A constitutive formulation of arterial mechanics including vascular smooth muscle tone

A pseudo-strain energy function (pseudo-SEF) describing the biomechanical properties of large conduit arteries under the influence of vascular smooth muscle (VSM) tone is proposed. In contrast to previous models that include the effects of smooth muscle contraction through generation of an active stress, in this study we consider the vascular muscle as a structural element whose contribution to load bearing is modulated by the contraction. This novel pseudo-SEF models not only arterial mechanics at maximal VSM contraction but also the myogenic contraction of the VSM in response to local increases in stretch. The proposed pseudo-SEF was verified with experimentally obtained pressure-radius curves and zero-stress state configurations from rat carotid arteries displaying distinct differences in VSM tone: arteries from normotensive rats displaying minimal VSM tone and arteries from hypertensive rats exhibiting significant VSM tone. The pressure-radius curves were measured in three different VSM states: fully relaxed, maximally contracted, and normal VSM tone. The model fitted the experimental data very well (r2 > 0.99) in both the normo- and hypertensive groups for all three states of VSM activation. The pseudo-SEF was used to illustrate the localized reduction of circumferential stress in the arterial wall due to normal VSM tone, suggesting that the proposed pseudo-SEF can be of general utility for describing stress distribution not only under passive VSM conditions, as most SEFs proposed so far, but also under physiological and pathological conditions with varying levels of VSM tone.     

Role of elastin anisotropy in structural strain energy functions of arterial tissue

The vascular wall exhibits nonlinear anisotropic mechanical properties. The identification of a strain energy function (SEF) is the preferred method to describe its complex nonlinear elastic properties. Earlier constituent-based SEF models, where elastin is modeled as an isotropic material, failed in describing accurately the tissue response to inflation–extension loading. We hypothesized that these shortcomings are partly due to unaccounted anisotropic properties of elastin. We performed inflation–extension tests on common carotid of rabbits before and after enzymatic degradation of elastin and applied constituent-based SEFs, with both an isotropic and an anisotropic elastin part, on the experimental data. We used transmission electron microscopy (TEM) and serial block-face scanning electron microscopy (SBFSEM) to provide direct structural evidence of the assumed anisotropy. In intact arteries, the SEF including anisotropic elastin with one family of fibers in the circumferential direction fitted better the inflation–extension data than the isotropic SEF. This was supported by TEM and SBFSEM imaging, which showed interlamellar elastin fibers in the circumferential direction. In elastin-degraded arteries, both SEFs succeeded equally well in predicting anisotropic wall behavior. In elastase-treated arteries fitted with the anisotropic SEF for elastin, collagen engaged later than in intact arteries. We conclude that constituent-based models with an anisotropic elastin part characterize more accurately the mechanical properties of the arterial wall when compared to models with simply an isotropic elastin. Microstructural imaging based on electron microscopy techniques provided evidence for elastin anisotropy. Finally, the model suggests a later and less abrupt collagen engagement after elastase treatment.     

Effect of elastin degradation on carotid wall mechanics as assessed by a constituent-based biomechanical model

Arteries display a nonlinear anisotropic behavior dictated by the elastic properties and structural arrangement of its main constituents, elastin, collagen, and vascular smooth muscle. Elastin provides for structural integrity and for the compliance of the vessel at low pressure, whereas collagen gives the tensile resistance required at high pressures. Based on the model of Zulliger et al. (Zulliger MA, Rachev A, Stergiopulos N. Am J Physiol Heart Circ Physiol 287: H1335-H1343, 2004), which considers the contributions of elastin, collagen, and vascular smooth muscle cells (VSM) in an explicit form, we assessed the effects of enzymatic degradation of elastin on biomechanical properties of rabbit carotids. Pressure-diameter curves were obtained for controls and after elastin degradation, from which elastic and structural properties were derived. Data were fitted into the model of Zulliger et al. to assess elastic constants of elastin and collagen as well as the characteristics of the collagen engagement profile. The arterial segments were also prepared for histology to visualize and quantify elastin and collagen. Elastase treatment leads to a diameter enlargement, suggesting the existence of significant compressive prestresses within the wall. The elastic modulus was more ductile in treated arteries at low circumferential stretches and significantly greater at elevated circumferential stretches. Abrupt collagen fiber recruitment in elastase-treated arteries leads to a much stiffer vessel at high extensions. This change in collagen engagement properties results from structural alterations provoked by the degradation of elastin, suggesting a clear interaction between elastin and collagen, often neglected in previous constituent-based models of the arterial wall.     

