Computational models of bone fracture healing and applications: a review

Monan Wang; Guodong Jiang; Haoyu Yang; Xin Jin

doi:10.1515/bmt-2023-0088

Article Publicly Available

Computational models of bone fracture healing and applications: a review

Monan Wang , Guodong Jiang , Haoyu Yang and Xin Jin

Published/Copyright: January 19, 2024

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From the journal Biomedical Engineering / Biomedizinische Technik Volume 69 Issue 3

Abstract

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.

Keywords: fracture healing; mathematical modeling; computational biology; bone engineering; fixation; clinical treatment strategies

Introduction

Bone is an essential part of the human body and assumes many roles in its functioning, the two most important of which are to provide structural support and physical protection for vital organs [1]. Bone’s growth plasticity – its capacity to change and adapt in response to pressure or mechanical forces – as well as its capacity for repair are directly correlated with an individual’s age. Bone is primarily made up of two substances: organic and inorganic matter. Children’s bones have more organic matter, so their bones are more resilient and elastic, less prone to fracture and heal faster after fracture [2], [3], [4]; older people have less organic matter, so their bones are more brittle, more prone to fracture and heal slower after fracture. Nowadays, the incidence of fractures has increased due to the increasing number of various means of transportation, recreational facilities and the aging of the population. According to reports, nonunion or incomplete healing occurs in 5–10 % of cases after fracture treatment [5], therefore, it is of great practical importance to conduct an in-depth study on fracture healing to understand the healing mechanism and to guide clinical treatment strategies.

Fracture healing is a complex biological process, and there are two ways of fracture healing, namely primary and secondary healing [6]. Primary healing of a fracture is an ideal state of healing, which can be called primary healing when there is no bone debris, blood clot, or inflammatory reaction between the two sections of the fracture, and the wound margin and wall of the broken end can be neatly aligned. Otherwise, secondary healing occurs. Secondary healing is a common form of healing in everyday life, and the secondary healing process has three overlapping phases: the inflammatory phase, the repair phase (soft healing tissue phase and hard healing tissue phase), and the remodeling phase [5, 7], [8], [9], [10] (Figure 1). Both mechanical stimulation and growth factors can have an impact on fracture healing. Appropriate mechanical stimulation will promote fracture healing and, conversely, inappropriate mechanical stimulation will result in delayed or even nonunion [11], [12], [13]. In addition to mechanical stimulation that can have an effect on fracture healing, biological factors also play an important role in the fracture healing process.

$Figure 1: Schematic diagram of secondary fracture healing process.$

Figure 1:

Schematic diagram of secondary fracture healing process.

Over the past few decades, as computer technology has advanced, the use of computer modeling to simulate the complex mechanical and biological events of fracture healing has become an increasingly powerful tool for studying bone healing, and these models are expected to guide the development and optimization of clinical treatment strategies to improve the healing environment for patients with nonunion and incomplete healing. This will not only reduce unnecessary medical costs, but will also reduce patient pain. The emergence of computer simulation has two implications. Firstly, the validation of experimental observations allows the development of a more comprehensive model describing the fracture healing process. These healing models can be used to study the effects of inter fragmentary movement, type and structure of the fixator, and growth factors on fracture healing and to explore the optimal healing environment. Secondly, due to the complexity of fracture healing and the diversity of time and scale of fracture healing, there are still some unknown mechanisms of fracture healing, and although they can be obtained from experimental data, the connections between these levels, such as the transduction of mechanical stimuli from the bone tissue level to the cellular or even intracellular level, are not fully understood. With the help of computerized healing models, these missing links can be established, and once validated, these models can provide valuable information [14].

Since Pauwels proposed the first framework describing the regulation of tissue differentiation by mechanical stimulation [15], numerous researchers have worked on the relationship between local mechanical stimulation and tissue formation at the fracture site [16], [17], [18], [19], and although these models represent a significant development in the field of bone healing, they are limited to mechanical factors and ignore the influence of biological factors on fracture healing, which allows the models to explain the problem in a very limited way. Therefore, more and more researchers have tried to develop computer models in a multiscale direction. The effect of growth factors was first added to computer models by Bailon-Plaza and Van der Meulen et al. [20]. They used partial differential equations to simulate the cell activities (migration, proliferation and differentiation) in the callus region. Subsequently, Geris, Peiffer, Calier et al. incorporated angiogenesis and oxygen factors into the model [21], [22], [23], and Trejob et al. investigated the effect of macrophages during inflammation of bone healing [24, 25]. Also, coupled models that combine mechanical stimuli with biological factors are beginning to emerge [26], [27], [28]. The complexity of the models gradually increased as the number of influencing factors in the models increased, and these models provided increasingly in-depth explanations of bone healing mechanisms.

We consult Wang and Yang’s review for an overview of bone healing models. They suggested classifying models into three groups: mechanoregulatory models, bioregulatory models, and coupled mechanobioregulatory models [29]. They detailed the bioregulatory models and the coupled mechanobioregulatory models in yet another review [11], emphasizing the crucial role that biochemical signals play in the healing of fractures. Our work focuses on a comprehensive analysis of model application, from which we draw conclusions about the limitations of the existing models and the challenges they face in real-world settings. In Section 2, we first summarize the developments in the field of computer models of bone healing over the past decades using Citespace analysis software, analyzing research highlights as well as trends. In Section 3, we provide an overview of some new computational models of bone healing in recent years (Tables 1 and 2), including mechanoregulatory computational healing models, bioregulatory mathematical healing models and coupled mechanobioregulatory computational healing models, followed by the analysis of their research methods and innovations, and a discussion of the limitations of these models and future directions for improvement. In Section 4, we provide a systematic summary of the application of the models in three areas: bone tissue engineering, fixator optimization, and clinical treatment strategies. We conclude the article with a discussion of the difficulties faced in the clinical translation of computer models.

Table 1:

Summary of mechanoregulatory healing models.

