A finite element analysis for monitoring the healing progression of fixator-bone system under three loading conditions

2.1 Three-dimensional model

Modelling CT images of the right lower limb used in this study were acquired in connection with a previous study [26]. The slice thickness of the CT images was 1.5 mm in a 512 × 512 matrix. The DICOM data set, which consists of 225 CT images, was imported into Mimics (software version 15.1, Materialise, Leuven, Belgium) to reconstruct the surface geometry of the tibia. Based on the bone density threshold value, a segmentation process was carried out on each slice of the CT data set to tibia bone.

The external fixator systems, the Orthofix frames were designed using 3D computer-aided design (CAD) software (Solidworks 2012, Dassault Systems Solidworks Corp, USA). To simulate bone fixation, two pins were fixed at the tibia diaphysis, the distance between the external and the bone was set at 40 mm. The tibia bone was cut in the middle, and all the bone models were then converted into a surface triangular mesh and saved in STL format.

2.2 Finite element model

The STL files of the bones were imported into ANSYS (ANSYS, USA) and meshed with tetrahedral elements. A mesh convergence study was performed using five different mesh sizes, from 1 to 5 mm. The optimum mesh size for the bone was 3 mm and 1 mm for the external fixator. The fixators were meshed with four noded tetrahedral linear elements. The total number of elements and nodes for the fixator model and bone were 624,000, 139,000 and 245,000, 75,000 respectively. The contact body between the external fixator and the bone was set with a friction coefficient of 0.4 based on a previous study [27]. Finite element mesh of fixator-bone is shown in Fig. 1.

A human tibia fracture takes around 12 weeks to heal [29]. As very little bone grows in the first two weeks, the Young’s modulus of callus in the first two weeks was mainly the connective tissue. The Young’s modulus for the soft callus (initial connective tissue) was about 10 MPa, around 0.0004 that of the intact bone [28,29]. At the completion of healing, the Young’s modulus of a healing callus is roughly 60% of that for the intact bone [30,31]. Detailed material properties of intact bone and fixator are shown in Table 1.

2.3 Loading and boundary conditions

For axial stiffness, the distal bone end was fixed and proximal bone end was subjected to a axial force (F = 600 N) (Fig. 2(a)). For torsional stiffness, the distal bone end was fixed and proximal bone end was subjected to a torque (T = 15 N ⋅ m/degree) (Fig. 2(b)). For 4-point bending stiffness, the two loads were applied on the position 90 mm away from the fracture gap (P = 500 N) (Fig. 2(c)). The degrees of freedom (DOFs) of the bone were fixed. For these three loading conditions, to avoid apparent stress concentration, the loads were applied onto the eight nodes surrounding the loading points. Loading and boundary condition is shown in Fig. 2.

3.1 Stress distribution

Figures 3(a)-(c) exhibit stress distributions of fixator-bone system under axial load, when callus are granulation tissue, immature bone and mature bone. Shown in Fig. 3(a), when the callus is granulation tissue, compared with fixator, the stiffness of bone is very low, and the most weight is born by the external fixator, the highest stress value was 7.957 × 10⁷ N/m² occurred in the contact surface between fixator and pin. According to Fig. 3(b), the callus gradually matured, the load progressively transfers from fixator to bone, fixator and bone both are bear load, the highest stress on the contact surface between the fixator and pin reduced, which is 3.3806 × 10⁷ N/m². For Fig. 3(c), when callus in the fracture gap matured, the bone in the fracture gap is matured bone tissue, the maximum stress value was 2.2355 × 10⁷ N/m², which occurred in the region of fracture gap, bone could bear the main load in the bone maturation stage, meaning fixator can be removed.

Figures 3(d)-(f) exhibits the stress distribution of fixator-bone system under torsional load during healing process. For Fig. 3(d) similar to axial load, under the action of 15 N.m torsional force, when the callus is granulation tissue, the maximum stress values reaches 2.483 × 10⁸ N/m², which occurred in the contact region of pin and fixator. When callus gradually matured as is shown in Fig. 3(e), compared with fixator, the bone bears a bigger load, meaning the load is transferred from fixator to bone. Maximum stress value of 3.1461 × 10⁸ N/m² occurs in the contact surface between bone and pin. When callus matured as is shown in Fig. 3(f), higher stress mainly concentrated in distal bone meaning bone bear bigger load than fixator, which indicates that the fixator could removed at this time. Due to the smaller diameter of the pin, maximum stress value of 8.8014 × 10⁸ N/m² still occurs at connect surface of bone and pin.

