Mathematical modelling of the atherosclerotic plaque formation

International audienceThis article is devoted to the construction of a mathematical model describing the early formation of atherosclerotic lesions. Following the work of El Khatib, Genieys and Volpert, we model atherosclerosis as an inflammatory disease. We consider that the inflammatory process starts with the penetration of Low Density Lipoproteins cholesterol in the intima. This phenomenon is related to the local blood flow dynamics. Using a system of reaction-diffusion equations, we first provide a one-dimensional model of lesion growth. Then we perform numerical simulations on a two-dimensional geometry mimicking the carotid artery. We couple the previous mathematical model with blood flow and we provide a model in which the lesion appears in the area of lower shear stress

Pulsatile non-newtonian blood flow in image-based models of carotid bifurcation

Present hemodynamical study is motivated by the ongoing clinical research at the University Hospital in Pilsen, Czech Republic. On the basis of provided CT scans, several carotid artery models were reconstructed and used for this numerical study of pulsatile blood flow. The blood is modelled as a shear-dependent incompressible fluid, motion of which is described by the non-linear system of Navier-Stokes equations coupled with the Carreau-Yasuda model. The mathematical model is solved using in-house software based on the principle of the SIMPLE algorithm and cell-centred finite volume method (FVM) formulated for hybrid unstructured tetrahedral grids. The discussion of obtained numerical results is performed with special emphasis placed on the analysis of velocity field and distribution of main hemodynamic factors such as cycle-averaged WSS and oscillatory shear index (OSI) in areas prone to atherosclerosis

Computationally Modelling Cholesterol Metabolism and Atherosclerosis

Cardiovascular disease (CVD) is the leading cause of death globally. The underlying pathological driver of CVD is atherosclerosis. The primary risk factor for atherosclerosis is elevated low-density lipoprotein cholesterol (LDL-C). Dysregulation of cholesterol metabolism is synonymous with a rise in LDL-C. Due to the complexity of cholesterol metabolism and atherosclerosis mathematical models are routinely used to explore their non-trivial dynamics. Mathematical modelling has generated a wealth of useful biological insights, which have deepened our understanding of these processes. To date however, no model has been developed which fully captures how whole-body cholesterol metabolism intersects with atherosclerosis. The main reason for this is one of scale. Whole body cholesterol metabolism is defined by macroscale physiological processes, while atherosclerosis operates mainly at a microscale. This work describes how a model of cholesterol metabolism was combined with a model of atherosclerotic plaque formation. This new model is capable of reproducing the output from its parent models. Using the new model, we demonstrate how this system can be utilized to identify interventions that lower LDL-C and abrogate plaque formation

A Two-Phase Model of Early Fibrous Cap Formation in Atherosclerosis

Atherosclerotic plaque growth is characterised by chronic inflammation that promotes accumulation of cellular debris and extracellular fat in the inner artery wall. This material is highly thrombogenic, and plaque rupture can lead to the formation of blood clots that occlude major arteries and cause myocardial infarction or stroke. In advanced plaques, vascular smooth muscle cells (SMCs) migrate from deeper in the artery wall to synthesise a cap of fibrous tissue that stabilises the plaque and sequesters the thrombogenic plaque content from the bloodstream. The fibrous cap provides crucial protection against the clinical consequences of atherosclerosis, but the mechanisms of cap formation are poorly understood. In particular, it is unclear why certain plaques become stable and robust while others become fragile and vulnerable to rupture. We develop a multiphase model with non-standard boundary conditions to investigate early fibrous cap formation in the atherosclerotic plaque. The model is parameterised using a range of in vitro and in vivo data, and includes highly nonlinear mechanisms of SMC proliferation and migration in response to an endothelium-derived chemical signal. We demonstrate that the model SMC population naturally evolves towards a steady-state, and predict a rate of cap formation and a final plaque SMC content consistent with experimental observations in mice. Parameter sensitivity simulations show that SMC proliferation makes a limited contribution to cap formation, and highlight that stable cap formation relies on a critical balance between SMC recruitment to the plaque, SMC migration within the plaque and SMC loss by apoptosis. The model represents the first detailed in silico study of fibrous cap formation in atherosclerosis, and establishes a multiphase modelling framework that can be readily extended to investigate many other aspects of plaque development

