Am J Physiol Heart Circ Physiol 292: H2687-H2697, 2007.
First published January 19, 2007; doi:10.1152/ajpheart.00606.2006
0363-6135/07 $8.00
Macromolecular transport in heart valves. III. Experiment and theory for the size distribution of extracellular liposomes in hyperlipidemic rabbits
Zhongqing Zeng,1
Patricia Nievelstein-Post,2
Yongyi Yin,3
Kung-Ming Jan,4
Joy S. Frank,2 and
David S. Rumschitzki1,4
1Department of Chemical Engineering and 3Department of Mechanical Engineering, City College of the City University of New York, New York, New York; 2Department of Medicine, UCLA School of Medicine, Los Angeles, California; and 4Department of Medicine, College of Physicians and Surgeons of Columbia University, New York, New York
Submitted 9 June 2006
; accepted in final form 16 January 2007
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ABSTRACT
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The heart valve leaflets of 29-day cholesterol-fed rabbits were examined by ultrarapid freezing without conventional chemical fixation/processing, followed by rotary shadow freeze-etching. This procedure images the leaflets' subendothelial extracellular matrix in extraordinary detail, and extracellular lipid liposomes, from 23 to 220 nm in diameter, clearly appear there. These liposomes are linked to matrix filaments and appear in clusters. Their size distribution shows 60.7% with diameters 23–69 nm, 31.7% between 70 and 119 nm, 7.3% between 120 and 169 nm, and 0.3% between 170 and 220 nm (superlarge) and suggests that smaller liposomes can fuse into larger ones. We couple our model from Part II of this series (Zeng Z, Yin Y, Jan KM, Rumschitzki DS. Am J Physiol Heart Circ Physiol 292: H2671–H2686, 2007) for lipid transport into the leaflet to the nucleation-polymerization model hierarchy for liposome formation proposed originally for aortic liposomes to predict liposome formation/growth in heart valves. Simulations show that the simplest such model cannot account for the observed size distribution. However, modifying this model by including liposome fusing/merging, using parameters determined from aortic liposomes, leads to predicted size distributions in excellent agreement with our valve data. Evolutions of both the liposome size distribution and total liposome mass suggest that fusing becomes significant only after 2 wk of high lumen cholesterol. Inclusion of phagocytosis by macrophages limits the otherwise monotonically increasing total liposome mass, while keeping the excellent fit of the liposome size distribution to the data.
aortic stenosis; kinetics of lipid accumulation in values; size distribution
LIPOPROTEINS, especially low-density lipoprotein (LDL), appear to deposit inside the leaflets mainly of the mitral and aortic valves before monocyte diapedesis, leading to valvular lesion formation. These processes bear a striking resemblance to the proven steps in the parallel development of atherosclerotic lesions in the aorta and other large arteries. Valve lesions eventually calcify and progress to aortic stenosis (2, 4, 9, 14, 19), the most common valvular disease in western countries. It is associated with significant morbidity and mortality, generally requiring valve replacement in advanced stages (2). Therefore, the mechanism for LDL's entrance into and retention in the leaflet's subendothelial matrix, specifically, the LDL-matrix interaction, is of particular importance for understanding the pathogenesis of such valve diseases.
In 1970, Walton et al. (19) showed that human heart valves, in particular the aortic aspect of the aortic valve and the ventricular aspect of the mitral valve, accumulate lipids and form lesions similar to early arterial atherogenesis. Simionescu et al. (16) used conventional freeze-fracture and thin-section electron microscopy to demonstrate the existence of lipid packets organized as small vesicles 100–300 nm in diameter in the ventricular aspect of atrioventricular valves of rabbits fed a high-cholesterol diet for as little as 2 wk. They labeled these extracellular densely packed uni- or multiphospholipid lamellar vesicles rich in unesterified cholesterol "extracellular liposomes." They found that the formation and deposition of such liposomes happened before monocyte diapedesis and foam cell formation and thus were prelesional ultrastructural changes that paralleled the proliferation of basal lamina-like material, microfibrils, and proteoglycans. Unfortunately, conventional chemical fixation, processing, and routine electron microscopic techniques caused proteoglycan collapse, thereby destroying the fine structure of the extracellular matrix and the liposomes bound to them. Frank and Fogelman (6) examined the intima from aortas of genetically hyperlipidemic (Watanabe heritable hyperlipidemic; WHHL) and cholesterol-fed normal (10–16 days) rabbits by using ultrarapid freezing without conventional chemical fixation followed by rotary shadow freeze-etching. They clearly demonstrated for the first time lipid-extracellular matrix association under in vivo conditions in the aortic intima. Their liposomes varied in size from 23 to 169 nm and were found mostly in clusters and linked to the matrix filaments. However, the size distribution of the liposomes found in the two rabbit populations differed markedly. Yin et al. (20) developed a hierarchy of nucleation/polymerization type models to describe the kinetics of extracellular lipid formation and growth, and even the simplest model in this hierarchy, a model with only one adjustable parameter, was able to fit the WHHL distribution. When coupled to Huang et al.'s (8) theory for the transport of LDL cholesterol from the lumen into the aortic wall, this simple model, with no further parameter adjustment, adequately modeled the cholesterol-fed data as well. However, the next model in the hierarchy appeared to do an even better job of fitting both distributions. Nievelstein-Post et al. (11) used the same ultrarapid freezing/rotary shadow etching technique as Frank and Fogelman (6) to find similar extracellular liposomes in the aortic valve's aortic aspect and the mitral valve's ventricular aspect from normal rabbits after 29 days of cholesterol feeding.
