Mathematical and numerical models for transfer of low-density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow

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.

How does the Nature of an Excipient and an Atheroma Influence Drug-Coated Balloon Therapy?

The advent of drug-eluting stents and drug-coated balloons have significantly improved the clinical outcome of patients with vascular occlusions. However, ischemic vascular disease remains the most common cause of death worldwide. Improving the current treatment modalities demands a better understanding of the processes which govern drug uptake and retention in blood vessels. In this study, we evaluated the influence of urea and butyryl-trihexyl citrate, as excipients, on the efficacy of drug-coated balloon therapy. An integrated approach, utilizing both in-vitro and in-silico methods, was used to quantify the tracking loss, vessel adhesion, drug release, uptake, and distribution associated with the treatment. Moreover, a parametric study was used to evaluate the potential influence of different types of lesions on drug-coated balloon therapy. Despite the significantly higher tracking loss (urea: 35.5% vs. butyryl-trihexyl citrate: 8.13%) observed in the urea-based balloons, the drug uptake was almost two times greater than with its hydrophobic counterpart. Non-calcified lesions were found to delay the transmural propagation of sirolimus while calcification was shown to limit the retentive potential of lesions. Ultimately this study helps to elucidate how different excipients and types of lesions may influence the efficacy of drug-coated balloon therapy.

Numerical study of the effect of hemodynamic variables on LDL concentration through the single layer of the Left Anterior Descending coronary artery (LAD) under the heart pulse

Heart attack is one of the most common causes of death in the world. Coronary artery disease is the most recognized cause of heart attack whose onset and progression have been attributed to low-density lipoprotein (LDL) passing through the wall of the artery. In this paper, hemodynamic variables as well as the concentration of LDL through the coronary porous artery at the Left Anterior Descending coronary artery (LAD), and its first diagonal branch (D1) under the heart motion investigated using computational simulation. The geometry that has been studied in this paper is the first bifurcation of Left Anterior Descending (LAD) that has been placed on a perimeter of hypothetical sphere representative of the heart geometry. Sinusoidal variations of sphere radii, simulated pulsating movement of the heart. Blood has been considered as a Newtonian and incompressible flow with pulsatile flow rate and real physiological profile. The plasma filtration boundary condition used over the walls in order to simulate the concentration of LDL to a one-layer artery wall. Variations in the concentration of LDL on the artery wall and its relation to oscillation on shear stress on the artery wall under different conditions are presented. Moreover, the effects of the pulsating inlet flow and dynamic movement of the artery are explored. The results declared that minimum shear stress and maximum LDL concentration take place at the bifurcation and on the myocardial wall which is in complete agreement with clinical studies. Furthermore, it has been shown that the heart pulse has a slight effect on the average time of concentration (0.1% increase); however, by analyzing all time steps, one could observe that the maximum concentration rises in some time steps; where this increases the possibility of LDL presence and helps them diffuse inside the artery wall.

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.

Effect of transport parameters on atherosclerotic lesion growth: A parameter sensitivity analysis

BACKGROUND

Atherosclerosis is a degenerative disease of the arterial wall. It results in the formation of progressively growing plaque lesions that can harden and narrow their host arteries. Current computational models of the inflammatory process that govern atherosclerosis growth are reliant on a number of parameters that can freely vary and whose precise values are not well known.

METHODS

To identify the significance of variation in such parameters, a parametric sensitivity study had been conducted on the blood density, blood viscosity, plasma viscosity and bulk flow low density lipoprotein (LDL) concentration. Using computational modeling, the significance of variation in these parameters was assessed on the transport of LDL. The simulation was performed via the 2^k factorial experimental design, which was conducted to identify the significance of the select parameters on the intima LDL concentration and endothelial LDL coverage area.

RESULTS

Results identified the blood viscosity and bulk flow LDL concentration are the dominant parameters for the atherosclerotic lesion growth. The coverage of LDL on the arterial wall surface was strongly dependent on the blood viscosity. The significance of these findings was discussed.

