Fracture path prediction in proximal femur using strain energy-based Dijkstra’s algorithm

Abstract

Proximal femoral fracture is a major issue associated with high mortality rates and entails significant rehabilitation costs. Conventional fracture prediction methods, primarily based on finite-element analysis (FEA), often suffer from high computational costs and limited clinical applicability. To address these limitations, this study proposes a novel fracture path prediction method using Dijkstra’s algorithm based on strain energy. The proposed method extracts the proximal femur from skeletal images, performs FEA to calculate strain energy, and applies Dijkstra’s algorithm to predict fracture paths by identifying regions of structural vulnerability. Compared with conventional methods, the proposed approach significantly reduces computational time while maintaining high accuracy. The predicted fracture paths demonstrate strong agreement with experimental results, validating the method’s effectiveness in replicating actual fracture mechanisms. This study provides an efficient and reliable tool for understanding proximal femoral fracture mechanisms and paves the way for the development of patient-specific preventive measures.

Graphical Abstract

Fracture path prediction in the proximal femur using Dijkstra’s algorithm.

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1. Introduction

Proximal femoral fractures are common among the elderly and are recognized as the major cause of disability and increased mortality worldwide (Citak et al., 2008; Sànchez-Riera et al., 2010). Such fractures significantly reduce mobility and the ability to live independently, reducing the quality of life (Mamarelis et al., 2020; Müller et al., 2018). Therefore, predicting proximal femoral fractures in advance and establishing prevention and treatment plans based on these predictions are clinically important. In particular, predicting fracture paths can help identify areas prone to fractures, thereby enabling the implementation of effective patient-specific preventive measures (Pérez-Cano et al., 2024). Furthermore, it can contribute to an improved understanding of bone damage mechanisms (Gustafsson et al., 2021). Therefore, fracture path prediction is necessary to provide rapid and accurate responses during treatment and recovery.

Various studies have used the extended finite-element method (XFEM) to predict fracture paths (Ali et al., 2014; Feerick et al., 2013; Giambini et al., 2016; Marco et al., 2018; Shu & Sugita, 2020). XFEM is a powerful technique that can simulate discontinuities such as cracks, rendering it possible to predict the initiation and growth of fractures. For instance, Feerick et al. (2013) performed three-dimensional (3D) simulations of crack propagation in the cortical bone tissue. Ali et al. (2014) simulated fractures under stance loading conditions using a computed tomography image-based femur. These studies reported reasonable accuracy when compared with actual experimental results. However, the overall fracture path could not be captured because the simulation was terminated early because of convergence issues.

Several finite element (FE)-based methods have been proposed to predict the overall fracture path (Gasser & Holzapfel, 2005; Dall’Ara et al., 2013; Hambli, 2013; Starvin et al., 2017; Gustafsson et al., 2021; Cui et al., 2022). Representative approaches include the partition of unity finite-element method (PUFEM) and incremental element deletion (IED)-based FE analysis. Gustafsson et al. (2021) applied PUFEM to predict fracture paths in the proximal femur and compared the results using experimental data. Cui et al. (2022) used IED–FE analysis to simulate fracture paths under different conditions, such as fall postures and varying cortical bone thicknesses. These methods demonstrate high accuracy in predicting complex proximal femoral fracture paths and effectively capture the overall fracture path. However, these methods require additional computations to model complex fracture paths, which necessitates an improvement in computational efficiency.

Complex calculations can be simplified using Dijkstra’s algorithm to improve computational efficiency (Noshita, 1985; Gunkel et al., 2012; Popa et al., 2022). Dijkstra’s algorithm is a graphical search method that efficiently determines the shortest path from the starting point to the target point (Dijkstra, 1959). The algorithm has been used to detect and analyze the paths of microcracks and has exhibited excellent results (Gunkel et al., 2012). Unlike FEM-based methods, which require repetitive computational processes to model complex fracture paths, Dijkstra's algorithm systematically evaluates multiple paths to efficiently derive the optimal fracture path. This indicates that the algorithm provides high computational efficiency, thereby enhancing its potential for clinical applications.

This study proposes a novel method for predicting fracture paths in the proximal femur using a strain energy-based Dijkstra’s algorithm. For this purpose, an FE model of the proximal femur was generated using clinical images. Strain energy was derived under loading conditions to simulate daily activities. A node–edge graph was created based on the strain energy, and edge weights were assigned using the strain energy values. Dijkstra’s algorithm was then used to explore the potential fracture paths according to the sum of the edge weights. The predicted path was compared with those reported in previous studies that performed compression experiments on the femurs of human cadavers.

