Deep residual-convolutional neural networks for event positioning in a monolithic annular PET scanner

Abstract

PET scanners based on monolithic pieces of scintillator can potentially produce superior performance characteristics (high spatial resolution and detection sensitivity, for example) compared to conventional PET scanners. Consequently, we initiated development of a preclinical PET system based on a single 7.2 cm long annulus of LYSO, called AnnPET. While this system could facilitate creation of high-quality images, its unique geometry results in optics that can complicate estimation of event positioning in the detector. To address this challenge, we evaluated deep-residual convolutional neural networks (DR-CNN) to estimate the three-dimensional position of annihilation photon interactions. Monte Carlo simulations of the AnnPET scanner were used to replicate the physics, including optics, of the scanner. It was determined that a ten-layer-DR-CNN was most suited to application with AnnPET. The errors between known event positions, and those estimated by this network and those calculated with the commonly used center-of-mass algorithm (COM) were used to assess performance. The mean absolute errors (MAE) for the ten-layer-DR-CNN-based event positions were 0.54 mm, 0.42 mm and 0.45 mm along the x (axial)-, y (transaxial)- and z- (depth-of-interaction) axes, respectively. For COM estimates, the MAEs were 1.22 mm, 1.04 mm and 2.79 mm in the x-, y- and z-directions, respectively. Reconstruction of the network-estimated data with the 3D-FBP algorithm (5 mm source offset) yielded spatial resolutions (full-width-at-half-maximum (FWHM)) of 0.8 mm (radial), 0.7 mm (tangential) and 0.71 mm (axial). Reconstruction of the COM-derived data yielded spatial resolutions (FWHM) of 1.15 mm (radial), 0.96 mm (tangential) and 1.14 mm (axial). These findings demonstrated that use of a ten-layer-DR-CNN with a PET scanner based on a monolithic annulus of scintillator has the potential to produce excellent performance compared to standard analytical methods.

1. Introduction

The first positron emission tomography (PET) scanners consisted of rings of discrete detectors, typically single crystals of bismuth germanate (BGO) (Bi₄Ge₃O₁₂) coupled to individual photomultiplier tubes (Hichwa 1985). Eventually, one-to-one coupling was replaced by the block detector concept (Casey and Nutt 1986), where a large number of discrete detector elements were coupled to relatively few photodetectors, often position-sensitive photomultiplier tubes (PSPMTs) and, more recently, arrays of silicon photomultipliers (SiPMs). Determination of the element struck by an annihilation photon is achieved by using a signal weighting method similar to that employed by Anger in early gamma cameras (Anger 1958). Alternatively, PET detectors can consist of monolithic pieces of scintillators coupled to arrays of photon detectors (Borghi et al 2016a, 2016b, Gonzalez et al 2016, Krishnamoorthy et al 2018). For example, the Bruker Albira preclinical PET scanner utilizes rings of eight continuous scintillator detector modules. Each module consists of a large, truncated pyramid shaped lutetium–yttrium oxyorthosilicate (LYSO) (Lu_2(1−x)Y_2xSiO₅) crystal coupled to arrays of SiPMs (Sanchez et al 2013). The reported spatial resolution of the scanner is ~1.5 mm; absolute detection sensitivity at scanner center is 2.0%. Additionally, the MOLECUBES-CUBE scanner uses detectors based on 8 mm thick LYSO coupled to SiPMs. Its spatial resolution in all three dimensions is ~1 mm; detection sensitivity is 12.4% (Krishnamoorthy et al 2018). Note that LYSO is often used in these applications due to its high light output relative to other scintillators such as BGO.

The potential to accurately identify the three-dimensional position of photon interaction points makes use of monolithic scintillator especially attractive in preclinical imaging applications requiring high spatial resolution. In pixelated detectors, scintillation light is confined to the physical extent of the detector element by reflective material separating the elements (figure 1(a)). Location of the interaction in the plane of the detector is estimated by calculating the center-of-mass (COM) of the two-dimensional light distribution detected by arrays of photosensors. This information is then utilized with a previously measured look-up table to determine the element that was struck by the annihilation photon. Thus, the intrinsic resolution of the scanner is limited by the physical size of the individual detector elements and errors in calculating the position of the light distribution centroid (Tomic et al 2005). Depth-of-interaction (DOI) of an annihilation photon in the detector can be estimated by comparing the intensity of light detected at both ends of the scintillator elements (Yang et al 2008). Hence, the number of photodetectors is doubled, increasing cost substantially. Other DOI methods for use with discrete scintillator elements include use of multilayer detectors (Liu et al 2001, Li et al 2018) and addition of triangular reflectors between the elements (Miyaoka and Lewellen 1998, Lewellen et al 2004, Ito et al 2010).

Figure 1.

Schematics showing (a) the confined light distribution in a pixelated scintillator and (b) the shape and peak intensity of interactions occurring in a continuous slab of scintillator at different interaction depths. Note that the shape of the photon distribution in the pixelated detector is not affected by interaction depth, unlike the continuous detector.

