Hyperelastic properties of human meniscal attachments

Meniscal attachments are ligamentous tissues anchoring the menisci to the underlying subchondral bone. Currently little is known about the behavior of meniscal attachments, with only a few studies quantitatively documenting their properties. The objective of this study was to quantify and compare the tensile mechanical properties of human meniscal attachments in the transverse direction, curve fit experimental Cauchy stress-stretch data to evaluate the hyperelastic behavior, and couple these results with previously obtained longitudinal data to generate a more complete constitutive model. Meniscal attachment specimens were tested using a uniaxial tension test with the collagen fibers oriented perpendicular to the loading axis. Tests were run until failure and load-optical displacement data was recorded for each test. The medial posterior attachment was shown to have a significantly greater elastic modulus (6.42±0.78 MPa) and ultimate stress (1.73±0.32 MPa) when compared to the other three attachments. The Mooney-Rivlin material model was selected as the best fit for the transverse data and used in conjunction with the longitudinal data. A novel computational approach to determining the transition point between the toe and linear regions is presented for the hyperelastic stress-stretch curves. Results from piece-wise non-linear longitudinal curve fitting correlate well with previous linear elastic and SEM findings. These data can be used to advance the design of meniscal replacements and improve knee joint finite element models.     

The non-linear response of a muscle in transverse compression: assessment of geometry influence using a finite element model

Most recent finite element models that represent muscles are generic or subject-specific models that use complex, constitutive laws. Identification of the parameters of such complex, constitutive laws could be an important limit for subject-specific approaches. The aim of this study was to assess the possibility of modelling muscle behaviour in compression with a parametric model and a simple, constitutive law. A quasi-static compression test was performed on the muscles of dogs. A parametric finite element model was designed using a linear, elastic, constitutive law. A multi-variate analysis was performed to assess the effects of geometry on muscle response. An inverse method was used to define Young's modulus. The non-linear response of the muscles was obtained using a subject-specific geometry and a linear elastic law. Thus, a simple muscle model can be used to have a bio-faithful, biomechanical response.     

Aortic valve leaflet mechanical properties facilitate diastolic valve function

This work was concerned with the numerical simulation of the behaviour of aortic valves whose material can be modelled as non-linear elastic anisotropic. Linear elastic models for the valve leaflets with parameters used in previous studies were compared with hyperelastic models, incorporating leaflet anisotropy with pronounced stiffness in the circumferential direction through a transverse isotropic model. The parameters for the hyperelastic models were obtained from fits to results of orthogonal uniaxial tensile tests on porcine aortic valve leaflets. The computational results indicated the significant impact of transverse isotropy and hyperelastic effects on leaflet mechanics; in particular, increased coaptation with peak values of stress and strain in the elastic limit. The alignment of maximum principal stresses in all models follows approximately the coarse collagen fibre distribution found in aortic valve leaflets. The non-linear elastic leaflets also demonstrated more evenly distributed stress and strain which appears relevant to long-term scaffold stability and mechanotransduction.     

Contribution of the nucleus to the mechanical properties of endothelial cells.

The cell nucleus plays a central role in the response of the endothelium to mechanical forces, possibly by deforming during cellular adaptation. The goal of this work was to precisely quantify the mechanical properties of the nucleus. Individual endothelial cells were subjected to compression between glass microplates. This technique allows measurement of the uniaxial force applied to the cell and the resulting deformation. Measurements were made on round and spread cells to rule out the influence of cell morphology on the nucleus mechanical properties. Tests were also carried out with nuclei isolated from cell cultures by a chemical treatment. The non-linear force-deformation curves indicate that round cells deform at lower forces than spread cells and nuclei. Finite-element models were also built with geometries adapted to actual morphometric measurements of round cells, spread cells and isolated nuclei. The nucleus and the cytoplasm were modeled as separate homogeneous hyperelastic materials. The models simulate the compression and yield the force-deformation curve for a given set of elastic moduli. These parameters are varied to obtain a best fit between the theoretical and experimental data. The elastic modulus of the cytoplasm is found to be on the order of 500N/m(2) for spread and round cells. The elastic modulus of the endothelial nucleus is on the order of 5000N/m(2) for nuclei in the cell and on the order of 8000N/m(2) for isolated nuclei. These results represent an unambiguous measurement of the nucleus mechanical properties and will be important in understanding how cells perceive mechanical forces and respond to them.     

