Keratoconus (KC) is an ectatic debilitating corneal disorder characterized by a progressive corneal thinning that results in corneal protrusion, irregular astigmatism, and decreased vision.^{1} Corneal elasticity and rigidity are severely affected in keratoconic eyes,^{2–4} which become more susceptible to the effect of any pressure, such as intraocular pressure. Consequently, the corneal shape is more easily distorted (corneal steepening and aberrometric increase in KC). This explains the usual significant increase in the anterior corneal irregularity and a deterioration of the visual quality in KC, aggravated by the high optical relevance of the first surface of the cornea.

Several grading systems have been described in the literature to classify the severity of KC.^{5–7} Most of these grading systems have been developed taking into account the visual performance of the patient, topographic morphology of the disease, the corneal keratometry readings, and corneal aberrometry^{8–10} and have been proven to be an essential tool in the therapeutic approach to the management of KC.

Nevertheless, there is a form of this disease, characterized by a milder modification in corneal topography and morphology but without the impairment of the visual function of the patient, that has been defined as an early KC, subclinical KC, or KC suspect. One of the main difficulties in relation to this entity is the lack of its clear definition in the literature.^{11} The subclinical KC is still considered the most important risk factor for developing post-LASIK ectasia,^{12,13} a devastating condition leading to a significant visual impairment of the patient. Thus, improving the screening strategies, tools, and techniques that allow us to identify those cases with the potential hazard of developing such a feared complication has become a major challenge within the ophthalmic community.

The topographic analysis of the anterior corneal surface is the main tool that has been used for the KC diagnosis and characterization for years. Several indices, both simple and compound, decision trees, and even neural networks based on the corneal topographic data and optical parameters have been developed to provide a more reliable tool to detect abnormal and borderline suspect corneas.^{14–29} For instance, the vertical coma of the corneal aberration is one of the simplest direct KC markers used in clinical practice.^{9,15}

Most of the corneal indices, published in the literature, are based on the elevation or curvature data of the cornea, as well as pachymetry^{30} or the epithelial thickness profile.^{31} In the case of the Placido-based topographers, these data are not obtained by direct (and verifiable) measurements but are an outcome of a mathematical processing of the image of the rings in the keratographic picture by more or less sophisticated (and in the case of commercial devices, by proprietary and not always transparent) algorithms.^{32–34} These procedures make important assumptions on the corneal shape (rotational symmetry, approximability by cubic splines, etc.) that are not always satisfied by a real corneal surface. Therefore, standard indices developed for KC detection based on curvature inherit unnecessarily the complexity of the currently used ring image-to-curvature conversion methods. They also might be affected by the unavoidable intrinsic errors appearing during such a conversion.^{35,36}

To overcome these shortcomings and to improve and complement the existing set of corneal disease markers, a set of new irregularity indices was recently introduced.^{37} These indices bypass the conversion to corneal power and directly use digitized Placido images.

The previous contribution^{37} had a methodological character, although some preliminary discussion of the performance of the indices was carried out there. The aim of the current study is to assess in a sample of normal, keratoconic, and keratoconic suspect eyes a simplified subset of the topographic indices proposed in that article, evaluating their potential as a tool for KC detection.

As a final remark, we should point out that any additional information about a cornea, such as its pachymetry, could improve considerably the screening capability of any marker. The indices analyzed here use only the data available to a Placido-based topographer, but the reader should bear in mind that this is still a dominant diagnostic technology in clinical practice. Thus, any improvement of the performance of these topographers is a primary interest especially in the field of refractive surgery.

#### METHODS

This case-series comparative study is composed of a total of 124 eyes of 106 patients. Two Spanish ophthalmologic centers participated in the recruitment of patients for this study, Vissum Alicante and Vissum Almería, forming part of the Thematic Network of the Cooperative Sanitary Research (RETIC) RD07/0062. All these cases were assigned to one of the following three groups depending on the presence or absence of KC: a control group, which included 50 eyes (from 50 patients); a KC group, which included a total of 50 eyes (from 32 patients); and a subclinical KC or KC suspect group, with a total of 24 eyes (from 24 patients).

