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A simplified application (APP) for the parametric design of screw-plate fixation of bone fractures
Journal of the Mechanical Behavior of Biomedical Materials 77 (2018) 642–648
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Journal of the Mechanical Behavior of Biomedical Materials journal homepage: www.elsevier.com/locate/jmbbm
A simplified application (APP) for the parametric design of screw-plate fixation of bone fractures
MARK
Chen-Yuan Chung Department of Mechanical Engineering, National Central University, No. 300, Zhongda Rd., Zhongli District, Taoyuan City 32001, Taiwan
A R T I C L E I N F O
A B S T R A C T
Keywords: Application (APP) Analytical approach Stress shielding Interfragmentary strain
Screw and plate fixation is commonly used to treat bone fractures. A prototype application (APP) for presurgical simulation was developed and validated by comparing it with current analytical approach and other models. In this APP, alternative plate designs and materials to limit the effects of stress shielding could be tested. In addition, the number and position of screws and the gap between bone and plate that achieved acceptable stability were predicted. The fixation stability providing a situation of interfragmentary strain between 2% and 10% is necessary for callus formation. However, improving the fixation stability leads to a stress shielding effect. The simultaneous alleviation of stress shielding and maintenance of stability are important in fracture healing. In this study, the feasibility of creating a specialized APP to evaluate different screw-plate configurations for diaphyseal femoral fractures was investigated. The ultimate goal is to extend this technique to computer-assisted preoperative planning for orthopedic surgery.
1. Introduction Screw-plate fixation is routinely used in the treatment of femoral fractures with the aim of maintaining blood supply to promote rapid healing, reduce the need for bone grafting, and decrease the risk of infection and refracture (Miclau and Martin, 1997). Effective preoperative planning includes choosing the proper plate length, screw number, and screw position (Mast et al., 1989). A dynamic compression plate (DCP), known as traditional fixation, is compressed directly onto the bone (Allgöwer et al., 1970). However, DCPs are currently being replaced by locking compression plates (LCPs) which have a smaller contact surface that serves to mitigate the interruption of blood supply (Miller and Goswami, 2007). To improve fixation, LCP is designed to contain a combination-hole system that can house either locking or conventional non-locking screws. Whereas conventional non-locking screws function by pressing the plate to the bone, locking screws keep a gap between the bone and plate to protect periosteal vascularity. Recently, non-contact plates (NCPs) have become available for use in the treatment of postoperative infection due to bacterial biofilm formation as well as in the treatment of fractures (Alemdar et al., 2015). Selection of screw-plate fixation depends on bone type and fracture type and is clinically important. The traditional plating fixations made of metal alloys such as stainless steel, cobalt-chrome, and titanium are accepted as biocompatible materials. Carbon fiber composite can be made more compliant than metal alloys by varying the orientation and number of
carbon fiber layers, making its elastic modulus similar to that of cortical bone (Tayton et al., 1982). A plate with high stiffness results in a large portion of the load being carried by the plate rather than by the underlying bone, which is referred to as stress shielding (Ramakrishna et al., 2001). This is a harmful long-term phenomenon that leads to bone loss because the stress in the bone falls below the normal values. Clinical research has shown that strain in the bone decreases when the plate carries the most of external load (Hastings, 1980). Stability determines the amount of strain, which influences the type of bone healing occurring at the fracture site (Egol et al., 2004). A proper strain level stimulates callus formation during fracture healing (Claes et al., 1998). Therefore, specially designed plates combined with novel screws have been proposed to enable interfragmentary motion required for the promotion of callus healing (Doornink et al., 2011; Bottlang and Feist, 2011). Recently, finite-element APPs have been used to provide new insights into many fields, such as electronic cooling (Segui, 2016), food processing (Carver, 2017), and automotive industry (Marra, 2017). This is the first investigation that uses an APP to demonstrate the presurgical evaluation of various screw-plate fixations for femoral fractures. The goal of this study is to develop an APP to provide information about the screw-plate configuration affecting the mechanical behavior of a hypothetical bone-implant system. This APP enables the parametric study of screw-plate fixation to alleviate the stress shielding and maintain the construct stability. This methodology could, in principle, be extended to computed tomography (CT)-based models and applied to planning and
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmbbm.2017.10.025 Received 27 February 2017; Received in revised form 23 September 2017; Accepted 18 October 2017 Available online 06 November 2017 1751-6161/ © 2017 Elsevier Ltd. All rights reserved.
Journal of the Mechanical Behavior of Biomedical Materials 77 (2018) 642–648
C.-Y. Chung
Fig. 1. (a) An idealized model of an implant plate attached to an intact bone. It is assumed that the external forces and moments are reduced to an equivalent axial load F , which acts at a distance αr0 from the centroidal axis of the bone. The neutral axis of the bone-plate system is located at distance zˆ from the centroidal axis of the bone. The centroidal axis of the implant plate is located at distance zP from the centroidal axis of the bone. (b) Finite element representation in Section 2.1 for validation of the analytical model. Fig. 2. Representation of finite element model mentioned in Section 2.2 for comparison with Fouad's model (2011). FE mesh of the bone and implant, as well as the applied boundary and load conditions are shown. A uniform pressure of 2.5 MPa, produced by a body weight of approximately 800 N is applied to the end surface of the bone; the opposite end was fixed. In the combined loading condition, a torque of 1 Nm is applied on the fracture site in addition to the pressure.
