# A Liquid-Solid Coupling Hemodynamic Model with Microcirculation Load
## 2 Model and Methods
### Calculation Examples and Parameters  
In order to verify the reliability and superiority of the hemodynamic model proposed, five calculation examples (see Table 1) were taken through changing the tube length, load conditions, and cardiac cycle. All the calculation examples were based on the physiological parameters of the descending aorta, and the transient flow was adopted for simulating actual the working state of a blood vessel. The fluid in the tube was simplified as the Newtonian fluid of blood parameters, with a density of 1050 kg/m\({}^{3}\) and a dynamic viscosity coefficient of 0.0035 Pa s [14]. The density of the flexible tube wall was 1120 kg/m\({}^{3}\), Young's modulus was 0.5 MPa [15], and Poisson's ratio is 0.49 [16]. The permeability of the porous media was \(9.5\times 10^{-9}\)m\({}^{2}\), and the porosity was 100%.  
All the calculation examples selected velocity inlet condition and free outlet, and the inlet condition and outlet condition are shown in Figure 4. The inlet condition was velocity inlet, which was used for simulating the process of cardiac impulse. Systole was 0-0.2 s, and the maximum systolic velocity was 0.8 m/s [17]. Diastole ranged from 0.2 s to the end of a cardiac cycle. Inlet velocity at this period was kept at 0 m/s to simulate the state of a closed heart valve without reflux. The average Reynolds number was about 661, which was based on the average velocity of the inlet. The flow regime was laminar flow. The outlet condition was pressure-free, _i.e._, the pressure was always kept at 0 Pa. This not only eliminated the influence of disturbance from outer pressure factors, but also achieved similarity with the physiological reality in the vein.  
For the two calculation examples where the heart rate was changed, this paper assumed the outflow after every heart pulse to be consistent. Thus one only has to change the length of the diastole to achieve a change in the cardiac cycle. As shown in Figure 4, for different cardiac cycle calculation examples, the systole was kept unchanged while the diastole changed. Accordingly, in the example of normal physiology, heart rate was 75/min and cardiac cycle was 0.8 s (0.2 s for systole, and 0.6 s for diastole); in the example of accelerated heart rate, heart rate was 100 min\({}^{-1}\) and cardiac cycle was 0.6 s  
\begin{table}
\begin{tabular}{c c c c} \hline
**Calculation Examples** & **Permeability of** & **Length of** & **Cardiac Cycle \(T\)** \\  & **Porous Media k** & **Flow Field** & **Cardiac Cycle \(T\)** \\ \hline Example 1. Normal physiology & \(9.5\times 10^{-9}\) & 1000 mm & 0.8 \\ Example 2. No seepage load & N/A & 1000 mm & 0.8 \\ Example 3. Short tube & \(9.5\times 10^{-9}\) & 200 mm & 0.8 \\ Example 4. Accelerated heart rate & \(9.5\times 10^{-9}\) & 1000 mm & 0.6 \\ Example 5. Decreased heart rate & \(9.5\times 10^{-9}\) & 1000 mm & 1 \\ \hline \end{tabular}
\end{table}
Table 1: Five calculation examples and their parameters.  
Figure 3: Grid division for the tube cross-section.  
(0.2 s for systole, and 0.4 s for diastole); in the example of decreased heart rate, heart rate was 60 min\({}^{-1}\) and cardiac cycle was 1 s (0.2 s for systole, and 0.8 s for diastole).