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).

On the time-dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. I. Short-time model for cleft exit region.

In this two-part study we shall quantitatively study, using time-dependent models, the hypothesis that transient open junctions associated with widely scattered endothelial cells undergoing mitosis are the structural equivalent for the large pore pathway via which macromolecules the size of albumin or larger cross the vascular endothelium. In an earlier steady-state model [Am. J. Physiol. 248, H945-960 (1985)], the authors demonstrated that such an open-junction pathway could quantitatively account for the regional differences in macromolecular permeability observed in various mammalian arteries in regions of enhanced cell turnover as indicated by 3H-thymidine although these cells were less than 1% of the population and the open junctions occupied less than 10(-5) of the endothelial surface. The time-dependent models described herein have been used to identify a time window and size of probe molecule wherein this hypothesis could be tested experimentally in the larger blood vessels. The first stages of these experiments have now been completed and provide convincing evidence that the junctions of virtually all endothelial cells in the M phase of the cell cycle are leaky to macromolecules (Lin et al., 1988). The statistical frequency of such leakage sites has also been determined. The time-dependent models developed herein contain two important refinements that were not contained in the earlier steady state model. First the finite resistance of the open cleft as a function of molecular size is accounted for by introducing a diffusion coefficient ratio Dj/Dz describing the relative resistance of the open cleft compared to the subendothelial tissue in the direction normal to the endothelial surface. Second the non-isotropy of the vessel wall due to the elastic lamina is considered by introducing a second diffusion coefficient ratio Dx/Dz describing the relative resistance in the lateral as compared to the normal direction. This second ratio can be as large as 100 for the arterial intima, but is of order unity for capillaries. In Part I a short time model is presented to describe the initial labeling of the open cleft and the subendothelial space in the vicinity of the cleft exit following the introduction of a tracer macromolecule. This model is valid for both larger vessels and capillaries since wall thickness and curvature and the interaction between leakage sites does not enter into the model description. In Part II (Wen et al., 1988) a long-time model is developed for larger vessels only which is valid for greater times including steady-state labeling.

A theoretical model to study the effect of convection and leaky junctions on macromolecule transport in artery walls.

A mathematical model is presented herein to determine the effect of convection on macromolecular transport across an artery wall due to transmural or osmotic pressure differences. The model is based on an extension of the leaky junction-cell turnover model of Weinbaum et al. (1985) to take into account a combined transport mechanism of convection and diffusion and also to provide the leaky junctions in the model with a finite resistance, thus allowing the results to be extended to intercellular clefts with a retarding extracellular matrix or to macromolecules whose dimensions are nearly the same as the junctional width. The results from this improved model show that the effect of pressure on transarterial macromolecular transport is important especially for cell turnover rates greater than 1% and that significant changes in the equilibrium balance of the cholesterol carrying LDL molecules in the arterial wall can occur due to a very small fraction of leaky junctions. At very high turnover rates (large fraction of leaky junctions) the effect of convection on macromolecular transport becomes dramatic and explains the very large increases in uptake observed experimentally after artificially inducing extensive endothelial damage.

Effect of cell turnover and leaky junctions on arterial macromolecular transport.

A new quantitative model is presented to explore the changes in vascular permeability that would result if the intercellular clefts around widely scattered endothelial cells were to become leaky to macromolecules in the range of roughly 4-10 nm during normal cell turnover. Although these open junctions occupy less than 10(-5) of the en face area of the endothelial surface, it is shown that the endothelial permeability can increase by 50-100% due to the experimentally observed regional variations in turnover in the larger arteries, whereas in the thinner walled veins and smaller arteries the subendothelial concentration is not significantly elevated. These results provide a very plausible explanation for the observed focal differences in the uptake of 125I-albumin and 131I-fibrinogen in blue and white areas and the nonselectivity of the local enhancement in uptake for these two molecules as a function of molecular size. The model has important implications for the localization of atherogenesis and the importance of endothelial cell turnover on the transport of proteins in vessels of all sizes.

Diffusion of macromolecules across the arterial wall in the presence of multiple endothelial injuries.

In this paper, the two-phase arterial wall model developed by Weinbaum and Caro [2] has been extended to obtain analytic solutions for the steady-state flux, uptake and concentration of macromolecules in the arterial wall due to the presence of periodically dispersed local sites of enhanced permeability. In the endothelial cell layer these sites are believed to be associated with the dying and regeneration of individual cells in the endothelial monolayer. Nir and Pfeffer [9] obtained similar solutions for a single dying cell in an otherwise undamaged endothelial cell layer. However this model requires that multiple cell turnover sites be spaced sufficiently far apart such that no interaction between neighboring sites takes place and hence cannot be applied to closely spaced endothelial injuries which have been observed experimentally in physiological studies. The theoretical predictions of the present model compare very favorably with experimental results for the enhanced uptake found in blue versus white areas reported in morphological studies of the endothelial surface (Bell, et al. [10, 11]).