1. Introduction
Keratoconus (KCN) is a noninflammatory disease typically affecting both eyes [
1]. Early detection of KCN and monitoring its progression are challenging tasks that require subjective evaluation of refractive corneal maps as well as other corneal parameters [
2]. While the shape of a normal cornea looks like a dome, the shape of a cornea with KCN looks like a cone.
Figure 1 shows the overall structure of a normal and KCN cornea.
Artificial intelligence (AI) techniques have shown promise for addressing several ocular conditions and have achieved accuracy that equals or esceeds expert ophthalmologist [
3,
4,
5,
6,
7,
8,
9,
10,
11]. Various machine learning techniques have been developed for KCN detection and refractive surgery screening [
12,
13,
14,
15] and are becoming a crucial tool to aid ophthalmologists in making a better KCN diagnosis. Machine learning has been applied to discriminate normal from KCN eyes based on corneal imaging parameters [
12,
15]. Most of these methods for KCN diagnosis rely on subjective evaluation of topographical maps [
16]. However, automated models may provide a more accurate and objective evaluation of KCN. Conventional machine learning models such as neural networks, binary decision trees, and Zernike polynomial have been applied to corneal topography indices for KCN diagnosis [
12,
17,
18,
19]. Maeda et al. [
20] applied one of the first supervised ML models to corneal topographic maps to classify KCN. Eleven topographic indices extracted from the topographic maps were fed to neural network to classify different grades of KCN. Mosa et al. [
21] applied ML to extract sixteen features from the four topographic maps including five indices from Pentacam measurements. The features were then fed to Support Vector Machine (SVM) and decision tree classifiers, where accuracies of 90% and 87.5% were achieved for both classifiers, respectively. Different ML techniques have been applied to corneal images collected based on OCTbased instrument (CASIA, Tomey, Japan) [
18]. A support vector machine (SVM) was applied on eight corneal parameters that provided a 94% accuracy on discriminating KCN from healthy eyes. Corneal parameters from 88 patients were analyzed with machine learning to detect subclinical KCN from healthy eyes [
17]. Using five parameters, the random forest method was able to achieve an area under the receiver operator characteristics curve (AUC) of 0.97 for detecting subclinical KCN. Most of these models have however used sole features from the anterior topography of corneal. Several follow up models have used features from the posterior of the cornea to evaluate KCN [
22,
23,
24]. However, from the machine learning perspective, it is ideal to evaluate a comprehensive set of features and select features that are most promising in classification.
Most of the current methods for automatic detection of KCN are supervised [
18,
22,
23,
24,
25,
26,
27]. Some models have achieved AUC up to 0.97 using Pentacam indices only and some have combined Pentacam indices with OCT parameters and have obtained AUROCs up to 1.0 [
26,
27]. Rare unsupervised models have been developed to identify KCN. Such an unsupervised approach requires no prelabeled data avoiding expert annotation biases [
28,
29,
30]. For instance, an unsupervised machine learning model was developed to identify KCN stages [
28]. Principal component analysis (PCA), manifold learning, and stochastic embedding was used to cluster 4 classes of KCN for CASIA OCT imaging system. The unsupervised methods for KCN detection may address of big data labeling challenge, as medical data labeling by a human expert, is one of the challenges of applying ML for the medical field [
31].
The main contributions of this study are: (i) to introduce a hybrid algorithm for an unsupervised learning model and optimization techniques for KCN detection; (ii) to evaluate the performance of the algorithm for KCN detection, and (iii) to assess the model based on two independent realworld datasets from different populations. The performance of the proposed model is evaluated based on five different metrics: precision, recall, FScore, purity, and accuracy. Furthermore, we evaluated the model based on two large dataset collected from two independent centers (Brazil and Japan).