A computational model for collagen fibre remodelling in the arterial wall

As the interaction between tissue adaptation and the mechanical condition within tissues is complex, mathematical models are desired to study this interrelation. In this study, a mathematical model is presented to investigate the interplay between collagen architecture and mechanical loading conditions in the arterial wall. It is assumed that the collagen fibres align along preferred directions, situated in between the principal stretch directions. The predicted fibre directions represent symmetrically arranged helices and agree qualitatively with morphometric data from literature. At the luminal side of the arterial wall, the fibres are oriented more circumferentially than at the outer side. The discrete transition of the fibre orientation at the media-adventitia interface can be explained by accounting for the different reference configurations of both layers. The predicted pressure-radius relations resemble experimentally measured sigma-shaped curves. As there is a strong coupling between the collagen architecture and the mechanical loading condition within the tissue, we expect that the presented model for collagen remodelling is useful to gain further insight into the processes involved in vascular adaptation, such as growth and smooth muscle tone adaptation.     

Effects of longitudinal stretch on VSM tone and distensibility of muscular conduit arteries

With progressing age, large arteries diminish their longitudinal stretch, which in extreme cases results in tortuosity. Increased age is also associated with loss of vessel distensibility. We measured pressure-diameter curves from muscular porcine carotid arteries ex vivo at different longitudinal stretch ratios (lambda(z) = 1.4 and 1.8) and under different vascular smooth muscle (VSM) conditions (fully relaxed, normal VSM tone, and maximally contracted). Distensibility was found to be halved by decreasing longitudinal stretch from lambda(z) = 1.8 to 1.4 at physiological pressures. This counterintuitive observation is possible because highly nonlinear elastic modulus of the artery and anisotropic properties. Furthermore, a significantly larger basal VSM contraction was observed at lambda(z) = 1.8 than 1.4, although this was clearly not related to a myogenic response during inflation. This dependence of VSM tone to longitudinal stretch may have possible implications on the functional characteristics of the arterial wall.     

Biomechanical and histological characteristics of passive esophagus: Experimental investigation and comparative constitutive modeling

Information on the passive biomechanical properties of two-layered esophagus is still limited, although this would enhance our understanding of its physiology/pathophysiology and help to address problems in surgery, medical-device applications, and for the optimal design of prostheses. In this study, rabbit esophagi were excised and dissected into mucosa–submucosa and muscle layers that were submitted to histological quantification of elastin and collagen content and orientation, as well as to inflation-extension testing and geometrical analysis, i.e. delineation of the zero-stress state serving as a reference configuration for biomechanical analysis. The pressure–radius data of both layers displayed a monotonically rising slope with inflating pressure, unlike the sigma shape characterizing elastin-rich tissues, for which biphasic constitutive models were initially postulated. Three phenomenological expressions of strain-energy function (SEF), commonly appearing in soft-tissue biomechanics literature, were used in an attempt to model the pseudoelastic response of esophageal tissue, namely the exponential Fung-type SEF, and the combined neo-Hookean (isotropic) or quadratic (anisotropic) and exponential Fung-type SEF. Accurate fits were attained for the pressure–radius–force data, spanning a wide range of longitudinal stretch ratios, when using the exponential form; the biphasic SEFs failed to generate improved fits, being also over-parameterized. According to the calculated material parameters, mucosa–submucosa was stiffer than muscle in both directions, justified by our histological observation of increased collagen content in that layer, and tissue was stiffer longitudinally, substantiated by the increased elastin and collagen contents and their preferential alignment towards that direction. Our results demonstrate that the passive response of esophagus is best modeled with an exponential Fung-type SEF.     

A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution

Accurate constitutive models are required to gain further insight into the mechanical behavior of cardiovascular tissues. In this study, a structural constitutive framework for cardiovascular tissues is introduced that accounts for the angular distribution of collagen fibers. To demonstrate its capabilities, the model is applied to study the biaxial behavior of the arterial wall and the aortic valve. The pressure-radius relationships of the arterial wall accurately describe experimentally observed sigma-shaped curves. In addition, the nonlinear and anisotropic mechanical properties of the aortic valve can be analyzed with the proposed model. We expect that the current model offers strong possibilities to further investigate the complex mechanical behavior of cardiovascular tissues, including their response to mechanical stimuli.     

