A collaborative approach of finite element method and machine learning algorithms for biomechanical analysis of implants used in tibial shaft fractures

Abstract

Background

Tibial fractures are among the most common complex orthopedic injuries. The mechanical strength and biomaterial properties of implants used in the treatment of such fractures directly affect the healing process. In this study, the mechanical effects of different implant designs and biomaterials on obliquely fractured tibia were analyzed. In addition, it was aimed to evaluate the data obtained from finite element analysis (FEA) with machine learning (ML) algorithms.

Methods

Seven implant models for tibial shaft fractures were analyzed using static structural simulations in Ansys Workbench. Implants and cortical screws were made of Ti–6Al–4 V alloy or 316 L stainless steel (SS), and axial loads of 600, 800, and 1000 N simulated single-leg stance. A dataset of 1008 points, including maximum stress and total displacement, was generated and used to train Multilayer Perceptron (MLP), Support Vector Machine (SVM) and Decision Tree (DT) models in WEKA.

Results

The mechanical behaviors of different implant and biomaterial combinations were compared, and the maximum stress value in implants with 316 L SS material properties was higher than the maximum stress value in Ti-6Al-4 V alloy implants. When the total displacement values ​​in the tibia fracture region were examined, 316 L SS implants gave better results. In the machine learning estimations, the SVM model outperformed the MLP and DT algorithms. For maximum stress prediction, SVM achieved an mean absolute error (MAE) of 0.24 and 0.41 for the training and test sets, respectively, while MLP and DT showed higher errors (3.27/4.02 and 10.99/14.90, respectively). Similarly, for total displacement prediction, SVM showed the lowest errors with MAE values of 0.0003 and 0.0015 for the training and test sets, whereas MLP and DT had higher MAE (0.0032/0.0040 and 0.0058/0.0072, respectively).

Conclusion

This study evaluated the effects of different implant designs and biomaterials on oblique tibial fractures and demonstrated that finite element results can be accurately predicted using machine learning models. The SVM algorithm showed superior performance, with prediction errors of approximately 0.24–0.41% MAE and 0.27–0.49% root mean square error (RMSE) for maximum stress, and 0.03–0.15% MAE and 0.03–0.23% RMSE for total displacement in the training and test sets, respectively. In comparison, MLP and DT exhibited higher errors. These findings highlight the potential of data-driven approaches in biomechanical analyses and their contribution to developing clinical decision support systems.

Introduction

Tibial shaft fractures are one of the most common orthopedic injuries and account for approximately 15% of all adult fractures [1, 2]. Tibial shaft fractures usually occur as a result of high-energy trauma such as falls from heights or motor vehicle accidents and, therefore, can lead to severe clinical consequences [3]. Such injuries can cause significant health problems not only in the acute phase but also in the long term. In particular, the negative consequences, such as prolonged hospitalization, the risk of permanent loss of function, and high healthcare costs, make these fractures extremely concerning from a clinical perspective.

There are various surgical and nonsurgical techniques used to fix a fractured tibial shaft (tibial diaphysis). The appropriate technique is selected depending on the type of fracture, its location, whether it is open or closed, the patient’s age, general health status, and the surgeon’s experience. The main techniques used to fix a fractured tibial shaft are; intramedullary nailing, plate and screw fixation, external fixation, and conservative treatment (immobilization with plaster or functional brace) in appropriate cases [4–7]. In this study, the plate and screw fixation method was used along the bone fracture area using Ti–6Al–4 V titanium alloy and 316 L SS implants, which have been extensively studied recently. In plate fixation, there are many factors that can affect the mechanical stability of plate fixation, such as plate placement, plate length, number of screws, plate and screw material. In addition to experimental studies in the literature, there are also FEA studies in which these factors are analyzed comprehensively [8, 9].

Cao et al. [2] investigated the effects of plate length and hole number on stress and displacement by simulating the adult knee under a loading condition of 2500 N with 60% distribution to the medial compartment in a tibial shaft fracture model using FEA. Sathone and Potdar [10] optimized implant design and material selection in their studies to improve the performance of implants used in tibial fractures. Kumar et al. [11] analyzed the effects of fracture intervals ranging from 1 mm to 10 mm on stress distribution in a magnesium alloy implant plate under 750 N loading conditions.

Many studies have proven experimental and numerical techniques to be helpful in the analysis of biomechanical systems. However, the main disadvantage of these techniques is that they are time-consuming and lead to increased computational costs. There are no comprehensive experimental biomechanical databases that specifically examine the strength and fixation of implants used in tibial fractures. Therefore, new approaches that combine numerical simulations that provide a wide database with machine learning techniques are needed. Thanks to these approaches, complex biomechanical problems can be solved in a much shorter time and with high accuracy.

ML algorithm-based analyses have begun to be used in many different disciplines, such as finance [12], mathematics [13], physics [14], energy [15], engineering [16] applications, and in recent years in biomechanics [17] research.

In biomechanical analyses, machine learning algorithms are used to estimate the amount of stress and deformation in the fracture area in implants, both shortening the analysis time and reducing the need for experimental studies. Machine learning methods help determine the most appropriate treatment or design approaches by creating simulations according to different material properties and loading scenarios. Thus, the dependency on physical tests is reduced, and the design or clinical decision processes are accelerated. Compared to traditional trial-and-error methods, the integration of these technologies reduces costs and makes the research and development process more innovative and efficient.

Titanium alloys offer high strength-to-weight ratio, excellent corrosion resistance, and biocompatibility, but are expensive and have higher stiffness than bone, potentially causing stress shielding. In contrast, 316 L SS is cost-effective, has good corrosion resistance, and a stiffness closer to bone, though it is heavier and less biocompatible than titanium. By comparing these two widely used biomaterials under realistic loading conditions, this study provides insights into optimal implant selection, implant design, and clinical decision-making for tibial fracture management.

