Elastic Modulus of Woven Bone: Correlation with Evolution of Porosity and X-ray Greyscale.

The woven bone created during the healing of bone regeneration processes is characterized as being extremely inhomogeneous and having a variable stiffness that increases with time. Therefore, it is important to study how the mechanical properties of woven bone are dependent on its microarchitecture and especially on its porosity and mineral content. The porosity and the x-ray greyscale of specimens taken from bone transport studies in sheep were assessed by means of ex vivo imaging. Our study demonstrates that the porosity of the woven bone in the distraction area diminishes during the healing process from 73.3% 35 days after surgery to 31.9% 525 days after surgery. In addition, the woven bone's porosity is negatively correlated with its Young's modulus. The x-ray greyscale, was measured as an indicator of the level of mineralization of the woven bone. Greyscale index has been demonstrated to be inversely proportional to porosity and to increase to up to 60-80% of the level in cortical bone. The results of this study may contribute to the development of micromechanical models of woven bone and improvements in in silico modelling.

Data-Driven Computational Simulation in Bone Mechanics.

The data-driven approach was formally introduced in the field of computational mechanics just a few years ago, but it has gained increasing interest and application as disruptive technology in many other fields of physics and engineering. Although the fundamental bases of the method have been already settled, there are still many challenges to solve, which are often inherently linked to the problem at hand. In this paper, the data-driven methodology is applied to a particular problem in tissue biomechanics, a context where this approach is particularly suitable due to the difficulty in establishing accurate and general constitutive models, due to the intrinsic intra and inter-individual variability of the microstructure and associated mechanical properties of biological tissues. The problem addressed here corresponds to the characterization and mechanical simulation of a piece of cortical bone tissue. Cortical horse bone tissue was mechanically tested using a biaxial machine. The displacement field was obtained by means of digital image correlation and then transformed into strains by approximating the displacement derivatives in the bone virtual geometric image. These results, together with the approximated stress state, assumed as uniform in the small pieces tested, were used as input in the flowchart of the data-driven methodology to solve several numerical examples, which were compared with the corresponding classical model-based fitted solution. From these results, we conclude that the data-driven methodology is a useful tool to directly simulate problems of biomechanical interest without the imposition (model-free) of complex spatial and individually-varying constitutive laws. The presented data-driven approach recovers the natural spatial variation of the solution, resulting from the complex structure of bone tissue, i.e. heterogeneity, microstructural hierarchy and multifactorial architecture, making it possible to add the intrinsic stochasticity of biological tissues into the data set and into the numerical approach.

An interspecies computational study on limb lengthening.

Distraction osteogenesis is a surgical technique that produces large volumes of new bone by gradually separating two osteotomized bone segments. A previously proposed mechanical-based model that includes the effect of pre-traction stresses (stress level in the gap tissue before each distraction step) during limb lengthening is used here. In the present work, the spatial and temporal patterns of tissue distribution during distraction osteogenesis in different species (sheep, rabbit) and in the human are compared numerically to predict experimental results. Interspecies differential characteristics such as size, distraction protocol, and rate of distraction, among others, are chosen according to experiments. Tissue distributions and reaction forces are then analysed as indicators of the healing pattern. The results obtained are in agreement with experimental findings regarding both tissue distribution and reaction forces. The ability of the model to qualitatively predict the two animal models and the human healing pattern in distraction osteogenesis indicates its potential in understanding the influence of mechanics in this complex process.

In silico dynamic characterization of the femur: Physiological versus mechanical boundary conditions.

It is established that bone tissue adapts and responds to mechanical loading. Several studies have suggested an existence of positive influence of vibration on the bone mass maintenance. Thus, some bone regeneration therapies are based on vibration of bone tissue under circumstances of disease to stimulate its formation. Frequency of loading should be properly selected and therefore a correct characterization of the dynamic properties of this tissue may be critical for the success of such orthopedic techniques. On the other hand, many studies implement vibration techniques with in silico models. Numerical results are exclusively dependent on properties of bone tissue, i.e. geometry, density distribution and stiffness, as well as boundary conditions. In the present study, the influence of boundary conditions and material properties on the dynamic characteristics of bone tissue was explored in a human femur. Bone shape and density were directly reconstructed from computer tomographies, whereas natural frequencies and modes of vibration were obtained for different boundary conditions including physiological and mechanical ones. Results of this study show the moderate effect of material properties compared to the much substantial effect of boundary conditions. A factor of 2 in the natural frequency was obtained depending on imposed boundary conditions, highlighting the importance in the selection of appropriate conditions in the analysis of the bone organ.

In silico design of magnesium implants: Macroscopic modeling.

Magnesium-based biomedical implants offer many advantages versus traditional ones although some challenges are still present. In this context, mathematical modeling and computational simulation may be a useful and complementary tool to evaluate in silico the performance of magnesium biomaterials under different conditions. In this paper, a phenomenologically-based model to simulate magnesium corrosion is developed. The model describes the physico-chemical interactions and evolution of species present in this phenomenon. A set of 7 species is considered in the model, which allows to simulate hydrogen release, pH evolution, corrosion products formation as well as degradation of magnesium. The model is developed under the continuum media theory and is implemented in a finite element framework. In the results section, the effect of model parameters on outcomes is firstly explored. Second, model results are qualitative validated versus two examples of application found in the literature. Two main conclusions are derived from this work: (i) the model captures well the experimental trends and allows to analyze the main variables present in magnesium corrosion, (ii) even though further validation is needed the model may be a useful standard in the design of degradable metal implants.

