Partner: V. Volpert


Ostatnie publikacje
1.El Khatib N., Genieys S., Kaźmierczak B., Volpert V., Reaction-diffusion model of artherosclerosis development, JOURNAL OF MATHEMATICAL BIOLOGY, ISSN: 0303-6812, DOI: 10.1007/s00285-011-0461-1, Vol.65, pp.349-374, 2012

Streszczenie:

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.

Słowa kluczowe:

Atherosclerosis, Reaction–diffusion equations, Nonlinear boundary conditions, Existence of travellingwaves, Numerical simulations

Afiliacje autorów:

El Khatib N.-other affiliation
Genieys S.-other affiliation
Kaźmierczak B.-IPPT PAN
Volpert V.-other affiliation
30p.
2.Khatib N.El., Génieys S., Kaźmierczak B., Volpert V., Mathematical modelling of atherosclerosis as an inflammatory disease, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, ISSN: 1364-503X, DOI: 10.1098/rsta.2009.0142, Vol.18, No.4, pp.345-352, 2009

Streszczenie:

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.

Słowa kluczowe:

mathematical modelling, biomathematics, partial differential equations, travelling waves, reaction–diffusion equations, atherosclerosis

Afiliacje autorów:

Khatib N.El.-other affiliation
Génieys S.-other affiliation
Kaźmierczak B.-IPPT PAN
Volpert V.-other affiliation
32p.
3.Kaźmierczak B., Volpert V., Travelling calcium waves in systems with non-diffusing buffers, MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, ISSN: 0218-2025, Vol.18, No.6, pp.883-912, 2008

Streszczenie:

The existence and structural stability of travelling waves of systems of the free cytosolic calcium concentration in the presence of immobile buffers are studied. The proof is carried out by passing to zero with the diffusion coefficients of buffers. Thus, its method is different from Ref. 13 where the existence is proved straightforwardly.

Słowa kluczowe:

Calcium dynamics, travelling waves, implicit function theorem

Afiliacje autorów:

Kaźmierczak B.-IPPT PAN
Volpert V.-other affiliation
4.Kaźmierczak B., Volpert V., Calcium waves in systems with immobile buffers as a limit of waves for systems with nonzero diffusion, NONLINEARITY, ISSN: 0951-7715, DOI: 10.1088/0951-7715/21/1/004, Vol.21, pp.71-96, 2008

Streszczenie:

We study the existence and properties of calcium waves in the presence of buffers. The model represents a reaction–diffusion system of equations with some diffusion coefficients equal to zero. They correspond to immobile buffers. The proof of the existence of travelling waves is carried out by passing to zero in the diffusion coefficients.

Afiliacje autorów:

Kaźmierczak B.-IPPT PAN
Volpert V.-other affiliation
5.Kaźmierczak B., Volpert V., Mechano-chemical calcium waves in systems with immobile buffers, ARCHIVES OF MECHANICS, ISSN: 0373-2029, Vol.60, No.1, pp.3-22, 2008

Streszczenie:

We study the existence of travelling waves of the free cytosolic calcium concentration in the presence of immobile buers. We also take into account the mechanochemical interaction. In the proof we use the a priori estimates for solutions of systems with diusing buers and pass to zero with their diusion coefficients.

Afiliacje autorów:

Kaźmierczak B.-IPPT PAN
Volpert V.-other affiliation
6.Kaźmierczak B., Volpert V., Travelling waves in partially degenerated reaction- diffusion systems, Mathematical Modelling of Natural Phenomena, Vol.2, pp.106-125, 2007
7.Belk M., Kaźmierczak B., Volpert V., Existence of reaction-diffusion-convection waves in unbounded cylinders, International Journal of Mathematics and Mathematical Sciences, Vol.2, pp.169-193, 2005