A mathematical framework to study the effects of growth factor influences on fracture healing

During fracture healing, multipotential stem cells differentiate into specialized cells responsible for producing the different tissues involved in the bone regeneration process. This cell differentiation has been shown to be regulated by locally expressed growth factors. The details of their regulatory mechanisms need to be understood. In this work, we present a two-dimensional mathematical model of the bone healing process for moderate fracture gap sizes and fracture stability. The inflammatory and tissue regeneration stages of healing are simulated by modeling mesenchymal cell migration; mesenchymal cell, chondrocyte and osteoblast proliferation and differentiation, and extracellular matrix synthesis and degradation over time. The effects of two generic growth factors on cell differentiation are based on the experimentally studied chondrogenic and osteogenic effects of bone morphogenetic proteins-2 and 4 and transforming growth factor-beta-1, respectively. The model successfully simulates the progression of healing and predicts that the rate of osteogenic growth factor production by osteoblasts and the duration of the initial release of growth factors upon injury are particularly important parameters for complete ossification and successful healing. This temporo-spatial model of fracture healing is the first model to consider the effects of growth factors. It will help us understand the regulatory mechanisms involved in bone regeneration and provides a mathematical framework with which to design experiments and understand pathological conditions.     

A hybrid bioregulatory model of angiogenesis during bone fracture healing

Bone fracture healing is a complex process in which angiogenesis or the development of a blood vessel network plays a crucial role. In this paper, a mathematical model is presented that simulates the biological aspects of fracture healing including the formation of individual blood vessels. The model consists of partial differential equations, several of which describe the evolution in density of the most important cell types, growth factors, tissues and nutrients. The other equations determine the growth of blood vessels as a result of the movement of leading endothelial (tip) cells. Branching and anastomoses are accounted for in the model. The model is applied to a normal fracture healing case and subjected to a sensitivity analysis. The spatiotemporal evolution of soft tissues and bone, as well as the development of a blood vessel network are corroborated by comparison with experimental data. Moreover, this study shows that the proposed mathematical framework can be a useful tool in the research of impaired healing and the design of treatment strategies.     

Influence of fracture gap size on the pattern of long bone healing: a computational study

Following fractures, bones restore their original structural integrity through a complex process in which several cellular events are involved. Among other factors, this process is highly influenced by the mechanical environment of the fracture site. In this study, we present a mathematical model to simulate the effect of mechanical stimuli on most of the cellular processes that occur during fracture healing, namely proliferation, migration and differentiation. On the basis of these three processes, the model then simulates the evolution of geometry, distributions of cell types and elastic properties inside a healing fracture. The three processes were implemented in a Finite Element code as a combination of three coupled analysis stages: a biphasic, a diffusion and a thermoelastic step. We tested the mechano-biological regulatory model thus created by simulating the healing patterns of fractures with different gap sizes and different mechanical stimuli. The callus geometry, tissue differentiation patterns and fracture stiffness predicted by the model were similar to experimental observations for every analysed situation.     

Molecular signaling in bone fracture healing and distraction osteogenesis

The process of fracture healing has been described in detail in many histological studies. Recent work has focused on the mechanisms by which growth and differentiation factors regulate the fracture healing process. Rapid progress in skeletal cellular and molecular biology has led to the identification of many signaling molecules associated with the formation of skeletal tissues, including members of the transforming growth factor-beta (TGF-beta) superfamily and the insulin-like growth factor (IGF) family. Increasing evidence indicates that they are critical regulators of cellular proliferation, differentiation, extracellular matrix biosynthesis and mineralization. Limb lengthening procedure (distraction osteogenesis) is a relevant model to investigate the in vivo correlation between mechanical stimulation and biological responses as the callus is stretched by a proper rate and rhythm of mechanical strain. This model also provides additional insights into the molecular and cellular events during bone fracture repair. TGF-beta 1 was significantly increased in both the distracted callus and the fracture callus. The increased level of TGF-beta 1, together with a low concentration of calcium and an enhanced level of collagen synthesis, was maintained in the distracted callus as long as mechanical strain was applied. Less mineralization is also associated with a low level of osteocalcin production. These observations provide further insights into the molecular basis for the cellular events during distraction osteogenesis.     

Regulation of fracture repair by growth factors.

