Aquaporin-1 shifts the critical transmural pressure to compress the aortic intima and change transmural flow: theory and implications

Aortic endothelial cell aquaporin-1 participates in transmural-pressure-driven water flow. This flow may affect the kinetics of lipid-vessel wall matrix binding that can trigger preatherosclerotic lesions. The theory explains how aquaporin-1 expression changes alter vessel wall properties (intimal compression) to lower or enhance this flow, suggesting potential interventions to slow atherogenesis.

Abstract

Transmural-pressure (ΔP)-driven plasma advection carries macromolecules into the vessel wall, the earliest prelesion atherosclerotic event. The wall's hydraulic conductivity, L_P, the water flux-to-ΔP ratio, is high at low pressures, rapidly decreases, and remains flat to high pressures (Baldwin AL, Wilson LM. Am J Physiol Heart Circ Physiol 264: H26–H32, 1993; Nguyen T, Toussaint, Xue JD, Raval Y, Cancel CB, Russell LM, Shou S, Sedes Y, Sun O, Yakobov Y, Tarbell JM, Jan KM, Rumschitzki DS. Am J Physiol Heart Circ Physiol 308: H1051–H1064, 2015; Tedgui A, Lever MJ. Am J Physiol Heart Circ Physiol. 247: H784–H791, 1984. Shou Y, Jan KM, Rumschitzki DS. Am J Physiol Heart Circ Physiol 291: H2758–H2771, 2006) due to pressure-induced subendothelial intima (SI) compression that causes endothelial cells to partially block internal elastic laminar fenestrae. Nguyen et al. showed that rat and bovine aortic endothelial cells express the membrane protein aquaporin-1 (AQP1) and transmural water transport is both transcellular and paracellular. They found that L_P lowering by AQP1 blocking was perplexingly ΔP dependent. We hypothesize that AQP1 blocking lowers average SI pressure; therefore, a lower ΔP achieves the critical force/area on the endothelium to partially block fenestrae. To test this hypothesis, we improve the approximate model of Huang et al. (Huang Y, Rumschitzki D, Chien S, Weinbaum SS. Am J Physiol Heart Circ Physiol 272: H2023–H2039, 1997) and extend it by including transcellular AQP1 water flow. Results confirm the observation by Nguyen et al.: wall L_P and water transport decrease with AQP1 disabling. The model predicts 1) low-pressure L_P experiments correctly; 2) AQP1s contribute 30–40% to both the phenomenological endothelial + SI and intrinsic endothelial L_P; 3) the force on the endothelium for partial SI decompression with functioning AQP1s at 60 mmHg equals that on the endothelium at ∼43 mmHg with inactive AQP1s; and 4) increasing endothelial AQP1 expression increases wall L_P and shifts the ΔP regime where L_P drops to significantly higher ΔP than in Huang et al. Thus AQP1 upregulation (elevated wall L_P) might dilute and slow low-density lipoprotein binding to SI extracellular matrix, which may be beneficial for early atherogenesis.

NEW & NOTEWORTHY

Aortic endothelial cell aquaporin-1 participates in transmural-pressure-driven water flow. This flow may affect the kinetics of lipid-vessel wall matrix binding that can trigger preatherosclerotic lesions. The theory explains how aquaporin-1 expression changes alter vessel wall properties (intimal compression) to lower or enhance this flow, suggesting potential interventions to slow atherogenesis.

atherosclerosis begins when transmural pressure (ΔP)-driven advection transports (9, 38) macromolecules, e.g., low-density lipoprotein (LDL) cholesterol (∼20–25 nm), from the lumen across focal leaks, rare (in rat ∼1 in 2,000–6,000; Refs. 14, 15, 16) endothelial cells (EC) with wide (enough) junctions (34), into the subendothelial intima (SI), where it spreads them radially from the leak. LDL reacts with the SI extracellular matrix (ECM) to trigger a cascade of events leading to observable lesions. Since water, plasma's major constituent, easily passes through the vastly more abundant normal EC tight junctions, ΔP drives far more water, albeit absent LDL, through normal than leaky junctions. This flow transports and dilutes SI LDL, potentially slowing LDL-ECM binding reaction kinetics and flushing unbound LDL from the SI and ultimately the wall. Starling's law gives the overall transwall water flux, J_v, as in Ref. 25: J_v = L_P(ΔP − σΔπ), with L_P the vessel wall hydraulic conductivity, σ the osmotic reflection coefficient, and Δπ the transwall osmotic pressure difference. Since large vessels are generally considered isotonic¹ (11), one neglects Δπ and J_v = L_P × ΔP. Several groups (1, 30), including ours (19, 23), have measured L_P of rabbit and rat aorta ex vivo over a range of ΔP and found L_P for a vessel with intact endothelium is high at low ΔP, decreases with increasing ΔP, and then remains essentially flat to very high ΔP. The same vessel's deendothelialized L_P is ΔP insensitive at approximately double its high-ΔP intact value. Ultrastructure studies (6) evince stark SI-media contrasts: a far sparser SI [∼95% SI void; (10)] vs. media ECM (<50% void; Ref. 9). Huang et al. (10) thus suggested and later experimentally confirmed (12) that high ΔP compresses the SI. This causes ECs to partially block internal elastic laminar (IEL) fenestral pores, which inhibits water flow and decreases L_P. Once the proteoglycans (PG) are fully compressed, stiffer collagen (CG) fibers resist further compression (L_P drop). Endothelial removal eliminates a resistance layer and fenestral blocking, making L_P ΔP insensitive. This paracellular flow theory (10) agrees with all experimental L_P (ΔP) (1, 30) measurements.

Aquaporin-1 (AQP1), a highly water-specific channel protein in otherwise hydrophobic membranes found in various ECs, epithelial, and other cells (20), allows high Δπ-driven water throughputs (∼3 × 10⁹ molecules·s⁻¹·channel⁻¹) at little or no ATP cost (18). Using immunohistochemical techniques, Nguyen et al. (19) showed AQP1 expression and distribution in rat and bovine aortic ECs (RAECs and BAECs) both in cultured monolayers and in whole rat aortas ex vivo, suggesting possible trans-EC AQP1 water transport. They lowered functioning AQP1 numbers using very low HgCl₂ exposures titrated to nontoxic, reversible levels that chemically block AQP1s and repeated these experiments using siRNA against AQP1, both in vitro (19) and, in a much more challenging procedure, on whole rat aortas ex vivo (36). In all cases they found significant vessel wall and endothelial L_P reduction with reduced functioning AQP1 (L_P dropped 22.1 ± 6% in BAECs with HgCl₂ and 56.4 ± 7.9% in RAECs with siRNA, both vs. control that indicated trans-EC flow). Studies with tracers that only cross the endothelium paracellularly show that these treatments cause insignificant junctional transport changes.

The ex vivo studies measured L_P (ΔP) of an excised vessel with functioning AQP1s, then with HgCl₂-blocked or knocked-down AQP1s, and then with the endothelium denunded, all on the same vessel. L_P dropped ∼32 ± 4, 11 ± 2, and 5 ± 3% for blocked AQP1s, at 60, 100, and 140 mmHg, respectively (19); they dropped 37 ± 13.0% at 60 and 8.7 ± 3.8% at 100 mmHg for the siRNA studies (36) (Tables 1 and 2). The wall's total resistance to flow, 1/L_P, is the sum of the resistance of the endothelium + SI and that of the media + IEL in series, and the latter is ΔP insensitive. Since all L_P measurements were taken on each vessel, one can calculate the drops in each constituent L_P for each vessel and average. The percent decreases in total wall L_P are then due to drops in endothelial + SI L_P of 51 ± 2.3, 21 ± 3.6, and 11 ± 6.5% at these three pressures. Clearly AQP1-mediated transcellular flow contributes significantly to endothelial L_P. However, it is confounding that this percent decrease is apparently strongly ΔP dependent, unlike for a simple material. Whereas L_P in the absence of blocker is pressure dependent from 60 to 100 mmHg, when the AQP1s are blocked, L_P seems ΔP independent at these pressures. Reasoning from Huang et al. (10), we hypothesize that, whereas in the absence of blocker, the SI reaches full compression between 80 and 100 mmHg, with AQP1 blocking or knockdown, the SI fully compresses at 60 mmHg or less.

Table 1.