2. Methodology

The objective of this study was to predict fracture paths in the proximal femur using Dijkstra’s algorithm based on strain energy (Figure 1). The study was conducted in two steps (Figure 2). The first step involved calculating the strain energy via FE analysis based on a preprocessed proximal femur image. The second step involved applying Dijkstra’s algorithm based on the calculated strain energy to predict the fracture path in the proximal femur.

The strain energy-based Dijkstra’s algorithm proposed in this study consists of the following steps. First, finite-element analysis (FEA) is performed to calculate the strain energy generated in each element. Next, the calculated strain energy is reversed to obtain the reversed strain energy (RSE), which is used as the edge weight. Based on the RSE values, the algorithm defines a node–edge graph where each element serves as a node, and connections between adjacent elements are defined as edges. The edge weight is set as the average RSE value of the two connected nodes. Finally, Dijkstra’s algorithm identifies the path with the minimum total weight between the starting node (proximal femur upper region) and the ending node (proximal femur lower region) on the graph.

2.1. FE analysis for strain energy calculation

In this step, the strain energy was calculated using FE analysis based on a preprocessed image of a proximal femur, which was obtained through medical image segmentation. The cortical and trabecular bones were segmented owing to their distinct mechanical properties. Preprocessing was performed using various image processing algorithms, and the strain energy was calculated using a three-step FE process (Yan Kang et al., 2003; Wang et al., 2016). First, an FE model was constructed by applying pixel-based modeling techniques to preprocessed images. The pixel-based approach has been used in various studies, as it enables the implementation of FEM on images of individuals with different bone mineral density (BMD; Bagheri & Rouhi, 2020; Mohammed et al., 2023; Vania et al., 2019). This technique accurately reflected the shape of each pixel, wherein each pixel represented by four vertices formed a square (Falcinelli & Whyne, 2020). The vertices were converted into unique nodes for each element, creating a two-dimensional (2D) quadrilateral element model (bilinear Lagrange 4-node elements). The generated elements overlapped, and this method has been successfully used in biomechanics to construct FEMs (Jang & Kim, 2008). Each node comprised two degrees of freedom in the x- and y-directions, and the element behavior was assumed to be under plane-stress conditions. The material properties of each element were assigned based on the segmentation results of the cortical and trabecular bones.

Second, FEA was performed to calculate the strain energy of the proximal femur. Strain energy affects crack initiation and propagation (Deng et al., 2021). Strain energy is generally considered to be highest in structurally weakest areas, which are known to correspond to actual crack paths (Fadeel et al., 2022; Henry et al., 2025). The analysis was conducted considering the boundary conditions in which compression was applied to the proximal femur, and the strain energy was calculated based on the analysis results, as follows:

where |${U_e}$| is the strain energy of each element, |${u_e}$| is the nodal displacement vector of the element, and |${S_e}$| is the stiffness matrix of the element.

Third, the pixel values of the trabecular bone in the preprocessed images were converted into RSE values, which is the reciprocal of the strain energy (SE). The standard Dijkstra’s algorithm works by finding the path with the minimum sum of edge weights. To utilize this mechanism, this study assigned RSE as the edge weight. This ensures that areas with higher strain energy are prioritized during pathfinding. RSE was used to compare the vulnerability of the proximal femur under external loading and predict the crack path by applying Dijkstra’s algorithm. RSE can be calculated as follows:

where |${R_{SE,i}}$| and |$S{E_i}$| are the RSE and strain energy of the ith element, respectively. Consequently, high strain energies were converted to low RSE values, whereas low strain energies were converted to high RSE values.

2.2. Fracture path prediction using Dijkstra’s algorithm

In this step, Dijkstra’s algorithm was applied to predict the fracture path based on strain energy via a two-step process. First, a node–edge graph was generated based on the image using RSE values. Each pixel in the RSE image was defined as a node in the graph, and each node was assigned an RSE value based on the location of its corresponding pixel. The connection between each node in the graph can be expressed as an edge, which connects two adjacent nodes and indicates the adjacency between pixels. The weight of an edge is defined as the average RSE value of two connected nodes. The edge weight can be calculated as follows:

where u and v denote arbitrary nodes and w is the weight of the edge connected to u and v.