In this report, we do a much more extensive study of the valve leaflets of these 29-day cholesterol-fed rabbits, examining a large number of matrix-associated lipid liposomes to determine the extracellular lipid liposome size distribution in these leaflets. We then test the Ansatz that liposome formation and growth are local lipid-matrix events that should be similar—with similar kinetics—in tissues with similar matrix structure. If that is the case, then one can investigate whether tissues or vessels that differ in their susceptibility to atherosclerosis can trace these differences simply to differences in the rate of the prelesion lipid liposome accumulation events in their walls. To test this latter hypothesis, one needs to solve the lipid delivery, i.e., transport, to the tissue for each vessel and couple it to the established liposome formation and growth kinetics. Before moving on to resistant vessels, the present study attempts to verify that liposome formation and growth follow similar kinetics in the large artery and in the valve leaflet, the other susceptible tissue. To do this, we couple the model for the transport of LDL cholesterol from the lumen into the valve leaflet developed in the first two papers in this series [Part I (21) and Part II (22); this issue] to Yin et al.'s hierarchy of nucleation/polymerization/merging from the arterial intima (20). We use this model, with all transport parameters fixed from Part II and all liposome kinetic parameters fixed from aortic intima liposome data, to simulate lipid infiltration into and liposome formation and growth in the rabbit's valve leaflets for 29-day feeding. We shall find that the theoretical particle size distribution agrees very well with the experiments.
This paper first presents the experimental methods, with results and discussion, before proceeding to the theoretical model, its results, and a comprehensive discussion of model results and their comparison with the experiments.
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EXPERIMENTAL MATERIALS AND METHODS
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Valvular preparation.
We report experiments carried out by P. Nievelstein-Post and J. S. Frank, a small fraction of which formed the basis of the study of Nievelstein-Post et al. (11), and, as such, the experimental procedure is identical. For completeness, we repeat it in brief here; for more details, see Ref. 11. All animal protocols conformed to institutional guidelines on the care and handling of animals. Normal New Zealand White rabbits (2.5 kg, 8 wk old) were fed for 29 days a chow diet containing 2% cholesterol (wt/wt) in soybean oil (10% wt/wt). The rabbits were killed with an overdose of pentobarbital, and their mitral and aortic valves were harvested within 20 min and incubated for 4 h in 125 µg/ml LDL-gold conjugate in PBS-NaHCO2 buffer supplemented with 4.5 mg/ml freshly thawed human lipoprotein-deficient serum for 4 h or with 300 µg/ml LDL for 2 h in a 5% CO2 incubator. [These incubations were needed so that Nievelstein-Post et al.(11) could identify some of the 23-nm circular objects in the replicas as LDL derived but are not important for the results discussed here.] Valve leaflets without any visible lesions and with the lesion prone side up were immediately placed on slabs of gelatin resting on aluminum support disks. The disks were shaped to fit on the cold stage of the Balzers 301 freeze-fracture apparatus and the freezing head of the Polarion quick-freeze apparatus. The freeing head with gelatin slab and adhered piece of valve was plunged onto an ultrapure copper block that had been precooled to 4 K by liquid helium (6, 11).
Freeze-fracture and replication.
Each frozen piece of valve leaflet was fractured very superficially (maximum between 7 and 20 µm), making it likely that the fracture plane often fell in the mechanically weak valve intima. One pass of the microtome arm produced fractures that were 80% within endothelial cells, and two or three passes of the knife produced a fracture plane in the intima. Fracturing was performed at –120°C and a vacuum of 1x10–7 Torr. For deep-etching the specimen stage was warmed to –90°C and maintained at this temperature for 
8 min. The fractured and etched surface of the valve was rotary replicated by depositing 2 nm of platinum/carbon from the electron-beam gun mounted at a 25° angle while the tissue rotated through 360° six times, followed by three short (5–8 s) bursts of carbon. The fractured and etched replica was digested overnight in Purex bleach, washed in distilled water, and picked up on 3-mm specimen grids (6, 11).
Electron microscopy.
The replicas were examined at 80 kV in a JEOL 100 CX electron microscope. The negatives were printed as negative images.
Experimental results.
As noted, these experiments represent a much wider range of those reported in Ref. 11. Figure 1 shows typical subendothelial regions of the aortic valve from cholesterol-fed rabbits. As shown by Nievelstein-Post et al. (11), the subendothelial extracellular matrix of heart valves, composed chiefly of collagen fibers, elastin microfibrils, and proteoglycans, is strikingly similar in structure to that of aortic intima. The freezing technique that avoids chemical fixation preserves the components of subendothelial extracellular matrix and displays them in extraordinary detail in Fig. 1. Abundant collagen fibers are linked together with an extensive heterogeneous network of likely proteoglycan filaments. All replicas studied showed some individual, but mostly clusters of, smooth surfaced extracellular lipid liposomes, with sizes ranging from 23 to 220 nm, enmeshed in the matrix at the contact site or the junction between the collagen fibers and the matrix filaments and in direct contact with them. The identification of these structures as extracellular lipid liposomes has been addressed previously (6, 11, 12). These nonuniformly distributed liposome clusters displayed some in the process of fusing into larger-size particles. No traces of blood or foam cells were found in examined areas of subendothelial matrix, indicating that liposome formation is an early prelesional change before monocyte diapedesis and foam cell formation.
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Fig. 1. Freeze-etch electron micrographs from the heart valves of normal rabbits fed with cholesterol for 29 days. A: at high magnification, thick rodlike collagen fibrils (long arrows) are periodically cross-banded by short filaments that connect them to each other and to the extracellular matrix network. Smooth surfaced liposomes are directly attached to the filament network and to many collagen fibrils (short arrow). Liposomes with different sizes tend to fuse into a larger one (arrowhead). B: liposome aggregates with different sizes are clearly evident in clusters and enmeshed in the extracellular matrix and collagen fibrils.
