1. Introduction
The knee is composed of the femur, tibia, fibula, and patella (or knee cap), the extensive ligaments, and two main muscle groups (
i.e., quadriceps and the hamstrings) [
1]. The knee joint has a crescent-shaped fibrocartilaginous structure, named meniscus, which fits into the joint between the tibia and the femur. It allows the bones to slide freely on each other in the joint. A pair of knees support nearly the entire weight of the human body, and help the body perform different mobility functions in the locomotion [
2]. The knee joint can afford moderate stress without much internal injury, but it is not able to withstand rotational forces that commonly occur in athletic activities. In addition to the sports injury, the degeneration or damage of cartilage in the articular surface may cause rheumatic disorders such as osteoarthritis [
3]. Screening of knee joint disorders at an early stage can provide preliminary information for physicians to undertake some appropriate therapy options in order to retard the degenerative process of arthritis [
1].
Detection of knee joint disorders requires a thorough medical history, physical examination, and may need some additional tests such as magnetic resonance imaging (MRI) or computed tomography (CT). MRI is commonly used to characterize the
in vivo orthopaedic condition of articular cartilage in the knee joint [
4]. It is a popular noninvasive method for the assessment of knee joint degeneration. Although MRI provides good image contrast that helps discriminate the articular cartilage and synovial fluid of the joint, it only displays anatomical morphology of the knee joint rather than the functional condition illustration during leg movement. In the standard CT scanning procedure, the X-ray beam moves in a circle around the body. CT images can provide much detailed information about the bone tissue and bone structure [
4]. However, CT scanning takes the risks of radiation exposure due to the cumulative number of X-ray examinations.
Arthroscopy is a common surgical procedure to visualize, diagnose, and sometimes treat the interior damage of a knee joint [
1]. In an arthroscopy examination, orthopaedic surgeons insert an arthroscope fiber through a small incision in the knee to inspect the structure inside the joint. Compared with the traditional open surgery, arthroscopy is a semi-invasive method that does not require to open up the joint fully. The knee arthroscopy just needs to make two small incisions: one for the arthroscope and the other for the surgical instruments to be used in the knee cavity. It could reduce recovery time and may also increase the rate of surgical success due to less trauma to the connective tissue. Although arthroscopy is widely used as the “gold standard” for relatively low-risk assessment of knee joint injury [
3], it is not suited for the patients whose knees are in a highly degenerated state due to ligamentous instability, meniscectomy, or patellectomy. In addition, arthroscopy cannot be used for routine examinations of articular cartilage because the same incision may not undergo repeated arthroscope invasions.
Vibration arthrometry is an alternative and useful technology for the screening of knee joint pathology [
5,
6]. The protocol usually requires the subject to perform voluntary leg movement, and uses accelerometers or electro-stethoscope sensors attached on skin of the knee to record the acceleration time series called vibroarthrographic (VAG) signal. The healthy articular surfaces are bathed by joint fluid and can prevent any collision in the knee joint during leg movements. For the patients who suffer from the degeneration of cartilages, the articular surface frictions are very likely to occur in walking and sporting activities [
3,
7]. Knee vibration arthrometry can provide more information about the stability behavior of a knee joint, and is used as a noninvasive detection technique to study knee joint disorders [
1].
The purpose of knee joint VAG signal analysis is to interpret distinct features in the temporal and frequency scales [
8,
9,
10,
11]. With the statistical features extracted [
12,
13,
14,
15], computational algorithms can be utilized to distinguish anomalous patterns associated with pathology [
10,
16,
17,
18]. The aim of the present study is to use the kernel density estimation method with two-dimensional Gaussian kernels to represent the knee joint VAG signals in the bivariate feature space. The classification task is then performed by using the maximal posterior probability decision criterion that can distinguish the VAG signals into the normal or pathological group.
