Modeling of long-term fatigue damage of soft tissue with stress softening and permanent set effects

One of the major failure modes of bioprosthetic heart valves is non-calcific structural deterioration due to fatigue of the tissue leaflets. Experimental methods to characterize tissue fatigue properties are complex and time-consuming. A constitutive fatigue model that could be calibrated by isolated material tests would be ideal for investigating the effects of more complex loading conditions. However, there is a lack of tissue fatigue damage models in the literature. To address these limitations, in this study, a phenomenological constitutive model was developed to describe the stress softening and permanent set effects of tissue subjected to long-term cyclic loading. The model was used to capture characteristic uniaxial fatigue data for glutaraldehyde-treated bovine pericardium and was then implemented into finite element software. The simulated fatigue response agreed well with the experimental data and thus demonstrates feasibility of this approach.     

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.     

On the Constitutive Model of Nitrogen-Containing Austenitic Stainless Steel 316LN at Elevated Temperature

The nitrogen-containing austenitic stainless steel 316LN has been chosen as the material for nuclear main-pipe, which is one of the key parts in 3rd generation nuclear power plants. In this research, a constitutive model of nitrogen-containing austenitic stainless steel is developed. The true stress-true strain curves obtained from isothermal hot compression tests over a wide range of temperatures (900–1250°C) and strain rates (10⁻³–10 s⁻¹), were employed to study the dynamic deformational behavior of and recrystallization in 316LN steels. The constitutive model is developed through multiple linear regressions performed on the experimental data and based on an Arrhenius-type equation and Zener-Hollomon theory. The influence of strain was incorporated in the developed constitutive equation by considering the effect of strain on the various material constants. The reliability and accuracy of the model is verified through the comparison of predicted flow stress curves and experimental curves. Possible reasons for deviation are also discussed based on the characteristics of modeling process.     

A simplified approach to quasi-linear viscoelastic modeling

The fitting of quasi-linear viscoelastic (QLV) constitutive models to material data often involves somewhat cumbersome numerical convolution. A new approach to treating quasi-linearity in 1-D is described and applied to characterize the behavior of reconstituted collagen. This approach is based on a new principle for including nonlinearity and requires considerably less computation than other comparable models for both model calibration and response prediction, especially for smoothly applied stretching. Additionally, the approach allows relaxation to adapt with the strain history. The modeling approach is demonstrated through tests on pure reconstituted collagen. Sequences of "ramp-and-hold" stretching tests were applied to rectangular collagen specimens. The relaxation force data from the "hold" was used to calibrate a new "adaptive QLV model" and several models from literature, and the force data from the "ramp" was used to check the accuracy of model predictions. Additionally, the ability of the models to predict the force response on a reloading of the specimen was assessed. The "adaptive QLV model" based on this new approach predicts collagen behavior comparably to or better than existing models, with much less computation.     

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.     

Some Thoughts about Bond Energies,Bond Lengths,and Force Constants

The bond energy (BE) of a polyatomic molecule cannot be measured and, therefore, determination of BEs can only be done within a model using a set of assumptions. The bond strength is reflected by the intrinsic BE (IBE), which is related to the intrinsic atomization energy (IAE) and which represents the energy of dissociation under the provision that the degree of hybridization is maintained for all atoms of the molecule. IBE and BE differ in the case of CC and CH bonds by the promotion, the hybridization, and the charge reorganization energy of carbon. Since the latter terms differ from molecule to molecule, IBE and BE are not necessarily parallel and the use of BEs from thermochemical models can be misleading. The stretching force constant is a dynamical quantity and, therefore, it is related to the bond dissociation energy (BDE). Calculation and interpretation of stretching force constants for local internal coordinate modes are discussed and it is demonstrated that the best relationship between BDEs and stretching force constants is obtained within the model of adiabatic internal modes. The valence stretching force constants are less suitable since they are related to an artificial bond dissociation process with geometrical relaxation effects suppressed, which leads to an intrinsic BDE (IBDE). In the case of AX_n molecules, symmetric coordinates can be used to get an appropriate stretching force constant that is related to the BE. However, in general stretching force constants determined for symmetry coordinates do not reflect the strength of a particular bond since the related dissociation processes are strongly influenced by the stability of the products formed.     

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.     

A nonlinear biphasic fiber-reinforced porohyperviscoelastic model of articular cartilage incorporating fiber reorientation and dispersion

A nonlinear biphasic fiber-reinforced porohyperviscoelastic (BFPHVE) model of articular cartilage incorporating fiber reorientation effects during applied load was used to predict the response of ovine articular cartilage at relatively high strains (20%). The constitutive material parameters were determined using a coupled finite element-optimization algorithm that utilized stress relaxation indentation tests at relatively high strains. The proposed model incorporates the strain-hardening, tension-compression, permeability, and finite deformation nonlinearities that inherently exist in cartilage, and accounts for effects associated with fiber dispersion and reorientation and intrinsic viscoelasticity at relatively high strains. A new optimization cost function was used to overcome problems associated with large peak-to-peak differences between the predicted finite element and experimental loads that were due to the large strain levels utilized in the experiments. The optimized material parameters were found to be insensitive to the initial guesses. Using experimental data from the literature, the model was also able to predict both the lateral displacement and reaction force in unconfined compression, and the reaction force in an indentation test with a single set of material parameters. Finally, it was demonstrated that neglecting the effects of fiber reorientation and dispersion resulted in poorer agreement with experiments than when they were considered. There was an indication that the proposed BFPHVE model, which includes the intrinsic viscoelasticity of the nonfibrillar matrix (proteoglycan), might be used to model the behavior of cartilage up to relatively high strains (20%). The maximum percentage error between the indentation force predicted by the FE model using the optimized material parameters and that measured experimentally was 3%.     

