^{1,2}

^{1,2}

^{1,2}

^{1}

^{2}

Femur parameters are key prerequisites for scientifically designing anatomical plates. Meanwhile, individual differences in femurs present a challenge to design well-fitting anatomical plates. Therefore, to design anatomical plates more scientifically, analyses of femur parameters with statistical methods were performed in this study. The specific steps were as follows. First, taking eight anatomical femur parameters as variables, 100 femur samples were classified into three classes with factor analysis and Q-type cluster analysis. Second, based on the mean parameter values of the three classes of femurs, three sizes of average anatomical plates corresponding to the three classes of femurs were designed. Finally, based on Bayes discriminant analysis, a new femur could be assigned to the proper class. Thereafter, the average anatomical plate suitable for that new femur was selected from the three available sizes of plates. Experimental results showed that the classification of femurs was quite reasonable based on the anatomical aspects of the femurs. For instance, three sizes of condylar buttress plates were designed. Meanwhile, 20 new femurs are judged to which classes the femurs belong. Thereafter, suitable condylar buttress plates were determined and selected.

Orthopaedic surgeons often use anatomical plates to treat bone fractures [

Anatomical information of the bone is the basis for the design of anatomical plates. Thus, analysis of bone parameters is very important and essential. In recent years, many scholars have carried out studies of bone parameters. Dong and Zheng [

The femur is the bone that is most commonly fractured. Thus, we will focus on analyses of femurs for anatomical plate design. While a customized plate is designed using an individual’s own anatomy, a general plate can be naturally designed using an average femur model of a specific population. The average model can be easily achieved with advanced statistical methods [

The main aims of this paper are twofold. First, it aims to classify femurs into different classes with advanced statistical methods. Second, it aims to design average anatomical plates with different sizes to be suitable for femurs in the different classes. Then, for a new femur, judge which class that femur would fall into based on Bayes discriminant analysis, thereby allowing a suitable anatomical plate for the new femur to be determined. To achieve these aims, statistic methods (such as factor analysis, Q-type cluster analysis, and Bayes discriminant analysis) and software (such as Mimics and Catia) were used. Experiment showed that femurs are rationally classified into three classes. The condylar buttress plate was taken as an example to illustrate the design of anatomical plates based on classified femurs.

To analyse anatomical information of femurs, anatomical parameters are necessary. These parameters include the height of the total femur (

Feature points in reference entity.

Points | Description |
---|---|

| The central point of femoral head. |

| The central point of femoral neck. |

| The central point of the interface between femoral trochanter and femoral shaft. |

| The central point of the interface between femoral shaft and femoral condyle. |

| The highest point of femoral trochanter. |

| The pit of medial condyle of femur. |

| The convex point of lateral condyle of femur. |

| The lowest point of medial condyle of femur. |

| The anterior medial condyle point. |

| The posterior medial condyle point. |

| The anterior lateral condyle point. |

| The posterior lateral condyle point. |

Description of femur parameters.

Parameters | Description |
---|---|

| The distance between |

| The angle between the line |

| The distance between |

| The average value of distances from |

| The average value of distances from |

| The vertical distance between |

| The distance between |

| The distance between |

Parameters defined on reference entity.

Reference entity

Anatomical parameters

In this study, a total of 100 unrelated healthy adults, who belonged to the Chinese Han ethnic group, were voluntarily enrolled. After the subjects were informed about associated risks, a questionnaire was given to obtain the subject’s age, sex, medical history, and physical activity, under the direction of a clinician. We adopted the exclusion criteria detailed elsewhere [

After the demographics and medical history were obtained, each subject’s femur CT images were imported into Mimics software version 15.0 (Materialise, Belgium). Thereafter, femoral contours were segmented and a three-dimensional (3D) model was calculated based on these contours. Finally, eight anatomical parameters of each reconstructed 3D model were measured and are depicted in Figure

Anatomical parameters of 100 femur samples.

Due to the fact that factor analysis will be used to analyse the femur parameters, Kaiser-Meyer-Olkin (KMO) measure [

KMO and Bartlett’s test.

KMO and Bartlett’s test | |
---|---|

Kaiser-Meyer-Olkin measure of sampling adequacy | 0.788 |

| |

Approx. chi-square | 1368.567 |

df | 28 |

Sig. | 0.000 |

In addition, histograms and normal curves for the variables are intuitively described in Figure

Histograms and normal curves of eight parameters.

The above tests showed that the sample femur data not only obeyed normal distribution but also had strong correlations. Thus, factor analysis, cluster analysis, and discriminant analysis could be conducted. As shown in Figure

Study workflow.

To express the method more concisely and clearly, the specific steps are listed as follows.

Factors are extracted and factor scores are calculated using factor analysis [

Using Q-type clustering [

Based on our knowledge of anatomy, average anatomical plates are designed for each class of femurs.

Based on Bayes discriminant analysis [

During factor analysis, important factors were determined using the PCA extraction method. The effect of the extracted components on the original variable is evaluated by eigenvalues (also called variance). The values of eight eigenvalues can be clearly seen in the scree plot (see Figure

Scree plot.

