Experimental, theoretical and numerical investigation of the nonlinear micromechanical properties of bone

Aging societies suffer from an increasing incidence of bone fractures. Bone strength depends on the amount of mineral measured by clinical densitometry, but also on the micromechanical properties of the bone hierarchical organization. A good understanding has been reached for elastic properties on several length scales, but up to now there is a lack of reliable postyield data on the lower length scales. In order to be able to describe the behavior of bone at the microscale, an anisotropic elastic-viscoplastic damage model was developed using an eccentric generalized Hill criterion and nonlinear isotropic hardening. The model was implemented as a user subroutine in Abaqus and verified using single element tests. A FE simulation of microindentation in lamellar bone was finally performed show-ing that the new constitutive model can capture the main characteristics of the indentation response of bone. As the generalized Hill criterion is limited to elliptical and cylindrical yield surfaces and the correct shape for bone is not known, a new yield surface was developed that takes any convex quadratic shape. The main advantage is that in the case of material identification the shape of the yield surface does not have to be anticipated but a minimization results in the optimal shape among all convex quadrics. The generality of the formulation was demonstrated by showing its degeneration to classical yield surfaces. Also, existing yield criteria for bone at multiple length scales were converted to the quadric formulation. Then, a computational study to determine the influence of yield surface shape and damage on the in-dentation response of bone using spherical and conical tips was performed. The constitutive model was adapted to the quadric criterion and yield surface shape and critical damage were varied. They were shown to have a major impact on the indentation curves. Their influence on indentation modulus, hardness, their ratio as well as the elastic to total work ratio were found to be very well described by multilinear regressions for both tip shapes. For conical tips, indentation depth was not a significant fac-tor, while for spherical tips damage was insignificant. All inverse methods based on microindentation suffer from a lack of uniqueness of the found material properties in the case of nonlinear material behavior. Therefore, monotonic and cyclic micropillar com-pression tests in a scanning electron microscope allowing a straightforward interpretation comple-mented by microindentation and macroscopic uniaxial compression tests were performed on dry ovine bone to identify modulus, yield stress, plastic deformation, damage accumulation and failure mecha-nisms. While the elastic properties were highly consistent, the postyield deformation and failure mech-anisms differed between the two length scales. A majority of the micropillars showed a ductile behavior with strain hardening until failure by localization in a slip plane, while the macroscopic samples failed in a quasi-brittle fashion with microcracks coalescing into macroscopic failure surfaces. In agreement with a proposed rheological model, these experiments illustrate a transition from a ductile mechanical behavior of bone at the microscale to a quasi-brittle response driven by the growth of preexisting cracks along interfaces or in the vicinity of pores at the macroscale. Subsequently, a study was undertaken to quantify the topological variability of indentations in bone and examine its relationship with mechanical properties. Indentations were performed in dry human and ovine bone in axial and transverse directions and their topography measured by AFM. Statistical shape modeling of the residual imprint allowed to define a mean shape and describe the variability with 21 principal components related to imprint depth, surface curvature and roughness. The indentation profile of bone was highly consistent and free of any pile up. A few of the topological parameters, in particular depth, showed significant correlations to variations in mechanical properties, but the cor-relations were not very strong or consistent. We could thus verify that bone is rather homogeneous in its micromechanical properties and that indentation results are not strongly influenced by small de-viations from the ideal case. As the uniaxial properties measured by micropillar compression are in conflict with the current literature on bone indentation, another dissipative mechanism has to be present. The elastic-viscoplastic damage model was therefore extended to viscoelasticity. The viscoelastic properties were identified from macroscopic experiments, while the quasistatic postelastic properties were extracted from micropillar data. It was found that viscoelasticity governed by macroscale properties has very little influence on the indentation curve and results in a clear underestimation of the creep deformation. Adding viscoplasticity leads to increased creep, but hardness is still highly overestimated. It was possible to obtain a reasonable fit with experimental indentation curves for both Berkovich and spherical indenta-tion when abandoning the assumption of shear strength being governed by an isotropy condition. These results remain to be verified by independent tests probing the micromechanical strength prop-erties in tension and shear. In conclusion, in this thesis several tools were developed to describe the complex behavior of bone on the microscale and experiments were performed to identify its material properties. Micropillar com-pression highlighted a size effect in bone due to the presence of preexisting cracks and pores or inter-faces like cement lines. It was possible to get a reasonable fit between experimental indentation curves using different tips and simulations using the constitutive model and uniaxial properties measured by micropillar compression. Additional experimental work is necessary to identify the exact nature of the size effect and the mechanical role of interfaces in bone. Deciphering the micromechanical behavior of lamellar bone and its evolution with age, disease and treatment and its failure mechanisms on several length scales will help preventing fractures in the elderly in the future.