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.     

Biaxial mechanical response of bioprosthetic heart valve biomaterials to high in-plane shear

Utilization of novel biologically-derived biomaterials in bioprosthetic heart valves (BHV) requires robust constitutive models to predict the mechanical behavior under generalized loading states. Thus, it is necessary to perform rigorous experimentation involving all functional deformations to obtain both the form and material constants of a strain-energy density function. In this study, we generated a comprehensive experimental biaxial mechanical dataset that included high in-plane shear stresses using glutaraldehyde treated bovine pericardium (GLBP) as the representative BHV biomaterial. Compared to our previous study (Sacks, JBME, v.121, pp. 551-555, 1999), GLBP demonstrated a substantially different response under high shear strains. This finding was underscored by the inability of the standard Fung model, applied successfully in our previous GLBP study, to fit the high-shear data. To develop an appropriate constitutive model, we utilized an interpolation technique for the pseudo-elastic response to guide modification of the final model form. An eight parameter modified Fung model utilizing additional quartic terms was developed, which fitted the complete dataset well. Model parameters were also constrained to satisfy physical plausibility of the strain energy function. The results of this study underscore the limited predictive ability of current soft tissue models, and the need to collect experimental data for soft tissue simulations over the complete functional range.     

A finite element study of invariant-based orthotropic constitutive equations in the context of myocardial material parameter estimation

A previous study investigated a number of invariant-based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. The study was based on the assumption of a homogeneous deformation. Here, we extend the previous study by performing an inverse finite element material parameter estimation. This ensures a more realistic deformation state and material parameter estimates. The constitutive relations were compared on the basis of (i) ‘goodness of fit’: how well they fit a set of six shear deformation tests and (ii) ‘variability’: how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive relations. It was found that a specific form of the polyconvex type as well as the exponential Fung-type equations were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.     

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 model of growth and rupture of abdominal aortic aneurysm

We present here a coupled mathematical model of growth and failure of the abdominal aortic aneurysm (AAA). The failure portion of the model is based on the constitutive theory of softening hyperelasticity where the classical hyperelastic law is enhanced with a new constant indicating the maximum energy that an infinitesimal material volume can accumulate without failure. The new constant controls material failure and it can be interpreted as the average energy of molecular bonds from the microstructural standpoint. The constitutive model is compared to the data from uniaxial tension tests providing an excellent fit to the experiment. The AAA failure model is coupled with a phenomenological theory of soft tissue growth. The unified theory includes both momentum and mass balance laws coupled with the help of the constitutive equations. The microstructural alterations in the production of elastin and remodeling of collagen are reflected in the changing macroscopic parameters characterizing tissue stiffness, strength and density. The coupled theory is used to simulate growth and rupture of an idealized spherical AAA. The results of the simulation showing possible AAA ruptures in growth are reasonable qualitatively while the quantitative calibration of the model will require further clinical observations and in vitro tests. The presented model is the first where growth and rupture are coupled.     

Three-dimensional mechanical properties of porcine coronary arteries: a validated two-layer model

The normal coronary artery consists of two mechanically distinct layers: intima-media and adventitia. The objective of this study is to establish a two-layer three-dimensional (3-D) stress-strain relation of porcine coronary arteries. Experimental measurements were made by a series of biaxial tests (inflation and axial extension) of intact coronary arteries and, subsequently, their corresponding intima-media or adventitia layer. The Fung-type exponential strain energy function was used to describe the 3-D strain-stress relation for each layer and the intact wall. A genetic algorithm was used to determine the material constants in the Fung-type constitutive equation by curve fitting the experimental data. Because one layer must be sacrificed before the other layer can be tested, the material property of the missing layer was computed from the material constants of the intact vessel and the tested layer. A total of 20 porcine hearts were used: one group of 10 hearts for the left anterior descending artery and another group of 10 hearts for the right coronary artery. Each group was further divided into two subgroups of five specimens tested for the intact wall and the intima-media layer and for the intact wall and the adventitia layer. Our results show statistically significant differences in the material properties of the two layers. The mathematical model was validated by experimental stress-strain data for individual layers. The validated 3-D constitutive model will serve as a foundation for formulation of layer-specific boundary value problems in coronary physiology and cardiology.     

