On constitutive relations and finite deformations of passive cardiac tissue: I. A pseudostrain-energy function

A three-dimensional constitutive relation for passive cardiac tissue is formulated in terms of a structurally motivated pseudostrain-energy function, W, while the mathematical simplicity of phenomenological approaches is preserved. A specific functional form of W is proposed on the basis of limited structural information and multiaxial experimental data. The material parameters are determined in a least-squared sense from both uniaxial and biaxial data. Our results suggest that (1) multiaxially-loaded cardiac tissue is nearly transversely-isotropic with respect to local muscle fiber directions, at least for a limited range of strain histories, (2) material parameters determined from uniaxial papillary muscle data result in gross underestimates of the stresses in multiaxially-loaded specimens, and (3) material parameters determined from equibiaxial tests predict the behavior of the tissue under various nonequibiaxial stretching protocols reasonably well.     

Biomechanical experiments on excised myocardium: theoretical considerations

Both the specific functional form of a constitutive relation and the particular material parameters must be determined from experiments. Constitutive formulations based on experimental data are valid, however, only when the experimentally measured stresses closely represent the actual distribution of stress in a specimen. In this paper, we present theoretical considerations to help experimentalists identify conditions under which biaxial data obtained from thin slabs of non-contracting myocardium satisfy the above criterion. In particular, we present both a general framework to investigate, and specific numerical examples to illustrate, the influence of certain distributions of muscle fibers, in-plane shearing strains and particular stretching protocols on the distribution of stress within biaxially tested myocardial specimens.     

A constitutive theory for biomembranes: application to epicardial mechanics.

We present a new theoretically motivated experimental approach for identifying the functional form of a constitutive relation for any nonlinear, anisotropic pseudoelastic biological membrane. The utility of this approach is illustrated by identifying, from biaxial data, a new constitutive relation for excised ventricular epicardium. Values of the associated material parameters are calculated and compared for right and left ventricular specimens. Based on our findings, we suggest that there are no significant differences in the biomechanical behavior of epicardium excised from the right and left ventricular free walls of canine hearts.     

Physical invariant strain energy function for passive myocardium

Principal axis formulations are regularly used in isotropic elasticity, but they are not often used in dealing with anisotropic problems. In this paper, based on a principal axis technique, we develop a physical invariant orthotropic constitutive equation for incompressible solids, where it contains only a one variable (general) function. The corresponding strain energy function depends on six invariants that have immediate physical interpretation. These invariants are useful in facilitating an experiment to obtain a specific constitutive equation for a particular type of materials. The explicit appearance of the classical ground-state constants in the constitutive equation simplifies the calculation for their admissible values. A specific constitutive model is proposed for passive myocardium, and the model fits reasonably well with existing simple shear and biaxial experimental data. It is also able to predict a set of data from a simple shear experiment.     

A constitutive model for two-dimensional soft tissues and its application to experimental data

A constitutive relation proposed by Shoemaker (Ph.D. dissertation, 1984) to model the mechanical behavior of membraneous or two-dimensional soft tissues is described. Experiments by Schneider (Ph.D. dissertation, 1982) on human skin and Lee et al. (Am. J. Physiol., 249, H222-H230, 1985) on canine pericardium, and the application of the constitutive model to biaxial stress-strain data from these experiments, are discussed. Some experimental data and predictions of the model obtained by curvefitting are presented for comparison. Values of material parameters are also presented. It is concluded that the constitutive model is well able to fit results of individual tests, and that its generality (judged by consistency of parameters from test to test of the same specimen), though not complete, does compare favorably with some other results presented in the literature.     

New interpretation of arterial stiffening due to cigarette smoking using a structurally motivated constitutive model

Cigarette smoking is the leading self-inflicted risk factor for cardiovascular diseases; it causes arterial stiffening with serious sequelea including atherosclerosis and abdominal aortic aneurysms. This work presents a new interpretation of arterial stiffening caused by smoking based on data published for rat pulmonary arteries. A structurally motivated "four fiber family" constitutive relation was used to fit the available biaxial data and associated best-fit values of material parameters were estimated using multivariate nonlinear regression. Results suggested that arterial stiffening caused by smoking was reflected by consistent increase in an elastin-associated parameter and moreover by marked increase in the collagen-associated parameters. That is, we suggest that arterial stiffening due to cigarette smoking appears to be isotropic, which may allow simpler phenomenological models to capture these effects using a single stiffening parameter similar to the approach in isotropic continuum damage mechanics. There is a pressing need, however, for more detailed histological information coupled with more complete biaxial mechanical data for a broader range of systemic arteries.     

