**Purpose.**:
To evaluate with the use of corneal topographic data the differences between total corneal power calculated using ray tracing (TCP) and the Gaussian formula (GEP) in normal eyes, eyes that previously underwent laser in situ keratomileusis/photorefractive keratectomy (LASIK/PRK), and theoretical models.

**Methods.**:
TCP and GEP using mean instantaneous curvature were calculated over the central 4-mm zone in 94 normal eyes, 61 myopic-LASIK/PRK eyes, and 9 hyperopic-LASIK/PRK eyes. A corneal model was constructed to assess the incident angles at the posterior corneal surface for both refracted rays and parallel rays. Corneal models with varying parameters were also constructed to investigate the differences between mean TCP and GEP (4-mm zone), and an optical design software validation was performed.

**Results.**:
The TCP values tended to be less than GEP in normal and myopic-LASIK/PRK eyes, with the opposite relationship in some hyperopic-LASIK/PRK eyes having the highest anterior surface curvature. The difference between TCP and GEP was a function of anterior surface instantaneous radii of curvature and posterior/anterior ratio in postrefractive surgery eyes but not in normal eyes. In model corneas, posterior incident angles with parallel rays were greater than those with refracted rays, producing an overestimation of negative effective posterior corneal power; differences in magnitude between TCP and GEP increased with decreasing ratio of posterior/anterior radii of curvature, consistent with clinical results.

**Conclusions.**:
In eyes after refractive surgery, calculating posterior corneal power using the Gaussian formula and its paraxial assumptions introduces errors in the calculation of total corneal power. This may generate errors in intraocular lens power calculation when using the Gaussian formula after refractive surgery.

^{ 1,2 }and the use of the standardized index of refraction of 1.3375, which does not account for the altered relationship between the anterior and posterior surfaces, is no longer valid.

^{ 3 }

^{ 4 –6 }

*n*= 1.336). TCP values over the central, paracentral, and peripheral zones are displayed. We recorded the average TCP over the central 4-mm area for each eye and used the index of refraction of the aqueous (n = 1.336) to convert ray traced focal length to power.

^{ 5 }by converting the average central instantaneous curvature (central 4-mm zone) displayed on the Galilei in diopters to anterior power by multiplying by 376/337.5. The F2 value was the posterior average central instantaneous curvature, for which the dioptric value displayed on the Galilei was calculated using the same paraxial formula with both the corneal (1.376) and the aqueous (1.336) indices of refraction. The pachymetric value used was the average over the central 4-mm area, as displayed on the Galilei. As with most corneal topographers, the posterior curvature is converted to diopters using the same formula as the anterior surface, assuming that incoming rays are parallel. It should also be noted that the GEP is referenced to the second principal plane, which is distinct from the TCP calculation, which is referenced to the anterior corneal surface.

*t*-test was used to compare the TCP and GEP, and correlation analysis was performed to assess the relationship between the differences of TCP and GEP and the anterior instantaneous radii of curvature as well as the posterior/anterior ratio. Statistical analysis was performed using statistical analysis software (SPSS, version 15.0; SPSS, Inc., Chicago, IL), and

*P*≤ 0.05 was considered statistically significant.

*P*= 0.543) in normal eyes, −0.232 (

*P*= 0.069) in myopic-LASIK/PRK eyes, and −0.313 (

*P*= 0.412) in hyperopic-LASIK/PRK eyes. If the postrefractive surgery eyes were grouped together, the Pearson correlation coefficient value was −0.504 (

*P*< 0.001). Note that without the single outlier in the normal population, the range of difference is <1 D in normal eyes and approximately 1.5 D in eyes after refractive surgery. The differences between TCP and GEP were also a function of posterior/anterior ratio in eyes after refractive surgery, whereas no relationship was found in normal eyes (Fig. 2). Differences were greatest at the lowest ratios in myopic LASIK/PRK eyes.