CONCLUSION

This statistical study identifies two dominating blood factors, LDL concentration and blood viscosity, and how they influence atherosclerosis which will serves as a guideline for further investigation on the atherosclerosis topic.

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.

Modeling of the early stage of atherosclerosis with emphasis on the regulation of the endothelial permeability

In this paper, we develop a mathematical model for the early stage of atherosclerosis, as a chronic inflammatory disease. It includes also processes that are relevant for the "thickening" of the vessel walls, and prepares a more complete model including also the later stages of atherosclerosis. The model consists of partial differential equations: Navier-Stokes equations modeling blood flow, Biot equations modeling the fluid flow inside the poroelastic vessel wall, and convection/chemotaxis-reaction-diffusion equations modeling transport, signaling and interaction processes initiating inflammation and atherosclerosis. The main innovations of this model are: a) quantifying the endothelial permeability to low-density-lipoproteins (LDL) and to the monocytes as a function of WSS, cytokines and LDL on the endothelial surface; b) transport of monocytes on the endothelial surface, mimicking the monocytes adhesion and rolling; c) the monocytes influx in the lumen, as a function of factor increasing monocytopoiesis; d) coupling between Navier-Stokes system, Biot system and convection/chemotaxis-reaction-diffusion equations. Numerical simulations of a simplified model were performed in an idealized two-dimensional geometry in order to investigate the dynamics of endothelial permeability, and the growth and spread of immune cells populations and their dependence in particular on low-density-lipoprotein and wall-shear stress.

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.

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.

Boundary layer considerations in a multi-layer model for LDL accumulation

Boundary layer effects for Low-Density Lipoprotein (LDL) concentration problems in a multi-layer artery model are analyzed in this work. Both a straight artery and aorta-iliac bifurcation are analyzed. Mass, momentum and species governing equations are based on the porous media theory and solved with the commercial finite-element based code COMSOL Multiphysics. For the straight artery, various inlet velocities, arterial sizes and intramural pressure values are investigated. Results are presented in terms of concentration profiles close to the lumen/endothelium interface and boundary layer thickness. It is shown that the boundary layer is affected by all of the three analyzed parameters. The results in this work will further clarify the concentration polarization effects imposed by the arterial wall.

Low-density lipoprotein accumulation within the right coronary artery walls for physiological and hypertension conditions

We demonstrate how low-density lipoproteins (LDL) are transported and accumulated through walls of the coronary artery. The result of modeling is a map of the LDL concentration on the patient specific vessel. It identifies places at high risk for plague growth. Using the geometry of the same patient we compare the results of two-layer and four-layer models of LDL transport.

Simulation of blood and oxygen distributions in a hepatic lobule with sinusoids obstructed by cancer cells

The liver is one of the common metastatic sites for many cancers. The obstruction of sinusoids by circulating tumor cells changes liver microenvironments and is thus considered a source of hepatic metastases. To date, few studies provide detailed information, either experimentally or theoretically, concerning the changes in blood and oxygen distributions induced by the obstruction of sinusoids. In this study, we utilized a 3D porous medium-vascular tree geometric structure to mimic the hepatic lobule and studied theoretical blood flow and oxygen transport in the lobule. The simulation was validated with data from the literature. Then, the distributions of blood and oxygen in the presence of the obstruction by cancer cells were simulated. The area and degree of the liver damage induced by the obstruction were analyzed by comparing the difference of liver microenvironments between physiological (non-blocked sinusoid) and pathological (fully or partially blocked sinusoid) conditions and the minimum cancer cell sizes causing liver damage for various obstruction positions were obtained. The work presented in this study can be used to predict the degree of liver damage induced by the local ischemia caused by the obstruction of sinusoids and to characterize the relationship between hepatic metastases and liver microenvironments.

