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On the influence of the wall shear stress vector form on hemodynamic indicators

  • Original Article
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Computing and Visualization in Science

Abstract

Hemodynamic indicators such as the averaged wall shear stress (AWSS) and the oscillatory shear index (OSI) are well established to characterize areas of arterial walls with respect to the formation and progression of aneurysms. Here, we study two different forms for the wall shear stress vector from which AWSS and OSI are computed. One is commonly used as a generalization from the two-dimensional setting, the latter is derived from the full decomposition of the wall traction force given by the Cauchy stress tensor. We compare the influence of both approaches on hemodynamic indicators by numerical simulations under different computational settings. Namely, different (real and artificial) vessel geometries, and the influence of a physiological periodic inflow profile. The blood is modeled either as a Newtonian fluid or as a generalized Newtonian fluid with a shear rate dependent viscosity. Numerical results are obtained by using a stabilized finite element method. We observe profound differences in hemodynamic indicators computed by these two approaches, mainly at critical areas of the arterial wall.

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Notes

  1. It is very common that the shear-thinning of blood is described by a more general Carreau–Yasuda model, having in comparison with the Carreau model one more additional material parameter. Nevertheless, in the case of blood, both models give the same quantitative and qualitative fits. Thus we use the simpler one.

  2. In general, blood viscosity is depending on many factors like hematocrit, pH, age, gender, etc., and thus different values of fitted material parameters can be found through the literature.

  3. Sometimes, the Navier–Stokes equations are expressed in terms of the kinematic viscosity \(\nu =\mu /\rho \). Nevertheless, in the case of non-dimensionalization, this issue is irrelevant due to the Reynolds number being then of the form \(\text {Re}={L^{*} U^{*}}{/}{\nu }\)

  4. The range in non-dimensionalized physical quantities shall be specified later.

  5. In general, the vessel cross-section may not posses a circular profile. In that case, the prescribed parabolic function needs to be properly scaled or has to have a suitable decay at the boundary of such a cross-section.

  6. Artery of our interest.

  7. For this particular case, 7 summands of the series are approximating the waveform accurately enough.

References

  1. Anand, M., Rajagopal, K.: A shear-thinning viscoelastic fluid model for describing the flow of blood. Int. J. Cardiovasc. Med. Sci. 4(2), 55–68 (2004)

    Google Scholar 

  2. Bochev, P., Dohrmann, C.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fluids 46, 183–201 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  4. Chien, S., Usami, S., Dellenback, R.J., Gregersen, M.I., Nanninga, L.B., Guest, M.M.: Blood viscosity: influence of erythrocyte aggregation. Science (New York, N.Y.) 157, 829–831 (1967)

    Article  Google Scholar 

  5. Chien, S., Usami, S., Dellenback, R.J., Gregersen, M.I., Nanninga, L.B., Guest, M.M.: Blood viscosity: influence of erythrocyte deformation. Science (New York, N.Y.) 157, 827–829 (1967)

    Article  Google Scholar 

  6. Cunningham, K.S., Gotlieb, A.I.: The role of shear stress in the pathogenesis of atherosclerosis. Lab. Investig. 85(1), 9–23 (2005)

    Article  Google Scholar 

  7. Galdi, G., Rannacher, R., Robertson, A., Turek, S.: Hemodynamical Flows: Modeling, Analysis and Simulation, vol. 37. Birkhäuser, Basel (2008)

    Book  MATH  Google Scholar 

  8. Gambaruto, A., Janela, a, Moura, A., Sequeira, A.: Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Math. Biosci. Eng. 8(2), 409–423 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Guzman, R.J., Abe, K., Zarins, C.K.: Flow-induced arterial enlargement is inhibited by suppression of nitric oxide synthase activity in vivo. Surgery 122(2), 273–280 (1997)

    Article  Google Scholar 

  10. Hazama, F., Kataoka, H., Yamada, E., Kayembe, K., Hashimoto, N., Kojima, M., Chiegon, K.: Early changes of experimentally induced cerebral aneurysms in rats. Light-microscopic study. Am. J. Pathol. 124, 399–404 (1986)

    Google Scholar 

  11. Kayembe, K.N., Sasahara, M., Hazama, F.: Cerebral aneurysms and variations in the circle of Willis. Stroke 15(5), 846–850 (1984)

