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Meridional analysis for calculating the expected spherocylindrical refraction in eyes with toric intraocular lenses

Fam, Han Bor FRCSE; Lim, Kooi Ling BOptom

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Journal of Cataract & Refractive Surgery: December 2007 - Volume 33 - Issue 12 - p 2072-2076
doi: 10.1016/j.jcrs.2007.07.034
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Power calculations for spherical intraocular lenses (IOLs) are optimized to increase the precision and predictability of outcomes.1–3 With toric IOLs, an additional parameter, the astigmatic error, is added to the IOL calculation. A standard IOL formula can be used to determine the spherical equivalent (SE) of the IOL to be implanted as well as the SE of the expected refraction. However, this does not describe the individual spherical and cylindrical components of the expected postoperative refraction that will assist the surgeon in selecting the ideal toricity of the IOL to be implanted.

The Holladay 1 IOL formula, first described in 1988,2 is a free and widely used method of calculating IOL power that has good precision over a wide range of axial lengths. This paper describes a method of calculating the expected spherocylindrical refraction in eyes with toric IOLs using the Holladay 1 formula and to compare it with the actual postoperative outcome.


The calculation of the expected spherocylindrical refraction was performed using the following steps:

  1. Calculate the surgically induced refractive change (SIRC) on the corneal plane.
  2. Combine the SIRC with the existing corneal cylinder to predict the postoperative corneal cylinder.
  3. Calculate the emmetropic IOL power required for both the flat and steep corneal meridians. The desired spherical and cylindrical IOL power for surgery can be determined.
  4. Calculate the expected refraction along the steep and flat corneal meridians with the desired IOL power using meridional power analysis.

Calculating the Surgically Induced Refractive Change

To refine the postoperative outcome, the SIRC on the cornea is calculated and factored into the existing corneal cylinder. The SIRC will influence the magnitude of the postoperative corneal cylinder and affect the choice of the IOL to be implanted.

The corneal SIRC is influenced by incision size, construction, and location.4–6 The magnitude of surgically induced corneal cylinder after phacoemulsification has been reported to range from 0.50 to more than 1.00 diopter (D).1–3 The prerequisite data required for this are the preoperative corneal cylinder and postoperative corneal cylinder. The calculation of the SIRC was described by Holladay et al.7

The SIRC is written in a spherocylindrical form of

where all cylindrical values (refraction and keratometry) are reported in negative cylinder values.

Combining the Surgically Induced Refractive Change and Corneal Cylinder to Predict the Postoperative Corneal Cylinder

Once the SIRC is known, it is combined with the existing corneal cylinder to determine the predicted corneal cylinder from the surgery.

The spherocylindrical power of the cornea is written as

where Max K is the maximum keratometry, Min K is the minimum keratometry, CK is the Min K − Max K, and AK is the meridian of flat keratometry.

Rearrange equation 1 and equation 2 into

where A2 > A1.

Using Holladay's method of combining obliquely crossed cylinders

Therefore, S3/C3 × A3 is the resultant corneal cylinder after combining the SIRC and keratometry, where C3 will be written in a negative cylinder form.

The meridional corneal power at the corneal plane will be

Meridional Intraocular Lens Power Calculation to Determine the Desired Spherical and Cylindrical Intraocular Lens Power

In this study, the IOL power is calculated using the Holladay 12 formula for both the steep (equation 10) and flat (equations 11) corneal meridians independently. However, the anterior chamber depth, ACDHolladay (ie, effective lens position [ELP]), is calculated from the mean keratometry of the cornea, not from the meridional keratometry. This is because the ELP signifies the estimated distance between the anterior cornea and the principal plane of the IOL8 and as such, the same IOL in the same eye cannot have more than 1 ELP. This is the same method used in present IOL calculations, in which the mean Max K and Min K are used to determine the ELPmean and expected SE refraction.


where raverage = 0.5(rsteep + rflat)

The emmetropic IOL power for the steep corneal meridian will then be

where L=AxL+TR−t; TR = 0.2 mm, t = 0, and AxL is the axial length of the eye.

With the calculated IOL power of the steep (IOLsteep) and flat (IOLflat) meridians determined, the desired toric IOL power will be

where IOLflat − IOLsteep = the toricity of the desired IOL (at the IOL plane) (in negative cylinder) and A3 = intended axis of the IOLdesired.

Expected Spherocylindrical Refraction Calculation with the Desired Intraocular Lens Power Using Meridional Power Analysis

By analyzing the cornea and IOL along the principal meridians, the expected refraction along each meridian can be determined.

For a toric IOL of SIOL/CIOL × AIOL (also in the negative cylinder form), the meridional IOL power at the IOL plane will be

The IOL SE will be

A3 = AIOL as the toric IOL is intended to be aligned to the corneal cylinder.

Using the Holladay 1 formula, the expected refraction can be determined for the steep and flat meridians using the radius of curvature of each corneal meridian, respectively. Again, the ACD is calculated as in equations 12 and 13.

For the steep meridian corneal power, the expected refraction (ERsteep) will be

By calculating ERsteep and ERflat, the expected sphere, cylinder, and axis can be determined as follows:

An example of detailed calculations using this method is presented in the Appendix.

The calculations above were used to determine the expected refraction in 7 eyes of 7 consecutive patients who had phacoemulsification with toric IOL implantation by the same surgeon (F.H.B.). The same type of toric IOL (AcrySof, Alcon) was implanted in all 7 eyes. The AcrySof toric comes in 3 models: SN60T3, SN60T4, and SN60T5, with 1.50 D, 2.25 D, and 3.00 D of cylindrical correction, respectively. Subjective refraction was obtained at the 1-month postoperative visit.

