## 2 Materials and Methods
### Risk of Second Primary Cancers  
The lungs, left and right breasts, thyroid, and esophagus were located close to the treatment volume, and, therefore, parts of these organs were exposed to high radiation doses of more than 2 Gy. The dose distribution within these organs was inhomogeneous. The linearity between the radiation dose and the risk of carcinogenesis exists for doses up to 2.0-2.5 Gy [19; 20; 21]. Non-linear models need to be used for estimating the second cancer risk from therapeutic exposures with high doses [19]. Several previously published non-linear models employed for radiotherapy-induced cancer risk assessments may ignore either the effect of the tumor dose fractionation or the impact of tissue repopulation between the consecutive fractions [19]. The lifetime attributable risk (LAR) for developing second primary cancers in the OARs due to 3D-CRT, IMRT, and VMAT for thymoma was estimated with the aid of a mechanistic non-linear model introduced by [PERSON] et al. [22]. The cell killing, dose fractionation, and the interfraction repair of the highly irradiated healthy organs are considered by the aforementioned model [22]. Differential dose-volume histograms (DVHs) with a bin width of 0.01 Gy were extracted from the treatment plans. These histograms were used to determine the organ equivalent dose (OED) with the following equation reported by [PERSON] et al. [22]:  
\[\text{OED}=\frac{1}{\text{V}_{\text{OAR}}}\,\sum_{\text{i}}\text{V}_{\text{D}_ {\text{i}}}\,\frac{\exp\left(-\text{a}_{\text{i}}^{{}^{\prime}}\,\text{D}_{ \text{i}}\right)}{\text{a}_{\text{i}}^{{}^{\prime}}\,\text{R}}\left[1-2\text{ R}+\text{R}^{2}\exp\left(\text{a}_{\text{i}}^{{}^{\prime}}\,\text{D}_{ \text{i}}\right)-\left(1-\text{R}\right)^{2}\,\exp\left(-\frac{\text{a}_{ \text{i}}^{{}^{\prime}}\,\text{R}}{1-\text{R}}\,\text{D}_{\text{i}}\right)\right] \tag{1}\]  
The V\({}_{\text{OAR}}\) and V\({}_{\text{D}_{\text{i}}}\) in the above equation refer to the total volume of each OAR and the OAR volume receiving a radiation equal to D\({}_{\text{i}}\), respectively. The parameter R denotes the organ-dependent repopulation ability, whereas the a\({}_{\text{i}}^{{}^{\prime}}\) is associated with the cell killing. The number of fractions (n) and the parameters a and \(\beta\) obtained by the linear-quadratic model were employed for calculating the a\({}_{\text{i}}^{{}^{\prime}}\) as follows [22]:  
\[\text{a}_{\text{i}}^{{}^{\prime}}=\text{a}+\beta\,\text{D}_{\text{i}}\,\text{ n}^{-1} \tag{2}\]  
It should be noted that the R value for the thyroid gland is missing from the publication of [PERSON] et al. [22]. The probability for developing secondary thyroid malignancies was assessed with the aid of the bell-shaped approach, assuming the lack of cell repopulation. The OED of the thyroid was calculated using the equation [22]:  
\[\text{OED}=\frac{1}{\text{V}_{\text{OAR}}}\,\sum_{\text{i}}\text{V}_{\text{D}_ {\text{i}}}\,\text{D}_{\text{i}}\,\text{exp}\big{(}-\text{a}_{\text{i}}^{{}^{ \prime}}\,\text{D}_{\text{i}}\big{)} \tag{3}\]  
The OED calculations were used to estimate the organ-specific LAR for cancer induction with the following equation:  
\[\text{LAR}=\beta_{\text{o}}\,\text{OED}\,\sum_{\text{age}_{\text{e,min}}}^{ 75}\exp\Big{[}\gamma_{\text{e}}\big{(}\text{age}_{\text{e}}-30\big{)}+\gamma_{ \text{a}}\,\ln\Big{(}\frac{\text{age}_{\text{a}}}{70}\Big{)}\Big{]}\frac{\text{ S}\big{(}\text{age}_{\text{e}}\big{)}}{\text{S}\big{(}\text{age}_{\text{e}} \big{)}} \tag{4}\]  
The \(\beta_{\text{o}}\) corresponds to the slope of the dose-response curve at low radiation doses, and \(\gamma_{\text{e}}\) and \(\gamma_{\text{a}}\) are correction factors related to the age of the exposed subject. The age\({}_{\text{e}}\) and age\({}_{\text{a}}\) is the patient's age at the time of radiotherapy and the attained age taken equal to 75 years, respectively. The S(age\({}_{\text{e}}\))/S(age\({}_{\text{e}}\)) is the probability for a healthy female to survive from age\({}_{\text{e}}\) to age\({}_{\text{a}}\).