# A mathematical model of the atherosclerosis development in thin blood vessels and its asymptotic approximation

## TL;DR

This paper introduces a new mathematical model for atherosclerosis in thin blood vessels, incorporating recent experimental findings on macrophages, and demonstrates that a simplified 2D model can accurately approximate the complex 3D model.

## Contribution

It proposes an improved mathematical model of atherosclerosis considering macrophage roles and provides an asymptotic approximation to simplify the complex 3D model.

## Key findings

- Existence and uniqueness of the model's positive solution are proven.
- Asymptotic approximation justifies replacing 3D with a 2D model.
- Simplified 2D model maintains sufficient accuracy.

## Abstract

Some existing models of the atherosclerosis development are discussed and a new improved mathematical model, which takes into account new experimental results about diverse roles of macrophages in atherosclerosis, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the nondimensionalisation, small parameters are found. Then asymptotic approximation for the solution is constructed and justified with the help of asymptotic methods for boundary-value problems in thin domains. The results argue for the possibility to replace the complex $3D$ (dimensional) mathematical model with the corresponding simpler $2D$ model with sufficient accuracy measured by these small parameters.