A new view of convective-diffusive transport processes in the arterial intima

In this paper a new theoretical framework is presented for analyzing the filtration and macromolecular convective-diffusive transport processes in the intimal region of an artery wall with widely dispersed macromolecular cellular leakage sites, as proposed in the leaky junction-cell turnover hypothesis of Weinbaum et al. In contrast to existing convection-diffusive models, which assume that the transport is either 1-D, or convection is primarily in a direction normal to the endothelial surface, the present model considers for the first time the nonuniform subendothelial pressure field that arises from the different hydraulic resistances of normal and leaky endothelial clefts and the special role of the internal elastic lamina (IEL) in modulating the horizontal transport of macromolecules after they have passed through the leaky clefts of cells that are either in mitosis or demonstrate IgG labeling. The new theory is able to quantitatively explain the growing body of recent experiments in which an unexpectedly rapid early-time growth of the leakage spot has been observed and the longer time asymptotic behavior in which the leakage spot appears to approach an equilibrium diameter. The new theory also predicts the observed doubling in macromolecular permeability between EBA labeled blue and white areas when the frequency of leakage sites is doubled. This frequency for doubling of permeability, however, is an order of magnitude smaller than predicted by the author's previous model, Tzeghai et al., in which only convection normal to the endothelial surface was considered and the pressure was uniform in the intima. The longer time model predictions are used to explain the time scale for the formation of liposomes in subendothelial tissue matrix in animal feeding experiments where it has been observed that the extracellular lipid concentration rises sharply prior to the entry of monocytes into the intima.     

In vitro study of LDL transport under pressurized (convective) conditions

It is difficult to assess the transport pathways that carry low-density lipoprotein (LDL) into the artery wall in vivo, and there has been no previous in vitro study that has examined transendothelial transport under physiologically relevant pressurized (convective) conditions. Therefore, we measured water, albumin, and LDL fluxes across bovine aortic endothelial cell (BAEC) monolayers in vitro and determined the relative contributions of vesicles, paracellular transport through "breaks" in the tight junction, and "leaky" junctions associated with dying or dividing cells. Our results show that leaky junctions are the dominant pathway for LDL transport (>90%) under convective conditions and that albumin also has a significant component of transport through leaky junctions (44%). Transcellular transport of LDL by receptor-mediated processes makes a minor contribution (<10%) to overall transport under convective conditions.     

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.     

The role of mitosis in LDL transport through cultured endothelial cell monolayers

We (7) have previously shown that leaky junctions associated with dying or dividing cells are the dominant pathway for LDL transport under convective conditions, accounting for >90% of the transport. We (8) have also recently shown that the permeability of bovine aortic endothelial cell monolayers is highly correlated with their rate of apoptosis and that inhibiting apoptosis lowers the permeability of the monolayers to LDL. To explore the role of mitosis in the leaky junction pathway, the microtubule-stabilizing agent paclitaxel was used to alter the rate of mitosis, and LDL flux and water flux (J(v)) were measured. Control monolayers had an average mitosis rate of 0.029%. Treatment with paclitaxel (2.5 μM) for 1.5, 3, 4.5, or 6 h yielded increasing rates of mitosis ranging from 0.099% to 1.03%. The convective permeability of LDL (P(e)) increased up to fivefold, whereas J(v) increased up to threefold, over this range of mitosis rates. We found strong correlations between the mitosis rate and both P(e) and J(v). However, compared with our previous apoptosis study (8), we found that mitosis was only half as effective as apoptosis in increasing P(e). The results led us to conclude that while mitosis-related leaky junctions might play a role in the initial infiltration of LDL into the artery wall, the progression of atherosclerosis might be more closely correlated with apoptosis-related leaky junctions.     

Internal elastic lamina affects the distribution of macromolecules in the arterial wall: a computational study