Model of dissolution in the framework of tissue engineering and drug delivery.

Dissolution phenomena are ubiquitously present in biomaterials in many different fields. Despite the advantages of simulation-based design of biomaterials in medical applications, additional efforts are needed to derive reliable models which describe the process of dissolution. A phenomenologically based model, available for simulation of dissolution in biomaterials, is introduced in this paper. The model turns into a set of reaction-diffusion equations implemented in a finite element numerical framework. First, a parametric analysis is conducted in order to explore the role of model parameters on the overall dissolution process. Then, the model is calibrated and validated versus a straightforward but rigorous experimental setup. Results show that the mathematical model macroscopically reproduces the main physicochemical phenomena that take place in the tests, corroborating its usefulness for design of biomaterials in the tissue engineering and drug delivery research areas.

Mechanical characterization via nanoindentation of the woven bone developed during bone transport.

Nanoindentation has been used successfully in the determination of the mechanical properties of bone. Its application in fracture healing provides information on the evolution of material properties of the woven bone during regeneration process. However, this technique has not been applied in assessing the mechanical properties of woven bone during distraction osteogenesis. Therefore, the aim of this work is to evaluate the spatial and temporal variations of the elastic modulus of the woven bone generated during the bone transport process. Callus samples were harvested from intervened animals at different time points during the bone transport process (35, 50, 79, 98, 161 and 525 days after surgery) for nanoindentation measurements. Results clearly showed that the mean elastic modulus of the woven bone increased during the bone transport process reaching 77% of value for cortical bone after 525 days (from 7GPa 35 days after surgery to 14GPa 525 days after surgery approximately). Woven bone generated during bone transport seems to present similar evolution of elastic modulus with time as values reported for fracture healing. Furthermore, different spatial variations of elastic modulus within the callus were found for different stages of the process.

Distraction osteogenesis device to estimate the axial stiffness of the callus in Vivo.

Knowing the evolution of callus stiffness is very important in distraction osteogenesis and bone healing. It allows the characterization of the bone maturation process and the assessment of the moment to retire the fixator. A new distractor device that monitors the callus axial stiffness is presented in this study. It quantifies the callus stiffness during the bone transport process with some advantages over previous methods to assess stiffness during simple distraction and bone healing. This device avoids a misalignment between bone segments, uses real load conditions, monitors forces continuously, does not involve radiation for patients, and allows the study of the complete distraction process, i.e., the distraction and consolidation phases. The device was calibrated in vitro simulating different real bone load conditions depending on the stage of the process. The stiffness of the callus could be estimated for values between 4.2 N/mm and 9066.8 N/mm. The average relative error in measurements carried out in in vitro calibration tests was 7.8% during the distraction phase and 9.5% during the consolidation phase. These results improve the accuracy and increase the callus stiffness range of estimation with respect to other devices in the literature. In addition, the device was used successfully in vivo in a preliminary experiment.

In Vivo Gait Analysis During Bone Transport.

The load bearing characteristics of the intervened limb over time in vivo are important to know in distraction osteogenesis and bone healing for the characterization of the bone maturation process. Gait analyses were performed for a group of sheep in which bone transport was carried out. The ground reaction force was measured by means of a force platform, and the gait parameters (i.e., the peak, the mean vertical ground reaction force and the impulse) were calculated during the stance phase for each limb. The results showed that these gait parameters decreased in the intervened limb and interestingly increased in the other limbs due to the implantation of the fixator. Additionally, during the process, the gait parameters exponentially approached the values for healthy animals. Corresponding radiographies showed an increasing level of ossification in the callus. This study shows, as a preliminary approach to be confirmed with more experiments, that gait analysis could be used as an alternative method to control distraction osteogenesis or bone healing. For example, these analyses could determine the appropriate time to remove the fixator. Furthermore, gait analysis has advantages over other methods because it provides quantitative data and does not require instrumented fixators.

In Vivo Mechanical Characterization of the Distraction Callus During Bone Consolidation.

Understanding the evolution of callus mechanical properties over time provides insights in the mechanobiology of fracture healing and tissue differentiation, can be used to validate numerical models, and informs clinical practice. Bone transport experiments were performed in sheep, in which a distractor type Ilizarov was implanted. The forces through the fixator evolution were measured and the callus stiffness was estimated from these forces. Computerized tomography images were taken and bone volume of the callus at different stages was obtained. The results showed that the maximum bone tissue production rate (0.146 cm³/day) was achieved 20 days after the end of the distraction phase. 50 days after the end of the distraction phase, the callus was ossified completely and had its maximum volume, 6-10 cm³ In addition, 80-90% of the load sustained by the operated limb was recovered and the callus stiffness increased exponentially until 5.4-11.4 kN/mm, still below 10% of the healthy level of callus stiffness. The effects of the bony bridging of the callus and the time of the fixator removal on callus force, stiffness and volume were analyzed. These outcomes allowed relating quantifiable biological aspects (callus volume and tissue production rate) with mechanical parameters (callus force and stiffness) using data from the same experiment.

