BASIC INVESTIGATION
New Approach for Correction of Error Associated With Keratometric Estimation of Corneal Power in Keratoconus Vicente J. Camps, PhD,* David P. Piñero, PhD,*†‡ Esteban Caravaca-Arens, Msc,* Dolores de Fez, PhD,* Rafael J. Pérez-Cambrodí, OD, PhD,†‡ and Alberto Artola, MD, PhD†
Purpose: The aim of this study was to obtain the exact value of the keratometric index (nkexact) and to clinically validate a variable keratometric index (nkadj) that minimizes this error.
Methods: The nkexact value was determined by obtaining differences (DPc) between keratometric corneal power (Pk) and Gaussian corneal power (PGauss ) equal to 0. The nkexact was defined as the c value associated with an equivalent difference in the magnitude of DPc for extreme values of posterior corneal radius (r2c) for each anterior corneal radius value (r1c). This nkadj was considered for the calculation of the adjusted corneal power (Pkadj). Values of r1c ˛ (4.2, 8.5) mm and r2c ˛ (3.1, 8.2) mm were considered. Differences of True Net Power with PGauss , Pkadj, and Pk(1.3375) were calc culated in a clinical sample of 44 eyes with keratoconus.
Results: nkexact ranged from 1.3153 to 1.3396 and nkadj from 1.3190 to 1.3339 depending on the eye model analyzed. All the nkadj values adjusted perfectly to 8 linear algorithms. Differences between Pkadj and PGauss did not exceed 60.7 D (Diopter). Clinically, nk = 1.3375 c was not valid in any case. Pkadj and True Net Power and Pk(1.3375) and Pkadj were statistically different (P , 0.01), whereas no differences were found between PGauss and Pkadj (P . 0.01). c Conclusions: The use of a single value of nk for the calculation of the total corneal power in keratoconus has been shown to be imprecise, leading to inaccuracies in the detection and classification of this corneal condition. Furthermore, our study shows the relevance of corneal thickness in corneal power calculations in keratoconus. Key Words: keratoconus, corneal power, keratometric index (Cornea 2014;33:960–967)
Received for publication April 4, 2014; revision received May 16, 2014; accepted May 20, 2014. Published online ahead of print July 2014. From the *Department of Optics, Pharmacology and Anatomy, Grupo de Óptica y Percepción Visual (GOPV), University of Alicante, Alicante, Spain; †Department of Ophthalmology (OFTALMAR), Medimar International Hospital, Alicante, Spain; and ‡Fundación para la Calidad Visual (FUNCAVIS), Alicante, Spain. The authors have no funding or conflicts of interest to disclose. Supplemental digital content is available for this article. Direct URL citations appear in the printed text and are provided in the HTML and PDF versions of this article on the journal’s Web site (www.corneajrnl.com). Reprints: Vicente J. Camps, PhD, Department of Optics, Pharmacology and Anatomy, University of Alicante, Crta San Vicente del Raspeig s/n, 03690 San Vicente del Raspeig, Alicante, Spain (e-mail: vicente.camps@ ua.es). Copyright © 2014 by Lippincott Williams & Wilkins
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n clinical practice, optical power of the cornea is usually estimated by assuming a single spherical surface model, and therefore only considering the radius of curvature of the anterior corneal surface and a fictitious refractive keratometric index (nk). This clinical simplification has been demonstrated to lead to relevant overestimations of the central corneal power in healthy corneas1–4 and after laser refractive surgery.5,6 Several recalculations of nk have been proposed to define a general valid algorithm for the estimation of corneal power in healthy eyes and in eyes that have undergone previous refractive surgery (radial keratotomy,7 photorefractive keratectomy,8–11 and laser in situ keratomileusis, LASIK).3,5,12–17 Our research group has recently published a series of articles reporting the differences obtained theoretically and clinically between the central corneal power estimated using a keratometric index (named Pk) and that calculated consid) in ering the curvature of both corneal surfaces (named PGauss c healthy18,19 and post-myopic LASIK corneas.16 All these studies concluded that the use of a single value of nk for the calculation of corneal power is imprecise in both kinds of population and can lead to clinically significant errors.16–19 Similarly, a variable keratometric index depending on the radius of curvature of the anterior corneal surface (adjusted keratometric index, nkadj) was proposed and validated as an approach to minimize the error associated with keratometric estimation of corneal power in healthy and post-LASIK eyes.16,18,19 Furthermore, an additional relevant finding of these studies was that the value 1.3375 was only valid in a very limited number of cases.16,18,19 In eyes with keratoconus, our research group has recently estimated in theoretical simulations and clinically validated the errors associated with the keratometric estimation of corneal power.20 In this study, theoretical differences between Pk and PGauss ranged from an underestimation of 21.2 Diopter (D) to c an overestimation of +3.1 D when Le Grand and Gullstrand eye models were considered for the simulations. For nk = 1.3375, differences between Pk(1.3375) and PGauss were found c to range from an underestimation of 20.3 D to an overestimation of +4.3 D. This was consistent with the clinical validation performed, showing always overestimations (range, +0.5 D to +2.5 D) of corneal power when the keratometric estimations were performed with nk = 1.3375.20 The aim of this study was to obtain the exact value of the keratometric index (named nkexact) using theoretical simulations, as in previous studies in healthy and post-LASIK corneas, which is needed to avoid the error associated with the keratometric estimation in different cases, and to obtain Cornea Volume 33, Number 9, September 2014
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and clinically validate a variable keratometric index depending on the radius of the anterior corneal surface (adjusted keratometric index, nkadj) that minimizes the keratometric error.
METHODS Theoretical Calculations The central corneal power was calculated using the classical keratometric index and also using the Gaussian equation that considers the contribution of both corneal surfaces. All calculations and simulations were performed using the Matlab software (Math Works Inc, Natick, MA). Differences among both types of central corneal power calculations were determined (equations 1 and 3) and modeled using regression analysis, as in previous studies of our research group16,18–20: DPc ¼ Pk 2 PGauss c nk 21 nc 2 na nha 2 nc ec nc 2 na nha 2 nc ¼ 2 þ 2 r1c r1c r2c nc r1c r2c (1) and also, k¼
r1c r2c
(2)
nk 2 1 ¼ DPc ¼Pk 2 PGauss c r1c nc 2 na nha 2 nc ec nc 2 nha nha 2 nc 2 þ r1c 2 · · r1c : r1c nc r1c k k (3)
Calculation of the Exact Keratometric Index The calculation of the exact keratometric index for each theoretical eye model (Le Grand and Gullstrand) was performed by making equation 1 or equation 3 equal to 0. Considering this, the following expressions were obtained16,18:
Different values of nc, nha, and ec were used as a function of the eye model considered (Le Grand eye model: nc = 1.3771, nha = 1.3374, and ec = 0.55 mm; Gullstrand eye model: nc = 1.376, nha = 1.336, and ec = 0.5 mm).
Calculation of the Adjusted Keratometric Index As in our previous studies,16,18,19 the adjusted keratometric index (nkadj) was defined as the value associated with an equivalent difference in the magnitude of DPc for extreme values of r2c for each r1c value and eye model. Specifically, for each r1c value considered, nkadj was obtained with the following equation DPc(r2cmin) = DPc(r2cmax). This nkadj was considered for the calculation of the adjusted corneal power (Pkadj) as follows: Pkadj ¼
nkadj 2 1 : r1c
(6)
Definition of the Range of Corneal Curvature in Keratoconus Eyes For our simulations, a range of curvature for the anterior and posterior corneal curvature was defined after reviewing in detail previous studies on keratoconus.21–24 In this review, only studies using the Scheimpflug imaging technology were considered because it has been demonstrated to be reliable for providing a measurement of r2c.21,25 According to these studies, the anterior corneal radius of curvature in keratoconus corneas ranged between 4.2 and 8.5 mm and the posterior corneal radius between 3.1 and 8.2 mm.22–24 Accordingly, k ratio was found to range between 0.963 and 1.556.
Clinical Validation Patients and Examination Protocol This study included a total of 44 eyes diagnosed with keratoconus that were reviewed at the Department of Ophthalmology (Oftalmar) of the Medimar International Hospital (Alicante, Spain). The inclusion criterion for the
2 ec nc þ ec n2c 2 ec nha 2 ec nc nha 2 n2c r1c þ n2c r2c þ nc nha r1c nc r2c
(4)
2 ec knc þ ec kn2c 2 ec knha 2 ec knc nha 2 n2c r1c þ kn2c r1c þ knc nha r1c : nc r2c
(5)
nk ¼
or
nk ¼
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study was the presence of keratoconus using the standard criteria for the diagnosis of this corneal condition: corneal topography revealing an asymmetric bowtie pattern with or without skewed axes and at least 1 keratoconus sign on slitlamp examination, such as stromal thinning, conical protrusion of the cornea at the apex, Fleischer ring, Vogt striae, or anterior stromal scar.26 The exclusion criteria were previous ocular surgery and any type of active ocular disease. Consent to include clinical information in scientific studies was obtained from all patients, following the tenets of the Helsinki Declaration. In addition, local ethics committee approval was obtained for this investigation.