Table 1 Material properties used in the FE model of bones and implants (Fouad, 2010, 2011).
Intact bone Fractured bone (callus) at 1% healing Titanium alloy screws Stainless steel plate
Young's modulus (GPa)
Poisson's ratio
20 0.02 110 210
0.3 0.3 0.3 0.3
σB = −
FEB MtB EB − . EB AB + EP AP EB IB + EP IP
(1)
The first and second terms of Eq. (1) represent the normal and bending stresses resulting from the axial compressive load and the bending moment, respectively. Here EB and EP are the Young's moduli of the bone and the plate, respectively. AB and AP are the cross-sectional areas of the bone and the plate, respectively. It is very important to note that IB and IP are the area moments of inertia of the bone and the plate, respectively, with respect to the neutral axis of the system. The moment M about the neutral axis produced by the axial load applied at an eccentric distance αr0 from the centroidal axis of the bone may be written as
evaluation before surgery.
2. Methods
M = F (αro − zˆ),
2.1. Analytical approach
(2)
and the location of the neutral axis of bending zˆ is given by
Fig. 1(a) shows an idealized bone-implant system including a plate attached to an intact femur, which is represented as hollow cylinder (Gray, 1918). The plate can be approximated by a rectangular cross section. Composite beam theory (Gere and Timoshenko, 1997; Carter and Vasu, 1981; Cordey et al., 2000) provides calculation of the stress in the bone at a distance tB from the neutral axis of the bone-implant system as
zˆ =
ρA zP . 1 + ρA
(3)
Using these relationships, the stress in the bone can be written as
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C.-Y. Chung
Fig. 3. User interface of the screw-plate fixation design APP. In this demo APP, users can change parameters, such as geometry, material, and applied load, to investigate stress and strain in the bone-implant system.
Bone stress can be generalized by Eq. (4) in terms of the following normalized values of structural stiffness and geometry:
Table 2 Input parameters and variables in this simplified APP. Its video is given in the online supplementary material.
ρI =
Bone Parameters Outer radius of bone (ro ) Inner radius of bone (rm ) Length of bone (L1) Length of fractured bone (L2 ) Gap between bone and plate (t )
EP IP E A r z z , ρ = P P , β = m , γ = P and α = , EB IB A EB AB ro ro ro
where IB and IP are the area moments of inertia of the bone and the plate, respectively, with respect to their centroidal axes; rm is the radius of the medullary cavity; and ro is the outside radius of the bone. Here, the location distance tB is determined by tB = z − zˆ . Its range is between ro − zˆ (immediately beneath the plate) and − ro − zˆ (farthest away from the plate). The stress in the plate at a distance tP from the neutral axis of the bone-implant system can be obtained from an equation similar to Eq. (1); thus,
Plate Parameters Plate width (Pw ) Plate height (Ph ) Plate length (Pl ) Screw Parameters
σP = − Screw length (Sl ) Screw radius (Sr ) Screw head radius (Shr ) Numbers of screw-pair (n ) Distance of central pair /2 (Sx1) Distance between adjacent screws (Sx2 ) Applied Load
FEP MtP EP − . EB AB + EP AP EB IB + EP IP
(5)
A similar expression can be derived for the stress in the plate
σP = −
F ⎡ ρA AP ⎢ 1 + ρA ⎣ −1
+
Pressure (P ) Torsional moment (T )
ρA γ ⎞ ⎛ 1 γ2 ⎞ ⎤ 1 tP ⎛ (1 + β 2) + (γ − 1)2 + . ⎜α − ⎟⎜ ⎟ 1 + ρA ⎠ ⎝ 4ρA 3 1 + ρA ⎠ ⎥ ro ⎝ ⎦
(6)
Material Parameters Young's modulus Poisson's ratio of Young's modulus Poisson's ratio of Young's modulus Poisson's ratio of Young's modulus Poisson's ratio of
In the analytical model, the plate had a length of 141 mm, width of 8 mm, and thickness of 4 mm. The intact bone had an outer diameter of 25 mm, cortical thickness of 5 mm, and length of 141 mm. One end of the bone was fixed and an eccentric compressive load of 800 N was applied to the opposite end at a distance of αro = 10mm from the centroidal axis of the bone. Analytical results (ai ) were compared with the finite element results ( fi ). The corresponding finite element model is shown in Fig. 1(b). The mean percentage error (MPE) in the predicted stresses was computed using the following equation: a −f n 100 % MPE = n ∑i = 1 i a i .
