Preliminary validation of an optimized algorithm for intraocular lens power calculation in keratoconus.

PURPOSE: This study aimed to evaluate the theoretical influence on intraocular lens power (PIOL) calculation of the use of keratometric approach for corneal power (Pc) calculation in keratoconus and to develop and validate an algorithm preliminarily to minimize this influence.
METHODS: Pcwas calculated theoretically with the classical keratometric approach, the Gaussian equation, and the keratometric approach using a variable keratometric index (nkadj) dependent on r1c(Pkadj). Differences in PIOLcalculations (ΔPIOL) using keratometric and Gaussian Pcvalues were evaluated. Preliminary clinical validation of a PIOLalgorithm using Pkadjwas performed in 13 keratoconus eyes.
RESULTS: PIOLunderestimation was present if Pcwas overestimated, and vice versa. Theoretical PIOLoverestimation up to -5.6 D and -6.2 D using Le Grand and Gullstrand eye models was found for a keratometric index of 1.3375. If nkadjwas used, maximal Δ PIOLwas ±1.1 D, with most of the values ≤±0.6 D. Clinically, PIOLunder- and over-estimations ranged from -1.1 to - 0.4 D. No statistically significant differences were found between PIOLobtained with Pkadjand Gaussian equation (P > 0.05).
CONCLUSION: The use of the keratometric Pcfor PIOLcalculations in keratoconus can lead to significant errors that may be minimized using a Pkadjapproach.

It has been demonstrated theoretically and clinically that differences (ΔPc) between the central corneal power(Pc) calculated with the classical keratometric approach (assumption of only one corneal surface and a fictitious index of refraction, keratometric index, (nk) (Pk) and that considering the curvature of both corneal surfaces and the Gaussian equation (PcGauss) can be significant and lead to errors in clinical practice.[12345] Specifically, the keratometric approach for estimating the Pc has been shown to be able to induce over- and under-estimations of intraocular lens power (PIOL) in a range between +0.14 D and −3.01 D.[6]

In simulations in normal and nonpathological corneas, Pk(nk= 1.3375) has been found to be able to overestimate PcGauss up to 2.50 D. Similarly, in eyes with previous myopic laser refractive surgery, Pk can theoretically overestimate PcGauss up to 3.50 D if nk= 1.3375 is used.[4] These theoretical outcomes were confirmed clinically using a commercially available Scheimpflug imaging-based topography system.[5] According to this, our research group proposed a variable termed keratometric index (adjusted keratometric index, nkadj) dependent on r1c as a simple option to calculate the Pc and to minimize the significant errors associated with the keratometric approach (named Pkadj).

In keratoconus, the use of the classical keratometric index of 1.3375 has shown to produce an overestimation of Pc in theoretical simulations and clinical measurements, with a range of overestimation among 0.5 and 2.5 D found in a sample of 44 keratoconic corneas evaluated with a Scheimpflug imaging-based system.[1] As the use of a single value of nk for the calculation of Pc has been demonstrated to be also imprecise in keratoconus, our research group developed eight different algorithms according to the severity of keratoconus to also obtain a variable called keratometric index (nkadj) and a calculation of Pkadj. This adjusted Pc minimized the error associated to the use of the keratometric approach for Pc calculation to a range of ±0.7 D.[1] However, the impact of the use of the classical and adjusted keratometric approach for Pc estimation has not been evaluated in keratoconus. The aim of the current study was to evaluate the theoretical influence on PIOL calculation of the error in the calculation of Pc(ΔPc) due to the use of the keratometric index (nk) in a preliminary sample of keratoconus eyes (no previous ocular surgeries) as well as the potential benefit of using our adjusted keratometric algorithms.

Methods

Pc was calculated for a range of anterior and posterior curvatures that can be found in keratoconus according to the peer-reviewed literature using nk and also using the Gaussian equation that considers the contribution of two corneal surfaces.[78] The nk values corresponding to the Gullstrand and Le Grand eye models (1.3315 and 1.3304, respectively) as well as the classical value of 1.3375 were used. Differences in PIOL calculation obtained with a simplified formula using the keratometric and Gaussian approaches to determine Pc were determined and modeled by regression analysis. All calculations and simulations were performed by means of Matlab software (MathWorks Inc., Natick, MA, USA).

