#### Abstract

To evaluate the accuracy of intraocular lens (IOL) power calculations using ray tracing software in patients who had undergone phototherapeutic keratectomy (PTK).

In this retrospective case series, 37 eyes of 22 patients (mean age: 69.4 years; range: 56 to 85 years) who underwent cataract surgery after PTK were reviewed. The prediction error, defined as the difference between the estimated postoperative spherical equivalent and the postoperative manifest refraction at the spectacle plane, was calculated using the following formulas: OKULIX (Tedics, Dortmund, Germany), PhacoOptics (IOL Innovations ApS, Aarhus, Denmark), Barrett True K No History (NH), and Camellin-Calossi. The PhacoOptics formula was used in three different ways: historical method (H), no history method (NH), and C-constant method (C). The median values of the arithmetic and absolute prediction errors among these six IOL calculation methods were compared.

The median arithmetic errors (in diopters [D]) and percentages of eyes within ±0.50 D of the absolute errors were as follows: OKULIX (0.33, range: −2.20 to 2.50, 30.6%), PhacoOptics (H) (−0.12, range: −3.28 to 4.85, 22.2%), PhacoOptics (NH) (−0.25, range: −2.08 to 1.70, 48.4%), PhacoOptics (C) (0.04, range: −1.40 to 2.18, 48.5%), Barrett True K (NH) (−0.35, range: −1.90 to 1.89, 48.6%), and Camellin-Calossi (−0.19, range: −1.78 to 1.47, 59.5%).

The PhacoOptics, especially the C-constant method (C), and Camellin-Calossi formulas were good options for calculating IOL powers in eyes that underwent PTK. **[ J Refract Surg. 2019;35(5):310–316.]**

Granular corneal dystrophy (GCD), a hereditary autosomal dominant disease, is a common corneal stromal dystrophy in Japan. In GCD, characteristics such as granular deposits, lattice deposits, and diffuse stromal haze are observed in sequence with progression.^{1} Band keratopathy is a chronic degenerative disease characterized by accumulation of whitish to grayish opacities of calcium hydroxyapatite in the superficial corneal layers. Band keratopathy, which usually is associated with chronic ocular inflammation or a related systemic disease, can cause irritation, glare, or recurrent corneal erosions in the eyes resulting from disruption of the corneal epithelium.^{2} When the opacities are in the central cornea, they may cause decreased vision. Phototherapeutic keratectomy (PTK) is performed to remove vision-threatening corneal opacities in eyes with GCD and band keratopathy.^{3} The intraocular lens (IOL) power calculations are less predictable in patients who underwent a previous PTK than in virgin corneas, because PTK induces corneal flattening (with a subsequent hyperopic shift).^{4}

In eyes after myopic refractive surgery such as photorefractive keratectomy (PRK) and laser in situ keratomileusis (LASIK), difficulties in power calculations arise from several factors. First, the corneal power is calculated inaccurately from the anterior radius alone when using conventional keratometry because the ratio between the anterior and posterior radii of the corneal curvature has changed considerably and the keratometric calibration index of 1.3375 is no longer applicable. In addition, most keratometers measure the central corneal radius of curvature in the 2.5- to 3.2-mm zone, which leads to overestimation of the refractive power of the flattened cornea. Second, formulas include errors when third-generation IOL formulas are used because these formulas calculate a falsely shallow pseudophakic anterior chamber depth due to the flattened cornea after PRK and LASIK, leading to an underestimated IOL power. We theorize that these sources of errors are almost the same as those that occur in eyes after PTK. We reported that the Camellin-Calossi formula^{5} provides the most predictable results compared with the SRK/T, Shammas-PL,^{6} Haigis-L,^{7} and OKULIX (Tedics, Dortmund, Germany) formulas.^{8}

The current study analyzed and compared the prediction errors of the IOL power calculations using ray tracing methods (OKULIX and PhacoOptics [IOL Innovations ApS, Aarhus, Denmark, version 1.10.100.2029, revision date 2017/2/9]) and a newer IOL power calculation method (Barrett True K method [Singapore, Asia-Pacific Association of Cataract and Refractive Surgeons]), in addition to the Camellin-Calossi formula that we previously reported as providing the best results.^{8}

### Patients and Methods

#### Patients

Thirty-seven eyes of 22 patients (16 eyes of 9 men, 21 eyes of 13 women) who underwent phacoemulsification and IOL implantation after PTK at Keio University School of Medicine from March 2010 to October 2014 were reviewed retrospectively. The criteria for patient inclusion were a postoperative corrected distance visual acuity of 20/40 or better after cataract surgery and had undergone cataract surgery longer than 3 months after PTK. The etiologies of the corneal opacities were limited to GCD type 2 (GCD2), band keratopathy, or a combination of GCD2 and band keratopathy. The exclusion criteria were that the axial length was unmeasurable by partial coherence interferometry (IOLMaster; Carl Zeiss Meditec, Jena, Germany), any vision-threatening ocular diseases except for cataracts, and corneal opacities unrelated to the original corneal diseases treated with PTK.

