Resolving inter-crystal scatter in a light-sharing depth-encoding PET detector

Abstract

Objective.

Inter-crystal scattering (ICS) in light-sharing PET detectors leads to ambiguity in positioning the initial interaction, which significantly degrades the contrast, quantitative accuracy, and spatial resolution of the resulting image. Here, we attempt to resolve the positioning ambiguity of ICS in a light-sharing depth-encoding detector by exploiting the confined, deterministic light-sharing enabled by the segmented light guide unique to Prism-PET.

Approach.

We first considered a test case of ICS between two adjacent crystals using an analytical and a neural network approach. The analytical approach used a Bayesian estimation framework constructed from a scatter absorption model - the prior - and a detector response model - the likelihood. A simple neural network was generated for the same scenario, to provide mutual validation for the findings. Finally, we generalized the solution to three-dimensional event positioning that handles all events in photopeak using a convolutional neural network with unique architecture that separately predicts the identity and depth-of-interaction (DOI) of the crystal containing the first interaction.

Main Results.

The analytical Bayesian method generated an estimation error of 20.5keV in energy and 3.1mm in DOI. Further analysis showed that the detector response model was sufficiently robust to achieve adequate performance via MLE, without prior information. We then found convergent results using a simple neural network. In the generalized solution using a convolutional neural network, we found crystal identification accuracy of 83% and DOI estimation error of 3.0mm across all events. Applying this positioning algorithm to simulated data, we demonstrated significant improvements in image quality over the baseline, centroid-based positioning approach, attaining 38.9% improvement in intrinsic spatial resolution and enhanced clarity in hot spots of diameters 0.8 mm to 2.5 mm.

Significance.

The accuracy of our findings exceeds those of previous reports in the literature. The Prism-PET light guide, mediating confined and deterministic light-sharing, plays a key role in ICS recovery, as its mathematical embodiment - the detector response model - was the essential driver of accuracy in our results.

1. Introduction

In positron emission tomography (PET), the total interaction cross-section of an incident annihilation photon in the scintillator crystal of the detector is dominated by Compton scatter. For instance, in lutetium yttrium oxyorthosilicate (LYSO) - a common scintillator material used in modern detectors - the cross section for Compton scatter is roughly double that for photoelectric absorption and accounts for 63.4% of the total cross-section [1]. Given that PET detector blocks generally have a length scale on the order of the attenuation length of 511 keV photons, it is highly likely that Compton-scattered photons will interact a second time in the detector block, as their attenuation length falls dramatically with increasing scatter angle in LYSO and other high-Z scintillator materials. These multi-interaction events lead to positioning ambiguity, especially in finely pixelated detectors, where the interactions are more likely to occur in different crystal elements - a phenomenon known as inter-crystal scatter (ICS) [2]. For instance, Lee and Lee reported that 42% of incident annihilation photons undergo ICS in an array of 1.5 × 1.5 × 20 mm³ lutetium oxyorthosilicate crystals - implying that approximately 66% of coincidence events are contaminated by ICS [3].

The positioning ambiguity caused by ICS leads to misidentification of the initial point of interaction within the detector for one or both annihilation photons of a given nuclear disintegration, resulting in activity spillover from the true Line of Response (LOR) into neighboring sinogram bins. This leads to clinically-impactful performance degradation of the imaging systems in terms of contrast, quantitation, and spatial resolution. The impact of ICS has been experimentally quantified, and consequences include: blurring in the system-level impulse response in terms of full width at half maximum (FWHM) and full width at tenth maximum (FWTM) [3, 4]; compression of peak-to-valley ratios across a hot rod phantom [5,6]; reduced visibility of hot and cold lesions [7]; broadening of the coincidence response function [4]; and diminished contrast-to-noise and contrast recovery coefficient [8].

Solutions to ICS-induced degradation in imaging performance generally require two components. First, the detector must resolve the multiple interactions of an ICS event in terms of energy and position - ideally in both the lateral and depth directions. Second, a positioning algorithm must be applied to the set of detected interactions to identify the initial interaction point in the detector. If executed correctly, positioning ambiguity in the ICS event is removed and the detected nuclear disintegration is assigned to the proper LOR.

Extensive research efforts have identified, analyzed, and validated algorithms that position a multi-interaction event according to the resolved positions and energies of its constituent single interactions. Most of these exploit the underlying physics of Compton scatter by using statistical correlations between the order of interactions and their relative energies and positions [8–11], as well as maximum likelihood approaches to identify the interaction sequence most consistent with the kinematics of Compton scatter [12]. Additionally, methods have been developed that avoid the explicit complexities of photon scattering and attenuation by allocating ICS events across possible LORs according to the distribution of single-crystal events [7] or utilizing Monte Carlo simulated data to train a neural network [3,13].

The performance of any positioning algorithm is critically dependent on the detector’s ability to resolve multiple interactions across 4 dimensions: lateral position in the plane normal to photon incidence, depth, and energy [9]. In detectors with true 1:1 coupling, where each pixelated detector element is read out exclusively by a dedicated photosensor, multiple interactions do not seriously degrade resolution beyond the standard single-interaction capabilities of the device. The critical concern is consistency of energy resolution across the lower energies involved in Compton scatter compared with photoelectric absorption. However, such detectors face a complexity versus spatial resolution tradeoff, as the number of readout channels (and associated circuitry) scales with the inverse square of the intrinsic spatial resolution for 2-dimensional positioning, or the inverse cube if 1:1 coupling extends into 3 dimensions. Therefore, most contemporary PET detectors, especially for ultra-high resolution applications, employ some degree of light-sharing to extend performance limits of spatial resolution while managing overall cost and complexity.

In these light-sharing designs, an individual scintillator crystal is optically coupled to multiple photodetectors. The energy and position of interactions within the crystal are encoded in the spatial pattern of coupled photodetector signals. Decoding the spatial pattern generally requires integrating information across the entire extent of the spatial pattern in terms of a sum (energy), weighted average (lateral position), or ratio (interaction depth). This approach enables gains in spatial resolution, as minuscule crystal elements can be deployed with pitch smaller than that of the photosensor by a factor of 2–3 [14]. Additionally, light-sharing allows for depth estimation as different photodetectors couple more strongly than others to the top versus the bottom end of the crystal, and thus their signal ratios correlate strongly with the depth of interaction (DOI) [15,16].

These benefits, however, come at a cost, as each photodetector must also couple to multiple crystal elements - either directly or indirectly via light-sharing. When ICS occurs among crystals that share a coupled photodetector, partial or complete overlap results between their spatial readout patterns, and standard decoding algorithms cannot resolve multiple interactions. For lateral position and energy, standard decoding algorithms return the averaged position and summed energy across interactions. The result is yet worse in the case of depth resolution, as the signal ratios used to estimate DOI become meaningless when readout patterns from multiple crystals are overlaid, resulting in a complete loss of depth resolution.

The severity of this multi-interaction blurring can be quantified by considering the number of crystals that contribute signal to each photodetector - a quantity we will refer to as the optical multiplexing ratio (OMR). The OMR for a particular detector configuration is estimated as the product of two parameters: the crystal-to-sensor ratio (CSR), given as the ratio between the number of crystals and photodetectors in a module, and the light-sharing ratio (LSR), given as the quantity of photodetectors over which each crystal distributes a significant fraction of its scintillation photons. Since both CSR and LSR tend to increase when narrower crystal elements are used in a light-sharing design, ultra-high resolution detectors must contend with significant OMRs. For instance, in a detector with a 4:1 CSR and a 4:1 LSR, each photodetector carries signal components from 16 different crystals - implying that ICS events are highly likely to generate overlapping readout patterns and consequent distortion in position and energy resolution.

Given this extremely challenging landscape, little progress has been made in addressing ICS in light-sharing detectors. Most efforts demonstrating improvements in imaging performance through ICS recovery utilize 1:1 coupling without light-sharing or assume that multi-interaction positions and energies are known [7–9,11,13,17–20]. Efforts to address ICS in light-sharing modules have generally been defined by a sensitivity versus spatial resolution tradeoff, as the ICS events are only identified, not positioned [21,22]. Therefore, improvements in image quality - such as enhanced peak-valley ratios across a hot-rod phantom - come with a proportional loss in sensitivity [5]. Recently, approaches utilizing neural networks and linear optimization have shown promise in both resolving and positioning ICS events in light-sharing detectors [3,23]. However, the accuracy of these methods in resolving the energies and position of the interacting crystals degrades sharply with increasing crystal-photodetector ratio. Further, any improvements in spatial resolution do not extend to radial blurring, as the detector and recovery methods lack DOI localization capability.

Prism-PET - a segmented light guide that enhances spatial and depth-of-interaction (DOI) resolution in multi-crystal, single-ended readout PET modules - is well positioned to address this challenge in a cost-effective manner [14, 24, 25]. Excellent event localization has been demonstrated for single-crystal events - where the incident photon deposits its energy within a single crystal - in Prism-PET modules with both 4:1 and 9:1 coupling between scintillator crystals and photodetectors [15,26–28].

This performance is enabled by a segmented light guide comprised of an array of prismatoid reflectors offset from the underlying array of photodetectors; this creates a unique readout pattern for each crystal in the detector block. We postulate that this controlled, confined, and deterministic light-sharing scheme may be leveraged to identify multi-crystal events, resolve the individual interaction coordinates, and accurately localize the first interaction.

2. Theory

In light-sharing depth-encoding detectors, the coordinates of interaction of the incident annihilation photon - spatial position and energy - must be decoded from the observed photodetector signal. For 2D pixelated arrays, interactions are aggregated by crystal, and their coordinates specified by the pair of vectors $(\mathbf{E},\mathbf{z})$, where $\mathbf{E}={\left[E_1,\cdots ,E_n\right]}^{\mathrm{T}}$ and $\mathbf{z}={\left[z_1,\cdots z_n\right]}^{\mathrm{T}}$ specify the total energy deposit and energy-averaged DOI, respectively, in each of the $n$ crystals of the array. For such detectors, there are well-known and validated decoding algorithms that estimate the interaction coordinates $(\mathbf{E},\mathbf{z})$ from the detector response $\mathbf{y}={\left[y_1,\cdots ,y_m\right]}^{\mathrm{T}}$ giving the observed signal on each of the $m$ photodetectors in the array [14]. However, such methods can estimate only a single set of interaction coordinates and are unable to resolve the multiple interactions of an ICS event - returning $(\mathbf{E},\mathbf{z})$ vectors with a non-zero entry for only one crystal.

