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Am J Physiol Heart Circ Physiol 287: H905-H913, 2004. First published March 11, 2004; doi:10.1152/ajpheart.00647.2003
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Internal elastic lamina affects the distribution of macromolecules in the arterial wall: a computational study

Shigeru Tada1 and John M. Tarbell2

1Energy Phenomena Laboratory, Mechanical Engineering and Science, Faculty of Engineering, Tokyo Institute of Technology, Ookayama, Tokyo 152-0033, Japan; and 2Department of Biomedical Engineering, The City College of New York, New York, New York 10031

Submitted 10 February 2003 ; accepted in final form 10 March 2004


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS AND DISCUSSION
 GRANTS
 REFERENCES
 
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).

smooth muscle cell; low-density lipoprotein; adenosine 5'-triphosphate; numerical analysis

IT HAS BEEN SUGGESTED that molecular transport to the blood vessel wall and accumulation within the wall play a major role in the development of atherosclerosis (21, 24). Many studies have focused on quantifying the arterial wall distribution of macromolecules such as low-density lipoprotein (LDL) and albumin (4, 6, 7, 9, 10, 12, 14, 34). However, a full description of the mechanisms and structures that control molecular transport in arteries remains elusive.

Molecular transport within the artery wall occurs by diffusion driven by concentration gradients across the wall, convection arising from pressure-driven transmural volume flow, and chemical reactions on the surfaces and within the cellular components of the wall. A typical mammalian artery consists of the following three structurally distinct regions: intima, media, and adventitia. The intima is separated from the medial layer by the internal elastic lamina (IEL), which is a fenestrated layer of elastic tissue. The medial layer is viewed as a porous heterogeneous medium consisting of a continuous extracellular matrix phase with embedded smooth muscle cells (SMCs). The adventitia is the outermost layer of the arterial wall typically comprised of loose connective tissue and fibroblasts. Molecules that have gained entry to the intima by crossing the endothelial cell layer pass through the fenestrae in the IEL and enter the media where diffusion and convection transport them toward the adventitia and embedded SMCs that may consume them. Molecules are also transported in the media from the adventitia because of the presence of vasa vasorum.

Many analytical and numerical works have explored the mechanism of transport of macromolecules within the artery wall (11, 15, 17, 18, 22, 25). Huang and Tarbell (15) characterized the transport and reaction processes for ATP and LDL in the media, which they modeled as a heterogeneous material consisting of a continuous interstitial porous media phase and an array of cylindrical SMCs embedded in the interstitial phase. They did not consider the entrance effects associated with the distribution of material in the media through fenestral pores in the IEL. Tada and Tarbell (26) studied the effects of the fenestral pores on the fluid shear stress distribution on the superficial layer of SMC and observed a significant entrance effect (up to 20 times higher shear stress on the first SMC resulting from the IEL), but they did not consider the associated biomolecular transport problem.

In the present study, a mathematical model is developed to describe fenestral pore entrance effects on the convective and diffusive mass transport processes in the media coupled to chemical reactions on SMC surfaces. The punctate solute flux across the IEL resulting from the fenestrations is hypothesized to alter the rates of uptake over the surface of the SMC residing in the vicinity of the IEL where the most active biochemical reactions are likely to occur. To test this hypothesis, we compare our numerical predictions of concentration profiles of both ATP and LDL with those without a complex entrance condition associated with IEL fenestrations that were obtained by Huang and Tarbell (15) and demonstrate distinctive differences.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS AND DISCUSSION
 GRANTS
 REFERENCES
 
Model description.

In the present study, two-dimensional (2D) numerical simulations of interstitial flow and transport in the tunica media were performed to investigate the influence of IEL fenestrations on uptake of biomolecules by SMCs distributed across the media.