In detectors based on monolithic plates of scintillator, DOI can be estimated using the shape of the light pulse produced by the interaction of the annihilation photon in the scintillator (Pani et al 2011). The width of the photon distributions broaden, and peak intensities falls as interaction points move farther away from the photon detectors (figure 1(b)). DOI can be estimated by comparing the width of the photon distribution to a previously measured table. Planar position is estimated by calculating the COM of the light distribution and applying a previously measured lookup table, similar to the method used in pixelated detectors. Spatial resolution is not limited by the size of detector elements as in pixelated systems. It is, however, affected by the accuracy of identifying annihilation photon interaction points in the scintillator. One of the most common drawbacks in using continuous scintillator is difficulty in accurately identifying event positions close to the edges of the detector due to distortion of the photon distribution caused by reflection or absorption in these regions. Another complication is the often-laborious task of calibrating the event positioning algorithms. This task usually requires the acquisition of data from numerous locations utilizing specialized radiation sources and analyses (Hunter et al 2009, Miyaoka et al 2010).

In addition to the analytic methods described above, statistical methods have also been used to estimate interaction points in monolithic scintillation detectors. Perhaps the most mature of these efforts is application of the k-Nearest Neighbor algorithm (k-NN) (Maas et al 2006, van Dam et al 2011, Borghi et al 2015). This method compares the light distribution of an unknown event with distributions from a large dataset of reference events. The reference events most similar to the unknown events are selected and used to estimate event position (k is the number of events used in the comparison). Since its initial application, a number of investigators have worked to continue the adaption of k-NN to position estimation. The downside to this method is the often extended amount of time required to calculate positions. Additionally, these networks require large reference data sets and are most appropriate for general classification tasks.

The maximum-likelihood (ML) algorithm has also been used to estimate event positions in continuous scintillator detectors (Ling et al 2006, 2007, Barrett et al 2009, van Dam et al 2013, España et al 2014, Lee and Lee 2015, Morrocchi et al 2016, Park and Lee 2019). Briefly, the ML method assumes a set of independent measurements of photon distributions made at some position (x or y) in the detector that can be represented by Gaussian functions with means μ_i, and standard deviations σ_i. The position p (x or y) of an event is determined by minimizing a likelihood estimator. This method has been shown to produce good spatial resolution results. For example, Miyaoka et al reported average resolutions of 1.05 mm (radial), 0.99 mm (tangential) and 1.24 mm (axial) full-width-at-half-maximum (FWHM) for cMiCe detectors (Miyaoka et al 2011).

Finally, neural networks (NN) designed to emulate the neurons of a human brain have also been applied to the task of estimating event positions in continuous scintillator detectors (Bruyndonckx et al 2004, Wang et al 2013, Conde et al 2016, Müller et al 2018, Iborra et al 2019). Artificial NNs (ANN) are perhaps the most common architecture applied to this application. ANNs consist of an input layer, a hidden layer and an output layer that are trained to recognize outputs from PSPMTs or SiPMs. This training usually requires a large data set that connects output signals with known event positions (Wang et al 2018). Convolutional NNs (CNNs) are used in computer vision and image processing applications due to their ability to identify image features, often in facial recognition applications (He et al 2016, 2019). CNNs have also been used to denoise CT and PET images (Zhao et al 2019, Gong et al 2019a) and estimate event positions in continuous scintillator (LaBella et al 2021). CNNs, like most NNs that utilize back-propagation during training, are subject to the vanishing gradient problem (VGP). The gradients carry information used to update the CNN kernel. As these gradients become very small, network parameter updates become almost insignificant during training, which inhibits learning. Hence, the VGP results in limiting a network’s ability to extract features. An effective method to address the VGP is to add layers and residual (also known as identity) connections, creating a deep, residual CNN (DR-CNN). A deep network contains multiple layers between the input and output layers. The residual connections allow information from earlier parts of the network to be passed to the deeper parts. In other words, these connections help reduce the VGP by short-circuiting shallow layers in favor of deeper ones. DR-CNNs have been applied to aid in evaluation of PET images to detect Parkinson’s disease (Zhao et al 2019), classify onco-PET images (Kawauchi et al 2020) and reconstruct PET images (Gong et al 2019b), but not as event position-estimators in annular PET scanners.

In this investigation, we explored application of deep CNNs to the estimation of the three-dimensional positions of annihilation photon interactions in a preclinical PET scanner based on a faceted monolithic annulus of LYSO (called AnnPET) currently under construction (Stolin et al 2016, 2017, Gonzalez et al 2018). The ability of these NNs to extract features from imagery is particularly well-suited to analyses of the unique photon distributions produced by AnnPET. For example, the optics at the interface of two facets perturbs photon transport to the SiPMs, distorting the resulting photon distributions. Additionally, the curved inner surface of the annulus reflects some photons back into the scanner to improve the overall light signal. These photons create non-uniform background signal. Thus, the goal of this investigation is to identify and evaluate a deep CNN architecture for use with a preclinical PET scanner based on a faceted, monolithic annulus of scintillator currently under construction by our group.

