Abstract
Atherosclerosis is a chronic inflammatory disease that occurs mainly in large and medium-sized elastic and muscular arteries. This pathology is essentially caused by the high concentration of low-density-lipoprotein (LDL) in the blood. It can lead to coronary heart disease and stroke, which are the cause of around 17.3 million deaths per year in the world. Mathematical modeling and numerical simulations are important tools for a better understanding of atherosclerosis and subsequent development of more effective treatment and prevention strategies. The atherosclerosis inflammatory process can be described by a model consisting of a system of three reaction-diffusion equations (representing the concentrations of oxidized LDL, macrophages and cytokines inside the arterial wall) with non-linear Neumann boundary conditions. In this work we prove the existence, uniqueness and boundedness of global solutions, using the monotone iterative method. Numerical simulations are performed in a rectangle representing the intima, to illustrate the mathematical results and the atherosclerosis inflammatory process.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The wall shear stress function, WSS, is computed using the solution of the blood flow model (for instance a generalized Navier-Stokes model).
- 2.
Let K be an arbitrary compact set and U an open subset of \(\varGamma _{end}\), taken as a very small neighbourhood of K, containing K. There exists a bump function \(\psi (x)\) which is equal to 1 on K and falls off rapidly to 0 outside of K, while still being smooth.
References
Mitchell, M.E., Sidawy, A.N.: The pathophysiology of atherosclerosis. Semin. Vasc. Surg. 3(11), 134–141 (1998)
Ross, R.: Atherosclerosis—an inflammatory disease. Mass. Med. Soc. 340(2), 115–126 (1999)
Mitrovska, S., Jovanova, S., Matthiesen, I., Libermans, C.: Atherosclerosis: Understanding Pathogenesis and Challenge for Treatment. Nova Science Publishers Inc., New York (2009)
Guretzki, H.J., Gerbitza, K.D., Olgemöller, B., Schleicher, E.: Atherogenic levels of low density lipoprotein alter the permeability and composition of the endothelial barrier. Elsevier 107(1), 15–24 (1994)
Mendis, S., Puska, P., Norrving, B.: Global Atlas on Cardiovascular Disease Prevention and Control. World Health Organization, Geneva (2011)
El Khatib, N., Génieys, S., Kazmierczak, B., Volpert, V.: Reaction-diffusion model of atherosclerosis development. J. Math. Biol. 65, 349–374 (2012)
Calvez, V., Ebde, A., Meunier, N., Raoult, A.: Mathematical and numerical modeling of the atherosclerotic plaque formation. ESAIM Proc. 28, 1–12 (2009)
Calvez, V., Houot, J., Meunier, N., Raoult, A., Rusnakova, G.: Mathematical and numerical modeling of early atherosclerotic lesions. ESAIM Proc. 30, 1–14 (2010)
Hao, W., Friedman, A.: The LDL-HDL profile determines the risk of atherosclerosis- a mathematical model. PLoS ONE 9(3), 1–15 (2014)
Liu, B., Tang, D.: Computer simulations of atherosclerosis plaque growth in coronary arteries. Mol. Cell. Biomech. 7(4), 193–202 (2010)
Filipovic, N., Nikolic, D., Saveljic, I., Milosevic, Z., Exarchos, T., Pelosi, G., Parodi, O.: Computer simulation of three-dimensional plaque formation and progression in the coronary artery. Comput. Fluids 88, 826–833 (2013)
Silva, T., Sequeira, A., Santos, R., Tiago, J.: Mathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flow. In: Hindawi Publishing Corporation Conference Papers in Mathematics (2013)
El Khatib, N., Génieys, S., Kazmierczak, B., Volpert, V.: Mathematical modeling of atherosclerosis as an inflammatory disease. Philos. Trans. R. Soc. A 367, 4877–4886 (2009)
Silva, T., Sequeira, A., Santos, R., Tiago, J.: Existence, uniqueness, stability and asymptotic behavior of solutions for a model of atherosclerosis, DCDS-S, AIMS, 9(1), 343–362 (2016)
Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)
Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Springer-Verlag, Berlin Heidelberg (1979)
Acknowledgements
FCT (Fundação para a Ciência e a Tecnologia, Portugal) through the grant SFRH/BPD/66638/2009, the project EXCL/MAT-NAN/0114/2012 and the Research Center CEMAT-IST are gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Japan
About this paper
Cite this paper
Silva, T., Tiago, J., Sequeira, A. (2016). Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis. In: Shibata, Y., Suzuki, Y. (eds) Mathematical Fluid Dynamics, Present and Future. Springer Proceedings in Mathematics & Statistics, vol 183. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56457-7_21
Download citation
DOI: https://doi.org/10.1007/978-4-431-56457-7_21
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-56455-3
Online ISBN: 978-4-431-56457-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)