In this study, an oblique fracture line with a 45° angle was defined in the middle of the tibial shaft, and analyses were performed using seven different implant models. Two different biomaterials, Ti–6Al–4 V and 316 L SS, were defined for implants and cortical screws. In order to simulate the axial compressive load on the knee of an adult during a single-leg stance to carry the whole body weight, three different human weights of 60 kg, 80 kg, and 100 kg were taken as references, and axial compressive forces of 600 N, 800 N and 1000 N were applied to the top point of the distal tibia to determine the stresses occurring in the implants and the total displacements occurring in the fracture region of the tibia. A data set was created with the results obtained from the numerical analyses, and MLP, SVM, and DT algorithms were trained to make maximum stress and total displacement estimates. 1008 data points were created for each algorithm, and 70% of the data were used for training and 30% for testing. In this article, the results of the three ML methods for maximum stress and total displacement estimates, which are the output parameters, are presented comparatively. The novelty of this work lies not only in the integration of high-fidelity FEA simulations with machine learning predictions but also in providing a systematic, data-driven framework that enables rapid, highly accurate, and cost-effective assessment of implant performance. This combined approach addresses the limitations of conventional experimental and computational studies, offering a powerful tool for optimizing implant design and material selection, and advancing evidence-based decision-making in orthopedic biomechanics.

Materials and methods

Finite element analysis

CT scan images were obtained from a healthy 30-year-old male, 180 cm in height and weighing 80 kg. No anatomical abnormalities were observed in his left tibia. CT data were acquired at 1-mm intervals, extending from 10 cm above the knee to 5 cm below the ankle. The DICOM-format CT images were imported into Mimics 15 (Materialize Company, Leuven, Belgium) software to reconstruct the geometric surface of the tibia. The resulting 3D model was saved in STL format and subsequently imported into Geomagic Studio 13.0 (3D system Inc., Rock Hill, SC, USA) for surface smoothing and removal of protruding triangles, before being exported to Hypermesh 12.0 (Altair Engineering, Inc., USA).

The 3D modeling of implants, and cortical screws was done with Ansys/SpaceClaim module, and static structural analysis were performed in an Ansys workbench (Software Corporation, Canonsburg, USA). 12th Gen Intel(R) Core(TM) i7-12700 H, 2300 Mhz, 14 Cores, 20 Logical Processors 2.30 GHz, 16.00 GB, 64-bit computer was used in modeling and analysis. All materials were assumed to be isotropic and linear elastic in the analysis. Two different biomaterials were defined for implants and cortical screws as Ti–6Al–4 V and 316 L SS. Due to their superior mechanical properties, outstanding corrosion resistance, surface smoothness, and proven biocompatibility, Ti–6Al–4 V alloy and 316 L SS are extensively utilized in biomedical applications. Accordingly, these materials were chosen as the primary focus of this study. The mechanical properties of all materials are shown in Table 1.

Table 1.

Mechanical properties of materials

Materials	Young’s modulus (GPa)	Poisson ratio	Yield strength (MPa)
Ti–6Al–4 V [18]	114	0.34	830
316 L SS [18, 19]	193	0.28	332
Tibia Cortical Bone [20, ]	20	0.3	120

A mesh convergence study was performed by comparing maximum von Mises stress in the implant and total displacement across the fracture region for the 4 mm, 2 mm, and 1 mm meshes. As the mesh was refined, the results stabilized; the fine (1 mm) mesh provided the most consistent predictions and was therefore used for all reported results. Mesh quality was monitored throughout the model; distorted elements were removed and remeshed to keep element quality within commonly accepted limits (low skewness and reasonable aspect ratios), ensuring robust contact behavior and strain resolution. The effect of mesh density on both maximum stress and total displacement is presented in Figs. 10, 11, 12, 13, 14 and 15, which illustrate decreasing variation with refinement and the improved agreement of the fine mesh with expected yield criteria. Mesh size selection balanced numerical accuracy with computational efficiency; finer meshes significantly increased runtime without materially changing the response beyond the converged values.

Fig. 10.

Maximum stress values ​​of the FEA model with coarse mesh

Fig. 11.

Maximum stress values ​​of the FEA model with medium mesh

Fig. 12.

Maximum stress values ​​of the FEA model with fine mesh

Fig. 13.

Total displacement values ​​of the FEA model with coarse mesh

Fig. 14.

Total displacement values ​​of the FEA model with medium mesh

Fig. 15.

Total displacement values ​​of the FEA model with fine mesh

All degrees of freedom at the distal end of the tibia were fully constrained to eliminate any rigid body movements throughout the analysis. In order to simulate the axial compressive load on the knee during a single-leg stance, representing the full body weight of an adult, three different human weights of 60 kg, 80 kg, and 100 kg were considered, corresponding to axial forces of 600 N, 800 N, and 1000 N applied to the tibial plateau in full extension. In the literature, there are also studies that applied different loading scenarios under various physiological conditions and for different body weights [22, 23]. Moreover, the application of a unidirectional axial force along the length of the tibia has been widely used to represent physiological loading in finite element analyses [10]. Although linear material behavior is assumed in the present model, these loading values provide a reasonable approximation of the stresses experienced under normal body weight conditions. It was assumed that direct contact with a frictional coefficient of 0.3 existed between the implants and bones [24–26], while a friction coefficient of 0.003 was assigned to the contact pair of the fracture surfaces [2, 27]. An oblique fracture line with an angle of 45° was created in the middle of the tibial shaft to simulate transverse fractures of the tibia. The fracture structure, boundary conditions, loading conditions, and mesh structures are given in Fig. 1.

Fig. 1.

Views of tibial shaft fracture, boundary conditions, and mesh structures in FEA

In this study, analyses were performed with three different length implant models (130 mm, 170 mm, and 210 mm) using a total of 4, 6, 8, and 10 screws equally on both sides of the fracture. All models used in the analysis are shown in Fig. 2. In naming the models, the number to the right of H in the coding represents the number of holes, and the number of screws to the right of S. For example, H9S8 represents an implant consisting of 9 holes and 8 screws. The plates had a width of 15 mm and a thickness of 3 mm, with adjacent holes spaced 20 mm apart. Cortical screws with a diameter of 4.5 mm and a core diameter of 3 mm were used, each with a length of 40 mm.

