Inclusion of Inter-crystal Scattering in PET: Analytical Models and Dedicated Reconstruction

TL;DR

This work presents a physics-based analytical model for inter-crystal scattering (ICS) in PET, deriving analytical forms for the ICS sensitivity image $s^\text{I}$ and the ICS system matrix $\mathbf{H}^\text{I}$ without needing to locate the Compton scattering site. The model is integrated into a joint list-mode MLEM framework with conventional (golden) events, enabling simultaneous use of ICS and golden data to boost sensitivity while preserving quantitative accuracy. Validation against Monte Carlo simulations and MERMAID small-animal PET data shows that including ICS reduces statistical noise and improves uniformity in low-count scenarios, albeit with a small reduction in recovery coefficients due to the less informative V-shaped LORs. The approach is adaptable to arbitrary PET geometries, avoids heavy training or data requirements, and is particularly advantageous when sensitivity is the limiting factor, such as in very small-animal imaging or low-dose studies.

Abstract

Inter-crystal scattering (ICS) in Positron Emission Tomography (PET) is commonly regarded as a degradation effect that might compromise the image spatial resolution. In parallel, the inclusion of ICS events has also been recognized as a potential approach to increase PET sensitivity, which could be especially beneficial in scenarios where the latter is a limiting factor, such as very small animal imaging. Several methods for the recovery of ICS events have been proposed, many of which aim to locate the first interaction, i.e., the Compton scattering site, usually limited by their success rate, computational burden or data and training dependency. Conversely, this work proposes a physics-based model for ICS events, leading to analytical expressions of the sensitivity image and the system matrix (required by statistical reconstruction algorithms), without the need to identify the original line of response. After validating the model, the work shows how ICS events can be integrated into a joint image reconstruction algorithm (based on list-mode MLEM) together with conventional PET events, for which dedicated analytical models are also developed. To assess the performance of the proposed approach, Monte-Carlo simulated and experimental data of an image quality phantom were obtained with the MERMAID small-fish PET scanner prototype. Both simulation and experimental results indicate that, while slightly decreasing the recovery coefficient values, the inclusion of ICS clearly reduces statistical noise and improves uniformity.

Figures (15)

• Figure 1: Scheme of an ics event with relevant quantities of the proposed analytical model. The red lines indicate the photon trajectories. The PET detector elements where the physical interactions occur are highlighted in light blue. In particular, the Compton scattering takes place in $\vec{r}_2$.
• Figure 2: Transversal ($XY$) and sagittal ($YZ$) projections over the analytical ics sensitivity image, computed for the MERMAID geometry with $3$ rotation steps and $10$ bed positions, using (\ref{['eq:sens2']}) and assuming an energy threshold of $180~\textrm{keV}$ over the ics events. The scale represent the probability of an annihilation emitted in a given group of voxels (e.g. a $Z$-row of voxels for the transversal projection) to be detected as an ics event in MERMAID.
• Figure 3: Transversal ($XY$) projections over two different analytical ics system matrix rows, computed for the MERMAID geometry using (\ref{['eq:sm4']}). The scale represents the probability of annihilation emitted in a $Z$-row of voxels to be detected in terms of the particular measurement element considered.
• Figure 4: Validation of the ics analytical sensitivity image model with no energy threshold over the ICS events (orange line) and with a threshold of $180~\textrm{keV}$ (blue line) against Monte Carlo simulations (red and violet crosses, respectively), (a) including ics ocurring in neighboring crystals, and (b) discarding them. Each simulated $X$-position represents the mean of five independent simulations; the displayed error bars correspond to the standard error of the mean.
• Figure 5: Results with the $1~\textrm{MBq}$mc-simulated cylinder source: (a) Transversal ($XY$) projection at iteration 10 for golden and ics events, with the color scale representing the activity in MBq recovered in the sum of voxels involved in the projection; (b) evolution of the total activity recovered in the golden and ics, only golden and only ics images by summing all the voxel values in the three cases.