2. Methods

2.1. Detector modeling

The AnnPET scanner was modeled using the GEometry ANd Tracking 4 (GEANT4) platform (Agostinelli et al 2003), with the G4SiPM simulation package (Niggemann et al 2015). This software has been validated in the simulation of the physics and optics of monolithic LYSO-SiPM based detectors (van der Laan et al 2010, Játékosa et al 2016, Ahmed et al 2020). The annular LYSO scintillator used in AnnPET was 7.2 cm long with a maximum outer diameter of 8.2 cm and inner diameter of 6 cm (appropriate for imaging of mice). Fourteen, 18.25 mm wide, facets were equidistantly positioned around its outer surface. The minimum transaxial scintillator thickness was ~1 cm (figure 2). The inner surface was covered with a specular reflector. Both ends of the annulus were coated with black paint. The reflective coating enhances the collection of photons, while the absorptive coating reduces reflection of photons at the ends of the detector, minimizing distortions of the photon distributions in these regions. Each facet was coupled to a simulated 4 × 16 array of 4 mm² SiPMs (4 × 4 Hamamatsu Photonics, S14160/S14161–4050HS MPPCs, pitch = 4.4 mm); photon detection efficiency (PDE) at 420 nm was ~50% (Hamamatsu Photonics 2020). The LYSO optical photon/energy conversion ratio was 32 000 photon MeV⁻¹, the pulse decay time was 42 ns (Mouhti et al 2019). The SiPMs were readout individually, replicating the actual AnnPET scanner. Thus, each facet produced sixty-four individual outputs. The SiPMs were coupled to the scintillator with an optical Silastic (Sylgard 184, Dow Silicones, Corp., Midland, MI). Its presence and properties were included in the simulation. To create the network training sets, a simulated beam of 511 keV photons was scanned across the surface of a single facet to ensure that the whole region was exposed. Data set sizes ranged from 0.5 to 3.7 M annihilation photons. For all simulation studies, a 200 keV energy threshold was applied to emulate the operating parameters of the AnnPET scanner.

Figure 2.

Schematic drawing of the AnnPET scanner.

There are three possible fates of an annihilation photon in scintillator: transit without interaction, photoelectric absorption (PA) and Compton scatter (CS). Transport of the optical photons produced by photoelectric and Compton scatter interactions were tracked until they exited the scintillator or were incident on the surfaces of the SiPMs. Intrinsic radiation present in LYSO and SiPM noise were not simulated. The number of optical photons striking the SiPMs on the primary interaction facet and a facet to each side of the primary facet was recorded to produce an optical photon distribution. Inclusion of a single facet to each side of the primary was based on a previous analysis that showed approximately 90% of the photons that exited the scintillator were confined to these three facets (Stolin et al 2017). Each optical photon distribution was converted to a 16 bit 12 × 16 Tagged Image File Format (TIFF) file.

Perhaps the main challenge of using optical photon distributions to estimate event position is the different interactions that can occur in the scintillator. As noted above, an annihilation photon can interact with the scintillator via PA or CS. In PA, the incoming photon transfers all of its energy to an electron, resulting in emission of a pulse of optical photons. This interaction produces a single, well-defined, optical photon distribution (figure 3(a)). In CS, the incoming photon transfers a fraction of its energy to an electron, resulting in a pulse of optical photons. The number of optical photons produced is smaller compared to that created by a photoelectric interaction since less energy was deposited in the scintillator. The scattered photon can either exit the scintillator without any further interaction in the scintillator (figure 3(b)) or interact via the photoelectric effect (figure 3(c)). In this case, a dual photon distribution is produced. It is possible for the scattered photon to undergo additional scattering interactions, but those events are relatively rare (Pino et al 2014). Note, that shallower events produce wider optical distributions. This information can be used to estimate DOI.

Figure 3.

The three most likely interactions of an annihilation photon in scintillator. (a) Photoelectric absorption, (b) Compton scatter with escape and (c) Compton scatter with photoelectric absorption. For each, the relative size and shape of the expected light distributions are shown. Shallower events produce wider photon distributions.

The geometry of AnnPET results in some unique advantages to the creation of preclinical PET images. For example, use of continuous scintillator makes it possible to achieve high spatial resolution, perhaps submillimeter, without using rings of individual pixelated detector modules consisting of arrays of very small scintillator elements, reducing cost. A downside to use of continuous detectors, however, are distortions of photon distributions at the edges of the scintillator due to optical effects (Yoshida et al 2011), making identification of event position difficult in these regions. The lack of detector module edges in the monolithic scintillator design eliminates this effect, potentially enhancing scanner performance. This geometry, however, produces its own optical aberrations that complicate event positioning. For example, photon distributions at the interface between two facets are distorted (figure 4), making the application of analytic methods like COM challenging. Furthermore, the small bore of the AnnPET scanner (6 cm) necessitates the ability to accurately measure DOI to reduce parallax distortions. Since AnnPET employs single-ended readout, analysis of photon distribution shape must be used to estimated DOI, which is difficult with analytic methods. Thus, to maximize the performance enhancements possible with the use of annular scintillator, it is likely that the application of a NN is necessary to produce accurate estimates of event position in the detector.

