Constitutive Model For Soft Tissue Research Articles

Difference analysis of phenomenological models with two variable forms for soft tissue quasi-static mechanical characterization

It was important to understand the accurate mechanical properties of soft tissue for the evaluation of its injury, provide reliable protective ways or design effective human resist-injury devices. There was little study that clarified the difference between phenomenological models based on strain invariant and the principal stretches variables respectively although some quasi-static constitutive models of soft tissue were developed. In this study, we enumerate several typical hyperelastic models and derive the tensor equation of stress-strain based on continuum mechanics to fit the experimental data of human brain specimens under multiple loading modes in previous studies and give the coefficient of determination based on the least square fitting. It was suggested that two variable forms of phenomenological models with only the first strain invariant are consistent under uniaxial compression and tension, but the Cauchy stress tensor expressed by strain is completely different under simple shear loading. Also, the shear stress derived from the constitutive model based on strain invariants and principal stretchs has multiple relationships related to shear strain. The results in this study would be used to understand the more accurate mechanical characterization of soft tissue, which will allow us to evaluate the injury and develop much accurate injury criteria for soft tissue.

Biomechanical analysis of sheep oesophagus subjected to biaxial testing including hyperelastic constitutive model fitting

High quality computational model of soft tissues is a function of accurate and reliable mechanical properties. Hyperelastic constitutive models are normally utilised in developing reliable computational models. Therefore, section of proper and reliable constitutive models for soft tissue is critical. This work presents the biomechanical properties of oesophagus subjected to biaxial mechanical tensile test. Additionally, six hyperelastic constitutive models commonly used for modelling behaviour of soft tissues were selected. The experimental data were then fitted on Fung, Choi-Vito, Holzapfel (2000), Holzapfel (2005), Polynomial (Anisotropic) and Four-Fiber Family hyperelastic constitutive models. The sheep oesophagus subjected to equi-biaxial tension has exhibited different stress magnitude in both longitudinal and circumferential directions. There is significant difference between circumferential and longitudinal stresses (p = 0.0034). The average circumferential and longitudinal stresses are recorded to be 82.87 ± 30.36 kPa and 41.42 ± 32.02 kPa, respectively (p = 0.0034). Between six hyperelastic constitutive models, it was observed that Four-Fiber model has produced better fit when compared to others. After fitting biaxial mechanical properties of oesophagus, it was found that the Four-fiber family hyperelastic constitutive model would best fit.

A new anisotropic soft tissue model for elimination of unphysical auxetic behaviour

Auxetic behaviour, the unphysical transverse expansion during uniaxial tension, is a common and undesirable feature of classical anisotropic hyperelastic constitutive models for soft tissue. In this study we uncover the underlying mechanism of such behaviour; high levels of in-plane compaction occurs due to increasing tension in strain-stiffening fibres, leading to unphysical out-of-plane expansion. We demonstrate that auxetic behaviour is primarily influenced by the ratio of fibre to matrix stiffness, and is accentuated by strain-stiffening fibres in a constant stiffness matrix (e.g., the widely used neo-Hookean matrix with exponentially stiffening fibres). We propose a new bilinear strain stiffening fibre and matrix (BLFM) model which allows close control of the fibre-matrix stiffness ratio, thereby robustly eliminating auxetic behaviour. We demonstrate that our model provides accurate prediction of experimentally observed out-of-plane compaction, in addition to stress-stretch anisotropy, for arterial tissue subjected to uniaxial tension testing.

Some unexpected predictions from strongly anisotropic hyperelastic constitutive models of soft tissue

It is shown that a widely used class of constitutive models for the mechanical response of elastic arteries, which includes the so-called HGO model, responds as if it were inextensible in simple tension in the zero limit of a non-dimensional ratio of material parameters. A significant auxetic response is predicted for an incompressible hyperelastic elastic sheet reinforced with inextensible cords. Thus, a significant lateral deformation of arterial specimens modelled by this class of materials should be observed in simple tension for small values of the non-dimensional parameter. However, such a response has not been observed experimentally. The analysis therefore suggests that predictions for this class of strongly anisotropic constitutive models for arteries should be treated with caution.

