#### Haigis-L Formula

By Wolfgang Haigis, MS, PhD

With its three constants a0, a1, and a2 properly optimized, the regular Haigis formula has a good and proven clinical performance irrespective of axial length (AL).^{1} Because corneal power is not used in this formula as a predictor of the effective lens position (ELP), it has the advantage that, for application after refractive surgery, only the keratometer index error must be corrected.

By comparing measurement results of the IOLMaster (Carl Zeiss Meditec) with those from the refractive history method with reliable data,^{2,3} a correction curve for the IOLMaster was derived. This curve translates the keratometric measurements of the IOLMaster into a corrected corneal power, which is then input into the regular Haigis formula. The combination of the correction curve and the regular Haigis formula has been given the name Haigis-L, which is implemented in the IOLMaster software from Version 4 onward. An advantage of this formula is that it does not require any historic patient data, relying fully on currently available measurements. Biometry and keratometry are performed for a postrefractive surgery patient just as for a so-called normal cataract patient; the only difference lies in the selection of the formula for IOL calculation: for the refractive patient, the Haigis-L is chosen, and for the so-called normal patient, the regular Haigis or any of the other popular IOL formulas is selected.

In 2008, our group published the first results of 187 cataract procedures after refractive laser surgery for the correction of myopia, performed by 57 surgeons from seven countries using this formula.4 In the meantime, we have collected results for 91 more eyes, giving a total of 278 eyes (222 previously myopic and 56 hyperopic). The myopic eyes were supplied by 64 surgeons with 35 different IOL types and the hyperopic eyes by 15 surgeons with 13 different IOL types. For all cases, the IOLMaster was used for biometry (axial length and anterior chamber depth [ACD]) and keratometry. IOL power was calculated with the Haigis-L formula, in most cases prospectively.

By comparison with the truly achieved stable postoperative refractions at distance BCVA, the following mean arithmetical prediction errors were derived: -0.08 ±0.71 D for myopic eyes and -0.06 ±0.77 D for hyperopic; the respective medians of the absolute prediction errors were 0.37 and 0.40 D. Correct refraction predictions within ±2.00 D, ±1.00 D, and ± 0.50 D were obtained in 98.6%, 82.9%, and 59.9% of the myopic and 96.4%, 82.1%, and 58.9% of the hyperopic eyes, respectively.

These results compare favorably with normal eyes, although the error margins in predicted refraction are somewhat higher after refractive surgery.

*Wolfgang Haigis, MS, PhD, is a Professor in the Department of Ophthalmology, University of Wuerzburg, Germany. Professor Haigis states that he is a consultant to Carl Zeiss Meditec. He may be reached at tel: +49 931 201 20640; fax: +49 931 201 20454; e-mail: w.haigis@augenklinik.uni-wuerzburg.de.*

- Haigis W. Intraocular lens calculation after refractive surgery. European Ophthalmic Review. 2012;6(1):21-24.
- Holladay JT. IOL calculations following RK. Refract Corneal Surg. 1989;5(3):203.
- Hoffer KJ. Intraocular lens power calculation for eyes after refractive keratotomy. J Refract Surg. 1995;11:490-493.
- Haigis W. IOL calculation after refractive surgery for myopia: the Haigis-L formula. J Cataract Refract Surg. 2008;34(10):1658-1663.

#### Hoffer Q Theoretical Vergence Formula

By Kenneth J. Hoffer, MD

The year 1974 brought many firsts for me: I implanted my first IOL; I began organizing my first scientific meeting for the American Intra-Ocular Implant Society, which eventually became the American Society of Cataract and Refractive Surgery (ASCRS); I began working on a scientific peerreviewed journal for the society (Journal of Cataract and Refractive Surgery); I performed the first A-scan ultrasound axial length measurement for IOL power calculation in the United States; and, as a result of the latter, I wrote my first IOL power calculation formula.^{1,2}

The original Hoffer formula built upon Professor M.C. Colenbrander’s formula. What made my variation unique from his and other existing formulas at that time, however, was that the desired postoperative refractive error could be added into the formula so the formula could be used to calculate the postoperative refractive error from any given IOL power. Also, the unit could be entered in diopters and AL and ACD could be entered in millimeters rather than meters. Eight years later, in 1982, I did a study that showed that the ACD and AL of the eye were directly proportional. I then updated my formula to calculate the ACD based on the AL (ACD = 2.92 x AL + 2.93).

