1.5. Computational Analyses
1.5.4 Arterial and Plaque Modelling
One of the most important key aspects of modelling stent biomechanics is the constitutive material behaviour of the arterial walls and stenosis. A review by Holzapfel and Ogden (2010) shows that the main constituents of the artery are collagen fibres, elastin and smooth muscle (Figure 1.16), making the arterial tissue exhibit highly non-linear, anisotropic and incompressible behaviour.
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Figure 1.16. Illustration of the main layers composing a human blood vessel wall (Holzapfel et al.,
2000).
In most cases arterial layers can be assumed as purely elastic and therefore characterised by hyperelastic strain energy potentials, but the fibre orientation and its interaction with the other constituents can be considered to refine the constitutive model. Also, the biomechanical behaviour of arterial layers is affected by age and disease, which results in a reduction of elasticity due to the degradation of the elastin. It was also pointed out that the understanding of arterial material behaviour is a key aspect in FE modelling of cardiovascular systems, such as the stent-artery system. Refinement in the constitutive modelling of the arterial layers will increase the potential of FE analyses in predicting physiological functional interactions. Holzapfel et al. (2005) measured the stress-strain behaviour of individual arterial layer (i.e. intima, media and adventitia) gathered
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from 13 diseased cadavers. Their results showed that the elasticity of the material in tension is highly heterogeneous among the patients and all the specimens exhibited hysteresis and strong anisotropic non-linear behaviour. A constitutive model, which was developed originally by Holzapfel et al. (2000), was used in their study to model the experimental stress-strain behaviour. The model consists of a strain energy potential, that is given by the equation:
1
) 3 1 ( ) 3 ( ln 2 1 1 1 exp 2 ) 3 ( 4 1 2 2 2 2 1 1 10 I I E J J D E k k k I C W , (1.2)whereC , 10 D, k , 1 k2 and are material parameters, I4 is the invariant of Cauchy-Green
deformation tensor and E represents the deformation of the fibre families. The Macauley bracket is indicated by the operator <>.
Results demonstrated that the Holzapfel-Gasser-Ogden (HGO) model was able to describe the stress-strain responses of all the layers. Holzapfel and Gasser (2007) analyzed the stress and deformation of a Left Anterior Descending (LAD) coronary artery, modelled as a double layered (consisting of media and adventitia), thick walled, anisotropic and inelastic circular tube. Results showed that the media was the layer that carried most of the stress (75% of the pressure load), while the adventitia took the load-carrying role at supraphysiological loading. Due to its high stiffness, the adventitia provided a support to soft artery layers during radial compression. From these work, it is clear that the mechanical properties of arterial tissues are heterogeneous, anisotropic and inelastic. As reviewed in Section 2.4.2, the arterial layers play an equally important role in the stent expansion. It is therefore important to understand to what extent arterial factors (i.e. anisotropy, inelasticity and layers’ stiffness) can affect the process of stent deployment.
For this purpose, modelling of the arterial and plaque inelasticity was carried out by Maher et al. (2011 and 2012) in two published papers. In the first paper, the inelasticity of the atherosclerotic
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plaque was tested experimentally under uniaxial compressive loading and a constitutive model was developed to account for the permanent deformation and stress softening of the plaque. From the experimental results, the inelastic behaviour of the plaque (i.e. plastic deformation and stress softening) could be observed and independent of the plaque composition. A material model was then developed to account for the stress softening, by introducing a continuum damage approach, and the plastic deformation, by means of an additive split of the stress tensor. FE results showed that the novel constitutive model produced a consistent quality fit of the experimental loading- unloading curves. In the second paper, Maher et al. (2012) presented an anisotropic inelastic model for the arterial tissue, using the same constitutive model developed in their first paper. Numerical simulations of the aorta and carotid arterial tissue were performed to simulate uniaxial tensile testing in both longitudinal and circumferential directions. Results were compared with their own experimental data, showing consistency between the model prediction and the experimental curves during uniaxial loading and unloading of the specimens in both directions. It was discussed that neglecting the inelasticity of the arterial tissues and stenosis would lead to an underestimation of stent expansion. It should be noted that the plaque and arterial plasticity is not included in the material formulation used in this thesis due to the level of complexity, but worth investigation in future studies.
Other interesting aspects that are relevant to stent simulations are the mechanisms of plaque rupture and arterial perforation, which have been the subjects of FE studies. As shown by the clinical studies (Section 2.2.2), stent deployment has been associated with arterial perforation and plaque rupture. The mechanisms of rupture of human soft tissues, such as the arterial layers and the stenotic plaque, have been capturing the interest of researchers for decades. Cheng et al. (1993) analysed the circumferential stress distribution in 24 plaques in order to understand the mechanisms of plaque rupture. Their findings showed that circumferential stress plays an important role in the plaque rupture, along with local variations in the plaque properties. Stress
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analysis of the atherosclerotic plaque has also been studied through imaging techniques, i.e. Magnetic Resonance Imaging (MRI), combined with FE Analysis by Li et al. (2006) (Figure 1.17). Their findings showed that plaques which underwent rupture exhibited considerably higher stresses (0.683MPa) compared to those not ruptured (i.e. 0.227MPa). Also the relative stiffness of fribrous cap to lipid pool can lead to high stress levels within the cap and should be considered as a predicting factor for the vulnerability of the plaque. A further investigation by Li et al. (2009) compared two possible mechanisms for plaque rupture, i.e. the shear stress and the pressure gradient. The shear stress plays a major role on the damage on the endothelium and fissure plaque, but according to these studies the pressure gradient is likely to be one of the main mechanical trigger for the plaque rupture.
Arterial failure was recently investigated by Khamdaengyodtai et al. (2012), who, in particular, analysed the effect of pressure on the failure prediction of arterial layers. The model included a 5- layered arterial wall which were affected by internal pressure, and results showed that the rupture occurs in the circumferential direction, starting from the inner part of the media layer and propagating towards the outside. Another numerical study on the fracture of human arteries was made by Ferrara et al. (2008), adopting a digitally reconstructed geometry of a damaged human artery. Results showed that stress concentrations are critically dependent on the geometry of the fibrous cap and the lipid pool, while the percentage of occlusion caused by the stenosis was not a remarkable factor. Furthermore, all ruptures originated from the internal intima and developed in longitudinal direction. Arterial dissection caused by angioplasty was also studied by Badel et al. (2014) using cohesive interface surfaces on a simplified arterial FE model. The analysis revealed several damage processes, i.e. detachment of the stenotic plaque and overt media dissection, which may lead to proliferation of restenosis. These studies prove that the stress concentration on the plaque obtained by FE analyses can be used as a predictor for plaque rupture and arterial damage. However, none of the studies modelled the actual failure process of the stenosis or the arterial
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layers, which may be considered as a challenge. In this thesis, the failure of the arterial layers and plaque was not modelled, but the stress distributions on the artery-plaque system were used to discuss the effect of stenting on the potential damage of blood vessels.
Figure 1.17. Stress contour of a carotid plaque (70% stenosis):, showing (a) the MRI image of the
plaque, (b) the histology for co-registration of plaque characterization, (c) the FEA model of the plaque and (d) the von Mises stress contour (Li et al., 2006).
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