A lattice-based approach to model distraction osteogenesis

Computational models of bone fracture healing and applications: a review

Fracture healing is a very complex physiological process involving multiple events at different temporal and spatial scales, such as cell migration and tissue differentiation, in which mechanical stimuli and biochemical factors assume key roles. With the continuous improvement of computer technology in recent years, computer models have provided excellent solutions for studying the complex process of bone healing. These models not only provide profound insights into the mechanisms of fracture healing, but also have important implications for clinical treatment strategies. In this review, we first provide an overview of research in the field of computational models of fracture healing based on CiteSpace software, followed by a summary of recent advances, and a discussion of the limitations of these models and future directions for improvement. Finally, we provide a systematic summary of the application of computational models of fracture healing in three areas: bone tissue engineering, fixator optimization and clinical treatment strategies. The application of computational models of bone healing in clinical treatment is immature, but an inevitable trend, and as these models become more refined, their role in guiding clinical treatment will become more prominent.

In vivo and in silico monitoring bone regeneration during distraction osteogenesis of the mouse femur

BACKGROUND AND OBJECTIVE

Distraction osteogenesis (DO) is a mechanobiological process of producing new bone by gradual and controlled distraction of the surgically separated bone segments. Mice have been increasingly used to study the role of relevant biological factors in regulating bone regeneration during DO. However, there remains a lack of in silico DO models and related mechano-regulatory tissue differentiation algorithms for mouse bone. This study sought to establish an in silico model based on in vivo experimental data to simulate the bone regeneration process during DO of the mouse femur.

METHODS

In vivo micro-CT, including time-lapse morphometry was performed to monitor the bone regeneration in the distraction gap. A 2D axisymmetric finite element model, with a geometry originating from the experimental data, was created. Bone regeneration was simulated with a fuzzy logic-based two-stage (distraction and consolidation) mechano-regulatory tissue differentiation algorithm, which was adjusted from that used for fracture healing based on our in vivo experimental data. The predictive potential of the model was further tested with varied distraction frequencies and distraction rates.

RESULTS

The computational simulations showed similar bone regeneration patterns to those observed in the micro-CT data from the experiment throughout the DO process. This consisted of rapid bone formation during the first 10 days of the consolidation phase, followed by callus reshaping via bone remodeling. In addition, the computational model predicted a faster and more robust bone healing response as the model's distraction frequency was increased, whereas higher or lower distraction rates were not conducive to healing.

CONCLUSIONS

This in silico model could be used to investigate basic mechanobiological mechanisms involved in bone regeneration or to optimize DO strategies for potential clinical applications.

Novel approach to estimate distraction forces in distraction osteogenesis and application in the human lower leg

PURPOSE

In distraction osteogenesis (DO) of long bones, new bone tissue is distracted to lengthen limbs or reconstruct bone defects. However, mechanical boundary conditions in human application such as arising forces are mainly based on limited empirical data. Our aim was the numerical determination of the callus distraction force (CDF) and the total distraction force (TDF) during DO in the tibia of adults to advance the understanding of callus tissue behavior and optimize DO procedures.

METHOD

We implemented a mathematical model based on an animal experiment to enable the calculation of forces arising while distracting callus tissue, excluding the influence of surrounding soft tissue (muscles, skin etc.). The CDF progression for the distraction period was calculated using the implemented model and varying distraction parameters (initial gap, area, step size, time interval, length). Further, we estimated the CDF based on reported forces in humans and compared the results to our model predictions. In addition, we calculated the TDF based on our CDF predictions in combination with reported resisting forces due to soft tissue presence in human cadavers. Finally, we compared the progressions to in vivo TDF measurements for validation.

