Long time evolution of atherosclerotic plaques

Simulation of plaque formation in a realistic geometry of a human aorta: effects of endothelial layer properties, heart rate, and hypertension

Nowadays, cardiovascular diseases are the most common cause of death worldwide. Besides, atherosclerosis is a cardiovascular disease that occurs with persistent narrowing of arteries, especially medium and large-sized arteries. Atherosclerosis begins with a local elevation in the permeability of the arterial wall as a result of endothelial inflammation. Subsequently, excess LDL permeates into the arterial wall. Then, through several chemical responses and reactions, foam cells are produced. These foam cells serve as a crucial indicator for assessing the development of atherosclerosis within the arteries. In this study, the effect of endothelial layer modeling, heart rate (HR) and hypertension on the foam cell accumulation is numerically investigated in a patient-specific geometry of the human thoracic aorta. Navier-Stokes, Darcy, and mass transfer equations are used to obtain the velocity and concentration field within the domain. Regarding the dependence of endothelial cell properties on time-averaged wall shear stress, it is observed that foam cells are mainly concentrated in the outer curvature of the aortic arch, downstream of the left subclavian artery. However, considering oscillatory-shear-rate as the determinant of endothelial cell properties leads to the accumulation of foam cells in the inner curvature of the descending aorta. Regarding the HR, with the increase of HR, the volume average concentration of the foam cell decreases. However, there is no substantial difference between the cases of different HRs. Moreover, foam cell concentration significantly increases in the hypertension case. This result implies that a slight increase in the blood pressure may induce irreparable problems in the circulatory system.

Impact of geometric and hemodynamic changes on a mechanobiological model of atherosclerosis

BACKGROUND AND OBJECTIVE

In this work, the analysis of the importance of hemodynamic updates on a mechanobiological model of atheroma plaque formation is proposed.

METHODS

For that, we use an idealized and axisymmetric model of carotid artery. In addition, the behavior of endothelial cells depending on hemodynamical changes is analyzed too. A total of three computational simulations are carried out and their results are compared: an uncoupled model and two models that consider the opposite behavior of endothelial cells caused by hemodynamic changes. The model considers transient blood flow using the Navier-Stokes equation. Plasma flow across the endothelium is determined with Darcy's law and the Kedem-Katchalsky equations, considering the three-pore model, which is also employed for the flow of substances across the endothelium. The behavior of the considered substances in the arterial wall is modeled with convection-diffusion-reaction equations, and the arterial wall is modeled as a hyperelastic Yeoh's material.

RESULTS

Significant variations are noted in both the morphology and stenosis ratio of the plaques when comparing the uncoupled model to the two models incorporating updates for geometry and hemodynamic stimuli. Besides, the phenomenon of double-stenosis is naturally reproduced in the models that consider both geometric and hemodynamical changes due to plaque growth, whereas it cannot be predicted in the uncoupled model.

CONCLUSIONS

The findings indicate that integrating the plaque growth model with geometric and hemodynamic settings is essential in determining the ultimate shape and dimensions of the carotid plaque.

Impact of hypertension and arterial wall expansion on transport properties and atherosclerosis progression

This study explored the impact of hypertension on atheroma plaque formation through a mechanobiological model. The model incorporates blood flow via the Navier-Stokes equation. Plasma flow through the endothelium is determined by Darcy's law and the Kedem-Katchalsky equations, which consider the three-pore model utilized for substance flow across the endothelium. The behaviour of these substances within the arterial wall is described by convection-diffusion-reaction equations, while the arterial wall itself is modelled as a hyperelastic material using Yeoh's model. To accurately evaluate hypertension's influence, adjustments were made to incorporate wall compression-induced wall compaction by radial compression. This compaction impacts three key variables of the transport phenomena: diffusion, porosity, and permeability. Based on the obtained findings, we can conclude that hypertension significantly augments plaque growth, leading to an over 400% increase in plaque thickness. This effect persists regardless of whether wall mechanics are considered. Tortuosity, arterial wall permeability, and porosity have minimal impact on atheroma plaque growth under normal arterial pressure. However, the atheroma plaque growth changes dramatically in hypertensive cases. In such scenarios, the collective influence of all factors-tortuosity, permeability, and porosity-results in nearly a 20% increase in plaque growth. This emphasizes the importance of considering wall compression due to hypertension in patient studies, where elevated blood pressure and high cholesterol levels commonly coexist.

