## 1 Introduction  
Breast Cancer (BC) is the most frequent neoplasm in women. In recent years, the life expectancy of patients with BC has increased due to screening-related earlier detection and treatment improvements. The current gold standard for early-stage BC is conserving surgery followed by external beam radiotherapy (RT). This adjuvant treatment may reduce the incidence of locoregional recurrence and BC deaths [1]. However, RT inevitably exposes structures proximal to the Planning Target Volume (PTV), such as the ipsilateral lung considered in this study. Although the 5-year survival increased to 85% or higher in many countries [2], exposure to ionizing radiation remains a concern. Epidemiological studies have shown correlations between exposure to ionizing radiation and the long-term carcinogenesis risk [3]. This type of risk depends on several factors [4]: attained patient's age, age of the patient at the time of exposure, total dose, dose fractionation schedule, and Organ At Risk (OAR) dose distribution. Therefore, it is important to calculate the specific risk for each patient based on their treatment plan and characteristics rather than generalizing it. [PERSON]'s mechanistic model [5,6] provides patient-, organ-, and treatment-specific risk estimation starting from biological considerations about cell killing, mutation, and repopulation. This model is based on the Life Span Study (LSS) [3], a comprehensive investigation of data from atomic bomb survivors. The model parameters are derived by aligning this data with epidemiological studies on Hodgkin's Disease (HD) [6, 7, 8, 9, 10, 11], as well as using population-specific statistical parameters. At Humanitas Istituto Clinico Catanese (H-ICC) in Misterbianco (Italy), Breast Cancer Radiotherapy (BC RT) treatments are typically carried out using two distinct fractionation schemes, based on clinical and radiobiological considerations: the Standard Fractionation (SF) approach, which delivers a total dose of 50 Gy in 25 daily fractions, and the Hypofraction (HF) approach, which deliver a total dose of 42.56 Gy in 16 daily fractions. Evidence from randomized trials and institutional series supports the effectiveness and safety of hypofractionation for the adjuvant treatment of regional lymph nodes [12]. Although hypofractionation is supported by evidence of safety and efficacy in the regional lymph node setting, it is not suitable for all patients. Those with postoperative complications, large breasts, or breast implants may face higher risks of late toxicity and cosmetic effects. This clinical variability justifies the need to investigate and compare both treatment approaches, as performed in the present study. Secondary tumors do not arise immediately following exposure to ionizing radiation; rather, a latency period exists between treatment and disease onset [13]. Several studies have retrospectively investigated the risk of RT-induced lung cancer in BC patients [14, 15, 16, 17, 18, 19]. However, past epidemiological data on cancer induction are not sufficient, because radiation therapy techniques have changed substantially in recent decades [8]. Risk estimation can be performed using models. The Linear No-Threshold (LNT) model, derived from the LSS on A-Bomb survivors, provides a dose-response relationship [3]. While the data support a linear response between 0 and 2 Gy, a flattening of the dose-response curve is observed at higher doses. Therefore, estimating carcinogenesis risk in RT, where higher doses and various fractionation schemes are used, requires a more appropriate dose-risk model. Several approaches have been proposed by the scientific community to address this issue [4]. [PERSON] et al. [20] found that cancer induction results from the interplay between the induction of carcinogenic mutations and radiation-induced cell kill. Starting from this concept, a competition model based on the Linear Quadratic (LQ) model was included in the UNSCEAR reports [21, 22]. [PERSON] et al.