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1Department of Chemical Engineering and 2Department of Mechanical Engineering, City College of the City University of New York; and 3Department of Medicine, College of Physicians and Surgeons of Columbia University, New York, New York
Submitted 9 June 2006 ; accepted in final form 5 January 2007
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ABSTRACT |
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 TOP ABSTRACT MATHEMATICAL MODELSRESULTS AND DISCUSSIONAPPENDIX: SOLUTION METHODSGRANTSREFERENCES |
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= 2.28 x 10–16 cm2) and LDL diffusion coefficient [D2(LDL) = 5.93 x 10–9 cm2/s], using one of Tompkins et al.'s profiles, and fixes them throughout. It accurately predicts Part I's rapid localized HRP leakage spot growth rate in rat leaflets that results from the intima's much sparser structure, dictating its far larger transport parameters [K
= 1.10 x 10–12 cm2, D1(LDL/HRP) = 1.02/4.09 x 10–7 cm2/s] than the middle layer. This contrasts with large arteries with similarly large HRP spots, since the valve has no internal elastic lamina. The model quantitatively explains all of Tompkins et al.'s monkey profiles with these same parameters. Different numbers and locations of isolated macromolecular leaks on both aspects and different section-leak(s) distances yield all profiles.
convection-diffusion; two-dimensional transport; variation in LDL profiles
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7 nm) largely cross the endothelia and transport into the walls of blood vessels by passing through transient, localized endothelial leaks, rather than uniformly, and spread rapidly in the subendothelial intima. These leaks are often attributable to widened junctions of endothelial cells (ECs) that are either dying or dividing (8, 15, 19–21), This so-called leaky junction-cell turnover hypothesis suggested that a tiny fraction of leaky vascular ECs could account for the experimentally observed local variations in endothelial macromolecular permeability in large arteries (19). Motivated by this earlier work in vessels (8), our experimental studies with horseradish peroxidase (HRP) in the first of the papers in this series (Ref. 46; Part I, this issue) sought to discover whether macromolecular leakage into the valve was similar. En face interrogation under light microscopy of aortic valve leaflets from normal Sprague-Dawley rats after short-time HRP circulation showed only very few, localized HRP leaks, seen as isolated brown spots in the endothelium after histochemical treatment (Fig. 1 in Ref. 46), rather than a uniform brown staining. For rats, this contradicts the earlier model's (34) first assumption and indicates that macromolecular transport in heart valves depends on the direction parallel to, as well as the direction normal to, the endothelium, i.e., transport is at least two-dimensional (2D).
Part I also examined the growth of these HRP leakage spots with increasing HRP circulation time (Fig. 2 in Ref. 46). This growth is extremely fast, from roughly 12- to almost 50-µm radius in 1 min [of which 25 s was needed for the tracer to reach the aorta (16) or valve (see 
Fig. 10) from the point of infusion in the femoral vein] and to 75 µm in 4 min. In fact, considering that the valve leaflet is closed, and thus subject to a transverse pressure gradient, for only 60–70% of the cardiac cycle, the spot growth is comparable to that in the aorta. It is unlikely that molecular diffusion alone can account for it without an unrealistically high diffusivity (at least 10–6 cm2/s). Hence, the model should allow for convection, to see whether it is indeed significant, as it turns out to be in large arteries (16). The different lumen pressures on the endothelia of the leaflet's aortic and ventricular aspects when the aortic valve is closed would provide the driving pressure for convection across the valve. The difference in the hydraulic conductivities of the (rare) leaky and the (vastly more numerous) normal EC junctions would result in a subendothelial pressure gradient. This gradient parallel to the endothelium could drive a convection that could account for the rapid growth of HRP leaky spots.