A lattice-based approach to model distraction osteogenesis.

Distraction osteogenesis is a well-known technique in which new bone tissue is created when a distraction displacement is applied through an external frame. This orthopedic process is nowadays focus of intense research, both experimentally and numerically, as there are still many aspects not well understood. The aim of this study is to simulate bone distraction by means of a combined discrete-continuum approach based on a lattice formulation. Existing computational models simulate the main processes of distraction osteogenesis from a continuum perspective, considering as state variables the population of cells and tissue distributions. Results of the continuum and lattice-based approaches are similar with respect to the global evolution of the different cells but rather different in terms of the type of ossification process. Differences in the size of the soft interzone in the gap have also been found. In addition, the discrete-continuum formulation allows including a more realistic approach of the migration/proliferation process with a discrete random walk model instead of the Fick's law used in continuum approaches. Also, blood vessel growth can be simulated explicitly in this model with the inclusion of the endothelial cells. Further study is needed to provide additional insights to understand coupled phenomena at different scales in the cell-tissue interactions. However this work provides a first preliminary step for improving multiscale models.

Three-dimensional simulation of mandibular distraction osteogenesis: mechanobiological analysis.

Distraction osteogenesis is a surgical process for reconstruction of skeletal deformities, which has been widely investigated from the clinical perspective. However, little has been analyzed about the capability of numerical models to predict the clinical outcome generated by distraction. Therefore, this article presents a finite element analysis of the mechanobiological behavior of a pediatric patient's mandible with hemifacial microsomia during the distraction process. It focuses on the three-dimensional simulation of a long bone defect in the ramus of the mandible and introduces additional aspects to be considered in the computational simulation as compared to the bidimensional simulation. The evolution of the different tissues within the gap is evaluated and in order to check the effectiveness of the model, the predicted numerical outcome will be compared from a qualitative point of view with radiographies provided by the surgeons. It is shown that the morphology of the mandible changed in a similar manner than that observed clinically. These results reveal that three-dimensional models are useful tools in the predictive assessment of mandibular distraction osteogenesis.

Effect of the fixator stiffness on the young regenerate bone after bone transport: computational approach.

Bone transport is a well accepted technique for the treatment of large bony defects. This process is mechanically driven, where mechanical forces play a central role in the development of tissues within the distracted gap. One of the most important mechanical factors that conditions the success of bone regeneration during distraction osteogenesis is the fixator stiffness not only during the distraction phase but also during the consolidation phase. Therefore, the aim of the present work is to evaluate the effect of the stiffness of the fixator device on the interfragmentary movements and the tissue outcome during the consolidation phase. A previous differentiation model (Claes and Heigele, 1999) is extended in order to take into account the different behaviors of the tissues in tension and compression. The numerical results that were computed concur with experimental findings; a stiff fixator promotes bone formation while the excessive motion induced by extremely flexible fixators is adverse for bony bridging. Experimental interfragmentary movement is similar to that computed numerically.

Biomechanical response of a mandible in a patient affected with hemifacial microsomia before and after distraction osteogenesis.

Distraction osteogenesis (DO) has gained wide acceptance in the craniofacial surgery due to the huge possibilities it offers. However this orthopaedic field is under continuous development as it still presents uncertainties. In this context, numerical modelling/analysis may help us to design patient specific treatments once they have been experimentally verified. This paper presents a finite element analysis of the biomechanical behavior of a patient's mandible with hemifacial microsomia (HFM) before and after distraction. In order to check the effectiveness of the clinical protocol, the predicted biomechanical response will also be compared with that of a symmetrical healthy mandible. Strain and displacement fields, masticatory forces as well as reaction forces at the condyles are evaluated in each mandible analyzed. The results show that the present model is a useful tool to understand the normal function of the mandible and to predict changes due to alterations in the mandible geometry, such as those occurring in hemifacial microsomia.

Modeling distraction osteogenesis: analysis of the distraction rate.

Distraction osteogenesis is a useful technique aimed at inducing bone formation in widespread clinical applications. One of the most important factors that conditions the success of bone regeneration is the distraction rate. Since the mechanical environment around the osteotomy site is one of the main factors that affects both quantity and quality of the regenerated bone, we have focused on analyzing how the distraction rate influences on the mechanical conditions and tissue regeneration. Therefore, the aim of the present work is to explore the potential of a mathematical algorithm to simulate clinically observed distraction rate related phenomena that occur during distraction osteogenesis. Improvements have been performed on a previous model (Gómez-Benito et al. in J Theor Biol 235:105-119, 2005) in order to take into account the load history. The results obtained concur with experimental findings: a slow distraction rate results in premature bony union, whereas a fast rate results in a fibrous union. Tension forces in the interfragmentary gap tissue have also been estimated and successfully compared with experimental measurements.