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New algorithm for toric intraocular lens power calculation considering the posterior corneal astigmatism

Canovas, Carmen PhD*; Alarcon, Aixa PhD; Rosén, Robert PhD; Kasthurirangan, Sanjeev PhD; Ma, Joseph J.K. MD, FRCSC; Koch, Douglas D. MD; Piers, Patricia PhD

Author Information
Journal of Cataract & Refractive Surgery: February 2018 - Volume 44 - Issue 2 - p 168-174
doi: 10.1016/j.jcrs.2017.11.008
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Abstract

The influence of posterior corneal astigmatism (PCA) was postulated by Javal over 120 years ago.1 Contemporary studies have documented its existence and evaluated its importance in managing astigmatism during cataract surgery.2–7 In 2012, Koch et al.8 showed the non-negligible refractive effect of PCA, which is against-the-rule (ATR) in over 80% of eyes and has an average magnitude of 0.3 diopter (D). Specifically, these authors confirmed that the magnitude of PCA was correlated with the magnitude of anterior corneal astigmatism for patients who had with-the-rule (WTR) astigmatism, whereas there was only a weak correlation for oblique astigmatism and no correlation for ATR astigmatism. In 2013, Ma and LawA presented wavefront aberrometry data at the American Society of Cataract and Refractive Surgery meeting that suggested that the vector addition of a correction factor representing the internal astigmatism of the aphakic eye, including PCA, might improve the prediction of absolute postoperative residual astigmatism, and recommended a vector addition of such a correction factor to reduce residual astigmatism with toric intraocular lens (IOL) implantation.

Because of its relevance in toric IOL power calculations, some new algorithms have been developed that incorporate the effect of PCA in toric IOL power determination.9–12 One such algorithm is the Baylor toric IOL nomogram.12 The Baylor nomogram recommends (1) that for corneas that have WTR anterior corneal astigmatism, to account for the increasing ATR effect of PCA as anterior corneal astigmatism increases, progressively raising the threshold for IOL toricity (eg, 1.7 D of anterior corneal astigmatism is the lower threshold for implanting a toric IOL that corrects 1.0 D at the spectacle plane, dropping to a threshold of 5.0 D for implanting a toric IOL that corrects 4.0 D at the spectacle plane) and (2) for corneas that have ATR anterior corneal astigmatism, consistently lowering the threshold for implanting a toric IOL by 0.7 D (eg, a 1.5 D toric IOL is used to correct astigmatism in the 0.8 to 1.29 D range). In addition, the Baylor nomogram incorporates other effects such as the systematic ATR shift that occurs with age.8

In this study, we assess a new PCA algorithm that is incorporated in the Tecnis toric online calculator.B This algorithm was developed by combining clinical study results of patients who had monofocal IOLs implanted with the research that led to the Baylor nomogram.12 A retrospective evaluation of more than 200 eyes implanted with monofocal IOLs showed an overall ATR shift when postoperative refraction was compared with keratometry (K) values. The magnitude of that shift depended on various factors, which are accounted for in the algorithm. The combination of this trend and those underpinning the Baylor nomogram12 are the basis of this PCA algorithm. The newly developed PCA algorithm can be added to the measured anterior corneal astigmatism using vector summation to estimate total preoperative corneal astigmatism, as suggested by Ma and Law.A With this mathematical approach, PCA can be incorporated in toric IOL power calculation formulas for any orientation of anterior corneal astigmatism. Therefore, eyes with oblique astigmatism could also benefit from better predictability because of the PCA incorporation. The aim of the current study was to test the accuracy of toric IOL power determination with this new PCA algorithm in a dataset independent from that used to develop the algorithm.

Patients and methods

Previous Prospective Clinical Study

The current retrospective study was performed with the data from the Investigational Device Exemption registration clinical trial (ClinicalTrials.gov identifier, NCT01098812C) of the Tecnis toric IOL (Model ZCT, Abbott Medical Optics, Inc.). It was a prospective multicenter 2-armed bilateral study with a 6-month follow-up of patients who received IOL implantation between March 31, 2010, and May 31, 2011, at 13 clinical sites in the United States and 1 site in Canada. A report of the design and results in this study has been published.13

The study included eyes receiving a toric IOL (Tecnis toric ZCT150, ZCT225, ZCT300, or ZCT400, Abbott Medical Optics, Inc.). Preoperatively, a complete ophthalmologic examination was performed in all patients including keratometry, ocular biometry, and IOL power calculation. Surgically induced astigmatism (SIA) was determined by the surgeons based on their experience. Postoperative measurements included residual refractive error and K readings.

