# Influence of Impeller Speed Patterns on Hemodynamic Characteristics and Hemolysis of the Blood Pump
## 2 Materials and Methods
### Cardiovascular System Simulation Model  
This study referred to the LPM calculation method of the cardiovascular circulatory system, proposed by [PERSON] and [PERSON] [22; 23], to simulate changes in physiological parameters in patients with heart failure assisted with a blood pump. Figure 6 shows that the model included left atrium, left ventricle, mitral valve, aortic valve, arterial system, and blood pump. The network model could simulate the opening and closing of the mitral and aortic valves of the human heart and analyze the effect of blood pump assistance on the physiological parameters of patients with heart failure. Figure 5: Comparison between the experimental and simulation results of the H-Q curve. Figure 6: Cardiovascular system model coupled with the hydraulic performance model of a blood pump. The hydraulic performance model reflecting the pulsation characteristics of the blood pump [20] was as follows:  
\[H=\beta\omega^{2}+R_{p}Q+R_{w}Q^{2}+L_{p}dQ/dt \tag{3}\]  
where \(H\) is the differential pressure between the outlet and the inlet of the blood pump, in mm Hg; \(\omega\) is the impeller speed of the blood pump, in rpm; and \(Q\) is the blood pump flow rate, in mL/s. Therefore, \(dQ/dt\) reflected the time-dependent characteristics of the blood flow rate through the blood pump. The coefficients \(\beta\), \(R_{p}\), \(R_{w}\), and \(L_{p}\) could be calculated from the experimental result, as shown in Table 2.  
According to [PERSON]'s law, the cardiovascular system model coupled with the hydraulic model of a blood pump was established, and the differential equation with \(x\) as the state variable was calculated, as shown in Equation (4). Detailed definitions of time-varying capacitance reflecting left ventricular characteristics \(C(t)\), mitral valve state function \(r(x)\), and function simulating left ventricular aspiration \(R_{su}\) can be referred to from previous studies [22; 23]. \[\left\{ \begin{array}{l}\frac{dx_{1}}{dt}=\frac{1}{C(t)}\bigg{[}\frac{1}{k_{M }}r(x_{2}-x_{1})-(\frac{dC(t)}{dt}+R)x_{1}-\frac{1}{k_{A}}(x_{1}-x_{4})-x_{6} \bigg{]}\\ \frac{dx_{2}}{dt}=\frac{1}{k_{S}C_{S}}(x_{3}-x_{2})-\frac{1}{C_{S}R_{M}}r(x_{ 2}-x_{1})\\ \frac{dx_{3}}{dt}=\frac{1}{k_{S}C_{S}}(x_{2}-x_{3})-\frac{1}{C_{S}}x_{5}\\ \frac{dx_{4}}{dt}=\frac{1}{C_{A}}\bigg{[}x_{6}-x_{5}+\frac{1}{k_{A}}r(x_{1}-x_ {4})\bigg{]}\\ \frac{dx_{3}}{dt}=\frac{1}{C_{S}}(x_{4}-x_{3})-\frac{C_{S}}{C_{S}}x_{5}\\ \frac{dx_{4}}{dt}=\frac{1}{C}(x_{1}-x_{4}-R^{*}x_{6}+\beta\omega^{2}+R_{w}x_{ 6}^{2})\end{array} \right. \tag{4}\]  
When the heart rate was 75 bpm, the cardiac cycle was 0.8 s. The myocardial systolic force and the slope of the left ventricular end-diastolic pressure-volume relationship curve were modified to simulate the symptoms of heart failure. In Equation (4), \(R^{*}=R_{su}+R_{i}+R_{o}+R_{p}\), \(L^{*}=L_{i}+L_{o}+L_{p}\), and the detailed definition of state equation parameters are shown in Table 3, \(x=[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}]^{T}=\big{[}LVP,LAP,AP,AOP,Q_{\text{ total}},Q_{pump}\big{]}^{T}\), and the initial value of the state variable was taken as \(x_{0}=[6.9\,\text{mm Hg},\,9.6\,\text{mm Hg},\,67\,\text{mm Hg},\,80\,\text{mm Hg},\,0\,\text{mL/s},\,0\,\text{mL/s}]^{T}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\beta\) & \(R_{p}\) & \(R_{w}\) & \(L_{p}\) \\ (mm Hg/\(\text{rpm}\)) & (mm Hg/s/mL) & (mm Hg/(s\({}^{2}\cdot\)mL)) & (mm Hg/s\({}^{2}\)/mL) \\ \hline
1.5674e\({}^{-5}\) & \(-0.1169\) & \(-0.0023\) & \(-0.0034\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hydraulic performance model coefficient of the blood pump. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\beta\) & \(R_{p}\) & \(R_{w}\) & \(L_{p}\) \\ (mm Hg/\(\text{rpm}\)) & (mm Hg/s/mL) & (mm Hg/(s\({}^{2}\cdot\)mL)) & (mm Hg/s\({}^{2}\)/mL) \\ \hline
1.5674e\({}^{-5}\) & \(-0.1169\) & \(-0.0023\) & \(-0.0034\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hydraulic performance model coefficient of the blood pump.