Calibration of hyperelastic constitutive models: the role of boundary conditions, search algorithms, and experimental variability.

The calibration of hyperelastic constitutive models of soft tissue and tissue surrogates is often treated as an exercise in curve-fitting to the average experimental response, and many of the complicating factors such as experimental boundary conditions and data variability are ignored. In this work, we focus on three questions that arise in this area: the ramifications of ignoring the experimental boundary conditions, the use of local optimizers, and the role of data variability. Using data from a uniaxial extension experiment on a tissue surrogate, we study how these three factors affect the calibration of isotropic hyperelastic constitutive models. Our results show that even with the simplest of constitutive models, it is necessary to look beyond a "good fit" to the average.

The elimination method of preloading force for soft tissue based on the linear loading region

Aiming at the problem of the influence of preloading force on its mechanical response in soft tissue compression experiments, an elimination method of preloading force based on linear loading region is proposed. Unconfined compression experiments under a variety of different preloading forces are performed. The influence of the preloading force on the parameters of constitutive model is analyzed. In the preload phase, the mechanical response of the soft tissue is taken as a linear model. The preloading force is eliminated by taking the preloading phase into account throughout the response process. According to five different preloading forces of the unconfined compression experiments, the elimination method is validated with two different constitutive models of soft tissue, and the error between the models obtained by the preloading force elimination method and the traditional method with the experimental results is compared. The results show that the error obtained by preloading force elimination method is significantly smaller than the traditional method. The preloading force elimination method can eliminate the influence of preloading force on mechanical response to a certain extent, and constitutive model parameters which are closer to the true properties of soft tissue can be obtained.

Poroelastic Modeling of Highly Hydrated Collagen Hydrogels: Experimental Results vs. Numerical Simulation With Custom and Commercial Finite Element Solvers.

This study presents a comparison between the performances of two Finite Element (FE) solvers for the modeling of the poroelastic behavior of highly hydrated collagen hydrogels. Characterization of collagen hydrogels has been a widespread challenge since this is one of the most used natural biomaterials for Tissue Engineering (TE) applications. V-Biomech® is a free custom FE solver oriented to soft tissue modeling, while Abaqus® is a general-purpose commercial FE package which is widely used for biomechanics computational modeling. Poroelastic simulations with both solvers were compared to two experimental protocols performed by Busby et al. (2013) and Chandran and Barocas (2004), also using different implementations of the frequently used Neo-Hookean hyperelastic model. The average differences between solvers outputs were under 5% throughout the different tests and hydrogel properties. Thus, differences were small enough to be considered negligible and within the variability found experimentally from one sample to another. This work demonstrates that constitutive modeling of soft tissues, such as collagen hydrogels can be achieved with either V-Biomech or Abaqus standard options (without user-subroutine), which is important for the biomechanics and biomaterials research community. V-Biomech has shown its potential for the validation of biomechanical characterization of soft tissues, while Abaqus' versatility is useful for the modeling and analysis of TE applications where other complex phenomena may also need to be captured.

Characterization of three-dimensional anisotropic heart valve tissue mechanical properties using inverse finite element analysis

Computational modeling has an important role in design and assessment of medical devices. In computational simulations, considering accurate constitutive models is of the utmost importance to capture mechanical response of soft tissue and biomedical materials under physiological loading conditions. Lack of comprehensive three-dimensional constitutive models for soft tissue limits the effectiveness of computational modeling in research and development of medical devices. The aim of this study was to use inverse finite element (FE) analysis to determine three-dimensional mechanical properties of bovine pericardial leaflets of a surgical bioprosthesis under dynamic loading condition. Using inverse parameter estimation, 3D anisotropic Fung model parameters were estimated for the leaflets. The FE simulations were validated using experimental in-vitro measurements, and the impact of different constitutive material models was investigated on leaflet stress distribution. The results of this study showed that the anisotropic Fung model accurately simulated the leaflet deformation and coaptation during valve opening and closing. During systole, the peak stress reached to 3.17MPa at the leaflet boundary while during diastole high stress regions were primarily observed in the commissures with the peak stress of 1.17MPa. In addition, the Rayleigh damping coefficient that was introduced to FE simulations to simulate viscous damping effects of surrounding fluid was determined.