Retrospective Study Analysis

In the current retrospective study, toric IOL power calculations were performed using an exact vergence formula (Holladay 114) residing in the Tecnis toric calculator (Johnson & Johnson Vision).B Preoperative biometric inputs, such as axial length (AL), preoperative corneal power at both principal meridians of the cornea, estimated SIA, implanted spherical equivalent, and toric IOL power orientation were used to predict refractive astigmatism. This prediction was performed with the PCA algorithm (referred to as the toric calculator + PCA) and without the PCA algorithm (referred to as the toric calculator). Calculations were also performed using postoperative K readings to avoid the potential effect of an inaccurate estimation of SIA. The toric IOL axis alignment at 6 months postoperatively, measured according to procedure described elsewhere,13 was considered in all calculations.

The error in predicted refractive astigmatism was considered as a measure of the prediction error of each method. This was calculated for all eyes as the vector difference between the predicted refractive astigmatism and the actual refractive astigmatism corrected for the corneal plane measured at 6 months postoperatively. The mean absolute error (MAE), median absolute error, and centroid error in predicted refractive astigmatism were calculated per calculation method. The proportion of eyes with absolute error in predicted refractive astigmatism within certain levels of astigmatism (0.50 D, 0.75 D, 1.0 D, and 1.50 D) were also evaluated.

Finally, for all calculation procedures, a subgroup analysis was performed by dividing eyes based on the orientation of anterior corneal astigmatism. Eyes were considered WTR when the anterior corneal steep meridian was between 60 degrees and 120 degrees, and ATR when the anterior corneal steep meridian was between 0 degree and 30 degrees or between 150 degrees or 180 degrees. Otherwise, the eye was included in the oblique group.

Comparative analyses were performed with Matlab (R2013a for Windows, Mathworks Inc.). The normality of data distributions was evaluated using the Kolmogorov-Smirnov test. The pairwise comparison of absolute error in predicted astigmatism was performed using the paired Student t test and the Wilcoxon ranked-sum test. The comparison of the proportion of eyes with absolute errors below or equal to 0.50 D, 0.75 D. 1.0 D, and 1.50 D was performed with the chi-square test. Statistical analysis of centroid errors was performed in accordance with the method described by Naeser and Hjortdal15 allowing statistical comparisons using the Hotelling T-squared distribution.

Results

The study evaluated 274 eyes receiving a toric IOL. The mean AL was 23.7 mm ± 1.0 (SD) and the mean IOL power was 20.6 ± 2.9 D. Table 1 shows the number of eyes per toric IOL model as well as the preoperative keratometric astigmatism, estimated SIA, and 6-month postoperative keratometric astigmatism values.

Table 1
Table 1:
Estimated SIA, and preoperative and postoperative keratometric astigmatism values per toric IOL model.

Absolute Error in Predicted Refractive Astigmatism

Table 2 shows the absolute errors in predicted refractive astigmatism for each calculation method. The median absolute error in predicted residual astigmatism was reduced when the PCA algorithm was incorporated in the calculation, independently of whether preoperative or postoperative K values were used (P < .001). With preoperative K values, the median absolute error decreased by 0.18 D with the incorporation of the PCA algorithm (P < .001). With postoperative K values, the median absolute error decreased by 0.10 D with the incorporation of the PCA algorithm (P < .001). When incorporating the PCA algorithm, there was no statistically significant difference between the median absolute error calculated with preoperative K values and the median absolute error calculated with postoperative K values (P = .09). However, without incorporating the PCA algorithm, there was a statistically significant difference (P = .04).

Table 2
Table 2:
Absolute errors and centroid errors in predicted refractive astigmatism per calculation method (ie, with and without the PCA algorithm) when preoperative and postoperative K values were used.

Figure 1 shows the distribution of the absolute error in predicted residual astigmatism with and without using the PCA algorithm. The incorporation of the PCA algorithm led to a significant increase in the number of eyes with residual astigmatism within 0.50 D, 0.75 D, 1.00 D, and 1.5 D (P = .03 for the group with postoperative K values; P < .001 for the group with preoperative K values). Specifically, the percentage of cases in which the absolute error was below 0.50 D increased with the use of the PCA algorithm by 12 and 19 percentage points with postoperative K values and preoperative K values, respectively.

Figure 1
Figure 1:
Distribution of absolute error in predicted refractive astigmatism with preoperative and postoperative K values with and without using the new PCA algorithm (PCA = posterior corneal astigmatism; SIA = surgically induced astigmatism).