The internal elastic lamina (IEL), which separates the arterial intima from the media, affects macromolecular transport across the medial layer. In the present study, we have developed a two-dimensional numerical simulation model to resolve the influence of the IEL on convective-diffusive transport of macromolecules in the media. The model considers interstitial flow in the medial layer that has a complex entrance condition because of the presence of leaky fenestral pores in the IEL. The IEL was modeled as an impermeable barrier to both water and solute except for the fenestral pores that were assumed to be uniformly distributed over the IEL. The media were modeled as a heterogeneous medium composed of an array of smooth muscle cells (SMCs) embedded in a continuous porous medium representing the interstitial proteoglycan and collagen fiber matrix. Results for ATP and low-density lipoprotein (LDL) demonstrate a range of interesting features of molecular transport and uptake in the media that are determined by considering the balance among convection, diffusion, and SMC surface reaction. The ATP concentration distribution depends strongly on the IEL pore structure because ATP fluid-phase transport is dominated by diffusion emanating from the fenestral pores. On the other hand, LDL fluid-phase transport is only weakly dependent on the IEL pore structure because convection spreads LDL laterally over very short distances in the media. In addition, we observe that transport of LDL to SMC surfaces is likely to be limited by the fluid phase (surface concentration less than bulk concentration), whereas ATP transport is limited by reaction on the SMC surface (surface concentration equals bulk concentration).     

Epithelial Fluid Transport: Protruding Macromolecules and Space Charges Can Bring about Electro-Osmotic Coupling at the Tight Junctions

The purpose of the present work is to investigate whether the idea of epithelial fluid transport based on electro-osmotic coupling at the level of the leaky tight junction (TJ) can be further supported by a plausible theoretical model. We develop a model for fluid transport across epithelial layers based on electro-osmotic coupling at leaky tight junctions (TJ) possessing protruding macromolecules and fixed electrical charges. The model embodies systems of electro-hydrodynamic equations for the intercellular pathway, namely the Brinkman and the Poisson-Boltzmann differential equations applied to the TJ. We obtain analytical solutions for a system of these two equations, and are able to derive expressions for the fluid velocity profile and the electrostatic potential. We illustrate the model by employing geometrical parameters and experimental data from the corneal endothelium, for which we have previously reported evidence for a central role for electro-osmosis in translayer fluid transport. Our results suggest that electro-osmotic coupling at the TJ can account for fluid transport by the corneal endothelium. We conclude that electro-osmotic coupling at the tight junctions could represent one of the basic mechanisms driving fluid transport across some leaky epithelia, a process that remains unexplained.     

Epithelial fluid transport: protruding macromolecules and space charges can bring about electro-osmotic coupling at the tight junctions

The purpose of the present work is to investigate whether the idea of epithelial fluid transport based on electro-osmotic coupling at the level of the leaky tight junction (TJ) can be further supported by a plausible theoretical model. We develop a model for fluid transport across epithelial layers based on electro-osmotic coupling at leaky tight junctions (TJ) possessing protruding macromolecules and fixed electrical charges. The model embodies systems of electro-hydrodynamic equations for the intercellular pathway, namely the Brinkman and the Poisson-Boltzmann differential equations applied to the TJ. We obtain analytical solutions for a system of these two equations, and are able to derive expressions for the fluid velocity profile and the electrostatic potential. We illustrate the model by employing geometrical parameters and experimental data from the corneal endothelium, for which we have previously reported evidence for a central role for electro-osmosis in translayer fluid transport. Our results suggest that electro-osmotic coupling at the TJ can account for fluid transport by the corneal endothelium. We conclude that electro-osmotic coupling at the tight junctions could represent one of the basic mechanisms driving fluid transport across some leaky epithelia, a process that remains unexplained.     

On the time dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. 2. Long time model for interaction between leakage sites

In Part 1 of this study (Weinbaum et al., 1988) a short time model has been proposed to describe the initial time dependent leakage of macromolecules at short distances (5 microns or less) from the exit of a transient open junction which the authors have hypothesized as a characteristic feature of endothelial cells in the process of turnover (Weinbaum et al., 1985). This open junction pathway has also been proposed (Weinbaum et al., 1988) to be the primary ultrastructural correlate of the 20 nm diameter large pore suggested by Renkin et al. (1977) using the predictions of cylindrical pore theory. The short time model in (Weinbaum et al., 1988), however, has major limitations in that it neglects the interaction between leakage sites, macromolecular entry through other pathways, the finite thickness of the vessel wall and the curvature of the cell perimeter. The longer time model developed herein will attempt to describe each of these features and also present an improved model and analytic solution for the steady state flux and uptake. In the previous steady state model developed by Weinbaum et al. (1985) the effect of the resistance of the transient open junctions and the non-isotropic diffusion in the underlying tissue due to the internal elastic lamina (IEL) were both neglected. New solutions are first presented which describe the effect of these important model refinements on the steady state macromolecular permeability of the major arteries. Time dependent solutions are then presented to predict the transient longer time labeling following the introduction of tracer macromolecules of varying size. These solutions and the corresponding short time solutions in Weinbaum et al. (1988) are the first solutions to our knowledge to describe the difficult time-dependent boundary value problem to determine how the channel exit concentration and flux at a leaky junction vary with time. This is accomplished by casting the boundary value problem in the form of an integral equation for the unknown flux at the cleft exit and then solving this problem using a specially designed numerical technique. The theoretical predictions are used to interpret the behavior of the localized leaks to HRP and albumin that have been reported in Stemerman et al. (1986) and our own recent experiments (Lin et al., 1988).     

