Concentration polarization of atherogenic lipids in the arterial system

Nomenclature c, Normalized LDL concentration (C*/C0); C0, incoming (bulk) LDL concentration (gr/cm3); Cw, LDL concentration on the luminal surface (gr/cm3); ,wC time average value of LDL concentration on the luminal surface (gr/cm3); D, diffusion coef-ficient of LDL (cm2/s); Q, blood flow rate (mL/s); 0R, average internal radius of the artery (cm); Re, Reynolds number (002/Run); Sc, Schmidt number (/Dn); t, normalized time (00*/tuR); u, normalized axial velocity (0*/uu); 0u, time a…     

Theoretical study on flow-dependent concentration polarization of low density lipoproteins at the luminal surface of a straight artery

It is suspected that physical and fluid mechanical factors play important roles in the localization of atherosclerotic lesions and intimal hyperplasia in man by affecting the transport of cholesterol in flowing blood to arterial walls. Hence, we have studied theoretically the effects of various physical and fluid mechanical factors such as wall shear rate, diffusivity of low density lipoproteins (LDL), and filtration velocity of water at the vessel wall on surface concentration of LDL at an arterial wall by means of a computer simulation of convective and diffusive transport of LDL in flowing blood to the wall of a straight artery under conditions of a steady flow. It was found that under normal physiologic conditions prevailing in the human arterial system, due to the presence of a filtration flow of water at the vessel wall, flow-dependent concentration polarization (accumulation or depletion) of LDL occurs at a blood/endothelium boundary. The surface concentration of LDL at an arterial wall takes higher values than that in the bulk flow in that vessel, and it is affected by three major factors, that is, wall shear rate, gamma w, filtration velocity of water at the vessel wall, Vw, and the distance from the entrance of the artery, L. It increases with increasing Vw and L, and decreasing gamma w hence the flow rate. Thus, under certain circumstances, the surface concentration of LDL could rise locally to a value which is several times higher than that in the bulk flow, or drop locally to a value even lower than a critical concentration for the maintenance of normal functions and survival of cells forming the vessel wall. These results suggest the possibility that all the vascular phenomena such as the localization of atherosclerotic lesions and intimal hyperplasia, formation of cerebral aneurysms, and adaptive changes of lumen diameter and wall structure of arteries and veins to certain changes in hemodynamic conditions in the circulation are governed by this flow-dependent concentration polarization of LDL which carry cholesterol.     

The effect of the endothelial glycocalyx layer on concentration polarisation of low density lipoprotein in arteries

It has been postulated that a flow-dependent (and hence spatially varying) low density lipoprotein (LDL) concentration polarisation layer forms on the luminal surface of the vascular endothelium. Such a layer has the potential to cause heterogeneity in the distribution of atherosclerotic lesions by spatially modulating the rate of LDL transport into the arterial wall. Theoretical analysis suggests that a transmural water flux which is spatially heterogeneous at the cellular scale can act to enhance LDL concentration polarisation in a shear dependent fashion. However, such an effect is only observed if a relevant Peclet number (i.e. the ratio of LDL convection to LDL diffusion) is of order unity or greater. Based on the diffusivity of LDL in blood plasma, such a Peclet number is found to be far less than unity, implying that the aforementioned enhancement and shear dependence will not occur. However, this conclusion ignores the existence of the endothelial glycocalyx layer (EGL), which may inhibit the diffusion of LDL near the luminal surface of the endothelium, and hence raise any Peclet number associated with the transport of LDL. The present study numerically investigates the effect of the EGL, as well as a heterogeneous transmural water flux, on arterial LDL concentration polarisation. Particular attention is paid to measures of LDL concentration polarisation thought relevant to the rate of transendothelial LDL transport. It is demonstrated that an EGL is unlikely to cause any additional shear dependence of such measures directly, irrespective of whether or not LDL can penetrate into the EGL. However, it is found that such measures depend significantly on the nature of the interaction between LDL and the EGL (parameterised by the height of the EGL, the depth to which LDL penetrates into the EGL, and the diffusivity of LDL in the EGL). Various processes may regulate the interaction of LDL with the EGL, possibly in a flow dependent and hence spatially non-uniform fashion. It is concluded that any such processes may be as important as vascular scale flow features in terms of spatially modulating transendothelial LDL transport via an LDL concentration polarisation mechanism.     

