An ODE Model of Early Stages of Atherosclerosis: Mechanisms of the Inflammatory Response

A mathematical model for treatment using chemo-immunotherapy

In this study, we investigated a mathematical model for chemoimmunotherapy (a combination of chemotherapy and immunotherapy) for brain cancer. In most cases, the standard protocol for cancer treatment is fixed in terms of treatment time intervals and dosages. We offer a wide range of non-fixed protocols, which essentially vary in terms of time intervals and dosages (i.e., personalised medicine). The functions that describe this treatment are explicit and analytical. Hence, the parameters of the function can be easily changed and a new protocol can be obtained. We compared different protocols and obtained an optimal solution. In addition, we applied the singular perturbed vector field (SPVF) method to determine the hierarchy of the system of equations, which enabled us to identify the equilibrium points of the mathematical model and investigate their stability.

Modeling cholesterol metabolism and atherosclerosis

Atherosclerotic cardiovascular disease (ASCVD) is the leading cause of morbidity and mortality among Western populations. Many risk factors have been identified for ASCVD; however, elevated low-density lipoprotein cholesterol (LDL-C) remains the gold standard. Cholesterol metabolism at the cellular and whole-body level is maintained by an array of interacting components. These regulatory mechanisms have complex behavior. Likewise, the mechanisms which underpin atherogenesis are nontrivial and multifaceted. To help overcome the challenge of investigating these processes mathematical modeling, which is a core constituent of the systems biology paradigm has played a pivotal role in deciphering their dynamics. In so doing models have revealed new insights about the key drivers of ASCVD. The aim of this review is fourfold; to provide an overview of cholesterol metabolism and atherosclerosis, to briefly introduce mathematical approaches used in this field, to critically discuss models of cholesterol metabolism and atherosclerosis, and to highlight areas where mathematical modeling could help to investigate in the future. This article is categorized under: Cardiovascular Diseases > Computational Models.

Mathematical Modeling of Severe Acute Respiratory Syndrome Coronavirus 2 Infection Network with Cytokine Storm, Oxidative Stress, Thrombosis, Insulin Resistance, and Nitric Oxide Pathways

Coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection is a systemic disease affecting not only the lungs but also multiple organ systems. Clinical studies implicate that SARS-CoV-2 infection causes imbalance of cellular homeostasis and immune response that trigger cytokine storm, oxidative stress, thrombosis, and insulin resistance. Mathematical modeling can offer in-depth understanding of the SARS-CoV-2 infection and illuminate how subcellular mechanisms and feedback loops underpin disease progression and multiorgan failure. We report here a mathematical model of SARS-CoV-2 infection pathway network with cytokine storm, oxidative stress, thrombosis, insulin resistance, and nitric oxide (NO) pathways. The biochemical systems theory model shows autocrine loops with positive feedback enabling excessive immune response, cytokines, transcription factors, and interferons, which can imbalance homeostasis of the system. The simulations suggest that changes in immune response led to uncontrolled release of cytokines and chemokines, including interleukin (IL)-1β, IL-6, and tumor necrosis factor α (TNFα), and affect insulin, coagulation, and NO signaling pathways. Increased production of NETs (neutrophil extracellular traps), thrombin, PAI-1 (plasminogen activator inhibitor-1), and other procoagulant factors led to thrombosis. By analyzing complex biochemical reactions, this model forecasts the key intermediates, potential biomarkers, and risk factors at different stages of COVID-19. These insights can be useful for drug discovery and development, as well as precision treatment of multiorgan implications of COVID-19 as seen in systems medicine.

Mathematical modeling of plaque progression and associated microenvironment: How far from predicting the fate of atherosclerosis?