Fig. 2.

The implant types and cortical screw

Machine learning

Machine learning algorithms refer to a computational process that is used to perform desired tasks through data without having to be pre-programmed to produce a specific result. These algorithms constantly improve themselves through iterative processes (i.e., experience) and thus become more efficient over time. During this process of iteration and improvement, algorithms adapt themselves to the task and begin to produce more accurate results. This adaptation process is called training, and during this process, the algorithm is presented with various input data samples along with the desired results. Machine learning algorithms optimize themselves with the data they receive during training. Thanks to this optimization, they not only produce accurate results with the data used in training but also have the ability to generalize to new, previously unseen data. The training process allows the algorithm to continuously improve itself as it works on the data, allowing it to process a more extensive dataset successfully. This is the “learning” part of machine learning. However, the training process of machine learning is not limited to a specific adaptation at the beginning. Just like humans, a good algorithm can constantly process new data and learn from its previous mistakes to perform better. This process is called “lifelong learning”. This feature allows machine learning algorithms to learn more complex and unknown data patterns, not just limited to the initial data. Thus, machine learning systems are constantly strengthened by data fed from their environment and become more efficient. The power of machine learning algorithms lies in these “learning” and “adaptation” processes that make them systems that can adapt to their environment, constantly improve themselves, and produce the best results according to changing conditions [28].

Machine learning is the process of developing algorithms that enable computers to learn by imitating human intelligence. Artificial intelligence is influenced by many disciplines, such as statistics, computer science, information theory, psychology, and philosophy [29–31]. The historical beginning of machine learning dates back to the 17th century. During this period, Pascal and Leibniz [32] developed the first mechanical machines that could imitate addition and subtraction operations. The term “machine learning” in its current meaning was introduced by Arthur Samuel; Samuel showed that computers could learn to play checkers [33]. An important milestone was the development of the multilayer perceptron by Werbos in 1975 [34]. Later, in 1986, Quinlan [35] developed decision trees, and in 1995, Cortes and Vapnik [36] developed support vector machines.

ML performance metrics

Three different performance metrics were included in this study to evaluate the performance of machine learning models in order to make more accurate maximum stress and total displacement estimations. These metrics were determined as linear correlation coefficient (R), MAE and RMSE.

R shows how well the model’s estimation is compatible with the real data, and the accuracy of the model’s estimations increases as the R value approaches 1. In other words, a high R indicates that the model gives more reliable results. MAE is another important metric that determines how close the predictions are to the actual values. A low MAE indicates that the model’s prediction errors are smaller, meaning that the model is more successful. RMSE is calculated by taking the square root of the mean square of the differences between the model’s predicted values ​​and the actual values. A low RMSE indicates that the model’s margin of error is smaller and, therefore, the predictions are more accurate.

When these three performance metrics are evaluated together, it is possible to analyze the prediction capabilities of machine learning models more objectively and comprehensively. Thus, the best-performing model can be selected, and the accuracy of biomechanical analyses can be increased.

The formulations of R, MAE, and RMSE statistical metrics for maximum stress and total displacement prediction results are as follows:

1

2

3

In these equations, represents the FEA data, is the estimated value, is the mean value of the FEA data, and n is the number of samples in the dataset.

MLP

The architecture of a MLP comprises an input layer, one or more hidden layers, and an output layer, each serving a distinct function in the overall operation of the network. The input layer receives and represents the initial data, while the hidden layers process this information, converting it into increasingly abstract representations. Finally, the output layer utilizes the transformed data to produce the network’s final prediction or decision. Thanks to these structures, MLP becomes a powerful and flexible model that can provide solutions to a wide variety of problems by switching between linear and non-linear relationships. With these features, MLP is widely used in different areas, especially classification and regression. Figure 3 shows the layers and structures in the MLP algorithm for maximum stress and total displacement estimations.

Fig. 3.

MLP network structure for maximum stress and total displacement predictions

The optimization process was performed by changing the network structures for maximum stress and total displacement estimations. To improve model accuracy and achieve more robust results, a detailed optimization process was performed. Tables 2 and 3 show the sixteen MLP architectures generated throughout this process, all of which were developed using WEKA. The depth and complexity of the model are regulated by changing the network structure. During the optimization process, the performance of each model was evaluated on the test data set. This evaluation was made based on important error metrics such as R, MAE, and RMSE. These metrics measure the differences between the values ​​predicted by the model and the actual values ​​and reveal the accuracy of the model. As a result of the tests, it was observed that Model 9 for maximum stress estimation and Model 15 for total displacement estimation provided lower error rates in terms of MAE and RMSE values ​​compared to other models, and these two models were determined as optimum models for the MLP approach. The superior performances achieved by Model 9 and Model 15 once again demonstrate the potential of the MLP method in biomechanical analyses.

Table 2.

MLP models for maximum stress

Model No	Network Structure	Training Set R MAE RMSE			Testing Set R MAE RMSE
1	7-2-1-1	0.9948	4.7896	5.9643	0.9950	5.0406	6.4151
2	7-2-2-1-1	0.9943	4.9539	6.2020	0.9939	5.4077	7.1568
3	7-3-1-1	0.9974	3.4182	4.2084	0.9974	3.4509	4.4544
4	7-3-3-1-1	0.9972	3.4213	4.3093	0.9966	3.8980	5.2435
5	7-4-1-1	0.9974	3.8227	4.5953	0.9961	4.5317	6.1161
6	7-4-4-1-1	0.9972	3.3972	4.3470	0.9960	4.3997	5.6763
7	7-5-1-1	0.9974	3.3900	4.2635	0.9966	4.1108	5.0526
8	7-5-5-1-1	0.9975	3.1679	4.1206	0.9960	4.3716	5.5911
9	7-6-1-1	0.9979	3.2668	4.0198	0.9979	3.3232	4.1875
10	7-6-6-1-1	0.9975	3.2689	4.0759	0.9957	4.6679	5.9176
11	7-7-1-1	0.9986	2.6956	3.2785	0.9973	3.6381	4.9342
12	7-7-7-1-1	0.9975	3.3451	4.1591	0.9954	4.8125	6.2512
13	7-8-1-1	0.9982	2.7669	3.5043	0.9969	3.9081	4.8616
14	7-8-8-1-1	0.9975	3.2374	4.1888	0.9967	4.0435	5.1581
15	7-9-1-1	0.9976	3.4062	4.0931	0.9974	3.6012	4.5441
16	7-9-9-1-1	0.9977	3.4022	4.2607	0.9958	4.7508	6.1153

Table 3.

