Bone morphological feature extraction for customized bone plate design

Fractures are difficult to treat because of individual differences in bone morphology and fracture types. Compared to serialized bone plates, the use of customized plates significantly improves the fracture healing process. However, designing custom plates often requires the extraction of skeletal morphology, which is a complex and time-consuming procedure. This study proposes a method for extracting bone morphological features to facilitate customized plate designs. The customized plate design involves three major steps: extracting the morphological features of the bone, representing the undersurface features of the plate, and constructing the customized plate. Among these steps, constructing the undersurface feature involves integrating a group of bone features with different anatomical morphologies into a semantic feature parameter set of the plate feature. The undersurface feature encapsulates the plate and bone features into a highly cohesive generic feature and then establishes an internal correlation between the plate and bone features. Using the femoral plate as an example, we further examined the validity and feasibility of the proposed method. The experimental results demonstrate that the proposed method improves the convenience of redesign through the intuitive editing of semantic parameters. In addition, the proposed method significantly improves the design efficiency and reduces the required design time.

• The plate features must be constructed according to anatomical morphological information.
• The feature parameters of the plate must be stratified to improve the adjustment efficiency.
To satisfy these two requirements, we propose an effective method for extracting bone morphological features and guiding a customized plate design.
The remainder of this paper is organized as follows. In "Overview" section provides a brief overview of the proposed method. The main steps of the proposed method (i.e., extracting the bone morphological features, constructing the plate feature model, and generating the customized plate) are presented in "Extraction of bone morphological features", "Construction of the plate feature model", and "Generation of the customized plate" sections, respectively. Finally, the conclusions are presented in "Discussion and conclusion" section. Figure 1 shows the extracted bone morphological features which were integrated into the plate feature model in this study. The steps for this method are as follows: www.nature.com/scientificreports/ This study uses two types of features from the above steps:

Overview
• Bone features, including features of the fracture and its morphological features, which consist of the following aspects. Fracture features are an important basis for judging fracture types, which are usually characterized by imaging fracture lines 27 . Morphological features describe the anatomical structure of the bone, which is usually extracted from parameters using a semantic form 28 . After confirming the fracture type of the patient, we only focused on bone morphology. • The plate feature includes a group of geometric shapes that embody the designer's intention and constitutes a certain set of engineering semantics 29 . In this way, the plate feature supports editing the plate in the semantic feature layer and avoids tedious operation on the geometric layer at the bottom.