Centroid Error in Predicted Refractive Astigmatism

Figure 2 shows the distribution of errors in predicted refractive astigmatism in the calculations with preoperative and postoperative K values using the PCA algorithm and without using the PCA algorithm. With the incorporation of the PCA algorithm, the centroid error was reduced from 0.50 @ 1 to 0.19 @ 3 when using preoperative K values and from 0.30 @ 0 to 0.02 @ 84 when using postoperative K values. All centroids were significantly different from zero (P < .001), except when the PCA algorithm was incorporated in the postoperative K values (P = .901). When compared pairwise with the PCA algorithm and without incorporating the PCA algorithm, the centroid differed significantly (P < .001) with the preoperative K values as well as with the postoperative K values.

Figure 2
Figure 2:
Double-angle plots of errors in predicted refractive astigmatism (difference between predicted and 6-month postoperative refractive astigmatism) without (top plots) and with (bottom plots) using the new PCA algorithm using preoperative and postoperative K values. Each ring corresponds to 0.50 D whereas the outer ring is for 3.00 D (SIA = surgically induced astigmatism; TC = toric calculator).

Anterior Corneal Astigmatism Orientation Analysis

Table 3 shows the centroid error in predicted residual astigmatism divided by the orientation of anterior corneal astigmatism as well as the number of eyes per group. The number of eyes with ATR or WTR astigmatism was similar for preoperative and postoperative K values. In both cases, the number of eyes with oblique astigmatism was substantially lower. In all cases, the incorporation of the PCA algorithm reduced the centroid error (P < .001). When the preoperative K values were used, the ATR and WTR centroid differed significantly from zero (P < .001 and P = .002), but not the oblique group (P = .914). When the postoperative K values were used with the PCA algorithm, the centroid errors were not significantly different from zero for any orientation of anterior corneal astigmatism (P = .915, P = .270, and P = .988, respectively).

Table 3
Table 3:
Centroid error in predicted refractive astigmatism per calculation method (ie, with and without the PCA algorithm), divided by the orientation of anterior corneal astigmatism when preoperative and postoperative K values were used.

Discussion

Several factors have been shown to contribute to a reduced predictability of refractive correction with toric IOLs, including inadequate management of corneal incisions,16 intraoperative misalignment of the IOL,17 postoperative IOL rotation in the capsular bag,18 and inaccurate calculation of IOL toricity.19 Several developments in surgical techniques and IOL designs have had the goal of minimizing the potential contribution of the first 3 factors mentioned above.17 However, the optimization of toric IOL power calculation remains a challenge. Several approaches have been used to incorporate the potential contribution of PCA in total corneal astigmatism and toric IOL power calculations.9–12 The aim of the current research was to show the potential benefit of using a new algorithm that incorporates estimated PCA in toric IOL power calculation and refractive astigmatic outcomes. Figure 3 shows the inputs and outputs of the toric IOL power calculation method we used herein (Tecnis toric calculatorB). Given the preoperative K reading, the estimates of PCA and SIA are added by vector summation to predict preoperative total corneal astigmatism. This value and other biometric parameters are used in a vergence formula that calculates different toric IOL powers and the corresponding predicted residual astigmatism for each of these.

Figure 3
Figure 3:
Diagram of the complete procedure for IOL power calculation when the PCA algorithm is included. Inputs are marked with dashed lines and intermediate outputs with continuous boxes (AL = axial length; IOL = intraocular lens; K = keratometry; SE = spherical equivalent; PCA = posterior corneal astigmatism; SIA = surgically induced astigmatism).

In the current study, we evaluated the distribution of the errors in predicted refractive astigmatism, defined as the difference between predicted and actual postoperative refractive astigmatism in eyes implanted with toric IOLs. Calculations were performed using the same vergence formulas as those included in the toric calculator with and without considering the new PCA algorithm. A higher proportion of eyes within lower prediction errors were found when the new PCA algorithm was used. Furthermore, a significant reduction was also observed in the mean magnitude of centroid error in predicted residual astigmatism. Specifically, the mean magnitude of centroid error was reduced from 0.50 D to 0.19 D when preoperative K values were considered, and from 0.30 D to 0.02 D when postoperative K values were used. In addition, the median absolute error in predicted astigmatism was the smallest (0.44 D) when PCA and postoperative K values were used. It is important to note that the use of postoperative K values for the analysis eliminates the effect of any incorrect estimates of SIA and, therefore, provides the most accurate estimation of the effect of the PCA algorithm in the toric IOL power calculations. These analyses confirm the benefit of incorporating this new algorithm considering the effect of PCA in IOL power calculations.

