Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials

A comprehensive study on the spherical indentation of hyperelastic soft materials is carried out through combined theoretical, computational, and experimental efforts. Four widely used hyperelastic constitutive models are studied, including neo-Hookean, Mooney–Rivlin, Fung, and Arruda–Boyce models. Through dimensional analysis and finite element simulations, we establish the explicit relations between the indentation loads at given indentation depths and the constitutive parameters of materials. Based on the obtained results, the applicability of Hertzian solution to the measurement of the initial shear modulus of hyperelastic materials is examined. Furthermore, from the viewpoint of inverse problems, the possibility to measure some other properties of a hyperelastic material using spherical indentation tests, e.g., locking stretch, is addressed by considering the existence, uniqueness, and stability of the solution. Experiments have been performed on polydimethylsiloxane to validate the conclusions drawn from our theoretical analysis. The results reported in this study should help identify the extent to which the mechanical properties of hyperelastic materials could be measured from spherical indentation tests.     

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.     

Simulated bioprosthetic heart valve deformation under quasi-static loading

For more than 40 years, the replacement of diseased natural heart valves with prosthetic devices has dramatically extended the quality and length of the lives of millions of patients worldwide. However, bioprosthetic heart valves (BHV) continue to fail due to structural failure resulting from poor tissue durability and faulty design. Clearly, an in-depth understanding of the biomechanical behavior of BHV at both the tissue and functional prosthesis levels is essential to improving BHV design and to reduce rates of failure. In this study, we simulated quasi-static BHV leaflet deformation under 40, 80, and 120 mm Hg quasi-static transvalvular pressures. A Fung-elastic material model was used that incorporated material parameters and axes derived from actual leaflet biaxial tests and measured leaflet collagen fiber structure. Rigorous experimental validation of predicted leaflet strain field was used to validate the model results. An overall maximum discrepancy of 2.36% strain between the finite element (FE) results and experiment measurements was obtained, indicating good agreement between computed and measured major principal strains. Parametric studies utilizing the material parameter set from one leaflet for all three leaflets resulted in substantial variations in leaflet stress and strain distributions. This result suggests that utilization of actual leaflet material properties is essential for accurate BHV FE simulations. The present study also underscores the need for rigorous experimentation and accurate constitutive models in simulating BHV function and design.     

A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue

The mechanical behavior of soft tissue demonstrates a number of complex features including nonlinearity, anisotropy, viscoelasticity, and growth. Characteristic features of the time-dependent and anisotropic behavior are related to the properties of various components of the tissue such as fibrous collagen and elastin networks, large proteins and sugars attached to these networks, and interstitial fluid. Attempts to model the elastic behavior of these tissues based on assumptions about the behavior of the underlying constituents have been reasonably successful, but the essential addition of viscoelasticity to these models has been met with varying success. Here, a new rheological network model is proposed using, as its basis, an orthotropic hyperelastic constitutive model for fibrous tissue and a viscoelastic reptation model for soft materials. The resulting model has been incorporated into numerical and computational models, and is shown to capture the mechanical behavior of soft tissue in various modes of deformation including uniaxial and biaxial tension and simple shear.     

Platelet activation of mechanical versus bioprosthetic heart valves during systole

Thrombus formation is a major concern for recipients of mechanical heart valves (MHVs), which requires them to take anticoagulant drugs for the rest of their lives. Bioprosthetic heart valves (BHVs) do not require life-long anticoagulant therapy but deteriorate after 10–15 years. The thrombus formation is initiated by the platelet activation which is thought to be mainly generated in MHVs by the flow through the hinge and the leakage flow during the diastole. However, our results show that the activation in the bulk flow during the systole phase might play an essential role as well. This is based on our results obtained by comparing the thrombogenic performance of a MHV and a BHV (as control) in terms of shear induced platelet activation under exactly the same conditions. Three different mathematical activation models including linear level of activation, damage accumulation, and Soares model are tested to quantify the platelet activation during systole using the previous simulations of the flow through MHV and BHV in a straight aorta under the same physiologic flow conditions. Results indicate that the platelet activation in the MHV at the beginning of the systole phase is slightly less than the BHV. However, at the end of the systole phase the platelet activation by the bulk flow for the MHV is several folds (1.41, 5.12, and 2.81 for linear level of activation, damage accumulation, and Soares model, respectively) higher than the BHV for all tested platelet activation models.     

