T. Christian Gasser

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

Failure properties of intraluminal thrombus in abdominal aortic aneurysm under static and pulsating mechanical loads

OBJECTIVES It has been suggested that mechanical failure of intraluminal thrombus (ILT) could play a key role in the rupture of abdominal aortic aneurysms (AAAs), and in the present study, this hypothesis has been investigated. An in vitro experimental approach has been proposed, which provides layer-specific failure data of ILT tissue under static and pulsatile mechanical loads. METHODS In total, 112 bone-shaped test specimens are prepared from luminal, medial, and abluminal layers of eight ILTs harvested during open elective AAA repair. Three different types of mechanical experiments, denoted as control test, ultimate strength test, and fatigue test were performed in Dulbeccos modified eagles medium (DMEM) supplemented with fetal calf serum, L-ascorbic acid, and antibiotics at 37 degrees C and pH 7.0. In detail, fatigue tests, which are experiments, where the ILT tissue is loaded in pulsatile manner, were carried out at three different load levels with a natural frequency of 1.0 Hz. RESULTS ILTs ultimate strength (156.5 kPa, 92.0 kPa, and 47.7 kPa for luminal, medial, and abluminal layers, respectively) and referential stiffness (62.88 kPa, 47.52 kPa, and 41.52 kPa, for luminal, medial, and abluminal layers, respectively) continuously decrease from the inside to the outside. ILT tissue failed within less than 1 hour under pulsatile loading at a load level of 60% ultimate strength, while a load level of about 40% ultimate strength did not cause failure within 13.9 hours. CONCLUSIONS ILT tissue is vulnerable against fatigue failure and shows significant decreasing strength with respect to the number of load cycles. Hence, after a reasonable time of pulsating loading ILTs strength is far below its ultimate strength, and when compared with stress predictions from finite element (FE) studies, this indicates the likelihood of fatigue failure in vivo. Failure within the ILT could propagate towards the weakened vessel wall behind it and could initialize AAA failure thereafter.

Blood flow and coherent vortices in the normal and aneurysmatic aortas: a fluid dynamical approach to intra-luminal thrombus formation

Abdominal aortic aneurysms (AAAs) are frequently characterized by the development of an intra-luminal thrombus (ILT), which is known to have multiple biochemical and biomechanical implications. Development of the ILT is not well understood, and shear–stress-triggered activation of platelets could be the first step in its evolution. Vortical structures (VSs) in the flow affect platelet dynamics, which motivated the present study of a possible correlation between VS and ILT formation in AAAs. VSs educed by the λ2-method using computational fluid dynamics simulations of the backward-facing step problem, normal aorta, fusiform AAA and saccular AAA were investigated. Patient-specific luminal geometries were reconstructed from computed tomography scans, and Newtonian and Carreau–Yasuda models were used to capture salient rheological features of blood flow. Particularly in complex flow domains, results depended on the constitutive model. VSs developed all along the normal aorta, showing that a clear correlation between VSs and high wall shear stress (WSS) existed, and that VSs started to break up during late systole. In contrast, in the fusiform AAA, large VSs developed at sites of tortuous geometry and high WSS, occupying the entire lumen, and lasting over the entire cardiac cycle. Downward motion of VSs in the AAA was in the range of a few centimetres per cardiac cycle, and with a VS burst at that location, the release (from VSs) of shear-stress-activated platelets and their deposition to the wall was within the lower part of the diseased artery, i.e. where the thickest ILT layer is typically observed. In the saccular AAA, only one VS was found near the healthy portion of the aorta, while in the aneurysmatic bulge, no VSs occurred. We present a fluid-dynamics-motivated mechanism for platelet activation, convection and deposition in AAAs that has the potential of improving our current understanding of the pathophysiology of fluid-driven ILT growth.

Biomechanical factors in the biology of aortic wall and aortic valve diseases

The biomechanical factors that result from the haemodynamic load on the cardiovascular system are a common denominator of several vascular pathologies. Thickening and calcification of the aortic valve will lead to reduced opening and the development of left ventricular outflow obstruction, referred to as aortic valve stenosis. The most common pathology of the aorta is the formation of an aneurysm, morphologically defined as a progressive dilatation of a vessel segment by more than 50% of its normal diameter. The aortic valve is exposed to both haemodynamic forces and structural leaflet deformation as it opens and closes with each heartbeat to assure unidirectional flow from the left ventricle to the aorta. The arterial pressure is translated into tension-dominated mechanical wall stress in the aorta. In addition, stress and strain are related through the aortic stiffness. Furthermore, blood flow over the valvular and vascular endothelial layer induces wall shear stress. Several pathophysiological processes of aortic valve stenosis and aortic aneurysms, such as macromolecule transport, gene expression alterations, cell death pathways, calcification, inflammation, and neoangiogenesis directly depend on biomechanical factors.

