Inverse finite element characterization of the human myometrium derived from uniaxial compression experiments / Von einaxialen Druckversuchen hergeleitete inverse Finite-Elemente-Charakterisierung des menschlichen Myometriums

Abstract The low strain-rate behavior of the human myometrium under compression was determined. To this end, uniaxial, unconstrained compression experiments were conducted on a total of 25 samples from three excised human uteri at strain rates between 0.001 s(-1) and 0.008 s(-1). A three-dimensional finite element model of each sample was created and used together with an optimization algorithm to find material parameters in an inverse estimation process. Friction and shape irregularities of samples were incorporated in the models. The uterine specimens in compression were modeled as viscoelastic, non-linear, nearly incompressible and isotropic continua. Simulations of uniaxial, frictionless compressions of an idealized cuboid were used to compare the resulting material parameters among each other. The intra- and inter-subject variability in stiffness of specimens was found to be large and to cover such a wide range that the effect of anisotropy which is of minor influence under compressive deformations in the first place could be neglected. Material parameters for a viscoelastic model based on a decoupled, reduced quadratic strain-energy function were presented for the uterine samples representing a median stiffness.     

Strain energy density as a rupture criterion for the kidney: impact tests on porcine organs, finite element simulation, and a baseline comparison between human and porcine tissues

High-velocity (up to 25 m/s) impact tests were performed on pig kidneys to characterize failure behavior at deformation rates associated with traumatic injury. Cylindrical tissue samples (n = 45) and whole perfused organs (n = 34) were impacted using both falling weights and a high-velocity pneumatic projectile impactor. Impact energy was incrementally increased until visible rupture occurred. The strain energy density failure threshold fell between 25 and 60 kJ/m3 for excised porcine tissue samples, and between 15 and 30 kJ/m3 for whole, perfused organs. The relationship between localized failure in whole organ impacts and tissue level failure thresholds observed in cylindrical tissue samples was explored using a detailed finite element model of the human kidney. The model showed good correlation between experimentally observed injury patterns and predicted strain energy density distributions within the renal parenchyma. Finally, to facilitate interpretation of the porcine renal impact results with regard to human trauma, quasi-static compression test results of freshly excised human kidney cortex samples (n = 30) were compared against similar tests on pig kidneys. Human tissues failed at Lagrange strain levels similar to porcine tissue (63+/-6.3%), but at 52% lower Lagrange stress (116+/-28 kPa), and 35% lower strain energy density (17.1+/-4.4 kJ/m3). Thus conservative interpretation of porcine test results is recommended.     

Viscoelastic properties of the passive mechanical behavior of the porcine carotid artery: Influence of proximal and distal positions

The viscoelastic properties of porcine carotid tissue are investigated in this work. Experimental uniaxial stress relaxation tests along the longitudinal and circumferential directions of the vessel were performed for carotid strips extracted from 10 vessels. Directional and local differences - distal versus proximal position - in the tissue behavior were investigated. The experimental tests reveal a highly anisotropic, non-linear viscoelastic response and local dependence of the samples. The carotid artery shows anisotropic relaxation behavior for both proximal and distal samples. The highest stress relaxation was found in the circumferential tensile test for the highest applied strain at the distal position. For the circumferential direction, the relaxation stress was higher than in the longitudinal being at its highest in the distal position. These facts show that the stress relaxation is higher in the distal than in the proximal position. However, there are no differences between both positions in the longitudinal direction. In addition, a constitutive law that takes into account the fundamental features, including non-linear viscoelasticity, of the arterial tissue is proposed. The present results are correlated with the purely elastic response and the microstructural analysis of the tissue by means of histological quantification presented in a previous study.     

Viscoelastic material model for the temporomandibular joint disc derived from dynamic shear tests or strain-relaxation tests

Viscoelastic material models for the temporomandibular joint disc, based upon strain relaxation, were considered to underestimate energy absorption for loads with time constants beyond the relaxation time. Therefore, the applicability of a material model that takes the viscous behavior at a wide range of frequencies into account was assessed. To that purpose a non-linear multi-mode Maxwell model was tested in cyclic large-strain compression tests. Its material constants were approximated from dynamic small-strain shear deformation tests. The storage and loss moduli as obtained from a disc sample could be approximated with a four-mode Maxwell model. In simulated large-strain compression tests it behaved similarly as observed from the experimental tests. The underestimation of energy dissipation, as obtained from a single-mode Maxwell model was considerably reduced, especially for deformations with a higher strain rate. Furthermore, in contrast to the latter it was able to predict the increase of the stress amplitude with the compression frequency much better. In conclusion, the applied four-mode Maxwell model, based upon dynamic shear tests, was considered more suitable to predict higher frequency viscoelastic response, for instance during shock absorption, than a model based upon strain-relaxation.     