Measured hydraulic conductivity with and without endothelium in rabbit and rat aortas

	Hydraulic Conductivity × 108, cm·s−1·mmHg−1
Data/Species/Pressure, mmHg	Intact endothelium	Denuded endothelium
Tedgui and Lever (30)/rabbit
70	4.00 ± 1.31	5.36 ± 1.62
180	2.44 ± 0.80	5.27 ± 0.84
Baldwin and Wilson (1)/rabbit
50	7.42 ± 4.00	7.41 ± 4.08
75	4.14 ± 1.67	10.19 ± 1.91
100	4.54 ± 0.89	10.00 ± 2.20
125	4.26 ± 0.51	9.66 ± 2.20
150	5.00 ± 0.66	10.53 ± 2.83
Shou et al. (23)/Sprague-Dawley rat
60	4.69 ± 1.20	5.35 ± 1.46
80	3.02 ± 1.34	5.01 ± 2.05
100	2.79 ± 0.72	4.89 ± 1.01
120	2.60 ± 0.73	4.76 ± 1.46
140	2.68 ± 0.49	5.05 ± 1.22
Nguyen et al. (19)/Sprague-Dawley rat
60	2.55 ± 0.36	4.06 ± 0.28
100	1.84 ± 0.14	3.44 ± 0.76
140	2.10 ± 0.23	3.86 ± 0.37

Values are means ± SD.

Table 2.

Data of Nguyen et al. (19): effect of AQP1 blocking on hydraulic conductivity of intact artery wall

	Hydraulic Conductivity, LPt × 108, cm·s−1·mmHg−1
Pressure, mmHg	Unblocked AQPs	Blocked AQPs
60	2.55 ± 0.36	1.74 ± 0.35
100	1.84 ± 0.14	1.64 ± 0.16
140	2.10 ± 0.23	2.00 ± 0.28

Values are means ± SD. AQP1, aquaporin-1.

To test this hypothesis, this paper extends the local filtration theory of Huang et al. (10) by including transcellular flow. It hypothesizes that decreasing the number of functioning EC AQP1s decreases the number of available water transport pathways and thus the intrinsic endothelial hydraulic conductivity L_Pe. A lower L_Pe decreases SI pressure (P_i^*) at fixed ΔP, i.e., it increases the force per unit area, the difference (P_L^* − P_i^*) between lumen and SI pressures, acting on the endothelium. A lower overall ΔP can thus compress the SI and cause partial fenestral blockage. We shall see if the theory explains the AQP-blocking effect on L_P(ΔP) observed by Nguyen et al. (19).

filtration model: paracellular flow presents the filtration model and infinite series solution for the SI and media pressures of Huang et al. (10) using their three suggested approximate matching schemes in the IEL fenestrae and compares with a new finite difference implementation with exact matching there. filtration model: transcellular flow extends the model of Huang et al. (10) by incorporating transcellular flow. Results and Discussion compares the new model's predicted L_P(ΔP) with experiment (19) and makes several predictions for future experiments, including those that may have clinical relevance.

FILTRATION MODEL: PARACELLULAR FLOW

Model Description

Since marcromolecular transport across the artery wall is advection dominated (9), it is critical to understand paracellular junctional (this section) and trans-EC AQP1 (see filtration model: transcellular flow) water flow in detail.

ΔP, the pressure difference between the inside (lumen) and outside (adventitia) of the vessel, drives water through the glycocalyx (GX) layer on the luminal EC surface into the SI through the inter-EC junctions along the EC perimeter. It then spreads radially parallel to the endothelium in the SI and enters the media through the IEL fenestrae (Fig. 1). SI compression at increased ΔP narrows the cross section for this SI flow and causes the endothelium to (partially) block the IEL fenestral entrance (10). The results in decreased SI permeability, K_P, and this alteration in flow can lead to significant SI pressure gradients that can radically change the head losses incurred in traversing the SI and fenestrae. To understand the effect of SI compaction on these flows, we consider a local (on the scale of a single EC) model for SI flow into and through the fenestral hole. The model differs from Huang et al. (10) in that it also models the EC surface GX layer and treats the fenestral flow exactly.

Fig. 1.

Schematic of periodic local wall unit around a fenestral pore including, from top, endothelial glycocalyx, endothelial cell, subendothelial intima (SI), internal elastic laminar (IEL), and media. SI under nondeformable endothelium is compressible; L_i0^* (L_i^*) is the uncompressed (compressed) SI thickness at 0 (elevated) transmural pressure. Water enters the SI through junctions/clefts surrounding the endothelial cell (EC) and the media through the IEL fenestra.

By abuse of geometry (circles do not tile the plane), Fig. 1 shows a representative local periodic wall unit of a circular cylinder of radius ξ_l^* = (R^∗ +ΔR^∗/2), where R^∗ is the radius of an assumed round EC and ΔR^∗ is the width of a normal intercellular junction. The fluid source along the wall unit's perimeter represents the normal EC junction. Since normal junctions vastly outnumber leaky junctions by a factor of 2,000–6,000 (14, 15, 16), they account for the overwhelming majority of water flow across the endothelium; this section thus models a cell with normal tight junctions. Figure 1 greatly exaggerates the vertical scale of the SI, which is of the order of 0.2–0.5 μm (9, 10) in healthy rat aorta. Experimental data (21) indicate that the number of fenestrae per EC is between 0.1 and 10; we take an average value of one fenestra as in Ref. 10. Thus a round fenestra of radius r_f^* is ideally placed at the unit's center concentric with the EC, an obvious idealization that preserves axisymmetry. Pressure loading acts on the assumed nondeformable endothelium to compress the SI from L_i0^* (initial thickness at zero transmural pressure) to L_i^*. The model treats the IEL as an impenetrable barrier of zero thickness except for its fenestral openings and neglects any nonuniform deformation of the endothelium due to spatial differences in the transendothelial pressure. Due to the media's high density, the model presumes that it undergoes no compression upon pressure loadings and assumes its filtration properties (e.g., K_P) are uniform.

Mathematical Formulation

Let j be a dummy index that takes values g for GX, i for SI, and m for media. Let U_j^*, W_j^* be the dimensional lateral and normal velocities in the r and z directions of a cylindrical coordinate system and P_j^* and K_Pj be the pressure and Darcy permeability, in region j of thickness L_j^*; μ is the fluid viscosity. We neglect pressure pulsatility; thus let the time-invariant lumen pressure be P_L^*. We introduce the following nondimensional (no superscript *) variables:

$$\begin{array}{l}r=\frac{r\ast }{r_{\text{f}}^{\ast }},\xi =\frac{{\xi }_I^{\ast }}{r_{\text{f}}^{\ast }},z_i=\frac{z_j^{\ast }}{L_j^{\ast }}\\ {\text{P}}_j=\frac{{\text{P}}_j^{\ast }}{{\text{P}}_{\text{L}}^{\ast }},U_j=\frac{U_j^{\ast }}{\frac{K_{{\text{P}}_j}}{\mu }\frac{{\text{P}}_{\text{L}}^{\ast }}{r_{\text{f}}^{\ast }}},W_j=\frac{W_j^{\ast }}{\frac{K_{\text{P}j}}{\mu }\frac{{\text{P}}_{\text{L}}^{\ast }}{{\text{L}}_j^{\ast }}}\end{array}$$

The continuity equations, in nondimensional form, for the three regions are:

$$h_j^2\left(\frac{\partial U_j}{\partial r}+\frac{U_j}{r}\right)+\frac{\partial W_j}{\partial z_j}=0\left(j=\text{g},\text{i},\text{m}\right)$$ (1)

where h_j = $\frac{L_j^{*}}{r_f^{*}}$ are the region thicknesses nondimensionalized by the fenestral radius. Porous media flow describes the water flow across the arterial wall: through the SI, made up of ECM of PG and CG fibers, and the media, consisting of smooth muscle cells, ECM, and elastic layers. Darcy's law: V^∗ = $\frac{-K_P}{\mu }▿P^{*}$ with an effective Darcy permeability for each region governs such flows. We use Darcy's, rather than Brinkman's, equation because it does not seem consistent to explicitly enforce no slip on the region boundaries while simultaneously lumping the no-slip on the far more ubiquitous fibers, elastic layers, and cells into the bulk parameters K_Pj. [A detailed analysis of this issue in such problems shows only a minor effect (9)]. Thus Eq. 1 becomes:

$$h_j^2\left(\frac{{\partial }^2{P}_j}{\partial r^2}+\frac{1}{r}\frac{\partial P_j}{\partial r}\right)+\frac{{\partial }^2{P}_j}{\partial z_j^2}=0\left(j=\text{g},\text{i},\text{m}\right)$$ (2)

The (nondimensional) boundary conditions for this system of coupled partial differential equations (PDEs) follow: (Note z_m^* = 0 is at the IEL, not at the EC, both modeled as infinitely thin.)

