Bone’s capability of “perfect” regeneration is unique, unlike other tissues it is capable of recovering its form without permanent scars. However not all fractures heal spontaneously, it has been found that 20 per 100,000 people per year will have delayed healing or a non-union, where the fractured bone fails to fuse (1). It is known that mechanical forces can influence the pathways through which healing occurs; several studies have shown that changes in the mechanical environment can modulate the time taken to heal, change the proportions of different tissue type as well as gene expression patterns of cells in the healing bone (2, 3). The exact mechanism through which mechanical stimuli are sensed and incorporated in the healing process is not fully understood and still remains an open question. Answering this question will lead to improved treatment methods for bone fracture repair, reducing the amount of time patients are hospitalized. To this end, we have compiled a summary of literature on the topic examining the experimental studies and numerical theories.

Depending on the stability of the bone fragments the healing can progress down two paths, rigidly fixed fragments with only a small fracture gap can heal through primary bone healing, where the bone remodeling units, responsible for the adaption of the cortical bone, bridge the gap; secondary fracture healing occurs when relative motion occurs between the bone fragments, causing a callus to form. Secondary fracture healing can be divided into three overlapping stages as described in Figure 1, the reactive, reparative, and remodeling phases. Immediately post-fracture there is an inflammatory response, termed the reactive phase, in which blood vessels, which have ruptured fill the injured area with blood forming clot called the fracture hematoma. The fracture hematoma is infiltrated by fibroblasts and small blood vessels, becoming granulation tissue. The initial callus is thus a mixture of hematoma, fibrous tissues, and infiltrating blood vessels. The reparative phase begins once bone and cartilage form, the bone is initially formed through intramembranous ossification initiating on the existing cortical bone and progressing with time toward the plane of the fracture, while the cartilage forms in regions of low oxygen tension (4, 5). Once the blood supply is sufficient enough, cartilage is calcified and converted into woven bone through endochondral ossification. The stability of a fracture influences the amount of intramembranous and endochondral ossification, with more cartilage being formed in less stable fractures and thus more endochondral ossification (6). Bony bridging, the union of the hard callus from either side of the fracture, occurs making the structure extremely stable. The remodeling phase begins during the reparative stage, with the bone structure being adapted back to its original load bearing form. The cortical bone is also remodeled with the cortical bone adjacent to the fracture becoming woven bone, likely as the vasculature within this bone is damaged during the fracture, causing hypoxia (7–9). Hypoxia has been shown to up regulate the formation, size, and activity of osteoclasts, the cells responsible for the resorption of bone (10). The remodeling phase concludes with the callus being completely remodeled into the shape of the original bone, recovering the original strength, and functionality. The events are driven through intercellular signaling, levels of oxygen tension and the mechanical environment directing mesenchymal stems cells to differentiate into osteoblasts, chondrocytes, or fibroblasts each of these cells being responsible for the production of particular tissues (7). In this review, we concentrate explicitly on the mechanical factors influencing bone regeneration, what has been experimentally observed as well as theories and models, which have been developed to explain this.

Mechanical loads are applied at the organ level and propagate to a level where cells can sense them, which in turn results in changes at the tissue level. Many studies have investigated applying different forces to fractured bones in vivo and quantifying the tissue produced or the mechanical competence of the bone. Loads applied to bone cause non-homogenous strains throughout the healing tissue, resulting in a variety of tissues forming. Experimentally, these strains cannot be quantified in vivo preventing the direct development of mechanobiological rules, however, the result of organ level loading is important for validating fracture healing models, as an accurate rule set should predict the outcomes of such studies. Experimental studies investigating the effect of inter-fragmentary movement (IFM) and fixator stability/stiffness are essentially referring to how loading effects fracture healing outcome. IFM is ambiguously used to describe either axial tension/compression of the bone defect, shear movement in the plane of the defect, relative axial rotation of the fragments, or a bending. Here, we will distinguish between these different modes using the following terms: inter-fragmentary compression (IFC), inter-fragmentary tension (IFT), inter-fragmentary bending (IFB), and inter-fragmentary shear (IFS). The different loading modes are illustrated in Figures 2A–E with the exception of IFT, which is just the opposite of IFC. The cases of shear movement and rotation both create a non-uniform shear loading within the tissue, therefore we combine both of these loading states as IFS.