Calculation of the Gaussian and keratometric intraocular lens power

The starting point of almost all theoretical formulas for PIOL calculation is the use of a simplified eye model, with thin cornea and lens models.[9] According to such scheme, the power of the IOL (PIOL) that replaces the lens can be easily calculated using the Gauss equations in paraxial optics:

In this equation, Pc represents the total Pc, effective lens position (ELP), the effective lens plane, axial length (AL), the AL, nha, the aqueous humor refractive index, nhv, the vitreous humor refractive index, and Rdes represents the postoperative desired refraction calculated at corneal vertex.

When a keratometric Pc(Pk) was used, the PIOL was defined as PIOLK, and when Gaussian Pc(PcGauss) was used, it was defined as PIOLGauss. The calculation of Pk and PcGauss has been described in detail in a previous article.[6] The corresponding equations were performed as follows:

It is important to note that, in equations 2 and 3, the Pc is referenced from different planes due to the one-surface and two-surface corneal models that were considered. However, the secondary principle plane for corneas in the normal range is only around a fraction of millimeter from the corneal vertex. Therefore, it is unable to introduce any significant bias in the calculations proposed.

We defined the k ratio as the relation between the anterior corneal radius and the posterior corneal radius (k = r1c/r2c). When this parameter was used in equation 3, we obtained the following expression:

In all these expressions, nk is the keratometric index, r1c is the anterior corneal surface radius, r2c is the posterior corneal radius, na is the refractive index of air, nc is the refractive index of the cornea, nha is the refractive index of the aqueous humor, and ec is the central corneal thickness.

Difference between the Gaussian and keratometric intraocular lens power

The difference between the keratometric and Gaussian PIOL calculation (ΔPIOL) was calculated using equations 2 and 4 as follows:

If the k ratio was used in equation 5, we obtained the following expression:

As can be seen in equations 5 and 6, ΔPIOL was not dependent on AL.

ΔPIOL was calculated for the range of corneal curvature defined for the keratoconus population. According to the peer-reviewed literature, we considered that the anterior corneal radius in the keratoconus population ranged between 4.2 and 8.5 mm, whereas the posterior corneal radius ranged between 3.1 and 8.2 mm.[12] Therefore, we assumed k ratio values ranging from 0.96 to 1.56 in our theoretical calculations.[2] It should be considered that differences among keratometric and Gaussian Pc are commonly zeroed by constant optimization in the range of corneal curvature of the normal healthy eyes, but not for eyes with significantly higher corneal curvature, as in keratoconus. In addition, we considered that ELP could vary between 2 and 6 mm in the calculations performed in the current study according to previous authors dealing with this issue.[610] The desired postoperative refraction was also modified in the calculations, performing an analysis of ΔPIOL for values of Rdes of 0, +1, and −1 D.

Difference between Gaussian and keratometric intraocular lens power calculation using the adjusted keratometric index

Using our eight algorithms[1] [Table 1] for adjusting the keratometric estimation of Pc, a new value named adjusted keratometric Pc(Pkadj) can be calculated using the classical keratometric Pc formula. Therefore, PIOLADJ was defined as the PIOL calculated from equation 2 using the nkadj value for the estimation of Pc(Pkadj). After that, ΔPIOL was also calculated considering the adjusted PIOL(PIOLADJ) and the Gaussian PIOL(PcGauss).

Table 1

Algorithms for n to obtain the adjusted keratometric power (P) using the Le Grand and Gullstrand eye models

Preliminary clinical validation

A preliminary validation of the PIOL calculation with the algorithm proposed in this study was performed in a sample of keratoconus eyes with AL between 21 and 27 mm. Specifically, 13 eyes of eight candidates for cataract surgery who were screened at the Department of Ophthalmology (Oftalmar) of the Vithas Medimar International Hospital (Alicante, Spain) were included. Eyes with other active ocular pathologies or previous ocular surgeries were excluded from the study. All patients were informed about the study and signed an informed consent document in accordance with the Declaration of Helsinki.

A comprehensive ophthalmologic examination was performed in all cases, which included optical biometry (IOLMaster, Carl Zeiss Meditec) and analysis of the corneal structure by means of a Scheimpflug photography-based tomographer, the Pentacam system (software version 1.14r01, Oculus Optikgeräte GmbH, Germany). PIOL calculation was performed with the IOL-Master software and also with our paraxial approximation using the nkadj(PIOLADJ) and the True Net Power (PIOLTrueNet). The True Net Power is the Pentacam system Pc calculated using the Gaussian equation PcGauss with the Gullstrand eye model neglecting the corneal thickness (ec).

