The power of the cornea is conventionally measured by manual keratometry or autokeratometry (K reading). The keratometer measures the anterior radius of corneal curvature and uses an arbitrary index of refraction of 1.3375 so that a radius of 7.5 mm would yield 45.0 diopters (D).^{1,2} Therefore, the K reading is the standardized power of the cornea and applicable only if the cornea thickness is 500 μm and the ratio of the back to front surface central radius is approximately 82%.^{3} Although the K reading does not represent the equivalent corneal power, current formulas for intraocular lens (IOL) power calculation in cataract surgery are based on K readings and are reliable for the majority of eyes with a physiologic and prolate cornea.^{3–5} This reliability can be obtained by adjusting formulas using many clinical cases. For example, the SRK/T formula was optimized using an iterative process on 5 data sets consisting of 1677 posterior chamber IOL cases.^{6} The Holladay 2 formula, which was used in this study, is based on previous observations from a 35 000 patient data set.^{7}

In corneal refractive surgery, the anterior surface is intended to be modified and there is no longer a physiologic ratio between the front and back corneal radii; this results in an unreliable K reading for IOL power calculation with current formulas.^{2,8} In addition, keratometers have an unmeasured central zone that is approximately 3.2 mm in diameter.^{2,9} Therefore, the central region of the cornea, which is more flattened or steepened by keratorefractive surgery, is essentially ignored.^{3}

The Pentacam (Oculus Optikgeräte GmbH) a rotating Scheimpflug camera that measures the entire central area of the cornea as well as the posterior corneal surface.^{2} Techniques that use the Scheimpflug camera for IOL power calculation in cataractous eyes that had previous myopic keratorefractive surgery have been developed. One is the true corneal net power, an approximation of the Gaussian equivalent power, which is the arithmetic sum of the front and back surface powers according to the Gaussian formula.^{5,10} The Gaussian equivalent power is significantly less than the K reading in virgin corneas and is not appropriate for current IOL formulas.^{2,3,11,12} However, studies^{13,14} have incorporated the true net power into the SRK/T formula without effective lens position (ELP) correction. In addition, the Holladay equivalent K reading (EKR) is an adjusted K reading for the difference in the posterior radius from normal.^{2,5} The Pentacam software (version 1.20r02) calculates the equivalent K reading with anterior and posterior powers by Snell’s law. There have been some discussions about the validity of the equivalent K reading in the Pentacam system.^{5,15,16}

The total corneal refractive power (TCRP) is the equivalent corneal power of the Scheimpflug camera using a ray-tracing method rather than the Gaussian formula.^{5,A} The Gaussian formula depends on the assumption of paraxial optics, but ray tracing does not rely on paraxial optics and hence produces an accurate method of measuring the corneal power with respect to both the anterior and posterior corneal curvature.^{10,A} To the best of our knowledge, the TCRP has not been used for IOL power calculation, especially in eyes with an altered anterior corneal curvature following myopic refractive surgery.

However, the equivalent corneal power would produce inaccurate results with current IOL power calculation formulas, which are accustomed to standard K readings. The equivalent corneal power should be customized for these formulas with some type of conversion. We postulated that each K reading would represent a specific equivalent corneal power for the formula. If we knew the relationship between them, we could pick an equivalent K reading for a given equivalent corneal power. We expected a simple relationship between them because current formulas are successfully adapted to K readings despite their arbitrariness. To verify our hypothesis, we evaluated the simple relationship in the normal cornea, rechecked it, and then expanded the range of its application in myopic keratorefractive surgery patients. Lastly, the converted power from TCRP was coupled with the Holladay 2 formula, and the accuracy of IOL power calculation was evaluated in cataract surgery cases after myopic keratorefractive surgery.

Patients and methods
This retrospective clinical practice study prospectively obtained approval of the Institutional Review Board, CHA University. It adhered to the tenets of the Declaration of Helsinki.

Measurement of Total Corneal Refractive Power
Each patient was seated with the head in a headrest and was asked to focus on the target at the center of the Scheimpflug system (Pentacam or Pentacam HR). Every Scheimpflug system in this study was calibrated and technically supported by the regional branch of Oculus. The operator moved the joystick until arrows on the display were aligned with the horizontal, vertical, and anterior–posterior axes. As soon as the image was aligned perfectly, the patient was asked to keep his or her eye open, after which the scanning process started. Automatic release was used to reduce variables. The 25-images mode was chosen so that the rotating camera acquired 25 scans in 1 second.