We therefore develop an analytical decoding algorithm for ICS events that estimates coordinates of interaction $(\mathbf{E},\mathbf{z})$ from the photodetector signal $\mathbf{y}$. Working in a Bayesian framework, we first develop a forward model of the system, mapping interaction coordinates to photodetector readout: $(\mathbf{E},\mathbf{z})\to \mathbf{y}$. Next, we model the spatial relation between scatter and absorption in an ICS event, using only the underlying physics and geometry. Using these two components, we then create a likelihood function for the observed detector response $ℒ(\mathbf{E},\mathbf{z})$ and a prior probability distribution for the ICS interaction coordinates $P_{\mathit{prior}}(\mathbf{E},\mathbf{z})$. Their product then estimates the posterior probability of the interaction coordinates $P_{\mathit{MAP}}(\mathbf{E},\mathbf{z})$ incorporating both the observed detector response and prior knowledge. A maximum a posteriori (MAP) estimate for the interaction coordinates is then obtained by maximizing the posterior probability over the $(\mathbf{E},\mathbf{z})$.

$$[\stackrel{\mathbf{ˆ}}{\mathbf{E}},\stackrel{\mathbf{ˆ}}{\mathbf{z}}{]}_{MAP}=\underset{\mathbf{E},\mathbf{z}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\left[ℒ(\mathbf{E},\mathbf{z})\cdot P_{\mathit{prior}}(\mathbf{E},\mathbf{z})\right]$$ (1)

2.1. Detector response model

2.1.1. General structure of the model.

We consider the Prism-PET detector module as a linear system, mapping energy deposit in $n$ scintillator crystals, $\mathbf{x}={\left[E_1/E_{\gamma },\dots ,E_n/E_{\gamma }\right]}^{\mathrm{T}}$, onto signal recorded on $m$ photodetectors, $\mathbf{y}={\left[y_1,\dots ,y_m\right]}^{\mathrm{T}}$, where $E_{\gamma }$ is the energy of the incident annihilation photon (511keV). This mapping is typically described by a simple matrix equation,

$$\mathbf{y}=\mathbf{A}\mathrm{x};\mathbf{A}=\left[\begin{array}{lll}\mid \hfill & \hfill & \mid \hfill \\ {\mathbf{a}}_1\hfill & \cdots \hfill & {\mathbf{a}}_n\hfill \\ \mid \hfill & \hfill & \mid \hfill \end{array}\right]$$ (2)

where the $j$^th column of the response matrix, ${\mathbf{a}}_j$, is given by the characteristic system response to a photoelectric event in the $j$^th crystal element.

In order to incorporate the depth-dependence of the response in a Prism-PET module, we make two modifications to this simple framework. First, the impulse response for a particular crystal is described by two characteristic response vectors $\left({\mathbf{a}}_j^{\downarrow },{\mathbf{a}}_j^{\uparrow }\right)$ instead of one, corresponding to the response from optical photons propagating towards opposite ends of the crystal; downwards, directly into the photodetector array, and upwards into the prismatoid light guide. Second, a depth-dependent weighting factor $\left(w_j^{\downarrow },w_j^{\uparrow }\right)$ is attached to each characteristic response to account for the attenuation of optical photons. The impulse response is then given as a weighted sum of characteristic response vectors,

$${\mathbf{y}}^{\left(j\right)}=w_j^{\downarrow }{\mathbf{a}}_j^{\downarrow }+w_j^{\uparrow }{\mathbf{a}}_j^{\uparrow }$$ (3)

where the weighting factors are given as

$$w_j^{\downarrow }=\mathrm{e}\mathrm{x}\mathrm{p}\frac{-\left(L-z_j\right)}{\lambda };w_j^{\uparrow }\mathrm{e}\mathrm{x}\mathrm{p}\frac{-z_j}{\lambda }$$ (4)

for crystals of length $L$ with an effective attenuation length $\lambda$ for optical photons. The full detector response can then be expressed in matrix form by packaging the characteristic response vectors into matrices $\left({\mathbf{A}}^{\downarrow },{\mathbf{A}}^{\uparrow }\right)$

$${\mathbf{A}}^{\downarrow }=\left[\begin{array}{ccc}\mid & & \mid \\ {\mathbf{a}}_1^{\downarrow }& \cdots & {\mathbf{a}}_n^{\downarrow }\\ \mid & & {\mid }^{\mathrm{\text{'}}}\end{array}\right];{\mathbf{A}}^{\uparrow }=\left[\begin{array}{ccc}\mid & & \mid \\ {\mathbf{a}}_1^{\uparrow }& \cdots & {\mathbf{a}}_n^{\uparrow }\\ \mid & & \mid \end{array}\right]$$ (5)

and the DOI-weighting factors into vectors.

$${\mathbf{w}}^{\downarrow }=\left[\mathrm{e}\mathrm{x}\mathrm{p}\frac{-\left(L-z_1\right)}{\lambda },\dots ,\mathrm{e}\mathrm{x}\mathrm{p}\frac{-\left(L-z_n\right)}{\lambda }\right];{\mathbf{w}}^{\downarrow }=\left[\mathrm{e}\mathrm{x}\mathrm{p}\frac{-z_1}{\lambda },\dots ,\mathrm{e}\mathrm{x}\mathrm{p}\frac{-z_n}{\lambda }\right]$$ (6)

The matrix equation for the response is then compactly expressed as

$$\mathbf{y}=\left[{\mathbf{A}}^{\downarrow }\cdot \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\mathbf{w}}^{\downarrow }\right)+{\mathbf{A}}^{\uparrow }\cdot \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\mathbf{w}}^{\uparrow }\right)\right]\cdot \mathbf{x}$$ (7)

where the response operator may be understood as a DOI-weighted sum of the characteristic response matrices.

The standard form of the linear system model for detector response [3] - dependent only on energy deposit - is easily obtained from (7) by writing the characteristic response of the $j$^th crystal as the expectation over $z_j$ of the depth-dependent impulse response, ${\stackrel{-}{\mathbf{a}}}^{\left(j\right)}=\mathbb{E}\left[{\mathbf{y}}^{\left(j\right)}\right]$. A simplified detector response matrix $\stackrel{-}{\mathbf{A}}$ is then formed from these DOI-averaged response vectors, and the system response given simply as

$$\stackrel{-}{\mathbf{y}}=\stackrel{-}{\mathbf{A}}\mathbf{x}$$ (8)

2.1.2. Quantifying the matrix elements.

Since the characteristic response vectors for downward and upwards propagation cannot be measured independently, their values are inferred from the DOI dependence of the impulse response vector. Using Monte Carlo simulation, the impulse response for each crystal ${\mathbf{y}}^{\left(j\right)}$ is simulated 1600 times, uniformly sampling the full range of DOI, $z_j$. Each component of the impulse response, $y_i^{\left(j\right)}$, is then modeled as a function of its DOI, $z_j$,

$$y_i^{\left(j\right)}\left(z_j;a_{i,j}^{\downarrow },a_{i,j}^{\uparrow }\right)=a_{i,j}^{\downarrow }\mathrm{e}\mathrm{x}\mathrm{p}\frac{-\left(L-z_j\right)}{\lambda }+a_{i,j}^{\uparrow }\mathrm{e}\mathrm{x}\mathrm{p}\frac{-z_j}{\lambda }$$ (9)

and the corresponding components of characteristic response $(a_{i,j}^{\downarrow },a_{i,j}^{\uparrow })$ are estimated as parameters of a non-linear least squares fit. In this way, the characteristic response vectors can be constructed for each crystal in the array, building up the entire detector response matrix.

This method may be applied in a brute-force manner to calculate the fitting parameters for each of the m × n matrix elements. However, such an approach is computationally expensive and, more importantly, fails to incorporate the key feature of Prism-PET: confined, prismatoid-mediated light-sharing. We will formalize this concept and illustrate that the global m × n detector response matrix is comprised of compact, localized coupling units.

2.1.3. Coupling Units.

The generalized response matrix $\mathbf{A}$ provides a simple and robust analytical framework. However, its global interconnectedness - where any crystal may couple to any photodetector with any coupling strength - is a poor description of the confined and deterministic response in the Prism-PET module. We tailor the model by contextualizing the arbitrary response $a_{i,j}$ within the specific geometry of segmented prismatoids, crystals, and photodetectors that comprise the detector.

We center our analysis around the prismatoids of the segmented light guide, as they mediate crystal-photodetector coupling. Consider a prismatoid in the detector module, indexed as $k$. It defines a set of coupled crystals $C_k=\left\{c_{kj}\right\}$ where the index $kj$ refers to the $j$^th crystal coupled to the $k$^th prismatoid. The collection of all such sets forms a partition of the set $C=\left\{c_1,\dots ,c_N\right\}$ of all crystals in the detector module. Each set $C_k$ is defined by the geometry of the prismatoid and is easily visualized, with 4 elements for center prismatoids, 2 for edge, and 5 for corner (figure 1). Stimulus within any element of the set results in optical photon output from all elements in the set, and only from those elements within the set. Thus we assume that each crystal in $C_k$ couples to the same set of photodetectors $P_k$ (note that the collection of all $P_k$ does not partition the complete set of photodetectors due to the 4:1 crystal: photodetector ratio). Unlike $C_k$, which is explicitly defined by the geometry of the device, $P_k$ is determined by the nature of optical photon transport across the gap between the scintillator crystals and the photodetector array. Thus, in estimating the members of $P_k$ we consider three factors. First, the optical coupler between crystal and photodetectors is chosen to minimize changes in the index of refraction. Second, the gap is relatively small at approximately 100μm. Third, photons emerging out the bottom of a crystal are broadly distributed (±90°) relative to normal incidence. Taken together, we consider the light transport across this gap as determined by a collection of point sources spread uniformly across the bottom of each crystal in $C_k$, projecting light downwards onto the photodetector array. The set of coupled photodetectors $P_k$ is then determined by the downwards projection of the footprint of the crystal group, plus a border determined by the lateral light spread.

Figure 1:

The light-sharing depth-encoding Prism-PET detector module, and its constituent coupling units - center, edge, and corner - depicted left-to-right.

