Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.     

Biomechanical properties of the pelvic floor muscles of continent and incontinent women using an inverse finite element analysis

Pelvic disorders can be associated with changes in the biomechanical properties in the muscle, ligaments and/or connective tissue form fascia and ligaments. In this sense, the study of their mechanical behavior is important to understand the structure and function of these biological soft tissues. The aim of this study was to establish the biomechanical properties of the pelvic floor muscles of continent and incontinent women, using an inverse finite element analysis (FEA). The numerical models, including the pubovisceral muscle and pelvic bones were built from magnetic resonance (MR) images acquired at rest. The numerical simulation of Valsalva maneuver was based on the finite element method and the material constants were determined for different constitutive models (Neo-Hookean, Mooney-Rivlin and Yeoh) using an iterative process. The material constants (MPa) for Neo-Hookean (c₁) were 0.039 ± 0.022 and 0.024 ± 0.004 for continent vs. incontinent women. For Mooney-Rivlin (c₁) the values obtained were 0.026 ± 0.010 vs. 0.016 ± 0.003, and for Yeoh (c₁) the values obtained were 0.031 ± 0.023 vs. 0.016 ± 0.002, (p < 0.05). Muscle displacements obtained in the numerical simulations of Valsalva maneuver were compared with the muscle displacements obtained through additional dynamic MRI. Incontinent women presented a higher antero-posterior displacement than the continent women. The results were also similar between MRI and numerical simulations (40.27% vs. 42.17% for Neo-Hookean, 39.87% for Mooney-Rivlin and 41.61% for Yeoh). Using an inverse FEA coupled with MR images allowed to obtain the in vivo biomechanical properties of the pelvic floor muscles, leading to a relationship between them for the continent and incontinent women in a non-invasive manner.     

The Polyvinyl Alcohol Sponge Model Implantation

Wound healing is a complicated, multistep process involving many cell types, growth factors and compounds^1-3. Because of this complexity, wound healing studies are most comprehensive when carried out in vivo. There are many in vivo models available to study acute wound healing, including incisional, excisional, dead space, and burns. Dead space models are artificial, porous implants which are used to study tissue formation and the effects of substances on the wound. Some of the commonly used dead space models include polyvinyl alcohol (PVA) sponges, steel wire mesh cylinders, expanded polytetrafluoroethylene (ePTFE) material, and the Cellstick^1,2.Each dead space model has its own limitations based on its material''s composition and implantation methods. The steel wire mesh cylinder model has a lag phase of infiltration after implantation and requires a long amount of time before granulation tissue formation begins¹. Later stages of wound healing are best analyzed using the ePTFE model^1,4. The Cellstick is a cellulose sponge inside a silicon tube model which is typically used for studying human surgery wounds and wound fluid². The PVA sponge is limited to acute studies because with time it begins to provoke a foreign body response which causes a giant cell reaction in the animal⁵. Unlike other materials, PVA sponges are easy to insert and remove, made of inert and non-biodegradable materials and yet are soft enough to be sectioned for histological analysis^2,5.In wound healing the PVA sponge is very useful for analyzing granulation tissue formation, collagen deposition, wound fluid composition, and the effects of substances on the healing process^1,2,5. In addition to its use in studying a wide array of attributes of wound healing, the PVA sponge has also been used in many other types of studies. It has been utilized to investigate tumor angiogenesis, drug delivery and stem cell survival and engraftment^1,2,6,7. With its great alterability, prior extensive use, and reproducible results, the PVA sponge is an ideal model for many studies^1,2.Here, we will describe the preparation, implantation and retrieval of PVA sponge disks (Figure 1) in a mouse model of wound healing.     