Anomalous rate dependence of the preconditioned response of soft tissue during load controlled deformation

It has been observed in load controlled laboratory tests of myocardium and skin that the tissues can exhibit a decrease in nonlinear stiffness with an increase in loading rate: the faster a test is performed, the more compliant is the preconditioned material behavior. This response seems to conflict with what is generally expected of soft tissues based on stretch or strain controlled tests, in which an increased rate of deformation results in a stiffer material response. It is hypothesized that this anomalous behavior has not been observed previously due to the small number of cyclic load controlled mechanical characterization tests that are geared specifically towards viscoelastic tissue response. The goal of this paper is to examine the preconditioned response of soft tissue to load controlled deformation using nonlinear viscoelastic material models including quasi-linear viscoelasticity, and to determine under what conditions this anomalous behavior becomes apparent. Results from this study suggest that this behavior is a true phenomenon unique to load controlled deformations that results from the interplay of nonlinear effects and creep behavior. These results call for increased attention to experimental parameters when testing and modeling nonlinear viscoelastic material behavior.     

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.     

A model for the compressible,viscoelastic behavior of human amnion addressing tissue variability through a single parameter

A viscoelastic, compressible model is proposed to rationalize the recently reported response of human amnion in multiaxial relaxation and creep experiments. The theory includes two viscoelastic contributions responsible for the short- and long-term time-dependent response of the material. These two contributions can be related to physical processes: water flow through the tissue and dissipative characteristics of the collagen fibers, respectively. An accurate agreement of the model with the mean tension and kinematic response of amnion in uniaxial relaxation tests was achieved. By variation of a single linear factor that accounts for the variability among tissue samples, the model provides very sound predictions not only of the uniaxial relaxation but also of the uniaxial creep and strip-biaxial relaxation behavior of individual samples. This suggests that a wide range of viscoelastic behaviors due to patient-specific variations in tissue composition can be represented by the model without the need of recalibration and parameter identification.     

The mechanical properties of rat spinal cord in vitro

Freshly excised rat spinal cords were tested in uniaxial tension, in vitro, at strain rates ranging from 0.002 to 0.2 s-1. Stress relaxation tests were performed for a range of strains from 2% to 5%, with the relaxation behaviour being recorded for a period of at least 30 min. Samples exhibited a characteristic "J" shaped non-linear stress-strain response, with stiffness increasing with applied strain. The cords were labelled with rows of small markers and the uniaxial tension tests were recorded via video. Subsequent image analysis enabled the distribution of strain on the cord surface to be determined. Viscoelastic models were developed to model the mechanical behaviour of the specimens and were found to adequately describe the material behaviour.     

The role of viscoelasticity of collagen fibers in articular cartilage: theory and numerical formulation

The relative importance of fluid-dependent and fluid-independent transient mechanical behavior in articular cartilage was examined for tensile and unconfined compression testing using a fibril reinforced model. The collagen matrix of articular cartilage was modeled as viscoelastic using a quasi-linear viscoelastic formulation with strain-dependent elastic modulus, while the proteoglycan matrix was considered as linearly elastic. The collagen viscoelastic properties were obtained by fitting experimental data from a tensile test. These properties were used to investigate unconfined compression testing, and the sensitivity of the properties was also explored. It was predicted that the stress relaxation observed in tensile tests was not caused by fluid pressurization at the macroscopic level. A multi-step tensile stress relaxation test could be approximated using a hereditary integral in which the elastic fibrillar modulus was taken to be a linear function of the fibrillar strain. Applying the same formulation to the radial fibers in unconfined compression, stress relaxation could not be simulated if fluid pressurization were absent. Collagen viscoelasticity was found to slightly weaken fluid pressurization in unconfined compression, and this effect was relatively more significant at moderate strain rates. Therefore, collagen viscoelasticity appears to play an import role in articular cartilage in tensile testing, while fluid pressurization dominates the transient mechanical behavior in compression. Collagen viscoelasticity plays a minor role in the mechanical response of cartilage in unconfined compression if significant fluid flow is present.     