Figures 3(g)-(i) exhibit stress distribution of fixator-bone system under bending load. When the bone just begins to heal as is shown in Fig. 3(g), the maximum stress value is 1.8789 × 10⁷ N/m², which also occurs on the surface of the pin. Because the callus at the fracture gap is a granulation tissue, the stiffness of bone is very low, so orthopedic patients are not suitable for bearing load at the early stage of healing process. When bone begins to grow, as is shown in Fig. 3(h), bending load is transferred from fixator to bone, higher stress appears at the distal and proximal bone surface near the pin. The maximum load value of 1.4448 × 10⁷ N/m² appears in the contact surface between bone and pin. After bone maturation, as is shown in Fig. 3(i), the stress on the bone is increased and bigger than that on the fixator, the maximum load value of 2.5624 × 10⁷ N/m² occurs at the fracture site.

Assessment of healing progress of fractured bones is important both in orthopedic practice and research. The healing status of fractured long bones is clinically assessed by radiological and manual methods, which have been proved subjective and inaccurate in determining whether a fracture has healed. The assessment of the healing process aims to monitor the material strength at the fracture site. As measuring the ultimate strength involves destructive test, which is not applicable in clinical practice, non-invasive monitoring techniques, such as measurement of bending or torsional stiffness and resonant frequencies, have been proposed for quantitative assessment of callus material properties [20–23]. Therefore, the aim of the study is to provide a new vitro-method to monitor the healing state of the fractured bone. The results in Figs. 3(f) and 3(i) show that orthopedics can predict the healing state of fractured bone through observing mechanical environment and determining the time of removing the fixator. The fixator device can be removed safely through other methods. Nevertheless, the method provided in this study can also help reduce radiation exposure. It is a non-invasive monitoring technique and still has clinical practical significance.

3.2 Displacement

Figure 4 shows maximum displacement of fixator-bone system, when callus is granulation tissue, higher displacements were expected in initial stage (when callus is granulation tissue) compared to second stage and third stage (when callus is immature bone or mature bone) under three loading states. In initial stage shown in Fig. 4(a), the strength of fractured bone is very low, axial force causes the largest deformation (1.913 mm) at proximal fractured site which harm the healing of fractured bone, and also generates a large deformation in contact surface between bone and pin. This information indicates that at the early stage of healing, it is not suitable for bearing large loads. In the second stage, with the growth of callus, the callus, proximal bone and distal bone gradually grow into an entirety. When the strength of bone increased, axial load transferred between fixator and the bone, and both fixator and bone support the load, the maximum displacement of proximal fracture site decreased to 0.52992 mm, the deformation of pin and fixator become smaller than in initial stage, as is shown in Fig. 4(b). When callus in the fracture gap matured shown in Fig. 4(c), the strength of the bone continues to increase, the maximum displacement of fractured site reduced to 0.34866 mm, at this time, bone could bear main load and pin deformation is lowest in this stage.

Figures 4(d)-(f) show the displacement of fixator-bone system under torsional load when callus is granulation tissue, immature bone and mature bone. In initial stage, as is shown in Fig. 4(d), bone strength is very low, torsional load causes a large deformation (3.8181 degree) at proximal bone, which brings about displacement dislocation at fracture site that harms the healing of fractured bone. Torsional load also gives rise to a large deformation on the pin surface due to a big gap in the load-carrying capacity between the fixator and bone. As the callus grows to the second stage, as is shown in Fig. 4(e), the callus, proximal bone and distal bone gradually grow into an entirety, and the bone strength increased, the maximum torsional displacement of fracture site decreased to 1.453 degree at second stage, due to the gap in the load-carrying capacity between the fixator and the bone decreased, the deformation of pin and fixator decreased. When callus is mature bone tissue, as is shown in Fig. 4(f), the maximum deformation is only 0.8658 degree, meaning the healing of bone has reached a certain extent and the strength of the bone is enough to support sufficient load. Thus, the deformation of the pin is also reduced.

