Effect of permeability on the initiation of Atherosclerosis modeled as an inflammatory process.

This work presents a mathematical model, based on partial differential equations, that analyzes the inflammatory stage of atherosclerosis. Four leading players are taken into consideration: Low Density Lipoproteins, oxidized Low Density Lipoproteins, immune cells and the inflammatory cytokines. In addition to this, the permeability of the endothelial layer is taken into account in the model. A stability analysis of the fixed points of the kinetic system is presented in details followed by the proof of existence of traveling wave solutions of the system of partial differential equations. The mathematical analysis leads to a biological interpretation. We distinguish three main cases of the disease state that correlate with the permeability of the endothelial layer. In fact, having a low permeability indicates the disease free state since no chronic inflammatory reaction occurs due to the non initiation of the inflammation. With intermediate permeability, a wave propagation corresponding to a chronic inflammatory reaction might occur whether the initial perturbation overcomes a threshold or not. With high permeability, even a small perturbation of the disease free state leads to a chronic inflammatory reaction represented by a wave propagation. We perform numerical simulations of the solutions to illustrate the biological results.

Mathematical modeling of atherogenesis: Atheroprotective role of HDL.

Atherosclerosis is a chronic inflammatory cardiovascular disease in which arteries harden through the build-up of plaques. This work is devoted to the mathematical modeling and analysis of the inflammatory process of atherosclerosis. We propose a mathematical model formed by three coupled partial differential equations of reaction-diffusion type. We take into account three key-role players: the inflammatory immune cells, the inflammatory cytokines and the oxidized low density lipoproteins. A stability analysis of the kinetic system is performed. It leads to the presence of three stable fixed points relevant to appropriate biological states of atherogenesis; no inflammation, stabilized inflammation (stable plaque) and advanced inflammation (vulnerable plaque). The cases that may occur are subject to the variation of the parameters values. A detailed discussion showing how the model fits the biological phenomena is then established. We investigate as well the existence of solutions of traveling waves type along with numerical simulations that show the wave propagation in different cases. This shows that the inflammatory process propagates inside the intima as a traveling wave. Then, we consider the effect of high density lipoprotein (HDL) on the atherosclerotic plaque formation. To do that, we elaborate a map that determines the level of risk of plaque formation with respect to the prevalence of HDL in the blood. These results confirm but also generalize previous results published in the literature. They also give a deeper understanding to the propagation of the inflammation inside the artery in terms of the interplay among the different main players in the whole process.

Modeling circadian clock-cell cycle interaction effects on cell population growth rates.

The circadian clock and the cell cycle are two tightly coupled oscillators. Recent analytical studies have shown counter-intuitive effects of circadian gating of the cell cycle on growth rates of proliferating cells which cannot be explained by a molecular model or a population model alone. In this work, we present a combined molecular-population model that studies how coupling the circadian clock to the cell cycle, through the protein WEE1, affects a proliferating cell population. We show that the cell cycle can entrain to the circadian clock with different rational period ratios and characterize multiple domains of entrainment. We show that coupling increases the growth rate for autonomous periods of the cell cycle around 24 h and above 48 h. We study the effect of mutation of circadian genes on the growth rate of cells and show that disruption of the circadian clock can lead to abnormal proliferation. Particularly, we show that Cry 1, Cry 2 mutations decrease the growth rate of cells, Per 2 mutation enhances it and Bmal 1 knockout increases it for autonomous periods of the cell cycle less than 21 h and decreases it elsewhere. Combining a molecular model to a population model offers new insight on the influence of the circadian clock on the growth of a cell population. This can help chronotherapy which takes benefits of physiological rhythms to improve anti-cancer efficacy and tolerance to drugs by administering treatments at a specific time of the day.

Inflammation propagation modeled as a reaction-diffusion wave.

Inflammation is a physiological process aimed to protect the organism in various diseases and injuries. This work presents a generic inflammation model based on the reaction-diffusion equations for the concentrations of uninflamed cells, inflamed cells, immune cells and the inflammatory cytokines. The analysis of the model shows the existence of three different regimes of inflammation progression depending on the value of a parameter R called the inflammation number. If R>1, then inflammation propagates in cell culture or tissue as a reaction-diffusion wave due to diffusion of inflammatory cytokines produced by inflamed cells. If 0

Reaction-diffusion model of atherosclerosis development.

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction-diffusion wave. The theoretical results are confirmed by the results of numerical simulations.

Macrophage recruitment and activation: a model for comparing resistance to Salmonella enteritidis in different broiler breeds.

