Growth mixture model of distraction osteogenesis: effect of pre-traction stresses

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.

Mechanobiology of Bone Consolidation During Distraction Osteogenesis: Bone Lengthening Vs. Bone Transport

Bone lengthening and bone transport are regeneration processes that commonly rely on distraction osteogenesis, a widely accepted surgical procedure to deal with numerous bony pathologies. Despite the extensive study in the literature of the influence of biomechanical factors, a lack of knowledge about their mechanobiological differences prevents a clinical particularization. Bone lengthening treatments were performed on sheep metatarsus by reproducing the surgical and biomechanical protocol of previous bone transport experiments. Several in vivo monitoring techniques were employed to build an exhaustive comparison: gait analysis, radiographic and CT assessment, force measures through the fixation, or mechanical characterization of the new tissue. A significant initial loss of the bearing capacity, quantified by the ground reaction forces and the limb contact time with the ground, is suffered by the bone lengthening specimens. The potential effects of this anomaly on the musculoskeletal force distribution and the evolution of the bone callus elastic modulus over time are also analyzed. Imaging techniques also seem to reveal lower bone volume in the bone lengthening callus than in the bone transport one, but an equivalent mineralization rate. The simultaneous quantification of biological and mechanical parameters provides valuable information for the daily clinical routine and numerical tools development.

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.

Simulating lateral distraction osteogenesis

Distraction osteogenesis is an effective method for generating large amounts of bone in situ for treating pathologies such as large bone defects or skeletal malformations, for instance leg-length discrepancies. While an optimized distraction procedure might have the potential to reduce the rate of complications significantly, our knowledge of the underlying mechanobiological processes is still insufficient for systematic optimization of treatment parameters such as distraction rate or fixation stiffness. We present a novel numerical model of lateral distraction osteogenesis, based on a mechanically well-controlled in vivo experiment. This model extends an existing numerical model of callus healing with viscoplastic material properties for describing stress relaxation and stimuli history-dependent tissue differentiation, incorporating delay and memory effects. A reformulation of appositional growth based non-local biological stimuli in terms of spatial convolution as well as remeshing and solution-mapping procedures allow the model to cope with severe mesh distortions associated with large plastic deformations. With these enhancements, our model is capable of replicating the in vivo observations for lateral distraction osteogenesis in sheep using the same differentiation rules and the same set of parameters that successfully describes callus healing in sheep, indicating that tissue differentiation hypotheses originally developed for fracture healing scenarios might indeed be applicable to distraction as well. The response of the model to modified distraction parameters corresponds to existing studies, although the currently available data is insufficient for rigorous validation. As such, this study provides a first step towards developing models that can serve as tools for identifying both interesting research questions and, eventually, even optimizing clinical procedures once better data for calibration and validation becomes available.

A Computed Microtomography Method for Understanding Epiphyseal Growth Plate Fusion

The epiphyseal growth plate is a developmental region responsible for linear bone growth, in which chondrocytes undertake a tightly regulated series of biological processes. Concomitant with the cessation of growth and sexual maturation, the human growth plate undergoes progressive narrowing, and ultimately disappears. Despite the crucial role of this growth plate fusion "bridging" event, the precise mechanisms by which it is governed are complex and yet to be established. Progress is hindered by the current methods for growth plate visualization; these are invasive and largely rely on histological procedures. Here, we describe our non-invasive method utilizing synchrotron X-ray computed microtomography for the examination of growth plate bridging, which ultimately leads to its closure coincident with termination of further longitudinal bone growth. We then apply this method to a dataset obtained from a benchtop micro computed tomography scanner to highlight its potential for wide usage. Furthermore, we conduct finite element modeling at the micron-scale to reveal the effects of growth plate bridging on local tissue mechanics. Employment of these 3D analyses of growth plate bone bridging is likely to advance our understanding of the physiological mechanisms that control growth plate fusion.

Novel systems for the application of isolated tensile, compressive, and shearing stimulation of distraction callus tissue

Background

Distraction osteogenesis is a procedure widely used for the correction of large bone defects. However, a high complication rate persists, likely due to insufficient stability during maturation. Numerical fracture healing models predict bone regeneration under different mechanical conditions allowing fixation stiffness optimization. However, most models apply a linear elastic material law inappropriate for the transient stresses/strains present during limb lengthening or segment transport. They are also often validated using in vivo osteotomy models lacking precise mechanical regulation due to the unavoidable stimulation of secondary interfragmentary motion during ambulation under finitely stiff fixation. Therefore, in order to create a robust numerical model of distraction osteogenesis, it is necessary to both characterize the new tissue’s viscoelasticity during distraction and determine the influence of strictly isolated stimulation in each loading mode (tension, compression, and shear) to account for potential differences in mechanical and histological response.