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150-nm, region of intense staining, followed by a step change to much more diffuse staining. This likely indicates the existence of a sparse layer immediately beneath the valve's endothelia where the HRP concentration is much higher than that in the matrix below it (Fig. 3 in Ref. 46). We called this layer the "valvular subendothelial intima," in analogy to the vascular subendothelial intima, and the balance of the interposed matrix the "middle layer." Nievelstein-Post et al.'s (22) electron micrograph of an ultrarapid freezing/rotary shadow etching of a normal rabbit valve's subendothelial space showed a clear step change in matrix porosity from a sparse, immediately subendothelial region of the same dimension to a much thicker, dense region. The much sparser structure of the valvular intima allows for a much larger available volume for fluid and macromolecules. In particular, the typical spacing between the dominant matrix fibers and collagens in the valve intima is 30–40 nm in Fig. 4 of Ref. 46, which is the same as that in the aortic intima (11, 16, 23). The fiber matrix theory that these references expound leads to ab initio calculated values for the intima transport parameters that are the same as those in the aortic intima. As we show here, even in the presence of convection, the existence of sparse intima regions is necessary to account for the observed rapid growth of HRP leaky spots. The new experimental results in Part I (46) guide us to advance a 2D convection-diffusion model in which the valvular intimae are much thinner and sparser than the balance of the interposed matrix. We use this model to explain the rapid growth of HRP spots in rat valves noted above. Whereas this model is based on rat (and some rabbit) valve data, we apply it to the monkey valve as well in an attempt to explain the observed large variations in Tompkins et al.'s transvalvular LDL concentration profiles in Fig. 1. We suggest that this variation might have resulted from the different number and position of leaky ECs on the valve's endothelia and from the position of the tissue sections examined relative to those leaks. Rather than using a very different set of parameters for each transvalvular LDL concentration profile as in Ref. 34, we use a single parameter set to rationally explain all of Fig. 1's experimental monkey profiles. The close agreement between theory and experiment for these profiles and for the HRP spot growth (46) strongly supports our model.
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MATHEMATICAL MODELS |
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TOPABSTRACTMATHEMATICAL MODELSRESULTS AND DISCUSSIONAPPENDIX: SOLUTION METHODSGRANTSREFERENCES |
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i
, t
P*
R
Superscript and Subscripts
Model description. Figure 2 illustrates the proposed aortic valve model. It is comprised of two endothelia, defining the aortic and ventricular aspects. Each endothelium contacts the lumen on the outside and a thin intima on the inside. A thick middle matrix layer lies between the two intimae. This sometimes mentioned (14, 28) but seldom emphasized valve intima (thickened when diseased; Refs. 6, 24) plays a critical role in short-time macromolecular transport in heart valves. Each of these matrix layers is assumed to be a uniform, isotropic, homogeneous, and porous medium. Since we have no reason to assume otherwise, we assume that the two valve intimae are morphologically and histologically identical.
A rat aortic valvular leaflet has an area of 1 mm2 covered with ECs of an average radius of 11.7 µm (46), i.e., 2,300 ECs/aspect. In the most lesion-susceptible regions of the rat aorta, approximately 1 in 2,000–6,000 ECs (19) (1/5,000 in rabbit; Ref. 1) has a junction that leaks at any given time. If the valve leaflet's leakage frequency is similar, one expects 0–2 leaks per aspect at any time, consistent with Part I's (46) HRP spot frequency in rat valves of 0–3 leaks per leaflet. We assume that these leaky cells can occur on the endothelia of both aspects, and Part I found these leaks more likely to occur near the line of coaptation. That the leakage frequency estimate is consistent with observation is interesting, but since modeling HRP spot growth only assumes the existence of a leak, the leakage frequency and location do not enter as long as leaks are not too close together (see next two paragraphs). Similarly, since the purpose of our application of the model to the monkey data is to present a plausible explanation of what may have occurred in Tompkins et al.'s (34) specific measurements and not an average behavior over a large number of measurements, the leakage frequency and distribution are, again, not germane.
