# A Novel Radial Artery P-S Curve Model Based on Radial Vibration of Vascular Wall
## 2 Models and Methods
### Model of Vascular Wall Movement  
Vascular wall movement and blood flow are closely coupled, and the pressure of the doctor's fingertip touching the radial artery in the clinic is the result of the direct effect of the vascular wall movement. To obtain the equation of motion of the arterial tube, a very small unit with the arc angle of the arterial tube as \(d\theta\) is selected and the force analysis is performed to establish the following model, as shown in Figure 2.  
Figure 1: Vascular model. In this model, \(x\) represents the axial direction, \(r\) represents the radial direction, \(T_{\rm C}\) and \(T_{L}\) represent the peripheral and axial tension of the vascular wall, \(F_{r}\) and \(F_{x}\) represent the radial and axial components of the blood viscous friction force on the vascular wall, and \(\xi\) and \(\zeta\) represent the radial and circumferential displacement of the vascular wall under the action of external force, respectively. Assuming that blood is a Newtonian fluid, according to the constitutive equation of Newtonian fluid [24], the viscous stress that acts on the inner surface of the blood vessel wall is shown in Formula (6). \[\left\{ \begin{array}{l}\tau_{rr}=2\eta\left(\frac{\partial\
u}{\partial r} \right)_{r=R}\\ \tau_{rx}=\eta\left(\frac{\partial\mu}{\partial r}+\frac{\partial\
u}{ \partial x}\right)_{r=R}\end{array} \right. \tag{6}\]  
Therefore, the radial and axial components of the viscous friction force are shown in Equation (7). \[\left\{ \begin{array}{l}F_{r}=2\eta\left(\frac{\partial\
u}{\partial r} \right)_{r=R}\cdot Rd\theta\cdot dx\\ F_{x}=\eta\left(\frac{\partial\mu}{\partial r}+\frac{\partial\
u}{\partial x} \right)_{r=R}\cdot Rd\theta\cdot dx\end{array} \right. \tag{7}\]  
In addition to the effect of blood friction, the blood vessel wall will also be subject to elastic tension and assuming that the Young's modulus of the blood vessel wall is \(E\), the Poisson ratio of the blood vessel is \(\sigma\), and the thickness of the blood vessel is \(h\), the axial strain and circumferential strain of the blood vessel wall are shown in Formula (8). \[\left\{ \begin{array}{l}\varepsilon_{x}=\frac{\partial\zeta}{\partial x}= \frac{T_{L}}{T_{L}}-\sigma\frac{T_{C}}{Eth}=\frac{1}{Eth}(T_{L}-\sigma T_{C}) \\ \varepsilon_{c}=\frac{2\pi(R+\xi)-2\pi R}{2\pi R}=\frac{\xi}{R}=\frac{1}{Eth}( T_{C}-\sigma T_{L})\end{array} \right. \tag{8}\]  
The circumferential and axial tension can be derived from Equation (9). \[\left\{ \begin{array}{l}T_{\rm C}=\frac{Eth}{1-\sigma^{2}}\left(\sigma\frac {\partial\zeta}{\partial x}+\frac{\zeta}{R}\right)\\ T_{L}=\frac{Eth}{1-\sigma^{2}}\left(\frac{\partial\zeta}{\partial x}+\sigma \frac{\zeta}{R}\right)\end{array} \right. \tag{9}\]  
Assuming that the vascular wall density is \(\rho\), the equation of radial motion of the vascular wall can be obtained according to the momentum theorem, as shown in Equation (10). \[\left(\rho hRd\theta\cdot dx\right)\cdot\frac{\partial^{2}\zeta}{\partial t^{ 2}}=(P-P_{e})Rd\theta\cdot dx-T_{C}\cdot\sin\frac{d\theta}{2}\cdot dx-(T_{C}+ \frac{\partial T_{C}}{\partial\theta}d\theta)\cdot\sin\frac{d\theta}{2}\cdot dx -F_{r} \tag{10}\]  
where \(P\) represents the internal pressure of the blood vessel wall, \(P_{e}\) represents the external pressure of the blood vessel wall and by substituting Formulas (7) and (9) into Formula (10),  
Figure 2: Micro element force model of blood vessel wall. and eliminating the area micro element \(Rd\theta\cdot dx\), the radial motion equation of the blood vessel wall can be obtained, as shown in Formula (11). \[\rho\frac{\partial^{2}\xi}{\partial t^{2}}=\frac{P-P_{e}}{h}-\frac{E}{1-\sigma^{ 2}}\bigg{(}\frac{\sigma}{R}\frac{\partial\zeta}{\partial x}+\frac{\xi}{R^{2}} \bigg{)}-\frac{2\eta}{h}\bigg{(}\frac{\partial v}{\partial r}\bigg{)}_{r=R} \tag{11}\]  
To simplify Equation (11) of the motion of the vascular wall, its various magnitudes are estimated.