## 2 Materials and Methods
### Regression Analysis  
The data in this study were analyzed using simple and multiple linear regression models. Regression analysis models the relationship between one output (dependent) variable and one or more input (independent) variable(s). Thus, linear regression is a technique that finds a line that best fits a dataset [18]. Simple linear regression model has one input feature as in (1):  
\[\hat{y}=a+b_{1}X_{1} \tag{1}\]  
where \(\hat{y}\) is the predicted output variable, \(X_{1}\) is the input variable, \(a\) is a scaler and \(b_{1}\) is the slope of the line. Multiple linear regression model has multiple input variables \(X_{1}\dots X_{m}\) as in (2):  
\[\hat{y}=a+b_{1}X_{1}+b_{2}X_{2}+\dots+b_{m}X_{m} \tag{2}\]  
In this study, linear regression analysis was performed between each patient-related parameter (BMI, weight and age) as input variables and each CT dose measure (CTDI\({}_{\text{vol}}\), DLP, ED and SSDE as output variables) for each of the four CT exams. In addition, multiple linear regression analysis was performed on all input variables \(X_{i}\) altogether with each output variable separately.  
In the evaluation process, several evaluation metrics are used to evaluate the resulting regression models, namely the coefficient of determination (\(R^{2}\)), mean square error (MSE), mean absolute error (MAE) and mean absolute percentage error (MAPE). These metrics give an intuition of how much the line of a model fits the data [19]. Moreover, the Pearson correlation coefficient (\(r\)) and the two-tailed \(p\)-value are calculated to assess the strength of the correlation between the input and output variables. The metrics \(R^{2}\), \(RMSE\), \(MAE\), \(MAPE\) and \(r\) are calculated as in the following Formulas (3)-(7) [19]:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}-\overline {y})^{2}}, \tag{3}\]  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**CT Exam**} & **No. of Patients** & \multirow{2}{*}{**Average Age (Range)**} & \multirow{2}{*}{**Average Weight (Range)**} & \multirow{2}{*}{**Average BMI (Range)**} \\  & **(Male, Female)** & & & \\ \hline Chest & 68 (37, 31) & 65.3 (21–96) & 66.4 (36–110) & 25.5 (15.6–41.4) \\ Cardiac Angiogram & 91 (49, 42) & 55.7 (33–85) & 83.8 (47–44) & 31.2 (19.6–51.6) \\ Cardiac Calcium Score & 94 (48, 46) & 56.1 (38–85) & 88.9 (52–144) & 31.5 (19.6–51.6) \\ Abdomen/Pelvis & 73 (31, 42) & 48.4 (20–92) & 72.9 (35–113) & 27.4 (16–39.7) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Population statistics of the dataset according to the groups of protocols.  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{n}}, \tag{4}\] \[MAE=\frac{\sum_{i=1}^{n}\left|y_{i}-\hat{y}_{i}\right|}{n},\] (5) \[MAPE=\frac{\sum_{i=1}^{n}\left|\frac{y_{i}-\hat{y}_{i}}{y_{i}} \right|}{n}, \tag{6}\]  
where \(y_{i}\) is the true value of the output variable, \(\hat{y}_{i}\) is the predicted value of the output variable, \(\overline{y}\) is the mean of \(y_{i}\) values and \(n\) is the number of data points.  
\[r=\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sqrt{\sum_{i= 1}^{n}(x_{i}-\overline{x})^{2}\cdot\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}} \tag{7}\]  
where \(x_{i}\) is the input variable sample and \(\overline{x}\) is the mean of \(x_{i}\) values.