MLP models for total displacement

Model No	Network Structure	Training Set R MAE RMSE			Testing Set R MAE RMSE
1	7-2-1-1	0.9950	0.01	0.0117	0.9951	0.0096	0.0116
2	7-2-2-1-1	0.9941	0.0078	0.0095	0.9956	0.0073	0.0092
3	7-3-1-1	0.9975	0.0038	0.005	0.9963	0.0045	0.0058
4	7-3-3-1-1	0.9974	0.0043	0.0052	0.9965	0.0047	0.0056
5	7-4-1-1	0.9979	0.0034	0.0046	0.9965	0.0044	0.0059
6	7-4-4-1-1	0.9976	0.0041	0.0050	0.9962	0.0046	0.0059
7	7-5-1-1	0.9987	0.0030	0.0038	0.9975	0.0038	0.0048
8	7-5-5-1-1	0.9977	0.0043	0.0051	0.9959	0.0050	0.0062
9	7-6-1-1	0.9982	0.0034	0.0044	0.9963	0.0042	0.0058
10	7-6-6-1-1	0.9984	0.0038	0.0045	0.9967	0.0043	0.0057
11	7-7-1-1	0.9983	0.0032	0.0041	0.9966	0.0042	0.0055
12	7-7-7-1-1	0.9981	0.0039	0.0048	0.9956	0.0050	0.0064
13	7-8-1-1	0.9986	0.0031	0.0038	0.9973	0.0037	0.0049
14	7-8-8-1-1	0.9981	0.0037	0.0045	0.9958	0.0049	0.0062
15	7-9-1-1	0.9991	0.0032	0.0040	0.9982	0.0037	0.0045
16	7-9-9-1-1	0.9979	0.0043	0.0052	0.9961	0.0046	0.0062

SVM

SVM is a powerful classification method based on statistical learning theory and is known for its high accuracy rates. SVM is extremely effective, especially in the classification of non-linear relationships, and is one of the most preferred methods among supervised learning techniques. Supervised learning is a type of learning in which the model is trained with labeled data and produces accurate results by making inferences from this data. In this context, SVM is often used as a powerful tool to deal with complex data sets and classification problems. One of the most distinctive features of SVM is that it can effectively deal with non-linear classification problems. In the supervised learning paradigm, SVM generally tries to find the best-dividing line between two classes. Unlike classical neural network methods, SVM focuses on reaching only a single optimum solution. This means that the model offers a more reliable and more understandable solution. In addition, its success in discovering and classifying non-linear relationships makes SVM a flexible and powerful model. Unlike neural networks, dimensionality problems do not usually arise during the training process of SVM. This feature provides a significant advantage when working with high-dimensional data. SVM, which does not encounter the difficulties encountered by neural networks in high-dimensional data, provides reliable results, especially when working with large data sets. This increases the generalization ability of the model and provides better adaptation to complex classification problems in the real world. In the diagram explaining the working mechanism of SVM in Fig. 4, two class examples are shown with red and purple circles. In this diagram, the boundary between both classes visually clarifies how SVM works. Support vectors are the points that intersect with the boundary line in this diagram, and the classification accuracy largely depends on these support vectors. Support vectors play a critical role in creating the model because they are the most important elements that determine the classification boundary. Margin is the distance between the support vectors on both sides of the classification boundary, and this distance is directly related to the accuracy of the model. The larger the margin, the higher the generalization power of the model. Therefore, the primary goal of SVM is to maximize this margin using support vectors. The accuracy of the classification boundary of SVM is measured by the distance between two parallel hyperplanes. This distance is mathematically defined as 2/||ω|| Here, ||ω|| represents the norm of the ω vector and ω represents the normal vector perpendicular to the hyperplane. This vector is the principal component that determines the classification boundary. Equation 1 gives the hyperplane equation. In the equation, x represents a sample point on the hyperplane, and b represents the bias term [37, 38]. The mathematical foundations of SVM provide a versatile solution by establishing a strong bridge between linear and non-linear classification problems. The success of SVM lies in its ability to work effectively not only with simple linear boundaries but also on complex data sets and non-linear relationships. Therefore, SVM continues to be a widely used and reliable method in the world of machine learning.

4

Fig. 4.

Working principle of SVM

Tables 4 and 5 show three different models developed using different kernel functions for maximum stress and total displacement estimations, respectively. The performance of each model was evaluated using R, MAE, and RMSE metrics, which measure the differences between the predicted values ​​and the actual values. Considering these metrics, it was determined that Model 3 made predictions with lower error values ​​than the other models and, therefore, was the model with the best performance. This result led to the selection of Model 3 as the optimum model because the prediction accuracy was maximized with a low error rate.

Table 4.

SVM models for maximum stress

Model No	Kernel Function	Training Set R MAE RMSE			Testing Set R MAE RMSE
1	Polynomial Kernel	0.9629	9.4699	16.9213	0.9641	11.5304	18.8240
2	Normalized Polynomial Kernel	0.9576	11.3346	17.3592	0.9702	13.0215	16.7682
3	Pearson VII Kernel	1	0.24180	0.27250	0.9999	0.41305	0.48607

Table 5.

