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Adaptive elasticity in cortical bone has traditionally been modeled using Strain Energy Density (SED). Recent studies have highlighted the importance of interstitial fluid in bone adaptation, yet no research has quantified the role of interstitial fluid pressure and its effects, specifically incorporating both SED and interstitial fluid pressure in the adaptation process. This study introduces a novel formulation combining theory of porous media and theory of adaptive elasticity that considers both SED and interstitial fluid's pressure in cortical bone adaptation. The formulation is solved using ANSYS Fluent and a MATLAB script, and sensitivity analyses were conducted, analyzing various porosities, loading magnitudes, anisotropic properties of cortical bone, and involvement coefficients of interstitial fluid's pressure. This study reveals that bones with different vascular porosities (PV) tend to achieve similar density distributions under uniform loading over time. This highlights the significant role of interstitial fluid pressure in accelerating the convergence to optimal bone properties, especially in specimens with larger PV porosities. The findings emphasize the importance of fluid pressure in bone remodeling, aligning with previous studies. Furthermore, this study demonstrates that considering transversely isotropic material properties can significantly alter the remodeling configuration compared to isotropic material properties. This highlights the importance of accurately representing the anisotropic nature of cortical bone in models to better predict its adaptive responses. However, aspects such as fluid density variations and bone geometry changes remain unexplored, suggesting directions for future research. Overall, this research enhances the understanding of cortical bone adaptation and its mechanical interactions., Competing Interests: Declaration of competing interest None Declared, (Copyright © 2024 Elsevier Ltd. All rights reserved.)

The primary function of the tetrapod jaw is to transmit jaw muscle forces to bite points. The routes of force transfer in the jaw have never been studied but can be quantified using load paths - the shortest, stiffest routes from regions of force application to support constraints. Here, we use load path analysis to map force transfer from muscle attachments to bite point and jaw joint, and to evaluate how different configurations of trabecular and cortical bone affect load paths. We created three models of the mandible of the Virginia opossum, Didelphis virginiana, each with a cortical bone shell, but with different material properties for the internal spaces: (1) a cortical-trabecular model, in which the interior space is modeled with bulk properties of trabecular bone; (2) a cortical-hollow model, in which trabeculae and mandibular canal are modeled as hollow; and (3) a solid-cortical model, in which the interior is modeled as cortical bone. The models were compared with published in vivo bite force and bone strain data, and the load paths calculated for each model. The trabecular model, which is preferred because it most closely approximates the actual morphology, was best validated by in vivo data. In all three models, the load path was confined to cortical bone, although its route within the cortex varied depending on the material properties of the inner model. Our analysis shows that most of the force is transferred through the cortical, rather than trabecular bone, and highlights the potential of load path analysis for understanding form-function relationships in the skeleton., Competing Interests: Competing interests The authors declare no competing or financial interests., (© 2024. Published by The Company of Biologists Ltd.)

An in silico model for the estimation of volumetric bone mineral density (vBMD) changes at a cortical bone site subjected to mechanobiological bone remodeling is proposed in this manuscript. Mechanisms of cell differentiation, receptor-ligand binding, mechanical signaling, and resorption or deposition of bone matrix were considered, therefore providing a comprehensive description of mechanobiological bone remodeling in the bone microenvironment and enabling the analysis of temporal evolution of disease or therapy scenarios. The proposed model is composed by five modules, namely, bone cells populations, mechanobiology, volume fractions and porosity, mineral density, and structural stiffness. The model is an extension of other models found in the literature because equations for the obtaining of cortical vBMD and the binding of parathyroid hormone (PTH) to parathyroid hormone 1 receptor are included. The proposed model showed a satisfactory agreement with the solutions of other in silico models found in the literature. Simulations of walking and running exercise routines were performed for the evaluation of model capability regarding the control of the numerical error and prediction of vBMD. The computational method used to solve the case study controlled the relative numerical error by less than 1 × 10-7 for approximately 1.7 × 106 time steps. The predicted values correlate with the concept of increasing BMD by vigorous physical activity; however, they contrast with the specific effect of physical activities on cortical vBMD., (Copyright © 2020 by ASME.)

