An optical system with regular (=low or 3rd order) astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances. The term comes from the Greek α- (a-) meaning “without” and στίγμα (stigma), “a mark, spot, puncture”.  The following article is from a dear friend, colleague, mentor, and co-author of past publications, Dr. Jack Holladay who has contributed immensely to our understanding of optics. 


Improving toric IOL outcomes,
Reducing a patient’s ocular astigmatism with toric IOLs is on the rise and deservedly so.
by Jack T. Holladay, MD, MSEE, FACS

Jack T. Holladay

The toricity on an IOL is manufactured to within 0.25 D, just as the spheroequivalent power. As with any emerging technology and procedure, there are some pitfalls along the way that need to be addressed. Specifically, in this series, I will discuss 1) measuring corneal astigmatism, 2) toric IOL calculators, 3) toric optimizer, 4) choosing the residual astigmatic target, 5) proper IOL alignment and centration, and 6) managing unfortunate outcomes of residual astigmatism and higher-order aberrations.

Measuring corneal astigmatism

Most surgeons have used a manual or automated keratometer in the past to get the preoperative keratometry readings used for IOL calculations. These instruments usually measure four points on a ring and produce the orthogonal steep and flat meridional power. For a 44 D cornea, the manual keratometer measures points that are 3.2 mm apart; however, the IOLMaster (Carl Zeiss Meditec) measures 2.5 mm apart, and the Lenstar (Haag-Streit) has two rings and measures 2.2 mm and 1.7 mm apart. As long as the astigmatism is perfectly symmetrical (symmetric bowtie shown in Figure 1a) and does not change radially, the result should be the same. Unfortunately, this is rarely the case. The astigmatism often changes as we move radially and is not symmetrical, and sometimes the flat and steep meridian are not even orthogonal (irregular astigmatism). In general, the smaller the sample zone, the smaller the magnitude of the measured astigmatism and axes will change if shaped like a crab claw (Figure 1b).

Figure 1a. Symmetric bowtie astigmatism.

Figure 1b. Irregular crab-claw astigmatism.
Images: Holladay JT

Topography and tomography do a much better job, when irregularity is present. Rather than four points on a ring, thousands of points are measured within a 3 mm to 4.5 mm zone, and a torus or toric ellipsoid is used to perform a least squares fit of the surface to all of the measured points. We first did this on the EyeSys many years ago and named it zonal effective refractive power (Eff RP) over a 3.0 mm zone (Figure 2, values in the first column at the bottom). This value can be compared with the simulated keratometry (Sim K) values in the second column. The Sim K values use the same ring as the manual keratometer and yield essentially the same values. Topography uses Purkinje-Sanson Image 1 (the first reflection of a Placido disk) to determine the curvatures of the surface as a mirror. The reflecting optics are identical to the refracting optics on the front surface, and the analysis can be very accurate, especially to determine the refractive astigmatism. The limitations of Placido-based topography are the central scotoma (unmeasured zone) of 1 mm to 2 mm, depending on the instrument and no measurement of the back surface power.


The back surface power of the cornea is negative and contributes 10% or less to the net corneal power. In the vast majority of patients, the back surface contribution of power never changes the net astigmatism or power of the cornea in a way that cannot be compensated by a constant multiplier. For example, keratometry readings of 44 D × 46 D (2 D of keratometric astigmatism) would convert to a net power of 42.80 × 44.75 (1.95 D of net astigmatism), using 1.3283 as the net corneal index of refraction and 1.3375 for the keratometric standardized index of refraction. Unless the patient has posterior keratoconus, the only clinical condition (other than trauma) that can have a measurable effect, the difference in the astigmatism from not measuring the back surface is negligible.

Figure 2. Holladay diagnostic summary map illustrating zonal refractive power. Notice the average Sim K reading is 43.75 D and the Eff RP is 43 D, 0.75 D steeper due to the limited sample at the 3 mm ring vs. the entire 3 mm zone. Also notice the steep and flat refractive power astigmatism is +0.88 @ 143, whereas the delta Sim K is +1.42 @ 97. In this case, the irregular astigmatism is secondary to over-wear of a rigid contact lens inducing warpage of the cornea.