We estimate the extent of this light spread on the photodetector array by considering a point source at the edge of a crystal emitting photons at ±85° relative to normal incidence. Given a crystal-photodetector gap of 0.1mm, the projection extends approximately 1.14mm from the edges of the crystals (0.1mm × tan85° ≈ 1.14mm). Thus the coupled photodetectors $P_k$ may be identified by drawing a 1.2mm border around the footprint of the crystal group $C_k$: those falling partially or fully within this border comprise the set $P_k$. However, for any prismatoid coupling scheme $k$ in the Prism-PET module - center, edge, or corner - inclusion of this lateral light-spread border does not expand the set of coupled photodetectors, as any photodetector within 1.2mm of the edges of crystal $c_{kj}$ sits directly below another crystal in the same group $C_k$.

This collection of a prism $k$, its coupled crystals $C_k$, and their coupled photodetectors $P_k$ comprises a coupling unit - the localized unit of response in a Prism-PET module. All such units may be mapped to one of three archetypes by simple translation or rotation: a center coupling unit comprised of 4 crystals and 4 photodetectors, an edge unit with 2 crystals and 2 photodetectors, and a corner unit with 5 crystals and 3 photodetectors (figure 1). Response to crystal stimulus within a given coupling unit is therefore given by a compact, localized response matrix of dimensions 4 × 4, 2 × 2 and 3 × 5 for center, edge, and corner coupling units, respectively. Further, symmetries within each coupling unit allow matrix elements to be recycled. Thus only 4 unique coupling elements are required for the center coupling unit, 2 for the edge, and 9 for the corner.

2.2. Scatter-absorption model

Next, we develop a simple analytic model for the spatial relation between the scatter and absorption of an incident annihilation photon in an ICS event, incorporating the geometric constraints of the device alongside the interaction physics. This enables us to contextualize the well-known physics of Compton scatter, develop a more precise understanding of the scatter trajectories that comprise ICS events, and ultimately construct an informed Bayesian prior for the analytical ICS decoding algorithm, which seeks to estimate the interaction coordinates from the observed photodetector readout.

Consider a 511keV photon, perpendicularly incident on a pixelated crystal array, that deposits its energy in two interactions across two distinct crystal volumes: an initial Compton scatter in the first volume followed by photoelectric absorption in the second (figure A1). The ICS geometry is specified by the sequence of interacted crystals, $\tau$, along with the geometric parameters of the detector module (e.g. crystal width, height). For instance, an ICS event with energy deposition in crystals $j$ and $k$ permits 2 ICS geometries: ${\tau }_1=\{j,k\}$ and ${\tau }_2=\{k,j\}$. The spatial relation between scatter and absorption interactions may then be described in terms of the DOI, $z_1$, and scatter angle, $\theta$, of the initial Compton interaction - the absorption DOI, $z_2$, may be inferred from $\left(z_1,\theta \right)$ and the scatter geometry, given long narrow crystals.

Figure A1:

Schematic description of the Scatter-absorption model. The spatial relation between scatter and absorption in an ICS event is modeled by requiring an initial Compton interaction followed by photoelectric absorption, with geometric constraints applied to each interaction. Parameters of the model are illustrated in the drawing.

We therefore develop a probability density function (PDF) for ICS events in a specified geometry over the space of initial DOI and scatter angle, $P_{ICS}\left(z_1,\theta \right)$. This distribution has three components: the probability of a Compton interaction along the incident trajectory $P_{\mathit{comp}}\left(z_1\right)$, the probability of scatter through angle $\theta$, and the subsequent probability of photoelectric absorption within the scatter volume $P_{\mathit{phot}}\left(z_1,\theta \right)$. The first, given by the Beer-Lambert relation, is proportional to

$$P_{\mathit{comp}}\left(z_1\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-z_1/{\lambda }_c\right)$$ (10)

where ${\lambda }_c$ is the attenuation length for Compton interactions, obtained from published databases [1] for 511keV photons in LYSO. The second is obtained from the Klein-Nishina differential scattering cross-section and is proportional to

$$P_{kn}(\theta )=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{(2-\mathrm{c}\mathrm{o}\mathrm{s}\theta {)}^3}(1+(2-\mathrm{c}\mathrm{o}\mathrm{s}\theta {)}^2-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta (2-\mathrm{c}\mathrm{o}\mathrm{s}\theta ))$$ (11)

where the details of the calculation are given in Appendix A.2. The final component may again be obtained from the Beer-Lambert relation as

$$P_{phot}\left(z_1,\theta \right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-r/{\lambda }_p\right)\times \left(1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{\Delta }r/{\lambda }_p\right)\right)$$ (12)

where $r$ is the trajectory length from the point of scatter to the absorption volume, $\Delta r$ is the trajectory length within the absorption volume, and ${\lambda }_p$ is the attenuation length for photoelectric absorption. Both $r$ and $\Delta r$ are functions of $\left(z_1,\theta \right)$ and may be calculated geometrically for a given scatter geometry, while ${\lambda }_p$ is dependent on the energy of the scattered photon, and therefore the scatter angle (details of the calculation for ${\lambda }_p$ are given in Appendix A.3). It is this term, $P_{phot}$, that contains the ICS geometry requirements, specified through the trajectory lengths $(r,\Delta r)$. The complete expression for the ICS distribution is then given as the product of these three components.

$$P_{ICS}\left(z_1,\theta \right)=P_{comp}\left(z_1\right)\times P_{kn}(\theta )\times P_{phot}\left(z_1,\theta \right)$$ (13)

The distribution may be written directly in terms of scatter and absorption DOI as $P_{ICS}\left(z_1,z_2\right)$ via change-of-variables, where the absorption DOI is written in terms of the scatter DOI, scatter angle, and the transverse distance w between the centers of the scatter and absorption volumes.

$$z_2=z_1+w\mathrm{t}\mathrm{a}\mathrm{n}(\pi /2-\theta )$$ (14)

2.3. Bayesian decoding algorithm

Using the detector response model, we developed a likelihood function to assess the statistical consistency between a hypothesized set of interaction coordinates $(\mathbf{E},\mathbf{z})$ and the observed photodetector signal $\mathbf{t}={\left[t_1,\dots ,t_m\right]}^{\mathrm{T}}$. Assuming the measured signals are subject to a zero-mean Gaussian electronic noise with variance ${\sigma }_i^2$, the signal on the $i$^th photodetector $t_i$ is described by the Gaussian distribution $𝒩\left(t_i\mid y_i(\mathbf{E},\mathbf{z}),{\sigma }_i^2\right)$ and the likelihood is given by

$$ℒ(\mathbf{E},\mathbf{z})=\prod _{i=1}^m\frac{1}{{\sigma }_i\sqrt{2\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{1}{2{\sigma }_i^2}{\left(t_i-y_i(\mathbf{E},\mathbf{z})\right)}^2\right]$$ (15)

where $y_i(\mathbf{E},\mathbf{z})$ is the modelled response to interaction coordinates $(\mathbf{E},\mathbf{z})$, and the noise is modeled as ${\sigma }_i^2=y_i$. An estimate for the interaction coordinates may be obtained by maximizing $ℒ$, yielding the maximum likelihood estimate (MLE). However, the stability of this estimation may be improved by incorporating prior knowledge, especially as the measured response vector consists of just a few noisy components, due to the confined nature of light-sharing in the Prism-PET module.

We therefore develop a prior probability distribution to ensure that the estimated interaction coordinates are consistent with the underlying physics. In constructing our prior from the scatter-absorption model described above, we note that the prior must depend only on the observed coordinates $\left(z_j,z_k\right)$, agnostic to their interaction sequence. In contrast, the scatter-absorption distribution requires sequenced coordinates $\left(z_1,z_2\right)$ pertaining to a particular ICS geometry, specified as a permutation $\tau$ of the interacted crystals. We therefore write the prior, $P_{\mathit{prior}}$, by expanding the scatter-absorption distribution $P_{ICS}$, across all possible interaction sequences $\left\{{\tau }_i\right\}$ as,

$$P_{prior}(\mathbf{E},\mathbf{z})=\frac{1}{N}\sum _{i=1}^N{P}_{ICS}(\mathbf{z}\mid {\tau }_i)$$ (16)

where we have assumed all interaction sequences are equally likely for a given set of interacted crystals. For a two-crystal event, where N = 2, the expansion amounts to a simple average of the ICS distributions generated by each possible interaction sequence. We also note that the energy dependence drops out of the right-hand side of (16) because the DOI vector, $\mathbf{z}$, together with the interaction sequence, $\mathrm{\tau }$, specifies the scatter angles, which in turn give the energies via Compton kinematics.

Combining the likelihood function with this prior probability distribution, we form the MAP objective for 2-crystal ICS events involving the set $\mathit{S}$ of interacting crystals as,

$$P_{MAP}(\mathbf{E},\mathbf{z}\mid S)=ℒ(\mathbf{E},\mathbf{z}{)}^{\left(1-\beta \right)}\times P_{\mathit{prior}}(\mathbf{E},\mathbf{z}\mid S{)}^{\beta }$$ (17)

where the parameter $\mathrm{\beta }$ enables us to tune the weighting of the prior, such that $\beta =0$ gives the Maximum Likelihood Estimate (MLE) while $\beta =1$ yields an estimate based exclusively on prior knowledge.

The MAP estimate for interaction coordinates was obtained by optimizing the natural logarithm of $P_{MAP}$ over the 4D parameter space of $\left\{E_j,E_k,z_j,z_k\right\}$.

$$\begin{gathered} [\stackrel{\mathbf{ˆ}}{\mathbf{E}},\stackrel{\mathbf{ˆ}}{\mathbf{z}}{]}_{MAP}=\underset{\mathbf{E},\mathbf{z}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\left[(1-\beta )\times \sum _i{\left(t_i-y_i\left(E_j,E_k,z_j,z_k\right)\right)}^2/\left(2{\sigma }_i^2\right)\right. \\ \left.-\beta \times \mathrm{l}\mathrm{o}\mathrm{g}{P}_{\mathit{prior}}\left(z_j,z_k\right)\right] \end{gathered}$$ (18)

A simplified estimate that does not consider depth-dependence and forgoes prior knowledge is obtained from (8), yielding an energy-based Maximum Likelihood Estimate (eMLE) for the interaction coordinates given as,

$$[\stackrel{\mathbf{ˆ}}{\mathbf{E}}{]}_{eMLE}=\underset{\mathbf{E}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\left[\sum _i{\left(t_i-{\bar{y}}_i\left(E_j,E_k\right)\right)}^2/\left(2{\sigma }_i^2\right)\right]$$ (19)

where ${\bar{y}}_i$ is the response predicted by the simplified detector response model (8).