A model for the nonlinear elastic response of large arteries

The nonlinear elastic response of large arteries subjected to finite deformations due to action of biaxial principal stresses, is described by simple constitutive equations. Generalized measures of strain and stress are introduced to account for material nonlinearity. This also ensures the existence of a strain energy density function. The orthotropic elastic response is described via quasi-linear relations between strains and stresses. One nonlinear parameter which defines the measures of strain and stress, and three elastic moduli are assumed to be constants. The lateral strain parameters (equivalent to Poisson's ratios in infinitesimal deformations) are deformation dependent. This dependence is defined by empirical relations developed via the incompressibility condition, and by the introduction of a fifth material parameter. The resulting constitutive model compares well with biaxial experimental data of canine carotid arteries.     

A planar biaxial constitutive relation for the luminal layer of intra-luminal thrombus in abdominal aortic aneurysms

The rupture risk of abdominal aortic aneurysms (AAAs) is thought to be associated with increased levels of wall stress. Finite element analysis (FEA) allows the prediction of wall stresses in a patient-specific, non-invasive manner. We have recently shown that it is important to include the intra-luminal thrombus (ILT), present in approximately 70% of AAA, into FEA simulations of AAA. All FEA simulations to date assume an isotropic, homogeneous material behavior for this material. The purpose of this work was to investigate the multi-axial biomechanical behavior of ILT and to derive an appropriate constitutive relation. We performed planar biaxial testing on the luminal layer of nine ILT specimens obtained fresh in the operating room (9 patients, mean age 71+/-4.5 years, mean diameter 5.9+/-0.4 cm), and a constitutive relation was derived from this data. Peak stretch and maximum tangential modulus (MTM) values were recorded for the equibiaxial protocol in both the circumferential (theta) and longitudinal (L) directions. Stress contour plots were used to investigate the presence of mechanical anisotropy, after which an appropriate strain energy function was fit to each of the specimen datasets. The peak stretch values for the luminal layer of the ILT were (mean+/-SEM) 1.18+/-0.02 and 1.13+/-0.02 in the theta and L directions, respectively (p=0.14). The MTM values were 20+/-2 and 23+/-3N/cm(2) in the theta and L directions, respectively (p=0.37). From these results and our observation of the symmetry of the stress contour plots for each specimen, we concluded that the use of an isotropic strain energy function for ILT is appropriate. Each specimen data set was then fit to a second-order polynomial strain energy function of the first invariant of the left Cauchy-Green strain tensor, resulting in an accurate fit (average R(2)=0.92+/-0.02; range 0.80-0.99). Comparison of our previously reported, uniaxially derived constitutive relation with the biaxially derived relation derived here shows large differences in the predicted mechanical response, underscoring the importance of the appropriate experimental methods used to derive constitutive relations. Further work is merited in an effort to produce more accurate predictions of wall stresses in patient-specific AAA, and viscoelastic behaviors of the ILT.     

Determination of material parameters of the two-dimensional Holzapfel–Weizsäcker type model based on uniaxial extension data of arterial walls

Soft tissues are anisotropic materials yet a majority of mechanical property tests have been uniaxial, which often failed to recapitulate the tensile response in other directions. This paper aims to study the feasibility of determining material parameters of anisotropic tissues by uniaxial extension with a minimal loss of anisotropic information. We assumed that by preselecting a certain constitutive model, we could give the constitutive parameters based on uniaxial extension data from orthogonal strip samples. In our study, the Holzapfel–Weizsäcker type strain energy density function (H–W model) was used to determine the material parameters of arterial walls from two fresh donation bodies. The key points we applied were the relationships between strain components in uniaxial tensile tests and the methods of stochastic optimisation. Further numerical experiments were taken. The estimate–effect ratio, defined by the number of data with the precision of estimation less than 0.5% over whole size of data, was calculated to demonstrate the feasibility of our method. The material parameters for Chinese aorta and pulmonary artery were given with the maximum root mean square (RMS) errors 0.042, and the minimal estimate–effect ratio in numerical experiments was 90.79%. Our results suggest that the constitutive parameters of arterial walls can be determined from uniaxial extension data, given the passive mechanical behaviour governed by H–W model. This method may apply to other tissues using different constitutive models.     