2. Features Extraction
Pentacam data comprises topography/tomography, pachymetry, anterior chamber angle, volume, height, corneal and lens densitometry, and other ocular indices. Central Corneal Thickness (CCT) is a fundamental pachymetric index that is the basis for identifying corneal thinning disorders. Furthermore, the thinnest corneal thickness (TCT) is a valuable diagnostic parameter in detecting primary ectatic disease, including Volume Parameters and pupil position. The conic shape parameters also describe corneal shape. Conic shape parameters are the minimum curvature radius of the cornea (MCR in mm) and corneal eccentricity (ECC), indicating the corneal threshold of the peripheral curvature from the apical radius defines the degree of asphericity. The corneal parameters obtained are considered the most discriminative ones when it comes to KCN severity level identification. The largest dataset corresponds to Topometric (corneal shape) and topographic (corneal superficial landscape) indices that Pentacam captures and evaluates objectively. In addition to topometric and topographic data, Pentacam also generates anterior elevation and Central Keratoconus Index (CKI, which includes KCN index, index of height asymmetry, index of height decentration, index of surface variance, index of vertical asymmetry). Other features calculated by the difference between observed parameters and ideal shapes are the Posterior Elevation (PE) of the cornea, representing the maximum PE in a zone above the standardized reference shape, which is typically a best fit sphere (BFS) or best fit torric ellipsoid (BFTE) [
22,
23,
24,
32,
33,
34].
3. Background
In this section, we will describe the Flower Pollination algorithm (FPA), kmeans, kmedoid, and Spectral Clustering algorithm briefly.
3.1. Flower Pollination Algorithm (FPA)
In the optimization context, metaheuristic algorithms are classified into three main categories: evolutionarybased algorithms (EA), trajectorybased algorithms (TAs), and swarm intelligence (SI) [
35]. In fact, all the metaheuristic types are natureinspired such as the flower pollination algorithm (FPA). This algorithm was proposed by Yang in 2012 [
36] for global optimization. The main idea of the FPA is inspired from the flowering plants’ pollination behavior. Flowers are the main element in plant reproduction, which produce large quantities of pollen to attract pollinators (such as birds, butterflies, and bees). Through these pollinators, pollen grains will be transferred from one plant to another, and in this process, pollination takes place, and the plant continues to reproduce. This type of pollination is called global pollination, and it occurs between one plant and another. There is another type of pollination called local pollination, which takes place in the same plant, and the process of pollen transfer is carried out by wind or rain.
Figure 2 shows some examples of pollinators.
Yang [
36] considered such concepts to model the fundamentals of FPA in optimizationoriented problems.
Table 1 presents the equivalence or relationship between flower context and optimization terms.
The FPA has been applied to several realworld problems including medical, security and privacy, image and signal processing, feature selection, and electrical and power systems, with promising outcomes [
37,
38,
39]. In this study, we have integrated FPA with kmeans to detect KCN from corneal imaging parameters.
3.2. KMeans
The
kmeans is one of the wellknown heuristic clustering algorithms [
40]. The primary aim of
kmeans is identifying similar samples in the dataset based on their particular characteristics (i.e., features). There are many approaches to calculate the similarity among samples [
41]. The
kmeans start with a random selection of sample as cluster center (i.e., cluster centroid) and then use similarity measures such as Euclidean distance to calculate the similarity or dissimilarity between the cluster center and the sample. After assigning the closest sample for each cluster, the centroids of the clusters are recalculated based on the average values for all sample features. The
kmeans process is stopped once the predefined number of iterations is meet or until no more changing of the clusters centroids values. The main steps of
kmeans process are implemented in Algorithm 1.
Algorithm 1. Psudo code of kmeans clustering heuristic algorithm. 
 1:
Input: Number of clusters k, row data (samples) S.  2:
Output: Matrix contains S assigned to K.  3:
Select random from S as for each K.  4:
while the number of iterations is not meet or no more changing of the clusters centroids values do  5:
Calculate the similarity between the clusters centroids and S.  6:
Assign S to the closest cluster centroid.  7:
Calculate the clusters centroids.  8:
end while

3.3. KMedoid
kmedoid is another clustering algorithm based on partitions. It selects the actual dataset and identifies the clusters based on the medoids. A medoid reflects the samples in the data collection that is most centrally located. The primary objective of the kmedoid is to minimize the dissimilarity among the samples in the same cluster and maximize the dissimilarity among the rest of samples in others clusters based on the medoid.
The mechanism of the
kmedoid algorithm is like the
kmeans algorithm. It also starts with a random selection of sample as initial medoids to from the clusters. The sample with the closest medoid are assigned to the same cluster. A new medoid then represents the cluster. The samples are iteratively allocated to the clusters with the closest medoid—the medoids’ positions changes with each iteration. The
kmedoid seeks to reduce the number of differences between each sample and its corresponding medoid [
42]. As the
kmeans algorithm, the
kmedoid process is stopped once the predefined number of iterations is meet or until no more changing of the clusters medoids values. Ultimately,
K clusters are created with center on medoids, and all samples are placed into the fitting cluster on the basis of the nearest medoid. The main steps of
kmedoid process are implemented in Algorithm 2.