Reference	Model type	Dimension	Material property	Mechanical stimuli	Cells	Callus growth
Ren et al. [40]	Single phase element model and fuzzy logic	3D	Linear elastic isotropic and homogeneous	Distortional strain and hydrostatic strain	–	–
Ghiasi et al. [41]	Biphasic finite element model and fuzzy logic	2D	Linear poroelastic	Shear strain and relative fluid/solid velocity	Mesenchymal stem cells	Fixed volume Various geometric dimensions
Ghiasi et al. [41]	Biphasic finite element model and fuzzy logic	2D	Linear poroelastic	Shear strain and relative fluid/solid velocity	Osteoblasts	Fixed volume Various geometric dimensions
Naveiro et al. [42]	Single phase element model diffusion model and mesh growing algorithm	2D	Poroelastic	Distortional strain and hydrostatic strain	Mesenchymal stem cells	Volume growth
Naveiro et al. [42]		2D	Poroelastic	Distortional strain and hydrostatic strain	Chondrocytes	Volume growth
Schwarzenberg et al. [43]	Single phase element model and spatial proximity functions	3D	Linear elastic	Distortional strain and hydrostatic strain	–	Volume growth
Ghimire et al. [44]	Biphasic finite element model	2D	Poroelastic	Octahedral shear strain and relative fluid velocity	Mesenchymal stem cells	Bone Cartilage formation
					Fibroblasts	Bone Cartilage formation
					Chondrocytes
					Osteoblasts
Miramini et al. [45]	Biphasic finite element model and probabilistic-based computational model	3D	Poroelastic	Octahedral shear strain and relative fluid velocity	–	Bone
Miramini et al. [45]		3D	Poroelastic	Octahedral shear strain and relative fluid velocity	–	Cartilage formation
Wang et al. [46]	Biphasic finite element model and fuzzy logic	3D	Poroelastic	Octahedral shear strain and fluid velocity	Mesenchymal stem cells	Volume growth
Pietsch et al. [49]	Single phase element model and interface-capturing techniques	2D	Linear elastic	Distortional strain and hydrostatic strain	–	Bone and cartilage formation
Wang et al. [48]	Single phase element model	2D	Linear elastic	Distortional strain and dilatational strain	–	–
Wang et al. [48]	Diffusion equations and fuzzy logic	2D	Linear elastic	Distortional strain and dilatational strain	–	–
Ghimire et al. [50]	Biphasic finite element model	2D	Poroelastic	Octahedral shear strain and fluid velocity	–	Bone formation

“–” represents that this healing process did not include in the healing model.

Table 2:

Summary of bioregulatory and coupled mechanobioregulatory healing models.

Reference	Model type	Biochemical signals	Mechanical stimuli	Cells	Angiogenesis
Vavva et al. [51]	Partial differential equations and discrete variable for angiogenesis	Chondrogenic growth factors	–	Mesenchymal stem cells Chondrocytes Osteoblasts Endothelial cells	Blood vessel growth Anastomosis
		Osteogenic growth factors
		Angiogenic growth factors
		Angiogenic growth factors
H. V. Kojouharov et al. [25]	Nonlinear ordinary differential equations	Inflammatory cytokines	–	Macrophages Mesenchymal stem cells	–
		Anti-inflammatory cytokines		Macrophages Mesenchymal stem cells

				Osteoblasts
Trejo et al. [24]	Nonlinear ordinary differential equations	Inflammatory cytokines	–	Unactivated macrophages	–
		Anti-inflammatory cytokines		Unactivated macrophages
				Classical macrophages
				Alternative macrophages
				Mesenchymal stem cells
				Osteoblasts
Zhang et al. [57]	Partial differential equations	Tumor necrosis factor-alpha	–	Macrophages	–
				Mesenchymal stem cells
				Chondrocytes
				Osteoblasts
				Fibroblasts
O’Reilly et al. [59]	Partial differential equations and discrete variable for angiogenesis	Oxygen concentration	Octahedral shear strain	Mesenchymal stem cells	Blood vessel growth
	Biphasic finite element model and lattice mode		Octahedral shear strain	Mesenchymal stem cells	Blood vessel growth
			Fluid velocity	Osteoblasts	Anastomosis
			Substrate stiffness	Adipocytes
			Oxygen tension	Chondrocytes
				Hypertrophic
				Chondrocytes
				Fibroblasts
Frame et al. [61]	Partial differential equations and finite element analysis	Chondrogenic growth factors	Shear strain	Mesenchymal stem cells	–
		Osteogenic growth factors		Mesenchymal stem cells
				Chondrocytes
				Osteoblasts
				Fibroblasts
Grivas et al. [28]	Partial differential equations and discrete variable for angiogenesis	Chondrogenic growth factors	Shear strain and fluid velocity	Mesenchymal stem cells Chondrocytes Osteoblasts Endothelial cells	Blood vessel growth Anastomosis
		Osteogenic growth factors
		Angiogenic growth factors
		Angiogenic growth factors

“–” represents that this healing process did not include in the healing model.

Analysis of research highlights based on CiteSpace

Bibliometrics is a visual quantitative analysis method that can use mathematical and statistical methods to describe and evaluate various external specialties of scientific literature, summarize the current status of research, and predict future trends within the scientific field. In this paper, Citespace is chosen as a literature analysis tool. Citespace is a JAVA-based application that can quickly find key information about research and new trends in specific disciplines in large volumes of literature, and the software is widely used to understand the research progress and current research frontiers in a particular discipline.

The terms “fracture healing” and “computer modeling” were used to search the Web of Science for this study. The search yielded 498 publications from 1991 to 2022 after reviews, conference articles, letters, and other non-relevant content were eliminated. A visual analysis of 498 documents using keywords as node types was performed with the help of Citespace software to make a keyword co-occurrence map. To show the changes in keyword types and the frequency of keywords over time, a timezone view was generated to visualize the searched literature quantitatively.

In Figure 2, a total of 558 nodes and 1,424 connecting lines were generated, with the nodes indicating research content and the connecting lines indicating the existence of an association between two contents. Among them, the keywords‘model’, ‘fracture healing’, ‘tissue differentiation’ and ‘finite element analysis’ have larger node rings, which indicates that these contents appear more frequently in all studies and are the hot contents in research and receive more and more attention.

Figure 2:

Keywords co-occurrence graph.

In Figure 3, the keyword centrality was calculated, and keywords with a centrality higher than 0.1 are the key nodes in the diagram. The higher the centrality of the keyword, the more important it is. Since there are many keyword nodes, we divided the time into six intervals with five years as a stage and used the following criteria to select the key nodes displayed in the intervals: 1. keywords ranked in the top five in frequency in each interval; 2. keywords with centrality above 0.1; 3. if there are no keywords with centrality above 0.1 in the interval, the keywords ranked in the top five in centrality are displayed.

Figure 3:

Keywords time zone map.

From Figures 2 and 3, we can summarize the field of fracture healing computer models. In the intervals from 1991 to 1995 and from 1996 to 2000, we can clearly see “fracture healing”, “finite element analysis”, “tissue differentiation”, etc. In the intervals from 2001 to 2005 and from 2006 to 2010, the keywords “bone regeneration”, “mesenchymal stem cells”, “biophysical stimuli”, etc. were particularly important. The distribution of the keywords shown in the graph is consistent with the fact that, after Pauwels, researchers have extensively used finite elements to analyze the effect of mechanical stimulation on bone healing and subsequently incorporated biological factors such as cellular activity, angiogenesis, and oxygen into the model. Of course, important elements such as “fixation”, “gap size”, and “callus” were also included. Unfortunately, as can be seen from the keyword time zone graph, the frequency of keywords appearing in the field of computational models of fracture healing after 2015 is generally low and the centrality is not high, which indicates that there have been few new studies in this field in recent years, the content is scattered, and the research is not very hot.