Concentration polarization of macromolecules in canine carotid arteries and its implication for the localization of atherogenesis

To test the hypothesis that concentration polarization of atherogenic lipids may occur in the arterial system and play an important role in the localization of atherogenesis, we measured in vitro the luminal surface concentration of bovine serum albumin (as a tracer macromolecule) in the canine carotid artery by directly taking liquid samples from the luminal surface of the artery. The experimental results show that the luminal surface albumin concentration, c(w), was higher than the bulk concentration, c(0) as predicted by our theory. The relative luminal surface albumin concentration, c(w)/c(0), decreased very sharply at low wall shear rate, G, but gradually approached the value of 1.0 asymptotically as G was increased. The experiment shows that water flux rate across the vessel wall, v(w), has a profound impact on concentration polarization. For instance, at G = 0 and 185 s(-1), when v(w) = 8.9 +/- 1.7 x 10(-6) cm/s, c(w) was 65% and 15% higher than c(0), respectively, meanwhile when v(w) = 4.8 +/- 0.6 x 10(-6)cm/s, c(w) was only 42% and 5% higher than c(0), respectively. The experiment also revealed that concentration polarization occurred in a thin layer close to the luminal surface of the artery. The thickness of this layer was water flux rate-dependent. The higher the water flux rate, the thicker was the layer. The present study therefore confirms that concentration polarization can indeed occur in the arterial system and our theoretical analysis is accurate in predicting this mass transfer phenomenon.     

狭窄血管远心端低密度脂蛋白浓度极化促进动脉粥样硬化形成

低密度脂蛋白（low density lipoprotein,LDL）浓度极化可能是动脉粥样硬化局灶性的重要原因,本文以狭窄血管远心端为研究对象,探讨LDL浓度极化对动脉粥样硬化发生、发展的影响。用数值计算模拟狭窄血管远心端LDL的壁面浓度分布,用激光扫描共聚焦显微镜测定狭窄血管远心端LDL沿z轴的浓度分布;用外科手术方法建立颈总动脉局部狭窄的实验模型,从整体动物水平观察LDL浓度极化对动脉粥样硬化形成的影响。数值计算和激光扫描共聚焦显微镜测定的结果表明,狭窄血管远心端存在显著的LDL浓度极化现象,且LDL壁面浓度与入口液流速度和狭窄程度有关：在相同的速率下,LDL壁面浓度在狭窄度为40％的圆管内最大;在狭窄程度相同的情况下,雷诺数（Re）为250时测得的LDL壁面浓度高于Re为500时测得的壁面浓度。整体动物实验表明,在狭窄血管远心端LDL浓度极化显著的区域形成明显的动脉粥样硬化病变,并且有大量的脂质沉积。以上结果提示,LDL浓度极化可能是导致动脉粥样硬化局灶性的重要因素。     

Comparative study of Newtonian and non-Newtonian simulations of drug transport in a model drug-eluting stent

To elucidate the difference between Newtonian and shear thinning non-Newtonian assumptions of blood in the analysis of DES drug delivery, we numerically simulated the local flow pattern and the concentration distribution of the drug at the lumen-tissue interface for a structurally simplified DES deployed in a curved segment of an artery under pulsatile blood flow conditions. The numerical results showed that when compared with the Newtonian model, the Carreau (shear thinning) model could lead to some differences in the luminal surface drug concentration in certain areas along the outer wall of the curved vessel. In most areas of the vessel, however, there were no significant differences between the 2 models. Particularly, no significant difference between the two models was found in terms of the area-averaged luminal surface drug concentration. Therefore, we believe that the shear thinning property of blood may play little roles in DES drug delivery. Nevertheless, before we draw the conclusion that Newtonian assumption of blood can be used to replace its non-Newtonian one for the numerical simulation of drug transport in the DES implanted coronary artery, other more complex mechanical properties of blood such as its thixotropic behavior should be tested.     

