Preliminary look at why some vessels get atherosclerosis and others don't.

Atherosclerosis is a disease mainly of large, high pressure arteries and of valves, typically sparing veins and small, low pressure arteries. We investigate the resistances of the vena cava and the pulmonary artery to the flow of water and the infiltration of solutes into the vessel walls and compare them with similar processes in the aorta. The goal is to see if differences in macromolecular transport from the blood into the vessel wall amongst vessels can explain their different susceptibilities to atherosclerosis.

Structural changes in rat aortic intima due to transmural pressure.

Huang et al. (1997) propose a new hypothesis and develop a mathematical model to explain rationally the in vitro and in situ measured changes (Tedgui and Lever, 1984; Baldwin and Wilson, 1993) in the hydraulic conductivity of the artery wall of rabbit aorta with transmural pressure. The model leads to the intriguing prediction that this hydraulic conductivity would decrease by one half if the thin intimal layer between the endothelium and the internal elastic lamina volume-compresses approximately fivefold. This paper presents the first measurements of the effect of transmural pressure on intimal layer thickness and shows that the intimal matrix is, indeed, surprisingly compressible. We perfusion-fixed rat thoracic aortas in situ with 2 percent glutaraldehyde solution at 0, 50, 100, or 150 mm Hg lumen pressure and sectioned for light and electron microscopic observations. Electron micrographs show a dramatic, nonlinear decrease in average intimal thickness, i.e., 0.62 +/- 0.26, 0.27 +/- 0.14, 0.15 +/- 0.10, and 0.12 +/- 0.07 (SD) micron for 0, 50, 100, and 150 mm Hg lumen pressure, respectively. The volume strain of the intima is more than 20 times greater than the radial strain of the artery wall due to hoop tension and two orders of magnitude greater than the consolidation of the artery wall as a whole assuming constant medial density (Chuong and Fung, 1984). Moreover, in both light and electron microscopic observations, it is easy to find numerous sites where the endothelium puckers into the fenestral pores at high lumen pressure, as predicted by the theory in Huang et al. (1997). In contrast, the average diameter of a fenestral pore increases only 10 percent as the lumen pressure is increased from 0 to 150 mm Hg. These results indicate that the thin intimal layer comprising less than 1 percent of the wall thickness can have a profound effect on the filtration properties of the wall due to the large change in Darcy permeability of the layer and the large reduction in the entrance area of the flow entering the fenestral pores, though the pores themselves experience only a minor enlargement due to hoop tension.

A fiber matrix model for the filtration through fenestral pores in a compressible arterial intima.

We advance a new hypothesis to explain the changes in hydraulic conductivity of an intact artery wall with transmural pressure previously observed by Tedgui and Lever [Am. J. Physiol. 247 (Heart Circ. Physiol. 16): H784-H791, 1984] and Baldwin and Wilson [Am. J. Physiol. 264 (Heart Circ. Physiol. 33): H26-H32, 1993]. This hypothesis suggests that compaction due to pressure loading of the proteoglycan matrix in the arterial intima near fenestral pores of the internal elastic lamina (IEL) leads to a narrowing of the pore entrance area and a large decrease in local intrinsic Darcy permeability of the matrix. To quantitatively assess the feasibility of this mechanism, a local two-dimensional model is proposed to study filtration flow in the vicinity of fenestral pores in a compressible intima. Using a heterogenous fiber matrix theory, we first predict the change in Darcy permeability with intimal thickness (Li). The model then calculates local velocity profiles and pressure distributions in the intima and media. The results show a marked nonlinear steepening of intimal pressure profiles near fenestral pores when the intima thins at higher luminal pressures. The predicted relative change in resistances of the IEL (with intima, R(I)) and of the media (Rm) shows a steep increase in R(I)/Rm when Li is <20% of its unstressed value. Numerical results also suggest that intimal compression has a limiting behavior in which the much stiffer collagen fibrils inhibit further compaction at high pressures after the proteoglycan matrix is maximally compressed. Predictions are also presented to show how different transmural pressures alter growth of an intimal horseradish peroxidase spot that derives from a localized (a single cell's boundary) endothelial leakage. Such a prediction is amenable to experimental verification.

A model for the initiation and growth of extracellular lipid liposomes in arterial intima.

There is considerable evidence that lipoprotein cholesterol, after crossing the arterial endothelium and entering the intima from the vascular lumen, lodges in extracellular lipid packets (labeled "liposomes") bound to the extracellular matrix. These liposomes appear to form by occasional attachment of a low-density lipoprotein (LDL) to the intimal matrix and to grow in place mainly by appending available free LDL. The liposome size distributions observed in chronically hypercholesteremic (WHHL) and in short-term cholesterol-fed rabbits are quite different. We propose a hierarchy of simple nucleation-polymerization models to describe liposome formation and growth. Even the simplest of these (with only one adjustable parameter) agrees extremely well with the WHHL data. In contrast, the cholesterol-fed rabbit data seem to result from the short-term nonuniform intimal history of LDL supply, which is a consequence of the focal nature of the transendothelial LDL flow through isolated transient leakyjunctions. The same models used for the WHHL data, together with this intimal nonuniformity, superimposed on a slow uniform transendothelial seepage also account very well for this cholesterol-fed rabbit data.

A fiber matrix model for the growth of macromolecular leakage spots in the arterial intima.

A new model is presented for the growth of cellular level macromolecular leakage spots in the arterial intima. The theoretical approach differs from the recent study by Yuan et al. [19] in that it directly models and calculates the intimal transport parameters based on Frank and Fogelman's [22] ultrastructural observations of the extracellular subendothelial proteoglycan matrix that their rapid freeze etching technique preserves (see Addendum). Using a heterogeneous fiber matrix theory, which includes proteoglycan and collagen components, the model predicts that the Darcy permeability Kp and macromolecular diffusivity D of the subendothelial intima is two orders of magnitude larger than the corresponding values measured in the media, and supports the observations in Lark et al. [24] that the proteoglycan structure of the intima differs greatly from that of the media. Numerical results show that convection parallel to the endothelium is a very significant transport mechanism for macromolecules in the intima in a large region of roughly 200 microns diameter surrounding the leaky cleft. The predictions of the new model for the early-time spread of the advancing convective-diffusive front from the leakage spots in the intima are in close agreement with our experimental measurements for the growth of HRP spots in [20]. The regions of high concentration surrounding the leaky cell, however, are much more limited and cover an area that is typically equivalent to 20 cells. This prediction is consistent with the recent measurements of Truskey et al. for LDL spot size in rabbit aorta [21] and the hypothesis advanced in [19] that there is a colocalization of subendothelial liposome growth and cellular level leakage. Finally, comparison of predicted and experimentally-measured average LDL concentration in leakage spots strongly suggests that there is significant local molecular sieving at the interface between the fenestral openings in the internal elastic lamina and the media.

Spectral properties of Eigen evolution matrices.

Eigen has employed deterministic kinetic-like equations to describe macromolecular replication and mutation leading to selection. The solutions to these equations and their physically interesting properties depend upon the spectrum of a type of matrix appearing in those equations. Below we explicitly solve for the spectrum of a fairly general class of such matrices. These solutions are obtained recursively for equations describing macromolecules of any length v.