Trabecular bone fracture healing simulation with finite element analysis and fuzzy logic

Trabecular bone fractures heal through intramembraneous ossification. This process differs from diaphyseal fracture healing in that the trabecular marrow provides a rich vascular supply to the healing bone, there is very little callus formation, woven bone forms directly without a cartilage intermediary, and the woven bone is remodelled to form trabecular bone. Previous studies have used numerical methods to simulate diaphyseal fracture healing or bone remodelling, however not trabecular fracture healing, which involves both tissue differentiation and trabecular formation. The objective of this study was to determine if intramembraneous bone formation and remodelling during trabecular bone fracture healing could be simulated using the same mechanobiological principles as those proposed for diaphyseal fracture healing. Using finite element analysis and the fuzzy logic for diaphyseal healing, the model simulated formation of woven bone in the fracture gap and subsequent remodelling of the bone to form trabecular bone. We also demonstrated that the trabecular structure is dependent on the applied loading conditions. A single model that can simulate bone healing and remodelling may prove to be a useful tool in predicting musculoskeletal tissue differentiation in different vascular and mechanical environments.     

A fuzzy logic controller.

This paper describes a fuzzy sets method which is very useful for handling uncertainties and essential for knowledge acquisition of a human expert. Kinetics of a reactor is often complex and not trivial to describe by mathematical equations. Reactor control by traditional control technology is therefore difficult. A novel technology is presented. In the following a fuzzy inference (approximate reasoning) is used for decision making in analogy to human thinking, facilitating a more sophisticated control. Readers of this paper do not need any advanced mathematics beyond the four basic operations in arithmetic (+, -, x, divided by) and using the maximum and minimum values. This fuzzy inference is introduced to construct a fuzzy logic controller which is suitable for a nonlinear, multivariable and time variant system applied to a bioreactor.     

Control model of human stance using fuzzy logic

 A control model of human stance is proposed based on knowledge from behavioral experiments and physiological systems. The proposed model is based on the control of global variables specific to body orientation and alignment, rather than on the control of the body’s center of mass within the base of support. Furthermore, the proposed control model is not based on purely inverted pendulum body mechanics where only motion at one joint is controlled, as for instance the ankle. In the proposed model, the degrees of freedom are controlled by using reciprocal and synergistic muscle actions at multiple joints. The control model is based on three sets of different global variables which act in parallel: (1) limb length and its derivative, (2) limb orientation and its derivative, and (3) trunk attitude and its derivative. An important feature of the control model is the use of fuzzy logic, which enables us to model experimental findings and physiological knowledge in a meaningful and explicit way using fuzzy if-then rules. In the control model, 36 fuzzy if-then rules are implemented and applied using a four-linked segment model consisting of a trunk, thigh, shank and foot. Uni- and biarticular limb muscles and trunk muscles are represented as torque actuators at each individual joint. In the model, three sets of global variables act in parallel and make corrective and coordinated responses to internal, self-induced perturbations. The data show that the use of global variables and fuzzy logic successfully enables us to model human standing with sway about a point of equilibrium. Small changes in, for example, total body sway are comparable to those seen during natural sway in human stance. The selected controllers—limb length, limb orientation and trunk attitude—seem to be appropriate for human stance control. Received: 30 October 1996/Accepted in revised form: 7 April 1997     

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.     

Fetal fracture healing in a lamb model.

A large animal model to assess fetal fracture repair and the ability to close excisional bony defects is presented. Incisional and excisional ulnar fractures were made in 14 midgestation fetal lambs, harvested at serial time points, and subjected to high-resolution low-kilovolt magnification radiographs, magnetic resonance imaging scans, and histologic analysis. Fetal fracture healing was characterized by early closure of excisional defects and rapid fracture healing with minimal or no soft-tissue inflammation or callus formation. Magnetic resonance imaging scans of the fractures revealed a characteristic pattern compatible with the histologic findings, namely, minimal inflammation in soft tissue adjacent to the fracture site. Histologic and magnification radiographic findings indicated that complete bony repair occurred within 21 days in incisional defects and within 40 days in excisional defects. In both cases, healed fetal bone resembled normal bone matrix. Excisional defects, including periosteum, of greater than three times the width of the bony cortex closed rapidly with virtually normal-appearing bony matrix and with minimal or no callus formation.     

A numerical model of the fracture healing process that describes tissue development and revascularisation

A dynamic model was developed to simulate complex interactions of mechanical stability, revascularisation and tissue differentiation in secondary fracture healing. Unlike previous models, blood perfusion was included as a spatio-temporal state variable to simulate the revascularisation process. A 2D, axisymmetrical finite element model described fracture callus mechanics. Fuzzy logic rules described the following biological processes: angiogenesis, intramembranous ossification, chondrogenesis, cartilage calcification and endochondral ossification, all of which depended on local strain state and local blood perfusion. In order to evaluate how the predicted revascularisation depended on the mechanical environment, we simulated two different healing cases according to two groups of transverse metatarsal osteotomies in sheep with different axial stability. The model predicted slower revascularisation and delayed bony bridging for the less stable case, which corresponded well to the experimental observations. A revascularisation sensitivity analysis demonstrated the potential of the model to account for different conditions regarding the blood supply.     