Extraction of bone morphological features
Bone sample data collection for PCA. This 16), in the same order as presented above after normalization.
To ensure that the collected data represent the general population and avoid severe deviations, the samples included in the analysis should be close to a normal distribution. The height of individuals follows a normal distribution, and the femoral length is often proportional to the height; therefore, we can reasonably assume that the femoral length should also follow a normal distribution 30 . Experiments with 100 samples show that the femoral length distribution is left skewed, with a skewness value of − 0.07, mean of 420.072 mm, and standard deviation of 22.968 mm. Notably, the deviation degree is small, and the collected samples can be considered as being reasonably representative of the general population. Furthermore, we used the Kaiser-Meyer-Olkin (KMO) test31 to compare simple and partial correlation coefficients between the variables. The KMO statistic was between 0 and 1. The closer the KMO value is to 1, the stronger the correlation between variables, and the more suitable the original variables for PCA. In the present study, the KMO statistic was 0.731. Furthermore, Bartlett's sphericity test 31 , which can also test the correlation between variables, indicates a significance value (i.e., Sig.) of 0.00, which is less than 0.05. These results indicate that the collected set of samples is particularly suitable for PCA requirements.
Extraction of the principal components. PCA is the most commonly used linear dimensionalityreduction method. It maps the high-dimensional data to a low-dimensional space via linear projection and extracts most information in the dimensionality (maximum variance) of the projected data to use fewer data dimensions while retaining as much feature information as possible from the original data. By mapping an n-dimensional feature to a k-dimensional feature, in which k is often less than n, the k-dimensional feature is reconstructed, which is called the principal component 31 . In this way, we can use the principal components to reflect the information of the original variable and replace the original variable for an in-depth study. The steps for feature extraction via PCA are as follows.
1. We calculated the covariance matrix, eigenvalues, and eigenvectors. The selection principles of the k eigenvectors are as follows: • The eigenvalue must be greater than 0.5, and ideally greater than 1.
• The cumulative contribution rate should be greater than 90% 31 .
2. We sort the eigenvalues and retain the eigenvectors corresponding to the first k largest eigenvalues. 3. We converted the original features into a new space constructed by the k feature vectors obtained above. www.nature.com/scientificreports/ Notably, the KMO statistic for parameters of the femoral shaft is 0.465, which is much lower than the average level of 0.731, suggesting that the correlation between the parameters of the femoral shaft is relatively small. Furthermore, there were fewer femoral shaft parameters. Therefore, extracting the principal components of the femoral shaft parameters is not required. Feature extraction of the proximal and distal femurs is described below.
1. For the proximal femur, the yellow line in Fig. 2a indicates the curve change of the eigenvalue with component number. Notably, the first eigenvalue is the largest and those after the fifth are smaller and only change slightly. Therefore, we selected the first four components. Table 1    www.nature.com/scientificreports/ of the PCA of proximal femoral parameters, i.e., the correlation between the principal components, p 1 , p 2 , p 3 , and p 4 , and the original variables, X 1 -X 9 .
2. For the distal femur, the blue line in Fig. 2a indicates a change in the eigenvalue with component number. The curve becomes less steep from the eighth component onward. Therefore, we selected the first seven components. Table 2 lists the principal component load matrix of the distal femur, B ij , where each element in the matrix is called b ij , the range of i is 1-7, and the range of j is 1-16. The column of each principal component constitutes the eigenvector corresponding to the principal component. Equation (2) shows the mathematical model of PCA, i.e., the correlation between the principal components, q 1 , q 2 , q 3 , q 4 , q 5 , q 6 , and q 7 , and the original variables, Y 1 -Y 16 .
In this experiment, as shown in Fig. 2b, the cumulative variance contribution rate 31 of the proximal femur was 90.474%, whereas the variance contribution rates of the first four components were 43.054%, 25.449%, 14.176%, and 7.795%, respectively. As shown in Fig. 2c, the cumulative variance contribution rate of the distal femur was 93.241%, whereas the variance contribution rates of the first seven components were 31.338%, 22.293%, 11.936%, 9.589%, 7.630%, 6.112%, and 4.343%, respectively. The extracted principal components can explain the original data well. In addition, to interpret the principal component representative of the morphological parameters, we adopted the maximum variance orthogonal rotation method 31 . Figure 3 shows the rotated principal components of the proximal and distal parameters. In Fig. 3a, Component 1 primarily explains D fh , D fn , L t , and H t1 ; Component 2, L fhs and L fn ; Component 3, H fh and A fn ; and Component 4, H t2 . In Fig. 3b, Component 1 primarily explains L f , L l , H m , and H l ; Component 2, A m and A l ; Component 3, L a and L p ; Component 4, A s and H tg ; Component 5, A a and A p ; Component 6, L df ; and, finally, Component 7, C f .
We further calculated the weighted sum of the extracted principal components to obtain the final evaluation value, which is the variance contribution rate of each principal component. We divided the femur samples into three categories based on the principal component composite scores. We used variance analysis to verify the rationality of the classification. The results confirmed that most morphological parameters were statistically significant in the differences between classes (i.e., Sig. less than 0.05), indicating classification rationality. This suggests that the extraction method used in this study is reasonable and effective.

