A Mathematical Model of Atherosclerosis with Reverse Cholesterol Transport and Associated Risk Factors

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.

Integrating Computational and Biological Hemodynamic Approaches to Improve Modeling of Atherosclerotic Arteries

Atherosclerosis is the primary cause of cardiovascular disease, resulting in mortality, elevated healthcare costs, diminished productivity, and reduced quality of life for individuals and their communities. This is exacerbated by the limited understanding of its underlying causes and limitations in current therapeutic interventions, highlighting the need for sophisticated models of atherosclerosis. This review critically evaluates the computational and biological models of atherosclerosis, focusing on the study of hemodynamics in atherosclerotic coronary arteries. Computational models account for the geometrical complexities and hemodynamics of the blood vessels and stenoses, but they fail to capture the complex biological processes involved in atherosclerosis. Different in vitro and in vivo biological models can capture aspects of the biological complexity of healthy and stenosed vessels, but rarely mimic the human anatomy and physiological hemodynamics, and require significantly more time, cost, and resources. Therefore, emerging strategies are examined that integrate computational and biological models, and the potential of advances in imaging, biofabrication, and machine learning is explored in developing more effective models of atherosclerosis.

HDL and plaque regression in a multiphase model of early atherosclerosis

Atherosclerosis is a chronic disease of the arteries characterised by the accumulation of lipids and lipid-engorged cells in the artery wall. Early plaque growth is aggravated by the deposition of low density lipoproteins (LDL) in the wall and the subsequent immune response. High density lipoproteins (HDL) counterbalance the effects of LDL by accepting cholesterol from macrophages and removing it from the plaque. In this paper, we develop a free boundary multiphase model to investigate the effects of LDL and HDL on early plaque development. We examine how the rates of LDL and HDL deposition affect cholesterol accumulation in macrophages, and how this impacts cell death rates and emigration. We identify a region of LDL-HDL parameter space where plaque growth stabilises for low LDL and high HDL influxes, due to macrophage emigration and HDL clearance that counterbalances the influx of new cells and cholesterol. We explore how the efferocytic uptake of dead cells and the recruitment of new macrophages affect plaque development for a range of LDL and HDL influxes. Finally, we consider how changes in the LDL-HDL profile can change the course of plaque development. We show that changes towards lower LDL and higher HDL can slow plaque growth and even induce regression. We find that these changes have less effect on larger, more established plaques, and that temporary changes will only slow plaque growth in the short term.

A stochastic lipid structured model for macrophage dynamics in atherosclerotic plaques

We propose to model certain aspects of the dynamics of a macrophage that moves randomly in a one dimensional space in arterial wall tissue and grows by accumulating localized lipid particles, thus reducing its motility. This phenomenon has been observed in the context of atherosclerotic plaque formation. For this purpose, we use a system of stochastic differential equations satisfied by the position and diffusion coefficient of a Brownian particle whose diffusion coefficient is modified at each visit to the origin and with a dumping coefficient. The novelty of the model, with respect to Bénichou et al. (Phys Rev E 85(2):021137, 2012), Meunier et al. (Acta Appl Math 161:107-126, 2019), is to include offloading of lipids through the dumping term. We find explicit necessary and sufficient conditions for macrophage trapping in the locally enriched region.

Steady solution and its stability of a mathematical model of diabetic atherosclerosis

Atherosclerosis is a leading cause of death worldwide. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation. Diabetic patients are twice as likely to have a heart attack or stroke. In this paper, we consider a simplified mathematical model for diabetic atherosclerosis involving LDL, HDL, glucose, insulin, free radicals (ROS), β cells, macrophages and foam cells, which satisfy a system of partial differential equations with a free boundary, the interface between the blood flow and the plaque. We establish the existence of small radially symmetric stationary solutions to the model and study their stability. Our analysis shows that the plague will persist due to hyperglycemia even when LDL and HDL are in normal range, hence confirms that diabetes increase the risk of atherosclerosis.

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.