of bone (EB ) bone (νB ) of plate (EP ) plate (νP ) of fractured bone (EF ) fractured bone (νF ) of screw (ES ) screw (νS )
−1
σB = −
ρA γ ⎞ ⎛ 1 ρ γ2 ⎞ ⎤ F ⎡ 1 t (1 + β 2)(1 + ρI ) + A + B ⎜⎛α − . ⎟⎜ ⎟ ⎢ 1 + ρA ⎠ ⎝ 4 1 + ρA ⎠ ⎥ AB 1 + ρA ro ⎝ ⎣ ⎦ (4)
i
2.2. Finite element analysis (FEA) A simplified 3D finite element (FE) model of a bone-implant system 644
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et al., 2003). The Poisson's ratio of 0.3 was assumed for all plates. The linear elastic material model including geometric nonlinearity due to large strains was considered in the computation. The model was meshed with tetrahedral elements. Nodes on the end surface of the bone were fixed (Behrens et al., 2008; Nasr et al., 2013). Although it was not the in vivo constraint condition, the local effects were reasonably confined to the region near the boundaries according to Saint-Venant's principle. For a patient weighing 800 N, an equivalent pressure of 2.5 MPa was applied to the other end surface of the bone (Benli et al., 2008; Fouad, 2011). In order to mimic the in vivo loading conditions of femur, a torque of 1 Nm was applied to the fractured bone, in addition to the pressure (Fouad, 2011; Carter et al., 1981). Contact pairs were defined at the interface between the plate and the screws, as well as between the bone and the screws. Additional contact between the bones and the plate was created when using a contacted plate. The lateral surface of hollow cylindrical bone was modified to be partially flat to ensure a complete contact with the plate. A penalty method available in COMSOL was used to implement contact constraints. An initial contact pressure was automatically determined using the solver configuration. All contacting surfaces were assigned a friction coefficient of 0.9 (Fouad, 2010, 2011). After the FE model was validated against published results (Fouad, 2011), it was turned into an APP (Fig. 3) using the application builder in COMSOL to demonstrate the mechanical behavior of the bone-implant system by changing the values of the editable parameters (Table 2). The purpose of the APP was to design the conceptual configuration of a screw-plate fixation for fractured bones. One of the four geometric parameters (i.e., t, Sx1, Sx2, and n) was altered in each design. Other parameters were the same as those used in Fouad's model (2011), except that the plate length was 130 mm. For example, the gap (t) between bone and plate varied within the range described in Ahmad et al. (2007). The APP simulated the bone-implant system when subjected to pressure and torque. The effect of stress shielding was represented by the following equation (Piao et al., 2014):
Fig. 4. Comparison of the analytical approach and finite element analysis under the eccentric axial loading condition. The values of MPE between these two methods were both less than 10% in the bone and plate.
was generated using COMSOL Multiphysics 5.2 (COMSOL, Inc., Burlington, MA). Its dimensions were the same as previously described by Fouad (2010). In the model, the femur had an outer diameter of 25 mm, cortical thickness of 5 mm, and length of 141 mm (Fig. 2). It was assumed that a simple transverse fracture, AO type 32-A3 (Rüedi and Murphy, 2000), with a 1 mm gap was filled with a callus during healing. The plate was modeled with dimensions of 70 mm length, 8 mm width and 4 mm thickness. The screws were 4 mm in diameter and 32 mm in length. The material properties of bone, fractured bone, screws, and plate were adopted from Benli et al. (2008) and Fouad (2010) and are summarized in Table 1. The material properties of different plates assigned to the FE model included titanium alloy and carbon fiber composite as alternative materials with elastic moduli of approximately 110 and 50 GPa, respectively (Ramakrishna et al., 2004). The carbon fiber composite was assumed to be isotropic (Saidpour, 2006) and to behave in an elasto-plastic manner (Veerabagu
η = (1 − σ / σ0) × 100%.
(7)
where η is the stress shielding rate, σ is the stress of bone after implantation of plate, and σ0 is the stress of bone at the same position before implantation of plate.
Fig. 5. Maximum stress of callus (fractured bone) predicted using the present FE model compared with that predicted using the FE model of Fouad (2011). Stainless steel was chosen as the material for both non-contacted and contacted plates. Percent differences between the present study and the reference (Fouad, 2011) were 8% and 9% for compressive and combined loading conditions, respectively, in the case of the non-contacted plate.
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Fig. 6. Effect of the gap between bone and plate on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where n = 2, Sx1 = 10 mm, and Sx2 = 15 mm.
Fig. 7. Effect of the half of distance between central pair of screws on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where n = 2, t = 0.5 mm, and Sx1 + Sx2 = 60 mm.
3. Results
combined static loading was applied to FE models constructed with different plate materials. The corresponding maximal stress and strain at the callus (fracture site) are provided by APP (Fig. 6). Generally, enlarging the gap between bone and plate increases the stress and strain at the callus. The maximum stress of callus oscillates in the range from 0.8 to 1.4 mm in gap between bone and carbon fiber composite plate. This is because when using APP, the automatic-fine mesh size varies with different gap designs. Similar investigations of the results obtained with changes in Sx1, Sx2, and n performed individually, and the effect of each parameter on stress and strain at the callus are shown in Figs. 7–9. Overall, plates with a lower Young's modulus produced higher stress and strain at the callus. Additionally, non-contacted plates generated higher stress and strain at the callus than the contacted plates did. Qualitatively similar results were previously reported by Benli et al. (2008) and Fouad (2010, 2011).
Fig. 4 shows the axial stresses in the bone and plate in the transverse direction as predicted by the analytical approach and FEA described in Section 2.1. The stresses computed using analytical formulas correlated well with those computed using FEA. The values of mean percentage error between these two methods were 9% (in bone) and −3% (in plate), respectively. The analytical approach to the stress prediction of bone-implant system was validated by the FEA. For verification of the bone-implant model at an early stage of healing (1%), the FE solutions agreed with the FE results reported by Fouad (2011). As shown in Fig. 5, the present predictions of the maximum von Mises stress at the callused bone under either a non-contacted or contacted plate were smaller than those reported by Fouad (2011). The difference in stress prediction of callused bone was larger with a contacted plate than with a non-contacted plate because this model and Fouad's model differed in node quantity, element shape, and contact algorithm. For parametric investigation of the gap (t) between bone and plate,
4. Discussion FEA should be of much use to guide the clinicians for investigating 646
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Fig. 8. Effect of the distance between the first and second left screws on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where n = 2, t = 0.5 mm, and Sx1 = 10 mm.