Stress distributions and material properties determined in articular cartilage from MRI-based finite strains

The noninvasive measurement of finite strains in biomaterials and tissues by magnetic resonance imaging (MRI) enables mathematical estimates of stress distributions and material properties. Such methods allow for non-contact and patient-specific modeling in a manner not possible with traditional mechanical testing or finite element techniques. Here, we employed three constitutive (i.e. linear Hookean, and nonlinear Neo-Hookean and Mooney-Rivlin) relations with known loading conditions and MRI-based finite strains to estimate stress patterns and material properties in the articular cartilage of tibiofemoral joints. Displacement-encoded MRI was used to determine two-dimensional finite strains in juvenile porcine joints, and an iterative technique estimated stress distributions and material properties with defined constitutive relations. Stress distributions were consistent across all relations, although the stress magnitudes varied. Material properties for femoral and tibial cartilage were found to be consistent with those reported in literature. Further, the stress estimates from Hookean and Neo-Hookean, but not Mooney-Rivlin, relations agreed with finite element-based simulations. A nonlinear Neo-Hookean relation provided the most appropriate model for the characterization of complex and spatially dependent stresses using two-dimensional MRI-based finite strain. These results demonstrate the feasibility of a new and computationally efficient technique incorporating MRI-based deformation with mathematical modeling to non-invasively evaluate the mechanical behavior of biological tissues and materials.     

A novel approach for extracting viscoelastic parameters of living cells through combination of inverse finite element simulation and Atomic Force Microscopy

Dynamic mechanical behaviour of living cells has been described by viscoelasticity. However, quantitation of the viscoelastic parameters for living cells is far from sophisticated. In this paper, combining inverse finite element (FE) simulation with Atomic Force Microscope characterization, we attempt to develop a new method to evaluate and acquire trustworthy viscoelastic index of living cells. First, influence of the experiment parameters on stress relaxation process is assessed using FE simulation. As suggested by the simulations, cell height has negligible impact on shape of the force–time curve, i.e. the characteristic relaxation time; and the effect originates from substrate can be totally eliminated when stiff substrate (Young’s modulus larger than 3 GPa) is used. Then, so as to develop an effective optimization strategy for the inverse FE simulation, the parameters sensitivity evaluation is performed for Young’s modulus, Poisson’s ratio, and characteristic relaxation time. With the experiment data obtained through typical stress relaxation measurement, viscoelastic parameters are extracted through the inverse FE simulation by comparing the simulation results and experimental measurements. Finally, reliability of the acquired mechanical parameters is verified with different load experiments performed on the same cell.     

The choice of a constitutive formulation for modeling limb flexion-induced deformations and stresses in the human femoropopliteal arteries of different ages

Open and endovascular treatments for peripheral arterial disease are notorious for high failure rates. Severe mechanical deformations experienced by the femoropopliteal artery (FPA) during limb flexion and interactions between the artery and repair materials play important roles and may contribute to poor clinical outcomes. Computational modeling can help optimize FPA repair, but these simulations heavily depend on the choice of constitutive model describing the arterial behavior. In this study finite element model of the FPA in the standing (straight) and gardening (acutely bent) postures was built using computed tomography data, longitudinal pre-stretch and biaxially determined mechanical properties. Springs and dashpots were used to represent surrounding tissue forces associated with limb flexion-induced deformations. These forces were then used with age-specific longitudinal pre-stretch and mechanical properties to obtain deformed FPA configurations for seven age groups. Four commonly used invariant-based constitutive models were compared to determine the accuracy of capturing deformations and stresses in each age group. The four-fiber FPA model most accurately portrayed arterial behavior in all ages, but in subjects younger than 40 years, the performance of all constitutive formulations was similar. In older subjects, Demiray (Delfino) and classic two-fiber Holzapfel–Gasser–Ogden formulations were better than the Neo-Hookean model for predicting deformations due to limb flexion, but both significantly overestimated principal stresses compared to the FPA or Neo-Hookean models.     

Synthesis and Characterization of PVA-HA-Silk Composite Hydrogel by Orthogonal Experiment

PVA-HA-Silk composite hydrogel was synthesized with polyvinyl alcohol (PVA),nano-hydroxyapatite (HA) and natural silk by using the method of repeated freezing and thawing.A series of tests were performed to study water content,stress relaxation behavior,elastic modulus,and creep characteristics of PVA-HA-Silk composite hydrogel.Orthogonal experimental design method was used to analyze the influence degree of PVA,HA and silk (three kinds of raw materials) on mechanical properties and water content of the PVA-HA-Silk composite hydrogel to select the best material ratio according to their overall performance.The results demonstrate that the mass percentage of PVA has the greatest impact on the water content,followed by HA and silk.Compression stress-strain variation of PVA-HA-Silk composite hydrogel presents a nonlinear relationship,which proves that it is a typical viscoelastic material.Comparing the mechanical properties of 16 formulas,the formula of PVA-HA-silk composite hydrogel with mass percentage of PVA 15％,HA 2.0％ and silk 1.0％ is the best.     