In the power distribution display of the Scheimpflug system software (version 1.18r04), the zone, apex, and total corneal refractive power options were selected and the zone diameter was set as 4.0 mm. Then, Km in “power calculations in actual zone” was read as the total corneal refractive power in the central 4.0 mm zone (TCRP4).

Basic Concept of the Total Corneal Refractive Power Method
If TCRP4pre is the correctly computed power of the cornea before refractive surgery, the K reading of the cornea before refractive surgery (Kpre) and TCRP4pre is as follows:

From the clinical history method, the power of the post-refractive surgery cornea (Kch) is given by^{4,17}

where R4c.pre is the refraction before refractive surgery measured at the apex-centered 4.0 mm zone of the corneal plane and R4c.post is refraction after refractive surgery measured at the apex-centered 4.0 mm zone of the corneal plane.

If the next equation is proven,

(TCRP4post is the TCRP4 of the cornea after refractive surgery)

Kch will be calculated from TCRP4post, where

Now, both effective cornea powers for pre-refractive and post-refractive surgery (Ktcrp4) can be generally expressed with TCRP4 by equations 1 and 4 :

Study Procedure
First, equation 1 was proved and the C constant of equation 1 was determined through a retrospective review of medical records of healthy subjects. Second, equation 3 was verified by reviewing the preoperative and postoperative records of patients who had photorefractive keratectomy (PRK) or laser in situ keratomileusis (LASIK) for myopia. Then, Ktcrp4 for post-refractive surgery was calculated with equation 5 , and the compatibility of Kch and Ktcrp4 was studied. Lastly, the cases of cataract surgery after PRK, LASIK, or laser-assisted subepithelial keratectomy (LASEK) for myopia were reviewed and the IOL prediction error with Ktcrp4 was estimated.

Verification of Equation 1 and Determination of the Constant
One eye of each healthy subject who had visited the Ian Eye Center was randomly selected and reviewed. The inclusion criteria were as follows: no history of corneal surgery or ocular surface abnormalities; mean K power of autokeratometry (Ka) ranging from 40.0 D to 45.0 D; difference between mean corneal front power of the Scheimpflug camera (Kp) and Ka less than 0.5 D; Ka greater than TCRP4. Because Kp is developed to produce the same value as Ka, any difference implies that an unreliable measurement was taken with the Scheimpflug camera or with autokeratometry. In addition, TCRP4 rarely exceeds Ka in healthy corneas because the true corneal power in the principle plane is less than the K reading with a keratometric index of 1.3375.^{11} Finally, healthy eyes of healthy subjects (20 subjects for each subset in the 20, 30, 40, 50, and 60 years of age range) who had visited between 2007 and 2012 were recruited.

To verify equation 1 , the agreement between Ka and TCRP4 was studied with the Bland-Altman plot. The C constant of equation 1 was calculated by averaging (Ka − TCRP4). It was confirmed with the Pearson product moment correlation that the C constant was independent of the patient’s age and corneal power.

Verification of Equation 3 and Calculation of Refraction for the 4.0 mm Zone
One eye of each patient who had PRK or LASIK at B&VIIT Eye Center between 2009 and 2011 was randomly selected and reviewed. The Allegretto Wave Eye-Q laser (Wavelight Laser Technologie AG) or the Amaris excimer laser (Schwind eye-tech-solutions GmbH and Co. KG) was used for the surgery. The wavefront-optimized treatment of the Allegretto Wave Eye-Q laser or the aberration-free treatment of Amaris excimer laser was implemented to preserve the cornea’s asphericity and avoid additional higher-order aberrations (HOAs).^{18,19} Data from more than 6 months after surgery were used. The eye was included in the study based on the following criteria: corrected visual acuity not worse than 20/20 preoperatively and postoperatively; cornea healthy before the surgery and not accompanied by any postoperative complication; difference between Kp and corresponding Ka less than 0.5 D preoperatively and postoperatively.