Numerical simulation of low-density lipoprotein mass transport in human arterial stenosis - Calculation of the filtration velocity

Accumulation of cholesterol and other atherogenic lipids such as low-density lipoprotein (LDL) in artery wall causes reduction of vessel diameter and artery stenosis. The study of the mass transfer of these large molecules in the wall with considering effective factors on lumen flow and different physiological factors is the subject considered nowadays. In this paper, results of two dimensional and axi-symmetric simulations of three different models of the artery with 60% stenosis under pulsatile blood flow are presented. Filtration velocity of LDL mass transport in the permeable artery wall and shear stress of blood flow are investigated using ADINA software Three different flow models are considered. In the first and second models, the filtration velocity considered as a given parameter and constant in arterial wall boundary, while in third model arterial wall considered as porous wall, the filtration velocity is calculated from pressure difference as an input parameter of the model. The results show that filtration velocity is strongly depend on geometry and it is not constant along the wall, contrary to simplified models. The results of concentration variations in lumen and wall illustrate the increase in near wall LDL concentration or concentration polarization.

Numerical simulation of haemodynamics and low-density lipoprotein transport in the rabbit aorta and their correlation with atherosclerotic plaque thickness

Two mechanisms of shear stress and mass transport have been recognized to play an important role in the development of localized atherosclerosis. However, their relationship and roles in atherogenesis are still obscure. It is necessary to investigate quantitatively the correlation among low-density lipoproteins (LDL) transport, haemodynamic parameters and plaque thickness. We simulated blood flow and LDL transport in rabbit aorta using computational fluid dynamics and evaluated plaque thickness in the aorta of a high-fat-diet rabbit. The numerical results show that regions with high luminal LDL concentration tend to have severely negative haemodynamic environments (HEs). However, for regions with moderately and slightly high luminal LDL concentration, the relationship between LDL concentration and the above haemodynamic indicators is not clear cut. Point-by-point correlation with experimental results indicates that severe atherosclerotic plaque corresponds to high LDL concentration and seriously negative HEs, less severe atherosclerotic plaque is related to either moderately high LDL concentration or moderately negative HEs, and there is almost no atherosclerotic plaque in regions with both low LDL concentration and positive HEs. In conclusion, LDL distribution is closely linked to blood flow transport, and the synergetic effects of luminal surface LDL concentration and wall shear stress-based haemodynamic indicators may determine plaque thickness.

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.

Low density lipoprotein transport through patient-specific thoracic arterial wall

BACKGROUND AND AIMS

The distribution of Low density lipoprotein (LDL) within the arterial wall is helpful in understanding the onset and development of atherosclerosis. The objective of the study is to study the transport and LDL distribution within patient-specific arterial wall using computational analysis under normal and hypertensive conditions.

METHODS

Patient specific model of the thoracic aorta is computationally examined. The arterial wall is treated macroscopically as homogeneous (one layered) porous media of variable thickness. The interfacial lumen-arterial wall (endothelium) coupling is achieved by the Kedem-Katchalsky equation.

RESULTS

High values of LDL are located at areas where WSS values range from 0.4 N/m² to 1.5 N/m² for normal conditions. In this case the Pearson correlation coefficient r between LDL values and WSS is equal to -0.655 denoting a strong negative linear correlation. In the case that hypertension takes place, high LDL values are located at areas where WSS values range from 0.59 N/m² to 1.7 N/m² and the corresponding Pearson correlation coefficient r is equal to -0.808 denoting a very strong negative linear correlation. For the same parabolic intake flow velocity profile, the luminal surface concentration of LDL is 0.2-2.1% higher than that of the bulk flow for the normal pressure and 0.4-3.4% higher than that of the bulk flow for the hypertensive pressure. For normal conditions, the concentration of LDL at the endothelium/media interface is considerably lower (almost 20 times) than the LDL concentration value at lumen/endothelium interface. For hypertensive conditions, the LDL concentration at the endothelium/media interface is only 4.5 times lower than the corresponding luminal (endothelium side) concentration. The lumen/endothelium side locations (mainly the concave parts) of low WSS - high LDL concentration values coincide with those of high wall-side LDL concentration.

CONCLUSIONS

The transport and LDL distribution is affected by elevated transmural pressure which causes higher LDL concentration. Thus, hypertensive conditions theoretically enhance atherosclerosis.