Quantitative computed tomography-based finite element (QCT-FE) modeling has potential to clarify the role of altered subchondral bone stiffness in osteoarthritis. The objective of this research was to evaluate different QCT-FE modeling and thresholding approaches to identify the method which best predicted experimentally measured local subchondral structural stiffness with highest explained variance and least error. Our results showed that separate modeling of proximal tibial cortical and trabecular bone offered little improvement in QCT-FE-predicted stiffness (0% to +3% improvement in explained variance) when compared to modeling the proximal tibia as a single structure. Based on the results of this study, we do not recommend separate modeling of cortical bone and trabecular bone when developing QCT-FE models of the proximal tibia for predicting subchondral bone stiffness.

Stress intensity factor and energy release rate are important parameters to understand the fracture behaviour of bone. The objective of this study is to predict stress intensity factor and energy release rate using finite element method, element-free Galerkin method, and extended finite element method and compare these results with the experimentally determined values. For experimental purpose, 20 longitudinally and transversely fractured single-edge notched bend specimens were prepared and tested according to ASTM standard. All specimens were tested using the universal testing machine. For numerical simulations (finite element method, element-free Galerkin method, and extended finite element method), two-dimensional model of cortical bone was developed by assuming plane strain condition. Material properties of the cortical bone were considered as anisotropic and homogeneous. The values obtained through finite element method, element-free Galerkin method, and extended finite element method are well corroborated to experimentally determined values and earlier published data. However, element-free Galerkin method and extended finite element method predict more accurate results as compared to finite element method. In the case of the transversely fractured specimen, the values of stress intensity factor and energy release rate were found to be higher as compared to the longitudinally fractured specimen, which shows consistency with earlier published data. This study also indicates element-free Galerkin method and extended finite element method predicted stress intensity factor and energy release rate results are more close to experimental results as compared to finite element method, and therefore, these methods can be used in the different field of biomechanics, particularly to predict bone fracture.

Finite element (FE) models based on quantitative computed tomography (CT) images are better predictors of bone strength than conventional areal bone mineral density measurements. However, FE models require manual segmentation of the femur, which is not clinically applicable. This study developed a method for automated FE analyses from clinical CT images. Clinical in-vivo CT images of 13 elderly female subjects were collected to evaluate the method. Secondly, proximal cadaver femurs were harvested and imaged with clinical CT (N = 17). Of these femurs, 14 were imaged with µCT and three had earlier been tested experimentally in stance-loading, while collecting surface deformations with digital image correlation. Femurs were segmented from clinical CT images using an automated method, based on the segmentation tool Stradwin. The method automatically distinguishes trabecular and cortical bone, corrects partial volume effect and generates input for FE analysis. The manual and automatic segmentations agreed within about one voxel for in-vivo subjects (0.99 ± 0.23 mm) and cadaver femurs (0.21 ± 0.07 mm). The strains from the FE predictions closely matched with the experimentally measured strains (R 2  = 0.89). The method can automatically generate meshes suitable for FE analysis. The method may bring us one step closer to enable clinical usage of patient-specific FE analyses., (Copyright © 2019 IPEM. Published by Elsevier Ltd. All rights reserved.)

The fundamental ultrasonic guided modes are consistently observed in long bones ex vivo and in vivo. However, the responses of ultrasonic guided waves to the changes of cortical thickness, cortical elastic parameters, and thickness of the overlying soft tissues are not comprehensively understood. This paper systematically presents a sensitivity analysis of leaky Lamb modes to the geometry and material characteristics of layered bone model by means of semi-analytical finite element modeling. The stratified bone model is consisted of a transversely isotropic cortex with an overlying soft tissue and underlying marrow. The study is important as it offers guidance to the parameter inversion process about the optimal selection of guided modes and regions of sensitivity for better inversion results.