A schematic illustration of the arterial media right beneath the subendothelial layer is shown in Fig. 1. The illustration shows a transverse sectional view of an artery wall. The left end in Fig. 1 corresponds to the IEL, and the right end corresponds to the interface between the medial layer and adventitia. The IEL separates the subendothelial intimal layer (not shown) from the medial layer and provides a complex entrance condition associated with the fenestral pores. In the present study, the medial layer is approximated as a heterogeneous medium composed of a periodic array of cylindrical SMCs (axis into the page) embedded in a continuous porous extracellular matrix phase that is modeled as a Darcy medium. Water and solutes are delivered to the medial layer through leaky fenestral pores in the IEL. Fenestral pores are modeled as circular openings distributed in a square array over the IEL. The IEL is assumed to be an impermeable wall except for its pore openings that provide transport pathways for solutes and water. Justification for this assumption is discussed extensively in previous studies (26, 33). SMCs are modeled as circular cylinders impenetrable by water flow because of the low hydraulic conductance of the cell membrane relative to that of interstitium (33). Modeling of the degradation of ATP on the SMC surface and permeation of LDL in SMC by bulk-phase endocytosis are described by Eqs. 10–15. In the present calculation, following Huang and Tarbell (15), these receptor-mediated processes are approximated as effective first-order reactions.



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  		Fig. 1. Schematic illustration of the arterial media underneath the subendothelial layer. Internal elastic lamina (IEL) has a single fenestral pore. The IEL is an impermeable barrier to water and solute flow except for its fenestral openings. Smooth muscle cells (SMCs) residing in the medial layer are modeled as circular cylinders of 4-µm diameter arranged in a square-array configuration. Between SMCs in the medial layer is the interstitial matrix phase containing proteoglycan and collagen fibers. U, flow velocity at the fenestral pore; Uo, superficial velocity; d, pore diameter; D, SMC diameter; B.C., boundary condition.

 
Under the assumptions described above, solutes are transported in the interstitial fluid (treated as a Newtonian fluid of viscosity µ) through an extracellular matrix modeled as a homogeneous porous medium having a Darcy permeability coefficient (Kp) and are depleted by first-order chemical reactions on SMC surfaces.

Mathematical formulation.

The governing equations for fluid flow in the media are Brinkman's (3) equation, a generalization of Darcy's law for flow in a saturated porous medium that allows for satisfaction of no-slip boundary conditions on impenetrable surfaces

(1)
and the equation of continuity

(2)
where {Delta} is the Laplacian, u is the superficial velocity vector, P is the pressure, µ is the viscosity of the fluid, and Kp is the Darcy permeability of the extracellular matrix. The term on the left-hand side of Eq. 1 is the pressure force that drives the flow; the first term on the right-hand side of Eq. 1 represents the viscous force that allows satisfaction of the no-slip condition; and the last term on the RHS (Darcy-Forchheimer term) characterizes viscous forces in the porous medium away from the solid boundaries. On the surfaces of both the IEL and SMCs, a nonslip boundary condition was applied. The velocity profile at the fenestral pore (U in Fig. 1) was assumed to be uniform. This assumption is a simple but reasonable approximation because the flow velocity at the pore has almost a flat distribution except in the immediate vicinity of the pore edge within the Brinkman boundary layer of thickness {delta} [~O(Kp)] (26). The prescription of the flow velocity at the fenestral pore (U) is described in Eq. 15.

Solute transport through the extracellular matrix is described by a convective-diffusion equation

(3)
where Kcf is the lag coefficient for convective transport in the fiber matrix, C is the interstitial macromolecule concentration, and Df is the effective diffusivity of solutes in the fiber matrix. The boundary condition on the surface of an SMC is

(4)
which assumes that the rate of disappearance of solute by surface reaction or cell permeation is a first process with rate constant kr (Eqs. 11 and 14). The surface of the IEL was assumed to be impermeable and nonreactive to solutes except at the fenestral openings. Thus the boundary conditions for solute concentration on the IEL are

(5)
everywhere except at the fenestral openings where

(6)

Model consistency.