Symmetry-breaking longitude bifurcations for a free boundary problem modeling small plaques in three dimensions

Atherosclerosis, one of the leading causes of death in USA and worldwide, begins with a lesion in the intima of the arterial wall, allowing LDL to penetrate into the intima where they are oxidized. The immune system considers these oxidized LDL as a dangerous substance and tasks the macrophages to attack them; incapacitated macrophages become foam cells and leads to the formation of a plaque. As the plaque continues to grow, it progressively restricts the blood flow, possibly triggering heart attack or stroke. Because the blood vessels tend to be circular, two-space dimensional cross section model is a good approximation, and the two-space dimensional models are studied in Friedman et al. (J Differ Equ 259(4):1227-1255, 2015) and Zhao and Hu (J Differ Equ 288:250-287, 2021). It is interesting to see whether a true three-space dimensional stationary solution can be developed. We shall establish a three-space dimensional stationary solution for the mathematical model of the initiation and development of atherosclerosis which involves LDL and HDL cholesterols, macrophages and foam cells. The model is a highly nonlinear and coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow. We establish infinite branches of symmetry-breaking stationary solutions which bifurcate from the annular stationary solution in the longitude direction.

A cancer model with nonlocal free boundary dynamics

Cancer cells at the tumor boundary move in the direction of the oxygen gradient, while cancer cells far within the tumor are in a necrotic state. This paper introduces a simple mathematical model that accounts for these facts. The model consists of cancer cells, cytotoxic T cells, and oxygen satisfying a system of partial differential equations. Some of the model parameters represent the effect of anti-cancer drugs. The tumor boundary is a free boundary whose dynamics is determined by the movement of cancer cells at the boundary. The model is simulated for radially symmetric and axially symmetric tumors, and it is shown that the tumor may increase or decrease in size, depending on the "strength" of the drugs. Existence theorems are proved, global in-time in the radially symmetric case, and local in-time for any shape of tumor. In the radially symmetric case, it is proved, under different conditions, that the tumor may shrink monotonically, or expand monotonically.

Data-driven causal model discovery and personalized prediction in Alzheimer's disease

With the explosive growth of biomarker data in Alzheimer’s disease (AD) clinical trials, numerous mathematical models have been developed to characterize disease-relevant biomarker trajectories over time. While some of these models are purely empiric, others are causal, built upon various hypotheses of AD pathophysiology, a complex and incompletely understood area of research. One of the most challenging problems in computational causal modeling is using a purely data-driven approach to derive the model’s parameters and the mathematical model itself, without any prior hypothesis bias. In this paper, we develop an innovative data-driven modeling approach to build and parameterize a causal model to characterize the trajectories of AD biomarkers. This approach integrates causal model learning, population parameterization, parameter sensitivity analysis, and personalized prediction. By applying this integrated approach to a large multicenter database of AD biomarkers, the Alzheimer’s Disease Neuroimaging Initiative, several causal models for different AD stages are revealed. In addition, personalized models for each subject are calibrated and provide accurate predictions of future cognitive status.

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.

Macrophage polarization as a potential therapeutic target for atherosclerosis: a dynamic stochastic modelling study

We proposed a dynamic stochastic mathematical model to evaluate the role of macrophage polarization in plaque development. The dynamic process of macrophages from proliferation to death was simulated under different lipid microenvironments. The probability of macrophage phenotypic switching was described using a Bernoulli distribution where the stochastic variable was determined by the local lipid level. Moreover, the interactions between macrophages and microenvironmental factors vary with macrophage phenotype. We investigated the distribution of key microenvironmental factors, the dynamics of macrophage polarization and its influence on foam cell formation. M1 macrophages were found to predominate in advanced plaque corresponding to the exacerbated inflammation observed in mice experiments. The imbalance between the deposition of oxidized low-density lipoprotein and phagocytic effects of macrophages governed the formation of foam cells. Furthermore, we simulated targeted therapies by either directly inhibiting the polarization probability to M1 macrophages or indirectly regulating macrophage polarization due to high-density lipoprotein levels. Comparison of simulation results with experimental findings in both therapies indicated that the intervention and regulation of macrophage polarization could influence plaque microenvironment and subsequently induce plaque regression, especially in the early stage. The proposed modelling system can facilitate the evaluation of novel therapies targeting macrophage polarization.