Fractured bones heal by a cascade of cellular events in which mesenchymal cells respond to unknown regulators by proliferating, differentiating, and synthesizing extracellular matrix. Current concepts suggest that growth factors may regulate different steps in this cascade (10). Recent studies suggest regulatory roles for PDGF, aFGF, bFGF, and TGF-beta in the initiation and the development of the fracture callus. Fracture healing begins immediately following injury, when growth factors, including TGF-beta 1 and PDGF, are released into the fracture hematoma by platelets and inflammatory cells. TGF-beta 1 and FGF are synthesized by osteoblasts and chondrocytes throughout the healing process. TGF-beta 1 and PDGF appear to have an influence on the initiation of fracture repair and the formation of cartilage and intramembranous bone in the initiation of callus formation. Acidic FGF is synthesized by chondrocytes, chondrocyte precursors, and macrophages. It appears to stimulate the proliferation of immature chondrocytes or precursors, and indirectly regulates chondrocyte maturation and the expression of the cartilage matrix. Presumably, growth factors in the callus at later times regulate additional steps in repair of the bone after fracture. These studies suggest that growth factors are central regulators of cellular proliferation, differentiation, and extracellular matrix synthesis during fracture repair. Abnormal growth factor expression has been implicated as causing impaired or abnormal healing in other tissues, suggesting that altered growth factor expression also may be responsible for abnormal or delayed fracture repair. As a complete understanding of fracture-healing regulation evolves, we expect new insights into the etiology of abnormal or delayed fracture healing, and possibly new therapies for these difficult clinical problems.     

Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth

Bone fractures heal through a complex process involving several cellular events. This healing process can serve to study factors that control tissue growth and differentiation from mesenchymal stem cells. The mechanical environment at the fracture site is one of the factors influencing the healing process and controls size and differentiation patterns in the newly formed tissue. Mathematical models can be useful to unravel the complex relation between mechanical environment and tissue formation. In this study, we present a mathematical model that predicts tissue growth and differentiation patterns from local mechanical signals. Our aim was to investigate whether mechanical stimuli, through their influence on stem cell proliferation and chondrocyte hypertrophy, predict characteristic features of callus size and geometry. We found that the model predicted several geometric features of fracture calluses. For instance, callus size was predicted to increase with increasing movement. Also, increases in size were predicted to occur through increase in callus diameter but not callus length. These features agree with experimental observations. In addition, spatial and temporal tissue differentiation patterns were in qualitative agreement with well-known experimental results. We therefore conclude that local mechanical signals can probably explain the shape and size of fracture calluses.     

Dysfunctional stem and progenitor cells impair fracture healing with age

Successful fracture healing requires the simultaneous regeneration of both the bone and vasculature; mesenchymal stem cells(MSCs) are directed to replace the bone tissue, while endothelial progenitor cells(EPCs) form the new vasculature that supplies blood to the fracture site. In the elderly, the healing process is slowed, partly due to decreased regenerative function of these stem and progenitor cells. MSCs from older individuals are impaired with regard to cell number, proliferative capacity, ability to migrate, and osteochondrogenic differentiation potential. The proliferation, migration and function of EPCs are also compromised with advanced age. Although the reasons for cellular dysfunction with age are complex and multidimensional, reduced expression of growth factors, accumulation of oxidative damage from reactive oxygen species,and altered signaling of the Sirtuin-1 pathway are contributing factors to aging at the cellular level of both MSCs and EPCs. Because of these geriatric-specific issues, effective treatment for fracture repair may require new therapeutic techniques to restore cellular function. Some suggested directions for potential treatments include cellular therapies, pharmacological agents, treatments targeting age-related molecular mechanisms, and physical therapeutics.Advanced age is the primary risk factor for a fracture, due to the low bone mass and inferior bone quality associated with aging; a better understanding of the dysfunctional behavior of the aging cell will provide a foundation for new treatments to decrease healing time and reduce the development of complications during the extended recovery from fracture healing in the elderly.     

Beneficial effects of moderate,early loading and adverse effects of delayed or excessive loading on bone healing

Fracture healing involves the differentiation and proliferation of cells in the callus and the synthesis and degradation of connective, cartilage and bone tissue. These processes are initiated and tightly regulated by growth factors and by the mechanical environment in the callus. In this work we incorporated the effects of mechanical stimulation on cell differentiation and ossification into a previously developed temporal-spatial model of growth factor mediated fracture healing. In particular, the stimulatory and inhibitory effects of dilatational and deviatoric strains were modeled. This predictive model was then calibrated and validated using well-defined in vivo experiments from the literature. As in the experiments, the results of the model demonstrated the beneficial and adverse effects of moderate and excessive loading, respectively, as well as the negative effects of delaying mechanical stimulation of rigidly fixed calluses. In addition, the model examined loading conditions and time points beyond those used in the experiments, providing a more complete and mechanistic characterization of the effects of loading in the biological tissue response associated with fracture healing.     