Viscoelastic properties of passive skeletal muscle in compression—Cyclic behaviour

Skeletal muscle relaxation behaviour in compression has been previously reported, but the anisotropic behaviour at higher loading rates remains poorly understood. In this paper, uniaxial unconfined cyclic compression tests were performed on fresh porcine muscle samples at various fibre orientations to determine muscle viscoelastic behaviour. Mean compression level of 25% was applied and cycles of 2% and 10% amplitude were performed at 0.2–80 Hz. Under cycles of low frequency and amplitude, linear viscoelastic cyclic relaxation was observed. Fibre/cross-fibre results were qualitatively similar, but the cross-fibre direction was stiffer (ratio of 1.2). In higher amplitude tests nonlinear viscoelastic behaviour with a frequency dependent increase in the stress cycles amplitude was found (factor of 4.1 from 0.2 to 80 Hz).The predictive capability of an anisotropic quasi-linear viscoelastic model previously fitted to stress-relaxation data from similar tissue samples was investigated. Good qualitative results were obtained for low amplitude cycles but differences were observed in the stress cycle amplitudes (errors of 7.5% and 31.8%, respectively, in the fibre/cross-fibre directions). At higher amplitudes significant qualitative and quantitative differences were evident. A nonlinear model formulation was therefore developed which provided a good fit and predictions to high amplitude low frequency cyclic tests performed in the fibre/cross-fibre directions. However, this model gave a poorer fit to high frequency cyclic tests and to relaxation tests. Neither model adequately predicts the stiffness increase observed at frequencies above 5 Hz.Together with data previously presented, the experimental data presented here provide a unique dataset for validation of future constitutive models for skeletal muscle in compression.     

Modelling and simulation of porcine liver tissue indentation using finite element method and uniaxial stress–strain data

We hypothesize that both compression and elongation stress–strain data should be considered for modeling and simulation of soft tissue indentation. Uniaxial stress–strain data were obtained from in vitro loading experiments of porcine liver tissue. An axisymmetric finite element model was used to simulate liver tissue indentation with tissue material represented by hyperelastic models. The material parameters were derived from uniaxial stress–strain data of compressions, elongations, and combined compression and elongation of porcine liver samples. in vitro indentation tests were used to validate the finite element simulation. Stress–strain data from the simulation with material parameters derived from the combined compression and elongation data match the experimental data best. This is due to its better ability in modeling 3D deformation since the behavior of biological soft tissue under indentation is affected by both its compressive and tensile characteristics. The combined logarithmic and polynomial model is somewhat better than the 5-constant Mooney–Rivlin model as the constitutive model for this indentation simulation.     

Material properties of the human calcaneal fat pad in compression: experiment and theory

The structural behaviour of the human heel pad has been studied extensively due to its ability to absorb shock, protect against excessive local stress, and reduce plantar pressures. However, the material properties of the tissue have not been adequately measured. These must be known in order to perform a finite element analysis of the effect of factors such as foot geometry and shoe/surface construction on heel pad function. Therefore, the purposes of this study were to (a) measure the viscoelastic behaviour of the fat pad in compression, and (b) to determine an appropriate constitutive equation to model the tissue. A series of unconfined compression tests were performed on 8 mm diameter cylinders of fat pad tissue, consisting of quasi-static, 175, 350 mm/s and stress-relaxation tests to 50% deformation. The tissue exhibited nonlinear, viscoelastic behaviour. No significant difference was found in the quasi-static behaviour between samples from different locations and orientations in the heel. The stress-relaxation tests were used to determine the time constant (τ₁=0.5 s), the 175 mm/s test to determine the relaxation coefficient (g₁=28), and the 350 mm/s compression test to determine the material constants (C₁₀₀=C₀₁₀=0.01, C₂₀₀=C₀₂₀=0.1 Pa) of a single-phase, hyperelastic, linear viscoelastic strain energy function (r²=0.98).     

The applicability of the time/temperature superposition principle to brain tissue

This paper deals with the mechanical characterization of brain tissue which behaves as a viscoelastic material. We focus on the linear viscoelastic behavior, which should apply for small strains at any strain rate, and demonstrate the applicability of the time/temperature superposition principle. This principle allows the opportunity to extend the range of shear rates for which the material is characterized, and makes the results applicable to impact conditions. This characterization of the linear behavior forms the basis for a further nonlinear characterization of the tissue.     