Figure 3. Holladay report on Pentacam with table of equivalent keratometry reading from 1 mm to 7 mm. Note there is a steady increase in zonal power of 0.8 D (41.8 to 42.6) and 0.4 D increase in astigmatism (0.0 D to 0.4 D). The changes are quite variable from patient to patient.

Tomography measures the elevations of the front and back surface directly with no central scotoma, because the camera is located peripherally. However, similar to measuring the curvatures of a spectacle lens with a Geneva lens clock, a surface must be fit to the elevations and then the power calculated, which is not trivial if the surface is irregular. In normal corneas with only slight irregularities, both tomography and topography determined over a 3 mm to 4.5 mm zone are superior to keratometry. For irregular corneas, measurements with these devices are imperative. In most of these instruments, these area values are referred to as zonal power and zonal astigmatism, like that shown in the Table in Figure 3 on the Pentacam by Oculus.

If one finds a significant difference between the 3.0 mm zonal astigmatism (central zone) and keratometric astigmatism (four points or Sim Ks), then irregular astigmatism is present and one must carefully review the topography/tomography. In summary, the 3.0 mm to 4.5 mm zonal values from topography/tomography will yield better results for both the power and astigmatism of the cornea than the four values from any keratometer.

Toric calculators

Toric calculators have been available since the early ’90s, first used with the toric STAAR Plate IOLs in the U.S. Later the Alcon Toric Calculator became available for its toric IOLs. There are a number of standalone third-party generic calculators, similar to that in the Holladay IOL Consultant, that also perform these calculations.

The commercial calculators have implemented the effect of the incision on the corneal astigmatism determining the crossed cylinder solution of the original astigmatism and the astigmatism induced by the surgical incision. These considerations have continued to improve the results. One source of error, however, is the value that the surgeon enters for the magnitude and axis of the surgically induced astigmatism (SIA). Although the average magnitude of the SIA for a group of surgeons is usually between 0.2 D and 0.7 D for small incisions (2.2 mm to 3.0 mm), in an individual patient the actual magnitude and axis vary widely within this range. Figure 4 shows actual data for 76 cases of the SIA for an excellent surgeon. The average SIA was 0.72 D, but there is a significant variation for each patient. For this surgeon, the variation at 90° and 0° is much greater than in the oblique meridians (45° and 135°). These variations at different axes have been documented in many studies. The meridian of the incision is normally within 7° of the target, but this imprecision also adds to the variability. Surgeons must measure pre- and postoperative astigmatism with the same instrument, to determine SIA as a function the incision meridian (Figure 4) for optimal results.

Figure 4. Double angle plot of surgically induced astigmatism. Note the variation in magnitude and variability as a function of the axis.

All of the toric calculators with which I am familiar, except the Holladay IOL Consultant program and the AMO Express Calculator, use an approximation method rather than the exact solution. This involves using a constant for the ratio of the IOL toricity to corneal astigmatism. The commercial ratios used are from 1.41 to 1.48 and are correct for a 22 D spheroequivalent power IOL and an average effective lens position (ELP) of 5.50 mm (equivalent A-constant of 118.9). The power of the IOL and depth within the eye (effective lens position, ELP) are the two direct factors that influence this ratio. As shown in Table 1, the exact ratio is greater the lower the power of the IOL and the deeper the IOL in the eye (ratio is 1.745 for a 10 D SEQ IOL at 6.5 mm ELP). In contrast, for a 46 D SEQ IOL at 4.00 mm behind the corneal vertex, the ratio is 1.121. Table 2 shows the amount of toricity in diopters necessary for a 10 D, 22 D, 34 D and 46 D IOL (the normal range of IOL powers) to correct 2 D of corneal astigmatism for the range of ELPs from 4.0 mm to 6.5 mm. The maximum difference is 1.25 D (2.24 D to 3.49 D). This is obviously a substantial difference when one is trying to eliminate only 2 D of astigmatism.