Residual stress in the adult mouse brain

This work provides direct evidence that sustained tensile stress exists in white matter of the mature mouse brain. This finding has important implications for the mechanisms of brain development, as tension in neural axons has been hypothesized to drive cortical folding in the human brain. In addition, knowledge of residual stress is required to fully understand the mechanisms behind traumatic brain injury and changes in mechanical properties due to aging and disease. To estimate residual stress in the brain, we performed serial dissection experiments on 500-mum thick coronal slices from fresh adult mouse brains and developed finite element models for these experiments. Radial cuts were made either into cortical gray matter, or through the cortex and the underlying white matter tract composed of parallel neural axons. Cuts into cortical gray matter did not open, but cuts through both layers consistently opened at the point where the cut crossed the white matter. We infer that the cerebral white matter is under considerable tension in the circumferential direction in the coronal cerebral plane, parallel to most of the neural fibers, while the cerebral cortical gray matter is in compression. The models show that the observed deformation after cutting can be caused by more growth in the gray matter than in the white matter, with the estimated tensile stress in the white matter being on the order of 100–1,000 Pa.     

Elastic response of the roots of field crops

The elasticity of root tips was investigated in experiments of the static clamped-beam type. The response was non-linear. The results were interpreted in terms of a generalized elastic modulus M , and two parameters, α and β, which account for the non-linearity.
Seminal axes and primary lateral roots of 16 crop species grown in the laboratory show significant inter-species differences but insignificant inter-variety differences in the elastic properties. A smaller set of results for field-grown roots with two tillage treatments shows that, in nearly all cases, the elastic modulus is smaller and the diameter is larger than with laboratory-grown roots. Environmental factors such as water potential, and nutrition have significant influences on the diameters and elastic behaviour of roots.     

Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules

Unexplained length-dependence of flexural rigidity and Young's modulus of microtubules is studied using an orthotropic elastic shell model. It is showed that vibration frequencies and buckling load predicted by the accurate orthotropic shell model are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is this inaccuracy of the isotropic beam model used by all previous researchers that leads to reported lower flexural rigidity and Young's modulus for shorter microtubules. In particular, much lower shear modulus and circumferential Young's modulus, which only weaken flexural rigidity of shorter microtubules, are responsible for the observed length-dependence of the flexural rigidity. These results confirm that longitudinal Young's modulus of microtubules is length-independent, and the observed length-dependence of the flexural rigidity and Young's modulus is a result of strongly anisotropic elastic properties of microtubules which have a length-dependent weakening effect on flexural rigidity of shorter microtubules.     

Influence of experimental protocols on the mechanical properties of the intervertebral disc in unconfined compression

The lack of standardization in experimental protocols for unconfined compression tests of intervertebral discs (IVD) tissues is a major issue in the quantification of their mechanical properties. Our hypothesis is that the experimental protocols influence the mechanical properties of both annulus fibrosus and nucleus pulposus. IVD extracted from bovine tails were tested in unconfined compression stress-relaxation experiments according to six different protocols, where for each protocol, the initial swelling of the samples and the applied preload were different. The Young's modulus was calculated from a viscoelastic model, and the permeability from a linear biphasic poroviscoelastic model. Important differences were observed in the prediction of the mechanical properties of the IVD according to the initial experimental conditions, in agreement with our hypothesis. The protocol including an initial swelling, a 5% strain preload, and a 5% strain ramp is the most relevant protocol to test the annulus fibrosus in unconfined compression, and provides a permeability of 5.0 ± 4.2e(-14)m(4)/N[middle dot]s and a Young's modulus of 7.6 ± 4.7 kPa. The protocol with semi confined swelling and a 5% strain ramp is the most relevant protocol for the nucleus pulposus and provides a permeability of 10.7 ± 3.1 e(-14)m(4)/N[middle dot]s and a Young's modulus of 6.0 ± 2.5 kPa.     

