# The Influence of Loading Program on the Stimulated Callus Mineralization
## 3 Results  
The results are presented in Figures 2-4 for three analyzed loading programs. On the right side, each of ten colored lines represent the Young's modulus distribution along the height of the analyzed rectangular section (Figure 1C) for the given time step (\(t=1,2,3,\ldots,10\)). The first line on the left represents the starting point (\(t=0\)) with the initial value of the callus elasticity modulus equal to 2 MPa. Moving to the right, along the \"time increase\" arrow, we observe an increase of the Young's modulus maximal value and the mean value in the callus. The three parameters of homogeneity are calculated for the final time step (\(t=10\) (time unit)). Figure 2: The optimal loading program and the resulting in-time Young’s modulus distribution along the height of the analyzed rectangular section, with parameters of homogeneity. Figure 4: The intermittent loading program with 10% residual load and the resulting in-time Young’s modulus distribution along the height of the analyzed rectangular section, with the parameters of the homogeneity. Figure 3: The intermittent loading program and the resulting in-time Young’s modulus distribution along the height of the analyzed rectangular section, with the parameters of the homogeneity. Figure 2 presents the outcome of a progressive remodeling process for the optimal loading program. The values of the bending moment, \(M\), and shear force, \(T\), in time, are calculated using the control algorithm described in [21; 22]. This control algorithm is used for the load parameters so as to improve homogeneity of the elasticity modulus with possibly maximal remodeling rate. At each time step, we require that the value of the stimulus parameter \(\psi\)  
\[\psi(z)=\frac{SED(z)}{\rho(z)}=\frac{\sqrt[3]{C}}{2}\cdot\frac{M^{2}z^{2}\cdot [E(z)]^{\frac{2}{3}}}{I_{\mathrm{E}}{}^{2}}+\frac{\sqrt[3]{C}(1+\
u)}{[E(z)]^{ \frac{4}{3}}}\cdot\frac{T^{2}[S_{\mathrm{E}}(z)]^{2}}{I_{\mathrm{E}}{}^{2}\cdot b ^{2}} \tag{8}\]  
is kept in a narrow vicinity of the optimal value, \(K_{\mathrm{over}}\), for all section points, which denotes that the remodeling rate remains close to the maximal possible value, \(K_{\mathrm{w}}\). Satisfying \(\min_{z}\psi(z)=(1-\alpha)K_{\mathrm{over}}\) and \(\max_{z}\psi(z)=(1+\alpha)K_{\mathrm{over}}\) (\(\alpha\) determines the vicinity), we come to a set of equations to determine the optimal values of the bending moment, \(M\), and the shear force, \(T\). They are related to the distribution of \(E(z)\) obtained at the preceding time steps, starting from \(E(z)=\underset{z}{\mathrm{const}}=2\) (MPa) at time \(t=0\). \[\frac{\sqrt[3]{C}}{2}\cdot\frac{M^{2}z_{\mathrm{min}}^{2}\cdot[E(z_{\mathrm{ min}})]^{\frac{2}{3}}}{I_{\mathrm{E}}{}^{2}}+\frac{\sqrt[3]{C}(1+\
u)}{[E(z_{ \mathrm{min}})]^{\frac{2}{3}}}\cdot\frac{T^{2}[S_{\mathrm{E}}(z_{\mathrm{min} })]^{2}}{I_{\mathrm{E}}{}^{2}\cdot b^{2}}=(1-\alpha)K_{\mathrm{over}} \tag{9}\]  
\[\frac{\sqrt[3]{C}}{2}\cdot\frac{M^{2}z_{\mathrm{max}}^{2}\cdot[E(z_{\mathrm{ max}})]^{\frac{2}{3}}}{I_{\mathrm{E}}{}^{2}}+\frac{\sqrt[3]{C}(1+\
u)}{[E(z_{ \mathrm{max}})]^{\frac{2}{3}}}\cdot\frac{T^{2}[S_{\mathrm{E}}(z_{\mathrm{max} })]^{2}}{I_{\mathrm{E}}{}^{2}\cdot b^{2}}=(1+\alpha)K_{\mathrm{over}} \tag{10}\]  
\(z_{\mathrm{max}}\) and \(z_{\mathrm{min}}\) denote coordinates of section points where the stimulus is maximal or minimal, respectively. With this optimal loading program it is possible to reach unrestricted value of the elasticity modulus with its distribution kept close to homogeneous. In Figure 2 (right plot), only the first 10 time steps are presented and the \(E_{\mathrm{max}}=233\) MPa, which fulfills the requirement of 200 MPa as an ending point of calculation. The left plots present the loading program as the time-variation of dimensionless bending moment and shear force. The dimensionless quantities are defined with the use of section dimensions \((b,h)\) and the initial value of the callus Young's modulus (\(E_{0}\)).