Fig. 9. Effect of the numbers of screw-pairs on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where t = 0.5 mm, Sx1 = 10 mm, and Sx2 = 10 mm.
Excel to offer predictions that are consistent with FEA calculations. In the same way, results from the analytical approach were validated by Tsai et al. (2013) using an FE model based on the X-ray film of the femur. Simple models towards the prediction of stress in the femur have been preliminarily evaluated for clinical purposes. The analysis revealed that the screw-plate implant experiences higher stress than the bone does, which reflects the stress shielding offered by the implant to the bone. As indicated in Fig. 5, the stress shielding rates in the present study were 16.2% and 37.4% for noncontacted and contacted plates, respectively, under compressive loading. The callus stress in the present study decreased more significantly than that in the reference (Fouad, 2011) when using the contacted plate instead of the non-contacted plate. It has been noted that plates with decreased rigidity accelerate healing and eliminate the risk of osteopenia due to stress shielding (Woo et al., 1977). Stability of the bone-implant system often influences the healing process, which is explained by the concept of interfragmentary strain (Perren, 2002). Strain between 2% and 10% provides a status of stability with enough
biomechanical responses of bone-implant system under physiological loading. Unfortunately, customized stress calculation by FEA in individual patients would be both time-consuming and technically demanding. The analytical approach predicts stress magnitude over a continuum of structures in parametric forms of equations. Differences in the predicted stress in the two methods (Fig. 4) occurred because the periphery of bone was not as circular in the FE model as in the analytical model. According to Saint-Venant's principle, the location of comparison was made throughout the middle region of the bone-implant length where uniform stress distribution was produced. Furthermore, the screws and their contact surfaces were not considered in the present analytical model. Therefore, a possible future work is how to establish an analytical model for consideration of the forces acting on plate through the screws (Ramakrishna et al., 2004). When the boneimplant system is subjected to combined loading conditions, the torque can be taken into consideration in the analytical model by superposition of stress components (Gere and Timoshenko, 1997). While not as rigorous as FEA, the analytical approach can be executed in Microsoft
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interfragmentary motion to stimulate callus formation (Egol et al., 2004). Bone healing may not occur when the strain at a fracture site exceeds 10%. A limitation of this study is that the nonlinear material constitutive relation for the callus subject to large strain was not taken into account in the current FE model. Generally, securing the LCP to bone with locking screws at a small distance between plate and bone enhances the fixation stability and suppresses interfragmentary strain (Miller and Goswami, 2007). Appropriate degrees of strain and stress promote bone healing. The reduction of stress shielding (increment in the stress of callus) enlarged the interfragmentary strain of callus. On the contrary, smaller strain kept below 10% may achieve stability. The results obtained with the present analysis consistently showed that noncontacted plates consisting of low-stiffness material provided relatively high bone stress. However, the mineralized callus may rupture if exposed to strain greater than 10%. The APP developed in this study could, in principle, be used to design screw-plate fixation for reducing stress shielding and providing stability. Surgical techniques such as the positions and number of screws have greatly influenced healing time and stability of fracture sites (Stoffel et al., 2003; Nasr et al., 2013). Some clinical and experimental studies indicated the influence of screw position in the implant plates on the stability of the bone-implant system (Field et al., 1999). The purpose of this APP was to investigate the stress and strain distribution in a simplified bone-implant system based on a modifiable parametric model. The present study did not intend to validate or suggest that a simplified bone-implant model is a universal model for orthopedic biomechanics. It should be possible to adapt this simplified model to CT-based models. In principle, an APP with specific functions can be applied to the development of preoperative planning simulations for orthopedic surgery. Such a specialized APP can be run by connecting to the COMSOL Server from web browsers (Dagastine et al., 2016) and using cloud-computing technology (Kaner et al., 2014). Future studies with CT-based FEA will further corroborate these findings. Clinicians can bring a customized APP on-site and arrange screw positions and numbers as well as the gap between bone and plate to perform a presurgical evaluation. Acknowledgements The author would like to acknowledge the grant received (grant number MOST 105-2218-E-008-005) from the Ministry of Science and Technology in Taiwan. Conflict of interest statement The author has no conflicts of interest related to this manuscript, commercial or otherwise. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jmbbm.2017.10.025. References Ahmad, M., Nanda, R., Bajwa, A.S., Candal-Couto, J., Green, S., Hui, A.C., 2007. Biomechanical testing of the locking compression plate: When does the distance between bone and implant significantly reduce construct stability? Inj. Int. J. Care Inj. 38, 358–364. Alemdar, C., Azboy, I., Atiç, R., Özkul, E., Gem, M., Kapukaya, A., 2015. Management of infectious fractures with “non-contact plate” (NCP) method. Acta Orthop. Belg. 81, 523–529. Allgöwer, M., Perren, S., Matter, P., 1970. A new plate for internal fixation—the dynamic compression plate (DCP). Inj. Int. J. Care Inj. 2, 40–47. Behrens, B.-A., Wirth, C.J., Windhagen, H., Nolte, I., Meyer-Lindenberg, A., Bouguecha, A., 2008. Numerical investigations of stress shielding in total hip prostheses. Proc.