Clinical Evaluation A comprehensive ophthalmologic examination was performed in all cases, including manifest refraction, corrected distance visual acuity, slit-lamp biomicroscopy, Goldmann tonometry, fundus evaluation, and corneal analysis by a Scheimpflug photography-based topography system (Pentacam system, software version 1.14r01; Oculus Optikgeräte GmbH, Germany). Specifically, the following parameters were recorded and analyzed with the Pentacam system: anterior (r1c) and posterior corneal radius (r2c) in the central 3-mm corneal area, anterior (ACA) and posterior corneal astigmatism (PCA) in the central 3-mm corneal area, anterior (QA) and posterior corneal asphericity (QP), corneal volume (VOL), and minimum (ecmin) and central corneal thickness (eccentral). The True Net Power equation 7 was also recorded, which is the corneal power calculated by the Pentacam System using the ) and the Gullstrand eye model, but Gaussian equation (PGauss c neglecting the corneal thickness (ec). True Net Power ¼
1:376 2 1 · 1000 r1c 1:336 2 1:376 þ · 1000: r2c
(7)
In addition to corneal parameters provided by the Pentacam system, the adjusted keratometric corneal power (Pkadj) was also calculated using equation (6) and the keratometric corneal power [Pk(1.3375)] using nk = 1.3375. Similarly, , Pkadj, and differences (DPc) of True Net Power with PGauss c
Pk(1.3375) were also calculated and analyzed. It should be mentioned that the Pentacam system has been shown to provide precise measurement of anterior and posterior radius of curvature.27–29
Statistical Analysis Statistical analysis was performed using the software SPSS version 19.0 for Windows (SPSS, Chicago, IL). Normality of all data distributions was first confirmed by the Kolmogorov–Smirnov test. When parametric statistics could be applied, the paired Student t test was used for comparing the corneal power values obtained with the different methods of calculation evaluated [(Pk(1.3375), Pkadj), True Net )], whereas the Wilcoxon test was used if Power, and (PGauss c parametric statistics could not be applied. Bland–Altman analysis was used for evaluating the agreement and interchangeability of the different methods of corneal power estimation.30 The limits of agreement were defined as mean 6 1.96 SD of the differences. Pearson or Spearman correlation coefficients, depending on whether normality condition could be assumed or not, were used to assess the correlation of DPc with other clinical parameters analyzed.
RESULTS Theoretical Study Exact (Nkexact) and Adjusted Keratometric Index (Nkadj) The value of nkexact considering all possible combinations of r1c 2 r2c (or all possible k values) ranged from 1.3153 to 1.3381 for the Gullstrand eye model and from 1.3170 to 1.3396 for the Le Grand eye model (see Table, Supplemental Digital Content 1, http://links.lww.com/ICO/A234). The value of nkadj ranged from 1.3190 to 1.3324 and from 1.3207 to 1.3339 for the Gullstrand and Le Grand eye models, respectively (Tables 1 and 2). All the nkadj values adjusted perfectly to 8 linear equations (R2 = 1) for each eye model, providing 8 theoretical algorithms for the calculation of corneal power only depending on r1c (Tables 1 and 2).
TABLE 1. nkadj Algorithms Developed Using the Gullstrand Eye Model for Different r1c and/or k Intervals and the Corresponding Theoretical Ranges for nkadj, Pkadj, and PGauss and Differences (DPc) Between Pkadj and PGauss c c r1c, mm
kmin, kmax
4.2, 4.8, 5.7, 6.3, 6.5, 6.9, 7.6, 7.9,
1.20, 1.17, 1.21, 1.05, 1.14, 1.03, 1.09, 0.96,
4.7 5.6 6.2 6.4 6.8 7.5 7.8 8.5
1.52 1.56 1.55 1.31 1.45 1.39 1.39 1.35
nkadj Algorithm 20.01217 20.01043 20.00926 20.00741 20.00792 20.00669 20.00643 20.00561
r1c r1c r1c r1c r1c r1c r1c r1c
+ + + + + + + +
1.3777 1.3774 1.3773 1.3770 1.3771 1.3767 1.3767 1.3768
nkadj 1.3205, 1.3190, 1.3199, 1.3296, 1.3243, 1.3266, 1.3266, 1.3291,
1.3266 1.3273 1.3245 1.3303 1.3266 1.3306 1.3279 1.3324
PGauss ,D c 67.5, 56.3, 50.9, 50.8, 47.0, 42.9, 41.2, 38.0,
78.5 68.6 57.7 53.2 51.0 48.6 43.9 42.8
Pkadj, D 68.2, 57.0, 51.6, 51.5, 47.4, 43.8, 41.9, 38.7,
77.8 68.2 56.9 52.4 50.2 47.9 43.1 42.1
DPc, D 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7,
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
Minimum and maximum nkadj, Pkadj, and PGauss values are bolded in the table. c
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TABLE 2. nkadj Algorithms Developed Using the Le Grand Eye Model for Different r1c and/or k Intervals and the Corresponding Theoretical Ranges for nkadj, Pkadj, and PGauss and Differences (DPc) Between Pkadj and PGauss c c r1c, mm
kmin, kmax
4.2, 4.8, 5.7, 6.3, 6.5, 6.9, 7.6, 7.9,
1.20, 1.17, 1.21, 1.05, 1.14, 1.03, 1.09, 0.96,
4.7 5.6 6.2 6.4 6.8 7.5 7.8 8.5
nkadj Algorithm 20.01207 20.01036 20.00919 20.00736 20.00777 20.00664 20.00643 20.00575
1.52 1.56 1.55 1.31 1.45 1.39 1.39 1.35
r1c r1c r1c r1c r1c r1c r1c r1c
+ + + + + + + +
1.3789 1.3787 1.3785 1.3782 1.3783 1.3780 1.3767 1.3780
PGauss ,D c
nkadj 1.3222, 1.3207, 1.3215, 1.3311, 1.3259, 1.3282, 1.3283, 1.3306,
1.3282 1.3290 1.3261 1.3318 1.3282 1.3322 1.3296 1.3339
67.8, 56.6, 51.1, 51.0, 47.2, 43.1, 41.4, 38.2,
Pkadj, D 68.5, 57.3, 51.9, 51.7, 47.6, 43.8, 42.1, 38.9,
78.8 69.2 57.9 53.4 51.1 48.9 44.1 43.0
78.1 68.5 57.2 52.7 50.4 48.1 43.4 42.3
DPc, D 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7,
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
Minimum and maximum nkadj, Pkadj, and PGauss values are bolded in the table. c
Differences Between Pkadj and PGauss c
Pkadj ranged from 38.7 to 77.8 D, whereas PGauss ranged c from 38.0 to 78.5 D for the Gullstrand eye model (Table 1). With the Le Grand eye model (Table 2), Pkadj was found to between 38.2 and range between 38.9 and 78.1 D and PGauss c 78.8 D. As shown in Tables 1 and 2, differences between (DPc) did not exceed 60.7 D. Pkadj and PGauss c
Clinical Validation The clinical study comprised 44 eyes of 27 patients with keratoconus, 12 women (44.4%) and 15 men (55.6%), with a mean age of 40.8 6 12.8 years (range, 14–73 years). The sample comprised 24 (54.5%) left and 20 (45.5%) right eyes. Main clinical features of the analyzed sample were already presented in the previous article.20 Because the True Net Power provided by the Pentacam system is calculated using the Gullstrand eye model, Pkadj was calculated considering the algorithm that was developed according to such eye model.
Exact (Nkexact) and Adjusted Keratometric Index (Nkadj) Considering that r1c ranged in the analyzed sample from 5.7 to 8.5 mm and that r2c ranged from 4.3 to 7.5 mm, values of nkexact from 1.3225 to 1.3314 were found (Table 3). All these curvature values were within ranges used in our previous theoretical simulations. The nkadj values obtained ranged from 1.3245 to 1.3291 (Table 3). These values were
also within the range for nkadj obtained in our previous simulations.
