Analysis of Mechanical Stability for External Fixation Device in the Case of Anterior-Posterior Bending

Analysis of mechanical stability for external fixation device Orthofix in the case of anterior-posterior bending is carried out in this paper. Device is applied to the lower leg for the case of unstable fracture. Real device is mea sured and 3D CAD model is developed. CAD model is used for numerical structural stress analysis which is car ried out using CATIA V5 software. Results for displacements are obtained for selected critical places on the device and for the place of fracture. In addition, values of principal and von Misses stresses are obtained and analyzed. Using obtained results, conclusions about mechanical stability of device are formulated.


INTRODUCTION
Basic goal of fracture fixation is to enable fractured bone to be used as soon as possible. Mechanical and biomechanical characteristics of fixation devices have a big influence on the speed and quality of healing process [1]. During healing process all forces generated during bone movement are transferred to the fixation device. Behavior of device is defined by its mechanical and biomechanical properties [2].
From the first fixation device up to today their types, methodology of selection, places of usage and types of application has changed during time. They are improved significantly thought history [3]. With a goal to improve fixation, and to adjust fixation device to the individual fracture types more research is focused to the analysis of mechanical properties of this types of devices using standard theoretical and experimental research usually used in mechanical engineering.
In order to improve necessary tests and make improvements, researchers seek to develop theoretical background of the fixation thematic based on the principles of structural mechanics [4]. Stiffness of the fixation device is defined according to specific type of load: axial load due to pressure force, bending and torsion [5]. Additionally, for the purposes of defining properties of these devices, researchers rely on force transducers [6]. In the context of biomechanical research, great focus is put on analyzing the influential construction parameters on the fixation device stability. These parameters include stiffness, maximum Von Mises stress for zones of interest as well as bearing capacity of the pin-bone connection, as shown in many experimental studies [7,8,9]. In recent period, conducted researches are not only based on experimental investigations, but also on benefits of 3D modeling and numerical analysis. This way, a more complete image and understanding of fixation device behavior is obtained [10,11,12].
Main focus is on the stiffness of a device used for fixation [13], because stiffness has the biggest influence on the quality and speed of healing. Usually most of the research are focused on one type of load with a goal to simplify numerical analysis [14,15].
All of the commercial fixation device, which are in use today, have undergone the biomechanical tests before usage. Biomechanical investigation of the Orthofix fixation device was not

DEVELOPMENT OF CAD/FEM MODEL
External fixation device Orthofix was developed after long research and development process carried out by professor Giovanni de Bastiani. He was researcher and professor at University of Verona (Italia) [16]. External fixation device Orthofix have a frame, coupling with spherical joint, four Schanz half pins and dynamic lever. This device is usually called Dynamic axial fixator (DAF). This name is connected to the dynamic lever. Using this lever, it is possible to adjust distance between broken fragments of bone ( Figure 1).
For Orthofix device, two types of materials are used, stainless steel and aluminum alloy. Black parts from Figure 1 are manufactured using T6-7075 aluminum alloy. Aluminum is non-toxic, non-magnetic and it does not have corrosion.
Also it is much lighter and cheaper in comparison to the other non-ferrous metals. Parts of the device in grey color ( Figure 1) are manufactured using AISI 304 austenite stainless steel. Main advantage of this stainless steel is their resistance to the intercrystalline corrosion. Main disadvantages of this stainless steel is lower value of yield strength due to the lower amount of carbon inside. Mechanical properties of Orthofix device parts are shown in Table 1 [17].
For development of CAD/FEM model of Ortofix device software package CATIA V5 is used. First step is to create CAD models of all parts of the device and then to create assembly. Parts are developed using Part design module of the same software. All parts are connected and assembled using Assembly design module.
Using developed CAD model next step is to create FEM model for all components and for assembly. First step in FEM model development is to create mesh by selecting appropriate finite element. Finite elements of linear and parabolic tetrahedron are used. Linear tetrahedron is used for couplings, spherical joints and screws. Parabolic tetrahedron is used for frame, bone and half pins.
After discretization, next step is to create connections between device parts. Fastened constrains are applied to the places where half pins are connected to the bone ( Figure 2a) and contact connections are applied between device parts (Figure 2b).  After connection definition, next step is to create supports. Supports are constrains in the same time ( Figure 3).
Next step in the development process of FEM model is to apply loads. For the case of anterior-posterior bending surface force load is applied at the place of fracture. Force load have a value of 500 N. Two supports are added at the two ends of the bones. Using this arrangement