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The present freeze-etching technique easily allows resolution of the smallest lipid particles (23 nm) present. Measurement of the sizes of all liposomes seen in 15 replicas from different areas results in the liposome size histogram in Fig. 2, which compares it with the corresponding histogram taken by Frank and Fogelman (6) with the same technique from the aortic intimae of 10- to 16-day cholesterol-fed normal rabbits. More than half (60.7%) of the lipid liposomes in the subendothelial valve matrix of the experimental hypercholesteremic rabbits had diameters between 20 and 69 nm (small), 31.7% between 70 and 119 nm (medium) and 7.3% between 120 and 169 nm (large). The valve has fewer small, but somewhat more medium and large, liposomes than the aortic intima of rabbits fed roughly half the time. However, unlike in the aortic intima, the valve replicas also displayed superlarge liposomes (170–220 nm), which, if the liposomes are assumed spherical, correspond to the lipid content of 398–875 LDL particles, amounting to 0.3% of the total valve liposomes. Such superlarge liposomes were not found in Ref. 6, even for genetically hypercholesteremic WHHL rabbits that were 2–8 mo old.
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Fig. 2. Extracellular lipid liposome size distribution in the immediately subendothelial extracellular matrix from cholesterol-fed rabbit heart valves (29 days) and aortas [10–16 days, by Frank and Fogelman (6)] measured from photographic prints. The percentages of superlarge liposomes with diameters of 170–220 nm from the liposome populations observed are 0.3% in the valve and 0% (Ref. 6) in the aorta.
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Discussion.
Simionescu et al. (16) first found that extracellular cholesterol-rich liposomes accumulated in the heart valves of hyperlipidemic rabbits. Filip et al. (5) studied cellular events in the development of valvular lesions induced by experimental hypercholesteremia in rabbits and hamsters. They showed that the progressive subendothelial accumulation of extracellular liposome-like structures, rich in LDL- and 
-VLDL-derived unesterified cholesterol and associated with the proliferation of a basal lamina-like material, was the earliest detectable physiological modification in the disease process. They also showed that diapedesis of blood monocytes followed in the same locations, leading to the formation of macrophages. These subendothelial extracellular phospholipid liposomes were believed to form by the interaction of LDL, which penetrated the endothelium from the cardiovascular lumen and deposited in the subendothelial intima with the matrix. Yin et al.'s (20) kinetic models for the initiation and growth of these extracellular lipid liposomes in the arterial intima were successful at explaining both short-term, feeding-induced as well as genetically hypercholesteremic lipid liposome size distributions with a simple model with only one adjustable parameter. In their view, the two distributions traced their different characters to the focal, rather than uniform, transendothelial transport of large particles such as LDL that is evident in the observed nonuniform clustering of liposomes, particularly at very short times (12) and is a natural implication of Huang et al.'s (8) macromolecular transport theory. This focal transport that Huang et al. describe assumes that large molecules traverse the endothelium at rare endothelial sites, many of which are associated with cells in turnover. This is consistent with the early endothelial response to injury hypothesis (10), whereby endothelium regenerated after denudation sequesters far more lipid in its intima than unregenerated or undamaged endothelium, since the dividing cells repairing the injury both synthesize new intima matrix and allow LDL transmigration. Although recognition of the nonuniform macromolecular transport in normal endothelium through rare, localized endothelial leaks is critical to explaining observations over times similar or short compared with the characteristic waiting time for an endothelial leak to open in a given neighborhood (a few hundred hours), it becomes progressively less relevant over the much longer timescales over which WHHL rabbits are hypercholesteremic.
Since the ventricular aspect of the mitral valve and the aortic aspect of the aortic valve are lesion prone (19), Nievelstein-Post et al.'s (11) and our investigations focus on these areas in 29-day cholesterol-fed rabbits. Nievelstein-Post et al.'s (11) and our experiments used ultrarapid freezing and rotary shadow freeze-etching to preserve the detailed ultrastructure of the subendothelial extracellular matrix and the embedded liposome particles and aggregates in close to the in vivo condition. They both show in a quasi-three-dimensional way the intricate ultrastructure of subendothelial extracellular matrix components consisting of collagen fibers, elastin microfibrils, and proteoglycans and their enmeshing of lipid that tend to bind to matrix and fuse to form extracellular lipid liposomes of various sizes, nearly identical to those found in the aortic intima (6). As in that case, the size of these liposomes relative to matrix dimensions strongly suggests liposome formation by aggregation in situ; replicas in Fig. 1 and Ref. 6 catch this fusion in the act.
This work broadens Nievelstein-Post et al.'s study (11) further in that it uses a large number of printed replicas from different areas to determine the particle size distribution of these extracellular lipid liposomes. As noted, observed liposome sizes range from 20 to 220 nm, with slightly fewer small (60.7% vs. 75%) and more medium (31.7% vs. 14%) and large (7.3% vs. 11%) liposomes than in the aortic intimae of 10- to 16-day cholesterol-fed normal rabbits. Unlike the aortic intimae, the valves also contain 0.3% superlarge liposomes, with lipid contents from 398–875 LDL particles. The similarities between the aortic and valve replicas suggests that the processes underlying the formation and growth of the extracellular lipid liposomes in both tissues are similar or even identical. However, since the geometry (e.g., 2 endothelia adjoining the lumen for the valve; 1 for the vessel) and overall structure, i.e., many tissue constituents, of the aortic wall and the leaflet differ substantially from one another (even though both have continuous endothelia and the local extracellular matrix components are quite similar), the transport processes delivering lumen lipid to the subendothelium where it can bind extracellular matrix are clearly quite different.