3. Methods

Next, we apply the theory developed above - broadly applicable to pixelated, mono-layer detectors - to the specific case of a Prism-PET module. Leveraging a simulated dataset, we devleop and validate several analytical and machine-learning approaches to ICS decoding and recovery, as depicted by flowchart in figure S1.

3.1. Detector module

The Prism-PET detector module used in this study maintains a 4:1 ratio between crystal elements and readout pixels via light sharing mediated by a segmented light guide affixed to the radiation-receiving end of the device. The 16 × 16 crystal array is comprised of 1.4 × 1.4 × 20mm³ lutetium-yttrium oxyorthosilicate (LYSO) scintillator crystals (Lu_1.8Y_.2SiO₅:Ce, density 7.1 g cm^–3) coated along their depth direction with BaSO₄ reflector. Optical signal from this 16×16 crystal array is read out from the bottom by an 8 × 8 array of silicon photomultipliers (SiPMs), with each 2 × 2 segment of the crystal array coupled to a unique SiPM. Coupling of the optical signal from the scintillator crystals is mediated by a segmented array of right-angle prismatoids atop the crystal array. Each prismatoid in the light guide is optically isolated from its neighbors via BaSO₄ and coupled to multiple crystals: prisms along the edges of the detector array are coupled to 2 crystals each, those at the corners coupled to 5, and the remainder in the center coupled to 4 crystals apiece. The layout of prisms, crystals, and SiPMs is such that each crystal is coupled directly to 1 SiPM below, and indirectly to 1 (edge), 3 (center), or 4 (corner) additional SiPMs via prismatoid-mediated light-sharing. Spatial offsets between prisms and SiPMs ensure all crystals coupled to a particular prism are themselves directly coupled to different SiPMs, thereby enabling a unique readout pattern for each crystal in the 16 × 16 array.

3.2. Simulation

The response of the Prism-PET detector module to incident 511 keV photon radiation was simulated using Monte Carlo methods. Interactions between the high-energy photons and the scintillator crystal array were implemented in GATE [29]. Interaction physics was governed by the Penelope package of low-energy electromagnetic models, including photoelectric absorption, Compton and Rayleigh scattering, electron ionization and scattering, bremsstrahlung, and atomic de-excitation. Range cuts for photons and electrons were set to 0.1 mm to ensure the point spread function (PSF) of the crystal incorporates blurring from secondary particles e.g. electrons, characteristic x-rays [30]. The position, time, and deposited energy of individual interactions were extracted in list mode from the GATE Hits file.

Following GATE simulation, the deposited energy, $E$ (keV), from each hit was mapped to a quantity, $N$, of scintillation photons by sampling a Normal distribution with mean and standard deviation given by

$${\mu }_N=\alpha E,{\sigma }_N=\gamma \sqrt{\alpha E}$$ (20)

where $\alpha$ is the scintillation yield and $\gamma$ the dimensionless resolution scale factor, as defined in the GATE user manual [29]. In this study, we set $\alpha =33.2$ photons/keV and $\gamma =5.05$ based on reported energy resolution of 8% at 662keV for LYSO [31].

The optical photons generated by an interaction in crystal $j$ were then multiplexed to a photodetector response pattern, where the signal on the $i$^th individual detector element is generated by sampling a Normal distribution with parameters given by

$${\mu }_i=N\cdot {\bar{s}}_{ijk},{\sigma }_i={\gamma }_{ijk}\sqrt{N\cdot {\bar{s}}_{ijk}}$$ (21)

where $N$ is the quantity of optical photons generated by the interaction, ${\bar{s}}_{ijk}$ the expected fraction of scintillation photons generated in depth bin k of crystal j which ultimately are absorbed onto photodetector element $i$, and ${\gamma }_{ijk}$ the corresponding resolution scale factor.

Values for the parameters ${\bar{s}}_{ijk},{\gamma }_{ijk}$, were estimated via Monte Carlo simulation using the TracePro platform. The setup and physical parameters of this simulation are described in detail in a previous publication [27]. The simulation was conducted by releasing 10000 optical photons from a point source at a particular depth, $k$, within a crystal, $j$. We defined 80 equally spaced depth bins along the length of each crystal (0.25mm per bin). Transport of each photon was tracked throughout the crystals, prismatoids, SiPM pixels, and optical coupling media, ultimately generating a normalized readout pattern giving the fraction of initial scintillation photons absorbed on each SiPM readout pixel, $s_{ijk}$. The simulation was repeated 20 times for each depth ($k$) and each crystal ($j$). The final ${\bar{s}}_{ijk}$ was taken as the mean of the $s_{ijk}$ from the 20 simulations. The $\gamma$ were calibrated based on measured standard deviation in $s_{ijk}$ at N = 10,000 photons.

3.3. Simulation validation

In order to validate the simulation against experiment, several characterization measurements were generated, paralleling methods used experimentally [24]. Flood maps were generated by exposing the detector module to a uniform flux of 10⁷ perpendicularly incident 511 keV photons. 2-D positions were calculated for each event as the energy-weighted centroid of the photodetector response pattern, following the standard Anger logic algorithm, and then histogrammed together to create the flood map.

DOI estimates were obtained by a centroid-based algorithm where the parameter $w=E_{\mathit{max}}/E_{\mathit{sum}}$, describing the contribution of the highest-intensity photodetector to the total signal, is assumed to vary monotonically with DOI for photoelectric events [16]. A linear calibration method was then applied to create $w$-DOI mappings for each crystal by modeling the edges of the w-histogram - corresponding to the top and bottom ends of the crystal - as a pair of error functions such that the mean and standard deviation of their derivatives - a pair of Gaussians - yield estimates of the location and resolution, respectively, at each end of the crystal [24].

Energy was calibrated in 3-D, with individual calibration factors computed for 8 depth bins, equally spaced along the length of the crystal, within each of the 256 crystals [24]. After applying these crytstal-specific, depth-corrected factors, the energies were histogrammed together and an energy resolution was computed by fitting a Gaussian to the 511 keV photopeak.

3.4. Analytical decoding algorithms

Using statistical methods and physical modeling, we developed two analytical approaches to resolve the multiple interactions of an ICS event: a maximum posterior estimate of the energy deposit and DOI in each interacting crystal, and a maximum likelihood estimate of only the energy deposit in each crystal. Both approaches, termed MAP and eMLE, respectively, aim to demonstrate that the photodetector response to an ICS event generated by a pair of crystals with highly overlapped response patterns may be accurately decoded within the context of confined and deterministic light-sharing mediated by the Prism-PET light guide. Thus, both algorithms were deployed on the specific decoding test case of nearest-neighbor ICS, where overlap between single-crystal response patterns is maximized.

An estimate of the interaction coordinates for a given photodetector response was obtained by optimizing the appropriate objective function (18,19) developed in Section 2.3. The optimization was implemented in the R programming language using the Improved Stochastic Ranking Evolution Strategy (ISRES) algorithm for non-linearly constrained global optimization from the NLopt non-linear optimization library [32,33]. The ISRES algorithm enabled robust evolution across the parameter space, and was not sensitive to variance in initial conditions. Computational speed was balanced with accuracy by stopping the optimization when the step size fell below 1keV in energy and 0.1mm in DOI, with the total iterations limited to 1 million per event. The average execution time for computation of the MAP-estimated interaction coordinates was 9.1sec per 100 events using an Apple MacBook Pro with a 2.3GHz Intel Core i5 CPU and 8GB RAM.

The accuracy of the MAP and eMLE estimations were evaluated on a labeled dataset of approximately 64000 simulated detector response patterns, generated by exposing the detector module to a uniform flux of 511 keV photons with incident angle distributed uniformly within ±40° of the normal, and subsequently extracting the nearest-neighbor ICS events. These were identified as photopeak events where the deposited energy per crystal exceeds 75keV in exactly 2 nearest-neighbor crystal elements. We set this threshold to isolate detectable ICS events generated by Compton scatter, while minimizing inclusion of photoelectric events where the photoelectron and characteristic x-ray deposit their energies in distinct crystal elements. Such events are fairly prominent in the ground-truth spectrum of interaction energies due to the KL and KM transitions in lutetium that fall within 53keV to 62keV [34].

3.5. Machine learning algorithms

We developed two different neural networks, termed Decoding-Net and Recovery-Net, each with a distinct purpose and scope. Both neural networks were constructed using Keras, a Python-based deep learning API, and operated on the TensorFlow machine learning platform. The execution of the processes was carried out using an NVIDIA GeForce RTX 3090 GPU.

3.5.1. Decoding-Net.

The Decoding-Net is an Artificial Neural Network (ANN) designed to estimate the energy deposit and DOI in individual crystals contributing to an ICS event. It is deployed on the decoding test case in parallel with the analytical MAP algorithm, enabling mutual validation between the two approaches. Its architecture is depicted in figure 2a, where a 4D input vector of photodetector signals is mapped along separate pathways to the energy deposit and DOI in each crystal. Multiple neural network layers are implemented along both the energy and DOI prediction pathways, where each layer, comprised of a different number of kernels, is followed by a rectified linear unit (ReLU) activation function. The mean squared error was selected as the loss function and adaptive moment estimation (Adam) was implemented as the optimizer to update network parameters in each training epoch.

Figure 2:

(a) Architecture of the Decoding-Net ANN deployed in parallel with the analytical Bayesian algorithm to estimate the coordinates of interaction $\left(E_1,E_2,z_1,z_2\right)$ in the two nearest-neighbor crystals contributing to an ICS event. (b) Architecture of the Recovery-Net CNN developed for 3-D positioning of all photopeak events, including PE and ICS. Input from the photodetector array is mapped along separate pathways to crystal selection and DOI estimation of the initial point of interaction.

A dataset of approximately 500000 detector response patterns, each labeled with the energy and DOI of the interacting crystals, was generated by exposing the detector module to a uniform flux of 511 keV photons with incident angle distributed uniformly within ±40° of the normal, and filtering for photopeak events where the deposited energy per crystal exceeds 75keV in exactly 2 nearest-neighbor crystal elements. Training was carried out on ⁷/₈ of the full dataset (~445000 events) with 10% reserved for validation, while the remaining ¹/₈ of the data (~64000 events) was used for testing.