Computational model of fracture healing

Mechanoregulatory computational healing models

Mechanical stimulation plays an important role in the process of bone healing, and changes in mechanical stimulation can lead to alterations in healing pathways [11], [12], [13, 30]. The local strain field in the callus has a modulating effect on tissue differentiation; the site of fracture break, the mode of fracture, the direction of interfragmentary motion, and the magnitude of displacement are all factors that determine the local mechanical stimuli in the callus. The role of fixation techniques following a fracture (external fixators and the increasingly popular internal fixation techniques) is to give adequate support to the fractured bone, provide stability to the injury site, and provide a certain growth environment for the regenerative healing of bone tissue. However, high-stiffness fixation has side effects on the healing process due to the high stress in the bone and pins [31]. Excessive flexibility can also lead to excessive bone movement, resulting in implant failure and healing failure. It is generally accepted that flexible fixation promotes the healing process, whereas rigid fixation and unstable fixation impede healing or even lead to bone nonunion [32], [33], [34]; appropriate axial motion at the fracture site can promote bone healing, while shear motion may cause delayed healing [35, 36]. The understanding of fracture healing has improved significantly over the past few decades as a result of research on mechanical stimuli, and mechanoregulatory healing models have been successfully applied to reproduce the course of fracture healing. However, these models have various drawbacks: 1) model parameters such as material properties are difficult to obtain; 2) the models have been simplified in terms of material model used and number of mechanical properties (number of tissues) simulated; 3) mechanical factors alone are considered, and biochemical factors are missing [29]. In addition, individual differences have an impact on the process of fracture healing, such as fracture shape [37], age, disease, smoking or not, etc. [38, 39]. These factors will greatly increase the prediction difficulty of the model. Ren et al. [40] used a fuzzy logic based mechanoregulatory computational model to simulate the bone healing process. They added different fracture geometries, intramedullary nail fixation and multiaxial loading (axial load and torsion) conditions to their model. In their simulations, delayed healing occurred for all types of fractures when torsional instability was present, as well as nonunion when realistic torsional loads with fixator mechanics were included in the model.

The effect of the initial healing stage on bone healing is usually ignored in simulations of fracture healing, such as the inflammatory phase, the granulation tissue formation and initial callus formation. Meanwhile, past models mainly considered tissue differentiation and bone remodeling [16, 20, 29], updating the material type of the unit according to different mechanical and biological factors. These models limited the development of callus to a predefined range without considering the callus growth itself. Therefore, in further studies, the shape, size, composition, and mechanical properties of the callus should be considered as important optimization targets. In recent years, several researchers have improved models for such problems. In order to study the modulation of initial parameters on bone healing, Ghiasi et al. [41] developed a finite element model to simulate the migration and diffusion coefficients of MSCs at different levels, Young’s modulus of granulation tissue, thickness of callus, and gap size of fragments. Their study quantified the impact of the initial healing stage on fracture healing outcome, which is of great significance for understanding the mechanobiological mechanisms of bone healing. Naveiro et al. [42] and Schwarzenberg et al. [43] have further explored the growth process of callus. The former applied a diffusion problem and a mesh-growing algorithm in a bone fragment model of fracture to allow the callus to grow naturally under established conditions until it heals. This method had been proved effective in transverse fractures, oblique fractures, and comminuted fractures, and was even able to connect isolated fragments in comminuted fractures. The latter added two spatial proximity functions to the existing mechanoregulatory model as a way to control the positioning of the callus within the healing domain, and in their simulations the growth of the callus was successfully restricted to near the fracture line with only a 2 % difference in volume size between the different domains.

In addition, the models in recent years have further supplemented the exploration of bone healing. Ghimire et al. [44] proposed an integrated computational model, which considered cell and growth factor transport under dynamic loading as well as MSC differentiation and histogenesis mediated by mechanical stimulation, to study the effect of dynamic loading on early fracture. Miramini et al. [45] argued that there is uncertainty in the factors affecting fracture healing, such as genetic factors, trauma characteristics and fixation methods, and therefore these uncertainties should be incorporated into the model. They developed a probability-based computational model, which presents the input parameters in the form of a distribution rather than discrete sets, to investigate the effect of uncertainty in key input parameters on the probability of osteogenesis. Wang et al. [46] created a dynamic model integrating a two-phase poroelastic finite element analysis with fuzzy logic control to mimic the healing process in various mechanical settings. They then added growth factor activity to the model previously published by Simon et al. [47] to investigate the effect of mechanical stability on growth factor activity [48]. In the new model, they used finite elements to calculate the spatiotemporal distribution of mechanical stimuli, partial differential equations to describe growth factor activity, and finally fuzzy logic rules to describe tissue differentiation. Pietsch et al. [49] proposed a new numerical model of bone healing using an interface capture technique from the field of fluid dynamics to describe tissue growth. Ghimire et al. [50] proposed a computational framework consisting of a coupled callus mechanics model and a mechano-regulation model to study the rate of bone formation under intramembranous and endochondral osteogenesis conditions.

Bioregulatory mathematical healing models

Many growth factors, in addition to mechanical stimuli, are involved in the healing of fractures (examples include transforming growth factor-β, platelet-derived growth factors, bone morphogenetic proteins, fibroblast growth factors, insulin-like growth factors, vascular endothelial growth factor, and oxygen). Wang and Yang’s review described the function of common growth factors in fracture healing [14]. These growth factors regulate activities including cell migration, proliferation, and differentiation, which have a significant impact on various stages of fracture healing and tissue development. They are secreted by various cells at various times and locations. The model becomes significantly more complex as a result of cellular behavior, growth factor interactions, and extracellular matrix formation and breakdown. Less consideration has been given to growth factors than to mechanical stimuli, despite their significance in the fracture healing process. Partial differential equations (as well as ordinary differential equations) are frequently used in bioregulatory models to simulate cellular activity and the regulatory effects of biochemical signals. Recent mathematical models that use nonlinear ordinary differential equations take into account the interactions between pro- and anti-inflammatory cytokines as well as the regulation of inflammation by macrophages and mesenchymal stem cells, reflecting cell proliferation and differentiation as well as the production and degradation of extracellular matrix. The spatio-temporal evolution of soft tissue, bone, and vascular networks is described in bioregulatory healing models consisting of partial differential equations.

