## 2 Materials and Methods
### Artificial Neural Network  
ANN is employed as a surrogated method to predict a specific response from a process. In ANN, artificial neurons are interconnected with an associated weight coefficient and are grouped in layers. The first layer is called an input layer connected to a one or more hidden layers, which use a transfer function to obtain a response, followed by an output layer that generally uses a linear transfer function. According to [21; 22], the hyperbolic transfer function is most suitable for hidden layers. Each layer contains several neurons. In the case of input and output layers, the quantity of neurons is equal to the number of input and output variables respectively, whereas in the case of the hidden layer, the number of neurons is defined in terms of the predictive accuracy level of the ANN model. To obtain the transfer function argument, the bias in the hidden layer must be considered, which is added to the weight coefficients. Under these conditions, the output of ANN is obtained by using Equation (1). \[Y=\sum_{s}\left\{Wo_{l,s}\left[\frac{2}{1+e^{-2\cdot\left(\sum_{k}W_{ls,k} \cdot In_{k}+b1_{s,s}\right)}}-1\right]\right\}+b2_{l,s} \tag{1}\]  
Here, \(Y\) is the predicted value, \(Wo\) is the weight coefficient of output layer, \(Wi\) is the weight coefficient associated to hidden layers, \(In\) is the input variable, \(b1\) is the bias associated to hidden layers, \(b2\) is the bias of the output layer, subscript \(l\) is the output \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Modulus of** & \multirow{2}{*}{**Poisson’s Ratio**} & \multirow{2}{*}{**Density (kg/m\({}^{3}\))**} \\  & **Elasticity (GPa)** & & \\ \hline Poly(lactic-co-glycolic) Acid (PLGA) & 2 & 0.3 & 1270 \\ Cortical bone & 14 & 0.3 & 1900 \\ Trabecular bone & 1.5 & 0.3 & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties assigned for finite element method (FEM) analysis. Figure 3: Material properties and boundary conditions for FEM analysis. number, subscript \(s\) is the hidden neuron number, and subscript \(k\) is the input number. In order to have input data that allow us to efficiently calculate the parameters of the ANN, each value was scaled using the min-max scale linear mapping, to ensure that they begin within the range of the transfer function [23]. Hence, normalized value \(A_{i}\) is calculated by using Equation (2):  
\[A_{i}=\frac{0.8\cdot(X_{i}-X_{min})}{X_{max}-X_{min}}+0.1 \tag{2}\]  
where \(X_{i}\) is the unnormalized value and \(X_{max}-X_{min}\) are the extreme values of the input data. Once the network structure is defined, database is used to adjust the weights and biases coefficients. This procedure, named training, is performed to minimize the error of the output of the function with respect to a known output value stored in the database. There are different methods for training neural networks in which the second order algorithms stand out due to the higher convergence speed and ability to handle complex problems with compact network sizes. Within these methods, the Levenberg-Marquardt is the most relevant for industrial uses and can be found in various applications. The method consists of generating an output, \(Z_{p}\), from calculating the dot product between the inputs and neuron's weight vectors, which acts on the activation function. Then, function \(f\) models the remaining neural network and produces the output, \(O_{p}\). The weight of the network is updated by using Equation (3)  
\[\Delta w=(H+\mu I)^{-1}g \tag{3}\]  
where \(g\) is the gradient vector, \(H\) the Hessian approximation matrix, \(I\) the identity matrix, and \(\mu\) the learning parameter [24; 25]. In this paper, we use, as a database for the ANN, the arrangement of DOE, taking the 80% data for training the network and correlating its output and FEM values as a criterion of adequacy of the model. The remaining data were used to test the ANN.