Data-driven, Energy-based method for estimation of Scattered events in Positron Emission Tomography

Abstract

Objective

Scattered events add bias in the reconstructed PET images. Our objective is the accurate estimation of the scatter distribution, required for an effective scatter correction.

Approach

In this paper, we propose a practical energy-based (EB) scatter estimation method that uses the marked difference between the energy distribution of the non-scattered and scattered events in the presence of randoms. In contrast to previous EB methods, we model the unscattered events using data obtained from measured point sources.

Main results

We demonstrate feasibility using Monte Carlo simulated as well as experimental data acquired on the long axial field-of-view (FOV) PennPET EXPLORER scanner. Simulations show that the EB scatter estimated sinograms, for all phantoms, are in excellent agreement with the ground truth scatter distribution, known from the simulated data. Using the standard NEMA image quality (IQ) phantom we find that both the EB and single scatter simulation (SSS) provide good contrast recovery values. However, the EB correction gives better lung residuals.

Significance

Application of the EB method on measured data showed, that the proposed method can be successfully translated to real-world PET scanners. When applied to a 20 cm diameter ×20 cm long cylindrical phantom the EB and SSS algorithms demonstrated very similar performance. However, on a larger 35 cm × 30 cm long cylinder the EB can better account for increased multiple scattering and out-of-FOV activity, providing more uniform images with 12%–36% reduced background variability. In typical PET ring sizes, the EB estimation can be performed in a matter of a few seconds compared to the several minutes needed for SSS, leading to efficiency advantages over the SSS implementation. as well.

1. Introduction

In Positron Emission Tomography (PET), background events lead to bias, cause artifacts, and impair the quantitative accuracy of the reconstructed images. Background coincidence events comprise of either two γ-photons originating from different annihilation events (random coincidences), or two γ-photons from the same annihilation, but one or both undergo one or more Compton interactions before entering the PET detector (scatter coincidences). While random coincidences are accurately estimated using the delayed window method, correction for scatter coincidences can still be challenging in certain imaging situations. Therefore, a reliable and robust method for estimating the scattered events’ contribution above the energy threshold is critical for reducing bias in the image.

The most successful and fairly standard method for scatter estimation is the Single Scatter Simulation (SSS) (Ollinger 1996, Watson et al. 1996). SSS assumes that only one of the two γ-photons scatters and only once. The method starts from an initial uncorrected image and uses the attenuation map to distribute scatter points within the object, then invokes the Klein-Nishina equation to estimate the relative amount of scattering in each LOR. In recent years the SSS method has been extended to include Time-of-Flight (TOF) information (Werner et al. 2006, Watson 2007) and to model double scattering (Tsoumpas et al. 2005, Watson et al. 2018). Due to its reasonably good accuracy, robustness, and speed SSS is the gold standard for routine scatter estimation in the clinic. However, SSS has two major shortcomings: a lack of multiple scatter modeling and the need for sinogram tail-fitting to scale the scatter estimate to an absolute value. These limitations become critical when imaging larger patients (Surti et al. 2007) who have a higher probability of multiple scattering, reduced counts in the sinogram tails due to attenuation, and cover the largest part of the transaxial Field-of-View (FOV), as well as in imaging situations with limited counts as seen in short dynamic studies or when isotopes with low positron yield are imaged. In addition, SSS also has difficulty in providing an accurate scatter estimate when there is a sharp change in activity concentration (Heußer et al. 2017, Lindemann et al. 2019, Magota et al. 2020).

Several other scatter estimation methods have been proposed and developed in the past (Zaidi 2000, Polycarpou et al. 2011). Such as methods using models of the scatter shape to estimate the scatter contribution in each Line-of-Response (LOR) of emission data. The above are either limited by their simplified assumptions (tail-fitting methods) (Karp et al. 1989, Cherry & Huang 1995) or their need to use prior phantom measurements to generally characterize the scatter shape (convolution subtraction methods) (Bailey & Meikle 1994). Multiple energy window-based methods (e.g., dual-energy window (Grootoonk et al. 1996, Adam et al. 2000), triple energy window (Shao et al. 1994, Deidda et al. 2019), estimation of trues (Adam et al. 2000)) or with fast Monte Carlo simulations (Guerin & El Fakhri 2011, Álvarez-Gómez et al. 2021) that rely on estimating the relative number of scattered and true events for each LOR from events collected in a few (2–3) discrete energy windows and require a calibration scale factor which varies as a function of activity distribution or object size (Adam et al. 2000). The underlying assumptions and practical implementation of these methods make them limited in accuracy for clinical applications. Full Monte Carlo (MC) methods providing an accurate scatter estimate have also been developed (Holdsworth et al. 2002), but these are computationally intensive and generally not fast enough for routine clinical use. More recently, scatter estimation using deep learning has shown promising results (Berker et al. 2018, Berker & Kachelrieß 2019, Maier et al. 2018); however, they rely on a large number of preexisting data for their training.