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.     

A theory for water and macromolecular transport in the pulmonary artery wall with a detailed comparison to the aorta

The pulmonary artery (PA) wall, which has much higher hydraulic conductivity and albumin void space and approximately one-sixth the normal transmural pressure of systemic arteries (e.g, aorta, carotid arteries), is rarely atherosclerotic, except under pulmonary hypertension. This study constructs a detailed, two-dimensional, wall-structure-based filtration and macromolecular transport model for the PA to investigate differences in prelesion transport processes between the disease-susceptible aorta and the relatively resistant PA. The PA and aorta models are similar in wall structure, but very different in parameter values, many of which have been measured (and therefore modified) since the original aorta model of Huang et al. (23). Both PA and aortic model simulations fit experimental data on transwall LDL concentration profiles and on the growth of isolated endothelial (horseradish peroxidase) tracer spots with circulation time very well. They reveal that lipid entering the aorta attains a much higher intima than media concentration but distributes better between these regions in the PA than aorta and that tracer in both regions contributes to observed tracer spots. Solutions show why both the overall transmural water flow and spot growth rates are similar in these vessels despite very different material transport parameters. Since early lipid accumulation occurs in the subendothelial intima and since (matrix binding) reaction kinetics depend on reactant concentrations, the lower intima lipid concentrations in the PA vs. aorta likely lead to slower accumulation of bound lipid in the PA. These findings may be relevant to understanding the different atherosusceptibilities of these vessels.     

Mass transport in a microchannel enzyme reactor with a porous wall: Hydrodynamic modeling and applications

A two-dimensional flow model, incorporating mass transport, has been developed to simulate a microchannel enzyme reactor with a porous wall. A two-domain approach based on the finite volume method was implemented. Two parameters are defined to characterize the mass transports in the fluid and porous regions: the porous Damkohler number and the fluid Damkohler number. For reactions close to first-order type (enzyme reactor), the concentration results are found to be well correlated by the use of a reaction–convection distance parameter which incorporates the effects of axial distance, substrate consumption and convection. The reactor efficiency reduces with reaction–convection distance parameter because of reduced reaction (or flux) due to the lower concentration. Increased fluid convection improves the efficiency but it is limited by the diffusion in the fluid region. The correlated results can find applications for the design of enzyme reactors with a porous wall.     

Permselectivity of the glomerular capillary wall to macromolecules. I. Theoretical considerations.

The transport of macromolecules across the renal glomerular capillary wall has been described theoretically using flux equations based on (a) restricted transport through small pores, and (b) the Kedem-Katchalsky formulation. The various assumptions and limitations inherent in these two approaches are discussed. To examine the coupling between macromolecular solute transport and the determinants of glomerular filtration rate, these flux equations were combined with mass balance relations which allow for variations in the transmembrane driving forces along a glomerular capillary. It was predicted, using both pore theory and the Kedem-Katchalsky equations, that fractional solute clearance should be strongly dependent on the determinants of glomerular filtration rate when convection and diffusion both contribute to solute transport. When convection becomes the sole mechanism for transcapillary solute transport, however, fractional solute clearance is essentially independent of changes in the determinants of glomerular filtration rate. Consequently, unless diffusion is absent, fractional solute clearances alone are insufficient to characterize the permselective properties of the glomerular capillary wall, since these values may be altered by changes in glomerular pressures and flows as well as changes in the properties of the capillary wall per se.     

