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Astigmatism analysis helps reveal induced error, improve outcomes
Using mathematical formulas and vector analysis to assess surgically induced refractive change, clinicians can determine the error they induce.
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Refractive surgeons seeking to reduce the amount of astigmatism induced by surgery and improve their surgical outcomes can do so by employing mathematical formulas designed to determine the amount of astigmatism induced by surgery in a single patient or in multiple patients, said Jack T. Holladay, MD, MSEE, FACS.
“You must analyze your astigmatism after surgery in order to improve your performance at surgery,” Dr. Holladay told Ocular Surgery News. “If you don’t analyze your outcomes, you don’t know how to change your surgical technique so that you reach the intended refraction for your patients.”
Dr. Holladay said the actual refractive outcome of surgery is often different from the intended outcome due to variables that include surgical technique, nomogram accuracy and laser configuration. To determine why a patient achieved a postoperative result different from the intended outcome, surgeons must determine the amount of error that was surgically induced.
The surgically induced refractive change (SIRC) is determined by measuring the difference between a patient’s preoperative refractive error and postoperative refractive error. While the formula for determining SIRC involves simple subtraction for spherical refractive errors or when the astigmatism is at the same axis, the calculation becomes more complex and requires vector analysis when the axis of astigmatism has changed from preop to postop.
Dr. Holladay said it is important for clinicians to understand the theories behind the SIRC method so that they can use the calculations in their own practices. While computer programs have been developed to analyze astigmatism change, Dr. Holladay said that few are accurate.
“Although there are several available computer programs that use the exact mathematical solution to calculate the SIRC, these approximation methods can lead to significant errors in the analysis and cause erroneous conclusions,” he said.
It is therefore “very important for the average clinician who performs refractive or cataract surgery to understand the SIRC formulas so that they can use them in their own practice and enhance their surgical performance,” he added.
Calculating with refraction
The first step in assessing astigmatic error after surgery is calculating the SIRC, Dr. Holladay said.
“Determining the SIRC requires that you take the entire refractive error of the eye after surgery, including sphere, cylinder and axis, and subtract from it the preoperative error of the eye. The outcome is the surgically induced refractive change,” he said.
The calculation of the SIRC can be expressed by this equation:
SIRC = postoperative refraction – preoperative refraction
The SIRC produces the same effect on an eye as spectacle correction, Dr. Holladay noted.
“Correction with a laser, IOL or other method does exactly the same thing to the eye as a pair of glasses – it cancels out the error of the eye,” he said.
When employing the SIRC formula, Dr. Holladay said that surgeons should base their calculations on the actual error of the eye, rather than the spectacle prescription, to avoid sign errors.
“If a patient’s spectacle prescription is –1 D, it means the error of the eye is actually +1 D too strong,” he said. Thus, the error of the eye in a patient with a –1 D spectacle prescription is actually +1 D. This means that this eye is 1 D stronger than emmetropia, with a total ocular refractive power of 61 D, vs. the standard 60 D (total power of cornea plus crystalline lens in a normal eye). Therefore, adding a spectacle prescription of –1 D to a patient with a +1 D error should result in zero, Dr. Holladay said.
In theory, the same outcome should be achieved in a LASIK patient who is a 1 D myope (with +1 D error of the eye) who receives –1 D of SIRC.
Vertexing focal length
Dr. Holladay cautioned that when calculating the SIRC for laser ablation procedures, surgeons must add an extra step – to include the vertex distance from the spectacles to the cornea – in order to achieve the desired result.
“If you are working on the cornea, at the corneal plane, the refractive values are slightly different than those at the spectacle plane, because you are further away from the focal point in front of the eye (minus spectacles),” Dr. Holladay explained.
Spectacles with –1 D of power provide a focal length of 1 m, he said. When that correction is moved back (about 1 cm) from the spectacle plane to the corneal plane, the focal length is 1 cm longer.
“You need to get a new focal length for the lens because the correction has changed,” Dr. Holladay said. “So, now, instead of treating that myope with –1 D, you should treat that myope with a value closer to –0.99 D … which is insignificant for refractions less than ± 4 D but is significant above this value.”
Dr. Holladay called switching from a spectacle plane focal length to a corneal plane focal length “vertexing sphereocylinder to the corneal plane.” He said the new focal length at the corneal plane can be determined by adding the focal length of the old lens at the spectacle plane (in the above example, 1 m) to the vertex distance (in the example, 1 cm) to get the new focal length at the corneal plane (in the example, a total of 101 cm). This equation can be expressed as follows:
focal length of new lens = focal length of old lens + vertex distance
Dr. Holladay noted that when performing this calculation, the vertex distance is considered to be negative when going from spectacle plane to corneal plane, just as the focal lengths for minus lenses are negative.
Additionally, when implanting a phakic IOL in the anterior or posterior chamber, the focal length must be vertexed from the spectacle plane to the IOL plane. The formula for the phakic IOL plane varies according to the position of the lens (whether it is iris-supported, anterior chamber or posterior chamber). It follows the same concept as the equation above for vertexing to the corneal plane.
Dr. Holladay said that it is “absolutely necessary” to vertex to the corneal plane when comparing the SIRC outcomes with refraction to SIRC outcomes with keratometry.
Calculating with keratometry
Keratometry is a second method for calculating the SIRC. It is important for surgeons to calculate SIRC measurements with both refraction and keratometry to validate their outcomes, Dr. Holladay said.
“When calculating the SIRC with corneal topography or with keratometry, you must subtract the preop K-reading from the postop K-reading to get your value,” Dr. Holladay said. Because keratometry and topography measurements are taken at the corneal plane, they do not need to be vertexed like refractive measurements at the spectacle plane, he noted.