Mathematical modeling contributes to pathophysiological research of atherosclerosis by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. In turn, the complexity of atherosclerosis is well suited to quantitative approaches as it provides challenges and opportunities for new developments of modeling. In this review, we summarize the current 'state of the art' on the mathematical modeling of the effects of biomechanical factors and microenvironmental factors on the plaque progression, and its potential help in prediction of plaque development. We begin with models that describe the biomechanical environment inside and outside the plaque and its influence on its growth and rupture. We then discuss mathematical models that describe the dynamic evolution of plaque microenvironmental factors, such as lipid deposition, inflammation, smooth muscle cells migration and intraplaque hemorrhage, followed by studies on plaque growth and progression using these modelling approaches. Moreover, we present several key questions for future research. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving plaque vulnerability and propose future potential direction in therapy for cardiovascular disease.

Mathematical modeling of atherogenesis: Atheroprotective role of HDL

Atherosclerosis is a chronic inflammatory cardiovascular disease in which arteries harden through the build-up of plaques. This work is devoted to the mathematical modeling and analysis of the inflammatory process of atherosclerosis. We propose a mathematical model formed by three coupled partial differential equations of reaction-diffusion type. We take into account three key-role players: the inflammatory immune cells, the inflammatory cytokines and the oxidized low density lipoproteins. A stability analysis of the kinetic system is performed. It leads to the presence of three stable fixed points relevant to appropriate biological states of atherogenesis; no inflammation, stabilized inflammation (stable plaque) and advanced inflammation (vulnerable plaque). The cases that may occur are subject to the variation of the parameters values. A detailed discussion showing how the model fits the biological phenomena is then established. We investigate as well the existence of solutions of traveling waves type along with numerical simulations that show the wave propagation in different cases. This shows that the inflammatory process propagates inside the intima as a traveling wave. Then, we consider the effect of high density lipoprotein (HDL) on the atherosclerotic plaque formation. To do that, we elaborate a map that determines the level of risk of plaque formation with respect to the prevalence of HDL in the blood. These results confirm but also generalize previous results published in the literature. They also give a deeper understanding to the propagation of the inflammation inside the artery in terms of the interplay among the different main players in the whole process.

Effects of the Haemodynamic Stimulus on the Location of Carotid Plaques Based on a Patient-Specific Mechanobiological Plaque Atheroma Formation Model

In this work, we propose a mechanobiological atheroma growth model modulated by a new haemodynamic stimulus. To test this model, we analyse the development of atheroma plaques in patient-specific bifurcations of carotid arteries for a total time of 30 years. In particular, eight geometries (left or right carotid arteries) were segmented from clinical images and compared with the solutions obtained computationally to validate the model. The influence of some haemodynamical stimuli on the location and size of plaques is also studied. Plaques predicted by the mechanobiological models using the time average wall shear stress (TAWSS), the oscillatory shear index (OSI) and a new index proposed in this work are compared. The new index predicts the shape index of the endothelial cells as a combination of TAWSS and OSI values and was fitted using data from the literature. The mechanobiological model represents an evolution of the one previously proposed by the authors. This model uses Navier-Stokes equations to simulate blood flow along the lumen in the transient mode. It also employs Darcy's law and Kedem-Katchalsky equations for plasma and substance flow across the endothelium using the three-pore model. The mass balances of all the substances that have been considered in the model are implemented by convection-diffusion-reaction equations, and finally the growth of the plaques has been computed. The results show that by using the new mechanical stimulus proposed in this study, prediction of plaques is, in most cases, better than only using TAWSS or OSI with a minimal and maximal errors on stenosis ratio of 2.77 and 32.89 %, respectively. However, there are a few geometries in which haemodynamics cannot predict the location of plaques, and other biological or genetic factors would be more relevant than haemodynamics. In particular, the model predicts correctly eleven of the fourteen plaques presented in all the geometries considered. Additionally, a healthy geometry has been computed to check that plaque is not developed with the model in this case.