The model in Fig. 2 presumes that the local thickness of the leaflet does not change appreciably over distances of several EC radii, so that one can take the endothelia to be parallel. The model is based on an axisymmetric cylindrical slab of radius and height L, the valvular leaflet's local thickness. This cylindrical assumption encompasses three cases (Fig. 2C), each with a different region assumed to be circular. 1) If there is a single leak located at or near the center of the aspect, the entire leaflet is assumed circular; is then the leaflet radius, 564 µm. 2) If there is more than one leak on an aspect, we divide the leaflet surface into (by abuse of geometry) circular periodic units whose area equals that of the leaflet surface divided by the instantaneous number of leaks, e.g., 400 µm for two leaks in an aspect. The spacing between adjacent leaks is 2, as in arteries (16, 19, 37, 45). 3) If the leak is far from the leaflet's center, the circular region surrounds the leak and extends to the edge of the leaflet, a distance away. As shown below, leaky junction influence on transport extends less than 15R1, 
175 µm in the direction parallel to the endothelium; thus the solutions are insensitive to the precise value when > 175 µm.
If the leaflet has no leaks, the problem reduces to one dimensional in z normal to the endothelial surface from the aortic z = 0 to the ventricular aspect z = L, the local leaflet thickness. With 
1 leak, the problem becomes 2D with dependence on r, parallel to the endothelial surface, from the center of the leaky cell to . Axisymmetry requires simultaneous leaks on both aspects that are close in r to both be located at r = 0. If they are far apart in r, their interaction is minimal and we can superimpose two single-leak solutions. Although this violates randomness, ventricular aspect leaks where convection opposes diffusion are far less important than aortic leaks where it reinforces diffusion.
Filtration model.
Over the cardiac cycle, the aortic valve periodically opens when the pressure in the left ventricle is higher than that in the aorta and closes otherwise. The pressure on the aortic valve leaflet's ventricular aspect is negligibly higher than on its aortic aspect when the valve is open (30–40% of the time), but is much lower (90 mmHg) when the valve is closed. As a simplification, we assume a steady, nonzero, time-averaged pressure difference of P* = 50 mmHg, below the time averaged aortic transmural pressure of 100 mmHg between the two aspects. This gradient drives the overall convection through the leaflet from the high-pressure aorta toward the lower-pressure ventricle. Water enters or leaves the leaflet through both normal and leaky endothelial junctions. The pressure drop across the leaky junction is much smaller than across the normal junction since its hydraulic conductivity (Lp) is much larger. Thus the pressure near the leaky junction is higher than that far from the leaky junction in the aortic aspect's intima, and vice versa in the ventricular aspect's intima. These radial pressure gradients and the intima's relatively large permeability drive radial intimal flows, giving big spots.
U* (radial velocity) and W* (normal velocity) [V* = (U*, W*)] represent the local volume-averaged superficial water velocities, equal to the fluid's volumetric flow rate divided by the total cross-sectional area including the area of the fluid and the tissue. V* is governed by Darcy's law, in r and z. Lp and Kp are the hydraulic conductivity and permeability, each carrying a subscript denoting the region the parameters represent. The subscripts v and a represent the ventricular and the aortic aspects. The subscripts nj (j = a, v) and lj (j = a, v) represent the normal and leaky junctions at both the aortic and the ventricular aspects. Lpnj is the homogenized, area-averaged value representing both normal junctions and cells. We shall be agnostic as to whether this solvent transport is purely around the ECs or through them as well, and thus assign the same value to the cell r < R1, where R1 is the average EC radius. (This complex topic is the subject of a forthcoming detailed experimental/theoretical investigation.) Clarification of possible transcellular flow may reduce the endothelial Lp for r < R1, but since the area fraction and Lpnj/Lplj are negligible, its water flux and influence on the problem are also negligible (see Fig. 11, curve 2a, for verification). A similar argument pertains to the mass transfer problem and, in particular, to Eqs. 20b and 22b below.
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P* is the difference between the lumen (Pa) and ventricular (Pv) pressures. One symbol represents the same dimensional (with *) and nondimensional (without *) variable. The coordinates are normalized by the average thickness, L, of the leaflet
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P* as follows:
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 | (3) |
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Each porous matrix layer satisfies continuity, constant fluid mass density, and Darcy's law:
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 | (5) |
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(periodicity or having reached the leaflet edge) give
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R = R2 – R1 be the average leaky junction width. The (linear) z boundary conditions are as follows. If there is a leak from R1 < r < R2 at z = 0,
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 | (8b) |
 | (9) |
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denote the endothelium with EG3 and normal junctions.
