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A 2D Mathematical Model of Blood Flow and its Interactions inan Atherosclerotic Artery

Published online by Cambridge University Press:  31 July 2014

S. Boujena ,
O. Kafi  and
N. El Khatib
S. Boujena
Affiliation:
Université Hassan II-Casablanca, Faculté des Sciences -Ain Chock-, B.P 5366. Maarif. Casablanca
O. Kafi
Affiliation:
Université Hassan II-Casablanca, Faculté des Sciences -Ain Chock-, B.P 5366. Maarif. Casablanca
N. El Khatib*
Affiliation:
Department of Computer Science and Mathematics, Lebanese American University - Byblos campus P.O. Box: 36, Byblos, Lebanon
*
⋆Corresponding author. E-mail: [email protected]
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Abstract

A stenosis is the narrowing of the artery, this narrowing is usually the result of theformation of an atheromatous plaque infiltrating gradually the artery wall, forming a bumpin the ductus arteriosus. This arterial lesion falls within the general context ofatherosclerotic arterial disease that can affect the carotid arteries, but also thearteries of the heart (coronary), arteries of the legs (PAD), the renal arteries... It cancause a stroke (hemiplegia, transient paralysis of a limb, speech disorder, sailing beforethe eye). In this paper we study the blood-plaque and blood-wall interactions using afluid-structure interaction model. We first propose a 2D analytical study of thegeneralized Navier-Stokes equations to prove the existence of a weak solution forincompressible non-Newtonian fluids with non standard boundary conditions. Then, coupled,based on the results of the theoretical study approach is given. And to form a realisticmodel, with high accuracy, additional conditions due to fluid-structure coupling areproposed on the border undergoing inetraction. This coupled model includes (a) a fluidmodel, where blood is modeled as an incompressible non-Newtonian viscous fluid, (b) asolid model, where the arterial wall and atherosclerotic plaque will be treated as nonlinear hyperelastic solids, and (c) a fluid-structure interaction (FSI) model whereinteractions between the fluid (blood) and structures (the arterial wall and atheromatousplaque) are conducted by an Arbitrary Lagrangian Eulerian (ALE) method that allowsaccurate fluid-structure coupling.

Keywords

stenosed arteryatheromatous plaqueblood flownon-Newtonian fluidfluid-structure interactiongeneralized Navier-Stokes equationsboundary conditions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 6: Blood flows , 2014 , pp. 46 - 68
Copyright
© EDP Sciences, 2014

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References

Gambaruto, A.M., Janela, J., Moura, A., Sequeira, A.. Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Mathematical Biosciences and Engineering, 8 (2011), no. 2, 409–423. Google Scholar
A. Quarteroni, L. Formaggia. Mathematical modelling and numerical simulation of the cardiovascular system. P.G. Ciarlet (ED.),Handbook of numerical analysis, vol XII, North-Holland, Amsterdam, (2004), 3–127.
Cioranescu, D.. Sur une classe de fluides non newtoniens. App. Math. and Opt, 3 (1977), no. 2/3, 263–282. CrossRefGoogle Scholar
Hecht, F., Martin-Borret, G., Thiriet, M.. Écoulement rhéofluidifiant dans un coude et une bifurcation plane symétrique. Application à l’écoulement sanguin dans la grande circulation. J. Phys., 3 (1996), France 6, 529–542. Google Scholar
F. Nobile. Numerical approximation of fluid-structure interaction problems with application to haemodynamics. EPFL. PhD thesis, Lausanne, 2001.
Holzapfel, G., Gasser, T., Ogden, R.. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity, 61 (2000), 1–48. CrossRefGoogle Scholar
Janela, J., Moura, A., Sequeira, A.. A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. Journal of Computational and Applied Mathematics, 234 (2010), 2783–2791. CrossRefGoogle Scholar
R. Temam. Navier-Stokes equations: theory and numerical analysis. Amsterdam; New York: North-Holland, 1977.
L. Ait Moudid. Couplage fluide-structure pour la simulation numérique des écoulements fluides dans une conduite à parois rigides ou élastiques, en présence d’obstacles ou non. Université d’Artois. PhD thesis, Compiègne, 2007.
Li, M.X., Beech-Brandt, J.J., John, L.R., Hoskins, P.R., Easson, W.J.. Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses. Journal of Biomechanics, 40 (2007), 3715–3724. CrossRefGoogle ScholarPubMed
N. El Khatib. Modélisation mathématique de l’athérosclérose. Université Claude Bernard – Lyon 1. PhD thesis, Lyon, 2009.
P.A. Raviart, J.M. Thomas. Introduction à l’analyse numérique des équations aux dérivées partielles. Masson, Paris, 1993.
Crosetto, P., Raymond, P., Deparis, S., Kontaxakis, D., Stergiopulos, N., Quarteroni, A.. Fluid-structure interaction simulations of physiological blood flow in the aorta. Computers and Fluids. Elsevier, 43 (2011), no. 1, 46–57. CrossRefGoogle Scholar
R. Aboulaich, S. Boujena, E. El Guarmah. A non linear diffusion model with non homogeneous boundary conditions in image restoration. Esc10 Milan, (2009), 22–26.
S. Boujena. Étude d’une classe de fluides non-Newtoniens, les fluides Newtoniens généralisés. Thèse de troisième cycle. Univ. Pierre et Marrie-Currie, Paris 6, 1986.
Papaioannou, T.G., Stefanadis, Christodoulos. Vascular wall shear stress: basic principles and methods. Hellenic J Cardiol 46, (2005), 9–15. Google ScholarPubMed
Y.C. Fung. Biomechanics: Mechanical properties of living tissues. Springer-Verlag, New York, 1993.
Li, Z.Y., Howarth, Simon P.S., Tang, Tjun, Gillard, Jonathan H.. How critical is fibrous cap thickness to carotid plaque stability? A flow-plaque interaction model. Stroke 37, (2006), 1195–1196. CrossRefGoogle ScholarPubMed