position of gamma rays along the z axis (DOI) was estimated using PCA. The distribution of photons was parameterized by the fitting ellipse, and seven features were calculated:

• Time difference between the first and last photon
• Length of the major and minor axes of the ellipse
• Rotation angle
• Mean, maximum, and minimum distance between two photons

We calculated a linear approximation between the first principal component and the true value to estimate the interaction position ${\hat{z}}_{{\rm{PCA}}}.$

The position resolution was evaluated from the FWHM, full width at tenth maximum (FWTM), and cumulative histogram at half maximum (CHHM) of the difference between the true and estimated positions.

Figure 4 shows training/validation loss at an SPTR of 0 ps. There is slight difference between the training and validation loss; therefore, the networks could not have been overfit.

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Figure 5 shows histograms of the xy plane estimation error ${x}_{true}-\hat{x}.$ from all the validation datasets, and figure 6 shows histograms of the z-axis estimation error ${z}_{true}-\hat{z}$ from all the validation datasets at SPTR of 0, 10, and 100 ps. On the xy plane, as the SPTR increases, both the histograms and cumulative histograms obtained from the MLP approach those obtained from the CoG. In contrast, along the z axis, the MLP histogram is symmetric, different from the PCA histogram, showing a difference between the proposed and comparison methods.

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Figure 7 shows the position resolution the xy plane and along the z axis according to SPTR. Unlike the MLP, the results using the CoG are not affected by the SPTR on the xy plane because its calculation does not consider time information. Along the z axis, the MLP is slightly affected by the SPTR but retrieves better results than PCA. Figure 8 shows the 2D histograms of the xy plane estimation error and the true position of z-axis by the CoG method and DNN at SPTRs of 0, 10, and 100 ps. The error of the CoG method tends to degrade around the incident surface; in contrast, the distribution of the MLP method tries to narrow down at all z-axes to minimalize the MSE of the training dataset.

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Figure 9 shows the position resolution on the xy plane and along the z axis according to the readout pitch. Both on the xy plane and along the z axis, the MLP is slightly affected by the readout pitch but retrieves better results than the CoG and PCA method.

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We validated a feasibility of 3D gamma-ray interaction position estimation in monolithic Cherenkov-based detectors using a deep neural network. In addition, we evaluated the influence of SPTR and readout pitch on position resolution. For the MLP architecture, several combinations of hidden layers and units were assessed. Considering the position resolution, we selected 3 hidden layers and 256 hidden units in this study. The ideal SPTR σ = 0 ps retrieved a negligible difference of FWHM on the xy plane between the methods based on MLP and CoG. In contrast, FWTMs and CHHMs on the xy plane were 1.81 and 0.28 mm for the MLP and 5.03 and 0.87 mm for the CoG method, respectively. For SPTR σ = 10 and 100 ps, the MLP retrieved larger FWHMs than the CoG method, and the CHHMs of the CoG were 1.34 and 1.07 times higher than those of the MLP for the two SPTR values, respectively. Both FWHM and FWTM of the CoG method showed better results than those of the MLP given its sharp distribution deviated from the Gaussian distribution. In contrast, the CHHMs of MLP retrieved better results than those of the CoG method because we evaluated the cumulative event considering histogram shapes. This may be because MLP removes anomalous photons more accurately than the anomaly detection method proposed by Ota et al (2018). These results indicate that estimation on the xy plane using MLP would be comparable to that obtained from the CoG method. According to figure 8, the error of the CoG method tends to degrade around the incident surface; in contrast, the distribution of the MLP method tries to narrow down all the DOIs to minimalize the MSE of the training dataset. In order to further improve the performance, we need to train the each DOI separately.

Along the z axis, for SPTR σ = 0, 10, and 100 ps, FWHMs of the PCA were 3.66, 3.07, and 2.34 times higher than those of the MLP, respectively. In addition, CHHMs of the PCA were 2.59, 2.00, and 1.58 times higher than those of the MLP, respectively. The SPTR had a negligible effect on the resolution of the z axis compared with that of the xy plane. Furthermore, it seems that the deep neural network expresses features not obtained by PCA. In fact, PCA can only describe linear relations between the features and vectors. However, the relation between the DOI and the time–space information of the detected Cherenkov photons may be nonlinear as described in section 1. Therefore, the nonlinear features of time–space information may deteriorate the performance. Actually, an opposite trend is observed with the PCA method. The position resolution was slightly affected by the readout pitch but retrieves better results than the PCA method. These results indicate that deep neural networks can be utilized for 3D reconstruction of interaction position.

In this study, Monte Calro simulation was not considered practical conditions, e.g. dark noise, afterpulsing, cross-talk, PDE. This is a limitation of this study. In addition, we only estimated the 3D interaction position, but time information such as the coincidence time resolution is also important for PET, and its estimation has been recently improved using a convolutional neural network (Berg and Cherry 2018). The 3D position information obtained from our proposed method can be used for correcting the time variation between interaction and photon detection and further improve the estimation of time information.

Although some researchers refer to deep neural networks as black boxes, they have been visualized and analyzed in detail (Gallagher and Downs 2003, Suzuki 2017). In future developments, we will conduct these visualizations on our proposed network. In addition, we will experimentally evaluate the proposed method using collimated gamma rays on experimental systems consisting of PbF₂ crystals and photodetector arrays such as multi-anode photomultiplier tubes or multipixel photon counters.

This study showed that it is feasible to estimate the interaction position of 511 keV photons in the Cherenkov-based detectors using deep neural networks. In the Monte Carlo simulation, compared to the existing estimation method based on CoG and PCA, the proposed method improves position resolution on the xy plane and along the z axis. In future work, we will experimentally evaluate the proposed method.

The authors would like to thank the staff of the Central Research Laboratory of Hamamatsu Photonics K. K.

The authors have no conflicts of interest to disclose.