To study the effect of each mode of movement demands the precise control and measurement of bone fragment movement requiring the use of some form of fixation. There are two categories of fixation used in biomechanical research, external and internal fixation, which can be seen in Figures 2F–I. External fixation is commonly used in large animal studies often with modifications to the fixator or to the surgical technique so as to change the stability of the fixator, or be instrumented to measure the fragment displacements (11). In contrast, internal fixators are simpler in design and technique. An intramedullary nail for example is guided by the medullary cavity and can be a single piece, which has made them much more commonly used in small animal studies (12–15). The fixation method has a large effect on the loads, which can be applied, external fixation allows for controlled movements or fixed forces, whereas, internal fixation typically limits the load to an applied force. The fragile nature of the soft tissue means that constant loading throughout a study is not always possible, thus some studies will define a maximum load and displacement. For example, Goodship and Kenwright (16) apply a 33% inter-fragmentary strain or a 360 N load, as initially such a load would induce strains in the hematoma that inhibited healing, whereas, once bony bridging occurs, this level of strain would damage the new bone.

There exists no standard methodology for loading during fracture healing. Studies can either use active loading like Goodship and Kenwright (16) passively allow a limited amount of movement as done by Claes et al. (11) or use a fixator structure or orientation with a different stiffness as done by Klein et al. (17) and Schell et al. (18). While it is often possible to compare initial loading of the callus, the loading is altered as the tissue distribution changes. This can lead to diverging results making comparisons between how bone heals in relation to loading difficult between studies. We attempt to provide a summary of how different organ level loadings affect the healing outcome and were possible highlight variations in fixation and loading between studies.

In this section, different theories for tissue differentiation are described followed by an overview of the simulations, which have been performed using them (summary of these data can be found in Table 4). Experimental evidence demonstrates that mechanical forces can direct the healing process, i.e., tissue differentiation is mechanobiologically regulated. There are several theories as to which mechanical quantities are the stimuli for differentiation such as SED, deviatoric and volumetric strain, or relative fluid flow between cells and the matrix. Due to the complicated geometries, which occur in fracture healing it is not possible to analytically apply these theories. Instead they are applied as components of simulations, we concentrate here on mechanically driven simulations, which typically consist of five parts summarized in Figure 3; the geometries of bone, defect and callus; the boundary conditions; finite element analysis used to determine the mechanical signal in the callus; the tissue differentiation rules, through which a new callus geometry is created; and finally most simulations consider an additional “biological aspect,” which adds a temporal scale and directs spatially where the ossification occurs. A simulation will go through several iterations changing the tissues composing the callus until a state of equilibrium is reached.

The callus geometries are typically assumed to be constant and ellipsoidal, so simulations of fracture healing start at the reparative phase once the soft callus has formed. The only exception to this is the model of Gomez-Benito et al. (44) who presented a model, which allows the callus boundaries to evolve over time. Boundary conditions are based upon the loading described by the experimental study the authors aim to replicate.

The majority of simulations included an additional level often described as the biological aspect. As Claes and Heigele (4) observed the front of healing bone follows a path, starting at the original cortical faces and advancing toward the fracture gap. The inclusion of biological aspects in the form of cell, revascularization, nutrient supply, or oxygen tension, allows ossification to follow such a path. These biological elements are typically modeled as diffusive processes (39), random walks (45, 74), or through logical association of neighboring elements (36, 42). While cell proliferation and revascularization are vital aspects of fracture healing, the interplay between mechanical forces and these processes are not fully understood, in addition there is limited experimental data available so the validation of such model becomes much more difficult.

