1. Introduction
Some authors have found that one of the most common facial lesions are mandibular fractures due to traffic accidents, violence, falls or sports [
1,
2,
3]. Currently, advances in the ability to treat mandibular fractures have been gained and, typically, titanium screws are used to hold a plate which fixes the fracture. Titanium is chosen because of its high stiffness and strength. However, this property makes the implant to support most of the mechanical load and cause a loss of bone mass in the vicinity of it, this effect is called stress shielding [
4,
5]. Other problems reported with titanium implants include palpability, pain and the need for subsequent removal. For these reasons, absorbable plate and screw systems are commonly used for maxillofacial surgery, as they prevent the stress shielding, reduce the risk of infection and avoid the need of secondary surgery for removing implants. However, there are also problems related with the use of absorbable systems such as incomplete screw insertion and screw breakage [
6,
7,
8,
9]. Therefore, the correct application of the insertion torque prevents the complication of screw breakage [
10]. While this insertion torque is applied, a tensile stress named pretension is generated in the screw body and, once the implant is in place, this is maintained. Thus, it is important to define the optimal pretension in plate and screw absorbable systems, which depends on the geometry of the screw, the contact angle, friction coefficient and mechanical properties of the parts [
11].
Most studies related to the optimization of screw design focused only on one or two parameters and do not take full advantage of finite element method (FEM) software. On the other hand, the total number of simulations to investigate all possible combinations of parameters can be very large, hence, the use of the design of experiment (DOE) layout is feasible to have an adequate number of them, in order to reduce experimental efforts [
12].
Moreover, artificial neural networks (ANNs) are now commonly used in many areas, such as engineering and medical science, as they are helpful models for, among other things, prediction and optimization. The ANNs function similar to a human brain, imitating the adjustments to the synaptic relationship between and among neurons in the learning process. An advantage of ANNs is the easy to use and accurate models obtained [
13].
Consequently, the aim of this paper is to find the optimal screw pretension (SP) for a mini plate and screw absorbable system for fracture fixation when desired value for maximum von Mises strain (MVMS) in absorbable screw is set and its geometric parameters are known. To achieve this, the methodology proposed in [
14] is applied. The inverse artificial neural network (ANNi) is a variant of the ANN that has been implemented in several processes to determine the optimal parameters successfully [
15]. The inputs of the data base used to train the ANN were constructed based on a DOE, whereas the output, MVMS, was calculated by the means of FEM.
3. Results
In this paper, we are varying geometric parameters and pretension of an absorbable mini screw used for bone fixation in an absorbable screw-plate system. The database, as observed in
Table 3, includes 32 configurations based in a DOE layout and was split into 2 parts. The 80% data correspond to learning stage and the other 20% were used for testing the network. The recorded output obtained from FEM analysis was the MVMS value. The strain concentrated regions were similar in all the simulations and MVMS for each configuration was found in the shaft, as shown in
Figure 5.
Inputs and outputs were normalized from 0.1 to 0.9 and, after that, the values of biases, weights and number of neurons in hidden layer were calculated in the training stage using Matlab tool for neural networks. The resulting configuration of the ANN was with 5 neurons in input layer, 3 neurons in hidden layer and 1 neuron in output layer.
Table 4 gives the obtained parameters for weights and biases that best fit the ANN model.
The MVMS can be calculated by Equation (7), which includes the coefficients obtained from the ANN.
where:
Figure 6 shows the comparison between MVMS from the numerical solution and predicted values from the ANN. Data were compared with linear regression model obtaining an R
2 of 0.9938. The obtained MSE was of 9.6925 × 10
−7.
The intercept-slope test was run with a confidence level of 99%. The results are presented in
Table 5. As is observed, the confidence level is reached due to a 0 and 1, found between intercept and slope limits. Based on these findings, it is concluded that the ANN proposed model is capable to accurately predict the MVMS.
Using (6), we calculate the relative importance for each input. All variables have a strong effect on the output. Furthermore, LS is the most influential parameter (37.85%), followed by ID (21.71%) and SP (13.95%), while TN (13.69%) and AT (12.8%) are the least. The effect of LS and ID on MVMS is due to the increasing or decreasing the contact area between the mini absorbable screw and bone. Thread number and angle of the thread shape have a relative low contribution to MVMS value.
Nevertheless, to ensure the success of mini absorbable screw plate system, the adequate SP must be applied on the mini screw to avoid breakage or migration due to inadequate installation.
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).
Equation (12) is used to involve
Ink = 4 as the input to be optimized.
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).
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
must be an absolute value.
Here:
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
was 1.0845 × 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 × 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 × 10
−3, as shown in
Figure 7.
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.
4. Conclusions
MVMS on absorbable mini screw for bone fixation was successfully predicted by ANN model consisting of 1 output, 3 neurons in the hidden layer, and 5 inputs. For training the network, a database was constructed based on the variation of geometric parameters and SP on the screw, according to the configuration of a full factorial design and MVMS as output, obtained by means of FEA.
Since strain is accepted as the mechanical stimuli for bone remodeling [
12] and knowing that limiting its maximum value within the elastic range of the material, we ensure contact between screw and bone threads avoiding problems of screw migration and breakage, the MVMS was recorded. Accordingly, the model proposed considers known values for
ID,
LS,
AT,
TN, and a desired outcome of MVMS for the screw. The relative importance of each input was calculated,
LS being the most influential parameter and, on the other hand, AT the one of less influence. However,
SP was chosen as the value to be found, when other inputs are well known, since it is of the utmost importance to avoid breakage or migration on absorbable mini screws due to inappropriate tightening.
This investigation was limited to avoid complexity by the assumption of constant bone properties, despite being considered an influence factor in absorbable mini screw behavior due to the variations that bone quality can have. In addition, the screw insertion process was not taken into account to reduce the factors considered and the material of screw remained constant.
Results indicates the ability of the ANN to predict the MVMS value and, also, the capability of ANNi to find an optimal value of one desired variable for a required MVMS.
For the mini screw of the absorbable implant system evaluated, as LS increases, the strain decreases. This is probably due to the increment of contact surface area between the mini screw and bone. The maximum stress occurs at the neck, near the location of MVMS. As expected, due to the founds in literature about screw breakage in
Section 1, the maximum stress occurs at the neck, near the location of MVMS.
Other authors have studied the effect of thread or head designs of bone fixation mini screws for both metallic and absorbable materials [
12,
29,
30,
31]. Nevertheless, most studies focus on optimizing one geometrical parameter, or evaluate its contribution to biomechanical response and do not consider the SP necessary for its correct installation, which would allow avoiding subsequent surgical intervention due to screw migration or breakage. This study performs a methodology to define the appropriate SP for a known material and geometrical design of a mini screw.
Furthermore, the developed methodology could be applied to find any desired input parameter. Actually, if a suitable multi-objective optimization algorithm is used, an optimal value for each geometrical variable could be obtained for a required MVMS and then its optimal SP could be found.