At the interfaces z = L1 and z = L1 + L2 between the middle layer and the aortic and ventricular aspects' intimae, pressure and normal velocity are continuous, giving matching conditions
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 | (13) |
Macromolecular transport model.
Macromolecular transport in heart valves satisfies conservation at constant mass density. The model neglects solute binding to extracellular matrix [weeks (8, 15, 19–21)], valid over the timescales, 30 min, of the experiments considered. It also neglects cellular uptake and metabolism of tracer: healthy intimal extracellular matrix is cell free, and the endothelial surface receptors are normally already completely bound with unlabeled LDL. Only small numbers of free receptors recycle to the cell surface on the experimental timescale. The conservation equation is
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 | (15) |
 | (16a) |
 | (16b) |
the nondimensional time, and Di the effective diffusion coefficient. fi is the retardation coefficient, the ratio of solute velocity to water velocity, and i is the volume fraction for macromolecules, the fractional volume available for macromolecules per unit total tissue volume. Peir and Peiz are the Péclet numbers in r and z, whose velocities, U* and W*, come from the filtration model.
Because the width of the leaky junction (20–100 nm; Ref. 7) is comparable to the diameters of macromolecules, e.g., LDL 23 nm, that traverse it, the macromolecular concentrations at the subendothelial and lumen sides of the leaky junction differ (40, 43). According to Tzeghai et al.'s (37) simple 1D, Cartesian, quasi-steady convection-diffusion model, shown in Fig. 2B, the nondimensional macromolecular flux ql through the leaky junction is (x is normal to the endothelial surface):
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Define the Biot number as the ratio Bij := kjL/D2 (j = a, v), of the rate of endothelial mass transfer to the rate at which diffusion sweeps it away; kj is the (uniform, except for the leaky junction) endothelial mass transfer coefficient for diffusive transport across the endothelia of both aspects. The z = 0 and z = 1 boundary conditions are as follows. If there is a leak at z = 0,
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Axisymmetry at r = 0 and periodicity at r = give
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When the tracer reaches the lumen adjacent to the valve leaflet, the tissue is tracer free:
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Constants and parameters.
Table 1 summarizes the baseline values of the constants/parameters in the models. There is far less experimental data and theoretical work on the valvular than the arterial endothelium. The endothelium is one of the largest "organs" in the body (17). As such, and because of the paucity of data, we assume identical endothelial parameters for both aspects, i.e.,
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 | (29) |
equal to 1 – the total fiber volume per unit total volume. Reference 16 also estimates due to the collagen. Curry (9) sets = exp[(1 + )(a/rf + 1)2], where a and rf are the radii of solute and fiber, respectively, when there is a single fiber present. Since both collagen and proteoglycan are present, we set equal to the product of two such factors, one for each fiber type. By varying the fiber spacing from 30 to 40 nm and the geometric parameters describing the proteoglycan structure over the ranges defined in Ref. 16, we find that 1 varies from 0.49 to 0.63; we work with 1 = 0.6. In Fig. 11, we demonstrate that using 0.5 (curve 2b) leads to a negligible change in the results, compared with curve 2. Because of the almost identical characteristic sizes that one measures from ultrarapid freezing/rotary shadow etchings for the valvular intima (22, 46) and the aortic intima (11, 16), this fiber matrix theory predicts that K*p1 (K*p3) and D*1 (D*3) are nearly equal to the corresponding aortic intima parameters. Part I (46) estimates the thickness, L1(L3), of the intima. Because of the paucity of the data, unless otherwise indicated we do not distinguish between the rat (16, 46) and squirrel monkey [Tompkins et al. (34)] aortic valve endothelial and intimal parameters. Reference 34 measured the volume fraction, 2, in vitro. Although, strictly speaking, Ref. 34 only measured the volume fraction of the whole valve leaflet, this value is very close to 2, because the valvular intima comprises a negligible fraction of the whole valvular leaflet. The retardation coefficient, f2, relates to the volume fraction via (9)
