## 2 Materials and Methods
### Statistical Properties of the Model Parameters  
Consider that the parameter vector \(\mathbf{\theta}\) given by (3) is a random variable following a multivariate normal distribution, which describes the statistical properties related to the inter-subject parametric variability in the population of patients with cancer.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  
The mean value is given by vector \(\overline{\theta}\) as  
\[\overline{\theta}=E\{\theta\}. \tag{5}\]  
The second statistical characteristic is the covariance matrix \(Q\in\mathbb{R}^{14\times 14}\) defined as  
\[Q=\text{cov}(\theta,\theta)=E\Big{\{}\big{(}\theta-\overline{\theta}\big{)} \big{(}\theta-\overline{\theta}\big{)}^{\text{T}}\Big{\}}. \tag{6}\]  
The diagonal elements of \(Q\) represent the variances of the individual parameters, hence defining the measure of parametric uncertainties. The off-diagonal elements represent the covariances between pairs of parameters, which quantify the degree to which these parameters change together.  
To generalize the theoretical analysis, we will introduce a new dimensionless random vector \(\xi\in\mathbb{R}^{m}\) following the standard normal distribution, which can be noted as  
\[E\{\xi\}=\mathbf{0},\ \ \text{cov}(\xi,\xi)=E\Big{\{}\xi\overline{\xi}^{\text{T}} \Big{\}}=I. \tag{7}\]  
Then, the parameter vector \(\theta\) will be related to random vector \(\xi\) via matrix \(R\in\mathbb{R}^{14\times m}\) and the nominal vector \(\overline{\theta}\) as  
\[\theta=\text{diag}\big{(}\overline{\theta}\big{)}R\xi+\overline{\theta}. \tag{8}\]  
Dimension \(m\in\mathbb{N}\) represents the number of random factors affecting the parameter vector \(\theta\). According to (8), the covariance matrix \(Q\) defined by (6) obtains  
\[Q=E\Big{\{}\big{(}\theta-\overline{\theta}\big{)}\big{(}\theta-\overline{ \theta}\big{)}^{\text{T}}\Big{\}}=\text{diag}\big{(}\overline{\theta}\big{)} RR^{T}\text{diag}\big{(}\overline{\theta}\big{)}. \tag{9}\]  
Considering the confidence level \(\alpha\), the random vector \(\xi\) will lie in the symmetrical confidence interval  
\[-\mathbf{1}\mathbf{r}_{1-\
icefrac{{-}}{{2}}}\leq\xi\leq\mathbf{1}\mathbf{r}_{1-\
icefrac{{ -}}{{2}}}, \tag{10}\]  
with \(1-\alpha\) probability, where \(r_{1-\
icefrac{{-}}{{2}}}\) is the \(1-\
icefrac{{-}}{{2}}\) quantile of the standard normal distribution and \(\mathbf{1}=\begin{pmatrix}1&1&\cdots&1\end{pmatrix}^{\text{T}}\) is the vector of ones.  
Based on (8), we can claim that the parameter vector \(\theta\) lies in the confidence interval  
\[-\text{diag}\big{(}\overline{\theta}\big{)}R\mathbf{1}\mathbf{r}_{1-\
icefrac{{-}}{ {2}}}+\overline{\theta}\leq\theta\leq\text{diag}(\overline{\theta})R\mathbf{1} \mathbf{r}_{1-\
icefrac{{-}}{{2}}}+\overline{\theta}. \tag{11}\]  
Then, the absolute interval parametric uncertainty obtains  
\[\Delta\theta=\text{diag}\big{(}\overline{\theta}\big{)}R\mathbf{1}\mathbf{r}_{1- \
icefrac{{-}}{{2}}}. \tag{12}\]  
Equation (12) will be used to determine the entries of matrix \(R\) based on the assumptions about the relative interval parametric uncertainties given in percent by \(\frac{\Delta\theta}{\overline{\theta}_{i}}\times 100\) as  
\[R_{ij}=\frac{\Delta\theta_{i}}{\overline{\theta}_{i}}\frac{1}{r_{1-\
icefrac{{ -}}{{2}}}}, \tag{13}\]  
where \(i=1,2,\ldots,14\) is the index of parameter and \(j=1,2,\ldots,m\), which increments if \(\frac{\Delta\theta_{i}}{\overline{\theta}_{i}}\
eq 0\).