Determination of a constitutive relation for passive myocardium: I. A new functional form

The specific aim of this study is to determine a constitutive relation for non-contracting myocardium in terms of a pseudostrain-energy function W whose form is guided by both theory and experiment. We assume that the material symmetry of myocardium is initially and locally transversely-isotropic, and seek a W which depends upon only two coordinate invariant measures of the finite deformation. The specific functional form of such a W is inferred directly from experimental protocols in which one invariant is held constant while the other is varied, and vice versa. On the basis of data from families of these "constant invariant" tests on thin slabs of myocardium taken from the mid-walls of six canine left ventricles, we propose a new polynomial form of W containing only five material parameters.     

Determination of material models for arterial walls from uniaxial extension tests and histological structure

An approach is proposed that allows the determination of material models from uniaxial tests and histostructural data including fiber orientation of the tissue. A combination of neo-Hookean and Fung-type strain-energy functions is utilized, and inequality constraints imposed on the constitutive parameters are derived providing strict local convexity and preferred fiber orientations. It is shown how the Fung-type model gets a pseudo-structural aspect inherent in the phenomenological model; a correlation between the fiber structure and the parameters of the Fung-type model is explicitly provided. In order to apply the proposed approach, quasi-static uniaxial extension tests of preconditioned prepared strips from the intima, media and adventitia of a human aorta with non-atherosclerotic intimal thickening are acquired in axial and circumferential directions; structural information from histological analyses for each aortic tissue are documented. Data reveal a remarkable thickness, load-bearing capacity and stiffness of the intimal samples in comparison with the media and adventitia. Constitutive parameters for each aortic tissue layer are determined by solving the constrained problem using a penalty function method; a new approach for the estimation of appropriate start values is proposed. Finally, the predictivity and efficacy of the material models is shown by comparing model data with data from the uniaxial extension tests and histological image analyses.     

Anisotropic AAA: Computational comparison between four and two fiber family material models

Abdominal aortic aneurysm (AAA) is a cardiovascular disease with high incidence among elderly population. Biomechanical computational analyses can provide fundamental insights into AAA pathogenesis and clinical management, but modeling should be sufficiently accurate. Several constitutive models of the AAA wall are present in the literature, and some of them seem to well describe the experimental behavior of the aneurysmatic human aorta. In this work we compare a two (2FF) and a four (4FF) fiber families constitutive models of the AAA wall. Both these models satisfactorily fit literature data from biaxial tests on the aneurysmatic tissue. To investigate the peculiar characteristics of these models, we considered the problem of AAA inflation, and solved it by implementing the constitutive equations in a finite element code. A 20% axial stretch was imposed to the aneurysm ends, to simulate the physiological condition. Although fitted on the same dataset, the two material models lead to considerably different outcomes. In particular, adopting a 4FF strain energy function (SEF), an increase of the circumferential stress values can be observed, while higher axial stresses are recorded for the 2FF model. These differences can be attributed to the intrinsic characteristics of the SEFs and to the effective stress field, with respect to the one experienced in biaxial experimental tests on which the fitting is based. In fact the two SEFs appear similar within the region of the stress-strain experimental data, but become different outside it, as in case of aneurysms, due to the effects of the data extrapolation process. It is suggested that experimental data should be obtained for conditions similar to those of the application for which they are intended.     

Human annulus fibrosus material properties from biaxial testing and constitutive modeling are altered with degeneration

The annulus fibrosus (AF) of the intervertebral disk undergoes large and multidirectional stresses and strains. Uniaxial tensile tests are limited for measuring AF material properties, because freely contracting edges can prevent fiber stretch and are not representative of in situ boundary conditions. The objectives of this study were to measure human AF biaxial tensile mechanics and to apply and validate a constitutive model to determine material properties. Biaxial tensile tests were performed on samples oriented along the circumferential–axial and the radial–axial directions. Data were fit to a structurally motivated anisotropic hyperelastic model composed of isotropic extra-fibrillar matrix, nonlinear fibers, and fiber–matrix interactions (FMI) normal to the fibers. The validated model was used to simulate shear and uniaxial tensile behavior, to investigate AF structure–function, and to quantify the effect of degeneration. The biaxial stress–strain response was described well by the model (R ²?>?0.9). The model showed that the parameters for fiber nonlinearity and the normal FMI correlated with degeneration, resulting in an elongated toe-region and lower stiffness with degeneration. The model simulations in shear and uniaxial tension successfully matched previously published circumferential direction Young’s modulus, provided an explanation for the low values in previously published axial direction Young’s modulus, and was able to simulate shear mechanics. The normal FMI were important contributors to stress and changed with degeneration, therefore, their microstructural and compositional source should be investigated. Finally, the biaxial mechanical data and constitutive model can be incorporated into a disk finite element model to provide improved quantification of disk mechanics.     