1) Axisymmetry at r = 0 and periodicity (and, therefore, no radial flux) at r = ξ_I require:

$$\frac{d{P}_j}{dr}=0\hspace{0.17em}\text{at}\hspace{0.17em}r=0\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}r={\xi }_I\left(j=\text{g},\text{i},\text{m}\right)$$ (3)

2) The pressure at the top of the GX layer equals the lumen pressure:

$$P_{\text{g}}=1\hspace{0.17em}\text{at}\hspace{0.17em}{z}_{\text{g}}=1$$ (4)

3) The adventitia defines the reference pressure:

$$P_{\text{m}}=0\hspace{0.17em}\text{at}\hspace{0.17em}{z}_{\text{m}}=-1$$ (5)

4) Mixed boundary conditions at the GX/EC boundary (z_i = 1): the EC is assumed impermeable to water (relaxed in filtration model: transcellular flow). The hydraulic conductivity of the normal junction (L_Pnj) and the pressure difference across it govern the amount of water entering the SI through it.

4a) On the EC: (0 < r ≤ R, z_i = 1)

$$\frac{d{\text{P}}_j}{d{z}_j}=0\hspace{0.17em}(j=\text{g},\text{i})$$ (6)

4b) In the normal junction: (R < r ≤ ξ_I, z_i = 1)

$$W_{\text{g}}=W_{\text{i}}$$ (7)

$$-W_{\text{i}}=\frac{L_{Pnj}\mu L_i^{*}\left(P_{\text{g}}-P_{\text{i}}\right)}{K_{\text{Pi}}}$$ (8)

Previous studies (39) that modeled the vessel wall over radial scales equivalent to 1,000 EC radii used L_Pnj to describe the area-averaged hydraulic conductivity of the endothelium (ECs and normal junctions), equivalent to L_Pe in this local model. Here we allow only junctional water transport, so L_Pe and our L_Pnj are related by the ratio of junctional to total endothelial areas; filtration model: transcellular flow includes a separate transcellular contribution, and both will contribute to an area averaged L_Pe.

5) Mixed boundary conditions at the SI/media boundary (z_i = 0): assume the IEL is an impenetrable barrier except for its fenestral openings. Thus water enters the media only through the fenestra where pressures and velocities are continuous. This requires

5a) In the fenstral hole: (0 < r ≤ 1, z_i = 0)

$$P_{\text{i}}=P_{\text{m}}$$ (9)

$$W_{\text{i}}=W_{\text{m}}.$$ (10)

5b) Outside the fenstral hole: (1 < r ≤ ξ_I, z_i = 0)

$$\frac{\text{d}{P}_j}{\text{d}{z}_j}=0\left(j=\text{i},\text{m}\right).$$ (11)

The model of Huang et al. (10) omitted the GX layer on the EC's luminal side. Instead of periodicity at r = ξ_I in the SI (Eq. 3), they assumed a ring source at the EC cleft and determined the unknown constant pressure P₀ at r = ξ_I, 0 ≤ z ≤ 1 by imposing flow incompressibility. With these simplifications, Huang et al. (10) found analytical solutions of Eq. 2 for the SI and media pressures by decomposing the pressures into orthogonal pieces in r, solving the z dependence of each piece and reassembling the infinite sums of zero order Bessel functions (J₀) (graphed in Fig. 2):

$$P_i\left(r,z_i\right)=P_0+P_0{\displaystyle \sum _{n=1}^{\infty }\left(A_n\frac{\cosh \left[h_i\sqrt{{\lambda }_n}\left(z_i-1\right)\right]}{\cosh \left(h_i\sqrt{{\lambda }_n}\right)}\cdot J_0\left(\sqrt{{\lambda }_n}r\right)\right)}$$ (12)

$$P_m\left(r,z_m\right)=P_0\left(z_m+1\right)C_0+P_0{\displaystyle \sum _{p=1}^{\infty }\left(C_p\frac{\sinh \left[h_m\sqrt{{\lambda }_p}\left(z_m+1\right)\right]}{\sinh \left(h_m\sqrt{{\lambda }_p}\right)}\cdot J_0\left(\sqrt{{\lambda }_p}r\right)\right)},$$ (13)

Fig. 2.

Local pressure distributions on the SI side (P_i: the top curve of a given color) and the medial side (P_m: the bottom curve of a given color) of the IEL for different SI thicknesses using the data of Tedgui and Lever (30) with the series solution of Huang et al. (10) using the three approximate boundary conditions (A) and the finite difference solution and exact boundary conditions (B). L_Pm and L_Pe are given in Tables 4 and 5. A: at 50 nm, all 3 approximate fenestral matching conditions: exact pressure matching only at the fenestral center, only on average across the fenestra or at 3 points in the fenestra. The top 3 curves use the most accurate 3-point condition. They are similar to the finite difference solution in B but show slight deviations from it even far from the fenestra. Both show a shift of the major pressure drop from the media to the SI as the SI thins.

where λ_n and λ_p are the roots of the eigenvalue equations J₀($\sqrt{{\lambda }_n}$ξ_I) = 0 (n = 1, 2, 3,.,∞) and J₁($\sqrt{{\lambda }_p}$ξ_I) = 0 (p = 1, 2, 3,.,∞). The constants A_n and C_p depend on the fenestral boundary conditions in Eqs. 9 and 10. Huang et al. (10) approximated these fenestral (0 < r ≤ 1, z_i = 0) conditions in the following three ways and Fig. 2 will show their effect on the pressure distribution:

1) The z-velocity, W_f, in the fenestra is uniform on 0 < r ≤ 1, z_i = 0 and the pressures in the SI and media exactly match only at the fenestra center, i.e., P_i = P_m at (r = 0, z_i = 0);

2) W_f is uniform on 0 < r ≤ 1, z_i = 0 and only the average pressure, P̄_j = 2∫₀¹P_j(r)rdr, j = i, m, across the fenestra matches in the fenestral hole, i.e., P̄_i = P̄_m at z_i = 0;

3) At z = 0, W_f(r) fits a cubic polynomial that satisfies dW_f/dr = 0 at r = 0 and the pressure is continuous only at three points in the fenestra, r = 0, 0.5, 0.9, i.e., W_f = (a₀ + c₀r² + d₀r³)P₀ and P_i = P_m at r = 0, 0.5, 0.9; z = 0.

Exact Numerical Solution of the Boundary Value Problem

We adopt a direct-discretization, finite difference approach using central difference formulas for nonuniform meshing (22) to solve the system of coupled PDEs in Eq. 2. Although the governing equations for the filtration problem are Laplace equations, the mixed boundary conditions (Eqs. 3–11) make it difficult to obtain an exact analytical solution. This numerical method, in principle, allows us to use the exact boundary conditions rather than the approximate ones employed by Huang et al. (10). Since the thickness of the SI (L_i^*) and the radius of the fenestral hole (r_f^*) are both small compared with both the radius of wall unit (ξ_I^*) and the length of media (L_m^*), we use nonuniform grids in the r- and z-directions. We expect a steep pressure gradient near the fenestral hole and thus form a very dense grid there with the smallest nondimensional grid size (r^∗/r_f^*) of 0.0005 (nondimensional r, z_g, z_i, and z_m vary from 0 to 15.0125, 0 to 1, 0 to 1, and 0 to −1, respectively) in the fenestral hole. To resolve the pressure variations in the normal junction [width ∼20 nm (10)], we use an extremely fine grid near the junction with the smallest grid size being 8 ×10⁻¹⁰. Since the coefficients (L_i^*, L_m^*, K_Pi, and K_Pm) in the velocity matching condition (Eq. 10) differ by several orders of magnitude between regions being matched (see Mathematical Formulation), one has to be very careful in selecting the mesh sizes in the z-directions near the fenestral hole. If the mesh size is not sufficiently small, the finite difference approximations used for the derivative dP/dz could lead to significant errors there. We chose the smallest grid size near the hole to be 0.0005 (SI) and 0.000001 (media). Similarly, the smallest grid sizes used in the z-directions close to the endothelial cell boundary are 0.0005 and 0.00006 in the SI and GX regions.

We use second order difference formulas to discretize the boundary conditions. This leads to a linear system of algebraic equations whose number equals the total number of mesh points. We solve the set of equations representing the three domains simultaneously using Matlab (Mathworks). To test the accuracy of the solution, we adopt a successive mesh-refinement procedure until the difference between two consecutive computations is in the fourth significant digit.

Constants and Parameters

Geometric parameters.

Most of the parameters used in this study, given in Table 3, are adopted from Huang et al. (10). They extracted an average spacing, δ_PG, between PG fibers of ∼30–40 nm from the freeze etchings of Frank and Fogelman (6). We use δ_PG0 = 40 nm for the relaxed SI. Huang et al. (9) developed a fiber matrix theory to calculate the effective radius (a^∗) of the PGs and the zero ΔP void fraction (${ϵ}_{\text{PG}0}$) for the PGs. We use the theory of Huang et al. (9) to estimate a^* ∼2.37 nm and ${ϵ}_{\text{PG}0}$ ∼98.83%. Using these values, we calculate the Darcy permeability of a fiber matrix of PGs in the SI using the Carman-Kozeny expression (2, 3, 4) as:

$$K_{\text{P}\left(\text{PG}\right)}=\frac{a^{{*}^2}{ϵ}_{\text{PG}0}^3}{4G\left(1-{ϵ}_{\text{PG}0}^2\right)},$$

Table 3.