Construction of the plate feature model
Our previous work was based on bone plate features based on the idea of features 32 . Plate features were formalized into a quadruple form, including constraint relationships, mapping relationships, and semantic parameters. Because its shape contained a feature point, feature curve, and surface, a plate was expressed at a higher level www.nature.com/scientificreports/ to avoid the tedious process of designing from the bottom; this improves the design efficiency and quality. By following the same rationale for this study, we enhanced the plate feature model by integrating the bone's morphological features to set the semantic parameters of the plate's features. The key steps included the representation of the relation, semantic feature parameters, and mapping relation.

Constraint relationships. The constraint relation of the surface features primarily includes structural
and dimensional constraints [33][34][35] . Typically, shape features represent the plates. The structural constraint is also referred to as the shape constraint. The shape contains points, curves, and surfaces. The curve is an important bridge from a point to a surface and its topological relation determines the feature contour shape of the surface. As a result, shape constraints are mainly reflected in the topological relationships between the feature curves. For the undersurface of the orthopedic plates, the feature curves primarily include the boundary curve (referred to as B) and internal auxiliary curve (referred to as I). For brevity, the numbers represent the position (separation or intersection) between the two feature curves, as shown in Fig. 4a. The topological relation between the boundary feature curve and the internal auxiliary feature curve is called the BI relation, based on the number of intersection points between the boundary feature curve and the internal auxiliary feature curve. The topological relation between any two internal auxiliary feature curves, called the II relation, is based on the position of the intersection points between the two internal auxiliary feature curves. Figure 4b-d shows the undersurface of the eagle-shaped plate constructed as a skeleton structure. The bone plate has a good mechanical effect 36 and is accepted as fracture types 31A3 and 32A3 37 . The feature curves contain the boundary feature curve (C 0 ) and the auxiliary feature curves. We used a boundary feature curve to represent the edge profiles. The auxiliary feature curve also resembles the skeleton and contains ridge curves (C 1 and C 2 ) and costal curves (from C 3 to C 11 ). The following adjacency matrix represents the structural constraints of the feature curves.  Semantic feature parameters. Size parameters include global feature parameters that reflect the common shape of the plate and local parameters, which are constructed by dividing the global parameters into several special feature parameters according to different angles. Furthermore, we constructed several generic feature parameters using the attributes of the global parameters. We also constructed several special feature parameters using the local parameter attributes. The semantic feature parameters primarily refer to those with certain semantics that support high-level operations 38 . The definition of feature parameters corresponds to the specific engineering semantics and functional information, facilitating surface feature instantiation.
As shown in Fig. 5, the size parameters of the undersurface of the eagle-shaped plate include the total length L, total width W, tail length l 1 , head length l 2 , local tail widths w 0 and w 1 , local head width w i (i = 2, 3, …, 8), local tail bump height h j (j = 0, 1), and local head bump height, h j (j = 2, 3, …, 8). Moreover, L and W are mainly determined by the bone morphological parameters, including femur length H f and femoral shaft coronal diameter D fs . L is divided into tail length and head length by inheritance; similarly, W is divided into tail width and head width by inheritance.
Mapping relations. Based on the concept of layering from whole to part, the mapping relation is divided into two levels 38 : • Primary level mapping refers to the correspondence between the feature parameters of the upper layer and the feature curves of the middle layer. • Secondary level mapping refers to the correspondence between the feature points of the middle and bottom layers.
Because this mapping relationship did not adequately incorporate bone morphology, the constructed plate feature was relatively independent and, as a result, did not fit well with the bone morphological features. In this study, based on the above research, we integrated the principal component into the primary mapping to form a new mapping relation, which is called component mapping (see Fig. 6a). The mapping relation is represented as , where F 1 is the primary mapping, F 2 is the secondary mapping, P 1 denotes the upper layer, P 2 denotes the middle layer, and P 3 denotes the bottom layer. The component mapping, −→ Y|X ∈ P 1 ,Y ∈ P 2 } , w h e r e P 4 i s t h e s e t o f p r i n c i p a l c o m p o n e n t s , P 4 = {p 1 , p 2 , p 3 , p 4 , q 1 , q 2 , q 3 , q 4 , q 5 , q 6 , q 7 } , extracted as described in "Extraction of bone morphological features" section. Here, we add the principal components to the mapping as weights. Figure 6b illustrates this, and Fig. 6c shows the principal component values. BI_21 BI_21  I  II_11 II_0 II_0 II_0 II_0 II_0 II_0 II_12 II_13 II_13  I  II_13 II_13 II_13 II_13 II_13 II_13 II_12 II_0 II_0  I  II_0 II_0 II_0 II_0 II_0 II_0 II_0 II_0  I  II_0 II_0 II_0 II_0 II_0 II_0 II_0  I  II_0 II_0 II_0 II_0 II_0 II_0  I  II_0 II_0 II_0 II_0 II_0  I  II_0 II_0 II_0 II_0  I  II_0 II_0 II_0  I  II_0 II_0  I  II_0  www.nature.com/scientificreports/