The Effect of a Spatially Heterogeneous Transmural Water Flux on Concentration Polarization of Low Density Lipoprotein in Arteries

Uptake of low density lipoprotein (LDL) by the arterial wall is likely to play a key role in atherogenesis. A particular process that may cause vascular scale heterogeneity in the rate of transendothelial LDL transport is the formation of a flow-dependent LDL concentration polarization layer on the luminal surface of the arterial endothelium. In this study, the effect of a spatially heterogeneous transmural water flux (that traverses the endothelium only via interendothelial cell clefts) on such concentration polarization is investigated numerically. Unlike in previous investigations, realistic intercellular cleft dimensions are used here and several values of LDL diffusivity are considered. Particular attention is paid to the spatially averaged LDL concentration adjacent to different regions of the endothelial surface, as such measures may be relevant to the rate of transendothelial LDL transport. It is demonstrated in principle that a heterogeneous transmural water flux can act to enhance such measures, and cause them to develop a shear dependence (in addition to that caused by vascular scale flow features, affecting the overall degree of LDL concentration polarization). However, it is shown that this enhancement and additional shear dependence are likely to be negligible for a physiologically realistic transmural flux velocity of 0.0439 μm s⁻¹ and an LDL diffusivity (in blood plasma) of 28.67 μm² s⁻¹. Hence, the results imply that vascular scale studies of LDL concentration polarization are justified in ignoring the effect of a spatially heterogeneous transmural water flux.     

The accelerated atherogenesis of venous grafts might be attributed to aggravated concentration polarization of low density lipoproteins: A numerical study

We hypothesize that after implantation the much elevated water filtration rate of venous grafts may cause aggravated concentration polarization of low density lipoproteins (LDLs), in turn lead to the accelerated atherogenesis of the grafts. To verify the hypothesis, we numerically simulated the transport of LDLs in various models of arterial bypasses with different grafts (veins or arteries) and geometrical configurations. The results showed that the venous grafts might endure abnormally high lipid infiltration/accumulation within the vessel wall due to severely elevated luminal surface LDL concentration. When compared to the conventional bypass models, the S-type bypass had the lowest luminal surface LDL concentration along its host artery floor, but the highest degree of risk to develop atherosclerotic lesions in its venous graft. Among the three conventional bypass models, the one with 30° anastomosis had the lowest risk to develop atherosclerosis in the venous graft. In conclusion, when compared with the bypass models with arterial grafts, the venous bypass models had rather high levels of LDL concentration polarization (c_w) in the vein grafts, especially at the early stages of implantation. This might result in high infiltration/accumulation of LDLs within the walls of the venous grafts, leading to a fast genesis/development of atherosclerosis there.     

Effect of non-Newtonian and pulsatile blood flow on mass transport in the human aorta