High-resolution peripheral quantitative computed tomography (HR-pQCT) derived micro-finite element (FE) modeling is used to evaluate mechanical behavior at the distal radius microstructure. However, these analyses typically simulate non-physiologic simplified platen-compression boundary conditions on a small section of the distal radius. Cortical and trabecular regions contribute uniquely to distal radius mechanical behavior, and various factors affect these regions distinctly. Generalized strength predictions from standardized platen-compression analyses may not adequately capture region specific responses in bone. Our goal was to compare load sharing within the cortical-trabecular compartments between the standardized platen-compression BC simulations, and physiologic BC simulations using a validated multiscale approach. Clinical- and high-resolution images were acquired from nine cadaveric forearm specimens using an HR-pQCT scanner. Multiscale FE models simulating physiologic BCs, and micro-FE only models simulating platen-compression BCs were created for each specimen. Cortical and trabecular loads (N) along the length of the distal radius micro-FE section were compared between BCs using correlations. Principal strain distributions were also compared quantitatively. Cortical and trabecular loads from the platen-compression BC simulations were strongly correlated to the physiologic BC simulations. However, a 30% difference in cortical loads distally, and a 53% difference in trabecular loads proximally was observed under platen BC simulations. Also, distribution of principal strains was clearly different. Our data indicated that platen-compression BC simulations alter cortical-trabecular load sharing. Therefore, results from these analyses should be interpreted in the appropriate mechanical context for clinical evaluations of normal and pathologic mechanical behavior at the distal radius., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

Loading-induced interstitial fluid flow in the microporosities of bone is critical for osteocyte mechanotransduction and for the maintenance of tissue health, enhancing convective transport in the lacunar-canalicular system. In recent studies, our group has reported alterations of bone's vascular porosity and lacunar-canalicular system microarchitecture in a rat model of postmenopausal osteoporosis. In this work, poroelastic finite element analysis was used to investigate whether these microstructural changes can affect interstitial fluid flow around osteocytes. Animal-specific finite element models were developed combining micro-CT reconstructions of bone microstructure and measures of the poroelastic material properties. These models were used to quantify and compare loading-induced fluid flow in the lacunar-canalicular system of ovariectomized and sham-operated rats. A parametric analysis was also used to quantify the influence of the lacunar-canalicular permeability and vascular porosity on the fluid velocity magnitude. Results show that mechanically-induced interstitial fluid velocity can be significantly reduced in the lacunar-canalicular system of ovariectomized rats. Interestingly, the vascular porosity is shown to have a major influence on interstitial fluid flow, while the lacunar-canalicular permeability influence is limited when larger than 10 -20 m 2 . Altogether our results suggest that microstructural changes associated with the osteoporotic condition can negatively affect interstitial fluid flow around osteocytes in the lacunar-canalicular system of cortical bone. This fluid flow reduction could impair mechanosensation of the osteocytic network, possibly playing a role in the initiation and progression of age-related bone loss and postmenopausal osteoporosis., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

Introduction: Previously, a finite element (FE) model of the proximal tibia was developed and validated against experimentally measured local subchondral stiffness. This model indicated modest predictions of stiffness (R 2 =0.77, normalized root mean squared error (RMSE%)=16.6%). Trabecular bone though was modeled with isotropic material properties despite its orthotropic anisotropy. The objective of this study was to identify the anisotropic FE modeling approach which best predicted (with largest explained variance and least amount of error) local subchondral bone stiffness at the proximal tibia., Methods: Local stiffness was measured at the subchondral surface of 13 medial/lateral tibial compartments using in situ macro indentation testing. An FE model of each specimen was generated assuming uniform anisotropy with 14 different combinations of cortical- and tibial-specific density-modulus relationships taken from the literature. Two FE models of each specimen were also generated which accounted for the spatial variation of trabecular bone anisotropy directly from clinical CT images using grey-level structure tensor and Cowin's fabric-elasticity equations. Stiffness was calculated using FE and compared to measured stiffness in terms of R 2 and RMSE%., Results: The uniform anisotropic FE model explained 53-74% of the measured stiffness variance, with RMSE% ranging from 12.4 to 245.3%. The models which accounted for spatial variation of trabecular bone anisotropy predicted 76-79% of the variance in stiffness with RMSE% being 11.2-11.5%., Conclusions: Of the 16 evaluated finite element models in this study, the combination of Synder and Schneider (for cortical bone) and Cowin's fabric-elasticity equations (for trabecular bone) best predicted local subchondral bone stiffness., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

