# Deep-Learning Based Positron Range Correction of PET Images
## 1 Introduction
### Positron Range Models  
PR can be modeled in many ways, which can be grouped into the following categories in order of increasing complexity:  
Isotropic and tissue-independent model: A uniform model is used for all voxels, assuming water (or soft tissue) as the medium through which positrons travel and annihilate. This is a fast and easy procedure to implement, and only requires the activity distribution as input. PR distributions for most common radionuclides used in PET in water- or plastic-equivalent materials can be measured [15] or computed using Monte Carlo (MC) simulations based on well-established libraries such as GEANT4, PENELOPE, and FLUKA [10, 16, 17]. The simulated spatial distributions can be fitted to an equation [9, 10]. These models can consider differences in the energy distribution of the positrons emitted by each radionuclide (see Figure 1a), but in general they are not quite realistic and can yield artefacts in heterogeneous media.  
Isotropic but tissue-dependent model: The positron range effect is modeled as a blurring kernel that depends on the material of the voxel from which the positron is emitted, irrespective of the surrounding media. In this case, each blurring kernel is homogeneous and isotropic. This correction requires the coregistration of the CT image with the activity distribution and it is expected to work well everywhere except near tissue boundaries. However, a non-negligible number of positrons close to the skin-air boundary can escape and using space-invariant filters can result in severe artefacts [19]. For instance,  
Figure 1: Simulated energy spectra of positrons emitted by \({}^{18}\)F and \({}^{68}\)Ga obtained with the code PenNuc [18] (**a**), and distribution of annihilation points for \({}^{18}\)F and \({}^{68}\)Ga in water and lung tissue (**b**).  
these artefacts may be observed in clinical \({}^{124}\)I PET imaging of thyroid glands close to the trachea [20].  
Anisotropic and tissue-dependent (approximated) model: To address the presence of inhomogeneous media, space-variant models have been proposed [21; 22]. The anisotropic PR has been modeled by anisotropically truncating an isotropic point probability density function based on tissue type, performing successive convolution operations of tissue-dependent PR kernels, or using the average of the fitting parameters of annihilation densities for the originating voxel and the target voxel [23]. These filters provide a fast and robust method for implementing a PR model, but due to the complexity of positron migration at irregular interfaces, it may not be always accurate.  
Anisotropic and tissue-dependent (full) model: In this final case, the blurring kernel takes into account the material from which the positron is emitted and all the different materials that the positron travels through until it annihilates. These models are accurate even when the activity is distributed at extreme tissue boundaries. Monte Carlo simulations, despite their large computational cost, can be used to generate these models [24], including when the PR is in the presence of magnetic fields [17; 25; 26].