Coronary artery plaque growth: A two-way coupled shear stress-driven model

Atherosclerosis in coronary arteries can lead to plaque growth, stenosis formation, and blockage of the blood flow supplying the heart tissue. Several studies have shown that hemodynamics play an important role in the growth of coronary artery plaques. Specifically, low wall shear stress (WSS) appears to be the leading hemodynamic parameter promoting atherosclerotic plaque growth, which in turn influences the blood flow and WSS distribution. Therefore, a two-way coupled interaction exists between WSS and atherosclerosis growth. In this work, a computational framework was developed to study the coupling between WSS and plaque growth in coronary arteries. Computational fluid dynamics (CFD) was used to quantify WSS distribution. Surface mesh nodes were moved in the inward normal direction according to a growth model based on WSS. After each growth stage, the geometry was updated and the CFD simulation repeated to find updated WSS values for the next growth stage. One hundred twenty growth stages were simulated in an idealized tube and an image-based left anterior descending artery. An automated framework was developed using open-source software to couple CFD simulations with growth. Changes in plaque morphology and hemodynamic patterns during different growth stages are presented. The results show larger plaque growth towards the downstream segment of the plaque, agreeing with the reported clinical observations. The developed framework could be used to establish hemodynamic-driven growth models and study the interaction between these processes.

Identifying important parameters in the inflammatory process with a mathematical model of immune cell influx and macrophage polarization

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage polarization, with the simplifying assumption that macrophages can be classified as M1 or M2. With this model, we were able to reproduce the expected timing of sequential influx of immune cells and mediators in a general inflammatory setting. We then fit this model to in vivo experimental data obtained from a mouse peritonitis model of inflammation, which is widely used to evaluate endogenous processes in response to an inflammatory stimulus. Model robustness is explored with local structural and practical identifiability of the proposed model a posteriori. Additionally, we perform sensitivity analysis that identifies the population of apoptotic neutrophils as a key driver of the inflammatory process. Finally, we simulate a selection of proposed therapies including points of intervention in the case of delayed neutrophil apoptosis, which our model predicts will result in a sustained inflammatory response. Our model can therefore provide hypothesis testing for therapeutic interventions that target macrophage phenotype and predict outcomes to be validated by subsequent experimentation.

Using experimental data and mathematical analysis, we develop a model for the inflammatory response that includes macrophage polarization between M1 and M2 phenotypes. Dysfunction of this phenotypic switch can disrupt the timely influx and egress of immune cells during the healing process and lead to chronic wounds or disease. The modulation of macrophage population has been suggested as a strategy to dampen inflammation in diseases that feature chronic inflammation, such as diabetes and atherosclerosis. It is therefore important that we learn more about which components of the system drive the population level switch in phenotype. Our model is able to reproduce the expected timing of sequential influx of neutrophils and macrophages in response to an inflammatory stimulus. Model parameters were estimated with weighted least squares fitting to in vivo experimental data from a mouse model of peritonitis while considering identifiability of parameter sets. We perform sensitivity analysis that identifies primary drivers of the system, and predict the effects of variations in these key parameters on immune cell populations.

Mathematical Modelling and Simulation of Atherosclerosis Formation and Progress: A Review

Cardiovascular disease (CVD) is a major threat to human health since it is the leading cause of death in western countries. Atherosclerosis is a type of CVD related to hypertension, diabetes, high levels of cholesterol, smoking, oxidative stress, and age. Atherosclerosis primarily occurs in medium and large arteries, such as coronary and the carotid artery and, in particular, at bifurcations and curvatures. Atherosclerosis is compared to an inflammatory disease where a thick, porous material comprising cholesterol fat, saturated sterols, proteins, fatty acids, calcium etc., is covered by an endothelial membrane and a fragile fibrous tissue which makes atheromatic plaque prone to rupture that could lead to the blockage of the artery due to the released plaque material. Despite the great progress achieved, the nature of the disease is not fully understood. This paper reviews the current state of modelling of all levels of atherosclerosis formation and progress and discusses further challenges in atherosclerosis modelling. The objective is to pave a way towards more precise computational tools to predict and eventually reengineer the fate of atherosclerosis.