High-Pressure Behavior and Phase Stability of Al5BO9, a Mullite-Type Ceramic Material

Phase stability, elastic behavior, and pressure-induced structural evolution of synthetic boron-mullite Al5BO9 (a = 5.6780(7), b = 15.035(6), and c =7.698(3) Å, space group Cmc21, Z = 4) were investigated up to 25.6(1) GPa by in situ single-crystal synchrotron X-ray diffraction with a diamond anvil cell (DAC) under hydrostatic conditions. No evidence of phase transition was observed up to 21.7(1) GPa. At 25.6(1) GPa, the refined unit-cell parameters deviated significantly from the compressional trend, and the diffraction peaks appeared broader than at lower pressure. At 26.7(1) GPa, the diffraction pattern was not indexable, suggesting amorphization of the material or a phase transition to a high-pressure polymorph. Fitting the P–V data up to 21.7(1) GPa with a second-order Birch–Murnaghan Equation-of-State, we obtained a bulk modulus KT0 = 164(1) GPa. The axial compressibilities, here described as linearized bulk moduli, are as follows: KT0(a) = 244(9), KT0(b) = 120(4), and KT0(c) = 166(11) GPa (KT0(a):KT0(b):KT0(c) = 2.03:1:1.38). The structure refinements allowed a description of the main deformation mechanisms in response to the applied pressure. The stiffer crystallographic direction appears to be controlled by the infinite chains of edge-sharing octahedra running along [100], making the structure less compressible along the a-axis than along the b- and c-axis.

Mechanisms of intrinsic beating variability in cardiac cell cultures and model pacemaker networks

Heart rate variability (HRV) exhibits fluctuations characterized by a power law behavior of its power spectrum. The interpretation of this nonlinear HRV behavior, resulting from interactions between extracardiac regulatory mechanisms, could be clinically useful. However, the involvement of intrinsic variations of pacemaker rate in HRV has scarcely been investigated. We examined beating variability in spontaneously active incubating cultures of neonatal rat ventricular myocytes using microelectrode arrays. In networks of mathematical model pacemaker cells, we evaluated the variability induced by the stochastic gating of transmembrane currents and of calcium release channels and by the dynamic turnover of ion channels. In the cultures, spontaneous activity originated from a mobile focus. Both the beat-to-beat movement of the focus and beat rate variability exhibited a power law behavior. In the model networks, stochastic fluctuations in transmembrane currents and stochastic gating of calcium release channels did not reproduce the spatiotemporal patterns observed in vitro. In contrast, long-term correlations produced by the turnover of ion channels induced variability patterns with a power law behavior similar to those observed experimentally. Therefore, phenomena leading to long-term correlated variations in pacemaker cellular function may, in conjunction with extracardiac regulatory mechanisms, contribute to the nonlinear characteristics of HRV.

Morphology–elasticity relationships using decreasing fabric information of human trabecular bone from three major anatomical locations

With improving clinical CT scanning technology, the accuracy of CT-based finite element (FE) models of the human skeleton may be ameliorated by an enhanced description of apparent level bone mechanical properties. Micro-finite element (μFE) modeling can be used to study the apparent elastic behavior of human cancellous bone. In this study, samples from the femur, radius and vertebral body were investigated to evaluate the predictive power of morphology–elasticity relationships and to compare them across different anatomical regions. μFE models of 701 trabecular bone cubes with a side length of 5.3 mm were analyzed using kinematic boundary conditions. Based on the FE results, four morphology–elasticity models using bone volume fraction as well as full, limited or no fabric information were calibrated for each anatomical region. The 5 parameter Zysset–Curnier model using full fabric information showed excellent predictive power with coefficients of determination ( r2adj ) of 0.98, 0.95 and 0.94 of the femur, radius and vertebra data, respectively, with mean total norm errors between 14 and 20%. A constant orthotropy model and a constant transverse isotropy model, where the elastic anisotropy is defined by the model parameters, yielded coefficients of determination between 0.90 and 0.98 with total norm errors between 16 and 25%. Neglecting fabric information and using an isotropic model led to r2adj between 0.73 and 0.92 with total norm errors between 38 and 49%. A comparison of the model regressions revealed minor but significant (p<0.01) differences for the fabric–elasticity model parameters calibrated for the different anatomical regions. The proposed models and identified parameters can be used in future studies to compute the apparent elastic properties of human cancellous bone for homogenized FE models.