WELL-POSEDNESS OF A MATHEMATICAL MODEL OF DIABETIC ATHEROSCLEROSIS WITH ADVANCED GLYCATION END-PRODUCTS

Atherosclerosis is a leading cause of death worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. The pathophysiology of diabetic vascular disease is generally understood. Dyslipidemia with increased levels of atherogenic LDL, hyperglycemia, oxidative stress and increased inflammation are factors that increase the risk and accelerate development of atherosclerosis. In a recent paper [53], we have developed mathematical model that includes the effect of hyperglycemia and insulin resistance on plaque growth. In this paper, we propose a more comprehensive mathematical model for diabetic atherosclerosis which include more variables; in particular it includes the variable for Advanced Glycation End-Products (AGEs)concentration. Hyperglycemia trigger vascular damage by forming AGEs, which are not easily metabolized and may accelerate the progression of vascular disease in diabetic patients. The model is given by a system of partial differential equations with a free boundary. We also establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.

Control of Cholesterol Metabolism Using a Systems Approach

Simple Summary

Cholesterol is the main sterol in mammals that is essential for healthy cell functionining. It plays a key role in metabolic regulation and signaling, it is a precursor molecule of bile acids, oxysterols, and all steroid hormones. It also contributes to the structural makeup of the membranes. Its homeostasis is tightly controlled since it can harm the body if it is allowed to reach abnormal blood concentrations. One of the diseases associated with elevated cholesterol levels being the major cause of morbidities and mortalities worldwide, is atherosclerosis. In this study, we have developed a model of the cholesterol metabolism taking into account local inflammation and oxidative stress. The aim was to investigate the impact of the interplay of those processes and cholesterol metabolism disturbances on the atherosclerosis development and progression. We have also analyzed the effect of combining different classes of drugs targeting selected components of cholesterol metabolism.

Abstract

Cholesterol is an essential component of mammalian cells and is involved in many fundamental physiological processes; hence, its homeostasis in the body is tightly controlled, and any disturbance has serious consequences. Disruption of the cellular metabolism of cholesterol, accompanied by inflammation and oxidative stress, promotes the formation of atherosclerotic plaques and, consequently, is one of the leading causes of death in the Western world. Therefore, new drugs to regulate disturbed cholesterol metabolism are used and developed, which help to control cholesterol homeostasis but still do not entirely cure atherosclerosis. In this study, a Petri net-based model of human cholesterol metabolism affected by a local inflammation and oxidative stress, has been created and analyzed. The use of knockout of selected pathways allowed us to observe and study the effect of various combinations of commonly used drugs on atherosclerosis. The analysis results led to the conclusion that combination therapy, targeting multiple pathways, may be a fundamental concept in the development of more effective strategies for the treatment and prevention of atherosclerosis.

WELL-POSEDNESS OF A MATHEMATICAL MODEL OF DIABETIC ATHEROSCLEROSIS

Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 21 million Americans have diabetes, a disease where patients' cells cannot efficiently take in dietary sugar, causing it to build up in the blood. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effect of diabetes on plaque growth. In this paper, we propose a mathematical model for diabetic atherosclerosis; the model is given by a system of partial differential equations with a free boundary. We establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.

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 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.

Disturbance of Mitochondrial Dynamics and Mitochondrial Therapies in Atherosclerosis

Mitochondrial dysfunction is associated with a wide range of chronic human disorders, including atherosclerosis and diabetes mellitus. Mitochondria are dynamic organelles that undergo constant turnover in living cells. Through the processes of mitochondrial fission and fusion, a functional population of mitochondria is maintained, that responds to the energy needs of the cell. Damaged or excessive mitochondria are degraded by mitophagy, a specialized type of autophagy. These processes are orchestrated by a number of proteins and genes, and are tightly regulated. When one or several of these processes are affected, it can lead to the accumulation of dysfunctional mitochondria, deficient energy production, increased oxidative stress and cell death—features that are described in many human disorders. While severe mitochondrial dysfunction is known to cause specific and mitochondrial disorders in humans, progressing damage of the mitochondria is also observed in a wide range of other chronic diseases, including cancer and atherosclerosis, and appears to play an important role in disease development. Therefore, correction of mitochondrial dynamics can help in developing new therapies for the treatment of these conditions. In this review, we summarize the recent knowledge on the processes of mitochondrial turnover and the proteins and genes involved in it. We provide a list of known mutations that affect mitochondrial function, and discuss the emerging therapeutic approaches.