The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta

The biomechanical response of normal and pathologic human abdominal aortic tissue to uniaxial loading conditions is insufficient for the characterization of its three-dimensional (3D) mechanical behavior. Planar biaxial mechanical evaluation allows for 3D constitutive modeling of nearly incompressible tissues, as well as the investigation of the nature of mechanical anisotropy. In the current study, 26 abdominal aortic aneurysm (AAA) tissue samples and 8 age-matched (> 60 years of age) nonaneurysmal abdominal aortic (AA) tissue samples were obtained and tested using a tension-controlled biaxial testing protocol. Graphical response functions (Sun et al., 2003. J. Biomech. Eng. 125, 372-380) were used as a guide to describe the pseudo-elastic response of AA and AAA. Based on the observed pseudo-elastic response, a four-parameter exponential strain energy function developed by Vito (1990. J. Biomech. Eng. 112, 153-159) was used from which both an individual specimen and group material parameter sets were determined for both AA and AAA. Peak Green strain values in the circumferential (Ethetatheta,max) and longitudinal (ELL,max) directions under an equibiaxial tension of 120 N/m were also compared. The strain energy function fit all of the individual specimens well with an average R2 of 0.95 +/- 0.02 and 0.90 +/- 0.02 (mean +/- SEM) for the AA and AAA groups, respectively. The average Ethetatheta,max at 200 N/m equibiaxial tension was found to be significantly smaller for AAAs as compared to AAs (0.07 +/- 0.01 versus 0.13 +/- 0.03, respectively; p < 0.01). There was also a pronounced increase in the circumferential stiffness for AAA tissue as compared to AA tissue, indicating a larger degree of anisotropy for this tissue as compared to age-matched AA tissue. We also observed that the four-parameter Fung-elastic model was not able to fit the AAA tissue mechanical response using physically realistic material parameter values. It was concluded that aneurysmal degeneration of the abdominal aorta is associated with an increase in mechanical anisotropy, with preferential stiffening in the circumferential direction.     

Biocomposite hydrogels with carboxymethylated, nanofibrillated cellulose powder for replacement of the nucleus pulposus

Biocomposite hydrogels with carboxymethylated, nanofibrillated cellulose (c-NFC) powder were prepared by UV polymerization of N-vinyl-2-pyrrolidone with Tween 20 trimethacrylate as a cross-linking agent for replacement of the native, human nucleus pulposus (NP) in intervertebral disks. The swelling ratios and the moduli of elasticity in compression of neat and biocomposite hydrogels were evaluated in dependence of c-NFC concentration (ranging from 0 to 1.6% v/v) and degree of substitution (DS, ranging from 0 to 0.23). The viscoelastic properties in shear and the material relaxation behavior in compression were measured for neat and biocomposite hydrogels containing 0.4% v/v of fibrils (DS ranging from 0 to 0.23), and their morphologies were characterized by cryo-scanning electron microscopy (cryo-SEM). The obtained results show that the biocomposite hydrogels can successfully mimic the mechanical and swelling behavior of the NP. In addition, the presence of the c-NFC shows lower strain values after cyclic compression tests and consequently creates improved material relaxation properties compared with neat hydrogels. Among the tested samples, the biocomposite hydrogel containing 0.4% v/v of c-NFC with a DS of 0.17 shows the closest behavior to native NP. Further investigation should focus on evaluation and improvement of the long-term relaxation behavior.     

Viscoelastic testing methodologies for tissue engineered blood vessels

In order to function in vivo, tissue engineered blood vessels (TEBVs) must encumber pulsatile blood flow and withstand hemodynamic pressures for long periods of time. To date TEBV mechanical assessment has typically relied on single time point burst and/or uniaxial tensile testing to gauge the strengths of the constructs. This study extends this analysis to include creep and stepwise stress relaxation viscoelastic testing methodologies. TEBV models exhibiting diverse mechanical behaviors as a result of different architectures ranging from reconstituted collagen gels to hybrid constructs reinforced with either untreated or glutaraldhyde-crosslinked collagen supports were evaluated after 8 and 23 days of in vitro culturing. Data were modeled using three and four-parameter linear viscoelastic mathematical representations and compared to porcine carotid arteries. While glutaraldhyde-treated hybrid TEBVs exhibited the largest overall strengths and toughness, uncrosslinked hybrid samples exhibited time-dependent behaviors most similar to native arteries. These findings emphasize the importance of viscoelastic characterization when evaluating the mechanical performance of TEBVs. Limits of testing methods and modeling systems are presented and discussed.     