Jack T. Holladay, MD, introduced his Holladay 1 formula in 1988 using another component to IOL power calculations, when, in addition to AL, he began using corneal power (K) to calculate the position of the IOL using the Fyodorov corneal height formula.^{2} At his instigation, I spent months and months trying to improve my formula to also include K. The result is a long, complicated formula that incorporates the factor of the tangent of K as well as the AL to generate an ELP; this is the Hoffer Q formula. In short, the Hoffer Q formula is the same as the 1974 Hoffer formula but using this additional Q formula to first calculate the ELP.

#### PROGRAMMING THE FORMULA: PEARLS AND WARNINGS

I am the first to admit that the Hoffer Q is a complicated formula, with certain factors that make it hard to program. For instance, the AL used in the Q formula does have a maximum value but obviously it is not limited in the vergence formula.

The Hoffer Q is notoriously the worst formula to program by yourself; almost everyone who has programmed it based on the original publication in 1993 has gotten it wrong.^{3} It is crucial to let me check the formula programming to ensure that is has been programmed properly. The best tip that I can give these practitioners is to use the Hoffer Q formula in the 1993 erratum rather than the original article,^{3} because the journal had left out a minus sign and a parenthesis in the original article. I wrote another erratum in 2007,^{3} listing common errors that practitioners make when programming the formula. This list is typically helpful for solving the majority of problems for most practitioners.

I would also warn practitioners to use the formula in a licensed instrument, such as an ultrasound device licensed to use the formula, the IOLMaster, or the Lenstar (Haag-Streit). Using the Hoffer Q in a licensed machine means that I have had access to the equipment and have run the formula through 500 eyes to ensure that it is programmed correctly and licensed properly. My original study indicated that the Hoffer Q was comparable to the Holladay 1 formula in the normal range of eyes (22.0–24.5 mm) but was less reliable in longer eyes (more than 24.5 mm.) It also indicated that it was more accurate in shorter eyes (less than 22.0 mm). I later proved this fact statistically (P<.05) in a study of 843 eyes shorter than 22.0 mm supplied to me by James P. Gills, MD (unpublished data). A recent study by Aristodemou et al, from the United Kingdom, on 8,108 eyes using the IOLMaster has statistically proven that this accuracy of the Hoffer Q in short eyes is real.^{4}

As far as postrefractive surgery eyes are concerned, the above obviously leads one to refrain form using the Hoffer Q formula in long eyes that have had myopic LASIK or PRK but definitely use it in eyes that have had hyperopic corneal refractive surgery.

I am currently working with H. John Shammas, MD, on the Hoffer*Shammas formula, but that is a work in progress.

*Kenneth J. Hoffer, MD, is Clinical Professor of Ophthalmology at the University of California, Los Angeles, and practices at the St. Mary’s Eye Center, Santa Monica, California. Dr. Hoffer may be reached at tel: +1 310 451 2020; e-mail: khoffermd@aol.com.*

- Hoffer KJ. Mathematics and computers in intraocular lens calculation. Am Intra-Ocular Implant Soc J. 1975;1(1):4-5.
- Holladay JT, Prager TC, Chandler TY, et al. A three-part system for refining intraocular lens power calculations. J Cataract Refract Surg. 1988;14(1):17-24.
- Hoffer KJ. The Hoffer Q formula: A comparison of theoretic and regression formulas. J Cataract Refract Surg. 1993;19:700-712; errata: 1994;20:677 and 2007;33(1):2-3.
- Aristodemou P, Knox Cartwright NE, Sparrow JM, Johnston RL. Formula choice: Hoffer Q, Holladay 1, or SRK/T and refractive outcomes in 8108 eyes after cataract surgery with biometry by partial coherence interferometry. J Cataract Refract Surg. 2011; 37:63–71.

#### Holladay 1 and 2: Theoretical Vergence Formulas

By Jack T. Holladay, MD

The Holladay 1 is a theoretical vergence formula that predicts ELP using a two-variable predictor with AL and K. First published in 1988,1 the Holladay 1 was the first twovariable predictor of ELP using a theoretical vergence formula, which has stood the test of time for the past 25 years.