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Journal of the Mechanical Behavior of Biomedical Materials journal homepage: www.elsevier.com/locate/jmbbm
A simplified application (APP) for the parametric design of screw-plate fixation of bone fractures
MARK
Chen-Yuan Chung Department of Mechanical Engineering, National Central University, No. 300, Zhongda Rd., Zhongli District, Taoyuan City 32001, Taiwan
A R T I C L E I N F O
A B S T R A C T
Keywords: Application (APP) Analytical approach Stress shielding Interfragmentary strain
Screw and plate fixation is commonly used to treat bone fractures. A prototype application (APP) for presurgical simulation was developed and validated by comparing it with current analytical approach and other models. In this APP, alternative plate designs and materials to limit the effects of stress shielding could be tested. In addition, the number and position of screws and the gap between bone and plate that achieved acceptable stability were predicted. The fixation stability providing a situation of interfragmentary strain between 2% and 10% is necessary for callus formation. However, improving the fixation stability leads to a stress shielding effect. The simultaneous alleviation of stress shielding and maintenance of stability are important in fracture healing. In this study, the feasibility of creating a specialized APP to evaluate different screw-plate configurations for diaphyseal femoral fractures was investigated. The ultimate goal is to extend this technique to computer-assisted preoperative planning for orthopedic surgery.
1. Introduction Screw-plate fixation is routinely used in the treatment of femoral fractures with the aim of maintaining blood supply to promote rapid healing, reduce the need for bone grafting, and decrease the risk of infection and refracture (Miclau and Martin, 1997). Effective preoperative planning includes choosing the proper plate length, screw number, and screw position (Mast et al., 1989). A dynamic compression plate (DCP), known as traditional fixation, is compressed directly onto the bone (Allgöwer et al., 1970). However, DCPs are currently being replaced by locking compression plates (LCPs) which have a smaller contact surface that serves to mitigate the interruption of blood supply (Miller and Goswami, 2007). To improve fixation, LCP is designed to contain a combination-hole system that can house either locking or conventional non-locking screws. Whereas conventional non-locking screws function by pressing the plate to the bone, locking screws keep a gap between the bone and plate to protect periosteal vascularity. Recently, non-contact plates (NCPs) have become available for use in the treatment of postoperative infection due to bacterial biofilm formation as well as in the treatment of fractures (Alemdar et al., 2015). Selection of screw-plate fixation depends on bone type and fracture type and is clinically important. The traditional plating fixations made of metal alloys such as stainless steel, cobalt-chrome, and titanium are accepted as biocompatible materials. Carbon fiber composite can be made more compliant than metal alloys by varying the orientation and number of
carbon fiber layers, making its elastic modulus similar to that of cortical bone (Tayton et al., 1982). A plate with high stiffness results in a large portion of the load being carried by the plate rather than by the underlying bone, which is referred to as stress shielding (Ramakrishna et al., 2001). This is a harmful long-term phenomenon that leads to bone loss because the stress in the bone falls below the normal values. Clinical research has shown that strain in the bone decreases when the plate carries the most of external load (Hastings, 1980). Stability determines the amount of strain, which influences the type of bone healing occurring at the fracture site (Egol et al., 2004). A proper strain level stimulates callus formation during fracture healing (Claes et al., 1998). Therefore, specially designed plates combined with novel screws have been proposed to enable interfragmentary motion required for the promotion of callus healing (Doornink et al., 2011; Bottlang and Feist, 2011). Recently, finite-element APPs have been used to provide new insights into many fields, such as electronic cooling (Segui, 2016), food processing (Carver, 2017), and automotive industry (Marra, 2017). This is the first investigation that uses an APP to demonstrate the presurgical evaluation of various screw-plate fixations for femoral fractures. The goal of this study is to develop an APP to provide information about the screw-plate configuration affecting the mechanical behavior of a hypothetical bone-implant system. This APP enables the parametric study of screw-plate fixation to alleviate the stress shielding and maintain the construct stability. This methodology could, in principle, be extended to computed tomography (CT)-based models and applied to planning and
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmbbm.2017.10.025 Received 27 February 2017; Received in revised form 23 September 2017; Accepted 18 October 2017 Available online 06 November 2017 1751-6161/ © 2017 Elsevier Ltd. All rights reserved.
Journal of the Mechanical Behavior of Biomedical Materials 77 (2018) 642–648
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Fig. 1. (a) An idealized model of an implant plate attached to an intact bone. It is assumed that the external forces and moments are reduced to an equivalent axial load F , which acts at a distance αr0 from the centroidal axis of the bone. The neutral axis of the bone-plate system is located at distance zˆ from the centroidal axis of the bone. The centroidal axis of the implant plate is located at distance zP from the centroidal axis of the bone. (b) Finite element representation in Section 2.1 for validation of the analytical model. Fig. 2. Representation of finite element model mentioned in Section 2.2 for comparison with Fouad's model (2011). FE mesh of the bone and implant, as well as the applied boundary and load conditions are shown. A uniform pressure of 2.5 MPa, produced by a body weight of approximately 800 N is applied to the end surface of the bone; the opposite end was fixed. In the combined loading condition, a torque of 1 Nm is applied on the fracture site in addition to the pressure.
Table 1 Material properties used in the FE model of bones and implants (Fouad, 2010, 2011).