3.5.2. Recovery-Net.

The Recovery-Net addresses the broader problem of localizing the initial point of interaction in 3D for both photoelectric (PE) and ICS events. Depicted in figure 2b, the architecture is a hybrid of the U-Net, artificial neural network, and autoencoder structures. The 64 photodetector signals were regarded as input, and the position and DOI value of the primary interacted scintillation crystal were the outputs. The modified U-Net part consisted of 10 convolutional layers where each applied 2 × 2 padded convolutions followed by a rectified linear unit (ReLU) activation function. The downsampling was achieved by 2×2 max-pooling layers with strides two during the contraction path, and the same size of up-sampling layers was used for the expansive path. The output dimension of the modified U-Net network was 8 × 8 × 4, which was loaded into separate paths for predicting the position and DOI of the primary interacted crystal. The artificial neural network, implemented for predicting crystal location, was composed of 1 hidden layer with 64 neurons followed by ReLU activation function, and one output layer with 256 neurons followed by softmax activation function, where the maximum probability output represented the position of the crystal. The DOI estimation was achieved by an autoencoder network, where the output dimension of the U-Net would be reshaped as 16 × 16 × 1 at first, and 3 × 3 padded convolutions were applied with stride two during the encoding part. The same kernel and stride sizes of transposed convolutions were used for the decoding part, where both layers implemented ReLU as the activation function. The predicted DOI value was generated from the last layer with a single neuron.

The model’s parameters were initialized using the He Normal Initialization method [35]. These parameters are subsequently updated during each training epoch through backpropagation to minimize the loss function values. Finally, the best-performing model will be saved when further improvement ceases on the validation dataset.

A much larger dataset was acquired for Recovery-Net, compared to Decoding-Net, comprised of 48 million detector response patterns (~190000 per crystal), each labeled with the index and DOI of the crystal containing the first interaction. As with the Decoding-Net, there were generated by exposing the detector module to a uniform flux of 511 keV photons with incident angle distributed uniformly within ±40° of the normal.

3.6. Intrinsic spatial resolution

To assess the impact of ICS recovery at a system level, we conducted an intrinsic spatial resolution measurement within our simulation environment, applying Recovery-Net positioning alongside two comparator algorithms: centroid-based and ideal detector positioning. Centroid-based positioning describes the classical approach lacking recovery, where the first-interacted crystal is identified from the centroid position on a segmented flood map, and DOI estimated by a linear scaling with the estimation parameter, w, defined by the contribution of the highest-intensity photodetector to the total signal [24]. Ideal detector positioning, based on ground-truth information from GATE, represents an attainable upper limit on positioning accuracy in detectors based on 2D pixelated crystal arrays. Here, the earliest interaction among crystals with a minimum of 75 keV in deposited energy defines the first-interacted crystal, and the energy-weighted interaction depth of all hits within that crystal gives the DOI estimate.

A point source of antiparallel 511 keV photons was stepped in 0.2 mm increments along the center of the field of view between two exactly opposed detector modules separated by 34 cm (the average diameter in our Prism-PET brain scanner [24]). Each positioning algorithm was applied to the coincidence events and an initial point of interaction was calculated using the estimated DOI and the midpoint of the identified first-interacted crystal. The entrance crystal (DOI-corrected) in each block was then estimated from the pair of initial interaction points in each module. Coincidence counts were then binned by entry crystal pairs, and count profiles were thus constructed between opposed crystal pairs.

We then quantified the FWHM of each count profile to estimate the contribution of the coincidence response function to the intrinsic spatial resolution of the detector under each positioning scheme [36]. The resultant improvement in intrinsic spatial resolution, compared to baseline centroid-based positioning, was quantified using the approach described by Lee et al. [37]. Lastly, the fraction of coincidence events assigned to an incorrect LoR, termed ‘Spillover’, is also quantified for each positioning method.

3.7. Image quality

The performance of the Recovery-Net at the scanner level was assessed by imaging a simulated hot spot phantom within the geometry of our proposed Prism-PET brain scanner [6,24]. The scanner was constructed of 12 rings, each with 40 detector modules arranged in an elliptical geometry with long and short diameters of 38.5cm and 29.1cm, respectively. An ultra-micro hot spot phantom - comprised of rods with diameters of 0.75mm to 2.4mm filled with 250kBq/cc of ¹⁸F - was then placed at the center of the transaxial FOV for a 50M count acquisition. Each of the three positioning algorithms described above was applied to coincidence events to identify the DOI-corrected, first-interacted crystals of each coincidence pair. The resulting LoRs were then used to build the list mode data. Images were reconstructed using open-source Customizable and Advanced Software for Tomographic Reconstruction (CASToR), employing 3D list-mode ordered subset expectation maximization (OSEM) with 20 iterations and 15 subsets [38]. A voxel size of 0.25mm × 0.25mm × 0.25mm was selected for the reconstruction.

4. Results

4.1. Detector response model

The detector response model was explored and validated through Monte Carlo simulation of the impulse response from a center crystal across the full range of DOI. As shown in figure 3, each component of the response exhibited a clear depth dependence determined by the spatial relation of the photodetector to the crystal within the coupling unit. The response component corresponding to the photodetector directly beneath the $j^{\mathit{th}}$ crystal, denoted $y_1^{(j)}$, increases monotonically with DOI, indicating the response is driven by the direct, downwards propagation of optical photons towards the photodetector. The other three components, $y_{2-4}^{(j)}$, exhibit the opposite trend, attaining maximal response at the entrance end of the crystal, nearest the prismatoid. Here, the photodetectors are offset from the crystal, indicating the importance of prismatoid-mediated light-sharing in their response.

Figure 3:

Estimating matrix elements of the detector response from the depth-dependence of the impulse response vector. (a) Schematic depiction of the interacted crystal (shaded grey), its coupled prismatoid, and the 4 photodetectors comprising its impulse response. (b) Monte Carlo simulated response on each photodetector with fitted curve given by the detector response model. The characteristic response vectors for downward and upward propagation were obtained as parameters of the non-linear fit.

These observations were developed and quantified by fitting the simulated impulse response with the detector response model, as described in (9). The depth-dependence of each response component was well-characterized by the modeled superposition of growing and decaying exponentials, as shown in figure 3b, enabling quantification of the response vectors $({\mathbf{a}}_j^{\downarrow },{\mathbf{a}}_j^{†})$ as parameters of a non-linear fit. Illustrated in figure S1, the characteristic response demonstrates the impact of the prismatoid on light-sharing patterns. The downwards response ${\mathbf{a}}_j^{\downarrow }$ does not interact with the prismatoid and therefore exhibits minimal light-sharing with 83% of its signal concentrated on the directly underlying photodetector. Light-sharing is enhanced in the upwards, prismatoid-mediated response ${\mathbf{a}}_j^{\uparrow }$ with approximately 52% of the total signal distributed across the three offset photodetectors. However, the other half of the signal from upwards propagation still lands on the directly-coupled photodetector, indicating that a significant fraction of light entering the prismatiod is reflected back down the original crystal.

The components of characteristic response $(a_{i,j}^{\downarrow },a_{i,j}^{\uparrow })$ describe the coupling relationship between a crystal-photodetector pair and form the elements of the response matrices. Additionally, the magnitude and ratio between the pair of matrix elements provide further insight, describing, respectively, the overall coupling strength and the relative weighting between upwards and downwards characteristic responses. The latter of these, quantified as the ratio $a_{i,j}^{\uparrow }/a_{i,j}^{\downarrow }$, connects the observed depth dependence with the light-sharing geometry of the coupling unit: for matrix elements corresponding to offset crystal-photodetector geometry $\left(a_{2-4,j}\right)$, the ratio is greater than one, ranging from 2.08 to 3.64, indicating that prismatoid-mediated propagtion drives the response. The opposite holds for $a_{1,j}$, where the photodetector sits directly beneath the crystal, and a ratio of 0.45 implies the downwards contribution is approximately double that of the upwards.

4.2. Scatter-absorption model

Key components and results of the scatter absorption model are illustrated in figure 4 for ICS between nearest-neighbor LYSO crystals with dimensions 1.5 × 1.5 × 20mm³. Without an absorption constraint, the distribution of Compton-scattered events over the trajectory space of initial DOI and scatter angle recapitulates the Beer-Lambert and Klein-Nishina relations, where the most likely trajectory is 30° forward scatter at the entrance end of the crystal (figure 4a). The absorption likelihood map, illustrated in figure 4b, is peaked at 150° and heavily biased towards backscatter, reflecting a precipitous drop in the photoelectric attenuation length with increasing scatter angle (figure 4d). As a result, the ICS distribution - formed by weighting the unconstrained distribution by the absorption likelihood - is pulled towards backscatter, with the likeliest trajectory described by 120° backscatter at 2mm penetration from the entrance end of the crystal (figure 4c). The total population of ICS events, however, is split in approximately equal proportion between forward ($\theta <90°$) and backward($\theta >90°$) scatter, in contrast to the 70/30 split obtained from the underlying Klein-Nishina distribution (figure 4e).

Figure 4:

Scatter-absorption model applied to nearest-neighbor ICS in 1.5 × 1.5 × 20 mm³ crystals, showing (a) the distribution of Compton-scattered events over scatter angle and DOI, (b) absorption probability for each scatter trajectory (c) distribution of ICS events, (d) attenuation length for photoelectric absorption, and (e) scatter angle distribution for ICS events compared to the Klein-Nishina distribution.

Deviation from the Klein-Nishina angular distribution has been noted in a previous study, where it was suggested that intra-crystal absorption of scatter near 0° and 180° reshapes the angular distribution of ICS events, shifting the forwards/backward proportion from 70/30 to 67/33 [9]. This deviation - obtained with larger 3×3×30mm³ crystals - is directionally similar, yet significantly less pronounced than the 50/50 split observed here in 1.5 × 1.5 × 20mm³ crystals, suggesting that crystal width plays a key role in shaping the ICS scatter angle distribution. Figure 5a explores this relationship, plotting the scatter angle distribution for ICS events between nearest-neighbor LYSO crystals of dimension w × w × 20mm³, where the crystal width, w, is varied from 1.5 to 10 mm (the Klein-Nishina distribution is replotted at the bottom for reference). As the crystal cross-section expands, and the path length from scatter interaction to the neighboring crystal increases, more and more of the low energy backscatter is lost to intra-crystal absorption, while the more penetrating forward-scattered photons have a greater probability of absorption, resulting in increasing forward bias in the scatter angle distribution, with the peak moving from 120° in 1.5mm² crystals to 60° in 5mm² crystals.

Figure 5:

Impact of crystal width on observed scatter angle and DOI separation in ICS events. (a) Scatter angle distribution for crystal widths 1.5 – 10 mm, plotted above the reference Klein-Nishina distribution. (b) Distribution of DOI separation for 1–5 mm crystals. (c) Total probability of forward scatter resulting in DOI separation greater than 1 mm for various crystal widths.