Figure 4.

Schematic showing light distributions created by photoelectric absorption at the center of a facet (a) and at an interface between two facets (b). Also shown are TIFF images of photon distributions from the simulation.

2.2. Statistical position estimation method: NN

The effectiveness of statistical-based data analysis methods, such as CNNs, in processing complex imagery has led to their use in facial recognition (He et al 2016). Since CNNs are usually applied to the processing of imagery instead of numerical data (Zhang et al 2018, He et al 2019, Zhang et al 2020), we converted the event-position-estimation task from a numerical analysis-based one to an image processing one by creating TIFF images of optical photon distributions detected by the arrays of SiPMs. Residual connections were added to enhance training to produce a deep residual-convolutional neural network (DR-CNN). The software was written using the PyTorch open source, machine learning library running on an Alienware computer with a NVIDIA RTX 3090 GPU (10496 CUDA cores).

Our networks used blocks with convolution layers that either maintained the number of features (the convo-identity block, (figure 5(a))) or doubled the number of features (the convo-pooling block, (figure 5(b))). The convolution layer used a nk × (3 × 3 × nf) kernel: nf is the number of features in the input tensor and nk is the number of features of the output tensor. Increasing the number of features improves the accuracy in classifying objects in the image, which results in increased event positioning accuracy. Convolution layers were followed by batch normalizations and leaky rectified linear unit (ReLU) activation layers that introduce nonlinearities to enhance network training efficiency. The convo-identity block had a residual (identity) connection (right side of the block (figure 5(a))). The left side of the block consisted of convolution layers using a stride and pad of 1. The right side of the convo-pooling block (figure 5(b)) had a residual (identity) connection with an average pooling layer to halve the output tensor dimensions, followed by a convolution layer. In both blocks, the output tensors from the convolution arm (left side of each block) and the residual connections (right side of each block) were summed feature wise (designated by the ⊕ symbol), to make the training more efficient, and the model more accurate. The final step was application of a standard ReLU to return some sparsity to the network and to deactivate some features not relevant to the image, making network training more efficient.

Figure 5.

Schematic diagrams of the (a) convo-identity and (b) convo-pooling blocks.

Figure 6 schematically shows the ten-layer-DR-CNN algorithm. The goal of the network is to utilize optical photon distributions captured by the detector to estimate the position of annihilation photon interaction points in the scintillator (x₁, y₁, z₁, x₂, y₂, z₂); which is required to ultimately produce PET images. The coordinates x₁, y₁, z₁ are the positions of the initial interaction point, x₂, y₂, z₂ are the coordinates of the second interaction point, which will only be non-zero in the case of a Compton scatter followed by PA (figure 3(c)) or a second scatter event. Event processing begins with input of a 12 × 16 TIFF image created by the detector simulation described above. An initial convolution was performed to convert the images into a 12 × 16 × 64 tensor appropriate for processing by the CNN. The ReLU layer was then used to activate feature neurons. The convo-identity block initialized these features. The subsequent three convo-pooling layers concentrate the information present in the 12 × 16 × 64 input tensor by reducing the size of the features, while increasing their number.

Figure 6.

Schematic diagram of ten-layer-DR-CNN algorithm showing the integration of the convolution blocks into the network.

The pooling layer down-samples the 2 × 2 × 512 tensor created by application of three convo-pooling layers. This process made the features more robust to changes of position in the image. Average and maximum pooling were applied in the next step. The pooled tensor was then converted to a vector and normalized to facilitate the final stage of the process, which was performed in the fully connected linear layer. This layer connected the 1024 features to the six position coordinates. Specifically, each feature in the flattened vector (A_i) has a separable weight (P_ij) and bias (b_j) related to the event position vector L_j; where j = 0 to 5, are estimates of the six event coordinates (x₁, y₁, z₁, x₂, y₂, z₂). All of the CNNs tested used this architecture with and without residual connections, and with more or less layers as noted.

The known position of each interaction was encoded in the labels of the TIFF files. During the training process, the algorithm used this information to adjust the convolution kernels, model weights (P_ij) and position biases (b_j) to minimize the value of the mean squared error (MSE) between the known and network-estimated positions (equation (1)).

$$\text{MSE}=\frac{1}{N}\sum _{i=0}^5\sum _{j=0}^{N-1}{\left(v_{o;ij}-v_{ij}\right)}^2,$$ (1)

where, v_o;ij s are the known coordinates of an event, i is the index designating the position on an event’s initial and secondary interactions (x₁, y₁, z₁, x₂, y₂, z₂) and j is the index designating one of the N events in the data set. The v_ij s are the network’s estimate of those coordinates. A study was performed to determine the minimum training data set size necessary to achieve a stable MSE. Training data sets ranged in size from 0.5 to 3.7 M events. As part of this study, the advantages of adding layers and residual connections were investigated by testing four deep CNNs: a five-layer network, a ten-layer networks (with and without residual connections) and a twenty-layer network. The errors calculated for each network were plotted as a function of training set size. During training, of all the networks the same parameters were used: batch size = 128, weight decay = 0.1, dropout factor = 0.5 and learning rate = 0.0001. Choice of these parameters was based on the results from preliminary testing. MSEs were calculated for a validation data set of 500 K events independent from the training data set. We chose to use TIFF files instead of conventional numeric matrices as input to the CNNs because the PyTorch software library has built-in functionality to create training and validation datasets from image files, and then link the known coordinates of events to the corresponding file for training. Thus, use of TIFF images made network training and validation more efficient to implement.