Fortunately, in Parts I (21) and II (22) of this series we have worked out and tested a theory that can quantitatively explain and predict what in the present setting one can call the lipid delivery processes into the valve leaflet. This theory again rests on the focal, rather than uniform, LDL transmigration due at least partly to cell turnover, either normally or as a response to injury (10). We now combine the result of these lipid delivery theories with Yin et al.'s (20) models for the formation and growth of extracellular lipid liposomes from free lipoproteins that have entered the subendothelial space. It should be noted that all parameters for the transport theories from Part II (22) are fixed, as are those of Yin et al.'s kinetic theories (20). We use this combined theory to explain/predict the liposome histogram from valve leaflets of 29-day cholesterol-fed rabbit and its difference from that of the aortas of the 10- to 16-day cholesterol-fed rabbits. In the course of this calculation, we note that the simplest model of Yin et al. (20) only took into account liposome formation plus a single mechanism of liposome growth—by an existing liposome appending a free LDL particle. This theory, when coupled to the nonuniform lipid delivery to the aortic intima through isolated endothelial leaks in Huang et al.'s models (8), was adequate to explain the difference in liposome size distributions between cholesterol-fed and WHHL genetically hypercholesteremic rabbits. However, this simplest theory did not allow for the formation of very large liposomes by the fusing/merging of neighboring, existing liposomes, despite the fact that the replicas clearly catch this process. Yin et al.'s (20) more sophisticated models did indeed include such processes and, with the penalty of an additional parameter, also explained the two distributions from the two rabbit populations. Here, as we shall see, fitting the valve distribution, including the generation of superlarge liposomes, will necessitate the use of the latter model that includes liposome merging because the simpler model, even with an arbitrary refitting of its kinetic parameter, cannot.
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MATHEMATICAL MODELS
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Introduction.
As Frank and Fogelman (6) demonstrated in the arterial intima with colloidal gold, the lipid in extracellular liposomes derives from plasma LDL and -VLDL. Walton et al. (19) demonstrated that the extracellular lipids deposited in human mitral and aortic valves also originate from plasma LDL. O'Brien et al. (13) examined lipoprotein accumulation in aortic valve lesions of humans and detected apolipoprotein (apo) B, A, and E, macrophages, and 
-actin-expressing cells in 18 aortic valves that ranged from normal to stenotic. They found a large amount of apoB in stenotic aortic valves, indicating that lipid accumulation occurred as a result of LDL retention in valve tissue.
Tompkins et al. (18) were the first to study the transport mechanism by which plasma LDL crosses the endothelium and accumulates in the subendothelial space of heart valves. They obtained transvalve 125I-labeled LDL profiles in various regions of squirrel monkey valve leaflets after 30-min 125I-labeled LDL circulation and found very disparate profile shapes and magnitudes between different leaflets and even between different regions of the same leaflet. Their simple one-dimensional diffusion-only model could mimic individual profiles, each with a separate set of parameters, but, not surprisingly, was too simple to explain the large observed variation. In Part I (21) we used short-circulation time horseradish peroxidase (HRP) studies to find isolated HRP spots, indicating that large molecules mainly traversed the valve endothelium through rare hot spots, and that these spots grew too fast to be consistent with any realistic diffusion-only model. In Part II (22) we set up a two-dimensional (2D) convection-diffusion model (Fig. 3A) that, using a single set of parameters, explained all of Tompkins et al.'s (18) transvalve profiles as well as our (22) independent HRP spot size growth extremely well. The crux of this 2D model was that water passes through all junctions, but the macromolecules crosses predominantly through the junctions around rare (e.g., mitotic, dying, or other) endothelial cells whose junctions are significantly widened. The subendothelial flow then spreads the macromolecules in directions parallel to the endothelium in the very thin, low-solid-fraction immediate subendothelial region, resulting in the observed rapid HRP spot growth, before they progress in the direction of the overall pressure gradient (present only when the valve is closed) from the aortic to the ventricular aspect. A radial pressure gradient arising from the different permeability of normal and leaky junctions drives this radial flow/advection. The mismatch in void fractions between the immediate subendothelial region and the bulk of the valve tissue keeps tracer initially adjacent to the endothelium, despite the absence of an internal elastic lamina to act as a barrier. The model quantitatively shows that the large variation in Tompkins et al.'s (18) profiles could be due to different numbers/distributions of leaks and different proximities of the sections examined to these leaks. We use this 2D model (Fig. 3B) to predict the time-dependent LDL concentration distribution in the valve's subendothelial region where it can bind to extracellular matrix. Note how the LDL concentration decays as a function of the distance r from the center of a leak to a small, leak-independent value at large r.
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Fig. 3. A: 2-dimensional convection-diffusion model of water filtration and of large molecule, e.g., low-density lipoprotein (LDL), transport in heart valves proposed by Zeng et al. (Ref. 22, Part II). NEC, normal endothelial cell; LEC, leaky endothelial cell; r, radial distance from center of a leaky endothelial cell; z, direction normal to the endothelium. B: LDL concentration in the subendothelial intima as a function of r. L, tissue LDL concentration; Cp, plasma LDL concentration; R1, radius of an endothelial cell. Modified from Figs. 2 and 8 in Part II (22).
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Yin et al. (20) proposed a mathematical model to describe liposome formation and growth in the arterial intima. The model postulated that liposome formation and growth resembles classic nucleation-polymerization processes and that large liposomes form in regions with prolonged high LDL concentrations, e.g., leak centers or regions where leaks overlap. Only small liposomes would form in low-LDL-concentration spot peripheries. We use Yin et al.'s (20) models to predict liposome formation and growth in valve leaflets from free lipid delivered to the tissue by Part II's (22) mechanism and compare with the data in Fig. 2. We note that Part II's transport parameters derive from rat data and the liposome data are for the rabbit. Liposomes have not been observed in fed rats, and extensive transport data, which require large numbers of animals, has only been taken in rats. Whereas it would be better to have both from the same species, this same approach successfully described liposome formation in the aortic intima.
Description of mathematical models.