A mechanical characterization of the porcine atria at the healthy stage and after ventricular tachypacing

Atrial fibrillation (AF) is a cardiac arrhythmia that highly increases the risk of stroke and is associated with significant but still unexplored changes in the mechanical behavior of the tissue. Planar biaxial tests were performed on tissue specimens from pigs at the healthy stage and after ventricular tachypacing (VTP), a procedure applied to reproduce the relevant features of AF. The local arrangement of the fiber bundles in the tissue was investigated on specimens from rabbit atria by means of circularly polarized light. Based on this, mechanical data were fitted to two anisotropic constitutive relationships, including a four-parameter Fung-type model and a microstructurally-motivated model. Accounting for the fiber-induced anisotropy brought average R(2) = 0.807 for the microstructurally-motivated model and average R(2) = 0.949 for the Fung model. Validation of the fitted constitutive relationships was performed by means of FEM simulations coupled to FORTRAN routines. The performances of the two material models in predicting the second Piola-Kirchhoff stress were comparable, with average errors <3.1%. However, the Fung model outperformed the other in the prediction of the Green-Lagrange strain, with 9.2% maximum average error. To increase model generality, a proper averaging procedure accounting for nonlinearities was used to obtain average material parameters. In general, a stiffer behavior after VTP was noted.     

A strain-hardening bi-power law for the nonlinear behaviour of biological soft tissues

Biological soft tissues exhibit a strongly nonlinear viscoelastic behaviour. Among parenchymous tissues, kidney and liver remain less studied than brain, and a first goal of this study is to report additional material properties of kidney and liver tissues in oscillatory shear and constant shear rate tests. Results show that the liver tissue is more compliant but more strain hardening than kidney. A wealth of multi-parameter mathematical models has been proposed for describing the mechanical behaviour of soft tissues. A second purpose of this work is to develop a new constitutive law capable of predicting our experimental data in the both linear and nonlinear viscoelastic regime with as few parameters as possible. We propose a nonlinear strain-hardening fractional derivative model in which six parameters allow fitting the viscoelastic behaviour of kidney and liver tissues for strains ranging from 0.01 to 1 and strain rates from 0.0151 s^?1 to 0.7 s^?1.     

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.     

Hyperelastic anisotropic microplane constitutive model for annulus fibrosus

In a recent paper, Peng et al. (2006, "An Anisotropic Hyperelastic Constitutive Model With Fiber-Matrix Interaction for the Human Annulus Fibrosis," ASME J. Appl. Mech., 73(5), pp. 815-824) developed an anisotropic hyperelastic constitutive model for the human annulus fibrosus in which fiber-matrix interaction plays a crucial role in simulating experimental observations reported in the literature. Later, Guo et al. (2006, "A Composites-Based Hyperelastic Constitutive Model for Soft Tissue With Application to the Human Fibrosis," J. Mech. Phys. Solids, 54(9), pp. 1952-1971) used fiber reinforced continuum mechanics theory to formulate a model in which the fiber-matrix interaction was simulated using only composite effect. It was shown in these studies that the classical anisotropic hyperelastic constitutive models for soft tissue, which do not account for this shear interaction, cannot accurately simulate the test data on human annulus fibrosus. In this study, we show that the microplane model for soft tissue developed by Caner and Carol (2006, "Microplane Constitutive Model and Computational Framework for Blood Vessel Tissue," ASME J. Biomech. Eng., 128(3), pp. 419-427) can be adjusted for human annulus fibrosus and the resulting model can accurately simulate the experimental observations without explicit fiber-matrix interaction because, in microplane model, the shear interaction between the individual fibers distributed in the tissue provides the required additional rigidity to explain these experimental facts. The intensity of the shear interaction between the fibers can be adjusted by adjusting the spread in the distribution while keeping the total amount of the fiber constant. A comparison of results obtained from (i) a fiber-matrix parallel coupling model, which does not account for the fiber-matrix interaction, (ii) the same model but enriched with fiber-matrix interaction, and (iii) microplane model for soft tissue adapted to annulus fibrosus with two families of fiber distributions is presented. The conclusions are (i) that varying degrees of fiber-fiber and fiber-matrix shear interaction must be taking place in the human annulus fibrosus, (ii) that this shear interaction is essential to be able to explain the mechanical behavior of human annulus fibrosus, and (iii) that microplane model can be fortified with fiber-matrix interaction in a straightforward manner provided that there are new experimental data on distribution of fibers, which indicate a spread so small that it requires an explicit fiber-matrix interaction to be able to simulate the experimental data.     

Constitutive model for brain tissue under finite compression

While advances in computational models of mechanical phenomena have made it possible to simulate dynamically complex problems in biomechanics, accurate material models for soft tissues, particularly brain tissue, have proven to be very challenging. Most studies in the literature on material properties of brain tissue are performed in shear loading and very few tackle the behavior of brain in compression. In this study, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10 s(-1) ramp rate and 20 s hold time has been developed. The assumption of quasi-linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C(10), C(01) and C(11)) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a two parameter Ogden model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06 s and 14 s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0 s(-1)) and the results in compression and their extrapolation to tension were compared against previous models.     