Northern hemispheric winter warming pattern after tropical volcanic eruptions: Sensitivity to the ozone climatology

An important key for the understanding of the dynamic response to large tropical volcanic eruptions is the warming of the tropical lower stratosphere and the concomitant intensification of the polar vortices. Although this mechanism is reproduced by most general circulation models today, most models still fail in producing an appropriate winter warming pattern in the Northern Hemisphere. In this study ensemble sensitivity experiments were carried out with a coupled atmosphere-ocean model to assess the influence of different ozone climatologies on the atmospheric dynamics and in particular on the northern hemispheric winter warming. The ensemble experiments were perturbed by a single Tambora-like eruption. Larger meridional gradients in the lower stratospheric ozone favor the coupling of zonal wind anomalies between the stratosphere and the troposphere after the eruption. The associated sea level pressure, temperature, and precipitation patterns are more pronounced and the northern hemispheric winter warming is highly significant. Conversely, weaker meridional ozone gradients lead to a weaker response of the winter warming and the associated patterns. The differences in the number of stratosphere-troposphere coupling events between the ensembles experiments indicate a nonlinear response behavior of the dynamics with respect to the ozone and the volcanic forcing.

An anisotropic elastic-viscoplastic damage model for bone tissue

A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain D(κ) , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.

Cement distribution, volume, and compliance in vertebroplasty: some answers from an anatomy-based nonlinear finite element study

STUDY DESIGN: The biomechanics of vertebral bodies augmented with real distributions of cement were investigated using nonlinear finite element (FE) analysis. OBJECTIVES: To compare stiffness, strength, and stress transfer of augmented versus nonaugmented osteoporotic vertebral bodies under compressive loading. Specifically, to examine how cement distribution, volume, and compliance affect these biomechanical variables. SUMMARY OF BACKGROUND DATA: Previous FE studies suggested that vertebroplasty might alter vertebral stress transfer, leading to adjacent vertebral failure. However, no FE study so far accounted for real cement distributions and bone damage accumulation. METHODS: Twelve vertebral bodies scanned with high-resolution pQCT and tested in compression were augmented with various volumes of cements and scanned again. Nonaugmented and augmented pQCT datasets were converted to FE models, with bone properties modeled with an elastic, plastic and damage constitutive law that was previously calibrated for the nonaugmented models. The cement-bone composite was modeled with a rule of mixture. The nonaugmented and augmented FE models were subjected to compression and their stiffness, strength, and stress map calculated for different cement compliances. RESULTS: Cement distribution dominated the stiffening and strengthening effects of augmentation. Models with cement connecting either the superior or inferior endplate (S/I fillings) were only up to 2 times stiffer than the nonaugmented models with minimal strengthening, whereas those with cement connecting both endplates (S + I fillings) were 1 to 8 times stiffer and 1 to 12 times stronger. Stress increases above and below the cement, which was higher for the S + I cases and was significantly reduced by increasing cement compliance. CONCLUSION: The developed FE approach, which accounts for real cement distributions and bone damage accumulation, provides a refined insight into the mechanics of augmented vertebral bodies. In particular, augmentation with compliant cement bridging both endplates would reduce stress transfer while providing sufficient strengthening.

Distortion-sensitive insight into the pretransitional behavior of 4- n -octyloxy-4′-cyanobiphenyl (8OCB)

Results of studies of the static and dynamic dielectric properties in rod-like 4-n-octyloxy-4'-cyanobiphenyl (8OCB) with isotropic (I)–nematic (N)–smectic A (SmA)–crystal (Cr) mesomorphism, combined with measurements of the low-frequency nonlinear dielectric effect and heat capacity are presented. The analysis is supported by the derivative-based and distortion-sensitive transformation of experimental data. Evidence for the I–N and N–SmA pretransitional anomalies, indicating the influence of tricritical behavior, is shown. It has also been found that neither the N phase nor the SmA phase are uniform and hallmarks of fluid–fluid crossovers can be detected. The dynamics, tested via the evolution of the primary relaxation time, is clearly non-Arrhenius and described via τ(T) = τc(T−TC)−phgr. In the immediate vicinity of the I–N transition a novel anomaly has been found: Δτ ∝ 1/(T − T*), where T* is the temperature of the virtual continuous transition and Δτ is the excess over the 'background behavior'. Experimental results are confronted with the comprehensive Landau–de Gennes theory based modeling.

