## 3 Materials and Methods
### Preparation of an Algorithm to Simulate the Bone Union Process  
Mechanobiology deals with the study of the impact of mechanical loads on biological processes in tissues, especially at the cellular and tissue level. Computational mechanobiology is a branch of this science that uses mathematical models and computer simulations to determine exactly how mechanical forces affect cell differentiation, tissue remodelling processes, and its response to loads. Mechanobiological models enable simulation of bone healing processes and other biological phenomena, eliminating the need to conduct expensive, time-consuming, and sometimes impossible physical experiments. As a result of them, we can predict how a given orthopaedic implant will behave in the patient's body, whether bone fusion will occur and whether it will be accompanied by the risk of damage. These models also allow for selecting the optimal solution in a specific clinical situation, minimizing the need to experiment on living organisms [45; 46; 47]. The only frequently used mechanobiological model is the [PERSON] model developed in 1984 based on [PERSON]'s theory. In this mathematical model, mechanical stimuli, such as uniaxial compression and tension, play a key role, causing deformations and determining the adaptation and remodelling of bone tissue [48]. According to [PERSON], there is a close relationship between the mechanical signal and the rate of bone remodelling. Osteocytes, or bone cells, respond to these stimuli, with strain energy density being a key controlling factor. Maximum loads, generating the highest strain density, result in intensive remodelling of bone tissue [30; 48]. [PERSON]'s concept assumes the existence of three zones of relationship between the mechanical stimulus and changes in the tissue. This relationship is linear. When the strain density drops below a certain value, osteoclastic processes, i.e., tissue resorption, begin. The next range of mechanical stimulus values leads to achieving biological balance in which the processes of bone tissue resorption and construction are balanced. This zone, called the lazy zone, occurs with small load changes. The formation of bone tissue takes place when the mechanical stimulus exceeds a certain value. Figure 11 graphically presents the occurrence of defined zones of dependencies between the mechanical stimulus and changes in the tissue [49]. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Linear Simulations**} \\ \hline
**Material** & \(E\) [**GPa**] & \(v\) [-] & _\(\rho\)_ [**kg/m\({}^{3}\)**]** \\ \hline
316L Steel & 200.0 & 0.29 & 7900 \\ \hline Cortical Bone & 17.0 & 0.30 & 1800 \\ \hline Trabecular Bone & 0.5 & 0.30 & 700 \\ \hline \hline  & \multicolumn{3}{c}{**Non-linear simulations**} \\ \hline
**Material** & \(A\) [**MPa**] & \(B\) [**MPa**] & \(n\) [-] & \(c\) [-] \\ \hline
316L Steel & 490 & 600 & 0.21 & 0.015 \\ \hline Cortical Bone & 150 & 58 & 0.21 & - \\ \hline  & _D\({}_{1}\)_ [-] & _D\({}_{2}\)_ [-] & _D\({}_{3}\)_ [-] & _D\({}_{4}\)_ [-] \\ \hline
316L Steel & 0.05 & 3.44 & 2.12 & 0.002 \\ \hline  & \multicolumn{3}{c}{**\(\epsilon_{\textbf{limit}}\)** [-]**} \\ \hline \hline \multicolumn{3}{c}{Cortical Bone} & 0.021 \\ \hline \hline \end{tabular} where: _E_—Young’s modulus, _v_—Poisson’s ratio, _p_—density, _A_—yield strength, _B_—strengthening constant, _n_—strengthening coefficient, _c_—strain rate coefficient, _D\({}_{1}\)_–_D\({}_{4}\)_—Johnson–[PERSON] damage constants, and _\(\epsilon_{\textbf{limit}}\)_—maximum strain value.  
\end{table} 
Table 1: Material parameters used for numerical analyses [42; 43; 44]. [PERSON]'s algorithm (Figure 12) allows predicting changes in bone tissue density based on the value of the mechanical stimulus that will be generated in the tissue due to external loads.