Figure 5b illustrates how the angular distribution, coupled with the scatter geometry, shapes the observable indicator of scatter direction: DOI separation. Defined as the spatial separation between scatter and absorption interactions in the depth direction, the DOI separation is calculated from the scatter angle and DOI, with positive values indicating forward scatter and negative values corresponding to backscatter. In narrow crystals, the DOI separation is tightly peaked about zero with a slight negative bias driven by the underlying angular distribution. As crystal width increases, the distribution broadens with increasing bias toward forward scatter. This broadening is largely driven by geometry, as the DOI separation for a given scatter angle scales directly with the crystal width. Indeed, the FWHM of the DOI separation distribution increases from 2.0mm to 9.1mm as crystal width expands from 1mm to 5mm, where the small non-proportionality is driven by the contraction in angular range at larger crystal dimensions.

In order for a DOI-based recovery algorithm to correctly and unambiguously identify the first interaction, the incident photon must scatter in the anticipated direction, and achieve a measurable DOI separation. Figure 5c shows the fraction of ICS events meeting these criteria for an assumed forward bias and a 1mm minimum DOI separation. In narrow crystals (width 1–2 mm), the large fraction of events with minimal DOI separation leads to unambiguous recovery rates below 40%. This rate does not rise appreciably above 50% until crystal width exceeds 4mm, where the scatter geometry permits a strong forward bias in angular distribution and sufficient space for the scatter trajectories to develop measurable DOI separation. Thus, the DOI separation may be an unreliable indicator of the initial point of interaction, especially in narrow crystals.

Figure 6a illustrates the prior probability density function (PDF) for the joint DOI distribution in nearest-neighbor ICS events, developed from the scatter-absorption model, as described in Section 2.3. The two DOI values are tightly correlated, and their marginal distributions each resemble a typical DOI profile, with a small buildup region at the entrance end of the crystal followed by exponential decay. To simplify representation, we applied a change-of-variables transformation,

$$\bar{z}=\left(z_j+z_k\right)/2,{\mathrm{\Delta }}_z=z_k-z_j$$ (22)

to represent the distribution in terms of the difference, ${\mathrm{\Delta }}_z$, and average, $\bar{z}$, of the two DOI values, $\left(z_j,z_k\right)$. As shown in figure 6b, the sum and average DOI are approximately independent descriptors of the prior, as the DOI separation remains tightly peaked around zero across the length of the crystal, while the DOI profile remains consistent across the narrow range of DOI separation. We therefore represented the prior in closed form as the product of these two univariate distributions, $P_{\bar{z}}$ and $P_{{\mathrm{\Delta }}_z}$_z, as

$$\begin{gathered} P_{\mathit{prior}}\left(\bar{z},{\mathrm{\Delta }}_z\right)=P_{\bar{z}}\times P_{{\mathrm{\Delta }}_z} \\ =\mathrm{e}\mathrm{x}\mathrm{p}[-\bar{z}/\lambda ]\frac{\bar{z}-c_1}{\sqrt{{\bar{z}}^2+c_2}}\times \mathrm{e}\mathrm{x}\mathrm{p}\left[-(\mathrm{\Delta }z{)}^2/2{\sigma }_{\mathrm{\Delta }z}^2\right] \end{gathered}$$ (23)

where the parameters $c_1,$ $c_2$ are used to tune the offset and width in the buildup region of the averaged DOI distribution, $\lambda$ describes the effective attenuation length in the DOI profile, and ${\sigma }_{\mathrm{\Delta }z}$ describes the width of the DOI separation distribution. These parameters may be tuned empirically to the calculated prior distribution, enabling simple and flexible application to the decoding problem across various detector geometries.

Figure 6:

Prior PDF describing the expected spatial relation between scatter and absorption in nearest-neighbor ICS events, given in terms of (a) the pair of DOI values and (b) their average and difference.

4.3. Simulation validation

The simulation-generated flood map, shown in figure 7, exhibits 256 well-defined spots, corresponding to photoelectric events in each crystal of the scintillator array. Asymmetric elongation in each spot reflects the prismatoid-mediated depth-dependence of the detector response. A crossed-box pattern, evident in the background, illustrates the ICS events, whose centroid position tends to fall along lines connecting the interacted crystals [39]. Qualitative comparison with corresponding, experimentally-generated flood maps shows good agreement in terms of both the spatial distribution and shape of the crystal spots [14,24]. A somewhat tighter grouping of crystal spots near their peaks, noted in the experimental flood maps, may be attributed to SiPM saturation effects. Since saturation was not included in the simulation, the w-parameter values tend to be higher than in experiment, keeping the crystal spots closer to the highest-intensity SiPM across the DOI range, resulting in a more uniform distribution of spots.

Figure 7:

Simulation validation. Benchmark measurements generated via simulation for comparison with experimental results, including the flood map (a), energy spectrum (b) with (blue line) and without (red line) DOI correction, and DOI distribution (c). The energy and DOI resolutions are indicated within the plots.

DOI histograms for representative crystals located at the center, edge, and corner of the detector block are shown in figure 7c, along with Gaussian fits to their derivatives at top and bottom ends of the crystal. The FWHM resolutions were 2.62 mm, 3.04 mm, and 3.26 mm for center, edge, and corner crystals, respectively, with an overall resolution of 2.67 mm for the entire block. These values align with experimental reports of 2.50 – 3.41 mm FWHM for a single block [24], and 2.50 – 2.57 mm FWHM for single center crystals [14,15]. The trend in resolution, progressively degrading from the center towards the periphery, also agrees with experimental reports [15].

Energy spectra for representative center, edge, and corner crystals are shown in figure 7b for both DOI-corrected (3-D) and uncorrected (2-D) energy calibration. The depth correction resulted in 1.0 – 1.3% improvement in energy resolution, reflecting a depth-dependence in the light collection efficiency. Disparities across center, edge, and corner crystals show degradation in resolution towards the periphery of the crystal array which is partially mitigated by depth-correction, suggesting that the depth-dependence in light collection efficiency varies with crystal location. These measurements demonstrate alignment between simulation and experiment, both directionally and quantitatively, with depth-corrected energy resolutions of 8.5 – 12.6% reported in the literature [14,24].

4.4. Decoding

The impact of prior information on the decoding accuracy of the analytical MAP algorithm is explored in figure 8, where the MAP estimation error in energy and DOI, relative to MLE, is shown for β values between zero and one. Introducing prior information leads to a ~50% reduction in error that remains stable for $\beta$ values of 0.1 to 0.7, yielding optimal MAP estimation errors of 20.5keV in energy and 3.1mm in DOI. Further upweighting of the prior, where $\beta$ > 0.7, leads to a rapid degradation in accuracy as estimation becomes exclusively based on the prior, disregarding the observed photodetector response. We note that the maximum $\beta$ value included in the plot is 0.98, as the energy estimations collapse to a uniform distribution at $\beta =1$, producing a FWHM error of several hundred keV.

Figure 8:

Relative error in MAP estimation plotted over the range of the prior weighting parameter $\beta$, where $\beta =0$ corresponds to a pure MLE estimate without prior knowledge. The MAP estimation is stable across a wide range of $\beta$ with an optimal weighting at $\beta =.6$ (indicated with dashed line).

Both the analytical MAP algorithm and the Decoding-Net were able to decode interaction coordinates in nearest-neighbor ICS events, estimating the energy and DOI in each interacting crystal with similar accuracy (figure 9). The MAP approach outperformed the Decoding-Net in predicting crystal energies, achieving an error distribution with full-width at half-maximum (FWHM) of 20.5 keV versus 26.2 keV for Decoding-Net (figure 9a). Both approaches were able to recapitulate the true energy distribution, with peaks at approximately 180keV and 320keV and a local minimum near 250keV. Decoding-Net achieved slightly better accuracy in estimating DOI, with a FWHM error of 2.9mm, compared to 3.1mm for the MAP algorithm (figure 9b). Both the MAP and Decoding-Net algorithms produced an exaggerated peak in the buildup region near the entrance-end of the crystal, with the Decoding-Net exhibiting a larger divergence from the true distribution. Both methods also exhibited distortions near the bottom of the crystal, indicating that the discontinuities at crystal edges present challenges to both algorithms.

Figure 9:

Performance of the MAP algorithm and Decoding-Net ANN on the decoding test case showing estimation error distributions for (a) energy and (b) DOI, in addition to overlaid distributions of true and estimated values for energy (c,e) and DOI (d,f).

The relationship between true and predicted values is illustrated in figure 10 for the MAP algorithm and Decoding-Net. Excellent linearity in energy is demonstrated for both approaches, with R² and slope values of approximately 1. A very slight degradation in linearity is observed for DOI estimation, with R² values of 0.92 and 0.93 and slope values of 0.94 and 0.93 for MAP and Decoding-Net, respectively (figure 10). The decoding results are summarized in table 1 for all decoding approaches.

Figure 10:

Linearity in energy and DOI estimation for analytical MAP algorithm (a,b) and the Decoding-Net ANN (c,d).

Table 1:

Performance of the analytical eMLE and MAP algorithms, and the Decoding-Net ANN, as evaluated on the decoding test case.

	Energy			DOI
	FWHM (keV)	Fitted line	R 2	FWHM (mm)	Fitted line	R 2
eMLE	25.2	y = 0.99x + 2.24	0.98	-	-	-
MAP	20.5	y = 1.01x − 2.84	0.99	3.1	y = 0.94x + 0.31	0.92
ANN	26.2	y = 0.98x + 3.89	0.98	2.9	y = 0.93x + 0.48	0.93

4.5. Recovery

The Recovery-Net achieved an overall crystal selection accuracy of 83.0%, correctly identifying the primary crystal for 100.0% of PE events and 64.7% of ICS events (table 2). This performance is stratified by coupling unit, as ICS accuracy degrades when the primary interaction falls in a corner or edge unit instead of a center unit. Accuracy of DOI estimation followed similar trends, attaining an overall error FWHM of 2.9mm in center crystals, with 2.3mm among PE events and 4.0mm among ICS events. The performance degradation at the periphery was significant and uneven, as the FWHM error for PE events increased to 3.0 mm in both edge and corner units, while the error in ICS events was 6.4 mm for edge units and 7.3 mm for corner units.

Table 2:

Performance of Recovery-Net CNN in positioning the initial point of interaction for all events in the photopeak.