Here, we present a practical extension of a generalized Energy-based (EB) scatter estimation method that was initially proposed by Popescu et al. (Popescu et al. 2006, Efthimiou, Karp & Surti 2021) for estimating the number of scattered events in a LOR based on the energy distribution of the events (Efthimiou, Karp & Surti 2021). The method takes advantage of the marked difference between the non-scattered and scattered events energy spectra. Compared to multiple energy window-based methods, our method utilizes the full energy spectrum of the data and benefits from the lack of pile-up effects and the improved energy resolution of the current generation of PET scanners. Our method, as implemented here, focuses on algorithm robustness and practicality for routine use. The work uses Monte Carlo (MC) simulations to validate and evaluate the performance of our method in realistic, challenging imaging scenarios. The simulations are based on a single ring of the PennPET Explorer (PPEx) scanner geometry with three different phantoms under various settings, to demonstrate proof of principle. We then performed phantom measurements with the PPEx scanner in both 5-ring and 6-ring configurations (112 – 136 cm axial field-of-view) and compared the EB estimation to 3D TOF-SSS that we routinely use for patient imaging on this scanner.

2. Materials and Methods

Each PET event consists of two annihilation γ-photons, detected in coincidence between two detectors with energies (E₁ and E₂). Unscattered γ-photons can deposit their full initial energy (E₀ = 511 keV) in the detector. However, after a Compton interaction within the patient or emission object, γ-photons deviate from their original flight path and lose part of their initial energy. Photons scattered at large angles can be effectively rejected with an energy cut. However, uncertainty in energy measurement broadens the detected energy distribution and leads to miss-classification of scattered photons near the 511 keV energy peak.

We can express the 2D coincidence energy histogram C(E₁, E₂), as the sum of four types of events:

$$C\left(E_1,E_2\right)=c_{0}U\left(E_1\right)U\left(E_2\right)+c_{1}U\left(E_1\right)S\left(E_2\right)+c_{2}S\left(E_1\right)U\left(E_2\right)+c_{3}S\left(E_1\right)S\left(E_2\right)$$ (1)

where U(E) and S(E) are the energy Probability Density Functions (PDF) of the unscattered and scattered photons, respectively. The four terms in eq. 1 represent events where i) both photons are unscattered (uSc event) with weight c₀, ii) only photon 1 or photon 2 is scattered (sSc₁ or sSc₂ event) with weight c₁ or c₂, and events where both photons scattered (dSc) with weight c₃. The number of unscattered events equals c₀ and the total number of scattered events is given by c_Sc = c₁ + c₂ + c₃. Figure 1.A shows a 1D energy spectrum (both photons comprising a coincident event tabulated together) from a Monte Carlo simulation for all scattered and unscattered events (prompts), as well as the two components individually (tagged in Monte Carlo).

Figure 1.

A. 1D energy histograms for unscattered events (uSc) in green, scattered events (sSc₁ + sSc₂ + dSc) in orange, and prompt events in blue. B. Decomposed 1D energy histograms for scattered (red) and unscattered (green) photons within all scattered events.

To simplify the extraction of the energy PDFs for scattered and unscattered photons, we assume the energies of the two annihilation photons in each coincident event to be uncorrelated, thereby avoiding fitting the 2D energy plane formed by the two photons’ energies (Hamill 2019, 2021). The 1D energy histogram for scattered events c_ScSc(E) can then be decomposed into two parts:

$${\sigma }_U(E)=p_{1}U(E)$$ (2)

$${\sigma }_S(E)=p_{2}S(E)$$ (3)

where σ_U is the energy histogram of the unscattered photons within the sSc events (illustrated in Fig. 1.B (green)), σ_S is the summed energy histogram of the scattered photons within the sSc events and all the photons within the dSc events and p₁, p₂ are their respective weights. The 1D energy spectrum for prompt events (C(E)) can then be defined as:

$$C(E)=\left(p_0+p_1\right)U(E)+p_{2}S(E)$$ (4)

where p₀ is the number of uSc events and hence (p₀ + p₁) represents the contribution of all unscattered photons from uSc and sSc events.

2.1. Estimation of energy PDFs using global energy spectra

For most of the energy range, the energy distribution of scattered photons depends on the amount of single and multiple scattering within the object or patient. However, the higher part of the energy spectrum (above the photopeak), is driven by the detector’s energy resolution (Adam et al. 2000). In order to obtain S(E) we first scale a model of unscattered photon energy spectrum to this region of the 1D prompt energy histogram. This scaled model then represents (p₀ + p₁)U(E). Subtracting it from C(E) (eq. 4) we get:

$$S(E)=\left(C(E)-\left(p_0+p_1\right)U(E)\right)/p_2$$ (5)

As a first approximation, we assume that the very high part of the energy histogram primarily contains unscattered photons and scattered photons with a small scattering angle that have been broadened by the detector’s energy resolution, modeled by a Gaussian function (u(E)). In this part of the energy range we can set a high energy window (HEW) with boundaries (he_l, he_u) and scale u(E) by a factor $p_0^{\prime }$ as:

$$\left(p_0+p_1\right)U(E)\approx p_0^{\prime }{u}_i(E)=\frac{{\sum }_{h{e}_l}^{h{e}_u}C(E)}{{\sum }_{h{e}_l}^{h{e}_u}{u}_i(E)}{u}_i(E)$$ (6)

However, the detector’s energy response is not the only factor in the energy shape of the unscattered photons. Inter-crystal scattering, partial energy deposition, energy sharing schemes, the effect of energy binning and other factors affect the accuracy of $p_0^{\prime }$ estimation. As such, a Gaussian model does not accurately describe the energy distribution of the unscattered photons. This can be seen in Fig 1.B where unscattered photons energy histogram extends below and above the ±4σ from the photopeak, which otherwise should not happen. Hence, our assumption of a Gaussian model for u(E) will negatively affect our scaling as described in eq. 6. Consequently, the scaling factor $p_0^{\prime }$ will not be close enough to the ideal p₀ + p₁ and, hence, the shape of S(E) as calculated from eq. 5 will not be correct.

