A 3-D “Super Surface” Combining Modern Intraocular Lens Formulas to Generate a “Super Formula” and Maximize Accuracy

Importance  Cataract surgery is the most common eye surgery. Calculating the most accurate power of the intraocular lens (IOL) is a critical factor in optimizing patient outcomes.

Objectives  To develop a graphical method for displaying IOL calculation formulas in 3 dimensions, and to describe a method that uses the most accurate and current information on IOL formulas, adjustments, and lens design to create one “super surface” and develop an IOL “super formula.”

Design, Setting, and Participants  A numerical computing environment was used to create 3-D surfaces of IOL formulas: Hoffer Q, Holladay I, Holladay I with Koch adjustment, Haigis, and SRK/T. The surfaces were then analyzed to determine where the IOL powers calculated by each formula differed by more than 0.5, 1.0, and 1.5 diopters (D) from each of the other formulas. Next, based on the current literature and empirical knowledge, a super surface was rendered that incorporated the ideal portions from 4 of the 5 formulas to generate a super formula. Last, IOL power values of a set of 100 eyes from consecutive patients at an eye institute were calculated using the 5 formulas and super formula. The study was performed from December 11, 2014, to April 20, 2015. Analysis was conducted from February 18 to May 6, 2015.

Main Outcomes and Measures  Intraocular lens power value in diopters and the magnitude of disparity between an existing individual IOL formula and our super formula.

Results  In the 100 eyes tested, the super formula localized to the correct portion of the super surface 100% of the time and thus chose the most appropriate IOL power value. The individual formulas deviated from the optimal super formula IOL power values by more than 0.5 D 30% of the time in Hoffer Q, 16% in Holladay I, 22% in Holladay I with Koch adjustment, 48% in Haigis, and 24% in SRK/T.

Conclusions and Relevance  A novel method was developed to represent IOL formulas in 3 dimensions. An IOL super formula was formulated that incorporates the ideal segments from each of the existing formulas and uses the ideal IOL formula for an individual eye. The expectation is that this method will broaden the conceptual understanding of IOL calculations, improve clinical outcomes for patients, and stimulate further progress in IOL formula research.

By the year 2020, the World Health Organization predicts that there will be 32 million cataract operations performed every year.¹ The power of the intraocular lens (IOL) that is to be implanted at the time of surgery is determined by several mathematical formulas. These modern formulas are very sophisticated and differ in their approach to determining the power of the IOL.^2-5

In a general sense, these formulas are also very good at predicting the appropriate IOL to meet a specific target refraction with a high degree of accuracy.^6,7 However, certain formulas have been found to be more accurate under specific conditions related to the input variables that are used, such as axial length and corneal power,^2,7-9 which is likely related to the different ways the formulas use these data to determine the theoretical effective lens position based on the input data.

In addition, there is literature that supports using a particular formula for negatively powered IOLs.^10,11 Also, adjustment factors have been used with specific formulas to improve the accuracy under specific conditions (eg, Koch adjustment).¹² Thus, to our knowledge, there is no IOL formula that has been demonstrated to be the most accurate under all situations.

Because there is no one formula that performs best, the surgeon is often left choosing a particular formula based on his or her knowledge of the strengths and weaknesses of all the formulas. Furthermore, the surgeon may not have access to all the formulas to make reasoned comparisons and choose the best formula for a given situation.

It would be ideal if a formula existed that used the most accurate portions of existing formulas to generate a “super formula.” Furthermore, the particular regions of this super formula would be based on the existing data from the peer-reviewed literature. In addition, the super formula could be adjusted and optimized as new data are generated over time. This article describes a method that uses the most accurate and current information on IOL formulas, adjustments, and lens design to create one super formula.

Box Section Ref ID

A literature review was performed to investigate the most accurate IOL formulas to date using the National Library of Medicine database with the following search terms in all languages: Hoffer Q, Holladay I, Holladay I with Koch adjustment, SRK/T, Haigis, IOL formula, and IOL calculation. Based on this review, specific formulas were chosen and agreed on by all of us.