3. Bivariate Probability Distribution Modeling
In order to estimate the distribution of the knee joint VAG signals in the bivariate feature space, we applied the multivariate kernel density estimation method. The kernel density estimation can be used to approximate arbitrary multimodal distributions without the assumption of underlying density functions [
22,
23]. In the present study, we used the kernel functions to estimate the class-conditional probability density of the bivariate features for the two VAG signal groups respectively. Let
be the vector of the VMS and FF features. Suppose that a particular VAG group
ω, (
,
and
indicating the normal and abnormal signal groups respectively) contains
K signals in total. The class-conditional probability density is then estimated as [
22]:
The bivariate Gaussian kernel function
is written as
The center of Gaussian kernel is
, which is the location of the
k-th VAG signal in the group
ω. The spread area of the Gaussian kernel is determined by the symmetric and positive definite covariance matrix, which is given by
where
and
represent the standard deviation of the VMS and FF features respectively,
denotes the correlation coefficient between these two features, and
g is the scaling factor that controls the bandwidth of the Gaussian kernel. In order to estimate the appropriate probability distributions in the bivariate feature space, we searched the scaling factor in the range from 0.01 to 1, and chose the optimal scaling factor
that helped minimize the class-conditional probability error or risk between the normal and abnormal signal groups.
5. Results
The estimated distributions of the normal and abnormal knee joint VAG signals in the bivariate VMS-FF feature space are illustrated in
Figure 1. It can be observed that the normal and abnormal VAG signals locate in the distinctive regions in the feature space. The majority of the normal signals aggregates in the area with the VMS value smaller than 0.012 and the FF value lower than 6. The normal signals recorded from the healthy subjects possess the unimodal probability density, whereas the abnormal signals associated with pathological conditions in the knee joint present more of multimodal characteristics.
Figure 1.
Distributions of the bivariate features for the normal and abnormal knee joint vibration signals. The distribution of the normal signals is shown with the cold color map (blue representing the highest density), whereas the distribution of the abnormal signals is illustrated with the hot color map (red representing the highest density).
Figure 1.
Distributions of the bivariate features for the normal and abnormal knee joint vibration signals. The distribution of the normal signals is shown with the cold color map (blue representing the highest density), whereas the distribution of the abnormal signals is illustrated with the hot color map (red representing the highest density).
The decision boundaries given by the FLDA, the SVM, and the MPP classifiers are shown in
Figure 2. The FLDA made a linear decision boundary close to the vertical line to split the bivariate VMS-FF feature space into two parts. The SVM and MPP produced the different nonlinear discriminant boundaries to separate the normal and abnormal VAG signals. The SVM provided the decision boundary similar to an S-shape curve based on the selected support vectors. The MPP treated the normal signal locating in the left-top corner separately and distinguished the other VAG signals based on the contours of the estimated bivariate feature distribution.
Figure 2.
Decision boundaries provided by the Fisher’s linear discriminant analysis (FLDA), the support vector machine (SVM) with polynomial kernels, and the maximum posteriori probability (MPP) estimation method.
Figure 2.
Decision boundaries provided by the Fisher’s linear discriminant analysis (FLDA), the support vector machine (SVM) with polynomial kernels, and the maximum posteriori probability (MPP) estimation method.
The FLDA and the SVM produced the same total classification accuracy of 81.33%, and the MPP method provided a higher correct accurate rate of 86.67%. The confusion matrices provided by the FLDA, SVM and MPP classifiers are listed in
Table 1,
Table 2 and
Table 3, respectively. The true negative rate of the FLDA (specificity: 0.8936) was worse than that of the SVM (specificity: 0.9149), whereas the FLDA provided a better true positive rate (sensitivity: 0.6786) than the SVM (sensitivity: 0.6429), with an additional abnormal VAG signal correctly classified. Compared with the FLDA and SVM classifiers, the MPP method was able to correctly distinguish more VAG signals for the both normal and abnormal groups, with the highest values of sensitivity (0.75) and specificity (0.9362). The area under the receiver operating characteristic (ROC) curve
of the SVM was 0.8533 (standard error, SE: 0.0467). Such results was slightly worse than that of the FLDA (
SE:
), because the FLDA successfully classified one more abnormal signal than the SVM. The best ROC curve was obtained with the MPP method, with the highest
value of 0.9096 (SE: 0.0332). It is very clear that the MPP method outperformed the linear FLDA and the SVM with nonlinear inner-product kernels for the knee joint VAG signal classification. Moreover, the results of the MPP method were better than those of the radial-basis function neural network with the features of form factors, skewness, kurtosis, entropy (
: 0.8172) [
12], or with the features of variance of mean-squared values and turns count with adaptive threshold (
: 0.857) [
13].
Table 1.
The confusion matrix and area under the receiver operating characteristic (ROC) curve of the Fisher’s linear discriminant analysis (FLDA). : area under ROC curve. SE: standard error.