RESULTS

Due to relaxation, a peak and resting CDF is observable for each distraction step. Our biomechanical results show a non-linear degressive increase of the resting and peak CDF at the beginning and a steady non-linear increase thereafter. The calculated resting and peak CDF in the tibial metaphysis ranged from 0.00075 to 0.0089 N and 0.22-2.6 N at the beginning as well as 20-25 N and 70-75 N at the end of distraction. The comparison to in vivo data showed the plausibility of our predictions and resulted in a 10-33% and 10-23% share of resting CDF in the total resting force for bone transport and elongation, respectively. Further, the percentage of peak CDF in total peak force was found to be 29-58% and 27-55% for bone transport and elongation, respectively. Moreover, our TDF predictions were valid based on the comparison to in vivo forces and resulted in a degressive increase from 6 to 125 N for the peak TDF and from 5 to 76 N for the resting TDF.

CONCLUSION

Our approach enables the estimation of forces arising due to the distraction of callus tissue in humans and results in plausible force progressions as well as absolute force values for the callus distraction force during DO. In combination with measurements of resisting forces due to the presence of soft tissue, the total distraction force in DO may also be evaluated. We thus propose the application of this method to approximate the behavior of mechanical callus properties during DO in humans as an alternative to in vivo measurements.

Multiscale modeling of bone tissue mechanobiology

Mechanical environment has a crucial role in our organism at the different levels, ranging from cells to tissues and our own organs. This regulatory role is especially relevant for bones, given their importance as load-transmitting elements that allow the movement of our body as well as the protection of vital organs from load impacts. Therefore bone, as living tissue, is continuously adapting its properties, shape and repairing itself, being the mechanical loads one of the main regulatory stimuli that modulate this adaptive behavior. Here we review some key results of bone mechanobiology from computational models, describing the effect that changes associated to the mechanical environment induce in bone response, implant design and scaffold-driven bone regeneration.

Comparison of methods for assigning the material properties of the distraction callus in computational models

In silico models of distraction osteogenesis and fracture healing usually assume constant mechanical properties for the new bone tissue generated. In addition, these models do not always account for the porosity of the woven bone and its evolution. In this study, finite element analyses based on computed tomography (CT) are used to predict the stiffness of the callus until 69 weeks after surgery using 15 CT images obtained at different stages of an experiment on bone transport, technique in which distraction osteogenesis is used to correct bone defects. Three different approaches were used to assign the mechanical properties to the new bone tissue. First, constant mechanical properties of the hard callus tissue and no porosity were assumed. Nevertheless, this approach did not show good correlations. Second, random variations in the elastic modulus and porosity of the woven bone were taken from previous experimental studies. Finally, the elastic properties of each element were assigned depending on gray scale in CT images. The numerically predicted callus stiffness was compared with previous in vivo measurements. It was concluded firstly that assignment depending on gray scale is the method that provides the best results and secondly that the method that considers a random distribution of porosity and elastic modulus of the callus is also suitable to predict the callus stiffness from 15 weeks after surgery. This finding provides a method for assigning the material properties of the distraction callus, which does not require CT images and may contribute to improve current in silico models.

Integration of in vitro and in silico Models Using Bayesian Optimization With an Application to Stochastic Modeling of Mesenchymal 3D Cell Migration

Cellular migration plays a crucial role in many aspects of life and development. In this paper, we propose a computational model of 3D migration that is solved by means of the tau-leaping algorithm and whose parameters have been calibrated using Bayesian optimization. Our main focus is two-fold: to optimize the numerical performance of the mechano-chemical model as well as to automate the calibration process of in silico models using Bayesian optimization. The presented mechano-chemical model allows us to simulate the stochastic behavior of our chemically reacting system in combination with mechanical constraints due to the surrounding collagen-based matrix. This numerical model has been used to simulate fibroblast migration. Moreover, we have performed in vitro analysis of migrating fibroblasts embedded in 3D collagen-based fibrous matrices (2 mg/ml). These in vitro experiments have been performed with the main objective of calibrating our model. Nine model parameters have been calibrated testing 300 different parametrizations using a completely automatic approach. Two competing evaluation metrics based on the Bhattacharyya coefficient have been defined in order to fit the model parameters. These metrics evaluate how accurately the in silico model is replicating in vitro measurements regarding the two main variables quantified in the experimental data (number of protrusions and the length of the longest protrusion). The selection of an optimal parametrization is based on the balance between the defined evaluation metrics. Results show how the calibrated model is able to predict the main features observed in the in vitro experiments.