This study adhered to the tenets of the Declaration of Helsinki and was approved by the Institutional Review Board for Human Studies of the Keio University School of Medicine. All patients provided written informed consent before the examinations and surgeries were performed.

#### Surgical Procedure

The PTK procedure was performed with the EC-5000 CX II excimer laser platform (Nidek, Aichi, Japan). The treatment zone was 6 mm, and a transition zone of 1 mm was set beyond the ablation zone in all cases. The average ablation depth was 102.87 µm (range: 51 to 130 µm) and the pulse frequency was 40 Hz. Twenty-one eyes had GCD2, 14 eyes had band keratopathy, and 2 eyes had both pathologies. The average ablation depth was 110 µm (range: 100 to 130 µm) in the cases with GCD2 and 91.2 µm (range: 51.0 to 110 µm) in the cases with band keratopathy. The difference between them reached significance (*P* < .05).

After PTK, the same experienced surgeons performed the cataract surgeries. The average sclerocorneal incision size for the cataract surgeries was 2.32 ± 0.23 mm (range: 2.2 to 2.75 mm). The IOLs were placed in the capsular bags in all cases. The implanted IOLs were the Tecnis ZCB00 IOL (Johnson & Johnson Vision, Inc., Santa Ana, CA) in 20 eyes, NY-60 IOL (Hoya Corporation, Tokyo, Japan) in 4 eyes, AcrySof SN60WF IOL (Alcon Laboratories, Inc., Fort Worth, TX) in 4 eyes, AcrySof SA60AT IOL (Alcon Laboratories, Inc.) in 4 eyes, YA65BB (Hoya Corporation) in 3 eyes, Tecnis ZA9003 IOL (Johnson & Johnson Vision, Inc.) in 1 eye, and NX70 IOL (Santen Pharmaceutical, Inc., Osaka, Japan) in 1 eye.

#### Examinations for IOL Power Calculations

In all patients, a full ophthalmic examination was performed before cataract surgery. Three experienced examiners used the autorefractor ARK-730A (Nidek) objective refractive data and performed the examinations preoperatively and 1 month postoperatively. The spherical equivalent was calculated from the objective refraction. The axial lengths for all IOL power calculations were measured using the IOLMaster (version 2.0) after PTK. The anterior chamber depth and the lens thickness for the PhacoOptics, Barrett True-K No History (NH) (*P* = .001 for both comparisons), and Camellin-Calossi formulas were measured by the UD-6000 biometer (Tomey, Aichi, Japan). The corneal thickness for the Camellin-Calossi formula was measured by the Pentacam (OCULUS Optikgeräte, Wetzlar, Germany).

The average corneal powers in a 3-mm zone for the Camellin-Calossi formula were measured using the ARK-10000 (Nidek). Anterior segment optical coherence tomography (SS-1000; Tomey) was used to measure both the corneal topography for the OKULIX ray tracing software and the conic coefficient of the front and back corneal surfaces and keratometry readings for the PhacoOptics and Barrett True K (NH) formulas.

The IOL power was calculated using the OKULIX ray tracing software, PhacoOptics ray tracing software, Barrett True K (NH) formula, and Camellin-Calossi formula, based on the preoperative data.

#### IOL Power Calculations

The OKULIX ray tracing software was used to calculate the monochromatic optical capacities of the human eye using numerical ray tracing combined with corneal topography. In the numerical ray tracing calculation, the individual rays undergo refraction on all IOLs and cornea surfaces. The refractions are calculated exactly using Snell's law. The rays can be calculated for any distance from the optical axis. The OKULIX ray tracing software requires corneal topography, axial length, and preinstalled IOL information.