The ELP in the two tables is not the lens constant for an IOL, but the actual position of the IOL in the patient’s eye. The lens constant is the average value of these ELPs from all the patients in the sample population for which the IOL was initially tested. Using an IOL with a lens constant of 5.50 mm, the actual ELP of this IOL in a specific patient may range from 4.0 mm in a nanophthalmic eye to 6.5 mm in a high myopic eye.

The only difference among modern IOL formulas today is the prediction of the specific ELP in a patient from the average value reported by the manufacturer that is used in the theoretical vergence formula. The actual vergence formula is more than 140 years old. Richard Binkhorst was the first to adjust the ELP for a specific patient for a given model IOL in 1981. He used the average ELP for a posterior chamber IOL in the sulcus (4.5 mm at that time) and scaled it up or down depending on the patient’s measured axial length compared with normal (23.5 mm). If the axial length were 10% longer than normal, he would use 4.95 (4.5 + 10% × 4.5) (one variable predictor). In 1988, we introduced the Holladay 1 with axial length and keratometry (two variable predictors) for predicting the ELP. Several subsequent two-variable predictors followed, such as the SRK/T and Hoffer Q. Tom Olsen in 1995 developed a four-variable predictor using axial length, keratometry, anatomic anterior chamber depth and lens thickness. In 1996, we introduced the Holladay 2, a seven variable predictor, using axial length, keratometry, anatomic anterior chamber depth, lens thickness, horizontal white-to-white (corneal diameter), age and refraction. Using any IOL formula (Haigis, Hoffer Q, SRK/T, Holladay 1, Olsen or Holladay 2) will yield slightly different spheroequivalent powers, more so the more unusual the eye, but the difference in the toricity needed to correct corneal astigmatism is negligible (< 0.20 D).

The approximation of using a constant ratio between the necessary IOL toricity and the corneal astigmatism, rather than the theoretical vergence formula, will cause errors. It is similar to trying to determine the necessary power of a secondary implant to correct a refractive surprise. The ratio of the refraction to the IOL power is about 1.5 for positive IOLs, if the cornea is 44 D, the IOL is at 5.50 mm, and the primary implant is about 22 D. Nevertheless, as most surgeons are aware, this is only an approximation and the exact calculation must use the refractive vergence formula to be precise. Simply be aware that the more the specific patient’s ocular parameters vary from normal, the more important it is to use the exact rather than the approximation solution. A quick check is to run the calculation for 10 D and 22 D IOL and see if the residual astigmatism changes. If it does not change, then the calculator is using the approximation method.

Part 2 of this article will cover toric optimizer, choosing the residual astigmatic target, proper IOL alignment and centration, and managing unfortunate outcomes of residual astigmatism and higher-order aberrations in the June 10 issue of Ocular Surgery News.

One advanced feature that is included in the Holladay IOL Consultant and will probably soon appear on commercial toric calculators is the toric incision location optimizer. If one enters the pre- and postop keratometry and has operated at many incision angles, then it is possible to determine the average magnitude of the surgically induced astigmatism (SIA) as a function of incision angle (Figure 1). With the patient’s original keratometry readings and knowledge of the magnitude of the SIA by location, it is possible to determine the precise location of the incision to eliminate or minimize the residual astigmatism. The program takes the patient’s original astigmatism and calculates the residual astigmatism for every possible angle of the surgical incision. The result is shown in Figure 2. Notice there are four locations (50°, 130°, 230° and 310°) for which the residual astigmatism is zero. This is usually the case, unless the step size of the toricity of the IOL is not small enough to correct fully the necessary corneal astigmatism, in which case there are two minima (in Figure 1 it would be 90° or 270° and the entire curve would be above zero residual astigmatism). In any case, the computer can always find the best result with the constraints present to give the surgeon the optimal locations for the incision.