Table 1.
The confusion matrix and area under the receiver operating characteristic (ROC) curve of the Fisher’s linear discriminant analysis (FLDA). : area under ROC curve. SE: standard error.
| actual group |
---|
Total number | positive | negative |
---|
| 28 | 47 |
---|
classified group | positive | 24 | 19 | 5 |
| negative | 51 | 9 | 42 |
sensitivity | |
specificity | |
± SE | |
Table 2.
The confusion matrix and area under the receiver operating characteristic (ROC) curve of the support vector machine (SVM) with polynomial kernels. : area under ROC curve. SE: standard error.
Table 2.
The confusion matrix and area under the receiver operating characteristic (ROC) curve of the support vector machine (SVM) with polynomial kernels. : area under ROC curve. SE: standard error.
| actual group |
---|
Total number | positive | negative |
---|
| 28 | 47 |
---|
classified group | positive | 25 | 18 | 7 |
| negative | 50 | 7 | 43 |
sensitivity | |
specificity | |
± SE | |
Table 3.
The confusion matrix and area under the receiver operating characteristic (ROC) curve of the maximal posterior probability (MPP) decision. : area under ROC curve. SE: standard error.
Table 3.
The confusion matrix and area under the receiver operating characteristic (ROC) curve of the maximal posterior probability (MPP) decision. : area under ROC curve. SE: standard error.
| actual group |
---|
Total number | positive | negative |
---|
| 28 | 47 |
---|
classified group | positive | 27 | 21 | 6 |
| negative | 48 | 4 | 44 |
sensitivity | |
specificity | |
± SE | |
Most of the misclassified normal signals were recorded from the elderly healthy subjects. These subjects were suspected in the course of chronic articular cartilage disorders, although there was no symptom found in the current physical examinations for them. By checking with the arthroscopy surgery records, we found that majority of the misclassified abnormal VAG signals were associated with the lateral or medial meniscus tears, and the most of the corrected distinguished abnormal signals were recorded from patients who suffered chondromalacia at different grades.
Figure 3 shows a pair of abnormal signal segment samples recorded from the patients with Grade I–II chondromalacia (upper trace) and lateral meniscus tear (lower trace), respectively. It can be observed that the morphological characteristics of the two abnormal signal segments were different. The signal with chondromalacia exhibited a higher degree of fluctuations, and the morphology of the signal with lateral meniscus tear was closer to that of the signal recorded from healthy adults.
Figure 3.
Segments of the knee joint vibration signals associated with (a) Grade I–II chondromalacia and (b) lateral meniscus tear.
Figure 3.
Segments of the knee joint vibration signals associated with (a) Grade I–II chondromalacia and (b) lateral meniscus tear.
6. Discussion and Conclusions
Computer-aided analysis of knee joint vibration signals helps improve the noninvasive detection and localization of articular cartilage disorders at an early stage. The present study used the kernel-based probability density modeling method and maximal posterior probability decision criterion to study the VAG signals. The estimated densities showed the distributions of the VAG signals in the bivariate feature space, and the contours of the normal and abnormal signal groups were more explicit than the scatter plot of these signals. Experiment results of the signal classifications, in terms of total accuracy and area under the receiver operating characteristic curve, demonstrated the merits and superiority of the maximal posterior probability decision criterion, compared with the Fisher’s linear discriminant analysis and support vector machine with polynomial kernels.
The present study still has some limitations because the number of the abnormal signals with chondromalacia and meniscus tears was not large enough to support the further study of the pathological subclasses. It is important to extract other morphological features to describe the dynamics characteristics of the abnormal VAG signals with chondromalacia and meniscus tears in the future work. The present protocol did not consider the knee flexion and extension under the loaded conditions, because some subjects with severe knee joint disorders were not able to perform the requested leg movement under loaded conditions. Further experiments will recruit more subjects with knee joint disorders at different stages and some subjects with knee pain but without knee injury. We are interested in studying the morphological features of VAG signals recorded from these subjects at different loaded conditions. We also plan to incorporate the methodology presented in this paper in a follow-up investigation to establish the feature distributions of the subjects with degenerative articular cartilage disorders at different stages. It is positively considered that the progressive changes of feature density distributions could provide useful information about the progress of articular cartilage degeneration, which could assist physicians and medical experts to assess the pathological conditions of patients.