Other studies have also shown the benefit of incorporating the contribution of the posterior corneal surface in monofocal and toric IOL power calculations.9–12,20,21 In a retrospective study that used only postoperative K values, Abulafia et al.20 found that the centroid errors in predicted residual astigmatism using different devices for measuring corneal power ranged from 0.53 D to 0.56 D with the Alcon and Holladay toric calculators. These centroid errors were significantly reduced with approaches considering the potential contribution of the posterior corneal surface, such as the Baylor nomogram (0.21 to 0.26 D) and the Barrett toric calculator (0.01 to 0.16 D). The same authors reported median absolute refractive errors between 0.65 D and 0.67 D for methods that did not incorporate the effect of PCA. The use of the Baylor nomogram or the Barrett toric calculator reduced the median absolute error to 0.50 D and 0.48 D, respectively. In a more recent study, Ferreira et al.22 reported a comparison between different toric calculation procedures. Calculation methods that do not account for PCA provided a systematic ATR centroid error (between 0.40 D and 0.43 D), which was reduced by the application of different nomograms to account for PCA such as the Baylor nomogram (0.35 D) or the Abulafia-Koch formula (between 0.34 D and 0.25 D), and by the use of the Barrett toric calculator (0.17 D). Interestingly, a ray-tracing software that uses measured posterior corneal curvatures yielded a higher prediction error than the Barrett calculator (0.32D). Similar trends were found for MAEs.

Therefore, these results are comparable to the outcomes of our study with the new PCA algorithm that resulted in a median absolute error of 0.44 D and a centroid error of 0.02 D when postoperative K values are included in the calculation. Eom et al.10 found that the median magnitude of the predicted residual astigmatism error calculated with the estimated total corneal astigmatism (0.30 D) was significantly smaller than that calculated with simulated K (0.50 D). These authors estimated the total corneal astigmatism by performing the vectorial sum of anterior corneal astigmatism and an estimated PCA.9 Recently, Reitblat et al.21 confirmed that the median simulated residual astigmatism was lower when based on vector summation of anterior corneal astigmatism and PCA than with calculations based on anterior corneal measurements only, application of the Baylor nomogram, true net power, and Scheimpflug-based total corneal power measurements (0.49 D versus 0.70 D, 0.60 D, 0.64 D, and 0.76 D, respectively). Therefore, the inclusion of the effect of PCA is crucial for optimizing IOL power calculations. Indeed, previous research has shown that PCA has the highest influence on the error in refractive astigmatism after toric IOL implantation.23 This study shows that our PCA algorithm led to a significant reduction in centroid error in predicted residual astigmatism, which was comparable to the results described in the literature.

This study also shows that the incorporation of PCA reduced the centroid error in predicted refractive astigmatism irrespective of the orientation of anterior corneal astigmatism. When postoperative K values were used, the centroid errors were 0.02 D, 0.11 D, and 0.01 D for ATR, oblique, and WTR eyes, respectively. Abulafia et al.20 reported that with the Barrett toric calculator, an ATR centroid error of 0.06 D for WTR anterior corneal astigmatism and a WTR centroid error of 0.16 D for ATR eyes, also using postoperative K values. This study concluded that the Barrett toric calculator showed a centroid error close to zero for eyes with WTR astigmatism and a minimal overcorrection for eyes with ATR corneal astigmatism that might result in minor WTR residual astigmatism. With the same calculation procedure, Ferreira et al.22 reported an ATR centroid error of 0.12 D and 0.21 D for ATR and WTR eyes, respectively. Our results for ATR and WTR were not statistically different from zero. Therefore, the new PCA algorithm fully compensates the average effect of PCA for these anterior corneal astigmatism orientations. Neither Abulafia at al.20 nor Ferreira et al.22 reported results for eyes with oblique astigmatism. In our study, the centroid error was 0.11 D WTR for this group, which was not statistically different from zero. However, the limited number included in this group compared with the others, which were approximately 3 times larger, might indicate that further studies with a greater number of eyes are required to get a better understanding of the accuracy of the PCA algorithm in this population.