A model for comparing resistance to Salmonella Enteritidis was evaluated in different broiler breeds. The recruitment and phagocytic activity of peritoneal macrophages were assessed in three different broiler breeds (A, B and C) which are farmed world-wide. Assessment was performed after three days of intraperitoneal (i.p.) administration of 3% Sephadex G-200 (10 ml), initiated at twenty-one days of age, followed by contact with i.p. live S. Enteritidis (10 ml, 1.2 x 10(8) colony forming units/ml) for 45 min. Assessment included determination of the number of i.p. macrophages recruited, the number of i.p. phagocytized S. Enteritidis cells per macrophage, the levels of degranulated i.p. beta-glucuronidase and beta-galactosidase, and the count of surviving S. Enteritidis cells. Confirmation of the significance of the model was obtained by comparing resistance to field infection by S. Enteritidis in the three broiler breeds. The recruitment of i.p. macrophages in response to challenge with Sephadex and S. Enteritidis was significantly higher (P < 0.05) in birds of breed A (mean cumulative i.p. macrophage count, in 10 fields of microscopic slide smear magnified at x1,000, was equal to 81.7), compared to recruitment in birds of breed B (33.3) or breed C (41.2). The mean number of phagocytized S. Enteritidis cells per i.p. macrophage in birds of breed A (2.68) was significantly higher (P < 0.05) than in breed B (0.83) and insignificantly higher (P > 0.05) than in breed C (2.35). In addition, the highest level of recruitment and phagocytic activity of macrophages, in birds of breed A, was associated with a higher significant mean i.p. beta-glucuronidase activity (10,425.5 units/ml) than in breed B (3,438.2 units/ml) or breed C (3,356.94 units/ml) (P < 0.05). Moreover, birds of breed A demonstrated a higher mean i.p. beta-galactosidase activity (2.225 units/ml) than birds of breed B (0.852 units/ml) or breed C (1.852 units/ml) (P > 0.05). The higher level of recruitment and activity of i.p. macrophages and the higher rate of degranulation of i.p. enzymes in breed A were associated with a greater number of surviving i.p. S. Enteritidis cells. In response to outbreaks of S. Enteritidis in the field, the average mortality was significantly higher in flocks of breed A (3.2%) than in flocks of breed B (1.2%) or breed C (0.96%) (P < 0.05). These data provide an indication of the significance of the model in reflecting the differences in resistance of S. Enteritidis of broiler breeds reared in a farm environment.

Calcium and carbonate supply in the shell gland of hens laying eggs with strong and weak shells and during and after a rest from lay.

The concentration of calcium-binding protein (CaBP) and the activities of calcium adenosine triphosphatase (Ca(2+)-ATPase) and carbonic anhydrase (CA) were determined in the shell gland mucosa of hens in two experiments. In Experiment 1, laying hens on a proprietary layer mash were compared with hens rested from lay by the feeding of whole grain barley. In Experiment 2 comparisons were made of laying hens fed the proprietary layer mash and producing eggs with either strong or weak shells. These latter comparisons were also made when the shell gland was quiescent or active with respect to daily eggshell formation. Feeding whole grain barley reduced egg production to zero after 11 days. This reduction in rate of lay was accompanied by significant reductions in all three markers, the effect on Ca(2+)-ATPase and CaBP being less than for CA. Control values were regained between 10 and 16 days after the barley was replaced with the layer mash. Relative shell strength and the physiological status of the shell gland with respect to time of daily eggshell formation had no significant effect on any marker in Experiment 2.

Mathematical modelling of atherosclerosis as an inflammatory disease.

Atherosclerosis is an inflammatory disease. The atherosclerosis process starts when low-density lipoproteins (LDLs) enter the intima of the blood vessel, where they are oxidized (ox-LDLs). The anti-inflammatory response triggers the recruitment of monocytes. Once in the intima, the monocytes are transformed into macrophages and foam cells, leading to the production of inflammatory cytokines and further recruitment of monocytes. This auto-amplified process leads to the formation of an atherosclerotic plaque and, possibly, to its rupture. In this paper we develop two mathematical models based on reaction-diffusion equations in order to explain the inflammatory process. The first model is one-dimensional: it does not consider the intima's thickness and shows that low ox-LDL concentrations in the intima do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system, which can lead to a travelling wave that can be initiated by certain conditions, such as infection or injury. High ox-LDL concentrations correspond to a monostable system, and even a small perturbation of the non-inflammatory case leads to travelling-wave propagation, which corresponds to a chronic inflammatory response. The second model we suggest is two-dimensional: it represents a reaction-diffusion system in a strip with nonlinear boundary conditions to describe the recruitment of monocytes as a function of the cytokines' concentration. We prove the existence of travelling waves and confirm our previous results, which show that atherosclerosis develops as a reaction-diffusion wave. The results of the two models are confirmed by numerical simulations. The latter show that the two-dimensional model converges to the one-dimensional one if the thickness of the intima tends to zero.