The inclusion in the KC group was based on the standard criteria for the diagnosis of this corneal condition and the absence of any previous surgical intervention that could have altered the corneal properties. The following signs were considered at diagnosis^{1}: corneal topography revealing an asymmetric bow tie pattern with or without skewed axes and at least one KC sign on slit lamp examination, such as stromal thinning, conical protrusion of the cornea at the apex, Fleischer ring, Vogt striae, or anterior stromal scar. In those patients wearing contact lenses for the correction of refractive error, only data obtained after an appropriate contact lens discontinuation were considered: at least 2 weeks for soft contact lenses and at least 4 weeks for rigid gas-permeable contact lenses. The exclusion criteria for the KC group were other ocular active pathology at the moment of diagnosis and the presence of an advanced KC (grade 4 according to the Alió-Shabayek grading system^{8}).

The group of normal eyes or control group only included eyes with no other ocular pathology, previous ocular surgery, or irregular corneal pattern. In this control group, only one eye from each patient was selected randomly (random sampling) for the inclusion in the study to avoid the potential bias introduced by the correlation between both eyes of the same patient.

The definition of KC suspect cases was based on the following clinical and topographic evaluation: no slit lamp findings; no scissoring on retinoscopy; and the presence of asymmetric bow tie (AB), inferior steepening (IS), skewed axes (SRAX), or asymmetric bow tie with skewed axes (AB/SRAX) pattern on topography.^{10}

All patients were informed about the study and signed an informed consent document in accordance with the Declaration of Helsinki.

##### Examination Protocol

The corneal topographic analysis was carried out with the CSO topography system (CSO, Firenze, Italy). This topographer analyzes a total of 6144 points of a corneal area enclosed in a circular annulus defined by an inner radius of 0.33 mm and an outer radius of 10 mm with respect to the corneal vertex. The software of this system, the EyeTop2005 (CSO), automatically performs the conversion of the corneal elevation profile into corneal wavefront data using the Zernike polynomials with an expansion up to the seventh order, although it allows to export the raw data (positions of the digitized mires) as an ASCII file. For the sake of reliability of the analysis of the indices, the standard KPI index, as well as the I-S index, has been stored for comparative purposes. Both indices are well known and precisely defined in the literature.^{7,34}

##### Definitions of the Corneal Indices

It is convenient to point out that, in the description of the indices, we skip the initial discretization step performed by every commercially available topographer using presumably standard and widely available edge-detection procedures when the high-contrast black-and-white images of the mires are converted into a discrete point set. Hence, the coordinates of these points along the edges of consecutive mires are taken as the input data. With this information, we have calculated the irregularity indices directly from these positions following the previously discussed methodology,^{37} which we briefly explain. From the original set of indices described,^{37} we used a subset of those with a better performance and higher robustness with respect to the misalignment of the eye, complemented with an additional index defined below.

The digitized points *Pj* captured by the camera of the Placido disk corneal topographer were grouped in *N* ≤ 15 mires. For the sake of precision, we assume that there were 256 points equally spaced along each ring, corresponding to the same number of semimeridians (a value found in a majority of existing devices). We used only data from complete rings. In all cases, we had complete data from at least nine mires, with a considerable loss of reliability starting from the 15th mire. This is the reason for limiting the number of rings to the maximum of 15. Cases exhibiting less than nine complete mires were discarded as unreliable.

The indices were defined according to the information obtained from the mires as follows.

For each in the range from 1 to *N*, the center *C*_{k} and radius *AR(k)* of the best-fit circle for the *k*-th mire were calculated using a standard least squares procedure.^{38} In practice, only the fourth mire (index *AR(4)*) was used in the combined model described, and thus, only its individual performance will be analyzed in the next section.