Mathematical model of atherosclerotic aneurysm

Atherosclerosis is a major cause of abdominal aortic aneurysm (AAA) and up to 80% of AAA patients have atherosclerosis. Therefore it is critical to understand the relationship and interactions between atherosclerosis and AAA to treat atherosclerotic aneurysm patients more effectively. In this paper, we develop a mathematical model to mimic the progression of atherosclerotic aneurysms by including both the multi-layer structured arterial wall and the pathophysiology of atherosclerotic aneurysms. The model is given by a system of partial differential equations with free boundaries. Our results reveal a 2D biomarker, the cholesterol ratio and DDR1 level, assessing the risk of atherosclerotic aneurysms. The efficacy of different treatment plans is also explored via our model and suggests that the dosage of anti-cholesterol drugs is significant to slow down the progression of atherosclerotic aneurysms while the additional anti-DDR1 injection can further reduce the risk.

Controlling the thickness of the atherosclerotic plaque by statin medication

Atherosclerosis, a chronic inflammatory disorder of the arterial wall, is a complex process whose dynamics are affected by multiple factors. The disease control consists of restraining it by administering statins. Slowing down or halting the plaque growth depends on the patient age at which the statin treatment begins and on the thickness of the intima-media (IMT) at that time. In this paper, we propose a mathematical model to estimate the sets of atherosclerosis states, from which the use of statins can restrain the disease. Our model is control-theoretic, and the estimated sets are the viability kernels, in the parlance of viability theory. To our best knowledge, this way of modelling the atherosclerosis progression is original. We compute two viability kernels, each for a different statin-treatment dose. Each kernel is composed of the vector [age, IMT] from which the disease can be restrained. By extension, the disease can’t be restrained from the kernel complements, this being mainly because of the disease and patient-age advancement. The kernels visualise tradeoffs between early and late treatments, which helps the clinician to decide when to start the statin treatment and which statin dose may be sufficient.

Bifurcation analysis of a free boundary model of the atherosclerotic plaque formation associated with the cholesterol ratio

The low-density lipoprotein (LDL)/high-density lipoprotein (HDL)-cholesterol ratio has been shown to have a high correlation with the cardiovascular risk assessment. Is it possible to quantify the correlation mathematically? In this paper, we develop a bifurcation analysis for a mathematical model of the plaque formation with a free boundary in the early stage of atherosclerosis. This bifurcation analysis, to the ratio of LDL/HDL, is based on explicit formulations of radially symmetric steady-state solutions. By performing the perturbation analysis to these solutions, we establish the existence of bifurcation branches and derive a theoretical condition that a bifurcation occurs for different modes. We also analyze the stability of radially symmetric steady-state solutions and conduct numerical simulations to verify all theoretical results.

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.

Analysis of a mathematical model of rheumatoid arthritis

Rheumatoid arthritis is an autoimmune disease characterized by inflammation in the synovial fluid within the synovial joint connecting two contiguous bony surfaces. The inflammation diffuses into the cartilage adjacent to each of the bony surfaces, resulting in their gradual destruction. The interface between the cartilage and the synovial fluid is an evolving free boundary. In this paper we consider a two-phase free boundary problem based on a simplified model of rheumatoid arthritis. We prove global existence and uniqueness of a solution, and derive properties of the free boundary. In particular it is proved that the free boundary increases in time, and the cartilage shrinks to zero as [Formula: see text], even under treatment by a drug. It is also shown in the reduced one-phased problem, with cartilage alone, that a larger prescribed inflammation function leads to a faster destruction of the cartilage.

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.

A lipid-structured model for macrophage populations in atherosclerotic plaques

Atherosclerosis is a chronic inflammatory disease driven by the accumulation of pro-inflammatory, lipid-loaded macrophages at sites inside artery walls. These accumulations lead to the development of atherosclerotic plaques. The rupture of plaques that contain lipid-rich necrotic cores can trigger heart attacks and strokes via occlusion of blood vessels. We construct and analyse a system of partial integro-differential equations that model lipid accumulation by macrophages, the generation of apoptotic cells and the formation of the necrotic core. The model accounts for the following cell behaviours: monocyte recruitment into the plaque and differentiation into macrophages; macrophage ingestion of low density lipoproteins (LDL) and of apoptotic cells and necrotic material; lipid offloading to high density lipoproteins (HDL); macrophage emigration; and apoptosis of macrophages and necrosis of apoptotic cells. With this model, we study how changes in parameters predict the characteristic features of plaque pathology. In particular, we find the qualitative form of lipid distribution across the macrophage population and show that high lipid loads can occur in the absence of LDL ingestion. We also demonstrate the importance of macrophage emigration in mitigating and resolving inflammation and plaque lipid accumulation.