Note that flow through the fenestral pores is modeled as 2D flow through a fenestral "plane slit." It is necessary to relate the 2D parameters to the three-dimensional (3D) circular pore parameters to ensure that the volumetric flow rate in 2D slits is equal to that in 3D circular pores. In the 3D model, circular fenestral pores are arranged in a square array, whereas in the idealized 2D model the fenestral "slits" are aligned along the 3D circular pore centers. Defining the 3D fenestral pore diameter and the width of the fenestral slit (2D pore diameter) as d and d2D, respectively, the width of the fenestral slit was specified so that the area per unit length of the 2D slit model is equivalent to that of the 3D circular pore model, namely,

(7)
where l is the distance between neighboring pores. Utilizing the definition of pore area fraction

(8)
d is obtained as a function of d2D and area fraction (f)

(9)
Further details underlying Eq. 7 are described in the paper by Tada and Tarbell (27).

Physiological parameters.

The physiological properties used in the present model were justified in greater detail (15, 26). Briefly, the SMC diameter (D) was taken as 4.0 µm based on data for the rabbit thoracic aorta (2), and the volume fraction of the SMC (F) was held constant at 0.4 (33). Therefore, the distance between neighboring SMCs was L = 5.6 µm. The fenestral pore spacing (l) was taken as a multiple of the distance between neighboring SMCs (L x n; n = 1, 2, 3...) so that the pores were centered on SMCs, as shown in Fig. 2. This alignment was chosen to simulate maximum transport to the first SMC. Once l was specified, d2D was computed from Eq. 8 with f specified; and d was then computed from Eq. 9. For the extracellular fiber matrix, the value of hydraulic permeability, Kp = 1.43 x 10–14 cm2, is consistent with the value used by Wang and Tarbell (33) based on the data of Tedgui and Lever (29) for the rabbit aorta. The superficial velocity of the fully developed interstitial flow, U0 = 5.8 x 10–6 cm/s, was obtained from Darcy's law (15). The lag coefficients in Eq. 3 for ATP and LDL are 1.21 and 2.00, respectively (8).



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  		Fig. 2. Various geometric configurations of fenestral pores and SMCs. When pore spacing (l) is larger, both the number of SMCs between neighboring pores and pore diameter (d) are proportionally increased to keep the value of the area fraction (f) constant. L, SMC spacing. A: 1 L. B: 2 L. C: 3 L. D: 4 L.

 
In modeling solute uptake by SMCs, we chose ATP and LDL as substances representing a broad range of molecular size. The degradation of ATP (hydrolysis of ATP to ADP) on the surface of SMC can be modeled using Michaelis-Menten kinetics with a rate

(10)
where Vmax is the maximum rate, Cs is the surface concentration, and km is the Michaelis constant. When Cs << km, as is often the case, then the reaction rate is pseudo-first order and Eq. 10 reduces to

(11)
where kr = Vmax/km. The effective reaction rate coefficient for ATP was taken as kr = 1.25 x 10–4 cm/s based on experimental data for Vmax and km (13). The effective diffusivity (Df) was set at 2.36 x 10–6 cm2/s (15).

The kinetics of receptor-mediated internalization of LDL by SMC have been described previously (15) and will only be outlined here. On the basis of classical steady-state enzyme-substrate interactions, internalization of the LDL-LDL receptor complex (LR) by SMC is described as

(12)
where k1 is the biomolecular rate constant for the association of LDL (L) and the LDL receptor (R) on the SMC surface, k2 is the first-order rate constant for the dissociation of the surface LDL-LDL receptor, and k3 is the first-order rate constant for the process of LDL-LDL receptor complex internalization. By assuming that the total receptor concentration is conserved, we can obtain the rate of internalization of LDL in the form of a Michaelis-Menten equation

(13)
where [Rt] is the total LDL receptor concentration. Following Truskey et al. (31) we assume that k3 >> k2 and Cs << (k2 + k3)/k1, and Eq. 13 is reduced to a first-order reaction with rate coefficient

(14)
The value of kr is estimated in Ref. 15 using data in Ref. 31 as 2.2 x 10–6 cm/s. The value of the effective diffusivity for LDL was taken as Df = 5.0 x 10–10 cm2/s, as estimated in Ref. 15 based on experimental data (13, 30, 31).