As described in Ref. 20, the simplest nucleation-polymerization model begins with (Eq. 1a) a free LDL (L) attaching to abundant extracellular matrix (concentration lumped into the rate constant k0) to form (nucleated) liposomes M1 composed of the lipid from a single LDL particle. This first-order "reaction" is the simplest way to model liposome initiation.
 | (1a) |
 | (1b) |
Equation 1b models liposome growth proceeding via an existing liposome Mj of size j appending a free LDL particle to become a liposome Mj+1 of size j + 1. Because of the lack of data and to minimize the number of free parameters, we assume that the rate constant, k1, for the "polymerization" reaction is independent of the number j of "monomers." In Ref. 20, this model with an appropriately chosen k1 produced essentially zero superlarge liposomes in the arterial intimae of even the 5-mo-old WHHL rabbits, as observed in Ref. 6. Since Fig. 2 contains a small amount of superlarge liposomes, we also consider one of Yin et al.'s (20) higher models that includes the fusing/merging of two existing liposomes to form a larger one, i.e.,
 | (1c) |
Again, we presume that the merging constant, km, for any pair of existing liposomes does not depend on their sizes, j or k. If the same symbol is used to represent both a species and its concentration, the kinetic equations corresponding to the above model are
 | (2a) |
 | (2b) |
where t is experimental time and n is a dummy variable. Simply setting km = 0 recovers the simple model's (Eqs. 1a–b) kinetics.
For km = 0, Yin et al. (20) found the simple analytic solution for an initially liposome-free system, i.e., Mj(t = 0) = 0 for all j, in terms of the integral 
, where s is the variable of integration of an arbitrary potentially time-varying local LDL concentration L(t) as
 | (3) |
As Fig. 3 clearly illustrates, even for a single leak, L(t), and therefore Mj(t) for all j, are functions of both position and time, with L(t) highest at the leaky junction and quickly decaying with distance from the leaky cell's perimeter. [At very long hypercholesteremic times, as in the WHHL rabbits, a large number of leaks have opened and closed and the spatial variation of integral of L(t) and thus of a(t) become minimal, but that is not the case here.] Consistent with this spatial dependence, we have experimentally demonstrated that, after 29 days of feeding, liposomes of varying sizes appear immediately subendothelially in clusters in heart valve leaflets, rather than uniformly distributed, and only small, individual liposomes emerge outside of these clusters. It is tempting to associate these clusters with the high lipid concentration regions of Fig. 3B and the relatively larger region of lower LDL concentration with the abundant smaller ones. We only allow LDL entering through leaks, and not lipid that crosses the endothelium uniformly (see Part II), to form liposomes. Moreover we assume that the consumption of free lipid by Eqs. 1a–b does not significantly change the LDL concentration distribution given by Fig. 3B.
Summing Eqs. 2a and b gives the rates of increase of total liposome number and mass as
 | (4a) |
 | (4b) |
Setting km = 0 yields the simple model's rates. The rate of increase, Eq. 4a, of the total liposome number for the simple model (km = 0) is simply controlled by k0 and the LDL concentration L(t), independently of k1. A merging event does not directly change the total liposome mass. As such, the evolution, Eq. 4b, of the total liposome mass does not depend explicitly on the merging constant km, and thus Eq. 4b takes exactly the same for both the simple and modified models. This does not mean that merging has no effect on the total liposome mass. On the contrary, liposome merging slows the increase in the total liposome mass by lowering the total liposome number (km term in Eq. 4a). For the simple model km = 0, one integrates Eqs. 4a and b to find
 | (5a) |
 | (5b) |
Methods of solution.
We perform the numerical simulation of liposome formation and growth in a region of leaflet containing 2,500 endothelial cells per face. From the numbers in Parts I (21) and II (22), this corresponds to roughly an entire rat leaflet, but only to a portion of a leaflet of a larger animal such as a rabbit. We choose a square mesh with 50 x 50 grid points corresponding to a valve area of 50 x 50 cells, whose size is large compared with the extent of elevated (10 cell radii) LDL concentration due to a single leak. We choose periodic boundary conditions for the cell so as to avoid having to deal with end effects that affect a negligible fraction of the region of interest. The simulation starts in a liposome-free intima, i.e., Mj(t = 0) = 0 for all j, because the feeding experiments above started in normal rabbits whose controls showed leaflets free of liposomes (11). We presume that leaky cells occur in this area at random locations in the square according to a Poisson process in time of occurrence [mean 1 leak per 4,000 cells per hour (1)] and in duration [mean 1 h (3)]. Once a leak has opened, the calculation in Part II suggests that the LDL distribution in the tissue due to spot leakage approaches steady state in a few minutes, a time much shorter than the time that the leak, on average, is open. As such, we use the LDL concentration profile from Fig. 3B that corresponds to a leak that has been open for 60 min for the entire time the leak is open. At each time point, there is a distribution of leaky sites, and we calculate the LDL concentration, L(t), at each mesh point by summing the LDL concentration there due to each active leak, each of which is just a function of the distance from that leak's center and from those of its periodic images in the neighboring squares. With this LDL concentration history as a function of position and time, we use the kinetic model (Eqs. 2a and b) to calculate at each mesh point at each time step the distribution of liposomes of different sizes. Although one can utilize Z-transforms to solve these kinetic equations, we simply numerically solve the first 875 of these equations, i.e., j = 875 [diameter (d) = 220 nm] corresponding to the upper bound of superlarge liposomes, directly at each mesh point for its corresponding LDL history. Calculations for j = 1,000 show Mj for j > 875 at least 100 times smaller than for j much less than 875, and thus larger j are not necessary at 29 days. Finally, at the end of the experimental time, t = 29 days, we calculate an aggregate liposome size distribution by summing the local distributions over the entire grid. We reparameterize the liposome size from monomer number j (proportional to liposome volume) to liposome diameter d by assuming spherical liposomes. Thus d3 = j x (23 nm)3, where 23 nm is the diameter of a single LDL particle (6).
Since the liposome size distributions have been culled from a number of prints of freeze etchings from portions of leaflets, and not from entire leaflets, the data only have meaning as percentages in each size range of the total number of liposomes, and not as absolute liposome numbers. As such, the y-axis in Fig. 2 represents percentages and the parameters in the model that affect predictions of percentages are k1/k0 and km/k0, with k0 affecting the timescale.
