# Prediction of Joint Space Narrowing Progression in Knee Osteoarthritis Patients

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Methodology

#### 2.1.1. Data Pre-Processing

#### 2.1.2. Data Clustering

_{l}∈ $\mathcal{P}$, the medical data from patient $p$ for the left $\left(l\right)$ knee, and with p

_{r}∈ $\mathcal{P}$ the medical data from patient $p$ for the right $\left(r\right)$ knee. The sum of the absolute sum of each leg was calculated:

Algorithm 1 Pseudoalgorithm of the clustering process |

Input: JSM measurements of the first five visits Output: Labeled data |

1. For each patient p ∈ $\mathcal{P}$:Calculate the differences between the consecutive JSM measurements: ${d}_{j}^{p}={m}_{j}^{p}-{m}_{i}^{p},\forall i\in \left[1,\dots ,n-1\right],j\in \left[2,\dots ,n\right]$ Calculate the sum of the absolute differences: ${{\displaystyle \sum}}_{k=2}^{n}\left|{d}_{k}^{p}\right|$ End for each2. For each clustering method $m$ examinedPerform clustering evaluation with Davies Boulding index and calculate the optimal number of clusters ${C}_{m}$. Perform clustering with ${C}_{m}$ clusters End for each3. Return labels and evaluate the clustered data |

#### 2.1.3. Feature Engineering

Algorithm 2 Pseudoalgorithm of the feature selection |

Input: Clinical data |

1. All features were normalized as described in the Pre-processing Section 2. For each feature j, Set ${V}_{j}=0$ End for each$3$. For each FS technique $i$For each feature jIf feature $j$ is in $FS{S}_{i}$,
$${V}_{j}={V}_{j}+1$$
End ifEnd for eachEnd for each4. Rank features to descending order with respect to ${V}_{j}$. In case of equality the ranking is shaped from the feature importance of the best performing FS technique. End |

#### 2.1.4. Data Classification

- Gradient boosting model (GBM) is an ensemble ML algorithm, which can be used for classification or regression predictive tasks. Weak learners are used from GBM to produce strong learners through a gradual, additive, and sequential process. Hence, for the development of a new improved tree a modified version of the initial training data set is fitted in GBM [30,31];
- Logistic regression (LR) describes the relationship of data to a dichotomous dependent variable. LR is based on the logistic function (1). This model is designed to describe the data with a probability in the range of 0 and 1 [32]:

- Neural networks (NNs), both shallow and deep NNs were employed. NNs are based on a supervised training procedure to generate a nonlinear model for prediction. They consist of layers (e.g., input layer, hidden layers, and output layer). Following a layered feedforward structure, the information is transferred unidirectionally from the input layer to output layer through the hidden layers [33,34,35];
- Support vector machines (SVMs) are another supervised learning model [40,41]. SVMs target to create the hyperplane, which is a decision boundary between two classes that enables the prediction of labels from one or more feature vectors. The main aim of SVMs is to maximize the class margin that is actually the distance between the closest points (support vectors) of each class [42].

#### 2.1.5. Post-Hoc Interpretation/Explainability

## 3. Evaluation

#### 3.1. Medical Data

#### 3.2. Evaluation Methodology

#### 3.3. Results and Discussion

#### 3.3.1. Clustering Results

#### 3.3.2. Feature Selection Results

#### 3.3.3. Classification Results

#### 3.3.4. Post-Hoc Explainability Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Clustering process of the proposed methodology. JSM: Joint space narrowing on medial compartment.

**Figure 3.**K-means clustering results of (

**a**) left knee with Davies Bouldin index; (

**b**) right knee with Davies Bouldin index; (

**c**) left knee with 2 clusters; (

**d**) right knee with 2 clusters; and (

**e**) left and right knees combined with Davies Bouldin index.

**Figure 4.**The first 100 features selected for the left (

**top**), the right knee (

**middle**), and both legs (

**down**).

**Figure 5.**The accuracy of models over test set for increasing number of features for the left leg. Results are shown with a step size of 5 (two features added at each step).

**Figure 6.**The accuracy of LR (logistic regression) and SVM at [155, 175] features over the test set for left leg. Results are shown with a step size of 1 (one feature added at each step).

**Figure 7.**The accuracy of prediction models over test set for various number of features for the right leg. Results are shown with a step size of 5 (two features added at each step).

**Figure 8.**The performance evaluation of (

**a**) LR in the range of 175–195 features and (

**b**) SVM in the range of 85–95 features. Results are shown with a step size of 1 (one feature added at each step).

**Figure 9.**The accuracy of prediction models over test set for various number of features for the left and right legs combined. Results are shown with a step size of 5 (two features added at each step).