THE MECHANICAL FACTORS INFLUENCING THE ASSESSMENT OF INTERMEDIATE STENOSIS SEVERITY EXPLAINED THROUGH FRACTIONAL FLOW RESERVE

Assessment of intermediate coronary lesions with diameter stenosis of 40% to 70% severity is being a challenge for cardiologist to identify potentially ischemic stenosis for revascularization and nonculprit stenosis which can be deferred from stenting. An invasive coronary angiography and intravascular ultrasound provide anatomic information of stenosis severity whereas an invasive fractional flow reserve index (FFR) provides the functional significance of the stenosis severity. The measurement of functional significance of stenosis severity minimizes the procedural complications such as coronary dissection, in stent restenosis etc. rather than anatomical significance measure. The FFR cutoff value of [Formula: see text]0.8 is used to distinguish ischemic and nonischemic stenosis. The FFR is clinically well validated even though it is influenced by the mechanical factors such as hyperemic flow and guide wire insertion. In recent times, noninvasive coronary computed tomography (CCTA) modality has become popular in the diagnosis of coronary artery disease. The CCTA permits the assessment of cross-sectional parameters such as minimum lumen area and lumen diameter, lesion length and plaque morphology. However, the CCTA provides limited information on the functional significance of stenotic lesions as compared to FFR. The purpose of this review is to discuss the mechanical factors influencing the invasive FFR while assessing the functional significance of intermediate stenosis severity. In addition, the hidden mechanical factors influencing the noninvasive CCTA assessment of stenosis severity will be discussed from the critical information obtained from FFR which could be beneficial for the clinician particularly in the assessment of intermediate stenosis severity.

Low-density lipoprotein transport through an arterial wall under hypertension - A model with time and pressure dependent fraction of leaky junction consistent with experiments

The influence of hypertension on low-density lipoproteins intake into the arterial wall is an important factor for understanding mechanisms of atherosclerosis. It has been experimentally observed that the increased pressure leads to the higher level of the LDL inside the wall. In this paper we attempt to construct a model of the LDL transport which reproduces quantitatively experimental outcomes. We supplement the well-known four-layer arterial wall model to include two pressure induced effects: the compression of the intima tissue and the increase of the fraction of leaky junctions. We demonstrate that such model can reach the very good agreement with experimental data.

Analysis of non-Newtonian effects on Low-Density Lipoprotein accumulation in an artery

In this work, non-Newtonian effects on Low-Density Lipoprotein (LDL) transport across an artery are analyzed with a multi-layer model. Four rheological models (Carreau, Carreau-Yasuda, power-law and Newtonian) are used for the blood flow through the lumen. For the non-Newtonian cases, the arterial wall is modeled with a generalized momentum equation. Convection-diffusion equation is used for the LDL transport through the lumen, while Staverman-Kedem-Katchalsky, combined with porous media equations, are used for the LDL transport through the wall. Results are presented in terms of filtration velocity, Wall Shear Stresses (WSS) and concentration profiles. It is shown that non-Newtonian effects on mass transport are negligible for a healthy intramural pressure value. Non-Newtonian effects increase slightly with intramural pressure, but Newtonian assumption can still be considered reliable. Effects of arterial size are also analyzed, showing that Newtonian assumption can be considered valid for both medium and large arteries, in predicting LDL deposition. Finally, non-Newtonian effects are also analyzed for an aorta-common iliac bifurcation, showing that Newtonian assumption is valid for mass transport at low Reynolds numbers. At a high Reynolds number, it has been shown that a non-Newtonian fluid model can have more impact due to the presence of flow recirculation.

