Corroboration of mechanobiological simulations of tissue differentiation in an in vivo bone chamber using a lattice-modeling approach

Application of finite element analysis to tissue differentiation and bone remodelling approaches and their use in design optimization of orthopaedic implants: A review

Post-operative bone growth and long-term bone adaptation around the orthopaedic implants are simulated using the mechanoregulation based tissue-differentiation and adaptive bone remodelling algorithms, respectively. The primary objective of these algorithms was to assess biomechanical feasibility and reliability of orthopaedic implants. This article aims to offer a comprehensive review of the developments in mathematical models of tissue-differentiation and bone adaptation and their applications in studies involving design optimization of orthopaedic implants over three decades. Despite the different mechanoregulatory models developed, existing literature confirm that none of the models can be highly regarded or completely disregarded over each other. Not much development in mathematical formulations has been observed from the current state of knowledge due to the lack of in vivo studies involving clinically relevant animal models, which further retarded the development of such models to use in translational research at a fast pace. Future investigations involving artificial intelligence (AI), soft-computing techniques and combined tissue-differentiation and bone-adaptation studies involving animal subjects for model verification are needed to formulate more sophisticated mathematical models to enhance the accuracy of pre-clinical testing of orthopaedic implants.

Influence of sequential opening/closing of interface gaps and texture density on bone growth over macro-textured implant surfaces using FE based mechanoregulatory algorithm

Intramedullary implant fixation is achieved through a press-fit between the implant and the host bone. A stronger press-fit between the bone and the prosthesis often introduces damage to the bone canal creating micro-gaps. The aim of the present investigation is to study the influences of simultaneous opening/closing of gaps on bone growth over macro-textured implant surfaces. Models based on textures available on CORAIL and SP-CL hip stems have been considered and 3D finite element (FE) analysis has been carried out in conjunction with mechanoregulation based tissue differentiation algorithm. Additionally, using a full-factorial approach, different combinations (between 5 µm to 15 µm) of sliding and gap distances at the bone-implant interface were considered to understand their combined influences on bone growth. All designs show an elevated fibrous tissue formation (10.96% at 5 µm to 29.38% at 40 µm for CORAIL based textured model; 11.45% at 5 µm to 32.25% at 40 µm for SP-CL based textured model) and inhibition of soft cartilaginous tissue (75.64% at 5 µm to 53.94% at 40 µm for CORAIL based model; 76.02% at 5 µm to 53.60% at 40 µm SP-CL based model) at progressively higher levels of normal micromotion, leading to a fragile bone-implant interface. These results highlight the importance of minimizing both sliding and gap distances simultaneously to enhance bone growth and implant stability. Further, results from the studies with differential texture density over CORAIL based implant reveal a non-linear complex relationship between tissue growth and texture density which might be investigated in a machine learning framework.

Initial mechanical conditions within an optimized bone scaffold do not ensure bone regeneration - an in silico analysis

Large bone defects remain a clinical challenge because they do not heal spontaneously. 3-D printed scaffolds are a promising treatment option for such critical defects. Recent scaffold design strategies have made use of computer modelling techniques to optimize scaffold design. In particular, scaffold geometries have been optimized to avoid mechanical failure and recently also to provide a distinct mechanical stimulation to cells within the scaffold pores. This way, mechanical strain levels are optimized to favour the bone tissue formation. However, bone regeneration is a highly dynamic process where the mechanical conditions immediately after surgery might not ensure optimal regeneration throughout healing. Here, we investigated in silico whether scaffolds presenting optimal mechanical conditions for bone regeneration immediately after surgery also present an optimal design for the full regeneration process. A computer framework, combining an automatic parametric scaffold design generation with a mechano-biological bone regeneration model, was developed to predict the level of regenerated bone volume for a large range of scaffold designs and to compare it with the scaffold pore volume fraction under favourable mechanical stimuli immediately after surgery. We found that many scaffold designs could be considered as highly beneficial for bone healing immediately after surgery; however, most of them did not show optimal bone formation in later regenerative phases. This study allowed to gain a more thorough understanding of the effect of scaffold geometry changes on bone regeneration and how to maximize regenerated bone volume in the long term.