The number of rows of SMC in the media was set to 20 (Fig. 1). Therefore, the thickness of the medial layer calculated from the SMC diameter (4 µm) and volume fraction (F = 0.4) is 112 µm, which is consistent with the observed medial layer thickness of 120~150 µm for rabbit thoracic aorta (2).

Besides the constants mentioned above, the fenestral pore diameter (d), the area fraction of fenestral pores (f), and the distance between the IEL and the upstream end of the most proximal SMC (a) are key parameters characterizing the entrance condition to the media. Experimental data in various animals show f in the range 0.002–0.02 and d in the range 0.4–2.1 µm (34). Therefore d (d2D) was varied in the range 0.4–3.2 µm (0.02–0.18 µm), and the area fraction was set in the range 0.004–0.016. The parameter a was estimated to be 0.36 µm (rabbit thoracic aorta; see Ref. 26). The complete specification of constants and values of key parameters required for numerical calculation are listed in Table 1.


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  		Table 1. Physiological parameters used in the numerical simulation

 
Computational method.

The finite element method software package FIDAP (Ver.8.62 parallel computing version; FLUENT, Lebanon, NH; www.fluent.com) was used for the numerical simulations. The set of governing equations (Eq. 1–Eq. 3) was transformed into algebraic equations using the finite element method. 2D, four-noded square elements were used for the present simulation. The resulting algebraic equations were solved by the direct Gaussian elimination method with a structured boundary-fitted coordinate (BFC) system. The BFC was generated by using a submodule of the FIDAP solver, FI-GEN. Velocity and mass concentration grid points were arranged on each nodal point of the coordinate system, whereas the pressure was solved at the centroid of the finite elements. Numerical simulations were carried out on the parallel supercomputing machine (Silicon Graphics Origin-2000) at the National Center for Supercomputing Applications (NCSA, Champaign, IL). The geometric configuration and the grid system of the present 2D model are shown in Figs. 1 and 3, respectively. Because the fenestral pores and SMCs were assumed to be arranged in spatially periodic square arrays for computational convenience, a symmetry boundary condition was applied on lateral surfaces of the rectangular computational domain, the width of which was one-half the distance between neighboring fenestral pores (l). On the surfaces of solid bodies, i.e., the bottom surface of the IEL and the periphery of SMCs, a no-slip boundary condition was taken. The flow velocity at the fenestral pore (U) was provided by using the relation

(15)
where U0 is the superficial velocity (see Fig. 1).



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  		Fig. 3. Computational grid configuration used in the present calculations. The grid system consists of between 20 and 80 subdomains on the half-space of each SMC with finer mesh resolution at surfaces of both the IEL and SMCs. The number of SMC subdomain elements is changed according to the spacing of neighboring SMCs.

 
At the downstream end, a gradient-free outlet boundary condition was applied for velocities. On the other hand, a bulk concentration of solute of 1.0 was specified on both the inlet (fenestral pore) and outlet flow boundaries because solutes enter the media from both the IEL fenestrae and the vasa vasorum.

Because the flow is dominated by fluid viscosity in the vicinity of solid boundaries, a finer numerical mesh perpendicular to solid surfaces was employed to ensure sufficiently high resolution of the interstitial flow, as shown in Fig. 3, bottom. In the present problem, the Brinkman boundary layer thickness ({delta}), over which the velocity undergoes 99% of its variation, is on the order of 5. The order of magnitude of {delta} is estimated to be 1 x 10–3D (D is the SMC diameter); hence, a minimum mesh size of 1 x 10–4D was taken in the proximity of both the SMC and IEL surfaces. After establishing that the numerical results were independent of mesh density, a computational mesh consisting of about 50,000~200,000 finite elements [60 (radial) x 40 (circumferential) x 20~80 (no. of cell units; see Figs. 2 and 3) + 2,000~10,000 (near the pore inlet for a high resolution at the pore)] was applied. As the convergence criterion for implicit Gaussian elimination iterations, a relative error of 1 x10–8 was applied on the values of velocity for every iteration.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS AND DISCUSSION
GRANTS
 REFERENCES
 
Numerical validation.