Relative Effects of Fluid Oscillations and Nutrient Transport in the In Vitro Growth of Valvular Tissues

Engineered valvular tissues are cultured dynamically, and involve specimen movement. We previously demonstrated that oscillatory shear stresses (OSS) under combined steady flow and specimen cyclic flexure (flex-flow) promote tissue formation. However, localized efficiency of specimen mass transport is also important in the context of cell viability within the growing tissues. Here, we investigated the delivery of two essential species for cell survival, glucose and oxygen, to 3-dimensional (3D) engineered valvular tissues. We applied a convective-diffusive model to characterize glucose and oxygen mass transport with and without valve-like specimen flexural movement. We found the mass transport effects for glucose and oxygen to be negligible for scaffold porosities typically present during in vitro experiments and non-essential unless the porosity was unusually low (<40%). For more typical scaffold porosities (75%) however, we found negligible variation in the specimen mass fraction of glucose and oxygen in both non-moving and moving constructs (p > 0.05). Based on this result, we conducted an experiment using bone marrow stem cell (BMSC)-seeded scaffolds under Pulsatile flow-alone states to permit OSS without any specimen movement. BMSC-seeded specimen collagen from the pulsatile flow and flex-flow environments were subsequently found to be comparable (p > 0.05) and exhibited some gene expression similarities. We conclude that a critical magnitude of fluid-induced, OSS created by either pulsatile flow or flex-flow conditions, particularly when the oscillations are physiologically-relevant, is the direct, principal stimulus that promotes engineered valvular tissues and its phenotype, whereas mass transport benefits derived from specimen movement are minimal.

Integrated Stent Models Based on Dimension Reduction: Review and Future Perspectives

Stent modeling represents a challenging task from both the theoretical and numerical viewpoints, due to its multi-physics nature and to the complex geometrical configuration of these devices. In this light, dimensional model reduction enables a comprehensive geometrical and physical description of stenting at affordable computational costs. In this work, we aim at reviewing dimensional model reduction of stent mechanics and drug release. Firstly, we address model reduction techniques for the description of stent mechanics, aiming to illustrate how a three-dimensional stent model can be transformed into a collection of interconnected one-dimensional rods, called a "stent net". Secondly, we review available model reduction methods similarly applied to drug release, in which the "stent net" concept is adopted for modeling of drug elution. As a result, drug eluting stents are described as a distribution of concentrated drug release sources located on a graph that fully represents the stent geometry. Lastly, new results about the extension of these model reduction approaches to biodegradable stents are also discussed.

Low-density lipoprotein accumulation within a carotid artery with multilayer elastic porous wall: fluid-structure interaction and non-Newtonian considerations

Low-density lipoprotein (LDL), which is recognized as bad cholesterol, typically has been regarded as a main cause of atherosclerosis. LDL infiltration across arterial wall and subsequent formation of Ox-LDL could lead to atherogenesis. In the present study, combined effects of non-Newtonian fluid behavior and fluid-structure interaction (FSI) on LDL mass transfer inside an artery and through its multilayer arterial wall are examined numerically. Navier-Stokes equations for the blood flow inside the lumen and modified Darcy's model for the power-law fluid through the porous arterial wall are coupled with the equations of mass transfer to describe LDL distributions in various segments of the artery. In addition, the arterial wall is considered as a heterogeneous permeable elastic medium. Thus, elastodynamics equation is invoked to examine effects of different wall elasticity on LDL distribution in the artery. Findings suggest that non-Newtonian behavior of filtrated plasma within the wall enhances LDL accumulation meaningfully. Moreover, results demonstrate that at high blood pressure and due to the wall elasticity, endothelium pores expand, which cause significant variations on endothelium physiological properties in a way that lead to higher LDL accumulation. Additionally, results describe that under hypertension, by increasing angular strain, endothelial junctions especially at leaky sites expand more dramatic for the high elastic model, which in turn causes higher LDL accumulation across the intima layer and elevates atherogenesis risk.

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.

Mass transport of low density lipoprotein in reconstructed hemodynamic environments of human carotid arteries: the role of volume and solute flux through the endothelium