A biomechanical analysis of venous tissue in its normal and post-phlebitic conditions

Although biomechanical studies of the normal rat vein wall have been reported (Weizsacker, 1988, Plante, 2002), there are no published studies that have investigated the mechanical effects of thrombus formation on murine venous tissue. In response to the lack of knowledge concerning the mechanical consequences of thrombus resolution, distinct thrombus-induced changes in the biomechanical properties of the murine vena cava were measured via biaxial stretch experiments. These data served as input for strain energy function (SEF) fitting and modeling (Gasser et al., 2006). Statistical differences were observed between healthy and diseased tissue with respect to the structural coefficient that represents the response of the non-collagenous, isotropic ground substance. Alterations following thrombus formation were also noted for the SEF coefficient which describes the anisotropic contribution of the fibers. The data indicate ligation of the vena cava leads to structural alterations in the ground substance and collagen fiber network.     

Computational study of growth and remodelling in the aortic arch

Opening angles (OAs) are associated with growth and remodelling in arteries. One curiosity has been the relatively large OAs found in the aortic arch of some animals. Here, we use computational models to explore the reasons behind this phenomenon. The artery is assumed to contain a smooth muscle/collagen phase and an elastin phase. In the models, growth and remodelling of smooth muscle/collagen depends on wall stress and fluid shear stress. Remodelling of elastin, which normally turns over very slowly, is neglected. The results indicate that OAs generally increase with longitudinal curvature (torus model), earlier elastin production during development, and decreased wall stiffness. Correlating these results with available experimental data suggests that all of these effects may contribute to the large OAs in the aortic arch. The models also suggest that the slow turnover rate of elastin limits longitudinal growth. These results should promote increased understanding of the causes of residual stress in arteries.     

Time-course of venous wall biomechanical adaptation in pressure and flow-overload: Assessment by a microstructure-based material model

Arteriovenous fistulae have been previously created by our group, through implantation of e-PTFE grafts between the carotid artery and jugular vein in healthy pigs, to gather comprehensive data on the time-course of the adapted geometry, composition, and biomechanical properties of the venous wall exposed to chronic increases in pressure and flow. The aim of this study was to mathematically assess the biomechanical adaptation of venous wall, by characterizing our previous in vitro inflation/extension testing data obtained 2, 4, and 12 weeks post-fistula, using a microstructure-based material model. Our choice for such a model considered a quadratic function for elastin with a four-fiber family term for collagen, and permitted realistic data characterization for both overloaded and control veins. As structural validation to the hemodynamically-driven differences in the material response, computerized histology was employed to quantitate the composition and orientation of collagen and elastin-fiber networks. The parameter values optimized showed marked differences among the overloaded and control veins, namely decrease in the quadratic function parameters and increase in the four-fiber family parameters. Differences among the two vein types were highlighted with respect to the underlying microstructure, namely the reduced elastin and increased collagen contents induced by pressure and flow-overload. Explicit correlations were found of the material parameters with the two basic scleroprotein contents, substantiating the material model used and the characterization findings presented. Our results are expected to improve the current understanding of the dynamics of venous adaptation under sustained pressure- and flow-overload conditions, for which data are largely unavailable and contradictory.     

A description of arterial wall mechanics using limiting chain extensibility constitutive models

Certain aspects of the mechanical response of arterial walls can be described using nonlinear elasticity theory. Uniaxial tests on vascular walls reveal nonlinear stress-strain behavior, with higher extensibility in the low stretch range and progressively lower extensibility with increasing stretch. This phenomenon is well known in the framework of rubber-like materials where it is called a strain-hardening or strain-stiffening effect. Constitutive models of incompressible hyperelasticity that take this into account include power-law models and limiting chain extensibility models. Our purpose in this paper is to bring to the attention of the biomechanics community some essential features of one such model of the latter type due to Gent. This model is compared with isotropic versions of biomechanical constitutive models by Takamizawa-Hayashi and Fung; the latter is a limiting version of a power-law material. Two particular problems are considered for which experimental data on arterial wall deformations are available. The first concerns small oscillations superposed on a large static stretch of a vertical string of arterial tissue. It is shown that the exponential model of Fung and the Gent model match well with the experimental data. The second problem is the extension of an internally pressurized circular cylindrical tube. It is shown that an inversion phenomenon observed experimentally for the human iliac artery can be described within a membrane theory by the Gent model whereas this cannot be described using the exponential model. The foregoing considerations are carried out for isotropic elastic materials in the absence of residual stress. Extensions to include anisotropy are also indicated.     

