# An Assisted Numerical Simulation Diagnosis Method for Atherosclerosis Based on Hemodynamics
## 2 Mathematical Modeling
### Discrete Method of Governing Equations  
There is elastic deformation in the blood vessel wall during blood flow, so using an elastic model to simulate the hemodynamic characteristics in the artery can more closely replicate the actual data. In the arbitrary Lagrange-Euler method, the computational grid can move in any form in space, that is, it can move independently of the physical and spatial coordinate systems. In this way, the moving interface of the object can be accurately described and the reasonable shape of the element can be maintained by specifying the appropriate mesh-motion form. The arbitrary Lagrange-Euler method adopted here overcomes the shortcomings of the single Lagrangian or single Euler method in solving the problem, and is an effective method to realize the coupling calculation of the blood fluid phase and solid wall phase in a moving grid.  
The equation of motion of a viscous incompressible fluid is written as follows:  
\[\frac{\partial v_{i}}{\partial t}+\big{(}v_{j}-\overline{v}_{j}\big{)}v_{i,j }-\gamma\big{(}v_{i,j}+v_{j,i}\big{)}_{,j}+\frac{1}{\rho}p_{,i}=0 \tag{11}\]  
\[v_{i,i}=0 \tag{12}\]  
The standard index notation is used here, where the subscript represents the tensor or vector component, the nonrepeating subscript of the same physical quantity represents the partial derivative, and the repeating subscript represents the sum: \(i,j=1,2,3\). For example, \(v_{j}\); \(j=1,2,3\) represents the velocity of the fluid in different directions and \(\overline{v}_{j}\); \(j=1,2,3\) represents the velocity of the moving area in different directions.  
Using the discrete scheme of forward difference for the time term in Equation (10), it can be obtained:  
\[\frac{v_{i}^{n+1}-v_{i}^{n}}{\Delta t}=-\left(v_{j}^{n}-\overline{v}_{j}^{n} \right)v_{i,j}^{n}-\frac{1}{\rho}p_{j,i}^{n+1}+\gamma\left(v_{i,j}^{n}+v_{j,i}^{ n}\right)_{,j} \tag{13}\]  
The Poisson equation of pressure can be obtained by taking the divergence of both sides of the equation and introducing the continuity equation:  
\[p_{,i}^{n+1}=\frac{\rho v_{i,j}^{n}}{\Delta t}-\rho\left(v_{j}^{n}-\overline{v }_{j}^{n}\right)v_{i,j}^{n}+\eta\left(v_{i,j}^{n}+v_{j,i}^{n}\right)_{,j} \tag{14}\]  
According to the obtained pressure field, the velocity field distribution can be obtained by substitution:  
\[v_{i}^{n+1}=v_{i}^{n}+\Delta t\left[-\left(v_{j}^{n}-\overline{v}_{j}^{n} \right)v_{i,j}^{n}-\frac{1}{\rho}p_{,i}^{n+1}+\gamma\left(v_{i,j}^{n}+v_{j,i} ^{n}\right)_{,j}\right] \tag{15}\]  
The crucial feature in fluid-structure coupling between the blood and the blood vessel wall is to realize data transfer at the coupling interface, which requires the following conditions to be met between the corresponding particles of blood and the blood vessel wall at the boundary of the coupling domain.  
The velocity continuity condition is satisfied: \(u=\frac{\partial\mathbf{x}_{s}}{\partial t}\); the condition of force continuity is satisfied: \(\sigma_{s}\cdot\mathbf{n}_{s}=-\sigma\cdot\mathbf{n}\); and the displacement continuity condition is satisfied: \(\mathbf{x}=\mathbf{x}_{s}\).