Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

Nonlinear elastic analysis of blood vessels

Based on the theory of Green and Adkins [9], a strain energy function is proposed to describe the nonlinear mechanical behavior of arteries. The arterial tissue is assumed to be a nonlinear elastic, incompressible material with local triclinicity and transverse isotropy. Although the arterial tissue shows viscous phenomena, experimental results have indicated that viscosity is only a second-order effect as compared to the nonlinear elasticity of the tissue. The advantage of the formulation presented herein is that it is relatively simple and results in an accurate stress-strain relation that can be used readily for the study of wave propagations in the blood vessels. For nonlinear finite strain elasticity of the order two, ten elastic constants are needed to describe the material nonlinearity of the arterial tissue. Based on the orthogonal, transverse isotropies and the incompressibility conditions, ten constraint equations may be established and the elastic constants can be uniquely determined by correlating with the experimental results. An example of calculating these elastic constants is made by using the experimental data of Patel, et al. [14-17] for the intercoastal arteries in living dogs. The predicted mechanical behavior of canine arteries is quite satisfactory as compared to the experimental data except when the longitudinal and the circumferential stretches exceed 1.60. However, such a strain magnitude may not be expected in in-vivo arteries because of the constraints of peripheral connecting tissues. Otherwise, the strain energy function including higher order strain terms should be used.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

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.     

Quantifying nonlinear anisotropic elastic material properties of biological tissue by use of membrane inflation

Determination of material parameters for soft tissue frequently involves regression of material parameters for nonlinear, anisotropic constitutive models against experimental data from heterogeneous tests. Here, parameter estimation based on membrane inflation is considered. A four parameter nonlinear, anisotropic hyperelastic strain energy function was used to model the material, in which the parameters are cast in terms of key response features. The experiment was simulated using finite element (FE) analysis in order to predict the experimental measurements of pressure versus profile strain. Material parameter regression was automated using inverse FE analysis; parameter values were updated by use of both local and global techniques, and the ability of these techniques to efficiently converge to a best case was examined. This approach provides a framework in which additional experimental data, including surface strain measurements or local structural information, may be incorporated in order to quantify heterogeneous nonlinear material properties.     

A finite deformation theory for cartilage and other soft hydrated connective tissues--I. Equilibrium results

The determination of valid stress-strain relations for articular cartilage under finite deformation conditions is a prerequisite for constructing models for synovial joint lubrication. Under physiological conditions of high strain rates and/or high stresses in the joint, large strains occur in cartilage. A finite deformation theory valid for describing cartilage, as well as other soft hydrated connective tissues under large loads, has been developed. This theory is based on the choice of a specific Helmholtz energy function which satisfies the generalized Coleman-Noll (GCN0) condition and the Baker-Ericksen (B-E) inequalities established in finite elasticity theory. In addition, the finite deformation biphasic theory includes the effects of strain-dependent porosity and permeability. These nonlinear effects are essential for properly describing the biomechanical behavior of articular cartilage, even when strain rates are low and strains are infinitesimal. The finite deformation theory describes the large strain behavior of cartilage observed in one-dimensional confined compression experiments at equilibrium, and it reduces to the linear biphasic theory under infinitesimal strain and slow strain rate conditions. Using this theory, we have determined the material coefficients of both human and bovine articular cartilages under large strain conditions at equilibrium. The theory compares very well with experimental results.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

An orthotropic viscoelastic material model for passive myocardium: theory and algorithmic treatment

This contribution presents a novel constitutive model in order to simulate an orthotropic rate-dependent behaviour of the passive myocardium at finite strains. The motivation for the consideration of orthotropic viscous effects in a constitutive level lies in the disagreement between theoretical predictions and experimentally observed results. In view of experimental observations, the material is deemed as nearly incompressible, hyperelastic, orthotropic and viscous. The viscoelastic response is formulated by means of a rheological model consisting of a spring coupled with a Maxwell element in parallel. In this context, the isochoric free energy function is decomposed into elastic equilibrium and viscous non-equilibrium parts. The baseline elastic response is modelled by the orthotropic model of Holzapfel and Ogden [Holzapfel GA, Ogden RW. 2009. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans Roy Soc A Math Phys Eng Sci. 367:3445–3475]. The essential aspect of the proposed model is the account of distinct relaxation mechanisms for each orientation direction. To this end, the non-equilibrium response of the free energy function is constructed in the logarithmic strain space and additively decomposed into three anisotropic parts, denoting fibre, sheet and normal directions each accompanied by a distinct dissipation potential governing the evolution of viscous strains associated with each orientation direction. The evolution equations governing the viscous flow have an energy-activated nonlinear form. The energy storage in the Maxwell branches has a quadratic form leading to a linear stress–strain response in the logarithmic strain space. On the numerical side, the algorithmic aspects suitable for the implicit finite element method are discussed in a Lagrangian setting. The model shows excellent agreement compared to experimental data obtained from the literature. Furthermore, the finite element simulations of a heart cycle carried out with the proposed model show significant deviations in the strain field relative to the elastic solution.     

