A Study of the Mechanical Characteristics of a Mandibular Parasymphyseal Fracture with Internal Fixation Device Subject to Variable Bite Forces: Finite Element Analysis

In recent times, research into mandibular fracture has gained momentum from advances in scanning techniques, software/algorithm developments and improvements, and numerical structural modeling using the finite-element method (FEM). In this work, the FEM is used to model a mandibular fracture (using an inhomogeneous and orthotropic jaw model) simulating the effect of different bite tasks/forces on the stability of the fixated fracture. Specifically, bilateral and unilateral clenches (using muscle data) were studied using a low-profile 3D 4 × 2 hole mini-plate deployed for fracture fixation. Here, the mandible bone was treated as orthotropic and spatially inhomogeneous. Although the results of stress and displacement analyses, for this fixation hardware, indicate sufficient fixation under normal biting conditions, the results show that the unilateral and ipsilateral bites develop, in general, the highest stresses or displacements. Such results can guide post-surgery recommendation on bite behavior.

methods for repairing mandibular fractures.
Open reduction and internal fixation is the most frequent treatment for mandibular fractures [1]. These procedures are largely considered successful, though studies have reported up to 30% rate of complications requiring plate removal (depending on the fracture site) [1] [2] [3] [4]. Methods outside the scope of the medical practitioner's expertise are necessary to understand the fundamental reasons by which a particular rigid internal fixation device may perform superior to another. Fixation of fractured mandibular bone segments is primarily of a structural nature involving various material properties, each exhibiting different mechanical responses (i.e. stress and strain under loading). The mechanical engineering discipline is suitable to further the efforts in developing improved plating systems and understanding the role of fracture mechanics in complications and removal after internal fixation.
A useful technique for analyzing complex structures is the finite element method (FEM), which essentially reduces a complex problem that might otherwise be impossible to solve analytically into numerous smaller and simpler problems generating a multitude of equations, or matrices, that can be solved numerically. Assembling each individual element matrix into the global stiffness matrix by applying the appropriate boundary conditions and reviewing the entire solution gives an understanding to the behavior of the structure. With the improvement in finite element analysis (FEA) software and the significant increase in capability of computing resources, FE modeling of biological tissues and physiological behavior is now becoming an established practice.
FEA offers insight into the mechanistic behavior of fixation plates used in rigid internal fixation and, if modeled carefully, could eventually become an accurate design tool. A literature investigation has sought to determine the appropriateness of fixation plating used in rigid internal fixation through FE analyses. Most studies have analyzed fixation by focusing on a single bite load, making it difficult to extrapolate findings to expected real world functioning. In addition, fracture contact conditions are largely ignored or greatly simplified. Surgeons will most often establish physical contact between the fractured mandibular bone pieces during surgery (reducing the fracture), allowing normal and frictional forces to mitigate some of the stresses and displacements that would otherwise be transferred entirely to the fixation hardware [5]. A study by Caraveo et al. [6] has determined the necessity to include frictional contact boundary conditions in the FEA analysis of the fracture region. Some of the studies that utilized one type of bite or bite task are [6] [7] [8].
One of the purposes of this investigation is to identify the effectiveness of a rigid internal fracture fixation device using FE simulations of the human mandible under variable bite forces. The primary bite forces are bilateral bite forces and unilateral bite forces both ipsilateral (interchangeably referred to the left side throughout this paper) and contralateral (right side) to the fracture site. A total of seven different bites and eleven different bite tasks are considered in this study. The authors hypothesize the biting tasks ipsilateral to the fracture will produce the greatest displacement in the fracture region.
Additionally, mechanical responses include stress measures in the plate-screw system as well as the cortical bone surrounding the screws. The commonly used von Mises criteria in FEA to evaluate mandibular bone stress studies is called into question since this measure is typically considered more suitable for ductile materials, whereas bone is characterized as having more of a brittle nature. The stress measure that captures the largest tensile value is the principal stress criteria in which the coordinate system is such that there are no shear stresses acting on the element and only normal stresses act in the three orthogonal directions. [8]. CAD model verification for this study was detailed in Chaudhary et al. [9] and Lovald et al. [7].

Material Properties
The mandible considered in this work consists of teeth, cortical bone, cancellous bone and dental segment. The dental segment consists of dentin, enamel, periodontal ligament. There are six symmetric volumes in the mandibular volume: the symphysis, parasymphysis, angle, ramus, condyle, and coronoid. That makes twelve the total number of mandibular volumes in the model. Each of these volumes was treated as orthotropic material, which is more accurate than the isotropic approximation. This required the specification of local coordinate systems into the FE software. Table 1 gives the material properties used in this study for simulating the fractured mandibular model. The orthotropic cortical bone values were taken from a study by Schwartz-Dabney et al. [10]. Isotropic properties for cancellous bone were taken from reference [11]. The properties for dentin are aligned with a study by Craig and Peyton [12]. Since dentin is much stiffer than other parts of the dental segment, its elastic properties were accounted for in the model. For the titanium plates, the properties were taken out of [13] which was a FE study of mandibular angle fractures. All bite load configurations utilized the same model; the only differences were the boundary condition and applied muscle load distribution corresponding to each respective bite task.