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RESULTS AND DISCUSSION
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The goal of this work was to test whether one set of kinetics can describe extracellular lipid liposome formation/growth in both the arterial intima and the valve leaflet. Since the rabbit feeding experiments raise lumen cholesterol as a means to induce liposome formation, they convolve these kinetics with LDL transport to the tissue. We test whether two kinetic models that were successful in characterizing liposome formation and growth in the aortic intima can, when combined with the valve's lipid transport from Part II of this series, also explain the different features of the valve liposome size distribution above. We then examine its predictions for the valve's lipid accumulation rate. If successful, this would lend credence to our plan, to be carried out in subsequent manuscripts, to investigate the differences in lipid accumulation rates of vessels of different atherosclerotic susceptibilities, assuming tissue-independent liposome formation/growth kinetics.
Simple model.
From the solution, Eq. 3, of the simple model, k1/k0 simply scales the absolute amount, Mj, of liposomes of size j but does not affect the size distribution. Hence, the liposome size distribution only depends on a(t). Since the normalized LDL concentration in Fig. 3B is the ratio of the LDL tissue and plasma concentrations, i.e., L(t)/Cp, and Cp and k1 appear as a product in a(t) in Eq. 3, any change in Cp can just be absorbed by k1. Thus the only model parameter that one needs to adjust to explore the model's potential to fit the data is k1 (at fixed k1/k0). Table 1 first compares the data with the model results, lumped, as are the experimental data, by diameter ranges rather than by absolute diameter, for the simple model with Cp = 839 mg/dl and k1 fit in Ref. 20 to WHHL rabbit aortic liposome data. It then adjusts k1.
Yin et al. (20) analyzed the size distributions, both without and with spatial variation, of the simple and more complex models in the arterial intima in great detail. Our present calculations all take spatial variation into account. The results for model prediction b in Table 1 using Yin et al.'s (20) parameters have fewer small, more medium, and many fewer large liposomes than observed in experiments and no superlarge liposomes, contrary to observation. As Table 1's model predictions b–d show, initially increasing k1 further suppresses small and raises medium, large, and superlarge liposomes for the allotted feeding time. Further increase in k1 (predictions d–f) eventually achieves the observed superlarge liposome percentage, but by this point the percentage of small liposomes is far below observation and those of medium (now decreasing with k1) and large liposomes are far above observation. We argue that this is due to the underlying kinetics, independent of the transport.
Yin et al. (20) first solved model Eqs. 1a–b in the absence of LDL spatial variation and found that the solution, expressed as (normalized) Mj vs. j, had the form of an inverted step function that moved to the right with increasing a(t), i.e., increasing time t and/or growth "reaction" rate constant k1. For j to the left of the near-vertical drop, the formation rate of liposomes Mj of size j from those of size j – 1 balances their consumption rate to form liposomes of size j + 1 and the number of Mj is and remains steady with further increase in a(t). To the right of the step, all larger liposomes are essentially unpopulated. The narrow width in the j-coordinate of the near-step transition region contains the few liposome sizes that are populated, but whose numbers are not yet steady. This means that for small a(t), nearly 100% of the liposomes are small. By the time a(t) is large enough that the step has reached the size where medium liposomes appear, the number (not percentage) of small liposomes has reached steady state, even though its percentage decreases as more medium-sized liposomes appear. When a(t) is large enough that the step has reached the size where the first large liposomes emerge, the number of small and medium liposomes, as well as their ratio, become steady and no longer change, although their combined percentages decrease as the number of large liposomes increases. Then the relationship of large vs. the sum of small plus medium proceeds as the medium vs. small relationship did previously, with the obvious extension to superlarge liposomes. With spatial variation (this study), these trends take place locally at each grid point, rather than uniformly, and thus these boundaries in the aggregate distribution are not as sharp. However, the basic trends are reflected in the decrease in the ratio of small- to medium-size liposomes with time before many larger liposomes have appeared. At long times after many leaks have occurred, the spatial nonuniformity of a(t) fades, and this ratio will eventually level off to the values found in the spatially uniform LDL theory (20).
Thus we conclude that, irrespective of the parameter values, the simple model produces either reasonable percentages of small and medium but far too few larger liposomes, or reasonable percentages of the largest at the expense of far too few of the smallest liposomes.
Model modified to include merging of liposomes.
As noted above, our freeze-etching electron micrographs of valvular intimae show that subendothelial liposomes can fuse to form larger-sized liposomes, as in Frank and Fogelman's (6) arterial intima study. Thus an obvious remedy to the failing of the simple model is to include liposome merging, Eq. 1c, and km terms in Eqs. 2a and b, to produce superlarge liposomes. Here, since the merging terms on the right of Eqs. 2a and b have no explicit L(t) dependence, unlike the simple model where all reaction terms are linear in L(t), a change in Cp or in any of the rate constants will affect the size distribution and its evolution nontrivially. We keep Yin et al.'s (20) k1 value, and use a value of km that is small enough so as not to significantly change Yin et al.'s (20) results for the arterial intima. The inclusion of liposome merging aims to produce amounts of large and superlarge liposomes comparable to the data, without radically changing the percentages of small and medium liposomes. Success would support the physical supposition that liposome formation/growth kinetics depend only on the similarity of the extracellular matrix and not on tissue type.
For small km, merging introduces a small large-liposome tail to the Mj vs. j number distribution responsible for an increase in large and superlarge liposomes and, to the left of the tail, a slight decrease in the smaller liposomes that will experience a net consumption due to merging. Figure 4 shows the results as percentages of the total for the modified model at 29 days. Curve 2 is the fit of the modified model for k0 = k1 = 0.043 h–1 and Cp = 839 mg/dl, as in the arterial study (20), and liposome merging constant km = 0.0002 h–1. It has slightly fewer small and large and slightly more medium and superlarge liposomes than the valve data. As km increases (curve 4) the percentage of all the medium liposomes decreases while those of the other sizes increase. Since there are far more j (and therefore far more liposomes) in the medium than in the small diameter range, merging depletes more medium than small liposomes to form large and superlarge liposomes and even the percentage (not number) of small liposomes increases because merging decreases the total number of liposomes. As in the simple model, lowering k1 improves the small-to-medium liposome ratio and hardly changes the merging, as curve 3 with k1 = 0.035 h–1, k0 and km unchanged, shows.