The best-fit circles for the mires yield the definition of the following primary indices (*PI*):

*PI*_{1}: the diameter of the set of centers *C*_{k} (i.e., the largest mutual distance within the set of centers *C*_{1}, *C*_{2},…, *C*_{N}), normalized by the total number of rings N:

*PI*_{2}: the total drift or the deviation in the consecutive centers *C*_{k}:

These two indices give global information about the deviation of the image of the rings from a concentric pattern. Index *PI*_{2} will not be used in the generalized linear model GPLI below, but it has one of the best discrimination abilities among the individual markers.

Data from mires were also fit with an ellipse, with the aim of capturing the spatial orientation and deformation of each mire (Fig. 1) by means of a simplification^{38–42} of efficient methods for computation of the best-fit ellipse, rendering the following asymmetry index:

*PI3*: the dispersion of the values of the axis ratios *r*_{k} = *a*_{k}/*b*_{k} ≥ 1 of the k-th best-fit ellipse by means of the following expressions:

We also carried out the standard linear regression of the coordinates of the centers *C*_{k} = (*X*_{k}*,Y*_{k}), yielding the coefficients for the linear fit *y = ax* + *b*. With this approach, high values of *a* correspond to a vertical alignment of the centers, so its value contains information about their spatial distribution (Fig. 2). These considerations motivate the following index (we use the name of an index defined previously^{37} but with a new meaning):

*SL*: the absolute value of the slope of the linear regression, *S**L* = | *a* |.

Each of these metrics can be used for KC detection (or at least, as a measure of corneal irregularity), but as it usually happens with the individual indices, none achieves the necessary sensitivity and specificity to meet the standards. For this reason, a combination was used to improve the detection efficiency. We added to our protocol of indices a new additional combined metric called GLPI, which takes continuous values between 0 and 100 (0% corresponding to a totally normal, and 100%, to a totally altered cornea).

GLPI: is a generalized linear (Placido-based) model combining four of the individual indices previously mentioned. Their linear combination (with fixed coefficients) is evaluated in the so-called probit link function.^{43,44} This yields a quantity between 0 and 1 that is multiplied by 100 for convenience. This value, in the interval [0,100], is a percentage of irregularity of the cornea. This definition of GLPI is slightly different from the one given previously^{37}; it has been modified to achieve a better accuracy with a smaller number of individual indices and also to include the new index *SL*:

, with

The methodology of construction of GLPI was explained in a previous work^{37}: coefficients in equation 1 were computed by means of a generalized linear regression applied to databases of normal and keratoconic eyes used in the study^{37} and rescaled to improve the GLPI grading ability.

Index *PI*_{2} is not used in the calculation of GLPI because of its high correlation with the index *PI*_{1}.

A summary of these indices is contained in Table 1.

##### Statistical Analysis

To determine the homogeneity of the sample, when divided into training and test sets, a Mann-Whitney-Wilcoxon nonparametric test^{45,46} was applied to each of the primary indices. Without assumption of normality, this test checks whether the two samples come from the same population (null hypothesis). It can also be used to analyze the discriminating ability of the indices, checking if it renders different values in each group.

In addition, Fisher exact test^{47,48} is a statistical method used when a dichotomous classification process is made. This test checks whether the classifier has enough discrimination ability, and it is valid for any sample size. The idea is to compare the expected proportions of false-/true-positives/negatives with the actual proportion obtained after classifying. This procedure has been used in this study to check if the true proportions of success of the primary indices when classifying normal and keratoconic eyes are independent, and consequently, if the primary indices show classification ability or not.

The K-fold cross validation is a standard statistical tool to assess the global accuracy of a regression or classification model.^{49,50} The main benefit of this method is that it makes use (independently) of the same data to fit the model and to check its performance, which is useful when the sample size is relatively small. The sample is divided into K groups of approximately equal size. Then the regression model is fit (or re-fit, if an initial model was specified) to the data using K-1 of the K subsets, and its accuracy is measured with the predicted values for the remaining group. When K becomes equal to the sample size, this scheme reduces to the well-known leave-one-out cross-validation method. This technique allows estimating the global accuracy of a classification method with only one data set but independently using subsets of the sample to fit and to validate the model.