In order to mitigate this problem we add an additional step in the estimation process. First, after correcting for the ¹⁷⁶Lu activity (see par 2.1.2) we subtract from C(E) the number of events that are above 4σ of the photopeak, prior to scaling u(E), as described above. Then we replace the scaled u(E) model with a photopeak shape measured with a low scattering, low randoms source (u_m(E)), demonstrated at Fig. 2. This could be the scanner normalization source, or an arrangement of point sources that provides an accurate representation of the energy response of unscattered photons. Finally, we scale it by the previously estimated $p_0^{\prime }$ and then subtract it from the C(E), to obtain an accurate estimate of S(E).

Figure 2.

Normalized energy spectra (u_m(E)) that we used to replace the idealized Gaussian photopeak (u(E)). The figure includes the measured and simulated point source histograms, that were used in this paper and the histogram from the thinnest aluminum filament of the sensitivity calculation, as a viable alternative.

2.1.1. Selection of upper threshold (he_u) for HEW

The selection of the (he_u) is critical for the success of our method. Among other factors, the exact value will depend on the energy binning and there might be more than one (he_u)value that can work for each data set and bin size. However, two conditions need to be met:

1. After subtraction of the modeled peak, the scattered photon energy PDF (S(E)) should not have negative values. Negative values of statistical nature, can be thresholded.

2. Visual inspection of the estimated scattered photon energy PDF (S(E)) should not have a peak in the 511 keV range.

2.1.2. Removal of random coincidences and ¹⁷⁶Lu contribution

In the simulated data we did not include the ¹⁷⁶Lu background activity, however, it was present in the measured data due to the LYSO crystals. Most of the coincidences from ¹⁷⁶Lu are part of random coincidences and we can subtract their total contribution to the prompt energy histogram by subtracting the energy histogram of the delayed events used to estimate randoms. For the residual prompt coincidences from 176Lu we performed an analytic simulation following a method presented previously (Alva-Sánchez et al. 2018, Domínguez-Jiménez & Alva-Sánchez 2021). The simulated histogram was fitted to events in the prompt energy histogram that lie above the 4σ level from the photopeak, and then subtracted from the prompt energy spectrum. Alternatively, one could acquire a blank scan on the scanner (no activity or object in the FOV) to directly estimate the contribution of ¹⁷⁶Lu.

2.2. Scatter calculation using the estimated energy PDFs

We use the moments method to estimate the scatter present in individual LOR in the collected data by fitting the estimated energy PDFs for scattered and unscattered photons (S(E) and U(E)). The raw, uncentered moments are computed easily and can reveal important aspects of the two distributions. Using different moments we can place restrictions on the location, scale or shape of a distribution.

The estimator with moments of order (m, n) of the prompt energy distribution can be directly calculated from the listmode data for each LOR (l) as ${\widehat{q}}_{mn;l,k}={\Sigma }^{N_{l,k}}{E}_1^m{E}_2^n$ where N_l,k are the total number of prompt events and k is the TOF bin. Equating this to the estimate from the model we have:

$$\begin{gathered} q_{mn;l,k}-{\widehat{w}}_{mn;l}/K={\int }_{\epsilon }{\int }_{\epsilon }{E}_1^m{E}_2^n{C}_{lk}\left(E_1,E_2\right)d{E}_{1}d{E}_2 \\ ={\sigma }_0{a}_m{a}_n+{\sigma }_1{a}_m{b}_n+{\sigma }_2{b}_m{a}_n+{\sigma }_3{b}_m{b}_n \end{gathered}$$ (7)

where, ${\widehat{w}}_{mn;l}={\Sigma }^{M_l}{E}_{1,i}^m{E}_{2,i}^n$ is the estimator calculated from the delayed events (M_l) in l, $a_m={\int }_{\epsilon }{E}^{m}U(E)dE$, $b_m={\int }_{\epsilon }{E}^{m}S(E)dE$ and K are the total number of TOF bins. We want to get the most information out of our moment conditions thus we would like to use the optimal weighting matrix that minimizes the estimator. Since the moments method requires the selection of an appropriate pair of moments. We used a data-driven approach to test a variety of combinations and we found that first and second moments work the best, and using a value beyond the third moment leads to poor results.

Equation 7 formulates a 4×4 system of equations for which we can calculate an exact solution using the Cramer’s method. The estimated number of scattered and unscattered events in a given LOR l and TOF bin k are then given by:

$$e_{Sc}(l,k)={\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)}_{l,k}\approx c_{Sc}(l,k)$$ (8)

$$e_U(l,k)={\left({\sigma }_0\right)}_{l,k}\approx c_U(l,k)$$ (9)

When solving for the σ₀ … σ₃ in Eq.7 it is critical to address the absence (i.e. value of zero) of any of the four σ components in a particular LOR. For example, this situation can arise in the NEMA scatter phantom where all the activity is located in the line source. Therefore, LORs that are at some distance from the source will not contain any unscattered events. Then the Cramer’s method will inflate the other components and assign negative values to the missing component(s). To avoid this, we run a minimization algorithm removing negative values and re-solving the system for the other components, until no negative values remain, or a stable solution has been found.

$$\begin{array}{l}\underset{\sigma }{\mathrm{\text{minimize}}}\Vert A^{mn}\sigma -Q^{mn}\Vert \\ \text{ subject to }{\sigma }_i\ge 0,i=1,\dots ,4.\end{array}$$ (10)

where Q^mn and A^mn are the estimators and integrals from Eq. 7, with the respective moments.

