## 2 Modeling of Cardiovascular System
### Windkessel Model  
The concept of the windkessel effect is shown in Figure 2a. The windkessel means air chamber in German, and is an air-spring type capacity mounted in the middle of the pipe to control the flow rate. The combination of air-spring effect and peripheral resistance in the windkessel can reproduce the properties of the elastic artery wall. The fluid dynamics model of the cardiovascular system lumped with the windkessel effect is called the windkessel model, which can be converted into an equivalent electrical circuit through fluid-electric analogy [19]. The windkessel model is classified according to the types of components, the 2-element model considering only resistance (\(R\)) and capacitance (\(C\)  
Figure 1: Human blood pressure waveform: **(a)** Wiggers diagram, redrawn based on https://commons. wikimedia.org/wiki/File:Wiggers_Diagram_2.svg; **(b)** ventricular pressure–volume loop. the 3-element model considering up to characteristic resistance of the vessel (\(R_{C}\)), and the 4-element model considering up to inertance of blood flowing through the vessel (\(L\)). The 4-element windkessel model shown in Figure 2b is known to be an effective simplified model to describe the characteristics of the cardiovascular system [18]. In this model, the inductor and characteristic resistance are connected in parallel; the capacitor and peripheral resistance are also connected in parallel, where the two groups are connected in series. Fluid-electric analogy is summarized in Table 1. Blood pressure is converted to voltage, fluid resistance to electrical resistance, blood inertia to inductance, and elastic artery wall compliance to capacitance. Each electrical element can be formulated by fluid-dynamic parameters as follows [20]. \[R=\frac{8\mu l}{\pi r^{4}}, \tag{1}\]  
\[L=\frac{9\rho l}{4\pi r^{2}} \tag{2}\]  
\[C=\frac{2\pi r^{3}l\Big{(}1-o^{2}\Big{)}}{Et}=\frac{3\pi r^{3}l}{2 Et} \tag{3}\]  
\[R_{C}=\sqrt{LC}=\frac{1}{\pi r^{2}}\sqrt{\frac{3\rho}{2}\frac{Et}{r}} \tag{4}\]  
Equations (1)-(4) show how to calculate electric parameters based on the fluid-electric analogy of the blood vessel to construct the windkessel model. Peripheral resistance \(R\) is determined by the viscosity of blood \(\mu\) and the diameter \(r\) and length \(l\) of the blood vessels. Inertance \(L\) is determined by the diameter and length of the blood vessels and the density of blood \(\rho\). Compliance \(C\) is determined \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fluid Dynamics** & **Electric Analogy** & **Cardiovascular Parameters** \\ \hline Pressure \(P\) [Pa] & Voltage \(V\) [V = J/C] & Blood pressure [mmHg] \\ Flow rate \(Q\) [m\({}^{3}\)/s] & Current \(I\) [A = C/s] & Blood flow rate [mL/s] \\ Volume \(V\) [m\({}^{3}\)] & Charge \(q\) [C] & Blood volume [mL] \\ Viscosity \(\mu\) [Pa-s] & Resistance \(R\) [\(\Omega\)] & Blood resistance \(R\) [Pa-s/m\({}^{3}\)] \\ Compliance \(\frac{I}{\pi}\) [1/Pa] & Capacitance \(C\) [F = C/V] & Vessel’s wall compliance \\ Inertance \(I\) [kg/m\({}^{4}\)] & Inductance \(L\) [H = V.s/A] & Blood inertia \\ [PERSON]’s law \(Q=\frac{\Delta P}{R}\) & [PERSON]’s law \(I=\frac{\Delta V}{K}\) & \(R=\frac{8\mu l}{\pi r^{4}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fluid mechanics, their electrical analogy and corresponding cardiovascular parameter. Figure 2: Windkessel model: (**a**) schematic diagram of the windkessel effect, redrawn based on [[https://commons.wikimedia.org/wiki/File:Windkessel_effect.svg](https://commons.wikimedia.org/wiki/File:Windkessel_effect.svg)]([https://commons.wikimedia.org/wiki/File:Windkessel_effect.svg](https://commons.wikimedia.org/wiki/File:Windkessel_effect.svg)); (**b**) 4-element windkessel model through fluid–electric analogy, \(V(t)\): input voltage, \(I(t)\): input current, \(R\): peripheral resistance, \(R_{C}\): characteristic resistance, \(L\): inertance, \(C\): compliance. by the diameter and length of the blood vessels and the thickness \(t\) and stiffness of the vascular walls \(E\).