A Transcritical Bifurcation at Infinity Governs Plaque Stability in a Lipid-Structured Macrophage Model of Atherosclerosis.

Abstract

Atherosclerotic plaques are fatty deposits in arterial walls and a major cause of heart attacks and strokes. Macrophage proliferation triggers plaque growth and instability, but the specific conditions that convert stable plaques into unstable ones remain unclear. To provide insight into the conditions for this transition, we apply bifurcation analysis to the lipid-structured atherosclerosis model proposed by Chambers et al. (Bull Math Biol 86(8):104, 2024). We demonstrate that, in the asymptotic regime where macrophage levels become large ( $M\to \infty$ ), the model exhibits an asymptotic fast-slow structure that does not hold outside this limit. Within this asymptotic regime, we reduce the full system onto a slow invariant manifold, providing a simplified yet accurate description of the dynamics near the critical proliferation-emigration threshold. We prove, via centre manifold theory, that the positive steady state loses stability through a transcritical bifurcation at infinity at the critical proliferation-emigration threshold ${\rho }_c=1+\gamma$ . The analysis reveals that the positive equilibrium branch approaches and exchanges stability with a boundary equilibrium at infinity, providing a rigorous dynamical explanation for the transition that was identified but left unexplored in the original study. Complementing this analytical contribution, we conduct a global sensitivity analysis using Partial Rank Correlation Coefficients (PRCC), identifying macrophage proliferation, emigration, and efferocytosis as the dominant regulators of plaque dynamics. Targeted parameter investigations near ${\rho }_c$ reveal a regulatory decoupling between macrophage accumulation and necrotic core growth, providing new biological insight into the behaviour of the model near the critical proliferation-emigration threshold.