For verifying equation 3 , the manifest refraction was converted to the refraction measured at the apex-centered 4.0 mm zone of corneal plane (R4c). Because the manifest refraction had been measured under mesopic conditions, it was presumed that the corresponding pupil size of the patients was approximately 6.5 mm.^{20} The total corneal refractive power of the apex-centered 6.5 mm zone (TCRP6.5) was obtained from the power distribution display of the Scheimpflug camera after the zone diameter in the lower box was edited as 6.5. Then, R4c was calculated with the manifest refraction measured at the corneal plane (Rc), TCRP4, and TCRP6.5, where

By the equation 6 ,

where Rc.pre and TCRP6.5pre are preoperative Rc and TCRP6.5, respectively, and Rc.post and TCRP6.5post are the postoperative Rc and TCRP6.5, respectively.

Rc was calculated from the refractive error at spectacle plane (R) with the formula^{1,17}

Then, equation 3 was confirmed by the Bland-Altman plot between (R4c.post − R4c.pre) and (TCRP4pre − TCRP4post). Finally, equation 4 was checked by the Bland-Altman plot between Kch and (TCRP4post + C).

Accuracy Test for Equation 5 with Cataract Surgery Cases after Refractive Surgery
The cases of cataract surgery after PRK, LASIK, and LASEK for myopia performed between 2009 and 2012 at several clinics were reviewed. The cases were included if there were no significant complications during cataract surgery; postoperative manifest refraction was taken after more than 30 days; the difference between Kp and Ka was less than 0.5 D.

The spherical aberration of the 6.0 mm total corneal wavefront aberration was calculated using the Zernike analysis of the Scheimpflug camera. The spherical equivalent (SE) of the predicted refraction (Rcalc) according to Ktcrp4 and the Holladay 2 formula was compared with the SE of the postoperative manifest refraction (Rtrue) through calculation of the arithmetic prediction error, Rtrue − Rcalc. The percentages of correct refraction predictions within ±0.5 D and ±1.0 D were calculated. The Holladay 2 formula is available as part of the Holladay IOL Consultant software.^{B} The “Previous RK, PRK, ALK, LASIK…” box was checked, and Ktcrp4 was input as the surgeon-entered K value for alternate K. The horizontal white-to-white length and ultrasound measurements of the phakic anterior chamber depth (ACD), lens thickness, and axial length (AL) were entered. The results of Ktcrp4 and Holladay 2 formula were compared with results obtained with the no-history method of Shammas and Shammas.^{21} The no-history method of Shammas and Shammas is 1 of the 5 most accurate methods of the 25 formula combinations and uses only postoperative biometry.^{8} For the Shammas method, the corrected K value (Shammas.cd) and the predictive refraction were calculated with the clinically derived method and the Shammas post-LASIK formula (Shammas-PL), respectively. In addition, the prediction error of the Holladay EKR with the Holladay 2 formula (“Previous RK, PRK, ALK, LASIK…” on) was calculated. In the Holladay report of the Pentacam system, the Holladay EKR is displayed for pupil-centered 4.5 mm zone.

Statistical Analysis
SPSS software (version 20, International Business Machines Corp.) was used for general statistical analysis. Bland-Altman plots and scatterplots were drawn and analyzed using Sigmaplot software (version 12.0, Systat Software, Inc.).

Results
Equation 1: Relationship between Keratometric Power of Autokeratometry and Total Corneal Refractive Power of the Scheimpflug Camera in Healthy Corneas
The mean age of 100 healthy patients (45 men and 55 women) was 44 years ± 15 (SD). The difference between Ka and TCRP4 in healthy corneas was independent of corneal power and patient’s age (Figures 1 and 2 ). The mean difference between Ka and TCRP4 was 0.7 ± 0.3 D (range 0.7 to 0.8 D; 95% confidence interval [CI]), and equation 1 was completed as:

Figure 1:
Bland-Altman plot of the difference between the mean keratometric power of autokeratometry (Ka) and the total corneal refractive power of apex-centered 4.0 mm zone (TCRP4) in healthy subjects; Ka − TCRP4 was not correlated with the mean of Ka and TCRP4 (Pearson r = 0.116, P =.251).

Figure 2:
Correlation between age and Ka − TCRP4. No significant correlation was found (Pearson r = 0.005, P =.963) (Ka = mean keratometric power of autokeratometry; TCRP4 = total corneal refractive power of apex-centered 4.0 mm zone).