2.3. Analytic position estimation method: COM

In addition to use of CNNs to estimate event positioning, an analytic method, calculation of COM (Poladyan et al 2020), was applied to the TIFF images. The optical photon distributions were projected onto x and y axes to obtain two, 1D distributions. The COMs along each axis were then calculated and used as estimates of the location of the event in the plane of the SiPM arrays (x–y plane). Depth of interaction (z-coordinate) was estimated by first calculating the ratio of the peak magnitude of the distribution (K) to the total number of detected photons within two standard deviations from the peak location (I). If two standard deviations exceeded the boundaries of the 12 × 16 image, the total number of counts in the image was used. This ratio (K/I) was used with equation (2) to calculate DOI (Pani et al 2011).

$$z(\text{mm})=30.0+\left(10.0\times \left(\frac{K}{I}\right)\right),$$ (2)

where, 30.0 mm is the distance from the center of the axis of the scanner (z = 0) to the inner surface of the scintillator and 10.0 is the distance from the inner surface to the front surface of the SiPMs.

2.4. Testing (ten-layer-DR-CNN-derived versus COM-calculated position estimates)

Based on the results of the investigation described in section 2.2, a ten-layer-DR-CNN was determined to be the best choice for use with AnnPET. Mean absolute errors (MAE) comparing known event positions to those estimated with the ten-layer-DR-CNN and those estimated using the COM algorithm were calculated using the equation:

$${\text{MAE}}_i=\frac{1}{N}\sum _{j=0}^{N-1}\left|v_{o;ij}-v_{ij}\right|,$$ (3)

where, v_o;ij s are the known event positions and the v_ij s are estimates of event positions from either the ten-layer-DR-CNN or COM calculation. The index i (i = 0 to 5) denotes the primary and secondary event coordinates (x₁, y₁, z₁, x₂, y₂, z₂), the index j denotes the event number, it spans the validation data set (N = 500 K).

2.5. Resolution measurements

To assess the effect of event positioning accuracy on reconstructed images, the spatial resolution of the simulated AnnPET scanner was measured using the methodology described in the NEMA NU 4-2008 protocol (NEMA 2008). Specifically, a simulated point source of annihilation photons was placed at the center of the scanner and at ¼ of the axial field-of-view (18 mm). For each axial position, the source was placed at four transaxial positions offset from the scanner’s central axis: 5, 10, 15 and 20 mm. Five hundred thousand coincident events were recorded at each location. Event positions were determined using the ten-layer-DR-CNN and COM methods to produce lines-of-response (LOR). An additional set of LORs was created using the known positions of the photon interactions in the scintillator. The LORs were sorted into sinograms with a bin size of 0.5 mm. These data were reconstructed using the 3D-filtered backprojection (3D-FBP) algorithm in the STIR software environment (Thielemans et al 2012) (0.25 mm image voxel size). Spatial resolution was measured from profiles drawn through images of the point source. A unique aspect of the use of the GEANT4-based simulations is that the actual coordinates of the annihilation photon interactions in the detector were known. The profiles measured on the reconstructed images produced from the known event position data are a delta function convolved with a Gaussian kernel that describes effects produced by the 3D-FBP reconstruction algorithm. Thus, if the COM and ten-layer-DR-CNN based images are deconvolved with the 3D-FBP kernel, the result could be used to measure the intrinsic spatial resolution of the scanner due only to the physics of the detector and the errors introduced by the event positioning method (COM or ten-layer-DR-CNN). The convolution/deconvolution process is simplified because the images of the point source can be described by a Gaussian distribution, and because the Fourier transform of a Gaussian function is a Gaussian function. Taking advantage of Fourier domain mathematics and knowledge that the FWHM of a Gaussian distribution is approximately equal to 2.355σ, where σ is the standard deviation of the distribution, the FWHM of the profile of an image of the point source can be expressed as:

$${\text{FWHM}}_{\text{Image}}^2={\text{FWHM}}_{\text{Intrinsic}}^2+{\text{FWHM}}_{3\text{D}-\text{FBP}}^2,$$ (4)

where, FWHM_Image is the measured FWHM of the profile drawn on a reconstructed image of the point source, FWHM_Intrinsic is the FWHM of an image of the point source produced without blurring caused by the reconstruction processes and FWHM_3D-FBP is the FWHM of the Gaussian convolution kernel that describes the blurring effects of the 3D-FBP reconstruction algorithm. FWHM_3D-FBP was measured from images of the point source reconstructed using the known endpoints of the LORs. The σs utilized to calculate FWHM_Image and FWHM_3D-FBP were obtained by fitting profiles of images of the point source.