To validate the present mass transport simulation code, the Sherwood number, Sh, which represents the dimensionless mass flux at the surface of an SMC

(16)
where KL is the mass transfer coefficient, R is the SMC radius (=D/2), Cb is the bulk concentration that is calculated as the average concentration over the interstitial phase surrounding an individual cell (shaded region in Fig. 3), and Cs is the SMC surface concentration, that was computed for a wide range of Peclet numbers

(17)
Pe is indicative of the relative importance of convective transport to diffusive transport. Sh was computed for fully developed interstitial flow (no IEL and uniform velocity and concentration at x = 0) as in the earlier paper of Huang and Tarbell (15). When Pe << 1, it is expected that diffusion is a dominant mechanism relative to convection. The comparison between the present calculations and the results of Huang and Tarbell (15) is shown in Fig. 4A. Shown is the dependence of Sh on Pe when the Damkohler number (Da; dimensionless reaction rate)

(18)
is 1.0. The results show a favorable agreement between the two studies and provide validation of the present model and methods. In the low Pe range, Sh is independent of Pe, and fluid-phase transport is dominated by diffusion. At higher Pe, Sh increases with Pe as convective transport becomes important. It is shown elsewhere (15) that the value of Da has very little influence on the value of Sh for a fixed value of Pe. That is, the surface boundary condition has a minor influence on the fluid-phase transport processes. It is also shown (15) that, when Sh >> Da, the surface reaction limits consumption, whereas fluid-phase transport is limiting when Sh << Da. For LDL in the present study, Pe = 4.60 and Da = 0.88, and for fully developed interstitial flow (far from the IEL) we expect Sh ~4 (Fig. 4A). Thus, for LDL, convection is important, and fluid-phase transport and surface reaction are both important to the overall consumption rate. On the other hand, for ATP, Pe = 5.96 x 10–4, Da = 0.011, and Sh ~2.8 (Fig. 4A). Therefore, the transport of ATP in the fluid phase is dominated by diffusion, and, because Da << Sh, the surface reaction of ATP controls the overall consumption rate.



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  		Fig. 4. A: computational results showing the variation of the Sherwood number (Sh) as a function of the Peclet number (Pe) for fully developed interstitial flow when the Damkohler number (Da) is 1.0. The results of Huang and Tarbell (17) are plotted for comparison with the results of the present model. B: profiles of low-density lipoprotein (LDL) concentration at transmural pressure differences of 70 and 160 mmHg compared with the experiments of Mayer et al. (20). A concentration-gradient-free boundary condition was applied at the downstream end (x ~100 µm), and the simulation and data were matched at the upstream end (relative concentration = 1).

 
Figure 4B shows the profiles of LDL concentration at transmural pressure differences of 70 and 160 mmHg compared with the experiments of Mayer et al. (20). To compare with experimental data, a concentration-gradient-free boundary condition was applied at the downstream end (x ~100 µm), and the simulation and data were matched at the upstream end (relative concentration = 1). The value of the computed concentration is a weighted average with respect to the area fraction of SMC (concentration in the cell) and the area fraction of the extracellular matrix (bulk concentration) in the computational cell unit depicted in Fig. 3. Moreover, computed points in Fig. 4B are values averaged over a distance of four columns of SMCs (~20 µm) for the comparison with the experiments. The predictions for 70 and 160 mmHg show good agreement with the experiments of Mayer et al. (20).

Characteristics of ATP transport in the media.

Figure 5 presents the influence of IEL fenestrations on the profile of SMC surface ATP concentration for two different values of fenestral pore area fraction (f). The value of the surface concentration represents an average with respect to all the cells in the same row. The ATP concentration is normalized to that at the fenestral pore entrance. Figure 5A shows concentration profiles for four different values of the pore diameter (d = 0.4~1.6 µm) for f = 0.004, and Fig. 5B is for d = 0.8~3.2 µm and f = 0.016. Concentration profiles for only eight rows of the SMC array are displayed to show details near the IEL. The concentration profiles without the IEL entrance condition are displayed for comparison. Note that the ATP concentration is proportional to the rate of ATP uptake by SMCs because the uptake rate is a first-order process. Consequently, the results indicate the degree of influence of fenestral pores on the uptake rate of SMCs exposed to interstitial flow.



