Model-Based Deep Learning PET Image Reconstruction Using Forward-Backward Splitting Expectation Maximisation

We propose a forward-backward splitting algorithm for maximum-a-posteriori (MAP) PET image reconstruction, which provides an elegant framework for integration of deep learning (DL) into model-based iterative reconstruction. The MAP reconstruction is split into three steps: regularisation, expectation maximisation (EM) and a weighted fusion. For regularisation, use of either a weighted Tikhonov prior (a hypothesis-driven prior based on Markov random fields) or a residual learning unit (a data-driven prior based on convolutional neural networks) were considered. For the latter, our proposed forward-backward splitting expectation maximisation (FBSEM), accelerated with ordered subsets (OS), was unrolled into a recurrent neural network in which the network parameters (including regularisation strength) are shared across all states and learned while PET images are reconstructed from emission data. FBSEM net was trained on simulated brain phantoms using PET-MRI (FBSEM-PM net) and PET-only (FBSEM-P net) data and was compared to OSEM, MR-guided MAPEM and a post-reconstruction U-Net denoiser, trained on the same PET dataset, for different count-levels. Our results showed that FBSEM-PM net accurately reconstructs test data without suppressing PET unique lesions or introducing MR unique lesions. It was also found that FBSEM-P net and U-Net both outperform MAPEM and OSEM reconstructions and achieve similar performance for moderate-to-high count levels. However, as the count level is reduced, the FBSEM-P net results in notably lower noise-induced hot/cold spots. These results therefore highlight the superiority of model-based DL reconstruction over conventional MAPEM and post-reconstruction DL-based denoising methods. Future work will include realistic 3D simulations and transfer learning for reconstruction of in-vivo PET and PET-MR data.