Algorithm 2. Psudo code of kmedoid clustering heuristic algorithm. 
 1:
Input: Number of clusters k, row data (samples) S.  2:
Output: Matrix contains S assigned to K.  3:
Select random from S as for each K.  4:
while the number of iterations is not meet or no more changing of the clusters centroids values do  5:
Calculate the similarity between the clusters medoids and S.  6:
Assign S to the closest cluster medoids.  7:
Calculate the clusters medoids.  8:
end while

3.4. Spectral Clustering Algorithm
The Laplacian matrix (LM) eigenvectors that correspond to clustering the samples in the dataset are used in the spectral clustering [
43]. The first step in the spectral clustering algorithm is to build an undirected graph based on the data points (i.e., features). The weight on edge is the similarity among the data points, and each vertex on the graph corresponds to a data point. To construct the similarity matrix, the researchers usually use the Gaussian kernel function. We can then get a matrix with degrees, whose main diagonal element is equal to the total of the row elements that fit the identity matrix. The Laplacian matrix is typically constructed in three ways: (1) denormalized LM, (2) normalized symmetric LM, and (3) normalized asymmetric LM. The eigenvector is utilized to calculate the first k eigenvalues of the LM. Then, by normalizing the LM, a new feature matrix is created. A sample is presented in each row of the feature matrix, clustered to produce clusters in the next step. The main steps of the Spectral Clustering algorithm process are implemented in Algorithm 3.
Algorithm 3. Psudo code of Spectral Clustering algorithm. 
 1:
Input: Let W be the weighted matrix, the number of clusters K, Similarity Matrix.  2:
Output: Matrix contains S assigned to K.  3:
Using the Gaussian kernel function to create the similarity matrix.  4:
Generate a symmetric normalized Laplacian matrix.  5:
Create the feature matrix from the feature vector corresponding to the first k eigenvalues.  6:
Normalized matrix, which includes n space points reduced to dimensions of k by normalize the feature matrix.  7:
apply cluster algorithm to cluster the samples (rows of the matrix).

4. Proposed Method
In this study, we integrate flower pollination algorithm with kmeans and evaluate the model based on realworld datasets. The proposed method includes five steps in which the output of each step is used as the input in the next step.
Figure 3 presents the diagram of the model that are described below.
4.1. Data Acquisition
The datasets utilized for this study consisted of Corneal parameters acquired with Pentacam device and OCT (CASIA, Tomey, Japan) instruments (Scheimpflug Imaging, Oculus, Germany) and OCT device from two independent cohorts in Brazil and Japan with different KCN severity levels. The severity levels were collected from the instrument without any expert intervention. The dataset from Brazil included 7339 corneal data collected from Department of Ophthalmology and Visual Sciences, Paulista Medical School, Federal University of São Paulo, São Paulo, Brazil hospital (3410 normal eyes and 3929 eyes at four different stages of KCN) in which the KCN severity level was obtained from To pographical KCN Classification (TKC) of the Pentacam instruments without expert intervention. The dataset from Japan included 9544 corneal data collected from Department of Ophthalmology, Jichi Medical University, Tochigi hospital (8539 normal eyes and 1005 eyes at four different stages of KCN) in which the KCN severity level was obtained from Ectais Screening Index (ESI) of the CASIA instruments without expert intervention. Data use agreements were signed among institutes and study was conducted in accordance with ethical standards in the declaration of Helsinki and its later amendments [
44]. The details of the number of cases are illustrated in
Table 2.