Blood Lipoproteins Shape the Phenotype and Lipid Content of Early Atherosclerotic Lesion Macrophages: A Dual-Structured Mathematical Model

Macrophages in atherosclerotic lesions exhibit a spectrum of behaviours or phenotypes. The phenotypic distribution of monocyte-derived macrophages (MDMs), its correlation with MDM lipid content, and relation to blood lipoprotein densities are not well understood. Of particular interest is the balance between low density lipoproteins (LDL) and high density lipoproteins (HDL), which carry bad and good cholesterol respectively. To address these issues, we have developed a mathematical model for early atherosclerosis in which the MDM population is structured by phenotype and lipid content. The model admits a simpler, closed subsystem whose analysis shows how lesion composition becomes more pathological as the blood density of LDL increases relative to the HDL capacity. We use asymptotic analysis to derive a power-law relationship between MDM phenotype and lipid content at steady-state. This relationship enables us to understand why, for example, lipid-laden MDMs have a more inflammatory phenotype than lipid-poor MDMs when blood LDL lipid density greatly exceeds HDL capacity. We show further that the MDM phenotype distribution always attains a local maximum, while the lipid content distribution may be unimodal, adopt a quasi-uniform profile or decrease monotonically. Pathological lesions exhibit a local maximum in both the phenotype and lipid content MDM distributions, with the maximum at an inflammatory phenotype and near the lipid content capacity respectively. These results illustrate how macrophage heterogeneity arises in early atherosclerosis and provide a framework for future model validation through comparison with single-cell RNA sequencing data.

Haemodynamic Effects on the Development and Stability of Atherosclerotic Plaques in Arterial Blood Vessel

Studying the formation and stability of atherosclerotic plaques in the hemodynamic field is essential for understanding the growth mechanism and preventive treatment of atherosclerotic plaques. In this paper, based on a multiplayer porous wall model, we established a two-way fluid-solid interaction with time-varying inlet flow. The lipid-rich necrotic core (LRNC) and stress in atherosclerotic plaque were described for analyzing the stability of atherosclerotic plaques during the plaque growth by solving advection-diffusion-reaction equations with finite-element method. It was found that LRNC appeared when the lipid levels of apoptotic materials (such as macrophages, foam cells) in the plaque reached a specified lower concentration, and increased with the plaque growth. LRNC was positively correlated with the blood pressure and was negatively correlated with the blood flow velocity. The maximum stress was mainly located at the necrotic core and gradually moved toward the left shoulder of the plaque with the plaque growth, which increases the plaque instability and the risk of the plaque shedding. The computational model may contribute to understanding the mechanisms of early atherosclerotic plaque growth and the risk of instability in the plaque growth.

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

Atherosclerotic plaques are fatty growths in artery walls that cause heart attacks and strokes. Plaque formation is driven by macrophages that are recruited to the artery wall. These cells consume and remove blood-derived lipids, such as modified low-density lipoprotein. Ineffective lipid removal, due to macrophage death and other factors, leads to the accumulation of lipid-loaded macrophages and formation of a necrotic lipid core. Experimental observations suggest that macrophage functionality varies with the extent of lipid loading. However, little is known about the influence of macrophage lipid loads on plaque fate. Extending work by Ford et al. (J Theor Biol 479:48-63, 2019) and Chambers et al. (A lipid-structured model of atherosclerosis with macrophage proliferation, 2022), we develop a plaque model where macrophages are structured by their ingested lipid load and behave in a lipid-dependent manner. The model considers several macrophage behaviours, including recruitment to and emigration from the artery wall; proliferation and apotosis; ingestion of plaque lipids; and secondary necrosis of apoptotic cells. We consider apoptosis, emigration and proliferation to be lipid-dependent and we model these effects using experimentally informed functions of the internalised lipid load. Our results demonstrate that lipid-dependent macrophage behaviour can substantially alter plaque fate by changing both the total quantity of lipid in the plaque and the distribution of lipid between the live cells, dead cells and necrotic core. The consequences of macrophage lipid-dependence are often unpredictable because lipid-dependent effects introduce subtle, nonlinear interactions between the modelled cell behaviours. These observations highlight the importance of mathematical modelling in unravelling the complexities of macrophage lipid accumulation during atherosclerotic plaque formation.