Computational analysis of coupled blood-wall arterial LDL transport

The transport of macromolecules, such as low density lipoproteins (LDLs), across the artery wall and their accumulation in the wall is a key step in atherogenesis. Our objective was to model fluid flow within both the lumen and wall of a constricted, axisymmetric tube simulating a stenosed artery, and to then use this flow pattern to study LDL mass transport from the blood to the artery wall. Coupled analysis of lumenal blood flow and transmural fluid flow was achieved through the solution of Brinkman's model, which is an extension of the Navier-Stokes equations for porous media. This coupled approach offers advantages over traditional analyses of this problem, which have used possibly unrealistic boundary conditions at the blood-wall interface; instead, we prescribe a more natural pressure boundary condition at the adventitial vasa vasorum, and allow variations in wall permeability due to the occurrence of plaque. Numerical complications due to the convection dominated mass transport process (low LDL diffusivity) are handled by the streamline upwind/Petrov-Galerkin (SUPG) finite element method. This new fluid-plus-porous-wall method was implemented for conditions typical of LDL transport in a stenosed artery with a 75 percent area reduction (Peclet number=2 x 10(8)). The results show an elevated LDL concentration at the downstream side of the stenosis. For the higher Darcian wall permeability thought to occur in regions containing atheromatous lesions, this leads to an increased transendothelial LDL flux downstream of the stenosis. Increased transmural filtration in such regions, when coupled with a concentration-dependent endothelial permeability to LDL, could be an important contributor to LDL infiltration into the arterial wall. Experimental work is needed to confirm these results.     

Channels in epithelial cell membranes and junctions.

Epithelia may be classified as "tight" or "leaky," depending on whether there is a significant pathway for transepithelial ion permeation via the junctions and bypassing the cells. The resistance of this paracellular channel may depend partly on structures visible in the electron microscope, partly on wall charge. Permeability determinations in the leaky junctions of gallbladder epithelium, using many different organic cations, suggest that the critical barriers barriers to ion permeation are 5--8 A in radius and bind cations by up to four strongly proton-accepting oxygens. The apical cell membrane of tight epithelia contains a Na+-selective channel that is blocked by amiloride and Ca2+, subject to negative feedback control by the Na+ pump in the basolateral membrane, and somehow promoted by aldosterone. To determine the permeabilities of these two channels (the junctional channel of leaky epithelia, and the Na+ channel of tight epithelia) to water and nonelectrolytes remains a major unsolved problem.     

Macromolecular transport in heart valves. I. Studies of rat valves with horseradish peroxidase

The present study aims to experimentally elucidate subtle structural features of the rat valve leaflet and the related nature of macromolecular transport across its endothelium and in its subendothelial space, information necessary to construct a rational theoretical model that can explain observation. After intravenous injection of horseradish peroxidase (HRP), we perfusion-fixed the aortic valve of normal Sprague-Dawley rats and found under light microscopy that HRP leaked through the leaflet's endothelium at very few localized brown spots, rather than uniformly. These spots grew nearly as rapidly with HRP circulation time before euthanasia as aortic spots, particularly when the time axis only included the time the valve was closed. These results suggest that macromolecular transport in heart valves depends not only on the direction normal to, but also parallel to, the endothelial surface and that convection, as well as molecular diffusion, plays an important role in macromolecular transport in heart valves. Transmission electron microscopy of traverse leaflet sections after 4-min HRP circulation showed a very thin ( approximately 150 nm), sparse layer immediately beneath the endothelium where the HRP concentration was much higher than that in the matrix below it. Nievelstein-Post et al.'s (Nievelstein-Post P, Mottino G, Fogelman A, Frank J. Arterioscler Thromb 14: 1151-1161, 1994) ultrarapid freezing/rotary shadow etching of the normal rabbit valve's subendothelial space supports the existence of this very thin, very sparse "valvular subendothelial intima," in analogy to the vascular subendothelial intima.     

Epithelial water transport in a balanced gradient system.

The relationship between epithelial fluid transport, standing osmotic gradients, and standing hydrostatic pressure gradients has been investigated using a perturbation expansion of the governing equations. The assumptions used in the expansion are: (a) the volume of lateral intercellular space per unit volume of epithelium is small; (b) the membrane osmotic permeability is much larger than the solute permeability. We find that the rate of fluid reabsorption is set by the rate of active solute transport across lateral membranes. The fluid that crosses the lateral membranes and enters the intercellular cleft is driven longitudinally by small gradients in hydrostatic pressure. The small hydrostatic pressure in the intercellular space is capable of causing significant transmembrane fluid movement, however, the transmembrane effect is countered by the presence of a small standing osmotic gradient. Longitudinal hydrostatic and osmotic gradients balance such that their combined effect on transmembrane fluid flow is zero, whereas longitudinal flow is driven by the hydrostatic gradient. Because of this balance, standing gradients within intercellular clefts are effectively uncoupled from the rate of fluid reabsorption, which is driven by small, localized osmotic gradients within the cells. Water enters the cells across apical membranes and leaves across the lateral intercellular membranes. Fluid that enters the intercellular clefts can, in principle, exit either the basal end or be secreted from the apical end through tight junctions. Fluid flow through tight junctions is shown to depend on a dimensionless parameter, which scales the resistance to solute flow of the entire cleft relative to that of the junction. Estimates of the value of this parameter suggest that an electrically leaky epithelium may be effectively a tight epithelium in regard to fluid flow.     

