# Influence of Impeller Speed Patterns on Hemodynamic Characteristics and Hemolysis of the Blood Pump
## 3 Results
### Eulerian Hemolysis Model  
High shear force in the flow field could cause damage to blood cells. Both turbulent shear stress and viscous shear stress were present in the flow field because of the irregular turbulent flow in the blood pump. Turbulent shear stress led to larger numerical simulation results; therefore, the viscous shear stress was used to calculate the hemolysis index [25; 26]. For incompressible Newtonian fluid, the viscous shear stress was proportional to the deformation rate of the fluid and consisted of two components, the normal stress component and the shear stress component [27], as follows:  
\[\left[ \begin{array}{ccc}\tau_{xx}&\tau_{xy}&\tau_{xz}\\ \tau_{yx}&\tau_{yy}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\tau_{zz}\end{array} \right]=2\eta\left[ \begin{array}{ccc }\frac{\partial u}{\partial x}&\frac{1}{2}(\frac{\partial u}{\partial y}+\frac {\partial v}{\partial x})&\frac{1}{2}(\frac{\partial u}{\partial z}+\frac{ \partial w}{\partial x})\\ \frac{1}{2}(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})& \frac{\partial v}{\partial y}&\frac{1}{2}(\frac{\partial v}{\partial z}+\frac {\partial w}{\partial y})\\ \frac{1}{2}(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x})& \frac{1}{2}(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})& \frac{\partial w}{\partial z}\end{array} \right]. \tag{7}\]  
In Equation (7), \(\eta\) is the fluid viscosity; \(u\), \(v\), and \(w\) are the velocity components of velocity vectors \(u\) in \(x\), \(y\), and \(z\) directions, respectively; and \(\tau_{xx}\), \(\tau_{yy}\), \(\tau_{zz}\), \(\tau_{xy}\), \(\tau_{xz}\), and \(\tau_{yz}\) are the six tensor components of viscous stress, respectively. Therefore, according to the calculation method of shear stress proposed by [PERSON] [28], the magnitude of shear stress was expressed:  
\[\tau=\sqrt{\frac{1}{6}\Big{[}\Big{(}\tau_{xx}-\tau_{yy}\Big{)}^{2}+\left(\tau_ {xx}-\tau_{zz}\right)^{2}+\Big{(}\tau_{yy}-\tau_{zz}\Big{)}^{2}\Big{]}+\Big{(} \tau_{xy}^{2}+\tau_{yz}^{2}+\tau_{xz}^{2}\Big{)}}. \tag{8}\]  
The hemolysis formed in the flow field was the result of shear stress and exposure time. The amount of blood damage was estimated by the power-law relationship using the shear force and exposure time. Therefore, the function model of shear force and exposure time was used to predict the blood damage. [PERSON] [29] proposed the following hemolysis prediction model based on the experimental results as follows:  
\[HI=C\tau^{\alpha}t_{exp}^{\beta}, \tag{9}\]  
where \(HI\) represents the hemolysis index, which is the damage degree of blood flow through the blood pump; \(\tau\) is the shear force acting on the blood cells; \(t_{\text{exp}}\) is the exposure time; and \(C\), \(\alpha\), and \(\beta\) are constants. In this study, the constants proposed by [PERSON] were chosen, which were \(3.62\times 10^{-7}\), \(0.785\), and \(2.416\), respectively. The Lagrangian method involved a large number of calculations for the flow field of the transient blood pump and could not directly obtain the region with blood damage in the flow field. However, the Euler method could provide the hemolysis value in the whole flow field of the blood pump. It was convenient and could also analyze the flow field intuitively. Moreover, it was also \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Patterns** & \begin{tabular}{c} **CO** \\ **(L/min)** \\ \end{tabular} & \begin{tabular}{c} **MAP** \\ **(mm Hg)** \\ \end{tabular} & \begin{tabular}{c} **PP** \\ **(mm Hg)** \\ \end{tabular} & \
\begin{tabular}{c} **EEP** \\ **(mm Hg)** \\ \end{tabular} &
\begin{tabular}{c} **SHE** \\ **(erg/cm3)** \\ \end{tabular} \\ \hline Constant & 5.2 & 97.2 & 5.0 & 98.2 & 1300.4 \\ Sinusoidal & 5.1 & 94.2 & 11.6 & 97.2 & 4054.9 \\ Pulsatile & 4.9 & 90.8 & 12.7 & 93.8 & 4036.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cardiac output indicators under different speed patterns of the blood pump. convenient to compare the influence of different working modes of the blood pump on the flow field and hemolytic characteristics.