The C constant could be deviated ±0.5 D from 0.7 D (Figure 1 ).

Equation 3: Statistically Equivalence of Refractive Changes at Central 4.0 mm of Cornea Plane and Changes in Total Corneal Refractive Power of Scheimpflug Camera
The medical records of 63 patients (30 cases of PRK and 33 cases of LASIK) were reviewed. The mean age was 26 ± 5 years, and the mean preoperative manifest refraction was −5.37 ± 1.93 D. The postoperative manifest refraction was taken after a median time of 284 days (interquartile range, 201 to 336 days). The mean difference between the refractive changes at the central 4.0 mm of the corneal plane and the TCRP4 changes was 0.0 ± 0.5 D (95% CI, −0.1 to 0.1 D), and the absolute value of the limits of agreement (LoA) was less than 1.0 D (Figure 3 ).

Figure 3:
Bland-Altman plot of the difference between postoperative changes in manifest refraction for the central 4.0 mm zone of corneal plane (R4c_{post-pre} ) and the total corneal refractive power of the apex-centered 4.0 mm zone (TCRP4_{pre-post} ).

Equation 4: Verification of Equation 1 in a Different Group Via Combination of Equation 3
The group for the equation 3 demonstration was applied again to prove equation 4 . By equation 3 , the refractive changes at the central 4.0 mm of the corneal plane could be replaced with the TCRP4 changes. If equation 1 and equation 3 were accurate, Kch and (TCRP4post + 0.7) in equation 4 would show no significant difference between them. The mean difference between Kch and (TCRP4post + 0.7) was −0.1 ± 0.5 D (95% CI, −0.2 to 0.0 D), and the LoA were −1.0 D and 0.8 D (Figure 4 ).

Figure 4:
Bland-Altman plot of the difference between the correct K value derived by the clinical history method (Kch) and the effective corneal power calculated with the total corneal refractive power of the apex-centered 4.0 mm zone (Ktcrp4). For the clinical history method, postoperative changes in manifest refraction for the central 4.0 mm zone of corneal plane were calculated.

Accuracy of Equation 5 with Holladay 2 Formula (Total Corneal Refractive Power Method) in Cataract Surgery after Corneal Refractive Surgery
In 18 cases of 14 patients, 9 cases (50%) were correctly predicted within ±0.50 D and 14 cases (78%) were within ±1.0 D by the Shammas method (Table 1 ). By the TCRP method, 15 cases (83%) and 17 cases (94%) of patients were properly predicted within ±0.5 D and ±1.0 D, respectively (Table 1 ). The difference between Ktcrp4 and Shammas.cd strongly correlated with the difference between Rcalc of Ktcrp4/Holladay 2 and Rcalc of Shammas.cd/Shammas-PL (Spearman ρ = −0.786, P <.001) (Figure 5 ). Neither method gave a hyperopic prediction error greater than 1.0 D (Table 1 ).

Figure 5:
Strong positive correlation between differences of cornea powers and predicted refraction (Rcalc). One outlier was observed (Case 11 in

Table 1 ). (Ktcrp4 = cornea power calculated with the TCRP4 method; Shammas.cd = corrected keratometry power of the clinically derived Shammas method; TCRP4 = total corneal refractive power of apex-centered 4.0 mm zone).

Table 1: Comparison of prediction errors between the Shammas method, the total corneal refractive power method, and the Holladay equivalent keratometry reading method.

In cases 3 and 14, the centers of the pupil in the Scheimpflug camera were erroneously measured and the Holladay EKRs were not reliable (Figure 6 ). Without both cases, 5 cases (31%) and 12 cases (75%) had a prediction error within ±0.5 D and ±1.0 D, respectively, when the Holladay EKR was joined with Holladay 2 formula (Table 1 ).

Figure 6:
Holladay equivalent K reading detail report of case 3. The red ring represents the 4.5 mm calculation zone of the Holladay equivalent K reading. The pupil edge (black dotted line ) is irregular, and the pupil center is erroneously displaced.