Parameters and constants used in the model

Constant/Parameter	Value	Reference
a*, nm	2.37	*
Lg*, nm	200	(40)
Li0*, nm	500	(10)
Lm*, μm	141 (rat), 125 (rabbit)	(10, 23)
R*, μm	12	(35)
rf*, μm	0.8	(10)
ΔR*, nm	20	(35)
KPg, cm2	4.08 × 10−14	†
KPi, cm2	2.20 × 10−12	(10)‡
δCG, nm	170.35	§
δPG, nm	40	(10)
ϵCG0	0.95	(9)
ϵPG0	0.9883	*
μ, kg·m−1·s−1	7.2 × 10−4	(8)

^*Calculated using fiber matrix theory as explained in Ref. 9.

^†Estimated as explained in the text.

^‡Uncompressed subendothelial intima (SI) value at 0 transmural pressure; corresponding values for compressed SI (L_i^*) are calculated as explained in the text and in Ref 10.

^§Calculated by assuming hexagonally packed collagen fibers of radius 20 nm (9) with an initial volume fraction of 5% (9).

where G, the Kozeny constant, is obtained as in Ref. 7. The spacing, δ_CG0, between radius 20 nm (9) CG fibers, assumed to form a parallel, triangular fiber array with a zero-ΔP volume fraction, ${ϵ}_{\text{CG}}$ ∼5% (9), is calculated as 170.35 nm. The correlation of Tsay and Weinbaum (31) gives the collagen matrix Darcy permeability, K_PCG, in terms of the average fiber radius and spacing:

$$K_{{\text{P}}_{\left(\text{PG}\right)}}={\left(\frac{r_{\text{CG}}^{*}}{a^{*}}\right)}^{0.377}{\left(\frac{{\delta }_{\text{PG}0}-2a^{*}}{{\delta }_{\text{CG}0}-2r_{\text{CG}}^{*}}\right)}^{2.377}.$$

We assume that the resistances due to the PG and CG fiber populations act as resistors in series and thus compute the overall SI Darcy permeability, (K_Pi), as in Ref. 13

$$\frac{1}{K_{\text{Pi}}}=\frac{1}{K_{{\text{P}}_{\left(\text{PG}\right)}}}+\frac{1}{K_{{\text{P}}_{\left(\text{CG}\right)}}}.$$

At zero ΔP, we calculate K_Pi = 2.20 × 10⁻¹² cm², in agreement with Huang et al. (10). As in Huang et al. (12), ΔP-induced SI compression alters the void fractions for the PG (${ϵ}_{\text{PG}}$) and CG fibers (${ϵ}_{\text{CG}}$) via the SI thickness, L_i^*, as:

$$\begin{array}{l}{ϵ}_{\text{PG}}=1-\frac{L_{io}^{*}}{L_i^{*}}\left(1-{ϵ}_{\text{PG0}}\right);\\ {ϵ}_{\text{CG}}=1-\frac{L_{io}^{*}}{L_i^{*}}\left(1-{ϵ}_{\text{CG0}}\right).\end{array}$$

At a given compression, one calculates the PG and CG void fractions and estimates the average spacing between these fibers assuming they form a parallel, triangular array. With this spacing, we compute K_Pi of the compressed SI as explained. As in Huang et al. (10), we find that K_Pi decreases rapidly as the SI starts compressing with pressure loading. K_Pi at L_i^* = 0.2L_i0^* (L_i^* = 0.1L_i0^*) is 2.06 × 10⁻¹³ cm² (4.7 × 10⁻¹⁴ cm²), an order (approximately 2 orders) of magnitude lower than its uncompressed value. This drastic drop in K_P significantly affects the pressure and velocity distributions in the vicinity of the fenestra, as explained in the results.

The major constituents of the GX layer are PGs, which include both glycosaminoglycan side chains and glycoproteins (28). Recent observations of Squire et al. (24) and Weinbaum et al. (33) suggest a bush-like GX structure with clusters of core proteins projecting normally from the EC surface. We model the GX as a fiber matrix with Darcy permeability, K_Pg, defined as in Ref. 31:

$$K_{\text{Pg}}=0.572a_f^2{\left(\frac{\Delta }{a_f}\right)}^{2.377},$$

where a_f is the fiber radius (6 nm; Ref. 24) and Δ is the open spacing between fibers [8 nm (24)], giving K_Pg = 4.08 × 10⁻¹⁴ cm². Using Δ = 20 nm, Dabagh et al. (5) calculated K_Pg as 3.6 × 10⁻¹³ cm². Assuming a regular two-dimensional hexagonal arrangement, we compute the fiber volume fraction to be 0.326 and thus a void fraction of 0.674. Using this void fraction and the relation of Zhang et al. (41) among fiber radius, void fraction, and the open flow area between fibers, Liu et al. (17) estimated K_Pg as 6.04 × 10⁻¹⁴ cm². Seki et al. (26) calculated the GX Darcy permeability considering flows perpendicular (3.16 × 10⁻¹⁴ cm²) and parallel (6.10 × 10⁻¹⁴ cm²) to a hexagonal array of cylindrical fibers. Our K_Pg value lies in this range. We assume that, since the GX lies between the lumen and the EC, increasing lumen pressure affects neither the GX nor its structural properties.

Hydraulic conductivities.

As described in the Introduction, the engineering quantity, hydraulic conductivity, L_P, the ratio of the fluid velocity to the driving pressure difference, is P independent for simple materials. We are concerned with the L_Ps of the endothelium, the normal junction, SI, IEL, media, endothelium + SI, IEL + media (i.e., the denuded vessel), and total arterial wall, denoted as: L_Pe, L_Pnj, L_Pi, L_PI, L_Pm, L_Pe+i, L_Pm+I, L_Pt, respectively. Tedgui and Lever (30) and Baldwin and Wilson (1) measured L_Pt and L_Pm+I for rabbit aorta over a range of ΔPs, while Shou et al. (23) and Nguyen et al. (19) did analogous experiments on rat aortas. The latter three groups did measurements at all ΔPs for the intact (L_Pt) and then for the denuded (L_Pm+I) aorta, all on each vessel. One can calculate the phenomenological L_Pe+i for each vessel. 1/L_P is a specific resistance. Since linear resistances in series add, the total resistance is related to the layer resistances by:

$$\frac{1}{L_{\text{Pt}}}=\frac{1}{L_{\text{Pe}+\text{i}}}+\frac{1}{L_{\text{Pm}+\text{I}}}$$ (14a)

$$\frac{1}{L_{\text{Pe}+\text{i}}}=\frac{1}{L_{\text{Pe}}}+\frac{1}{L_{\text{Pi}}}$$ (14b)

$$\frac{1}{L_{\text{Pm}+\text{I}}}=\frac{1}{L_{\text{Pm}}}+\frac{1}{L_{\text{P}\text{I}}}$$ (14c)

K_Pm = L_Pm × μ × L_m^*, where μ is the viscosity of water and L_m^*, the media thickness, is 125 μm (141 μm) for rabbit (rat) aorta. Note that the rat aorta media thickness measurement is larger because Shou et al. (23) measured it on vessels fixed after excision rather than on vessels fixed in situ, where it is still tethered and in a stretched state inside the animal. Excision releases this stretching, causing the vessel to retract and, by mass conservation, become thicker. At steady state (for fixed geometry, even in the unsteady case), fluid incompressibility requires the water flow across each arterial layer to be the same. Thus, as explained by Huang et al. (10), by matching water fluxes across the IEL and using Eqs. 20 and 21 from Ref. 10 with the average L_Pm+1 for each data set given in Table 1, we obtain L_Pm = 11 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹ for the data of Tedgui and Lever (30). This gives us K_Pm as 8.38 × 10⁻¹⁵ cm², which is very close to 6.09 × 10⁻¹⁵ cm² used by Tada et al. (27). Table 4 lists the calculated values of L_Pm for the other data sets.

Table 4.