Figure 1. Double angle plot of surgically induced astigmatism. Note the variation in magnitude and variability as a function of the axis.
Images: Holladay JT

Choosing the optimal residual astigmatism

At the beginning of your experience with a toric IOL, you may be uncomfortable placing the incision at any meridian (some surgeons are only comfortable temporally or superiorly), so it will not be possible to completely eliminate the final residual ocular astigmatism. For example, you may have a choice between a 1.50 D toricity yielding 0.50 D × 90° of residual astigmatism and the 2.25 D toricity yielding 0.12 D × 180°. If the patient had with-the-rule astigmatism initially and you were taught never to flip the axis of astigmatism, many surgeons would choose the first option and leave more residual astigmatism: This is incorrect. Flipping the axis is only of concern in glasses, in which meridional aniseikonia and spatial distortion from spectacles occur due to the base curves, power, astigmatism and vertex distance. Because the goal of a toric IOL is spectacle independence, without glasses the only difference is the size of the blur circle in the conoid of Sturm, so the choice should always be to minimize the residual astigmatism. A patient with 0.50 D of with-the-rule or against-the-rule astigmatism has the same blur on the retina. One should always try to minimize the residual astigmatism to provide the best overall quality of vision. Only if the choice is exactly equal would you choose the IOL with lower toricity. If postoperatively the axis is flipped and the patient needs a spectacle, consider leaving out the cylinder and giving the spheroequivalent prescription. It is paramount that you always choose the toric IOL that achieves the lowest residual astigmatism to attain the best uncorrected vision, regardless of the axis.


Proper IOL alignment, centration

The correct toric alignment is always with the steepest meridian of the postoperative cornea, because the reference marks on the IOL are always on the lowest power meridian. Any deviation from aligning the lowest power meridian of the IOL with the steepest axis of the cornea will result in greater amounts of residual astigmatism. Predicting this axis with the cross-cylinder solution of the SIA and original corneal astigmatism before the cataract incision is very accurate provided the original corneal astigmatism is regular and the magnitude and location of the SIA induced from the incision are precise. Because the SIA is on the order of 0.5 D or less in most small-incision surgery, the change in the axis of astigmatism will be more with lower amounts of original astigmatism than in more astigmatic corneas. Patients with 1.00 D of original corneal astigmatism will have more change in the resulting cross cylinder magnitude and axis from 0.50 D of SIA than 2.00 D of original corneal astigmatism.

Because the toric IOL normally centers in the bag, it will be on the optical axis that is co-linear with the optical axis of the cornea, which in turn is concentric with the limbus (Figure 3). Just like the crystalline lens, the cornea and IOL form an optical axis. The human eye is turned temporally ~5.2° (horizontal angle alpha) and up ~1.3° (vertical angle alpha) to place the image point of fixation on the foveola. The visual axis is the line from the point of fixation through the nodal point (near the posterior pole of the crystalline lens or IOL) to the foveola (Figure 4). The cornea and crystalline lenses are therefore tilted by angle alpha, relative to the visual axis, which induces small amounts of astigmatism and coma. It also results in a decentration of the IOL temporally relative to the cornea. For spherical surfaces (or nearly spherical as with a prolate ellipsoid cornea), this astigmatism and coma are on the order of 0.25 D to 0.50 D and have little effect on the final refraction. For toric surfaces (cornea and IOL), the tilt and decentration result in secondary astigmatism, coma and other higher-order aberrations. For corneal astigmatism greater than 3 D, the induced aberrations are often the limiting factor in the visual quality. As with diffractive multifocal IOLs, the optimal location of the IOL is centered on the 3 mm to 4 mm pupil as shown in Figure 5. Therefore, the IOL should be slightly nasal in the bag to achieve pupillary centration. In most cases, the haptics have to be vertical or slightly oblique to achieve this location. If the haptics are horizontal, self-centering lenses will move back to the center of the bag and appear temporal to the pupil within a few minutes to hours. For toric IOLs, due to this decentration relative to the cornea, it is usually impossible to align the IOL toric marks with the marks on the limbus; at best they can be parallel to this axis as shown in Figure 3.