Nonlinear dynamics of early atherosclerotic plaque formation may determine the efficacy of high density lipoproteins (HDL) in plaque regression

We use a computational model to explore the effect of foam cell accumulation on plaque regression following an increase in high density lipoprotein (HDL) influx into the plaque. Atherosclerotic plaque formation is the outcome of cellular and cytokine responses to low density lipoproteins (LDL) that penetrate the artery wall following an injury to the endothelium and become modified. We modelled the cells and cytokines that are most important in plaque formation using partial differential equations. The model includes monocytes and macrophages, foam cells, macrophage chemoattractants, endothelium-stimulating cytokines, modified low density lipoproteins (mod LDL) and HDL. We included interactions both at the endothelium surface and inside the artery wall. The model predicts that when HDL influx into a well-established plaque with large numbers of foam cells is increased, the plaque may not regress but may continue to grow at a slower rate. If HDL influx is increased when a model plaque is recently established and has fewer foam cells, then the plaque does regress. If modLDL influx into the plaque is lowered at the same time that HDL influx increased or the capacity of the HDL to remove cholesterol from foam cells is increased, then the plaque is more likely to regress. The predictions of the model are in qualitative agreement with experimental studies in mice and rabbits. The results suggest that the intrinsic dynamics of reverse cholesterol transport by HDL are important in determining the success of HDL raising in promoting plaque regression.

A multiphysics approach for modeling early atherosclerosis

This work is devoted to the development of a mathematical model of the early stages of atherosclerosis incorporating processes of all time scales of the disease and to show their interactions. The cardiovascular mechanics is modeled by a fluid-structure interaction approach coupling a non-Newtonian fluid to a hyperelastic solid undergoing anisotropic growth and a change of its constitutive equation. Additionally, the transport of low-density lipoproteins and its penetration through the endothelium is considered by a coupled set of advection-diffusion-reaction equations. Thereby, the permeability of the endothelium is wall-shear stress modulated resulting in a locally varying accumulation of foam cells triggering a novel growth and remodeling formulation. The model is calibrated and applied to an murine-specific case study, and a qualitative validation of the computational results is performed. The model is utilized to further investigate the influence of the pulsatile blood flow and the compliance of the artery wall to the atherosclerotic process. The computational results imply that the pulsatile blood flow is crucial, whereas the compliance of the aorta has only a minor influence on atherosclerosis. Further, it is shown that the novel model is capable to produce a narrowing of the vessel lumen inducing an adaption of the endothelial permeability pattern.

How radiation influences atherosclerotic plaque development: a biophysical approach in ApoE⁻/⁻ mice

Atherosclerosis is the development of lipid-laden plaques in arteries and is nowadays considered as an inflammatory disease. It has been shown that high doses of ionizing radiation, as used in radiotherapy, can increase the risk of development or progression of atherosclerosis. To elucidate the effects of radiation on atherosclerosis, we propose a mathematical model to describe radiation-promoted plaque development. This model distinguishes itself from other models by combining plaque initiation and plaque growth, and by incorporating information from biological experiments. It is based on two consecutive processes: a probabilistic dose-dependent plaque initiation process, followed by deterministic plaque growth. As a proof of principle, experimental plaque size data from carotid arteries from irradiated ApoE[Formula: see text] mice was used to illustrate how this model can provide insight into the underlying biological processes. This analysis supports the promoting role for radiation in plaque initiation, but the model can easily be extended to include dose-related effects on plaque growth if available experimental data would point in that direction. Moreover, the model could assist in designing future biological experiments on this research topic. Additional biological data such as plaque size data from chronically-irradiated mice or experimental data sets with a larger variety in biological parameters can help to further unravel the influence of radiation on plaque development. To the authors' knowledge, this is the first biophysical model that combines probabilistic and mechanistic modeling which uses experimental data to investigate the influence of radiation on plaque development.

A biochemical and mechanical model of injury-induced intimal thickening

In this paper, we investigate an axisymmetric model of intimal thickening using hyperelasticity theory. Our model describes the growth of the arterial intima due to cell proliferation which, in turn, is driven by the release of a cytokine such as platelet-derived growth factor (PDGF). With the growth rate tied to both local stress and the local concentration of PDGF, we derive a quadruple free boundary problem with different regions of the vessel wall characterized by different homeostatic stress. We compare our model predictions to rabbit and rodent models of atherosclerosis and find that in order to achieve the growth rates reported in the experiments, growth must be mainly cytokine induced rather than stress induced. Our model is also able to reproduce Glagov remodelling where, as a vessel becomes more diseased, the lumen expands before rapidly contracting.