    Article  Google Scholar 

  12. Ku, D.N.: Blood flow in arteries. Annu. Rev. Fluid Mech. 29(1), 399–434 (1997)

    Article  MathSciNet  Google Scholar 

  13. Ku, D.N., Giddens, D.P., Zarins, C.K., Glagov, S.: Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arterioscler. Thromb. Vasc. Biol. 5(3), 293–302 (1985)

    Article  Google Scholar 

  14. Malek, A., Alper, S., Izumo, S.: Hemodynamic shear stress and its role in atherosclerosis. J. Am. Med. Assoc. 282(21), 2035–2042 (1999)

    Article  Google Scholar 

  15. Málek, J., Nečas, J., R\(\mathring{{\rm u}}\)žička, M.: On the non-Newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3(1), 35–63 (1993)

  16. Ramalho, S., Moura, A.: Sensitivity to outflow boundary conditions and level of geometry description for a cerebral aneurysm. Int. J. Numer. Methods Biomed. Eng. 28, 697–713 (2012)

    Article  MathSciNet  Google Scholar 

  17. Samady, H., Eshtehardi, P., McDaniel, M.C., Suo, J., Dhawan, S.S., Maynard, C., Timmins, L.H., Quyyumi, A.A., Giddens, D.P.: Coronary artery wall shear stress is associated with progression and transformation of atherosclerotic plaque and arterial remodeling in patients with coronary artery disease. Circulation 124(7), 779–788 (2011)

  18. Schenk, O., Bollhöfer, M., Römer, R.: On large scale diagonalization techniques for the Anderson model of localization. SIAM Rev. 50(1), 91–112 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sforza, D., Putman, C., Cebral, J.: Hemodynamics of cerebral aneurysms. Annu. Rev. Fluid Mech. 41, 91–107 (2009)

    Article  MATH  Google Scholar 

  20. Soulis, J.V., Giannoglou, G.D., Chatzizisis, Y.S., Seralidou, K.V., Parcharidis, G.E., Louridas, G.E.: Non-Newtonian models for molecular viscosity and wall shear stress in a 3D reconstructed human left coronary artery. Med. Eng. Phys. 30(1), 9–19 (2008)

    Article  Google Scholar 

  21. Thiriet, M.: Biology and Mechanics of Blood Flows. Springer, New York (2008)

    Book  MATH  Google Scholar 

  22. Thurston, G.: Viscoelasticity of human blood. Biophys. J. 12(4), 1205–1217 (1972)

    Article  Google Scholar 

  23. Thurston, G.: Rheological parameters for the viscosity viscoelasticity and thixotropy of blood. Biorheology 16(3), 149–162 (1979)

    Article  Google Scholar 

  24. Turek, S., Rivkind, L., Hron, J., Glowinski, R.: Numerical study of a modified time-stepping \(\theta \)-scheme for incompressible flow simulations. J. Sci. Comput. 28(2–3), 533–547 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wasserman, S.M., Topper, J.N.: Adaptation of the endothelium to fluid flow: in vitro analyses of gene expression and in vivo implications. Vasc. Med. 9(1), 35–45 (2004)

    Article  Google Scholar 

  26. Yeleswarapu, K., Kameneva, M., Rajagopal, K.: The flow of blood in tubes: theory and experiment. Mech. Res. 25(3), 257–262 (1998)

    Article  MATH  Google Scholar 

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Acknowledgements

This work has been supported by the Austrian Science Fund (FWF) under the grant SFB Mathematical Optimization and Applications in Biomedical Sciences, and by Graz University of Technology.

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  1. Institute of Computational Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    L. John, P. Pustějovská & O. Steinbach

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  1. L. John
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  2. P. Pustějovská
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  3. O. Steinbach
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Correspondence to O. Steinbach.

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Communicated by Gabriel Wittum.

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John, L., Pustějovská, P. & Steinbach, O. On the influence of the wall shear stress vector form on hemodynamic indicators. Comput. Visual Sci. 18, 113–122 (2017). https://doi.org/10.1007/s00791-017-0277-7

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  • DOI: https://doi.org/10.1007/s00791-017-0277-7

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