4.2. Data PreProcessing
Both datasets were examined and cases with examination error greater than zero were excluded from the study. Moreover, the cases of
$KCN$ progression according to
$TKC$ classification were considered with the highest grade (e.g.,
$KC$ 3–4 was assigned to the severity level of
$KC4$). We included 50 raw Corneal Shape, Thickness, and Elevation Parameters and excluded all parameters that were generated by the Pentacam instrument. To assess
KCN, i.e.,
$ISV$: Index of Surface Variance,
$IVA$: index of vertical asymmetry,
$KI$: Keratoconus Index,
$CKI$: Central Keratoconus Index,
$IHA$: index of Height Asymmetry,
$IHD$: index of Height Decentration and Rmin: (the minimum sagittal curvature). Five, three and two classes were investigated in this study where the 5class problem refers to healthy versus 4 levels of
KCN severity (
Table 2). The 2class problem included tackling the healthy cases versus all other abnormal cases of
KCN. Finally, we constructed a 3class problem subset which was generated from the original Brazil dataset which includes 3410 normal cases, 1650 cases of moderate
KCN (
KC1 and
KC2) and 2278 cases of severe
KCN (
KC3 and
KC4). All parameters were exported from the Pentacam to CSV file format and they were changed to Mat files for further processing.
4.3. Feature Extraction
Although using a comprehensive set of features may seem ideal in machine learning applications, but irrelevant features may confuse the classifier and degrade the accuracy. As such, feature selection is a critical step in data mining pipelines.
4.4. Clustering
Clustering algorithms try to group similar samples in the same cluster. As such, possible solution (to the problem) and a fitness function to evaluate the solution are two major components in the clustering process [
45,
46,
47]. The possible solution is the first step after preprocessing phase is providing feasible solutions for a particular problem while the fitness function evaluates these solutions in the clustering process. These components are discussed in the following subsections.
4.4.1. Representation of Solutions
Possible solution is presented by a vector indexing by number (n) of samples (s), where n is total number of $KCN$ features datasets. Each element of that vector is presented any cluster (k) number within the range $[1,k]$.
4.4.2. Fitness Function
In this section, we explain the fitness function to evaluate features quality of each solutions. The fitness function which is used in this work is the similarity measures. Usually, a clustering algorithm is used a different technique to minimize the distance between the centroids of the clusters and samples [
48]. The main steps to evaluate solution quality of the proposed clustering method can be summarized as the following.
For each cluster:
 1.
Find the cluster centroid;
 2.
Find the distance between input feature sample and the cluster centroid.
We can calculate the cluster centroid of cluster
j. as shown in Equation (
1).
where
$D{s}_{ji}$ refers to a binaryvalued vector,
n represents the total number of dataset, and
${K}_{j}$ refers to the cluster
j centroid. The
$D{s}_{ji}$ defined as follows:
Finally, the average distance of input samples for all clusters must be calculated. In this work, the average distance of samples to the cluster centroid measure (
$Av{e}_{cluster}$) is defined as shown in Equation (
3):
where
${n}_{j}$ represents the total sample number for cluster
j.
$D({K}_{j},{s}_{j}$ refers to the total distance between the input sample and the cluster centroid
j with the cluster
j. The main objective of the proposed method is to find the best solution by minimize the
$Av{e}_{cluster}$ value for each solution. That means, the proposed method assigns the input data to the optimal cluster or near optimal.
Figure 4 shows the fitness process.
5. Experiments and Results
To evaluate the performance of the proposed method (FPAkmeans) five quality assessment measures are calculated: namely recall, accuracy, purity, precision, fmeasure. These measures are calculated using the following:
Accuracy:
where
${T}_{a}$,
${T}_{r}$,
${F}_{a}$, and
${F}_{r}$ represent true acceptance, true reject, false acceptance, and false reject, respectively.
Precision:
where
${L}_{{i}_{j}}$ is the number of tasks from class
i correctly identified in cluster
j, and
$\left(\right)$ is the numbers of tasks (samples) in cluster
j.
Purity:
where
$max(i,j)$ stands for the maximum number of correct label assignment (i.e., correct classification) from class
i in cluster
j.
Recall:
where
${T}_{i}$ is the total number of tasks (samples) from class
i.
Figure 5 shows comparison of results between the proposed method (FPA
Kmeans) and standard Kmeans, Kmedoids, and Spectral clustering approaches.
Table 3 presents the experiment results with 25 runs concerning standard Kmeans, Kmedoids, and Spectral cluster. It is clear to notify that, the proposed hybrid method (FPA
Kmeans) achieves almost the best results overall measures for all dataset with 2, 3, and 5 classes. The best results obtained with 2 classes dataset where the FPA
Kmeans achieves 96.03, 96.29, 96.06, 96.17, and 96.03 for accuracy, Precision, Recall, Fmeasure, and Purity, respectively. Same scenario with 3 classes dataset the FPA
Kmeans achieves 71.02, 53.53, 75.37, 64.64, and 80.85 for accuracy, Precision, Recall, Fmeasure, and Purity, respectively. With 5 classes dataset the FPA
Kmeans almost achieves the best results except with Recall and Purity measures the proposed method obtained second ranked after the Spectral cluster. With such these results the FPA
Kmeans suitable for keratoconus detection and can be improved to get the best results.