The role of fiber-matrix interactions in a nonlinear fiber-reinforced strain energy model of tendon

The objective of this study was to develop a nonlinear and anisotropic three-dimensional mathematical model of tendon behavior in which the structural components of fibers, matrix, and fiber-matrix interactions are explicitly incorporated and to use this model to infer the contributions of these structures to tendon mechanical behavior. We hypothesized that this model would show that: (i) tendon mechanical behavior is not solely governed by the isotropic matrix and fiber stretch, but is also influenced by fiber-matrix interactions; and (ii) shear fiber-matrix interaction terms will better describe tendon mechanical behavior than bulk fiber-matrix interaction terms. Model versions that did and did not include fiber-matrix interaction terms were applied to experimental tendon stress-strain data in longitudinal and transverse orientations, and the R2 goodness-of-fit was evaluated. This study showed that models that included fiber-matrix interaction terms improved the fit to longitudinal data (R2(toe) = 0.88, R2(Lin) = 0.94) over models that only included isotropic matrix and fiber stretch terms (R2(Toe) = 0.36, R2(Lin) = 0.84). Shear fiber-matrix interaction terms proved to be responsible for the best fit to data and to contribute to stress-strain nonlinearity. The mathematical model of tendon behavior developed in this study showed that fiber-matrix interactions are an important contributor to tendon behavior The more complete characterization of mechanical behavior afforded by this mathematical model can lead to an improved understanding of structure-function relationships in soft tissues and, ultimately, to the development of tissue-engineered therapies for injury or degeneration.     

Isotropy and anisotropy of the arterial wall

The passive biomechanical response of intact cylindrical rat carotid arteries is studied in vitro and compared with the mechanical response of rubber tubes. Using true stress and natural strain in the definition of the incremental modulus of elasticity, the tissue wall properties are analyzed over wide ranges of simultaneous circumferential and longitudinal deformations. The type of loading chosen is 'physiological' i.e. symmetric: the cylindrical segments are subjected to internal pressure and axial prestretch without torsion or shear. Several aspects pertaining to the choice of parameters characterizing the material are discussed and the analysis pertaining to the deformational behavior of a hypothetical compliant tube with Hookean wall material is presented. The experimental results show that while rubber response can be adequately represented as linearly elastic and isotropic, the overall response of vascular tissue is highly non-linear and anisotropic. However, for states of deformation that occur in vivo, the elasticity of arteries is quite similar to that of rubber tubes and as such the arterial wall may be viewed as incrementally isotropic for the range of deformations that occur in vivo.     

Mechanical properties of elastin along the thoracic aorta in the pig

Understanding the mechanical environment of each component within the arterial wall is fundamental for understanding vascular growth and remodelling and for engineering artificial vascular conduits. We have investigated the mechanical status of arterial elastin by measuring the circumferential mechanical properties of purified elastin as function of position along the descending thoracic aorta of the pig. The tensile circumferential secant modulus, E(sec), measured in uniaxial mechanical tests, increased 30% (P<0.001), from a value of 0.88 MPa in the proximal tissue near the aortic arch to 1.14 MPa in the distal tissue near the diaphragm, indicating the stiffness of the elastin sample increased with position. Breaking stress was 54% higher in the distal tissue compared to the proximal (P<0.001), but the breaking stretch ratio did not change. E(sec) correlated with the ratio of radius to wall thickness measured in the no load state, r(nl)/h(nl), suggesting that the rise in stiffness was linked to ring morphology. The higher stiffness and strength of the distal tissue might be explained by a higher proportion of circumferentially oriented fibres in the distal tissue, which would indicate that the elastin meshwork in the thoracic aorta may become progressively anisotropic with distance from the heart. The ratio r(nl)/(h(nl)E (sec))rose only 7%, which suggests that the in vivo circumferential strain on the elastin may be constant along the pig thoracic aorta. The positional variation in elastin's properties should be taken into account in mechanical studies on purified elastin and in mathematical models of aorta mechanics.     

Linked mechanical and biological aspects of remodeling in mouse pulmonary arteries with hypoxia-induced hypertension

Supravalvular aortic stenosis (SVAS) is associated with decreased elastin and altered arterial mechanics. Mice with a single deletion in the elastin gene (ELN(+/-)) are models for SVAS. Previous studies have shown that elastin haploinsufficiency in these mice causes hypertension, decreased arterial compliance, and changes in arterial wall structure. Despite these differences, ELN(+/-) mice have a normal life span, suggesting that the arteries remodel and adapt to the decreased amount of elastin. To test this hypothesis, we performed in vitro mechanical tests on abdominal aorta, ascending aorta, and left common carotid artery from ELN(+/-) and wild-type (C57BL/6J) mice. We compared the circumferential and longitudinal stress-stretch relationships and residual strains. The circumferential stress-stretch relationship is similar between genotypes and changes <3% with longitudinal stretch at lengths within 10% of the in vivo value. At mean arterial pressure, the circumferential stress in the ascending aorta is higher in ELN(+/-) than in wild type. Although arterial pressures are higher, the increased number of elastic lamellae in ELN(+/-) arteries results in similar tension/lamellae compared with wild type. The longitudinal stress-stretch relationship is similar between genotypes for most arteries. Compared with wild type, the in vivo longitudinal stretch is lower in ELN(+/-) abdominal and carotid arteries and the circumferential residual strain is higher in ELN(+/-) ascending aorta. The increased circumferential residual strain brings the transmural strain distribution in ELN(+/-) ascending aorta close to wild-type values. The mechanical behavior of ELN(+/-) arteries is likely due to the reduced elastin content combined with adaptive remodeling during vascular development.     