Anterior (mm) | Posterior (mm) | Ratio (posterior/anterior) | |
---|---|---|---|

Normal eyes (n = 94) | 7.69 ± 0.24 (7.25–8.28) | 6.27 ± 0.25 (5.71–7.04) | 0.82 ± 0.02 (0.73–0.87) |

Myopic-LASIK/PRK eyes (n = 61) | 8.29 ± 0.34 (7.46–9.08) | 6.34 ± 0.26 (5.60–6.81) | 0.76 ± 0.03 (0.69–0.83) |

Hyperopic-LASIK/PRK eyes (n = 9) | 7.46 ± 0.14 (7.30–7.68) | 6.40 ± 0.17 (6.20–6.63) | 0.86 ± 0.02 (0.82–0.91) |

TCP (D) | GEP (D) | Difference (D) | |
---|---|---|---|

Normal eyes (n = 94) | 42.27 ± 1.33 (39.26–44.96) | 42.71 ± 1.33 (39.65–45.29) | −0.44 ± 0.20 (−0.89 to 0.72) |

Myopic-LASIK/PRK eyes (n = 61) | 38.65 ± 1.82 (34.48–42.86) | 39.20 ± 1.72 (35.47–43.40) | −0.55 ± 0.29 (−1.37 to 0.08) |

Hyperopic-LASIK/PRK eyes (n = 9) | 44.41 ± 1.11 (42.82–45.64) | 44.33 ± 0.87 (43.02–45.64) | 0.08 ± 0.47 (−0.84 to 0.71) |

^{ 7,8 }However, the dual Scheimpflug topographer used in this study also calculates total corneal power using the ray tracing method. To the best to our knowledge, this is the first study to compare the differences between values for total corneal power calculated using ray tracing and the Gaussian formula.

^{ 6,7 }Using optical coherence tomography (OCT), in normal eyes, the total corneal power calculated by the summation of the anterior and posterior corneal powers underestimated the Atlas SimK (Humphrey Atlas; Carl Zeiss Meditec, Jena, Germany) by 1.13 D.

^{ 9 }The contribution of corneal thickness in the Gaussian formula is around 0.1 D, indicating that the Gaussian formula using the OCT would have underestimated the SimK by approximately 1.23 D. These reported differences between the SimK and the equivalent corneal power calculated with the Gaussian formula are consistent with our finding of 1.30 D using the Galilei (Table 3).

Study | Corneas | Device for SimK | Device for GEP | Difference SimK-GEP (D) |
---|---|---|---|---|

Borasio et al^{ 6 } | Normal | Topcon (Oakland, NJ) | Pentacam | 1.30 |

Savini et al^{ 7 } | Normal | TMS-2 (Tomey Corporation, Nishi-Ku, Nagoya, Japan) | Pentacam | 1.20 |

Keratron (Optikon, Rome, Italy) | Pentacam | 1.29 | ||

Pentacam | Pentacam | 1.25 | ||

Tang et al^{ 8 } | Normal | Humphrey Atlas | OCT* | 1.13 |

Current study | Normal | Galilei | Galilei | 1.30 |

^{ 10 }pointed out that the commonly used index of refraction of 1.3375 gives the power at the posterior vertex of the cornea, and an index of 1.3315 proposed by Olsen

^{ 11 }gives the power at the second principal plane, which is approximately 0.8 D less than at the posterior vertex. Estimated corneal power is further reduced by about another 0.5 D

^{ 9 }when the recently reported lower posterior/anterior ratio of 0.813 is used

^{ 12 }instead of the Gullstrand ratio of 0.883 (6.8/7.7). However, because of variation in the population in the ratio of posterior to anterior corneal radius of curvature, especially in eyes after corneal refractive surgery, a single index of refraction is not sufficient, and the accuracy of SimK in estimating the total corneal power is poor.

^{ 13 }which produces a power difference of <0.1 D. This magnitude of difference is small in comparison with the mean differences of ≥0.4 D between TCP and GEP found in healthy clinical subjects and those after myopic refractive surgery. It is important to note that, to the best of our knowledge, posterior corneal power is not accurately represented in any corneal topographer or anterior segment imaging device because radius of curvature is converted to diopters using a paraxial formula that does not account for a Snell's law refraction, as has been described for the anterior surface.

^{ 6 }In addition, the rays propagating to the posterior surface have already been refracted by the anterior surface; therefore, the “effective” posterior power will be less than what is calculated using parallel incident rays and a paraxial formula.