 | (30) |
w2 is the fractional volume, 0.5 (32), available for water in the middle layer. If one neglects the very small pressure difference between the ventricle and aorta during systole, one can roughly estimate P* from the aorta's diastolic blood pressure multiplied by the diastole fraction of the cardiac cycle. We have devised a fitting scheme, based on one of the transvalvular LDL concentration profiles in Fig. 1, for the remaining three valvular transport parameters, ka (kv), K*p2, and D*2. According to our 2D model's Ansatz, once fitted, these parameters are fixed, at least for any aortic valve from the same species of (normal) animal. Our working hypothesis is that the observed variations between the transvalvular LDL concentration profiles in Fig. 1 result simply from the number and location of leaky cells in the valvular endothelia and the location, relative to such leaks, from which the tissue sections were taken. If, for a particular set of serial sections, there happened to have been no leaks in the endothelia of either aspect of that aortic valve, the corresponding transvalvular LDL concentration profile would be the lowest possible. This follows because, other than through leaky junctions, LDL has difficulty traversing the normal endothelium. The corresponding profile should have a slightly higher concentration at z = 1 than at z = 0 because of the small overall convection toward the ventricular aspect that is present. Profile 9, the lowest profile, in Fig. 1 is the most likely representative of this case. Conveniently, in the absence of leaks in either aspect, the filtration and transport models reduce to 1D, with only a single spatial variable z. All dependent variables become independent of r, i.e., 
/r 
0, and Eqs. 4, 6, and 14 simplify dramatically to
 | (31) |
 | (32) |
 | (33) |
/r 0. It is obvious that Wi and Pi are uniform and linear functions of z, respectively, and lend themselves to easy analytical solutions with the corresponding boundary and matching conditions. With the substitution of this velocity into the convection-diffusion equation, Eq. 33, the concentration distribution lends itself to numerical solution by the standard Crank-Nicolson implicit finite difference method (30). (The matching conditions make it difficult to ascertain self-adjointness, which would be necessary for a direct analytic solution in terms of separation of variables, and the Laplace transform solution has far too many terms to be enlightening.) Thus one can obtain the remaining three valvular transport parameters, ka (kv), K*p2, and D*2, by fitting the 1D convection-diffusion model, Eqs. 31–33, to concentration profile 9 in Fig. 1. In this fitting, the search for the least square error between the theoretical and experimental values is a complicated global minimization process. There are no explicit functional expressions to describe the relationship between the square error and the values of the three independent variables, and the square error may have a large number of local minima. We choose Bremermann's method of unconstrained global optimization (2, 3) to find optimized values of ka (kv), K*p2, and D*2 that minimize the square error. This method is based on the chemotactic behavior of bacteria and efficiently and rapidly finds very good maxima or minima of a real-valued function of many variables, even when the function has many local maxima and/or minima.
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RESULTS AND DISCUSSION |
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TOPABSTRACTMATHEMATICAL MODELSRESULTS AND DISCUSSIONAPPENDIX: SOLUTION METHODSGRANTSREFERENCES |
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2 = 0.03–0.16 (34), is much smaller than that in the valvular intima, 1 = 0.6 (44), provides further support. Since K*p2 is much smaller than K*p1, the filtrating velocity, i.e., the overall z-convection, in the valve is small, because of the dense structure of the middle layer's matrix. In fact, Tompkins et al. (34) neglected convection in their 1D model (which also predated the discovery of nonuniform leakage/leaky junctions and the role of convection on macromolecular transport in the aorta), yet each of their individual fitted curves matched the shape of the corresponding data well. Unfortunately, their model required different parameters for each data set.