Component plot in rotated space.

Based on the rotated component matrix (see Table

Rotated component matrix.

| Rotated component | |
---|---|---|

Component | ||

1 | 2 | |

| | |

| | |

| | |

| | |

| | |

| | 0 |

| | |

| | |

Extraction method: PCA.

Rotation method: varimax with Kaiser normalization.

To further present the factors’ explanatory ability for the original variables, the percent of variance was used. The higher the percent of variance, the stronger the explanatory ability the extracted factors have. According to the accumulative variance contribution shown in Table

Total variance explained.

Component | Initial eigenvalues | Extraction sums of squared loadings | Rotation sums of squared loadings | ||||||
---|---|---|---|---|---|---|---|---|---|

Total | % of variance | Cumulative % | Total | % of variance | Cumulative % | Total | % of variance | Cumulative % | |

1 | 5.884 | 73.548 | 73.548 | 5.884 | 73.548 | 73.548 | 5.884 | 73.546 | 73.546 |

2 | 1.085 | 13.561 | 87.110 | 1.085 | 13.561 | 87.110 | 1.085 | 13.564 | 87.110 |

3 | 0.603 | 7.542 | 94.651 | ||||||

4 | 0.244 | 3.053 | 97.704 | ||||||

5 | 0.112 | 1.396 | 99.101 | ||||||

6 | 0.049 | 0.615 | 99.715 | ||||||

7 | 0.015 | 0.190 | 99.905 | ||||||

8 | 0.008 | 0.095 | 100.000 |

Extraction method: PCA.

According to the dendrogram (Figure

Descriptive statistic report for three classes of femurs.

Ward method | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

| Mean | 416.9869 | 119.2710 | 80.4011 | 21.4592 | 15.3946 | 308.3487 | 60.1973 | 68.5485 |

| 52 | 52 | 52 | 52 | 52 | 52 | 52 | 52 | |

Std. deviation | 22.00512 | 4.53646 | 4.41977 | 1.07996 | 0.72218 | 17.90407 | 2.99996 | 3.66439 | |

| |||||||||

| Mean | 383.5337 | 122.3039 | 70.3901 | 18.4421 | 13.3384 | 285.3189 | 53.9300 | 61.1437 |

| 38 | 38 | 38 | 38 | 38 | 38 | 38 | 38 | |

Std. deviation | 16.04435 | 3.98275 | 2.36992 | 0.80654 | 0.57340 | 13.11820 | 2.23255 | 2.74853 | |

| |||||||||

| Mean | 416.4560 | 133.4690 | 76.5051 | 20.9436 | 14.6423 | 316.1370 | 58.2902 | 66.0311 |

| 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

Std. deviation | 17.68810 | 5.54150 | 5.44721 | 1.36577 | 0.75898 | 18.12024 | 3.08894 | 3.47538 | |

| |||||||||

Total | Mean | 404.2216 | 121.8433 | 76.2073 | 20.2611 | 14.5380 | 300.3762 | 57.6250 | 65.4829 |

| 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |

Std. deviation | 25.28023 | 6.04407 | 6.09063 | 1.75628 | 1.17593 | 20.11964 | 4.01587 | 4.79921 |

The dendrogram for hierarchical cluster.

Although significant differences between different classes can be seen in Table

ANOVA.

| Sum of squares | df | Mean square | | Sig. |
---|---|---|---|---|---|

| |||||

Between groups | 26234.001 | 2 | 13117.000 | 34.354 | 0.000 |

Within groups | 37035.897 | 97 | 381.813 | ||

Total | 63269.898 | 99 | |||

| |||||

| |||||

Between groups | 1703.713 | 2 | 851.856 | 43.198 | 0.000 |

Within groups | 1912.830 | 97 | 19.720 | ||

Total | 3616.543 | 99 | |||

| |||||

| |||||

Between groups | 2201.375 | 2 | 1100.687 | 72.575 | 0.000 |

Within groups | 1471.113 | 97 | 15.166 | ||

Total | 3672.487 | 99 | |||

| |||||

| |||||

Between groups | 205.028 | 2 | 102.514 | 99.103 | 0.000 |

Within groups | 100.339 | 97 | 1.034 | ||

Total | 305.366 | 99 | |||

| |||||

| |||||

Between groups | 92.950 | 2 | 46.475 | 102.578 | 0.000 |

Within groups | 43.948 | 97 | 0.453 | ||

Total | 136.898 | 99 | |||

| |||||

| |||||

Between groups | 14404.542 | 2 | 7202.271 | 27.215 | 0.000 |

Within groups | 25670.659 | 97 | 264.646 | ||

Total | 40075.201 | 99 | |||

| |||||

| |||||

Between groups | 867.309 | 2 | 433.655 | 57.679 | 0.000 |

Within groups | 729.282 | 97 | 7.518 | ||

Total | 1596.591 | 99 | |||

| |||||

| |||||

Between groups | 1207.178 | 2 | 603.589 | 54.563 | 0.000 |

Within groups | 1073.034 | 97 | 11.062 | ||

Total | 2280.213 | 99 |

Usually, the type of anatomical plate used for the treatment of a femoral fracture is decided according to the surgeon’s clinical experience. A condylar buttress plate [

Parameter values of three condylar buttress plates.