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  		Fig. 5. Profiles of ATP concentration on SMCs in the direction of the interstitial flow for different values of the fenestral pore diameter (d) and area fraction (f). Each value of ATP concentration is an average value over the row of SMCs. A: concentration profile for d = 0.4~1.6 µm and f = 0.004. B: concentration profile for d = 0.8~3.2 µm and f = 0.016.

 
Fenestrations in the IEL reduce the ATP concentration compared with vessels without the IEL. Moreover, the effect of the size of the fenestral pore on the ATP concentration distribution is reduced as the value of the area fraction (f) increases. These trends can be understood conceptually if we note that fluid-phase transport for ATP is dominated by diffusion, not convection (Pe = 5.96 x 10–4). Therefore, each fenestral pore creates a diffusion plume projecting in the media. For fixed-pore area fraction (f), the pores are spaced further apart as pore diameter (d) increases (see Fig. 2). For larger d then, there are more SMC between neighboring pores to consume ATP, and the net result is ATP concentration drops as d increases.

Characteristics of LDL transport in the media.

Figure 6 presents the influence of IEL fenestrations on the profile of SMC surface LDL concentration for two values of the fenestral pore area fraction (f). The value of LDL concentration is normalized to that at the fenestral pore inlet and represents an average with respect to all cells in the same row. The SMC surface LDL concentration (proportional to the SMC uptake rate) decays continuously with increasing distance from the IEL. Unlike ATP (Fig. 5), however, levels of surface LDL concentration are hardly affected by the pore diameter (d) or the area fraction (f). These trends can be understood conceptually by realizing that fluid-phase transport for LDL is dominated by convection (Pe = 4.60). Once fluid enters the media through a fenestral pore, it is convected strongly in the transverse direction (y-axis in Fig. 3), and this effectively eliminates the discrete nature of the fenestral pore entrance condition over very short distances in the x direction. The discussion of Fig. 7 further illuminates this point.



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  		Fig. 6. Profiles of LDL concentration on SMCs in the direction of the interstitial flow for different values of the fenestral pore diameter (d) and area fraction (f). Each value of LDL concentrations is an average value over the row of SMCs. A: concentration profile for d = 0.4~1.6 µm and f = 0.004. B: concentration profile for d = 0.8~3.2 µm and f = 0.016.

 


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  		Fig. 7. Concentration contour plots (A and B) and streamline plot (C) in the media near the fenestral pore for area fraction f = 0.004. A: ATP concentration contour lines for d = 1.6 µm. B: LDL concentration contour lines for d = 1.6 µm. The interval of concentration between neighboring contour lines takes the same value (=0.02) in A and B. The highest level of the contour line is at a concentration (C) of 1.0 at the pore inlet (indicated by open arrow). C: streamline plots for d = 1.6 µm.

 
Figure 7 shows more clearly the difference between ATP and LDL transport processes. Figure 7, A and B, shows the contours of ATP and LDL concentration, respectively, for f = 0.004 and d = 1.6 µm. The intervals of the contour level are the same in Fig. 7, A and B. The highest value of the contour is 1.0 at the fenestral pore for both A and B. Figure 7C shows the streamline pattern that applies to both ATP and LDL. The lateral spread of the flow after entering the media through the fenestral pore is readily apparent in the streamline patterns in Fig. 7C. It is also clear in Fig. 7B for LDL (Pe = 4.60) that the first concentration contour is convected laterally, whereas in Fig. 7A for ATP (Pe = 5.96 x 10–4) the concentration contours are almost perpendicular to the SMC surfaces, a pattern that is characteristic of a diffusion-dominated transport process.

