# An Assisted Numerical Simulation Diagnosis Method for Atherosclerosis Based on Hemodynamics
## 2 Mathematical Modeling
### Three-Dimensional Fluid-Structure Interactions  
The blood vessel wall has elastic deformation in the process of blood flow, so it is more accurate to simulate the hemodynamic characteristics in the artery with elastic model. The arbitrary Lagrange-Euler method adopted in this paper 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. Blood flow in the human vasculature is a classic flow-structure interaction problem. Along with heart beats, blood interacts with blood vessel walls at the junction. Therefore, compatible coordination relationships need to be satisfied in the respective control equations for fluids and solids. The blood flow inside blood vessels follows the law of conservation of momentum and conservation of mass. Before the numerical simulation of the blood flow problem, a reasonable flow model based on mathematical expressions should be established to describe the movement of blood. The basic equations of liquid-solid two-phase flow are established by averaging the conservation equations of each phase. The mass conservation equation of liquid phase and solid phase is:  
\[\frac{\partial\rho}{\partial t}+\
abla\bullet(\rho\overset{\rightarrow}{v})=0 \tag{1}\]  
For the liquid phase, the mass conservation equation of the liquid phase is:  
\[\frac{\partial\alpha_{l}\rho_{l}}{\partial t}+\
abla\bullet(\alpha_{l}\rho_{ l}\overset{\rightarrow}{v}_{l})=0 \tag{2}\]  
For the solid phase, the mass conservation equation of solid phase is:  
\[\frac{\partial\alpha_{s}\rho_{s}}{\partial t}+\
abla\bullet(\alpha_{s}\rho_{ s}\overset{\rightarrow}{v}_{s})=0 \tag{3}\]  
where \(\alpha_{l}\) is the liquid phase volume fraction, \(\alpha_{s}\) is the solid phase volume fraction, \(\rho_{l}\) is the liquid density, \(\rho_{s}\) is the solid density, \(\overset{\rightarrow}{v}_{l}\) is the liquid velocity vector, and \(\overset{\rightarrow}{v}_{s}\) is the solid velocity vector. The motion of the fluid phase can be described by the continuity equation and N-S equation as follows:  
\[\frac{\partial v}{\partial t}+(v\cdot\
abla)v=-\frac{1}{\rho}\
abla p+\gamma \
abla^{2}v+f, \tag{4}\]  
For the liquid phase, the momentum conservation equation of the liquid phase is:  
\[\frac{\partial}{\partial t}(\alpha_{l}\rho_{l}\overset{\rightarrow}{v}_{l} )+\
abla\bullet\left(\alpha_{l}\rho_{l}\overset{\rightarrow}{v}_{l}\right) \overset{\rightarrow}{v}_{l}=-\frac{1}{\rho_{l}}\
abla p+\gamma\
abla^{2} \overset{\rightarrow}{v}_{l}+f_{sl} \tag{5}\]  
For the solid phase, the momentum conservation equation of solid phase is:  
\[\frac{\partial}{\partial t}(\alpha_{l}\rho_{s}\overset{\rightarrow}{v}_{s} )+\
abla\bullet\left(\alpha_{s}\rho_{s}\overset{\rightarrow}{v}_{s}\right) \overset{\rightarrow}{v}_{s}=-\frac{1}{\rho_{s}}\
abla p+\gamma\
abla^{2} \overset{\rightarrow}{v}_{s}+f_{ls} \tag{6}\]where \(p\) is the blood pressure, \(\rho\) is the blood density, \(\gamma\) is the kinematic viscosity, \(f_{sl}\) is the force of solid phase to liquid phase, and \(f_{ls}\) is the force of liquid phase to solid phase. The arterial vascular wall is composed of three layers, namely, the inner, middle, and outer membranes, which are featured by a complex structure and nonlinear elasticity. The nonuniformity and anisotropy of the tube wall created difficulties for the study. In previous studies, the vast majority of scholars regarded the vessel wall as a uniformly isotropic linear elastomer. The results have shown that this assumption is reliable in a small deformation range.