## 3 Establishment of Bone Density Prediction Models and Analysis of Parameter Effects
### Results and Discussion  
#### 3.2.1 Cortical Bone Density Prediction Modeling and Analysis  
The Design-Expert software is used to fit the experimental data to obtain a predictive model of cortical bone density:  
\[ \begin{array}{l}\gamma_{1}=0.4445+0.0026A-0.0118B+0.0417C+1.23\times 10^{-4}AB+ 1.25\times 10^{-4}AC\\ +7.5\times 10^{-4}BC-1.5\times 10^{-5}A^{2}+6.2\times 10^{-5}B^{2}-0.2455C^{2} \end{array} \, \tag{3}\]  
In this equation, \(Y_{1}\) denotes the quadratic regression equation for cortical bone density, \(A\) represents filling density, \(B\) denotes material flow, and \(C\) represents layer thickness. The ANOVA results presented in Table 6 indicate that the \(p\)-values for all three parameters (filling density, material flow rate, and layer thickness) are below 0.0001, indicating a highly significant impact of these parameters on the regression equation for cortical bone density. Additionally, the \(p\)-value for the interaction between the filling density and material flow rate is also less than 0.0001, highlighting a substantial effect on the predictive model for cortical bone density. The lack-of-fit F-statistic of 0.9306 for this regression suggests that the cortical bone prediction model fits the observed data well. Moreover, the coefficient of determination, \(R^{2}=0.9967\), signifies excellent predictive power, indicating that the model explains 99.67% of the variability in cortical bone density. The adjusted \(AdjR^{2}=0.9953\) further confirms the model's robustness, suggesting a reliable predictive capacity for cortical bone density. As depicted in Figure 5a, the normal probability plot of the residuals for the \(Y_{1}\) prediction model reveals that the residuals are roughly aligned along a straight line. This alignment suggests that the residuals adhere to a normal distribution, thereby affirming the validity and applicability of the \(Y_{1}\) model. Moreover, Figure 5b illustrates the scatter plot comparing the predicted and actual values of the \(Y_{1}\) model. It is evident that the data points closely align with the reference line, indicating a high degree of correlation between the model's predictions and the observed values. This observation underscores the high predictive accuracy of the cortical bone density prediction model. Figure 6a displays the surface plot illustrating the interaction effect of the filling density and flow rate on density. From this, we ascertain the impact of four boundary conditions \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Typology** & \begin{tabular}{c} **Sum of Squared** \\ **Deviations** \\ \end{tabular} & \begin{tabular}{c} **Degrees of** \\ **Freedom** \\ \end{tabular} & \begin{tabular}{c} **Mean Square** \\ **Error** \\ \end{tabular} & \
\begin{tabular}{c} **\(F\)-Value** \\ \end{tabular} &
\begin{tabular}{c} _\(p\)-Value** \\ \end{tabular} & **Significance** \\ \hline Model & 0.1680 & 9 & 0.0187 & 11,958.13 & \(<\)0.0001 & significance \\ \(A\)—Filling density & 0.1364 & 1 & 0.1364 & 87,412.05 & \(<\)0.0001 & significance \\ \(B\)—Material flow & 0.0311 & 1 & 0.0311 & 19,950.51 & \(<\)0.0001 & significance \\ \(C\)—Layer thickness & 0.0001 & 1 & 0.0001 & 43.31 & \(<\)0.0001 & significance \\ \(AB\) & 0.0003 & 1 & 0.0003 & 192.30 & \(<\)0.0001 & significance \\ \(AC\) & \(1.250\times 10^{-7}\) & 1 & \(1.250\times 10^{-7}\) & 0.0801 & 0.7829 & \\ \(BC\) & \(1.125\times 10^{-6}\) & 1 & \(1.125\times 10^{-6}\) & 0.7208 & 0.4157 & \\ \(A^{2}\) & \(5.818\times 10^{-6}\) & 1 & \(5.818\times 10^{-6}\) & 3.73 & 0.0823 & \\ \(B^{2}\) & \(6.568\times 10^{-6}\) & 1 & \(6.568\times 10^{-6}\) & 4.21 & 0.0673 & \\ \(C^{2}\) & 0 & 1 & 0 & 10.62 & 0.0086 & \\ Residual & 0 & 10 & \(1.561\times 10^{-6}\) & & & \\ Misfit term & 0 & 5 & \(2.055\times 10^{-6}\) & 0.9306 & 0.2446 & insignificance \\ Pure error & \(5.333\times 10^{-6}\) & 5 & \(1.067\times 10^{-6}\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis of variance for \(Y_{1}\) regression equation. Figure 5: Residual analysis of \(Y_{1}\) regression equation.