A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials

An orthotropic continuum model with substructure evolution for describing bone remodeling: an interpretation of the primary mechanism behind Wolff's law

We propose a variational approach that employs a generalized principle of virtual work to estimate both the mechanical response and the changes in living bone tissue during the remodeling process. This approach provides an explanation for the adaptive regulation of the bone substructure in the context of orthotropic material symmetry. We specifically focus upon the crucial gradual adjustment of bone tissue as a structural material that adapts its mechanical features, such as materials stiffnesses and microstructure, in response to the evolving loading conditions. We postulate that the evolution process relies on a feedback mechanism involving multiple stimulus signals. The mechanical and remodeling behavior of bone tissue is clearly a complex process that is difficult to describe within the framework of classical continuum theories. For this reason, a generalized continuum elastic theory is employed as a proper mathematical context for an adequate description of the examined phenomenon. To simplify the investigation, we considered a two-dimensional problem. Numerical simulations have been performed to illustrate bone evolution in a few significant cases: the bending of a rectangular cantilever plate and a three-point flexure test. The results are encouraging because they can replicate the optimization process observed in bone remodeling. The proposed model provides a likely distribution of stiffnesses and accurately represents the arrangement of trabeculae macroscopically described by the orthotropic symmetry directions, as supported by experimental evidence from the trajectorial theory.

Gear-shaped carbonate apatite granules with a hexagonal macropore for rapid bone regeneration

Synthetic bone grafts are in high demand owing to increased age-related bone disorders in the global aging population. Here, we report fabrication of gear-shaped granules (G-GRNs) for rapid bone healing. G-GRNs possessed six protrusions and a hexagonal macropore in the granular center. These were composed of carbonate apatite, i.e., bone mineral, microspheres with ∼1-μm micropores in the spaces between the microspheres. G-GRNs formed new bone and blood vessels (both on the granular surface and within the macropores) 4 weeks after implantation in the rabbit femur defects. The formed bone structure was similar to that of cancellous bone. The bone percentage in the defect recovered to that in a normal rabbit femur at week-4 post-implantation, and the bone percentage remained constant for the following 8 weeks. Throughout the entire period, the bone percentage in the G-GRN-implanted group was ∼10% higher than that of the group implanted with conventional carbonate apatite granules. Furthermore, a portion of the G-GRNs resorbed at week-4, and resorption continued for the following 8 weeks. Thus, G-GRNs are involved in bone remodeling and are gradually replaced with new bone while maintaining a suitable bone level. These findings provide a basis for the design and fabrication of synthetic bone grafts for achieving rapid bone regeneration.

Prediction of osteoporotic degradation of tibia human bone at trabecular scale

A theoretical numerical model is proposed to predict patient dependent osteoporotic bone degradation. The model parameters are identified through a particle swarm optimization algorithm and based on individual patient high resolution peripherical quantitative computer tomography (HRpQCT) scan data. The degradation model is based on cellular activity initiated by the elastic strain energy developed in the bone microstructure through patient's body weight. The macro (organ scale) and meso (trabecular scale) scale analyses are carried out and predicted bone volume fraction and microstructure evolution are compared with in-vivo experimental bone degradation for four elderly women over a period of 10 years. A significant correlation (r > 0.9) is observed between the model predictions and in-vivo experiments in all cases with an average deviation error of 1.46%. The model can easily be extended to other patients and provide good predictions for different population categories such as ethnicity, gender, age, etc.

A Proposal for a Novel Formulation Based on the Hyperbolic Cattaneo’s Equation to Describe the Mechano-Transduction Process Occurring in Bone Remodeling

In this paper, we propose a model for the mechanical stimulus involved in the process of bone remodeling together with its evolution over time. Accumulated evidence suggests that bone remodeling could be interpreted as a feedback control process in which the mechanical state of the bone tissue is monitored, then appropriate signals are derived from the daily mechanical usage of the bone, these signals are transmitted into the surrounding region, and then they are detected by other agents whose purpose is to adapt the bone mass to the mechanical requirements of the environment. Therefore, we employ the diffusion equation for mass transport which is improved with Cattaneo’s correction to model the stimulus. This last improvement considers the effects of relaxation and non-locality, which we believe play essential roles in signaling messengers transport phenomena and are essential to match the evidence that suggests time-dependent excitations provide a more significant response at specific frequencies. To illustrate this particular behavior, numerical simulations have been performed in a 2D framework. The results fit the central aspect addressed, related to the dependency of the time of the adaptive process of bone, suggesting that our model is promising and deserves further investigation, both theoretical and experimental.