2.3. Monte Carlo simulated data

We performed simulations using the GATE Monte Carlo simulation toolbox (v.8.1.p01) (Jan et al. 2004). The scanner model was based on a single ring of the PennPET Explorer (PPEx) scanner geometry (Viswanath et al. 2018, Karp et al. 2020). In brief, the scanner is comprised of 3.86 × 3.86 × 19 mm³ LYSO crystals grouped in 8 × 8 blocks or tiles, with a 4 × 7 arrangement of these blocks forming a single module. A single ring of the scanner has 18 modules, a diameter of 76.4 cm and an axial length of 23.0 cm. The simulated energy and timing resolutions were 11% and, 250 ps, respectively. The coincidence timing window was set at 4 ns and randoms were estimated in a delayed coincidence window with a 500 ns delay. All data were collected and reconstructed in a 440 – 660 keV energy window. The simulated energies were binned in 5.11 keV bins, same as those obtained from the PPEx scanner. In this work, the simulations did not include patient bed and end-shielding.

2.3.1. Phantoms and acquisitions for evaluation

Figure 3 illustrates the three phantoms that we used in this study. We simulated all phantoms using the e⁺ emitting ¹⁸F source, which models the positron range. The NEMA image quality phantom was first simulated in a standard imaging setup with a background activity concentration of 4.4 kBq/cc and hot sphere uptake of 4:1 relative to background. The second simulation of this phantom increased the background activity concentration to 11.9 kBq/cc with a hot sphere uptake of 120:1 relative to background in order to study the impact of increased randoms ratio as well as very high contrast objects on scatter estimation.

Figure 3.

Illustration of simulated phantoms used in this study. The NEMA IQ phantom is illustrated on the left, the cold slab phantom is in the middle, and the bladder insert phantom is on the right, the cold slab phantom is at the center. The phantoms are demonstrated approximately in-scale.

The cold slab phantom is a 35 cm diameter × 70 cm long polyethylene cylinder with a 2.54 cm thick rectangular polyethylene slab placed at the center of the cylinder and running axially through its full length. The diameter of this phantom represents a problematic setup for SSS where tail fitting can be challenging with reduced counts, while the length of the phantom exceeded that of the simulated scanner leading to the presence of out-of-FOV scattered events. We simulated this phantom for a combination of three different activity levels and imaging times representing varying level of collected counts and randoms ratio.

The bladder phantom is an oval-like cylinder with large and small diameter values of 30.5 cm and 12 cm and a length of 120 cm. In the middle of the phantom, a cylindrical insert is placed with high activity contrast (> 150×) which represents a full hot bladder. We simulated this phantom in two bed positions; one with the bladder located at the center of the scanner FOV and, the other where the center of the insert was located 6 cm outside the scanner’s axial FOV. The presence of a very hot uptake region (bladder) both within the FOV and just outside the FOV represents challenging scenarios for SSS where the EB scatter estimation may have an advantage.

2.3.2. Image reconstruction and scatter estimation

All simulated data were reconstructed with the STIR image reconstruction toolkit (Thielemans et al. 2012, Ovtchinnikov et al. 2020), using OSEM with 11 subsets and 2 full iterations for the NEMA IQ phantom and 1 full iteration for the cold slab and bladder phantoms. We used fewer iterations in the latter cases since the phantoms are quite large in size and have uniform activity distribution. The voxel size was 1.93 × 1.93 × 1.98 mm³ and the grids were restricted near the boundaries of each phantom, to speed up reconstruction. The NEMA IQ phantom is post-filtered with a 3D Gaussian filter of 0.8 × 0.8 × 0.8 mm³.

The default scatter estimation algorithm within STIR is a nonTOF 2D-SSS. Hence, for comparison purposes we estimated both TOF and nonTOF scatter using the EB method. We performed non-TOF image reconstruction in projection space using nonTOF SSS (4 iterations) and EB scatter estimates, while the TOF (Efthimiou et al. 2019, Wadhwa et al. 2020, Efthimiou, Kratochwil, Gundacker, Polesel, Salomoni, Auffray & Pizzichemi 2021) reconstruction used listmode data and the TOF EB scatter estimate. In addition, an ideal data set in which all non-random scattered events were excluded was also reconstructed, using both TOF and nonTOF algorithms. The images from this ideal data set provide the benchmark for our comparisons.

Due to the low frequency nature of the spatial distribution of scattered events, the scanner geometry was down-sampled within the EB scatter estimation in order to achieve better noise characteristics in the emission data . Down-sampling of the scanner geometry is a common practice in most scatter estimation algorithms including SSS. The estimated 3D EB scatter sinograms in down-sampled space are then up-sampled back into the simulated scanner template using full 3D interpolation. In this study, we considered two down-sampled geometries: d1) with 36 and d2) with 72 detectors per scanner ring. The total number of detector rings was fixed at 7 (from the original 56) and we down-sampled to 17 TOF bins when performing TOF scatter estimation.