Intact bone Fractured bone (callus) at 1% healing Titanium alloy screws Stainless steel plate
Young's modulus (GPa)
Poisson's ratio
20 0.02 110 210
0.3 0.3 0.3 0.3
σB = −
FEB MtB EB − . EB AB + EP AP EB IB + EP IP
(1)
The first and second terms of Eq. (1) represent the normal and bending stresses resulting from the axial compressive load and the bending moment, respectively. Here EB and EP are the Young's moduli of the bone and the plate, respectively. AB and AP are the cross-sectional areas of the bone and the plate, respectively. It is very important to note that IB and IP are the area moments of inertia of the bone and the plate, respectively, with respect to the neutral axis of the system. The moment M about the neutral axis produced by the axial load applied at an eccentric distance αr0 from the centroidal axis of the bone may be written as
evaluation before surgery.
2. Methods
M = F (αro − zˆ),
2.1. Analytical approach
(2)
and the location of the neutral axis of bending zˆ is given by
Fig. 1(a) shows an idealized bone-implant system including a plate attached to an intact femur, which is represented as hollow cylinder (Gray, 1918). The plate can be approximated by a rectangular cross section. Composite beam theory (Gere and Timoshenko, 1997; Carter and Vasu, 1981; Cordey et al., 2000) provides calculation of the stress in the bone at a distance tB from the neutral axis of the bone-implant system as
zˆ =
ρA zP . 1 + ρA
(3)
Using these relationships, the stress in the bone can be written as
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Fig. 3. User interface of the screw-plate fixation design APP. In this demo APP, users can change parameters, such as geometry, material, and applied load, to investigate stress and strain in the bone-implant system.
Bone stress can be generalized by Eq. (4) in terms of the following normalized values of structural stiffness and geometry:
Table 2 Input parameters and variables in this simplified APP. Its video is given in the online supplementary material.
ρI =
Bone Parameters Outer radius of bone (ro ) Inner radius of bone (rm ) Length of bone (L1) Length of fractured bone (L2 ) Gap between bone and plate (t )
EP IP E A r z z , ρ = P P , β = m , γ = P and α = , EB IB A EB AB ro ro ro
where IB and IP are the area moments of inertia of the bone and the plate, respectively, with respect to their centroidal axes; rm is the radius of the medullary cavity; and ro is the outside radius of the bone. Here, the location distance tB is determined by tB = z − zˆ . Its range is between ro − zˆ (immediately beneath the plate) and − ro − zˆ (farthest away from the plate). The stress in the plate at a distance tP from the neutral axis of the bone-implant system can be obtained from an equation similar to Eq. (1); thus,
Plate Parameters Plate width (Pw ) Plate height (Ph ) Plate length (Pl ) Screw Parameters
σP = − Screw length (Sl ) Screw radius (Sr ) Screw head radius (Shr ) Numbers of screw-pair (n ) Distance of central pair /2 (Sx1) Distance between adjacent screws (Sx2 ) Applied Load
FEP MtP EP − . EB AB + EP AP EB IB + EP IP
(5)
A similar expression can be derived for the stress in the plate
σP = −
F ⎡ ρA AP ⎢ 1 + ρA ⎣ −1
+
Pressure (P ) Torsional moment (T )
ρA γ ⎞ ⎛ 1 γ2 ⎞ ⎤ 1 tP ⎛ (1 + β 2) + (γ − 1)2 + . ⎜α − ⎟⎜ ⎟ 1 + ρA ⎠ ⎝ 4ρA 3 1 + ρA ⎠ ⎥ ro ⎝ ⎦
(6)
Material Parameters Young's modulus Poisson's ratio of Young's modulus Poisson's ratio of Young's modulus Poisson's ratio of Young's modulus Poisson's ratio of
In the analytical model, the plate had a length of 141 mm, width of 8 mm, and thickness of 4 mm. The intact bone had an outer diameter of 25 mm, cortical thickness of 5 mm, and length of 141 mm. One end of the bone was fixed and an eccentric compressive load of 800 N was applied to the opposite end at a distance of αro = 10mm from the centroidal axis of the bone. Analytical results (ai ) were compared with the finite element results ( fi ). The corresponding finite element model is shown in Fig. 1(b). The mean percentage error (MPE) in the predicted stresses was computed using the following equation: a −f n 100 % MPE = n ∑i = 1 i a i .