The paired t test was used to test the expected and actual postoperative outcomes. A P value less than 0.05 was considered statistically significant.


Table 1 shows the parameters of the eyes before and after toric IOL implantation. Five eyes (71.4%) were within ±0.25 D of the SE expected refraction, and all eyes were within ±0.50 D. Four eyes (57.1%) were within ±0.25 D of the expected cylinder, and all eyes were within ±0.50 D.

Table 1
Table 1:
Preoperative and postoperative keratometry measurements. The expected refraction was calculated using equation 19.

Figure 1 shows the efficacy of the toric IOL in reducing the postoperative cylindrical refraction. The mean preoperative keratometric cylinder was 2.07 D ± 0.97 (SD) (range 1.25 to 3.75 D) and the mean postoperative cylinder by subjective refraction, 0.68 ± 0.40 D (range 0.50 to 1.50 D). The mean expected cylindrical refraction was 0.62 ± 0.46 D (range 0.30 to 1.60 D). There was a highly significant reduction in the cylinder after toric IOL implantation (P<.001, paired t test). There was no statistically significant difference between the magnitude of expected cylindrical refraction and the actual postoperative cylinder by subjective refraction (P = .604).

Figure 1
Figure 1:
Doubled-angle polar plot showing the magnitude and direction of the preoperative cylinder (▴) and postoperative cylinder (•).


With conventional spherical power-only IOLs, the surgeon need only select 1 IOL parameter. With toric IOLs, the surgeon has a further task of choosing the IOL toricity that will compensate for that of the postoperative cornea. This is important in present-day cataract surgery in which postoperative success is measured not only in surgical terms but also in refractive outcome.

Intraocular lens power optimization propounded by surgeons and IOL manufacturers involves refining the prediction of the ELP to reduce the prediction error of the IOL calculation method. The ELP is represented in the third-generation IOL formula as part of the refractive vergence calculation.2,3 Synonymous with the concept of using the double-K method in IOL power calculation after laser in situ keratomileusis,9,10 taking into account 2 keratometric values to determine the ELP and corneal power has been shown to improve IOL power selection. The advantage of this method is that the estimated postoperative cylindrical power can be determined during the IOL selection process and hence aid the surgeon in selecting the desired toricity of the IOL. The mean of the calculated steep and flat meridian IOL powers will be equal to the SE power (as in normal IOL calculations using the mean keratometry).

The fundamental requirement in toric IOL implantation is that the IOL and corneal cylinder are well aligned. It is well known that a rotational misalignment of 15 degrees will cause a 50% loss of efficacy at the corneal plane. Some rotational misalignment could have contributed to the disagreement between the axis of the expected and actual cylindrical refraction in eye 2 and eye 7. Nevertheless, the high predictability and good refractive outcomes using toric IOLs could make the use of toric IOLs the standard of care in cataract surgery.

Although this study is limited by the small number of eyes reported, we proved that the meridional analysis method used in calculating the expected refraction correlates closely with the actual postoperative outcome. Meridional analysis of IOL calculations can improve the decision-making process in choosing the appropriate toric IOL power. Among methods in current practice is to empirically select the IOL toricity based on the rule-of-thumb that 1.00 D of IOL power equates to approximately 0.67 D of refractive power at the corneal plane.11 Although the rule-of-thumb provides a useful estimate, it is far from ideal as it does not take into account the actual optical system of the cornea, ACD, IOL position, and surgically induced astigmatism. Future work will focus on testing the meridional power-analysis method in larger groups of patients having toric IOL implantation.


1. Olsen T. Improved accuracy of intraocular lens power calculation with the Zeiss IOLMaster. Acta Ophthalmol Scand. 2007;85:84-97.
2. Holladay JT, Prager TC, Chandler TY, et al. A three-part system for refining intraocular lens power calculations. J Cataract Refract Surg. 1988;14:17-24.
3. 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.
4. Borasio E, Mehta JS, Maurino V. Surgically induced astigmatism after phacoemulsification in eyes with mild to moderate corneal astigmatism: temporal versus on-axis clear corneal incisions. J Cataract Refract Surg. 2006;32:565-572.
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6. Barequet IS, Yu E, Vitale S, et al. Astigmatism outcomes of horizontal temporal versus nasal clear corneal incision cataract surgery. J Cataract Refract Surg. 2004;30:418-423.
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For patient 4 (from Table 1), the preoperative corneal power can be written as 44.00/−1.75 × 101.

With a surgically induced cylinder of +0.125/−0.25 × 70, the combined SIRC and corneal cylinder will be as follows:

S1 = +0.125, C1 = −0.50, and A1 = 70; S2 = 44.00, C2 = −1.75, and A2 = 101

Using equations 3 to 9, S3 = 44.07, C3 = −1.88, and A3 = 98 (in negative cylinder form). Therefore, the steep meridian corneal curvature, rsteep = 7.658 mm @ 8 and the radius of the flat meridian, rflat = 8.00 mm @ 98 (equations 10 and 11).

With AxL = 24.08 mm and Surgeon Factor = 1.62, ELPHolladay = 3.91 and ACD = 5.53 (equations 12 and 13).

The emmetropic IOL power for the steep corneal meridian (equation 14) will then be

Based on the calculated powers, the IOLdesired of +19.50/+2.25 was chosen (the AcrySof range of toric IOLs, as used in this series, is marked with the spherical equivalent power/toricity).

The meridional power of IOLdesired is IOLsteep = 18.375 @ 8 and IOLflat = 20.625 @ 98. After equation 19,

Therefore, the expected refraction for this eye is +0.19 −0.32 × 98.

© 2007 by Lippincott Williams & Wilkins, Inc.