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.     

Finite element modelling of implant designs and cortical bone thickness on stress distribution in maxillary type IV bone

The aims of this study were to examine the effect of implant neck design and cortical bone thickness using 3D finite element analysis and to analyse the stability of clinical evidence based on micromotion and principal stress. Four commercial dental implants for a type IV bone and maxillary segments were created. Various parameters were considered, including the osseointegration condition, loading direction and cortical bone thickness. Micromotion and principal stresses were used to evaluate the failure of osseointegration and bone overloading, respectively. It was found that the maximum stress of the peri-implant bone decreased as cortical bone thickness increased. The micromotion level in full osseointegration is less than that in non-osseointegration and it also decreases as cortical bone thickness increases. The cortical bone thickness should be measured before surgery to help select a proper implant. In the early stage of implantation, the horizontal loading component induces stress concentration in bone around the implant neck more easily than does the vertical loading component, and this may result in crestal bone loss.     

Micromechanical response of mineral and collagen phases in bone

We report the first simultaneous quantification of Young's modulus in the separate material phases of bone: collagen and carbonated hydroxyapatite. High-energy X-ray scattering and in situ loading revealed macroscopic, mineral, and collagen Young's moduli (90% confidence limit) for a canine fibula equaled 24.7(0.2) GPa, 38.2(0.5) GPa {for 00.4 and 43.6(1.4) GPa for 22.2}, and 18(1.2) GPa, respectively. The mineral contained compressive residual stresses on the order of -60 to -80 MPa before loading and had a stress enhancement (ratio of internal to applied stress) between 2.0 and 2.3. The diffraction peak width increased with increasing applied stress, mainly along the bone's longitudinal direction, and peak widths returned to pre-deformation values when load was removed. In a second fibula section from the same animal, the mineral's internal stress changed from -50 MPa (22.2 reflection) to -75 MPa (00.4) just after removal from formalin to -10 MPa after eight hours immersion in phosphate-buffered saline; the corresponding change in collagen D-spacing DeltaD/D equaled 4.2x10(-3).     

Modeling microdamage behavior of cortical bone

Bone is a complex material which exhibits several hierarchical levels of structural organization. At the submicron-scale, the local tissue porosity gives rise to discontinuities in the bone matrix which have been shown to influence damage behavior. Computational tools to model the damage behavior of bone at different length scales are mostly based on finite element (FE) analysis, with a range of algorithms developed for this purpose. Although the local mechanical behavior of bone tissue is influenced by microstructural features such as bone canals and osteocyte lacunae, they are often not considered in FE damage models due to the high computational cost required to simulate across several length scales, i.e., from the loads applied at the organ level down to the stresses and strains around bone canals and osteocyte lacunae. Hence, the aim of the current study was twofold: First, a multilevel FE framework was developed to compute, starting from the loads applied at the whole bone scale, the local mechanical forces acting at the micrometer and submicrometer level. Second, three simple microdamage simulation procedures based on element removal were developed and applied to bone samples at the submicrometer-scale, where cortical microporosity is included. The present microdamage algorithm produced a qualitatively analogous behavior to previous experimental tests based on stepwise mechanical compression combined with in situ synchrotron radiation computed tomography. Our results demonstrate the feasibility of simulating microdamage at a physiologically relevant scale using an image-based meshing technique and multilevel FE analysis; this allows relating microdamage behavior to intracortical bone microstructure.     

Fatigue of immature baboon cortical bone

Strain-controlled uniaxial fatigue and monotonic tensile tests were conducted on turned femoral cortical bone specimens obtained from baboons at various ages of maturity. Fatigue loading produced a progressive loss in stiffness and an increase in hysteresis prior to failure, indicating that immature primate cortical bone responds to repeated loading in a fashion similar to that previously observed for adult human cortical bone. Bone fatigue resistance under this strain controlled testing decreased during maturation. Maturation was also associated with an increase in bone dry density, ash fraction and elastic modulus. The higher elastic modulus of more mature bone meant that these specimens were subjected to higher stress levels during testing than more immature bone specimens. Anatomical regions along the femoral shaft exhibited differences in strength and fatigue resistance.     