Another difference in the transport characteristics of ATP and LDL is related to the surface reaction (uptake) rate relative to the rate of transport to the surface (Da/Sh). As shown in Fig. 4, the range of Sh is limited to values of 2~5, whereas Da = 0.88 for LDL and Da = 0.011 for ATP. Therefore, Da/Sh << 1 for ATP, which implies a reaction-limited process (Cs/Cb {approx} 1), but Da/Sh {approx} O(1) for LDL, which implies transport limitation. Figure 8 shows values of the ratio of the average surface concentration (Cs) to the average bulk concentration (Cb) for each SMC row. For ATP, Cs/Cb {approx} 1.0 except on the first SMC where the entrance effect of the fenestral pore alters the concentration distribution. This indicates that ATP uptake throughout the media is reaction limited. On the other hand, the value of Cs/Cb approaches 0.83 for LDL, indicating a modest limitation of uptake by fluid-phase transport.



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  		Fig. 8. Ratio of the surface concentration (Cs) to the bulk concentration (Cb) as a function of SMC cell number. Values of Cs and Cb are averaged over all SMCs residing in the same row.

 
The SMC geometric configuration in the medial layer and the volume fraction of SMC in the media (F) may affect macromolecular transport and uptake in the medial layer. The data of Lever and Jay (19), based on available space for Cr-EDTA, imply F = 0.6 for the carotid artery, and the data of Baldwin et al. (1) imply that the value of F could be far higher than 0.4, the representative value taken in the present study. To address the effects of higher values of F, Fig. 9 shows Cs/Cb of both ATP and LDL for different values of the volume fraction of SMC: F = 0.4, 0.6, and 0.7. The dependence of Cs/Cb on F is insignificant for both ATP and LDL in this range of F. However, a change in the geometric configuration of SMCs in the media alters both levels and profiles of the concentration ratio (Cs/Cb) along the interstitial flow direction. In Fig. 9, "staggered" indicates that cells are aligned in a configuration where alternate layers are displaced parallel to the IEL by a distance equal to one-half the spacing between SMCs (=L/2). The rate of LDL uptake by SMCs was enhanced for SMCs in the staggered configuration by ~12% compared with the square-array configuration when F = 0.7, whereas ATP uptake was hardly affected by the geometric configuration. In contrast, for F = 0.4, no difference was found in the Cs/Cb profile for LDL or ATP between the staggered- and square-array configurations (data not shown). Thus the LDL concentration profile that depends strongly on the local interstitial flow field is altered with cell geometric configuration for higher volume fractions of SMC.



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  		Fig. 9. Ratio of the surface concentration (Cs) to the bulk concentration (Cb) for different values of SMC geometric configuration and volume fraction (F) as a function of SMC cell number. Values of Cs and Cb are averaged over all SMCs residing in the same row.

 
Macromolecular permeability coefficient of IEL.

Figure 10 shows the dependence of IEL permeability to ATP and LDL on the fenestral pore diameter. The IEL permeability is defined as

(19)
where FC is convective mass flux at the fenestral pore

(20)
FD is diffusive mass flux at the fenestral pore

(21)
and C0 is solute concentration at the pore entrance. The fenestral pore diameter (d) was varied from 0.4 to 1.6 µm under the condition of constant fenestral pore area fraction (f = 0.004). Permeability of ATP showed much stronger dependence on pore size than LDL, varying by 50-fold over a range of pore diameters for which LDL permeability varied by only 3-fold. This reflects the differences between diffusion-dominated transport for ATP and convection-dominated transport for LDL that were discussed in regard to Figs. 5 and 6.



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  		Fig. 10. Dependence of IEL permeability to ATP and LDL on the fenestral pore diameter for fenestral pore area fraction f = 0.004. Circles, 2-dimensional slit model with SMC arrays at F = 0.4; lines, 3-dimensional (3D) circular pore model with an effective medium approximation for the media.