Field Equations and Boundary Conditions
Multiple bite forces were used in this FEA. Muscle force vectors that were experimentally derived for a specific bite are distributed around the mandible [14].
Each force has a direction, area of attachment, and magnitude. The condyles and occlusal surface at loading point are restrained from movement. The equations of equilibrium used in this mandibular FEA are: In the realm of small deformation theory, the strain ε ij at a material point is related to the gradients of displacement as follows: For an orthotropic material deforming elastically, the stress-strain constitutive relation, or Hooke's Law, is as follows: where E i , G ij , and v ij are the different elastic material constants (9 total). In this last equation, a repeated index does not imply summation.
For an isotropic material, Hooke's Law reduces to two material constants (two from G, λ and ν): where ( )( )

Frictional Contact Boundary Conditions
Frictional contact boundary elements were applied to the surfaces of the fracture. Double-sided surface-to-surface contact boundary elements were utilized since it was not known initially which surface initiated contact with the other surface for each load configuration. The same coefficient of friction utilized in a previous study by the authors (μ = 0.4) [6], and obtained from an experimental study [15], is used for this model. ANSYS contact elements used in the ANSYS FE models were TARGE170 and CONTA174 ( Figure 2 shows these elements for a cross-section of the mandible, specifically the cortical bone, representing one of the fracture surfaces).
The algorithm chosen for the contact elements was the Augmented Lagrangian Method which requires more iterations than the Pure Penalty Method but was chosen for better conditioning in the event of element distortion. For contact analysis, the traction vector for contact has three components: one normal (Pressure or P) and two tangential (frictional) shear stresses (in the local y and z directions here, i.e. τ y and τ z ). The contact pressure is defined using Lagrange multiplier as it is directly related to the contact normal stiffness (K n ) and the contact gap size (u n ). The Lagrange multiplier at iteration i (λ i ) is computed locally (for each element) and iteratively. The frictional stress is obtained by Coulomb's law: where 1 n i τ − is the frictional stress in direction i = 1, 2 at the end of the previous substep, τ is the equivalent stress, K s is the tangential contact stiffness, i u ∆ is the slip increment in direction i over the current substep, u ∆ is the equivalent slip increment over the current substep, and μ is the coefficient of friction. The last equation is obtained from an ANSYS manual.

Mastication Forces
The magnitude and direction of muscle forces during the simulated bite were obtained from Nelson et al. [16]. Nelson's work is seen as an authority in muscle data forces and is the standard in most literature that have attempted to simulate bite forces in previous FEA studies [6] [7] [8] [14]. The data from this reference pertains to the bite of a healthy adult with an intact mandible. It is estimated that the bite force of a patient with a fractured mandible is 60% of that of a healthy adult [17]. The bite force data was modified accordingly in this study. The muscle attachment areas on the mandible were obtained from the literature [18].
The resultant muscle forces are described by the force vector and a central muscle insertion point. These direction cosines describing the direction of muscle forces are given in Table 2. The magnitude of each resultant muscle force is  in this study; symmetrical bilateral clenching and unsymmetrical unilateral bite forces, both ipsilateral and contralateral to the fracture region. The three symmetrical bilateral clenching tasks are: incisal (I), intercuspal (IC), and the bilateral molar clenches (BM). The incisal clench involves an anterior clenching emphasis and is constrained at both incisors in the tooth appropriate tooth reaction direction. The intercuspal clench is medially concentrated in the mandible and restraints are applied at two bilateral tooth locations (1 st premolar and 1 st molar). Bilateral molar clenching tasks concentrate forces in the posterior occlusal region and are restrained in the 1 st and 2 nd molars for each analysis.
Asymmetrical unilateral clenching tasks simulate bite loading that consists of working side muscles and balancing side muscles [16]. Each mastication task is evaluated both ipsilateral and contralateral to the fracture site. The unilateral clenching tasks include the right and left side unilateral canine clench (LUC and RUC) restrained at each respective canine. The unilateral molar clench is configured for the left (LUM) and right (RUM) side as well as being constrained at both the 1 st and 2 nd molars making for a total of four unilateral molar clenching tasks. The scaling factors, EMG Mi , for masticatory muscle forces for each of the bilateral clenching tasks, and the unilateral clenching tasks, are listed in Table 3 and Table 4.    Table 5.