To investigate the effects of both non-Newtonian behavior and the pulsation of blood flow on the distributions of luminal surface LDL concentration and oxygen flux along the wall of the human aorta, we numerically compared a non-Newtonian model with the Newtonian one under both steady flow and in vivo pulsatile flow conditions using a human aorta model constructed from MRI images. The results showed that under steady flow conditions, although the shear thinning non-Newtonian nature of blood could elevate wall shear stress (WSS) in most regions of the aorta, especially areas with low WSS, it had little effect on luminal surface LDL concentration (c(w)) in most regions of the aorta. Nevertheless, it could significantly enhance c(w) in areas with high luminal surface LDL concentration through the shear dependent diffusivity of LDLs. For oxygen transport, the shear thinning non-Newtonian nature of blood could slightly reduce oxygen flux in most regions of the aorta, but this effect became much more apparent in areas with already low oxygen flux. The pulsation of blood flow could significantly reduce c(w) and enhance oxygen flux in these disturbed places. In most other regions of the aorta, the oxygen flux was also significantly higher than that for the steady flow simulation. In conclusion, the shear shining non-Newtonian nature of blood has little effect on LDL and oxygen transport in most regions of the aorta, but in the atherogenic-prone areas where luminal surface LDL concentration is high and oxygen flux is low, its effect is apparent. Similar is for the effect of pulsatile flow on the transport of LDLs. But, the pulsation of blood flow can apparently affect oxygen flux in the aorta, especially in areas with low oxygen flux.     

Effect of the endothelial glycocalyx layer on arterial LDL transport under normal and high pressure

To quantitatively investigate the role of the endothelial glycocalyx layer (EGL) in protecting the artery from excessive infiltration of atherogenic lipids such as low density lipoproteins (LDLs), a multilayer model with the EGL of an arterial segment was developed to numerically simulate the flow and the transport of LDLs under normal and high pressure. The transport parameters of the layers of the model were obtained from the hydrodynamic theory, the stochastic theory, and from the literature. The results showed that the increase in the thickness of the EGL could lead to a sharp drop in LDL accumulation in the intima. A partial damage to the EGL could compromise its barrier function, hence leading to enhanced infiltration/accumulation of LDLs within the wall of the arterial model. Without the EGL, hypertension could lead to a significantly enhanced LDL transport into the wall of the model. However, the intact EGL could protect the arterial wall from hypertension so that the LDL concentration in the intima layer was almost the same as that under normal pressure conditions. The results also showed that LDL concentration within the arterial wall increased with Φ (the fraction of leaky junctions) on the intima layer. The increase in LDL concentration with Φ was much more dramatic for the model without the EGL. For instance, without the EGL, a Φ of 0.0005 could lead LDL concentration within the arterial wall to be even higher than that predicted for the EGL intact model with a Φ of 0.002. In conclusion, an intact EGL with a sufficient thickness may act as a barrier to LDL infiltration into the arterial wall and has the potential to suppress the hypertension-driven hike of LDL infiltration/accumulation in the arterial wall.     

Flow-dependent concentration polarization of plasma proteins at the luminal surface of a semipermeable membrane

The effect of steady shear flow on concentration polarization of plasma proteins and lipoproteins at the luminal surface of a semipermeable vessel wall was studied experimentally using suspensions of these molecules in a cell culture medium and a semipermeable membrane dialysis tube which served as a model of an implanted vascular graft or an artery. The study was carried out by flowing a cell culture medium containing fetal calf serum or bovine plasma lipoproteins or bovine albumin through a 7.5 mm diameter, 60 mm-long dialysis tube in steady flow under a physiologic mean arterial perfusion pressure of 100 mmHg, and measuring the filtration velocity of water (cell culture medium) at the vessel wall which varied as a consequence of the change in concentration of plasma protein particles at the luminal surface of the semipermeable membrane dialysis tube. It was found that for perfusates containing plasma proteins and/or lipoproteins, filtration velocity of water was the lowest in the absence of flow, and it increased or decreased as the flow rate (hence wall shear rate) increased or decreased from a certain non-zero value, indicating that surface concentration of protein particles varied reversibly as a direct function of flow rate. It was also found that at particle concentrations equivalent to those found in a culture medium containing serum at 5% by volume, plasma lipoproteins which were much smaller in number and lower in concentration but larger in size than albumin, had a much larger effect on the filtration velocity of water than albumin. These findings were very much the same as those previously obtained with a cultured endothelial cell monolayer, strongly suggesting that the flow-dependent variation in filtration velocity of water at a vessel wall results from a physical phenomenon, that is, flow-dependent concentration polarization of low density lipoproteins at the luminal surface of the endothelial cell monolayer.     

