Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T07:20:29.579Z Has data issue: false hasContentIssue false
  • English
  • Français

Blood Coagulation Simulations using a ViscoelasticModel

Published online by Cambridge University Press:  31 July 2014

A. Sequeira  and
T. Bodnár
A. Sequeira*
Affiliation:
Department of Mathematics and CEMAT/IST, Instituto Superior Técnico, University of Lisbon Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
T. Bodnár
Affiliation:
Faculty of Mechanical Engineering, Czech Technical University in Prague Karlovo Náměstí 13, 121 35 Prague 2, Czech Republic
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

This paper presents numerical results based on a macroscopic blood coagulation modelcoupled with a non-linear viscoelastic model for blood flow. The system of governingequations is solved using a central finite-volume scheme for space discretization and anexplicit Runge-Kutta time-integration. An artificial compressibility method is used toresolve pressure and a non-linear TVD filter is applied for stabilization. A simple testcase of flowing blood over a clotting surface in a straight 3D vessel is solved. This workpresents a significant extension of the previous studies [10] and [9].

Keywords

blood coagulationclotviscoelastic modelnumerical simulations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 6: Blood flows , 2014 , pp. 34 - 45
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anand, M., Rajagopal, K., Rajagopal, K.R.. A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood. J. of Theoretical Medicine, 5 (2003), no. 3-4, 183-218. CrossRefGoogle Scholar
Anand, M., Rajagopal, K., Rajagopal, K.R.. A model for the formation and lysis of blood clots. Pathophysiology Haemostasis Thrombosis, 34 (2005), 109–120. CrossRefGoogle ScholarPubMed
Anand, M., Rajagopal, K., Rajagopal, K.R.. A model for the formation, growth, and lysis of clots in quiescent plasma. A comparison between the effects of antithrombin III deficiency and protein C deficiency. J. of Theoretical Biology, 253 (2008), 725–738. CrossRefGoogle ScholarPubMed
Anand, M., Rajagopal, K.R.. A shear-thinning viscoelastic fluid model for describing the flow of blood. Int. J. of Cardiovascular Medicine and Science, 4 (2004), no. 2, 59–68. Google Scholar
Anand, M., Rajagopal, K.R.. A mathematical model to describe the change in the constitutive character of blood due to platelet activation. C. R. Méchanique, 330 (2002), 557–562. CrossRefGoogle Scholar
Ataullakhanov, F.I., Zarnitsina, V.I., Pokhilko, A.V., Lobanov, A.I., Morozova, O.L.. A spatio-temporal dynamics of blood coagulation and pattern formation. A theoretical approach. International Journal of Bifurcation and Chaos, 12 (2002), no. 9, 1985–2002. CrossRefGoogle Scholar
T. Bodnár. On the use of non-linear TVD filters in finite-volume simulations. In: Algoritmy 2012, Proceedings of Contributed Papers and Posters, Slovak University of Technology, Faculty of Civil Engineering, Bratislava (2012), 190–199.
Bodnár, T., Příhoda, J.. Numerical simulation of turbulent free-surface flow in curved channel. Journal of Flow, Turbulence and Combustion, 76 (2006), no. 4, 429–442. CrossRefGoogle Scholar
Bodnár, T., Rajagopal, K.R., Sequeira, A.. Simulation of the three-dimensional flow of blood using a shear-thinning viscoelastic fluid model. Mathematical Modelling of Natural Phenomena, 6 (2011), no. 5, 1–24. CrossRefGoogle Scholar
Bodnár, T., Sequeira, A.. Numerical simulation of the coagulation dynamics of blood. Computational and Mathematical Methods in Medicine, 9 (2008), no. 2, 83–104. CrossRefGoogle Scholar
T. Bodnár, A. Sequeira. Numerical Study of the Significance of the Non-Newtonian Nature of Blood in Steady Flow Through a Stenosed Vessel. In: Advances in Mathematical Fluid Mechanics. (edited by R. Rannacher & A. Sequeira), 83–104, Springer Verlag (2010).
T. Bodnár, A. Sequeira, L. Pirkl. Numerical Simulations of Blood Flow in a Stenosed Vessel under Different Flow Rates using a Generalized Oldroyd - B Model. In: Numerical Analysis and Applied Mathematics, Vols 1 and 2. Melville, New York: American Institute of Physics, vol. 2 (2009), 645–648.