Agreement Between Methods of Corneal Power Estimation Linear dependence was found between Pkadj and True Net Power (Pkadj = 20.28 + 1.01 True Net Power, R2 = 0.99). Statistically significant differences were found between Pkadj and True Net Power (Wilcoxon test, P , 0.01). A very strong and statistically significant correlation was found between these 2 corneal power values (r = 0.99, P , 0.01), as shown in Figure 1. The Bland–Altman method showed a mean difference between Pkadj and True Net Power of 0.18 D, with limits of agreement of 20.53 D and +0.89 D (Fig. 2). A linear dependence was also found between Pkadj and PGauss c (Pkadj = 20.16 + 1.004 PGauss , R2 = 0.99). No statistically c significant differences were found between Pkadj and PGauss c (Wilcoxon test, P = 0.70), with a very strong and statistically significant correlation between them (r = 0.996, P , 0.01). According to the Bland–Altman method, the range of agreewas 0.04 D, with limits of ment between Pkadj and PGauss c agreement of 20.63 D and +0.70 D (Fig. 3). Statistically significant differences were found between (Wilcoxon test, P , 0.01), with True Net Power and PGauss c a very strong and statistically significant correlation between them (r = 0.999, P , 0.01). The Bland–Altman method showed that the range of agreement between True Net Power was 0.13 D, with limits of agreement of +0.17 D and PGauss c and +0.09 D. In addition, statistically significant differences were found between Pk(1.3375) and Pkadj (Wilcoxon test,
TABLE 3. nkexact and nkadj for Different Intervals of r1c, and the Difference Between Them in Terms of Corneal Power (DPc) for the Sample of Keratoconus Eyes Analyzed r1c, mm 5.7, 6.3, 6.5, 6.9, 7.6, 7.9,
6.2 6.4 6.8 7.5 7.8 8.5
No. Patients 1 3 9 14 9 8
kmin, kmax 1.21, 1.05, 1.14, 1.03, 1.09, 0.96,
1.55 1.31 1.45 1.39 1.39 1.35
nkexact 1.3240 1.3250, 1.3225, 1.3249, 1.3264, 1.3265,
1.3273 1.3292 1.3308 1.3308 1.3314
nkadj 1.3245 1.3303 1.3250, 1.3266, 1.3266, 1.3291,
1.3266 1.3300 1.3279 1.3324
DPc, D 0.1 0.2, 0.7 20.6, 0.6 20.2, 0.4 20.4, 0.2 20.3, 0.7
Minimum and maximum nkexact and nkadj values are bolded in the table.
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found to correlate significantly with r1c (r = 0.62, P , 0.01), r2c (r = 0.54, P , 0.01), k (r = 20.50, P , 0.01), QCP (r = 0.50, P , 0.01), and QCA (r = 0.61, P , 0.01). Similarly, nPc between Pk(1.3375) and Pkadj correlated significantly with r2c (r = 20.55, P , 0.01), r1c (r = 20.44, P , 0.01), and QCP (r = 20.40, P , 0.01).
DISCUSSION
FIGURE 1. Scattergram showing the relationship between Pkadj (D) and True Net Power (D) in our clinical sample.
P , 0.01), with a very strong and statistically significant correlation (r = 0.99, P , 0.01) (Fig. 4). The Bland–Altman method showed a mean difference value between Pk(1.3375) and Pkadj of 1.30 D, with limits of agreement of +0.56 D and +2.04 D (Fig. 5). Note that there are fewer points in the Figures 4 and 5 because when Pkadj and Pk(1.3375) are calculated, only r1c value is required, and in our keratoconus population, some r1c values are repeated for different patients with keratoconus. Therefore, the Pkadj or Pk(1.3375) value is the same and the points appear overlapped.
Correlation of nPc With Other Clinical Variables The k ratio showed a moderate correlation with nPc between Pkadj and True Net Power (r = 0.62, P , 0.01) as (r = 0.58, P , 0.01). well as with nPc between Pkadj and PGauss c , it was Regarding nPc between True Net Power and PGauss c
The use of a single value of nk for the calculation of the total corneal power in keratoconus has been shown to be imprecise, leading the clinician to inaccuracies in the detection and classification of this corneal condition.20 In the first part of this study, the exact nk values avoiding the error of the keratometric approach were calculated considering the different combinations of anterior and posterior corneal curvature that can be found in keratoconus.22–24 Specifically, theoretical simulations showed that this exact nk value ranged between 1.3153 and 1.3381 for the Gullstrand eye model, and between 1.3170 and 1.3396 for the Le Grand eye model. Furthermore, the nk value of 1.3375 that is widely used in the clinical setting was found to be only valid for the combinations of curvatures r1c = 8.0/r2c = 8.2 mm and r1c = 8.3/r2c = 8.2 mm. For the remaining r1c/r2c combinations, nk = 1.3375 was not a valid keratometric index. All these results were similar to those found in a previous study of our research group calculating the range of nkexact in healthy eyes (Gullstrand model: 1.3163–1.3367; Le Grand model: 1.3179–1.3383)18 and in eyes with previous myopic laser refractive surgery (Gullstrand model: 1.2984–1.3367; Le Grand model: 1.3002–1.3382).16 It should be mentioned that only the upper limit of nkexact range was slightly higher in our simulations in keratoconus compared with that previously reported in simulations in healthy eyes.18 Clinically, the range found for nkexact in our sample of eyes with keratoconus was within the range defined in the theoretical simulations performed in the first part of this study. Considering the Gullstrand eye model, the clinical values of nkexact ranged from 1.3225 to 1.3314. This range was slightly smaller than that obtained in our simulations, and this may be due to the limitation of our sample that did not include cases with very severe or incipient keratoconus. Indeed, in
FIGURE 2. Bland–Altman plot showing differences between Pkadj (D) and True Net Power (D) against the mean value of both. The upper and lower lines represent the limits of agreement calculated as mean of differences 6 1.96 SD.
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FIGURE 3. Bland–Altman plot showing differences between Pkadj (D) and PGauss (D) against the mean c value of both. The upper and lower lines represent the limits of agreement calculated as mean of differences 6 1.96 SD.
our sample, 31 eyes had overall keratometric readings roughly 45 D or less, with 8 cases between 40 D and 43.25 D. Similarly, nk = 1.3375 was found to be not valid in any case of our sample of eyes with keratoconus. As with healthy18 or post-LASIK corneas,16 we attempted to define a variable nk value (nkadj) depending on r1c allowing a minimization of the difference (DPc) between ) corneal powers. The keratometric (Pk) and Gaussian (PGauss c mathematical reason for evaluating the differences obtained for extreme values of r2c for each r1c interval is that these selected extreme values of r2c assures that DPc is #0.7 D. In the case of keratoconus population, 8 different ranges of r1c were required to achieve this condition (Tables 1 and 2) independent of the r1c 2 r2c combination, in contrast to healthy18 and post-LASIK corneas16 where only 1 interval of r1c was required. Because the corneal curvature can vary significantly in both corneal surfaces in keratoconus, 8 different algorithms for the calculation of nkadj had to be developed to be used for different ranges of r1c/r2c (Tables 1 and 2). Thus, differences did not exceed 0.7 D, which was between Pkadj and PGauss c assumed to be an acceptable level of error. With these algorithms, nkadj was found to range from 1.3190 to 1.3324 using
FIGURE 4. Scattergram showing the relationship between Pk(1.3375) (D) and Pkadj (D) in our clinical sample. 2014 Lippincott Williams & Wilkins
the Gullstrand eye model, and from 1.3207 to 1.3339 using the Le Grand eye model. As may be expected, these intervals for nkadj differed from those obtained with the nkadj algorithms previously developed and reported for healthy18 and post-LASIK corneas.16 The differences obtained in Pkadj calculation between Gullstrand and Le Grand models were not clinically relevant. A mean difference of 0.2 D was obtained. Consequently, one group of equations from one eye model can be used for predicting corneal power associated with the other eye model. In addition to the development of the algorithm for nkadj in keratoconus, a clinical validation of such an approach was performed using a sample of 44 eyes with keratoconus in which the range for nkadj was from 1.3291 (r1c = 8.5 mm) to 1.3245 (r1c = 5.7 mm). This clinical validation revealed a strong correlation between the True Net Power provided by the Pentacam system and Pkadj, but with statistically significant and clinically relevant differences between them as evidenced with the Bland–Altman analysis. The limits of agreement between True Net Power and Pkadj were 20.53 and +0.89 D and therefore with potential differences higher than the theoretical prediction of 0.7 D. Specifically, the difference between True Net Power and Pkadj was above 0.7 D in only 3 cases (7%), with values of 0.5 D or below in 77% of cases. However, when central corneal thickness was considwas calculated, the level of agreement with ered and PGauss c Pkadj was clearly better. Indeed, a stronger correlation was , with no statistically signififound between Pkadj and PGauss c cant differences between them. Similarly, the level of agreement between corneal power calculation methods was within the expected range of error according to our simulations, with limits of agreement of 20.63 and +0.70 D. Also, the differwas of 0.5 D or below in 89% ence between Pkadj and PGauss c of cases. This is consistent with the level of agreement achieved in normal eyes with a Pkadj algorithm defined by our research group.19 The better level of agreement of Pkadj with PGauss rather than with the True Net Power shows the c relevance of corneal thickness in corneal power calculations in keratoconus and reveals the importance of using PGauss c instead of the True Net Power in corneal power calculations in keratoconus. This may be due to the more significant variability of pachymetry in keratoconus.21–24 Furthermore, we studied the influence of pachymetry on Pkadj calculation, www.corneajrnl.com |
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FIGURE 5. Bland–Altman plot showing differences between Pk(1.3375) (D) and Pkadj (D) against the mean value of both. The upper and lower lines represent the limits of agreement calculated as mean of differences 6 1.96 SD.