SVM models for total displacement

Model No	Kernel Function	Training Set R MAE RMSE			Testing Set R MAE RMSE
1	Polynomial Kernel	0.9919	0.0070	0.0090	0.9886	0.0077	0.0104
2	Normalized Polynomial Kernel	0.9722	0.0117	0.0171	0.9877	0.0094	0.0116
3	Pearson VII Kernel	1	0.0003	0.0003	0.9994	0.0015	0.0023

The general form of the Pearson VII function, which is frequently preferred in curve fitting operations, is shown in Eq. 5.

5

Here, H represents the height of the peak at the center of the function x₀, while x represents the independent variable. Other parameters that determine the shape of the function, σ and ω, control the half-width of the peak and the tail factor, respectively. These parameters are adjusted to optimize the fit of the function to the data. However, for a function to be accepted as a valid kernel function, the corresponding kernel matrix must be symmetric and positive semi-definite. In order to demonstrate that the Pearson VII function satisfies these conditions, this function is discussed in detail within the scope of Eq. 6 in the study conducted by Üstün et al. [39].

6

In this context, the fundamental change is the removal of the peak offset term x₀ of the Pearson VII function and the fixation of the peak height H as 1. This simplification is achieved without compromising the generality of the function. In this way, when the Pearson VII function is used as a kernel function, a symmetric matrix whose diagonal elements are 1 and which takes values ​​between 0 and 1 for all x_i and x_j pairs is obtained. The results show that the Pearson VII function maintains its robustness and has an equivalent or superior matching capacity compared to classical kernel functions. This feature provides better generalization performance, especially in machine learning algorithms such as SVM. Therefore, the Pearson VII function has the potential to increase model accuracy by providing a high level of fit to the data set in kernel-based learning algorithms [40].

DT

Decision trees are a widely preferred method in data analysis and classification problems. This preference stems from the simple and understandable structure of the decision tree. The visual representation of the model allows decision-making processes to be transparent and accessible. In this way, evaluations of the accuracy, errors, and results of the model can be made more easily and effectively. Users can easily follow the logic of the decisions made in each branch of the tree, which makes the decision tree a valuable tool, especially for data analysts and business decision-makers. Additionally, decision trees facilitate the visualization of complex data in a more comprehensible and structured form. Data follows a series of conditions and results by branching according to the tree structure, which makes it easier to understand how the model works. Visualization provides a great advantage, especially for users who want to see the relationships and distinctions in the data clearly. Thus, the model’s decisions become more transparent, and the evaluation of the model’s performance becomes more effective. Decision trees facilitate decision-making processes and also provide a powerful solution for important classification and regression problems. With the tests performed on each feature of the data, it becomes clear which results the model will produce under which conditions. This allows users to examine the data more deeply, especially in analyses performed on large data sets. Decision trees can model even non-linear relationships well, and this feature makes them a more flexible tool. The effective use of the decision tree makes data analysis and classification problems more accessible and understandable thanks to the simplicity of the model and the advantages of visualization. This allows users to make decision processes faster and more accurately and also plays an important role in managing complex data. In Tables 6 and 7, three different models were developed using different decision tree functions for maximum stress and total displacement estimates, respectively. The performance of each model was evaluated using R, MAE, and RMSE metrics, which measure the differences between the estimated ​​and actual values. Considering these measurements, it was determined that Model 3 made estimates with lower error values ​​compared to the other models and, thus, was the model that exhibited the best performance.

Table 6.

DT models for maximum stress

Model No	Decision Tree Model	Training Set R MAE RMSE			Testing Set R MAE RMSE
1	Decision Stump	0.6643	34.0116	43.2462	0.6152	38.3400	48.7267
2	REP Tree	0.9532	12.9661	17.4929	0.9251	18.1732	23.8364
3	M5P	0.9663	10.9960	14.8955	0.9701	11.1617	15.4503

Table 7.

DT models for total displacement

Model No	Decision Tree Model	Training Set R MAE RMSE			Testing Set R MAE RMSE
1	Decision Stump	0.8154	0.0335	0.0406	0.7242	0.0408	0.0469
2	REP Tree	0.9868	0.0077	0.0114	0.9805	0.0096	0.0132
3	M5P	0.9948	0.0058	0.0072	0.9942	0.0054	0.0073

The M5P algorithm [41], which is an extended version of the M5 algorithm [42], consists of four basic steps. In the first step, the input domain is divided into various subdomains to create a decision tree. Each subdomain is structured to minimize internal variability; thus, more accurate and meaningful decisions are made. This division process is based on a specific division criterion that aims to minimize the variance in each subdomain. In the process of subdomain generation, the variability of samples arriving at each node is quantified using standard deviation, which is subsequently employed to determine information gain [43]. During the construction of the tree, the expected error reduction at each node is evaluated, and the most appropriate modeling performance is targeted accordingly. In this evaluation process, the standard deviation reduction (SDR) factor plays an important role. Through the assessment of data variability at the nodes, SDR determines which division yields more meaningful and accurate results. Its primary purpose is to increase the overall accuracy of the model by minimizing the error rate at each node. This approach allows systematic improvement of the model performance at each level of the decision tree. The working principle of the M5P algorithm is visualized in the flow diagram presented in Fig. 5.

Fig. 5.

Flowchart of M5P algorithm [44]

The M5P algorithm seeks to minimize variability within each region while enhancing the accuracy of the tree by partitioning the input space into meaningful sub-regions. The incorporation of the SDR factor is intended to improve the model’s decision-making process at each stage. The formula for calculating SDR is presented in Eq. 7.

7

At this stage, the symbol S represents the set of data records reaching the relevant node; Si represents the subsets formed as a result of the division made on the basis of a particular attribute. Additionally, the term ‘sd’ represents the standard deviation of the data points within these sets [42]. After constructing the decision tree, linear regression models are developed for each subspace in the second step. These models aim to establish the most suitable relationship between the variables based on the data within their respective subspaces.