The accumulation of low density lipoprotein (LDL) in the arterial intima is a critical step in the initiation and progression of atheromatous lesions. In this study we examine subject-specific LDL transport into the intima of carotid bifurcations in three human subjects using a three-pore model for LDL mass transfer. Subject-specific carotid artery computational models were derived using magnetic resonance imaging (MRI) to obtain the geometry and phase-contract MRI (PC-MRI) to acquire pulsatile inflow and outflow boundary conditions for each subject. The subjects were selected to represent a wide range of anatomical configurations and different stages of atherosclerotic development from mild to moderate intimal thickening. A fluid-solid interaction (FSI) model was implemented in the computational fluid dynamics (CFD) approach in order to consider the effects of a compliant vessel on wall shear stress (WSS). The WSS-dependent response of the endothelium to LDL mass transfer was modeled by multiple pathways to include the contributions of leaky junctions, normal junctions, and transcytosis to LDL solute and plasma volume flux from the lumen into the intima. Time averaged WSS (TAWSS) over the cardiac cycle was computed to represent the spatial WSS distribution, and wall thickness (WTH) was determined from black blood MRI (BBMRI) so as to visualize intimal thickening patterns in the bifurcations. The regions which are exposed to low TAWSS correspond to elevated WTH and higher mass and volume flux via the leaky junctions. In all subjects, the maximum LDL solute flux was observed to be immediately downstream of the stenosis, supporting observations that existing atherosclerotic lesions tend to progress in the downstream direction of the stenosis.

A combined numerical and experimental framework for determining permeability properties of the arterial media

The medial layer of the arterial wall may play an important role in the regulation of water and solute transport across the wall. In particular, a high medial resistance to transport could cause accumulation of lipid-carrying molecules in the inner wall. In this study, the water transport properties of medial tissue were characterised in a numerical model, utilising experimentally obtained data for the medial microstructure and the relative permeability of different constituents. For the model, a new solver for flow in porous materials, based on a high-order splitting scheme, was implemented in the spectral/hp element library nektar++ and validated. The data were obtained by immersing excised aortic bifurcations in a solution of fluorescent protein tracer and subsequently imaging them with a confocal microscope. Cuboidal regions of interest were selected in which the microstructure and relative permeability of different structures were transformed to a computational mesh. Impermeable objects were treated fictitiously in the numerical scheme. On this cube, a pressure drop was applied in the three coordinate directions and the principal components of the permeability tensor were determined. The reconstructed images demonstrated the arrangement of elastic lamellae and interspersed smooth muscle cells in rat aortic media; the distribution and alignment of the smooth muscle cells varied spatially within the extracellular matrix. The numerical simulations highlighted that the heterogeneity of the medial structure is important in determining local water transport properties of the tissue, resulting in regional and directional variation of the permeability tensor. A major factor in this variation is the alignment and density of smooth muscle cells in the media, particularly adjacent to the adventitial layer.

Modelling and simulation of low-density lipoprotein transport through multi-layered wall of an anatomically realistic carotid artery bifurcation

A high concentration of low-density lipoprotein (LDL) is recognized as one of the principal risk factors for development of atherosclerosis. This paper reports on modelling and simulations of the coupled mass (LDL concentration) and momentum transport through the arterial lumen and the multi-layered arterial wall of an anatomically realistic carotid bifurcation. The mathematical model includes equations for conservation of mass, momentum and concentration, which take into account a porous layer structure, the biological membranes and reactive source/sink terms in different layers of the arterial wall, as proposed in Yang & Vafai (2006). A four-layer wall model of an arterial wall with constant thickness is introduced and initially tested on a simple cylinder geometry where realistic layer properties are specified. Comparative assessment with previously published results demonstrated proper implementation of the mathematical model. Excellent agreement for the velocity and LDL concentration distributions in the arterial lumen and in the artery wall are obtained. Then, an anatomically realistic carotid artery bifurcation is studied. This is the main novelty of the presented research. We find a strong dependence between underlying blood flow pattern (and consequently the wall shear stress distributions) and the uptake of the LDL concentration in the artery wall. The radial dependency of interactions between the diffusion, convection and chemical reactions within the multi-layered artery wall is crucial for accurate predictions of the LDL concentration in the media. It is shown that a four-layer wall model produced qualitatively good agreement with the experimental results of Meyer et al. (1996) in predicting levels of LDL within the media of a rabbit aorta under identical transmural pressure conditions. Finally, it is demonstrated that the adopted model represents a good initial platform for future numerical investigations of the initial stage of atherosclerosis for patient-specific geometries.

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.