Since ultrasound can significantly promote an increase in the levels of cytokines, fibroblast growth factor and vascular endothelial growth factor (VEGF), which can promote the formation of blood vessels, Vavva et al. [51], in order to explain the effect of ultrasound on angiogenesis, extended the model of Peiffer et al. [21] and proposed a hybrid mathematical model consisting of partial differential equations describing the spatio-temporal evolution of the soft tissue, bone and vascular network. In this model, based on previous in vitro experimental studies performed on human cells [52, 53], ultrasound was used as the main factor affecting VEGF, and different ultrasound intensities were simulated by imposing different sound pressure boundaries. This model provides novel insights into the effects of ultrasound on bone regeneration and angiogenesis.

Immune cells and the cytokines they release play a key role in bone healing [54], especially macrophages and mesenchymal stem cells (MSCs), which have been found to play a decisive role in controlling inflammation by clearing debris, releasing anti-inflammatory cytokines and suppressing functions of inflammatory immune cell respectively [55, 56]. However, the exact function of immune cells and the way they affect fracture healing are not completely understood. H. V. Kojouharov et al. [25] proposed a new mathematical model consisting of nonlinear ordinary differential equations to study the early inflammatory effects in the process of bone healing, which combines immune cells, histiocytes and their regulatory signals. Their numerical simulations showed that the use of anti-inflammatory cytokines at the beginning of the healing process may accelerate the healing time. The optimal dose depends on each type of fracture and high concentrations of inflammatory cytokines will negatively affect the healing time of the fracture.

In the work of H. V. Kojouharov et al. [25], the first attempt was made to include macrophage in a mathematical model simulating the fracture healing process, and Trejo et al. [24] extended the previous work to include two different phenotypes of macrophages (classical and alternatively activated macrophages), proposing a new mathematical model consisting of nonlinear ordinary differential equations to further study the macrophage mediated inflammation in the early stage of fracture healing. For the study of the effect of inflammation on the fracture healing process, additional molecular-cellular interactions between the inflammatory and repair phases should be added to the model in the future and the remodeling phase should be included as well.

Zhang et al. [57] also investigated the inflammatory process in the early phases of healing. To study the impact of tumor necrosis factor (TNF) on early fracture healing in both normal and diabetic settings, researchers created a numerical model made up of partial differential equations that describe the reactive action of cells and cytokines in fracture scabs. They came to the conclusion that there is an ideal level of TNF-α concentration that can positively influence the differentiation and added value of MSCs and efficiently promote early healing, whereas too high (as in the case of diabetes) or too low concentrations of TNF-αcan impair the healing process.

Among the recently published bioregulatory healing models, there are some shortcomings: 1) The exact mechanisms, functions and cellular interactions of, for example, immune cells and macrophages are not fully understood; 2) In these bioregulatory healing models, only biological factors in the healing process are considered, without taking into account mechanical stimuli that are important for fracture healing. The direct effect of mechanical stimuli on the cell differentiation process is not clarified in these models; 3) In these examples the simplified geometry is more a consecuences of the methods used to solve the equations (Figure 4). Other models have considered more realistic geometries; 4) For some of the latest bioregulatory healing models, there is still a lack of relevant experiments to validate them. At the current stage, the comparison between simulation predictions and validation provided by experimental studies is still an obstacle to the development of this field; 5) In these bioregulatory healing models, only one or several factors are considered, and other factors that may affect the healing outcome are not included in the models. In summary, there is still much room for expansion and refinement of the bioregulatory healing model, which provides a direction for future research on bioregulatory healing models for fracture healing.

Figure 4:

Two dimensional model of callus, 1 represents periosteal callus, 2 represents intercortical callus, 3 represents endosteal callus, 4 represents cortical bone [21, 58].

Coupled mechanobioregulatory computational healing models

It was a significant step to include the impacts of biochemical components in the simulation of fracture healing, but as was described in Section 3.1, the effects of mechanical stimulation must also be taken into account. Researchers started experimenting with coupled models after Bailon-Plaza and van der Meulen [20] expanded the earlier bioregulatory model to include mechanical stimulation.

Angiogenesis and oxygen supply are key regulators in the process of tissue regeneration, but there have been few studies of progenitor cells in vascularly impaired fractures subjected to changes due to this altered condition. Daniel J. Kelly [59] and his colleagues, in order to test the hypothesis that local oxygen tension can regulate chondrocyte hypertrophy and endochondral ossification within the bone scab of long bone fractures, added specific rules for oxygen levels associated with chondrocyte hypertrophy and endochondral osteogenesis to their previous computational model of mesenchymal stem cell differentiation [60]. They took three models with different degrees (mild, moderate, and severe) of vascular damage and compared the results of the simulation with experimental observations of fracture healing in mice with ischemic limbs. The updated algorithm was able to simulate the healing pattern of normal and angiogenically damaged fracture repair.

Successfully modeling bone tissue evolution is important for predicting multiple biological processes such as bone reconstruction, fracture healing and implant osseointegration. Jamie Frame et al. [61] proposed a new multi-tissue model based on the previous mechano-biological model of bone formation by Schmitt et al. [62], including some new mechano-biological criteria, such as distinction of several tissue types, strain-dependent tissue growth and strain-controlled tissue maturation. In their work, bone remodeling and bone healing processes are integrated in a numerical model where they are controlled by the same mechanical stimuli. The model shows potential in designing and implementing implants and describing fracture healing and bone remodeling processes.

In a previous study [51], Vavva first considered the beneficial effects of ultrasound on associated angiogenesis and proposed a bioregulatory model providing predictions of bone healing. Next, Vavva and Carlier et al. [28] proposed a mechanobioregulatory model of bone healing in the presence of ultrasound in conjunction with the mechanical environment of the regenerative process, presenting and discussing numerical simulations with and without the presence of ultrasound, illustrating the effect of progenitor cell origin on healing patterns and healing rates, while demonstrating the beneficial effects of ultrasound on bone repair.

In the coupled mechanobioregulatory mathematical healing models, the mechanical stimuli to which the callus is subjected are simulated and calculated using a finite element approach, and the cellular activity is simulated using a coupled system of partial differential equations with mechanical stimuli added as a parameter to the equations. Although these coupled mechanobioregulatory healing models take both mechanical stimuli and biological factors into account, these models are still deficient. First, the predictions provided by these models need to be experimentally validated, however, there are a number of models that have not been related to the appropriate in vitro or in vivo experiments can be used to verify their accuracy, and comparisons between the predictions provided by simulations and experimental validation remain an obstacle to advancing the field. In addition to this, the shape of callus model is still not a real callus shape, but a simplified shape to reduce computational costs, and similarly, the finite element models do not adequately simulate the real human condition. Although the effect of each simplification is considered small enough to have no significant effect on the predicted results, it still does not fully simulate the realistic situation of fracture healing.