A 3D in Silico Multi-Tissue Evolution Model Highlights the Relevance of Local Strain Accumulation in Bone Fracture Remodeling

Since 5–10% of all bone fractures result in non-healing situations, a thorough understanding of the various bone fracture healing phases is necessary to propose adequate therapeutic strategies. In silico models have greatly contributed to the understanding of the influence of mechanics on tissue formation and resorption during the soft and hard callus phases. However, the late-stage remodeling phase has not been investigated from a mechanobiological viewpoint so far. Here, we propose an in silico multi-tissue evolution model based on mechanical strain accumulation to investigate the mechanobiological regulation of bone remodeling during the late phase of healing. Computer model predictions are compared to histological data of two different pre-clinical studies of bone healing. The model predicted the bone marrow cavity re-opening and the resorption of the external callus. Our results suggest that the local strain accumulation can explain the fracture remodeling process and that this mechanobiological response is conserved among different mammal species. Our study paves the way for further understanding of non-healing situations that could help adapting therapeutic strategies to foster bone healing.

Orthotropic bone remodelling around uncemented femoral implant: a comparison with isotropic formulation

Peri-prosthetic bone adaptation has usually been predicted using subject-specific finite element analysis in combination with remodelling algorithms and assuming isotropic bone material property. The objective of the study is to develop an orthotropic bone remodelling algorithm for evaluation of peri-prosthetic bone adaptation in the uncemented implanted femur. The simulations considered loading conditions from a variety of daily activities. The orthotropic algorithm was tested on 2D and 3D models of the intact femur for verification of predicted results. The predicted orthotropic directionality, based on principal stress directions, was in agreement with the trabecular orientation in a micro-CT data of proximal femur. The validity of the proposed strain-based algorithm was assessed by comparing the predicted results of the orthotropic model with those of the strain-energy-density-based isotropic formulation. Despite agreement in cortical densities [Formula: see text], the isotropic remodelling algorithm tends to predict relatively higher values around the distal tip of the implant as compared to the orthotropic model. Both formulations predicted 4-8% bone resorption in the proximal femur. A linear regression analysis revealed a significant correlation [Formula: see text] between the stresses and strains on the cortex of the proximal femur, predicted by the isotropic and orthotropic formulations. Despite reasonable agreement in peri-prosthetic bone density distributions, the quantitative differences with isotropic model predictions highlight the combined influences of bone orthotropy and mechanical stimulus in the adaptation process.

Bio-Inspired Design of a Porous Resorbable Scaffold for Bone Reconstruction: A Preliminary Study

The study and imitation of the biological and mechanical systems present in nature and living beings always have been sources of inspiration for improving existent technologies and establishing new ones. Pursuing this line of thought, we consider an artificial graft typical in the bone reconstruction surgery with the same microstructure of the bone living tissue and examine the interaction between these two phases, namely bone and the graft material. Specifically, a visco-poroelastic second gradient model is adopted for the bone-graft composite system to describe it at a macroscopic level of observation. The second gradient formulation is employed to consider possibly size effects and as a macroscopic source of interstitial fluid flow, which is usually regarded as a key factor in bone remodeling. With the help of the proposed formulation and via a simple example, we show that the model can be used as a graft design tool. As a matter of fact, an optimization of the characteristics of the implant can be carried out by numerical investigations. In this paper, we observe that the size of the graft considerably influences the interaction between bone tissue and artificial bio-resorbable material and the possibility that the bone tissue might substitute more or less partially the foreign graft for better bone healing.

A two-layer elasto-visco-plastic rheological model for the material parameter identification of bone tissue

The ability to measure bone tissue material properties plays a major role in diagnosis of diseases and material modeling. Bone's response to loading is complex and shows a viscous contribution to stiffness, yield and failure. It is also ductile and damaging and exhibits plastic hardening until failure. When performing mechanical tests on bone tissue, these constitutive effects are difficult to quantify, as only their combination is visible in resulting stress-strain data. In this study, a methodology for the identification of stiffness, damping, yield stress and hardening coefficients of bone from a single cyclic tensile test is proposed. The method is based on a two-layer elasto-visco-plastic rheological model that is capable of reproducing the specimens' pre- and postyield response. The model's structure enables for capturing the viscously induced increase in stiffness, yield, and ultimate stress and for a direct computation of the loss tangent. Material parameters are obtained in an inverse approach by optimizing the model response to fit the experimental data. The proposed approach is demonstrated by identifying material properties of individual bone trabeculae that were tested under wet conditions. The mechanical tests were conducted according to an already published methodology for tensile experiments on single trabeculae. As a result, long-term and instantaneous Young's moduli were obtained, which were on average 3.64 GPa and 5.61 GPa, respectively. The found yield stress of 16.89 MPa was lower than previous studies suggest, while the loss tangent of 0.04 is in good agreement. In general, the two-layer model was able to reproduce the cyclic mechanical test data of single trabeculae with an root-mean-square error of 2.91 ± 1.77 MPa. The results show that inverse rheological modeling can be of great advantage when multiple constitutive contributions shall be quantified based on a single mechanical measurement.

