Nonlinear Coupled Equations Free-Boundary Problem in Atherosclerosis: Symmetry Analysis of Solutions and Numerical Simulation

Atherosclerosis, a chronic inflammatory cardiovascular disease closely related to plaque formation during arteriosclerosis, poses a significant threat to global health. To deepen the understanding of the multifaceted interactions driving atherosclerosis progression and provide theoretical support for designing targeted therapeutic strategies, this study establishes a nonlinear coupled atherosclerotic free-boundary model integrating inflammatory immune cells, cytokines, and oxidized low-density lipoprotein. By applying the compression mapping principle, the local and global existence and uniqueness of solutions are proven, while revealing certain symmetries in the model’s solution structure. Under specific assumptions, a quasi-steady-state approximate model is derived, and the existence of its solution is demonstrated. Through numerical simulations of the quasi-steady-state approximate model using the finite difference method, the temporal and spatial evolution of pro-inflammatory macrophages and oxidized low-density lipoprotein is analyzed. The findings highlight the model’s strength in capturing the intricate dynamics of atherosclerosis, uncovering underlying mechanisms and identifying therapeutic targets. By evaluating inflammatory dynamics across plaque types and stenosis levels, the experimental design further validated the model’s ability to replicate clinical processes and reinforced its predictive accuracy. Notably, in the process of model analysis and solution, symmetries in the equations and boundary conditions play a crucial role in determining the solution properties. However, the current one-dimensional model has limitations. Future research should focus on developing higher-dimensional models and integrating more influencing factors to enhance the model’s clinical applicability and deepen the understanding of this complex disease.