We began a second study in 1993, enlisting 35 surgeons from all over the world to provide data for 35 short (less than 21 mm), 35 long (greater than 26 mm), and 35 normal eyes that would be representative of 35,000 cataract cases, as it takes approximately 1,000 eyes to get 35 within the correct range. We analyzed 10 variables and found that seven of these related to the ELP: AL, K, horizontal corneal diameter (white-towhite), ACD, lens thickness, refraction, and age. We introduced this seven-variable predictor of ELP using the theoretical vergence formula in 1995 and named it the Holladay 2.^{2-20}

In the most recent Leaming Survey (2011), conducted by Richard J. Duffey, MD, and David V. Leaming, MD, 50% of all US cataract surgeons reported that they used either the Holladay 1 or 2 formula for IOL power calculation. When all seven of the variables are entered into the Holladay 2, it is accurate in eyes from 15 to 35 mm in length.

Other authors of formulas are beginning to use more of the seven variables we have used since 1995 (as many as five thus far, by Thomas Olsen, MD, PhD); however, the Holladay 2 remains the most comprehensive. Since 1995, more than 20 articles have found the Holladay 2 formula to be the most accurate or the second-most accurate formula for IOL power calculation.2-20 Most authors usually find their own formulas to be the best on the data sets and, since the Holladay 2 was used to determine the formula, we are very happy to usually be No. 2.

*Jack T. Holladay, MD, is a Clinical Professor of Ophthalmology at Baylor College of Medicine in Houston. Dr. Holladay states that he is a consultant to Abbott Medical Optics Inc., AcuFocus, Oculus Optikgeräte GmbH, WaveTec, and Carl Zeiss Meditec. He may be reached at fax: +1 713 669 9153; e-mail: holladay@docholladay.com.*

- Holladay JT, Prager TC, Chandler TY, et al. A three-part system for refining intraocular lens power calculations. J Cataract Refract Surg. 1988;14(1):17-24.
- Holladay JT. Accuracy of Scheimpflug Holladay equivalent keratometry readings after corneal refractive surgery. J Cat Ref Surg. 2010;36:182-183.
- Trivedi RH, Wilson ME, Reardon W. Accuracy of the Holladay 2 intraocular lens formula for pediatric eyes in the absence of preoperative refraction. J Cataract Refract Surg. 2011;37(7):1239-1243.
- Ghanem AA, El-Sayed HM. Accuracy of intraocular lens power calculation in high myopia. Oman J Ophthalmol. 2010;3(3):126-130.
- Bang S, Edell E, Yu Q, Pratzer K, Stark W. Accuracy of intraocular lens calculations using the IOLMaster in eyes with long axial length and a comparison of various formulas. Ophthalmology. 2011;118(3):503-506.
- Holladay JT, Hill WE, Steinmueller A. Corneal power measurements using Scheimpflug imaging in eyes with prior corneal refractive surgery. J Refract Surg. 2009;25:862-868.
- Terzi E, Wang L, Kohnen T. Accuracy of modern intraocular lens power calculation formulas in refractive lens exchange for high myopia and high hyperopia. J Cataract Refract Surg. 2009;35(7):1181-1189.
- Haigis W. IOL calculation for high ametropia. Ophthalmologe. 2008;105(11):999-1004.
- Lüchtenberg M, Kuhli-Hattenbach C, Fronius M, et al. Predictability of intraocular lens calculation using the Holladay II formula after in-the-bag or optic captured posterior chamber intraocular lens implantation in paediatric cataracts. Ophthalmologica. 2008;222(5):302-307.
- Hu BJ, Zhao SZ, Tseng P. Intraocular lens power calculation in cataract phacoemulsification after refractive surgery. Zhonghua Yan Ke Za Zhi. 2006;42(10):888-891.
- Narváez J, Zimmerman G, Stulting RD, Chang DH. Accuracy of intraocular lens power prediction using the Hoffer Q, Holladay 1, Holladay 2, and SRK/T formulas. J Cataract Refract Surg. 2006;32(12):2050-2053.
- Chan CC, Hodge C, Lawless M. Calculation of intraocular lens power after corneal refractive surgery. Clin Experiment Ophthalmol. 2006;34(7):640-644.
- Leccisotti A. Refractive lens exchange in keratoconus. J Cataract Refract Surg. 2006;32(5):742-746.
- Wang L, Booth MA, Koch DD. Comparison of intraocular lens power calculation methods in eyes that have undergone laser-assisted in-situ keratomileusis. Trans Am Ophthalmol Soc. 2004;102:189-96; discussion 196-197.
- Wang L, Booth MA, Koch DD. Comparison of intraocular lens power calculation methods in eyes that have undergone LASIK. Ophthalmology. 2004;111(10):1825-1831.
- Packer M, Brown LK, Hoffman RS, Fine IH. Intraocular lens power calculation after incisional and thermal keratorefractive surgery. J Cataract Refract Surg. 2004;30(7):1430-1434.
- Preetha R, Goel P, Patel N, et al. Clear lens extraction with intraocular lens implantation for hyperopia. J Cataract Refract Surg. 2003;29(5):895-899.
- Gimbel HV, Sun R. Accuracy and predictability of intraocular lens power calculation after laser in situ keratomileusis. J Cataract Refract Surg. 2001;27(4):571-576.
- Seitz B, Langenbucher A. Intraocular lens calculations status after corneal refractive surgery. Curr Opin Ophthalmol. 2000;11(1):35-46.
- Kolahdouz-Isfahani AH, Rostamian K, Wallace D, Salz JJ. Clear lens extraction with intraocular lens implantation for hyperopia. J Refract Surg. 1999;15(3):316-323. Erratum in: J Refract Surg. 1999;15(6):620.