**Figure 10.**The accuracy of LR, RF (random forest), and SVM from 25 to 35 features over the test set for left and right legs combined. Results are shown with a step size of 1 (one feature added at each step).

**Figure 11.**The box plot of the prediction models based on their performance for the right and left legs combined.

**Figure 12.**The distribution of the features’ impact on LR model output for the OAI (osteoarthritis initiative) dataset with 29 features across all instances.

**Figure 13.**The average impact magnitude of 29 features on the LR model output for the OAI dataset for all instances.

Category | Description | Number of Features |
---|---|---|

Anthropometrics | Includes measurements of participants such as height, weight, BMI (body mass index), etc. | 37 |

Behavioral | Questionnaire results which describe the participants’ social behaviour | 61 |

Symptoms | Includes variables of participants’ arthritis symptoms and general arthritis or health-related function and disability | 108 |

Quality of life | Variables which describe the quality level of daily routine | 12 |

Medical history | Questionnaire results regarding a participant’s arthritis-related and general health histories and medications | 123 |

Medical imaging outcome | Variables which contain medical imaging outcomes (e.g., osteophytes and joint space narrowing (JSN)) | 21 |

Nutrition | Variables resultfrom the use of the modified Block Food Frequency questionnaire | 224 |

Physical exam | Variables of participants’ measurements, performance measures, and knee and hand exams | 115 |

Physical activity | Questionnaire data results regarding household activities, leisure activities, etc. | 24 |

Total number of features: | 725 |

Clustering Method | Parameters |
---|---|

K-Means | City block distance, 5 replicates |

K-Medoids | City block distance, 5 replicates |

Hierarchical | Agglomerative cluster tree, Chebychev distance, farthest distance between clusters, 3 maximum number of clusters |

**Table 3.**Hyper parameter settings for tuning. GBM: Gradient Boosting Model; LR: Logistic Regression; NN: Neural Networks; NBG: Naïve Bayes Gaussian; RF: Random Forest; SVM: Support Vector Machine.

Classification Model | Hyper Parameters Tuning |
---|---|

GBM | The number of boosting stages to perform from 10 to 500 with 10 step size The maximum depth of the individual regression estimators from 1 to 10 with 1 step size The minimum number of samples required to split an internal node: 2, 5 and 10 The minimum number of samples required to be at a leaf node: 1, 2 and 4 The number of features to consider when looking for the best split: $\sqrt{{n}_{features}}$ or ${\mathrm{log}}_{2}\left({n}_{features}\right)$ |

LR | The inverse of regularization strength was tested on 0.001, 0.01, 0.1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Algorithm to use in the optimization problem was set to 4 different solvers that handle L2 or no penalty, such as ‘newton-cg’, ‘lbfgs’, ‘sag’ and ‘saga’ A binary problem is fit for each label or the loss minimized is the multinomial loss fit across the entire probability distribution, even when the data is binary With and without reusing the solution of the previous call to fit as initialization |

NN | Both shallow and deep structures were investigated Hidden layers varying from 1 to 3 with different number of nodes per layer (50, 100, 200) Activator function: Relu and $tanh$ Solver for weight optimization: adam, stochastic gradient descent, stochastic gradient-based optimizer proposed by Kingma, Diederik, and Jimmy Ba and an optimizer in the family of quasi-Newton methods L2 penalty (regularization term) parameter: 0.0001 and 0.05 The learning rate schedule for weight updates was set as a constant learning rate given by the given number and as adaptive by keeping the learning rate constant to the given number as long as training loss keeps decreasing. |

NBG | - |

RF | The number of trees in the forest from 10 to 500 with 10 step size The maximum depth of the tree from 1 to 10 with 1 step size The minimum number of samples required to split an internal node: 2, 5 and 10 The minimum number of samples required to be at a leaf node: 1, 2 and 4 The number of features to consider when looking for the best split: $\sqrt{{n}_{features}}$ or ${\mathrm{log}}_{2}\left({n}_{features}\right)$ With and without bootstrap |

SVM | The regularization parameter was tested on 0.001, 0.01, 0.1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Kernel type was set to linear, polynomial, sigmoid and radial basis functions |

**Table 4.**Clustering results of each case with the Davies Bouldin index. The best results are indicated with bold.

Clustering Method | Number of Clusters | Cluster Elements | ||||
---|---|---|---|---|---|---|