Nonlinear mechanical property of tracheal cartilage: a theoretical and experimental study

BACKGROUND: Despite being the stiffest airway of the bronchial tree, the trachea undergoes significant deformation due to intrathoracic pressure during breathing. The mechanical properties of the trachea affect the flow in the airway and may contribute to the biological function of the lung. METHOD: A Fung-type strain energy density function was used to investigate the nonlinear mechanical behavior of tracheal cartilage. A bending test on pig tracheal cartilage was performed and a mathematical model for analyzing the deformation of tracheal cartilage was developed. The constants included in the strain energy density function were determined by fitting the experimental data. RESULT: The experimental data show that tracheal cartilage is a nonlinear material displaying higher strength in compression than in tension. When the compression forces varied from -0.02 to -0.03N and from -0.03 to -0.04N, the deformation ratios were 11.03+/-2.18% and 7.27+/-1.59%, respectively. Both were much smaller than the deformation ratios (20.01+/-4.49%) under tension forces of 0.02 to 0.01N. The Fung-type strain energy density function can capture this nonlinear behavior very well, whilst the linear stress-strain relation cannot. It underestimates the stability of trachea by exaggerating the displacement in compression. This study may improve our understanding of the nonlinear behavior of tracheal cartilage and it may be useful for the future study on tracheal collapse behavior under physiological and pathological conditions.     

Multiaxial mechanical behavior of the porcine anterior lens capsule

The biomechanics of the lens capsule of the eye is important both in physiologic processes such as accommodation and clinical treatments such as cataract surgery. Although the lens capsule experiences multiaxial stresses in vivo, there have been no measurements of its multiaxial properties or possible regional heterogeneities. Rather all prior mechanical data have come from 1-D pressure–volume or uniaxial force-length tests. Here, we report a new experimental approach to study in situ the regional, multiaxial mechanical behavior of the lens capsule. Moreover, we report multiaxial data suggesting that the porcine anterior lens capsule exhibits a typical nonlinear pseudoelastic behavior over finite strains, that the in situ state is pre-stressed multiaxially, and that the meridional and circumferential directions are principal directions of strain, which is nearly equibiaxial at the pole but less so towards the equator. Such data are fundamental to much needed constitutive formulations.     

Continuum description of the Poisson׳s ratio of ligament and tendon under finite deformation

Ligaments and tendons undergo volume loss when stretched along the primary fiber axis, which is evident by the large, strain-dependent Poisson?s ratios measured during quasi-static tensile tests. Continuum constitutive models that have been used to describe ligament material behavior generally assume incompressibility, which does not reflect the volumetric material behavior seen experimentally. We developed a strain energy equation that describes large, strain dependent Poisson?s ratios and nonlinear, transversely isotropic behavior using a novel method to numerically enforce the desired volumetric behavior. The Cauchy stress and spatial elasticity tensors for this strain energy equation were derived and implemented in the FEBio finite element software (www.febio.org). As part of this objective, we derived the Cauchy stress and spatial elasticity tensors for a compressible transversely isotropic material, which to our knowledge have not appeared previously in the literature. Elastic simulations demonstrated that the model predicted the nonlinear, upwardly concave uniaxial stress–strain behavior while also predicting a strain-dependent Poisson?s ratio. Biphasic simulations of stress relaxation predicted a large outward fluid flux and substantial relaxation of the peak stress. Thus, the results of this study demonstrate that the viscoelastic behavior of ligaments and tendons can be predicted by modeling fluid movement when combined with a large Poisson?s ratio. Further, the constitutive framework provides the means for accurate simulations of ligament volumetric material behavior without the need to resort to micromechanical or homogenization methods, thus facilitating its use in large scale, whole joint models.     

Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus.

Accurate tissue stress predictions for the annulus fibrosus are essential for understanding the factors that cause or contribute to disc degeneration and mechanical failure. Current computational models used to predict in vivo disc stresses utilize material laws for annular tissue that are not rigorously validated against experimental data. Consequently, predictions of disc stress resulting from physical activities may be inaccurate and therefore unreliable as a basis for defining mechanical-biologic injury criteria. To address this need we present a model for the annulus as an isotropic ground substance reinforced with two families of collagen fibers, and an approach for determining the material constants by simultaneous consideration of multiple experimental data sets. Two strain energy functions for the annulus are proposed and used in the theory to derive the constitutive equations relating the stress to pure stretch deformations. These equations are applied to four distinct experimental protocols and the material constants are determined from a simultaneous, nonlinear regression analysis. Good agreement between theory and experiment is achieved when the invariants are included within multiple, separate exponentials in the strain energy function.     

