## 3 Establishment of Bone Density Prediction Models and Analysis of Parameter Effects
### Results and Discussion
#### 3.2.2 Predictive Modeling and Analysis of Cancellous Bone Density  
A predictive model of cancellous bone density can be obtained by Design-Expert software:  
\[ \begin{array}{l}Y_{2}=0.0749-0.0004A-0.0017B+0.0573C+1.24\times 10^{-3}AB+2.50 \times 10^{-3}AC\\ +1.25\times 10^{-3}BC+6.4647\times 10^{-6}A^{2}+9.5455\times 10^{-6}B^{2}-0.1545 C^{2}\end{array} \, \tag{4}\]  
In this equation, \(Y_{2}\) denotes the quadratic regression equation for cancellous bone density, \(A\) represents filling density, \(B\) denotes material flow, and \(C\) represents layer thickness. Similarly, the analysis of variance (ANOVA) for the regression equation of cancellous bone density is presented in Table 7. The ANOVA table shows that the \(p\)-values for the filling density and material flow rate are both less than 0.0001, indicating a highly significant influence of these two parameters on the regression equation for cancellous bone density. The \(p\)-value for layer thickness, which is less than 0.05, also signifies that its effect on the regression equation is statistically significant. Among the interaction terms involving these parameters, the \(p\)-value for the interaction between the filling density and material flow rate is less than 0.0001, emphasizing that this interaction term significantly affects the predictive model of cancellous bone density. However, the other interaction terms do not demonstrate statistically significant effects. Figure 6: Effect of process parameters’ interaction on cortical bone density. (**a**) Response of filling density versus material flow rate interaction; (**b**) response plot of interaction between filling density and layer thickness; (**c**) response plot of layer thickness versus material flow interaction. The lack-of-fit F-statistic for this regression, \(F\) = 0.7226, suggests that the cancellous bone prediction model fits the observed data well. The coefficient of determination \(R^{2}\) = 0.9991 indicates exceptional predictive power, signifying that the model explains 99.91% of the variability in cancellous bone density. Moreover, the adjusted \(AdjR^{2}\) = 0.9976 confirms that the cancellous bone prediction model effectively fits the data and can predict changes in the dependent variables. As illustrated in Figure 7a, the normal probability plot of the residuals from the \(Y_{2}\) prediction model reveals a near-linear distribution. This indicates that the residuals are normally distributed, supporting the applicability and validity of the \(Y_{2}\) model. Additionally, as depicted in Figure 7b, the scatter plot of predicted versus actual values for the \(Y_{2}\) model shows that the data points closely align with the reference line. This close alignment signifies that the model's predicted values are in close proximity to the observed values, confirming the high predictive accuracy of the cancellous bone density model. Figure 8a presents a surface plot illustrating the interaction effect of the filling density and flow rate on the density of the print block. The analysis reveals that among the four boundary conditions, the combination of a high filling density and high flow rate (40%, 100%) results in the highest density. This is followed by a high filling density with a lower