Simulation of discontinuous damage incorporating residual stresses in circumferentially overstretched atherosclerotic arteries
Introduction
The high number of deaths caused by cardiovascular diseases makes the biomechanical description and simulation of blood vessels more and more important. Numerous material models have been proposed to describe the hyperelastic behavior of arterial walls in the physiological range of deformations. As one of the first models taking some kind of anisotropy into account we should mention the model of Fung et al. [8]. Holzapfel et al. [10] proposed a model which is formulated in the framework of the invariant theory (see e.g., papers by Boehler [6] and Betten [5]). In their model, the orthotropy of arteries is represented by superposing two transversely isotropic layers having the same material parameters for each fiber direction. Thereby, a weak interaction between these two fiber families is assumed. Due to the fact that this model involves a polyconvex stored energy, the existence of minimizers is ensured as well as material stability. For the definition of polyconvexity and its relationship to other generalized convexity conditions we refer to the paper by Ball [1].
Balzani et al. [3] introduced another polyconvex model, which satisfies a priori the condition of a stress-free reference configuration, also in the framework of the invariant theory. Furthermore, the model is adjustable to real biological soft tissues via simple “hand-fitting”, which means that no computational optimization scheme is required. This model is used in this contribution and adjusted to the stress–strain response in uniaxial tension tests of a human abdominal aorta.
In papers by Holzapfel et al. [10] and Gasser and Holzapfel [9] it is pointed out that damage is observed in experiments when arteries are overstretched. Generally, in anisotropic damage mechanics, damage tensors of second or fourth order are used, which leads to complicated functions containing numerous material parameters. For an overview of damage mechanics with respect to engineering applications see e.g., the work by Lemaitre and Desmorat [14]. Based on the introduction of a scalar-valued damage variable by Schröder et al. [17] a one-dimensional damage approach is extended to finite strains and embedded into the concept of internal variables, cf. the paper by Simo [18]. In order to obtain a thermodynamically consistent model we utilize the formalism described by Lemaitre and Chaboche [13]. The basic underlying assumption in this model is that damage occurs only in the fiber direction, which means that the damage function is applied only to the anisotropic part of the stored energy.
It is reported e.g., by Vaishnav and Vossoughi [19] that axial segments of arterial walls spring open when they are sliced in a radial direction. Therefore, the unloaded configuration cannot be stress-free, which means that residual stresses must exist in this state. As often stated in the literature, an artery opened by a radial cut can be assumed to be stress-free (see e.g., papers by Chuong and Fung [7] or Humphrey and Delange [12]). The straightforward approach to determining residual stresses is to consider the open artery and to perform a simulation in which the artery is closed. A method for the incorporation of residual stresses based on this approach is proposed by Balzani et al. [4]. This is the method applied in this contribution.
The degradation of the cardiovascular system results mostly from atherosclerotic degenerations of the blood vessels, which lead to the development of atherosclerotic plaques reducing the arterial lumen. A frequently applied treatment is to dilate the vessel lumen by balloon-angioplasty: a balloon-catheter is placed in the stenotic artery and inflated. This overstretches the artery circumferentially. In the present contribution, we simulate numerically this overstretching by means of the finite-element method.
This paper is organized as follows: in Section 2 we briefly recall the basic terminology of continuum mechanics needed for the following sections. Section 3 explains the utilized model consisting of a damage function which is used in combination with a polyconvex hyperelastic stored energy. Section 4 describes the method for including residual stresses and explains details of the procedure. In Section 5, the hyperelastic stored energy is adjusted to the physiological stress–strain response of the media and adventitia of a human abdominal aorta and an atherosclerotic artery is discretized for a finite-element calculation of its circumferential overstretching state. Section 6 summarizes the contribution.
Section snippets
Continuum mechanics and coordinate-invariant representation
Let be the body of interest in the reference configuration parametrized in X, and let be the considered body in the current configuration parametrized in x. The nonlinear deformation map at time maps points onto points . The deformation gradient F and the strain measure, the right Cauchy–Green tensor C, are defined bywhere the hat-symbol denotes functional dependencies. Due to the fact that the determinant of F represents the change of
Anisotropic model for arterial tissues
From the mechanical point of view, biological soft tissues in arterial walls are basically composed of two fiber families embedded in an isotropic matrix substance. We consider polyconvex stored energies, because we can thus satisfy the necessary condition for the existence of deformations which minimize the elastic potential, i.e. the sequential weak lower semicontinuity (see Ball [1], [2]). In addition, a smooth polyconvex energy function implies a Legendre–Hadamard elliptic function leading
Incorporation of residual stresses
When axial segments of arteries are slit, two phenomena are generally observed: if they are cut transverse to the axial direction they shrink, and when they are sliced in radial direction they spring open. Due to these observations there must be some residual stresses in the closed unloaded configuration because otherwise the artery would not deform when cut. In the literature (e.g., the paper by Chuong and Fung [7]) it is stated that the open artery obtained by cutting in radial direction can
Numerical simulation of an overstretched atherosclerotic artery
Due to atherosclerotic plaques, evolving from accretion of fatty substances at the inner side of the arterial wall and from atherosclerotic intimal change, a reduction of the arterial lumen occurs, which is often referred to as stenosis. In addition, such atherosclerotic plaques lead to the rigidification of the affected arterial layers. To increase the lumen again, catheter-based methods are the mostly used methods of treatment nowadays. In most cases a balloon-angioplasty in combination with
Conclusion and outlook
This paper has dealt with the modeling of discontinuous damage in overstretched arteries, whereby the presence of residual stresses in the unloaded state is taken into account. For the representation of the damages, an anisotropic damage model accounting for an initial damage state differing from the undeformed reference configuration was applied. Hereby, the assumption that arterial walls are undamaged in the physiological domain could be realized. The damage model was used in combination with
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2019, Journal of the Mechanical Behavior of Biomedical MaterialsCitation Excerpt :Models can indeed help to clarify whether the underlying damage mechanisms are consistent with the observed evidence, especially those obtained from independent measurements (i.e., not involved in model calibration). Several structurally-based constitutive models have been proposed to incorporate microscale damage of collagen fibers (Balzani et al., 2006, 2012; Ehret and Itskov, 2009; Calvo et al., 2007; Gasser and Holzapfel, 2002; Liao and Belkoff, 1999; Natali et al., 2005; Tang et al., 2009; Schmidt et al., 2014). Without the claim of being exhaustive, recent reviews on this topic are provided by Gasser et al. (2017) and Li (2016).