Anisotropic viscoelastic properties of cortical bone

Relaxation Young's modulus of cortical bone was investigated for two different directions with respect to the longitudinal axis of bone (bone axis, BA): the modulus parallel (P) and normal (N) to the BA. The relaxation modulus was analyzed by fitting to the empirical equation previously proposed for cortical bones, i.e., a linear combination of two Kohlraush-Williams-Watts (KWW) functions (Iyo et al., 2003. Biorheology, submitted): E(t)=E0 (A1 exp[-(t/tau1)beta]+(1-A1) exp[-(t/tau2)gamma]), [0 < A1, beta, gamma < 1], where E0 is the initial modulus value E0. Tau1 and tau2(>tau1) are characteristic times of the relaxation, A1 is the fractional contribution of the fast relaxation (KWW1 process) to the whole relaxation process, and beta and gamma are parameters describing the shape of the relaxation modulus. In both P and N samples, the relaxation modulus was described well by the empirical equation. The KWW1 process of a P sample almost completely coincided with that of an N sample. In the slow process (KWW2 process), there was a difference between the relaxation modulus of a P sample and that of an N sample. The results indicate that the KWW1 process in the empirical equation represents the relaxation in the collagen matrix in bone and that the KWW2 process is related to a higher-order structure of bone that is responsible for the anisotropic mechanical properties of bone.     

Computed tomographic measurement of cortical bone geometry

In this study digital images of bone cross-sections obtained by computed tomography were analyzed with an automated outlining method. It was shown that unbiased cross-sectional geometric measurements of cortical bone could be obtained if the periosteal and endosteal surfaces were defined at separate thresholds. Use of different threshold levels for these two surfaces resulted in errors of 2.6% for periosteal diameters, 7.4% for endosteal diameters and 7.3% for cortical area. If incorrect thresholds were used, cortical thickness measurements can have errors as high as 30%. In addition, simulated variation in medullary fat content did not affect measurement of medullary dimensions.     

Measurement of microstructural strain in cortical bone

It is well known that mechanical factors affect bone remodeling such that increased mechanical demand results in net bone formation, whereas decreased demand results in net bone resorption. Current theories suggest that bone modeling and remodeling is controlled at the cellular level through signals mediated by osteocytes. The objective of this study was to investigate how macroscopically applied bone strains similar in magnitude to those that occur in vivo are manifest at the microscopic level in the bone matrix. Using a digital image correlation strain measurement technique, experimentally determined bone matrix strains around osteocyte lacuna resulting from macroscopic strains of approximately 2,000 microstrain (0.2%) reach levels of over 30,000 microstrain (3%) over fifteen times greater than the applied macroscopic strain. Strain patterns were highly heterogeneous and in some locations similar to observed microdamage around osteocyte lacuna indicating the resulting strains may represent the precursors to microdamage. This information may lead to a better understanding of how bone cells are affected by whole bone functional loading.     

Multi-scale characterization of swine femoral cortical bone

Multi-scale experimental work was carried out to characterize cortical bone as a heterogeneous material with hierarchical structure, which spans from nanoscale (mineralized collagen fibril), sub-microscale (single lamella), microscale (lamellar structures), to mesoscale (cortical bone) levels. Sections from femoral cortical bone from 6, 12, and 42 months old swine were studied to quantify the age-related changes in bone structure, chemical composition, and mechanical properties. The structural changes with age from sub-microscale to mesoscale levels were investigated with scanning electron microscopy and micro-computed tomography. The chemical compositions at mesoscale were studied by ash content method and dual energy X-ray absorptiometry, and at microscale by Fourier transform infrared microspectroscopy. The mechanical properties at mesoscale were measured by tensile testing, and elastic modulus and hardness at sub-microscale were obtained using nanoindentation. The experimental results showed age-related changes in the structure and chemical composition of cortical bone. Lamellar bone was a prevalent structure in 6 months and 12 months old animals, resorption sites were most pronounced in 6 months old animals, while secondary osteons were the dominant features in 42 months old animals. Mineral content and mineral-to-organic ratio increased with age. The structural and chemical changes with age corresponded to an increase in local elastic modulus, and overall elastic modulus and ultimate tensile strength as bone matured.     