Introducing mesoscopic information into constitutive equations for arterial walls

We propose a new elastic constitutive law for arterial tissue in which the limiting polymeric chain extensibility of both collagen and elastin fibres is accounted for. The elastic strain-energy function is separated additively into two parts: an isotropic contribution associated with the matrix (incorporating the elastin fibre network) and an anisotropic one associated with the collagen fibres. Information on the limiting extensibility in each case provides some mesoscopic input into the model. The (logarithm-based) model is compared with the Fung-Demiray exponential model and certain other recently proposed models. Some aspects of the elastic response under extension and inflation of a thin-walled circular cylindrical tube (the artery) are then examined and compared with the corresponding response of a rubber-like tube. We point out that our model, when both isotropic and anisotropic terms are included, can be developed to accommodate changing mechanical properties associated with degradation of the elastin and collagen by considering the material constants that define the limit of chain extensibility to evolve in time.     

A finite element model of stress-mediated vascular adaptation: application to abdominal aortic aneurysms

Despite rapid expansion of our knowledge of vascular adaptation, developing patient-specific models of diseased arteries is still an open problem. In this study, we extend existing finite element models of stress-mediated growth and remodelling of arteries to incorporate a medical image-based geometry of a healthy aorta and, then, simulate abdominal aortic aneurysm. Degradation of elastin initiates a local dilatation of the aorta while stress-mediated turnover of collagen and smooth muscle compensates the loss of elastin. Stress distributions and expansion rates during the aneurysm growth are studied for multiple spatial distribution functions of elastin degradation and kinetic parameters. Temporal variations of the degradation function are also investigated with either direct time-dependent degradation or stretch-induced degradation as possible biochemical and biomechanical mechanisms for elastin degradation. The results show that this computational model has the capability to capture the complexities of aneurysm progression due to variations of geometry, extent of damage and stress-mediated turnover as a step towards patient-specific modelling.     

Dynamics of intestinal propulsion

A biomechanical model and mathematical formulation of the problem of propulsion of a solid non-deformable pellet by an isolated segment of the gut are presented. The organ is modeled as a soft orthotropic cylindrical biological shell. Its wall is reinforced by transversely isotropic muscle fibers of orthogonal type of weaving embedded in a connective tissue stroma. The mechanical properties of the wall are assumed to be nonlinear, deformations are finite. The longitudinal smooth muscle syncitium possesses anisotropic and the circular muscle syncytium has anisotropic electrical properties. Their electromechanical activity is under control of a pacemaker, which is represented by interstitial cells of Cajal. The model describes the dynamics of the generation and propagation of mechanical waves of contraction-relaxation along the surface of the bioshell and propulsion of the pellet. The governing system of equations was solved numerically. The combined finite-difference and finite-element method was used. The results demonstrate that pendular movements alone provide an aboral transit, without mixing though, of the bolus. Non-propagating segmental contractions show small amplitude librations of the pellet without its visible propulsion. Only the coordinated activity of both smooth muscle layers in a form of the peristaltic reflex provides physiologically significant simultaneous propulsion and mixing of the intraluminal content (pellet).     

A finite element model of stress-mediated vascular adaptation: application to abdominal aortic aneurysms

Despite rapid expansion of our knowledge of vascular adaptation, developing patient-specific models of diseased arteries is still an open problem. In this study, we extend existing finite element models of stress-mediated growth and remodelling of arteries to incorporate a medical image-based geometry of a healthy aorta and, then, simulate abdominal aortic aneurysm. Degradation of elastin initiates a local dilatation of the aorta while stress-mediated turnover of collagen and smooth muscle compensates the loss of elastin. Stress distributions and expansion rates during the aneurysm growth are studied for multiple spatial distribution functions of elastin degradation and kinetic parameters. Temporal variations of the degradation function are also investigated with either direct time-dependent degradation or stretch-induced degradation as possible biochemical and biomechanical mechanisms for elastin degradation. The results show that this computational model has the capability to capture the complexities of aneurysm progression due to variations of geometry, extent of damage and stress-mediated turnover as a step towards patient-specific modelling.