Erythropoietin (EPO): EPO-receptor signaling improves early endochondral ossification and mechanical strength in fracture healing

Beyond its role in the regulation of red blood cell proliferation, the glycoprotein erythropoietin (EPO) has been shown to promote cell regeneration and angiogenesis in a variety of different tissues. In addition, EPO has been indicated to share significant functional and structural homologies with the vascular endothelial growth factor (VEGF), a cytokine essential in the process of fracture healing. However, there is complete lack of information on the action of EPO in bone repair and fracture healing. Therefore, we investigated the effect of EPO treatment on bone healing in a murine closed femur fracture model using radiological, histomorphometric, immunohistochemical, biomechanical and protein biochemical analysis. Thirty-six SKH1-hr mice were treated with daily i.p. injections of 5000 U/kg EPO from day 1 before fracture until day 4 after fracture. Controls received equivalent amounts of the vehicle. After 2 weeks of fracture healing, we could demonstrate expression of the EPO-receptor (EPOR) in terminally differentiating chondrocytes within the callus. At this time point EPO-treated animals showed a higher torsional stiffness (biomechanical analysis: 39.6+/-19.4% of the contralateral unfractured femur) and an increased callus density (X-ray analysis (callus density/spongiosa density): 110.5+/-7.1%) when compared to vehicle-treated controls (14.3+/-8.2% and 105.9+/-6.6%; p<0.05). Accordingly, the histomorphometric examination revealed an increased fraction of mineralized bone and osteoid (33.0+/-3.0% versus 28.5+/-3.6%; p<0.05). Of interest, this early effect of the initial 6-day EPO treatment had vanished at 5 weeks after fracture. We conclude that EPO-EPOR signaling is involved in the process of early endochondral ossification, enhancing the transition of soft callus to hard callus.     

Rice callus extracts for enhancing skin wound healing

Enhanced cell migration in the course of wound healing is required to repair damaged skin. We investigated the effects of rice callus extracts on the migration of skin keratinocytes. Rice callus extracts were obtained by using three different methods: pressurized hot water, crude ethanol, and liquid-liquid extractions. The extract obtained by using crude ethanol extraction was more effective in the migration of skin cells than that obtained by pressurized hot water extraction (PHWE). The crude ethanol extract (CEE) was further partitioned by performing liquid-liquid extraction. As phenolics are inhibitory compounds affecting cell migration, we analyzed the total phenolic content of the rice callus extracts. The level of phenolics in the n-hexane partitioned extract (n-HPE) of CEE was lower than that in all other partitioned extracts. The n-HPE was most effective in enhancing cell migration. We analyzed wound healingrelated factors including platelet derived growth factor B (PDGF-B), transforming growth factor beta 1 (TGF-β1), heparin-binding epidermal growth factor (HB-EGF), and fibroblast growth factor 2 (FGF-2) after the treatment of n-HPE. Most of the expressions of cell migration-related growth factors increased, but HB-EGF dramatically increased (6.5-fold) in keratinocytes treated with n-HPE. The results indicate that n-HPE contains more stimulating growth factors or proteins and less cell migration inhibiting factors than the other tested extracts; thus, n-HPE treatment produced the greatest enhancement of cell migration.     

Prediction of the Time Course of Callus Stiffness as a Function of Mechanical Parameters in Experimental Rat Fracture Healing Studies - A Numerical Study

Numerous experimental fracture healing studies are performed on rats, in which different experimental, mechanical parameters are applied, thereby prohibiting direct comparison between each other. Numerical fracture healing simulation models are able to predict courses of fracture healing and offer support for pre-planning animal experiments and for post-hoc comparison between outcomes of different in vivo studies. The aims of this study are to adapt a pre-existing fracture healing simulation algorithm for sheep and humans to the rat, to corroborate it using the data of numerous different rat experiments, and to provide healing predictions for future rat experiments. First, material properties of different tissue types involved were adjusted by comparing experimentally measured callus stiffness to respective simulated values obtained in three finite element (FE) models. This yielded values for Young’s moduli of cortical bone, woven bone, cartilage, and connective tissue of 15,750 MPa, 1,000 MPa, 5 MPa, and 1 MPa, respectively. Next, thresholds in the underlying mechanoregulatory tissue differentiation rules were calibrated by modifying model parameters so that predicted fracture callus stiffness matched experimental data from a study that used rigid and flexible fixators. This resulted in strain thresholds at higher magnitudes than in models for sheep and humans. The resulting numerical model was then used to simulate numerous fracture healing scenarios from literature, showing a considerable mismatch in only 6 of 21 cases. Based on this corroborated model, a fit curve function was derived which predicts the increase of callus stiffness dependent on bodyweight, fixation stiffness, and fracture gap size. By mathematically predicting the time course of the healing process prior to the animal studies, the data presented in this work provides support for planning new fracture healing experiments in rats. Furthermore, it allows one to transfer and compare new in vivo findings to previously performed studies with differing mechanical parameters.     