The PhacoOptics ray tracing software is also an IOL power calculation and management system that is supported by precise and paraxial ray tracing where optical methods are necessary and empirical methods when these are necessary. Olsen and Hoffmann partially released their formula in a patent application^{9} and a medical report^{10}; however, the full formula has not been published. PhacoOptics is available even in eyes that underwent LASIK and those with keratoconus, which have significant higher order aberrations. The PhacoOptics ray tracing software includes “the K-reading assistant for post-LASIK cases,” which was applied in our study in three ways. First, the historical method (H) helps to calculate the effective curvature of the anterior surface and the true ratio between the posterior and anterior surfaces of the post-LASIK cornea, and it supports the double-K method for the estimation of the effective lens position. Second, with the NH method, PhacoOptics can back-calculate the pre-LASIK cornea using the anterior and posterior corneal curvatures and the post-LASIK refraction. It also supports the double-K method. Third, with the C-constant method (C), PhacoOptics uses the concept of the C-constant approach^{10} for accurate prediction of the IOL position. The IOL position can be predicted from the preoperative anterior chamber depth and lens thickness, thus skipping the K-reading and axial length dependence found in the standard IOL calculations. **Figure A** (available in the online version of this article) shows the three different calculation methods in PhacoOptics formula.

The Barrett True K formula uses the Barrett Universal II formula, which is a modified version of the original universal theoretical formula.^{11} The Barrett True K formula calculates a modified K-value for eyes that underwent refractive surgery and requires the measured keratometries and the refractive values before and after laser treatment. If the refractive data are unavailable, the formula can calculate the estimated change in the manifest refraction using an internal regression formula. Thus, the Barrett True K formula is available without the induced change in manifest refraction as the NH method.^{12} The Barrett True K and Universal II formulas can predict the effective lens position based on a unique theory; however, the details regarding the design of the Barrett True K and Universal II formulas have not been published.

The Camellin-Calossi formula,^{5} a modification of the Binkhorst II formula, was proposed to calculate the IOL powers for eyes that underwent a myopic laser surgery. This method does not require the preoperative PTK data. The radius of the posterior corneal surface curvature can be calculated based on the measurement of the anterior corneal surface curvature and the corneal thickness in a 3-mm circular zone and the central cornea. The IOL power calculation software (IOL Station; Nidek) was used to calculate the Camellin-Calossi formula.

#### Evaluation of Predictability of the IOL Power Calculation Methods

The prediction error was calculated by subtracting the postoperative manifest refraction at the spectacle plane from the estimated postoperative spherical equivalent for each formula. The median values of the arithmetic and absolute prediction errors were compared statistically because the errors were not normally distributed. The percentages of eyes that were within ±0.50 and ±1.00 diopter (D) of the targeted correction and the accuracy of the IOL calculation methods between GCD2 and band keratopathy also were compared.

#### Statistical Analysis

The Wilcoxon one-sample signed-rank test was used to assess the median value of the arithmetic prediction errors produced by the various formulas. The Wilcox-on signed-rank test and adjusted multiplicity with the Bonferroni method were used to compare the median values of the arithmetic and absolute prediction errors with these IOL calculation methods. The chi-square test was used to compare the percentages of eyes with prediction errors within ±0.50 D, within −1.00 to 0.50 D, and within ±1.00 D in the six formulas.

All statistical analyses were performed using the SPSS Statistics software (version 24; IBM Japan, Ltd., Tokyo, Japan). A *P* value of less than .05 was considered statistically significant.

### Results

**Table 1** shows the patient demographic data before cataract surgery. The mean spherical equivalent values before and after cataract surgery were 1.53 ± 2.97 D (range: −10.50 to 5.75 D) and −1.46 ± 1.10 D (range: −3.88 to 0.13 D), respectively. The mean corrected distance visual acuity values before and after cataract surgery were 0.16 ± 0.15 D (range: −0.08 to 0.40 D) and 0.04 ± 0.13 D (range: −0.08 to 0.30 D), respectively.

Table 1: Preoperative Patient Characteristics |

**Table 2** shows the arithmetic and absolute prediction errors in each formula. The median values of the prediction errors did not differ significantly from zero except for the Barrett True K (NH), which produced slightly myopic prediction errors (*P* < .05). The median absolute prediction errors for the Camellin-Calossi, PhacoOptics (C), Barrett True K (NH), PhacoOptics (NH), OKULIX, and PhacoOptics (H) formulas were 0.46, 0.51, 0.52, 0.53, 0.64, and 0.94 D, respectively.

