Specimen-specific multi-scale model for the anisotropic elastic constants of human cortical bone

The anisotropic elastic constants of human cortical bone were predicted using a specimen-specific micromechanical model that accounted for structural parameters across multiple length scales. At the nano-scale, the elastic constants of the mineralized collagen fibril were estimated from measured volume fractions of the constituent phases, namely apatite crystals and Type I collagen. The elastic constants of the extracellular matrix (ECM) were predicted using the measured orientation distribution function (ODF) for the apatite crystals to average the contribution of misoriented mineralized collagen fibrils. Finally, the elastic constants of cortical bone tissue were determined by accounting for the measured volume fraction of Haversian porosity within the ECM. Model predictions using the measured apatite crystal ODF were not statistically different from experimental measurements for both the magnitude and anisotropy of elastic constants. In contrast, model predictions using common idealized assumptions of perfectly aligned or randomly oriented apatite crystals were significantly different from the experimental measurements. A sensitivity analysis indicated that the apatite crystal volume fraction and ODF were the most influential structural parameters affecting model predictions of the magnitude and anisotropy, respectively, of elastic constants.     

Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone

We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori-Tanaka scheme and the Hashin-Rosen bounds. The methods concur for specific engineering moduli such as the axial Young's modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.     

Maximum effect of the heterogeneity of tissue mineralization on the effective cortical bone elastic properties

Biomechanics and Modeling in Mechanobiology - The mineralization level is heterogeneous in cortical bone extracellular matrix as a consequence of remodeling. Models of the effective elastic...     

A continuous wave technique for the measurement of the elastic properties of cortical bone

A continuous wave technique is described for measuring the nine independent orthotropic elastic coefficients from a single cubic specimen. The side dimensions of this cubic specimen are on the order of 5 mm. Because of the small size of the specimen, the spatial resolution of material inhomogeneity using this technique is quite good. Although it is possible to apply this technique to any elastic material such as woods or metals, the elastic properties of human and canine cortical femora are presented here. The orthotropic elastic coefficients and the variation of these coefficients are presented as a function of anatomical position.     

An alternative ultrasonic method for measuring the elastic properties of cortical bone

We studied the elastic properties of bone to analyze its mechanical behavior. The basic principles of ultrasonic methods are now well established for varying isotropic media, particularly in the field of biomedical engineering. However, little progress has been made in its application to anisotropic materials. This is largely due to the complex nature of wave propagation in these media. In the present study, the theory of elastic waves is essential because it relates the elastic moduli of a material to the velocity of propagation of these waves along arbitrary directions in a solid. Transducers are generally placed in contact with the samples which are often cubes with parallel faces that are difficult to prepare. The ultrasonic method used here is original, a rough preparation of the bone is sufficient and the sample is rotated. Moreover, to analyze heterogeneity of the structure we measure velocities in different points on the sample. The aim of the present study was to determine in vitro the anisotropic elastic properties of cortical bones. For this purpose, our method allowed measurement of longitudinal and transverse velocities (C(L) and C(T)) in longitudinal (fiber direction) and the radial directions (orthogonal to the fiber direction) of compact bones. Young's modulus E and Poisson's ratio nu, were then deduced from the velocities measured considering the compact bone as transversely isotropic or orthotropic. The results are in line with those of other methods.     

Simple constitutive model for a cortical bone

This short study presents a simple, one-dimensional constitutive model for the cortical bone with haversian structure. The model is developed within the general framework of the continuum damage theory. The kinetic equation is derived (rather than assumed a priori) through consideration of the irreversible changes of the mesostructure. As a consequence the analytical results closely approximate experimental measurements even though the theory does not introduce a single experimentally unidentifiable material parameter.     

A damage model for nonlinear tensile behavior of cortical bone.

To describe the time-dependent nonlinear tensile behavior observed in experimental studies of cortical bone, a damage model was developed using two internal state variables (ISV's). One ISV is a damage parameter that represents the loss of stiffness. A rule for the evolution of this ISV was defined based on previously observed creep behavior. The second ISV represents the inelastic strain due to viscosity and internal friction. The model was tested by simulating experiments in tensile and bending loading. Using average values from previous creep studies for parameters in the damage evolution rule, the model tended to underestimate the maximum nonlinear strains and to overestimate the nonlinear strain accumulated after load reversal in the tensile test simulations. Varying the parameters for the individual tests produced excellent fits to the experimental data. Similarly, the model simulations of the bending tests could produce excellent fits to the experimental data. The results demonstrate that the 2-ISV model combining damage (stiffness loss) with slip and viscous behavior could capture the nonlinear tensile behavior of cortical bone in axial and bending loading.     

A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

At its highest level of microstructural organization—the mesoscale or millimeter scale—cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.     