Effects of transmural pressure and wall shear stress on LDL accumulation in the arterial wall: a numerical study using a multilayered model

The accumulation of low-density lipoprotein (LDL) is recognized as one of the main contributors in atherogenesis. Mathematical models have been constructed to simulate mass transport in large arteries and the consequent lipid accumulation in the arterial wall. The objective of this study was to investigate the influences of wall shear stress and transmural pressure on LDL accumulation in the arterial wall by a multilayered, coupled lumen-wall model. The model employs the Navier-Stokes equations and Darcy's Law for fluid dynamics, convection-diffusion-reaction equations for mass balance, and Kedem-Katchalsky equations for interfacial coupling. To determine physiologically realistic model parameters, an optimization approach that searches optimal parameters based on experimental data was developed. Two sets of model parameters corresponding to different transmural pressures were found by the optimization approach using experimental data in the literature. Furthermore, a shear-dependent hydraulic conductivity relation reported previously was adopted. The integrated multilayered model was applied to an axisymmetric stenosis simulating an idealized, mildly stenosed coronary artery. The results show that low wall shear stress leads to focal LDL accumulation by weakening the convective clearance effect of transmural flow, whereas high transmural pressure, associated with hypertension, leads to global elevation of LDL concentration in the arterial wall by facilitating the passage of LDL through wall layers.     

Large eddy simulation of LDL surface concentration in a subject specific human aorta

The development of atherosclerosis is correlated to the accumulation of lipids in the arterial wall, which, in turn, may be caused by the build-up of low-density lipoproteins (LDL) on the arterial surface. The goal of this study was to model blood flow within a subject specific human aorta, and to study how the LDL surface concentration changed during a cardiac cycle. With measured velocity profiles as boundary conditions, a scale-resolving technique (large eddy simulation, LES) was used to compute the pulsatile blood flow that was in the transitional regime. The relationship between wall shear stress (WSS) and LDL surface concentration was investigated, and it was found that the accumulation of LDL correlated well with WSS. In general, regions of low WSS corresponded to regions of increased LDL concentration and vice versa. The instantaneous LDL values changed significantly during a cardiac cycle; during systole the surface concentration was low due to increased convective fluid transport, while in diastole there was an increased accumulation of LDL on the surface. Therefore, the near-wall velocity was investigated at four representative locations, and it was concluded that in regions with disturbed flow the LDL concentration had significant temporal changes, indicating that LDL accumulation is sensitive to not only the WSS but also near-wall flow.     

Low-density lipoprotein concentration in the normal left coronary artery tree

Background

The blood flow and transportation of molecules in the cardiovascular system plays a crucial role in the genesis and progression of atherosclerosis. This computational study elucidates the Low Density Lipoprotein (LDL) site concentration in the entire normal human 3D tree of the LCA.

Methods

A 3D geometry model of the normal human LCA tree is constructed. Angiographic data used for geometry construction correspond to end-diastole. The resulted model includes the LMCA, LAD, LCxA and their main branches. The numerical simulation couples the flow equations with the transport equation applying realistic boundary conditions at the wall.

Results

High concentration of LDL values appears at bifurcation opposite to the flow dividers in the proximal regions of the Left Coronary Artery (LCA) tree, where atherosclerosis frequently occurs. The area-averaged normalized luminal surface LDL concentrations over the entire LCA tree are, 1.0348, 1.054 and 1.23, for the low, median and high water infiltration velocities, respectively. For the high, median and low molecular diffusivities, the peak values of the normalized LDL luminal surface concentration at the LMCA bifurcation reach 1.065, 1.080 and 1.205, respectively. LCA tree walls are exposed to a cholesterolemic environment although the applied mass and flow conditions refer to normal human geometry and normal mass-flow conditions.

