# The Influence of Loading Program on the Stimulated Callus Mineralization
## 1 Introduction
This theory for bone remodeling is adjusted and used as a new approach to describe the change in the callus density because during the mineralization process, the properties of the material (callus) change as an effect of external loading. The mechanical stimulus theory states that there are some quantities, that can be calculated, which elicit or accelerate a physiological activity or response of the tissue. In this work, we focus on the physiological response of the callus tissue to external load action, exhibited in the corresponding change of its density. There are many factors considered as the mechanical stimuli: strain, stress, and deformation energy [13]. The mechanical stimulus theory for bone remodeling was proposed by [PERSON] et al. [14]. This theory states that the strain energy density (SED) is the most favorable factor that induces bone growth. The remodeling theory includes mass density, remodeling constants, reference values of the stimulus and a percentage value denoting the width of the lazy zone region [13]. One may notice that the mathematical description is divided into three parts. In the first one the negative change of bone density indicates bone resorption due to under-load, i.e., the stimulus is too small to induce bone growth. The second part describes the lazy zone where no growth and no loss occur. And the third part with the positive growth ratio, indicates bone growth and, therefore, tissue density increase. This most commonly used approach takes into account 3 stages: underload resorption, equilibrium, and bone growth (apposition) [13; 15; 16; 17]. It is not the only possible formulation. [PERSON] [10] proposed the mathematical description of overload resorption but it results in the function's discontinuity, which can hardly be supported experimentally or clinically. This approach was later adopted by other authors [3; 11] and mainly used to describe and predict the density of the bone tissue around the dental implants. [PERSON] [12] proposed the quadratic equation for the continuous description of bone remodeling but entirely omitted the lazy zone (equilibrium state). The \"lazy zone\" effect, initially proposed by [PERSON] et al. in 1977 [18] based on experimental investigation, was verified by [PERSON] et al. [19] to be a relevant factor of the bone remodeling process, which should be included in the simulation. Some authors claim, based on the experimental study, that the idea of the \"lazy zone\" is controversial and they state that it is not valid [20]. However, in their study, [PERSON] et al. included only a narrow group of post-menopausal women, thus the results might be questionable. To sum up, only a few models proposed in the literature take into account bone resorption due to overload [3; 10; 12] and there are no such studies in relation to the callus material, which in the process of mineralization transforms into a compact bone tissue. Due to these facts, the aim of this work is to provide a new approach based on the mechanical stimulus theory to describe the change of the callus density in the process of healing. With the use of a mathematical model for callus remodeling [21; 22], the analyses of various loading programs are performed. In this work, we mainly focus on the elastic modulus, \(E\), of the callus tissue, i.e., we aim at obtaining its sufficiently high value and homogeneous distribution. For the initial model validation, the medical data of the fractured mandible is analyzed. Firstly, the callus region based on the computer tomography data using Mimics software is marked, then the density measurements are performed. The aim of this analysis is to determine whether it is possible to determine the modulus of elasticity of the callus and compare its distribution qualitatively with the numerical analysis results.