Converged L_Pm values

Data	LPm × 108, cm·s−1·mmHg−1
Tedgui and Lever (30)	11.00
Baldwin and Wilson (1)	19.87
Shou et al. (23)	10.37
Nguyen et al. (19)	7.50

The by-nature pressure-independent intrinsic hydraulic conductivity of the endothelium, L_Pe, is unknown. We assume that at very low ΔP the SI is fully expanded and there is no fenestral blocking. Therefore, the combination of SI K_P and the squeezing flow into the IEL fenestra should only account for a small part of the total resistance. [Eq. 14b guarantees L_Pe ≥ L_Pe+i at ΔP(min).] Thus one can initially estimate L_Pe by L_Pe+i for the least compressed SI configuration for which L_Pt data are available. One uses this value to solve the flow problem subject to boundary conditions Eqs. 3–11 and then calculates L_Pe+i from flow incompressibility, i.e., the overall water flow across the vessel wall must match the water flow across the endothelium as well as across the IEL:

$$L_{\text{Pe}+\text{i}}\left({\overline{P}}_g{|}_{z_g=0}-{\overline{P}}_i{|}_{z_i=0}\right)\pi {\xi }_I^2=\frac{2\pi K_{Pi}}{\mu L_i^{*}}{\displaystyle {\int }_0^1\frac{d{P}_i}{d{z}_i}}{|}_{z_i=0}rdr$$ (15)

where,

$${\overline{P}}_g{|}_{z_{g=0}}=\frac{2}{{\xi }_I^2}{\displaystyle {\int }_0^{{\xi }_I}{P}_g{|}_{z_{g=0}}rdr\hspace{0.17em}\text{and}\hspace{0.17em}}{\overline{P}}_i{|}_{z_{i=0}}=\frac{2}{{\xi }_I^2}{\displaystyle {\int }_0^{{\xi }_I}{P}_i}{|}_{z_i=0}rdr$$ (16)

If this L_Pe+i does not match the experimental value in Table 1 with which we started, we adjust L_Pe and iterate until the two L_Pe+i values match. This approach is only slightly more complicated than that in Ref. 10 due to the GX layer here. L_Pe is the ratio of junction to total area times L_Pnj:

$$L_{{\text{P}}_{\text{e}}}=\frac{L_{P_{nj}}2\pi R^{*}\left(\Delta R^{*}/2\right)}{\pi {\xi }_I^{*2}}.$$

Table 5 gives the converged results for L_Pe for the available data sets. We use these converged values of L_Pe (or, in this section, L_Pnj) for all further calculations and predictions.

Table 5.

Converged L_Pe values

Data	LPe × 108, cm·s−1·mmHg−1
Tedgui and Lever (30)	16.41
Baldwin and Wilson (1)	35.80
Shou et al. (23)	16.52
Nguyen et al. (19)	8.29

One test of the model is to compare the computed and experimental total wall L_Pt data. One way to find L_Pt is to insert L_Pe+i from Eq. 15 and an averaged L_Pm+I from the corresponding experimental data into Eq. 14a, as done in Ref. 10. However, L_Pm+I only enters the model calculation in determining K_Pm. The calculated media pressure field at each ΔP leads to a phenomenological L_Pm+I for Eq. 14a via an equation analogous to Eq. 15. This L_Pm+I, the media + IEL conductivity in an intact wall, might neither equal the averaged experimental value used to find K_Pm nor even be ΔP independent, as is L_Pm+I for the denuded wall. In fact, L_Pm+I calculated this way turns out to be nearly constant at low ΔP, but decreases significantly at higher ΔP. A better method for finding a fully model-generated value for L_Pt is to use fluid incompressibility to equate the total flow across the wall in terms of L_Pt with the calculated flow through the fenestral hole:

$$L_{{\text{P}}_{\text{t}}}\pi {\xi }_I^2=\frac{2\pi K_{P_i}}{\mu L_i^{*}}{\displaystyle {\int }_0^1\frac{d{P}_i}{d{z}_i}}{|}_{z_i=0}rdr$$ (17)

Results and Discussion

Pressure drop across the IEL.

Figure 2, A and B, depicts the pressures, nondimensionalized by the luminal pressure P_L^* (the adventitial pressure is the zero reference pressure), above and below the IEL as a function of r^∗/r_f^* for various intimal compressions. Figure 2A uses the analytical series solution with the approximate boundary conditions of Huang et al. (10) (see Mathematical Formulation) and Fig. 2B the finite difference solution with the exact boundary conditions. As in Huang et al. (10), Fig. 2A uses a ring source at the normal junction instead of Eqs. 7 and 8. Figure 2B properly matches the pressure and velocity in the fenestral hole (r^∗/r_f^* ≤ 1) and includes the GX layer and its matching conditions in Eqs. 7 and 8. In Fig. 2, A and B, the pressure decreases from the edge of the wall unit to the center on the SI side as the fluid approaches the fenestral hole and then decreases as r increases on the media side as the fluid exiting the fenestra spreads. Both sets of curves display a qualitative change in the pressure as the SI thins in response to increasing transmural pressure. For L_i^* ≥200 nm, the pressure in the SI is almost flat; most of the pressure drop occurs in the media. As the SI thins with increasing ΔP, say at L_i^* ∼50 nm, most of the pressure drop shifts to the SI within several pore radii from the fenestral opening. Thus the flow resistance shifts with pressure from the medial spreading flow to the entrance flow into an IEL fenestra that is partially blocked by an EC and water incurs a sharp pressure drop (50-nm curve in Fig. 2B) in traversing the SI.

Figure 2A is nearly identical to that of Huang et al. (10) with small differences likely due to different numbers of terms retained (∼50 in Ref. 10 vs. 200 here) and a different matrix inversion tool used. Figure 2, A and B, agrees well except in and near the fenestral hole, especially for compressed intimae where the predictions are off by ∼10% and in the junction region where they differ by roughly 3%. These two regions are where the pressures vary significantly in z and, for thinner SI, in r across the fenestra or junction. Except in these regions, the characteristic z-scale, L_i0, is far smaller than the characteristic r-scale, ξ_I, i.e., the z variation of the dynamic variables should be very small (not of leading order in L_i0/ξ_I <1). This scale separation fails and z-variation is significant in these regions, since the far smaller region width replaces ξ_I as the r-scale. The third, three-point-matching approximation of Huang et al. (10) is closer to the exact solution than their other two schemes (see Fig. 2 legend), as anticipated. To find the dimensional pressures, the next section uses the theory of Huang et al. (10) to first predict the extent of SI compression for a given P_L^*.

Pressure-dependent hydraulic conductivity.

To compare the predicted L_Pt with the four sets of measured L_Pt(ΔP) for the intact arterial wall, we model the SI matrix as a Hookean spring. The spring responds linearly to the force per unit area, P_L^* (P̄_g − P̄_i), on the endothelium with a SI compression, L_i^*/L_i0^*

$$P_L^{*}\left({\overline{P}}_g-{\overline{P}}_i\right)=k\left(1-\frac{L_i^{*}}{L_{i0}^{*}}\right),$$ (18)

where k is the ΔP-independent SI elastic coefficient (spring constant) [mmHg]. Using Eqs. 17 and 16, we calculate L_Pt and P̄_g − P̄_i at different SI thicknesses, L_i^*. For each ΔP, we find the SI thickness for which the calculated L_Pt matches experiment. For these L_Pt, P_L^*, and L_i^*, the best fit slope of transendothelial pressure vs. endothelial displacement (Eq. 18) is k.

Figure 3 plots the SI thickness, L_i^*/L_i0^*, vs. lumen pressure, P_L^*, from Eq. 18 using the ks determined from four different L_Pt(ΔP) data sets and all four curves compare well with (within the error bars of) the data of Huang et al. (12) for L_i^*/L_i0^* of rat aorta fixed at four different ΔPs. Given the complexity of measuring the thickness of the extremely irregular SI layer and the simplicity of our model, the model seems to capture the important physiology that governs SI compression. SI thickness decreases nearly linearly with ΔP for ΔP [less than ∼60 mmHg for data of Baldwin and Wilson (1) and Nguyen et al. (19) and less than ∼80 mmHg for data of Tedgui and Lever (30) and Shou et al. (23)], implying the transendothelial pressure difference is essentially proportional to ΔP there. Further increase in luminal pressure causes a nonlinear flattening of the L_i^*/L_i0^* until a critical ΔP beyond which no further compaction occurs. This explains why L_Pt becomes ΔP independent beyond this critical pressure. Our/Huang et al.’s (10) model predicts the critical thickness, L_ic^* (that corresponds to the critical pressure), relative to its unstressed value, L_i0^*, to be ∼14%/13% for the data of Tedgui and Lever (30) and 16%/16% for Baldwin and Wilson (1); our L_ic^*/L_i0^* is 16% for Shou et al. (23) and 15% for Nguyen et al. (19), which both postdate Huang et al. (10).

Fig. 3.