Modelling of the heart and pericardium at end-diastole

Herein we present a refined version of Vito's two-sphere static model of the heart with pericardium and discuss its possible applications. The improvements we make on Vito's model are: (i) Vito assumed that the elastic materials which constitute the model 'heart' and 'pericardium' are isotropic; we relax this assumption to that of transverse-isotropy. (ii) Our analysis, which does not assume the existence of stored-energy functions, links the model directly to empirical stress-strain relations of suitable biaxial uniform-extension tests; two such stress-strain relations (one for the pericardium, one for the myocardium, both of which may be described by the same equation except for difference in the values of response parameters) now define the model completely, so we avoid altogether the difficult task of determining full-fledged constitutive equations for the pericardium and myocardium. As for applications, we contend that the concentric spheres in static equilibrium can be taken as a model of the left ventricle and pericardium at end-diastole. We show that the model when equipped with suitable stress-strain relations does give good fit to the pressure-volume data which Spotnitz et al. (1966, Circulation Res., 18, 49-66) obtained from excised canine left ventricles and to the pericardium data which Pegram et al. (1975, Circulation Res., 9, 707-714) obtained from closed chest, anaesthetized dogs. Three different empirical formulae were tried in the data-fitting as the equation that describes the requisite stress-strain relations. The 'exponential law' gave the best results.     

Simulation of axoplasmic transport

We have analysed a kinetic model of axonal transport by simulating experimental tracer profiles. The existence of three phases of axoplasmic transport is assumed: fast anterograde, slow anterograde and retrograde. Each phase has its characteristic velocity. Transported materials are postulated to shift between these phases. Also catabolism and sequestration of material is allowed for in our model. Thus, we have set up equations which contain axonal transport, diffusion and cross-over terms. The rate constants of material shifts were determined by computer fitting to experimental data. Best-fitted values of the rate constants for transfer of material between the fast and slow phases were both 2 X 10(-5) sec-1, while the rate constants for transfer between the fast and retrograde phases were both 1 X 10(-5) sec-1. The rate constant of material loss from the slow phase to the extracellular space was 1 X 10(-6) sec-1. The material shift between the slow and retrograde phases was negligibly small. These data show that there is exchange of material between the fast and slow phases and between the fast and retrograde phases. However, there is no significant exchange between the slow and retrograde phases. Diffusion was found to have only a minor effect on the profiles. The velocity of the fast anterograde track in cold-blooded animals was predicted to be around 200 mm/day, or, in other words, to be close to experimentally observed values of the fast anterograde component of axonal transport.     

The validation of a generalized Hooke's law for coronary arteries

The exponential form of constitutive model is widely used in biomechanical studies of blood vessels. There are two main issues, however, with this model: 1) the curve fits of experimental data are not always satisfactory, and 2) the material parameters may be oversensitive. A new type of strain measure in a generalized Hooke's law for blood vessels was recently proposed by our group to address these issues. The new model has one nonlinear parameter and six linear parameters. In this study, the stress-strain equation is validated by fitting the model to experimental data of porcine coronary arteries. Material constants of left anterior descending artery and right coronary artery for the Hooke's law were computed with a separable nonlinear least-squares method with an excellent goodness of fit. A parameter sensitivity analysis shows that the stability of material constants is improved compared with the exponential model and a biphasic model. A boundary value problem was solved to demonstrate that the model prediction can match the measured arterial deformation under experimental loading conditions. The validated constitutive relation will serve as a basis for the solution of various boundary value problems of cardiovascular biomechanics.     

On true and apparent Michaelis constants in enzymology. I. Differences

Differences between both true and apparent rate constants and Michaelis constants have been examined. Rate constants of elementary stages of real mechanisms are true ones. True Michaelis constant Km is expressed by equation Km = (k(-1) + k2)/k. True constants may be determined for reliable mechanism only for which the equation of initial rate was obtained which displays physical sense of these constants and permits to find the method of their calculation. The true constant values are independent of concentration of reactants, activators, inhibitors, extraneous agents and pH. The apparent rate constants are such constants of the composite reaction which are observed when this reaction is described by the equation of simple reaction. Michaelis constant calculated by a half of the ultimate constant is an apparent constant. The apparent constants may be functions of several true rate constants and/or concentrations of reacting substances. The evident physical sense of apparent constants being absent, only formal relation between the reaction rate and reactant concentration independent of the investigated mechanism is provided.     

A linearized and incompressible constitutive model for arteries

In many biomechanical studies, blood vessels can be modeled as pseudoelastic orthotropic materials that are incompressible (volume-preserving) under physiological loading. To use a minimum number of elastic constants to describe the constitutive behavior of arteries, we adopt a generalized Hooke's law for the co-rotational Cauchy stress and a recently proposed logarithmic-exponential strain. This strain tensor absorbs the material nonlinearity and its trace is zero for volume-preserving deformations. Thus, the relationships between model parameters due to the incompressibility constraint are easy to analyze and interpret. In particular, the number of independent elastic constants reduces from ten to seven in the orthotropic model. As an illustratory study, we fit this model to measured data of porcine coronary arteries in inflation-stretch tests. Four parameters, n (material nonlinearity), Young's moduli E? (circumferential), E? (axial), and E? (radial) are necessary to fit the data. The advantages and limitations of this model are discussed.