The fitted value of D*2 for LDL in the squirrel monkey valve's middle layer is 10 times that, 5.4 x 10–10 cm2/s (LDL) (36), estimated in the arterial media from a 1D model including both convection and diffusion. Paradoxically, the middle layer hydraulic conductivity, L*p2 = K*p2/(µL2) = 0.42 x 10–8 cm·s–1·mmHg–1, appears smaller than the measured hydraulic conductivity, L
, of rat (4.76 x 10–8 cm·s–1·mmHg–1; Ref. 27) or rabbit (5.36 x 10–8 cm·s–1·mmHg–1; Ref. 33) aorta with denuded endothelium, i.e., media plus a very small internal elastic layer (IEL) correction. These numbers require discussion. Slight overestimate of L due to media hydration (27) or due to erroneously setting the valve's and aorta's endothelial Lp equal, despite known differences between the two [e.g., their ECs align differently with flow (4)] is unlikely to be responsible. That L L
3 x 10–8 cm·s·mmHg–1 > L
, where L is the Lp of intact aorta, negates the former, while small errors in the latter are likely negligible since the leaflet's middle layer resistance dwarfs its endothelial resistance. This Kp2 does not appear to be in error: one can try using the aortic medial Kp value instead of K in the calculation of LDL profiles corresponding to Tompkins et al.'s measurements below. The magnitude of transvalvular convection becomes so large that it sweeps all calculated profiles discussed below significantly toward the ventricular aspect, where they become far too high. The disparity in Kp and in LDL diffusivities does not contradict Weind et al.'s (42) finding that the effective oxygen diffusivity in the aorta, femoral vein, and valve leaflet are all in the same range. Their measured D values derive from measurements of oxygen arriving at a sensor at one end of a porcine leaflet interpreted as solely arriving due to diffusion, i.e., using models that do not include convection. The fit diffusivity then actually represents the combined effects of diffusion and convection, and its value being similar in the arteries and the leaflet is consistent with one parameter being higher and the other lower in the leaflet, by roughly the same factor.
The paradox may simply be due to species dependence of the precise parameter values. Alternatively, it is possible that both the D and Kp ratios are indeed approximately correct. The valve middle layer's matrix may be slightly denser (on average over its denser lamina fibrosa and ventricularis and looser lamina spongiosa) than the aortic media's. LDL space measurements in these tissues are not sufficiently sharp to address this, but it appears plausible. Tedgui and Lever (32) find that the aortic media's albumin space is 0.08, whereas others (5, 12, 35) give values of 0.09–0.17 for it. Tompkins et al. (34) find that the LDL space in the valve middle layer is in the range 0.03–0.16. Despite its higher density, Dm of the aortic media can still be smaller than D2 of the valvular middle layer because the aortic media is composed of repeated layers of elastic tissue, separated by smooth muscle cells (SMCs) and extracellular matrix. As such, a large LDL particle diffusing normal to the elastic sheets likely needs to find each elastic sheet's fenestrae before it can traverse the sheet. Moreover, these particles likely cannot diffuse through SMC, but rather must diffuse around them. As such, the diffusion path through the aortic media may be significantly longer than the medial thickness, thereby resulting in an effective diffusivity that is smaller than that in the elastic-free and nearly cell-free valvular middle layer, despite the former having the slightly higher LDL space. All of the calculations below employ the parameter values fit above.
Pressure and velocity distribution.
Figure 4 shows the detailed three-dimensional (3D) pressure distribution near the leaky junction for r*/R1 
0–15 and z 0–0.02 near the intima at the aortic aspect for the case of one leak at the aortic aspect and no leaks at the ventricular aspect. Because the leaky junction has a much larger hydraulic conductivity than the normal junction, the pressure drop across the leaky junction is much smaller than across the normal junctions and therefore the absolute pressure inside the leaflet's aortic aspect is highest at the leaky site. This value decays to a plateau in the radial direction (parallel to the endothelium), where the effect of the leaky junction vanishes. This radial pressure gradient drives a strong radial convection that rapidly advects tracer, yielding the observed rapidly growing tracer spots. The pressure decreases monotonically in the z-direction normal to the endothelium toward the ventricular aspect. Figure 4 illustrates that the pressure in the intima has only a slight, nearly linear z dependence far from the leak. This is the direct result of the scale separation there, i.e., that L1/ = O(10–4) which leads to a boundary layer-type result for Laplace's equation in the intima that the pressure is linear in z to leading (first) order in L1. (This scaling is not valid near the leak where variables change in the r direction over scales, 20–30 nm, comparable to the junction thickness and no longer large compared with L1.) In contrast, the pressure varies appreciably in z in the thick, high-resistance middle layer, where L2/ = O(1) and reflects the overall water flow through the leaflet.
