Strain rate-dependent viscohyperelastic constitutive modeling of bovine liver tissue

The mechanical response of most soft tissue is considered to be viscohyperelastic, making the development of accurate constitutive models a challenging task. In this article, we present a constitutive model for bovine liver tissue that utilizes a viscous dissipation potential, and use it to model the response of bovine liver tissue at strain rates ranging from 0.001 to 0.04 s⁻¹. On the material modeling front of this study, the free energy is assumed to depend on the right Cauchy–Green deformation tensor, whereas a separate rate-dependent viscous potential is posited to characterize viscoelasticity. This viscous dissipation component is a function of the time rate of change of the right Cauchy–Green deformation tensor. On the experimental front, no-slip uniaxial compression experiments are conducted on bovine liver tissue at various strain rates. A numerical correction approach is used to account for the no-slip edge conditions, and the constitutive model is fit to the resulting corrected stress–strain data. The complete derivation of the material model, its implementation in the finite element software package ABAQUS, and a validation study are presented in this article. The results show that bovine liver tissue exhibits a strong strain-rate dependence even at the low strain rates considered here and that the proposed constitutive model is able to accurately describe this response.     

Transversely isotropic properties of porcine liver tissue: experiments and constitutive modelling

Knowledge of the biomechanical properties of soft tissue, such as liver, is important in modelling computer aided surgical procedures. Liver tissue does not bear mechanical loads, and, in numerical simulation research, is typically assumed to be isotropic. Nevertheless, a typical biological soft tissue is anisotropic. In vitro uniaxial tension and compression experiments were conducted on porcine cylindrical and cubical liver tissue samples respectively assuming a simplistic architecture of liver tissue with its constituent lobule and connective tissues components. With the primary axis perpendicular to the cross sectional surface of samples, the tissue is stiffer with tensile or compressive force in the axial direction compared to that of the transverse direction. At 20% strain, about twice as much force is required to elongate a longitudinal tissue sample than that of a transverse sample. Results of the study suggest that liver tissue is transversely isotropic. A combined strain energy based constitutive equation for transversely isotropic material is proposed. The improved capability of this equation to model the experimental data compared to its previously disclosed isotropic version suggests that the assumption on the fourth invariant in the constitutive equation is probably correct and that anisotropy properties of liver tissue should be considered in surgical simulation.     

A mesostructurally-based anisotropic continuum model for biological soft tissues--decoupled invariant formulation

Characterising and modelling the mechanical behaviour of biological soft tissues is an essential step in the development of predictive computational models to assist research for a wide range of applications in medicine, biology, tissue engineering, pharmaceutics, consumer goods, cosmetics, transport or military. It is therefore critical to develop constitutive models that can capture particular rheological mechanisms operating at specific length scales so that these models are adapted for their intended applications. Here, a novel mesoscopically-based decoupled invariant-based continuum constitutive framework for transversely isotropic and orthotropic biological soft tissues is developed. A notable feature of the formulation is the full decoupling of shear interactions. The constitutive model is based on a combination of the framework proposed by Lu and Zhang [Lu, J., Zhang, L., 2005. Physically motivated invariant formulation for transversely isotropic hyperelasticity. International Journal of Solids and Structures 42, 6015-6031] and the entropic mechanics of tropocollagen molecules and collagen assemblies. One of the key aspects of the formulation is to use physically-based nanoscopic quantities that could be extracted from experiments and/or atomistic/molecular dynamics simulations to inform the macroscopic constitutive behaviour. This effectively couples the material properties at different levels of the multi-scale hierarchical structure of collagenous tissues. The orthotropic hyperelastic model was shown to reproduce very well the experimental multi-axial properties of rabbit skin. A new insight into the shear response of a skin sample subjected to a simulated indentation test was obtained using numerical direct sensitivity analyses.     