A Multiphase Model of Growth Factor-Regulated Atherosclerotic Cap Formation

Atherosclerosis is characterised by the growth of fatty plaques in the inner (intimal) layer of the artery wall. In mature plaques, vascular smooth muscle cells (SMCs) are recruited from the adjacent medial layer to deposit a cap of fibrous collagen over the fatty plaque core. The fibrous cap isolates the thrombogenic content of the plaque from the bloodstream and prevents the formation of blood clots that cause myocardial infarction or stroke. Despite the important protective role of the cap, the mechanisms that regulate cap formation and maintenance are not well understood. It remains unclear why certain caps become stable, while others become vulnerable to rupture. We develop a multiphase PDE model with non-standard boundary conditions to investigate collagen cap formation by SMCs in response to growth factor signals from the endothelium. Diffusible platelet-derived growth factor (PDGF) stimulates SMC migration, proliferation and collagen degradation, while diffusible transforming growth factor (TGF)-$\beta$ stimulates SMC collagen synthesis and inhibits collagen degradation. The model SMCs respond haptotactically to gradients in the collagen phase and have reduced rates of migration and proliferation in dense collagenous tissue. The model, which is parameterised using a range of in vivo and in vitro experimental data, reproduces several observations from studies of plaque growth in atherosclerosis-prone mice. Numerical simulations and model analysis demonstrate that a stable cap can be formed by a relatively small SMC population and emphasise the critical role of TGF-$\beta$ in effective cap formation and maintenance. These findings provide unique insight into the cellular and biochemical mechanisms that may lead to plaque destabilisation and rupture. This work represents an important step towards the development of a comprehensive in silico plaque

The influence of anesthesia and fluid-structure interaction on simulated shear stress patterns in the carotid bifurcation of mice

Background: Low and oscillatory wall shear stresses (WSS) near aortic bifurcations have been linked to the onset of atherosclerosis. In previous work, we calculated detailed WSS patterns in the carotid bifurcation of mice using a Fluid-structure interaction (FSI) approach. We subsequently fed the animals a high-fat diet and linked the results of the FSI simulations to those of atherosclerotic plaque location on a within-subject basis. However, these simulations were based on boundary conditions measured under anesthesia, while active mice might experience different hemodynamics. Moreover, the FSI technique for mouse-specific simulations is both time- and labor-intensive, and might be replaced by simpler and easier Computational Fluid Dynamics (CFD) simulations. The goal of the current work was (i) to compare WSS patterns based on anesthesia conditions to those representing active resting and exercising conditions; and (ii) to compare WSS patterns based on FSI simulations to those based on steady-state and transient CFD simulations. Methods: For each of the 3 computational techniques (steady state CFD, transient CFD, FSI) we performed 5 simulations: 1 for anesthesia, 2 for conscious resting conditions and 2 more for conscious active conditions. The inflow, pressure and heart rate were scaled according to representative in vivo measurements obtained from literature. Results: When normalized by the maximal shear stress value, shear stress patterns were similar for the 3 computational techniques. For all activity levels, steady state CFD led to an overestimation of WSS values, while FSI simulations yielded a clear increase in WSS reversal at the outer side of the sinus of the external carotid artery that was not visible in transient CFD-simulations. Furthermore, the FSI simulations in the highest locomotor activity state showed a flow recirculation zone in the external carotid artery that was not present under anesthesia. This recirculation went hand in hand with locally increased WSS reversal. Conclusions: Our data show that FSI simulations are not necessary to obtain normalized WSS patterns, but indispensable to assess the oscillatory behavior of the WSS in mice. Flow recirculation and WSS reversal at the external carotid artery may occur during high locomotor activity while they are not present under anesthesia. These phenomena might thus influence plaque formation to a larger extent than what was previously assumed. (C) 2016 Elsevier Ltd. All rights reserved

Macrophage anti-inflammatory behaviour in a multiphase model of atherosclerotic plaque development

Atherosclerosis is an inflammatory disease characterised by the formation of plaques, which are deposits of lipids and cholesterol-laden macrophages that form in the artery wall. The inflammation is often non-resolving, due in large part to changes in normal macrophage anti-inflammatory behaviour that are induced by the toxic plaque microenvironment. These changes include higher death rates, defective efferocytic uptake of dead cells, and reduced rates of emigration. We develop a free boundary multiphase model for early atherosclerotic plaques, and we use it to investigate the effects of impaired macrophage anti-inflammatory behaviour on plaque structure and growth. We find that high rates of cell death relative to efferocytic uptake results in a plaque populated mostly by dead cells. We also find that emigration can potentially slow or halt plaque growth by allowing material to exit the plaque, but this is contingent on the availability of live macrophage foam cells in the deep plaque. Finally, we introduce an additional bead species to model macrophage tagging via microspheres, and we use the extended model to explore how high rates of cell death and low rates of efferocytosis and emigration prevent the clearance of macrophages from the plaque

From PK/PD to QSP: Understanding the Dynamic Effect of Cholesterol-Lowering Drugs on Atherosclerosis Progression and Stratified Medicine

Current computational and mathematical tools are demonstrating the high value of using systems modeling approaches (e.g. Quantitative Systems Pharmacology) to understand the effect of a given compound on the biological and physiological mechanisms related to a specific disease. This review provides a short survey of the evolution of the mathematical approaches used to understand the effect of particular cholesterol-lowering drugs, from pharmaco-kinetic (PK) / pharmaco-dynamic (PD) models, through physiologically base pharmacokinetic models (PBPK) to QSP. These mathematical models introduce more mechanistic information related to the effect of these drugs on atherosclerosis progression and demonstrate how QSP could open new ways for stratified medicine in this field