2.4. Experimental data

For the experimental evaluation of our energy-based method we acquired data on PPEx using two cylindrical phantoms: a 35 cm diameter and 30 cm long uniform cylinder filled with 275 MBq and scanned for 15 minutes on the PPEx scanner with 5 rings (total axial length of 112 cm), and a 20 cm diameter × 20 cm long uniform cylinder with a 2.5 cm thick rectangular (20 × 20 cm²) cold slab placed axially at the center of the cylinder. The cold slab phantom was filled with 275 MBq and imaged for 20 minutes on an extended PPEx scanner with 6-rings (total axial length of 136 cm). The TOF resolution of PPEx is 250 ps and both data acquisitions were performed with a fully open acceptance angle. PPEx acquires data in list-mode where the energies of both photons forming an event are written out in 5.11 keV bins. For the experimental data we used only the d1 down sampling geometry for EB scatter estimation.

A total of 5.3 × 10⁹ prompt events were collected with the 35×30 cylinder out of which 3.7 × 10⁹ events were random coincidences (estimated with delayed window data). This data set was also sub-sampled into four smaller data sets in order to investigate the impact of reduced counts on the scatter estimation: 58 × 10⁶ prompts (41.5 × 10⁶ randoms), 291 × 10⁶ prompts (207 × 10⁶ randoms), 581 × 10⁶ prompts (415 × 10⁶ randoms), and 1160 × 10⁶ prompts (829 × 10⁶ randoms). The total counts collected with the cold slab phantom were 876 × 10⁶ prompts (281 × 10⁶ randoms).

2.4.1. Image reconstruction

Image reconstruction was performed using a TOF LM-OSEM with 25 subsets and 5 full iterations (Popescu et al. 2004). Images were generated with voxel size of 2 × 2 × 2 mm³. This algorithm uses blob based basis functions optimized in size and grid spacing for the spatial resolution and noise characteristics of the scanner (Matej & Lewitt 1996). The blobs suppress image noise while preserving signal; hence, no post-filtering is required. Normalization correction factors are generated by calculating the ratio of the collected data to an analytic rotating line sinogram, followed by modest smoothing (Casey & Hoffman 1986). The reference scatter estimation method is a 3D TOF-SSS method (Accorsi et al. 2004, Werner et al. 2006), which is routinely used with data acquired from the PPEx scanner.

3. Results

3.1. Simulation statistics

The output statistics and key information for the simulated phantoms are summarized in Table 1. The presented statistics of the simulated data were extracted directly from the output files, without following the NEMA protocol.

Table 1.

Basic statistics of the simulated data used in this study. The randoms’ fraction (Delayed/Prompts) as given here is the ground truth.

Phantom ID	Acquisition time	Total Activity	Prompts	Randoms ratio
Simulations
NEMA 4:1	150 s	43.1 MBq	81.1 × 106	11.6%
NEMA 120:1	400 s	115 MBq	48.8 × 107	24.7%
Cold Slab	30 s	30 MBq	1.9 × 106	18.6%
	60 s	60 MBq	9.4 × 106	31.3%
	30 s	300 MBq	53.0 × 106	69.5%
Bladder (inside)	300 s	60 MBq	78.0 × 106	31.5%
Bladder (outside)	350 s		74.3 × 106	35.7%
Measurements
35×30 Cylinder	15 min	275 MBq	5.29 × 109	70.7%
20×25 Cold slab	20 min	275 MBq	8.76 × 108	32.1%

3.2. Accuracy of EB scatter estimate relative to SSS and ground truth (GT)

In Fig. 4 ground truth (GT) and EB estimated scatter sinograms in the down-sampled space (d1), and associated profiles are demonstrated. From the profiles, we can see that the estimates for scattered and unscattered events, without and with TOF are in excellent agreement with the GT, especially in the tail regions. Also, we can see that the estimated scatter sinograms have more noise relative to the GT, as uncertainty in the estimation is added as statistical noise. The SFs for each simulation are summarized in Table 2. The relative error (RE) between the estimated number of scattered counts and the GT was below 3.1% for EB (nonTOF and TOF) and −10.5% for 2D-SSS. In the case of NEMA 120:1, the RE was below 5% for the EB estimates and about −14% for the 2D-SSS.

Figure 4.

Ground truth scatter sinograms taken from the Monte Carlo data (left column), nonTOF energy-based scatter estimation sinograms (middle column), and line profiles drawn through the second view, are illustrated for four simulated phantoms. In this example downsampling level 1, the direct segment and a central axial position, was used.

Table 2.

Ground truth and estimated scatter fractions for every simulation.