Bayesian calibration of hyperelastic constitutive models of soft tissue

There is inherent variability in the experimental response used to characterize the hyperelastic mechanical response of soft tissues. This has to be accounted for while estimating the parameters in the constitutive models to obtain reliable estimates of the quantities of interest. The traditional least squares method of parameter estimation does not give due importance to this variability. We use a Bayesian calibration framework based on nested Monte Carlo sampling to account for the variability in the experimental data and its effect on the estimated parameters through a systematic probability-based treatment.We consider three different constitutive models to represent the hyperelastic nature of soft tissue: Mooney–Rivlin model, exponential model, and Ogden model. Three stress–strain data sets corresponding to the deformation of agarose gel, bovine liver tissue, and porcine brain tissue are considered. Bayesian fits and parameter estimates are compared with the corresponding least squares values. Finally, we propagate the uncertainty in the parameters to a quantity of interest (QoI), namely the force-indentation response, to study the effect of model form on the values of the QoI. Our results show that the quality of the fit alone is insufficient to determine the adequacy of the model, and due importance has to be given to the maximum likelihood value, the landscape of the likelihood distribution, and model complexity.

Towards a pancreatic surgery simulator based on model order reduction

In this work a pancreatic surgery simulator is developed that provides the user with haptic feedback. The simulator is based on the use of model order reduction techniques, particularly Proper Generalized Decomposition methods. The just developed simulator presents some notable advancements with respect to existing works in the literature, such as the consideration of non-linear hyperelasticity for the constitutive modeling of soft tissues, an accurate description of contact between organs and momentum and energy conserving time integration schemes. Pancreas, liver, gall bladder, and duodenum are modeled in the simulator, thus providing with a very realistic and immersive perception to the user.

A Bayesian approach to selecting hyperelastic constitutive models of soft tissue

Abstract Hyperelastic constitutive models of soft tissue mechanical behavior are extensively used in applications like computer-aided surgery, injury modeling, etc. While numerous constitutive models have been proposed in the literature, an objective method is needed to select a parsimonious model that represents the experimental data well and has good predictive capability. This is an important problem given the large variability in the data inherent to soft tissue mechanical testing. In this work, we discuss a Bayesian approach to this problem based on Bayes factors. We propose a holistic framework for model selection, wherein we consider four different factors to reliably choose a parsimonious model from the candidate set of models. These are the qualitative fit of the model to the experimental data, evidence values, maximum likelihood values, and the landscape of the likelihood function. We consider three hyperelastic constitutive models that are widely used in soft tissue mechanics: Mooney–Rivlin, Ogden and exponential. Three sets of mechanical testing data from the literature for agarose hydrogel, bovine liver tissue, porcine brain tissue are used to calculate the model selection statistics. A nested sampling approach is used to evaluate the evidence integrals. In our results, we highlight the robustness of the proposed Bayesian approach to model selection compared to the likelihood ratio, and discuss the use of the four factors to draw a complete picture of the model selection problem.

Visco-hyperelastic constitutive modeling of soft tissues based on short and long-term internal variables.

BackgroundDifferential-type and integral-type formulations are two common approaches in modeling viscoelastic materials. A differential-type theory is often derived from a Helmholtz free energy function and is usually more suitable for the prediction of strain-rate dependent mechanical behavior during rapid loading, while an integral-type theory usually captures stress relaxation more efficiently than a differential-type theory. A modeling approach is needed to predict the viscoelastic responses during both rapid loading and relaxation phases.MethodsA constitutive modeling methodology based on the short and long-term internal variables was proposed in the present study in order to fully use the better features of the two types of theories. The short-term variables described the loading rate, while the long-term variables involving time constants characterized loading history and stress relaxation.ResultsThe application of the methodology was demonstrated with particular formulations for ligament and articular cartilage. Model parameters were calibrated for both tissues with experimental data from the literature. It was found that the proposed model could well predict a wide range of strain-rate dependent load responses during both loading and relaxation phases.ConclusionIntroducing different internal variables in terms of their time scales reduced the difficulties in the material characterization process and enabled the model to predict the experimental data more accurately, in particular at high strain-rates.