The receiver operating characteristic (ROC) curve analysis is a well-established tool for assessing the discriminating capability of a model. We present the results of this analysis for the redefined primary indices *SL* and *AR(4)*. The ROC curves for the rest of the indices can be found in the literature.^{37}

Finally, the standard F_{1} score^{51} was computed for the used indices (including KPI for comparison). This score is another tool to evaluate the discriminating ability of a classifier. It is defined as the harmonic mean of its precision (the number of true-positives divided by the total number of positives) and its sensitivity (the number of true-positives divided by the sum of true-positives and false-negatives). F_{1} score reaches its best value at 1 and worst score at 0.

#### RESULTS

The primary indices have been computed for all three groups in the database, and their means and SDs were calculated (Table 2). The classification ability of the primary indices was assessed in different ways. First, according to the Mann-Whitney-Wilcoxon tests, most of the indices are able to discriminate between the three groups (Table 3), except for *PI2* and *AR(4)*, which, being appropriate for discrimination between keratoconic eyes (KC) and the rest of the eyes, do not perform well discriminating between normal (N) and KC suspect (KS) eyes. In addition, Fisher exact test for all these indices indicated that the true proportions of positives within the N and KC groups differ (with a significance level of 0.05), so they actually have sensitivity to detect irregularities. Moreover, the ROC curves for *SL* and *AR(4)* illustrate the discrimination ability of these indices (Fig. 3); the values of A_{z}ROC (area under the ROC curve) for all the indices appear in Table 4.

Table 2 Image Tools |
Table 3 Image Tools |
Table 4 Image Tools |

Figure 3 Image Tools |

Concerning the combined indices, GLPI index computed using the whole database was able to reach the accuracy value 1 (perfect classifying capability between N and KC groups). The estimations rendered by the K-fold cross-validation method for different values of *K* are shown in Table 5, exhibiting consistent accuracy values between 0.94 and 0.95.

It is well known that the vertical coma (computed as the absolute value of the Zernike coefficient *Z*_{3}^{−1}) is a simple marker for detecting KC.^{9,15} It is actually very close in spirit to our irregularity index *SL*; both measure the vertical asymmetry, although *SL* follows the ideology of using only straightforward calculations from the mire images. For comparative reasons, the vertical coma has also been computed for all three groups in our database. According to a previous analysis,^{37} a suitable cutoff value for the vertical coma to discriminate between KC and N eyes is 3.59 × 10^{−5}. With this threshold, 8% of the eyes in the KC group of our database were classified as regular and 4% of N eyes were classified as irregular, which is a good performance. However, within the KC suspect group (KS), the vertical coma was able to classify only 29% of those corneas as irregular. To achieve a success rate of 0.79 within this group (the same as *SL*; Table 6), the cutoff value has to be set approximately to 2.00 × 10^{−5}, yielding that 22% of N eyes are classified as irregular. This is a much lower accuracy in comparison with *SL*.

There is a clear similarity in the philosophy of the construction of the KPI and the GLPI indices. Both are compound indices, indicating a degree of certainty of detection of a corneal irregularity, with moderate to severe cones receiving a KPI score of 100%.^{7,52,53} Both indices are derived by means of a variation of discriminant analysis applied to a control group of patients, although GPLI, unlike the KPI, uses only the primary information provided by the keratoscope.