The investigation of bone fracture healing under intramembranous and endochondral ossification

After trauma, fractured bone starts healing directly through bone union or indirectly through callus formation process. Intramembranous and endochondral ossification are two commonly known mechanisms of indirect healing. The present study investigated the bone fracture healing under intramembranous and endochondral ossification by developing theoretical models in conjunction with performing a series of animal experiments. Using experimentally determined mean bone densities in sheep tibia stabilized by the Locking Compression Plate (LCP) fixation system, the research outcomes showed that intramembranous and endochondral ossification can be described by Hill Function with two unique sets of function parameters in mechanical stimuli mediated fracture healing. Two different thresholds exist within the range of mechanical simulation index which could trigger significant intramembranous and endochondral ossification, with a relatively higher bone formation rate of endochondral ossification than that of intramembranous ossification. Furthermore, the increase of flexibility of the LCP system and the use of titanium LCP could potentially promote uniform bone formation across the fracture gap, ultimately better healing outcomes.

Predicting Bone Formation in Mesenchymal Stromal Cell-Seeded Hydrogels Using Experiment-Based Mathematical Modeling

In vitro bone formation by mesenchymal stromal cells encapsulated in type-1 collagen hydrogels is demonstrated after a 28-day in vitro culture period. Analysis of the hydrogels is carried out by X-ray microcomputed tomography, histology, and immunohistochemistry, which collectively demonstrates that bone formation in the hydrogels was quantifiably proportional to the initial collagen concentration, and subsequently the population density of seeded cells. This was established by varying the initial collagen concentration at a constant cell seeding density (3 × 10⁵ cells/0.3 mL hydrogel), and separately varying cell seeding density at a constant collagen concentration (1 mg/mL). Using these data, a mathematical model is presented for the total hydrogel volume and mineralization volume based on the observed linear contraction dynamics of cell-seeded collagen gels. The model parameters are fitted by comparing the predictions of the mathematical model for the hydrogel and mineralized volumes on day 28 with the experimental data. The model is then used to predict the hydrogel and mineralization volumes for a range of hydrogel collagen concentrations and cell seeding densities, providing comprehensive input/output descriptors for generating mineralized hydrogels for bone tissue engineering. It is proposed that this quantitative approach will be a useful tool for generating in vitro manufactured bone tissue, defining input parameters that yield predictable output measures of tissue maturation. Impact statement This article describes a simple yet powerful quantitative description of in vitro tissue-engineered bone by combining experimental data with mathematical modeling. The overall aim of the article is to examine what is currently known about cell-mediated collagen contraction, and demonstrate that this phenomenon can be exploited to tailor bone formation by choosing a specific set of input parameters in the form of cell seeding density and collagen hydrogel concentration. Our study utilizes a clinically relevant cell source (human mesenchymal stem cells) with a biomaterial that has received regulatory approval for use in humans (collagen type 1), and hence could be useful for clinical applications, as well as furthering our understanding of cell/extracellular matrix interactions in determining in vitro bone tissue formation.

The recent advances in the mathematical modelling of human pluripotent stem cells

Human pluripotent stem cells hold great promise for developments in regenerative medicine and drug design. The mathematical modelling of stem cells and their properties is necessary to understand and quantify key behaviours and develop non-invasive prognostic modelling tools to assist in the optimisation of laboratory experiments. Here, the recent advances in the mathematical modelling of hPSCs are discussed, including cell kinematics, cell proliferation and colony formation, and pluripotency and differentiation.

Continuum Modeling and Simulation in Bone Tissue Engineering

Bone tissue engineering is currently a mature methodology from a research perspective. Moreover, modeling and simulation of involved processes and phenomena in BTE have been proved in a number of papers to be an excellent assessment tool in the stages of design and proof of concept through in-vivo or in-vitro experimentation. In this paper, a review of the most relevant contributions in modeling and simulation, in silico, in BTE applications is conducted. The most popular in silico simulations in BTE are classified into: (i) Mechanics modeling and scaffold design, (ii) transport and flow modeling, and (iii) modeling of physical phenomena. The paper is restricted to the review of the numerical implementation and simulation of continuum theories applied to different processes in BTE, such that molecular dynamics or discrete approaches are out of the scope of the paper. Two main conclusions are drawn at the end of the paper: First, the great potential and advantages that in silico simulation offers in BTE, and second, the need for interdisciplinary collaboration to further validate numerical models developed in BTE.