Mechanical responses of the periodontal ligament based on an exponential hyperelastic model: a combined experimental and finite element method

The V–W exponential hyperelastic model is adopted to describe the instantaneous elastic response of the periodontal ligament (PDL). The general theoretical framework of constitutive modeling is described based on nonlinear continuum mechanics, and the elasticity tensor used to develop UMAT subroutine is formulated. Nanoindentation experiment is performed to characterize mechanical properties of an adult pig PDL specimen. Then the experiment is simulated by using the finite element (FE) analysis. Meanwhile, the optimized material parameters are identified by the inverse FE method. The good agreement between the simulated results and experimental data demonstrates that the V–W model is capable of describing the mechanical behavior of the PDL. Therefore, the model and its implementation into FE code are validated. By using the model, we simulate the tooth movement under orthodontic loading to predict the mechanical responses of the PDL. The results show that local concentrations of stress and strain in the PDL are found.     

Specimen-specific predictions of contact stress under physiological loading in the human hip: validation and sensitivity studies

Hip osteoarthritis may be initiated and advanced by abnormal cartilage contact mechanics, and finite element (FE) modeling provides an approach with the potential to allow the study of this process. Previous FE models of the human hip have been limited by single specimen validation and the use of quasi-linear or linear elastic constitutive models of articular cartilage. The effects of the latter assumptions on model predictions are unknown, partially because data for the instantaneous behavior of healthy human hip cartilage are unavailable. The aims of this study were to develop and validate a series of specimen-specific FE models, to characterize the regional instantaneous response of healthy human hip cartilage in compression, and to assess the effects of material nonlinearity, inhomogeneity and specimen-specific material coefficients on FE predictions of cartilage contact stress and contact area. Five cadaveric specimens underwent experimental loading, cartilage material characterization and specimen-specific FE modeling. Cartilage in the FE models was represented by average neo-Hookean, average Veronda Westmann and specimen- and region-specific Veronda Westmann hyperelastic constitutive models. Experimental measurements and FE predictions compared well for all three cartilage representations, which was reflected in average RMS errors in contact stress of less than 25 %. The instantaneous material behavior of healthy human hip cartilage varied spatially, with stiffer acetabular cartilage than femoral cartilage and stiffer cartilage in lateral regions than in medial regions. The Veronda Westmann constitutive model with average material coefficients accurately predicted peak contact stress, average contact stress, contact area and contact patterns. The use of subject- and region-specific material coefficients did not increase the accuracy of FE model predictions. The neo-Hookean constitutive model underpredicted peak contact stress in areas of high stress. The results of this study support the use of average cartilage material coefficients in predictions of cartilage contact stress and contact area in the normal hip. The regional characterization of cartilage material behavior provides the necessary inputs for future computational studies, to investigate other mechanical parameters that may be correlated with OA and cartilage damage in the human hip. In the future, the results of this study can be applied to subject-specific models to better understand how abnormal hip contact stress and contact area contribute to OA.     

Determination of the axial and circumferential mechanical properties of the skin tissue using experimental testing and constitutive modeling

The skin, being a multi-layered material, is responsible for protecting the human body from the mechanical, bacterial, and viral insults. The skin tissue may display different mechanical properties according to the anatomical locations of a body. However, these mechanical properties in different anatomical regions and at different loading directions (axial and circumferential) of the mice body to date have not been determined. In this study, the axial and circumferential loads were imposed on the mice skin samples. The elastic modulus and maximum stress of the skin tissues were measured before the failure occurred. The nonlinear mechanical behavior of the skin tissues was also computationally investigated through a suitable constitutive equation. Hyperelastic material model was calibrated using the experimental data. Regardless of the anatomic locations of the mice body, the results revealed significantly different mechanical properties in the axial and circumferential directions and, consequently, the mice skin tissue behaves like a pure anisotropic material. The highest elastic modulus was observed in the back skin under the circumferential direction (6.67 MPa), while the lowest one was seen in the abdomen skin under circumferential loading (0.80 MPa). The Ogden material model was narrowly captured the nonlinear mechanical response of the skin at different loading directions. The results help to understand the isotropic/anisotropic mechanical behavior of the skin tissue at different anatomical locations. They also have implications for a diversity of disciplines, i.e., dermatology, cosmetics industry, clinical decision making, and clinical intervention.     