A comparison of the new indices with the KPI and I-S renders some interesting conclusions. For the KC suspect group (KS), their values are summarized in Table 7. For the KPI, we used the standard cutoff reported in the literature, considering values equal to or greater than 23 as anomalous (the first two rows in Table 7 fall within the KPI range for N eyes, whereas the last two rows correspond to anomalous ones); in the case of the I-S index, values equal to or greater than 1.5 were considered anomalous (now, the first two columns in Table 7 correspond to N eyes, according to the I-S index, and the last two columns correspond to anomalous eyes). It follows from Table 7 that KPI was able to detect only six (25%) of 24 KC suspect eyes, whereas I-S was able to detect 12 (50%) of 24; moreover, eight (33.3%) of 24 cases were not detected by either index, and only four (16.7%) of 24 cases are detected by both indices simultaneously.

Finally, Table 6 shows that the classification power of GLPI and KPI are very similar in all three groups: N eyes, KC eyes, and KC suspect eyes. Index *SL*, exhibiting a reasonable behavior within the group of N eyes, has a slightly lower KC detection capability than either GLPI or KPI. However, within the crucial group of KS eyes, both GLPI (accuracy of 0.21) and KPI (accuracy of 0.29) have rather poor results, whereas the accuracy of *SL* is very acceptable (accuracy of 0.79). The values of F_{1} score confirm these results (Table 8): GLPI and KPI have very similar scores in both cases, but because index *SL* has a higher sensitivity with KS eyes, we achieve a better overall performance with our combined approach.

This suggests the following clinical procedure to examine an individual eye. First, one computes GLPI (which has a high performance, close to the KPI performance in all three groups) as the main diagnostic tool. If the value of GLPI suggests a regular cornea, we look at *SL*; if it renders values above the normal threshold of 1, we classify the patient as a possible KC suspect, requiring further careful examination by the clinician before considering him or her as a candidate for, say, refractive surgery.

#### DISCUSSION

The Placido-based anterior corneal topography is an affordable and valuable tool for screening for KC.^{1} Moderate and advanced KC can be reliably diagnosed by this method, complemented with the biomicroscopic, retinoscopic, and pachymetric study.^{1} Much more challenging is the detection of this ectatic disorder in its very early or preclinical stages. In the last years, much effort has been devoted to improve the analysis of the corneal topography data to increase the ability to diagnose early clinical and subclinical KC cases. The importance of an early detection of such cases lies, in particular, in screening out the candidates for the refractive surgery procedures in these weakened and altered corneas. In this sense, a variety of indices or markers have been proposed in the last three decades. The most well-known and widely used ones are the Rabinowitz and Rabinowitz/McDonnell indices (K, I-S, KISA%) and the Klyce/Maeda indices (KPI, KCI%), along with the vertical coma,^{9,15} although some others have also been defined.^{34} Almost all of them, in accordance with the standard definition of KC, are based on a combination of pachymetry and curvature and corneal power maps obtained by means of corneal topography devices. However, at least in the devices based on Placido disk technology, the corneal power value is a product of a mathematical processing of the raw data, usually obtained under certain *a priori* assumptions and by proprietary methods, as previously explained. This was one of the motivations for the introduction of new corneal irregularity indices^{37} for the Placido disk topographers, defined and analyzed in this work. All of them use exclusively the primary data, that is, the image of the reflection of the mires on the anterior surface of the cornea, bypassing the need to calculate the altimetric or curvature data. It should be stressed that these new indices require only elementary arithmetic manipulation of the digitized images of the mires and do not intend to imitate the reconstruction of the altimetry or local curvature of the cornea.^{33} The aim of the current study was to evaluate in an available sample of N, keratoconic, and preclinical keratoconic eyes these new topographic indices derived directly from the analysis of the digitized images of the Placido rings and to assess the potential of these indices as a tool for KC detection. We insist that the primary purpose of our markers was not to replace but to complement the standard indices (KPI, KISA%, and others), eventually providing the clinician with additional information, especially in the borderline and preclinical KC situations, by detecting an irregular cornea, independently of the type of irregularity it presents.