Phantom ID	GT	2D-SSS	nonTOF EB	TOF EB
NEMA 4:1	28.2%	25.2%	28.9%	29.0%
NEMA 120:1	28.7%	24.7%	30.1%	29.4%
Cold Slab 30s 30MBq	42.0%	37.1%	41.3%	44.8%
Cold Slab 60s 60MBq	42.0%	36.5%	41.4%	42.9%
Cold Slab 30s 300MBq	42.0%	35.0%	38.9%	41.2%
Bladder (inside)	37.2%	27.5%	35.3%	35.7%
Bladder (outside)	38.6%	29.2%	36.8%	37.3%

In the case of the cold slab phantom, 2D-SSS systematically underestimates the SF. The EB method leads to SF values that are much closer to the GT values even at very low counts (≈ 2 × 10⁶ prompts for 30s acquisition with 30MBq activity) and high randoms ratio (30s acquisition with 300MBq activity). These general findings for estimate SF also apply to the hot bladder phantom results, where the 2D-SSS underestimates scatter while the EB method is very close to the GT values.

3.3. Quantitative accuracy of reconstructed images from simulated data

3.3.1. Application to a standard imaging phantom and the impact of down-sampling on image accuracy

Following the NEMA guidelines, we calculated the %Δ_lung residuals and contrast recovery coefficient (CRC) for all spheres in the NEMA IQ phantom. In Fig. 5 we show the central reconstructed transverse slice of the NEMA 4:1 phantom with quantitative results from the %Δ_lung and the three more challenging spheres (37,28 and 10 mm in diameter).

Figure 5.

(Top row) Central transverse slices from reconstructed images of the simulated NEMA 4:1 IQ phantom. Images are shown for the ideal data set (non-random, scattered events excluded), full data set corrected with the 2D-SSS estimate and corrected with the EB estimate. (Bottom row) Lung residual and CRC values for three spheres from the NEMA 4:1 and 120:1 IQ phantoms. Results are shown for the ideal image as well as images using the 2D-SSS estimate for scatter as well as the EB scatter estimate in the two downsampled spaces (d1 and d2).

As expected, visually, none of the reconstructed images of the NEMA 4:1 phantom exhibit any issues. However, quantitatively the %Δ_lung values of the EB image are closer to those of the ideal image. On the other hand, the residuals of images corrected with the 2D-SSS method are ≈ 40% larger than the ideal. In terms of CRC, in most cases, EB.d1 outperforms 2D-SSS, while the benefits of EB.d2 are not clear. We believe large objects may benefit from the reduced interpolation needed to place the EB scatter estimated in d2 space back into the scanner space. However, the fewer events present in each downsampled bin can impact the accuracy of the EB estimation.

3.3.2. Application to large imaging objects and the impact of counts and randoms ratio on image accuracy

The cold slab phantom is a very challenging case for the tail fitting within SSS since the phantom with its 35 cm diameter fills a large portion of the transverse FOV. In addition, as simulated here, the phantom is longer than the scanner axial FOV, leading to scatter contribution from activity lying outside the imaging FOV that is not modeled with SSS. As shown in Fig. 6 2D-SSS is not able to provide a reasonable scatter estimate, leading to a reconstructed image with leftover activity in the cold slab. On the other hand, images corrected using the EB method are very close to the ideals.

Figure 6.

(Top row) Reconstructed transverse images of the simulated cold slab phantom (60 s, 60 MBq) with excluded non-random scattered events (ideal) and, corrected with 2D-SSS and EB estimated sinograms. (Bottom row) Horizontal and vertical profiles drawn across and over the cold slab, for all three phantom settings, are shown here.

In the case of 30 s 30 MBq simulation with ≈ 2 × 10⁶ prompts, the relative bias of the corrected image to the ideal was 9.8% and 23.4% for nonTOF EB and 2D-SSS, respectively. In the 60 s, 60 MBq case, the corresponding values were 6.3% and 21.7% and in 30 s, 300 MBq 10% and 25%, respectively. Using the TOF reconstruction and TOF EB scatter estimation (not shown), the relative bias in the images was 6.12%, 8.9%, 12.8%.

3.3.3. Impact of a hot activity source on image accuracy

The bladder phantom represents the combined challenge of providing accurate scatter estimates for a very hot source of activity placed inside or outside the FOV with large amounts of out-of-FOV background. Fig 7A shows reconstructed images from this hot bladder phantom study.

Figure 7.

(A) Reconstructed nonTOF images of the bladder phantom inside and 6 cm outside the FOV for an an ideal data set (left column), using 2D-SSS scatter estimate for scatter correction (middle column), and using EB scatter estimate for scatter correction (right column). (B) Profiles drawn through the background areas as indicated by the blue rectangles in (A).

With the bladder insert inside the FOV, the 2D-SSS corrected image shows non-uniformity in the background that the EB does not have. With the bladder positioned right outside the FOV (6 cm away), we see the impact of 2D-SSS not modeling the out-of-FOV activity, leading to increased non-uniformity that is not present in the EB corrected image. The image profiles shown in Fig. 7B demonstrate the improved accuracy of the EB method relative to 2D-SSS.

We calculated the relative bias (RB) between the two correction methods and the ideal dataset in the background area using two large ROIs on either side of the bladder insert. We found RB values of 8.3% and 14% for the EB and 2D-SSS, respectively. At the same time, the RB for hot bladder insert was −0.01% and 0.01% for the EB and 2D-SSS.

As the RB values display, both scatter correction methods do sufficiently for a hot object in the center of the FOV. However, for regions closer to the edge of the scanner, the out-of-FOV activity becomes a challenging factor for SSS. The above is accentuated further when the insert is placed outside the scanner boundary. The relative bias in the background is 6.7% and 24.7% for the EB and 2D-SSS, respectively. The above is also visually illustrated by the profiles in Fig. 7.