A comparative analysis of our estimations with those obtained with the other established formulas was performed using the statistical software SPSS version 19.0 for Windows (IBM, Armonk, NY, USA). Normality of data distributions was first evaluated by means of the Shapiro–Wilk test. The Mann–Whitney U-test was used for analyzing the statistical significance of differences between PIOL calculations, whereas the Bland–Altman method was used for evaluating the interchangeability of such calculations. In addition, Pearson's correlation coefficients were used to assess the correlation between differences among calculations and different clinical parameters.

Results

Relationship between ΔPIOL and ΔPc

For all possible combinations of r1c and r2c, Pk(1.3375) ranged from 80.4 D to 39.7 D. If Le Grand or Gullstrand eye models were used, Pk(1.3304) ranged from 78.7 D to 38.9 D and Pk(1.3315) from 78.9 D to 39 D, respectively. PcGauss ranged from 78.9 D to 38.2 D and from 78.5 D to 37.9 D for Le Grand and Gullstrand eye models, respectively. If nkadj was used, Pkadj ranged from 38.9 D to 78.1 D for the Le Grand eye model and between 38.7 D and 77.8 D if the Gullstrand eye model was used. Considering the keratometric Pc, the PIOL(PIOLk) was calculated (equation 2) for each r1c/r2c potential combination in keratoconus. If the Le Grand eye model was used (nk= 1.3304), PIOLk ranged between − 32.7 D and 20.5 D and between −35.2 D and 19.5 D if nk= 1.3375 was used. For the Gullstrand eye model (nk= 1.3315), PIOLk ranged between −33.86 D and 19.9 D, and if nk= 1.3375 was used, PIOLk ranged between −36 D and 19 D. When the Gaussian Pc was used, we obtained PIOLGauss values ranging from −32.96 D to 21.36 D and from −33.17 D to 21.1 D for Le Grand and Gullstrand eye models, respectively [Table 2]. When Pkadj was used, PIOLADJ ranged between −31.9 D and 20.5 D and between −32.1 D and 20.2 D for the Le Grand and Gullstrand eye models, respectively. Differences between PIOLADJ and PIOLGauss were calculated and are summarized in Table 3.

Table 2

Maximum and minimum ranges of keratometric corneal power and keratometric intraocular lens power when Le Grand and Gullstrand eye models were used, considering the range of anterior and posterior corneal curvatures reported in the peer-reviewed literature for keratoconus

Table 3

Comparative analysis of differences between the intraocular lens power estimated using the adjusted keratometric power (PIOL) and that obtained using the Gaussian corneal power (PIOLGauss) with the Gullstrand and Le Grand eye models

Table 4 summarizes the Δ PIOL data obtained for the range of anterior corneal curvature in keratoconus (r1c, from 4.2 to 8.5 mm) using the Le Grand and Gullstrand eye models and different values of nk. The edges of the interval shown for each value of ΔPIOL and ΔPc corresponded to the values associated to the extreme values of the keratoconus range defined for r2c, from 3.1 mm to 8.2 mm. As shown in Table 4, there were many over- and under-estimations of Pc when PIOLk was compared to PIOLGauss, although more underestimations were present with the Gullstrand eye model. The largest overestimation was found for the combination of r1c= 7.9 mm with r2c= 8.2 mm (unlikely corneal curvature combination), with values of + 1.0 D and + 1.4 D for the Le Grand and Gullstrand eye models (nk= 1.3304 and nk= 1.3315), respectively. The lowest underestimation was found for r1c= 4.7 mm combined with r2c= 3.1 mm, with values of − 3.5 D and − 4.3 D for the Le Grand and Gullstrand eye models, respectively.