Pseudoelasticity of excised visceral pleura

A pseudostrain-energy function is proposed for describing the behavior of excised sheets of canine visceral pleura. Pseudoelastic material constants are determined from experimental biaxial data by employing a nonlinear least-squares algorithm. The agreement between theory and experiment is shown to be quite good. Furthermore, the visceral pleura studied appears to be inelastic and to exhibit in-plane isotropy. Comparison with previous works is discussed.     

A method for planar biaxial mechanical testing that includes in-plane shear.

A limitation in virtually all planar biaxial studies of soft tissues has been the inability to include the effects of in-plane shear. This is due to the inability of current mechanical testing devices to induce a state of in-plane shear, due to the added cost and complexity. In the current study, a straightforward method is presented for planar biaxial testing that induces a combined state of in-plane shear and normal strains. The method relies on rotation of the test specimen's material axes with respect to the device axes and on rotating carriages to allow the specimen to undergo in-plane shear freely. To demonstrate the method, five glutaraldehyde treated bovine pericardium specimens were prepared with their preferred fiber directions (defining the material axes) oriented at 45 deg to the device axes to induce a maximum shear state. The test protocol included a wide range of biaxial strain states, and the resulting biaxial data re-expressed in material axes coordinate system. The resulting biaxial data was then fit to the following strain energy function W: [equation: see text] where E'ij is the Green's strain tensor in the material axes coordinate system and c and Ai are constants. While W was able to fit the data very well, the constants A5 and A6 were found not to contribute significantly to the fit and were considered unnecessary to model the shear strain response. In conclusion, while not able to control the amount of shear strain independently or induce a state of pure shear, the method presented readily produces a state of simultaneous in-plane shear and normal strains. Further, the method is very general and can be applied to any anisotropic planar tissue that has identifiable material axes.     

Finite element implementation of a generalized Fung-elastic constitutive model for planar soft tissues

Numerical simulations of the anisotropic mechanical properties of soft tissues and tissue-derived biomaterials using accurate constitutive models remain an important and challenging research area in biomechanics. While most constitutive modeling efforts have focused on the characterization of experimental data, only limited studies are available on the feasibility of utilizing those models in complex computational applications. An example is the widely utilized exponential constitutive model proposed by Fung. Although present in the biomechanics literature for several decades, implementation of this model into finite element (FE) simulations has been limited. A major reason for limited numerical implementations are problems associated with inherent numerical instability and convergence. To address this issue, we developed and applied two restrictions for a generalized Fung-elastic constitutive model necessary to achieve numerical stability. These are (1) convexity of the strain energy function, and (2) the condition number of material stiffness matrix set lower than a prescribed value. These constraints were implemented in the nonlinear regression used for constitutive model parameter estimation to the experimental biaxial mechanical data. We then implemented the generalized Fung-elastic model into a commercial FE code (ABAQUS, Pawtucket, RI, USA). Single element and multi-element planar biaxial test simulations were conducted to verify the accuracy and robustness of the implementation. Results indicated that numerical convergence and accurate FE implementation were consistently obtained. The present study thus presents an integrated framework for accurate and robust implementation of pseudo-elastic constitutive models for planar soft tissues. Moreover, since our approach is formulated within a general FE code, it can be straightforwardly adopted across multiple software platforms.     

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.     

Experimentally tractable,pseudo-elastic constitutive law for biomembranes: I. Theory

Although visco-elastic in general, the stress-strain relation of biomembranes is one-to-one or pseudo-elastic when being loaded after preconditioning. This pseudo-elastic relation is hypoelastic (i.e., it is not hyperelastic), yet much of the stress response can be characterized by a scalar function omega that represents the work done (per unit reference volume) on the specimen during loading. (Since a pseudo-strain-energy function W is optimized to fit the test data and not the work done, omega is not equal to W in general.) The remaining part tR of the stress response does no work during loading. With biaxial testing, omega can be definitively determined from data. Moreover, for tests with the stretch directions coaxial to the axes of anisotropy, tR can be accurately characterized by a scalar function omega that depends on the strain. This paper is part 1 of 2 with "I. Theory" and "II. Application."