Another evaluation measure is the plotting final the clustring
Figure 6 shows a scattering plotting for the used datasets.
Table 4 shows the comparison with previous machine learningbased studies for KCN detection. It should be noted that most of the previous studies used supervised machine learning classifiers [
18,
20], apart from [
28], where unsupervised machine leaning was used. The unsupervised machine learning used in [
28], included PCA, manifold learning and tSNE applied on 420 parameters obtained from CASIA corneal machine. Unlike [
28], our study flower pollination algorithm (FPA) and the kmeans algorithm and applied it on 50 features of the corneal shape.
Although FPAkmeans outperforms competing methods, it takes longer to reach the optimal solution and is more difficult to predict its generalizability in dealing with novel datasets. Additionally, the computational complexity of the FPAkmeans algorithm was calculated without considering the computational complexity of the objective function here thus not reflecting the overall computational complexity of the problem that may vary based on different applications. The time complexity of FPAkmeans calculated here does not take into account the time complexity of calculating the object function, which varies depending on the complexity of the problem.
6. Conclusions and Future Work
In this paper a new unsupervised machine learning model based on kmeans optimization through flower pollination algorithm for detecting KCN from corneal imaging data is proposed. We showed that this model outperforms other unsupervised models including simple kmeans based on scenarios with two, three, and five clusters. As the clustering approach does not require prelabeled data, the model is resistant to expert labeling bias. Validated model could be used in research and clinical practice to monitor the status of the eyes with KCN.
Two different datasets are used in this work consisted of Corneal parameters acquired with Pentacam device (Scheimpflug Imaging, Oculus, Germany) from two independent cohorts in Brazil and Japan with different keratoconus severity levels.
To evaluate the performance of the proposed method (FPAkmeans) five measures are calculated: namely recall, accuracy, purity, precision, fmeasure. Also, the performance of the proposed method is compared with three differnt clustering methods which are standard Kmeans, Kmedoids, and Spectral cluster. The proposed method achieved better results for all datasets used.
There are several aspects of this study that can be improved in future studies. First, other independent datasets are warranted to investigate the generalization of findings. Second, the proposed model may be compared against emerging deep learning models. Thirds, improving the computational aspects of the proposed model to work in near realtime situations.
Author Contributions
Conceptualization, S.Y., R.M.H. and H.T.; methodology, Z.A.A.A. and A.H.A.T.; software, Z.A.A.A., and A.K.A.; validation, Z.A.A.A., R.M.H. and H.J.M.; formal analysis, Z.A.A.A., A.H.A.T. and A.K.A.; investigation, Z.A.A.A. and S.Y.; resources, S.Y., R.M.H., H.T., J.A.M.F. and M.C.; data curation, R.M.H., H.T., J.A.M.F. and M.C.; writing—original draft preparation, Z.A.A.A., A.H.A.T. and A.K.A.; writing—review and editing, S.Y., A.H.A.T., A.L. and R.M.H.; visualization, H.J.M. and A.K.A.; supervision, S.Y.; project administration, S.Y.; funding acquisition, S.Y. All authors have read and agreed to the published version of the manuscript.
Funding
The APC was funded by the Department of Ophthalmology of the University of Tennessee Health Science Center (UTHSC) in Memphis through Siamak Yousefi’s laboratory.
Institutional Review Board Statement
The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Ethic Committee: CEPUNIFESP of Federal University of São Paulo (Protocol code CAAE: 28091019.3.0000.5505 20/08/2021).
Informed Consent Statement
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement
The first dataset used in this study from Brazil was collected from the Department of Ophthalmology and Visual Sciences, Paulista Medical School, Federal University of São Paulo, São Paulo, Brazil. The second dataset from Japan was collected from the Department of Ophthalmology, Jichi Medical University, Tochigi hospital.
Acknowledgments
The authors would like to acknowledge University of Tennessee for supporting research under grant number TS1234.
Conflicts of Interest
The authors declare that they have no conflicts of interests.
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