# Modelling the inflammatory process in atherosclerosis: a nonlinear renewal equation

## TL;DR

This paper develops a nonlinear renewal equation model to describe macrophage dynamics in atherosclerosis, incorporating lipoprotein interactions, and analyzes its long-term behavior and stability.

## Contribution

It introduces a novel population-structured model coupling renewal equations with lipoprotein dynamics, providing mathematical analysis of existence, uniqueness, and asymptotic behavior.

## Key findings

- Proved global existence and uniqueness of solutions.
- Analyzed long-term dynamics in plaque rupture scenarios.
- Established steady-state behavior of the model.

## Abstract

We present here a population structured model to describe the dynamics of macrophage cells. The model involves the interactions between modified LDL, monocytes/macrophages, cytokines and foam cells. The key assumption is that the individual macrophage dynamics depends on the amount of lipoproteins it has internalized. The obtained renewal equation is coupled with an ODE describing the lipoprotein dynamics. We first prove global existence and uniqueness for the nonlinear and nonlocal system. We then study long time asymp-totics in a particular case describing silent plaques which undergo periodic rupture and repair. Finally we study long time asymptotics for the nonlinear renewal equation obtained when considering the steady state of the ODE. and we prove that ....