On modelling and analysis of healthy and pathological human mitral valves: Two case studies

Biomechanical data and related constitutive modelling of the mitral apparatus served as a basis for finite element analyses to better understand the physiology of mitral valves in health and disease. Human anterior and posterior leaflets and chordae tendinae from an elderly heart showing no disease and a hypertrophic obstructive cardiomyopathic heart (HOCM) were mechanically tested by means of uniaxial cyclic extension tests under quasi-static conditions. Experimental data for the leaflets and the chordae tendinae showed highly nonlinear mechanical behaviours and the leaflets were anisotropic. The mitral valve from the HOCM heart exhibited a significantly softer behaviour than the valve from the healthy one. A comparison with porcine data was included because many previous mitral modelling studies have been based on porcine data. Some differences in mechanical response were observed. Material parameters for hyperelastic, transversely isotropic constitutive laws were determined. The experimental data and the related model parameters were used in two finite element studies to investigate the effects of the material properties on the mitral valve response during systole. The analyses showed that during systole the mitral valve from the HOCM heart bulged into the left atrium by taking on the shape of a balloon, whereas the anterior leaflet of the healthy valve remained in the left ventricle.     

In vitro technique in estimation of passive mechanical properties of bovine heart part II. Constitutive relation and finite element analysis

A pseudo-elastic constitutive equation describing the mechanical properties of bovine myocardium was developed. The myocardium was modeled as a hyperelastic transversely isotropic material with a minimum viscoelastic loses. The material parameters for the proposed constitutive equations were determined using GA regression technique. In this work, the development of a constitutive equation based on principal stretch ratios is explained. The predictive capability of proposed model was compared against the experimental data obtained from part one. Finally, the constitutive equations were implemented into a commercial finite element program and the results of the mathematical model and FEM were compared with the experimental data.     

Constitutive formulation and analysis of heel pad tissues mechanics

This paper presents a visco-hyperelastic constitutive model developed to describe the biomechanical response of heel pad tissues. The model takes into account the typical features of the mechanical response such as large displacement, strain phenomena, and non-linear elasticity together with time-dependent effects. The constitutive model was formulated, starting from the analysis of the complex structural and micro-structural configuration of the tissues, to evaluate the relationship between tissue histology and mechanical properties. To define the constitutive model, experimental data from mechanical tests were analyzed. To obtain information about the mechanical response of the tissue so that the constitutive parameters could be established, data from both in vitro and in vivo tests were investigated. Specifically, the first evaluation of the constitutive parameters was performed by a coupled deterministic and stochastic optimization method, accounting for data from in vitro tests. The comparison of constitutive model results and experimental data confirmed the model's capability to describe the compression behaviour of the heel pad tissues, regarding both constant strain rate and stress relaxation tests. Based on the data from additional experimental tests, some of the constitutive parameters were modified in order to interpret the in vivo mechanical response of the heel pad tissues. This approach made it possible to interpret the actual mechanical function of the tissues.     

A coupled electromechanical model for the excitation-dependent contraction of skeletal muscle

This work deals with the development and implementation of an electromechanical skeletal muscle model. To this end, a recently published hyperelastic constitutive muscle model with transversely isotropic characteristics, see Ehret et al. (2011), has been weakly coupled with Ohm’s law describing the electric current. In contrast to the traditional way of active muscle modelling, this model is rooted on a non-additive decomposition of the active and passive components. The performance of the proposed modelling approach is demonstrated by the use of three-dimensional illustrative boundary-value problems that include electromechanical analysis on tissue strips. Further, simulations on the biceps brachii muscle document the applicability of the model to realistic muscle geometries.     

Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons

The aim of this work is to investigate the instantaneous mechanical response of tendons by the use of an anisotropic elasto-damage constitutive model. This study addresses the analysis of the mechanical behaviour of healthy tendons during physiological loading and to degeneration phenomena. These are correlated with aging or traumatic events such as chronic or acute overloading during sporting activities. Histo-morphometric considerations suggest the adoption of a transversally isotropic constitutive model that describes the anisotropy of the material. The non-linearity of its overall mechanical response is taken into account by using a hyperelastic approach and also evaluates softening behaviour related to damage phenomena. The values of the parameters adopted within the analytical model are estimated both for human tendons previously subjected to cyclical loading and for specimens not subjected to cyclical loading. The results obtained by adopting this analytical model are compared with the experimental data in order to evaluate the capability of the model to describe the mechanical response of the tissue.     