Figure 5 presents the 2D pressure profiles in the valve intima at z = 0 and 1 as a function of radial distance from the center of the leaky cell for various cases of numbers of leaks in the two endothelia, including the no-leak case, the one aortic leak case, and the one ventricular leak case.
Figure 6 plots the corresponding radial velocity profiles (viz., Darcy's law, Eq. 5). Numerical calculations shows that these curves only depend on the leak situation at the nearby aspect; a change in the leak number at the faraway aspect yields changes in the given curves that are far too small to be seen in the figures. For example, curve 1 in Fig. 5 can represent the local pressure distribution at z = 0 not only for one leak at the aortic and zero leaks at the ventricular aspect, but also for one leak at each aspect. As Fig. 5 illustrates, the filtration model is symmetric about P = 0.5 and z = 0.5, since a simple substitution of z and Pi with 1 – z and 1 – Pi in the model for the case of either one or zero leaks on both aspects leaves the model unchanged. The same substitution for the case of one leak at one aspect and zero leaks at the other simply switches the location of the leak to the other aspect. Curves 3 and 4 in Fig. 5 are straight lines parallel to the x-axis, indicating an r-independent pressure and thus that there is no radial velocity or convective transport. This serves as a check that the 2D calculation reduces under these conditions to the 1D profile that was invoked in the parameter fit above. The zenith and nadir of pressures in curves 1 and 2 in Fig. 5 are at the border of the leaky cell, r*/R1 = 1, and correspond to the maximum/minimum of the corresponding radial velocity curves in Fig. 6. A leaky junction at the aortic aspect acts like a fluid source because the pressure has a local maximum there and as a fluid sink at the ventricular aspect, where it has a local minimum. The signs of the velocities in Fig. 6 reflect this. The radial pressure gradient in curves 1 and 2 in Fig. 5, i.e., the corresponding radial velocity in Fig. 6, is large within a radius of 6–8R1, with its largest (discontinuous) value, 10–4 cm/s, at r*/R1 = 1, and decays to a tiny amount, 10–7 cm/s, at r*/R1 = 15. Thus the effect of leaky junction on the radial convection in the intima acts at most within a circle of this radius.
Solution of mass transfer problem.
Figure 7 is the LDL concentration distribution in the vicinity of the leaky cell near the aortic aspect (r*/R1 0–15, z 0–0.02) 30 min after exposure of an initially tracer-free leaflet to a unit step change in lumen tracer concentration at both leaflet aspects. The leaflet has one leak at the aortic aspect and no leaks at the ventricular aspect. Because of the enhanced width of the leaky junction, macromolecular transport through the leaky junction is much larger than through the normal junctions that, in view of Tompkins et al. (34), is not zero. Thus there is a tracer concentration maximum at r*/R1 = 1 and z = 0. For any fixed normal distance z the concentration is the highest at r*/R1 = 1 and decays very rapidly in the radial direction in the intima. This rapid decay is due not only to the large difference in the rate at which LDL crosses the endothelium between the leak and the nonleak parts of the endothelium but also to the dilution of the LDL as it is advected radially in the intima by water crossing the endothelium everywhere. Note also that the same separation of scales that led us to conclude that the pressure was linear in z in the intima to leading order in the small parameter L1/ far from the leak also implies that the C is independent of z in the intima to leading order in L1/ there. The numerics illustrate this. The intima C clearly depends on r and is largest near the leak.