However, at this stage, a pruning process is applied to prevent over-learning. Over-learning causes the model to exhibit poor performance on new and unseen data due to over-fitting to the training data. This situation occurs especially when the SDR value of the linear model at the root of the subtree is low, but the expected error in other parts of the subtree is higher. Although the pruning process limits over-learning by reducing the complexity of the model, it can sometimes cause sudden transitions or incompatibilities between adjacent linear models. This can lead to unstable predictions.

In order to prevent such sharp transitions and prediction imbalances, a correction process is performed in the last step. This process involves combining linear models from the leaf to the root to produce the final model output of the leaf. In this process, the predicted value is filtered as it moves upwards to avoid sharp transitions. This makes the model output more consistent and balanced. As a result, over-learning is reduced, and the generalization performance of the model is improved. The output of this smoothing process is combined with the prediction from the linear regression model at the relevant node, as shown in Eq. 8.

8

In this context, E’ represents the predicted value passed to the next higher node, reflecting the outcome of predictions made at the higher-level node. Conversely, e denotes the predicted value passed down to the current node, representing the results of the predictions at the child node. A stands for the predicted value generated by the model at the current node, which is the value that best reflects the training data at that specific node. Additionally, n signifies the number of training examples reaching the child node, influencing the dataset size at that node and the accuracy of predictions based on this data. Finally, k is a constant parameter, typically used to enhance model accuracy or to balance specific calculations [43]. These parameters are essential in optimizing model performance and ensuring accurate predictions at each node of the decision tree, minimizing issues such as overfitting. Proper calculation of each node’s prediction directly impacts the overall performance and generalization capability of the tree.

Figure 6 shows the DT structure created for maximum stress and total displacement estimations in the M5P algorithm. In this structure, force serves as the root of the tree, with subsequent branches created according to the young modulus, force, and screw nodes. The algorithm iterates through these nodes, continuing until it reaches the terminal leaves (LM 1–13) at the base of the decision tree.

Fig. 6.

Graphical view of the M5P tree

Results

Evaluation of FEA results

This study was performed using Ansys/Workbench, a powerful software for engineering simulations and analyses. By applying axial compressive forces of 600 N, 800 N, and 1000 N to the apex of the distal tibia in full extension, the stresses occurring in the implants and the total displacements occurring in the fracture region of the tibia were examined. For example, Figs. 7 and 8 show the von Mises stress distributions in 5 frames for Ti-6Al-4 V alloy and 316 L SS, respectively, with a coarse mesh structure (4 mm) under an axial load of 1000 N. In the FEM analyses, stress concentration is observed around the hole in the center of the plate, namely in the tibial fracture region. The stress distributions occurred as compressive stresses in the medial plane and tensile stresses in the lateral plane. Since the total displacement value was small in the analyses, the total displacement image is presented in Fig. 9 by enlarging it 41 times from an accurate scale.

Fig. 7.

von Mises stress of the FEA (Ti-6Al-4 V)

Fig. 8.

von Mises stress of the FEA (316 L SS)

Fig. 9.

Total displacement of the FEA (with a coarse mesh structure under an axial load of 1000 N)

In Figs. 10, 11 and 12, the maximum stress values obtained from the FEA are presented separately for coarse, medium, and fine mesh structures. The graphs illustrate the maximum stress values of seven different implant configurations manufactured from Ti-6Al-4 V alloy and 316 L stainless steel under axial loads of 600 N, 800 N, and 1000 N. As the mesh density increased, a reduction in the maximum stress values was observed, indicating an improvement in model accuracy. Across all loading conditions, implants manufactured from 316 L stainless steel consistently exhibited higher maximum stress values compared to those manufactured from Ti-6Al-4 V alloy.

When comparing implant configurations, the highest stress concentration was recorded in model H7S4, whereas the lowest stress was obtained in model H11S10. Among the implants with identical screw numbers (H7S6, H9S6, and H11S6), stress values increased with increasing plate length. A similar trend was observed in the comparison of H9S8 and H11S8, where longer plates resulted in higher stress levels.

The yield strength values for Ti-6Al-4 V and 316 L stainless steel are 830 MPa and 332 MPa, respectively. For Ti-6Al-4 V implants, the maximum stress values in all models remained well below the yield strength across all mesh types and loading conditions, indicating that no material failure would occur. For 316 L stainless steel implants, stress levels remained below the yield strength under 600 N and 800 N loads in all mesh structures. However, under the 1000 N load, stress values exceeded the yield strength in coarse and medium mesh analyses, suggesting insufficient accuracy in these mesh densities. In contrast, the fine mesh analysis produced stress values below the yield strength for both materials in all implant models, confirming that fine mesh provides the most reliable representation of the real structure.

Figures 13, 14 and 15 present the total displacement values obtained from FEA for coarse, medium, and fine mesh structures, respectively. The graphs show the total displacement of seven implant configurations made from Ti-6Al-4 V alloy and 316 L stainless steel under axial loads of 600 N, 800 N, and 1000 N. As shown in the figures, implants made of 316 L stainless steel consistently exhibited lower total displacement values compared to Ti-6Al-4 V alloy implants across all mesh types and loading conditions. Among the implant configurations, total displacement decreased as the implant length and the number of screws increased. This trend was observed in both Ti-6Al-4 V and 316 L stainless steel implants.

For example, when comparing implants H7S4 and H7S6, which have the same length but a different number of screws, the total displacement in H7S6 was lower, indicating that increasing the number of screws enhances the rigidity of the fixation. Across all models and mesh densities, fine mesh analyses yielded the most accurate representation of the displacement behavior, showing systematically lower deformation values than coarse or medium meshes. Overall, the total displacement results indicate that longer implants with more screws provide higher structural stiffness, and 316 L stainless steel implants demonstrate superior performance in minimizing deformation under axial loading.