and all the algorithms were recalculated considering ecmin = 385 mm and ecmax = 603 mm. The differences obtained in Pkadj never exceeded 0.1 D. These results were similar to those obtained previously in normal18 and post-LASIK16 populations. In addition to the analysis of agreement between methods of calculation of central corneal power in keratoconus, the correlation of differences between them with several other clinical variables was investigated. Specifically, the was found to difference of Pkadj with True Net Power or PGauss c be significantly correlated with the k ratio. This was expected and highlights the relevance of the relationship of the central curvature of both corneal surfaces in the calculation of total corneal power in keratoconus corneas. Therefore, it is erroneous to estimate the central corneal power in such cases without considering the contribution of this relationship. This was the main reason for our interest in developing an algorithm for estimating the corneal power only considering the radius of the anterior corneal surface and indirectly the contribution of corneal thickness and the relationship between anterior and posterior corneal curvatures by introducing some constant numerical factors. Indeed, our Pkadj approach is an option for an acceptable estimation of the central corneal power when a topography system providing information of the posterior corneal surface is not available in our clinical setting. Therefore, our algorithms can be used in combination with any device providing reliable measurements of the anterior corneal curvature in millimeters. One limitation of this study is the use of paraxial optics, not considering the effect of asphericity in DPc and nk, and the effect of the spherical aberration in corneal power calculations. In normal eyes, differences of up to 2.5 D between paraxial versus ray tracing have been reported.15 In keratoconus, ray tracing studies have only been performed to date to simulate specific effects of keratoconus corneal irregularity on visual performance.31,32 However, the error associated with the use of the keratometric approach in such corneal conditions and how to minimize it have not been evaluated. It should be considered that keratometry is the most widely used parameter in clinical practice to characterize the corneal power. Keratometry is based on an approximation using paraxial optics, and for this reason, we have performed our study
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using paraxial optics and therefore calculating the central corneal power. In conclusion, the use of a single value of nk for the estimation of total corneal power calculation in keratoconus is imprecise. The error associated with the keratometric approach in keratoconus can be minimized by using an adjusted nk, consisting of a variable nk depending on the radius of the anterior corneal surface. The use of the nkadj may avoid incorrect approaches for keratoconus detection, may provide more exact determination of corneal astigmatism and intraocular lens power calculation, and may even allow the clinician to perform an improved contact lens fitting. All these potential benefits of nkadj should be confirmed in future studies. REFERENCES 1. Fam HB, Lim KL. Validity of the keratometric index: large populationbased study. J Cataract Refract Surg. 2007;33:686–691. 2. Olsen T. On the calculation of power from curvature of the cornea. Br J Ophthalmol. 1986;70:152–154. 3. Borasio E, Stevens J, Smith GT. Estimation of true corneal power after keratorefractive surgery in eyes requiring cataract surgery: BESSt formula. J Cataract Refract Surg. 2006;32:2004–2014. 4. Ho JD, Tsai CY, Tsai RJ, et al. Validity of the keratometric index: evaluation by the Pentacam rotating Scheimpflug camera. J Cataract Refract Surg. 2008;34:137–145. 5. Jin H, Holzer MP, Rabsilber T, et al. Intraocular lens power calculation after laser refractive surgery: corrective algorithm for corneal power estimation. J Cataract Refract Surg. 2010;36:87–96. 6. Liu Y, Wang Y, Wang Z, et al. Effects of error in radius of curvature on the corneal power measurement before and after laser refractive surgery for myopia. Ophthalmic Physiol Opt. 2012;32:355–361. 7. Camellin M, Calossi A. A new formula for intraocular lens power calculation after refractive corneal surgery. J Refract Surg. 2006;22:187–199. 8. Hugger P, Kohnen T, La Rosa FA, et al. Comparison of changes in manifest refraction and corneal power after photorefractive keratectomy. Am J Ophthalmol. 2000;129:68–75. 9. Holladay JT. Corneal topography using the Holladay Diagnostic Summary. J Cataract Refract Surg. 1997;23:209–221. 10. Royston JM, Dunne MC, Barnes DA. Measurement of the posterior corneal radius using slit lamp and Purkinje image techniques. Ophthalmic Physiol Opt. 1990;10:385–388. 11. Royston JM, Dunne MC, Barnes DA. Measurement of posterior corneal surface toricity. Optom Vis Sci. 1990;67:757–763. 12. Tang M, Li Y, Avila M, et al. Measuring total corneal power before and after laser in situ keratomileusis with high-speed optical coherence tomography. J Cataract Refract Surg. 2006;32:1843–1850.
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Cornea Volume 33, Number 9, September 2014 13. Hamed AM, Wang L, Misra M, et al. A comparative analysis of five methods of determining corneal refractive power in eyes that have undergone myopic laser in situ keratomileusis. Ophthalmology. 2002;109:651–658. 14. Jarade EF, Tabbara KF. New formula for calculating intraocular lens power after laser in situ keratomileusis. J Cataract Refract Surg. 2004; 30:1711–1715. 15. Gobbi PG, Carones F, Brancato R. Keratometric index, videokeratography, and refractive surgery. J Cataract Refract Surg. 1998;24:202–211. 16. Camps VJ, Piñero DP, Mateo V, et al. Algorithm for correcting the keratometric error in the estimation of the corneal power in eyes with previous myopic laser refractive surgery. Cornea. 2013;32:1454–1459. 17. Edmund C. Posterior corneal curvature and its influence on corneal dioptric power. Acta Ophthalmol (Copenh). 1994;72:715–720. 18. Camps VJ, Pinero-Llorens DP, de Fez D, et al. Algorithm for correcting the keratometric estimation error in normal eyes. Optom Vis Sci. 2012;89: 221–228. 19. Piñero DP, Camps VJ, Mateo V, et al. Clinical validation of an algorithm to correct the error in the keratometric estimation of corneal power in normal eyes. J Cataract Refract Surg. 2012;38:1333–1338. 20. Piñero DP, Camps VJ, Caravaca-Arens E, et al. Estimation of the central corneal power in keratoconus: theoretical and clinical assessment of the error of the keratometric approach. Cornea. 2014;33:274–279. 21. Montalbán R, Alió JL, Javaloy J, et al. Intrasubject repeatability in keratoconus-eye measurements obtained with a new Scheimpflug photography-based system. J Cataract Refract Surg. 2013;39:211–218. 22. Montalbán R, Alio JL, Javaloy J, et al. Comparative analysis of the relationship between anterior and posterior corneal shape analyzed by Scheimpflug photography in normal and keratoconus eyes. Graefes Arch Clin Exp Ophthalmol. 2013;251:1547–1555.
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23. Montalbán R, Alio JL, Javaloy J, et al. Correlation of anterior and posterior corneal shape in keratoconus. Cornea. 2013;32:916–921. 24. Piñero DP, Alió JL, Alesón A, et al. Corneal volume, pachymetry, and correlation of anterior and posterior corneal shape in subclinical and different stages of clinical keratoconus. J Cataract Refract Surg. 2010; 36:814–825. 25. Montalbán R, Piñero DP, Javaloy J, et al. Intrasubject repeatability of corneal morphology measurements obtained with a new Scheimpflug photography-based system. J Cataract Refract Surg. 2012;38:971–977. 26. Rabinowitz YS. Keratoconus. Surv Ophthalmol. 1998;42:297–319. 27. Sideroudi H, Labiris G, Giarmoulakis A, et al. Repeatability, reliability and reproducibility of posterior curvature and wavefront aberrations in keratoconic and cross-linked corneas. Clin Exp Optom. 2013;96: 547–556. 28. Szalai E, Berta A, Hassan Z, et al. Reliability and repeatability of swept-source Fourier-domain optical coherence tomography and Scheimpflug imaging in keratoconus. J Cataract Refract Surg. 2012; 38:485–494. 29. Labiris G, Giarmoukakis A, Sideroudi H, et al. Variability in Scheimpflug image-derived posterior elevation measurements in keratoconus and collagen-crosslinked corneas. J Cataract Refract Surg. 2012;38:1616– 1625. 30. Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet. 1986;1:307– 310. 31. Uçakhan OO. Predicted corneal visual acuity in keratoconus as determined by ray tracing. Acta Ophthalmol Scand. 2003;81:264–270. 32. Tan B, Baker K, Chen YL, et al. How keratoconus influences optical performance of the eye. J Vis. 2008;8:1–10.
