Analytic timing calculations and timing limits with prompt photons, high-aspect-ratio crystals, and complex TOF-kernels in TOF-PET

TL;DR

The paper addresses the challenge of predicting timing performance in TOF-PET detectors that mix scintillation and prompt Cherenkov light, especially in high-aspect-ratio crystals. It develops a modular analytic framework that combines closed-form light transport, prompt-photon statistics, DOI bias, and photodetector response to produce full first-photon and coincidence timing distributions, evaluated with three metrics and the CRLB. Validation against analytic approximations and Monte Carlo simulations shows good agreement across materials, crystal thicknesses, and detector types, while highlighting the impact of prompt photons and non-Gaussian tails on timing performance. The approach offers rapid (sub-second) timing predictions for detector configurations, enabling efficient parametric studies to guide fast-timing detector development for TOF-PET.

Abstract

Modeling the timing performance of light-based radiation detectors accurately is essential for optimizing time-of-flight positron emission tomography (TOF-PET). We present an analytic framework that combines existing models to predict the timing behavior of high-aspect ratio crystals, including contributions from prompt photons such as Cherenkov radiation. This framework is built on a closed-form solution for optical light transport, convolved with the photodetector response and photon production characteristics. Using conditional and joint probability distributions, we compute the first-photon arrival time distribution for hybrid detectors with scintillation and Cherenkov light. The detection time distribution is then self-convolved to derive the time delay spectra and three timing metrics are used to characterize complex TOF kernels. Additionally, we perform Cramér-Rao Lower Bound calculations with and without depth-of-interaction bias to evaluate the theoretical timing limits. Our analytic predictions align well with Monte Carlo simulations for BGO detectors under varying crystal thicknesses and single photon time resolution considering a digital photodetector. We show that the TOF shape is significantly affected by prompt photon statistics, crystal thickness, scintillation yield, and photodetector properties resulting in distinct metric-dependent timing performance. The proposed model enables rapid timing predictions for polished crystals, with the calculation time of a detector configuration in under a second, allowing for comprehensive parametric studies. This makes it a powerful tool for guiding detector development in fast-timing applications.

Figures (7)

• Figure 1: Photon detection time probability distribution for one prompt photon for different DOIs and SPTR. The scale of the x-axis is increased for higher SPTR values (bottom figures) to cover the full range. The spikes in the detection time for all DOIs without SPTR are due to limited DOI sampling (first DOI = 0.25 mm, last DOI = 19.75 mm, $dDOI$ = 0.5 mm).
• Figure 2: Top: Detection time probability distribution of the first detected scintillation photon for different number of M detected photons in linear scale for the first nanosecond (left) and logarithmic scale for the first 10 ns (right). Bottom left: detection time probability distribution of the first photon considering only BGO scintillation (black), only one detected prompt photon (red), an average of $\mu=2.7$ detected prompt photons with underlying Poisson fluctuations (blue), and combined BGO scintillation with Cherenkov photons (magenta). Bottom right: Coincidence time delay probability distribution considering the same cases.
• Figure 3: Left: Comparison of time resolution results of the analytic framework (this study) with approximations VINOGRADOV_2018_NIMA for different materials. The input parameters of each material is specified in table \ref{['tab:crystals']} and, if modified, indicated in the figure. The initial photon time density (IPTD) is calculated as outlined in Gundacker_2020_PMB. The analytic approximation is 3.33 times the inverse square root of the IPTD. The CRLB and CTR (SNR) are calculated for each material with and without the presence of prompt photons on top of scintillation. Right: CTR comparison of Monte-Carlo simulations Gundacker_2020_PMB (FWHM) and our analytic framework for two different BGO geometries as function of the SiPM SPTR.
• Figure 4: Calculated time resolution of BGO for different crystal thickness.
• Figure 5: Calculated time resolution metrics as function of the mean number of detected prompt photons on top of the BGO scintillation emission for 3 mm (top) and 20 mm (bottom) thick BGO crystals. The inset shows the graphs in linear scale up to 200 ps.