Figures 4(g)-(i) show the displacement of fixator-bone system under bending load when callus is granulation tissue, immature bone and mature bone. In the initial stage, when callus is granulation tissue, as is shown in Fig. 4(g), bending load causes a large lateral dislocation (0.15794 mm) at fracture site that is not good for bone healing. Due to the lower strength of bone and the higher strength of fixator, the deformation of the pin is higher than that in other stages. With the grow of callus, after the completion of bridging fracture site as is shown in Fig. 4(h), the callus, proximal bone and distal bone gradually grow into an entirety. The strength of fractured bone greatly increased, lateral displacement occurs at fractured site decreases to 0.11609 mm and the deformation of pin and fixator also decreased. After complete maturation of bone callus, as is shown in Fig. 4(i), the deformation area and the deformation value of the fracture site continued to decrease, the maximum displacement value reduced to 0.059783 mm and the deformation of fixator and pin also decreased. This indicates that it is not easy to check fracture dislocation phenomenon in the last stage.

Although the three loads are different, similar results in three groups of granulation tissue, immature bone and mature bone under three loads can be concluded as follows:

◦

In Figs. 4(a),(d) and (g), when callus is granulation tissue, the strength of fractured bone is very low. Three forces all cause the large deformation at proximal fractured site which harm the healing of fractured bone, and due to a big gap in the load-carrying capacity between the fixator and the bone, the deformation of pin surface is higher than that in other stages.

◦

In Figs. 4(b),(e) and (h), with the growth of callus, the strength of bone increased, both fixator and bone bear load, which resulted in a decrease in the displacement at the fixator and pin surface. At this time, the callus, proximal bone and distal bone gradually grow into an entirety. The displacement at the fracture site also decreased.

◦

In Figs. 4(c),(f) and (i), when callus grew into mature bone, the strength of bone increased slowly, and the deformation of the fixator-bone system reached into the least, compared to other stages. At this time, bone healing is complete and bone-fixator system is stable.

4 Discussion

External fixators have been widely used for the treatment of lower limb dislocations because this method is minimally invasive, maintains alignment and provides adequate stabilization [32,33]. Casts are still used in some cases of lower limb dislocation, but the use of external fixators has received attention as they allow some micro-motion and can improve healing time [34,35]. Stable fixation is important as it will minimize long-term morbidity and hastens soft tissue healing. As bone regeneration is known to be influenced by mechanical loading [36–39], the in vivo mechanical environment inside and around the fixator may influence micro-motion of fixator-bone system [40–42]. Since this environment is largely controlled by the fixator during bone healing process, a quantification of the in vivo mechanical environment of fixator-bone system is important. However, only a few studies have monitored micro-motion of fixator-bone system in the healing process.

Finite element simulation analysis can help us understand the biomechanical environment of bone and predict the load transfer between fixator and bone in the healing process. Under action three kinds of loads, the load distribution process between fixator and bone are similar. At the beginning stage, the callus tissues consist of fluid components and have very low stiffness. Due to bone stiffness being very low in comparison to fixator, the load is mainly carried by the fixator. The maximum stress occurs on the contact surface between the fixator and the pin. Such a high value of stress for fixator-bone system in the first stage is the reason why full weight bearing during treatment is never allowed in the early stages of treatment. In the second stage, the callus grew into immature bone tissue and the stiffness of the bone increased. Load distribution between the fixator and bone, and stress began to transfer from fixator to bone. In the last stage, stress was mainly concentrated in the bone, which predicted fixator can be removed and bone can bear a considerable part of the load. The results of this study are in line with the in vivo test and simulation results obtained by previous researchers [43,44].

Under three kinds of loads, the maximum displacements of the fixator-bone system are different at three different stages. The three load conditions have similar displacement distribution. In the first stage, the maximum displacement occurs at the fracture site and proximal bone. The displacement of the first stage is larger than that of the second and the third stages. With the increase of callus stiffness, the stiffness of bone increased, the ability of bone bearing loads also increased. At the beginning of healing, bone is the most prone to dislocation. This indicates that the fixator cannot bear the weight of the human body in the early stage of healing. Compared to torsional load and bending load, the axial stiffness of bone is greater than that of torsional and bending load. Bending load and torsional load are likely to cause lateral displacement.

The aim of this research is to develop finite element model for clinicians and compare biomechanical environment of bone-fixator system under three healing stages. The process of bone growth was represented by the increase of the Young’s modulus in the different healing time regardless of the irregularity of callus shape. These limitations, however, do not jeopardize the importance and correctness of the results obtained in this paper. This study still provides useful information for orthopedic doctors.