Mechanical properties of the adventitia are largely determined by the organization of collagen fibers. Measurements on the waviness and orientation of collagen, particularly at the zero-stress state, are necessary to relate the structural organization of collagen to the mechanical response of the adventitia. Using the fluorescence collagen marker CNA38-OG488 and confocal laser scanning microscopy, we imaged collagen fibers in the adventitia of rabbit common carotid arteries ex vivo. The arteries were cut open along their longitudinal axes to get the zero-stress state. We used semi-manual and automatic techniques to measure parameters related to the waviness and orientation of fibers. Our results showed that the straightness parameter (defined as the ratio between the distances of endpoints of a fiber to its length) was distributed with a beta distribution (mean value 0.72, variance 0.028) and did not depend on the mean angle orientation of fibers. Local angular density distributions revealed four axially symmetric families of fibers with mean directions of 0°, 90°, 43° and −43°, with respect to the axial direction of the artery, and corresponding circular standard deviations of 40°, 47°, 37° and 37°. The distribution of local orientations was shifted to the circumferential direction when measured in arteries at the zero-load state (intact), as compared to arteries at the zero-stress state (cut-open). Information on collagen fiber waviness and orientation, such as obtained in this study, could be used to develop structural models of the adventitia, providing better means for analyzing and understanding the mechanical properties of vascular wall.

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Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy.

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.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2010.0058

Constitutive modelling of arteries

Many soft tissues are naturally made of a matrix and fibres that present some privileged directions. They are known to support large reversible deformations. The mechanical behaviour of these tissues highlights different phenomena as hysteresis, stress softening or relaxation. A hyperelastic constitutive equation is typically the basis of the model that describes the behaviour of the material. The hyperelastic constitutive equation can be isotropic or anisotropic, it is generally expressed by means of strain components or strain invariants. This paper proposes a review of these constitutive equations.

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Hyperelastic Energy Densities for Soft Biological Tissues: A Review

Many treatments for cardiovascular diseases include an endovascular insertion of stents or stent grafts into arteries, a procedure which may cause high tissue stresses and even damage in the arterial wall. In order to study such problems by using finite element methods, both appropriate constitutive models and experimental data on human tissue samples are required. Layer-specific experimental data for human tissue tested up to the supra-physiological loading range are rare in the literature. In this study, intact and layer-separated experimental data from uniaxial extension tests are presented for human thoracic and abdominal aortas with non-atherosclerotic intimal thickening undergoing supra-physiological loading. A novel pseudo-elastic damage model, proposed to describe discontinuous softening in aortic arterial tissues, is fit to the obtained experimental data. Fitting of the model with and without consideration of damage accumulation in the non-collagenous matrix material reveals that tissue damage is primarily related to the collagen fiber fabric. By employing the fit model, the effect of aortic tissue pre-conditioning on the material parameters from the resulting data fits is evaluated. Histological examination of the collagen fibers under different applied stretches is used to gain more insights into the structural changes of the tissue under supra-physiological loading.

Layer-specific damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening.

The mechanical characteristics of skin are extremely complex and have not been satisfactorily simulated by conventional engineering models. The ability to predict human skin behaviour and to evaluate changes in the mechanical properties of the tissue would inform engineering design and would prove valuable in a diversity of disciplines, for example the pharmaceutical and cosmetic industries, which currently rely upon experiments performed in animal models. The aim of this study was to develop a predictive anisotropic, hyperelastic constitutive model of human skin and to validate this model using laboratory data. As a corollary, the mechanical characteristics of human and murine skin have been compared. A novel experimental design, using tensile tests on circular skin specimens, and an optimisation procedure were adopted for laboratory experiments to identify the material parameters of the tissue. Uniaxial tensile tests were performed along three load axes on excised murine and human skin samples, using a single set of material parameters for each skin sample. A finite element model was developed using the transversely isotropic, hyperelastic constitutive model of Weiss et al. (1996) and was embedded within a Veronda–Westmann isotropic material matrix, using three fibre families to create anisotropic behaviour. The model was able to represent the nonlinear, anisotropic behaviour of the skin well. Additionally, examination of the optimal material coefficients and the experimental data permitted quantification of the mechanical differences between human and murine skin. Differences between the skin types, most notably the extension of the skin at low load, have highlighted some of the limitations of murine skin as a biomechanical model of the human tissue.

The development of accurate, predictive computational models of human tissue, such as skin, to reduce, refine or replace animal models and to inform developments in the medical, engineering and cosmetic fields, is a significant challenge but is highly desirable. Concurrent advances in computer technology and our understanding of human physiology must be utilised to produce more accurate and accessible predictive models, such as the finite element model described in this study.

An anisotropic, hyperelastic model for skin: experimental measurements, finite element modelling and identification of parameters for human and murine skin.