A mechanistic model for atherosclerosis and its application to the cohort of Mayak workers

We propose a stochastic model for use in epidemiological analysis, describing the age-dependent development of atherosclerosis with adequate simplification. The model features the uptake of monocytes into the arterial wall, their proliferation and transition into foam cells. The number of foam cells is assumed to determine the health risk for clinically relevant events such as stroke. In a simulation study, the model was checked against the age-dependent prevalence of atherosclerotic lesions. Next, the model was applied to incidence of atherosclerotic stroke in the cohort of male workers from the Mayak nuclear facility in the Southern Urals. It describes the data as well as standard epidemiological models. Based on goodness-of-fit criteria the risk factors smoking, hypertension and radiation exposure were tested for their effect on disease development. Hypertension was identified to affect disease progression mainly in the late stage of atherosclerosis. Fitting mechanistic models to incidence data allows to integrate biological evidence on disease progression into epidemiological studies. The mechanistic approach adds to an understanding of pathogenic processes, whereas standard epidemiological methods mainly explore the statistical association between risk factors and disease outcome. Due to a more comprehensive scientific foundation, risk estimates from mechanistic models can be deemed more reliable. To the best of our knowledge, such models are applied to epidemiological data on cardiovascular diseases for the first time.

NUMERICAL MODELING OF ATHEROSCLEROSIS LESION EVOLUTION IN TIME I. INITIAL STAGE OF THE DISEASE

Atherosclerosis is one of the main causes of death in the developed world. The disease, which is an inflammatory disease, has been the focus of many studies. A few studies attempted to model atherosclerosis lesion development mathematically while no attention has been paid to the multistage nature of the disease. The present study provides a mathematical model for atherosclerosis evolution by focusing on the inflammatory responses of the initial stage of the disease. In the model, the inflammatory response in type I lesion, which includes endothelium dysfunction, LDL oxidation, monocytes entry, foam cell formation and intima property changes, are coupled with the transport equations of blood and LDL in lumen and arterial wall. The innovation of the model is determination of the duration of the initial stage of lesion propagation for a specific patient while the presence of leaky junction in endothelial layer and LDL oxidation in the intima layer are considered. The greatest advantage of the study in comparison with previous studies is to provide a model for the initiating stage of the atherosclerosis development so that a more precise result of the disease evolution is obtained.

Mathematical modeling and simulation of the evolution of plaques in blood vessels

In this paper, a model is developed for the evolution of plaques in arteries, which is one of the main causes for the blockage of blood flow. Plaque rupture and spread of torn-off material may cause closures in the down-stream vessel system and lead to ischemic brain or myocardial infarctions. The model covers the flow of blood and its interaction with the vessel wall. It is based on the assumption that the penetration of monocytes from the blood flow into the vessel wall, and the accumulation of foam cells increasing the volume, are main factors for the growth of plaques. The dynamics of the vessel wall is governed by a deformation gradient, which is given as composition of a purely elastic tensor, and a tensor modeling the biologically caused volume growth. An equation for the evolution of the metric is derived quantifying the changing geometry of the vessel wall. To calculate numerically the solutions of the arising free boundary problem, the model system of partial differential equations is transformed to an ALE (Arbitrary Lagrangian-Eulerian) formulation, where all equations are given in fixed domains. The numerical calculations are using newly developed algorithms for ALE systems. The results of the simulations, obtained for realistic system parameters, are in good qualitative agreement with observations. They demonstrate that the basic modeling assumption can be justified. The increase of stresses in the vessel wall can be computed. Medical treatment tries to prevent critical stress values, which may cause plaque rupture and its consequences.