Table 4

Summary of the differences between the keratometric and Gaussian intraocular lens power (ΔPIOL) obtained within the keratoconus range of anterior corneal curvature (r1c: from 4.2 to 8.5 mm) for Le Grand and Gullstrand eye models as well as for the different keratometric index values used (nk: 1.3304, 1.3315, and 1.3375)

When nk= 1.3375 was used in both eye models, an underestimation of PIOLk over PIOLGauss was observed in almost all cases. The magnitude of this underestimation was higher than 0.5 D in almost all possible combinations of r1c and r2c. The maximum underestimation was found again for the combination of r1c= 4.7 mm with r2c= 3.1 mm, with values of −5.6 D and −6.2 D for the Le Grand and Gullstrand eye models, respectively.

All these trends for ΔPIOL were modeled by means of linear regression analysis. Specifically, a predictive linear equation (R2: 0.99) relating ΔPIOL and k ratio as a function of r1c in 0.1-mm steps was found for the two eye models used in this study [Tables 4 and 5]. Likewise, ΔPIOL data could also be adjusted by a quadratic expression (R2: 0.99) dependent on r2c [Fig. 1]. As an example, ΔPIOL data corresponding to r1c= 4.2 mm using the Gullstrand eye model and nk= 1.3375 could be adjusted to the quadratic expression ΔPIOL= −1.5562 r22c+ 15.578 r2c− 38.3007, where r2c is expressed in millimeters [Fig. 1]. The equivalent equation depending on k was ΔPIOL= −13.7170 k + 13.6189 [Table 5].

Table 5

Linear equations (all R2: 0.99) relating ΔPIOL and k ratio as a function of r1c in 0.1 mm steps using the Gullstrand and Le Grand eye models

Figure 1

Relationship between ΔPIOL using the Gullstrand eye model and nk= 1.3375 and the curvature of the posterior corneal surface (r2c). This relation could be adjusted to a quadratic expression dependent on r2c(R2: 0.99)

Relationship between ΔPIOL and effective lens position

The dependency of ΔPIOL variation with ELP was analyzed. In our calculations, the value of ELP was considered to be equal to the anatomical anterior chamber deptha of the two eye models used (3.05 and 3.10 mm for Le Grand and Gullstrand eye models, respectively). Additional calculations were performed considering a range of variation of ELP between 2 and 6 mm, with no variation in the rest of parameters. When ELP = 2 mm was used in our model instead of the anatomical value, differences in ΔPIOL calculation did not become clinically significant in both Le Grand and Gullstrand eye models, with the largest variation of ΔPIOL reaching 0.15 D. When ELP = 6 mm was used, a maximum variation of ΔPIOL of 0.6 D was found in both Le Grand and Gullstrand eye models when r1c= 4.7 mm and r2c= 3.1 mm or 3.5 mm, with most of the rest of combinations providing variations of <0.5 D.

Relationship between ΔPIOL and Rdes

For a range of Rdes between − 1 D and +1 D and keeping constant the other parameters, the variation of ΔPIOL was of 0.02 D or less in comparison with the values obtained for Rdes= 0 D.

ΔPIOL using nkadj for minimizing ΔPc

If nkadj derived from our eight algorithms [Table 1] was used for the calculation of keratometric Pc and then for the calculation of PIOLk, a maximal error of ± 1.1 D in ΔPIOL was observed independently from the eye model used, r1c and Rdes. Considering that 1 D of variation of PIOL induces approximately 0.9 D of change in patients' refraction at the corneal vertex, ΔPIOL obtained was clinically acceptable, with most of the simulations not exceeding ± 0.60 D for most r1c-r2c combinations. Only ΔPIOL was maximal for the extreme values [Table 3].

This study comprised 13 eyes of eight patients with keratoconus (four eyes of women [30.8%] and nine eyes of men [69.2%] with a mean age of 41.1 years ± 19.1, range from 20 to 69 years). The sample comprised seven left eyes (53.8%) and six right eyes (46.2%). Mean anterior and posterior corneal radius of curvature were 7.28 mm (standard deviation [SD]: 0.64; median: 7.27; range: 6.30–8.26 mm) and 6.67 mm (SD: 0.99; median: 6.37; range: 5.58–8.45 mm), respectively. Mean central and minimum corneal thicknesses were 497.5 μm (SD: 44.7; median: 510.0; range: 419.0–510.0 μm) and 476.0 μm (SD: 51.7; median: 480.0; range: 385.0–539.0 μm), respectively. The location of the cone was inferior in all cases. According to the Amsler-Krumeich classification system, a total of eight eyes (61.5%) had keratoconus Grade I, four eyes (30.8%) had Grade II, and one eye (7.7%) had keratoconus Grade III.