Aim

Two electromechanical fixators with integrated load cells were designed to precisely perform and monitor in vivo lateral distraction and isolated stimulation in sheep tibiae using a mobile, hydroxyapatite-coated titanium plate. The novel surgical procedure circumvents osteotomy, eliminating the undesirable and unquantifiable mechanical stimulation during ambulation.

Methods

After a 10-day post-surgery latency period, two 0.275 mm distraction steps were performed daily for 10 days. The load cell collected data before, during, and after each distraction step and was terminated after no less than one minute from the time of distraction. A 7-day consolidation period separated the distraction phase and 18-day stimulation phase. Stimulation was carried out in isolated tension, compression, or shear while recording force/time data. Each stimulation session consisted of 120 cycles with a magnitude of either 0.1 mm or 0.6 mm in the tension and compression groups and 1.0 mm in the shear group. The animals were euthanized after a 3-day holding period following stimulation.

Results

Our initial results show that the tissue progressively stiffens and maintains an increasingly large residual traction. The force curves during compressive stimulation show a progressive drift from compression toward tension. We hypothesize that this behavior may be due to the preferential flow of fluid outward from the tissue and a greater resistance to reabsorption during the plate’s return to the starting position.

Bone fracture healing in mechanobiological modeling: A review of principles and methods

Bone fracture is a very common body injury. The healing process is physiologically complex, involving both biological and mechanical aspects. Following a fracture, cell migration, cell/tissue differentiation, tissue synthesis, and cytokine and growth factor release occur, regulated by the mechanical environment. Over the past decade, bone healing simulation and modeling has been employed to understand its details and mechanisms, to investigate specific clinical questions, and to design healing strategies. The goal of this effort is to review the history and the most recent work in bone healing simulations with an emphasis on both biological and mechanical properties. Therefore, we provide a brief review of the biology of bone fracture repair, followed by an outline of the key growth factors and mechanical factors influencing it. We then compare different methodologies of bone healing simulation, including conceptual modeling (qualitative modeling of bone healing to understand the general mechanisms), biological modeling (considering only the biological factors and processes), and mechanobiological modeling (considering both biological aspects and mechanical environment). Finally we evaluate different components and clinical applications of bone healing simulation such as mechanical stimuli, phases of bone healing, and angiogenesis.

In silico bone mechanobiology: modeling a multifaceted biological system

Mechanobiology, the study of the influence of mechanical loads on biological processes through signaling to cells, is fundamental to the inherent ability of bone tissue to adapt its structure in response to mechanical stimulation. The immense contribution of computational modeling to the nascent field of bone mechanobiology is indisputable, having aided in the interpretation of experimental findings and identified new avenues of inquiry. Indeed, advances in computational modeling have spurred the development of this field, shedding new light on problems ranging from the mechanical response to loading by individual cells to tissue differentiation during events such as fracture healing. To date, in silico bone mechanobiology has generally taken a reductive approach in attempting to answer discrete biological research questions, with research in the field broadly separated into two streams: (1) mechanoregulation algorithms for predicting mechanobiological changes to bone tissue and (2) models investigating cell mechanobiology. Future models will likely take advantage of advances in computational power and techniques, allowing multiscale and multiphysics modeling to tie the many separate but related biological responses to loading together as part of a larger systems biology approach to shed further light on bone mechanobiology. Finally, although the ever‐increasing complexity of computational mechanobiology models will inevitably move the field toward patient‐specific models in the clinic, the determination of the context in which they can be used safely for clinical purpose will still require an extensive combination of computational and experimental techniques applied to in vitro and in vivo applications. WIREs Syst Biol Med 2016, 8:485–505. doi: 10.1002/wsbm.1356

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Influence of high-frequency cyclical stimulation on the bone fracture-healing process: mathematical and experimental models