	Crystal ID accuracy (%)			DOI FWHM (mm)			Distribution of events (%)
	PE	ICS	Both	PE	ICS	Both	PE	ICS	Both
Center	100.0	65.6	82.9	2.3	4.0	2.9	42.6	42.1	84.7
Edge	100.0	60.1	83.7	3.0	6.4	3.5	6.7	4.6	11.3
Corner	99.9	54.9	83.3	3.0	7.3	3.6	2.6	1.5	4.1
All	100.0	64.7	83.0	2.4	4.2	3.0	51.8	48.2	100

To connect the accuracy in crystal selection with the error in DOI estimation, we calculated the Euclidean error distance in the transverse and depth directions, along with the combined error in three dimensions. An empirical cumulative distribution function (CDF) was then calculated for each error distance, as shown in figure 11, enabling insight into the recovery component of the point spread function (PSF). For instance, the error radius required to capture 90% of predicted values in a center crystal is 2mm in the transverse plane, 2.3mm along the DOI axis, and 4.1mm in the combined 3D space of crystal index and DOI. Thus, a proper three-dimensional understanding of the error distribution is needed to understand the full impact of recovery on imaging performance, especially in detectors with long crystals and conformal geometry that require 3D localization of the incident annihilation photons to avoid parallax errors.

Figure 11:

Cumulative distribution of recovery error in the (a) transverse direction, (b) depth direction, and (c) three-dimensionally, combining transverse and depth errors.

The improvement in depth decoding enabled by Recovery-Net is illustrated in figure 12a by comparing the DOI distribution obtained with the standard centroid approach, against that generated by the Recovery-Net. The centroid-generated method estimates DOI from a parameter, $w$, given as the ratio of maximum to the total signal recorded on the photodetector array [14,24]. However, the characteristic variation of $w$ with DOI breaks down in ICS, leading to a DOI distribution that extends far beyond the ends of the crystal. The distribution of Recovery-Net estimated DOI values, however, is contained within the crystal bounds and reproduces the exponential decay of the true distribution.

Figure 12:

Impact of recovery algorithm on DOI. (a) DOI distribution obtained naively using the standard center-of-gravity approach (CoG - blue curve) and the distribution obtained with the Recovery-Net CNN (red curve), overlaid on true distribution. (b) Distribution of DOI error for events with a correct and incorrect identification of the first-interacted crystal.

4.6. Intrinsic spatial resolution

Figure 13a shows the count profiles generated by three positioning schemes: centroid, Recovery-Net, and ideal detector. The improved positioning accuracy of Recovery-Net over centroid-based positioning reduced count spillover into neighboring LoRs from 63.8% to 47.0%, resulting in a sharpened count profile (table 3). The ideal detector scenario further reduced spillover to 15.6%, resulting in a count profile with a peak 58% higher than the Recovery-Net peak. However, increases in count profile peak - or, equivalently, decreases in spillover - do not translate to proportional improvements in intrinsic spatial resolution. Despite a 1/3 reduction in spillover counts, the ideal detector achieves only a 7% increase in resolution over the Recovery-Net. This indicates that recovery and positioning of ICS events by Recovery-Net enables significant improvement in spatial resolution, approaching the attainable upper limit.

Figure 13:

(a) Count profiles obtained by three positioning approaches: centroid-based (no recovery), Recovery-Net, and ideal detector (attainable upper limit). Displayed profile corresponds to single pair of exactly opposed crystals, each in the center of a Prism-PET detector module. (b) Line profiles through 1.0 mm and 1.5 mm spots in reconstructed image of ultra-micro hot spot phantom. (c) Reconstructed images of ultra-micro hot spot phantom under centroid, Recovery-Net, and ideal detector positioning schemes.

Table 3:

Impact of Recovery-Net positioning on intrinsic spatial resolution.

	FWHM (mm)	Spillover (%)	Improvement (%)
Centroid (no recovery)	0.82	63.8	-
Recovery-Net	0.75	47.0	38.9
Ideal detector	0.72	15.6	45.9

4.7. Image quality

Figure 13c presents reconstructed images of the activity distribution in an ultra-micro hot rod phantom, generated using centroid, Recovery-Net, and ideal detector positioning algorithms. ICS recovery through Recovery-Net results in significant, visible improvement in image quality. Specifically, Recovery-Net increases clarity of the smallest spots and demonstrates potential to resolve 0.8 mm spots, approaching the lower limit indicated by the ideal detector image. Additionally, Recovery-Net achieves a visible reduction of intensity in regions without activity for all rod sizes. We quantified this improvement in contrast by plotting line profiles through the 1.5 mm and 1.0 mm spots (figure 13b). The calculated peak-to-valley ratios for centroid, Recovery-Net, and ideal detector positioning were, respectively, 1.42, 1.78, and 2.53 for 1.0 mm rods, and 3.00, 4.23, and 8.33 for 1.5 mm rods - representing increases of 25% and 41% for 1.0 mm and 1.5 mm spots between centroid and Recovery-Net.

5. Discussion

In this paper, we addressed the ICS problem in light-sharing depth-encoding detectors. Specifically, we demonstrated that the multiple interactions of an ICS event can be resolved while preserving depth encoding in a light-sharing detector with a high crystal- to-sensor (CSR) ratio. Our solution was enabled jointly by hardware through PrismPET and analytical and computational methods. The overarching aim was to develop 3D event positioning that handles all events in the photopeak, both PE and ICS.

To this end, we took the approach of solving a test case to demonstrate a proof of concept and then upscaled it using deep learning to create a generalized solution. First, we developed an analytical Bayesian algorithm to resolve the multiple interactions of an ICS event to obtain the interaction coordinates of each crystal. This was done for a special case epitomizing the ICS challenge in a light-sharing detector where there is scatter between adjacent crystals producing completely overlapping response patterns. An ANN was also developed, paralleling the MAP, and demonstrated equivalent performance, thus providing bidirectional validation for the two approaches. Then, having established the feasibility of resolving multiple interactions in a light-sharing detector, we developed a deep learning approach using a convolutional neural network for the 3D positioning of all events in the photopeak.

The maximum a posteriori (MAP) objective was generated from a scatter absorption model - the prior, and the detector response model - the likelihood. The algorithm achieved excellent performance representing the highest maximum accuracy reported in the literature. We found an energy error of 20.5keV and a DOI error of 3.1mm. Excellent linearity was obtained between predicted and true values, with R² values of 0.99 and 0.92 for energy and DOI, respectively. These results compare favorably to recent findings in the field, where an energy R² of 0.73 was reported in a light-sharing detector module of equivalent crystal dimensions and CSR to that used in our study [3]. It should be noted that this significantly lower correlation was obtained without any prior assumptions about the configuration of interacted crystals, increasing the difficulty of the problem. However, even when the CSR is lowered to 1:1 - effectively eliminating any ambiguity in identifying the pair of interacted crystals - the R² still falls short of our result, at 0.91.

We also examined how error in energy and DOI estimates varied across different prior weightings. Here, we found that including the prior resulted in a 50% reduction in error for both Energy and DOI estimates, and that this finding was stable across a wide range of prior weighting. Therefore, we were able to quantify and demonstrate how using the prior improves the quality of the estimation. The value of the prior highlights the importance of incorporating the context for the specific scatter geometry at hand. Our derived prior, for example, was very different from the original Klein-Nishina equation. This consideration has not been fully explored in other works to date.

In addition, we found that, even without any prior weighting, the accuracy is acceptable, with FWHM errors roughly doubling relative to their optimized values with prior information included. Thus, we can also rule out over-dependence on the prior. Following the Bayesian framework, it is the combination of the prior and the likelihood that drives the accuracy. However, while the prior makes significant improvements to the accuracy, the likelihood is the absolutely essential part of the equation. Without the input generated from the likelihood, the error rises precipitously. It is apparent that the likelihood here is the primary driver of accuracy. In this model, the likelihood is based on the response model, which is an embodiment of the deterministic light-sharing patterns given by the prismatoid light guide in Prism-PET.

We applied the ANN in the same test case of scatter across a pair of adjacent crystals, and found its performance similar to that of the MAP. The results from the MAP and the ANN are thus mutually supportive, indicating that in one direction, the models underlying the MAP are valid, and conversely that the neural network is not overfitting and is reproducible. This parity implies that both analytical and machine-learning algorithms provide a valid generalization pathway to full photopeak 3D positioning. However, scalability of the MAP algorithm is limited by an excessive number of model parameters and variations on the prior to account for different scatter geometries. In addition, since the analytical approach requires event-by-event optimization of a higher-dimensional objective function, deploying the algorithm at scale becomes computationally prohibitive. Therefore, the ANN provides a proof of concept for using a machine learning approach to address the same problem.

We next deployed the CNN in order to scale up the solution to the full photopeak and localize the primary interaction. We found an overall accuracy of 83.0% in crystal identification and DOI estimation error of 3.0mm, incorporating both ICS and PE events across the entire detector module. The metrics here also compare favorably with other published findings, representing the highest crystal selection accuracy ever reported in a light-sharing detector [3].

Given that the goal is 3D localization of the initial interaction, we unified the errors measured separately in transverse and depth directions by calculating an empirical CDF of the 3D positioning error, thus allowing insight into the recovery component of the point-spread function. We also show that the CNN is able to recover the expected distribution of the DOI in a way that tightly mirrors the ground-truth distribution. In contrast, depth estimation ability is essentially lost when standard decoding methods are applied to ICS events. Our CNN, therefore, represents not just a marginal improvement in DOI accuracy, but a recovery of the depth encoding capability itself

The performance of the CNN in accurately localizing the initial point of interaction may be attributed to its unique branched architecture, where crystal index and DOI are estimated separately following several shared convolutional layers. This partial decoupling of depth estimation from crystal identification enabled DOI estimates that were only marginally impaired by crystal misidentification (figure 11b). This counterintuitive result illustrates an advantage of the CNN and reflects the importance of ”intelligent design” in artificial intelligence - that is, if we understand something about the system, then we can leverage this knowledge in the model design. In this case, we know that crystal location and DOI within that crystal influence output in different ways, so it makes sense to model them differently, with crystal identification being the first-order effect and the DOI representing a higher-order effect.