An underestimation was always present when PIOL1.3375k was compared with PIOLGauss, ranging from −0.9 D to −2.9 D. Differences between PIOL1.3375k and PIOLGauss were statistically significant (P < 0.05, unpaired Student's t-test). A very strong and statistically significant correlation was found between PIOL1.3375k and the PIOLGauss (r = 0.99, P < 0.01). Likewise, strong and statistically significant correlations of ΔPIOL with r2c(r = 0.96, P < 0.01), r1c(r = 0.84, P < 0.01), and central corneal thickness (r = 0.73, P < 0.01) were found. Furthermore, a good correlation of ΔPIOL with anterior corneal astigmatism (r = 0.64, P < 0.05), AL (r = 0.64, P < 0.05), and minimum corneal thickness (r = 0.57, P < 0.05) was found. The Bland–Altman method revealed the presence of a mean difference between PIOL1.3375k and PIOLGauss of −1.79 D, with limits of agreement of −0.59 and −3.00 D. Fig. 2a shows the Bland–Altman plot corresponding to this agreement analysis.

Figure 2

Bland–Altman plots of the comparative analyses performed in the current study. (a) Comparison between the PIOL obtained using the classical keratometric approach (PIOL1.3375k) and that obtained using the Gaussian equation (PIOLGaussian). (b) Comparison between the PIOL obtained using the adjusted keratometric approach (PIOLadjk) and that obtained using the Gaussian equation (PIOLGaussian). (c) Comparison between the PIOL obtained using the adjusted keratometric approach (PIOLadjk) and that obtained using the True Net estimation (PIOLTrue Net). (d) Comparison between the PIOL obtained using the True Net approach (PIOLadjTrue Net) and that obtained using the Gaussian equation (PIOLGaussian). Upper and lower lines represent the limits of agreement calculated as mean of differences ± 1.96 standard deviation

PIOLAdj under- and over-estimated PIOLGauss in a magnitude ranging from −1.1 to −0.4 D (within the limits established theoretically). No statistically significant differences between PIOLAdj and PIOLGauss were found (P > 0.05, unpaired Student's t-test). Likewise, a very strong and statistically significant correlation was found between PIOLAdj and PIOLGauss (r = 0.99, P < 0.01). Only ΔPIOL was found to correlate significantly with r2c, being this correlation of moderate strength (r = 0.51, P > 0.05). The Bland–Altman method revealed the presence of a mean difference between PIOLAdj and PIOLGauss of −0.31 D, with limits of agreement of −1.34 and 0.72 D [Fig. 2b].

When PIOLAdj was compared with PIOLTrueNet, under- and over-estimations ranging between − 1.3 and 0.2 D were found. Differences between these two PIOL values were statistically significant (P < 0.01, unpaired Student's t-test), with a very strong and statistically significant correlation between them (r = 0.99, P < 0.01). These differences correlated moderately with r2c(r = 0.55, P > 0.05). The Bland–Altman method showed a mean difference between PIOLAdj and PIOLTrueNet of −0.48 D, with limits of agreement of −1.53 and 0.57 D [Fig. 2c].

An overestimation was always present when PIOLTrueNet was compared with PIOLGauss, ranging from 0.1 D to 0.2 D. Differences between these two PIOL values were statistically significant (P < 0.01, unpaired Student's t-test). A very strong and statistically significant correlation was found between PIOLTrueNet and PIOLGauss (r = 1, P < 0.01). Furthermore, significant correlations of Δ PIOL with r2c(r = 0.92, P < 0.01), r1c(r = 0.93, P < 0.01), and central corneal thickness (r = 0.65, P < 0.05) were found. The Bland–Altman method revealed the presence of a mean difference between PIOLTrueNet and PIOLGauss of 0.17 D, with limits of agreement of 0.12 D and 0.22 D. Fig. 2d shows the Bland–Altman plot corresponding to this agreement analysis.

