## 3 Establishment of Bone Density Prediction Models and Analysis of Parameter Effects  
### Bone Density Prediction Modeling  
The response surface method (RSM) finds broad application in engineering disciplines, encompassing areas such as structural optimal design, reliability evaluation, dynamic analyses, and the development of predictive models [25]. When conducting optimization designs, it is essential to select representative discrete sampling points. Among various \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Filling Density** & **Layer Thickness** & **Material Flow** & **Printing Temperature** \\  & **(\%)** & **(mm)** & **(\%)** & **(\({}^{\circ}\)C)** \\ \hline Numerical value & 100 & 0.2 & 100 & 210 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Default values for process parameters when selecting printing speed as a variable parameter. Figure 4: Material flow versus specimen density. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Printing Speed (mm/s)**} & **Specimen Density (g/cm\({}^{3}\))** & **Printing Time (min)** \\ \hline
30 & 1.246 & 66 \\
50 & 1.245 & 41 \\
70 & 1.246 & 30 \\
90 & 1.244 & 24 \\
110 & 1.242 & 21 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Measurements of specimen density and printing time at different extrusion speeds. experimental design methods, the central composite design (CCD) is the most commonly employed technique. This design incorporates three categories of test points. The specific composition of these test points is as follows:  
\[N=m_{c}+m_{r}+m_{o}, \tag{1}\]  
In this equation, \(m_{0}\) denotes the center point, \(m_{c}\) denotes the cubic point, and \(m_{r}\) denotes the axial point, where the distances of the axial points from the center point are all \(\alpha\). This \(\alpha\) is called the asterisk arm. In this paper, a bone density prediction model is developed, with bone density serving as the response variable and printing parameters, specifically the filling density (%), material flow rate (%), and layer thickness (mm), acting as the influencing factors. The model considers three influencing factors (\(k=3\)). An experimental design is adopted with eight levels for each factor (\(m_{c}=8\)) and six center points (\(m_{r}=6\)). Since the center point in this modeling serves primarily to estimate the experimental error, a total of six center points (\(m_{0}=6\)) are selected, and the star arm length is set to unity (\(\alpha=1\)), facilitating the estimation of the experimental error and the robustness of the model. After gathering the data samples, selecting an appropriate response surface model is essential to derive the regression model that relates inputs to outputs. Due to its straightforward structure and superior capability in handling both linear and non-linear relationships, the quadratic polynomial method is favored in this context. Accordingly, this method has been adopted as the form of the bone density prediction model in this study. The mathematical representation of the model is as follows:  
\[y=\beta_{0}+\sum_{i=1}^{k}\beta_{i}x_{i}+\sum_{i=1}^{k}\beta_{ii}x_{i}^{2}+ \sum_{i}\sum_{j}\beta_{ij}x_{i}x_{j}+\epsilon, \tag{2}\]  
In this equation, \(y\) represents the response value of the model; \(\beta\) denotes the matrix of regression coefficients, which includes terms for linear, squared, and interaction effects. Specifically, \(\beta_{i}\), \(\beta_{ii}\), and \(\beta_{ij}\) are the coefficients corresponding to linear, squared, and interaction factor effects, respectively. \(x_{i}\) and \(x_{j}\) stand for the values of the design parameters, and \(\epsilon\) represents the error term, capturing any unexplained variability in the response. According to the RSM, selecting the central composite design requires determining the range and levels of printing parameters first. The data for the filling density, material flow rate, and layer thickness are input into the experimental design software Design-Expert12.0.