of bone (EB ) bone (νB ) of plate (EP ) plate (νP ) of fractured bone (EF ) fractured bone (νF ) of screw (ES ) screw (νS )
−1
σB = −
ρA γ ⎞ ⎛ 1 ρ γ2 ⎞ ⎤ F ⎡ 1 t (1 + β 2)(1 + ρI ) + A + B ⎜⎛α − . ⎟⎜ ⎟ ⎢ 1 + ρA ⎠ ⎝ 4 1 + ρA ⎠ ⎥ AB 1 + ρA ro ⎝ ⎣ ⎦ (4)
i
2.2. Finite element analysis (FEA) A simplified 3D finite element (FE) model of a bone-implant system 644
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et al., 2003). The Poisson's ratio of 0.3 was assumed for all plates. The linear elastic material model including geometric nonlinearity due to large strains was considered in the computation. The model was meshed with tetrahedral elements. Nodes on the end surface of the bone were fixed (Behrens et al., 2008; Nasr et al., 2013). Although it was not the in vivo constraint condition, the local effects were reasonably confined to the region near the boundaries according to Saint-Venant's principle. For a patient weighing 800 N, an equivalent pressure of 2.5 MPa was applied to the other end surface of the bone (Benli et al., 2008; Fouad, 2011). In order to mimic the in vivo loading conditions of femur, a torque of 1 Nm was applied to the fractured bone, in addition to the pressure (Fouad, 2011; Carter et al., 1981). Contact pairs were defined at the interface between the plate and the screws, as well as between the bone and the screws. Additional contact between the bones and the plate was created when using a contacted plate. The lateral surface of hollow cylindrical bone was modified to be partially flat to ensure a complete contact with the plate. A penalty method available in COMSOL was used to implement contact constraints. An initial contact pressure was automatically determined using the solver configuration. All contacting surfaces were assigned a friction coefficient of 0.9 (Fouad, 2010, 2011). After the FE model was validated against published results (Fouad, 2011), it was turned into an APP (Fig. 3) using the application builder in COMSOL to demonstrate the mechanical behavior of the bone-implant system by changing the values of the editable parameters (Table 2). The purpose of the APP was to design the conceptual configuration of a screw-plate fixation for fractured bones. One of the four geometric parameters (i.e., t, Sx1, Sx2, and n) was altered in each design. Other parameters were the same as those used in Fouad's model (2011), except that the plate length was 130 mm. For example, the gap (t) between bone and plate varied within the range described in Ahmad et al. (2007). The APP simulated the bone-implant system when subjected to pressure and torque. The effect of stress shielding was represented by the following equation (Piao et al., 2014):
Fig. 4. Comparison of the analytical approach and finite element analysis under the eccentric axial loading condition. The values of MPE between these two methods were both less than 10% in the bone and plate.
was generated using COMSOL Multiphysics 5.2 (COMSOL, Inc., Burlington, MA). Its dimensions were the same as previously described by Fouad (2010). In the model, the femur had an outer diameter of 25 mm, cortical thickness of 5 mm, and length of 141 mm (Fig. 2). It was assumed that a simple transverse fracture, AO type 32-A3 (Rüedi and Murphy, 2000), with a 1 mm gap was filled with a callus during healing. The plate was modeled with dimensions of 70 mm length, 8 mm width and 4 mm thickness. The screws were 4 mm in diameter and 32 mm in length. The material properties of bone, fractured bone, screws, and plate were adopted from Benli et al. (2008) and Fouad (2010) and are summarized in Table 1. The material properties of different plates assigned to the FE model included titanium alloy and carbon fiber composite as alternative materials with elastic moduli of approximately 110 and 50 GPa, respectively (Ramakrishna et al., 2004). The carbon fiber composite was assumed to be isotropic (Saidpour, 2006) and to behave in an elasto-plastic manner (Veerabagu
η = (1 − σ / σ0) × 100%.
(7)
where η is the stress shielding rate, σ is the stress of bone after implantation of plate, and σ0 is the stress of bone at the same position before implantation of plate.
Fig. 5. Maximum stress of callus (fractured bone) predicted using the present FE model compared with that predicted using the FE model of Fouad (2011). Stainless steel was chosen as the material for both non-contacted and contacted plates. Percent differences between the present study and the reference (Fouad, 2011) were 8% and 9% for compressive and combined loading conditions, respectively, in the case of the non-contacted plate.
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Fig. 6. Effect of the gap between bone and plate on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where n = 2, Sx1 = 10 mm, and Sx2 = 15 mm.
Fig. 7. Effect of the half of distance between central pair of screws on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where n = 2, t = 0.5 mm, and Sx1 + Sx2 = 60 mm.
3. Results
combined static loading was applied to FE models constructed with different plate materials. The corresponding maximal stress and strain at the callus (fracture site) are provided by APP (Fig. 6). Generally, enlarging the gap between bone and plate increases the stress and strain at the callus. The maximum stress of callus oscillates in the range from 0.8 to 1.4 mm in gap between bone and carbon fiber composite plate. This is because when using APP, the automatic-fine mesh size varies with different gap designs. Similar investigations of the results obtained with changes in Sx1, Sx2, and n performed individually, and the effect of each parameter on stress and strain at the callus are shown in Figs. 7–9. Overall, plates with a lower Young's modulus produced higher stress and strain at the callus. Additionally, non-contacted plates generated higher stress and strain at the callus than the contacted plates did. Qualitatively similar results were previously reported by Benli et al. (2008) and Fouad (2010, 2011).
Fig. 4 shows the axial stresses in the bone and plate in the transverse direction as predicted by the analytical approach and FEA described in Section 2.1. The stresses computed using analytical formulas correlated well with those computed using FEA. The values of mean percentage error between these two methods were 9% (in bone) and −3% (in plate), respectively. The analytical approach to the stress prediction of bone-implant system was validated by the FEA. For verification of the bone-implant model at an early stage of healing (1%), the FE solutions agreed with the FE results reported by Fouad (2011). As shown in Fig. 5, the present predictions of the maximum von Mises stress at the callused bone under either a non-contacted or contacted plate were smaller than those reported by Fouad (2011). The difference in stress prediction of callused bone was larger with a contacted plate than with a non-contacted plate because this model and Fouad's model differed in node quantity, element shape, and contact algorithm. For parametric investigation of the gap (t) between bone and plate,
4. Discussion FEA should be of much use to guide the clinicians for investigating 646
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Fig. 8. Effect of the distance between the first and second left screws on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where n = 2, t = 0.5 mm, and Sx1 = 10 mm.