Region-specific constitutive modeling of the plantar soft tissue

Recent research has shown that hyperelastic properties of the plantar soft tissue consisting of adipose tissue and fibrous septa change from region to region. However, relatively little research has been conducted to develop analytical or computational models to describe the region-specific behavior of the plantar soft tissue. The objective of the research is to develop a region-specific constitutive model of the plantar soft tissue. Plantar soft tissue specimens were dissected from six regions [subcalcaneal (CA), sublateral (LA), subnavicular (Nav), 1st, 3rd, and 5th submetatarsal (M1, M3, M5)] from cadaveric foot samples, and a picrosirius red staining technique was used to visualize the collagen fibers in fibrous septa. The volume fractions of adipose tissue and fibrous septa and the volume fractions of the principal orientations of the fibrous septa were calculated with the intensity gradient method. Region-specific constitutive models were then developed in finite element analysis considering the microstructure of the plantar soft tissue. The hyperelastic region specific material properties of the plantar soft tissue were validated with experimental unconfined compression tests and indentation tests from the literature. The results show that the models give reasonable predictions of the stiffness of the soft tissue within a standard deviation of the tests. The region-specific constitutive models help to explain how changes in the constituents are related to mechanical behavior of the soft tissue on a region specific basis.     

A visco-hyperelastic-damage constitutive model for the analysis of the biomechanical response of the periodontal ligament

The periodontal ligament (PDL), as other soft biological tissues, shows a strongly non-linear and time-dependent mechanical response and can undergo large strains under physiological loads. Therefore, the characterization of the mechanical behavior of soft tissues entails the definition of constitutive models capable of accounting for geometric and material non-linearity. The microstructural arrangement determines specific anisotropic properties. A hyperelastic anisotropic formulation is adopted as the basis for the development of constitutive models for the PDL and properly arranged for investigating the viscous and damage phenomena as well to interpret significant aspects pertaining to ordinary and degenerative conditions. Visco-hyperelastic models are used to analyze the time-dependent mechanical response, while elasto-damage models account for the stiffness and strength decrease that can develop under significant loading or degenerative conditions. Experimental testing points out that damage response is affected by the strain rate associated with loading, showing a decrease in the damage limits as the strain rate increases. These phenomena can be investigated by means of a model capable of accounting for damage phenomena in relation to viscous effects. The visco-hyperelastic-damage model developed is defined on the basis of a Helmholtz free energy function depending on the strain-damage history. In particular, a specific damage criterion is formulated in order to evaluate the influence of the strain rate on damage. The model can be implemented in a general purpose finite element code. The accuracy of the formulation is evaluated by using results of experimental tests performed on animal model, accounting for different strain rates and for strain states capable of inducing damage phenomena. The comparison shows a good agreement between numerical results and experimental data.     

On the implementation of a wrinkling,hyperelastic membrane model for skin and other materials

A number of researchers have studied the mechanical properties of skin and developed constitutive models to describe its behaviour. Typically, many of these studies have concentrated on the uniaxial tensile behaviour of the skin, on the grounds that it will wrinkle under in-plane compression and have minimal stiffness. However, although there is a substantial body of literature on wrinkling models, the practical implementation of such a model of skin in a finite element setting has not been widely addressed. This paper presents computational details of a wrinkling, hyperelastic membrane model and aspects of its implementation and areas requiring further research are discussed. The model is based on an Ogden constitutive model, which provides accurate results at moderate strains, but it would be straightforward to implement other constitutive models such as the Fung or Arruda–Boyce models using a similar approach. Example results are presented which demonstrate that the model can provide a good approximation to experimental data. The model has many other possible applications, both for biological materials and for other thin hyperelastic membranes.     

An approach to the mechanical constitutive modelling of arterial tissue based on homogenization and optimization

This paper is concerned with characterizing the quasistatic mechanical behaviour of arterial tissue undergoing finite deformation through hyperelastic constitutive functions. Commonly the parameters of constitutive functions are established by a process of optimization based on experimental data. Instead we construct a finite element model of a representative volume element of the material and compute its homogenized response to a range of deformations. These data are then used to provide objective functions for optimizing the parameters of two analytical models from the literature.     

Mechanical properties of brain tissue in tension

This paper contains experimental results of in vitro, uniaxial tension of swine brain tissue in finite deformation as well as proposes a new hyper-viscoelastic constitutive model for the brain tissue. The experimental results obtained for two loading velocities, corresponding to strain rates of 0.64 and 0.64 x 10(-2)s(-1), are presented. We believe that these are the first ever experiments of this kind. The applied strain rates were similar to those applied in our previous study, focused on explaining brain tissue properties in compression. The stress-strain curves are convex downward for all extension rates. The tissue response stiffened as the loading speed increased, indicating a strong stress-strain rate dependence. Swine brain tissue was found to be considerably softer in extension than in compression. Previously proposed in the literature brain tissue constitutive models, developed based on experimental data collected in compression are shown to be inadequate to explain tissue behaviour in tension. A new, non-linear, viscoelastic model based on the generalisation of the Ogden strain energy hyper-elastic constitutive equation is proposed. The new model accounts well for brain tissue deformation behaviour in both tension and compression (natural strain in <-0.3,0.2>) for strain rates ranging over five orders of magnitude.     