Nonlinear relationship between SI thickness and transmural pressure and comparison with the experimental measurements of Huang et al. (12) of SI thickness in rat aorta fixed in situ at different transmural pressures. Tables 4 and 5 give L_Pm and L_Pe for the 4 data sets. L_i0 is initial SI thickness. For each data set, we determine the SI spring constant as explained in the text and plot Eq. 18 with that value to get a continuous curve. All curves show an almost linear initial decline with pressure that quickly levels off as partial fenestral blockage occurs and maximal compression obtains.

With the use of Eq. 17, Fig. 4 plots the calculated L_Pt for the intact and average L_Pm+I for the denuded vessels as functions of ΔP, compares with each of the four different experimental data sets, and fits a value of k for each data set. Since the rat data L_Pt and L_Pm+I of Shou et al. (23) and rabbit data of Tedgui and Lever (30) overlap, we assume the same L_i0^* = 500 nm for both rabbit and rat data. Agreement is good for all data sets. When the vessel is intact, L_Pt is nearly constant until ∼60 mmHg, after which it drops nonlinearly by ∼40% until it reaches the critical ΔP where the force/area on the endothelium reaches its critical limit for maximal SI compression. L_Pt remains constant for ΔP ≥100 mmHg. L_Pm+I of the denuded vessel is ΔP independent since there is no SI in the denuded vessel to compress and no ECs to block IEL fenestrae. As in Huang et al. (10), this theory explains the observed shapes of the L_Ptand L_Pm+I vs. ΔP curves, including the marked drop in L_Pt over a 60–100 mmHg dynamic pressure range.

Fig. 4.

Comparison of the model's ΔP-dependent (independent) hydraulic conductivity of the intact (denuded) artery wall with the experiments and resulting spring constants, k, from Tedgui and Lever (30) (A) and Baldwin and Wilson (1) (B) on rabbit aortas and Shou et al. (23) (C) and Nguyen et al. (19) (D) on rat aortas. Diamonds (squares) are experimental data for denuded (intact) ECs; dashed and solid lines are corresponding model predictions. Tables 4 and 5 give L_Pm and L_Pe for each of these data sets. The data, and thus the derived k, in C are far different from the other experiments. The high k in A means a high ΔP is needed for maximum compression; the paucity of data means a lower k and ΔP would also fit.

As noted, unlike our use of Eq. 17 for L_Pt, Huang et al. (10) used their calculated pressure field to estimate L_Pe+i (Eq. 10 in Ref. 10) and used it with a constant, data set-specific L_Pm+I value in Eq. 14a to predict L_Pt vs. ΔP. We find (not shown) their method slightly (∼2%) underpredicts L_Pt at low SI compressions but overpredicts it (by 8–20% for L_i^*/L_i0^* ∼0.2-0.15) at high compressions. Our method finds spring constants for Tegui and Lever (30), Shou et al. (23), and Nguyen et al. (19), all ∼30 mmHg (32.7, 27.66, and 30.6 mmHg), but lower (23.68) than the rabbit data of Baldwin and Wilson (1), whose L_Pt are uniformly approximately double the L_Pts of the other rat and rabbit data sets. For the data of Tedgui and Lever (30)/Baldwin and Wilson (1), the method of Huang et al. (10) gave lower ks of 22/16 mmHg. A higher k means a stiffer spring, i.e., the same level of compression requires a higher force/area on the endothelium. Since both models predict roughly the same critical SI thickness, our prediction with a higher k corresponds to a higher critical transendothelial pressure difference, which, given differing L_Pes, may or may not mean a higher ΔP to achieve maximal SI compression. Our predicted critical ΔPs are 124/96 mmHg for the data (Fig. 4, A and B) of Tedgui and Lever (30)/Baldwin and Wilson (1). Huang et al.’s (10) predictions for the same data sets are 135/82 mmHg. We predict critical ΔPs of 100/90 mmHg for the data of Shou et al. (23)/Nguyen et al. (19).

FILTRATION MODEL: TRANSCELLULAR FLOW

Nguyen et al. (19) and Xue (36) observed marked decreases in L_Pe in culture and in L_Pt and L_Pe+i in whole rat aortas ex vivo with AQP1 blocking or knockdown. They infer a significant functional role of AQP1-mediated transcellular flow in determining and controlling transmural water transport through rat arterial walls. What is confounding is the fact that their observed drops in L_Pt with AQP1 blocking differ significantly at different ΔPs. The SI compaction theory of Huang et al. (10) (see filtration model: paracellular flow) yields a critical average force per unit area (P̄_g^* − P̄_i^*) on the endothelium to compress the SI, which occurs with normal AQP1s over the dynamic ΔP range of ∼60–100 mmHg. We propose that blocking AQP1 channels decreases the available pathways for water transport across the endothelium, thereby decreasing L_Pe and consequently P_i^*. At fixed ΔP, this increases the average force per unit area (P̄_g^* − P̄_i^*) acting on the endothelium and shifts a larger fraction of ΔP from the media to the endothelium. The critical force per unit area acting on the endothelium thus obtains at a lower ΔP, which should shift the dynamic range for SI compression to lower ΔP. At high ΔP, where (average) partial fenestral blocking takes place even at normal functioning AQP1 levels, reducing AQP1 function would only lower L_Pe, which should only minimally lower L_Pt, as Nguyen et al. (19) observed. To test this hypothesis, we incorporate transcellular flow into the model in filtration model: paracellular flow and use it to quantitatively explain the contribution of AQP1s to the ΔP-independent intrinsic endothelial L_Pe and the ΔP-dependent vessel wall L_Pt and phenomenological L_Pe+1.

Mathematical Formulation

Transcellular water flow alters only the endothelial (z_i = 1) boundary condition (see filtration model: paracellular flow) from EC impermeability to a nonzero EC hydraulic conductivity, L_PEC, due to AQP1. The endothelium's intrinsic L_Pe is now the area average of L_PEC and the junctional L_Pnj. The junctional boundary conditions, Eqs. 7 and8, do not change, but on the EC (z = 1;0 < r ≤ R) Eq. 19 replaces Eq. 6:

$$-W_j=\frac{L_{{\text{P}}_{\text{EC}}}\mu L_i^{*}\left({\text{P}}_{\text{g}}-{\text{P}}_{\text{i}}\right)}{K_{\text{Pi}}}at{z}_{\text{i}}=1\hspace{0.17em}\left(j=\text{g},\text{i}\right);$$ (19)

$$L_{{\text{P}}_{\text{e}}}\pi {\xi }_I^{*2}=L_{{\mathrm{P}}_{\mathrm{E}\mathrm{C}}}\pi R^{*2}+L_{{\text{P}}_{\text{nj}}}2\pi R^{*}\left(\Delta R^{*}/2\right).$$ (20)

Equation 20 allocates half the junctional area to the cell in question and half, by symmetry, to its neighboring cells. With only this change, we solve Eq. 2 by finite difference exactly as in filtration model: paracellular flow.

Constants and Parameters

The only new parameter not appearing in Constants and Parameters in filtration model: paracellular flow is L_PEC. To determine the AQP1 fraction of L_Pe, whose intrinsic value we calculated from experiment, we take L_PEC to be various fractions of L_Pe. Equation 20 gives the (now lower) L_Pnj corresponding to each assumed fraction.

Results and Discussion

Pressure-dependent intact aorta hydraulic conductivity, L_Pt, for unblocked AQP1s.

Functioning, unblocked AQP1s provide pathways, in parallel with the intercellular junctions, for substantial transendothelial water flow. To predict L_Pt (ΔP), we fix L_Pe, vary the fraction of L_Pe due to L_PEC and calculate the intact wall L_Pt for various SI thicknesses. For each L_PEC fraction, we use the procedure in Presssure-dependent hydraulic conductivity to find the compression that gives L_Pt equal to the measured value in, e.g., the unblocked data of Nguyen et al. (19) at that ΔP. The slope of the resulting compressions vs. the transendothelial force/area curve is the SI ECM spring constant (k); Eq. 18 gives the corresponding SI thickness at intermediate ΔP. With these inputs one solves the model for the pressure field, uses Eq. 17 to find L_Pt, and plots L_Pt vs. ΔP for each assumed AQP1 fraction.

