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Penalised PET image reconstruction algorithms are often accelerated during early iterations with the use of subsets. However, these methods may exhibit limit cycle behaviour at later iterations due to variations between subsets. Desirable converged images can be achieved for a subclass of these algorithms via the implementation of a relaxed step size sequence, but the heuristic selection of parameters will impact the quality of the image sequence and algorithm convergence rates. In this work, we demonstrate the adaption and application of a class of stochastic variance reduction gradient algorithms for PET image reconstruction using the relative difference penalty and numerically compare convergence performance to BSREM. The two investigated algorithms are: SAGA and SVRG. These algorithms require the retention in memory of recently computed subset gradients, which are utilised in subsequent updates. We present several numerical studies based on Monte Carlo simulated data and a patient data set for fully 3D PET acquisitions. The impact of the number of subsets, different preconditioners and step size methods on the convergence of regions of interest values within the reconstructed images is explored. We observe that when using constant preconditioning, SAGA and SVRG demonstrate reduced variations in voxel values between subsequent updates and are less reliant on step size hyper-parameter selection than BSREM reconstructions. Furthermore, SAGA and SVRG can converge significantly faster to the penalised maximum likelihood solution than BSREM, particularly in low count data.

Purpose Nuclear medicine imaging modalities like computed tomography (CT), single photon emission CT (SPECT) and positron emission tomography (PET) are employed in the field of theranostics to estimate and plan the dose delivered to tumors and the surrounding tissues and to monitor the effect of the therapy. However, therapeutic radionuclides often provide poor images, which translate to inaccurate treatment planning and inadequate monitoring images. Multimodality information can be exploited in the reconstruction to enhance image quality. Triple modality PET/SPECT/CT scanners are particularly useful in this context due to the easier registration process between images. In this study, we propose to include PET, SPECT and CT information in the reconstruction of PET data. The method is applied to Yttrium-90 ($$^{90}$$ 90 Y) data. Methods Data from a NEMA phantom filled with $$^{90}$$ 90 Y were used for validation. PET, SPECT and CT data from 10 patients treated with Selective Internal Radiation Therapy (SIRT) were used. Different combinations of prior images using the Hybrid kernelized expectation maximization were investigated in terms of VOI activity and noise suppression. Results Our results show that triple modality PET reconstruction provides significantly higher uptake when compared to the method used as standard in the hospital and OSEM. In particular, using CT-guided SPECT images, as guiding information in the PET reconstruction significantly increases uptake quantification on tumoral lesions. Conclusion This work proposes the first triple modality reconstruction method and demonstrates up to 69% lesion uptake increase over standard methods with SIRT $$^{90}$$ 90 Y patient data. Promising results are expected for other radionuclide combination used in theranostic applications using PET and SPECT.

Emission-based attenuation correction (AC) meth-ods offer the possibility of overcoming quantification errors induced by conventional MR-based approaches in PET/MR imaging. However, the joint problem of determining AC and the activity of interest is strongly ill-posed in non-TOF PET. This can be improved by exploiting the extra information arising from low energy window photons, but the feasibility of this approach has only been studied with relatively simplistic analytic simulations so far. This manuscript aims to address some of the remaining challenges needed to handle realistic measurements; in particular, the detection efficiency (“normalisation”) estimation for each energy window is investigated. An energy-dependent detection efficiency model is proposed, accounting for the presence of unscattered events in the lower energy window due to detector scatter. Geometric calibration factors are estimated prior to the reconstruction for both scattered and unscattered events. Different reconstruction methods are also compared. Results show that geometric factors differ markedly between the energy windows and that our analytical model correspond in good approximation to Monte Carlo simulation; the multiple energy window reconstruction appears sensitive to input/model mismatch. Our method applies to Monte Carlo generated data but can be extended to measured data. This study is restricted to single scatter events.

Application of stochastic variance reduction algorithms to iterative PET reconstruction. We investigated the SAGA and SVRG algorithms for non-TOF PET image reconstruction. Both similated data and a patient data sets were used in the analysis. We found that the stochastic algorithms can improve convergence rate and eliminate behaviour, commonly know as limit cycle behaviour, from PET reconstruction within 5 epochs.

Several groups have considered the option of using multiple-energy window emission data to improve reconstruction accuracy for PET/MR imaging. However, the feasibility of this approach has only been studied with analytic simulations. To be able to handle realistic measurements a remaining challenge is related to the detection efficiencies (“normalisation”) for the different energy windows, which likely differ for scattered and unscattered events given the heterogeneity of photon energies, angles of incidence, and points of origination. In addition, the lower energy windows will potentially contain events that have been affected by scatter in the detector. This manuscript investigates the problem of estimating the normalisation factors for PET data acquired from multiple energy windows.

Penalised PET image reconstruction methods are often accelerated with the use of only a subset of the data at each update. It is known that many subset algorithms, such as Ordered Subset Expectation Maximisation, do not converge to a single solution but to a limit cycle, which can lead to variations between subsequent image estimates. A new class of stochastic variance reduction optimisation algorithms have been recently proposed for general optimisation problems. These methods aim to reduce the subset update variance by incorporating previous subset gradients into the update direction computation. This work applies three of these algorithms to iterative PET penalised reconstruction and exhibits superior performance to standard deterministic reconstruction methods after only a few epochs.

Statistical PET image reconstruction methods are often accelerated by the use of a subset of available projections at each iteration. It is known that many subset algorithms, such as ordered subset expectation maximisation, will not converge to a single solution but to a limit cycle. Reconstruction methods exist to relax the update step sizes of subset algorithms to obtain convergence, however, this introduces additional parameters that may result in extended reconstruction times. Another approach is to gradually decrease the number of subsets to reduce the effect of the limit cycle at later iterations, but the optimal iteration numbers for these reductions may be data dependent. We propose an automatic method to increase subset sizes so a reconstruction can take advantage of the acceleration provided by small subset sizes during early iterations, while at later iterations reducing the effects of the limit cycle behaviour providing estimates closer to the maximum a posteriori solution. At each iteration, two image updates are computed from a common estimate using two disjoint subsets. The divergence of the two update vectors is measured and, if too great, subset sizes are increased in future iterations. We show results for both sinogram and list mode data using various subset selection methodologies.