In our sample, in addition to the reduction in centroid error in predicted residual astigmatism with the PCA algorithm, an increase in the percentage of eyes with a low absolute error in predicted astigmatism was also observed. This confirms the improvement in the refractive predictability with the new PCA algorithm, generating an increase in the percentage of cases with absolute error below 0.50 D between 12% and 19%, depending on whether preoperative or postoperative K values were included in the calculation. A total of 45% (PCA not considered) and 57% (PCA considered) of eyes in the group with postoperative K values of our sample showed an absolute error in predicted astigmatism below 0.50 D, respectively. Abulafia et al.20 found percentages of eyes with absolute errors below 0.50 D that ranged from 26.5% to 38.2% for the Alcon and Holladay toric calculators using the keratometric data provided by different types of devices. These percentages improved with the use of the Baylor nomogram and Barrett toric calculator, ranging from 39.7% to 75.0%.20

Although PCA has a significant effect on toric IOL power calculation, other factors have also been shown to influence the outcomes with toric IOLs, such as the accuracy of SIA estimation23 and effective lens position (ELP).24 The latter is included in all the calculations presented in this study because an exact vergence formula is used. In our sample, toric IOL power calculations were also performed including the postoperative K readings to analyze the potential effect of SIA on the error in predicted residual astigmatism. In these cases, the mean magnitude of centroid error in predicted residual astigmatism was significantly reduced with the use of the new PCA algorithm from 0.30 D to nearly zero (0.02 D), as was also reported for the Abulafia-Koch formula.9 Likewise, the percentage of eyes with an absolute error in predicted astigmatism below 0.50 D improved from 45% to 57% with the use of the new PCA algorithm. This confirms that in addition to the inclusion of the PCA effect in toric IOL power calculations, an accurate estimation of SIA is also required. Visser et al.25 suggested that the inclusion of SIA into the toric IOL power calculations increases their effectiveness. These authors recommended the incorporation of 0.0 D, −0.30 D, or −0.50 D of SIA for 2.2 mm, 3.4 mm, or 5.4 mm superior incisions, respectively. Eom et al.26 have recently developed a new algorithm for optimizing IOL power calculation that not only considers the effect of PCA, but also the contribution of incision-induced PCA and ELP.

Traditionally, SIA has been calculated as the mean magnitude of individual SIA vectors. However, Koch and Wang27 pointed out that the correct method for calculating SIA is to use the centroid value of a double-angle plot. This will correctly incorporate the effect of vector magnitude and orientation of the SIA vectors. In their study of SIA in 25 eyes, the mean vector magnitude was 0.39 D, whereas the centroid was much lower at 0.13 @ 95. In clinical practice, outcomes could potentially be improved if surgeons determine their own average SIA. Online tools are available to assist in this process.D

In conclusion, these results show that the new PCA algorithm, in combination with an exact vergence IOL power calculation formula, increases the predictability of the refractive correction with toric IOLs. Specifically, this PCA algorithm, based on vector summation, decreases the error in the prediction of refractive astigmatism in eyes implanted with toric IOLs.

What Was Known

  • Posterior corneal astigmatism has a non-negligible effect in toric IOL power determination.

What This Paper Adds

  • The use of a new online algorithm that incorporates the effect of the PCA significantly reduced the astigmatic prediction error after toric IOL implantation.
  • The improved predictability of the new algorithm did not depend on the orientation of anterior corneal astigmatism.

References

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Disclosures

Drs. Canovas, Alarcon, Rosén, Kasthurirangan, and Piers are employees of Johnson & Johnson Vision Care, Inc. Drs. Ma and Koch are consultants to Johnson & Johnson Vision Care, Inc.

Other cited material

A. Ma JJK, Law C, “Preoperative, Intraoperative and Postoperative Wavefront Aberrometry and Internal Astigmatism in a Cataract Population,” presented at the ASCRS Symposium on Cataract, IOL and Refractive Surgery, San Francisco, California, USA, April 2013. Abstract available at: https://www.eyeworld.org/2013-ascrs-4. Accessed November 25, 2017
B. Johnson & Johnson Vision. TECNIS® IOL Calculator Platform. Available at: https://www.amoeasy.com/calc(bD1lbiZjPTA1MA==)/landingpage.htm. Accessed November 25, 2017
C. U.S. National Insitutes of Health Clinical Trials. Clinical Evaluation of a 1-Piece Intraocular Lens. NCT01098812. Available at: https://clinicaltrials.gov/ct2/show/NCT01098812. Accessed November 25, 2017
D. Hill W. The Surgically Induced Astigmatism (SIA) Calculator. Available at: http://www.doctor-hill.com/iol-main/toric_sia_calculator.htm. Accessed November 25, 2017
© 2018 by Lippincott Williams & Wilkins, Inc.