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New Approach for Correction of Error Associated With Keratometric Estimation of Corneal Power in Keratoconus Vicente J. Camps, PhD,* David P. Piñero, PhD,*†‡ Esteban Caravaca-Arens, Msc,* Dolores de Fez, PhD,* Rafael J. Pérez-Cambrodí, OD, PhD,†‡ and Alberto Artola, MD, PhD†
Purpose: The aim of this study was to obtain the exact value of the keratometric index (nkexact) and to clinically validate a variable keratometric index (nkadj) that minimizes this error.
Methods: The nkexact value was determined by obtaining differences (DPc) between keratometric corneal power (Pk) and Gaussian corneal power (PGauss ) equal to 0. The nkexact was defined as the c value associated with an equivalent difference in the magnitude of DPc for extreme values of posterior corneal radius (r2c) for each anterior corneal radius value (r1c). This nkadj was considered for the calculation of the adjusted corneal power (Pkadj). Values of r1c ˛ (4.2, 8.5) mm and r2c ˛ (3.1, 8.2) mm were considered. Differences of True Net Power with PGauss , Pkadj, and Pk(1.3375) were calc culated in a clinical sample of 44 eyes with keratoconus.
Results: nkexact ranged from 1.3153 to 1.3396 and nkadj from 1.3190 to 1.3339 depending on the eye model analyzed. All the nkadj values adjusted perfectly to 8 linear algorithms. Differences between Pkadj and PGauss did not exceed 60.7 D (Diopter). Clinically, nk = 1.3375 c was not valid in any case. Pkadj and True Net Power and Pk(1.3375) and Pkadj were statistically different (P , 0.01), whereas no differences were found between PGauss and Pkadj (P . 0.01). c Conclusions: The use of a single value of nk for the calculation of the total corneal power in keratoconus has been shown to be imprecise, leading to inaccuracies in the detection and classification of this corneal condition. Furthermore, our study shows the relevance of corneal thickness in corneal power calculations in keratoconus. Key Words: keratoconus, corneal power, keratometric index (Cornea 2014;33:960–967)
Received for publication April 4, 2014; revision received May 16, 2014; accepted May 20, 2014. Published online ahead of print July 2014. From the *Department of Optics, Pharmacology and Anatomy, Grupo de Óptica y Percepción Visual (GOPV), University of Alicante, Alicante, Spain; †Department of Ophthalmology (OFTALMAR), Medimar International Hospital, Alicante, Spain; and ‡Fundación para la Calidad Visual (FUNCAVIS), Alicante, Spain. The authors have no funding or conflicts of interest to disclose. Supplemental digital content is available for this article. Direct URL citations appear in the printed text and are provided in the HTML and PDF versions of this article on the journal’s Web site (www.corneajrnl.com). Reprints: Vicente J. Camps, PhD, Department of Optics, Pharmacology and Anatomy, University of Alicante, Crta San Vicente del Raspeig s/n, 03690 San Vicente del Raspeig, Alicante, Spain (e-mail: vicente.camps@ ua.es). Copyright © 2014 by Lippincott Williams & Wilkins
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I
n clinical practice, optical power of the cornea is usually estimated by assuming a single spherical surface model, and therefore only considering the radius of curvature of the anterior corneal surface and a fictitious refractive keratometric index (nk). This clinical simplification has been demonstrated to lead to relevant overestimations of the central corneal power in healthy corneas1–4 and after laser refractive surgery.5,6 Several recalculations of nk have been proposed to define a general valid algorithm for the estimation of corneal power in healthy eyes and in eyes that have undergone previous refractive surgery (radial keratotomy,7 photorefractive keratectomy,8–11 and laser in situ keratomileusis, LASIK).3,5,12–17 Our research group has recently published a series of articles reporting the differences obtained theoretically and clinically between the central corneal power estimated using a keratometric index (named Pk) and that calculated consid) in ering the curvature of both corneal surfaces (named PGauss c healthy18,19 and post-myopic LASIK corneas.16 All these studies concluded that the use of a single value of nk for the calculation of corneal power is imprecise in both kinds of population and can lead to clinically significant errors.16–19 Similarly, a variable keratometric index depending on the radius of curvature of the anterior corneal surface (adjusted keratometric index, nkadj) was proposed and validated as an approach to minimize the error associated with keratometric estimation of corneal power in healthy and post-LASIK eyes.16,18,19 Furthermore, an additional relevant finding of these studies was that the value 1.3375 was only valid in a very limited number of cases.16,18,19 In eyes with keratoconus, our research group has recently estimated in theoretical simulations and clinically validated the errors associated with the keratometric estimation of corneal power.20 In this study, theoretical differences between Pk and PGauss ranged from an underestimation of 21.2 Diopter (D) to c an overestimation of +3.1 D when Le Grand and Gullstrand eye models were considered for the simulations. For nk = 1.3375, differences between Pk(1.3375) and PGauss were found c to range from an underestimation of 20.3 D to an overestimation of +4.3 D. This was consistent with the clinical validation performed, showing always overestimations (range, +0.5 D to +2.5 D) of corneal power when the keratometric estimations were performed with nk = 1.3375.20 The aim of this study was to obtain the exact value of the keratometric index (named nkexact) using theoretical simulations, as in previous studies in healthy and post-LASIK corneas, which is needed to avoid the error associated with the keratometric estimation in different cases, and to obtain Cornea Volume 33, Number 9, September 2014
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and clinically validate a variable keratometric index depending on the radius of the anterior corneal surface (adjusted keratometric index, nkadj) that minimizes the keratometric error.
METHODS Theoretical Calculations The central corneal power was calculated using the classical keratometric index and also using the Gaussian equation that considers the contribution of both corneal surfaces. All calculations and simulations were performed using the Matlab software (Math Works Inc, Natick, MA). Differences among both types of central corneal power calculations were determined (equations 1 and 3) and modeled using regression analysis, as in previous studies of our research group16,18–20: DPc ¼ Pk 2 PGauss c nk 21 nc 2 na nha 2 nc ec nc 2 na nha 2 nc ¼ 2 þ 2 r1c r1c r2c nc r1c r2c (1) and also, k¼
r1c r2c
(2)
nk 2 1 ¼ DPc ¼Pk 2 PGauss c r1c nc 2 na nha 2 nc ec nc 2 nha nha 2 nc 2 þ r1c 2 · · r1c : r1c nc r1c k k (3)
Calculation of the Exact Keratometric Index The calculation of the exact keratometric index for each theoretical eye model (Le Grand and Gullstrand) was performed by making equation 1 or equation 3 equal to 0. Considering this, the following expressions were obtained16,18:
Different values of nc, nha, and ec were used as a function of the eye model considered (Le Grand eye model: nc = 1.3771, nha = 1.3374, and ec = 0.55 mm; Gullstrand eye model: nc = 1.376, nha = 1.336, and ec = 0.5 mm).
Calculation of the Adjusted Keratometric Index As in our previous studies,16,18,19 the adjusted keratometric index (nkadj) was defined as the value associated with an equivalent difference in the magnitude of DPc for extreme values of r2c for each r1c value and eye model. Specifically, for each r1c value considered, nkadj was obtained with the following equation DPc(r2cmin) = DPc(r2cmax). This nkadj was considered for the calculation of the adjusted corneal power (Pkadj) as follows: Pkadj ¼
nkadj 2 1 : r1c
(6)
Definition of the Range of Corneal Curvature in Keratoconus Eyes For our simulations, a range of curvature for the anterior and posterior corneal curvature was defined after reviewing in detail previous studies on keratoconus.21–24 In this review, only studies using the Scheimpflug imaging technology were considered because it has been demonstrated to be reliable for providing a measurement of r2c.21,25 According to these studies, the anterior corneal radius of curvature in keratoconus corneas ranged between 4.2 and 8.5 mm and the posterior corneal radius between 3.1 and 8.2 mm.22–24 Accordingly, k ratio was found to range between 0.963 and 1.556.
Clinical Validation Patients and Examination Protocol This study included a total of 44 eyes diagnosed with keratoconus that were reviewed at the Department of Ophthalmology (Oftalmar) of the Medimar International Hospital (Alicante, Spain). The inclusion criterion for the
2 ec nc þ ec n2c 2 ec nha 2 ec nc nha 2 n2c r1c þ n2c r2c þ nc nha r1c nc r2c
(4)
2 ec knc þ ec kn2c 2 ec knha 2 ec knc nha 2 n2c r1c þ kn2c r1c þ knc nha r1c : nc r2c
(5)
nk ¼
or
nk ¼
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study was the presence of keratoconus using the standard criteria for the diagnosis of this corneal condition: corneal topography revealing an asymmetric bowtie pattern with or without skewed axes and at least 1 keratoconus sign on slitlamp examination, such as stromal thinning, conical protrusion of the cornea at the apex, Fleischer ring, Vogt striae, or anterior stromal scar.26 The exclusion criteria were previous ocular surgery and any type of active ocular disease. Consent to include clinical information in scientific studies was obtained from all patients, following the tenets of the Helsinki Declaration. In addition, local ethics committee approval was obtained for this investigation.
