## 3 Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Intercept**} & \multicolumn{2}{c}{**Slope**} \\ \hline \(a_{lower}\) & \(a_{upper}\) & \(b_{lower}\) & \(b_{upper}\) \\ \hline \(-0.00081748\) & 0.0004325 & 0.9852 & 1.0206 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Intercept-slope test for ANN. Figure 6: Comparison of FEM and ANN values of MVMS for absorbable mini screw. Once ANN model is obtained, it is possible to apply the ANNi to estimate the optimal conditions for value of SP [14; 24; 28]. The optimization was proposed considering that all the design factors are known, except for SP. Equation (1) can be expressed as shown in Equation (11). \[\text{MVMS}=b2_{1,1}-\sum_{s}Wo_{1,s}+\sum_{s}\biggl{[}\frac{2\cdot Wo_{1,s}}{1+ e^{-2(b1_{1s}+\sum_{k}Wi_{k}\cdot In_{k})}}\biggr{]} \tag{11}\]  
Equation (12) is used to involve \(I_{nk\ =\ 4}\) as the input to be optimized. \[\text{MVMS}=b2_{1,1}-\sum_{s}Wo_{1,s}+\sum_{s}\biggl{[}\frac{2\cdot Wo_{1,s}}{ 1+e^{-2(b1_{1s}+Wi_{k}\cdot In_{k}\rightarrow+\sum_{k\
eq 4}Wi_{k}\cdot In_{k})} }\biggr{]} \tag{12}\]  
The term of the left side in Equation (12) is moved to the right side and now the equation is expressed as function of SP as in Equation (13). \[f(\text{SP})=-\text{MVMS}+b2_{1,1}-\sum_{s}Wo_{1,s}+\sum_{s}\biggl{[}\frac{2 \cdot Wo_{1,s}}{1+e^{-2(b1_{1s}+Wi_{k}\cdot\text{SP}\sum_{k\
eq 4}Wi_{k}\cdot In _{k})}}\biggr{]} \tag{13}\]  
A desired value of MVMS will be reached if the result of Equation (13) is zero and thus the value of SP which satisfies this condition is also obtained. This is a typical optimization problem, and an algorithm is required to solve it. For this purpose, the differential evolution algorithm was employed. The design factors used in the ANNi and the desired value of MVMS are presented in Table 6. These values were established based on the geometry of a commercial mini absorbable screw. Equation (14) defines the function which will be minimized. To avoid values less than zero, the result of \(f(\text{SP})\) must be an absolute value. \[f(\text{SP})=\biggl{|}-a+\frac{2W_{o(1,1)}}{1+e^{b+1.9743\text{SP}}}+\frac{2W_ {o(1,2)}}{1+e^{c-1.5513\text{SP}}}+\frac{2W_{o(1,3)}}{1+e^{d-1.9943\text{SP}}} \biggr{|} \tag{14}\]  
Here:  
\[a=\text{MVMS}-b2_{(1,1)}+\sum_{s}Wo_{1,s} \tag{15}\]  
\[b=-2(W_{i(1,1)}ID+W_{i(1,2)}LS+W_{i(1,3)}AT+W_{i(1,5)}DN+b1_{(1,1)} \tag{16}\]  
\[c=-2(W_{i(2,1)}ID+W_{i(2,2)}LS+W_{i(2,3)}AT+W_{i(2,5)}DN+b1_{(1,2)} \tag{17}\]  
Figure 6:0  
The differential evolution algorithm was applied through 2000 generations with a population of 50 specimens. The scale factor and the crossover rate parameter were set to 0.4 and 0.8, respectively. The obtained result of \(f(\text{SP})\) was 1.0845 \(\times\) 10\({}^{-4}\), whereas the optimal value of SP was 14.9742 N. The value of MVMS calculated by means of Equation (7) was 19.1 \(\times\) 10\({}^{-3}\) m. Moreover, a FEM analysis was performed using the design factors of the ANNi and its optimal SP, obtaining a MVMS value of 19.21 \(\times\) 10\({}^{-3}\), as shown in Figure 6:1 \begin{table}
\begin{tabular}{c c c c c} \hline \hline _ID (mm)_ & _LS (mm)_ & _AT (\({}^{\circ}\))_ & _TN_ & **MVMS (\(\times\)10\({}^{-3}\)m)** \\ \hline
1.2 & 3.6 & 85 & 5 & 19 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Design factors and MVMS for ANNi. The difference between calculated MVMS and the proposed target was 0.52% for ANN, whereas for FEM analysis was 1.09%. Moreover, the error was 0.57% comparing MVMS for ANN and FEM analysis. These results assess the capability of the ANN to accurately predict the MVMS. As shown in previous FEM analyses for other configurations, the strain concentrated region was found in the shaft with the maximum strain located at the neck (radius). Similarly, the maximum stress occurs at the unthreaded area (shaft). Minimum stress and strain are placed at the top of the screw head.