Modelling the inflammatory process in atherosclerosis: a nonlinear renewal equation

Nicolas Meunier (1) , Nicolas Muller (2)

1 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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 ....

Mots clés: Renewal equation; long-time asymptotic; existence theory; structured popula- tions

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Meunier, Nicolas Muller. Modelling the inflammatory process in atherosclerosis: a nonlinear renewal equation. Acta Applicandae Mathematicae, 2019, 161 (1). ⟨hal-01497868⟩