Improving smooth muscle cell exposure to drugs from drug-eluting stents at early time points: a variable compression approach

The emergence of drug-eluting stents (DES) as a viable replacement for bare metal stenting has led to a significant decrease in the incidence of clinical restenosis. This is due to the transport of anti-restenotic drugs from within the polymer coating of a DES into the artery wall which arrests the cell cycle before restenosis can occur. The efficacy of DES is still under close scrutiny in the medical field as many issues regarding the effectiveness of DES drug transport in vivo still exist. One such issue, that has received less attention, is the limiting effect that stent strut compression has on the transport of drug species in the artery wall. Once the artery wall is compressed, the stents ability to transfer drug species into the arterial wall can be reduced. This leads to a reduction in the spatial therapeutic transfer of drug species to binding sites within the arterial wall. This paper investigates the concept of idealised variable compression as a means of demonstrating how such a stent design approach could improve the spatial delivery of drug species in the arterial wall. The study focused on assessing how the trends in concentration levels changed as a result of artery wall compression. Five idealised stent designs were created with a combination of thick struts that provide the necessary compression to restore luminal patency and thin uncompressive struts that improve the transport of drugs therein. By conducting numerical simulations of diffusive mass transport, this study found that the use of uncompressive struts results in a more uniform spatial distribution of drug species in the arterial wall.

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

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

Patient-specific computational modeling of subendothelial LDL accumulation in a stenosed right coronary artery: effect of hemodynamic and biological factors

Atherosclerosis is a systemic disease with local manifestations. Low-density lipoprotein (LDL) accumulation in the subendothelial layer is one of the hallmarks of atherosclerosis onset and ignites plaque development and progression. Blood flow-induced endothelial shear stress (ESS) is causally related to the heterogenic distribution of atherosclerotic lesions and critically affects LDL deposition in the vessel wall. In this work we modeled blood flow and LDL transport in the coronary arterial wall and investigated the influence of several hemodynamic and biological factors that may regulate LDL accumulation. We used a three-dimensional model of a stenosed right coronary artery reconstructed from angiographic and intravascular ultrasound patient data. We also reconstructed a second model after restoring the patency of the stenosed lumen to its nondiseased state to assess the effect of the stenosis on LDL accumulation. Furthermore, we implemented a new model for LDL penetration across the endothelial membrane, assuming that endothelial permeability depends on the local lumen LDL concentration. The results showed that the presence of the stenosis had a dramatic effect on the local ESS distribution and LDL accumulation along the artery, and areas of increased LDL accumulation were observed in the downstream region where flow recirculation and low ESS were present. Of the studied factors influencing LDL accumulation, 1) hypertension, 2) increased endothelial permeability (a surrogate of endothelial dysfunction), and 3) increased serum LDL levels, especially when the new model of variable endothelial permeability was applied, had the largest effects, thereby supporting their role as major cardiovascular risk factors.

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.

Three-dimensional modeling of oxidized-LDL accumulation and HDL mass transport in a coronary artery: a proof-of-concept study for predicting the region of atherosclerotic plaque development

Low density lipoprotein (LDL) has a significant role on the atherosclerotic plaque development, while the concentration of high density lipoproteins (HDL) is considered to play an atheroprotective role according to several biochemical mechanisms. In this work, it is the first time that both LDL and HDL concentrations are taken into account in order to predict the regions prone for plaque development. Our modeling approach is based on the use of a realistic three-dimensional reconstructed pig coronary artery in two time points. Biochemical data measured in the pig were also included in order to develop a more customized model. We modeled coronary blood flow by solving the Navier-Stokes equations in the arterial lumen and plasma filtration in the arterial wall using Darcy's Law. HDL transport was modeled only in the arterial lumen using the convection-diffusion equation, while LDL transport was modeled both in the lumen and the arterial wall. An additional novelty of this work is that we model the oxidation of LDL taking into account the atheroprotective role of HDL. The results of our model were in good agreement with histological findings demonstrating that increased oxidized LDL is found near regions of advanced plaques, while non-oxidized LDL is found in regions of early plaque types.