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Use of non-Gaussian time-of-flight kernels for image reconstruction of Monte Carlo simulated data of ultra-fast PET scanners

Efthimiou, Nikos; Thielemans, Kris; Emond, Elise; Cawthorne, Chris; Archibald, Stephen J.; Tsoumpas, Charalampos


Nikos Efthimiou

Kris Thielemans

Elise Emond

Chris Cawthorne

Charalampos Tsoumpas


Introduction: Time-of-flight (TOF) positron emission tomography (PET) scanners can provide significant benefits by improving the noise properties of reconstructed images. In order to achieve this, the timing response of the scanner needs to be modelled as part of the reconstruction process. This is currently achieved using Gaussian TOF kernels. However, the timing measurements do not necessarily follow a Gaussian distribution. In ultra-fast timing resolutions, the depth of interaction of the γ-photon and the photon travel spread (PTS) in the crystal volume become increasingly significant factors for the timing performance. The PTS of a single photon can be approximated better by a truncated exponential distribution. Therefore, we computed the corresponding TOF kernel as a modified Laplace distribution for long crystals. The obtained (CTR) kernels could be more appropriate to model the joint probability of the two in-coincidenceγ-photons. In this paper, we investigate the impact of using a CTR kernel vs. Gaussian kernels in TOF reconstruction using Monte Carlo generated data. Materials and methods: The geometry and physics of a PET scanner with two timing configurations, (a) idealised timing resolution, in which only the PTS contributed in the CTR, and (b) with a range of ultra-fast timings, were simulated. In order to assess the role of the crystal thickness, different crystal lengths were considered. The evaluation took place in terms of Kullback–Leibler (K-L) distance between the proposed model and the simulated timing response, contrast recovery (CRC) and spatial resolution. The reconstructions were performed using STIR image reconstruction toolbox. Results: Results for the idealised scanner showed that the CTR kernel was in excellent agreement with the simulated time differences. In terms of K-L distance outperformed the a fitted normal distribution for all tested crystal sizes. In the case of the ultra-fast configurations, a convolution kernel between the CTR and a Gaussian showed the best agreement with the simulated data below 40 ps timing resolution. In terms of CRC, the CTR kernel demonstrated improvements, with values that ranged up to 3.8% better CRC for the thickest crystal. In terms of spatial resolution, evaluated at the 60th iteration, the use of CTR kernel showed a modest improvement of the peek-to-valley ratios up to 1% for the 10-mm crystal, while for larger crystals, a clear trend was not observed. In addition, we showed that edge artefacts can appear in the reconstructed images when the timing kernel used for the reconstruction is not carefully optimised. Further iterations, can help improve the edge artefacts.


Efthimiou, N., Thielemans, K., Emond, E., Cawthorne, C., Archibald, S. J., & Tsoumpas, C. (in press). Use of non-Gaussian time-of-flight kernels for image reconstruction of Monte Carlo simulated data of ultra-fast PET scanners. EJNMMI Physics, 7(1), Article 42.

Journal Article Type Article
Acceptance Date May 20, 2020
Online Publication Date Jun 19, 2020
Deposit Date Jun 23, 2020
Publicly Available Date Oct 27, 2022
Journal EJNMMI Physics
Electronic ISSN 2197-7364
Publisher SpringerOpen
Peer Reviewed Peer Reviewed
Volume 7
Issue 1
Article Number 42
Keywords Monte Carlo; Positron emission tomography; Photon travel spread; Depth of interaction; Fast timing
Public URL
Publisher URL
Additional Information Received: 27 August 2019; Accepted: 20 May 2020; First Online: 19 June 2020; : Not applicable.; : Not applicable.; : The authors declare that they have no competing interests.


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© The Author(s). 2020 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,<br /> which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate<br /> credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were<br /> made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless<br /> indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your<br /> intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly<br /> from the copyright holder. To view a copy of this licence, visit

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