# The Influence of Loading Program on the Stimulated Callus Mineralization
## 2 Material and Methods  
### Geometry of the Callus Tissue  
The proposed callus remodeling model [21; 22] describes density change based on the mechanical stimulus theory using strain energy density (SED) in the stimulus definition. It should be clarified that the proposed model of tissue remodeling is used to analyze density changes only in the narrow section of the broken mandible in the process of stimulated healing and not in the mandible bone tissue. This section is a thin layer of the callus tissue of a shape similar to a rectangle and of 0.1-1.0 mm thickness (Figure 1), possibly stimulated during healing time. The section is subjected to loading as an effect of biting and muscular forces action. The result of load reduction to its centroid are mainly bending moment and shear force. In order to investigate the complex (not uniaxial) state of stresses and strains, a rectangular cross-section (\(b\times h\)) loaded with bending moment and transverse force is considered. The initial value of the Young's modulus (\(E_{0}\)) is set to 2 MPa [23] and its initially uniform distribution is assumed all over the section. Due to non-uniform distribution of the stimulus, non-uniform healing is observed. The Young's modulus which indicates that a bone may be loaded physiologically is set to 200 MPa after Knets [24]. Various loading programs may be considered for in-time change of the bending moment and shear force. The goal is to propose such a program that results in the fastest mineralization during mandible healing. Figure 1: The mandible cross-section for the fracture modeling (**A**) and the callus tissue (**B**) simplified to a rectangular cross-section (\(b\times h\)) (**C**), which is loaded with the bending moment (M) and shear force (T). In the literature, a bone is considered to be an orthotropic or anisotropic material. However, in most finite element simulations, bone material (cortical tissue) is assumed to exhibit isotropic behavior [15; 17; 25; 26; 27; 28; 29; 30; 31]. There are mainly three reasons for that. Firstly, the relations between Young's modulus and bone mass density are established experimentally mainly based on assumptions of material isotropy. Secondly, due to complex bone internal architecture, it is difficult to determine the orientation of the material properties in a bone. Thirdly, [PERSON] [32] conducted a finite element comparative analysis for the mandible with two different material properties (i.e., isotropic and orthotropic) and only a small difference was found between the results. From the biological point of view, in the process of mineralization, the callus tissue differentiates into mature bone tissue that exhibits the features of the cortical bone [2; 33; 34]. For these reasons, the isotropic material behavior for callus tissue is assumed in our analyses. It is worth mentioning that there are known various relations between tissue density and its Young's modulus [13; 35; 36; 37; 38]. Some mathematical relations between the bone mass density and the mechanical properties (Young's modulus) of the bone were experimentally established by other authors [25; 31; 37]. In this work, the elasticity modulus \(E\) (MPa) of the callus tissue is related to tissue mass density \(\rho\) (g/cm\({}^{3}\)):  
\[E=C\cdot\rho^{\gamma} \tag{1}\]  
where \(C\) and \(\gamma\) are the constants to be experimentally determined [39; 40; 41]. In this study \(C=3790\) MPa-cm\({}^{9}\)/g\({}^{3}\) and \(\gamma=3\) are accepted following [13; 14]. The callus tissue is modeled as a linear, elastic, and isotropic material for which constitutive relation is described with the use of two constants (the Young's modulus, \(E\), and the Poisson ratio, \(\
u\)). According to [PERSON]'s theory [9], a bone can adapt its internal architecture of the trabeculae together with its mechanical properties with regard to various mechanical stimuli. The strain energy density (SED) is widely used by other authors to predict and evaluate the remodeling process [3; 10; 11; 13; 20].