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.

A Control-Theoretic Model of Atherosclerosis

We propose a control-theoretic aggregate model of the progression of atherosclerosis plaque, a chronic inflammatory disease of the arterial wall, to study the basic features of this disease. In the model, we exploit the role of inflammation in the disease progression, and use statins-drugs commonly recommended in atherosclerosis-to control this progression. We use a logistic function to allow for constrained growth of plaque. In the model, both the patient's age and overall health impact the plaque growth and its sensitivity to statins. The model parameters are estimated using original data, or calibrated using published research as well as our own clinical and laboratory studies. We contend that our model helps to gauge the statins' impact on a patient's plaque thickness, hence the disease's progression and cardiovascular risk, without requiring artery scans.

Regulation and role of endoglin in cholesterol-induced endothelial and vascular dysfunction in vivo and in vitro

Our objective was to investigate the effect of cholesterol [hypercholesterolemia and 7-ketocholesterol (7K)] on endoglin (Eng) expression and regulation with respect to endothelial or vascular dysfunction in vivo and in vitro. In vivo experiments were performed in 2-mo-old atherosclerosis-prone apolipoprotein E-deficient/LDL receptor-deficient (ApoE^-/-/LDLR^-/-) female mice and their wild-type C57BL/6J littermates. In in vitro experiments, human aortic endothelial cells (HAECs) were treated with 7K. ApoE^-/-/LDLR^-/- mice developed hypercholesterolemia accompanied by increased circulating levels of P-selectin and Eng and a disruption of NO metabolism. Functional analysis of the aorta demonstrated impaired vascular reactivity, and Western blot analysis revealed down-regulation of membrane Eng/Smad2/3/eNOS signaling in ApoE^-/-/LDLR^-/- mice. 7K increased Eng expression via Krüppel-like factor 6 (KLF6), liver X nuclear receptor, and NF-κB in HAECs. 7K-induced Eng expression was prevented by the treatment with 2-hydroxypropyl-β-cyclodextrin; 8-{[5-chloro-2-(4-methylpiperazin-1-yl) pyridine-4-carbonyl] amino}-1-(4-fluorophenyl)-4, 5-dihydrobenzo[g]indazole-3-carboxamide; or by KLF6 silencing. 7K induced increased adhesion and transmigration of monocytic human leukemia promonocytic cell line cells and was prevented by Eng silencing. We concluded that hypercholesterolemia altered Eng expression and signaling, followed by endothelial or vascular dysfunction before formation of atherosclerotic lesions in ApoE^-/-/LDLR^-/- mice. By contrast, 7K increased Eng expression and induced inflammation in HAECs, which was followed by an increased adhesion and transmigration of monocytes via endothelium, which was prevented by Eng inhibition. Thus, we propose a relevant role for Eng in endothelial or vascular dysfunction or inflammation when exposed to cholesterol.-Vicen, M., Vitverova, B., Havelek, R., Blazickova, K., Machacek, M., Rathouska, J., Najmanová, I., Dolezelova, E., Prasnicka, A., Sternak, M., Bernabeu, C., Nachtigal, P. Regulation and role of endoglin in cholesterol-induced endothelial and vascular dysfunction in vivo and in vitro.

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.

Coupled Modeling of Lipid Deposition, Inflammatory Response and Intraplaque Angiogenesis in Atherosclerotic Plaque

We propose a multiphysical mathematical model by fully coupling lipid deposition, monocytes/macrophages recruitment and angiogenesis to investigate the pathophysiological responses of an atherosclerotic plaque to the dynamic changes in the microenvironment. The time evolutions of cellular (endothelial cells, macrophages, smooth muscle cells, etc.) and acellular components (low density lipoprotein, proinflammatory cytokines, extravascular plasma concentration, etc.) within the plaque microenvironment are assessed quantitatively. The thickening of the intima, the distributions of the lipid and inflammatory factors, and the intraplaque hemorrhage show a qualitative consistency with the MRI and histology data. Models with and without angiogenesis are compared to demonstrate the important role of neovasculature in the accumulation of blood-borne components in the atherosclerotic lesion by extravasation from the leaky vessel wall, leading to the formation of a lipid core and an inflammatory microenvironment, which eventually promotes plaque destabilization. This model can serve as a theoretical platform for the investigation of the pathological mechanisms of plaque progression and may contribute to the optimal design of atherosclerosis treatment strategies, such as lipid-lowering or anti-angiogenetic therapies.