PREDICTION OF THE TREND OF BONE FRACTURE HEALING BASED ON THE RESULTS OF THE EARLY STAGES SIMULATIONS: A FINITE ELEMENT STUDY

To date, several studies have implied the importance of early stage mechanical stability in the bone fracture healing process. This study aimed at finding a correlation between the predicted different tissue phenotypes in the early stages of healing and the ultimate healing outcome. For this purpose, the process of fracture healing was numerically simulated employing an axisymmetric bi-phasic finite element (FE) model for three initial gap sizes of 1, 3 and 6[Formula: see text]mm and four initial interfragmentary strains (IFS) of 7%, 11%, 15% and 19%. The model was validated with experimental and other numerical studies from the literature. Results of this study showed that the amount of cartilage and fibrous tissue observed in the early stage after fracture can be used to qualitatively assess the outcome of complete bone healing process. Greater amount of cartilage in early stage of healing process yielded faster callus maturation, and delayed maturation of callus was predicted in the case of high fibrous tissue production. Results of this study can be used to provide an estimation of the performance of different fixation systems by considering the amounts of cartilage and fibrous tissues observed in the early stage of healing.

Biomechanical behavior of mandibles reconstructed with fibular grafts at different vertical positions using finite element method

BACKGROUND

For large mandibular defects, surgical reconstruction using microvascular fibular grafts has advantages over other alternatives in terms of blood supply and good quality of grafted bone. However, the fibular segment is usually lower in height than that of the original mandible, meaning that the vertical positioning of the fibular graft is variable, with different biomechanical consequences on the reconstructed mandible.

OBJECTIVES

To use finite element method (FEM) to evaluate stress distribution and displacement of a reconstructed mandible versus an intact mandible under occlusal loads.

METHODS

A three-dimensional intact edentulous mandibular bone (Model I) and a reconstructed mandible bone with fibular graft were created from CBCT images. Calculation models were generated with fibular bone graft extracted from the reconstructed mandible of identical length placed into a mimicked defect area on the right-hand side of the mandible at three different vertical positions: superior (Model II), intermediate (Model III), and inferior (Model IV). Forces were applied at lower left first molar region and lower left central incisor area. Von Mises stresses and mandibular displacement were calculated as outcome measurements during loadings.

RESULTS

Maximum stress and strain within the reconstructed mandible were identified at the posterior border of the graft and the contralateral condyle. Maximum displacement occurred near the interface of fibular graft and anterior segment of the mandible. Stress distribution in the graft under functional loads is much higher than that in the residual mandibular segments from Models II to IV. The combined average maximum stress from anterior and posterior loads is 10.66 times higher in the mandible with inferiorly positioned graft (Model IV), 8.72 times for superior graft (Model II), and 3.68 times for intermediate graft (Model III) than that in the control group (Model I). The worst displacement result during functional loadings was in the group with fibular graft located at the inferior border of the mandible.

CONCLUSIONS

The position of fibular graft placed in the surgical resection site has significant effects on the mechanical behavior of the reconstructed mandible. The fibular graft aligned with the inferior border of the mandible, the most common site designated location by clinicians, has the worst effects on the stress distribution and displacement to the mandibular under functional loads. The fibular graft placed at the intermediate location has the best biomechanics and provides favorable condition for subsequent prosthetic reconstruction.

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.

The Use of Finite Element Analyses to Design and Fabricate Three-Dimensional Scaffolds for Skeletal Tissue Engineering

Computational modeling has been increasingly applied to the field of tissue engineering and regenerative medicine. Where in early days computational models were used to better understand the biomechanical requirements of targeted tissues to be regenerated, recently, more and more models are formulated to combine such biomechanical requirements with cell fate predictions to aid in the design of functional three-dimensional scaffolds. In this review, we highlight how computational modeling has been used to understand the mechanisms behind tissue formation and can be used for more rational and biomimetic scaffold-based tissue regeneration strategies. With a particular focus on musculoskeletal tissues, we discuss recent models attempting to predict cell activity in relation to specific mechanical and physical stimuli that can be applied to them through porous three-dimensional scaffolds. In doing so, we review the most common scaffold fabrication methods, with a critical view on those technologies that offer better properties to be more easily combined with computational modeling. Finally, we discuss how modeling, and in particular finite element analysis, can be used to optimize the design of scaffolds for skeletal tissue regeneration.

