## 1 Introduction  
It is well-known that protons have low lateral scattering and therefore deposit energy more locally when compared to photons [1; 2; 3; 4; 5]. Hence, the concept of proton range monitoring and verification is important to enhance the effectiveness of proton therapy treatment [6]. The dose deposition of protons increases with depth, and near the end of their range, it reaches its maximum, commonly referred to as the Bragg peak. Considering this, protons have a lower exit dose when compared to photons, which are used in conventional radiation therapy [7; 8; 9]. Therefore, protons are highly sensitive to spatial uncertainties [10], which makes range monitoring and verification crucial in therapeutic applications. Considering proton range monitoring, several strategies have been proposed and investigated, with their own specific advantages and disadvantages [11]. Proton radiography and tomography have both shown promising results for proton range verification, but scattering reduces the resolution of the acquired images [12]. Other direct and cost-effective techniques that were reported in previous investigations are the ionoacoustics technique [13] and secondary electron bremsstrahlung (SEB) [14]. In addition, several investigators have recently studied Prompt Gamma Imaging (PGI) [15, 16] for proton range verification and reported promising results. Another important technique for proton range monitoring and verification is auto-activation Positron Emission Tomography (PET) imaging, which is a non-invasive technique. The proton range is estimated by measuring annihilation photons generated by positron emitting radioisotopes such as \({}^{15}\)O, \({}^{11}\)C and \({}^{13}\)N. These positron emitting radioisotopes are generated as a result of nuclear reactions induced by incident protons. In 1969, [PERSON] et al. [17] first proposed the use of the auto-activation PET technique for hadron range monitoring and verification. Later, the proposed auto-activation PET technique was experimentally investigated at a heavy ion therapy facility at the Lawrence Berkeley national laboratory [18]. Furthermore, this technique was found to be promising for monitoring the delivered proton dose in therapeutic applications [19]. There are various PET system configurations that have been used to measure annihilation photons generated by positron emitting radioisotopes in the auto-activation PET technique with the ultimate goal of proton range monitoring and verification [20, 21, 22, 23, 24]. For example, full-ring scanners [21, 22], which were found to have high detection efficiency but at the same time to incorporate a full ring- ring PET system into beam delivery, which is technically tedious particularly when there are geometric constraints. Non-ring PET systems were developed to circumvent the drawback of full-ring scanners by providing a higher degree of freedom for better patient positioning and installation [25, 26, 27, 28, 29]. These non-ring PET systems include the DoPET (Dosimetry-PET) system [26], which is a compact prototype in-beam PET scanner [27], a compact planar PET prototype [28] and a modularized acquisition-based dual-head PET with wider detectors [29]. The main drawbacks associated with these non-ring PET systems are mainly the limited angular coverage and artifacts in the reconstructed images. Another notable system that has higher spatial resolution and sensitivity compared to previously introduced systems would be the positron emission mammography (PEM) system, which is a dedicated PET system for detecting breast cancer [30]. The PEM system was first proposed by [PERSON] et al.