A two-phase model of early fibrous cap formation in atherosclerosis

Atherosclerotic plaque growth is characterised by chronic, non-resolving inflammation that promotes the 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) are recruited 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 dangerously 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 data from a range of in vitro and in vivo studies, 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 demonstrate that stable cap formation relies primarily on a critical balance between the rates of SMC recruitment to the plaque, chemotactic SMC migration within the plaque and SMC loss by apoptosis or phenotype change. This 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.

Beyond Statins: Emerging Evidence for HDL-Increasing Therapies and Diet in Treating Cardiovascular Disease

Coronary heart disease continues to be the leading cause of death in the United States. Current attempts to treat atherosclerosis and coronary artery disease often involve pharmaceutical and surgical treatments. While these treatments are successful in managing the pain from coronary heart disease, they do little to prevent or stop it. There are a number of clinical strategies that are currently being researched to treat atherosclerosis through HDL-increasing therapies. These clinical studies have shown positive effects through nutritional intervention, exercise, stress reduction, and tobacco and alcohol cessation. These treatment options are explored in greater detail, including their potential to halt and even reverse atherosclerosis. The results from these recent studies and how they relate to the mechanism of reverse cholesterol transport are also critically examined. Reverse cholesterol transport is a multistep process resulting in the net movement of cholesterol from peripheral tissues back to the liver via the plasma. The mechanism of reverse cholesterol transport is also further explored in this review.

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.

A mathematical model of aortic aneurysm formation

Abdominal aortic aneurysm (AAA) is a localized enlargement of the abdominal aorta, such that the diameter exceeds 3 cm. The natural history of AAA is progressive growth leading to rupture, an event that carries up to 90% risk of mortality. Hence there is a need to predict the growth of the diameter of the aorta based on the diameter of a patient’s aneurysm at initial screening and aided by non-invasive biomarkers. IL-6 is overexpressed in AAA and was suggested as a prognostic marker for the risk in AAA. The present paper develops a mathematical model which relates the growth of the abdominal aorta to the serum concentration of IL-6. Given the initial diameter of the aorta and the serum concentration of IL-6, the model predicts the growth of the diameter at subsequent times. Such a prediction can provide guidance to how closely the patient’s abdominal aorta should be monitored. The mathematical model is represented by a system of partial differential equations taking place in the aortic wall, where the media is assumed to have the constituency of an hyperelastic material.

Modeling Granulomas in Response to Infection in the Lung

Alveolar macrophages play a large role in the innate immune response of the lung. However, when these highly immune-regulatory cells are unable to eradicate pathogens, the adaptive immune system, which includes activated macrophages and lymphocytes, particularly T cells, is called upon to control the pathogens. This collection of immune cells surrounds, isolates and quarantines the pathogen, forming a small tissue structure called a granuloma for intracellular pathogens like Mycobacterium tuberculosis (Mtb). In the present work we develop a mathematical model of the dynamics of a granuloma by a system of partial differential equations. The 'strength' of the adaptive immune response to infection in the lung is represented by a parameter α, the flux rate by which T cells and M1 macrophages that immigrated from the lymph nodes enter into the granuloma through its boundary. The parameter α is negatively correlated with the 'switching time', namely, the time it takes for the number of M1 type macrophages to surpass the number of infected, M2 type alveolar macrophages. Simulations of the model show that as α increases the radius of the granuloma and bacterial load in the granuloma both decrease. The model is used to determine the efficacy of potential host-directed therapies in terms of the parameter α, suggesting that, with fixed dosing level, an infected individual with a stronger immune response will receive greater benefits in terms of reducing the bacterial load.

Free boundary problems in biology

In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.