These formulas, with or without adjustment factors, were then entered as part of a complex mathematical and programming algorithm into a numerical computing environment. This code helped to generate a representation of each of the IOL formulas as a 3-D surface. The surfaces were generated individually because each formula uses input variables in a different method to arrive at the mathematical solution to determine the appropriate IOL power. The first-generation SRK I formula¹³ is shown graphically as a 3-D surface in Figure 1A. The third-generation SRK/T formula⁵ (Figure 1B) adds theoretical portions to the formula to improve its accuracy, which increases the complexity of the surface when compared with the SRK I surface (Figure 1C).

The Hoffer Q, Holladay I, Haigis, and SRK/T formulas are superimposed graphically in Figure 2. The Koch adjustment was also used with the Holladay I formula.¹²

The next step was to unify the ideal, most accurate output portions of each IOL formula into a single graphical representation and generate a “super surface.” A set of specific criteria was used to generate this single surface that used the specific areas of the surface that were deemed to be the most accurate based on our consensus on existing data from the peer-reviewed literature.^2-5,12 A mathematical and programming algorithm was written to help generate this 3-D super surface. Once this super surface was generated, a super formula was derived that was composed of the ideal portions of the existing formulas based on a range of input parameters most suitable to each formula.

Finally, a set of 100 eyes from consecutive patients was evaluated using the existing IOL formulas and adjustments. Next, the super formula was used to determine the most appropriate IOL power value based on the specific input criteria for each eye. The IOL power assignments based on an individual IOL formula was compared with IOL power assignments based on automatic selection by the super formula. In addition, we calculated the percentage of eyes for which the predicted IOL using any of the Hoffer Q, Holladay I, Holladay I with Koch adjustment, Haigis, and SRK/T formulas differed from the predicted IOL using the super formula by a specified amount (0.5 and 1.0 diopters [D]) to reflect the frequency at which a potentially significant clinical choice would have to be made by the surgeon. All research was performed in accordance with the Declaration of Helsinki¹⁴ and all local, regional, and national law. The Johns Hopkins Medicine Institutional Review Board granted approval for this study. The study was performed from December 11, 2014, to April 20, 2015. Analysis was conducted from February 18 to May 6, 2015.

Based on the peer-reviewed literature, the following criteria were used. For eyes with a measured axial length of 21.49 mm or less, the Hoffer Q formula was used.⁷ The Holladay I formula with the Koch adjustment was used for eyes with an axial length greater than 25 mm.¹² The Haigis formula was used when negatively powered IOLs were indicated.^10,11 Finally, we chose the Holladay I formula for all other eyes.

Although the existing formulas gave similar results over a range of input parameters, they also diverged significantly at specified ranges of input parameters (Figure 2). This divergence results in clinical dilemmas for surgeons and potential suboptimal results for patients. The Ladas-Siddiqui graph (Figure 3) highlights areas of clinical agreement and disparity between formulas at specified ranges of corneal power and axial length. Figure 3A demonstrates that all formulas differ from at least 1 of the other 4 by greater than 0.5 D over the entire range of input parameters. When the tolerance for divergence between formulas is increased to 1.0 D in predicted IOL power (a clinically undesirable level), there are areas of correspondence between all formulas tested (Figure 3B), which increase further when tolerance is raised to 1.5 D (Figure 3C). Thus, resolving these areas of discrepancy is of high clinical relevance and requires not only access to each formula but also a detailed knowledge of their particular strengths and weaknesses. Depicting comparisons between IOL formulas in this manner shows the specific range of input parameters over which the formulas diverge, enabling more precise understanding of the differences between formulas.

Based on our interest and more detailed knowledge of the strengths and weaknesses of each formula, we unified the ideal, most accurate output portions of each IOL formula into a combined super formula. The graphical results of this super formula are presented in Figure 4, showing a stepwise development and evolution of a singular multifaceted surface, including the most accurate portions of multiple individual formulas based on specified ranges of input variables.