Keratometry measures the central power of the corneal surface, which is 45 D in a normal eye, Dr. Holladay said. Using the SIRC formula above, a patient with a preoperative K value of 45 D and a postoperative K value of 44 D will have a SIRC value of –1 D because SIRC = postoperative refraction – preoperative refraction or 44 D – 45 D = –1 D. Thus, the SIRC of –1D ascertained by keratometry matches the SIRC of –1 D as calculated by refraction above.
“In a perfect world, the SIRC from K-readings would always match the SIRC from refraction after you have vertexed to the corneal plane,” Dr. Holladay said. “But they are rarely exactly the same due to measurement errors.”
Keratometry measurements are seldom accurate in the corneas of patients who have undergone corneal refractive surgery, Dr. Holladay said. As a result, the surgeon should rely more on refractive measurements than corneal measurements in these patients, he said.
Assessing residual cylinder
If a patient’s postoperative refraction is not plano, the patient may be left with a spherical error.
For example, if the myopic patient in the example above, instead of achieving a postoperative refraction of 0 D, is left with –0.25 D of sphere after surgery, the SIRC can be ascertained by plugging the patient’s postoperative and preoperative refractive errors into the universal formula: SIRC = postoperative refraction – preoperative refraction. The myopic patient was –1 D (with an error of +1 D) before surgery, and the SIRC is –0.75 D of undercorrection.
“The SIRC shows that the laser only did –0.75 D and not the full prescription of –1 D,” Dr. Holladay said. “So, there’s a quarter diopter of sphere left over.”
Assessing cylinder residual axis
If an astigmatic patient started out with with cylinder at 180° and achieves an unintended postoperative outcome of cylinder at 145°, calculating the SIRC requires the subtraction of vectors rather than simple mathematical subtraction, Dr. Holladay said.
“We can’t take 145° and subtract it directly from 180°,” he said. “Arithmetic analysis won’t work because the axes are not lined up; they are oriented differently. So you need to employ vector analysis.”
In an article in the Journal of Cataract and Refractive Surgery in 1992, Dr. Holladay listed the “10 steps for obliquely crossed cylinder,” a sequence of formulas for performing vector analysis, which is needed to determine the SIRC of a patient with misaligned axes.
After following the 10 steps to determine obliquely crossed cylinder, Dr. Holladay said, the patient’s SIRC will be a new axis, different from both the postoperative and preoperative axes of cylinder.
Source: Holladay JT | ||||
Plotting astigmatic errors
If a surgeon notes a pattern of residual astigmatism in his or her patients causing actual refractive outcomes to differ from intended outcomes, Dr. Holladay said that there could be a problem with the surgeon’s technique, nomogram, laser alignment or laser calibration.
Surgeons can determine a trend in their ablation patterns by plotting the SIRC of multiple patients on a polar plot, he said.
“The best way to display results for a group of patients with astigmatism is on a double-angle polar plot,” Dr. Holladay said. “When you plot inconsistencies on this type of plot, patterns that look random become clustered. You are able to get a sense of the trend in error of astigmatism across your patients.”
Dr. Holladay and colleagues developed the double-angle polar plot specifically for astigmatism analysis.
“To apply conventional geometry, trigonometry and vector analysis to astigmatism, the angles of astigmatism must be doubled so that 0° and 180° are equivalent,” Dr. Holladay explained. “Once the transformation has been performed, all the standard formulas can be used and produce the correct singular value for the SIRC.”
| Examples of polar plotting of refractive results |
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| The double-angle polar plot was developed by Dr. Holladay and colleagues specifically for astigmatism analysis. Source: Holladay JT |
Determining centroid values
The radial axes of a double-angle polar plot are oriented at 45° (at 12 o’clock), 90° (at 9 o’clock), 135° (at 6 o’clock) and 0° and 180° (at 3 o’clock). Plotting each patient’s SIRCs on this graph, the astigmatic errors often appear as a cluster formation, Dr. Holladay said.
“At the center of the cluster is the centroid,” he said. “The centroid is the average of each of the individual errors.”
By determining the centroid of a cluster, the surgeon can determine the average SIRC for all of his or her refractive cases.
“This information is highly valuable,” Dr. Holladay said. “Once you determine the average deviation of your procedures from actual to intended outcomes, you are able to readjust your surgical settings and prevent astigmatism from occurring again.”
For Your Information:References:
- Jack T. Holladay, MD, MSEE, FACS, can be reached at Holladay LASIK Institute, Bellaire Triangle Building, 6802 Mapleridge, Suite 200, Bellaire, TX 77401; (713) 668-7337; (713) 668-7336; e-mail:docholladay@docholladay.com, Web: www.docholladay.com.
- Holladay JT. Refractive power calculations for intraocular lenses in the phakic eye. Am J Ophthalmol. 1993;116:63-66.
- Holladay JT, Cravy TV, Koch DD. Calculating the surgically induced refractive change following ocular surgery. J Cataract Refract Surg. 1992;18:429-443.
- Holladay JT, Dudeja DR, Koch DD. Evaluating and reporting astigmatism for individual and aggregate data. J Cataract Refract Surg. 1998;24:57-65.
- Holladay JT, Moran JR, Kezirian GM. Analysis of aggregate surgically induced refractive change, prediction error, and intraocular astigmatism. J Cataract Refract Surg. 2001;27:61-79.
- Nicole Nader is an OSN Staff Writer who covers all aspects of ophthalmology, specializing in QOV, pediatrics/strabismus and neuro-ophthalmology.