Differential distribution of V-type H+-ATPase and Na+/K+-ATPase in the branchial chamber of the palaemonid shrimp Macrobrachium amazonicum

Increased fragility fracture risk with improper healing is a frequent and severe complication of insulin resistance (IR). The mechanisms impairing bone health in IR are still not fully appreciated, which gives importance to studies on bone pathologies in animal models of diabetes. Mice deficient in leptin signaling are widely used models of IR and its comorbidities. Leptin was first recognized as a hormone, regulating appetite and energy balance; however, recent studies have expanded its role showing that leptin is a link between insulin-dependent metabolism and bone homeostasis. In the light of these findings, it is intriguing to consider the role of leptin resistance in bone regeneration. In this study, we show that obese diabetic mice lacking leptin receptor (db/db) are deficient in postnatal regenerative osteogenesis. We apply an ectopic osteogenesis and a fracture healing model, both showing that db/db mice display compromised bone acquisition and regeneration capacity. The underlying mechanisms include delayed periosteal mesenchymatic osteogenesis, premature apoptosis of the cartilage callus and impaired microvascular invasion of the healing tissue. Our study supports the use of the db/db mouse as a model of IR associated bone-healing deficits and can aid further studies of mesenchymatic cell homing and differentiation, microvascular invasion, cartilage to bone transition and callus remodeling in diabetic fracture healing.     

Modelling the early phases of bone regeneration around an endosseous oral implant

The objective of this study was to see whether a mathematical model of fracture healing was able to mimic bone formation around an unloaded screw-shaped titanium implant as it is well-believed that both processes exhibit many biological similarities. This model describes the spatio-temporal evolution of cellular activities, ranging from mesenchymal stem cell migration, proliferation, differentiation to bone formation, which are initiated and regulated by the growth factors present at the peri-implant site. For the simulations, a finite volume code was used and adequate initial and boundary conditions were applied. Two sets of analyses have been performed, in which either initial and boundary condition or model parameter values were changed with respect to the fracture healing model parameter values. For a number of combinations, the spatio-temporal evolution of bone density was well-predicted. However reducing cell proliferation rate and increasing osteoblast differentiation and osteogenic growth factor synthesis rates, the simulation results were in agreement with the experimental data.     

A finite element inverse analysis to assess functional improvement during the fracture healing process

Assessment of the restoration of load-bearing function is the central goal in the study of fracture healing process. During the fracture healing, two critical aspects affect its analysis: (1) material properties of the callus components, and (2) the spatio-temporal architecture of the callus with respect to cartilage and new bone formation. In this study, an inverse problem methodology is used which takes into account both features and yields material property estimates that can analyze the healing changes. Six stabilized fractured mouse tibias are obtained at two time points during the most active phase of the healing process, respectively 10 days (n=3), and 14 days (n=3) after fracture. Under the same displacement conditions, the inverse procedure estimations of the callus material properties are generated and compared to other fracture healing metrics. The FEA estimated property is the only metric shown to be statistically significant (p=0.0194) in detecting the changes in the stiffness that occur during the healing time points. In addition, simulation studies regarding sensitivity to initial guess and noise are presented; as well as the influence of callus architecture on the FEA estimated material property metric. The finite element model inverse analysis developed can be used to determine the effects of genetics or therapeutic manipulations on fracture healing in rodents.     

Simulation of fracture healing incorporating mechanoregulation of tissue differentiation and dispersal/proliferation of cells

Modelling the course of healing of a long bone subjected to loading has been the subject of several investigations. These have succeeded in predicting the differentiation of tissues in the callus in response to a static mechanical load and the diffusion of biological factors. In this paper an approach is presented which includes both mechanoregulation of tissue differentiation and the diffusion and proliferation of cell populations (mesenchymal stem cells, fibroblasts, chondrocytes, and osteoblasts). This is achieved in a three-dimensional poroelastic finite element model which, being poroelastic, can model the effect of the frequency of dynamic loading. Given the number of parameters involved in the simulation, a parameter variation study is reported, and final parameters are selected based on comparison with an in vivo experiment. The model predicts that asymmetric loading creates an asymmetric distribution of tissues in the callus, but only for high bending moments. Furthermore the frequency of loading is predicted to have an effect. In conclusion, a numerical algorithm is presented incorporating both mechanoregulation and evolution of cell populations, and it proves capable of predicting realistic difference in bone healing in a 3D fracture callus.     