Left | Right | Both | Left | Right | Both | |

K-Means | $4$ | $4$ | $2$ | $\left[2763,209,62,28\right]$ | $\left[2733,199,84,50\right]$ | [2822,209] |

K-Medoids | $4$ | $4$ | $2$ | $\left[2763,209,62,28\right]$ | $\left[2733,199,84,50\right]$ | $\left[2822,209\right]$ |

Hierarchical | $4$ | $3$ | $2$ | $\left[2960,74,24,4\right]$ | $\left[2989,68,9\right]$ | $\left[3016,15\right]$ |

Clustering Method | Number of Clusters | Cluster Elements | ||||
---|---|---|---|---|---|---|

Left | Right | Both | Left | Right | Both | |

K-Means | $2$ | $2$ | $2$ | [2763,299] | [2764,302] | [2822,209] |

K-Medoids | $2$ | $2$ | $2$ | $\left[2763,299\right]$ | $\left[2764,302\right]$ | $\left[2822,209\right]$ |

Hierarchical | $2$ | $2$ | $2$ | $\left[3034,28\right]$ | $\left[2989,77\right]$ | $\left[3016,15\right]$ |

**Table 6.**Maximum, minimum, and mean accuracy of prediction models over the tested set for the left leg. The best results are indicated with bold.

Prediction Model | Maximum Accuracy | Minimum Accuracy | Mean Accuracy | Standard Deviation |
---|---|---|---|---|

Gradient Boosting | 0.72611 | 0.56688 | 0.66707 | 0.02622 |

Logistic Regression | 0.77707 | 0.60510 | 0.71540 | 0.03353 |

NNs (Neural Networks) | 0.75796 | 0.62420 | 0.68234 | 0.02933 |

Naïve Bayes Gaussian | 0.68153 | 0.59236 | 0.62794 | 0.02301 |

Random Forest | 0.70064 | 0.61783 | 0.65989 | 0.01616 |

SVM | 0.76433 | 0.63057 | 0.70377 | 0.02783 |

**Table 7.**Maximum, minimum, and mean accuracy of prediction models over the tested set for the right leg. The best results are indicated with bold.

Prediction Model | Maximum Accuracy | Minimum Accuracy | Mean Accuracy | Standard Deviation |
---|---|---|---|---|

Gradient Boosting | 0.72611 | 0.61783 | 0.67172 | 0.02445 |

Logistic Regression | 0.77070 | 0.63057 | 0.70691 | 0.03560 |

NNs | 0.76433 | 0.58599 | 0.69983 | 0.03858 |

Naïve Bayes Gaussian | 0.72611 | 0.50955 | 0.62774 | 0.03926 |

Random Forest | 0.71975 | 0.61783 | 0.67577 | 0.02217 |

SVM | 0.77707 | 0.60510 | 0.68598 | 0.03929 |

**Table 8.**Maximum, minimum, and mean accuracy of prediction models over the tested set for the left and right legs combined. The best results are indicated with bold.

Prediction Model | Maximum Accuracy | Minimum Accuracy | Mean Accuracy | Standard Deviation |
---|---|---|---|---|

Gradient Boosting | 0.81746 | 0.69841 | 0.74591 | 0.02449 |

Logistic Regression | 0.83333 | 0.65873 | 0.76503 | 0.03725 |

NNs | 0.79365 | 0.64286 | 0.73870 | 0.03470 |

Naïve Bayes Gaussian | 0.76984 | 0.50000 | 0.63300 | 0.06331 |

Random Forest | 0.79365 | 0.61111 | 0.70755 | 0.05645 |

SVM | 0.82540 | 0.64286 | 0.74928 | 0.04223 |

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**MDPI and ACS Style**

Ntakolia, C.; Kokkotis, C.; Moustakidis, S.; Tsaopoulos, D.
Prediction of Joint Space Narrowing Progression in Knee Osteoarthritis Patients. *Diagnostics* **2021**, *11*, 285.
https://doi.org/10.3390/diagnostics11020285

**AMA Style**

Ntakolia C, Kokkotis C, Moustakidis S, Tsaopoulos D.
Prediction of Joint Space Narrowing Progression in Knee Osteoarthritis Patients. *Diagnostics*. 2021; 11(2):285.
https://doi.org/10.3390/diagnostics11020285

**Chicago/Turabian Style**

Ntakolia, Charis, Christos Kokkotis, Serafeim Moustakidis, and Dimitrios Tsaopoulos.
2021. "Prediction of Joint Space Narrowing Progression in Knee Osteoarthritis Patients" *Diagnostics* 11, no. 2: 285.
https://doi.org/10.3390/diagnostics11020285