Finite Element Based Nonlinear Normalization of Human Lumbar Intervertebral Disc Stiffness to Account for Its Morphology

Disc degeneration, usually associated with low back pain and changes of intervertebral stiffness, represents a major health issue. As the intervertebral disc (IVD) morphology influences its stiffness, the link between mechanical properties and degenerative grade is partially lost without an efficient normalization of the stiffness with respect to the morphology. Moreover, although the behavior of soft tissues is highly nonlinear, only linear normalization protocols have been defined so far for the disc stiffness. Thus, the aim of this work is to propose a nonlinear normalization based on finite elements (FE) simulations and evaluate its impact on the stiffness of human anatomical specimens of lumbar IVD. First, a parameter study involving simulations of biomechanical tests (compression, flexion/extension, bilateral torsion and bending) on 20 FE models of IVDs with various dimensions was carried out to evaluate the effect of the disc's geometry on its compliance and establish stiffness/morphology relations necessary to the nonlinear normalization. The computed stiffness was then normalized by height (H), cross-sectional area (CSA), polar moment of inertia (J) or moments of inertia (Ixx, Iyy) to quantify the effect of both linear and nonlinear normalizations. In the second part of the study, T1-weighted MRI images were acquired to determine H, CSA, J, Ixx and Iyy of 14 human lumbar IVDs. Based on the measured morphology and pre-established relation with stiffness, linear and nonlinear normalization routines were then applied to the compliance of the specimens for each quasi-static biomechanical test. The variability of the stiffness prior to and after normalization was assessed via coefficient of variation (CV). The FE study confirmed that larger and thinner IVDs were stiffer while the normalization strongly attenuated the effect of the disc geometry on its stiffness. Yet, notwithstanding the results of the FE study, the experimental stiffness showed consistently higher CV after normalization. Assuming that geometry and material properties affect the mechanical response, they can also compensate for one another. Therefore, the larger CV after normalization can be interpreted as a strong variability of the material properties, previously hidden by the geometry's own influence. In conclusion, a new normalization protocol for the intervertebral disc stiffness in compression, flexion, extension, bilateral torsion and bending was proposed, with the possible use of MRI and FE to acquire the discs' anatomy and determine the nonlinear relations between stiffness and morphology. Such protocol may be useful to relate the disc's mechanical properties to its degree of degeneration.

Experimental validation of a nonlinear μ FE model based on cohesive-frictional plasticity for trabecular bone

Trabecular bone is a porous mineralized tissue playing a major load bearing role in the human body. Prediction of age-related and disease-related fractures and the behavior of bone implant systems needs a thorough understanding of its structure-mechanical property relationships, which can be obtained using microcomputed tomography-based finite element modeling. In this study, a nonlinear model for trabecular bone as a cohesive-frictional material was implemented in a large-scale computational framework and validated by comparison of μFE simulations with experimental tests in uniaxial tension and compression. A good correspondence of stiffness and yield points between simulations and experiments was found for a wide range of bone volume fraction and degree of anisotropy in both tension and compression using a non-calibrated, average set of material parameters. These results demonstrate the ability of the model to capture the effects leading to failure of bone for three anatomical sites and several donors, which may be used to determine the apparent behavior of trabecular bone and its evolution with age, disease, and treatment in the future.

Nonlinear quasi-static finite element simulations predict in vitro strength of human proximal femora assessed in a dynamic sideways fall setup

Osteoporotic proximal femur fractures are caused by low energy trauma, typically when falling on the hip from standing height. Finite element simulations, widely used to predict the fracture load of femora in fall, usually include neither mass-related inertial effects, nor the viscous part of bone's material behavior. The aim of this study was to elucidate if quasi-static non-linear homogenized finite element analyses can predict in vitro mechanical properties of proximal femora assessed in dynamic drop tower experiments. The case-specific numerical models of thirteen femora predicted the strength (R2=0.84, SEE=540 N, 16.2%), stiffness (R2=0.82, SEE=233 N/mm, 18.0%) and fracture energy (R2=0.72, SEE=3.85 J, 39.6%); and provided fair qualitative matches with the fracture patterns. The influence of material anisotropy was negligible for all predictions. These results suggest that quasi-static homogenized finite element analysis may be used to predict mechanical properties of proximal femora in the dynamic sideways fall situation.