Discussion

In the present study, we have demonstrated that the use of keratometric Pc in PIOL calculations can lead to significant errors in such population with a theoretical simulation using the range of corneal curvature in keratoconus. Specifically, an underestimation of PIOLk with respect to PIOLGauss was present due to an overestimation of the Pc and vice versa. This difference in the calculation of PIOL(ΔPIOL) has been demonstrated to be dependent on the nk value, k ratio (consequently on r1c and r2c) as well as on the theoretical eye model used for calculations. The nk values derived from the Le Grand and Gullstrand eye models (1.3304 and 1.3315, respectively) were shown to generate over- and under-estimations of PIOL(PIOLk with respect to PIOLGauss), with more trend to underestimations. The maximum over- and under-estimations were + 1.4 D and + 1.0 D and − 3.5 D and − 4.3 D for Le Grand and Gullstrand eye models, respectively. Furthermore, underestimations were always present when nk= 1.3375 was used, with a maximum value of −6.2 D for the Gullstrand eye model and −5.6 D for the Le Grand eye model. All these outcomes are similar to those found in normal healthy eyes,[6] although underestimations are higher in the keratoconus population. For example, when nk= 1.3375 is used in a normal eye, the maximum underestimation of PIOL is −3.01 D and −2.77 D for Gullstrand and Le Grand eye models,[6] respectively, instead of the values of −6.2 and −5.6 D found in keratoconus.

As in normal healthy eyes, for each value of r1c in 0.1-mm steps within the range of curvature for the keratoconus population,[71112] a linear equation dependent on k ratio as well as a quadratic equation dependent on r2c allows to obtain a highly accurate prediction of ΔPIOL [Table 5]. These equations may be useful to calculate the magnitude of the error associated to the use of a specific keratometric Pc in PIOL calculation (PIOLk). The consistency of our simulation model was studied by analyzing the dependency of ΔPIOL on ELP or Rdes. This analysis revealed that the variation in ΔPIOL was not clinically significant for a range of ELP between 2 and 6 mm or for an interval of Rdes ranging from + 1 to −1 D.

With the aim of minimizing the error associated to the use of the classical keratometric approach of Pc estimation, the variations of ΔPIOL were also analyzed when using the correction of the keratometric power with the algorithm developed by our research group consisting on the use of a variable keratometric index (nkadj) depending on r1c [Table 1].[1] Using this algorithm, the theoretical differences between PIOLAdj and PIOLGauss never exceeded ± 1.1 D, independently of the r1c value or theoretical eye model used. This error range was not clinically significant for most of the r1c/r2c combinations at the corneal vertex plane. Therefore, PIOLAdj can be considered a useful algorithm to be used in keratoconus for PIOL calculation when posterior corneal curvature data are not available.

Besides this theoretical analysis, a preliminary clinical validation with a reduced number of keratoconus eyes was performed in which PcGauss ranged between 40 and 52 D. Using PIOLk with nk= 1.3375, PLIOk was found to underestimate significantly PIOLGauss in a range between −0.9 and −2.9 D (P < 0.05, unpaired Student's t-test). The Bland–Altman method confirmed the clinical relevance of this underestimation, with a mean difference of − 1.79 D, and limits of agreement of −0.59 and −3.00 D. Differences between PIOL1.3375k and PIOLGauss were found to be in relation with r2c(r = 0.96, P < 0.01), r1c(r = 0.84, P < 0.01), and central corneal thickness (r = 0.73, P < 0.01). Therefore, in keratoconus, the contribution of the combined effect of posterior corneal curvature and corneal thickness to the total Pc seems to be clinically relevant and should be considered when the value of Pc is used in PIOL calculations. This is in agreement with previous studies clinically evaluating the impact of using the keratometric Pc for PIOL calculation in keratoconus.[131415] Park et al.[13] found that, in patients with posterior keratoconus, PIOL calculation from conventional keratometry may be inaccurate, and secondary piggyback IOL procedure may be needed after cataract surgery. Thebpatiphat et al.[14] in a retrospective cases series evaluating 12 keratoconus eyes undergoing cataract surgery concluded that IOL calculation was more predictable in mild keratoconus than in moderate and severe diseases. It should be considered that an increase in posterior corneal curvature and a decrease in central corneal thickness are present in more severe keratoconus cases.[7]