Human cortical bone: the SiNuPrOs model

Many models that have been developed for cortical bone oversimplify much of the architectural and physical complexity. With SiNuPrOs model, a more complete approach is investigated: it is multiscale because it contains five structural levels and multi physic because it takes into account simultaneously structure (with various properties: elasticity, piezoelectricity, porous medium), fluid and mineralization process modelization. The multiscale aspect is modeled by using 18 structural parameters in a specific application of the mathematical theory of homogenization and 10 other physical parameters are necessary for the multi physic aspect. The modelization of collagen as a piezoelectric medium has needed the development of a new behaviour law allowing a better simulation of the effect of a medium considered as evolving during a mineralization process. Then the main interest of SiNuPrOs deals with the possibility to study, at each level of the cortical architecture, either the elastic properties or the fluid motion or the piezoelectric effects or both of them. All these possibilities constitute a very large work and all this mass of information (fluid aspects, even at the nanoscopic scale, piezoelectric phenomena and simulations) will be presented in several papers. This first one is only devoted to the presentation of this model with an application to the computation of elastic properties at the macroscopic scale. The computational methods have been packed into software also called SiNuPrOs and allowing a large number of predictive simulations corresponding to various different configurations.     

Human cortical bone: the SiNuPrOs model

Many models that have been developed for cortical bone oversimplify much of the architectural and physical complexity. With SiNuPrOs model, a more complete approach is investigated: it is multiscale because it contains five structural levels and multi physic because it takes into account simultaneously structure (with various properties: elasticity, piezoelectricity, porous medium), fluid and mineralization process modelization. The multiscale aspect is modeled by using 18 structural parameters in a specific application of the mathematical theory of homogenization and 10 other physical parameters are necessary for the multi physic aspect. The modelization of collagen as a piezoelectric medium has needed the development of a new behaviour law allowing a better simulation of the effect of a medium considered as evolving during a mineralization process.

Then the main interest of SiNuPrOs deals with the possibility to study, at each level of the cortical architecture, either the elastic properties or the fluid motion or the piezoelectric effects or both of them. All these possibilities constitute a very large work and all this mass of information (fluid aspects, even at the nanoscopic scale, piezoelectric phenomena and simulations) will be presented in several papers. This first one is only devoted to the presentation of this model with an application to the computation of elastic properties at the macroscopic scale.

The computational methods have been packed into software also called SiNuPrOs and allowing a large number of predictive simulations corresponding to various different configurations.     

The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus

The elastic moduli of human subchondral, trabecular, and cortical bone tissue from a proximal tibia were experimentally determined using three-point bending tests on a microstructural level. The mean modulus of subchondral specimens was 1.15 GPa, and those of trabecular and cortical specimens was 4.59 GPa and 5.44 GPa respectively. Significant differences were found in the modulus values between bone tissues, which may have mainly resulted from the differences in the microstructures of each bone tissue rather than in the mineral density. Furthermore, the size-dependency of the modulus was examined using eight different sizes of cortical specimens (heights h = 100-1000 microns). While the modulus values for relatively large specimens (h greater than 500 microns) remained fairly constant (approximately 15 GPa), the values decreased as the specimens became smaller. A significant correlation was found between the modulus and specimen size. The surface area to volume ratio proved to be a key variable to explain the size-dependency.     

Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method

A two-level micromechanical model of cortical bone based on a generalized self-consistent method was developed to take into consideration the transversely isotropic elasticity of many microstructural features in cortical bone, including Haversian canals, resorption cavities, and osteonal and interstitial lamellae. In the first level, a single osteon was modeled as a two-phase composite such that Haversian canals were represented by elongated pores while the surrounding osteonal lamellae were considered as matrix. In the second level, osteons and resorption cavities were modeled as multiple inclusions while interstitial lamellae were regarded as matrix. The predictions of cortical bone elasticity from this two-level micromechanical model were mostly in agreement with experimental data for the dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. However, variation in cortical bone elastic constants was greater in experimental data than in model predictions. This could be attributed to variations in the elastic properties of microstructural features in cortical bone. The present micromechanical model of cortical bone will be useful in understanding the contribution of cortical bone porosity to femoral neck fractures.     

Micromechanical modeling of elastic properties of cortical bone accounting for anisotropy of dense tissue

The paper analyzes the connection between microstructure of the osteonal cortical bone and its overall elastic properties. The existing models either neglect anisotropy of the dense tissue or simplify cortical bone microstructure (accounting for Haversian canals only). These simplifications (related mostly to insufficient mathematical apparatus) complicate quantitative analysis of the effect of microstructural changes – produced by age, microgravity, or some diseases – on the overall mechanical performance of cortical bone. The present analysis fills this gap; it accounts for anisotropy of the dense tissue and uses realistic model of the porous microstructure. The approach is based on recent results of Sevostianov et al. (2005) and Saadat et al. (2012) on inhomogeneities in a transversely-isotropic material. Bone?s microstructure is modeled according to books of Martin and Burr (1989), Currey (2002), and Fung (1993) and includes four main families of pores. The calculated elastic constants for porous cortical bone are in agreement with available experimental data. The influence of each of the pore types on the overall moduli is examined.     