Effect of interfragmentary gap on the mechanical behavior of mandibular angle fracture with three fixation designs: A finite element analysis

BACKGROUND

The aim of this study was to simulate stress and strain distribution numerically on a normal mandible under physiological occlusal loadings. The results were compared with those of mandibles that had an angle fracture stabilized with different fixation designs under the same loadings. The amount of displacement at two interfragmentary gaps was also studied.

MATERIALS AND METHODS

A three-dimensional (3D) virtual mandible was reconstructed with an angle fracture that had a fracture gap of either 0.1 or 1 mm. Three types of plate fixation designs were used: Type I, a miniplate was placed across the fracture line following the Champy technique; Type II, two miniplates were used; and Type III, a reconstruction plate was used on the inferior border of the mandible. Loads of 100 and 500 N were applied to the models. The maximum von Mises stress, strain, and displacement were computed using finite element analysis. The results from the control and experimental groups were analyzed and compared.

RESULTS

The results demonstrated that high stresses and strains were distributed to the condylar and angular areas regardless of the loading position. The ratio of the plate/bone average stress ranged from 215% (Type II design) to 848% (Type I design) irrespective of the interfragmentary gap size. With a 1-mm fracture gap, the ratio of the plate/bone stress ranged from 204% (Type II design) to 1130% (Type I design). All strains were well below critical bone strain thresholds. Displacement on the cross-sectional mapping at fracture interface indicated that uneven movement occurred in x, y, and z directions.

CONCLUSIONS

Interfragmentary gaps between 0.1 and 1 mm did not have a substantial effect on the average stress distribution to the fractured bony segments; however, they had a greater effect on the stress distribution to the plates and screws. Type II fixation was the best mechanical design under bite loads. Type I design was the least stable system and had the highest stress distribution and the largest displacement at the fracture site.

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

For further resources related to this article, please visit the WIREs website.

The influence of hydrostatic pressure on tissue engineered bone development

The hydrostatic pressure stimulation of an appropriately cell-seeded porous scaffold within a bioreactor is a promising method for engineering bone tissue external to the body. We propose a mathematical model, and employ a suite of candidate constitutive laws, to qualitatively describe the effect of applied hydrostatic pressure on the quantity of minerals deposited in such an experimental setup. By comparing data from numerical simulations with experimental observations under a number of stimulation protocols, we suggest that the response of bone cells to an applied pressure requires consideration of two components; (i) a component describing the cell memory of the applied stimulation, and (ii) a recovery component, capturing the time cells require to recover from high rates of mineralisation.

A Mechanobiology-based Algorithm to Optimize the Microstructure Geometry of Bone Tissue Scaffolds

Complexity of scaffold geometries and biological mechanisms involved in the bone generation process make the design of scaffolds a quite challenging task. The most common approaches utilized in bone tissue engineering require costly protocols and time-consuming experiments. In this study we present an algorithm that, combining parametric finite element models of scaffolds with numerical optimization methods and a computational mechano-regulation model, is able to predict the optimal scaffold microstructure. The scaffold geometrical parameters are perturbed until the best geometry that allows the largest amounts of bone to be generated, is reached. We study the effects of the following factors: (1) the shape of the pores; (2) their spatial distribution; (3) the number of pores per unit area. The optimal dimensions of the pores have been determined for different values of scaffold Young's modulus and compression loading acting on the scaffold upper surface. Pores with rectangular section were predicted to lead to the formation of larger amounts of bone compared to square section pores; similarly, elliptic pores were predicted to allow the generation of greater amounts of bone compared to circular pores. The number of pores per unit area appears to have rather negligible effects on the bone regeneration process. Finally, the algorithm predicts that for increasing loads, increasing values of the scaffold Young's modulus are preferable. The results shown in the article represent a proof-of-principle demonstration of the possibility to optimize the scaffold microstructure geometry based on mechanobiological criteria.