$${\text{FWHM}}_{\text{Intrinsic}}=\sqrt{{\text{FWHM}}_{\text{Image}}^2-{\text{FWHM}}_{3\text{D}-\text{FBP}}^2}.$$ (5)

3. Results

Figure 7 shows a plot of MSE versus training set size for the four networks tested. The error plateaus at approximately 3 million events for each data set. Thus, a network trained with 3 million events was used for all subsequent estimates of event position. A training session using 3 million events took 7 h for the five-layer network 10 h for the ten-layer networks (with and without residual connections) and 13 h for the twenty-layer network. Based on the relatively short training time and very good event positioning accuracy, it was decided that a ten-layer-DR-CNN was the most appropriate CNN to use with AnnPET. Hence, all subsequent results were obtained using this network.

Figure 7.

Plot of MSE (calculated using a 500 K validation data set) versus the size of the training data set used to train the four networks.

GEANT4 simulations showed that ~45% of the photons that entered the detector interacted with the scintillator (55% exited the detector without interaction). Of these events, ~44% interacted via a single PA, ~19% via single and multiple Compton scatter and ~37% by first a Compton scatter and then PA of the scattered photon. Note that if the 200 keV lower energy threshold is not used, the results are ~35% of the annihilation photons interacted via a single PA, ~43% via single and multiple Compton scatter and ~22% by first a Compton scatter and then PA of the scattered photon. These results are consistent with calculations for a ~1 cm thick piece of LYSO (Hunter et al 2013, Phunpueok et al 2014 and Yawai et al 2017) (the minimum thickness of the AnnPET annulus is ~1 cm). Positioning errors for each of the three types of events photoelectric absorption (PA) (figure 3(a)), CS with escape (figure 3(b)) and Compton scattering with photoelectric absorption (CS + PA) (figure 3(c)), as well as the combination of all three types of interactions are shown in tables 1 and 2. These results demonstrate that the ten-layer-DR-CNN produces more accurate estimates of event position than the COM method. The spatial distributions of these errors were visualized by creating heatmaps in the x–y and x–z planes of the detector (figure 8).

Table 1.

MAE (mm) for calculations of the primary annihilation photon interaction point in the scintillator using the COM technique.

	x 1	y 1	z 1
PA	1.00	0.61	2.81
CS	1.67	0.76	2.53
CS + PA	1.48	1.63	2.78
Combined	1.22	1.04	2.79

Table 2.

MAE (mm) for calculations of the primary annihilation photon interaction point in the scintillator using the ten-layer-DR-CNN.

	x 1	y 1	z 1
PA	0.33	0.21	0.22
CS	0.35	0.23	0.35
CS + PA	0.85	0.66	0.77
Combined	0.54	0.42	0.45

Figure 8.

Heatmaps showing MAE (mm) as a function of position in the x–y and x–z planes: (a) and (b) were calculated using the COM method; (c) and (d) from the ten-layer-DR-CNN.

Spatial resolutions measured from the reconstructed images of the point source as a function of transaxial source offset are shown in figure 9. The plots in figure 10 show the intrinsic spatial resolution of the scanner (FWHM_Intrinsic) as a function of transaxial source offset. The results for the resolution measurements with the source shifted along the scanner’s axis by 18 mm were virtually identical to those shown in figures 9 and 10.

Figure 9.

Plots of spatial resolution measured from images reconstructed using the known event positions, ten-layer-DR-CNN and COM-estimated positions in the radial (a), tangential (b) and axial (c) directions, as a function of source offset.

Figure 10.

Plots of intrinsic spatial resolution (FWHM_Intrinsic) in the radial (a), tangential (b) and axial (c) directions, as a function of source offset.

4. Discussion

To capitalize and expand on the potential advantages of the use of continuous scintillator detectors, we are constructing a preclinical PET scanner based on a monolithic, faceted annulus of scintillator, called AnnPET (Stolin et al 2017). The unique optics of this system, however, present challenges to determining the three-dimensional positions of annihilation photon interactions in the scintillator. Accurately positioning these events are important for producing high resolution images. To achieve this task, we converted the event positioning task from an analytic computational one using the COM algorithm to an image recognition task using CNNs.

In this initial stage of the creation of the AnnPET scanner, we created software to simulate the characteristics of its detector systems. Use of a digital model facilitated detailed analyses of factors contributing to accurate estimates of annihilation event position. Output from the scanner simulation was first used to train the networks. The size of the training data set size has an impact on the accuracy of the model prediction. Therefore, we performed a study to determine the most appropriate data size. It was found that 3 million was adequate for producing accurate position estimates. The results shown in figure 7 also illustrated the consequences of increasing the depth of the CNN by increasing layers, and by adding residual connections. Specifically, increasing the number of layers from five to ten resulted in a substantial increase in event positioning accuracy (~20%). Adding residual connections to this network increased accuracy by ~5%. Doubling the number of layers of this network from ten to twenty improved accuracy by only ~1.0% at the cost of extending training time by ~30%. Thus, we chose to use the ten-layer, DR-CNN because of its shorter training time and good accuracy.