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Fig. 4. Liposome size distributions at 29 days. Curve 1, experimental data; curve 2, modified model with k1 = 0.043 h–1, km = 0.0002 h–1; curve 3, modified model with k1 = 0.035 h–1, km = 0.0002 h–1; curve 4, modified model with k1 = 0.043 h–1, km = 0.0002 h–1 and with phagocytosis kd = 0.00097 h–1. The percentages of superlarge liposomes with diameter of 170–220 nm for curves 1–4 are 0.3%, 0.71%, 0.43%, and 0.33%, respectively.
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Figure 5 shows the evolution of the size distribution (number vs. j) with feeding time corresponding to curve 2 at this small km value, and Table 2 compares this evolution in percentages between the simple and merging models. Unlike the simple model, where, for each j the number Mj of liposomes of size j initially increases with feeding time and then reaches a steady value, Fig. 5 shows for j less than 35 in 29 days that, with merging, it goes through a maximum. Mj then decreases slowly as the total number of liposomes, all of which are available to merge with those of size j, increases. As in the simple model, we expect the percentage of each lump for small-km merging to also eventually decrease, but 4 wk (Table 2) is apparently too soon to see this. In the simple model, large liposomes start to become appreciable at 3 wk, roughly corresponding to the local Mj vs. j inverted step function having just crossed j = 142, i.e., d = 120 nm. For small km, in the first 2 wk small and medium liposomes are being populated, but starting in the third week merging depletes their numbers. Figure 5 shows the development of the large liposome tale. Consistent with slightly perturbing the moving step picture of the simple model, with small-km merging, it still takes 3 wk for the large and superlarge liposomes to comprise 1% of the total. However, the large j tail makes their amounts far larger than for the simple model (0 superlarge) and shows superlarge early on, long before a perturbed Mj vs. j inverted step comes anywhere near j = 404 (d = 170 nm). Their value (0.19%) approaches the data (0.30%) by 3 wk.
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Fig. 5. Evolution of the continuous liposome size distribution Mj vs. j for the modified model (km = 0.0002 h–1) over 4 wk of feeding.
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Table 2 hints that merging should not affect the fit (20) of the small and medium liposomes and, in fact, should improve the fit of the large liposomes for the 10- to 16-day feeding study of the aortic intima, yet it can fit the 29-day valve data including producing superlarge liposomes. For longer feeding times, e.g., hyperlipidemic WHHL rabbits, liposome merging likely also produces small amounts of superlarge liposomes. These either may not have been detected because of the small number of replicas examined in Ref. 6 or may have been depleted by a mechanism, e.g., macrophage phagocytosis discussed below, not included in Eqs. 2a and b. Thus this kinetic model, coupled to tissue-specific transport models, adequately describes feeding-induced liposome formation/growth in both the aorta and the valve.
The quantity that is likely most relevant in triggering the recruitment of blood-borne monocytes to enter the subendothelial matrix is the total liposome mass. Its rate of increase likely determines whether monocyte-derived macrophages will succeed in cleaning up accumulated lipid in the valve or fail and become foam cells. Figure 6A shows the evolution of the total liposome mass (Eqs. 4a and b) for the simple (km = 0) and modified (km = 0.0002 h–1) models in the presence of the spatially nonuniform LDL transport. Since the present models do not include any mechanism for depleting formed liposomes, both kinetics yield monotonically increasing total liposome masses, but, as noted, the rate of increase is faster without merging. This difference again only becomes noticeable in the third week of feeding. To get some guidance as to this growth, we turn to Eqs. 5a and b, which suggest that, even when including spatial variation, the total liposome mass for the simple model might roughly follow a parabolic shape with time, and a plot of 
vs. t would then be roughly linear. Despite the fact that the analysis leading to Eqs. 5a and b does not hold with merging, Fig. 6B replots the data from Fig. 6A in this form for both models. For short cholesterol feeding time, the curve is roughly flat, i.e., the total mass of liposomes grows roughly linearly with time when the models hardly differ. After 9 days, Fig. 6B indeed becomes linear, i.e., the growth becomes parabolic, for both models, but we have not been able to predict the slope of the plot's straight-line portion analytically.
Phagocytosis by macrophages: liposome number decay.
The above models predict that the total liposome mass increases monotonically, without bound, with cholesterol feeding time (Fig. 6A), and lowering lumen LDL concentration, Cp, and thus L(t), only changes the magnitude, and not its sign, of this growth. These models adequately describe the short-term (29 day) liposome formation and growth in feeding-induced hyperlipidemic rabbits in the present study. But this unbounded growth suggests similar liposome accumulation even for low and normocholesteremic individuals, albeit at longer times, as for hypercholesteremic ones; that is, low and normocholesteremic individuals would have nontrivial liposome distributions equivalent to younger hypercholesteremic individuals. This contradicts experiments on valves (11) and arteries (6) that found no visible liposomes in control/normal rabbits. Moreover, Stary (17) found that early plaques develop in the arteries of breast-fed infants but quickly disappear after they are weaned. Therefore, there is biological recovery that models of Eqs. 2a and b do not yet include. Filip et al. (5) found that, as in the aorta of hyperlipidemic rabbits (7, 15), blood-borne monocytes adhere to endothelial cells situated at regions of high subendothelial liposome accumulation in heart valves, transmigrate there, and transform into macrophages that devour lipid deposits by phagocytosis. Our models clearly require a liposome depletion mechanism to be applicable to longer times/real situations. Such liposome number decay would remove small liposomes in normocholesteremic individuals and otherwise limit the growth of the total liposome mass.