Bodnár, T., Sequeira, A., Prosi, M.. On the Shear-Thinning and Viscoelastic Effects of Blood Flow under Various Flow Rates. Applied Mathematics and Computation, 217 (2011), 5055–5067. CrossRefGoogle Scholar
Butenas, S., Mann, K. G.. Blood coagulation. Biochemistry (Moscow), 67 (2002), no. 1, 3–12. Translated from Biokhimiya, 67 (2002), no. 1, 5–15. CrossRefGoogle ScholarPubMed
Engquist, B., Lötstedt, P., Sjögreen, B.. Nonlinear filters for efficient shock computation. Mathematics of Computation, 52 (1989), no. 186, 509–537. CrossRefGoogle Scholar
A. Fasano, R.F. Santos, A. Sequeira. Blood coagulation: a puzzle for biologists, a maze for mathematicians. In D. Ambrosi, A. Quarteroni, and G. Rozza (Eds.) Modeling of Physiological Flows. vol. 5, Modeling, Simulation & Applications, ch. 3 (2012) 41–75, Springer.
G.P. Galdi, R. Rannacher, A. M. Robertson, S. Turek (Eds.) Hemodynamical Flows: Modeling, Analysis and Simulations. (Oberwolfach Seminars), (2008) Birkhäuser Verlag.
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. Jameson. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. In AIAA 10th Computational Fluid Dynamics Conference, Honolulu (1991). AIAA Paper 91–1596.
A. Jameson, W. Schmidt, E. Turkel. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping scheme. In: AIAA 14th Fluid and Plasma Dynamics Conference, Palo Alto (1981), AIAA paper 81–1259.
Janela, J., Moura, A., Sequeira, A.. Absorbing boundary conditions for a 3D non- Newtonian fluid-structure interaction model for blood flow in arteries. International Journal of Engineering Science, 48 (2010), no. 11, 1332–1349. CrossRefGoogle Scholar
R. Keslerová. Numerical study of effect of stress tensor for viscous and viscoelastic fluids flow. In: A. Cangiani, R.L. Davidchack, E. Georgoulis, A.N. Gorban J. Levesley, and M.V. Tretyakov (Eds.) Numerical Mathematics and Advanced Applications (2013), 529– 538. Springer. Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011.
Keslerová, R., Kozel, K.. Numerical solution of laminar incompressible generalized newtonian fluids flow. Applied Mathematics and Computation, 217 (2011), no. 11, 5125–5133. CrossRefGoogle Scholar
Kuharsky, A.L., Fogelson, A.L.. Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition. Biophysical Journal, 80 (2001), 1050–1074. CrossRefGoogle Scholar
Mann, K.G., Brummel-Ziedins, K., Orfeo, T., Butenas, S.. Models of blood coagulation. Blood Cells, Molecules, and Diseases, 36 (2006), 108–117. CrossRefGoogle ScholarPubMed
L. Pirkl, T. Bodnár, K. Tuma. Viscoelastic fluid flows at moderate Weissenberg numbers using Oldroyd type model. In: AIP Conference Proceedings, vol. 1389 (2011), 102–105, American Institute of Physics.
Rajagopal, K.R., Srinivasa, A.R.. A thermodynamic frame work for rate type fluid models. Journal of Non-Newtonian Fluid Mechanics, 80 (2000), 207–227. CrossRefGoogle Scholar
Riha, P., Wang, X., Liao, R., Stoltz, J.F.. Elasticity and fracture strain of blood clots. Clinical Hemorheology and Microcirculation, 21 (1999), no. 1, 45–49. Google ScholarPubMed
A.M. Robertson, A. Sequeira, M. Kameneva. Hemorheology. In: Hemodynamical Flows: Modelling, Analysis and Simulation, vol 37, (2008), 63–120, Birkhäuser .
A.M. Robertson, A. Sequeira, R.G. Owens. Rheological models for blood. In:Cardiovascular Mathematics. Modeling and Simulation of the Circulatory System (MS&A) (2009), 211–241, Springer-Verlag .
Sequeira, A., Santos, R.F., Bodnár, T.. Blood coagulation dynamics: Mathematical modeling and stability results. Mathematical Biosciences and Engineering, 8 (2011), no. 2, 425–443. CrossRefGoogle ScholarPubMed
Shyy, W., Chen, M.-H., Mittal, R., Udaykumar, H.S.. On the suppression of numerical oscillations using a non-linear filter. Journal of Computational Physics, 102 (1992), 49–62. CrossRefGoogle Scholar
Zarnitsina, V.I., Pokhilko, A.V., Ataullakhanov, F. I.. A mathematical model for the spatio-temporal dynamics of intrinsic pathway of blood coagulation - I. The model description. Thrombosis Research, 84 (1996), no. 4, 225–236. CrossRefGoogle Scholar