Modelling the mechanical properties of single suspension-cultured tomato cells

BACKGROUND AND AIMS: The relationship between composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood, partly because intrinsic properties of walls such as Young's modulus cannot be obtained readily. The aim of this work is to show that Young's modulus of walls of single suspension-cultured tomato cells can be determined by modelling force-deformation data. METHODS: The model simulates the compression of a cell between two flat surfaces, with the cell treated as a liquid-filled sphere with thin compressible walls. The cell wall and membrane were taken to be permeable, but the compression was so fast that water loss could be neglected in the simulations. Force-deformation data were obtained by compressing the cells in micromanipulation experiments. RESULTS:Good fits were obtained between the model and low-strain experimental data, using the modulus and initial inflation of the cell as adjustable parameters. The mean Young's modulus for 2-week-old cells was found to be 2.3 +/- 0.2 GPa at pH 5. This corresponds to an instantaneous bulk modulus of elasticity of approx. 7 MPa, similar to a value found by the pressure probe method. However, Young's modulus is a better parameter, as it should depend only on the composition and structure of the cell wall, not on bulk cell behaviour. This new method has been used to show that Young's modulus of cultured tomato cell walls is at its lowest at pH 4.5, the pH optimum for expansin activity. CONCLUSIONS:The linear elastic model is very suitable for estimating wall Young's modulus from micromanipulation experiments on single tomato cells. This is a powerful method for determining cell wall material properties.     

Determination of Poisson's ratio of articular cartilage by indentation using different-sized indenters

Articular cartilage is often characterized as an isotropic elastic material with no interstitial fluid flow during instantaneous and equilibrium conditions, and indentation testing commonly used to deduce material properties of Young's modulus and Poisson's ratio. Since only one elastic parameter can be deduced from a single indentation test, some other test method is often used to allow separate measurement of both parameters. In this study, a new method is introduced by which the two material parameters can be obtained using indentation tests alone, without requiring a secondary different type of test. This feature makes the method more suitable for testing small samples in situ. The method takes advantages of the finite layer effect. By indenting the sample twice with different-sized indenters, a nonlinear equation with the Poisson's ratio as the only unknown can be formed and Poisson's ratio obtained by solving the nonlinear equation. The method was validated by comparing the predicted Poisson's ratio for urethane rubber with the manufacturer's supplied value, and comparing the predicted Young's modulus for urethane rubber and an elastic foam material with modulii measured by unconfined compression. Anisotropic and nonhomogeneous finite-element (FE) models of the indentation were developed to aid in data interpretation. Applying the method to bovine patellar cartilage, the tissue Young's modulus was found to be 1.79 +/- 0.59 MPa in instantaneous response and 0.45 +/- 0.26 MPa in equilibrium, and the Poisson's ratio 0.503 +/- 0.028 and 0.463 +/- 0.073 in instantaneous and equilibrium, respectively. The equilibrium Poisson's ratio obtained in our work was substantially higher than those derived from biphasic indentation theory and those optically measured in an unconfined compression test. The finite element model results and examination of viscoelastic-biphasic models suggest this could be due to viscoelastic, inhomogeneity, and anisotropy effects.     

The effects of modeling simplifications on craniofacial finite element models: the alveoli (tooth sockets) and periodontal ligaments

Several finite element models of a primate cranium were used to investigate the biomechanical effects of the tooth sockets and the material behavior of the periodontal ligament (PDL) on stress and strain patterns associated with feeding. For examining the effect of tooth sockets, the unloaded sockets were modeled as devoid of teeth and PDL, filled with teeth and PDLs, or simply filled with cortical bone. The third premolar on the left side of the cranium was loaded and the PDL was treated as an isotropic, linear elastic material using published values for Young's modulus and Poisson's ratio. The remaining models, along with one of the socket models, were used to determine the effect of the PDL's material behavior on stress and strain distributions under static premolar biting and dynamic tooth loading conditions. Two models (one static and the other dynamic) treated the PDL as cortical bone. The other two models treated it as a ligament with isotropic, linear elastic material properties. Two models treated the PDL as a ligament with hyperelastic properties, and the other two as a ligament with viscoelastic properties. Both behaviors were defined using published stress-strain data obtained from in vitro experiments on porcine ligament specimens. Von Mises stress and strain contour plots indicate that the effects of the sockets and PDL material behavior are local. Results from this study suggest that modeling the sockets and the PDL in finite element analyses of skulls is project dependent and can be ignored if values of stress and strain within the alveolar region are not required.     