The MAEs of position estimates calculated using the COM method (table 1) and the ten-layer-DR-CNN (table 2), illustrate the challenges that different types of photon interactions in the scintillator present for the analytic method compared to the NN. Positioning errors are caused not only by the nature of the annihilation photon interactions in the scintillator. Location of the event also plays a role because of the spatial dependence of the optics in the scintillator (figure 4). The heatmaps in figure 8 illustrate these effects. For example, figure 8(a) shows that calculations of the y-coordinate with the COM algorithm was most challenging at the corners adjoining two facets (left and right edges of the heatmap) (MAE = ~5 mm). Note that the errors at the ends of the annulus are not noticeably greater than in the central areas of a facet (MAE = ~1.5 mm). These results are likely due to application of light absorbent coating to these surfaces that prevented photon reflection. The accuracy of the K/I-based DOI calculation appears to have a dependence on depth (z-coordinate) (figure 8(b)). Specifically, at shallow interaction depths (<4 mm), where the scintillation light has enough space to distribute over a number of SiPMs, sampling of the distribution by the SiPM arrays was sufficient to produce good estimates of depth (MAE = ~1 mm). Beyond this region, error increased as a function of depth. This behavior was likely a result of the fact that at depths greater than ~4 mm, the width of photon distributions become narrower such that the sampling of the light by the SiPM arrays was reduced, diminishing accuracies of the COM and K/I calculations (equation 2). When events occur deep in the scintillator (>8 mm) this effect became more apparent, illustrated by the regularly spaced bands of increased error along the inner edge (x-axis) of the scintillator (figure 8(b)). These areas correspond to the 0.4 mm gaps between SiPMs where optical photons are not detectable.

The heatmaps in figures 8(c) and (d) show that the MAE, while smaller than that observed for COM, also exhibits some spatial dependance. For example, the error was slightly elevated at edges joining adjacent facets. As described above, the optical photon distributions in these regions are distorted (figure 4(b)), and, while the network ‘learned’ to adapt to these effects, there was enough variety in the nature of the distortions to create some small uncertainties in estimates of the y-coordinate of the event positions (maximum of ~1.2 mm). The photon distributions in these regions are distorted by the junction of two flat surfaces (figure 4(b)). In some cases these distortions create a photon distribution similar to one created by a CS + PA event, causing confusion as to whether an event is a PA at a facet interface or a CS + PA interaction. This ambiguity is perhaps a contributing factor to the position errors at facet interfaces (figure 8(c)) and for CS + PA events (table 2). As with the COM estimates, error at the upper and lower ends of the annulus (x-axis) were not elevated due to the use of photon absorptive coating. The error in DOI (z-coordinate) exhibited a spatial relationship (figure 8(d)), though it was different from that observed for COM. Specifically, there was no band of smaller errors for shallow events. Instead, there was a slight, near monotonic, error increase as annihilation photons interacted closer to the SiPM arrays. This effect was produced by narrowing of the optical photon distribution for deeper events (closer to the SiPMs) compared to shallower ones, which reduced sampling of the photon signal. The lower sampling increased uncertainty in estimates of optical photon distribution shape and DOI. Unlike the COM calculations of depth, there was only subtle structure in the errors close to the SiPM surfaces caused by gaps between devices, which demonstrates the flexibility of the ten-layer-DR-CNN to account for loss of photon signal at gaps between SiPMs. Accurate estimates of event DOI are especially important for AnnPET due to its small-bore size, making it especially susceptible to parallax effect artifacts.

AnnPET’s design is similar to that of the annular scanner reported by Xu et al (2019). This system used a solid annulus of LYSO (inner diameter = 48.5 mm, outer diameter = 58.5 mm and length = 25.1 mm). Scintillation light was collected at both ends of the annulus with a ring of SiPMs. Unlike AnnPET, there were no facets machined on the outer edge of the annulus. Event positioning in the detector was determined using cylindrical coordinates instead of Cartesian coordinates (θ and z (our x-coordinate)). DOI (r-coordinate) in the scintillator was not estimated. Both parameters were computed using an ANN. Due to the differences in coordinate system and the lack of DOI information, it was only possible to compare errors in estimates of the axial position. The reported error was 1.03 mm, compared our combined error of 0.54 mm.

Positioning errors calculated for the ten-layer-DR-CNN were also lower than those reported by investigators applying NNs to event positioning in parallelepiped blocks of scintillator. For example, Sanaat and Zaidi reported in-plane errors of 0.96 mm and 1.02 mm for depth estimates using a deep NN (Sanaat and Zaidi 2020). Wang, et al reported a ~2 mm error for in-plane estimates of event position (DOI error was also ~2 mm (Wang et al 2013)). Our results also compare well with x–y plane positioning errors reported from a simulation study of the effect of SiPM size, PDE and number of readout channels on position estimates in a 50 × 50 × 16 mm³ block of LYSO using the nearest neighbor algorithm (Stockhoff et al 2019). Specifically, our in-plane event positioning error is equivalent to that reported in this work, but our DOI estimates are approximately a factor of three more accurate.