Discussion
The new equivalent K reading (Ktcrp4) for the IOL power calculation formula was obtained with the simple addition of 0.7 D to TCRP4 (equation 5 ). The Ktcrp4 could apply to the healthy cornea (equation 1 ) and the modified cornea with PRK or LASIK for myopia correction (equation 4 ). Refraction changes with PRK or LASIK were well reflected by TCRP4 (equation 3 ). When Ktcrp4 was incorporated in the Holladay 2 formula in eyes after myopic PRK, LASIK, or LASEK, the calculated IOL power was accurate enough to comply with a benchmark standard.^{22}

The TCRP method was devised to be compatible with current IOL calculation formulas that depend on the K reading. The standard manual keratometer measures a 3.2 mm ring area for a 44.0 D cornea.^{2} The size of the measured ring area changes according to the radius of curvature of the cornea because a fixed size ring is emitted and mirrored by the anterior surface of the cornea.^{23} The 4.0 mm zone of TCRP4 embraces the group of various sizes of the ring area but is still close to the measured ring area of the standard keratometer.

For the accuracy of TCRP to be guaranteed, the mean corneal front power of the Pentacam (Kp) should be within ±0.5 D of the mean keratometric power of autokeratometry (Ka). According to the manufacturer of the Pentacam Scheimpflug camera (software version 1.18r04), Kp is the anterior simulated K reading on a ring that is 15 degrees around the corneal apex and developed to accord with Ka. If the Scheimpflug camera measures at the same region with standard keratometry, Kp would be equal to Ka. Actually, Kp was not statistically different from Ka in a recent study.^{24} Because the TCRP method is an adaptation of Ka, the alignment between the Scheimpflug camera and keratometry was critical. In addition, the Scheimpflug camera measurement might be imperfect as a result of blinking or extraneous light influences.^{25} Recently, fluorescein staining of the tear film was reported to cause more intense backscattering of light, resulting in a measurement error with the Pentacam Scheimpflug camera.^{26} Therefore, Kp should be reviewed carefully when pathologic conditions of the cornea affecting backscattering are suspected. Even in the healthy cornea, a significant Kp outlier can be encountered as a measurement error.^{24}

The C constant in equation 1 was determined with good accuracy; 95% of the difference from the C constant was as small as 0.5 D (Figure 1 ). Because Kp was allowed to differ from Ka up to 0.5 D, 0.5 D was the least deviation to be obtained. To the contrary, 95% of difference (LoA) for equation 3 validation might reach 1.0 D (Figure 3 ). For the verification of equation 3 , the direct measurement of refraction in the central 4.0 mm zone was technically impossible; therefore, TCPR4 and TCRP6.5 were used to calculate R4c (equation 6 ). Because of this limitation, the LoA for equation 3 validation seemed to increase but the mean changes in R4c were statistically equal to those in TCRP4 (Figure 3 ). Equation 4 was verified with the group of patients for used in the equation 3 demonstration (Figure 4 ). Because equation 4 was derived from equations 1 and 3 , the validity of equation 4 meant that equation 1 was adequate not only in the specified group but in another group (ie, that used for equation 3 ).

Finally, equations 1 and 4 were generalized into equation 5 . By equation 5 , a proper K reading for the current IOL formula could be acquired from the TCRP4 of a healthy or excimer laser–modified cornea. By equation 3 , which implied that the TCRP4 was a correct refractive power of the cornea, equation 5 could be validated beyond the included corneal power in equation 1 (40.0 to 45.0 D). If equation 3 is true, equation 5 may be applicable in a variety of altered corneas, such those after post-hyperopic PRK or LASIK, those with keratoconus, or those with trauma.

However, TCRP varies according to the type of instrument used. Equation 1 meant the real corneal power was 0.7 D less than the K reading in healthy corneas. If the corneal power is calculated with the Gaussian formula in a 7.5 mm anterior corneal radius, a 0.822 ratio of the back to front surface central radius, and 550 μm thickness, the Gaussian equivalent power would be 43.8 D whereas the K reading would be 45.0 D.^{2} Because the theoretical corneal power is 1.2 D less with the Gaussian formula, the corneal power by the Gaussian formula would be smaller than that calculated with TCRP. In contrast to this conjecture, ray tracing calculates a lower power of the cornea than the Gaussian formula for normal eyes with the Galilei dual Scheimpflug analyzer (Ziemer Ophthalmic Systems AG).^{10} Therefore, our results must be considered with respect to the Pentacam system only.