Second, Dijkstra’s algorithm was applied to search for the path with the lowest sum of edge weights. This algorithm is widely used to determine the shortest paths in global positioning systems and networks (Dijkstra, 1959). The fracture path in the proximal femur was extracted by searching for the shortest path in the node–edge graph, where the RSE values were reflected. The shortest path can be calculated as follows:

where u and v denote arbitrary nodes in the graph that connect an edge, w is the weight of the edge, and |${P^*}$| represents path P along which the sum of the edge weights is minimized.

Typically, Dijkstra’s algorithm searches for a path using edge weights that connect the start and end points. The shortest path is updated whenever a path with the lowest sum of the edge weight is identified. The algorithm sets the upper part of the proximal femur as the start range and the lower part as the end node to search for a path. However, because of the geometry of the proximal femur, the distance between the start and end ranges is not uniform and may distort the total weight of the path. For instance, if the distance between two ranges is short, the total path weight decreases; conversely, the total path weight increases if the distance is long. To prevent this distortion, the total path weight was divided by the Euclidean distance between the start and end points to obtain the adjusted path weight, as follows:

where |$Cost$| denotes the adjusted path weight obtained by considering the Euclidean distance, |$cost$| indicates the original path weight, and (⁠|${x_1}$|⁠,|$\,\,{y_1}$|⁠) and (⁠|${x_2}$|⁠,|$\,\,{y_2}$|⁠) represent the coordinates of the start and end points, respectively.

3. Verification

This study used a 2D image of an artificial proximal femur (94.2 mm × 104.4 mm) with a clinical resolution of 600 µm. As shown in Figure 3, the image includes the major bone structures of the femur based on Wolff’s law (Principal and Secondary Compressive Groups, Principal and Secondary Tensile Groups) and accurately represents the structural behavior of the bone under external loads; consequently, it has been widely implemented in several studies (Lee et al., 2015; Kim & Jang, 2016; Yoon et al., 2020). In this image, the cortical and trabecular bones are segmented according to the steps described in Section 2.1.

In this study, a pixel-based FE modeling technique was used to construct a FE model. The element size of the FE model was set to 600 µm, which corresponded to the resolution of the image. Each pixel was converted into a 4-node 2D element (PLANE42 in ANSYS), with the aspect ratio of the elements set to 1 to minimize the mesh distortion and improve the analysis accuracy (Bekdaş & Öztorun, 2015). The FE model comprised 27 617 nodes and 14 927 elements. The elastic modulus of the cortical bone elements was set to 22.5 GPa (Kim et al., 2018). Conversely, the elastic modulus of the trabecular bone elements was assigned using the transformation formula (Equation 6) proposed in a previous study based on the density of each element (Kim et al., 2018), as follows:

where |${E_i}$| and |${\rho _i}$| are the elastic modulus and density of the i_th element, respectively. The values of the Poisson's ratio for both the cortical and trabecular bones were set to 0.3 (Wirtz et al., 2000).

This study selected two conditions: single-leg and double-leg standing, as representative fracture loads that can occur in the proximal femur for FEA. Additionally, the loading conditions used in previous studies were applied to validate the strain energy-based Dijkstra’s algorithm (Hambli & Allaoui, 2013; Pérez-Cano et al., 2023). A distributed load of 1000 N was applied at angles of 20° and 6° relative to the vertical axis through the center of the femoral head for the single- and double-leg stances, respectively (Figure 4). The boundary conditions constrained all the lowest bottom nodes of the proximal femur. The strain energy of each element was calculated under these conditions. Dijkstra’s algorithm was then used to predict the fracture paths, wherein the strain energy values were inverted to generate RSE images. These images were used to obtain a node–edge graph. The edge weights represented the vulnerability of each element under external loads, and the algorithm identified the shortest path that minimized the sum of these weights.

The results of the predicted fracture path were compared with those obtained from previous experiments on the single-leg stance (Hambli & Allaoui, 2013) and double-leg stance (Pérez-Cano et al., 2023). The two experiments involved compression tests on ten and eight human cadaveric femurs. In the single-leg stance experiment, the distal end of the femur was attached using epoxy resin. A compressive load was applied to the top surface of the femoral head at an angle of 20° relative to the shaft axis. In the double-leg stance experiment, the upper and lower parts of the femur were attached using epoxy resin. A compressive load was applied at an angle of 6° in the proximal direction along the longitudinal axis of the bone diaphysis. In previous studies, proximal femurs were scanned after the fracture occurred. To compare the predicted and actual fracture paths, the results were compared based on the regions of the proximal end segment of the femur (Figure 5).