As Yin et al. (20) did in their arterial study, we crudely model phagocytosis by macrophages by including a first-order decay in the number of liposomes of each size, i.e., by subtracting a liposome decay term kdMj into the right side of Eqs. 2a and b (for Eq. 2a, j = 1). Yin et al. (20) solve the simple model with this modification and uniform LDL supply analytically, but we restrict our attention to the version with merging and spatially nonuniform LDL transport. We invoke a single decay rate constant (kd) for all liposome sizes since the entire liposome is ingested, presumably independently of its size. Phagocytosis will decrease the rates of growth of the total liposome number and mass by subtracting 
from Eq. 4a and 
from Eq. 4b. We choose phagocytosis constant kd = 0.00097 h–1, whose true size depends on the concentration of circulating monocytes, estimated in the arterial study (20) based on the WHHL data and Stary's (17) observations. No data are available for the valve, and so we, consistently, presume the kinetics is tissue independent. Figure 7 plots the growth of the total liposome mass for 60 days of feeding from the merging model with and without phagocytosis by macrophages. As in Fig. 6A, but now extended to 60 days of feeding, the model with merging but no phagocytosis predicts a monotonic increase in the total liposome mass with cholesterol feeding time. In contrast, phagocytosis slows down this growth. For this very low value of kd, the corresponding curve has a convex shape that is nearly identical to the phagocytosis-free curve for the first 3 wk before phagocytosis becomes appreciable. Later, as phagocytosis plays an ever more important role with increasing total liposome numbers by slowing down the growth in total liposome mass, the curve becomes concave and plateaus at long times (not shown). The phagocytosis rate constant kd determines the position of the curve's inflexion point. The value that we have chosen is small enough that phagocytosis does not change the liposome size distribution significantly in 29 days of cholesterol feeding, which maintains a nice fit to the data in Fig. 4. At the longer times characteristic of the WHHL rabbits used in Ref. 6, phagocytosis is appreciable.
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Fig. 7. Growth of the total liposome mass,  , for 60 days of feeding from the merging model (km = 0.0002 h–1) with and without phagocytosis (kd = 0.00097 h–1).
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Summary and conclusions.
Simionescu et al. (16) used conventional freeze-fracture and thin-section electron microscopy to show the presence of uni- or multilamellar lipid packet vesicles, or "extracellular liposomes," with diameters of 100–300 nm in the atrioventricular valves of normal rabbits fed a high-cholesterol diet for 2 wk. Nievelstein-Post et al. (11) first used a freeze-etch technique to preserve the detailed ultrastructure of the subendothelial extracellular matrix and the liposomes embedded therein in heart valves. We report on the results of a large number of such replicas and find the extracellular lipid liposome size distribution in rabbit valve leaflets after 29 days of feeding. The observed smooth-surfaced liposomes (Fig. 1) are in direct contact with matrix filaments. When compared with a similar study in aorta of rabbits after 10–16 days of cholesterol feeding (6), these heart valves have fewer small- and more medium-sized liposomes. They also show superlarge liposomes not found in the aorta (6).
We then combine the theory for lipid delivery to the valve's subendothelial tissue from Parts I (21) and II (22) of this series with Yin et al.'s (20) nucleation/polymerization theories for the formation and growth of extracellular lipid liposomes developed for the arterial intima. The transport theory (Fig. 3) showed a high lipid concentration near the center of a localized leak that decayed with distance from it. The Ansatz for the combination is that liposome formation and growth is tissue independent, depending only on local lipid and (similar) matrix concentrations. Simulations showed that the simplest kinetics, with liposome growth only by appending free LDL, could not produce the observed distributions even with nonuniform LDL transport.
Freeze etchings from both the aortic intima (6) and the valve leaflet (Fig. 1) clearly show that existing liposomes can also merge to form larger liposomes. By including this additional mechanism of liposome growth with a very small merging constant of km = 0.0002 h–1, the spatially dependent modified model's liposome size distribution agrees well with our experimental data (Fig. 4). Over the 29 days of feeding, the percentage of small liposomes decreases and those of other sizes increase monotonically. Large and superlarge liposomes only begin to reach significant percentages after 2 wk of feeding. Intuition developed from the study of spatially homogeneous versions of the kinetic models lends insight into these results. For km = 0.0002 h–1, liposome merging noticeably first alters the evolutions of the liposome size distribution and of its total mass (Fig. 6) from their simple model values in the third week of feeding. The model's total liposome mass growth unrealistically accelerates with time because it increases with the total number of liposomes. Further inclusion of phagocytosis by macrophages allows liposome cleanup that is known to occur in vivo. A small decay constant (kd = 0.00097 h–1) from the aorta (20) slows the growth of the total liposome mass for times longer than 1 month and causes it to plateau at much longer times (Fig. 7). It does not affect the fit at 29 days of feeding (Fig. 4).
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GRANTS
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This work was supported by National Science Foundation Grant CTS-0077520 and National Institutes of Health Grant 5-RO1-HL-067383.
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ACKNOWLEDGMENTS
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P. Nievelstein-Post was a UCLA postdoctoral student in 1989–1991.
Present address of Y. Yin: Dept. of Pathology, Albert Einstein College of Medicine, Bronx, NY 10461.
Present address of P. Nievelstein-Post: Dept. of Science, University College Utrecht of Utrecht University, The Netherlands.
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FOOTNOTES
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Address for reprint requests and other correspondence: D. Rumschitzki, Dept. of Chemical Engineering, CCNY, CUNY, New York, NY 10031 (e-mail: david{at}ccny.cuny.edu)
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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Copyright © 2007 by the American Physiological Society.
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ABSTRACT
EXPERIMENTAL MATERIALS AND...
vs. t.