Modified Bilston nonlinear viscoelastic model for finite element head injury studies

This paper proposes a modified nonlinear viscoelastic Bilston model (Bilston et al., 2001, Biorheol., 38, pp. 335-345). for the modeling of brain tissue constitutive properties. The modified model can be readily implemented in a commercial explicit finite element (FE) code, PamCrash. Critical parameters of the model have been determined through a series of rheological tests on porcine brain tissue samples and the time-temperature superposition (TTS) principle has been used to extend the frequency to a high region. Simulations by using PamCrash are compared with the test results. Through the use of the TTS principle, the mechanical and rheological behavior at high frequencies up to 10(4) rads may be obtained. This is important because the properties of the brain tissue at high frequencies and impact rates are especially relevant to studies of traumatic head injury. The averaged dynamic modulus ranges from 130 Pa to 1500 Pa and loss modulus ranges from 35 Pa to 800 Pa in the frequency regime studied (0.01 rads to 3700 rads). The errors between theoretical predictions and averaged relaxation test results are within 20% for strains up to 20%. The FEM simulation results are in good agreement with experimental results. The proposed model will be especially useful for application to FE analysis of the head under impact loads. More realistic analysis of head injury can be carried out by incorporating the nonlinear viscoelastic constitutive law for brain tissue into a commercial FE code.     

Digital image correlation and finite element modelling as a method to determine mechanical properties of human soft tissue in vivo

The mechanical properties of human soft tissue are crucial for impact biomechanics, rehabilitation engineering, and surgical simulation. Validation of these constitutive models using human data remains challenging and often requires the use of non-invasive imaging and inverse finite element (FE) analysis. Post-processing data from imaging methods such as tagged magnetic resonance imaging (MRI) can be challenging. Digital image correlation (DIC), however, is a relatively straightforward imaging method. DIC has been used in the past to study the planar and superficial properties of soft tissue and excised soft tissue layers. However, DIC has not been used to non-invasive study of the bulk properties of human soft tissue in vivo. Thus, the goal of this study was to assess the use of DIC in combination with FE modelling to determine the bulk material properties of human soft tissue. Indentation experiments were performed on a silicone gel soft tissue phantom. A two camera DIC setup was then used to record the 3D surface deformation. The experiment was then simulated using a FE model. The gel was modelled as Neo-Hookean hyperelastic, and the material parameters were determined by minimising the error between the experimental and FE data. The iterative FE analysis determined material parameters (μ=1.80 kPa, K=2999 kPa) that were in close agreement with parameters derived independently from regression to uniaxial compression tests (μ=1.71 kPa, K=2857 kPa). Furthermore the FE model was capable of reproducing the experimental indentor force as well as the surface deformation found (R²=0.81). It was therefore concluded that a two camera DIC configuration combined with FE modelling can be used to determine the bulk mechanical properties of materials that can be represented using hyperelastic Neo-Hookean constitutive laws.     

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.     

Experimental measurement and modeling analysis on mechanical properties of incudostapedial joint

The incudostapedial (IS) joint between the incus and stapes is a synovial joint consisting of joint capsule, cartilage, and synovial fluid. The mechanical properties of the IS joint directly affect the middle ear transfer function for sound transmission. However, due to the complexity and small size of the joint, the mechanical properties of the IS joint have not been reported in the literature. In this paper, we report our current study on mechanical properties of human IS joint using both experimental measurement and finite element (FE) modeling analysis. Eight IS joint samples with the incus and stapes attached were harvested from human cadaver temporal bones. Tension, compression, stress relaxation and failure tests were performed on those samples in a micro-material testing system. An analytical approach with the hyperelastic Ogden model and a 3D FE model of the IS joint including the cartilage, joint capsule, and synovial fluid were employed to derive mechanical parameters of the IS joint. The comparison of measurements and modeling results reveals the relationship between the mechanical properties and structure of the IS joint.     