Bifurcation and dynamics in a mathematical model of early atherosclerosis: How acute inflammation drives lesion development

We present here a mathematical model describing the primary mechanisms that drive the early stages of atherosclerosis. This involves the interactions between modified low density lipoprotein (LDL), monocytes/macrophages, cytokines and foam cells. This model suggests that there is an initial inflammatory phase associated with atherosclerotic lesion development and a longer, quasi-static process of plaque development inside the arterial wall that follows the initial transient. We will show results that show how different LDL concentrations in the blood stream and different immune responses can affect the development of a plaque. Through numerical bifurcation analysis, we show the existence of a fold bifurcation when the flux of LDL from the blood is sufficiently high. By analysing the model presented in this paper, we gain a greater insight into this inflammatory response qualitatively and quantitatively.

Using a Systems Pharmacology Approach to Study the Effect of Statins on the Early Stage of Atherosclerosis in Humans

More than 100,000 people have participated in controlled trials of statins (lowering cholesterol drugs) since the introduction of lovastatin in the 1980s. Meta-analyses of this data have shown that statins have a beneficial effect on treated groups compared to control groups, reducing cardiovascular risk. Inhibiting the HMG-CoA reductase in the liver, statins can reduce cholesterol levels, thus reducing LDL levels in circulation. Published data from intravascular ultrasound studies (IVUS) was used in this work to develop and validate a unique integrative system model; this consisted of analyzing control groups from two randomized controlled statins trials (24/97 subjects respectively), one treated group (40 subjects, simvastatin trial), and 27 male subjects (simvastatin, pharmacokinetic study). The model allows to simulate the pharmacokinetics of statins and its effect on the dynamics of lipoproteins (e.g., LDL) and the inflammatory pathway while simultaneously exploring the effect of flow-related variables (e.g., wall shear stress) on atherosclerosis progression.

Athero-protective effects of High Density Lipoproteins (HDL): An ODE model of the early stages of atherosclerosis

We present an ODE model which we use to investigate how High Density Lipoproteins (HDL) reduce the inflammatory response in atherosclerosis. HDL causes atherosclerotic plaque stabilisation and regression, and is an important potential marker and prevention target for cardiovascular disease. HDL enables cholesterol efflux from the arterial wall, macrophage and foam cell emigration, and has other athero-protective effects. Our basic inflammatory model is augmented to include several different ways that HDL can act in early atherosclerosis. In each case, the action of HDL is represented via a parameter in the model. The long-term model behaviour is investigated through phase plane analysis and simulations. Our results indicate that only HDL-enabled cholesterol efflux can stabilise the internalised lipid content in the lesion so that it does not continue to grow, but this does not reduce macrophage numbers which is required to stabilise the lesion or prevent rupture. HDL-enabled macrophage emigration guarantees lesion stabilisation by maintaining stable macrophage content.

Mathematical modelling of atheroma plaque formation and development in coronary arteries

Atherosclerosis is a vascular disease caused by inflammation of the arterial wall, which results in the accumulation of low-density lipoprotein (LDL) cholesterol, monocytes, macrophages and fat-laden foam cells at the place of the inflammation. This process is commonly referred to as plaque formation. The evolution of the atherosclerosis disease, and in particular the influence of wall shear stress on the growth of atherosclerotic plaques, is still a poorly understood phenomenon. This work presents a mathematical model to reproduce atheroma plaque growth in coronary arteries. This model uses the Navier-Stokes equations and Darcy's law for fluid dynamics, convection-diffusion-reaction equations for modelling the mass balance in the lumen and intima, and the Kedem-Katchalsky equations for the interfacial coupling at membranes, i.e. endothelium. The volume flux and the solute flux across the interface between the fluid and the porous domains are governed by a three-pore model. The main species and substances which play a role in early atherosclerosis development have been considered in the model, i.e. LDL, oxidized LDL, monocytes, macrophages, foam cells, smooth muscle cells, cytokines and collagen. Furthermore, experimental data taken from the literature have been used in order to physiologically determine model parameters. The mathematical model has been implemented in a representative axisymmetric geometrical coronary artery model. The results show that the mathematical model is able to qualitatively capture the atheroma plaque development observed in the intima layer.

