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Structural and Numerical Models for the (Visco)elastic Response of Arterial Walls with Residual Stresses

  • Chapter
Biomechanics of Soft Tissue in Cardiovascular Systems

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 441))

Abstract

In this chapter we focus attention on the description of structural and numerical models for the elastic and viscoelastic response of arterial walls with residual stresses. We start by reviewing briefly the arterial histology and describing the mechanical characteristics of arterial components. We also present a fully automatic technique for identifying the orientations of cellular nuclei.

Particular attention is concentrated on multi-layer models for predicting reliably the passive elastic and viscoelastic three-dimensional stress and deformation states of arterial walls under various loading conditions. All the models proposed are well suited for FE implementation. Each arterial layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous constituent of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the unstressed configuration of the material corresponds to an open sector of a tube. The viscoelastic model admits hysteresis loops that are known to be relatively insensitive to strain rate, an essential mechanical feature of muscular arteries. The concept of internal variables is introduced in order to replicate the characteristic dissipative mechanism. We summarize the equations that provide the general continuum description of the deformation and the hyperelastic stress response of arterial walls, which are assumed to behave isochorically. One particular simple mixed FE method is discussed in detail (leading to the Q1/P0-element). This approach circumvents numerical difficulties that arise from the overstiffening of the system associated with the analysis of isochoric deformations. Stiffness matrices and some insights into solution methods for nonlinear dynamic problems are provided. Three numerical examples are included to show the performance of the structural arterial models and to document FE results that are in good qualitative agreement with experimental data. The first example is concerned with a FE analysis of the mechanical behavior of an artery during clamping. The remaining two examples are concerned with investigation of the characteristic viscoelastic behavior of a healthy young artery under static and dynamic boundary loadings.

In the last section a layer-specific FE-model of balloon angioplasty is described. The model makes use of an in vitro MRI of a human stenotic post-mortem artery and mechanical tests of the corresponding vascular tissues under supra-physiological loadings. The three-dimensional FE realization considers the balloon-artery interaction and accounts for vessel-specific axial in situ pre-stretches. The proposed approach provides a tool that has the potential to improve procedural protocols and the design of interventional instruments on a lesion-specific basis, and to determine post-angioplasty mechanical environments, which may be correlated with restenosis responses.

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Authors and Affiliations

  1. Institute for Structural Analysis - Computational Biomechanics, Graz University of Technology, Schiesstattgasse 14-B, A-8010, Graz, Austria

    Gerhard A. Holzapfel

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  1. Graz University of Technology, Austria

    Gerhard A. Holzapfel

  2. University of Glasgow, UK

    Ray W. Ogden

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Holzapfel, G.A. (2003). Structural and Numerical Models for the (Visco)elastic Response of Arterial Walls with Residual Stresses. In: Holzapfel, G.A., Ogden, R.W. (eds) Biomechanics of Soft Tissue in Cardiovascular Systems. International Centre for Mechanical Sciences, vol 441. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2736-0_4

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  • DOI: https://doi.org/10.1007/978-3-7091-2736-0_4

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