#### Olsen Formula

By Thomas Olsen, MD, PhD

The calculation of IOL power in an eye that has undergone corneal refractive surgery is not a straightforward task. However, if you understand why the calculation may go wrong, you will have a fair chance of getting a good result using modern technology.

It is important to understand that the calculation errors of post-LASIK eyes are the sum of the traditional errors plus the errors associated with the abnormal corneal shape. The post-LASIK eye exposes some of the weaknesses of the traditional IOL power calculation formulas.

#### ERRORS OF TRADITIONAL FORMULAS

Many of the old thin-lens IOL power calculation formulas, such as the Binkhorst, Hoffer Q, Holladay 1, and SRK/T use only the K reading and the AL as input.^{1} When these formulas are used in post-LASIK cases, two problems occur.

**Problem No. 1.** The flatness of the K reading makes the formula believe the ELP is more anterior than it really is. With a shallow ELP, the formula will predict a low IOL power, causing a hyperopic surprise. Figure 1 explains Fyodorov’s concept of the corneal height, which is the ELP principle of the SRK/T and the Holladay 1 formulas.

**Problem No. 2.** A simple K reading measures only the curvature of the anterior surface of the cornea. However, most keratometers calculate the power according to the formula K = (1.3375-1)/r, where r is the anterior radius in meters and 1.3375 is the fictitious refractive index of the cornea, as if the cornea were a thin lens. Therefore, in order for the anterior curvature measurement to be translated into the true power of the cornea, the following must be calculated: the curvature of the posterior surface, the corneal thickness, and the true refractive index of the cornea (1.376). In normal corneas, it is reasonable to assume that there is a constant ratio (called the Gullstrand ratio from the Gullstrand exact schematic eye) between the anterior and posterior curvatures. For the post-LASIK cornea, however, the Gullstrand ratio is reduced, and the keratometerreported K reading will overestimate the true corneal power. Again, the formula will think the IOL power should be reduced, and the patient ends up with a hyperopic error.

Modern Scheimpflug imaging and other tomography techniques provide more insight into the variation of the Gullstrand ratio in normal and post-LASIK eyes. Figure 2 shows the results of Pentacam (Oculus Optikgeräte GmbH) measurement of the anterior and posterior corneal curvatures, giving the distribution of the Gullstrand ratio in a normal population and in a post-LASIK population (average ablation, -7.00 D). The reduction of the Gullstrand ratio is dependent on the amount of laser ablation (Figures 3 and 4). For example, if the Gullstrand ratio is 0.7 (ie, as a result of a 7.00 D ablation; Figure 3), the difference between the true corneal power and the K reading is approximately 2.00 D. However, for the normal cornea (Gullstrand ratio, 0.83), the difference is not nil. This is because the conventional K reading carries an inborn overestimation of the corneal power of approximately 1.00 D.

The effect of the two sources of error on IOL power prediction with the SRK/T formula is shown in Figure 5. The K reading error and the ELP error are the same magnitude, and both tend to give a hyperopic error. The total prediction error is about one-third the amount of the excimer laser ablation if not corrected for.