Enhancing risk prediction capabilities in patients with carotid artery disease using a 2-level computational approach

One of the main causes of death worldwide is carotid artery disease, which causes increasing arterial stenosis and may induce a stroke. To address this problem, the scientific community aims to improve our understanding of the underlying atherosclerotic mechanisms, as well as to make it possible to forecast the progression of atherosclerosis. Additionally, over the past several years, developments in the field of cardiovascular modeling have made it possible to create precise three-dimensional models of patient-specific main carotid arteries. The aforementioned 3D models are then implemented by computational models to forecast either the progression of atherosclerotic plaque or several flow-related metrics which are correlated to risk evaluation. A precise representation of both the blood flow and the fundamental atherosclerotic process within the arterial wall is made possible by computational models, therefore, allowing for the prediction of future lumen stenoses, plaque areas and risk prediction. This work presents an attempt to integrate the outcomes of a novel plaque growth model with advanced blood flow dynamics where the deformed luminal shape derived from the plaque growth model is compared to the actual patient-specific luminal model in terms of several hemodynamic metrics, to identify the prediction accuracy of the aforementioned model. Pressure drop ratios had a mean difference of <3%, whereas OSI-derived metrics were identical in 2/3 cases.Clinical Relevance-This establishes the accuracy of our plaque growth model in predicting the arterial geometry after the desired timeline.

Accurate adventitia reconstruction; significant or not in atherosclerotic plaque growth simulationsƒ A comparative study

A reform in the diagnosis and treatment process is urgently required as carotid artery disease remains a leading cause of death in the world. To this purpose, all computational techniques are now being applied to enhancing the most cutting-edge diagnosis techniques. Computational modeling of plaque generation and evolution is being refined over the past years to forecast the atherosclerotic progression and the corresponding risk in patient-specific carotid arteries. A prerequisite to their implementation is the reconstruction of the precise three-dimensional models of patient-specific main carotid arteries. Even with the most sophisticated algorithms, accurate reconstruction of the arterial vessel is frequently difficult. Furthermore, there are several works of plaque growth modeling that ignore the reconstruction of the artery's outer layer in favor of a virtual one. In this paper, we investigate the importance of an accurate adventitia layer in plaque growth modeling. This is done as a comparative study by implementing a novel plaque growth model in two reconstructed carotid arterial segments using either their realistic or virtual adventitia layer as input. The results indicate that accurate adventitia reconstruction is of minor importance regarding species distributions and plaque growth in carotid segments, which initially did not contain any plaque regions.Clinical Relevance- The findings of this comparative study emphasize the importance of precise adventitia geometry in plaque growth modeling. As a result, this work sets a higher standard for publishing new plaque growth models.