A Nonlinear Constitutive Model for Stress Relaxation in Ligaments and Tendons

A novel constitutive model that describes stress relaxation in transversely isotropic soft collagenous tissues such as ligaments and tendons is presented. The model is formulated within the nonlinear integral representation framework proposed by Pipkin and Rogers (J. Mech. Phys. Solids. 16:59?C72, 1968). It represents a departure from existing models in biomechanics since it describes not only the strain dependent stress relaxation behavior of collagenous tissues but also their finite strains and transverse isotropy. Axial stress?Cstretch data and stress relaxation data at different axial stretches are collected on rat tail tendon fascicles in order to compute the model parameters. Toward this end, the rat tail tendon fascicles are assumed to be incompressible and undergo an isochoric axisymmetric deformation. A comparison with the experimental data proves that, unlike the quasi-linear viscoelastic model (Fung, Biomechanics: Mechanics of Living Tissues. Springer, New York, 1993) the constitutive law can capture the observed nonlinearities in the stress relaxation response of rat tail tendon fascicles.     

人体下颌骨的弹性

本文应用应变电测技术研究了人体下颌骨的弹性。结果表明,其力学特性与其位置,方问密切关系,说明人体下颌骨是一种各向异性的非均质的复合材料。     

Noninvasive Determination of Ligament Strain with Deformable Image Registration

Ligament function and propensity for injury are directly related to regional stresses and strains. However, noninvasive techniques for measurement of strain are currently limited. This study validated the use of Hyperelastic Warping, a deformable image registration technique, for noninvasive strain measurement in the human medial collateral ligament using direct comparisons with optical measurements. Hyperelastic Warping determines the deformation map that aligns consecutive images of a deforming material, allowing calculation of strain. Diffeomorphic deformations are ensured by representing the deformable image as a hyperelastic material. Ten cadaveric knees were subjected to six loading scenarios each. Tissue deformation was documented with magnetic resonance imaging (MRI) and video-based experimental measurements. MRI datasets were analyzed using Hyperelastic Warping, representing the medial collateral ligament (MCL) with a hexahedral finite element (FE) model projected to a manually segmented ligament surface. The material behavior was transversely isotropic hyperelastic. Warping predictions of fiber stretch were strongly correlated with experimentally measured strains (R ² = 0.81). Both sets of measurements were in agreement with previous ex vivo studies. Warping predictions of fiber stretch were insensitive to bulk:shear modulus ratio, fiber stiffness, and shear modulus in the range of +2.5SD to −1.0SD. Correlations degraded when the shear modulus was decreased to 2.5SD below the mean (R ² = 0.56), and when an isotropic constitutive model was substituted for the transversely isotropic model (R ² = 0.65). MCL strains in the transitional region near the joint line, where the material behavior and material symmetry are more complex, showed the most sensitivity to changes in shear modulus. These results demonstrate that Hyperelastic Warping requires the use of a constitutive model that reflects the material symmetry, but not subject-specific material properties for accurate strain predictions for this application. Hyperelastic Warping represents a powerful technique for noninvasive strain measurement of musculoskeletal tissues and has many advantages over other image-based strain measurement techniques.     

A New Method to Determine Rate-dependent Material Parameters of Corneal Extracellular Matrix

The cornea protects internal ocular contents against external insults while refracting and transmitting the incoming light onto the lens. The biomechanical properties of the cornea are largely governed by the composition and structure of the stromal layer which is an extracellular matrix composed of collagen fibrils embedded in a hydrated soft matrix. The mechanical behavior of the corneal stroma has commonly been characterized using uniaxial tensile tests and inflation experiments. In the present study, unconfined compression experiments were used to investigate the influence of loading rates on compressive behavior of nineteen porcine corneal specimens. The experiments were performed at ramp displacement rates 0.15 μm/s (eight samples), 0.5 μm/s (six samples), and 1.0 μm/s (five samples). For all tests, a maximum compressive strain of 50% (five strain increments of 4% followed by three strain increments of 10%) was selected. The experimental data was analyzed by a transversely isotropic biphasic model and material parameters, i.e., the in-plane Young’s modulus, the out-of-plane Young’s modulus, and the permeability coefficient were calculated. It was observed that while the permeability coefficient decreased exponentially with increasing compressive strain, the in-plane and out-of-plane Young’s moduli increased exponentially with increasing strain. Furthermore, it was found that the equilibrium stress was almost rate independent.     