A transversely isotropic visco-hyperelastic constitutive model for soft tissues

A transversely isotropic visco-hyperelastic constitutive model is provided for soft tissues, which accounts for large deformations, high strain rates, and short-term memory effects. In the first part, a constitutive model for quasi-static deformations of soft tissues is presented, in which a soft tissue is simulated as a transversely isotropic hyperelastic material composed of a matrix and reinforcing fibers. The strain energy density function for the soft tissue is additively decomposed into two terms: a neo-Hookean function for the base matrix, and a polyconvex polynomial function of four invariants for the fibers. A comparison with existing experimental data for porcine brain tissues and bovine pericardium shows that this new model can well represent the quasi-static mechanical behavior of soft tissues. In the second part, a viscous potential is proposed to describe the rate-dependent short-term memory effects, resulting in a visco-hyperelastic constitutive model. This model is tested for a range of strain rates from 0.1 /s to 90 /s and for multiple loading scenarios based on available experimental data for porcine and human brain tissues. The model can be applied to other soft tissues by using different values of material and fitting parameters.

Transversely Isotropic Hyperelastic Constitutive Models for Plastic Thermoforming Simulation

For several years, modeling hyperelasticity has been focused on and leaded to a large choice of strain energy potential forms. Since then, many advances have been made in constitutive modeling of rubber like materials. These models are nowadays widely used in many applications like constitutive modeling of soft tissues in biomechanics problems or plastic thermoforming simulation. In this work, constitutive modeling of TPO sheets for thermoforming application is considered. Experimental measurements have shown that the material is transversely isotropic. To take into account this anisotropy, we implemented some new transversally isotropic hyperelastic constitutive models in Abaqus software with the help of user subroutines. Furthermore, different particular forms of the strain energy potential are investigated and their hyperelatic constants are fitted to the measurement data from tensile tests performed in different directions. Based on the results of these investigations, a transversely isotropic form of the energy potential derived from the Yeoh constitutive model is adopted and several tests are analyzed for validation purpose. The chosen model is a good compromise that achieves accurate predictions with limited amount of tests and limited identification efforts. Another key finding of this work is the influence of the anisotropy on the thermoformed parts.

Recent Advances in Computational Mechanics of the Human Knee Joint

Computational mechanics has been advanced in every area of orthopedic biomechanics. The objective of this paper is to provide a general review of the computational models used in the analysis of the mechanical function of the knee joint in different loading and pathological conditions. Major review articles published in related areas are summarized first. The constitutive models for soft tissues of the knee are briefly discussed to facilitate understanding the joint modeling. A detailed review of the tibiofemoral joint models is presented thereafter. The geometry reconstruction procedures as well as some critical issues in finite element modeling are also discussed. Computational modeling can be a reliable and effective method for the study of mechanical behavior of the knee joint, if the model is constructed correctly. Single-phase material models have been used to predict the instantaneous load response for the healthy knees and repaired joints, such as total and partial meniscectomies, ACL and PCL reconstructions, and joint replacements. Recently, poromechanical models accounting for fluid pressurization in soft tissues have been proposed to study the viscoelastic response of the healthy and impaired knee joints. While the constitutive modeling has been considerably advanced at the tissue level, many challenges still exist in applying a good material model to three-dimensional joint simulations. A complete model validation at the joint level seems impossible presently, because only simple data can be obtained experimentally. Therefore, model validation may be concentrated on the constitutive laws using multiple mechanical tests of the tissues. Extensive model verifications at the joint level are still crucial for the accuracy of the modeling.

Identifying a Minimal Rheological Configuration: A Tool for Effective and Efficient Constitutive Modeling of Soft Tissues

We describe a modeling methodology intended as a preliminary step in the identification of appropriate constitutive frameworks for the time-dependent response of biological tissues. The modeling approach comprises a customizable rheological network of viscous and elastic elements governed by user-defined 1D constitutive relationships. The model parameters are identified by iterative nonlinear optimization, minimizing the error between experimental and model-predicted structural (load-displacement) tissue response under a specific mode of deformation. We demonstrate the use of this methodology by determining the minimal rheological arrangement, constitutive relationships, and model parameters for the structural response of various soft tissues, including ex vivo perfused porcine liver in indentation, ex vivo porcine brain cortical tissue in indentation, and ex vivo human cervical tissue in unconfined compression. Our results indicate that the identified rheological configurations provide good agreement with experimental data, including multiple constant strain rate load/unload tests and stress relaxation tests. Our experience suggests that the described modeling framework is an efficient tool for exploring a wide array of constitutive relationships and rheological arrangements, which can subsequently serve as a basis for 3D constitutive model development and finite-element implementations. The proposed approach can also be employed as a self-contained tool to obtain simplified 1D phenomenological models of the structural response of biological tissue to single-axis manipulations for applications in haptic technologies.

