Geometric and finite element modeling of bones and implants for simulations of atlantoaxial dorsal fixation and finite element analyses for the bone stresses were performed using Abaqus software program (AbaqusⓇ, Version 6.10; Abaqus, Inc., Providence, RI, USA).
Geometric modeling of bones and implants
Bones used in this study were simplified to platy or arch-shaped structures. Arch-shaped bone was modeled based on a three-dimensional (3D) computed tomography (CT) image for the dorsal arch of the atlas of a toy breed dog (a 9-year-old male Maltese dog) without any cervical diseases. The 3D CT image had been taken using a four-row multidetector CT scanner (Hi Speed QX/I; GE Medical Co., Milwaukee, WI, USA) and Medicine (DICOM) viewer (OsiriX MD 4.1.2; OsiriXPixmeo, Geneva, Switzerland). Characteristics of the arch-shaped bone were as follows: 1) the bone got slightly thicker as it went laterally; 2) the bone had a notch with steep oblique angle in the craniomedial part; 3) the bone had cranial protrusions on both sides of the notch; and 4) the lateral part of the bone beyond the notch had a gentle oblique angle. Details for the arch-shaped bone are shown in Fig. 3. Two types of implants (wire implant and band implant) were modeled and applied to bones. The diameter of the wire implant was 0.8 mm. The thickness of the band implant was also 0.8 mm.
Finite Element modeling
3D FEMs for simulations of dorsal fixation were generated using hexahedral and pentahedral elements (Fig. 4). On average, 36,023 nodes and 28,557 elements were used in each simulation. Bones were defined as cortical bones. Cancellous bones in the internal matrix were ignored. Implants were made of stainless steel. Material properties of bones and implants are shown in Table 1. They were assumed to be homogenous and linear isotropic. The friction coefficient value for bone-steel is 0.37 . Material properties of bones were adapted from a previous study of human atlas  because of the lack of material definition for canine spine. Material properties of implants were obtained from the literature on stainless steel [31, 32].
Two studies were performed and referred to as project A and project B (Fig. 5 and Fig. 6). Project A using platy bones was carried out to explain stress-related results of project B. Project B using arch-shaped bones was implemented to simulate the clinical situation of dorsal fixation.
1. Project A
In project A, influences of various conditions such as bone shapes, implant types, and positioning of implants on the platy bones were investigated. Conditions were varied in five models. Model 1 was designed based on the width between wires. Platy bones with a width of 10 mm, a length of 8 mm, and a thickness of 1 mm were used and wire implants were applied to bones. The width between wires was 2 mm in model 1.1, 4 mm in model 1.2, and 6 mm in model 1.3. Model 2 was designed based on the length of the bones. Platy bones with a width of 10 mm and a thickness of 1 mm were used. Wire implants with a width of 4 mm between wires were applied to bones. The length of bones was 8 mm in model 2.1, 9 mm in model 2.2, and 10 mm in model 2.3. Model 3 was designed based on the thickness of bones. Platy bones with a width of 10 mm and a length of 10 mm were used. Wire implants with a width of 6 mm between wires were applied to bones. The thickness of bones was 1 mm in model 3.1, 2 mm in model 3.2, and 3 mm in model 3.3. Model 4 was designed based on the oblique angle of notches. Platy bones with a notch were used. They had a width of 10 mm and a thickness of 1 mm. Wire implants with a width of 4 mm between wires were applied to oblique planes of notches. The oblique angle was 15° in model 4.1, 30° in model 4.2, and 45° in model 4.3. Model 5 was designed based on types of implants. Platy bones with a width of 10 mm, a length of 10 mm, and a thickness of 2 mm were used. Wire implants with a width of 6 mm between wires and band implant with a width of 6 mm were applied in model 5.1 and model 5.2, respectively (Fig. 5).
2. Project B
In project B, influences of types and positioning of implants on arch-shaped bones were investigated. Four models (referred to as model 6) were used in project B. Wire implants were applied within notches of arch-shaped bones in models 6.1 and 6.2 and beyond the notch in model 6.3. The width between wires was 2 mm in model 6.1, 4 mm in model 6.2, and 6 mm in model 6.3. Band implant with a width of 6 mm was applied in model 6.4 (Fig. 6).
Finite element analyses
Arbitrary tension forces of 20 N were applied to implants in project A while 20 N, 40 N, and 80 N of tension forces were applied to implants in project B for simulations of the dorsal fixation. The maximum bone stresses under these forces did not exceed the yield stress of cortical bone. Considering our bone elastic modulus (10 GPa), the bone yield stress in this study was 66.4 MPa [yield stress (MPa) = 24.4 + 4.20 × elastic modulus (GPa)] . If the maximum bone stress gets close to the yield stress, the bone will be vulnerable to bone fracture. The maximum von Mises stress in the bone for each model was measured and compared to each other to find out which models had lower maximum stresses.