Conclusion

The relationship between WSS and luminal surface concentration of LDL indicates that LDL is elevated at locations where WSS is low. Concave sides of the LCA tree exhibit higher concentration of LDL than the convex sides. Decreased molecular diffusivity increases the LDL concentration. Increased water infiltration velocity increases the LDL concentration. The regional area of high luminal surface concentration is increased with increasing water infiltration velocity. Regions of high LDL luminal surface concentration do not necessarily co-locate to the sites of lowest WSS. The degree of elevation in luminal surface LDL concentration is mostly affected from the water infiltration velocity at the vessel wall. The paths of the velocities in proximity to the endothelium might be the most important factor for the elevated LDL concentration.     

Study of stress concentration in the walls of the bovine coronary arterial branch

The intramural stress concentration in the arterial wall is studied at the bovine circumflex coronary arterial branch. The material properties, geometry, and strains in the arterial branch are determined from experiments. The stresses are determined using a finite element analysis. The arterial branch is modeled as two interesecting thin cylindrical shells incorporating local variations in the branch geometry, thickness, and material properties. The artery is considered orthotropic and loaded with an incremental pressure of 40 mmHg. The highest intramural stresses are found to be localized at the proximal and distal regions of the ostium and are not significantly affected by the elastic properties. The stresses are 3 to 4 times greater in the branch at the inner surface than in the straight segment. The strains are twice as large at the branch than in the straight segment. We speculate that this stress concentration could injure the artery and make the branch region susceptible to atherosclerosis.     

Computational modeling of coupled blood-wall mass transport of LDL: effects of local wall shear stress

The work herein represents a novel approach for the modeling of low-density lipoprotein (LDL) transport from the artery lumen into the arterial wall, taking into account the effects of local wall shear stress (WSS) on the endothelial cell layer and its pathways of volume and solute flux. We have simulated LDL transport in an axisymmetric representation of a stenosed coronary artery, where the endothelium is represented by a three-pore model that takes into account the contributions of the vesicular pathway, normal junctions, and leaky junctions also employing the local WSS to yield the overall volume and solute flux. The fraction of leaky junctions is calculated as a function of the local WSS based on published experimental data and is used in conjunction with the pore theory to determine the transport properties of this pathway. We have found elevated levels of solute flux at low shear stress regions because of the presence of a larger number of leaky junctions compared with high shear stress regions. Accordingly, we were able to observe high LDL concentrations in the arterial wall in these low shear stress regions despite increased filtration velocity, indicating that the increase in filtration velocity is not sufficient for the convective removal of LDL.     

Effects of transmural pressure on the transport and distribution of low density lipoproteins in the arterial wall

In an attempt to investigate the effects of transmural pressure on LDL transport and distribution across the arterial wall, uptake of labeled LDL has been measured in excised rabbit thoracic aorta, held at in vivo length and pressurized to 70 or 160 mmHg. The transmural distribution of LDL concentration across the wall was determined by examining serial frozen sections cut parallel to the luminal surface at 20 microns intervals from the intima to adventitia. The LDL concentration observed in the first luminal section at 160 mmHg was 20-fold higher than that obtained at 70 mmHg. The LDL concentrations decreased in the subsequent sections of the first half of the media and became similar, in the outer half of the media, to the values observed under normal pressure. These results might provide an account of one of the mechanisms involved in the deleterious effects of hypertension in atherogenesis.     

Numerical simulation of LDL mass transfer in a common carotid artery under pulsatile flows

Symmetrical 30-60% stenosis in carotid artery with a semi-permeable wall under steady/unsteady flows for Newtonian/non-Newtonian fluids is investigated numerically. The results show that the unsteadiness of blood flow, blood pressure rise and LDL component size increase the luminal concentration, LC, of the surface. The maximum LC occurring immediately after the separation point and the non-Newtonian fluid predicts higher LDL accumulation. LC decreased as the recirculation length is increased and reaches maximum at 40% stenosis. This process is used to estimate the time-dependent growth of the arterial wall.     

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

Abstract

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.