Modeling mechanical signals on the surface of µCT and CAD based rapid prototype scaffold models to predict (early stage) tissue development

In the field of tissue engineering, mechano-regulation theories have been applied to help predict tissue development in tissue engineering scaffolds in the past. For this, finite element models (FEMs) were used to predict the distribution of strains within a scaffold. However, the strains reported in these studies are volumetric strains of the material or strains developed in the extracellular matrix occupying the pore space. The initial phase of cell attachment and growth on the biomaterial surface has thus far been neglected. In this study, we present a model that determines the magnitude of biomechanical signals on the biomaterial surface, enabling us to predict cell differentiation stimulus values at this initial stage. Results showed that magnitudes of the 2D strain--termed surface strain--were lower when compared to the 3D volumetric strain or the conventional octahedral shear strain as used in current mechano-regulation theories. Results of both µCT and CAD derived FEMs from the same scaffold were compared. Strain and fluid shear stress distributions, and subsequently the cell differentiation stimulus, were highly dependent on the pore shape. CAD models were not able to capture the distributions seen in the µCT FEM. The calculated mechanical stimuli could be combined with current mechanobiological models resulting in a tool to predict cell differentiation in the initial phase of tissue engineering. Although experimental data is still necessary to properly link mechanical signals to cell behavior in this specific setting, this model is an important step towards optimizing scaffold architecture and/or stimulation regimes.

Investigation of the in vitro culture process for skeletal-tissue-engineered constructs using computational fluid dynamics and experimental methods

The in vitro culture process via bioreactors is critical to create tissue-engineered constructs (TECs) to repair or replace the damaged tissues/organs in various engineered applications. In the past, the TEC culture process was typically treated as a black box and performed on the basis of trial and error. Recently, computational fluid dynamics (CFD) has demonstrated its potential to analyze the fluid flow inside and around the TECs, therefore, being able to provide insight into the culture process, such as information on the velocity field and shear stress distribution that can significantly affect such cellular activities as cell viability and proliferation during the culture process. This paper briefly reviews the CFD and experimental methods used to investigate the in vitro culture process of skeletal-type TECs in bioreactors, where mechanical deformation of the TEC can be ignored. Specifically, this paper presents CFD modeling approaches for the analysis of the velocity and shear stress fields, mass transfer, and cell growth during the culture process and also describes various particle image velocimetry (PIV) based experimental methods to measure the velocity and shear stress in the in vitro culture process. Some key issues and challenges are also identified and discussed along with recommendations for future research.

Mechanical regulation of bone regeneration: theories, models, and experiments

How mechanical forces influence the regeneration of bone remains an open question. Their effect has been demonstrated experimentally, which has allowed mathematical theories of mechanically driven tissue differentiation to be developed. Many simulations driven by these theories have been presented, however, validation of these models has remained difficult due to the number of independent parameters considered. An overview of these theories and models is presented along with a review of experimental studies and the factors they consider. Finally limitations of current experimental data and how this influences modeling are discussed and potential solutions are proposed.

The role of oxygen as a regulator of stem cell fate during fracture repair in TSP2-null mice

It is often difficult to decouple the relative importance of different factors in regulating MSC differentiation. Genetically modified mice provide model systems whereby some variables can be manipulated while others are kept constant. Fracture repair in thrombospondin-2 (TSP2)-null mice is characterized by reduced endochondral ossification and enhanced intramembranous bone formation. The proposed mechanism for this shift in MSC fate is that increased vascular density and hence oxygen availability in TSP2-null mice regulates differentiation. However, TSP2 is multifunctional and regulates other aspects of the regenerative cascade, such as MSC proliferation. The objective of this study is to use a previously developed computational model of tissue differentiation, in which substrate stiffness and oxygen tension regulate stem cell differentiation, to simulate potential mechanisms which may drive alterations in MSC fate in TSP2-null mice. Four models (increased cell proliferation, increased numbers of MSCs in the marrow decreased cellular oxygen consumption, and an initially stiffer callus) were not predictive of experimental observations in TSP2-null mice. In contrast, increasing the rate of angiogenic progression led to a prediction of greater intramembranous ossification, diminished endochondral ossification, and a reduced region of hypoxia in the fracture callus similar to that quantified experimentally by the immunohistochemical detection of pimonidazole adducts that develop with hypoxia. This study therefore provides further support for the hypothesis that oxygen availability during early fracture healing is a key regulator of MSC bipotential differentiation, and furthermore, it highlights the advantages of integrating computational models with genetically modified mouse studies for further elucidating mechanisms regulating stem cell fate.