When an adjusted keratometric index (nkadj) was used to obtain Pkadj in the calculation of PIOLAdj, differences with PIOLGauss did not exceed ± 1.1 D (range from 0.4 to −1.1 D) as the theoretical analysis predicted, obtaining an Δ PIOL between −0.1 and 0.4 D in 61.5% of cases. These differences between PIOLAdj and PIOLGauss did not reach statistical significance (P > 0.05), but the Bland–Altman analysis revealed a mean difference of −0.31 D, with clinically relevant limits of agreement (−1.34 and 0.72 D). The correlation between PIOLAdj and PIOLGauss was strong (r = 0.99, P < 0.01), being only the posterior corneal radius the main factor interfering in this relationship (r = 0.51, P > 0.05). This result supposes an improvement compared to those obtained when Pc is calculated with the classical nk= 1.3375, and differences among PIOLAdj and PIOLGauss can be considered acceptable in most of the cases. When PIOLAdj and PIOLTrueNet were compared, differences among them were found to be statistically significant (P < 0.01, unpaired Student's t-test), with clinically relevant differences in the Bland–Altman analysis (mean difference: −0.48 D, limits of agreement: −1.53 and 0.57 D). Likewise, differences between PIOLTrueNet and PIOLGauss were also statistically significant (P < 0.01, unpaired Student's t-test), but not clinically relevant. This suggests that corneal thickness has a limited effect on the calculation of Pc in keratoconus and therefore the use of the true net power in keratoconus can be considered as acceptable for clinical purposes. Specifically, the influence of central corneal thickness was studied considering a range of this parameter in keratoconus between 200 μm and 600 μm. The maximum errors considering corneal thickness in the calculation of PIOL were 0.4 and −0.1 D for Le Grand and Gullstrand eye models, respectively. Consequently, the clinical relevance of corneal thickness variations in our model seemed to be limited for the range of thickness of the keratoconus population. On the other hand, the study is based on two theoretical eye models, providing very similar results of ΔPIOL. The choice of one model or another is therefore not decisive and has minimal clinical relevance in keratoconus eyes.

It should be acknowledged that there are some potential weaknesses in this study: the use of paraxial optics, not considering the effect of asphericity, the effect of variations in corneal thickness, and the use of a limited number of theoretical eye models for the simulations. Future studies evaluating the validity of our model for nonparaxial optics as well as evaluating whether there is an improvement with clinical relevance when using a more complex optical estimation are required. In addition, we have not evaluated the impact of the adjustment developed for PIOL calculation in a prospective study and consequently the prediction error was not evaluated. Once the potential benefit of using our adjustment is demonstrated, a future study will be conducted to compare the prediction error with our formula and other commonly used formulas. Before beginning a prospective study involving a modification of the PIOL calculation, we prefer to confirm the potential improvement theoretically and if confirmed to conduct the corresponding prospective study. For this reason, the sample size was limited and therefore this study can be only considered as a preliminary experience to evaluate the potential applicability of the algorithms developed. Furthermore, it should be mentioned that only one case of severe keratoconus was included in the clinical validation and therefore the conclusions of the study cannot be extrapolated to this type of cases. Severe keratoconus cases should be included in the future prospective studies, clinically validating the algorithms for PIOL calculation.

Conclusion

We have shown that the use of a single value of nk in keratoconus for the calculation of PIOL can lead to inaccuracies that could explain the refractive surprises in keratoconus population and after cataract surgery. These inaccuracies in PIOL calculations can be minimized theoretically using a variable nk depending on the radius of curvature of the anterior corneal surface with a maximum error in most of the cases of approximately 0.6 D and over 1 D in very few cases. A preliminary clinical validation of this model has been performed, with results very close to those predicted theoretically. Our nkadj algorithm for Pc estimation in keratoconus may be especially useful in those clinical settings in which topographic devices providing posterior corneal surface data are not available, although a clinical validation with a larger sample size including severe keratoconus cases should be performed to obtain consistent conclusions. Our theoretical models of correction of the error introduced by nk and their clinical implications in PIOL calculations should be evaluated with clinical data in the future to validate their significance and applicability to other ectatic diseases or previous ocular surgeries as cross-linking or intracorneal ring segment implantation in keratoconus.

The research leading to these results has received funding from the Generalitat Valenciana (Valencian Community, Spain) under the grant for emergent research groups with reference GV2014/086.

Financial support and sponsorship

The research leading to these results has received funding from the Generalitat Valenciana (Valencian Community, Spain) under the grant for emergent research groups with reference GV2014/086.

Conflicts of interest

There are no conflicts of interest.