3.4. Quantitative accuracy of reconstructed images from measured data

3.4.1. Large uniform cylinder

In Fig 8 we show transverse and coronal slices of the reconstructed images from the 35 cm diameter uniform cylinder. The images show that the TOF 3D-SSS algorithm over-estimates the scatter near the center leading to non-uniformity in the reconstructed image. On the other hand, the EB scatter estimation leads to improved uniformity that is also demonstrated in the horizontal transverse profile as shown in this figure. In addition, we measured the phantom’s background variability (BV), using fifty 1 cm in diameter circular ROIs placed in a cross formation. The spacing between the ROIs was 4 cm and they were drawn over three transverse slices (10 cm apart). BV was calculate as described in the NEMA protocol with the above ROIs. The BV results show that the images generated using the EB scatter estimate have improved (lower) BV relative to the images generated using scatter estimated from the TOF 3D-SSS algorithm and this improvement is present in all data sets all the way down to 58 × 10⁶ prompt events.

Figure 8.

(A) Transverse and coronal slices from the the reconstructed images of the 35 cm diameter × 30 cm long uniform cylinder using the TOF EB and TOF 3D SSS algorithm for scatter estimation, (B) horizontal profiles drawn through the transverse images, and (C) background variability (BV %) calculated from the reconstructed images over the prompt events.

3.4.2. Cold slab phantom

Figure 9 shows transverse and coronal slices and profiles of the reconstructed images from the 20 cm diameter cylinder with a cold slab. This phantom was smaller in size than the one we used in the Monte Carlo simulations (par. 3.3.2), leading to a good estimation of scatter both with the EB method and the 3D TOF SSS algorithm. This is clearly demonstrated in the radial sinogram profiles shown in this figure, where both algorithms produce accurate scatter estimates.

Figure 9.

(A) Transverse and coronal slices from the the reconstructed images of the 20 cm diameter × 20 cm long cold slab phantom using the TOF EB and TOF 3D SSS algorithm for scatter estimation, (B) horizontal profiles drawn through the transverse images(slices 345 and 385), and (C) radial profile showing the prompt-delay events and overlaid scatter profiles as estimated with the two scatter algorithms.

4. Discussion

This paper presents the methodology and results from a data-driven, energy-based (EB) method to estimate the scatter coincidences present in PET data. In addition to accuracy, our method emphasizes robustness for routine use and speed, both of which will be especially relevant when imaging with long axial FOV PET scanners (Spencer et al. 2020, Karp et al. 2020, Prenosil et al. 2021). The EB method was validated with Monte Carlo simulated data of a single ring of PennPET Explorer (PPEx) geometry, using several challenging phantoms and acquisition settings. The method was also applied to experimental data acquired on the PPEx scanner. The simulation studies focused on performance and were compared with the ground truth or ideal (no scattered events present) data sets and, as well as scatter estimates obtained with a 2D SSS algorithm (as implemented in STIR). The results from measurements were compared with a state-of-the-art TOF 3D SSS algorithm that is routinely used on the PPEx scanner.

Simulation results show that scatter sinograms estimated with the EB algorithm are in excellent agreement with the ground truth obtained from the Monte Carlo data. In addition, reconstructed images corrected with the EB method have reduced bias and better contrast than those corrected with 2D-SSS. Moreover, EB corrected images are in very good agreement with those reconstructed from ideal data. Using the NEMA phantom, we show improvements in the Δ_lung residuals over a wide range of hot sphere contrasts. Finer data downsampling during scatter estimation did not show any clear advantage with the NEMA IQ phantom. With the challenging cold-slab phantom, we see that even with increased multiple scatter and significant out-of-FOV activity, the EB method provides images in excellent agreement with the ideal images and have a significantly lower bias (up to 40%) when compared to images corrected using the 2D-SSS algorithm. These results were further reinforced with the long, hot bladder phantom, where the reduction in relative bias is 27% and 40%, respectively when the bladder is positioned inside or outside the imaging FOV.

Finally, our simulation results show that the accuracy of the EB estimation depends on the amount of data per sinogram bin, but the effect is not detrimental to the image quality, at least for the range we investigated. With the cold slab phantom, the relative bias is < 10% with the EB method when using 2 × 10⁶ prompt events, while with 2D-SSS, the bias is higher (25%). Further optimization of downsampling during EB scatter estimation could potentially lead to even better performance at these low counts levels.

We translated and tested our EB scatter estimation method using a 35 × 30 cm² cylindrical phantom that was imaged on the 5-ring PPEx scanner. Large phantoms have increased amount of scatter, and a limited tail region in the emission sinogram that makes tail-fitting in the standard SSS algorithm very challenging. Our results clearly show that EB correction is more accurate than 3D TOF-SSS, which over-corrected the reconstructed images in the center of the phantom. In addition, the improved accuracy of the EB method is demonstrated with reduced counts as well. We used background variability to quantify this improvement, and our results show that the image corrected with EB was 36% better than that obtained with SSS (at the highest count levels). Even with the smallest data subset of 58 × 10⁶ prompt events, EB outperforms SSS. The physical cold-slab cylinder at our disposal for experimental measurements was smaller than the one we used in the simulations. Therefore, we see that both algorithms can produce accurate scatter sinograms, and the reconstructed images’ differences are negligible.