The Holladay 2 is a fourth-generation formula and uses measurements of corneal power, corneal diameter, ACD, lens thickness, refraction, AL, and age to refine the ELP.^{7,27} Because the third-generation formula assumes that the ELP is related to central corneal power, the artifact of low corneal power after myopic corneal refractive surgery will cause the formula to presume a falsely shallow ELP and recommend a lower IOL power than required.^{7,28} To avoid this pitfall, the Holladay 2 formula uses the mean power of the normal cornea for the ELP calculation in consideration of other factors by checking “Previous RK [radial keratometry], PRK, ALK [automated lamellar keratoplasty], LASIK…” box.^{7,29} In addition, the Holladay 2 formula^{21} is recommended for the accurate estimation in cases outside the usual range of corneal power or AL.^{7,27,C} A patient who has myopic LASIK frequently has a longer AL.

Our TCRP method was tested for various ranges of age (35 to 62 years) and AL (24.63 to 30.46 mm) in eyes after PRK, LASIK, and LASEK surgery. Diverse types of IOL (eg, 1-piece monofocal, 3-piece monofocal, multifocal, and toric) were included. A benchmark standard of refractive outcomes after normal cataract surgery without previous refractive surgery is 85% of patients achieving a final SE within ±1.0 D and 55% achieving within ±0.5 D.^{22} In a recent study,^{8} the Shammas method predicted 53.5% of eyes within ±0.5 D and 80.9% of eyes within ±1.0 D of the target refraction. When the prediction accuracy of the Shammas method was similar with a previous study,^{8,22} the TCRP method satisfied the benchmark standard with good predictability. The predicted refraction of each method was well correlated with each corneal power except in case 11, which had a very long AL. Because the TCRP method was better than the Shammas method in case 11, the Holladay 2 formula may be more accurate than the Shammas-PL formula in patients with a very long AL (Table 1 ).

However, a small number of cases were used to test the accuracy of the TCRP method. This could lead to a selection bias; thus, a larger scale study is should be performed. Because the prediction accuracy of the Shammas method in our cases was as good as the one previously reported,^{8} serious bias is not expected.

The Ktcrp4 in equation 5 was an equivalent K reading for a given equivalent corneal power, TCRP4. However, the Ktcrp4 was different from the Holladay equivalent K reading of the Scheimpflug camera (Table 1 ). Although the Holladay EKR represents the corneal zone in the pupil center, TCRP4 represents the zonal power around the corneal apex. The Holladay EKR shows low repeatability because of the dynamic nature of the pupil as the pupil center changes position.^{30} Sometimes, the Pentacam Scheimpflug camera measured the pupil center incorrectly (Figure 6 and Table 1 ). In addition, the radius of the corneal back surface should be preserved in the preoperative state for the Holladay EKR calculation.^{2} However, Ktcrp4 may be applicable in subclinical or manifest ectasia of the posterior cornea if TCRP4 would accurately reflect corneal refractive power.

The TCRP method seemed to work well in corneas with high spherical aberration (Table 1 ). Because optimized laser profiles were used to keep normal corneal asphericity and HOAs in verification of equation 3 , this equation might not be valid for abnormally oblate corneas treated with old laser profiles. In our case series, spherical aberration up to 1.8 μm did not affect the accuracy of the TCRP method (case 1 in Table 1 ).

In conclusion, TCRP4 was successfully converted into an equivalent K reading by the simple addition of 0.7 D in the corneal status after myopic keratorefractive correction. The TCRP4 method might be applicable to various corneal profiles if the TCRP represents the true refractive power of the cornea.

What Was Known
The total corneal refractive power of the Pentacam Scheimpflug camera is calculated by ray tracing to provide an equivalent corneal power but has not been applicable for IOL power calculation because current formulas are optimized to the standard K reading.
What This Paper Adds
By addition of 0.7 D to the total corneal refractive power at the 4.0 mm zone of the Scheimpflug camera, the total corneal refractive power was converted to the equivalent K reading in the normal and modified corneas after myopic keratometric surgery.
After myopic keratometric surgery, the IOL power was successfully calculated using the Holladay 2 formula and the corneal power that was derived from the total corneal refractive power at the 4.0 mm zone.
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