Evaluation of ML results

The open source WEKA software written in Java programming language developed for machine learning was used to estimate the stress values ​​occurring in implants and the total displacement values ​​occurring in tibia fractures by creating a data set from the values ​​obtained from finite element analyses. Estimates were made with MLP, SVM, and DT algorithms from machine learning techniques. The deep learning structure of MLP, flexible decision rules of DT, and the ability of SVM to distinguish in high-dimensional spaces play a critical role in successfully modeling the hidden patterns and non-linear relationships in the data set.

The linear black line in Figs. 16, 17 and 18 represents the actual maximum stress values, while the green, red, and purple circles represent the maximum stress values ​​estimated by the MLP, SVM, and DT methods, respectively. In these figures, the closest positioning of the circles to the black linear lines shows the estimation performance of the ML methods.

Fig. 16.

Maximum stress prediction results with MLP algorithm

Fig. 17.

Maximum stress prediction results with SVM algorithm

Fig. 18.

Maximum stress prediction results with DT algorithm

Figure 16 illustrates the results of maximum stress estimation for both the training and test sets using the MLP method. The MLP method showed an R value of 0.9979, an MAE of 3.2668, and an RMSE of 4.0198 for the training set. Similarly, for the test set, the method yielded an R value of 0.9979, an MAE of 3.3232, and an RMSE of 4.1875. Figure 17 presents the maximum stress estimation results using the SVM technique. The training set results from SVM showed outstanding accuracy, with an R value of 1, an MAE of 0.24180, and an RMSE of 0.27250, as demonstrated by the perfect alignment of the red circles with the linear black line. The test set results for SVM were also excellent, with an R value of 0.9999, an MAE of 0.41305, and an RMSE of 0.48607. Finally, Fig. 18 shows the maximum stress estimation outcomes for the training and test sets using the DT algorithm. When the training set results are analyzed, the DT algorithm is estimated to have 0.9663 R, 10.990 MAE, and 14.8955 RMSE. When the test set error rates are analyzed, the DT method leads to 0.9701 R, 11.1617 MAE, and 15.4503 RMSE.

Figure 19 shows the maximum stress estimation errors obtained by the MLP, SVM, and DT methods. The error expression given on the Y-axis gives the difference between the estimated and actual maximum stress values. The estimation errors caused by the MLP, SVM, and DT techniques are shown in green, red, and purple lines in Fig. 19 for both training and test sets, respectively. If the deviation from the zero point shown on the Y-axis is significant, the ML technique performs a poor estimation. On the other hand, a low deviation from the zero point indicates superior estimation performance. The error fluctuations caused by DT appear to be larger than the other ML methods used, and DT shows worse estimation performance compared to MLP and SVM.

Fig. 19.

Maximum stress prediction errors with MLP, SVM, and DT algorithms

Figure 20 shows the fit of the models created with the actual maximum stress values ​​for 20 randomly selected samples from the test data set. It can be determined that the DT algorithm has difficulty in creating a good prediction model for some data points. It can be said that the MLP algorithm creates a slightly better prediction model compared to the DT method. The actual maximum stress shown with the black line closely matches the SVM model defined with the red line, indicating the superior prediction performance of the SVM model. The linear black lines in Figs. 21,22 and 23 represent the total displacement values, while the green, red, and purple circles represent the total displacement values ​​estimated by the MLP, SVM, and DT methods, respectively. In these figures, the closest positioning of the circles to the black linear lines indicates the prediction performance of the ML methods

Fig. 20.

Comparison of maximum stress with predictions of generated models

Fig. 22.

Total displacement prediction results with SVM algorithm

Fig. 23.

Total displacement prediction results with DT algorithm

Fig. 21.

Total displacement prediction results with MLP algorithm

Figure 21 presents the total displacement estimation results for both the training and test sets using the MLP method. For the training set, the MLP method achieved an R value of 0.9991, an MAE of 0.0032, and an RMSE of 0.0040. For the test set, the MLP method resulted in an R value of 0.9982, an MAE of 0.0037, and an RMSE of 0.0045. Figure 22 shows the total displacement estimation results for the training and test sets using the SVM technique. The SVM model demonstrated exceptionally high accuracy and low error rates for the training set, with an R value of 1, an MAE of 0.0003, and an RMSE of 0.0003, as evidenced by the perfect alignment of the red circles with the linear black line. For the test set, the SVM method achieved an R value of 0.9994, an MAE of 0.0015, and an RMSE of 0.0023. Figure 23 displays the total displacement estimation results for the training and test sets using the DT algorithm When the training set results are analyzed, the DT algorithm is estimated to be 0.9948 R, 0.0058 MAE, and 0.0072 RMSE. When the test set error rates are analyzed, the DT method leads to 0.9942 R, 0.0054 MAE, and 0.0073 RMSE.

Figure 24 presents the total displacement estimation errors for the MLP, SVM, and DT methods. The error values on the Y-axis indicate the difference between the estimated and actual total displacement values. The errors for the MLP, SVM, and DT techniques are depicted by green, red, and purple lines, respectively, for both the training and test sets. A considerable deviation from the zero point on the Y-axis signifies poor estimation performance, while a smaller deviation from the zero point highlights superior estimation accuracy. The error fluctuations caused by DT appear to be larger than the other ML methods used, and DT shows worse estimation performance compared to MLP and SVM.

Fig. 24.

Total displacement prediction errors with MLP, SVM, and DT algorithms

Figure 25 shows the fit of the models created with the actual total displacement values ​​for 20 randomly selected samples from the test data set. It can be observed that the DT algorithm encounters challenges in generating an accurate prediction model for specific data points. The MLP algorithm, however, produces a slightly better prediction model than the DT method. The actual total displacement, represented by the black line, closely matches the SVM model, shown by the red line, demonstrating the enhanced prediction performance of the SVM model.

Fig. 25.

