Gerhard A. Holzapfel

A new Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. 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 natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

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.

Constitutive modelling of passive myocardium: a structurally based framework for material characterization

In this paper, we first of all review the morphology and structure of the myocardium and discuss the main features of the mechanical response of passive myocardium tissue, which is an orthotropic material. Locally within the architecture of the myocardium three mutually orthogonal directions can be identified, forming planes with distinct material responses. We treat the left ventricular myocardium as a non-homogeneous, thick-walled, nonlinearly elastic and incompressible material and develop a general theoretical framework based on invariants associated with the three directions. Within this framework we review existing constitutive models and then develop a structurally based model that accounts for the muscle fibre direction and the myocyte sheet structure. The model is applied to simple shear and biaxial deformations and a specific form fitted to the existing (and somewhat limited) experimental data, emphasizing the orthotropy and the limitations of biaxial tests. The need for additional data is highlighted. A brief discussion of issues of convexity of the model and related matters concludes the paper.

A structural model for the viscoelastic behavior of arterial walls : Continuum formulation and finite element analysis

In this paper we present a two-layer structural model suitable for predicting reliably the passive (unstimulated) time-dependent three-dimensional stress and deformation states of healthy young arterial walls under various loading conditions. It extends to the viscoelastic regime a recently developed constitutive framework for the elastic strain response of arterial walls (see Holzapfel et al. (2001)). The structural model is formulated within the framework of nonlinear continuum mechanics and is well-suited for a finite element implementation. It has the special merit that it is based partly on histological information, thus allowing the material parameters to be associated with the constituents of each mechanically-relevant arterial layer. As one essential ingredient from the histological information the constitutive model requires details of the directional organization of collagen fibers as commonly observed under a microscope. We postulate a fully automatic technique for identifying the orientations of cellular nuclei, these coinciding with the preferred orientations in the tissue. The biological material is assumed to behave incompressibly so that the constitutive function is decomposed locally into volumetric and isochoric parts. This separation turns out to be advantageous in avoiding numerical complications within the finite element analysis of incompressible materials. For the description of the viscoelastic behavior of arterial walls we employ the concept of internal variables. The proposed viscoelastic model admits hysteresis loops that are known to be relatively insensitive to strain rate, an essential mechanical feature of arteries of the muscular type. To enforce incompressibility without numerical difficulties, the finite element treatment adopted is based on a three-field Hu-Washizu variational approach in conjunction with an augmented Lagrangian optimization technique. Two numerical examples are used to demonstrate the reliability and efficiency of the proposed structural model for arterial wall mechanics as a basis for large scale numerical simulations.

Constitutive modelling of arteries

This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thick-walled circular cylindrical tube in order to illustrate its extension–inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

Anisotropic Mechanical Properties of Tissue Components in Human Atherosclerotic Plaques

Knowledge of the biomechanical properties of human atherosclerotic plaques is of essential importance for developing more insights in the pathophysiology of the cardiovascular system and for better predicting the outcome of interventional treatments such as balloon angioplasty. Available data are mainly based on uniaxial tests, and most of the studies investigate the mechanical response of fibrous plaque caps only. However, stress distributions during, for example, balloon angioplasty are strongly influenced by all components of atherosclerotic lesions. A total number of 107 samples from nine human high-grade stenotic iliac arteries were tested; associated anamnesis of donors reported. Magnetic resonance imaging was employed to test the usability of the harvested arteries. Histological analyses has served to characterize the different tissue types. Prepared strips of 7 different tissue types underwent cyclic quasistatic uniaxial tension tests in axial and circumferential directions; ultimate tensile stresses and stretches were documented. Experimental data of individual samples indicated anisotropic and highly nonlinear tissue properties as well as considerable interspecimen differences. The calcification showed, however a linear property, with about the same stiffness as observed for the adventitia in high stress regions. The stress and stretch values at calcification fracture are smaller (179 +/- 56 kPa and 1.02 +/- 0.005) than for each of the other tissue components. Of all intimal tissues investigated, the lowest fracture stress occurred in the circumferential direction of the fibrous cap (254.8 +/- 79.8 kPa at stretch 1.182 +/- 0.1). The adventitia demonstrated the highest and the nondiseased media the lowest mechanical strength on average.

Comparison of a Multi-Layer Structural Model for Arterial Walls With a Fung-Type Model, and Issues of Material Stability

The goals of this paper are (i) to re-examine the constitutive law for the description of the (passive) highly nonlinear and anisotropic response of healthy elastic arteries introduced recently by the authors, (ii) to show how the mechanical response of a carotid artery under inflation and extension predicted by the structural model compares with that for a three-dimensional form of Fung-type strain-energy function, (iii) to provide a new set of material parameters that can be used in a finite element program, and (iv) to show that the model has certain mathematical features that are important from the point of view of material and numerical stability.