Dynamic response of the brain with vasculature: a three-dimensional computational study

To date, the influence of the vasculature on the dynamic response of the brain has not been studied with a complete three-dimensional (3D) finite element head model. In this study, short duration rotational (10,000 rad/s(2) with a duration of 5 ms) and translational (100G with a duration of 5 ms) acceleration impulses were applied to the 3D finite element models to study the dynamic response of the brain. The hypothesis of this study was that due to the convoluted organization and non-linear material properties of cerebral vasculature, the difference in maximum principle strain between models with and without vasculature should be minimal. The effects of non-linear material properties and the convoluted structure of the vasculature were examined by comparing the results from the 3D finite element models. The peak average strain reduction in a model with non-linear elastic vasculature and a model with linear elastic vasculature compared to a model without vasculature was 2% and 5%, respectively, indicating that the influence of the vasculature on the dynamic response of the brain is minimal.     

Relationship between structural modeling and hyperelastic material behavior: application to CNS white matter

 Recent measurements of the material properties of brain tissue allow an examination of the underlying microstructural basis in both physiological and pathophysiological conditions. The purpose of this study is to develop a mathematical relationship between microstructurally based models of the central nervous system (CNS) white matter and equivalent hyperelastic material models. For simplicity, time dependent material behavior is not included in this formulation. The microstructural representation is used to formulate structural property relationships for highly oriented white matter, and is mathematically compared to one isotropic and two anisotropic hyperelastic formulations. For the anisotropic characterizations, the population of axons in the white matter is assumed to align along one preferred direction of the material, yielding a transversely isotropic formulation. Relatively simple strain–energy functions incorporating material anisotropy provide sufficient flexibility to model the nonlinear behavior predicted from structurally based models, although the tangential stiffness of the hyperelastic approaches does not follow completely the behavior of the structurally based formulations. This analysis is an initial step towards linking microstructural aspects of the tissue to material models commonly used for large deformations, and may be an important step in relating predicted tissue deformation to the deformation and stress of cellular and subcellular structures. Received: 15 October 2001 / Accepted: 30 September 2002 Funds for this work were provided by CDC grant R49/CCR312712 and NIH grants P50 NS08803, NICHD RO1 41699, and NINDS RO1 35712.     

Measurement of the circumferential mechanical properties of the umbilical vein: experimental and numerical analyses

Coronary artery disease is responsible for almost 30% of all deaths worldwide. The saphenous vein and umbilical vein (UV) are the most common veins using for treatment as a coronary artery bypass graft (CABG). The mechanical properties of UV belonging to its long-term patency for CABG are very important. However, there is a lack of knowledge on the linear elastic and nonlinear hyperelastic mechanical properties of the UV. In this study, three stress definitions (second Piola–Kichhoff stress, engineering stress and true stress) and four strain definitions (Almansi–Hamel strain, Green–St Venant strain, engineering strain and true strain) are used to determine the elastic modulus, maximum stress and strain of eight human UVs under circumferential loading. The nonlinear mechanical behaviour of the UV is computationally investigated using Mooney–Rivlin hyperelastic model. A numerical finite element analysis is also carried out to simulate the constitutive modelling versus its numerical results. The results show that the Almansi–Hamel strain definition overestimates the elastic modulus while Green–St Venant strain definition underestimates the elastic modulus at different stress definitions. The true stress–true strain definition, which gives more accurate measurements of the tissue's response using the instantaneous values, reveals the Young's modulus and maximum stress of 2.18 and 6.01 MPa, respectively. The Mooney–Rivlin material model is well represented by the nonlinear mechanical behaviour of the UV. The findings of this study could have implications not only for understanding the extension and rupture mechanism of UV but also for interventions and surgeries, including balloon angioplasty, bypass and stenting.     

Assessment of trace elements in human brain using inductively coupled plasma mass spectrometry