To further assess the super formula, we used it to analyze 100 consecutive eyes and compared the predicted IOL power with the results of the 5 formulas. The super formula chose the most appropriate IOL formula based on our specific selection criteria in all 100 patients (eTable in the Supplement). Furthermore, it shows the number of times each of the other formulas deviated by 0.5 and 1.0 D from the super formula. The individual formulas deviated from the optimal super formula IOL power values by more than 0.5 D 30% of the time in Hoffer Q, 16% in Holladay I, 22% in Holladay I with Koch adjustment, 48% in Haigis, and 24% in SRK/T. The individual formulas deviated from the optimal super formula IOL power values by more than 1.0 D 12% of the time in Hoffer Q, 5% in Holladay I, 2% in Holladay I with Koch adjustment, 8% in Haigis, and 1% in SRK/T. Given that the super formula incorporates detailed understanding based on the literature of the most appropriate formula to use in given situations, the eTable in the Supplement reflects the frequency at which a single formula could deviate from best practices.

Calculation of the most accurate IOL power value (and subsequent selection of the lens) can be a clinically challenging problem, especially when a surgeon is presented with multiple choices and may be unsure which one formula is the most accurate. This common issue is becoming more critically important with premium lenses and increased patient expectations. This article presents a graphical solution for visualizing and characterizing existing IOL formulas. Furthermore, we have demonstrated that surgeons can use this graphical solution to combine the ideal portions of these existing formulas based on the literature evidence, resulting in a super formula.

This method solves an important clinical problem of IOL selection as it currently stands. Perhaps more important, this is not a static solution but rather something that can evolve and improve over time. For instance, as new input parameters, adjustments, measurement devices, or formulas are developed, they can be compared empirically and statistically with the current version of the super formula, leading to refinements in the super formula itself.

In addition, we have demonstrated that we can use this graphical solution to compare formulas, which enables identification of specific subsets of input parameters over which formulas diverge from each other. This more precise understanding will enable more targeted assessment of comparative IOL formula accuracy, allowing such studies to be performed with fewer eyes and less resources. For instance, new or existing formulas would only need to be compared graphically to determine input parameters leading to a clinically significant difference, and only eyes with those parameters would need to be analyzed. The same approach could also be applied to newly proposed input parameters.

In addition, the process of optimizing IOL prediction for an individual or a group of surgeons could be significantly refined in a similar manner. For example, actual clinical results may diverge from the applied formula over only a select range of input parameters. By graphing these results against the chosen formula, such as the super formula, these ranges of parameters leading to clinical variability can be identified, enabling use of surgeon optimization only for applicable patients.

We have devised a novel method of graphically representing IOL formula output that broadens the conceptual understanding of IOL calculations. We have devised a novel IOL super formula that incorporates ideal portions of existing formulas as validated by the literature. The super formula represents a method and algorithm for determining the ideal IOL power for an individual eye. We expect this approach to IOL formulas will not only provide better understanding of the subject but also improve clinical outcomes for patients and stimulate further progress in the field of IOL research.

Submitted for Publication: May 12, 2015; accepted August 21, 2015.

Corresponding Author: John G. Ladas, MD, PhD, Wilmer Eye Institute, Johns Hopkins Medical Institutions, Smith Bldg 5011, 400 N Broadway, Baltimore, MD 21231 (jladas@marylandeye.com).

Published Online: October 15, 2015. doi:10.1001/jamaophthalmol.2015.3832.

Author Contributions: Dr Ladas had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.

Study concept and design: Ladas, Siddiqui, Jun.

Acquisition, analysis, or interpretation of data: All authors.

Drafting of the manuscript: Ladas, Siddiqui, Jun.

Critical revision of the manuscript for important intellectual content: All authors.

Statistical analysis: Siddiqui.

Administrative, technical, or material support: Devgan, Jun.

Study supervision: Ladas, Devgan, Jun.

Conflict of Interest Disclosures: All authors have completed and submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest and none were reported.