Factors influencing arterial wall mass transport

Arterial wall mass transport has particularly attracted attention because it may be implicated in the development of arterial disease, including arteriosclerosis. A short review is presented of the structure of the arterial wall and of studies of mass transport within it. Recent findings confirm that mass transport occurs across the entire arterial wall apparently from the lumen to the adventitial lymphatics. Evidence has emerged of inhomogeneity of the distribution volume for extracellular tracers in different layers of the wall. An attempt is made to interpret results which indicate that distension per se of arteries and increase of medial smooth muscle tone tend to compact the medial interstitium whereas pressure driven convection across the wall tends to expand that tissue. These findings imply a potentially important role of endothelial permeability, smooth muscle tone and luminal pressure in influencing solute transport in the wall and wall transport properties.     

Effects of arterial pressure on endothelial transport of macromolecules

The effects of variations in transmural pressure over a range of 0 to 200 mmHg on transendothelial transport of macromolecules were studied in the canine common carotid artery. The uptake of 125I-albumin per unit artery weight increased with rising pressure. There was no significant difference in albumin permeability per unit luminal surface area between 0 and 100 mmHg, but permeability nearly doubled when pressure was raised to 200 mmHg. The contribution of an increased rate of transendothelial vesicle diffusion, as evaluated from the experimental determination of the ratio of attached-to-free vesicles and theoretical modeling, was found to be negligible. The reduction in transendothelial vesicle diffusion distance due to pressure-induced thinning of the peripheral zone contributes to a 25% increase in permeability. With the use of colloidal Ag and Au of various sizes, vesicle loading of particles with diameters greater than or equal to 15 nm was found to be severely restricted at transmural pressure less than or equal to 100 mmHg, but it was significantly enhanced at 200 mmHg, when particles as large as 25 nm became detectable in endothelial vesicles and subendothelial space. This hypertension-induced increase in macromolecular transport across the endothelium may cause an overloading of the arterial wall with low-density lipoproteins and play a significant role in atherogenesis.     

The Role of the Tight Junction in Paracellular Fluid Transport across Corneal Endothelium. Electro-osmosis as a Driving Force

The mechanism of epithelial fluid transport is controversial and remains unsolved. Experimental difficulties pose obstacles for work on a complex phenomenon in delicate tissues. However, the corneal endothelium is a relatively simple system to which powerful experimental tools can be applied. In recent years our laboratory has developed experimental evidence and theoretical insights that illuminate the mechanism of fluid transport across this leaky epithelium. Our evidence points to fluid being transported via the paracellular route by a mechanism requiring junctional integrity, which we attribute to electro-osmotic coupling at the junctions. Fluid movements can be produced by electrical currents. The direction of the movement can be reversed by current reversal or by changing junctional electrical charges by polylysine. Aquaporin 1 (AQP1) is the only AQP present in these cells, and its deletion in AQP1 null mice significantly affects cell osmotic permeability but not fluid transport, which militates against the presence of sizable water movements across the cell. By contrast, AQP1 null mice cells have reduced regulatory volume decrease (only 60% of control), which suggests a possible involvement of AQP1 in either the function or the expression of volume-sensitive membrane channels/transporters. A mathematical model of corneal endothelium predicts experimental results only when based on paracellular electro-osmosis, and not when transcellular local osmosis is assumed instead. Our experimental findings in corneal endothelium have allowed us to develop a novel paradigm for this preparation that includes: (1) paracellular fluid flow; (2) a crucial role for the junctions; (3) hypotonicity of the primary secretion; (4) an AQP role in regulation and not as a significant water pathway. These elements are remarkably similar to those proposed by the Hill laboratory for leaky epithelia.