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.     

Impact of the porous microstructure on the overall elastic properties of the osteonal cortical bone

Mechanical properties of bones are largely determined by their microstructure. The latter comprises a large number of diverse pores. The present paper analyzes a connection between structure of the porous space of the osteonal cortical bone and bone's overall anisotropic elastic moduli. The analysis is based on recent developments in the theory of porous materials that predict the anisotropic effective moduli of porous solids in terms of pores' shapes, orientations and densities. Bone's microstructure is modeled using available micrographs. The calculated anisotropic elastic constants for porous cortical bone are, mostly, in agreement with available experimental data. The influence of each of the pore types on the overall moduli is examined. The results of the analysis can also be used to estimate the extent of mineralization (hydroxyapatite content) if the overall porosity and the effective moduli are known and, vice versa, to estimate porosity from the measured moduli and the extent of mineralization.     

Spatial distribution of acoustic and elastic properties of human femoral cortical bone

The purpose of this study is to quantify the spatial distribution of acoustic velocities and elastic properties (elastic constants) on Human femoral cortical bone. Four cross sections (average thickness of 2.09+/-0.27 mm) have been cut transversally between 40% and 70% of the total length and between them parallelepiped samples in each quadrant have been cut. Ultrasonic technique in transmission with immersion focused transducers at 5 MHz and contact transducers 2.25 MHz were used on the cross sections and parallelepiped samples, respectively. The first technique allows relative spatial distribution of velocities to be obtained, meanwhile the second technique allows the direct assessment of elastic constants. For both techniques, bulk velocities were found to be lower at the posterior side with an increase along the length (from 40% to 70% total length) (p < 0.05). Densities and elastic constants show equivalent pattern of variation. These variations are mainly due the cortical porosity related to vascularisation environment. The spatial distribution of velocities exhibits significant radial variation from the endosteal to the periosteal region. This is in agreement with variation of the porosity at that location. Same range of velocities was obtained with both techniques. The range of longitudinal velocities values varies from 3548 to 3967 m/s and between 18.5 and 33.1 GPa for the bulk velocities and axial elastic constants, respectively. Our results are within the range with those found in the literature. However, it must be noted that the range of acoustic and elastic properties variation is concerning the same bone. So, our new results show the ability of the technique to quantify accurately local variation of acoustic and elastic properties (within the section and along the length) of human cortical bone. Furthermore, our immersion technique could be used to assess the spatial distribution of the elastic constants with the knowledge of spatial distribution of densities.     

A mechanostatistical approach to cortical bone remodelling: an equine model

In this study, the development of a mechanostatistical model of three-dimensional cortical bone remodelling informed with in vivo equine data is presented. The equine model was chosen as it is highly translational to the human condition due to similar Haversian systems, availability of in vivo bone strain and biomarker data, and furthermore, equine models are recommended by the US Federal Drugs Administration for comparative joint research. The model was derived from micro-computed tomography imaged specimens taken from the equine third metacarpal bone, and the Frost-based ‘mechanostat’ was informed from both in vivo strain gauges and biomarkers to estimate bone growth rates. The model also described the well-known ‘cutting cone’ phenomena where Haversian canals tunnel and replace bone. In order to make this model useful in practice, a partial least squares regression (PLSR) surrogate model was derived based on training data from finite element simulations with different loads. The PLSR model was able to predict microstructure and homogenised Young’s modulus with errors less than 2.2 % and \(0.6\,\% \), respectively.     

The multiscale meso-mechanics model of viscoelastic cortical bone

Biomechanics and Modeling in Mechanobiology - Cortical bone is a complex hierarchical structure consisting of biological fiber composites with transversely isotropic constituents, whose...     

Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale

This paper addresses the relationships between the microscopic properties of bone and its elasticity at the millimetre scale, or mesoscale. A method is proposed to estimate the mesoscale properties of cortical bone based on a spatial distribution of acoustic properties at the microscopic scale obtained with scanning acoustic microscopy. The procedure to compute the mesoscopic stiffness tensor involves (i) the segmentation of the pores to obtain a realistic model of the porosity; (ii) the construction of a field of anisotropic elastic coefficients at the microscopic scale which reflects the heterogeneity of the bone matrix; (iii) finite element computations of mesoscopic homogenized properties. The computed mesoscopic properties compare well with available experimental data. It appears that the tissue anisotropy at the microscopic level has a major effect on the mesoscopic anisotropy and that assuming the pores filled with an incompressible fluid or, alternatively, empty, leads to significantly different mesoscopic properties.