## 2 Materials and Methods
### Relation between TVRR and a Patient's Clinical Variables  
The relation between the target volume reduction rate and a patient's clinical variables such as age, total dose (Gy), gender, number of treatment fractions, dose per fraction, and patient weight was analyzed. Tables 1 and 2 record the different variables in the patient dataset. Thus, the range of the variables may differ substantially between patients. Hence, using the original scale will by default apply more weight on the variables with an extensive range. Therefore, feature rescaling to independent variables or features of data, in data pre-processing were applied. The purpose of employing feature rescaling is to ensure a patient's different clinical variables are on almost the same scale, in order that each variable is equally weighted in importance. The standardization outcome is that the variables will be rescaled to ensure the mean and the standard deviation are 0 and 1, respectively.  
\[\mathrm{D_{s}}=(\mathrm{D}-\mathrm{D_{m}})/\mathrm{D_{std}} \tag{2}\]  
where \(\mathrm{D}\) is the original data, \(\mathrm{D_{s}}\) the scaled data, and \(\mathrm{D_{m}}\) and \(\mathrm{D_{std}}\) the mean and standard deviation of each variable, respectively.  
The significance of each patient's clinical variable was examined independently using a statistical F-test. These were based on the hypothesis that the variable values put together with the tumor volume reduction rate (TVRR) values are drawn from populations with the same mean, against the alternative hypothesis that the population means are not all the same. A small \(p\)-value of the test statistic implies that the corresponding variable is essential. The output score is -log(\(p\)). Therefore, a considerable score value indicates that the corresponding variable is important. The mixture technique used in this study relies on defining the Nano-particles (NPs) as a mixture with its parent volume (the volume the NPs are embedded within). This technique was reported to overestimate the dose in some  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Patient** & **Total Dose Gy** & **Age (Years)** & **Gender** & **No. of Treatment** & **Dose per** & **Weight** & **Volume** \\  & & & & **Fractions** & **Fraction (Gy)** & **(kg)** & **Change \%** \\ \hline Patient 1 & 50.4 & 71 & Female & 28 & 1.8 & 45 & 27.93 \\ \hline Patient 2 & 50 & 60 & Male & 25 & 2 & 84 & 15 \\ \hline Patient 3 & 50.4 & 58 & Female & 28 & 1.8 & 87 & 21.02 \\ \hline Patient 4 & 50.4 & 45 & Female & 28 & 1.8 & 69 & 24.38 \\ \hline Patient 5 & 50.4 & 53 & Female & 28 & 1.8 & 50 & 3.03 \\ \hline Patient 6 & 50.4 & 50 & Male & 28 & 1.8 & 77 & \(-\)0.43 \\ \hline Patient 7 & 50 & 67 & Female & 25 & 2 & 70 & 14.65 \\ \hline Patient 8 & 50.4 & 50 & Male & 28 & 1.8 & - & 1.37 \\ \hline Patient 9 & 50 & 74 & Male & 25 & 2 & - & 2.23 \\ \hline Patient 10 & 50.4 & 75 & Male & 28 & 1.8 & 60 & 39.75 \\ \hline Patient 11 & 50 & 56 & Male & 25 & 2 & 72 & \(-\)19.5 \\ \hline Patient 12 & 50.4 & 56 & Male & 28 & 1.8 & - & \(-\)1.51 \\ \hline Patient 13 & 59.4 & 59 & Male & 33 & 1.8 & 50 & 19.07 \\ \hline Patient 14 & 45 & 39 & Female & 25 & 1.8 & - & \(-\)42.81 \\ \hline Patient 15 & 45 & 71 & Female & 25 & 1.8 & 42 & 28.5 \\ \hline Patient 16 & 50.4 & 37 & Male & 25 & 1.8 & 60 & 15.94 \\ \hline Patient 17 & 50.4 & 59 & Male & 28 & 1.8 & 76 & 20.99 \\ \hline Patient 18 & 50.4 & 68 & Male & 28 & 1.8 & 83 & 56.9 \\ \hline Patient 19 & 45 & 38 & Female & 25 & 1.8 & 56 & \(-\)7.81 \\ \hline Patient 20 & 50 & 71 & Male & 25 & 2 & 88 & 0.15 \\ \hline \hline \hline \end{tabular}
\end{table}
Table 2: Tumor volume change for all patients.  
cases [21], but is faster to compute, easier to implement, and is useful when the objective is to compare different types of NPs and provide a general understanding of their effect on dosimetric quantities. The major drawback of this technique is that it does not allow studying the size or shape of the NPs, which is out of this study's scope.