Cortical Bone Stress -Von
For illustrative purposes, contour plots of the three stress measures of interest for an incisal clench (INC) and the left unilateral molar clench restrained at the first molar (LUM1) are shown in Figure 3. For the incisal biting task, the stress distribution contours appear to be very similar, however, the maximum 1 st principal stress location is at the anterior (with respect to fracture plane) bone segment surrounding the superior screw hole proximal to the fracture, while the maximum Von Mises Stress value can be found in the posterior bone section   surrounding the proximal inferior screw hole (indicated with red arrow). The minimum 3 rd principal stress value is also located at the postero-inferior screw location. There is a 4.2% difference in peak stress values for the 1 st principal stress from the Von Mises stress and a 12.4% difference for the 3 rd principal stress for this bite task. Not that peak stress values in cortical bone in this study should be interpreted in a relative fashion due to simplifications to the screw geometry and the interface conditions between the screw bodies and the bone. The intercuspal clench was restrained at each 2 nd premolar for the ICPM case and at the 1 st molars for the ICP1M case. For both clenching tasks the maximum peak nodal Von Mises stress and 3 rd principal stress values occur at the postero-inferior proximal screw-hole vicinity; whereas the maximum 1 st principal for both intercuspal cases occur at the postero-superior screw site. Similarly, the peak Von Mises, 1 st principal, and 3 rd principal stresses follow the same pattern for the bilateral molar clenches for both tooth restraint configurations. ing tasks occur at the proximal postero-inferior screw region, except for the right unilateral molar clenches, these occur at the proximal postero-superior screw vicinity. With the exception of the incisal clenching task, whose peak Von Mises stress is found in the area of the proximal antero-superior screw hole, the remaining bilateral clenching tasks exhibit peak Von Mises stress value near the postero-inferior screw hole.

Displacement Results
Relative displacement along the fracture provides insight into the behavior of the bone-hardware deformation behavior at the fracture site. The points of interest are at the superior and inferior nodes corresponding to the midpoint of the fracture plane. The relative displacement is measured at two coincident nodes on each side of the fracture. The x-axis is taken to be the transverse direction across the fracture, along the antero-posterior axis (see Figure 5). The y-axis is in the vertical direction parallel to the fracture plane. The z-axis is parallel to the fracture plane in the buccal-lingual direction and perpendicular to the x-axis. For illustration, displacement contour plots of x, y, z, and resultant magnitude displacements are respectively displayed in Figure 5 for a BM2 clench.

Superior Nodal Relative Displacement
The bilateral bite task that produced the greatest displacement along the x-axis is the ICPM bite task while the least displacement of the superior nodes is generated by the BM2 task ( Table 7). The INC bite causes slightly more displacement than the IC1M clench, and the BM1 clench relative separation displacement in the x-direction is only 0.1% greater BM2 task. In general, the right unilateral clenching tasks produced greater relative displacement of the fracture in the ho-   computations demonstrate the LUC clench causing the least amount of relative displacement. A RUM2 clenching bite results in the greatest relative translation along the y-axis for the superior nodes. The RUC bite provided for the least displacement in the z direction while its counterpart task (LUC), resulted in the greatest z-axis relative translation. The resultant relative magnitude displacement observation yields the least relative translation by the ICPM clenching task and the largest relative resultant displacement magnitude is given by the LUC biting task.

Inferior Nodal Relative Displacement
The minimum translation in all directions at the inferior node occurs for the INC bite force ( Table 8). The largest translation in x-axis is shown for the ICPM task, followed by the ipsilateral molar clenching tasks. Ipsilateral molar clenching tasks also generated the most relative translation along the vertical direction.
Maximum relative displacement along the z-axis was produced by the LUM2 Tasks. Table 9 averages the resultant relative translation produced by each clenching task. The INC task generates the smallest relative displacement while the LUM2 case produces the largest overall translation.   [22]. In simulations of the human mandible with and without an endosseous implant, Chen et al. [23] report stress values in principal, Von Mises and dilatational stress values. Simsek et al. [24]

Bone Healing and Displacement
Complete bone healing is the goal of internal fixation devices, and the stability provided by the fixation system is vital to facilitate the process.

Boundary Conditions and Muscle Forces
One of the greatest challenges in evaluating mandibular fracture FEA has been the determination of muscle forces boundary conditions [32]. Several studies,  [20]. This still limits the degree of freedom that the articular disk normally provides at the TMJ. The justification for this is based on St. Venant's Principle which states that stresses far away from a load are unaffected by the local stress at the point of loading. Another method is to represent the mandible without the articular disk and use "gap" elements that determine the boundary at which displacement is not to exceed; and used in conjunction with spring elements. This is probably a closer approximation but still does not take into account the friction encountered with rotation. For this treatment, the articular disk is assumed to behave in a uni-axial fashion and it is unclear as to what the contact interaction is with the glenoid fossa (i.e. friction, elastic properties) [25] [26] [35]. Yet another method is to apply spring elements on nodes fixed on a symmetry plane and compute reaction forces prior to applying the necessary balancing forces at the condyle [36]. The model may be underconstrained with this treatment.
Further concerns in modeling biting loads are the boundary conditions applied at the teeth. For the unilateral or incisal clenching tasks, the point of contact is only one tooth. This type of loading best represents some sort of small object (e.g. a peanut) such that the muscles are fully engaged at the initiation of mastication. The bilateral clenching tasks assume bite contacts which are symmetrical and bilateral with occlusal contact only occurring at two teeth. However, this still offers some qualitative inference for the patient that is simply clenching unconsciously while awake or asleep. There is no literature, to the author's knowledge that has taken into consideration the occlusal contact relationship with the upper teeth for any bite clench. This study is conservative in that fixation such as arch bars, or other devices which will limit mastication functioning, are not considered in this model.

Conclusion
The inclusion of multiple forces contributes to the understanding of the fixation device under multiple force loads allowing the identification potential failure