Viscoelastic behavior of discrete human collagen fibrils

Whole tendon and fibril bundles display viscoelastic behavior, but to the best of our knowledge this property has not been directly measured in single human tendon fibrils. In the present work an atomic force microscopy (AFM) approach was used for tensile testing of two human patellar tendon fibrils. Fibrils were obtained from intact human fascicles, without any pre-treatment besides frozen storage. In the dry state a single isolated fibril was anchored to a substrate using epoxy glue, and the end of the fibril was glued on to an AFM cantilever for tensile testing. In phosphate buffered saline, cyclic testing was performed in the pre-yield region at different strain rates, and the elastic response was determined by a stepwise stress relaxation test. The elastic stress-strain response corresponded to a second-order polynomial fit, while the viscous response showed a linear dependence on the strain. The slope of the viscous response showed a strain rate dependence corresponding to a power function of powers 0.242 and 0.168 for the two patellar tendon fibrils, respectively. In conclusion, the present work provides direct evidence of viscoelastic behavior at the single fibril level, which has not been previously measured.     

基于可行方向法的脂肪组织本构模型参数反求优化

目的研究脂肪组织在中等应变率下本构模型及其参数反求。方法基于脂肪组织力学性能实验,通过有限元方法重构脂肪组织压缩实验,并对常见表征脂肪组织的本构模型进行参数筛选。结合最优化方法中的可行方向法(method of feasible direction,MFD),进行中应变率下脂肪组织本构模型相关参数的反求。结果中应变率(260 s~(-1))下黏弹性本构模型相比Ogden本构模型更适合表征脂肪组织的力学响应,并反求得到适用于仿真的本构模型参数。结论中等应变率下黏弹性本构模型更适合表征脂肪组织力学响应。研究结果为汽车碰撞有限元仿真中探究人体脂肪组织对人体损伤的影响提供参考。     

Experimental and numerical analyses of indentation in finite-sized isotropic and anisotropic rubber-like materials

Indentation tests perpendicular to the major plane of a material have been proposed as a means to index some of its in-plane mechanical properties. We showed the feasibility of such tests in myocardial tissue and established its theoretical basis with a formulation of small indentation superimposed on a finitely stretched half-space of isotropic materials. The purpose of this study is to better understand the mechanics of indentation with respect to the relative effects of indenter size, indentation depth, and specimen size, as well as the effects of material properties. Accordingly, we performed indentation tests on slabs of silicone rubber fabricated with both isotropic, as well as transversely isotropic, material symmetry. We performed indentation tests in different thickness specimens with varying sizes of indenters, amounts of indentation, and amounts of in-plane stretch. We used finite-element method simulations to supplement the experimental data. The combined experimental and modeling data provide the following useful guidelines for future indentation tests in finite-size specimens: (i) to avoid artifacts from boundary effects, the in-plane specimen dimensions should be at least 15 times the indenter size; (ii) to avoid nonlinearities associated with finite-thickness effects, the thickness-to-radius ratio should be >10 and thickness to indentation depth ratio should be >5; and (iii) we also showed that combined indentation and inplane stretch could distinguish the stiffer direction of a, transversely isotropic material.     