The main failure factors of cortical bone are aging or osteoporosis, accident and high energy trauma or physiological activities. However, the mechanism of damage evolution coupled with yield criterion is considered as one of the unclear subjects in failure analysis of cortical bone materials. Therefore, this study attempts to assess the structural response and progressive failure process of cortical bone using a brittle damaged plasticity model. For this reason, several compressive tests are performed on cortical bone specimens made of bovine femur, in order to obtain the structural response and mechanical properties of the material. Complementary finite element (FE) model of the sample and test is prepared to simulate the elastic-to-damage behavior of the cortical bone using the brittle damaged plasticity model. The FE model is validated in a comparative method using the predicted and measured structural response as load-compressive displacement through simulation and experiment. FE results indicated that the compressive damage initiated and propagated at central region where maximum equivalent plastic strain is computed, which coincided with the degradation of structural compressive stiffness followed by a vast amount of strain energy dissipation. The parameter of compressive damage rate, which is a function dependent on damage parameter and the plastic strain is examined for different rates. Results show that considering a similar rate to the initial slope of the damage parameter in the experiment would give a better sense for prediction of compressive failure., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

The effects of formalin fixation on bone material properties remain debatable. In this study, we collected 36 fresh-frozen cuboid-shaped cortical specimens from five male bovine femurs and immersed half of the specimens into 4% formalin fixation liquid for 30 days. We then conducted three-point bending tests and used both beam theory method and an optimization method combined with specimen-specific finite element (FE) models to identify material parameters. Through the optimization FE method, the formalin-fixed bones showed a significantly lower Young's modulus (-12%) compared to the fresh-frozen specimens, while no difference was observed using the beam theory method. Meanwhile, both the optimization FE and beam theory methods revealed higher effective failure strains for formalin-fixed bones compared to fresh-frozen ones (52% higher through the optimization FE method and 84% higher through the beam theory method). Hence, we conclude that the formalin fixation has a significant effect on bovine cortical bones at small, elastic, as well as large, plastic deformations.

One of the key functions of load-bearing biological materials, such as bone, dentin and sea shell, is to protect their inside fragile organs by effectively damping dynamic impact. How those materials achieve this remarkable function remains largely unknown. Using systematic finite element analyses, we study the stress wave propagation and attenuation in cortical bone at the nanoscale as a model material to examine the effects of protein viscosity, mineral fraction and staggered architecture on the elastic wave decay. It is found that the staggered arrangement, protein viscosity and mineral fraction work cooperatively to effectively attenuate the stress wave. For a typical mineral volume fraction and protein viscosity, an optimal staggered nanostructure with specific feature sizes and layouts is able to give rise to the fastest stress wave decay, and the optimal aspect ratio and thickness of mineral platelets are in excellent agreement with experimental measurements. In contrary, as the mineral volume fraction or the protein viscosity goes much higher, the structural arrangement is seen having trivial effect on the stress wave decay, suggesting that the damping properties of the composites go into the structure-insensitive regime from the structure-sensitive regime. These findings not only significantly add to our understanding of the structure-function relationship of load-bearing biological materials, and but also provide useful guidelines for the design of bio-inspired materials with superior resistance to impact loading., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