A fully three dimensional finite-strain damage model for fibrous soft tissue is developed. The model assumes uncoupled contributions for the matrix and collagen fibers, and uncoupled bulk and deviatoric response over any range of deformations. A simple isotropic damage mechanism within the framework of continuum damage mechanics has been used to describe the softening behavior under deformation for the matrix. On the other hand, statistical aspects related to the length distribution of the reinforcing fibers lead to a damage model for the reinforcing material. As a result, a general theoretical framework for constitutive modeling of biological soft tissue with continuum damage is obtained. A theoretical example consisting of a biaxial test of a soft tissue reinforced with two families of collagen fibers has been considered to demonstrate the capabilities of the proposed model and to study the sensitivity to changes in the statistical parameters associated with the reinforcing material. Also, a preliminary numerical example is included to demonstrate the model on a inhomogeneous boundary value problem. Results show that the model is able to capture the typical stress–strain behavior observed in fibrous soft tissue and seems to confirm the soundness of the proposed formulation.

A stochastic-structurally based three dimensional finite-strain damage model for fibrous soft tissue

A micromechanically based constitutive model for fibrous tissues is presented. The model considers the randomly crimped morphology of individual collagen fibers, a morphology typically seen in photomicrographs of tissue samples. It describes the relationship between the fiber endpoints and its arc-length in terms of a measurable quantity, which can be estimated from image data. The collective mechanical behavior of collagen fibers is presented in terms of an explicit expression for the strain-energy function, where a fiber-specific random variable is approximated by a Beta distribution. The model-related stress and elasticity tensors are provided. Two representative numerical examples are analyzed with the aim of demonstrating the peculiar mechanism of the constitutive model and quantifying the effect of parameter changes on the mechanical response. In particular, a fibrous tissue, assumed to be (nearly) incompressible, is subject to a uniaxial extension along the fiber direction, and, separately, to pure shear. It is shown that the fiber crimp model can reproduce several of the expected characteristics of fibrous tissues.

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A constitutive model for fibrous tissues considering collagen fiber crimp

In coronary artery bypass graft (CABG) surgery the involved tissues are overstretched, which may lead to intimal hyperplasia and graft failure. We propose a computational methodology for the simulation of traditional CABG surgery, and analyze the effect of two clinically relevant parameters on the artery and graft responses, i.e., incision length and insertion angle for a given graft diameter. The computational structural analyses are based on actual three-dimensional vessel dimensions of a human coronary artery and a human saphenous vein. The analyses consider the structure of the end-to-side anastomosis, the residual stresses and the typical anisotropic and nonlinear vessel behaviors. The coronary artery is modeled as a three-layer thick-walled tube. The finite element method is employed to predict deformation and stress distribution at various stages of CABG surgery. Small variations of the arterial incision have relatively big effects on the size of the arterial opening, which depends solely on the residual stress state. The incision length has a critical influence on the graft shape and the stress in the graft wall. Stresses at the heel region are higher than those at the toe region. The changes in the mechanical environment are severe along all transitions between the venous tissue and the host artery. Particular stress concentrations occur at the incision ends. The proposed computational methodology may be useful in designing a coronary anastomotic device for reducing surgical trauma. It may improve the quantitative knowledge of vessel diseases and serve as a tool for virtual planning of vascular surgery.

A procedure to simulate coronary artery bypass graft surgery

The uterine cervix has to provide mechanical resistance to ensure a normal development of the fetus. This is guaranteed by the composition of its extracellular matrix, which functions as a fiber-reinforced composite. At term a complex remodeling process allows the cervical canal to open for birth. This remodeling is achieved by changes in the quality and quantity of collagen fibers and ground substance and their interplay, which influences the biomechanical behavior of the cervix but also contributes to pathologic conditions such as cervical incompetence (CI). We start by reviewing the anatomy and histological composition of the human cervix, and discuss its physiologic function and pathologic condition in pregnancy including biomechanical aspects. Established diagnostic methods on the cervix (palpation, endovaginal ultrasound) used in clinics as well as methods for assessment of cervical consistency (light-induced fluorescence, electrical current, and impedance) are discussed. We show the first clinical application of an aspiration device, which allows in vivo testing of the biomechanical properties of the cervix with the aim to establish the physiological biomechanical changes throughout gestation and to detect pregnant women at risk for CI. In a pilot study on nonpregnant cervices before and after hysterectomy we found no considerable difference in the biomechanical response between in vivo and ex vivo. An outlook on further clinical applications during pregnancy is presented.

Fernando Cacho

Papers

A stochastic-structurally based three dimensional finite-strain damage model for fibrous soft tissue

A constitutive model for fibrous tissues considering collagen fiber crimp

A procedure to simulate coronary artery bypass graft surgery

In vivo characterization of the mechanics of human uterine cervices