The crack path in the proximal femur was predicted using the strain energy-based Dijkstra’s algorithm and compared with that predicted using the conventional Dijkstra’s algorithm. In the conventional algorithm, bone density values were used as edge weights instead of strain energy to predict the fracture path in the proximal femur. The start node and end node were set as the proximal femur upper region (femoral head) and lower region (femoral neck), respectively, as in the proposed algorithm.

In this study, the time required to predict the fracture path was analyzed by separately measuring the time required for FE analysis (Step 1) and fracture path search (Step 2), which were added to determine the total time (Figure 2). All calculations were performed on a workstation equipped with an Intel Core™ i9-12900K processor, 3.20 GHz, and 64 GB RAM. Image segmentation, FE modeling, and fracture path predictions were performed using custom MATLAB R2022a code, whereas strain energy calculations were conducted using ANSYS Mechanical APDL 2022 R1.

4. Results

Figure 6 shows the strain energy distribution observed in the proximal femur during the single- and double-leg stances. In the single-leg stance, the strain energy gradually increased along the microstructure, from the femoral head to the femoral neck; the maximum strain energy was 1.05 μJ at the interior of the femoral neck. In the double-leg stance, the strain energy was evenly distributed between the greater trochanter and the femoral neck; the maximum strain energy was 3.41 μJ, which was higher compared with that calculated under the single-leg stance condition. The femoral neck served as the primary load-bearing structure in both stances, exhibiting a pattern in which the strain energy was concentrated in that area.

Figure 7 compares the fracture paths predicted using the BMD-based conventional method and the SE-based proposed method. First, the fracture path according to the load conditions of the SE-based proposed method, the predicted fracture path started at the femoral head and ended between the femoral head and neck in the single-leg stance. The actual fracture path shown in the experiment initiates locally at the superior cortex of the femoral head and progresses vertically into the inferior cortex. The predicted fracture path in the single-leg stance matched the fracture region observed in the experiment. In the double-leg stance, the predicted fracture path began at the exterior of the femoral neck, propagated diagonally, and ended in the interior of the femoral neck. The results shown in the experiment exhibited fractures in the neck region. The fracture path initiates near the base of the femoral head and extends down towards the lesser trochanter. The predicted fracture path in the double-leg stance was consistent with the fracture region observed in the experiment, and the morphology of the crack path exhibited similar patterns. The BMD-based conventional method showed a pattern of generating paths along areas of low bone density. The fracture path predicted by the conventional Dijkstra’s algorithm ended between the femoral neck and the greater trochanter. However, these results did not match the results observed in actual experiments and also showed different results from the fracture paths derived by the strain energy-based Dijkstra’s algorithm.

Figure 8 shows a comparison of the strain energy distribution and predicted fracture paths with and without considering the cortical bone. When the cortical bone was considered, the highest strain energy was observed at the medial femoral neck in both stances; the predicted fracture paths exhibited a linear pattern. By contrast, when the cortical bone was not considered, the strain energy was distributed along the load-susceptible locations, and the fracture path appeared diagonally along the microstructure.

The total time required for fracture path prediction using the proposed method was 127 s. Among these, the FE modeling and analysis time was 11 s, whereas 116 s were required to generate the node − edge graph and execute Dijkstra’s algorithm.

5. Discussion

Predicting fracture paths in the proximal femur is essential to understanding fracture mechanisms and enabling patient-specific treatment and prevention strategies. Previous studies have successfully simulated various fracture patterns and processes by simulating damage mechanisms in the proximal femur (Ali et al., 2014; Dall'Ara et al., 2013; Hambli, 2013; Marco et al., 2018). However, the existing methods require extensive computations to model the fracture path, resulting in a long calculation time. To address this issue, this study developed a novel method for fracture path prediction using a strain energy-based Dijkstra’s algorithm. This method efficiently predicts fracture paths by using an algorithm optimized for the shortest path search. The results confirmed that the predicted fracture path matched the observed fracture region in the human cadaveric femur. These findings indicate that the proposed method is effective and reliable for predicting fracture paths.