Figure 3. Optical axis and visual axis of the eye at the corneal plane. In this right eye, the limbal center is 0.37 mm temporal to the Purkinje-Sanson I image (vertex normal, almost exactly the visual axis). This angle is referred to as angle alpha and is 0.6 mm temporal to the visual axis on the average. Angle kappa is the angle between the pupil center (small blue dot between optical center and vertex normal) and the visual axis, which is usually one-half angle alpha or 2.6° horizontally.

Figure 4. Optical and visual axis in horizontal cross section. Angle alpha is the angle between the geometric centers of the cornea and crystalline lens (or IOL) forming the optical axis and visual axis of the eye. The average value is temporally ~5.2° (horizontal angle alpha) and up ~1.3° (vertical angle alpha) to place the image point of fixation on the foveola.

Figure 5. A centered multifocal IOL. Note diffraction rings are exactly concentric with the pupil.

All of these factors result in the toric IOL being placed within ~10° of the optimal axis, but decentration and tilt are rarely considered even by the most meticulous surgeons. New intraoperative instruments such as the ORange from WaveTec have helped reduce any rotational error, but centering the IOL on the pupil is much more difficult because there are no central rings on the optic as with diffractive multifocal IOLs. Intraoperative measurements will become more robust and accurate as the technology improves. These instruments will continue to improve the outcomes of the toric IOL alignment as well as eliminating spheroequivalent refractive surprises.

Managing unfortunate outcomes

When the final refractive outcome has greater than expected residual astigmatism, the first step is to assure that the toric IOL is not tilted or decentered significantly. The second step is to measure the postoperative corneal astigmatism and make sure that the alignment marks on the toric IOL are parallel with the steepest meridian of the cornea. If it is not aligned with the postoperative steep meridian, then the treatment is to rotate the toric IOL to this meridian. Back toric calculators also will be available that will take the postoperative refraction and keratometry to calculate the axis of the toric IOL to confirm that it is the same as the axis measured at the slit lamp. If the axis at the slit lamp and the axis determined by the back calculator do not agree, then something else is wrong, such as decentration, tilt or mislabeled toric IOL.

If the toric IOL is aligned at the steep meridian of the cornea, the residual refractive astigmatic axis will be at or 90° away from the steep meridian of the cornea, depending on whether it was slightly under- or overcorrected by the toric IOL (and whether one uses plus or minus cylinder for the refraction). A wavefront may be obtained, and if the coma, secondary astigmatism and other higher-order aberrations are significant and they are not on the topographic wavefront, then the IOL is decentered or tilted. In most of these cases, if the wavefront is stable and repeatable, a wavefront-guided ablation may be performed to achieve the best vision. Moving the IOL would be unpredictable for correcting the higher-order aberrations.

Toric IOLs are here to stay, are on the rise, and will soon be the dominant form of astigmatism correction with IOL implantation. Following these recommendations should help the surgeon achieve the full potential of these lenses and provide our patients with the best visual outcome.


  • Solomon R, Donnenfeld, Perry H, Stein J, Su M, Holladay J. Argon laser iridoplasty to improve visual function following multifocal IOL implantation. Presented at: American Academy of Ophthalmology; November 2007; New Orleans. Presented at: Association for Research in Vision and Ophthalmology; April-May 2008: Fort Lauderdale, Fla.
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  • Jack T. Holladay, MD, MSEE, FACS, can be reached at Holladay Consulting Inc., P.O. Box 717, Bellaire, TX 77402; fax: 713-669-9153; email:; website:
  • Disclosure: Dr. Holladay is a consultant to AcuFocus, Allergan, AMO, Nidek, Oculus, WaveTec and Zeiss.
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  • Jack T. Holladay, MD, MSEE, FACS, can be reached at Holladay Consulting Inc., P.O. Box 717, Bellaire, TX 77402-0717; fax: 713-669-9153; email:; website:
  • Disclosure: Dr. Holladay is a consultant to AcuFocus, Allergan, AMO, Nidek, Oculus, WaveTec and Zeiss.