## 2 Materials and Methods
### Risk Estimation
\(\mu\) is the modifying function; it introduces carcinogenesis risk age dependence:  
\[\mu(age_{e},age_{a})=exp(\gamma_{e}(age_{e}-30)+\gamma_{a}ln\big{(}\frac{age_{a }}{70}\big{)}), \tag{5}\]  
where \(\gamma_{e}\) and \(\gamma_{a}\) are two organ-specific parameters derived from the LSS study [3], \(age_{e}\) is the age of the patient during exposure, and \(age_{a}\) is the attained patient's age. Based on published data, the parameters of the mechanistic model employed for lung cancer risk assessments were \(\alpha\) (Gy\({}^{-1}\)) = 0.061, \(\alpha\)/\(\beta\) = 3, R = 0.84, \(\beta_{0}\) (10\({}^{4}\) PY Gy)\({}^{-1}\) = 8.0 (5.5; 11), \(\gamma_{e}\) (Y\({}^{-1}\)) = 0.002, \(\gamma_{a}\) (Y\({}^{-1}\)) = 4.23 [6; 7; 8; 9; 10; 11]. [PERSON]'s model concludes with the calculation of the Excess Absolute Risk (\(EAR\)); however, it can be extended further to derive two additional risk metrics: the Lifetime Attributable Risk (\(LAR\)) and the Relative Risk (\(RR\)), as conducted by [PERSON] et al. [35]. These metrics account not only for the radiation-induced risk but also incorporate the patient's survival probability and the baseline (intrinsic) risk of developing the disease independently of RT. By weighting the \(EAR\) with the patient's survival probability between the exposure age (\(age_{e}\)) and attained age (\(age_{a}\)), the \(LAR\) for developing secondary malignancies can be determined as follows:  
\[LAR=\sum_{age_{e}+L}^{75}EAR(OED,age_{e},age_{a})\frac{S(age_{a})}{S(age_{e})}, \tag{6}\]where _S(age\({}_{a}\))JS(age\({}_{e}\))_ represents the regional survival probability, as provided by ISTAT (Istituto Nazionale di Statistica), for a healthy individual living in Sicily (Italy) to survive from _age\({}_{e}\)_ to _age\({}_{a}\)_[36]. L is a free cancer risk interval or latency period (5 years for BC) [37]. The probability of survival to age L, defined as _S(age\({}_{e}\) + 5)/S(age\({}_{e}\))_, was calculated using internal H-ICC statistics from a cohort of 915 patients. The resulting survival probability is 98.60% for individuals younger than 65 years at exposure, and 96.34% for all other cases. In this study, the expected age was fixed at 75 years, based on the following considerations: (i) the Schneider model is derived from A-bomb survivor data, which assume an average age at exposure of 30 years and an expected age of 70 years; (ii) statistically robust data on the Lifetime Intrinsic Risk (LIR) of developing malignancies, provided by ISTAT, are available up to age 75. Beyond this age, a marked decline in the year-to-year survival probability (from _age\({}_{i}\)_ to _age\({}_{i+1}\)_) is observed (Figure 1), which compromises the reliability of LIR estimates. Independent of the RT treatment, for each person, there is a certain organ-, gender-, and age-dependent LIR for developing a malignancy in the radiosensitive structure of interest. In this study, LIR values were obtained using lung cancer incidence data for Sicilian women resident in Catania provided by the Integrated Cancer Registry CT-ME-EN (AOU Policlinico \"G. Rodolico--San Marco\", Catania, Italy). LIR values are equal to 5.37%, 5.34%, 5.26%, 5.10%, 4.70%, 3.99%, 3.18%, 2.16%, and 1.17% or exposure ages ranging from 30 to 70 years in 5-year increments. LAR values, obtained using the [PERSON] model, can be compared with the LIR through the patient-specific Relative Risk (_RR_) calculation (Equation (7)), which quantifies the increase in baseline cancer probability for an irradiated individual compared to an unexposed one. This metric enables an evaluation of which fractionation scheme (SF or HF) is associated with a higher risk of radiation-induced cancer. \[\textbf{RR}=\frac{\textbf{LAR}+\textbf{LIR}}{\textbf{LIR}}, \tag{7}\]