In the z-direction there is a sharp discontinuity in LDL concentration at the intima-middle layer interfaces because of the mismatch in the void space between these regions and the definition of concentration on a per-unit total volume (and not per-unit void) basis that we use throughout. The matching conditions, Eqs. 25 and 26, express this and lead to step changes in the concentration distribution according to the ratio 1/2. For any fixed radial position r, the concentration is the highest at z = 0 and 1 and decays to its lowest point in the middle layer, as in earlier (10, 42) diffusion-only oxygen transport studies. This minimum is near, but generally not exactly (because of the symmetry breaking directionality of the transvalvular convection), z = 0.5 for the cases of zero or one leak at both aspects, near z = 1 for the case of only one leak at the aortic aspect and no leaks at the ventricular aspect, and near 0 for the opposite case. The influence of the leaky junction on the concentration distribution appears to be limited to within 
10–11R1 in the radial direction. At further distances the concentration distribution decreases monotonically and is almost identical to the case of no leaks at both aspects. That is, far from the leak in the radial direction the leaflet does not know that there is a leak.
Figure 8 plots the LDL concentration at t = 30 min as a function of radial distance at fixed z = 0 and 1 for the same four leak arrangements handled in Fig. 5. Again, as in Fig. 5, these curves depend strongly only on the leak situation at the proximal aspect and a change in the leak status of the far aspect changes the curves imperceptibly. These curves illustrate the effect of the leaky junction on the mass transport 30 min after the step change in lumen LDL concentration. As discussed above, a leak in the aortic aspect causes a radial flow, and consequently mass transport, in the intima away from the leak. Curve 1, at z = 0 for one leak in the aortic aspect, illustrates this case. In contrast, curve 3 represents a leak in the ventricular aspect, where convection sweeps tracer toward and exits through the leaky junction into the lumen. As expected, curves 1 and 3 have their maxima at the leaky site, r*/R1 = 1. Since the lumen LDL concentration is higher than that of the tissue, convection at the aortic aspect sweeps tracer in the same direction as diffusive transport, whereas convection at the ventricular aspect sweeps it in the direction opposite to diffusion. Therefore, curve 1 is higher than curve 3 near r*/R1 = 1. Both curves exceed the leak-free curves 2 and 4 because of the enhanced transport through the leak. Curve 1 (3) decays to curve 4 (2) at fixed z = 0 (1) at large r*/R1 for the case of no leaks at either aspect, suggesting that the influence of the leaky junction on the tracer's concentration profile at 30 min is confined to 10–11R1. Curves 2 and 4 are straight lines parallel to the x-axis. Curve 2 slightly exceeds curve 4, because of the small normal convection. Similarly, in Fig. 3 the value at z = 1 is a little higher than that at z = 0.
Growth of macromolecular leakage spots.
Experiments described in Part I (46) measured the radii of HRP leakage spots as a function of HRP circulation time in the rat valve leaflet. The maximum circulation time was 4 min since for longer times HRP, with a diameter of only 6 nm (16), penetrates the normal junctions. This results in a difficulty in distinguishing leaky spots from background. As in similar experiments in the aorta, one does not know a priori the critical concentration Cc of HRP that is detectable as a dark spot. This threshold depends on the complicated conditions of the biochemical staining reaction and photographic exposure. Another issue is that it takes a finite time for HRP to reach the aortic valve from the point of HRP injection, the femoral vein. Consequently, there is an unknown time lag, t0, between the theoretical and the experimental initial times that is not directly available from the experimental data. For the aorta, a trial and error variation of the theoretical delay matched the data when t0 was 25 s (16). This procedure calculated the intimal HRP concentration as a function of r/R1, for different values of the time t – t0 over which we have solved the initial value problem corresponding to the time since the tracer reached the leaflet. For each of a series of values for t0, we determined Cc as the HRP concentration at t that corresponded to the experimental radius of the experimental time t = 30 s. We then assume that Cc is the same for all four experimental circulation times and therefore simply read off the values of r/R1 at which the horizontal line at Cc intersects to intimal profiles from the t = 60-, 120-, and 240-s curves. We then pick the value of 0 < t0 < 30 s giving the curve that best fits the data. Because of the proximity of the aortic valve to the aorta, we expect this procedure applied here to yield essentially the same value.
Wherea
(or L
) and ka (or kv) zero in the region 0 < r < R1 or with