A numerical model of the fracture healing process that describes tissue development and revascularisation

A dynamic model was developed to simulate complex interactions of mechanical stability, revascularisation and tissue differentiation in secondary fracture healing. Unlike previous models, blood perfusion was included as a spatio-temporal state variable to simulate the revascularisation process. A 2D, axisymmetrical finite element model described fracture callus mechanics. Fuzzy logic rules described the following biological processes: angiogenesis, intramembranous ossification, chondrogenesis, cartilage calcification and endochondral ossification, all of which depended on local strain state and local blood perfusion. In order to evaluate how the predicted revascularisation depended on the mechanical environment, we simulated two different healing cases according to two groups of transverse metatarsal osteotomies in sheep with different axial stability. The model predicted slower revascularisation and delayed bony bridging for the less stable case, which corresponded well to the experimental observations. A revascularisation sensitivity analysis demonstrated the potential of the model to account for different conditions regarding the blood supply.     

Mobilization of endogenous stem cell populations enhances fracture healing in a murine femoral fracture model

Background aimsDelivery of bone marrow–derived stem and progenitor cells to the site of injury is an effective strategy to enhance bone healing. An alternate approach is to mobilize endogenous, heterogeneous stem cells that will home to the site of injury. AMD3100 is an antagonist of the chemokine receptor 4 (CXCR4) that rapidly mobilizes stem cell populations into peripheral blood. Our hypothesis was that increasing circulating numbers of stem and progenitor cells using AMD3100 will improve bone fracture healing.MethodsA transverse femoral fracture was induced in C57BL/6 mice, after which they were subcutaneously injected for 3 d with AMD3100 or saline control. Mesenchymal stromal cells, hematopoietic stem and progenitor cells and endothelial progenitor cells in the peripheral blood and bone marrow were evaluated by means of flow cytometry, automated hematology analysis and cell culture 24 h after injection and/or fracture. Healing was assessed up to 84 d after fracture by histomorphometry and micro–computed tomography.ResultsAMD3100 injection resulted in higher numbers of circulating mesenchymal stromal cells, hematopoietic stem cells and endothelial progenitor cells. Micro-computed tomography data demonstrated that the fracture callus was significantly larger compared with the saline controls at day 21 and significantly smaller (remodeled) at day 84. AMD3100-treated mice have a significantly higher bone mineral density than do saline-treated counterparts at day 84.ConclusionsOur data demonstrate that early cell mobilization had significant positive effects on healing throughout the regenerative process. Rapid mobilization of endogenous stem cells could provide an effective alternative strategy to cell transplantation for enhancing tissue regeneration.     

Identification of tissue differentiation rates in a mechanobiological model of fracture healing

As a basis for model-based analysis of the processes in secondary fracture healing, a dynamical model is presented that characterises the physiological status in the fracture area by the location-dependent composition of tissues. Five types of tissue are distinguished: connective tissue, cartilage, bone, haematoma and avascular bone. A rule base is given that describes dynamical tissue differentiation processes. The rules consider not only a mechanical stimulus but also osteogenic and a vasculative factors as biological stimuli. Within this model structure, it is possible, e.g., to distinguish intramembranous from endochondral ossification processes. An objective function is introduced to assess accordance between the model-based simulation results and reference healing stages. By minimising this objective function, relevant tissue differentiation rates can be determined. For a reference process of secondary fracture healing it could be shown that the intramembranous ossification rate of 0.313%/day (from connective tissue to bone) is much smaller than the endochondral ossification rate of 1.136%/day (from cartilage to bone). In order to verify the model approach, it is transferred to simulate long bone distraction. Results show that healing patterns of bone distraction can be predicted. Using this method, it is possible to identify model parameters for individual subjects. This will allow a patient-specific analysis of tissue healing processes in future.     

A computational model to explore the role of angiogenic impairment on endochondral ossification during fracture healing

While it is well established that an adequate blood supply is critical to successful bone regeneration, it remains poorly understood how progenitor cell fate is affected by the altered conditions present in fractures with disrupted vasculature. In this study, computational models were used to explore how angiogenic impairment impacts oxygen availability within a fracture callus and hence regulates mesenchymal stem cell (MSC) differentiation and bone regeneration. Tissue differentiation was predicted using a previously developed algorithm which assumed that MSC fate is governed by oxygen tension and substrate stiffness. This model was updated based on the hypothesis that cell death, chondrocyte hypertrophy and endochondral ossification are regulated by oxygen availability. To test this, the updated model was used to simulate the time course of normal fracture healing, where it successfully predicted the observed quantity and spatial distribution of bone and cartilage at 10 and 20 days post-fracture (dpf). It also predicted the ratio of cartilage which had become hypertrophic at 10 dpf. Following this, three models of fracture healing with increasing levels of angiogenic impairment were developed. Under mild impairment, the model predicted experimentally observed reductions in hypertrophic cartilage at 10 dpf as well as the persistence of cartilage at 20 dpf. Models of more severe impairment predicted apoptosis and the development of fibrous tissue. These results provide insight into how factors specific to an ischemic callus regulate tissue regeneration and provide support for the hypothesis that chondrocyte hypertrophy and endochondral ossification during tissue regeneration are inhibited by low oxygen.     