A mathematical analysis of physiological and morphological aspects of wound closure

A computational algorithm to study the evolution of complex wound morphologies is developed based on a model of wound closure by cell mitosis and migration due to Adam [Math Comput Model 30(5–6):23–32, 1999]. A detailed analysis of the model provides estimated values for the incubation and healing times. Furthermore, a set of inequalities are defined which demarcate conditions of complete, partial and non-healing. Numerical results show a significant delay in the healing progress whenever diffusion of the epidermic growth factor responsible for cell mitosis is slower than cell migration. Results for general wound morphologies show that healing is always initiated at regions with high curvatures and that the evolution of the wound is very sensitive to physiological parameters.      

Adipose tissue extract shows potential for wound healing: in vitro proliferation and migration of cell types contributing to wound healing in the presence of adipose tissue preparation and platelet rich plasma

Growth factors are the key elements in wound healing signaling for cell migration, differentiation and proliferation. Platelet-rich plasma (PRP), one of the most studied sources of growth factors, has demonstrated to promote wound healing in vitro and in vivo. Adipose tissue is an alternative source of growth factors. Through a simple lipoaspirate method, adipose derived growth factor-rich preparation (adipose tissue extract; ATE) can be obtained. The authors set out to compare the effects of these two growth factor sources in cell proliferation and migration (scratch) assays of keratinocyte, fibroblast, endothelial and adipose derived stem cells. Growth factors involved in wound healing were measured: keratinocyte growth factor, epidermal growth factor, insulin-like growth factor, interleukin 6, platelet-derived growth factor beta, tumor necrosis factor alfa, transforming growth factor beta and vascular endothelial growth factor. PRP showed higher growth factor concentrations, except for keratinocyte growth factor, that was present in adipose tissue in greater quantities. This was reflected in vitro, where ATE significantly induced proliferation of keratinocytes at day 6 (p < 0.001), compared to plasma and control. Similarly, ATE-treated fibroblast and adipose stem cell cultures showed accelerated migration in scratch assays. Moreover, both sources showed accelerated keratinocyte migration. Adipose tissue preparation has an inductive effect in wound healing by proliferation and migration of cells involved in wound closure. Adipose tissue preparation appears to offer the distinct advantage of containing the adequate quantities of growth factors that induce cell activation, proliferation and migration, particularly in the early phase of wound healing.     

Evaluation of residual stresses due to bone callus growth: a computational study

Mechanical environment in callus is determinant for the evolution of bone healing. However, recent mechanobiological computational works have underestimated the effect that growth exerts on the mechanical environment of callus. In the present work, we computationally evaluate the significance of growth-induced stresses, commonly called residual stresses, in callus. We construct a mechanobiological model of a callus in the metatarsus of a sheep in two different stages: one week and four weeks after fracture. The magnitude of stresses generated during callus growth is compared with the magnitude of stresses when only external loads are applied to the callus. We predict that residual stresses are relevant in some areas, mainly located at the periosteal side far from the fracture gap. Therefore, the inclusion of these residual stresses could represent a significant impact on the callus growth and predict a different evolution of biological processes occurring during bone healing.     

Fracture healing as a post-natal developmental process: molecular,spatial, and temporal aspects of its regulation

Fracture healing is a specialized post-natal repair process that recapitulates aspects of embryological skeletal development. While many of the molecular mechanisms that control cellular differentiation and growth during embryogenesis recur during fracture healing, these processes take place in a post-natal environment that is unique and distinct from those which exist during embryogenesis. This Prospect Article will highlight a number of central biological processes that are believed to be crucial in the embryonic differentiation and growth of skeletal tissues and review the functional role of these processes during fracture healing. Specific aspects of fracture healing that will be considered in relation to embryological development are: (1) the anatomic structure of the fracture callus as it evolves during healing; (2) the origins of stem cells and morphogenetic signals that facilitate the repair process; (3) the role of the biomechanical environment in controlling cellular differentiation during repair; (4) the role of three key groups of soluble factors, pro-inflammatory cytokines, the TGF-beta superfamily, and angiogenic factors, during repair; and (5) the relationship of the genetic components that control bone mass and remodeling to the mechanisms that control skeletal tissue repair in response to fracture.