Fig. 9. Effect of the numbers of screw-pairs on (a) maximum von Mises stress of fractured bone and (b) maximum axial tensile strain of fractured bone, where t = 0.5 mm, Sx1 = 10 mm, and Sx2 = 10 mm.
Excel to offer predictions that are consistent with FEA calculations. In the same way, results from the analytical approach were validated by Tsai et al. (2013) using an FE model based on the X-ray film of the femur. Simple models towards the prediction of stress in the femur have been preliminarily evaluated for clinical purposes. The analysis revealed that the screw-plate implant experiences higher stress than the bone does, which reflects the stress shielding offered by the implant to the bone. As indicated in Fig. 5, the stress shielding rates in the present study were 16.2% and 37.4% for noncontacted and contacted plates, respectively, under compressive loading. The callus stress in the present study decreased more significantly than that in the reference (Fouad, 2011) when using the contacted plate instead of the non-contacted plate. It has been noted that plates with decreased rigidity accelerate healing and eliminate the risk of osteopenia due to stress shielding (Woo et al., 1977). Stability of the bone-implant system often influences the healing process, which is explained by the concept of interfragmentary strain (Perren, 2002). Strain between 2% and 10% provides a status of stability with enough
biomechanical responses of bone-implant system under physiological loading. Unfortunately, customized stress calculation by FEA in individual patients would be both time-consuming and technically demanding. The analytical approach predicts stress magnitude over a continuum of structures in parametric forms of equations. Differences in the predicted stress in the two methods (Fig. 4) occurred because the periphery of bone was not as circular in the FE model as in the analytical model. According to Saint-Venant's principle, the location of comparison was made throughout the middle region of the bone-implant length where uniform stress distribution was produced. Furthermore, the screws and their contact surfaces were not considered in the present analytical model. Therefore, a possible future work is how to establish an analytical model for consideration of the forces acting on plate through the screws (Ramakrishna et al., 2004). When the boneimplant system is subjected to combined loading conditions, the torque can be taken into consideration in the analytical model by superposition of stress components (Gere and Timoshenko, 1997). While not as rigorous as FEA, the analytical approach can be executed in Microsoft
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interfragmentary motion to stimulate callus formation (Egol et al., 2004). Bone healing may not occur when the strain at a fracture site exceeds 10%. A limitation of this study is that the nonlinear material constitutive relation for the callus subject to large strain was not taken into account in the current FE model. Generally, securing the LCP to bone with locking screws at a small distance between plate and bone enhances the fixation stability and suppresses interfragmentary strain (Miller and Goswami, 2007). Appropriate degrees of strain and stress promote bone healing. The reduction of stress shielding (increment in the stress of callus) enlarged the interfragmentary strain of callus. On the contrary, smaller strain kept below 10% may achieve stability. The results obtained with the present analysis consistently showed that noncontacted plates consisting of low-stiffness material provided relatively high bone stress. However, the mineralized callus may rupture if exposed to strain greater than 10%. The APP developed in this study could, in principle, be used to design screw-plate fixation for reducing stress shielding and providing stability. Surgical techniques such as the positions and number of screws have greatly influenced healing time and stability of fracture sites (Stoffel et al., 2003; Nasr et al., 2013). Some clinical and experimental studies indicated the influence of screw position in the implant plates on the stability of the bone-implant system (Field et al., 1999). The purpose of this APP was to investigate the stress and strain distribution in a simplified bone-implant system based on a modifiable parametric model. The present study did not intend to validate or suggest that a simplified bone-implant model is a universal model for orthopedic biomechanics. It should be possible to adapt this simplified model to CT-based models. In principle, an APP with specific functions can be applied to the development of preoperative planning simulations for orthopedic surgery. Such a specialized APP can be run by connecting to the COMSOL Server from web browsers (Dagastine et al., 2016) and using cloud-computing technology (Kaner et al., 2014). Future studies with CT-based FEA will further corroborate these findings. Clinicians can bring a customized APP on-site and arrange screw positions and numbers as well as the gap between bone and plate to perform a presurgical evaluation. Acknowledgements The author would like to acknowledge the grant received (grant number MOST 105-2218-E-008-005) from the Ministry of Science and Technology in Taiwan. Conflict of interest statement The author has no conflicts of interest related to this manuscript, commercial or otherwise. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jmbbm.2017.10.025. References Ahmad, M., Nanda, R., Bajwa, A.S., Candal-Couto, J., Green, S., Hui, A.C., 2007. Biomechanical testing of the locking compression plate: When does the distance between bone and implant significantly reduce construct stability? Inj. Int. J. Care Inj. 38, 358–364. Alemdar, C., Azboy, I., Atiç, R., Özkul, E., Gem, M., Kapukaya, A., 2015. Management of infectious fractures with “non-contact plate” (NCP) method. Acta Orthop. Belg. 81, 523–529. Allgöwer, M., Perren, S., Matter, P., 1970. A new plate for internal fixation—the dynamic compression plate (DCP). Inj. Int. J. Care Inj. 2, 40–47. Behrens, B.-A., Wirth, C.J., Windhagen, H., Nolte, I., Meyer-Lindenberg, A., Bouguecha, A., 2008. Numerical investigations of stress shielding in total hip prostheses. Proc.
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