A mechano-regulation model for tissue differentiation during fracture healing: analysis of gap size and loading

Bone has a capability to repair itself when it is fractured. Repair involves the generation of intermediate tissues, such as fibrous connective tissue, cartilage and woven bone, before final bone healing can occur. The intermediate tissues serve to stabilise the mechanical environment and provide a scaffold for differentiation of new tissues. The repair process is fundamentally affected by mechanical loading and by the geometric configuration of the fracture fragments. Biomechanical analyses of fracture healing have previously computed the stress distribution within the callus and identified the components of the stress tensor favouring or inhibiting differentiation of particular tissue phenotypes. In this paper, a biphasic poroelastic finite element model of a fracture callus is used to simulate the time-course of tissue differentiation during fracture healing. The simulation begins with granulation tissue (post-inflammation phase) and finishes with bone resorption. The biomechanical regulatory model assumes that tissue differentiation is controlled by a combination of shear strain and fluid flow acting within the tissue. High shear strain and fluid flows are assumed to deform the precursor cells stimulating formation of fibrous connective tissue, lower levels stimulate formation of cartilage, and lower again allows ossification. This mechano-regulatory scheme was tested by simulating healing in fractures with different gap sizes and loading magnitudes. The appearance and disappearance of the various tissues found in a callus was similar to histological observation. The effect of gap size and loading magnitude on the rate of reduction of the interfragmentary strain was sufficiently close to confirm the hypothesis that tissue differentiation phenomena could be governed by the proposed mechano-regulation model.     

Segmentation of protein surfaces using fuzzy logic

An algorithm has been developed that can be used to divide triangulated molecular surfaces into distinct domains on the basis of physical and topographical molecular properties. Domains are defined by a certain degree of homogeneity concerning one of these properties. The method is based on fuzzy logic strategies, thus taking into consideration the smooth changes of the properties considered along complex macromolecular surfaces. Scalar qualities assigned to every node point on a triangulated surface are translated into linguistic variables, which can then be processed using a special fuzzy dissimilarity operator.Possible applications are demonstrated using surface segmentation for properties like electrostatic potential, lipophilicity and shape for the analysis of serine proteinase substrate/inhibitor specificity.     

Dynamic analysis of a simplified bone model during the process of fracture healing

The dynamic analysis of fracture healing is tackled numerically by means of a bone model which uses the finite element method. The model is of non-uniform cross-sectional area and moment of inertia. Shear and rotatory inertia are taken into account. Considerable variation of the upper natural frequencies is observed as the healing process progresses. The practical implications, as well as present limitations, of the technique are examined.     

Biologic adjuvants for fracture healing

Bone tissue has an exceptional quality to regenerate to native tissue in response to injury. However, the fracture repair process requires mechanical stability or a viable biological microenvironment or both to ensure successful healing to native tissue. An improved understanding of the molecular and cellular events that occur during bone repair and remodeling has led to the development of biologic agents that can augment the biological microenvironment and enhance bone repair. Orthobiologics, including stem cells, osteoinductive growth factors, osteoconductive matrices, and anabolic agents, are available clinically for accelerating fracture repair and treatment of compromised bone repair situations like delayed unions and nonunions. Preclinical and clinical studies using biologic agents like recombinant bone morphogenetic proteins have demonstrated an efficacy similar or better than that of autologous bone graft in acute fracture healing. A lack of standardized outcome measures for comparison of biologic agents in clinical fracture repair trials, frequent off-label use, and a limited understanding of the biological activity of these agents at the bone repair site have limited their efficacy in clinical applications.     

Muscle-bone interactions during fracture healing

Although it is generally accepted that the rate and strength of fracture healing is intimately linked to the integrity of surrounding soft tissues, the contribution of muscle has largely been viewed as a vascular supply for oxygen and nutrient exchange. However, more is becoming known about the cellular and paracrine contributions of muscle to the fracture healing process. Research has shown that muscle is capable of supplying osteoprogenitor cells in cases where the periosteum is insufficient, and the muscular osteoprogenitors possess similar osteogenic potential to those derived from the periosteum. Muscle’s secrotome includes proteins capable of inhibiting or enhancing osteogenesis and myogenesis following musculoskeletal injury and can be garnered for therapeutic use in patients with traumatic musculoskeletal injuries. In this review, we will highlight the current knowledge on muscle-bone interaction in the context of fracture healing as well as concisely present the current models to study such interactions.