A continuum theoretical model and finite elements simulation of bacterial flagellar filament phase transition

The Bacterial flagellar filament can undergo a polymorphic phase transition in response to both mechanical and chemical variations in vitro and in vivo environments. Under mechanical stimuli, such as viscous flow or forces induced by motor rotation, the filament changes its phase from left-handed normal (N) to right-handed semi-coiled (SC) via phase nucleation and growth. Our detailed mechanical analysis of existing experiments shows that both torque and bending moment contribute to the filament phase transition. In this paper, we establish a non-convex and non-local continuum model based on the Ginzburg-Landau theory to describe main characteristics of the filament phase transition such as new-phase nucleation, growth, propagation and the merging of neighboring interfaces. The finite element method (FEM) is adopted to simulate the phase transition under a displacement-controlled loading condition (rotation angle and bending deflection). We show that new-phase nucleation corresponds to the maximum torque and bending moment at the stuck end of the filament. The hysteresis loop in the loading and unloading curves indicates energy dissipation. When the new phase grows and propagates, torque and bending moment remain static. We also find that there is a drop in load when the two interfaces merge, indicating a concomitant reduction in the interfacial energy. Finally, the interface thickness is governed by the coefficients of the gradient of order parameters in the non-local interface energy. Our continuum theory and the finite element method provide a method to study the mechanical behavior of such biomaterials.     

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.     

Fully non-linear hyper-viscoelastic modeling of skeletal muscle in compression

Understanding the behavior of skeletal muscle is critical to implementing computational methods to study how the body responds to compressive loading. This work presents a novel approach to studying the fully nonlinear response of skeletal muscle in compression. Porcine muscle was compressed in both the longitudinal and transverse directions under five stress relaxation steps. Each step consisted of 5% engineering strain over 1 s followed by a relaxation period until equilibrium was reached at an observed change of 1 g/min. The resulting data were analyzed to identify the peak and equilibrium stresses as well as relaxation time for all samples. Additionally, a fully nonlinear strain energy density–based Prony series constitutive model was implemented and validated with independent constant rate compressive data. A nonlinear least squares optimization approach utilizing the Levenberg–Marquardt algorithm was implemented to fit model behavior to experimental data. The results suggested the time-dependent material response plays a key role in the anisotropy of skeletal muscle as increasing strain showed differences in peak stress and relaxation time (p < 0.05), but changes in equilibrium stress disappeared (p > 0.05). The optimizing procedure produced a single set of hyper-viscoelastic parameters which characterized compressive muscle behavior under stress relaxation conditions. The utilized constitutive model was the first orthotropic, fully nonlinear hyper-viscoelastic model of skeletal muscle in compression while maintaining agreement with constitutive physical boundaries. The model provided an excellent fit to experimental data and agreed well with the independent validation in the transverse direction.     

Nonlinear Anisotropic Elastic Properties of the Canine Aorta

A nonlinear theory of large elastic deformations of the aortic tissue has been developed. The wall tissue has been considered to be incompressible and curvilinearly orthotropic. The strain energy density function for the tissue is expressed as a polynomial in the circumferential and longitudinal Green-St. Venant strains. Limiting application to states of strains wherein the geometric axes are the principal axes and truncating the energy expression to include terms with highest degrees 2, 3, and 4, three expressions with 3, 7, and 12 constitutive constants are obtained. Results of application of these expressions to data from three series of in vitro and in vivo experiments involving 31 dogs have been presented. Whereas all the three expressions are found to be applicable to various degrees, the third-degree expression for the strain energy density function with seven constitutive constants is particularly recommended for general use.     

An elasto-viscoplastic model to describe the ratcheting behavior of articular cartilage

In the present work, a constitutive model for articular cartilage is proposed in finite elasto-viscoplasticity. For simplification, articular cartilage is supposed to be a typical composite composed of a soft basis and a fiber assembly. The stress tensor and free energy function are hence accordingly divided into two components. The high nonlinear stress-strain response is assumed to be mainly related to the fiber assembly and described by an exponential-type hypoelastic relation. Ratcheting is considered according to the viscoplasticity, the evolution rule of which is deduced from the dissipative inequality by the co-directionality hypotheses. Then, the capability of the proposed model is validated by comparing its predictions with related experimental observations. Results show that the ratcheting behavior and stress-strain hysteresis loops are reasonably captured by the proposed model.     

Adjoint multi-start-based estimation of cardiac hyperelastic material parameters using shear data

Cardiac muscle tissue during relaxation is commonly modeled as a hyperelastic material with strongly nonlinear and anisotropic stress response. Adapting the behavior of such a model to experimental or patient data gives rise to a parameter estimation problem which involves a significant number of parameters. Gradient-based optimization algorithms provide a way to solve such nonlinear parameter estimation problems with relatively few iterations, but require the gradient of the objective functional with respect to the model parameters. This gradient has traditionally been obtained using finite differences, the calculation of which scales linearly with the number of model parameters, and introduces a differencing error. By using an automatically derived adjoint equation, we are able to calculate this gradient more efficiently, and with minimal implementation effort. We test this adjoint framework on a least squares fitting problem involving data from simple shear tests on cardiac tissue samples. A second challenge which arises in gradient-based optimization is the dependency of the algorithm on a suitable initial guess. We show how a multi-start procedure can alleviate this dependency. Finally, we provide estimates for the material parameters of the Holzapfel and Ogden strain energy law using finite element models together with experimental shear data.