From a speed perspective, the EB algorithm as implemented in this work is very fast: on an i7 (6th gen) laptop, with simulated and measured (not shown here) data of a single ring of the PPEx scanner, the 3D nonTOF EB estimation was performed in about 8 s, while the TOF estimate took 40 s. STIR’s 2D-SSS for the same geometry took about 12 min for four iterations. We executed the EB estimation of the five-ring geometry on a cluster with 56 processes (using OpenMP) due to the many segments and axial positions. Approximately 3 min were required for processing 291 × 10⁶ prompt events and about 10 minutes for 5.3 × 10⁹ prompt events. These processing speeds are achieved due to our efficient data organization in hybrid sinogram-listmode files, which allows parallel processing of small, organized, listmode files.

The central assumption in our work is the use of a global energy spectrum (collected over all LORs) to estimate a scatter photon energy PDF that is applied individually for all photons over all LORs. This provides a mean estimate and represents a trade-off since the estimation of scattered photons energy PDF for each LOR will be error-prone due to meager statistics within each LOR (and TOF bin). However, the use of a high energy threshold for collected data (typically ≥ 440 keV) for modern L(Y)SO scanners with 10 – 12% energy resolution reduces these variations in the PDF. Our results show that this assumption does not limit the performance of the method, for the imaging setups and count levels, studied in this work. However, an incorrect estimate of Sc(E) can propagate errors in the EB estimation at very low count levels. As an alternative, one could consider a hybrid approach, where SSS can be used to generate the scatter energy distributions, which can then be used by the EB method to estimate the full scatter distribution. However, this is beyond the scope of the current work but will be evaluated in the future.

Finally, all our work (simulations and experiments) uses list-mode data that provide energy information for both photons within an event. While this may not always be standard information from all PET vendors, to our knowledge, at least two major PET vendors do provide this information in their list data.

Overall, through simulation and measurements, we have developed and rigorously tested a new data-driven EB method for scatter estimation that performs at least as well as SSS for simple imaging tasks. However, in more challenging situations, such as when imaging larger objects, reduced counts, and the presence of out-of-FOV activity, the EB method out-performs SSS. The method only requires emission data and a simple estimate of the unscattered photon energy spectrum that can be acquired as a point source data acquisition on the scanner as part of a routine quality control (QC) process. Together with its speed advantage, the proposed method can impact clinical PET/CT imaging and other situations where a lack of CT image affects accurate scatter estimation with SSS (e.g., PET/MR imaging). Finally, the method can also be extended to provide accurate correction of prompt gamma coincidences that can arise when imaging non-standard PET isotopes such as ¹²⁴I, ⁸²Rb, and ⁸⁶Y (Conti & Eriksson 2016).

Another interesting application of our developed method may lie in obtaining real-time, semi-quantitative PET images. From eq. 9 we can see that a direct estimate of the non-random unscattered events can be obtained on-the-fly without any further processing (unscattered without randoms). Alternatively, a simple reformulation of eq. 7:

$$\begin{gathered} q_{mn;l,k}={\int }_{\epsilon }{\int }_{\epsilon }{E}_1^m{E}_2^n{C}_{lk}\left(E_1,E_2\right)d{E}_{1}d{E}_2 \\ ={\sigma }_0{a}_m{a}_n+{\sigma }_1{a}_m{b}_n+{\sigma }_2{b}_m{a}_n+{\sigma }_3{b}_m{b}_n \end{gathered}$$ (11)

would lead to an estimate of all unscattered events (randoms and non-randoms), that can be separately corrected for randoms using standard methods within image reconstruction. As a test, we reconstructed the NEMA IQ phantom (4:1) data set using these two alternative formulations, and Fig. 10 shows images without scatter correction but reconstructed from the estimated unscattered events with and without randoms. When compared to Fig. 5 we can see that these images are not as good as the ideal and fully corrected images, but quantitatively the accuracy is much better than an uncorrected image. We can envisage cases where a quick, even potentially real-time (Ozoemelam et al. 2020), semi-quantitative image is desired or necessary, and where this method can offer better approximations than leaving the background uncorrected. In literature, researchers have described PET guidance to navigate needles to targets (Venkatesan et al. 2011, Mauri et al. 2019) or in-beam PET online monitoring (Crespo et al. 2007, Kraan et al. 2015).

Figure 10.

Reconstructed images of the simulated NEMA 4:1 phantom using TOFOSEM (11 subsets and 2 full iterations, post-filtered with 0.8×0.8×0.8 Gaussian kernel) using EB estimation of unscattered events with and without randoms. In addition, contract recovery for different cases is given.

5. Conclusion

In this paper, we proposed a method for energy-based (EB) scatter estimation that is fast and robust, and suitable for use with long axial FOV PET scanners. We demonstrated the proof of principle through realistic simulations with challenging phantoms and showed that the proposed EB estimates produce quantitatively accurate images. EB performed better than SSS with large phantoms and large amounts of out-of-FOV activity. Furthermore, using measured data acquired on the 5- and 6-ring PennPET Explorer scanner, we showed that the EB performs as well or better than a state-of-the-art TOF 3D-SSS. The difference between the two algorithms was more pronounced for the larger, 35×30 cylindrical phantom.