The strain energy quantitatively represents the degree to which each element resists the applied load, rendering it useful for evaluating the impact of the load on specific regions. This study analyzed the strain energy distribution under single- and double-leg stance conditions. In the single-leg stance, a high strain energy was observed in the femoral head and neck. This was because the body weight was concentrated on one leg, causing the load to tilt laterally from the axial direction of the femur. In this position, the femoral head primarily absorbed and transferred the load to the femoral neck. By contrast, in the double-leg stance, higher strain energy was observed in the femoral neck and greater trochanter. This was because the load was evenly distributed in both legs. Consequently, the femoral head experienced a less concentrated load, leading to a more uniform compressive deformation throughout the proximal femur. In particular, the smaller angle of 6° caused the compressive and bending loads to act directly on the femoral neck. This resulted in higher strain energy in the femoral neck. In both stances, the maximum strain energy was observed in the interior of the femoral neck, indicating that the femoral neck functioned as the primary structural region that was directly subjected to body weight loading. Typically, the femoral neck is vulnerable to complex loads, such as compression and bending, rendering it susceptible to fractures (Auger et al., 2022; Kersh et al., 2018; Nawathe et al., 2015). Consequently, the observed distribution of strain energy aligns with previous research findings that the femoral neck is the primary location for the occurrence of fractures (Deng et al., 2021). Therefore, the FE analysis performed in this study was validated as accurate and reliable.

The proposed method exhibited high similarity with the results of compression experiments performed using human cadaveric femurs. This was because the method selected paths based on regions with low resistance to loads, as indicated by the strain energy. Dijkstra’s algorithm generated the shortest path along the edges between elements with high strain energy. Consequently, the fracture path was formed around the regions vulnerable to the load. This approach produced results similar to the actual fracture paths observed in the proximal femur, suggesting that the strain energy-based path prediction can effectively reflect the actual fracture pattern.

The proposed method effectively predicted the fracture paths observed in the experiments and demonstrated fracture regions that were similar to the actual ones. However, the predicted paths exhibit limitations in accurately matching the actual fracture paths. This is because the proximal femur model cannot accurately reflect the anatomical characteristics of an individual. The proximal femur comprises various structural characteristics such as gender, age, and bone density (Noble et al., 1995). Moreover, even in experiments involving cadavers, unique differences are observed in the characteristics of each proximal femur. Therefore, the model used in this study did not reflect individual characteristics, resulting in differences between the predicted and actual paths.

In this study, compressive loads were applied to the proximal femur including the cortical bone to analyze the strain energy. The highest strain energy was observed in the cortical bone. Typically, the cortical bone has higher stiffness than the trabecular bone, enabling it to withstand greater maximum stress. However, the cortical bone exhibits characteristics close to brittleness, rendering it susceptible to unexpected failure (Keaveny & Hayes, 1993; Osterhoff et al., 2016). It is difficult to predict the fracture path in the cortical bone because it fractures without significant deformation when subjected to a load. In this study, the predicted fracture path appeared to be linear when the cortical bone was considered. This was because the high strain energy in the cortical bone influenced Dijkstra’s algorithm, preventing the relatively lower strain energy of the trabecular bone from being fully reflected. By contrast, excluding the cortical bone facilitated a sufficient representation of the strain energy in the trabecular bone. Consequently, the fracture path exhibited a diagonal pattern according to the microstructure and strain energy distribution. Therefore, these results indicate that fracture path prediction is highly influenced by the cortical bone, which is consistent with previous findings indicating that identifying the fracture path within the cortical bone is difficult (Gustafsson et al., 2021).

The conventional Dijkstra’s algorithm tended to generate paths along regions of low bone density. This result was attributed to the intrinsic characteristic of the Dijkstra algorithm, which searches for the path with the minimum weight. However, paths based on bone density showed low similarity to actual experimental fracture paths and failed to sufficiently reflect structural vulnerabilities. By contrast, the strain energy-based Dijkstra’s algorithm predicted fracture paths that closely resembled actual experimental results. This demonstrates that strain energy serves as a critical parameter for reflecting structural vulnerability. These findings indicate that the proposed algorithm provides higher predictive reliability than the conventional Dijkstra’s algorithm and effectively reproduces the fracture mechanisms.

The proposed method requires 127 s to predict the fracture path. This is significantly shorter than that of the conventional FEM-based techniques, which involve multiple iterations. Conventional methods perform recalculation at each iteration to simulate the crack propagation, and a minimum of 28 iterations are necessary to achieve high accuracy (Xu et al., 2023). Assuming that each iteration requires an average of 180 s, a minimum of 5040 s is required for the calculation. By contrast, the proposed method calculates the strain energy distribution in a single FE analysis and applies Dijkstra’s algorithm to predict the fracture path. Therefore, the proposed method demonstrated improved time efficiency in comparison with the conventional methods.