Accurate positioning of events in the detector is perhaps the main contributor to producing good spatial resolution in reconstructed PET images. To assess the effect of the more accurate event positioning achieved with the ten-layer-DR-CNN, the NEMA NU4-2008 spatial resolution measurement protocol was performed using simulated scanner data. The results shown in figure 9 demonstrate the higher resolution (smaller FWHM) achievable with the use of the ten-layer-DR-CNN compared to the COM algorithm. For example, the ten-layer-DR-CNN FWHMs, at a source offset of 5 mm, were 0.79 mm (radial), 0.70 mm (tangential) and 0.71 mm (axial). While for the images reconstructed using COM-derived LORs, the FWHMs were 1.14 mm (radial), 0.96 mm (tangential) and 1.14 mm (axial). Perhaps the most striking aspect of the plots in figure 9 is the sharp increase of FWHM in the radial direction produced using the COM-derived LORs (figure 9(a)). This behavior is due to the relatively large error in estimating DOI and the resulting distortion of the images by the parallax effect. In contrast, the ten-layer-DR-CNN FWHMs are relatively constant as a function of source position due to the low positioning error along the z-axis. These results emphasize the point that, for a preclinical PET scanner with a small bore like AnnPET, the ability to accurately estimate DOI is extremely important to maintain uniform resolution.

The FWHMs for images using the known event positions to create LORs were 0.39 mm (radial), 0.44 mm (tangential) and 0.63 mm (axial). Comparison of these findings to the ones described above for the DR-CNN and COM-derived images illustrates the degradation of resolution induced by the 3D-FBP reconstruction process. Specifically, histogramming data into 3D sinograms with 0.5 mm bins, Ramp filtering and backprojecting into 0.25 mm voxels introduced image blurring. It was possible to remove this effect by deconvolving the Gaussian distributed image profiles with the 3D-FBP blurring kernel. The intrinsic FWHMs for AnnPET resulting from these analyses are shown in figure 10. The ten-layer-DR-CNN FWHMs_Intrinsic for a point source with a 5 mm offset were 0.69 mm (radial), 0.54 mm (tangential) and 0.33 mm (axial) for images created with ten-layer-DR-CNN-derived LORs. These results demonstrate the very high resolution potentially achievable with a PET scanner based on a faceted, monolithic annulus of scintillator used with DR-CNNs. For the images reconstructed using the COM-derived LORs, the FWHMs_Intrinsic were 1.08 mm (radial), 0.85 mm (tangential) and 0.95 mm (axial). While slightly reduced in magnitude compared to the FWHMs_Image, the relationship between radial resolution and source position for the COM results remained virtually unchanged.

As with many studies of the effects of physical phenomenon, this investigation possesses some limitations. For example, the presence of radioactive ¹⁷⁶Lu in LYSO and the inherent noise present in SiPMs were not included in the detector model. These effects produce a relatively uniform background signal that could introduce additional error into the estimation of event positions. Indeed, once the system is complete, measurements of the background signals produced by these effects will be measured and incorporated into our detector model. The potential advantage of use of DR-CNNs is that they can be presented with a training set that includes these effects.

5. Conclusions

The potential to achieve high resolution and accurate DOI estimates make detectors based on continuous scintillator excellent candidates for use as building blocks for preclinical PET scanners. Our group is working to extend application of continuous scintillator to creation of a preclinical scanner using a single annular piece of faceted scintillator. Among the number of challenges related to this task is the accurate positioning of annihilation photon interaction points in the scintillator necessary to produce high resolution images. To achieve this goal, we first determined that a ten-layer-DR-CNN, typically utilized for facial recognition, provided the best trade-off between accuracy and training efficiency for application to AnnPET. This method possessed the flexibility to adapt to signals produced by various types of annihilation photon interactions in the scintillator, as well as the optics of a faceted annulus. For example, we were able to achieve less than ~0.5 mm error for the determination of event positions, resulting in reconstructed spatial resolution of ~0.6 mm FWHM compared to >1 mm FWHM for images produced with COM-based event positioning. The radial resolution of COM-based images demonstrated parallax artifacts due to suboptimal estimates of DOI, not present in the DR-CNN-based images. The simulation created in this study will be used to simplify the calibration procedure since it is capable of replicating our scanner geometry and response. The AnnPET calibration procedure will first require validation of the simulation with actual data from the scanner, and then creation of a method to harmonize the simulation and actual data. While coincidence timing information could be incorporated into our network to aid in estimation of time-of-fight information, there would be little benefit for AnnPET given its small-bore size (6 cm diameter). Finally, the results in this study were obtained utilizing a simulation of the AnnPET scanner that did not include all of the potential sources of signal background present in the actual system. Future versions of the simulation will incorporate these effects once the scanner is completed and the characteristics of the background signal have been identified.