A Spatially Resolved and Quantitative Model of Early Atherosclerosis

Abstract

Atherosclerosis is a major burden for all societies and there is a great need for a deeper understanding of involved key inflammatory, immunological and biomechanical processes. A decisive step for the prevention and medical treatment of atherosclerosis is to predict what conditions determine whether early atherosclerotic plaques continue to grow, stagnate or become regressive. The driving biological and mechanobiological mechanisms that determine the stability of plaques are yet not fully understood. We develop a spatially resolved and quantitative mathematical model of key contributors of early atherosclerosis. The stability of atherosclerotic model plaques is assessed to identify and classify progression-prone and progressionresistant atherosclerotic regions based on measurable or computable in vivo inputs, such as blood cholesterol concentrations and wall shear stresses. The model combines Darcy's law for the transmural flow through vessels walls, the Kedem-Katchalsky equations for endothelial fluxes of lipoproteins, a quantitative model of early plaque formation from a recent publication and a novel submodel for macrophage recruitment. The parameterization and analysis of the model suggest that the advective flux of lipoproteins through the endothelium is decisive, while the influence of the advective transport within the artery wall is negligible. Further, regions in arteries with an approximate wall shear stress exposure below 20 % of the average exposure and their surroundings are potential regions where progressionprone atherosclerotic plaques develop.

Wall shear stress and its role in atherosclerosis

“…In the ideal state of straight vessels, the magnitude of shear stress can be estimated to be directly proportional to blood flow and viscosity, and inversely proportional to the cubic power of vessel diameter ( 11 ). Understanding the complex fluid dynamics of arteries will contribute to the future simulation-based CFD studies, especially in the context of atherosclerosis ( 16 ). Because of the limited time or space resolution of the imaging systems, examining the atherosclerosis in details in vivo process has been difficult ( 17 ).…”

Modeling cholesterol metabolism and atherosclerosis

“…Similarly, Lasiello and colleagues created a multilayer model of LDL accumulation (Iasiello et al, 2018). Most significantly, however, Thon and colleagues This work focuses on the arterial penetration of HDL and recruitment of macrophages as contributors to atherosclerosis (Thon et al, 2019). An area which had been overlooked by modeling.…”

A Lipid-Structured Model of Atherosclerotic Plaque Macrophages with Lipid-Dependent Kinetics

“…Most work to date has focussed on modelling the inflammatory response of macrophages in the early plaque. Published approaches include spatiallyaveraged ODE models (Bulelzai and Dubbeldam, 2012;Cohen et al, 2014;Islam and Johnston, 2016;Thon et al, 2018;Lui and Myerscough, 2021), spatially-resolved PDE models (El Khatib et al, 2007;Calvez et al, 2009;Little et al, 2009;Yang et al, 2016;Chalmers et al, 2017;Thon et al, 2019;Silva et al, 2020) and agent-based models (Bhui and Hayenga, 2017). To account for macrophage lipid ingestion it is common to assume that the modelled macrophages have two sub-populations: those with little or no internalised lipid (usually termed macrophages) and those with lots of internalised lipid (usually termed foam cells) (Calvez et al, 2009;Bulelzai and Dubbeldam, 2012;Cilla et al, 2014;Hao and Friedman, 2014;Islam and Johnston, 2016;Yang et al, 2016;Chalmers et al, 2017;Silva et al, 2020).…”

“…Lipid-dependent macrophage behaviour can be implicitly incorporated in this model framework by assuming that these sub-populations have, for example, different rates of lipid consumption (Calvez et al, 2009;Bulelzai and Dubbeldam, 2012;Cilla et al, 2014;Hao and Friedman, 2014;Islam and Johnston, 2016;Silva et al, 2020), migration (Calvez et al, 2009;Cilla et al, 2014;Yang et al, 2016;Chalmers et al, 2017;Silva et al, 2020) or apoptosis (Hao and Friedman, 2014;Islam and Johnston, 2016;Silva et al, 2020). An alternative approach is to model macrophages as a single population and track the population's total ingested lipid content (Little et al, 2009;Cohen et al, 2014;Thon et al, 2018Thon et al, , 2019Lui and Myerscough, 2021). Here, lipid-dependent effects can be included at the population-level by assuming that macrophage behaviour depends on the average ingested lipid load (Thon et al, 2018(Thon et al, , 2019.…”

“…An alternative approach is to model macrophages as a single population and track the population's total ingested lipid content (Little et al, 2009;Cohen et al, 2014;Thon et al, 2018Thon et al, , 2019Lui and Myerscough, 2021). Here, lipid-dependent effects can be included at the population-level by assuming that macrophage behaviour depends on the average ingested lipid load (Thon et al, 2018(Thon et al, , 2019. A more natural means to model lipid accumulation in plaque macrophage populations, including with lipid-dependent effects, is to use a structured population model in which macrophages are classified by their internalised lipid content.…”