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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.

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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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