A constitutive model for the warp-weft coupled non-linear behavior of knitted biomedical textiles

Knitted textiles have been used in medical applications due to their high flexibility and low tendency to fray. Their mechanics have, however, received limited attention. A constitutive model for soft tissue using a strain energy function was extended, by including shear and increasing the number and order of coefficients, to represent the non-linear warp-weft coupled mechanics of coarse textile knits under uniaxial tension. The constitutive relationship was implemented in a commercial finite element package. The model and its implementation were verified and validated for uniaxial tension and simple shear using patch tests and physical test data of uniaxial tensile tests of four very different knitted fabric structures. A genetic algorithm with step-wise increase in resolution and linear reduction in range of the search space was developed for the optimization of the fabric model coefficients. The numerically predicted stress–strain curves exhibited non-linear stiffening characteristic for fabrics. For three fabrics, the predicted mechanics correlated well with physical data, at least in one principal direction (warp or weft), and moderately in the other direction. The model exhibited limitations in approximating the linear elastic behavior of the fourth fabric. With proposals to address this limitation and to incorporate time-dependent changes in the fabric mechanics associated with tissue ingrowth, the constitutive model offers a tool for the design of tissue regenerative knit textile implants.

On constitutive modelling of porous neo-Hookean composites

In this paper a hyperelastic constitutive model is developed for neo-Hookean composites with aligned continuous cylindrical pores in the finite elasticity regime. Although the matrix is incompressible, the composite itself is compressible because of the existence of voids. For this compressible transversely isotropic material, the deformation gradient can be decomposed multiplicatively into three parts: an isochoric uniaxial deformation along the preferred direction of the material (which is identical to the direction of the cylindrical pores here); an equi-biaxial deformation on the transverse plane (the plane perpendicular to the preferred direction); and subsequent shear deformation (which includes “along-fibre” shear and transverse shear). Compared to the multiplicative decomposition used in our previous model for incompressible fibre reinforced composites [Guo, Z., Peng, X.Q., Moran, B., 2006, A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. J. Mech. Phys. Solids 54(9), 1952–1971], the equi-biaxial deformation is introduced to achieve the desired volume change. To estimate the strain energy function for this composite, a cylindrical composite element model is developed. Analytically exact strain distributions in the composite element model are derived for the isochoric uniaxial deformation along the preferred direction, the equi-biaxial deformation on the transverse plane, as well as the “along-fibre” shear deformation. The effective shear modulus from conventional composites theory based on the infinitesimal strain linear elasticity is extended to the present finite deformation regime to estimate the strain energy related to the transverse shear deformation, which leads to an explicit formula for the strain energy function of the composite under a general finite deformation state.

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.

A Theoretical Framework to Analyze Bend Testing of Soft Tissue

It has been hypothesized that repetitive flexural stresses contribute to the fatigue-induced failure of bioprosthetic heart valves. Although experimental apparatuses capable of measuring the bending properties of biomaterials have been described, a theoretical framework to analyze the resulting data is lacking. Given the large displacements present in these bending experiments and the nonlinear constitutive behavior of most biomaterials, such a formulation must be based on finite elasticity theory. We present such a theory in this work, which is capable of fitting bending moment versus radius of curvature experimental data to an arbitrary strain energy function. A simple finite element model was constructed to study the validity of the proposed method. To demonstrate the application of the proposed approach, bend testing data from the literature for gluteraldehyde-fixed bovine pericardium were fit to a nonlinear strain energy function, which showed good agreement to the data. This method may be used to integrate bending behavior in constitutive models for soft tissue.