Comparison of total displacement with predictions of generated models

Discussion

The results of this study highlight the combined influence of plate length, screw number, and implant material on the mechanical performance of fixation systems for oblique tibial shaft fractures. The finding that the H7S4 model produced the highest stress concentrations is consistent with the observations of Sathone and Potdar [10], who emphasized that insufficient fixation length and screw number may lead to mechanical failure. Conversely, the H11S10 model, which provided the most extensive fixation, achieved the lowest stress levels, confirming the stabilizing effect of increased screw density and plate coverage. Similar conclusions were drawn by Cao et al. [2] and further supported by Tümer et al. [45], as well as other experimental studies [46–48], all of which underlined the importance of optimal implant design in reducing peak stress concentrations at the fracture site. In particular, the improved stability observed in the H11S10 configuration underscores the biomechanical advantage of employing longer plates with multiple screw fixations.

From a materials perspective, the comparison between Ti–6Al–4 V and 316 L stainless steel implants revealed distinct mechanical behaviors. Ti–6Al–4 V implants, despite exhibiting lower stress values within the implant itself, allowed greater displacement in the fracture region due to their lower modulus of elasticity. In contrast, as reported by Tümer et al. [45], 316 L stainless steel implants provided greater construct rigidity and less displacement, although at the expense of higher stress levels in the plate. These findings are also consistent with the biomechanical investigations of Sathone and Potdar [10], who noted that titanium alloys were associated with improved load sharing and reduced risk of stress shielding, while stainless steel implants offered stronger initial stability but carried a higher risk of implant fatigue under repetitive loading. Therefore, as highlighted by Cao et al. [2], implant selection involves a trade-off between minimizing implant stresses and ensuring sufficient rigidity at the fracture site. Ti–6Al–4 V may be preferable in cases where gradual load transfer to the bone is desired to promote long-term healing, whereas 316 L stainless steel may be more suitable for immediate post-operative stability in high-demand patients.

Displacement analysis revealed significant differences between Ti–6Al–4 V and 316 L stainless steel implants. Stainless steel implants consistently exhibited lower displacement, reflecting their higher modulus of elasticity and greater rigidity. These findings align with the work of Tümer et al. [45], who reported that 316 L SS implants provide superior mechanical stability in terms of minimizing deformation at the fracture site. Ti–6Al–4 V implants allowed slightly higher displacement, which may facilitate gradual load transfer to the bone and promote secondary healing, as discussed by Sathone and Potdar [10]. However, excessive displacement in shorter or less rigid constructs could compromise fixation and delay recovery.

The dual consideration of stress and displacement illustrates a trade-off in implant selection. Titanium alloy implants reduce stress concentration within the plate and allow load sharing with bone, potentially decreasing stress shielding and improving long-term bone remodeling [10]. Stainless steel implants, while offering immediate mechanical stability due to reduced displacement, are subjected to higher peak stresses, which could increase the risk of fatigue failure under repetitive loading, a finding supported by previous FEA studies [45]. Therefore, the choice of implant material and configuration should balance the need for immediate stability and the biological requirements for bone healing.

The ML findings obtained in this study are consistent with similar studies in the literature. SVM demonstrates superior generalization performance compared to MLP and DT on datasets with limited samples and defined in high-dimensional feature space [49, 50]. SVM’s structure, which focuses only on support vectors, makes the model more resistant to overfitting and enables it to produce reliable classification results even on small datasets [36].

Limitations

Despite the strengths of the finite element approach, several limitations should be acknowledged. First, the tibial model included only cortical bone, and the cancellous bone structure was excluded to simplify the model geometry and reduce computational cost. Although cortical bone carries the majority of axial loads, cancellous bone contributes to the overall load distribution, and its absence may have influenced the predicted displacement values. Second, bone tissue was modeled as an isotropic material, whereas in reality it exhibits anisotropic and heterogeneous properties. This simplification, while common in finite element studies, may limit the accuracy of stress and strain predictions. Third, the simulations applied only static axial loads, without accounting for the complex, multi-directional, and time-dependent loading conditions encountered during gait or daily activities. Finally, implant–bone interface conditions were idealized, neglecting potential loosening or imperfect screw–bone contact that may occur clinically. These limitations should be considered when interpreting the results, and future work should incorporate anisotropic bone properties, cancellous bone modeling, and dynamic physiological loading to achieve more comprehensive predictions.

Conclusions

The effects of different implant designs and biomaterials on oblique tibial fractures were investigated using the finite element method. Artificial intelligence algorithms were developed to make low-cost and practical predictions with the data obtained from the analyses. In this article, three different ML models, namely MLP, SVM, and DT, were trained and tested. In the ML models, seven input parameters were used: mesh size, plate length, modulus of elasticity, Poisson’s ratio, number of screws, number of holes, and force, and two output parameters were used: maximum stress and total displacement. Of the 1008 data points generated for each algorithm, 70% were allocated for training and 30% for testing. The SVM algorithm showed much superior performance compared to the MLP and DT algorithms in maximum stress and total displacement predictions. With the SVM model, especially in total displacement estimation, very low values ​​of 0.0003 and 0.0015 MAE, 0.0003 and 0.0023 RMSE were achieved for the training and test sets, respectively. SVM estimated the maximum stress to be 0.24180 and 0.41305 MAE, 0.27250, and 0.48607 RMSE for the training and test sets, respectively.

Machine learning algorithms analyze large datasets and provide practical solutions to biomechanical problems thanks to their high-accuracy modeling capabilities. The results of this study indicate that implant geometry, material selection, and mesh characteristics significantly influence stress distribution and displacement in fracture fixation, highlighting the importance of careful implant design. Furthermore, the combined use of finite element analysis and AI models like SVM can accelerate the development of optimized fixation strategies while reducing experimental costs. With the combined work of artificial intelligence and data science techniques, it will be possible to develop effective biomechanical solutions in the future, where test costs will decrease.

Author contributions

HBM: Methodology, Investigation, Original Draft, Writing, Resources, Review & Editing.

Funding

Declaration.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval and consent to participate

The informed consent has been signed by all the participants and the study has been approved by the Human Research Ethics Committees of Tokat Gaziosmanpaşa University (No. E-33490967-044-651584). The present work was performed in accordance with the Declaration of Helsinki.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Clinical trial number

Clinical trial number: not applicable.