Clinical Evaluation A comprehensive ophthalmologic examination was performed in all cases, including manifest refraction, corrected distance visual acuity, slit-lamp biomicroscopy, Goldmann tonometry, fundus evaluation, and corneal analysis by a Scheimpflug photography-based topography system (Pentacam system, software version 1.14r01; Oculus Optikgeräte GmbH, Germany). Specifically, the following parameters were recorded and analyzed with the Pentacam system: anterior (r1c) and posterior corneal radius (r2c) in the central 3-mm corneal area, anterior (ACA) and posterior corneal astigmatism (PCA) in the central 3-mm corneal area, anterior (QA) and posterior corneal asphericity (QP), corneal volume (VOL), and minimum (ecmin) and central corneal thickness (eccentral). The True Net Power equation 7 was also recorded, which is the corneal power calculated by the Pentacam System using the ) and the Gullstrand eye model, but Gaussian equation (PGauss c neglecting the corneal thickness (ec). True Net Power ¼
1:376 2 1 · 1000 r1c 1:336 2 1:376 þ · 1000: r2c
(7)
In addition to corneal parameters provided by the Pentacam system, the adjusted keratometric corneal power (Pkadj) was also calculated using equation (6) and the keratometric corneal power [Pk(1.3375)] using nk = 1.3375. Similarly, , Pkadj, and differences (DPc) of True Net Power with PGauss c
Pk(1.3375) were also calculated and analyzed. It should be mentioned that the Pentacam system has been shown to provide precise measurement of anterior and posterior radius of curvature.27–29
Statistical Analysis Statistical analysis was performed using the software SPSS version 19.0 for Windows (SPSS, Chicago, IL). Normality of all data distributions was first confirmed by the Kolmogorov–Smirnov test. When parametric statistics could be applied, the paired Student t test was used for comparing the corneal power values obtained with the different methods of calculation evaluated [(Pk(1.3375), Pkadj), True Net )], whereas the Wilcoxon test was used if Power, and (PGauss c parametric statistics could not be applied. Bland–Altman analysis was used for evaluating the agreement and interchangeability of the different methods of corneal power estimation.30 The limits of agreement were defined as mean 6 1.96 SD of the differences. Pearson or Spearman correlation coefficients, depending on whether normality condition could be assumed or not, were used to assess the correlation of DPc with other clinical parameters analyzed.
RESULTS Theoretical Study Exact (Nkexact) and Adjusted Keratometric Index (Nkadj) The value of nkexact considering all possible combinations of r1c 2 r2c (or all possible k values) ranged from 1.3153 to 1.3381 for the Gullstrand eye model and from 1.3170 to 1.3396 for the Le Grand eye model (see Table, Supplemental Digital Content 1, http://links.lww.com/ICO/A234). The value of nkadj ranged from 1.3190 to 1.3324 and from 1.3207 to 1.3339 for the Gullstrand and Le Grand eye models, respectively (Tables 1 and 2). All the nkadj values adjusted perfectly to 8 linear equations (R2 = 1) for each eye model, providing 8 theoretical algorithms for the calculation of corneal power only depending on r1c (Tables 1 and 2).
TABLE 1. nkadj Algorithms Developed Using the Gullstrand Eye Model for Different r1c and/or k Intervals and the Corresponding Theoretical Ranges for nkadj, Pkadj, and PGauss and Differences (DPc) Between Pkadj and PGauss c c r1c, mm
kmin, kmax
4.2, 4.8, 5.7, 6.3, 6.5, 6.9, 7.6, 7.9,
1.20, 1.17, 1.21, 1.05, 1.14, 1.03, 1.09, 0.96,
4.7 5.6 6.2 6.4 6.8 7.5 7.8 8.5
1.52 1.56 1.55 1.31 1.45 1.39 1.39 1.35
nkadj Algorithm 20.01217 20.01043 20.00926 20.00741 20.00792 20.00669 20.00643 20.00561
r1c r1c r1c r1c r1c r1c r1c r1c
+ + + + + + + +
1.3777 1.3774 1.3773 1.3770 1.3771 1.3767 1.3767 1.3768
nkadj 1.3205, 1.3190, 1.3199, 1.3296, 1.3243, 1.3266, 1.3266, 1.3291,
1.3266 1.3273 1.3245 1.3303 1.3266 1.3306 1.3279 1.3324
PGauss ,D c 67.5, 56.3, 50.9, 50.8, 47.0, 42.9, 41.2, 38.0,
78.5 68.6 57.7 53.2 51.0 48.6 43.9 42.8
Pkadj, D 68.2, 57.0, 51.6, 51.5, 47.4, 43.8, 41.9, 38.7,
77.8 68.2 56.9 52.4 50.2 47.9 43.1 42.1
DPc, D 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7,
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
Minimum and maximum nkadj, Pkadj, and PGauss values are bolded in the table. c
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TABLE 2. nkadj Algorithms Developed Using the Le Grand Eye Model for Different r1c and/or k Intervals and the Corresponding Theoretical Ranges for nkadj, Pkadj, and PGauss and Differences (DPc) Between Pkadj and PGauss c c r1c, mm
kmin, kmax
4.2, 4.8, 5.7, 6.3, 6.5, 6.9, 7.6, 7.9,
1.20, 1.17, 1.21, 1.05, 1.14, 1.03, 1.09, 0.96,
4.7 5.6 6.2 6.4 6.8 7.5 7.8 8.5
nkadj Algorithm 20.01207 20.01036 20.00919 20.00736 20.00777 20.00664 20.00643 20.00575
1.52 1.56 1.55 1.31 1.45 1.39 1.39 1.35
r1c r1c r1c r1c r1c r1c r1c r1c
+ + + + + + + +
1.3789 1.3787 1.3785 1.3782 1.3783 1.3780 1.3767 1.3780
PGauss ,D c
nkadj 1.3222, 1.3207, 1.3215, 1.3311, 1.3259, 1.3282, 1.3283, 1.3306,
1.3282 1.3290 1.3261 1.3318 1.3282 1.3322 1.3296 1.3339
67.8, 56.6, 51.1, 51.0, 47.2, 43.1, 41.4, 38.2,
Pkadj, D 68.5, 57.3, 51.9, 51.7, 47.6, 43.8, 42.1, 38.9,
78.8 69.2 57.9 53.4 51.1 48.9 44.1 43.0
78.1 68.5 57.2 52.7 50.4 48.1 43.4 42.3
DPc, D 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7, 20.7,
0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
Minimum and maximum nkadj, Pkadj, and PGauss values are bolded in the table. c
Differences Between Pkadj and PGauss c
Pkadj ranged from 38.7 to 77.8 D, whereas PGauss ranged c from 38.0 to 78.5 D for the Gullstrand eye model (Table 1). With the Le Grand eye model (Table 2), Pkadj was found to between 38.2 and range between 38.9 and 78.1 D and PGauss c 78.8 D. As shown in Tables 1 and 2, differences between (DPc) did not exceed 60.7 D. Pkadj and PGauss c
Clinical Validation The clinical study comprised 44 eyes of 27 patients with keratoconus, 12 women (44.4%) and 15 men (55.6%), with a mean age of 40.8 6 12.8 years (range, 14–73 years). The sample comprised 24 (54.5%) left and 20 (45.5%) right eyes. Main clinical features of the analyzed sample were already presented in the previous article.20 Because the True Net Power provided by the Pentacam system is calculated using the Gullstrand eye model, Pkadj was calculated considering the algorithm that was developed according to such eye model.
Exact (Nkexact) and Adjusted Keratometric Index (Nkadj) Considering that r1c ranged in the analyzed sample from 5.7 to 8.5 mm and that r2c ranged from 4.3 to 7.5 mm, values of nkexact from 1.3225 to 1.3314 were found (Table 3). All these curvature values were within ranges used in our previous theoretical simulations. The nkadj values obtained ranged from 1.3245 to 1.3291 (Table 3). These values were
also within the range for nkadj obtained in our previous simulations.