Spatial orientation of collagen fibers in the abdominal aortic aneurysm’s wall and its relation to wall mechanics

Collagen is the most abundant protein in mammals and provides the abdominal aortic aneurysm (AAA) wall with mechanical strength, stiffness and toughness. Specifically, the spatial orientation of collagen fibers in the wall has a major impact on its mechanical properties. Apart from valuable microhistological information, this data can be integrated by histomechanical constitutive models thought to improve biomechanical simulations, i.e. to improve the biomechanical rupture risk assessment of AAAs. Tissue samples (n = 24) from the AAA wall were harvested during elective AAA repair, fixated, embedded, sectioned and investigated by polarized light microscopy. The birefringent properties of collagen were reinforced by picrosirius red staining and the three-dimensional collagen fiber orientations were identified with a universal rotary stage. Two constitutive models for collagen fibers were used to integrate the identified structural information in a macroscopic AAA wall model. The collagen fiber orientation in the AAA wall was widely dispersed and could be captured by a Bingham distribution function (κ(1) = 11.6, κ(2) = 9.7). The dispersion was much larger in the tangential plane than in the cross-sectional plane, and no significant difference between the medial and adventitial layers could be identified. The layered directional organization of collagen in normal aortas was not evident in the AAA. The collagen organization identified, combined with constitutive descriptions of collagen fibers that depend on its orientation, explain the anisotropic (orthotropic) mechanical properties of the AAA wall. The mechanical properties of collagen fibers depend largely on their undulation, which is an important structural parameter that requires further experimental investigation.

Nonlinear elasticity of biological tissues with statistical fibre orientation

The elastic strain energy potential for nonlinear fibre-reinforced materials is customarily obtained by superposition of the potentials of the matrix and of each family of fibres. Composites with statistically oriented fibres, such as biological tissues, can be seen as being reinforced by a continuous infinity of fibre families, the orientation of which can be represented by means of a probability density function defined on the unit sphere (i.e. the solid angle). In this case, the superposition procedure gives rise to an integral form of the elastic potential such that the deformation features in the integral, which therefore cannot be calculated a priori. As a consequence, an analytical use of this potential is impossible. In this paper, we implemented this integral form of the elastic potential into a numerical procedure that evaluates the potential, the stress and the elasticity tensor at each deformation step. The numerical integration over the unit sphere is performed by means of the method of spherical designs, in which the result of the integral is approximated by a suitable sum over a discrete subset of the unit sphere. As an example of application, we modelled the collagen fibre distribution in articular cartilage, and used it in simulating displacement-controlled tests: the unconfined compression of a cylindrical sample and the contact problem in the hip joint.

A numerical model to study the interaction of vascular stents with human atherosclerotic lesions.

A methodology is proposed that identifies optimal stent devices for specific clinical criteria. It enables to predict the effect of stent designs on the mechanical environment of stenotic arteries. In particular, we present a numerical study which is based on the interaction of a vascular stent with a patient-specific, atherosclerotic human iliac lesion of type V. The stress evolution in four different tissue components during and after stenting is investigated. The geometric model of the artery is obtained through MRI, while anisotropic material models are applied to describe the behavior of tissues at finite strains. In order to model the observed fissuring and dissection of the plaque under dilation, the undeformed configuration of the arterial wall incorporates two initial tears. The 3D balloon-stent-artery interaction problem is modeled by means of a contact algorithm, which is based on a C2-continuous surface parametrization, hence avoiding numerical instabilities of standard facet-based techniques. In the simulations three different stent designs are studied. The performance of each stent is characterized by scalar quantities relating to stress changes in the artery, contact forces, and changes in lumen area after stenting. The study concludes by suggesting two optimal stent designs for two different clinically relevant parameters.