Mechanical regulation of skeletal development

Development of the various components of a normal skeleton requires highly regulated signalling systems that co-ordinate spatial and temporal patterns of cell division, cell differentiation, and morphogenesis. Much work in recent decades has revealed cascades of molecular signalling, acting through key transcription factors to regulate, for example, organized chondrogenic and osteogenic differentiation. It is now clear that mechanical stimuli are also required for aspects of skeletogenesis but very little is known about how the mechanical signals are integrated with classic biochemical signalling. Spatially organized differentiation is vital to the production of functionally appropriate tissues contributing to precise, region specific morphologies, for example transient chondrogenesis of long bone skeletal rudiments, which prefigures osteogenic replacement of the cartilage template, compared with the production of permanent cartilage at the sites of articulation. Currently a lack of understanding of how these tissues are differentially regulated hampers efforts to specifically regenerate stable bone and cartilage. Here, we review current research revealing the influence of mechanical stimuli on specific aspects of skeletal development and refer to other developing systems to set the scene for current and future work to uncover the molecular mechanisms involved. We integrate this with a brief overview of the effects of mechanical stimulation on stem cells in culture bringing together developmental and tissue engineering aspects of mechanoregulation of cell behavior. A better understanding of the molecular mechanisms that link mechanical stimuli to transcriptional control guiding cell differentiation will lead to new ideas about how to effectively prime stem cells for tissue engineering and regenerative therapies.

The emergence of mechanoregulated endochondral ossification in evolution

The differentiation of skeletal tissue phenotypes is partly regulated by mechanical forces. This mechanoregulatory aspect of tissue differentiation has been the subject of many experimental and computational investigations. However, little is known about what factors promoted the emergence of mechanoregulated tissue differentiation in evolution, even though mechanoregulated tissue differentiation, for example during development or healing of adult bone, is crucial for vertebrate phylogeny. In this paper, we use a computational framework to test the hypothesis that the emergence of mechanosensitive genes that trigger endochondral ossification in evolution will stabilise in the population and create a variable mechanoregulated response, if the endochondral ossification process enhances fitness for survival. The model combines an evolutionary algorithm that considers genetic change with a mechanoregulated fracture healing model in which the fitness of animals in a population is determined by their ability to heal their bones. The simulations show that, with the emergence of mechanosensitive genes through evolution enabling skeletal cells to modulate their synthetic activities, novel differentiation pathways such as endochondral ossification could have emerged, which when favoured by natural selection is maintained in a population. Furthermore, the model predicts that evolutionary forces do not lead to a single optimal mechanoregulated response but that the capacity of endochondral ossification exists with variability in a population. The simulations correspond with many existing findings about the mechanosensitivity of skeletal tissues in current animal populations, therefore indicating that this kind of multi-level models could be used in future population based simulations of tissue differentiation.

Role of mathematical modeling in bone fracture healing

Bone fracture healing is a complex physiological process commonly described by a four-phase model consisting of an inflammatory phase, two repair phases with soft callus formation followed by hard callus formation, and a remodeling phase, or more recently by an anabolic/catabolic model. Data from humans and animal models have demonstrated crucial environmental conditions for optimal fracture healing, including the mechanical environment, blood supply and availability of mesenchymal stem cells. Fracture healing spans multiple length and time scales, making it difficult to know precisely which factors and/or phases to manipulate in order to obtain optimal fracture-repair outcomes. Deformations resulting from physiological loading or fracture fixation at the organ scale are sensed at the cellular scale by cells inside the fracture callus. These deformations together with autocrine and paracrine signals determine cellular differentiation, proliferation and migration. The local repair activities lead to new bone formation and stabilization of the fracture. Although experimental data are available at different spatial and temporal scales, it is not clear how these data can be linked to provide a holistic view of fracture healing. Mathematical modeling is a powerful tool to quantify conceptual models and to establish the missing links between experimental data obtained at different scales. The objective of this review is to introduce mathematical modeling to readers who are not familiar with this methodology and to demonstrate that once validated, such models can be used for hypothesis testing and to assist in clinical treatment as will be shown for the example of atrophic nonunions.

