Models foam cell

Leonard and Lemlich [36] were the first to obtain such expressions. Krotov [7] corrected them employing a compact tetradecahedron as an individual foam cell model. A qualitatively similar Q(n) dependence was obtained earlier but the exact value of the proportionality coefficient was not obtained [37]. 

For a dodecahedral foam cell model under the condition that A = const and as = ay (but JlnAVAWr dlnNJNyJT), the value of the proportionality coefficient is less than 6. Using other foam cell models the proportionality coefficient in Eq. (6.50) changes slightly. 

This formula is analogous to Eq. (10.8). Similar equations can be derived for other foam cell models. 

Figure 7.1. A foam cell model (a) pentagonal dodecahedron, (b) section of the Plateau border...

Probucol. Probucol is an antioxidant that is effective in lowering LDL cholesterol. Whereas probucol was known to lower cholesterol after relatively simple clinical trials (160), its mechanism of action as an antioxidant in the treatment of atherosclerosis is quite novel. Probucol has been shown to have the abiUty to produce regression of atherosclerotic lesions in animal models (161). Probucol therefore represents a novel class of pharmaceutical agent for the treatment of atherosclerosis. This effect occurs mechanistically, in part, by preventing oxidation of LDL, a necessary step in foam cell formation. This antioxidant activity has been shown in laboratory experiments and its activity in lowering LDL cholesterol in human studies is well documented (162). 

Chloramines change LDL charge characteristics, inducing uncontrolled uptake of modified LDL by macrophages and the formation of cholesterol-engorged foam cells [160]. The ability of HOC1 to modify LDL was confirmed in a model system [161]. 

The rat carotid artery injured by a balloon catheter has been widely used as a model of angioplasty. The rat model is a proliferation model without foam cells (93). This form of injury causes immediate coagulation and thrombosis cascade in which platelets adhere, spread, and degranulate on the denuded surface of the artery, and approximately 24 hours later SMC begin to proliferate. Liposomal BPs, clodronate, and alendronate were injected to male sabra rats, 15 and 3mg/kg, respectively (52,69,76). Marked neointimal formation and decreased luminal area were observed in control animals. Neointima/media (N/M) ratio was 1.3 0.2, and luminal stenosis was 44 3%. LC and LA suppressed intimal growth when administered intravenously on day -1 and day 6. N/M ratios were reduced by 60% and 69% for LC and LA, respectively. 

PP bead foams were subjected to oblique impacts, in which the material was compressed and sheared. This strain combination could occur when a cycle helmet hit a road surface. The results were compared with simple shear tests at low strain rates and to uniaxial compressive tests at impact strain rates. The observed shear hardening was greatest when there was no imposed density increase and practically zero when the angle of impact was less than 15 degrees. The shear hardening appeared to be a unique function of the main tensile extension ratio and was a polymer contribution, whereas the volumetric hardening was due to the isothermal compression of the cell gas. Foam material models for finite element analysis needed to be reformulated to consider the physics of the hardening mechanisms, so their predictions were reliable for foam impacts in which shear occurred. 16 refs. 

The effect of gas compression on the uniaxial compression stress-strain curve of closed-cell polymer foams was analysed. The elastic contribution of cell faces to the compressive stress-strain curve is predicted quantitatively, and the effect on the initial Young s modulus is said to be large. The polymer contribution was analysed using a tetrakaidecahedral cell model. It is demonstrated that the cell faces contribute linearly to the Young s modulus, but compressive yielding involves non-linear viscoelastic deformation. 3 refs. 

The non-isothermal viscoelastic cell model was used to study foam growth in the continuous extrusion of low density foam sheet. Surface escape of blowing agent was successfully incorporated to describe the foaming efficiency. Reasonable agreement was obtained with experimental data for HCFC-22 blown LDPE foam in the sub-centimetre thickness domain. 11 refs. 

In a subsequent theoretical study, Stamenovic [60] obtained an expression for the shear modulus independent of foam geometry or deformation model. The value of G was reported to depend only on the capillary pressure, which is the difference between the gas pressure in the foam cells and the external pressure, again for the case of <)> ca 1. Budiansky et al. [61] employed a foam model consisting of 3D dodecahedral cells, and found that the ratio of shear modulus to capillary pressure was close to that obtained by Princen, but within the experimental limits given by Stamenovic and Wilson. 

The viscous properties of HIPEs and high gas fraction foams have also been studied extensively, using a two dimensional, monodisperse, hexagonal cell model. Khan and Armstrong [52] showed that, under steady shear flow (i.e. beyond the yield point of the system), the foam viscosity was inversely proportional to shear rate. At high rates of shear, a constant viscosity value was approached. Gas fraction, <)>, was assumed to be very close to unity. 

There are two types of foams closed cell foams and open cell (or reticulated) foams. In open foams, air or other fluids are free to circulate. These are used for filters and as skeletons. They are often made by collapsing the walls of closed cell foams. Closed cell foams are much stiffer and stronger than open cell foams because compression is partially resisted by increased air pressure inside the cells. Figure 19.1 shows that the geometry of open and closed cell foams can modeled by Kelvin tetrakaidecahedra. 