A novel technique for the assessment of mechanical properties of vascular tissue

Accurate estimation of mechanical properties of the different atherosclerotic plaque constituents is important in assessing plaque rupture risk. The aim of this study was to develop an experimental set-up to assess material properties of vascular tissue, while applying physiological loading and being able to capture heterogeneity. To do so, a ring-inflation experimental set-up was developed in which a transverse slice of an artery was loaded in the radial direction, while the displacement was estimated from images recorded by a high-speed video camera. The performance of the set-up was evaluated using seven rubber samples and validated with uniaxial tensile tests. For four healthy porcine carotid arteries, material properties were estimated using ultrasound strain imaging in whole-vessel-inflation experiments and compared to the properties estimated with the ring-inflation experiment. A 1D axisymmetric finite element model was used to estimate the material parameters from the measured pressures and diameters, using a neo-Hookean and Holzapfel–Gasser–Ogden material model for the rubber and porcine samples, respectively. Reproducible results were obtained with the ring-inflation experiment for both rubber and porcine samples. Similar mean stiffness values were found in the ring-inflation and tensile tests for the rubber samples as 202 kPa and 206 kPa, respectively. Comparable results were obtained in vessel-inflation experiments using ultrasound and the proposed ring-inflation experiment. This inflation set-up is suitable for the assessment of material properties of healthy vascular tissue in vitro. It could also be used as part of a method for the assessment of heterogeneous material properties, such as in atherosclerotic plaques.

     

Time-dependent circumferential deformation of cortical bone upon internal radial loading

Short and long duration tests were conducted on hollow femoral bone cylinders to study the circumferential (hoop) creep response of cortical bone subjected to an intramedullary radial load. It was hypothesized that there is a stress threshold above which nonlinear creep effects dominate the mechanical response and below which the response is primarily determined by linear viscoelastic material properties. The results indicate that a hoop stress threshold exists for cortical bone, where creep strain, creep strain rate and residual strain exhibited linear behavior at low hoop stress and nonlinear behavior above the hoop stress threshold. A power-law relationship was used to describe creep strain as a function of hoop stress and time and damage morphology was assessed.     

Dynamic, regional mechanical properties of the porcine brain: indentation in the coronal plane

Stress relaxation tests using a custom designed microindentation device were performed on ten anatomic regions of fresh porcine brain (postmortem time <3 h). Using linear viscoelastic theory, a Prony series representation was used to describe the shear relaxation modulus for each anatomic region tested. Prony series parameters fit to load data from indentations performed to ～10% strain differed significantly by anatomic region. The gray and white matter of the cerebellum along with corpus callosum and brainstem were the softest regions measured. The cortex and hippocampal CA1/CA3 were found to be the stiffest. To examine the large strain behavior of the tissue, multistep indentations were performed in the corona radiata to strains of 10%, 20%, and 30%. Reduced relaxation functions were not significantly different for each step, suggesting that quasi-linear viscoelastic theory may be appropriate for representing the nonlinear behavior of this anatomic region of porcine brain tissue. These data, for the first time, describe the dynamic and short time scale behavior of multiple anatomic regions of the porcine brain which will be useful for understanding porcine brain injury biomechanics at a finer spatial resolution than previously possible.     

Application of the Tsai-Wu quadratic multiaxial failure criterion to bovine trabecular bone

As a first step toward development of a multiaxial failure criterion for human trabecular bone, the Tsai-Wu quadratic failure criterion was modified as a function of apparent density and applied to bovine tibial trabecular bone. Previous data from uniaxial compressive, tensile, and torsion tests (n = 139 total) were combined with those from new triaxial tests (n = 17) to calibrate and then verify the criterion. Combinations of axial compression and radial pressure were used to produce the triaxial compressive stress states. All tests were performed with minimal end artifacts in the principal material coordinate system of the trabecular network. Results indicated that the stress interaction term F12 exhibited a strong nonlinear dependence on apparent density (r2 > 0.99), ranging from -0.126 MPa-2 at low densities (0.29 g/cm3) to 0.005 MPa-2 at high densities (0.63 g/cm3). After calibration and when used to predict behavior of new-specimens without any curve-fitting, the Tsai-Wu criterion had a mean (+/- SD) error of -32.6 +/- 10.6 percent. Except for the highest density triaxial specimens, most (15/17 specimens) failed at axial stresses close to their predicted uniaxial values, and some reinforcement for transverse loading was observed. We conclude that the Tsai-Wu quadratic criterion, as formulated here, is at best only a reasonable predictor of the multiaxial failure behavior of trabecular bone, and further work is required before it can be confidently applied to human bone.