 
As a reference, results of a 3D pore (circular pore) model simulations are shown together with the 2D slit model results. In the 3D calculations, the medial layer was assumed to be a homogeneous reactive continuum having the following effective physiological properties: permeability, reaction rate coefficient, and diffusivity, as described by Huang and Tarbell (15). Values of the LDL permeability coefficient of the IEL for the 3D pore model show good agreement with those of the 2D slit model, whereas, for ATP, values for the 2D slit model overestimate by about two to five times those of the 3D pore model. This discrepancy may be attributed to the presence of enhanced diffusive spreading of ATP at the 2D slit opening. The slit provides a larger volume within a short distance of the opening than the circular pore, leading to a larger diffusive flux for a given pore area.

It is interesting to note that the LDL permeability coefficient of the IEL at the smallest pore size, 2.93 x 10–8 cm/s, is of similar magnitude to measured values of vessel wall permeability to LDL in the intact arteries of live animals [e.g., 1.9 x 10–8 cm/s measured by Truskey et al. (32) in rabbit arteries]. It has been assumed widely that artery wall permeability to LDL and other macromolecules is controlled by the endothelial layer permeability. The present calculations suggest, however, that the IEL may contribute significantly to measured values of artery permeability to LDL and that endothelial permeability to macromolecules may be higher than estimated from whole artery measurements. This implies that in vitro measurements of endothelial transport properties utilizing cultured endothelial cells (e.g., Refs. 5 and 16) may provide more reasonable estimates of endothelial transport properties than previously believed.

In conclusion, the present 2D interstitial flow and mass transport model incorporated, for the first time, the presence of an IEL that provided a complex fluid entrance condition associated with leaky fenestrations. The medial layer was approximated as a heterogeneous medium consisting of an array of cylindrical SMCs embedded in a continuous, porous interstitial matrix phase. The fenestral pores were modeled as circular openings distributed uniformly over the IEL. The IEL was assumed to be an impermeable wall except at its fenestral pores, which allowed fluid and solute passage. SMCs were assumed to be cylinders impermeable to fluid but which allow solute molecules to permeate across the surface membrane by bulk-phase endocytosis (LDL) or to be consumed on the surface (ATP).

In an earlier computational study (28), we showed that the complex fluid flow entrance condition associated with fenestral pores in the IEL could have a significant influence on the local shear stress on SMCs right beneath the fenestral pore. For reasonable physiological parameters, the shear stress could be elevated 10-fold above the level that would be obtained for a uniform entrance flow without an IEL. The present study was undertaken to investigate the influence of this complex entrance condition on mass transport to SMCs, near the intimal-medial boundary. This is an important region of the artery wall in regard to the development of intimal hyperplasia and the migration of SMCs from the media to the intima.

We observed that the medial concentration distribution and associated SMC uptake were affected dramatically by the IEL pore structure for a low-molecular-weight solute (ATP) that was transported through the interstitial phase predominantly by a diffusive mechanism (low Pe; Fig. 5). On the other hand, the concentration distribution and SMC uptake of a high-molecular-weight solute (LDL) in which interstitial phase transport was dominated by convection (high Pe) were not influenced significantly by the IEL pore structure (Fig. 6). Extrapolating these results to a molecule of importance in intima hyperplasia, the SMC mitogen and chemoattractant, platelet-derived growth factor (mol wt {approx}30 kDa) is expected to be transported by a diffusion-dominated mechanism and, based on our calculations, would therefore be influenced greatly by the IEL pore structure.


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 ABSTRACT
 METHODS
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This work was supported by National Heart, Lung, and Blood Institute Grants HL-35549 and HL-57093.


   ACKNOWLEDGMENTS
 
We are grateful to T. Matsuo for help with coding part of the computer simulation program.


   FOOTNOTES
 

Address for reprint requests and other correspondence: J. M. Tarbell, Dept. of Biomedical Engineering, Steinman Hall 2F, The City College of New York/CUNY, Convent Av. 140th St., New York, NY 10031 (E-mail: tarbell{at}ccny.cuny.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


   REFERENCES
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 ABSTRACT
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 REFERENCES
 

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