Recent brain research reveals a major role of trace elements in various diseases such as multiple sclerosis, Alzheimer's and Wilson's disease. The majority of published tissue concentrations dates back decades, and was assessed with various methods. Little is known about hemispherical differences, the correlation of trace elements or age-dependent changes in the human brain. Thus, the aim of this study was to examine trace element concentrations in different human brain regions after whole brain formalin fixation.549 samples of 13 brain regions were investigated in 11 deceased subjects without known history of brain pathology. Regional wet-to-dry mass ratios and concentrations of iron, copper, magnesium, manganese, calcium and zinc were determined using inductively coupled plasma mass spectrometry.Cortical gray matter revealed higher water content (wet-to-dry mass ratios 5.84–6.40) than white matter regions (wet-to-dry mass ratios 2.95–3.05). Element concentrations displayed specific regional differences. Good linear correlation of concentrations between elements was found for iron/copper as well as for manganese/magnesium (Spearman's rank correlation coefficient 0.74 and 0.65, respectively). Significant inter-hemispherical differences were found for copper in occipital white matter, for magnesium and calcium in putamen and for iron and copper in temporal white matter. An age dependent increase was seen in cortical gray matter for calcium, for magnesium in all regions except in cortical gray matter, for copper in substantia nigra and for zinc in occipital cortex.The presented trace element concentrations can serve as a fundamental basis for further brain research. Wet-to-dry mass ratios allow a comparison with reference data from other studies.     

Unconfined creep compression of chondrocytes

The study of single cell mechanics offers a valuable tool for understanding cellular milieus. Specific knowledge of chondrocyte biomechanics could lead to elucidation of disease etiologies and the biomechanical factors most critical to stimulating regenerative processes in articular cartilage. Recent studies in our laboratory have suggested that it may be acceptable to approximate the shape of a single chondrocyte as a disc. This geometry is easily utilized for generating models of unconfined compression. In this study, three continuum mechanics models of increasing complexity were formulated and used to fit unconfined compression creep data. Creep curves were obtained from middle/deep zone chondrocytes (n = 15) and separately fit using the three continuum models. The linear elastic solid model yielded a Young's modulus of 2.55+/-0.85 kPa. The viscoelastic model (adapted from the Kelvin model) generated an instantaneous modulus of 2.47+/-0.85 kPa, a relaxed modulus of 1.48+/-0.35 kPa, and an apparent viscosity of 1.92+/-1.80 kPa-s. Finally, a linear biphasic model produced an aggregate modulus of 2.58+/-0.87 kPa, a permeability of 2.57 x 10(-12)+/-3.09 m(4)/N-s, and a Poisson's ratio of 0.069+/-0.021. The results of this study demonstrate that similar values for the cell modulus can be obtained from three models of increasing complexity. The elastic model provides an easy method for determining the cell modulus, however, the viscoelastic and biphasic models generate additional material properties that are important for characterizing the transient response of compressed chondrocytes.     

Modelling of non-linear elastic tissues for surgical simulation

Realistic modelling of the interaction between surgical instruments and human organs has been recognised as a key requirement in the development of high-fidelity surgical simulators. Primarily due to computational considerations, most of the past real-time surgical simulation research has assumed linear elastic behaviour for modelling tissues, even though human soft tissues generally possess non-linear properties. For a non-linear model, the well-known Poynting effect developed during shearing of the tissue results in normal forces not seen in a linear elastic model. Using constitutive equations of non-linear tissue models together with experiments, we show that the Poynting effect results in differences in force magnitude larger than the absolute human perception threshold for force discrimination in some tissues (e.g. myocardial tissues) but not in others (e.g. brain tissue simulants).     

Fitted hyperelastic parameters for Human brain tissue from reported tension,compression, and shear tests

The mechanical properties of human brain tissue are the subject of interest because of their use in understanding brain trauma and in developing therapeutic treatments and procedures. To represent the behavior of the tissue, we have developed hyperelastic mechanical models whose parameters are fitted in accordance with experimental test results. However, most studies available in the literature have fitted parameters with data of a single type of loading, such as tension, compression, or shear. Recently, Jin et al. (Journal of Biomechanics 46:2795−2801, 2013) reported data from ex vivo tests of human brain tissue under tension, compression, and shear loading using four strain rates and four different brain regions. However, they do not report parameters of energy functions that can be readily used in finite element simulations. To represent the tissue behavior for the quasi-static loading conditions, we aimed to determine the best fit of the hyperelastic parameters of the hyperfoam, Ogden, and polynomial strain energy functions available in ABAQUS for the low strain rate data, while simultaneously considering all three loading modes. We used an optimization process conducted in MATLAB, calling iteratively three finite element models developed in ABAQUS that represent the three loadings. Results showed a relatively good fit to experimental data in all loading modes using two terms in the energy functions. Values for the shear modulus obtained in this analysis (897−1653 Pa) are in the range of those presented in other studies. These energy-function parameters can be used in brain tissue simulations using finite element models.