Mechanical stimulation affects the evolution of healthy and fractured bone. However, the effect of applying cyclical mechanical stimuli on bone healing has not yet been fully clarified. The aim of the present study was to determine the influence of a high-frequency and low-magnitude cyclical displacement of the fractured fragments on the bone-healing process. This subject is studied experimentally and computationally for a sheep long bone. On the one hand, the mathematical computational study indicates that mechanical stimulation at high frequencies can stimulate and accelerate the process of chondrogenesis and endochondral ossification and consequently the bony union of the fracture. This is probably achieved by the interstitial fluid flow, which can move nutrients and waste from one place to another in the callus. This movement of fluid modifies the mechanical stimulus on the cells attached to the extracellular matrix. On the other hand, the experimental study was carried out using two sheep groups. In the first group, static fixators were implanted, while, in the second one, identical devices were used, but with an additional vibrator. This vibrator allowed a cyclic displacement with low magnitude and high frequency (LMHF) to be applied to the fractured zone every day; the frequency of stimulation was chosen from mechano-biological model predictions. Analysing the results obtained for the control and stimulated groups, we observed improvements in the bone-healing process in the stimulated group. Therefore, in this study, we show the potential of computer mechano-biological models to guide and define better mechanical conditions for experiments in order to improve bone fracture healing. In fact, both experimental and computational studies indicated improvements in the healing process in the LMHF mechanically stimulated fractures. In both studies, these improvements could be associated with the promotion of endochondral ossification and an increase in the rate of cell proliferation and tissue synthesis.

Computational modeling of bone density profiles in response to gait: a subject-specific approach

The goal of this study is to explore the potential of computational growth models to predict bone density profiles in the proximal tibia in response to gait-induced loading. From a modeling point of view, we design a finite element-based computational algorithm using the theory of open system thermodynamics. In this algorithm, the biological problem, the balance of mass, is solved locally on the integration point level, while the mechanical problem, the balance of linear momentum, is solved globally on the node point level. Specifically, the local bone mineral density is treated as an internal variable, which is allowed to change in response to mechanical loading. From an experimental point of view, we perform a subject-specific gait analysis to identify the relevant forces during walking using an inverse dynamics approach. These forces are directly applied as loads in the finite element simulation. To validate the model, we take a Dual-Energy X-ray Absorptiometry scan of the subject's right knee from which we create a geometric model of the proximal tibia. For qualitative validation, we compare the computationally predicted density profiles to the bone mineral density extracted from this scan. For quantitative validation, we adopt the region of interest method and determine the density values at fourteen discrete locations using standard and custom-designed image analysis tools. Qualitatively, our two- and three-dimensional density predictions are in excellent agreement with the experimental measurements. Quantitatively, errors are less than 3% for the two-dimensional analysis and less than 10% for the three-dimensional analysis. The proposed approach has the potential to ultimately improve the long-term success of possible treatment options for chronic diseases such as osteoarthritis on a patient-specific basis by accurately addressing the complex interactions between ambulatory loads and tissue changes.

Using distraction forces to drive an autodistractor during limb lengthening

Distraction osteogenesis can result in high forces developing in the limb. To determine and control the distraction forces (DF), a motorized distractor driven by feedback from DF was developed and used to lengthen the tibiae of 6 sheep undergoing distraction osteogenesis. The forces were measured continually, and, in 4 of the sheep, a force threshold was set, above which an increase in rate was initiated. The rate kept increasing to a set limit if forces remained above the threshold; otherwise, the rate was decreased. Radiographs were acquired biweekly, and muscle samples were analyzed from both the operated and contralateral limbs upon termination of the experiment. Results demonstrated a drop in DF associated with increased lengthening rate, attributed to separation of the callus as indicated by radiography. Histological evidence of muscle damage generally correlated with higher DF levels. There was a significant decrease in muscle fiber diameter in lengthened relative to contralateral limbs. Collectively, the results demonstrated the use of a force-driven distraction system and support the need for considering force data in regulating distraction rates to achieve optimal clinical outcomes.

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 design of treatment strategies in wound healing and bone fracture healing

Wound and bone fracture healing are natural repair processes initiated by trauma. Over the last decade, many mathematical models have been established to investigate the healing processes in silico, in addition to ongoing experimental work. In recent days, the focus of the mathematical models has shifted from simulation of the healing process towards simulation of the impaired healing process and the in silico design of treatment strategies. This review describes the most important causes of failure of the wound and bone fracture healing processes and the experimental models and methods used to investigate and treat these impaired healing cases. Furthermore, the mathematical models that are described address these impaired healing cases and investigate various therapeutic scenarios in silico. Examples are provided to illustrate the potential of these in silico experiments. Finally, limitations of the models and the need for and ability of these models to capture patient specificity and variability are discussed.