Viscoelastic characterization of peripapillary sclera: material properties by quadrant in rabbit and monkey eyes

In this report we characterize the viscoelastic material properties of peripapillary sclera from the four quadrants surrounding the optic nerve head in both rabbit and monkey eyes. Scleral tensile specimens harvested from each quadrant were subjected to uniaxial stress relaxation and tensile ramp to failure tests. Linear viscoelastic theory, coupled with a spectral reduced relaxation function, was employed to characterize the viscoelastic properties of the tissues. We detected no differences in the stress-strain curves of specimens from the four quadrants surrounding the optic nerve head (ONH) below a strain of 4 percent in either the rabbit or monkey. While the peripapillary sclera from monkey eyes is significantly stiffer (both instantaneously and in equilibrium) and relaxes more slowly than that from rabbits, we detected no differences in the viscoelastic material properties (tested at strains of 0-1 percent) of sclera from the four quadrants surrounding the ONH within either species group.     

A model of fluid-biofilm interaction using a Burger material law

A two-dimensional finite element model of the biofilm response to flow was developed. The numerical code sequentially coupled the fluid dynamics of turbulent, incompressible flow with the mechanical response of a single hemispherical biofilm cluster (approximately 100 microm) attached to the flow boundary. A non-linear Burger material law was used to represent the viscoelastic response of a representative microbial biofilm. This constitutive law was incorporated into the numerical model as a Prony series representation of the biofilm's relaxation modulus. Model simulations illuminated interesting details of this fluid-structure interaction. Simulations revealed that softer biofilms (characterized by lower elastic moduli) were highly susceptible to lift forces and consequently were subject to even greater drag forces found higher in the velocity field. A bimodal deformation path due to the two Burger relaxation times was also observed in several simulations. This suggested that interfacial biofilm may be most susceptible to hydrodynamically induced detachment during the initial relaxation time. This result may prove useful in developing removal strategies. Additionally, plots of lift versus drag suggested that the deformation paths taken by viscoelastic biofilms are largely insensitive to specific material coefficients. Softer biofilms merely seem to follow the same path (as a stiffer biofilm) at a faster rate. These relationships may be useful in estimating the hydrodynamic forces acting on an attached biofilm based on changes in scale and cataloged material properties.     

Possible Taxonomic Trends in the Success of Primary Aggregate Formation in Marine Sponge Cell Cultures

A large number of novel compounds with significant medical potential have been isolated from sponges, motivating efforts to develop techniques for the sustainable cultivation of sponge biomass. To date, 33 sponges from nine different orders have been examined to assess their ability to be cultured in vitro. However, little consideration has been given to the relationships between these sponges; only one report has considering the phylogenetic relationships between the species. On the basis of morphological data, no taxonomic specificity was apparent as an indicator for the successful cultivation of the sponges. As the systematic classification of the Demospongiae is poorly understood, we collated available information on the success of in vitro sponge cell cultivation reports and examined the phylogenetic relationships of these sponges through the use of 18S and 28S rDNA sequence data. Based on molecular data, the ability of sponges to form primary aggregates from the dissociated cells of marine demosponges indicates that taxonomic trends may exist, emphasizing the need to better characterize sponges being investigated for biotechnological applications. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The orthotropic viscoelastic behavior of aortic elastin

In this paper, we studied the viscoelastic behaviors of isolated aortic elastin using combined modeling and experimental approaches. Biaxial stress relaxation and creep experiments were performed to study the time-dependent behavior of elastin. Experimental results reveal that stress relaxation preconditioning is necessary in order to obtain repeatable stress relaxation responses. Elastin exhibits less stress relaxation than intact or decellularized aorta. The rate of stress relaxation of intact and decellularized aorta is linearly dependent on the initial stress levels. The rate of stress relaxation for elastin increases linearly at stress levels below about 60 kPa; however, the rate changes very slightly at higher initial stress levels. Experimental results also show that creep response is negligible for elastin, and the intact or decellularized aorta. A quasi-linear viscoelasticity model was incorporated into a statistical mechanics based eight-chain microstructural model at the fiber level to simulate the orthotropic viscoelastic behavior of elastin. A user material subroutine was developed for finite element analysis. Results demonstrate that this model is suitable to capture both the orthotropic hyperelasticity and viscoelasticity of elastin.