## 1 Introduction  
In modern medicine, radiation therapy remains a cornerstone in cancer treatment. As the most widely established modality, it benefits from decades of clinical research and ex tensive infrastructure in radiotherapy centers worldwide. Over time, the field has evolved from individualized anatomical treatments toward a more sophisticated, personalized approach that integrates technological advancements for precise dose delivery with biological insights into tumor characteristics and resistance mechanisms [1]. Photon therapy is a conventional method that uses ionizing X-ray radiation to target and eradicate cancerous tissues. The primary mechanism involves damaging the DNA of malignant cells, leading to cell death and ultimately controlling or eliminating tumor growth. However, a significant limitation of X-ray therapy is its inability to selectively target tumors at varying depths while sparing adjacent healthy tissue. This issue arises from the inherent nature of photon energy deposition in tissue, where, after a brief accumulation, the radiation dose increases with depth, potentially resulting in unintended damage to healthy tissues beyond the tumor site [2]. To address this limitation, particle beam therapy, using protons and heavy ions, has emerged as a more precise modality, offering enhanced protection for normal tissues by concentrating energy deposition at a specific depth. Proton therapy, a leading form of particle therapy, was first introduced in a clinical setting at Loma Linda University Medical Center (CA) in 1990. By the end of 2022, over 300,000 patients had been treated with proton therapy [3]. Currently, more than 140 proton therapy facilities are operating worldwide, with more than 20 additional centers under construction [4]. Proton therapy has gained attention because of its superior dose-distribution properties. As protons travel through tissue, their velocity decreases as energy is lost through inelastic Coulombic interactions with atomic nuclei. Near the end of their path, protons deposit most of their energy, resulting in a low entrance dose, followed by a sharp increase at the maximum peak, and a steep fall-off at the end of the range. This peak, known as the \"Bragg peak\", occurs near the end of the proton beam range, which is determined by the energy of the protons [5]. As illustrated in Figure 1, which compares the depth-dose distribution characteristics of the X-rays and the proton beam, the peak of the proton depth-dose distribution occurs at a specific depth. The position of this peak can be adjusted by varying the proton energy to target a desired depth. In addition, the Bragg peak profile characterizes the depth-dose distribution of a proton beam, highlighting key dosimetric parameters. The range at 100% dose (RBP) marks the location of the peak dose deposition. The range at 80% dose (R80) is commonly used to define the clinical target position due to its stability and reproducibility [6]. The distal fall-off width (W80-20), defined as the distance between the 80% and 20% dose levels on the distal edge of the Bragg peak, reflects the sharpness of dose fall-off and is critical for assessing beam precision and minimizing dose to surrounding healthy tissues. This unique feature allows for precise targeting of the tumor while protecting surrounding healthy tissue [7], potentially reducing collateral damage to healthy cells. With the advancement of pencil beam scanning, protons can now be precisely directed using intensity-modulated proton therapy (IMPT) [8]. This advanced technique enables controlled energy deposition within the tumor at a predetermined depth or over a narrow range, minimizing exposure to healthy tissues around the tumor. These characteristics make proton and other charged-particle therapies more advantageous for cancer radiation therapy than conventional photon beams. Additionally, from a radiobiological perspective, charged particle beams provide a high linear energy transfer (LET) around the Bragg peak [9].