Table 2: Arithmetic and Absolute Prediction Errors for Each IOL Power Calculation Formula |

**Figure 1A** shows the distributions of the arithmetic prediction errors. The PhacoOptics (NH) formula had significant (*P* = .024 for both comparisons) myopic shifts of the distributions of the arithmetic prediction errors compared with the PhacoOptics (C) formula. The OKULIX ray tracing software showed significant (*P* = .006 for both comparisons) hyperopic shifts compared with the PhacoOptics software (NH). The Barrett True K (NH) formula showed significant (*P* = .001 for both comparisons) myopic shifts compared with the OKULIX, PhacoOptics (NH) (*P* = .030 for both comparisons), and PhacoOptics (C) (*P* = .001 for both comparisons) formulas.

The Camellin-Calossi formula had the smallest absolute prediction error of the six formulas and was significantly (*P* = .048) better than that of the OKULIX formula (**Figure 1B**).

**Figure 2** shows the distribution of the absolute prediction errors within ±0.50 and ±1.00 D. The Camellin-Calossi formula provided significantly better results than the OKULIX (*P* = .020) and PhacoOptics (H) (*P* = .005) formulas in the percentages of the errors within ±0.50 D. The Barrett True K (NH) formula also provided significantly better results than the PhacoOptics (H) formula (*P* = .031) in the percentages of the errors within ±0.50 D. The PhacoOptics (NH), PhacoOptics (C), Barrett True K (NH), and Camellin-Calossi formulas provided significantly better results (*P* = .010, .030, .048, .030, respectively) than the PhacoOptics (H) formula in the percentages of the errors within ±1.00 D.

### Discussion

Reduced accuracy of the IOL power calculations in eyes that underwent a previous PTK is an unsolved issue in cataract surgery. Various methods have been proposed to address the lack of predictability in eyes that underwent LASIK or PRK.^{5–7,13–17} However, few reports have been published regarding IOL power calculations after PTK.^{8,18,19} The purpose of the current study was to evaluate several ray tracing methods and compare them with the Camellin-Calossi formula. Our results showed that the PhacoOptics, particularly the C-constant method, and the Camellin-Calossi formula, were predictable IOL calculation methods in eyes that underwent a previous PTK.

The Camellin-Calossi formula, which is a modified version of the Camellin formula,^{5,20} was developed for use in normal eyes and those that underwent a previous refractive surgery.^{5,21} The formula estimates the curvature of the posterior corneal surface by measuring the curvature of the anterior corneal surface, the corneal thickness in a 3-mm circular zone, and the central cornea with ultrasonic pachymetry. We concluded that the Camellin-Calossi formula produced good results even in eyes that underwent a previous PTK. Furthermore, Suto et al.^{22} first reported that the accuracy of the Camellin-Calossi formula was comparable to that of the SRK/T and Haigis-L formulas for normal cataractous eyes.In that study, the mean arithmetic prediction error of the Camellin-Calossi formula was −0.01 ± 0.56 D (range: −1.04 to +1.22 D) compared with the median −0.19 D (range: −1.78 to 1.48 D in our sample). These results indicated that the Camellin-Calossi formula has broad utility for eyes treated with laser and for virgin cataractous eyes.

Over the past decade, only a few studies have reported the accuracy of the IOL calculation software using ray tracing.^{23–25} The ray tracing method is not subject to the problems that cause IOL power calculation inaccuracies after corneal excimer surgery for the following reasons: (1) ray tracing does not depend on a fictitious index to calculate the corneal power of the index of refraction error, (2) ray tracing can calculate the effective lens position without relying on the anterior corneal curvature, unlike the third-generation formulas, and (3) ray tracing can be performed for eyes with any corneal diameter because it uses different optical zones. Although some studies have reported the efficacy of ray tracing methods in eyes after excimer laser refractive surgery,^{26,27} few have reported the results in eyes after PTK. We first evaluated the accuracy of the OKULIX ray tracing software in eyes that underwent PTK; however, the results were unfavorable.

In the current study, the OKULIX ray tracing software showed a hyperopic outcome, which was similar to our previous study,^{8,26} in eyes after PTK or myopic LASIK. These results suggested that the OKULIX ray tracing software tends to show a hyperopic shift in eyes with a flattened cornea after corneal laser surgery.

PhacoOptics is based on paraxial and exact tracing using optical formulas when optical methods are necessary and empirical methods when these are necessary. To our knowledge, the current study is the first in the peer-reviewed literature to compare the PhacoOptics formula with other ray tracing calculation methods for eyes after PTK. In this study, the prediction error of the PhacoOptics (H) formula was unfavorable.