Generation of the customized plate
We edited the semantic parameters to generate customized features of the eagle-shaped plate using Microsoft Visual C++ 2008 and Dassault Systèmes CATIA V5R22. We modified tail length l 1 by changing the feature points of the tail. We also modified the local width w i (i = 1, 2, 3, …, 8) and height h i (i = 1, 2, 3, …, 8) by changing the boundary feature points in the head. Figure 7a shows how the entity of the eagle-shaped plate was generated by stretching the undersurface; Fig. 7b shows that the entity parameters include the tail thickness t 1 , head thickness t 2 , and hole parameters (ød 1 , ød 2 , ød 3 , and s 1 ); Fig. 7c shows the eagle-shaped plates with different parameters.   www.nature.com/scientificreports/ Figure 7d shows good adhesion between the designed eagle-shaped plate and femur. To prove this further, we calculated the distances between the feature points on the femoral contact area and the undersurface of the plate. The results show that every distance deviation was controlled within 0.16 mm; this shows that the plate designed in this study matches well with the bone and thus guarantees the design requirement of the bone plate. Table 3 presents the comparison results between the proposed method and existing design methods. Notably, this method constructs a customized bone plate by blending bone morphological features into the semantic layer of the plate feature model; thus, design efficiency can be improved by avoiding the design from scratch while ensuring a match between the bone plate and the bone.

Discussion and conclusion
Conventional design methods for customized plates heavily rely on interactions with medical professionals. They also have a low level of reusability and often result in large deviations from the design objectives. In this study, we extracted bone morphological features using the PCA method and incorporated them as weights into the mapping relation to construct plate feature models. The main contributions of this study are as follows: • First, to the best of our knowledge, this is the first study in which the bone's morphological features and plate features are fully encapsulated with an internal correlation mechanism. Undoubtedly, this greatly improved the matching degree between the plate and bone. The construction of the feature parameter set of the plate inherits the existing features and thus avoids unnecessary duplication of features. • For the second contribution, we obtained different plate features by adjusting the upper-level feature parameters. This improves the flexibility and applicability of the design, reducing the repetitive tasks required in conventional methods and shortening the plate design cycle.
The experimental results show that the constructed plate feature model is competitive in design convenience because it facilitates quick and efficient intuitive editing of the feature parameters. The proposed method can be easily extended to other types of bones and plates and can provide insights, methods, and techniques for other fields of customized design.
Notably, we did not introduce the feature parameter modification sequence of the plate feature model in this study. To address this potential disadvantage, future studies should prioritize setting the feature parameters. Further consideration should also be given to the hardness parameters of the bone plate and appropriate materials should be reasonably selected. More exciting research can be conducted on a feature template that can be constructed by integrating a group of bone morphological features with the same fracture type and different anatomical morphology into a set of semantic feature parameters of the plate object. Table 3. Comparison between the proposed method and existing methods.