Confocal arthroscopy-based patient-specific constitutive models of cartilaginous tissues - II: prediction of reaction force history of meniscal cartilage specimens

The theoretical framework developed in a companion paper (Part I) is used to derive estimates of mechanical response of two meniscal cartilage specimens. The previously developed framework consisted of a constitutive model capable of incorporating confocal image-derived tissue microstructural data. In the present paper (Part II) fibre and matrix constitutive parameters are first estimated from mechanical testing of a batch of specimens similar to, but independent from those under consideration. Image analysis techniques which allow estimation of tissue microstructural parameters form confocal images are presented. The constitutive model and image-derived structural parameters are then used to predict the reaction force history of the two meniscal specimens subjected to partially confined compression. The predictions are made on the basis of the specimens' individual structural condition as assessed by confocal microscopy and involve no tuning of material parameters. Although the model does not reproduce all features of the experimental curves, as an unfitted estimate of mechanical response the prediction is quite accurate. In light of the obtained results it is judged that more general non-invasive estimation of tissue mechanical properties is possible using the developed framework.     

Large deformation shear properties of liver tissue

Characterization of the mechanical properties of soft biological tissues is important for establishing the mechanical tolerances of the tissues, and for input to computational models. In this work, the viscoelastic properties of bovine liver tissue in shear loading have been measured using relaxation and constant shear rate loading. The tissue is nonlinearly viscoelastic for strains greater than 0.2%, has a yield strain of approximately 10, and shows moderate strain-rate sensitivity. The response can be modelled using a nonlinear viscoelastic differential model previously developed for brain tissue.     

Finite element modeling of 3D human mesenchymal stem cell-seeded collagen matrices exposed to tensile strain

The use of human mesenchymal stem cells (hMSCs) in tissue engineering is attractive due to their ability to extensively self-replicate and differentiate into a multitude of cell lineages. It has been experimentally established that hMSCs are influenced by chemical and mechanical signals. However, the combined chemical and mechanical in vitro culture conditions that lead to functional tissue require greater understanding. In this study, finite element models were created to evaluate the local loading conditions on bone marrow-derived hMSCs seeded in three-dimensional collagen matrices exposed to cyclic tensile strain. Mechanical property and geometry data used in the models were obtained experimentally from a previous study in our laboratory and from mechanical testing. Eight finite element models were created to simulate three-dimensional hMSC-seeded collagen matrices exposed to different levels of cyclic tensile strain (10% and 12%), culture media (complete growth and osteogenic differentiating), and durations of culture (7 and 14 days). Through finite element analysis, it was determined that globally applied uniaxial tensile strains of 10% and 12% resulted in local strains up to 18.3% and 21.8%, respectively. Model results were also compared to experimental studies in an attempt to explain observed differences between hMSC response to 10% and 12% cyclic tensile strain.     

Simulation of planar soft tissues using a structural constitutive model: Finite element implementation and validation

Computational implementation of physical and physiologically realistic constitutive models is critical for numerical simulation of soft biological tissues in a variety of biomedical applications. It is well established that the highly nonlinear and anisotropic mechanical behaviors of soft tissues are an emergent behavior of the underlying tissue microstructure. In the present study, we have implemented a structural constitutive model into a finite element framework specialized for membrane tissues. We noted that starting with a single element subjected to uniaxial tension, the non-fibrous tissue matrix must be present to prevent unrealistic tissue deformations. Flexural simulations were used to set the non-fibrous matrix modulus because fibers have little effects on tissue deformation under three-point bending. Multiple deformation modes were simulated, including strip biaxial, planar biaxial with two attachment methods, and membrane inflation. Detailed comparisons with experimental data were undertaken to insure faithful simulations of both the macro-level stress–strain insights into adaptations of the fiber architecture under stress, such as fiber reorientation and fiber recruitment. Results indicated a high degree of fidelity and demonstrated interesting microstructural adaptions to stress and the important role of the underlying tissue matrix. Moreover, we apparently resolve a discrepancy in our 1997 study (Billiar and Sacks, 1997. J. Biomech. 30 (7), 753–756) where we observed that under strip biaxial stretch the simulated fiber splay responses were not in good agreement with the experimental results, suggesting non-affine deformations may have occurred. However, by correctly accounting for the isotropic phase of the measured fiber splay, good agreement was obtained. While not the final word, these simulations suggest that affine fiber kinematics for planar collagenous tissues is a reasonable assumption at the macro level. Simulation tools such as these are imperative in the design and simulation of native and engineered tissues.