Pentagonal dodecahedron [76-80], compact tetradecahedron [73,80,82] and minimal tetrakaidecahedron [67,68] are most often used as models of foam cells in the calculation of foam electrical conductivity and hydroconductivity, foam dispersity and in the process of adsorption accumulation of foam. 

The shape of foam films and border profiles in large interval of foam expansion ratio from 10 to 1500 has been experimentally studied in [83], A regular pentagonal dodecahedron made up of transparent organic glass with an elastic rubber balloon inside it which took the shape of a sphere at inflation (Fig. 1.10) was used as a model of foam cell. 

The edges of this dodecahedron sized a - 8.5 cm. When the volume of the rubber balloon at inflation became bigger than the volume of the sphere inscribed in the dodecahedron, the balloon was deformed by the dedecahedron faces and took a shape close to the respective shape of a bubble in a monodisperse dodecahedral foam with a definite expansion ratio. The expansion ratio of the foam was determined by the volume of liquid (surfactant solution or black ink in the presence of sodium dodecylsulphate) poured into the dodecahedron. An electric bulb fixed in the centre of the balloon was used to take pictures of the model of the foam cell obtained. The film shape and the projection of the borders and vertexes on the dodecahedron face are clearly seen in Fig. 1.10. 

Considering a pentagonal dodecahedron model for foam cells Budansky and Kimmel [25] have derived a value for G which is between those obtained from the above discussed two models. 

In a polyhedral foam the liquid is distributed between films and borders and for that reason the structure coefficient B depends not only on foam expansion ratio but also on the liquid distribution between the elements of the liquid phase (borders and films). Manegold [5] has obtained B = 1.5 for a cubic model of foam cells, assuming that from the six films (cube faces) only four contribute to the conductivity. He has also obtained an experimental value for B close to the calculated one, studying a foam from a 2% solution of Nekal BX. Bikerman [7] has discussed another flat cell model in which a raw of cubes (bubbles) is shifted to 1/2 of the edge length and the value obtained was B = 2.25. A more detailed analysis of this model [45,46] gives value for B = 1.5, just as in Manegold s model. 

Modified (oxidized) lipid species have been identified in the plasma lipoproteins and aortic plaque of atherosclerotic humans (74, 77) and animal models (78, 79). Furthermore, the presence of COPs in circulating lipoprotein has been demonstrated in healthy humans (80) and monkeys. (73) Oxidized LDL has been proposed to have a role in foam cell formation (81) as well as having various proatherogenic properties, such as cytotoxicity and chemotactic activity (73, 82). 

As with cell shapes of a real foam, cell sizes in this material can also be characterized only by nominal (effective) values. The actual effective values depend, first, on the observation method (whether direct — macroscopic, or indirect — adsorption, volumetric, picnometric, etc.). Secondly, they depend on the particular simplified model of the structure and cell shape and thirdly on the method of processing the measured data. 

To calculate the foam cell size distribution function we consider, following Mihira an isolated cell foam structure model (Fig. 24). Let r be a true cell radius, and s the radius of sectional circles on the cut surface X, f and s their mean values, of and of their mean square deviations, and f(r) and f(s) their distribution functions. We will denote by x the depth of a cell dissected by the plane X (Fig. 24) and calculate the probability P(r,x) of cells having a radius in the range from r to (r + dr) and a depth from x to (x + dx). The probability P(r) for the cells dissected by the plane X to have a radius r is ... 

Fig. 4 The lipid influx/efflux rheostat model maintains lipid uptake and export mechanisms in a balance. ATP synthase is regulated by apoA-I or apoE leading to enhanced conversion of ATP to ADP. The absence of apoA-I would lead to enhanced sinking in phagocytosis since actin can bind ATP, polymerize, and form F-actin which is essential for type 11 phagocytosis. Hence apoA-I could lead to increased influx. On the other hand, apoA-I binds to ABCAl leading to enhanced lipid efflux. Dysfunction of this equilibrium may lead to severe disturbances of cellular lipid traffic. This is obvious in Tangier disease patients where ABCAl is inoperative and apoA-/-dependent cholesterol is absent. Cholesterol influx, however, is enhanced due to apoA-Z-dependent stimulation of ATP synthase B leading to cholesteryl ester formation and enhanced foam cell formation...

Actual foam contains bubbles whose shape is intermediate between spheres and polyhedra. Such foam is said to be cellular [214, 280]. The distinction between the cellular and polyhedral kinds of foam is rather conventional and is determined by very low moisture contents (of the order of some tenth of per cent). Nevertheless, the polyhedral model of foam cells is used rather frequently [38,125,244,438,480],... 

According to (7.1.15), the capillary rarefaction depends on the mean curvature s of the internal foam surface at the nodes of the foam structure. This variable was calculated in [378] for a monodisperse foam with cells modeled by pentagonal dodecahedra ... 

Preliminary remarks. Models of the foam cell. The polyhedral shape of foam cells is the limit shape as the foam multiplicity grows infinitely. At the same time, this is a rather convenient structural model for actual foam with finite multiplicity. A polyhedron constructed of liquid films must satisfy the following two rules, stated by Plateau [9, 379, 407] ... 

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