Regarding the primary corneal indices defined by our research group, statistically significant differences between the control and the KC groups were found for all indices. Therefore, the primary indices defining different features of the Placido disk images reflected on the cornea were able to discriminate between N and KC corneas. A careful observation of the ranges of values of the primary indices in the analyzed groups reveals that there was a relevant area of overlapping for all parameter ranges of both groups. Therefore, these two primary indices showed the best discriminating ability among N and KC eyes. Index *PI2* represents a measurement of the dispersion in the location of the centers of the fitted circles to the mires projected on the cornea, considering the diameter of the set of centers as well as their drift.^{37} Therefore, it characterizes the behavior of the centers of mass of each ring. The new *SL* is an indicator of the global asymmetry of the mires. Specifically, this index measures the slope of the regression line for the centers of the mires. In summary, the direct analysis of the asymmetry of the digitized Placido disks projected on the cornea by means of a corneal topography device allows an effective discrimination between normal and KC corneas.

In the case of the combined index, an excellent discriminating performance of the GLPI (which can be interpreted as a percentage of irregularity) was observed. It was a perfect classifier between keratoconic and N eyes and yielded results comparable to those of the KPI when discriminating between the N and subclinical KC eyes. Furthermore, a combination of GLPI with *SL* allows achieving an excellent capability of detection of irregular corneas, considering as irregular both the keratoconic and the preclinical keratoconic ones, as Table 6 shows. More specifically, all eyes in the KC group, as well as most of the eyes in the preclinical KC group, were classified by this combination of indices as irregular corneas. Thus, the use of the primary corneal indices characterizing the asymmetry of the mires seems to be especially useful for KC detection, whereas their combination yields a classification method with excellent discrimination ability between the three groups.

Along with the high sensitivity, other advantages of the corneal indices used in the current study over the standard approaches are (1) their independence from the proprietary algorithms of conversion of the raw ring images into curvature and corneal power and (2) the mathematical simplicity, with consequent very basic computational requirements. It is convenient to remark that these indices can be easily adapted to any particular commercially available Placido disk topographer; keep in mind that these devices are simple, relatively affordable, and easy to use and represent a vast majority of the topographic devices available in clinical practice. We should also point out that, at this stage, we aimed primarily at a gradation of irregularities on the anterior corneal surface rather than the discrimination between types of pathologies, in analogy to the standard indices for the detection of KC (such as KPI, I-S, or KISA%).

Currently, studies are being conducted to confirm the effectiveness of the defined indices in the detection and characterization of other corneal conditions. The correlation of these indices with higher order corneal aberrations and other optical quality parameters should be also investigated in the future.

In conclusion, the analysis of the digitized images of the Placido disks projected on the cornea is a valid and effective tool for the KC and preclinical KC screening that can be used additionally to the existing keratometric criteria. At this stage of our study, we can recommend them as a complementary screening tool designed to alert the clinician especially in the borderline cases of irregular corneas for which a more exhaustive examination is recommended. We believe that these indices, easily implemented in any Placido-based topographer by a simple software update, could endow a clinician with an additional layer of evidence in the examination of the anterior face of a cornea for possible pathologies.

Andrei Martinez-Finkelshtein

Department of Mathematics

University of Almería, 04120

Almería, Spain

e-mail: andrei@ual.es

##### ACKNOWLEDGMENTS

*The authors have no proprietary or commercial interest in the medical devices that are involved in this article. This study has been supported in part by the Thematic Network of the Cooperative Sanitary Research (RETIC) RD07/0062 from the Spanish Institute of Health “Carlos III.” AM-F and GCdL are partially supported by the Research Project FIS PI10/01843 from the Spanish Institute of Health “Carlos III.” AM-F and DR-L are also supported in part by the research group FQM-229 from Junta de Andalucía and by the project MTM2011-28952-C02-01 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund. DR-L is partially supported by the FPU Program from the Ministry of Education of Spain. In addition, AM-F is partially supported by the Excellence Grant P09-FQM-4643 from Junta de Andalucía.*

*Received September 19, 2012; accepted December 3, 2012.*