DETERMINATION OF STRESS, DISPLACEMENT, AND STIFFNESS
Numerical structural analysis gives values of displacements, principal stress and von Mises stress. Value of equivalent von Mises stress is used very often in solid body mechanics. It can be calculated using following equation [19]: Displacements are measured at the place of applied load. This values of displacement are used for stiffness calculation. Stiffness is relation between displacements and loads at the place where load is applied.
Stiffness of device according to the AP bending (C s ) can be calculated using following equation [20]: where: F s -bending force (N), δ s -displacement at the place of load (mm).
Stiffness of fracture can be calculated as relation between loads and relative displacement for selected points [21]: Relative displacements of pair (r D(x) , r D(y) , r D(z) ) selected points at the end of proximal (upper) and distal (lower) bone segment in x, y, and z direction are defined as: [22]: Figure 6 shows displacement vectors for selected points in the case of maximal loads. Direction and intensity of displacement vector can be  (Table 3).

RESULTS
For stiffness analysis in the case of AP bending, middle point at the place of load in the direction of y axis is selected. Using relations (4) relative displacements are calculated for selected points of proximal and distal bone segment for which vector of relative displacement at the place of fracture gives maximal values.
Values of displacements for the case of maximal value of load F p = 500N are given in Table 3.
In the case of maximal load, maximal displacement is at the place of fracture in lower segment of bone model. Its value is 2.53 mm. Maximal value of displacement on the fixation device is at the ends of half pins. Its value is 1.26 mm ( Figure 5).
Stress intensity are depending on the shape of the design (shape of the parts and its placement inside the design). Most critical places on the fixation device are place of contact between screw and spherical joint, between sphere neck and coupling and between sphere and metal shell ( Figure 6).
Intensity and direction of principal stresses are analyzed at three critical places for the case of maximal AP bending load (Figure 7). Results are shown using Table 4.

DISCUSSION
Structural analysis is showed that in the case of maximal load, maximal displacement is at the place of fracture in lower segment of bone model. Its value is 2.53 mm. Maximal value of displacement on the fixation device is at the ends of half pins. Its value is 1.26 mm. This values are in the range of allowed values for this type of device. Studies [10,13] also showed similar displacement values for a similar configuration of the external fixation device loaded with the same load case.
Using values of displacement at the place of fracture, stiffness of the fracture is calculated and its value is 283.28 N/mm. Stiffness of the device is calculated using values of displacements at the place of load, its value is 235.84 N/mm. Values of stiffness are also in the range of allowed values for this type of design. Studies [10,20] also showed similar stiffness values for a similar configuration of the external fixation device loaded with the same load case. During structural analysis for AP bending load it can be noticed that lower bone segment has bigger displacements in comparison to the upper segment of bone. Difference between displacement of upper and lower bone segment is around 1 mm. This is happening because of long screws and bone asymmetry which results with the supports which lies in different level. Lower bone segment is much thinner in comparison to upper bone segment, because of that screw length is shorter in lower bone segment. This have effect on design stiffness and values of displacements. This needs to be tested in future research. It will be good to test is it possible to reduce difference in displacements using different half pins placements or using more half pins.
Intensity and direction of principal stresses are analyzed at three critical places for the case of maximal AP bending load. Selected critical places on the fixation device are place of contact between screw and spherical joint, between sphere neck and coupling and between sphere and metal shell. Bigger value of stress (216.12 MPa) is at the place number 3 (MM3). This value is lower than value of allowed stress for materials of coupling and half pins.

CONCLUSIONS
With a goal to reduce number of errors during development and design of external fixation device and to reduce number of problems associated with healing process it is necessary to collect all data and important parameters which can have influence on that errors and problems. Important parameters are displacements at the place of fracture and on the device, stiffness of the device, stresses at the critical places, loads of device after its application on the patient. In some cases, other parameters can be taken in consideration also.
In this paper, mechanical stability analysis of external fixation device Orthofix is carried out for the case of anterior-posterior bending. In the first step CAD model of the device is created. Developed CAD model is used to create FEM model. FEM model is used for analysis of the movement of bone at the place of fracture, to analyze mechanical properties of the device and to calculate stiffness of the device. Taking in consideration all data it can be concluded that external fixation device Orthofix have good mechanical stability for the case of AP bending load. Also there is a possibility to improve device using new advance materials or by device redesign.