Application of fracture healing computer model

Computational models of fracture healing have been developed for decades and have made tremendous progress. Although the complexity of the fracture healing process poses a great challenge for the clinical application of computational models, and indeed there are currently few, if any, examples of these models in clinical use, they have served as an inspiration for research in many fields. Computational models have the potential to reveal unknown mechanisms in the bone healing process and improve our understanding of the healing process, while also promising to guide the design of clinical treatment protocols. In this section, we summarize the application of computational models of bone healing in three areas: bone tissue engineering, fixator design, and clinical treatment strategies (Figure 11).

Bone tissue engineering and regenerative medicine

Current methods for the treatment of critical-sized bone defects are mainly autologous and allogeneic bone grafts. However, they both have limitations, with autologous bone grafts bringing varying degrees of complications such as inflammation and infection [63] as well as pressure on treatment costs [64], while allogeneic bone grafting procedures are associated with slower and incomplete bone fusion and regeneration, which can easily lead to graft failure [65]. Bone tissue engineering is a potentially effective method to address this problem, which has a lower risk of infection, no significant complications, and better biocompatibility [66, 67]. Scaffolds are an important part of bone tissue engineering (Figure 5). However, the design of scaffolds is difficult; they need to not only help the injury site to withstand the load, but also be able to accelerate cell growth, angiogenesis and bone tissue regeneration at the injury site. The current design of scaffolds is mainly based on a trial-and-error approach, in which repeated in vivo and in vitro experiments are performed and existing designs are improved based on the results, which is costly. Many researchers have applied computer models of bone healing as an aid in scaffold design to study the mechanical behavior, porosity, microgeometry, biochemical signals, and cell seeding of scaffolds, and the application of mechanically regulated models has been reviewed by Boccaccio et al. [68], with further examples provided in this review.

Figure 5:

Scaffold-based tissue engineering approach and process [69].

Structural design of scaffolds

In the study of structural and mechanical properties of scaffolds, the mechanoregulatory model of Prendergast et al. is widely used [19], which describes tissue differentiation using a biphasic finite element model that characterizes the cell differentiation stimulus S by using the shear strain in the solid phase and the fluid velocity in the interstitial liquid phase as biophysical stimuli, as follows:

(1) S = γ a + v b

where γ is octahedral shear strain, ν is interstitial fluid flow velocity, a = 3.75   % and b = 3   μ m s − 1 are empirical constants.

Boccaccio et al. conducted a series of studies on the design of bone tissue engineering scaffolds using the theory of Prendergast. In 2016, they proposed an optimization algorithm to study the microstructure of scaffolds by combining finite element models, numerical optimization methods, and mechanoregulatory healing models [70]. They investigated the effects of scaffold pore shape, spatial distribution, and the number of pores per unit area on the performance of the scaffold, and determined the optimal dimensions of the pores by comparing different values of scaffold Young’s modulus and compressive load applied to the surface of scaffold. Subsequently, Boccaccio et al. continuously extended and upgraded the above model to investigate the optimal porosity distribution in functional graded scaffolds [71], the optimal geometry of scaffolds with rhombicuboctahedrom unit cell [72], the optimal scaffold geometry and the optimal loading value that allows for maximum bone formation [73], and their results were in high agreement with experimental data, demonstrating that the bone healing model has considerable potential in scaffold design.

In scaffold design, bone healing models are complementary to experimental studies, and these models are often used to simulate the bone healing and tissue growth process after scaffold optimization. Researchers can set up scaffolds with different shapes, stiffnesses, permeabilities, and loading conditions, and then use bone healing models to simulate the healing process separately, thereby evaluating the performance of the target scaffold (Figure 6). Chen et al. [74] investigated the interaction between biodegradable scaffold degradation and tissue growth. They optimized the topology of the scaffold to obtain a series of scaffold structures with different combinations of stiffness and permeability, and used the random walk model [75] and the mechanoregulatory healing model [19] to simulate the fracture healing process to explore the important influence of the structural design of the scaffold on tissue regeneration. Zhao et al. [76] investigated bone tissue formation in ideal porous hydrogels under a range of mechanical loading conditions (mechanical compression, liquid irrigation, and a combination of both) by adding cell differentiation, proliferation, migration, and apoptosis to the mechanically regulated model of Prendergast [19]. In 2019, Koh et al. [77] proposed a finite element method for cartilage tissue regeneration based on mechanical regulation theory to predict the optimal mechanical properties of scaffolds. Montaño et al. [78] combined the mechanical conditioning algorithm of Prendergast et al. [19] with a load adaptation algorithm to develop a computational framework to design and optimize the microstructure of irregular load adapted scaffolds in bone tissue engineering. Irregular load adapted scaffolds showed superior performance over conventional scaffolds under various boundary and loading conditions, which not only produced more bone but also were stronger than conventional scaffolds. Metz et al. [79] coupled a mechanical-biological bone regeneration model [80] with an automatic parametric scaffold design generation to investigate for the first time the time-dependence of the optimized scaffolds, comparing the bone healing results of scaffolds with various geometries and initial mechanical stimulation. They found that some scaffolds that performed very well immediately after implantation had instead low bone growth after 60 days, and therefore, they concluded that the mechanical properties calculated immediately after scaffold implantation could not be used as an indicator of successful bone regeneration. Their study emphasizes the need for long-term regeneration-promoting effects of scaffolds, which also has a guiding role in scaffold design.

Figure 6:

Schematic of the algorithm utilized to determine the best geometry and the best load value for scaffolds for bone tissue engineering [73]. δ denotes the difference between the value of the pressure applied to the surface of the bracket and the ideal value of the pressure, and ε denotes the preset difference.

Addition of biochemical signals

In addition to mechanical conditions, the process of bone tissue reconstruction involves the biological microenvironment as well as a series of biological events [81]. Currently, vascularization is an important factor in the treatment of critical-sized bone defects. In the absence of adequate vascularization, cells inside the implant will become necrotic and lead to failure of bone regeneration. Calier et al. [82] used a multiscale model of fracture healing [23] to study the complex interplay of angiogenesis, oxygen supply, growth factor production, and cell proliferation in critical-sized bone defects. Their model successfully simulated atrophic and oligotrophic nonunion, where the mechanism of action causing the nonunion was the delayed vascularization of the central region of the bone scab creating hypoxic conditions, resulting in cell death as well as failure of bone healing. Also using a multiscale bone healing model, Sun et al. developed a three-dimensional multiscale system model incorporating growth factor release, osteogenic differentiation and proliferation, angiogenesis, and nutrient transport, and investigated the effects of scaffold pore size, porosity, and the combined use of growth factors BMP2, Wnt, and VEGF on angiogenesis and bone formation [83]. O’Reilly and Kelly [84] simulated tissue formation during the treatment of osteochondral defects with cell-loaded scaffolds using a bone healing model that included oxygen tension factors [85].