A Layer-Specific Three-Dimensional Model for the Simulation of Balloon Angioplasty using Magnetic Resonance Imaging and Mechanical Testing

AbstractA detailed understanding of the mechanical procedure of balloon angioplasty requires three-dimensional (3D) modeling and efficient numerical simulations. We have developed a 3D model for eight distinct arterial components associated with specific mechanical responses. The 3D geometrical model is based on in vitro magnetic resonance imaging of a human stenotic postmortem artery and is represented by nonuniform rational B-spline surfaces. Mechanical tests of the corresponding vascular tissues provide a fundamental basis for the formulation of large strain constitutive laws, which model the typical anisotropic, highly nonlinear, and inelastic mechanical characteristics under supraphysiological loadings. The 3D finite-element realization considers the balloon–artery interaction and accounts for vessel-specific axial in situ prestretches. 3D stress states of the investigated artery during balloon expansion and stent deployment were analyzed. Furthermore, we studied the changes of the 3D stress state due to model simplifications, which are characterized by neglecting axial in situ prestretch, assuming plane strain states, and isotropic material responses, as commonly utilized in previous works. Since these simplifications lead to maximum stress deviations of up to 600%—where even the stress character may interchange—the associated models are, in general, inappropriate. The proposed approach provides a tool that has the potential (i) to improve procedural protocols and the design of interventional instruments on a lesion-specific basis, and (ii) to determine postangioplasty mechanical environments, which may be correlated with restenosis responses.

Mechanics of biological tissue

Tissue Growth and Remodelling.- Towards a Theory of Vascular Growth and Remodeling.- Complementary Roles of Theoretical Modeling and Computer-controlled Experimentation in Vascular Growth and Remodeling.- On the Modeling of Growth and Adaptation.- Growth in Soft Biological Tissue and Residual Stress Development.- Characterization and Modeling of Growth and Remodeling in Tendon and Soft Tissue Constructs.- Modeling and Simulation of Remodeling in Soft Biological Tissues.- Anisotropic Remodelling of Biological Tissues.- A Mechanobiological Formulation of Bone Healing.- Continuum Models of Growth with Emphasis on Articular Cartilage.- Micromechanics, Cells and Matrix.- Tensile Properties and Local Stiffness of Cells.- Microfluid-dynamics in Three-dimensional Engineered Cell Systems.- Nonlinear Constitutive Models for Cochlear Outer Hair Cells.- Prediction of Changes in Cell-substrate Contact under Cyclic Substrate Deformation Using Cohesive Zone Modelling.- Micromechanics and Macromechanics of the Tensile Deformation of Nacre.- Arteries in Health and Disease.- Mechanical Properties of Atherosclerotic Tissues.- Towards a Computational Methodology for Optimizing Angioplasty Treatments with Stenting.- Computational Modeling of Stented Arteries: Considerations for Evolving Stent Designs.- Simulation of In-stent Restenosis for the Design of Cardiovascular Stents.- Material Property Alterations with Early Atheroma in an Animal Model.- Microscopic Analysis of Residual Stress and Strain in the Aortic Media Considering Anisotropy of Smooth Muscle Layer.- Parameter Identification in Arteries Using Constraints.- Collagen Organization and Biomechanics of the Arteries and Aneurysms of the Human Brain.- Image-based Simulation of Blood Flow and Arterial Wall Interaction for Cerebral Aneurysms.- Biological Tissues.- A Framework for Soft Tissue and Musculo-skeletal Modelling: Clinical Uses and Future Challenges.- Invariant Formulation for Dispersed Transverse Isotropy in Tissues of the Aortic Outflow Tract.- Mathematical Modelling of Cardiac Mechanoenergetics.- Creep and Relaxation in Ligament: Theory, Methods and Experiment.- Viscoelastic Constitutive Law Based on the Time Scale of the Mechanical Phenomena.- A Coupled FE Analysis of the Intervertebral Disc Based on a Multiphasic TPM Formulation.- Is the Free Energy of Hydrogel the Sum of Elastic Energy and Ionic Energy?.- In Vivo Experiments to Characterize the Mechanical Behavior of the Human Uterine Cervix.- Viscoelastic Response of Vocal Fold Tissues and Scaffolds at High Frequencies.- An Alternative Fabric-based Yield and Failure Criterion for Trabecular Bone.- Image-based Analysis.- Functional Micro-imaging at the Interface of Bone Mechanics and Biology.- Strain Measurement Using Deformable Image Registration.- Image-based Hierarchical Analysis and Design of Tissue Engineering Scaffolds.

SECTION 10.11 – Biomechanics of Soft Tissue

This chapter discusses the biomechanics of soft tissue. An efficient constitutive formulation approximates all types of soft tissues with a reasonable accuracy over a large strain range. This chapter discusses a constitutive equation with only a few material parameters involved that allow for an “explanation” of the material response of tissues in terms of their structure. In addition, the constitutive formulation is fully three-dimensional and consistent with both mechanical and mathematical requirements, applicable for arbitrary geometries, and suitable for use within the context of finite element methods in order to solve complex initial boundary-value problems is discussed. The presented general model is a fully three-dimensional material description of soft tissues for which nonlinear continuum mechanics is used as the fundamental basis. It has the special feature that it is based partly on histological information. The general model describes the highly nonlinear and anisotropic behavior of soft tissues as composites reinforced by two families of collagen fibers. The constitutive framework is based on the theory of the mechanics of fiber-reinforced composites and is suitable to describe a wide variety of physical phenomena of soft tissues.