Agreement Between Methods of Corneal Power Estimation Linear dependence was found between Pkadj and True Net Power (Pkadj = 20.28 + 1.01 True Net Power, R2 = 0.99). Statistically significant differences were found between Pkadj and True Net Power (Wilcoxon test, P , 0.01). A very strong and statistically significant correlation was found between these 2 corneal power values (r = 0.99, P , 0.01), as shown in Figure 1. The Bland–Altman method showed a mean difference between Pkadj and True Net Power of 0.18 D, with limits of agreement of 20.53 D and +0.89 D (Fig. 2). A linear dependence was also found between Pkadj and PGauss c (Pkadj = 20.16 + 1.004 PGauss , R2 = 0.99). No statistically c significant differences were found between Pkadj and PGauss c (Wilcoxon test, P = 0.70), with a very strong and statistically significant correlation between them (r = 0.996, P , 0.01). According to the Bland–Altman method, the range of agreewas 0.04 D, with limits of ment between Pkadj and PGauss c agreement of 20.63 D and +0.70 D (Fig. 3). Statistically significant differences were found between (Wilcoxon test, P , 0.01), with True Net Power and PGauss c a very strong and statistically significant correlation between them (r = 0.999, P , 0.01). The Bland–Altman method showed that the range of agreement between True Net Power was 0.13 D, with limits of agreement of +0.17 D and PGauss c and +0.09 D. In addition, statistically significant differences were found between Pk(1.3375) and Pkadj (Wilcoxon test,
TABLE 3. nkexact and nkadj for Different Intervals of r1c, and the Difference Between Them in Terms of Corneal Power (DPc) for the Sample of Keratoconus Eyes Analyzed r1c, mm 5.7, 6.3, 6.5, 6.9, 7.6, 7.9,
6.2 6.4 6.8 7.5 7.8 8.5
No. Patients 1 3 9 14 9 8
kmin, kmax 1.21, 1.05, 1.14, 1.03, 1.09, 0.96,
1.55 1.31 1.45 1.39 1.39 1.35
nkexact 1.3240 1.3250, 1.3225, 1.3249, 1.3264, 1.3265,
1.3273 1.3292 1.3308 1.3308 1.3314
nkadj 1.3245 1.3303 1.3250, 1.3266, 1.3266, 1.3291,
1.3266 1.3300 1.3279 1.3324
DPc, D 0.1 0.2, 0.7 20.6, 0.6 20.2, 0.4 20.4, 0.2 20.3, 0.7
Minimum and maximum nkexact and nkadj values are bolded in the table.
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found to correlate significantly with r1c (r = 0.62, P , 0.01), r2c (r = 0.54, P , 0.01), k (r = 20.50, P , 0.01), QCP (r = 0.50, P , 0.01), and QCA (r = 0.61, P , 0.01). Similarly, nPc between Pk(1.3375) and Pkadj correlated significantly with r2c (r = 20.55, P , 0.01), r1c (r = 20.44, P , 0.01), and QCP (r = 20.40, P , 0.01).
DISCUSSION
FIGURE 1. Scattergram showing the relationship between Pkadj (D) and True Net Power (D) in our clinical sample.
P , 0.01), with a very strong and statistically significant correlation (r = 0.99, P , 0.01) (Fig. 4). The Bland–Altman method showed a mean difference value between Pk(1.3375) and Pkadj of 1.30 D, with limits of agreement of +0.56 D and +2.04 D (Fig. 5). Note that there are fewer points in the Figures 4 and 5 because when Pkadj and Pk(1.3375) are calculated, only r1c value is required, and in our keratoconus population, some r1c values are repeated for different patients with keratoconus. Therefore, the Pkadj or Pk(1.3375) value is the same and the points appear overlapped.
Correlation of nPc With Other Clinical Variables The k ratio showed a moderate correlation with nPc between Pkadj and True Net Power (r = 0.62, P , 0.01) as (r = 0.58, P , 0.01). well as with nPc between Pkadj and PGauss c , it was Regarding nPc between True Net Power and PGauss c
The use of a single value of nk for the calculation of the total corneal power in keratoconus has been shown to be imprecise, leading the clinician to inaccuracies in the detection and classification of this corneal condition.20 In the first part of this study, the exact nk values avoiding the error of the keratometric approach were calculated considering the different combinations of anterior and posterior corneal curvature that can be found in keratoconus.22–24 Specifically, theoretical simulations showed that this exact nk value ranged between 1.3153 and 1.3381 for the Gullstrand eye model, and between 1.3170 and 1.3396 for the Le Grand eye model. Furthermore, the nk value of 1.3375 that is widely used in the clinical setting was found to be only valid for the combinations of curvatures r1c = 8.0/r2c = 8.2 mm and r1c = 8.3/r2c = 8.2 mm. For the remaining r1c/r2c combinations, nk = 1.3375 was not a valid keratometric index. All these results were similar to those found in a previous study of our research group calculating the range of nkexact in healthy eyes (Gullstrand model: 1.3163–1.3367; Le Grand model: 1.3179–1.3383)18 and in eyes with previous myopic laser refractive surgery (Gullstrand model: 1.2984–1.3367; Le Grand model: 1.3002–1.3382).16 It should be mentioned that only the upper limit of nkexact range was slightly higher in our simulations in keratoconus compared with that previously reported in simulations in healthy eyes.18 Clinically, the range found for nkexact in our sample of eyes with keratoconus was within the range defined in the theoretical simulations performed in the first part of this study. Considering the Gullstrand eye model, the clinical values of nkexact ranged from 1.3225 to 1.3314. This range was slightly smaller than that obtained in our simulations, and this may be due to the limitation of our sample that did not include cases with very severe or incipient keratoconus. Indeed, in
FIGURE 2. Bland–Altman plot showing differences between Pkadj (D) and True Net Power (D) against the mean value of both. The upper and lower lines represent the limits of agreement calculated as mean of differences 6 1.96 SD.
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FIGURE 3. Bland–Altman plot showing differences between Pkadj (D) and PGauss (D) against the mean c value of both. The upper and lower lines represent the limits of agreement calculated as mean of differences 6 1.96 SD.
our sample, 31 eyes had overall keratometric readings roughly 45 D or less, with 8 cases between 40 D and 43.25 D. Similarly, nk = 1.3375 was found to be not valid in any case of our sample of eyes with keratoconus. As with healthy18 or post-LASIK corneas,16 we attempted to define a variable nk value (nkadj) depending on r1c allowing a minimization of the difference (DPc) between ) corneal powers. The keratometric (Pk) and Gaussian (PGauss c mathematical reason for evaluating the differences obtained for extreme values of r2c for each r1c interval is that these selected extreme values of r2c assures that DPc is #0.7 D. In the case of keratoconus population, 8 different ranges of r1c were required to achieve this condition (Tables 1 and 2) independent of the r1c 2 r2c combination, in contrast to healthy18 and post-LASIK corneas16 where only 1 interval of r1c was required. Because the corneal curvature can vary significantly in both corneal surfaces in keratoconus, 8 different algorithms for the calculation of nkadj had to be developed to be used for different ranges of r1c/r2c (Tables 1 and 2). Thus, differences did not exceed 0.7 D, which was between Pkadj and PGauss c assumed to be an acceptable level of error. With these algorithms, nkadj was found to range from 1.3190 to 1.3324 using
FIGURE 4. Scattergram showing the relationship between Pk(1.3375) (D) and Pkadj (D) in our clinical sample. 2014 Lippincott Williams & Wilkins
the Gullstrand eye model, and from 1.3207 to 1.3339 using the Le Grand eye model. As may be expected, these intervals for nkadj differed from those obtained with the nkadj algorithms previously developed and reported for healthy18 and post-LASIK corneas.16 The differences obtained in Pkadj calculation between Gullstrand and Le Grand models were not clinically relevant. A mean difference of 0.2 D was obtained. Consequently, one group of equations from one eye model can be used for predicting corneal power associated with the other eye model. In addition to the development of the algorithm for nkadj in keratoconus, a clinical validation of such an approach was performed using a sample of 44 eyes with keratoconus in which the range for nkadj was from 1.3291 (r1c = 8.5 mm) to 1.3245 (r1c = 5.7 mm). This clinical validation revealed a strong correlation between the True Net Power provided by the Pentacam system and Pkadj, but with statistically significant and clinically relevant differences between them as evidenced with the Bland–Altman analysis. The limits of agreement between True Net Power and Pkadj were 20.53 and +0.89 D and therefore with potential differences higher than the theoretical prediction of 0.7 D. Specifically, the difference between True Net Power and Pkadj was above 0.7 D in only 3 cases (7%), with values of 0.5 D or below in 77% of cases. However, when central corneal thickness was considwas calculated, the level of agreement with ered and PGauss c Pkadj was clearly better. Indeed, a stronger correlation was , with no statistically signififound between Pkadj and PGauss c cant differences between them. Similarly, the level of agreement between corneal power calculation methods was within the expected range of error according to our simulations, with limits of agreement of 20.63 and +0.70 D. Also, the differwas of 0.5 D or below in 89% ence between Pkadj and PGauss c of cases. This is consistent with the level of agreement achieved in normal eyes with a Pkadj algorithm defined by our research group.19 The better level of agreement of Pkadj with PGauss rather than with the True Net Power shows the c relevance of corneal thickness in corneal power calculations in keratoconus and reveals the importance of using PGauss c instead of the True Net Power in corneal power calculations in keratoconus. This may be due to the more significant variability of pachymetry in keratoconus.21–24 Furthermore, we studied the influence of pachymetry on Pkadj calculation, www.corneajrnl.com |
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FIGURE 5. Bland–Altman plot showing differences between Pk(1.3375) (D) and Pkadj (D) against the mean value of both. The upper and lower lines represent the limits of agreement calculated as mean of differences 6 1.96 SD.
