# Numerical Simulation and Analysis of Heart-Aorta Fluid-Structure Interaction Based on S-ALE Method
## 3 Results and Discussion
### Effect of Changes in Blood Viscosity on the Aortic Hemodynamic Response  
This study used viscosity coefficient as the input variable, with peak blood velocity, flow rate, and WSS as output variables. Through numerical simulations of five independent samples, quantitative response relationships between these hemodynamic parameters were established. Figure 10 shows the effect of changes in blood viscosity on aortic hemodynamic responses. From the figure, it can be observed that there is an approximately linear relationship between the viscosity coefficient and hemodynamic parameters. The Pearson correlation coefficient \(r\) was adopted to quantify the correlation between output and input variables, with an absolute value of \(r\) closer to 1 indicating stronger linear correlation and closer to 0 indicating weaker linear correlation. The mathematical expression is as follows:  
\[r=\frac{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(y_{i}-\overline{y} \right)}{\sqrt{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}\cdot\sum_{i=1 }^{n}\left(y_{i}-\overline{y}\right)^{2}}} \tag{14}\]  
where \(n\) is the sample size, \(x_{i}\) is the viscosity coefficient value of the \(i\)-th sample, \(\overline{x}\) is the sample mean of the viscosity coefficient, \(y_{i}\) is the observed value of the \(i\)-th sample, and \(\overline{y}\) represents the mean of the observed sample values. The correlation coefficients between the outputs and inputs in sections 1-4 are summarized in Table 3. The results showed that the peak WSS exhibited the strongest correlation, with its coefficient value approaching nearly 1. In contrast, weaker correlations were observed for both peak velocity and peak flow rate. Figure 9: Comparison of hemodynamic parameter errors between numerical simulation and experiment data. (**a**) Section-averaged peak velocity; (**b**) section-averaged peak flow rate; (**c**) section-averaged peak WSS. To further analyze the sensitivity of the viscosity coefficient to hemodynamic parameters, the slope of the linear regression model was adopted as a quantitative sensitivity indicator. Linear fitting was performed for the five sample groups, with the mathematical expression given by the following:  
\[y=mx+b \tag{15}\]  
where \(m\) is the slope and \(b\) is the intercept. The optimal slope \(m\) and intercept \(b\) were determined through least squares fitting by minimizing the residual sum of squares:  
\[\left\{ \begin{array}{l}m=\frac{n\sum\limits_{i=1}^{n}x_{i}y_{i}-\sum\limits_{ i=1}^{n}x_{i}\sum\limits_{i=1}^{n}y_{i}}{n\sum\limits_{i=1}^{n}x_{i}^{2}-\left( \sum\limits_{i=1}^{n}x_{i}\right)^{2}}\\ b=\frac{\sum\limits_{i=1}^{n}y_{i}-m\sum\limits_{i=1}^{n}x_{i}}{n}\end{array} \right. \tag{16}\]  
Due to the dimensional differences among blood velocity, flow rate, and WSS, the fitted slope \(m\) was normalized through dimensionless processing, calculated as follows:  
\[\varepsilon=\frac{m}{\overline{y}} \tag{17}\]  
The sensitivity analysis revealed that under different viscosity coefficients, the fitted slopes for both peak velocity and flow rate approached 0, with the fitted curves approximating straight lines parallel to the \(x\)-axis. This indicated that changes in the viscosity coefficient had no significant effect on peak velocity and flow rate in sections 1-4. In contrast, the WSS showed a clear increasing trend. In our study, WSS was calculated by multiplying the viscosity coefficient with the velocity gradient of the section. The blood velocity tends to stabilize in sections 1-4, as shown in Figure 10a, which implies that the viscosity coefficient has no effect on the change  
Figure 10: Effect of various in blood viscosity on aortic hemodynamic response. (**a**) Section-averaged peak velocity; (**b**) section-averaged peak flow rate; (**c**) section-averaged peak WSS.