As previously stated, the H method requires the pre-PTK K-reading, which is used to predict the anterior chamber depth. This result indicated that the pre-PTK K-reading is unreliable due to the corneal surface opacities associated with GCD2 and band keratopathy even when using anterior segment optical coherence tomography, which is less affected by opacities than Scheimpflug-based topography such as the Pentacam.

However, the C-constant method showed the best result among the three variations of PhacoOptics. This method is supported by the concept of the C-constant,^{10} which was developed as an unbiased method to predict the IOL position from the accurate measurements of the anterior chamber depth and preoperative dimension and position of the lens according to the simple formula: IOLc = preoperative ACD + C × LT, where IOLc is the center of the IOL, LT is the preoperative lens thickness, and C is a constant related to the IOL type determined as the mean value in a representative sample. This concept has the advantage that it is dependent on the anterior segment length and independent of the K-readings. We presume that this is the reason why the PhacoOptics (C) formula showed good results.

The Barrett True K (NH) formula adopts an internal regression formula to estimate the change in the manifest refraction; thus, it can be available even though the refractive change after PTK is unavailable. In the current study, although only the Barrett True K (NH) formula had a significantly negative median prediction error compared to zero, the percentage of eyes within ±0.50 D of the refractive prediction error was comparable to the PhacoOptics (NH), PhacoOptics (C), and Camellin-Calossi formulas. Abulafia et al.^{12} reported the accuracy of the Barrett True K formula for eyes after LASIK or PRK performed to treat myopia. In that study, the median value of the arithmetic refractive error of the Barrett True K (NH) formula was −0.20 D (range: −1.55 to 1.28 D), with a tendency toward myopic outcomes as in our study. This result suggested that surgeons should be alert to the tendency for a myopic shift.

Our study had some limitations, and further investigation is warranted. First, the study had a small sample size and included eyes with long and short axial lengths. It is necessary to obtain a relatively large sample. Second, this study dealt with only two specific etiologies of corneal opacity. Third, there were a variety of biometry methods and implanted IOLs due to the nature of a retrospective study. In this study, the anterior chamber depth and axial length were measured by the contact ultrasound biometry because the non-contact ultrasound biometry was not available during this study period. The contact ultrasound biometry may cause measurement errors of the anterior chamber depth because of compression. A test is needed that produces zero average prediction errors in normal eyes with each IOL model; otherwise, the results in eyes that are not normal may simply be biased by missing the zero-offset adjustment in a collective of normal eyes. In a future study, we will use a simple measurement device for examination and only one IOL with zero-offset adjustment.

The current study showed that the PhacoOptics, especially using the C-constant method, and the Camellin-Calossi formulas may be good options for calculating IOL powers in eyes that underwent various corneal ablative surgeries, if they are available.

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Preoperative Patient Characteristics

Age (y) | 69.4 ± 7.1 | 56 to 85 |

Axial length (mm) | 23.37 ± 1.50 | 20.51 to 29.19 |

Anterior chamber depth (mm) | 2.72 ± 0.44 | 2.02 to 4.09 |

Lens thickness (mm) | 4.92 ± 0.50 | 3.79 to 5.89 |

Corneal refractive power (D) | 42.53 ± 1.48 | 39.97 to 47.12 |

Sphere value (D) | 2.22 ± 2.99 | −10.00 to 6.50 |

Cylinder value (D) | −1.36 ± 0.68 | −3.00 to 0.00 |

Arithmetic and Absolute Prediction Errors for Each IOL Power Calculation Formula

OKULIX | 0.33 | −2.20 to 2.50 | 0.64 | 0.15 to 2.50 | 36 |

PhacoOptics (H) | −0.12 | −3.28 to 4.85 | 0.94 | 0.09 to 4.85 | 27 |

PhacoOptics (NH) | −0.25 | −2.08 to 1.70 | 0.53 | 0.04 to 2.08 | 31 |

PhacoOptics (C) | 0.04 | −1.40 to 2.18 | 0.51 | 0.02 to 2.18 | 33 |

Barrett True K (NH) | −0.35^{a} | −1.90 to 1.89 | 0.52 | 0.01 to 1.90 | 37 |

Camellin-Calossi | −0.19 | −1.78 to 1.47 | 0.46 | 0.00 to 1.78 | 37 |