A biomathematical model of atherosclerosis in mice

Atherosclerosis is one of the leading causes of death worldwide. Biomathematical modelling of the underlying disease and therapy processes might be a useful aid to develop and improve preventive and treatment concepts of atherosclerosis. We here propose a biomathematical model of murine atherosclerosis under different diet and treatment conditions including lipid modulating compound and antibiotics. The model is derived by translating known biological mechanisms into ordinary differential equations and by assuming appropriate response kinetics to the applied interventions. We explicitly describe the dynamics of relevant immune cells and lipid species in atherosclerotic lesions including the degree of blood vessel occlusion due to growing plaques. Unknown model parameters were determined by fitting the predictions of model simulations to time series data derived from mice experiments. Parameter fittings resulted in a good agreement of model and data for all 13 experimental scenarios considered. The model can be used to predict the outcome of alternative treatment schedules of combined antibiotic, immune modulating, and lipid lowering agents under high fat or normal diet. We conclude that we established a comprehensive biomathematical model of atherosclerosis in mice. We aim to validate the model on the basis of further experimental data.

Prevalence of traditional cardiovascular risk factors for coronary artery disease and elevated fibrinogen among active military personnel in Republic of Serbia: A cross-sectional study

Background

It is well known that less than 1% of the population achieves ideal cardiovascular health, and 65% of patients do not have their conventional risk biomarkers under control. Military service has its own particularities that may contribute to cardiovascular risk.

Methods

To define the preventive strategy goals, we analysed the prevalence of traditional cardiovascular risk factors for coronary artery disease and elevated fibrinogen among active military personnel in the Republic of Serbia.

Results

The cross-sectional study included 738 individuals older than 20 years, mostly between 31 and 40 years old. The mean value of SBP for the whole group was 122.39± 9.42 mmHg, and for the DBP, it was 79.94±6.56 mmHg. Among active military personnel, 72.7% (533) had prehypertension, and 13.8% (101) was hypertensive. Both body mass and BMI index among the observed age subgroups were found to increase with the age of the patients and cholesterol values. HDL cholesterol values also differed statistically significantly between age subgroups, with the proportion of individuals with HDL less than 1.5 mmol/L in all subgroups being about 85%, the only in the 41-50 age group was lower, 76.4%. LDL cholesterol and the proportion of individuals who had LDL 3.5 increases with the age of patients, and an identical trend was recorded with triglycerides. With ageing, fibrinogen levels increased.

Conclusions

Those findings considering cardio and cerebrovascular risk factors would help create a new approach for primary prevention for these categories of individuals.

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.

The Interdependency and Co-Regulation of the Vitamin D and Cholesterol Metabolism

Vitamin D and cholesterol metabolism overlap significantly in the pathways that contribute to their biosynthesis. However, our understanding of their independent and co-regulation is limited. Cardiovascular disease is the leading cause of death globally and atherosclerosis, the pathology associated with elevated cholesterol, is the leading cause of cardiovascular disease. It is therefore important to understand vitamin D metabolism as a contributory factor. From the literature, we compile evidence of how these systems interact, relating the understanding of the molecular mechanisms involved to the results from observational studies. We also present the first systems biology pathway map of the joint cholesterol and vitamin D metabolisms made available using the Systems Biology Graphical Notation (SBGN) Markup Language (SBGNML). It is shown that the relationship between vitamin D supplementation, total cholesterol, and LDL-C status, and between latitude, vitamin D, and cholesterol status are consistent with our knowledge of molecular mechanisms. We also highlight the results that cannot be explained with our current knowledge of molecular mechanisms: (i) vitamin D supplementation mitigates the side-effects of statin therapy; (ii) statin therapy does not impact upon vitamin D status; and critically (iii) vitamin D supplementation does not improve cardiovascular outcomes, despite improving cardiovascular risk factors. For (iii), we present a hypothesis, based on observations in the literature, that describes how vitamin D regulates the balance between cellular and plasma cholesterol. Answering these questions will create significant opportunities for advancement in our understanding of cardiovascular health.

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.

