# A Novel Radial Artery P-S Curve Model Based on Radial Vibration of Vascular Wall
## 2 Models and Methods
### Model of Elastic Deformation of the Vascular Wall  
When the doctor examines the pulse, the acupressure feels the strength of the pulse from shallow to deep to find the strongest point of the pulse, and in this process, the vascular wall will deform with the increasing external force. When the extravascular pressure \(P_{e}\) is less than the intravascular pressure \(P\), the vascular wall expands due to the action of the internal pressure of the blood. When the extravascular pressure \(P_{e}\) is equal to the intravascular pressure \(P\), the vascular wall returns to its natural state. When the extravascular pressure \(P_{e}\) is greater than the intravascular pressure \(P\), the vascular wall collapses. Since the blood is an incompressible viscous fluid, the flow in the radial artery is laminar, satisfying [PERSON]'s law [25]. Therefore, the pressure change in the flow is shown in Equation (19).  
\[P=\Delta P=\frac{Q\cdot\Re\eta l}{\pi R^{4}} \tag{19}\]where \(Q\) represents the flow rate, \(\eta\) represents the blood viscosity coefficient, and \(l\) represents the axial length of the vascular micron. If \(F\) is used to represent the strength of the pulse vibration felt by the fingertips of doctors, that is, the combined force of the internal pressure of blood vessels and the radial stress of the vascular wall deformation, the pulse strength \(F\) can be divided into the following situations:  
1. \(P_{e}<P\):  
\[F=P-F_{\rm C}=\frac{Q\cdot 8\eta l}{\pi R^{4}}-\frac{R}{\hbar}(P-P_{e})\cdot\sin \left(\frac{\theta}{2}\right) \tag{20}\]  
From Formula (20), it can be observed that as the extravascular pressure \(P_{e}\) increases, the radial stress \(F_{\rm C}\) of the vascular wall gradually decreases, and \(F\) gradually increases.  
2. \(P_{e}=P\):  
\[F=P-F_{\rm C}=\frac{Q\cdot 8\eta l}{\pi R^{4}} \tag{21}\]  
At this time, due to the balance of extravascular pressure and intravascular pressure, the vascular wall stress disappears, and the pulse strength \(F\) reaches its maximum, as shown in Formula (21).  
3. \(P_{e}>P\):  
The vascular wall collapses, assuming that \(L_{1}\) and \(L_{3}\) are fixed boundaries, \(L_{2}\) is the force boundary, and the natural state radius is R. The vascular collapse model is shown in Figure 3.  
When the vascular wall collapses due to extravascular pressure, the blood flow rate at the collapse increases with the degree of collapse. According to [PERSON]'s Equation (22), the flow weight potential energy at the same location is the same. The greater the velocity of the fluid, the smaller the pressure at the corresponding position. Therefore, when the vascular wall collapses, the intravascular pressure at the site of the collapse decreases.  
\[P_{1}+\frac{1}{2}\rho v_{1}^{2}+G_{1}=P_{2}+\frac{1}{2}\rho v_{2}^{2}+G_{2} \tag{22}\]  
The combined force \(F\) is shown in Equation (23).  
\[F=P_{2}+F_{\rm C}=P_{2}+\frac{R}{\hbar}(P_{e}-P_{2})\cdot\sin\left(\frac{ \theta}{2}\right) \tag{23}\]  
From Formula (23), it can be observed that when the vascular wall collapses, the intravascular pressure \(P_{2}\) decreases, the vascular radius \(R\) decreases, and the extravascular pressure \(P_{e}\) increases, eventually leading to a decrease in \(F\).  
In summary, with the increase in the extravascular pressure \(P_{e}\), the change trend of pulse strength \(F\) gradually increases first. When the intravascular pressure and extravascular pressure reach equilibrium, \(F\) reaches the maximum value. As the \(P_{e}\) continues to increase, the vascular wall collapses, and \(F\) will gradually decrease.