Among the growth factors that affect bone formation, bone morphogenetic protein 2 (BMP-2) is a widely and intensively studied direct inducer of osteoblast differentiation. Not only does it promote osteogenic differentiation and promote extracellular matrix production [86], but also BMP2 treatment strategies have similar healing success rates as bone grafting, but with less risk [87], and its promotion of fracture healing has been demonstrated in a study by Kim H et al. in 2013 [88]. Ribeiro et al. [89] proposed a mechanobiological model to study the effect of BMP-2 on the healing of large bone defects. They applied quantitative experimental data on the effect of BMP-2 on cellular activity to a mechanobiomodulation model, which was subsequently used to simulate the healing of large bone defects under three conditions: normal healing, implantation of empty hydrogel, and hydrogel soaked with BMP-2. Borgiani et al. [90] combined finite element and agent-based computer modeling to investigate the underlying mechanisms behind the pattern of bone tissue formation during the BMP-2 enhanced healing of critical size bone defects. Their simulations showed that BMP-2 has a strong chemotactic effect on bone marrow mesenchymal stem cells and that this chemotaxis, which regulates cellular dynamics and promotes early periosteal bridging, is a key mechanism behind bone tissue formation.

Fabrication of scaffolds

Additive manufacturing techniques for bone tissue engineering scaffolds include stereolithography, selective laser sintering, fused deposition modeling, and 3D printing. Peroco et al. [91] applied the mechanical conditioning model of Prendergast [19] to the optimization of fused deposition modeling techniques to determine the optimal spacing between columnar scaffold fibers in bone tissue engineering. Calier et al. [92] combined computational models of bone regeneration [23, 93] with 3D bioprinting techniques to describe a new integrated approach for the biomanufacturing of bone tissue engineering and successfully bioprinted such implants with specific spatial structures that are well suited to improve healing efficiency.

Optimum design of fixator

Internal fixation

Fixators are a common technique for the treatment of various fracture problems, as they not only provide stability to the injured area, but also this stability provides a certain growth environment for the regenerative healing of the bone tissue. However, implants like plates, screws and intramedullary nails will face considerable complexity in vivo, such as the material properties of the implant, its structure and implant/bone interactions, and because of this, it seems relevant to evaluate and optimize the performance of internal fixation devices using a computer model of bone healing. Mehboob et al. [94] combined a finite element model with a mechanical conditioning algorithm [95] to investigate the effect of stainless steel intramedullary nails/flexible composite intramedullary nails and reamed intramedullary nails/unreamed intramedullary nails on fracture healing. Their results showed that flexible intramedullary nails have better biocompatibility, while unreamed intramedullary nails seem to better facilitate fracture healing due to their ability to provide better blood supply conditions. Subsequently, the same authors combined a rejection coefficient algorithm (RC algorithm) for assessing the shape of the callus with a three-dimensional model and a mechanical conditioning algorithm to investigate the micromovements of the fracture gap after the implantation of glass/polypropylene fiber composite intramedullary nails [96], glass/polypropylene composite bone plates [97] and composite prostheses, respectively [98]. Liu et al. [99] combined experimental studies with a mechanical conditioning algorithms to develop a mathematical model to explore the configuration and postoperative loading of the metacarpal locking plate that facilitates indirect healing (Figure 7). Fu et al. [100] used a fuzzy logic-based mechanical conditioning tissue differentiation algorithm to study experimental data from a sheep osteotomy healing model in order to investigate the effect of the degree of dynamization of fixation stiffness (i.e., the relative change from rigid to flexible fixation) on bone healing. They simulated the bone healing process from fully rigid fixation to reduced 90 % stiffness fixation, and the results showed that the degree and timing of the dynamization of fixation stiffness can affect the healing process, and that moderate dynamization can enhance the recovery of fractured bone.

Figure 7:

Research methodology of fixator design [99].

Although plates and screws are commonly used to treat fractures, such treatment usually requires a second surgery to remove the implant, a process that faces problems such as infection, which not only poses a risk to the treatment but also causes secondary injuries to the patient. Therefore, biodegradable implants have been widely studied as alternatives. Regarding the effect of biodegradable implants on fracture healing, Mehboob et al. evaluated the healing performance of biodegradable composite bone plates in tibial fractures in combination with a mechanical conditioning model [101, 102], and optimized the design parameters of functional gradient biodegradable composite bone plates in combination with the Taguchi method [103]. In addition, Kowsar et al. [104] developed a three-dimensional finite element model of flexural bone and combined it with mechanical conditioning theory [105] to investigate the effects of biodegradable plates with different elastic moduli and degradation modes (both linear and nonlinear) on the fracture healing process. Their study showed that the healing process can be accelerated if the initial elastic modulus of the bone plate is increased, while a good healing rate will be maintained if the degradable bone plate has a sufficiently high initial elastic modulus and a sufficiently long degradation period. Vautrin et al. [106] developed a finite element model including biodegradation and bone healing algorithms and applied it to an orthognathic surgery scenario to simulate bone healing as well as time evolution of the biodegradation process of two different stiffness bone plates.

External fixation

The treatment of fracture healing with external fixators seems to have more potential [107, 108] than the healing rate and the possible risk of infection associated with implantation of plates, for example, which can be used to treat fractures by non-surgical methods. External fixators can be classified according to their structure as linear, circular, and hybrid fixators (as shown in Figure 8I–K), among which the Ilizarov is widely used, which not only limits the movement of the fracture site well but also has good stability [109]. Ganadhiepan et al. [110], in order to determine the effect of early weight-bearing on osteocytes and growth factors under different fracture gap sizes, axial loads, and preload of the wire of the circular fixator, developed a computational model of the fracture site using the poroelastic formula. Their study indicated that physiologically relevant dynamic loading can significantly increase chondrocyte and growth factor concentrations and promote secondary fracture healing. The same authors [111] then developed a computational framework including angiogenesis and mechano-regulation mediated fracture healing process to determine the optimal level of weight bearing under Ilizarov fixator treatment. Their study not only demonstrated the existence of optimal weight-bearing that enhances healing without inhibiting angiogenesis, but they also gave recommendations on when and how much weight should be applied.

$Figure 8: Internal and external fixators in fracture treatment. (A) Intramedullary nail fixation. (B) Laterally based plate fixation. (C) Lateral plate fixation. (D) Lateral plate fractured (arrow). (E) Two plates fixation and bone shortening. (F) A blade plate and nonlocking screws fixation. (G) The blade and screws fall out (arrow). (H) An anatomic specific plate fixation with locking screws [112]. (I) Linear fixator. (J) Circular fixator. (K) Hybrid fixator [109].$

Figure 8:

Internal and external fixators in fracture treatment. (A) Intramedullary nail fixation. (B) Laterally based plate fixation. (C) Lateral plate fixation. (D) Lateral plate fractured (arrow). (E) Two plates fixation and bone shortening. (F) A blade plate and nonlocking screws fixation. (G) The blade and screws fall out (arrow). (H) An anatomic specific plate fixation with locking screws [112]. (I) Linear fixator. (J) Circular fixator. (K) Hybrid fixator [109].