Kinetic modeling of low density lipoprotein oxidation in arterial wall and its application in atherosclerotic lesions prediction

Oxidation of low-density lipoprotein (LDL) is one of the major factors in atherogenic process. Trapped oxidized LDL (Ox-LDL) in the subendothelial matrix is taken up by macrophage and leads to foam cell generation creating the first step in atherosclerosis development. Many researchers have studied LDL oxidation using in vitro cell-induced LDL oxidation model. The present study provides a kinetic model for LDL oxidation in intima layer that can be used in modeling of atherosclerotic lesions development. This is accomplished by considering lipid peroxidation kinetic in LDL through a system of elementary reactions. In comparison, characteristics of our proposed kinetic model are consistent with the results of previous experimental models from other researches. Furthermore, our proposed LDL oxidation model is added to the mass transfer equation in order to predict the LDL concentration distribution in intima layer which is usually difficult to measure experimentally. According to the results, LDL oxidation kinetic constant is an important parameter that affects LDL concentration in intima layer so that existence of antioxidants that is responsible for the reduction of initiating rates and prevention of radical formations, have increased the concentration of LDL in intima by reducing the LDL oxidation rate.

Numerical and analytical study of an atherosclerosis inflammatory disease model

We study a reaction-diffusion mathematical model for the evolution of atherosclerosis as an inflammation process by combining analytical tools with computer-intensive numerical calculations. The computational work involved the calculation of more than sixty thousand solutions of the full reaction-diffusion system and lead to the complete characterisation of the ω-limit for every initial condition. Qualitative properties of the solution are rigorously proved, some of them hinted at by the numerical study.

Mathematical model of intimal thickening in atherosclerosis: vessel stenosis as a free boundary problem

Atherosclerosis is an inflammatory disease of the artery characterized by an expansion of the intimal region. Intimal thickening is usually attributed to the migration of smooth muscle cells (SMCs) from the surrounding media and proliferation of SMCs already present in the intima. Intimal expansion can give rise to dangerous events such as stenosis (leading to stroke) or plaque rupture (leading to myocardial infarction). In this paper we propose and study a mathematical model of intimal thickening, posed as a free boundary problem. Intimal thickening is driven by damage to the endothelium, resulting in the release of cytokines and migration of SMCs. By coupling a boundary value problem for cytokine concentration to an evolution law for the intimal area, we reduce the problem to a single nonlinear differential equation for the luminal radius. We analyze the steady states, perform a bifurcation analysis and compare model solutions to data from rabbits whose iliac arteries are subject to a balloon pullback injury. In order to obtain a favorable fit, we find that migrating SMCs must enter the intima very slowly compared to cells in dermal wounds. This cell behavior is indicative of a weak inflammatory response which is consistent with atherosclerosis being a chronic inflammatory disease.

ARTreat Project: three-dimensional numerical simulation of plaque formation and development in the arteries

Atherosclerosis is a progressive disease characterized by the accumulation of lipids and fibrous elements in arteries. It is characterized by dysfunction of endothelium and vasculitis, and accumulation of lipid, cholesterol, and cell elements inside blood vessel wall. In this study, a continuum-based approach for plaque formation and development in 3-D is presented. The blood flow is simulated by the 3-D Navier-Stokes equations, together with the continuity equation while low-density lipoprotein (LDL) transport in lumen of the vessel is coupled with Kedem-Katchalsky equations. The inflammatory process was solved using three additional reaction-diffusion partial differential equations. Transport of labeled LDL was fitted with our experiment on the rabbit animal model. Matching with histological data for LDL localization was achieved. Also, 3-D model of the straight artery with initial mild constriction of 30% plaque for formation and development is presented.

From inflammation to wound healing: using a simple model to understand the functional versatility of murine macrophages

Macrophages are fundamental cells of the innate immune system. Their activation is essential for such distinct immune functions as inflammation (pathogen-killing) and tissue repair (wound healing). An open question has been the functional stability of an individual macrophage cell: whether it can change its functional profile between different immune responses such as between the repair pathway and the inflammatory pathway. We studied this question theoretically by constructing a rate equation model for the key substrate, enzymes and products of the pathways; we then tested the model experimentally. Both our model and experiments show that individual macrophages can switch from the repair pathway to the inflammation pathway but that the reverse switch does not occur.