The proposed method offers the advantage of simplified procedures compared with existing methods. The traditional XFEM is effective in accurately predicting complex fracture paths. However, XFEM requires reflecting structural changes at each stage of fracture progression, necessitating iterative computational processes. This leads to high demands for computational resources and time. The proposed strain energy-based Dijkstra’s algorithm addresses these issues by processing strain energy data calculated using FEA in a single computational step to predict fracture progression. By eliminating iterative computational processes, it significantly reduces computational resource and time requirements. Liu et al. (2024) have implemented procedural simplification through the application of Dimensional Transfer Learning in manufacturing processes, thereby reducing computation time while maintaining solution quality. Zhang et al. (2024) improved both computation time and accuracy of time-window-partition-based algorithms through procedural simplification.

The proposed method explores crack paths by considering all elements within the entire structure without the need to explicitly define the starting point of the crack. By contrast, XFEM requires the initial crack starting point to be explicitly specified to simulate the fracture path. This necessitates user intervention during the setup phase, and the chosen crack location can significantly influence the results, potentially affecting the accuracy and reliability of the prediction.

Biomechanical studies are generally conducted in three stages before being applied clinically. The first stage is the proof-of-concept phase, where mechanical validation demonstrates that realistic mechanisms and processes can be accurately represented. The second stage involves a preliminary study to evaluate the predictive validity by comparing predicted outcomes with experimental data from the population. The final stage involves rigorous validation studies to demonstrate the reliability of the model through quantitative validation on individual samples. This study corresponds to the proof-of-concept phase. In this study, mechanical validation of a strain energy-based Dijkstra’s algorithm was conducted. The study results showed that the proposed algorithm predicted fracture paths similar to the experimental results. Based on this study, further evaluation of the methodology’s applicability in 3D models is planned. In the final stage, quantitative validation on individual samples will be conducted to demonstrate the reliability of the algorithm.

This study had several limitations. First, the fracture behavior was analyzed under only two loading conditions (single-leg and double-leg stance). By contrast, daily activities include complex and dynamic loading conditions such as abduction, adduction, twisting, and running. These activities generate multidirectional forces that can significantly affect stress distribution and fracture mechanics. Therefore, additional studies incorporating a wider range of loading conditions are needed to improve the generalizability of the results. Second, this study used a 2D proximal femur model with clinical resolution to analyze structural behavior. Although this approach is computationally efficient, it cannot fully represent the complex 3D geometry of bones or accurately capture microstructural characteristics, potentially causing discrepancies in fracture predictions. High-resolution 3D imaging should be used in future analyses to better capture the complex structural behavior of bones. Third, the FEM did not fully represent the characteristics of the actual proximal femur specimens used in the experiments. Important factors such as bone density, material properties, and geometric variations of the experimental specimens were not completely integrated into the model. Future analyses should incorporate these characteristics for more accurate comparisons. Fourth, the predicted fracture paths did not fully match the actual fracture paths observed in the experiments. These discrepancies may have resulted from individual anatomical differences, such as variations in sex, age, bone quality, and medical history, all of which influence the mechanical behavior of the femur. Finally, this study used only 2D images, whereas bones are inherently 3D structures that experience complex stress distributions and structural behavior in multiple directions. Future studies should adopt high-resolution 3D models and simulations that account for individual variability, complex loading conditions, and multidirectional forces to improve the predictive accuracy and applicability of the algorithm.

6. Conclusions

The fracture paths in the proximal femur predicted using the proposed strain energy-based Dijkstra’s algorithm matched the experimental results. This method offers advantages such as reduced computation time and simplified processes compared with the existing prediction methods. The findings highlight the potential of the proposed method to improve prediction accuracy; however, further validation is necessary before clinical application. Specifically, future research should focus on stepwise validation, including comprehensive testing under diverse loading conditions, high-resolution imaging, and integration of individual anatomical differences, to ensure reliability and generalizability. This study’s results can serve as a foundation for new directions in fracture prediction research.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Author Contributions

Jee A Baik: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing—Original Draft, Visualization. Jun Won Choi: Methodology, Software, Investigation, Writing—Review & Editing, Visualization. Jung Jin Kim: Conceptualization, Validation, Resources, Writing—Review & Editing, Supervision.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1I1A3043967). Also, we would like to acknowledge the technical support from Ansys Korea.

References