In this study, the hyperelastic models of demineralized and deproteinized bovine cortical femur bone were investigated and appropriate models were developed. Using uniaxial compression test data, the strain energy versus stretch was calculated and the appropriate hyperelastic strain energy functions were fitted on data in order to calculate the material parameters. To obtain the mechanical behavior in other loading conditions, the hyperelastic strain energy equations were investigated for pure shear and equi-biaxial tension loadings. The results showed the Mooney-Rivlin and Ogden models cannot predict the mechanical response of demineralized and deproteinized bovine cortical femur bone accurately, while the general exponential-exponential and general exponential-power law models have a good agreement with the experimental results. To investigate the sensitivity of the hyperelastic models, a variation of 10% in material parameters was performed and the results indicated an acceptable stability for the general exponential-exponential and general exponential-power law models. Finally, the uniaxial tension and compression of cortical femur bone were studied using the finite element method in VUMAT user subroutine of ABAQUS software and the computed stress-stretch curves were shown a good agreement with the experimental data., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

Purpose: The aim was to study by computer simulations the insulating role of the reactive zone surrounding a cortical osteoid osteoma (OO) in terms of electrical and thermal performance during radiofrequency ablation (RFA)., Material and Methods: We modelled a cortical OO consisting of a nidus (10 mm diameter) enclosed by a reactive zone. The OO was near a layer of cortical bone 1.5 mm thick. Trabecular bone partially surrounds the OO and there was muscle around the cortical bone layer. We modelled RF ablations with a non-cooled-tip 17-gauge needle electrode (300 s duration and 90 °C target temperature). Sensitivity analyses were conducted assuming a reactive zone electrical conductivity value (σrz) within the limits of the cortical and trabecular bone, i.e. 0.02 S/m and 0.087 S/m, respectively. In this way we were really modelling the different degrees of osteosclerosis associated with the reactive zone., Results: The presence of the reactive zone drastically reduced the maximum temperature reached outside it. The temperature drop was proportional to the thickness of the reactive zone: from 68 °C when it was absent to 44 °C when it is 7.5 mm thick. Higher nidus conductivity values (σn) implied higher temperatures, while lower temperatures meant higher σrz values. Changing σrz from 0.02 S/m to 0.087 S/m reduced lesion diameters from 2.4 cm to 1.8 cm., Conclusions: The computer results suggest that the reactive zone plays the role of insulator in terms of reducing the temperature in the surrounding area.

Accurate and reliable assessment of bone quality requires predictive methods which could probe bone microstructure and provide information on bone mechanical properties. Multiscale modelling and simulation represent a fast and powerful way to predict bone mechanical properties based on experimental information on bone microstructure as obtained through X-ray-based methods. However, technical limitations of experimental devices used to inspect bone microstructure may produce blurry data, especially in in vivo conditions. Uncertainties affecting the experimental data (input) may question the reliability of the results predicted by the model (output). Since input data are uncertain, deterministic approaches are limited and new modelling paradigms are required. In this paper, a novel stochastic multiscale model is developed to estimate the elastic properties of bone while taking into account uncertainties on bone composition. Effective elastic properties of cortical bone tissue were computed using a multiscale model based on continuum micromechanics. Volume fractions of bone components (collagen, mineral, and water) were considered as random variables whose probabilistic description was built using the maximum entropy principle. The relevance of this approach was proved by analysing a human bone sample taken from the inferior femoral neck. The sample was imaged using synchrotron radiation micro-computed tomography. 3-D distributions of Haversian porosity and tissue mineral density extracted from these images supplied the experimental information needed to build the stochastic models of the volume fractions. Thus, the stochastic multiscale model provided reliable statistical information (such as mean values and confidence intervals) on bone elastic properties at the tissue scale. Moreover, the existence of a simpler "nominal model", accounting for the main features of the stochastic model, was investigated. It was shown that such a model does exist, and its relevance was discussed.

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.