Substrate stiffness and oxygen as regulators of stem cell differentiation during skeletal tissue regeneration: a mechanobiological model

Extrinsic mechanical signals have been implicated as key regulators of mesenchymal stem cell (MSC) differentiation. It has been possible to test different hypotheses for mechano-regulated MSC differentiation by attempting to simulate regenerative events such as bone fracture repair, where repeatable spatial and temporal patterns of tissue differentiation occur. More recently, in vitro studies have identified other environmental cues such as substrate stiffness and oxygen tension as key regulators of MSC differentiation; however it remains unclear if and how such cues determine stem cell fate in vivo. As part of this study, a computational model was developed to test the hypothesis that substrate stiffness and oxygen tension regulate stem cell differentiation during fracture healing. Rather than assuming mechanical signals act directly on stem cells to determine their differentiation pathway, it is postulated that they act indirectly to regulate angiogenesis and hence partially determine the local oxygen environment within a regenerating tissue. Chondrogenesis of MSCs was hypothesized to occur in low oxygen regions, while in well vascularised regions of the regenerating tissue a soft local substrate was hypothesised to facilitate adipogenesis while a stiff substrate facilitated osteogenesis. Predictions from the model were compared to both experimental data and to predictions of a well established computational mechanobiological model where tissue differentiation is assumed to be regulated directly by the local mechanical environment. The model predicted all the major events of fracture repair, including cartilaginous bridging, endosteal and periosteal bony bridging and bone remodelling. It therefore provides support for the hypothesis that substrate stiffness and oxygen play a key role in regulating MSC fate during regenerative events such as fracture healing.

Cell-biomaterial mechanical interaction in the framework of tissue engineering: insights, computational modeling and perspectives

Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.

Simulation of fracture healing in the tibia: mechanoregulation of cell activity using a lattice modeling approach

In this study, a three-dimensional (3D) computational simulation of bone regeneration was performed in a human tibia under realistic muscle loading. The simulation was achieved using a discrete lattice modeling approach combined with a mechanoregulation algorithm to describe the cellular processes involved in the healing process-namely proliferation, migration, apoptosis, and differentiation of cells. The main phases of fracture healing were predicted by the simulation, including the bone resorption phase, and there was a qualitative agreement between the temporal changes in interfragmentary strain and bending stiffness by comparison to experimental data and clinical results. Bone healing was simulated beyond the reparative phase by modeling the transition of woven bone into lamellar bone. Because the simulation has been shown to work with realistic anatomical 3D geometry and muscle loading, it demonstrates the potential of simulation tools for patient-specific pre-operative treatment planning.

Inter-species investigation of the mechano-regulation of bone healing: comparison of secondary bone healing in sheep and rat

Inter-species differences in regeneration exist in various levels. One aspect is the dynamics of bone regeneration and healing, e.g. small animals show a faster healing response when compared to large animals. Mechanical as well as biological factors are known to play a key role in the process. However, it remains so far unknown whether different animals follow at all comparable mechano-biological rules during tissue regeneration, and in particular during bone healing. In this study, we investigated whether differences observed in vivo in the dynamics of bone healing between rat and sheep are only due to differences in the animal size or whether these animals have a different mechano-biological response during the healing process. Histological sections from in vivo experiments were compared to in silico predictions of a mechano-biological computer model for the simulation of bone healing. Investigations showed that the healing processes in both animal models occur under significantly different levels of mechanical stimuli within the callus region, which could explain histological observations of early intramembranous ossification at the endosteal side. A species-specific adaptation of a mechano-biological model allowed a qualitative match of model predictions with histological observations. Specifically, when keeping cell activity processes at the same rate, the amount of tissue straining defining favorable mechanical conditions for the formation of bone had to be increased in the large animal model, with respect to the small animal, to achieve a qualitative agreement of model predictions with histological data. These findings illustrate that geometrical (size) differences alone cannot explain the distinctions seen in the histological appearance of secondary bone healing in sheep and rat. It can be stated that significant differences in the mechano-biological regulation of the healing process exist between these species. Future investigations should aim towards understanding whether these differences are due to differences in cell behavior, material properties of the newly formed tissues within the callus and/or differences in response to the mechanical environment.

Variability observed in mechano-regulated in vivo tissue differentiation can be explained by variation in cell mechano-sensitivity

Computational simulations of tissue differentiation have been able to capture the main aspects of tissue formation/regeneration observed in animal experiments-except for the considerable degree of variability reported. Understanding and modelling the source of this variability is crucial if computational tools are to be developed for clinical applications. The objective of this study was to test the hypothesis that differences in cell mechano-sensitivity between individuals can explain the variability of tissue differentiation patterns observed experimentally. Simulations of an experiment of tissue differentiation in a mechanically loaded bone chamber were performed. Finite element analysis was used to determine the biophysical environment, and a lattice-modelling approach was used to simulate cell activity. Differences in cell mechano-sensitivity among individuals were modelled as differences in cell activity rates, with the activation of cell activities regulated by the mechanical environment. Predictions of the tissue distribution in the chambers produced the two different classes of results found experimentally: (i) chambers with a layer of bone across the chamber covered by a layer of cartilage on top and (ii) chambers with almost no bone, mainly fibrous tissue and small islands of cartilage. This indicates that the differing cellular response to the mechanical environment (i.e., subject-specific mechano-sensitivity) could be a reason for the different outcomes found when implants (or tissue engineered constructs) are used in a population.