Clinical treatment strategy

Distraction osteogenesis

Another application of computer models of fracture healing is to develop or optimize clinical treatment strategies in conjunction with different pathological conditions. Distraction osteogenesis (DO) is a common clinical treatment for skeletal deformities and bone defects by generating appropriate mechanical stimulation to induce bone tissue formation (Figure 9). However, the optimal parameters of DO in clinical applications remain vague, such as fixation stiffness and distraction rate. A number of researchers have used computer models of bone healing to simulate DO as a useful addition to human experiments.

Figure 9:

The distraction process of tibial lengthening: (A) Application of distractor; (B) start of distraction; (C) end of distraction; (D, E) consolidation phase without anydistraction until bone in the distraction gap consolidates; (F) removal of distractor [113].

The rate of distraction is one of the key factors for the success of DO. If the rate of distraction is too high or too low, it will affect the process of bone healing. Isaksson et al. [114] used mechanical conditioning algorithm of Prendergast to simulate the tissue distribution in time and space during distraction osteogenesis, and their study demonstrated the critical effect of distraction rate on bone tissue formation. What is the optimal draft rate for the process of DO? Boccaccio et al. [115] developed the mechanical conditioning model of Prendergast [19], to examine the effect of draft rate on mandibular fracture regeneration. Of the four study scenarios they set up (0.6 mm/day for 10 days, 1.2 mm/day for five days, 2 mm/day for three days, and 3 mm/day for two days), a retraction rate of 1.2 mm/day was considered to be the optimal retraction rate. Similarly, Romo et al. [116] extended the previous model of bone healing [117] with a series of studies on DO, and they similarly concluded that a distraction rate of 1 mm/day was the most effective for tissue regeneration (Figure 10). However, the above studies have certain drawbacks, and there is also some doubt as to whether these parameters are optimal due to the different age, gender, and site of injury in each individual. Subsequently, Romo et al. added angiogenesis and cell migration processes to the model [118] and also considered the effect of stress accumulation in DO [119] in an attempt to model the process of DO from a more comprehensive perspective.

Figure 10:

Flow chart of iterative computational simulation that involves several sequential steps [122]: (a) initialization (given the geometry, mesh and initial conditions), (b) a poroelastic finite element analysis to compute the mechanical stimulus (the distortional and dilatational strains are computed), (c) cell differentiation regulated by the resulting stimulus, which follows the modified diagram of Claes and Heigele [16], (d) matrix production is calculated as well as the material properties of each element, (e) MSC proliferation and two diffusion analyses to simulate MSC migration and the advance of the ossification front [117] and (f) a remeshing step to adapt the mesh to the large geometry changes that occur during the process of distraction [116].

In addition to the retraction rate, Boccaccio [120] studied tissue differentiation after mandibular osteotomy under masticatory forces using the mechanical conditioning model of Prendergast, and they predicted 7–8 days as the optimal latency time (time from osteotomy day to first device traction), a result that is consistent with the study of Conley and Legan et al. [121]. Romo et al. [122] developed the tissue differentiation model [16] for the fixator during DO. Their study showed that a rigid fixator can promote bone formation, while excessive motion caused by an extremely flexible fixator would be detrimental to bone bridging. Also, they applied a computer model to a real patient with hemifacial shortening [123] and demonstrated the potential of the model in calculating device removal time, appropriate retraction rate, and designing stable fixators by comparing simulated data with clinical treatment. In addition, Niemeyer et al. [124] proposed a new computational model for simulating lateral distraction osteogenesis based on the tissue differentiation hypothesis of Claes and Heigele and the bone healing model of Simon [16, 125, 126]. Their model is able to simulate the main features of DO and give different treatment options.

Pathological fracture problems

In addition to DO, bone healing models have contributed to the study of pathological fracture problems. Geris et al. [127] applied their previously developed biomodulation model [22] describing fracture healing to the study of the occurrence of atrophic bone discontinuity and used it to design some possible treatment strategies. The same authors also applied the bone healing model [26] to the prediction of normal healing and overload-induced bone nonunion [27]. Carlier et al. [128] used a computational model of bone regeneration [23, 93] to examine the effect of Nf1 mutations on fracture healing. Their model predicted not only the formation of malunion but also various congenital pseudarthrosis of the tibia (CPT) phenotypes.

Medication

Currently, a wide variety of drugs are applied in the treatment of skeletal diseases, such as antibiotics, anticancer, anti-inflammatory, growth factors, enzymes, antibodies, bioactive proteins, cellular and non-viral genes [129], however, many therapeutic strategies are still flawed. The application of computational models is expected to be of great help in the development of drug treatment strategies. It is generally accepted that the potential of computational models in drug development includes 1) the application of models to characterize how lead compounds affect intracellular signaling, 2) the application of models to predict cellular phenotypes from signaling information, and 3) to simulate clinical outcomes [130]. During the early inflammatory phase of bone healing, anti-inflammatory drugs are used to improve the bone healing process, and Kojouharov and Trejo et al. developed a bone healing model that simulates the early inflammatory process [24, 25], and used it to study the evolution of bone healing after administration of anti-inflammatory drugs and to explore possible therapeutic strategies for drug-accelerated bone healing. This provides good ideas for a better understanding of the bone healing process and for the development of new therapeutic strategies.

Figure 11:

Applications of bone healing computer model in different aspects.

Discussion

Computational models are powerful tools, and increasingly sophisticated models help to more fully understand the complex process of bone healing and to obtain rich biomechanical information, and the models’ prediction of clinical outcomes helps in the development and optimization of surgical protocols. However, it is undeniable that the current computer models of bone healing have certain limitations.

First of all, fracture healing is a complex process, and limited by existing biological knowledge as well as computer technology, numerous bone healing models have been simplified to some extent, such as using elastic or poroelastic models to represent the healing tissue and simplifying the fracture gap to a simple transverse geometry, etc. Naveiro et al. used a new method to simulate the initial healing tissue growth [37]; however, it was only successfully tested in an axisymmetric two-dimensional model and was not extended to more complex three-dimensional models. The simplification of the model depends on what problem the model is proposed to explain, then the problem here is that the model can explain factors that are taken into account but not those that are not, such as the absence of the effect of growth factors in mechanically regulated models. Also, the current models generally do not take into account the individual patient, whose age, presence of genetic disorders, smoking, etc., can have an impact on the bone healing process. Therefore, the question to be pondered is how the models can strike a balance between simplicity and validity given the existing known conditions.