How the biologic aspect influences the course of fracture healing can vary, Lacroix et al. (39) considered cells diffusing within the callus and scaled the tissue stiffness according to cell destiny within an element. An issue exists with this approach, as the maturity of the tissue is determined purely by the cell number within and the tissue phenotype by the mechanical stimuli this allows mature cartilage to switch directly to mature bone. Kelly and Prendergast (43) similarly applied a scaling, but considered multiple cell phenotypes and so multiple tissues within a single element removing this issue. Shefelbine et al. (42) considered nutrient supply to be the critical biological factor in bone development, and so bone could only form in areas with good vascularization. This was again used by Wehner et al. (57) and Simon et al. (58). Chen et al. (49) added an additional level to this considering nutrient diffusion from the developing vasculature.

Burke and Kelly (62) proposed a theory in which tissue differentiation is indirectly driven by mechanical forces. Revascularization is allowed on in elements where the deviatoric strain is below a level of 6%. The blood vessels were assumed to diffuse into the tissue. ∫Ωγ<0.06dVdtdx=∫Ωγ<0.060.5×ΔVdx Where V is the vascularity, γ is the deviatoric strain, and Ω is the computational domain with all elements where this strain was <6%. The oxygen is then assumed to diffuse from the vasculature without any dependence on the local mechanical environment. ∫ΩdO2dtdx=∫ΩDΔO2−Q⋅nmaxndx

Here O₂ is the oxygen concentration, D is the diffusion coefficient of oxygen in the tissue, Q is the oxygen consumption rate of cells in the tissue, and n represents the number of cells in the element and n^max the maximum cell density. The tissues then differentiated based on the oxygen tension and the stiffness of neighboring elements as described in Figure 4E.

Several models exist which consider purely biological factors in fracture healing, Bailon-Plaza and Van Der Meulen (75) first proposed a model for bone regeneration, which included diffusion of stem cells, and various growth factors, but no mechanical feedback. Geris et al. (76) applied this model to simulate the healing of tibial fractures in mice, finding that the model could predict the course of healing, but was sensitive to initial levels of growth factor production. Geris et al. (56) investigated if mechanical regulation of angiogenesis and growth factor production could improve this model and account for load induced non-unions, and concluded that mechanical feedback for both angiogenesis and osteogenesis was required to correctly predict unions and non-unions. Later changes to this model have excluded mechanics and focused on more detailed representations of angiogenesis (77, 78).

These theories consider cell density in homogenous tissue elements and apply rules for motion, differentiation, and proliferation to the cell population based upon tissue level stimuli. In vivo, the structure of tissues within callus is microscopically heterogeneous, thus mechanical stimuli at the tissue level are not easily translated to the cellular level. While at the tissue level, strains and perfusion of interstitial fluid (ISF) in the callus can be accurately determined, the stimulation they cause at the cellular level will be different for each cell within a tissue element due to the heterogeneity. This has not yet been quantified in vivo, however, fluid dynamic studies of perfusion bioreactors can lend insights as to the heterogeneity of mechanical stimuli when fluid is perfused through a structure at physiological rates. Zermatten et al. (79) used high resolution micro computed tomography (micro-CT) images of bone tissue engineering scaffolds. Fluid dynamic simulations of medium perfusion showed the wall shear stress within a scaffold had a wide range of values. It has been known for some time that in vitro osteoblastic differentiation and bone formation are augmented with ISF flow (80), however, application of this biological information in silico will require more detailed models of the tissue structure.

The features all these biological aspects share are the boundary conditions, considering the periosteum of the cortical fragments and the surrounding tissue at the source of cells, nutrients, or vascular tissue. The propagation of these are mainly all diffusive processes, though for ease of implementation and to reduce computational power required implementations have varied, the random walk used by Pérez and Prendergast (45) and fuzzy logic of Shefelbine et al. (42) allowed diffusive behavior without the computational cost of solving the diffusion equation numerically.

Mechanical forces and bone regeneration are intrinsically linked. While there has been a vast amount of research on this topic, experimental comparison between studies is extremely difficult. The greatest difficulty comes from non-standard experimental setups, specifically, different defect sizes and use of modified and non-standard fixators. While great efforts have been made to characterize the movements and stiffness of devices, without knowing the true bone geometry and exact tissue composition it is difficult to determine the precise strains tissues are experiencing. These factors play a part in the validation of simulations using this data, as differences in geometry between simulation and experiments should cause different mechanical stimuli.