Classes | Means (mm) | |||
---|---|---|---|---|

| | | | |

| 160.0 | 14.0 | 42.0 | 2.0 |

| 130.0 | 10.0 | 38.0 | 1.8 |

| 150.0 | 12.0 | 40.0 | 2.0 |

Classified condylar buttress plates.

Parameters of condylar buttress plates

Classified condylar buttress plates

For a new femur, to judge which class the new femur belongs to, Bayes discriminant analysis was used in this study. Based on the classification function coefficients in Table

For

For

For

Classification function coefficients.

| Ward method | ||
---|---|---|---|

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

(Constant) | | | |

Values of

Classification results.

Classification | ||||||
---|---|---|---|---|---|---|

Ward Method | Predicted group membership | Total | ||||

| | | ||||

Original | Count | 1 | 52 | 0 | 0 | 52 |

2 | 0 | 38 | 0 | 38 | ||

3 | 0 | 0 | 10 | 10 | ||

Ungrouped cases | 7 | 11 | 2 | 20 | ||

% | 1 | 100.0 | 0.0 | 0.0 | 100.0 | |

2 | 0.0 | 100.0 | 0.0 | 100.0 | ||

3 | 0.0 | 0.0 | 100.0 | 100.0 | ||

Ungrouped cases | 35.0 | 55.0 | 10.0 | 100.0 |

To design anatomical plates (such as condylar buttress plates) for femurs more scientifically, femur parameters were analysed with statistical methods in this study. Femurs were classified into three classes based on factor analysis and Q-type cluster analysis. Then, three average condylar buttress plates, one for each class of femur, were designed. For a new femur, Bayes discriminant analysis was used to judge which class the new femur fell into. A total of 20 new femurs had their class assigned and suitable plates were determined. Experiments showed that our classification of femurs was rational and provides a scientific basis for the design of anatomical plates. The contributions of the method in this paper are twofold:

Femur parameters were classified into three classes with factor analysis and Q-type cluster analysis. In factor analysis, “size factor” and “angle factor” were extracted with the PCA method. This simplification was appropriate according to our knowledge of human femur anatomy. Through analysis of the classified femurs, a pattern was found, a relationship between the “size factor” and “angle factor” relative to a given person’s heights. For tall people, the “size factor” is generally larger than average, while the “angle factor” (a manifestation of polarity) is typically below average, with only a small portion of tall people having an above average value. For short people, the “size factor” is generally below average, while the “angle factor” is above the average.

Taking condylar buttress plates as an example, three average plates, one per femur class, were designed based on the average parameters of each class. With this system, for a new femur, if we want to select a well-fitting condylar buttress plate, we only need to judge which class the new femur falls into. One nice thing about this design is that the selected condylar buttress plate can be better contoured and bent to follow the anatomy of the new femur. Thus, reshaping and trimming of the selected plate during surgery can be minimized or even avoided to an extent.

The analysis of femur parameters has several benefits for research. The average model of each class of femurs can be used as the starting point for optimizing an anatomical plate. In addition, the quantitative ratio of femurs of different classes can help to optimize the quantities of different sized plates that are manufactured. Specifically, in this study, the quantitative ratio of

However, there are some deficiencies in this study. The first is that the number of femur samples was relatively small. Although 100 samples were sufficient to describe the integral anatomy of femurs in a population, a larger sample size, as well as continued scientific and scholarly discourse, is still essential and necessary. The second is that due to the study being limited to a specific population group, characteristics of a new population group should be calculated starting from the beginning. Fortunately, with the development of advanced digital calculation methods, the calculation processes involved in this study are not a significant technical hurdle.

In summary, femur parameters were classified into three classes based on factor analysis and Q-type cluster analysis. Condylar buttress plates with three different sizes, one for each class of femur, were designed. Meanwhile, a new femur could be analysed and assigned to its appropriate class. Finally, the most suitable condylar buttress plate was selected based on the assigned class. Considering the potential value of this study, assessment and optimization of the biomechanical properties of the designed condylar buttress plates with finite element analysis still need to occur. In addition, due to space limitations, only condylar buttress plates were used to illustrate the design of anatomical plates based on our classification scheme. Thus, further experimentation needs to be more extensive, for example, analysing other types of long bones in humans to be able to scientifically design other types of anatomical plates or even intramedullary nails.

Informed consent was obtained from all individual participants included in the study.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by Natural Science Foundation of Jiangsu Province of China (Grant no. BK20141158), Natural Science Foundation of China (Grant no. 61472118), Science and Technology Program of Jiangsu Province of China (Grant no. BE2014048), and Research & Innovation Project for Postgraduates in Jiangsu Province (Grant no. KYZZ15_0152).