The light guide plays a critical role in the ability to solve this problem as evidenced by the importance of the likelihood in the MAP algorithm, and by our results with the energy-only Maximum Likelihood Estimate (eMLE). The essential role of the likelihood in driving accuracy of the MAP estimation indicates that measured outcomes are well-predicted by the characteristic response patterns of the detector, which, in turn, are mediated by the light guide. The reliability of these response patterns is enabled by confined and deterministic light-sharing, inherent to the Prism-PET design, that minimizes variability in detector response. This stands in contrast to detectors equipped with a uniform glass light guide, where light sharing is unrestricted and stochastic, resulting in characteristic response patterns that are weak predictors of observed outcomes and therefore restricting the potential of a Maximum Likelihood approach.

In an effort to make a more direct comparison with other detectors that have a uniform light guide, we constructed an ‘energy-only’ maximum likelihood and found that energy estimates were only marginally less accurate than our other solutions, with an energy error of 25.2keV, and remained superior to comparators, achieving an R² of 0.98. This approach strips down the full Bayesian MAP algorithm to isolate the confined, deterministic light-sharing at the core of Prism-PET technology, taking away prior knowledge and depth-encoding ability. The high performance of the eMLE in relation to other light-sharing detectors with identical crystal dimensions and CSR, therefore, indicates that the essential nature of the prismatoid-mediated light sharing in enabling ICS decoding and recovery

Of note, the theoretical portion of this work is broadly applicable to pixelated, mono-layer, detector modules. The detector response, built on a standard linear response model, may be adapted to a wide array of pixellated designs. Specifically, the matrix elements, $a_{ij}^{\uparrow ,\downarrow }$, may be adjusted based on the configuration or presence of reflectors on the entrance end of the crystal array, while the optical photon attenuation parameter, $\lambda$, may be tuned to match the crystal geometry and optical properties (e.g. depolishing). If $\lambda$ is allowed to grow arbitrarily large, effectively removing depth-dependence from the response, the model reduces to the familiar matrix equation [23]. The scatter-absorption model is built on the well-characterized physics of annihilation photon transport in matter, and is therefore relatively insensitive to the optical properties of the device. Adaptation to any pixelated detector module is straightforward; requiring changes only in the geometric parameters of the crystals and the interaction cross sections.

As reflected by recent interest in the field of explainable AI, when problems become more complex, and AI approaches more sophisticated, it is increasingly important to maintain transparency and interpretability in implementing AI solutions [40,41]. We similarly prioritized the transparency of our machine learning approaches by using an analytical approach derived from fundamental physical and mathematical principles to first demonstrate that a solution is feasible in a special case when enabled by the unique light-sharing pattern characteristic of the Prism-PET module. Then, only when we were able to show parity between the analytical solution and neural network did we feel sufficiently confident to deploy a more complex deep-learning solution for the full problem.

We believe that the theoretical and computational foundations developed in this work pave the way to significant improvements in the imaging performance of PET systems based on pixelated, mono-layer detector modules - particularly those with depth-encoding capability. In such systems, ICS-induced mispositioning of singles leads to spillover of coincidence counts into neighboring LoRs, degrading spatial and contrast resolution. Through simulation of a full Prism-PET brain scanner, we demonstrate that our proposed Recovery-Net algorithm addresses these issues, demonstrating a 38.9% improvement in intrinsic spatial resolution, enhanced discernability of tiny spots (1.0 mm and 0.8 mm diameter), and increase of 41% in peak-valley ratio across 1.5 mm spots.

A major limitation of the work presented here is that the method is tested only on a simulation dataset, while the most relevant benchmarks require performance evaluations with real sensor data. Propagation of optical photons is particularly sensitive to small variations in conditions and the details of the scanner setup; therefore simulations are especially limited in describing these phenomena. However, due to the confined and deterministic nature of the light sharing in Prism-PET, we were able to demonstrate alignment between the simulation and experimental data. Additionally, there are several sources of randomness and bias in real data that are not captured in the simulation, e.g., pileup, and subject scatter. And, there are inherently unpredictable stochastic effects in real-world imaging that cannot be modeled, such as misalignment between crystals, inhomogeneity in SiPM gain, and malfunctions in the individual components of the detector. These can only be known by empirically measuring the properties of the device and are not addressed here. The scope of this paper is centered around theory and proof-of-concept. Adaptation of these concepts to real sensor data and experimental validation will be presented in future works.

6. Conclusion

We developed a solution for the ICS problem through the application of statistical and computational methods that were able to capitalize on the unique physical design of the Prism-PET light guide that promotes confined and deterministic light sharing while enabling depth encoding. The findings were concordant between a Bayesian estimation model and neural network applied to a special test case of scatter between adjacent crystals, and were generalizable to 3D event positioning for the full photopeak. We were able to demonstrate best-in-field results for energy and DOI accuracy. Thus, we find that Prism-PET presents a feasible solution to the ICS problem and offers a significant advance in PET imaging capabilities.

Appendix A. Scatter-absorption model

Appendix A.1. ${\mathit{P}}_{\mathit{comp}}(\mathit{z})$

The principle of linear attenuation gives the cumulative probability density of a Compton interaction along a crystal of length $L$ as

$$F(z)=\frac{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-z/{\lambda }_c\right)}{1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-L/{\lambda }_c\right)}$$ A.1

where $z$ is the penetration depth along the direction of incidence and ${\lambda }_c$ is the attenuation length for Compton interactions. A probability density function is then given by differentiation.

$$f(z)=\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-z/{\lambda }_c\right)}{{\lambda }_c\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-L/{\lambda }_c\right)\right]}$$ A.2

Interpolation from NIST data gives the Compton cross section as 7.3×10⁻² cm² g⁻¹ for 511keV photons in LYSO [1]. Thus, ${\lambda }_c$ ≈ 19.2mm for 511 keV photons in LYSO of density of 7.1gcm⁻³.

Appendix A.2. ${\mathit{P}}_{\mathit{k}\mathit{n}}(\mathit{\theta })$

A probability density for Compton scatter over scatter angle $\theta$ is obtained from the Klein-Nishina differential scattering cross-section given in terms of cross-section per unit solid angle

$$\frac{d\sigma }{d\mathrm{\Omega }}=\frac{1}{2}{r}_e^2{P}^2\left[P+\frac{1}{P}-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \right]$$ (A.3)

where $r_e$ is the classical electron radius. The parameter $P$ gives the energy of the scattered photon as a fraction of the energy of the incident photon $E_{{\gamma }^{\mathrm{\text{'}}}}/E_{\gamma }$ as

$$P\left(E_{\gamma },\theta \right)=\frac{1}{1+\left(E_{\gamma }/m_e{c}^2\right)(1-\mathrm{c}\mathrm{o}\mathrm{s}\theta )}$$ (A.4)

where $m_e$ is the electron mass. Replacing $d\mathrm{\Omega }$ in the above expression with $2\pi \sin \theta d\theta$ gives the cross section per unit scattering angle as

$$\frac{d\sigma }{d\theta }=\pi r_e^2{P}^2\mathrm{s}\mathrm{i}\mathrm{n}\theta \left[P+\frac{1}{P}-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta \right]$$ (A.5)

which is then converted to a probability density by normalization with the total scattering cross section ${\sigma }_c$. Integrating the differential cross-section over all angles, we obtain the total cross-section as

$${\sigma }_c=2\pi r_e^2\left\{\frac{1+k}{k^2}\left[\frac{2(1+k)}{1+2k}-\frac{\mathrm{l}\mathrm{n}(1+2k)}{k}\right]+\frac{\mathrm{l}\mathrm{n}(1+2k)}{2k}-\frac{1+3k}{(1+2k{)}^2}\right\}$$ (A.6)

where $k=E_{\gamma }/m_e{c}^2$. Thus, the probability density function for the scatter angle is given by

$$f(\theta )=\frac{1}{{\sigma }_c}\frac{d\sigma }{d\theta }$$ (A.7)

For 511keV photons $k=1,$ $P=(2-\mathrm{c}\mathrm{o}\mathrm{s}\theta {)}^{-1}$ and the PDF for scatter angle becomes

$$f(\theta )\approx \frac{.87\mathrm{s}\mathrm{i}\mathrm{n}\theta }{(2-\mathrm{c}\mathrm{o}\mathrm{s}\theta {)}^3}\left(1+(2-\mathrm{c}\mathrm{o}\mathrm{s}\theta {)}^2-{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta (2-\mathrm{c}\mathrm{o}\mathrm{s}\theta )\right)$$ (A.8)

Appendix A.3. ${\mathit{\lambda }}_{\mathit{p}}(\mathit{\theta })$

The attenuation length was estimated by first fitting a curve to NIST data for the photoelectric absorption cross section ${\sigma }_p$ as

$${\sigma }_p\approx 5\times {10}^5{E}_{{\gamma }^{\mathrm{\text{'}}}}^{-2.64}$$ (A.9)

where $E_{{\gamma }^{\mathrm{\text{'}}}}$ is the energy of the Compton-scattered photon in $\mathrm{k}\mathrm{e}\mathrm{V}$ and ${\sigma }_p$ is given in units of ${\mathrm{c}\mathrm{m}}^2{\mathrm{g}}^{-1}$ [1]. Fitting was done for $E_{{\gamma }^{\mathrm{\text{'}}}}\in \left[100,511\right]\mathrm{k}\mathrm{e}\mathrm{V}$. Given ${\sigma }_p$, the attenuation length in $\mathrm{c}\mathrm{m}$ for photoelectric absorption is then

$${\lambda }_p={\left({\sigma }_p\rho \right)}^{-1}=2\times {10}^{-6}{\rho }^{-1}{E}_{{\gamma }^{\mathrm{\text{'}}}}^{\mathrm{2,64}}$$ (A.10)

where $\rho$ is the density of the scintillator crystal (7.1gcm⁻³ for LYSO). The energy of the scattered photon may be written in terms of the scatter angle as

$$E_{{\gamma }^{\mathrm{\text{'}}}}(\theta )=\frac{E_{\gamma }}{1+\left(E_{\gamma }/m_e{c}^2\right)(1-\mathrm{c}\mathrm{o}\mathrm{s}\theta )}$$ (A.11)

and the attenuation length finally becomes

$${\lambda }_p(\theta )=2\times {10}^{-6}{\rho }^{-1}{\left(\frac{E_{\gamma }}{1+\left(E_{\gamma }/m_e{c}^2\right)(1-\mathrm{c}\mathrm{o}\mathrm{s}\theta )}\right)}^{2.64}$$ (A.12)

For 511keV photons incident on LYSO with density 7.1gcm⁻³, the attenuation length in cm is estimated as

$${\lambda }_p(\theta )\approx 4(2-\mathrm{c}\mathrm{o}\mathrm{s}\theta {)}^{-2.64}$$ (A.13)