Figure 5 shows L_Pt vs. ΔP for various (open, functioning) AQP1 fractions of L_Pe at fixed k. These curves all have the same L_Pe but different ratios of para-to-transcellular flow. Identical L_Pm+I and L_Pe and experimental L_Pt force all curves to have the same low and high ΔP plateaus. The SI streamlines (not shown) without AQPs are parallel to the EC outside of the junction and fenestra; with AQPs, streamlines emanate from the EC and cause the adjacent streamlines from the junction to slope downwards around the EC-emanating ones as they approach the fenestra. Increasing the AQP1 fraction from 0 to 40% at fixed L_Pe flattens L_Pt(ΔP) in the dynamic range 60–100 mmHg since transcellular flow partially relieves the pressure difference across the EC for a given ΔP. Viewed differently, the shorter flow path traversed by the transcelluar vs. junctional water flow means less head loss across the EC. Thus one needs a higher ΔP to achieve the same trans-EC force/area at the same (e.g., for a single k for all curves, maximal) SI compression corresponding to the data points. As we shall see, such changes in ΔP for maximal compression may be very important. (In the next section we see this effect is much more pronounced with L_Pe not fixed.) Conversely, one can require all curves to look similar by adjusting k for each data set so they also all have the same ΔP for maximal compression (not shown). An AQP1 increase then lowers the trans-EC pressure, i.e., force/area, corresponding to the same L_Pt (i.e., essentially the same compression); this means a lower k. For 40% AQP1, k is 1) 27.98, 2) 19.17, 3) 23.84, and 4) 27.9 mmHg, indeed below the k values in Fig. 4 for 0% AQP1(no transcellular flow).

Fig. 5.

Effect of various aquaporin-1 (AQP1) fractions (open and functioning; L_PEC) at fixed total L_Pe = 8.29 × 10⁻⁸ cm·s⁻¹·mmHg and k on the ΔP-dependent hydraulic conductivity of an intact artery wall and comparison with the experimental data of Nguyen et al. (19). For each L_Pe curve, Eq. 20 determines L_Pnj. Symbols: experimental data; lines: model predictions. L_Pm and L_Pe determine the low ΔP and the experiment the high ΔP plateaus, both the same for all curves. The more of L_Pe that L_PEC accounts for the milder the drop between plateaus and thus the higher the ΔP for maximal SI compression.

Pressure-dependent intact aortic hydraulic conductivity, L_Pt, for blocked AQP1s.

Submilimolar concentrations of HgCl₂ block trans-AQP1 channels in red blood cells (32). Nguyen et al. (19) measured L_Pt at three ΔP values, flushed with 5 μM HgCl₂ to block EC AQP1s, remeasured L_Pt(ΔP), denuded, and measured L_Pm+I(ΔP), all on the same vessel. Titration curves on red blood cells suggests that 5 μM HgCl₂ blocks <100% and possibly as low as approximately one-third of AQP1s (37). However, since a transcellular pathway likely consists of more than one AQP1 in series, the number of blocked transcellular pathways may greatly exceed the number of blocked AQP1s. We get a lower bound on the percent AQP1 contribution to L_Pe by assuming 5 μM blocks all (100%) transcellular pathways and comparing model predictions for various assumed AQP1 fractions with the data of Nguyen et al. (19). Specifically, for an unblocked L_Pe from above with an assumed L_PEC (and L_Pnj from Eq. 20), we shut all AQP1s (set L_PEC = 0, i.e., 100% blocking), which lowers L_Pe, and compare the model's L_Pt(ΔP) prediction with the blocked data of Nguyen et al. (19). (Note that to predict AQP1 blocking's effect on L_Pt, one must use the same k for both unblocked and blocked AQP1 L_Pt). If these L_Pt(ΔP) are above the blocked data (≤100% blocking would worsen the comparison), we raise the AQP1 fraction; if they are below the blocked data, we either decrease the AQP1 fraction or assume <100% blocking to match the data.

Figure 6 compares the data of Nguyen et al. (19) with the model-generated L_Pt(ΔP) with HgCl₂ for various L_PEC fractions. The figure includes the intact-unblocked, intact-blocked, and denuded vessel data and the corresponding theoretical curves, with the intact-unblocked curve computed for an AQP1 fraction of 40% of L_Pe (Fig. 5). These predictions show a significant decrease in blocked-AQP1 intact aorta L_Pt with increased AQP1 fraction, including a small decrease after SI compression. More interesting, the higher this fraction the lower is the critical ΔP to achieve the force/area for maximal SI compression: the curves in Fig. 6 retain their shape but shift down (decreased L_Pe) and to the left as the AQP1 (L_PEC) fraction that is blocked increases. The curves for 30–40% AQP1 fraction, 100% blocked, match the vessel data. The in vitro data of Nguyen et al. (19) show AQP1 blocking reduced water flux across BAECs by ∼22.1 ± 6% but AQP1 knockdown lowered RAEC monolayer L_Pe by 56.4%, a small portion of which (∼17%) may be due to reduced junctional transport (19); thus 30–40% AQP1 is not unreasonable.

Fig. 6.

Effect of AQP blocking for different AQP1 fractions of L_Pe at fixed, unblocked L_Pe on the ΔP-dependent hydraulic conductivity of an intact artery wall, including a comparison with the experimental data of Nguyen et al. (19). Symbols: experimental data; lines: model predictions. Blocking AQP1s (solid lines) for a given AQP1 fraction leaves L_Pnj unchanged but shuts off (sets to 0) L_PEC and thus lowers L_Pe. L_Pnj for all curves is the same as that in Fig. 5; solid curve is the same as the 40% AQP curve in Fig. 4D. The larger the AQP fraction of L_Pe, the larger the HgCl₂ effect; 30–40% AQP1 fit the data best.

EXPERIMENTAL PREDICTIONS.

At 60 mmHg, the theory predicts a fully compressed SI with HgCl₂ but only a partially compressed SI and no fenestral blockage without it. Direct SI thickness measurement (under way) for vessels fixed under pressure as in Ref. 12, either with normal, elevated, or blocked AQP1s, will show if, as predicted, there is a large difference at 50 mmHg but not at 100 and 150 mmHg, between aortas fixed with or without HgCl₂. Our theory calculates that the force/area acting on the endothelium at 60 mmHg with functioning AQPs is the same, with the same slight SI compression and absence of fenestral blockage, as that acting at 43 mmHg with blocked AQP1s. However, HgCl₂ treatment still lowers L_Pe. The theory thus predicts L_Pt at 43 mmHg with HgCl₂ is 2.11 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹, between L_Pt with and without HgCl₂ at 60 mmHg (1.75 vs. 2.55 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹). Even with HgCl₂, sufficiently lowering ΔP should decompress the SI and raise L_Pt: the model predicts that, with HgCl₂, reducing ΔP from 60 to 20 mmHg increases L_Pt from 1.75 to 2.185 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹. Nguyen et al. (19) found L_Pt rose to 1.9 ± 0.2 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹ (SE) at 20 mmHg with HgCl₂, consistent with this prediction.

Nguyen et al. (19) found, with HgCl₂, L_Pt = 2.00 ± 0.28 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹ at ΔP = 140 mmHg, slightly (reproducibly, but not statistically significantly) higher than L_Pt at 60 and 100 mmHg. Hints of a small rise at high ΔP also appear in HgCl₂-free studies (1, 23, 19). These may simply be the result of added stress on the cannulation ties. Toxicity is unlikely: reducing the Hg²⁺-cis bond with 2-mercaptoethanol recovered L_Pt values to within 5% of baseline at the tested ΔP = 60 and 100 mmHg. Any conceivable EC toxicity at 140 mmHg would have caused a far larger L_Pt jump than observed. Since HgCl₂ raises the force/area on the EC at fixed ΔP, high ΔP (140 mmHg) and HgCl₂ might stretch EC junctions slightly, thus slightly elevating L_Pnj, L_Pe, and L_Pt. The model calculates that the force/area on the endothelium with HgCl₂ at P = 140 mmHg equals that without HgCl₂ at ∼200 mmHg. L_Pt measurement without HgCl₂ at 100, 140, and 200 mmHg and with HgCl₂ at 140 mmHg on the same vessel would confirm whether without HgCl₂ L_Pt rises significantly (stretching) at 200 vs. 140 mmHg and whether L_Pt at 140 mmHg with HgCl₂ only falls short of L_Pt at 200 mmHg without HgCl₂ by an amount due to setting L_PEC to zero at fixed SI compression. With no change in L_Pnj, the model [AQP1 30–40% of L_Pe; L_Pe derived from the data of Nguyen et al. (19) at 140 mmHg] predicts a 200 mmHg no blocker L_Pt = 2.18 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹.

Effect of AQP1 upregulation on force/area on EC.

Since decreasing functioning AQP1s lowers the critical ΔP for SI compression, could increasing functioning AQP1s decompress the SI and substantially raise L_Pt in the physiological range? Increased transmural water flow might slow LDL retention kinetics. Figure 7 shows how upregulating AQP1s (transcellular flow) would change the force/area on the ECs. Each curve in Fig. 7 starts out nearly proportional to ΔP since L_Pe is fixed for each curve and early compression causes no fenestral blocking. At a certain ΔP, SI compression starts to cause fenestral blockage, which lowers L_Pe+i and L_Pt (the intrinsic L_Pe is fixed) and nearly balances the higher driving ΔP; this flattens the curve. Beyond maximal SI compression, the geometry and L_Pt remain fixed and the material becomes simple, i.e., the transendothelial force/area (proportional to the transendothelial or transmural flow) is linear in ΔP. An increase in functioning AQP1s (trans-EC flow pathways) shifts the transendothelial force/area curves in Fig. 7 to the right, meaning a given force/area on the EC, e.g., the critical force/area for SI compression, requires a higher, potentially superphysiological, ΔP. This may be beneficial in flushing unbound lipid from the wall.