Investigation on the load-displacement curves of a human healthy heel pad: In vivo compression data compared to numerical results

The aims of the present work were to build a 3D subject-specific heel pad model based on the anatomy revealed by MR imaging of a subject's heel pad, and to compare the load-displacement responses obtained from this model with those obtained from a compression device used on the subject's heel pad. A 30 year-old European healthy female (mass=54kg, height=165cm) was enrolled in this study. Her left foot underwent both MRI and compression tests. A numerical model of the heel region was developed based on a 3D CAD solid model obtained by MR images. The calcaneal fat pad tissue was described with a visco-hyperelastic model, while a fiber-reinforced hyperelastic model was formulated for the skin. Numerical analyses were performed to interpret the mechanical response of heel tissues. Different loading conditions were assumed according to experimental tests. The heel tissues showed a non-linear visco-elastic behavior and the load-displacement curves followed a characteristic hysteresis form. The energy dissipation ratios measured by experimental tests (0.25±0.02 at low strain rate and 0.26±0.03 at high strain rate) were comparable with those evaluated by finite element analyses (0.23±0.01 at low strain rate and 0.25±0.01 at high strain rate). The validity and efficacy of the investigation performed was confirmed by the interpretation of the mechanical response of the heel tissues under different strain rates. The mean absolute percentage error between experimental data and model results was 0.39% at low strain rate and 0.28% at high strain rate.     

A stochastic visco-hyperelastic model of human placenta tissue for finite element crash simulations

Placental abruption is the most common cause of fetal deaths in motor-vehicle crashes, but studies on the mechanical properties of human placenta are rare. This study presents a new method of developing a stochastic visco-hyperelastic material model of human placenta tissue using a combination of uniaxial tensile testing, specimen-specific finite element (FE) modeling, and stochastic optimization techniques. In our previous study, uniaxial tensile tests of 21 placenta specimens have been performed using a strain rate of 12/s. In this study, additional uniaxial tensile tests were performed using strain rates of 1/s and 0.1/s on 25 placenta specimens. Response corridors for the three loading rates were developed based on the normalized data achieved by test reconstructions of each specimen using specimen-specific FE models. Material parameters of a visco-hyperelastic model and their associated standard deviations were tuned to match both the means and standard deviations of all three response corridors using a stochastic optimization method. The results show a very good agreement between the tested and simulated response corridors, indicating that stochastic analysis can improve estimation of variability in material model parameters. The proposed method can be applied to develop stochastic material models of other biological soft tissues.     

Constitutive Modeling of Rate-Dependent Stress–Strain Behavior of Human Liver in Blunt Impact Loading

An understanding of the mechanical deformation behavior of the liver under high strain rate loading conditions could aid in the development of vehicle safety measures to reduce the occurrence of blunt liver injury. The purpose of this study was to develop a constitutive model of the stress–strain behavior of the human liver in blunt impact loading. Experimental stress and strain data was obtained from impact tests of 12 unembalmed human livers using a drop tower technique. A constitutive model previously developed for finite strain behavior of amorphous polymers was adapted to model the observed liver behavior. The elements of the model include a nonlinear spring in parallel with a linear spring and nonlinear dashpot. The model captures three features of liver stress–strain behavior in impact loading: (1) relatively stiff initial modulus, (2) rate-dependent yield or rollover to viscous “flow” behavior, and (3) strain hardening at large strains. Six material properties were used to define the constitutive model. This study represents a novel application of polymer mechanics concepts to understand the rate-dependent large strain behavior of human liver tissue under high strain rate loading. Applications of this research include finite element simulations of injury-producing liver or abdominal impact events.

Constitutive modelling of inelastic behaviour of cortical bone

A visco-elasto-plastic constitutive model is formulated for investigating the mechanics of cortical bone tissue, accounting for an anisotropic configuration and post-elastic and time-dependent phenomena. The constitutive model is developed with reference to experimental data obtained from literature on the behaviour of cortical bone taken from multiple samples. Regarding the constitutive model, a specific procedure based on a coupled deterministic and stochastic method is applied in order to determine the values of the constitutive parameters with regard to human samples. The procedure entails processing of data deduced from mechanical tests to achieve relationships between permanent and total strain, elastic modulus and strain rate, and creep elastic modulus and time. Numerical results obtained by using a finite element model are compared with tensile experimental data on cortical bone including the post-elastic range and creep phenomena. The model shows an excellent capability to describe the tensile behaviour of the cortical bone for the specific mechanical condition analysed.