and all the algorithms were recalculated considering ecmin = 385 mm and ecmax = 603 mm. The differences obtained in Pkadj never exceeded 0.1 D. These results were similar to those obtained previously in normal18 and post-LASIK16 populations. In addition to the analysis of agreement between methods of calculation of central corneal power in keratoconus, the correlation of differences between them with several other clinical variables was investigated. Specifically, the was found to difference of Pkadj with True Net Power or PGauss c be significantly correlated with the k ratio. This was expected and highlights the relevance of the relationship of the central curvature of both corneal surfaces in the calculation of total corneal power in keratoconus corneas. Therefore, it is erroneous to estimate the central corneal power in such cases without considering the contribution of this relationship. This was the main reason for our interest in developing an algorithm for estimating the corneal power only considering the radius of the anterior corneal surface and indirectly the contribution of corneal thickness and the relationship between anterior and posterior corneal curvatures by introducing some constant numerical factors. Indeed, our Pkadj approach is an option for an acceptable estimation of the central corneal power when a topography system providing information of the posterior corneal surface is not available in our clinical setting. Therefore, our algorithms can be used in combination with any device providing reliable measurements of the anterior corneal curvature in millimeters. One limitation of this study is the use of paraxial optics, not considering the effect of asphericity in DPc and nk, and the effect of the spherical aberration in corneal power calculations. In normal eyes, differences of up to 2.5 D between paraxial versus ray tracing have been reported.15 In keratoconus, ray tracing studies have only been performed to date to simulate specific effects of keratoconus corneal irregularity on visual performance.31,32 However, the error associated with the use of the keratometric approach in such corneal conditions and how to minimize it have not been evaluated. It should be considered that keratometry is the most widely used parameter in clinical practice to characterize the corneal power. Keratometry is based on an approximation using paraxial optics, and for this reason, we have performed our study
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using paraxial optics and therefore calculating the central corneal power. In conclusion, the use of a single value of nk for the estimation of total corneal power calculation in keratoconus is imprecise. The error associated with the keratometric approach in keratoconus can be minimized by using an adjusted nk, consisting of a variable nk depending on the radius of the anterior corneal surface. The use of the nkadj may avoid incorrect approaches for keratoconus detection, may provide more exact determination of corneal astigmatism and intraocular lens power calculation, and may even allow the clinician to perform an improved contact lens fitting. All these potential benefits of nkadj should be confirmed in future studies. REFERENCES 1. Fam HB, Lim KL. Validity of the keratometric index: large populationbased study. J Cataract Refract Surg. 2007;33:686–691. 2. Olsen T. On the calculation of power from curvature of the cornea. Br J Ophthalmol. 1986;70:152–154. 3. Borasio E, Stevens J, Smith GT. Estimation of true corneal power after keratorefractive surgery in eyes requiring cataract surgery: BESSt formula. J Cataract Refract Surg. 2006;32:2004–2014. 4. Ho JD, Tsai CY, Tsai RJ, et al. Validity of the keratometric index: evaluation by the Pentacam rotating Scheimpflug camera. J Cataract Refract Surg. 2008;34:137–145. 5. Jin H, Holzer MP, Rabsilber T, et al. Intraocular lens power calculation after laser refractive surgery: corrective algorithm for corneal power estimation. J Cataract Refract Surg. 2010;36:87–96. 6. Liu Y, Wang Y, Wang Z, et al. Effects of error in radius of curvature on the corneal power measurement before and after laser refractive surgery for myopia. Ophthalmic Physiol Opt. 2012;32:355–361. 7. Camellin M, Calossi A. A new formula for intraocular lens power calculation after refractive corneal surgery. J Refract Surg. 2006;22:187–199. 8. Hugger P, Kohnen T, La Rosa FA, et al. Comparison of changes in manifest refraction and corneal power after photorefractive keratectomy. Am J Ophthalmol. 2000;129:68–75. 9. Holladay JT. Corneal topography using the Holladay Diagnostic Summary. J Cataract Refract Surg. 1997;23:209–221. 10. Royston JM, Dunne MC, Barnes DA. Measurement of the posterior corneal radius using slit lamp and Purkinje image techniques. Ophthalmic Physiol Opt. 1990;10:385–388. 11. Royston JM, Dunne MC, Barnes DA. Measurement of posterior corneal surface toricity. Optom Vis Sci. 1990;67:757–763. 12. Tang M, Li Y, Avila M, et al. Measuring total corneal power before and after laser in situ keratomileusis with high-speed optical coherence tomography. J Cataract Refract Surg. 2006;32:1843–1850.
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Cornea Volume 33, Number 9, September 2014 13. Hamed AM, Wang L, Misra M, et al. A comparative analysis of five methods of determining corneal refractive power in eyes that have undergone myopic laser in situ keratomileusis. Ophthalmology. 2002;109:651–658. 14. Jarade EF, Tabbara KF. New formula for calculating intraocular lens power after laser in situ keratomileusis. J Cataract Refract Surg. 2004; 30:1711–1715. 15. Gobbi PG, Carones F, Brancato R. Keratometric index, videokeratography, and refractive surgery. J Cataract Refract Surg. 1998;24:202–211. 16. Camps VJ, Piñero DP, Mateo V, et al. Algorithm for correcting the keratometric error in the estimation of the corneal power in eyes with previous myopic laser refractive surgery. Cornea. 2013;32:1454–1459. 17. Edmund C. Posterior corneal curvature and its influence on corneal dioptric power. Acta Ophthalmol (Copenh). 1994;72:715–720. 18. Camps VJ, Pinero-Llorens DP, de Fez D, et al. Algorithm for correcting the keratometric estimation error in normal eyes. Optom Vis Sci. 2012;89: 221–228. 19. Piñero DP, Camps VJ, Mateo V, et al. Clinical validation of an algorithm to correct the error in the keratometric estimation of corneal power in normal eyes. J Cataract Refract Surg. 2012;38:1333–1338. 20. Piñero DP, Camps VJ, Caravaca-Arens E, et al. Estimation of the central corneal power in keratoconus: theoretical and clinical assessment of the error of the keratometric approach. Cornea. 2014;33:274–279. 21. Montalbán R, Alió JL, Javaloy J, et al. Intrasubject repeatability in keratoconus-eye measurements obtained with a new Scheimpflug photography-based system. J Cataract Refract Surg. 2013;39:211–218. 22. Montalbán R, Alio JL, Javaloy J, et al. Comparative analysis of the relationship between anterior and posterior corneal shape analyzed by Scheimpflug photography in normal and keratoconus eyes. Graefes Arch Clin Exp Ophthalmol. 2013;251:1547–1555.
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23. Montalbán R, Alio JL, Javaloy J, et al. Correlation of anterior and posterior corneal shape in keratoconus. Cornea. 2013;32:916–921. 24. Piñero DP, Alió JL, Alesón A, et al. Corneal volume, pachymetry, and correlation of anterior and posterior corneal shape in subclinical and different stages of clinical keratoconus. J Cataract Refract Surg. 2010; 36:814–825. 25. Montalbán R, Piñero DP, Javaloy J, et al. Intrasubject repeatability of corneal morphology measurements obtained with a new Scheimpflug photography-based system. J Cataract Refract Surg. 2012;38:971–977. 26. Rabinowitz YS. Keratoconus. Surv Ophthalmol. 1998;42:297–319. 27. Sideroudi H, Labiris G, Giarmoulakis A, et al. Repeatability, reliability and reproducibility of posterior curvature and wavefront aberrations in keratoconic and cross-linked corneas. Clin Exp Optom. 2013;96: 547–556. 28. Szalai E, Berta A, Hassan Z, et al. Reliability and repeatability of swept-source Fourier-domain optical coherence tomography and Scheimpflug imaging in keratoconus. J Cataract Refract Surg. 2012; 38:485–494. 29. Labiris G, Giarmoukakis A, Sideroudi H, et al. Variability in Scheimpflug image-derived posterior elevation measurements in keratoconus and collagen-crosslinked corneas. J Cataract Refract Surg. 2012;38:1616– 1625. 30. Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet. 1986;1:307– 310. 31. Uçakhan OO. Predicted corneal visual acuity in keratoconus as determined by ray tracing. Acta Ophthalmol Scand. 2003;81:264–270. 32. Tan B, Baker K, Chen YL, et al. How keratoconus influences optical performance of the eye. J Vis. 2008;8:1–10.
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