Fig. 7.

Effect of increasing functioning AQP1 number, i.e., fraction of L_Pe, on the force/area acting on the endothelium as a function of ΔP. Baseline results assume that AQP1s are 40% of L_Pe. The corresponding L_PEC = 3.32 × 10⁻⁸ cm·s⁻¹·mmHg⁻¹ (Eq. 20), L_Pnj = 2.99× 10⁻⁵ cm·s⁻¹·mmHg⁻¹ and k = 27.9 mmHg; L_Pnj and k are fixed for different L_PEC fractions of L_Pe. All curves rise linearly as rising ΔP compresses the linear spring until partial fenestral blockage redistributes that total ΔP. Break in slope of all curves occurs at the same force/area on the endothelium, indicating maximal compression, after which no further compression occurs, the material becomes simple and the curves again rise linearly.

Effect of AQP1 upregulation on L_Pt.

The model predicts (Fig. 8) that increasing transcellular transport of an intact vessel wall by increasing the number of functioning EC AQP1s significantly increases L_Pt before full SI compression. As Fig. 8 depicts, more AQP1s make the L_Pt(ΔP) curve flatter (as already seen in Fig. 5) and shifts it up and to the right; the inverse of AQP1 lowering seen in Fig. 6. This shift raises the critical ΔP [for the data of Nguyen et al. (19)] to 90, 95, 104, 110, and 127 mmHg for successive 25% increases in absolute AQP1 expression. Thus the more active the transcellular pathways, the higher the ΔP needed to achieve the critical force/area for maximal SI compression, i.e., the farther right in ΔP the shift in the dynamic range of L_Pt. This prediction suggests a possible intervention: increase vessel wall L_Pt by decompressing the SI in the physiological ΔP range. We anticipate L_Pt measurements on vessels with upregulated EC AQP1s will show a significant L_Pt rise in the region, ΔP ∼70–95 mmHg, where the dynamic regime shifts, and only a small change beyond 110 mmHg, where the SI is fully compressed even with upregulated AQP1 (unpublished observations, Raval C).

Fig. 8.

Effect of increasing the number of functioning AQP1s, i.e., L_PEC, at fixed L_Pnj, on L_Pt of an intact artery wall. Baseline results assume that AQP1s are 40% of L_Pe; from Eq. 20, L_PEC = 3.32 × 10⁻⁸ cm·s⁻¹·mmHg and L_Pnj = 2.99 × 10⁻⁵ cm·s⁻¹·mmHg⁻¹. L_Pnj and k (=27.9 mmHg) are the same for all curves. Increasing AQP1 shifts the critical ΔP for SI compression and thus the dynamic range of L_Pt to higher ΔP s, potentially into the physiological regime. A decompressed SI substantially raises L_Pt.

CONCLUSION

Water flux across the artery wall plays an important role in prelesion atherosclerosis. Fluid mechanics models for these flows in a layered wall comprised of an endothelium, a compressible subendothelial intima, a fenestrated internal elastic laminar, and a media only appear to admit analytic solution for approximate descriptions of the flow in the fenestra (10). By using a numerical procedure that solves for the flow with a more realistic fenestral description, we improved on this approach and confirmed that pressure-induced intima compression can partially block IEL fenestrae and drastically lower wall L_Pt. Based on recent experiments (19), we then extended this model to include, for the first time, the role of cell membrane AQP1 in transendothelial water transport. We first found that reallocating a portion of the endothelial hydraulic conductivity from the intercellular junction to the cell body smoothed the intima's pressure-induced compression. Then, rather than reallocating the water permeability, we tested the effects of changing AQP1 number at fixed junction permeability. We thus explained the perplexing HgCl₂-induced changes in the pressure dependence of vessel wall hydraulic conductivity in Ref. 19 by showing that an increase (decrease) in the number of transcellular AQP1 pathways shifts to higher (lower) pressures the dynamic pressure range over which L_Pt drops from its uncompressed to its maximally compressed subendothelial intima value. Thus blocking AQP1s causes SI compression at lower ΔP than with normal, functioning AQP1s and, together with the lower intrinsic L_Pe, significantly reduces L_Pt in the shifted pressure regime. The agreement with Nguyen et al. (19) not only supports the hypothesis that AQP1s provide a significant transcellular water transport pathway but also quantifies this contribution by estimating that AQP1 is responsible for 30–40% of transendothelial water transport at normal aortic AQP1 levels. We also calculated that the force acting on the endothelium at 60 mmHg with functioning AQP1s is same as that at 43 mmHg for blocked AQP1s. This suggests that the SI in the presence of HgCl₂ should decompress at ΔPs <43 mmHg to a level that the untreated SI is at 60 mmHg (although L_Pt would still be lower with blocker at 43 mmHg than without blocker at 60 mmHg due to the blocker-reduced L_Pe). Data at 20 mmHg support this prediction. Since it may be clinically more desirable to increase, rather than to decrease L_Pt, we also examined the case of elevated endothelial AQP1 expression. In addition to its prediction of how AQP1 numbers directly affect SI thickness at different ΔPs, a strong test of the model is its prediction that raising endothelial AQP1 drastically increases the critical ΔP for the L_Pt drop with SI compression from 80 to 100 mmHg to well into the physiological regime (>100 mmHg). This could lead to a substantially higher physiological L_Pt and thus transmural water flow. Higher flows may slow lipid accumulation rates that can lead to early atherosclerotic lesions. Such new understandings of the nature of the transendothelial water flow may suggest new potential therapeutic targets for slowing the early progress of atherosclerosis.

GRANTS

This work was supported by National Heart, Lung, and Blood Institute Grant 1R01-HL-067383 and the National Science Foundation Grants IOS-0922051 and CTS-0077520.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: S.J. performed calculations; S.J. and D.S.R. constructed the model and analyzed data; S.J., K.-M.J., and D.S.R. interpreted results of calculations; S.J. prepared figures; S.J. and D.S.R. drafted manuscript; S.J. and D.S.R. edited and revised manuscript; S.J., K.-M.J., and D.S.R. approved final version of manuscript; K.-M.J. and D.S.R. conception and design of research.

Glossary

a^∗

Effective radius of proteoglycan aggregates

hj

Ratio of thickness of region j to radius of fenestral pore

j

Region of artery wall (j = g, i, m for GX, SI, and media, respectively)

J_v

Water flux

k

Elastic coefficient of SI

K_Pj

Darcy permeability of region j

L_i0^*

Thickness of SI at zero luminal pressure

L_ic^*

Critical thickness of SI layer

L_j^*

Thickness of region j

L_P

Hydraulic conductivity

L_Pe

Intrinsic hydraulic conductivity of the endothelium

L_PEC

Hydraulic conductivity of the endothelium attributed to its AQP1s

L_Pe+i

Hydraulic conductivity of the endothelium + SI

L_Pi

Hydraulic conductivity of SI

L_PI

Hydraulic conductivity of IEL

L_Pm

Hydraulic conductivity of media

L_Pm+I

Hydraulic conductivity of IEL + media

L_Pj

Hydraulic conductivity of normal junction surrounding an EC

L_Pt

Total hydraulic conductivity of the vessel wall

P_j^*

Dimensional pressure in region j

P_j

Dimensionless pressure in region j

P̄_j^*

Dimensional average pressure in region j

P̄_j

Dimensionless average pressure in region j

P_L^*

Dimensional lumen pressure

P_Lc^*

Dimensional critical pressure

r^∗

Dimensional radial coordinate

r

Dimensionless radial coordinate

r_f^*

Radius of fenestra

U_j^*

Dimensional lateral velocity in region j

U_j

Dimensionless lateral velocity in region j

R^∗

Dimensional radius of EC

R

Dimensionless radius of EC

V

Velocity vector

W_f^*

Dimensional water velocity through fenestra

W_j

Dimensionless normal velocity in region j

z^∗

Dimensional normal coordinate

z

Dimensionless normal coordinate

ΔR^∗

Dimensional width of normal junction

ΔR

Dimensionless width of normal junction

ΔP

Dimensionless pressure drop across the vessel wall

Δπ

Dimensionless osmotic pressure difference across a membrane

μ

Viscosity of fluid

ξ_l^*

Dimensional radius of periodic wall unit

ξ_I

Dimensionless radius of periodic wall unit