## 2 Clinical Dataset, Experimental Setup, and Stochastic Sensitivity Analysis Procedure
### Stochastic Sensitivity Analysis Procedure  
The stochastic sensitivity analysis to inlet boundary conditions is carried out using the gPC method in its nonintrusive form. In the gPC approach, the dependence between a stochastic variable of interest, \(Y\), and the vector of independent uncertain parameters \(\mathbf{\eta}\) is expressed by means of a polynomial expansion [57]. So, by employing term-base indexing:  
\[Y(\psi)=\sum_{i=0}^{\infty}a_{i}\Theta_{i}(\mathbf{\eta}(\psi)) \tag{1}\]  
where \(\psi\) is an aleatory event, \(\Theta_{i}(\mathbf{\eta})\) is the \(i\)-th gPC polynomial, and \(a_{i}\) is the coefficient of the aforementioned variable within the expansion. To practically obtain the response surface, it is necessary to truncate the previous sum from Equation (1). This finite limit (\(N\)) is here determined as follows:  
\[N=\prod_{j=1}^{K}(R_{j}+1)-1 \tag{2}\]  
where \(K\) is equal to the number of uncertainty variables of the problem, whereas \(R_{j}\) is the highest polynomial order selected for the variable of index \(j\). To calculate the coefficients \(a_{i}\), the following mathematical expression is used:  
\[a_{i}=\frac{\langle Y,\Theta_{i}\rangle}{\langle\Theta_{i},\Theta_{i}\rangle}= \frac{1}{\langle\Theta_{i},\Theta_{i}\rangle}\int_{\text{supp}\,\mathbf{\eta}}Y \Theta_{i}w(\mathbf{\eta})\text{d}\mathbf{\eta} \tag{3}\]  
where \(w(\mathbf{\eta})\) is the weight function of \(\Theta_{i}\). The Gaussian quadrature rule is herein used to calculate the coefficients \(a_{i}\). The polynomial family \(\Theta_{i}\) is a priori specified. Its optimal choice has a weight function in Equation (3) similar to the PDF assumed for the uncertain parameters. Partial sensitivities are computed through the Sobol indices [58], which measure the direct influence of individual input variables or couples of them. The Sobol index \(I_{i}\) is defined as the ratio between the variance only due to the \(i\)-th uncertain input parameter or to combinations of them, \(\sigma_{i}^{2}\), and the total variance \(\sigma^{2}\), as follows:  
\[I_{i}=\frac{\sigma_{i}^{2}}{\sigma^{2}} \tag{4}\]  
In the present work, the uncertain parameters of the experimental setup are the following: the cardiac-cycle period, \(T\), the stroke volume, \(SV\), and the length of the connecting pipe between the pump and the phantom, \(L\). As conducted in [31], the PDFs for the parameters \(T\) and \(SV\) are derived fitting clinical data from 23 patients with beta distributions:  
\[g(y;\alpha,\beta)=A(1-y)^{\alpha}(1+y)^{\beta} \tag{5}\]  
where \(A\) is a constant introduced to normalize the integration of \(g(y)\) over the interval \((-1,1)\). Exponents \(\alpha\) and \(\beta\) are the coefficients to be calibrated to match the input data. Considering as ranges of variation of the parameters \(T\in[0.6,1.4]\) s and \(SV\in[20,170]\) mL, we obtained \(\alpha=5.5\) and \(\beta=1.3\) for \(T\) and \(\alpha=4.3\) and \(\beta=2.6\) for \(SV\). As for the third uncertain parameter, its range of variation is \(L/D\in[0,70]\), the pipe diameter being \(D=1\) cm. The lower limit is such that it provides the plug flow condition, and the upper limit is the length necessary for carrying out MRI measurements on the phantom if needed (the MCL is MRI-compatible, except for the pump that should be placed outside the MRI-dedicated room). We assume a uniform PDF distribution for \(L/D\), obtained by using \(\alpha=1\) and \(\beta=1\) in Equation (5). We choose the Jacobi polynomial family for the gPC expansion, which is the optimal one when dealing with beta PDF distributions. For all the uncertainty parameters, the expansion in Equation (1) is truncated to the third order. Thus, 4 quadrature points are needed for \(T\), \(SV\), and \(L/D\), which are summarized in Table 1. Figure 4 shows the 16 inletflow-rate waveforms for all the tensor-product combinations of \(T\) and \(SV\). The flow-rate waveforms are obtained by properly scaling the one in Figure 1b.