Role of vascular smooth muscle cell phenotypic switching in plaque progression: A hybrid modeling study

Growing genetic lineage mapping experiments have definitively shown a wide-ranging plasticity of vascular smooth muscle cells (VSMCs) in atherosclerotic plaque and suggested that VSMCs can modulate their phenotypes in response to plaque microenvironment. Here, a multiscale hybrid discrete-continuous (HDC) modeling system is established to investigate the complex role of VSMC phenotypic switching within atherosclerotic lesions. The cellular behaviors of VSMCs and macrophages, including proliferation, migration, phenotypic transformation and necrosis, are determined by cellular automata (CA) rules in discrete model. While the dynamics of plaque microenvironmental factors, such as lipid, extracellular matrix (ECM) and chemokines, are described by continuous reaction-diffusion equations in macroscopy. The simulation results demonstrate how the VSMC activities change the extracellular microenvironment and consequently affect the plaque morphology and stability. The regulation of VSMC phenotypes can affect not only the plaque morphology (necrotic core size and fibrous cap thickness) but also the deposition and distribution of microenvironmental factors (lipoprotein, ECM, and chemokines). In addition, it is found that plaque vulnerability can be inhibited by blocking VSMC transdifferentiation to a macrophage-like state and promoting it to a myofibroblastic phenotype, which suggests that targeting VSMC phenotypic switching could be a potential and promising therapeutic strategy for atherosclerosis.

Simulation of atherosclerotic plaque growth using computational biomechanics and patient-specific data

Atherosclerosis is the one of the major causes of mortality worldwide, urging the need for prevention strategies. In this work, a novel computational model is developed, which is used for simulation of plaque growth to 94 realistic 3D reconstructed coronary arteries. This model considers several factors of the atherosclerotic process even mechanical factors such as the effect of endothelial shear stress, responsible for the initiation of atherosclerosis, and biological factors such as the accumulation of low and high density lipoproteins (LDL and HDL), monocytes, macrophages, cytokines, nitric oxide and formation of foams cells or proliferation of contractile and synthetic smooth muscle cells (SMCs). The model is validated using the serial imaging of CTCA comparing the simulated geometries with the real follow-up arteries. Additionally, we examine the predictive capability of the model to identify regions prone of disease progression. The results presented good correlation between the simulated lumen area (P < 0.0001), plaque area (P < 0.0001) and plaque burden (P < 0.0001) with the realistic ones. Finally, disease progression is achieved with 80% accuracy with many of the computational results being independent predictors.

A multiphase model of growth factor-regulated atherosclerotic cap formation

Atherosclerosis is characterised by the growth of fatty plaques in the inner artery wall. In mature plaques, vascular smooth muscle cells (SMCs) are recruited from adjacent tissue to deposit a collagenous cap over the fatty plaque core. This cap isolates the thrombogenic plaque content from the bloodstream and prevents the clotting cascade that leads to myocardial infarction or stroke. Despite the protective role of the cap, the mechanisms that regulate cap formation and maintenance are not well understood. It remains unclear why some 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 diffusible growth factor signals from the endothelium. Platelet-derived growth factor stimulates SMC migration, proliferation and collagen degradation, while transforming growth factor (TGF)-[Formula: see text] 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 in vivo and in vitro experimental data, reproduces several observations from plaque growth in mice. Numerical and analytical results demonstrate that a stable cap can be formed by a relatively small SMC population and emphasise the critical role of TGF-[Formula: see text] in effective cap formation. These findings provide unique insight into the mechanisms that may lead to plaque destabilisation and rupture. This work represents an important step towards the development of a comprehensive in silico plaque model.

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.

A Spatially Resolved and Quantitative Model of Early Atherosclerosis

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 progression-resistant 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 progression-prone atherosclerotic plaques develop.

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.

New models of atherosclerosis and multi-drug therapeutic interventions

Motivation

Atherosclerosis is amongst the leading causes of death globally. However, it is challenging to study in vivo or in vitro and no detailed, openly-available computational models exist. Clinical studies hint that pharmaceutical therapy may be possible. Here, we develop the first detailed, computational model of atherosclerosis and use it to develop multi-drug therapeutic hypotheses.

Results

We assembled a network describing atheroma development from the literature. Maps and mathematical models were produced using the Systems Biology Graphical Notation and Systems Biology Markup Language, respectively. The model was constrained against clinical and laboratory data. We identified five drugs that together potentially reverse advanced atheroma formation.