Validation of models with experimental data is crucial for all simulation work, and is the distinguishing factor between simulation and animation. What is clear is that in the field of bone regeneration, comparing the results of in silico studies with in vivo studies is not trivial. It is impossible to measure every factor simultaneously, yet models for fracture healing have become increasingly complicated. More factors are considered such as cellular events, revascularization, and even protein secretion. While validation of such models is not impossible it requires large cross-sectional studies, which have yet to be performed. To compound this, current models of bone regeneration all consider tissues as homogenous continuum, while woven bone is a macroscopically porous structure. The theories of Carter et al. (73), Claes and Heigele (4), and Prendergast et al. (72) are based on comparing the results of continuum finite element analyzes with histological data, and correlating the areas where bone has formed with the local mechanical environment. The histological data were the result of cross-sectional studies, so while they could observe the formation of woven bone, they could not see how individual structures evolved. One solution to this problem lies in in vivo micro-CT. Micro-CT uses two dimensional X-ray images of a sample taken from multiple directions to reconstruct a three dimensional image of the sample. The application of in vivo micro-CT means time-lapsed longitudinal studies can be performed, where the same animals can be imaged at several time points. These images can then be compared, highlighting the changes in the regenerating tissue. When combined with a well-defined loading regime an equivalent finite element analysis can be performed using the micro-CT images as a basis. The results of which can be compared to the changes in tissue observed in vivo, allowing the validation or falsification of current theories for tissue differentiation.

The development and maturation of the callus (reactive and reparative phase) is a single aspect of fracture healing, eventually the bone must remodel to its original form, while Lacroix and Prendergast (37) include this, it has never been shown that this is the correct mechanism. Schell et al. (8) have shown that early in the healing process osteoclasts are present, and that cortical bone closest to the defect gets remodeled into woven bone, then back to cortical. With in vivo micro-CT based studies this remodeling can be quantified. With quantitative data remodeling theories can be corroborated. Such a study could provide a window into the relationship between cortical remodeling and trabecular remodeling and what causes the differentiation between the two.

Subject specific simulations are a necessity, allowing direct comparison between simulations and experimental results on a sample by sample basis. Currently, no study has performed an animal specific simulation and determined if the simulation produces results without a significant difference to the in vivo results. This will remove the any error associated with averaging experimental results so they can be compared to simulations using idealized geometry. Additionally simulations using realistic geometry and loading conditions will allow the effects of the non-uniform bone geometry and strain distribution within the callus to be quantified. In Figure 5D, we can see clearly how asymmetric the callus is, and that on the left side of the bone cartilage has formed within the osteotomy gap whereas on the right side of the image the cartilage is bridging the larger hard callus. The asymmetry of the bone, callus, and loading cannot be captured in axisymmetric models and only partially using idealized models of the geometry. This geometric information will be the product of studies using in vivo micro-CT. What parameters should be compared between such models is an open question, which must be addressed first. Work already presented in the field of bone remodeling and adaptation (85) provides a basis for what can be measured, and models for bone remodeling such as Schulte et al. (86) may also be incorporated in simulations, representing the remodeling phase.

This review has presented how the mechanical models and experiments of fracture healing have developed since Pauwels (68) first proposed his theory. The differences between the types of data produced by simulations and experimental studies remains an obstacle for advancing the field. Existing rules without exception have all been derived from two dimensional continuum finite element models, though associating the simulated mechanical stimuli to tissue growth seen in histological section from cross-sectional a study. As more detailed, quantitative and longitudinal data are being gathered experimentally these rules must be re-examined, their accuracy assessed using longitudinal time-lapsed data. With simulations there is a need for simulations to move away from simplified representations of the geometry with continuum material properties, toward real bone microstructural geometries measured through micro-CT, so as to allow direct and quantitative comparison of their predicted tissue distribution directly to the results of in vivo studies, rather than a visual comparison with single histological slices.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.