## 1 Introduction  
Prostate cancer (PCa) represents one of the most common malignancies affecting males globally [1,2]. The global incidence of PCa is increasing with the aging of the population [3,4]. Because of the heterogeneity of PCa in terms of severity, ranging from indolent to highly aggressive, PCa patients are usually classified into risk groups (low-, intermediate-, and high-risk) before definitive treatment. According to the D'Amico and the National Comprehensive Cancer Network (NCCN) risk criteria, the risk classification is based on the prostate-specific antigen (PSA) level, clinical T (cT) stage, and biopsy Gleason score (GS) [5,6]. Both radiotherapy (RT) and radical prostatectomy (RP) have played an important role in the definitive treatment of non-metastatic localized or locally advanced PCa and can decrease the cause-specific mortality (CSM) rate [7]. High-risk PCa cases, which account for 20-30% of PCa diagnoses, are associated with biochemical recurrence (BCR), metastases, and CSM [8, 9, 10]. Although RT with androgen deprivation therapy (ADT) and RP with or without adjuvant therapy are the main treatments for high-risk PCa, no clear consensus has been reached yet on the optimal treatment recommendations in clinical guidelines [6,11].  
Carbon-ion radiotherapy (CIRT) was first used for localized prostate cancer in 1995 by the National Institute for Quantum Science and Technology in Chiba, Japan (formerly known as the National Institute of Radiological Science) [12]. The unique physical and biological properties of CIRT offer potential advantages over conventional RT, including enhanced cytotoxic effects on cancer cells and superior dose distribution [12, 13, 14, 15, 16, 17]. Favorable clinical outcomes, such as lower rates of BCR and CSM, have been observed in high-risk PCa patients treated with CIRT in combination with ADT [13, 14, 15, 16, 17]. Furthermore, CIRT combined with long-term ADT has yielded more favorable treatment outcomes compared with treatments involving RT and RP in high-risk PCa patients [14]. Despite these promising results, a subset of high-risk PCa patients remains at elevated risk for BCR, a critical clinical concern due to its strong correlation with overall mortality [14,18, 19, 20]. Consequently, more precise risk stratification for high-risk PCa beyond the conventional D'Amico and NCCN criteria is required to accurately assess BCR risk in individual patients. Conventional high-risk PCa classifications encompass a diverse spectrum of disease, necessitating refined predictive tools. Previously, we examined whether the Candidio nomogram, which was originally developed to predict 10-year BCR risk in all-risk PCa patients treated with conventional RT, can also predict outcomes after CIRT [13, 21, 22, 23]. Although our previous study showed better discrimination even in high-risk PCa patients treated with CIRT based on the Candidio risk classification, the risk categories and modality used differ between the Candidio classification and our studies [13].  
Therefore, a novel prediction model is needed to predict BCR-free survival in high-risk PCa patients after CIRT and ADT. The aim of this study was to develop a new tool for predicting 5- and 7-year BCR-free survival in high-risk PCa patients after CIRT.