#### MY PREFERRED METHOD

If available, pre-LASIK K readings and amount of laser correction should be recorded. The historic method is not regarded as reliable, and therefore we use these data only to check whether our measurements are consistent. As with any other case, the patient should have optical biometry; I prefer the Lenstar LS900 biometer because this instrument measures all intraocular distances (corneal thickness, ACD, lens thickness, and AL) with the accuracy of laser biometry. Additionally, the pupil diameter is also measured.

Next, the patient is examined with the Pentacam to determine the anterior and posterior curvatures of the cornea. I like to inspect the readings of the central cornea to check what the effective central reading might be. Useful tools of the Pentacam are its power distribution (Figure 6) and Fourier transformation (Figure 7) functions. These tools allow the surgeon to extract the effective radius in the pupillary area of the cornea.

#### COMMENT

Difficulties with reading the effective central part of the cornea add error to any formula. Keratometers usually measure the cornea in a noncentral ring of about 3 mm (about 2.5 mm with the IOLMaster). This is not adequate for a cornea with central flattening, and, in fact, all keratometers and topographers are blind to the central, most significant area of the cornea. Again, the potential error on the IOL power calculation is hyperopic.

When information on the effective central anterior curvature of the cornea and the posterior curvature are available, we enter these values directly in the PhacoOptics software (PhacoOptics.com) using the Olsen formula for IOL power calculation.^{2}

The advantage of the Olsen formula is that equivalent K readings are not needed, as the program accepts direct input of front and back corneal curvature. Moreover, the Olsen formula uses an improved algorithm for the prediction of the IOL position, making it possible to base the ELP on the anterior segment anatomy only (ie, preoperative ACD and lens thickness) and avoiding troublesome K reading dependence (eyetube.net/?v=fuhal).

In 24 eyes of 19 post-LASIK patients for which the Olsen formula was used to calculate IOL power, we found a mean refractive prediction error of 0.41 ±0.68 D (range, -0.92 to 1.63 D; Figure 8). There was no correlation with the Gullstrand ratio. The one patient with the highest error (1.63 D) was later reexamined and found to have a central cornea that was flatter than was measured at the preoperative visit, possibly caused by wearing a contact lens a few days before. The overall prediction accuracy is not as accurate as in normal eyes (and will never be), but we think the results are encouraging for our approach to be a feasible one.

#### NEW DEVELOPMENTS: RAY TRACING

As described above, much of the inaccuracy of IOL power calculations in post-LASIK eyes is due to the problem of getting reliable information on the effective refractive power of the cornea. To this end, various models are often used to fit the corneal measurement to certain well-defined shapes, such as ellipsoids, Zernike polynomials, and Fourier transformations. In physical optics, ray tracing is widely used and recognized as the most effective tool in optical design. The eye is an optical device, so why isn’t ray tracing the method of choice in our hands? The answer, perhaps, is because this technology is still evolving.

We have recently begun a study to import the Pentacam raw height data into the Zemax software (Radiant Zemax, LLC) for optical engineering.^{3} To import the data, we had to transform the matrix array of individual measurement points into a polygonal shape that could then be imported by Zemax. When the data are successfully transformed, it is possible to construct the cornea as a meshwork of miniature triangles, suitable for exact ray tracing (Figure 9). The fascinating thing about Zemax and other ray tracing software programs is that it is possible to compose and analyze the entire optical system and ask for optimal performance of any element. For example, how should the IOL be designed to fit the optical properties of the cornea (Figure 10)? For the post-LASIK cornea, it might be possible to solve for the higher-order aberrations of the selected IOL needed to correct for the HOA of the cornea. We expect this technology to be further elaborated in the future.

*Thomas Olsen, MD, PhD, is a Professor of Ophthalmology at the University Eye Clinic, Aarhus Hospital, in Denmark. Dr. Olsen states that he is a shareholder of IOL Innovations, manufacturer of the PhacoOptics program. He may be reached at tel: +45 89 493 228; fax: +45 86 121 653; e-mail: tkolsen@ dadlnet.dk.*

- Olsen T. Calculation of intraocular lens power: a review. Acta Ophthalmol Scand. 2007;85(5):472-485.
- Olsen T. Prediction of the effective postoperative (intraocular lens) anterior chamber depth. J Cataract Refract Surg. 2006;32(3):419-424.
- Olsen T, Funding M. Ray tracing analysis of the IOL power in situ. J Cataract Refract Surg. 2012;38:641-647.