Finite element method (FEM), mechanobiology and biomimetic scaffolds in bone tissue engineering

Techniques of bone reconstructive surgery are largely based on conventional, non-cell-based therapies that rely on the use of durable materials from outside the patient's body. In contrast to conventional materials, bone tissue engineering is an interdisciplinary field that applies the principles of engineering and life sciences towards the development of biological substitutes that restore, maintain, or improve bone tissue function. Bone tissue engineering has led to great expectations for clinical surgery or various diseases that cannot be solved with traditional devices. For example, critical-sized defects in bone, whether induced by primary tumor resection, trauma, or selective surgery have in many cases presented insurmountable challenges to the current gold standard treatment for bone repair. The primary purpose of bone tissue engineering is to apply engineering principles to incite and promote the natural healing process of bone which does not occur in critical-sized defects. The total market for bone tissue regeneration and repair was valued at $1.1 billion in 2007 and is projected to increase to nearly $1.6 billion by 2014.Usually, temporary biomimetic scaffolds are utilized for accommodating cell growth and bone tissue genesis. The scaffold has to promote biological processes such as the production of extra-cellular matrix and vascularisation, furthermore the scaffold has to withstand the mechanical loads acting on it and to transfer them to the natural tissues located in the vicinity. The design of a scaffold for the guided regeneration of a bony tissue requires a multidisciplinary approach. Finite element method and mechanobiology can be used in an integrated approach to find the optimal parameters governing bone scaffold performance.In this paper, a review of the studies that through a combined use of finite element method and mechano-regulation algorithms described the possible patterns of tissue differentiation in biomimetic scaffolds for bone tissue engineering is given. Firstly, the generalities of the finite element method of structural analysis are outlined; second, the issues related to the generation of a finite element model of a given anatomical site or of a bone scaffold are discussed; thirdly, the principles on which mechanobiology is based, the principal theories as well as the main applications of mechano-regulation models in bone tissue engineering are described; finally, the limitations of the mechanobiological models and the future perspectives are indicated.

Tissue differentiation in an in vivo bioreactor: in silico investigations of scaffold stiffness

Scaffold design remains a main challenge in tissue engineering due to the large number of requirements that need to be met in order to create functional tissues in vivo. Computer simulations of tissue differentiation within scaffolds could serve as a powerful tool in elucidating the design requirements for scaffolds in tissue engineering. In this study, a lattice-based model of a 3D porous scaffold construct derived from micro CT and a mechano-biological simulation of a bone chamber experiment were combined to investigate the effect of scaffold stiffness on tissue differentiation inside the chamber. The results indicate that higher scaffold stiffness, holding pore structure constant, enhances bone formation. This study demonstrates that a lattice approach is very suitable for modelling scaffolds in mechano-biological simulations, since it can accurately represent the micro-porous geometries of scaffolds in a 3D environment and reduce computational costs at the same time.

Computational simulation methodologies for mechanobiological modelling: a cell-centred approach to neointima development in stents

The design of medical devices could be very much improved if robust tools were available for computational simulation of tissue response to the presence of the implant. Such tools require algorithms to simulate the response of tissues to mechanical and chemical stimuli. Available methodologies include those based on the principle of mechanical homeostasis, those which use continuum models to simulate biological constituents, and the cell-centred approach, which models cells as autonomous agents. In the latter approach, cell behaviour is governed by rules based on the state of the local environment around the cell; and informed by experiment. Tissue growth and differentiation requires simulating many of these cells together. In this paper, the methodology and applications of cell-centred techniques--with particular application to mechanobiology--are reviewed, and a cell-centred model of tissue formation in the lumen of an artery in response to the deployment of a stent is presented. The method is capable of capturing some of the most important aspects of restenosis, including nonlinear lesion growth with time. The approach taken in this paper provides a framework for simulating restenosis; the next step will be to couple it with more patient-specific geometries and quantitative parameter data.