A review of recent developments in mathematical modeling of bone remodeling

In this article, we summarize the developments in the mathematical modeling of the mechanics of bone and related biological phenomena. We will devote special attention to the results of the last 10–15 years, although we will cover some relevant classical work to better frame the more recent researches. We will propose a division of the literature based on the main aim of the model (mechanical/biomathematical) and the type of biological phenomena considered (stimulus, growth, cell population dynamics). Finally, we will suggest some possible directions for future investigations.

Computational study of estimating 3D trabecular bone microstructure for the volume of interest from CT scan data

Inspired by the self-optimizing capabilities of bone, a new concept of bone microstructure reconstruction has been recently introduced by using 2D synthetic skeletal images. As a preliminary clinical study, this paper proposes a topology optimization-based method that can estimate 3D trabecular bone microstructure for the volume of interest (VOI) from 3D computed tomography (CT) scan data with enhanced computational efficiency and phenomenological accuracy. For this purpose, a localized finite element (FE) model is constructed by segmenting a target bone from CT scan data and determining the physiological local loads for the VOI. Then, topology optimization is conducted with multiresolution bone mineral density (BMD) deviation constraints to preserve the patient-specific spatial bone distribution obtained from the CT scan data. For the first time, to our knowledge, this study has demonstrated that 60-μm resolution trabecular bone images can be reconstructed from 600-μm resolution CT scan data (a 62-year-old woman with no metabolic bone disorder) for the 4 VOIs in the proximal femur. The reconstructed trabecular bone includes the characteristic trabecular patterns and has morphometric indices that are in good agreement with the anatomical data in the literature. As for computational efficiency, the localization for the VOI reduces the number of FEs by 99%, compared with that of the full FE model. Compared with the previous single-resolution BMD deviation constraint, the proposed multiresolution BMD deviation constraints enable at least 65% and 47% reductions in the number of iterations and computing time, respectively. These results demonstrate the clinical feasibility and potential of the proposed method.

Computational Investigation on the Biomechanical Responses of the Osteocytes to the Compressive Stimulus: A Poroelastic Model

Osteocytes, the major type of bone cells embedded in the bone matrix and surrounded by the lacunar and canalicular system, can serve as biomechanosensors and biomechanotranducers of the bone. Theoretical analytical methods have been employed to investigate the biomechanical responses of osteocytes in vivo; the poroelastic properties have not been taken into consideration in the three-dimensional (3D) finite element model. In this study, a 3D poroelastic idealized finite element model was developed and was used to predict biomechanical behaviours (maximal principal strain, pore pressure, and fluid velocity) of the osteocyte-lacunar-canalicular system under 150-, 1000-, 3000-, and 5000-microstrain compressive loads, respectively, representing disuse, physiological, overuse, and pathological overload loading stimuli. The highest local strain, pore pressure, and fluid velocity were found to be highest at the proximal region of cell processes. These data suggest that the strain, pore pressure, and fluid velocity of the osteocyte-lacunar-canalicular system increase with the global loading and that the poroelastic material property affects the biomechanical responses to the compressive stimulus. This new model can be used to predict the mechanobiological behaviours of osteocytes under the four different compressive loadings and may provide an insight into the mechanisms of mechanosensation and mechanotransduction of the bone.

Multiphysics of bone remodeling: A 2D mesoscale activation simulation

In this work, we present an evolutive trabecular model for bone remodeling based on a boundary detection algorithm accounting for both biology and applied mechanical forces, known to be an important factor in bone evolution. A finite element (FE) numerical model using the Abaqus/Standard® software was used with a UMAT subroutine to solve the governing coupled mechanical-biological non-linear differential equations of the bone evolution model. The simulations present cell activation on a simplified trabeculae configuration organization with trabecular thickness of 200µm. For this activation process, the results confirm that the trabeculae are mainly oriented in the active direction of the principal mechanical stresses and according to the principal applied mechanical load directions. The trabeculae surface activation is clearly identified and can provide understanding of the different bone cell activations in more complex geometries and load conditions.