Availability and implementation

The map is available in the Supplementary Material in SBGN-ML format. The model is available in the Supplementary Material and from BioModels, a repository of SBML models, containing CellDesigner markup.

Supplementary information

Supplementary data are available at Bioinformatics online.

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.

Effect of Transmural Transport Properties on Atheroma Plaque Formation and Development

We propose a mathematical model of atheroma plaque initiation and early development in coronary arteries using anisotropic transmural diffusion properties. Our current approach is on the process on plaque initiation and intimal thickening rather than in severe plaque progression and rupture phenomena. The effect of transport properties, in particular the anisotropy of diffusion properties of the artery, on plaque formation and development is investigated using the proposed mathematical model. There is not a strong influence of the anisotropic transmural properties on LDL, SMCs and collagen distribution and concentrations along the artery. On the contrary, foam cells distribution strongly depends on the value of the radial diffusion coefficient of the substances [Formula: see text] and the ratio [Formula: see text]. Decreasing [Formula: see text] or diffusion coefficients ratio means a higher concentration of the foam cells close to the intima. Due to the fact that foam cells concentration is associated to the necrotic core formation, the final distribution of foam cells is critical to evolve into a vulnerable or fibrotic plaque.

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.

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.

RADIAL-BASED SIGNAL-PROCESSING COMBINED WITH METHODS OF MACHINE LEARNING

The present paper describes a novel approach to performing feature extraction and classification in possibly layered circular structures, as seen in two-dimensional cutting planes of three-dimensional tube-shaped objects. The algorithm can therefore be used to analyze histological specimens of blood vessels as well as intravascular ultrasound (IVUS) datasets. The approach uses a radial signal-based extraction of textural features in combination with methods of machine learning to integrate a priori domain knowledge. The algorithm in principle solves a two-dimensional classification problem that is reduced to parallel viable time series analysis. A multiscale approach hereby determines a feature vector for each analysis using either a Wavelet-transform (WT) or a S-transform (ST). The classification is done by methods of machine learning — here support vector machines. A modified marching squares algorithm extracts the polygonal segments for the two-dimensional classification. The accuracy is above 80% even in datasets with a considerable quantity of artifacts, while the mean accuracy is above 90%. The benefit of the approach therefore mainly lies in its robustness, efficient calculation, and the integration of domain knowledge.

A new correlation for inclusion of leaky junctions in macroscopic modeling of atherosclerotic lesion initiation

Vascular endothelium cells are the main barriers between vessel wall and blood flow; they play an essential role in the progression of atherosclerosis. Various experimental and computational studies have been carried out to identify the pathways and mechanisms by which Low Density Lipoprotein (LDL) transfers through the endothelium cells. The most conventional hypothesis in LDL transfer is the presence of leaky junctions. Leaky junctions are large pores in endothelium cells associated with cell mitosis or apoptosis. Although some studies have microscopically modeled leaky junctions, none however have evaluated their effects in a macroscopic level modeling. In this study, a new approach is proposed to consider the presence of the leaky junction as the main pathway in LDL transport from the lumen into the arterial wall. LDL transport in macroscopic scale is simulated in a simplified axisymmetric model and Staverman filtration coefficient (SFC) is used as a measurement criterion for estimating the amount of leaky junctions. According to the results, decreasing SFC corresponds to decreasing the resistance of endothelium cells. In other words, an increase in the number of leaky junctions causes an increase in the LDL concentration inside the arterial wall. Additionally, a new correlation is presented for evaluating the fraction of leaky junctions in the endothelial cells by comparing the results of macroscopic and microscopic models. This correlation accredits each SFC to a specified fraction of leaky junction in the endothelial cells. Therefore, it can be used for the inclusion of leaky junctions in the macroscopic modeling without incorporating any of the complications that are raised by the microscopic modeling studies. This correlation has important implications in the modeling of the atherosclerosis lesions propagation.

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.