Modeling Plaque Fissuring and Dissection during Balloon Angioplasty Intervention

Balloon angioplasty intervention is traumatic to arterial tissue. Fracture mechanisms such as plaque fissuring and/or dissection occur and constitute major contributions to the lumen enlargement. However, these types of mechanically-based traumatization of arterial tissue are also contributing factors to both acute procedural complications and chronic restenosis of the treatment site. We propose physical and finite element models, which are generally useable to trace fissuring and/or dissection in atherosclerotic plaques during balloon angioplasty interventions. The arterial wall is described as an anisotropic, heterogeneous, highly deformable, nearly incompressible body, whereas tissue failure is captured by a strong discontinuity kinematics and a novel cohesive zone model. The numerical implementation is based on the partition of unity finite element method and the interface element method. The later is used to link together meshes of the different tissue components. The balloon angioplasty-based failure mechanisms are numerically studied in 3D by means of an atherosclerotic-prone human external iliac artery, with a type V lesion. Image-based 3D geometry is generated and tissue-specific material properties are considered. Numerical results show that in a primary phase the plaque fissures at both shoulders of the fibrous cap and stops at the lamina elastica interna. In a secondary phase, local dissections between the intima and the media develop at the fibrous cap location with the smallest thickness. The predicted results indicate that plaque fissuring and dissection cause localized mechanical trauma, but prevent the main portion of the stenosis from high stress, and hence from continuous tissue damage.

A constitutive model for vascular tissue that integrates fibril, fiber and continuum levels with application to the isotropic and passive properties of the infrarenal aorta

A fundamental understanding of the mechanical properties of the extracellular matrix (ECM) is critically important to quantify the amount of macroscopic stress and/or strain transmitted to the cellular level of vascular tissue. Structural constitutive models integrate histological and mechanical information, and hence, allocate stress and strain to the different microstructural components of the vascular wall. The present work proposes a novel multi-scale structural constitutive model of passive vascular tissue, where collagen fibers are assembled by proteoglycan (PG) cross-linked collagen fibrils and reinforce an otherwise isotropic matrix material. Multiplicative kinematics account for the straightening and stretching of collagen fibrils, and an orientation density function captures the spatial organization of collagen fibers in the tissue. Mechanical and structural assumptions at the collagen fibril level define a piece-wise analytical stress-stretch response of collagen fibers, which in turn is integrated over the unit sphere to constitute the tissues macroscopic mechanical properties. The proposed model displays the salient macroscopic features of vascular tissue, and employs the material and structural parameters of clear physical meaning. Likewise, the constitutive concept renders a highly efficient multi-scale structural approach that allows for the numerical analysis at the organ level. Model parameters were estimated from isotropic mean-population data of the normal and aneurysmatic aortic wall and used to predict in-vivo stress states of patient-specific vascular geometries, thought to demonstrate the robustness of the particular Finite Element (FE) implementation. The collagen fibril level of the multi-scale constitutive formulation provided an interface to integrate vascular wall biology and to account for collagen turnover.

Analysis of aortic wall stress and rupture risk in patients with abdominal aortic aneurysm with a gender perspective.

OBJECTIVE The most commonly used predictor of rupture of an abdominal aortic aneurysm (AAA) is the diameter; however, this does not estimate the true risk for each patient. Why women with AAAs have an increased growth rate, weaker aortic wall, and increased risk for rupture is yet unclear. It is likely that geometrical and biomechanical properties contribute to found gender differences. Several studies have shown that peak wall stress (PWS) and peak wall rupture risk (PWRR), predicted by a finite element (FE) analysis of AAAs derived from computed tomography (CT), is a better predictor of rupture than maximum diameter. The purpose of this study was to investigate if women with AAAs have an increased PWS and PWRR using an FE model compared to men. METHOD Fifteen men and 15 women (AAAs 4-6 cm) were included. AAA geometry was derived from CT scans, and PWS and PWRR were estimated using the FE method. Comparisons were made by t test and Mann-Whitney test. RESULTS Mean age (women 73 years old vs men 71 years old) and mean AAA diameter was similar (49.7 mm vs 50.1 mm) for women and men. PWS did not differ for women 184 and men 198 kPa. PWRR was 0.54 (0.28-0.85) for women and 0.43 (0.24-0.66) for men, P = .06. CONCLUSION This is the first analysis of stress and strength of the aneurysm wall with a gender perspective. The reported higher rupture risk for women has previously not been tested with geometrical and biomechanical properties. PWS did not differ, but the PWRR was slightly higher in women. However, the difference did not reach statistical significance, probably due to the small sample size. In summary, the results in the present study suggest that differences in biomechanical properties could be a contributing explanation for the higher rupture risk reported for female patients with AAAs.