Age-dependent fatigue behaviour of human cortical bone

Despite a general understanding that bone quality contributes to skeletal fragility, very little information exits on the age-dependent fatigue behavior of human bone. In this study four-point bending fatigue tests were conducted on aging bone in conjunction with the analysis of stiffness loss and preliminary investigation of nanoindentation based measurements of local tissue stiffness and histological evaluation of resultant tensile and compressive damage to identify the damage mechanism responsible for the increase in age-related bone fragility. The results obtained show that there is an exponential decrease in fatigue life with age, and old bone exhibits different modulus degradation profiles than young bone. In addition, this study provides preliminary evidence indicating that during fatigue loading, younger bone formed diffuse damage, lost local tissue stiffness on the tensile side. Older bone, in contrast, formed linear microcracks lost local tissue stiffness on the compressive side. Thus, the propensity of aging human bone to form more linear microcracks than diffuse damage may be a significant contributor to bone quality, and age related fragility in bone.     

Mathematical modelling of bone adaptation of the metacarpal subchondral bone in racehorses

In Thoroughbred racehorses, fractures of the distal limb are commonly catastrophic. Most of these fractures occur due to the accumulation of fatigue damage from repetitive loading, as evidenced by microdamage at the predilection sites for fracture. Adaptation of the bone in response to training loads is important for fatigue resistance. In order to better understand the mechanism of subchondral bone adaptation to its loading environment, we utilised a square root function defining the relationship between bone volume fraction \((f_{BM} )\) and specific surface \((S_v )\) of the subchondral bone of the lateral condyles of the third metacarpal bone (MCIII) of the racehorse, and using this equation, developed a mathematical model of subchondral bone that adapts to loading conditions observed in vivo. The model is expressed as an ordinary differential equation incorporating a formation rate that is dependent on strain energy density. The loading conditions applied to a selected subchondral region, i.e. volume of interest, were estimated based on joint contact forces sustained by racehorses in training. For each of the initial conditions of \(f_{BM} \) we found no difference between subsequent homoeostatic \(f_{BM} \) at any given loading condition, but the time to reach equilibrium differed by initial \(f_{BM} \) and loading condition. We found that the observed values for \(f_{BM} \) from the mathematical model output were a good approximation to the existing data for racehorses in training or at rest. This model provides the basis for understanding the effect of changes to training strategies that may reduce the risk of racehorse injury.     

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.     

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.     

Piezo1 opposes age-associated cortical bone loss

As we age, our bones undergo a process of loss, often accompanied by muscle weakness and reduced physical activity. This is exacerbated by decreased responsiveness to mechanical stimulation in aged skeleton, leading to the hypothesis that decreased mechanical stimulation plays an important role in age-related bone loss. Piezo1, a mechanosensitive ion channel, is critical for bone homeostasis and mechanotransduction. Here, we observed a decrease in Piezo1 expression with age in both murine and human cortical bone. Furthermore, loss of Piezo1 in osteoblasts and osteocytes resulted in an increase in age-associated cortical bone loss compared to control mice. The loss of cortical bone was due to an expansion of the endosteal perimeter resulting from increased endocortical resorption. In addition, expression of Tnfrsf11b, encoding anti-osteoclastogenic protein OPG, decreases with Piezo1 in vitro and in vivo in bone cells, suggesting that Piezo1 suppresses osteoclast formation by promoting Tnfrsf11b expression. Our results highlight the importance of Piezo1-mediated mechanical signaling in protecting against age-associated cortical bone loss by inhibiting bone resorption in mice.