But as the number of variables in the model rises, the number of equations in the model rises as well (7 equations in the work of Bailon-Plaza and van der Meulen [20], 12 equations in the work of Geris et al. [22], 11 equations in the work of Peiffer et al. [21], 17 equations in the work of Carlier et al. [23], 16 equations in the work of Carlier et al. [131]), making the model more intricate and challenging to solve. Additionally, the size and complexity of the finite element mesh affect the accuracy of the finite element model, and these factors directly relate to computing cost [132]. This puts a strain on the computational speed and effectiveness of the bone healing model, in addition to requiring a more powerful computer. The deployment of the model in routine clinical practice will undoubtedly be hampered if the calculation of the model takes too long. Currently, a statistical modeling technique is offered by the development of machine learning technology to increase the effectiveness of finite element computations [132], [133], [134]. Liu et al. [135] used a combination of numerical simulation and machine learning to train a machine learning algorithm using the healing results predicted by a finite element model as input to establish a statistical relationship between the characteristic parameters and the healing results, with the aim of improving the efficiency of computational modeling. Their method provides a new way of thinking for predicting complex healing processes. In future research, the partial differential equations in the model can be integrated into the algorithm to improve computational efficiency. Likewise, deep learning can be taken into consideration.

The second issue is the lack of quality of the model data. Limited by experimental conditions as well as materials, the source of data in many models is data from other in vivo experiments, while different species are used for the experimental subjects, and although the same treatment strategy may have less disparity between some species [136], it is still unknown whether the mechanisms between these animals and humans are similar. In the model of Wang et al. [48], the parameters of the partial differential equations in their model were estimated from in vivo and in vitro experiments in different species, and these factors are bound to negatively affect the results of the study and make the model predictions inaccurate. Numerous researchers have used mice and sheep to study the effects of drugs, mechanical stimuli, biochemical factors, and other factors on fracture healing. The experimental data obtained in mice and sheep may not be accurate due to the different mechanisms of life action in different species, and the application to human experiments is equally unknown.

In the meantime, the existing model adopts a wide range of pertinent parameters needed for modeling, and the correctness of model is significantly impacted by the uncertainty of the parameters. On the one hand, this uncertainty may result in inaccurate forecast outcomes and reduced model stability. On the other hand, it will be more challenging to estimate the parameters if there is uncertainty in several model parameters. The uncertainty of parameters in a complex model may also spread to other components, increasing the likelihood of model mistakes and the likelihood that the model and reality may diverge. Miramini et al. [45] adopted a probabilistic technique to incorporate parameter uncertainty in the prediction of cell differentiation and tissue creation during fracture healing in light of the diversity of factors affecting fracture healing. In contrast to deterministic research, they studied the association between mechanical stimulation inside the fracture gap and the degree of weight bearing, bone-plate distance, and plate working length by entering the parameters as a distribution rather than a single data set. This is the first time that probabilistic methods have been used to predict cell differentiation and tissue formation during fracture healing, despite the fact that they have recently been applied to the reliability assessment of orthopedic structures [137], [138], [139], [140], [141] and the design of novel orthopedic components [142, 143]. Although their work still has to be verified with a significant amount of experimental data, it is a commendable effort to point the way for model improvement.

Finally, bone healing models have been successfully applied to design in vivo and in vitro experiments [144], [145], [146]. Benito et al. designed controlled experiments using sheep in order to determine the effect of high-frequency and low-amplitude cyclic displacements of fracture fragments on the bone healing process [147]. A static fixator was implanted in the control group of sheep, and a fixator with a vibrator was used in the stimulated group of sheep. From the experimental results, it was observed that the bone healing process improved in the stimulated group. Lesage et al., on the other hand, provided a computerized in vitro strategy where they developed a signal transduction network model using a knowledge-based and data-driven modeling technique, which opens up new avenues for the development of targeted therapies for osteoarthritis [148]. However, one of the key factors in the translation of computational models to the clinic is the need for comprehensive validation and assessment of the accuracy and reliability of existing computer models of bone healing (including the matching of model predictions to experimental results, sensitivity analysis, robustness analysis, etc.). On the one hand, we need to assess whether the disease mechanisms of animals in animal models match those of humans. On the other hand, we need to comprehensively evaluate the validity and application prospects of computational models.

The modeling and simulation study of bone fracture healing is a complex process that requires not only an in-depth understanding of the mechanism of bone healing and delicate settings of material properties, boundary conditions, loading conditions, etc., but also the physical condition and age of the fracture subject, such as diabetes, osteoporosis, and hypertension. Therefore, with the gradual advancement of mechanobiology of bone healing and the development of computer capabilities, there will be more and more comprehensive and complex models in the future, which will integrate knowledge from various directions such as mechanobiology and materials science to provide a more accurate description of the physiological process of bone healing at multiple scales.

In addition, once the models are fully validated and evaluated, more accurate multiscale models will have a tremendous impact on clinical treatment. At the same time, the model will combine imaging technology, bio-detection sensing technology, and advanced implant manufacturing to provide intelligent, integrated, and efficient treatment to patients during fracture treatment. Intelligent means using imaging technology such as computed tomography to understand the fracture condition, using bio-detection sensing technology to complete the assessment of the patient’s physical condition, inputting the patient’s basic information of age, gender and disease into the system through the input module. And coding the above imaging and detection technology into the same system is integrated, and then the system will give the best treatment plan according to the patient’s information, which is efficient. It not only shortens the doctor’s testing and diagnosis time, but also reduces the patient’s pain and shortens the healing cycle.

Conclusions

In this paper, we have reviewed recent computer models of bone healing according to the classification of mechanically adjusted, biologically adjusted, and coupled models, and analyzed the advantages and disadvantages of these models. We also summarize the applications of bone healing models in three areas: bone tissue engineering, fixator design, and clinical treatment strategies. Finally, we discuss the difficulties in translating bone healing models to clinical applications. In terms of the application of the model, although there are few or no examples of its application to practice, computer models of bone healing have been of great guidance in many areas of research, and increasingly mature models are expected to have a profound impact on the field of bone healing in the future.

Corresponding author: Monan Wang, School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, Heilongjiang, China, E-mail: mnwang@hrbust.edu.cn

Funding source: Natural Science Foundation of Heilongjiang Province of China

Award Identifier / Grant number: ZD2019E007

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: No. 61972117

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors state no conflict of interest.
Research funding: This article was supported by the National Natural Science Foundation of China (No. 61972117) and the Natural Science Foundation of Heilongjiang Province of China (ZD2019E007).
Data availability: Not applicable.

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Received: 2023-03-02

Accepted: 2023-12-12

Published Online: 2024-01-19

Published in Print: 2024-06-25

Articles in the same Issue

https://doi.org/10.1515/bmt-2023-0088

Keywords for this article

fracture healing; mathematical modeling; computational biology; bone engineering; fixation; clinical treatment strategies