What is the corneal power? Most clinicians will ask for the “K-reading,” neglecting the fact that the keratometer does not measure the power directly. What the keratometer really does is measure the size of the Purkinje I image reflected from the front surface of the cornea in a paracentral ring of approximately 3 mm and calculate the radius of curvature of the convex mirror constituted by the tear film of the cornea. The next step in the classic K-reading is to regard the cornea as a “thin lens” and calculate the dioptric power “K” as a function of the radius according to the paraxial equation for a single refracting surface:
where r is the radius in meters, nair
is the refractive index of air (= 1.0), and ncornea
is the assumed keratometric refractive index of the cornea (= 1.3375 in conventional keratometers). According to this model, a corneal radius of 7.5 mm equals a corneal power of 45.00 D.
There are several known problems with the corneal power measured in this way. First, the cornea is not a thin lens with just one surface, but rather a thick lens with two surfaces that both contribute to the total corneal spherical and cylindrical power. Second, the assumed refractive index of 1.3375 overestimates the corneal power calculated by Gaussian optics (the paraxial power), assuming the Gullstrand schematic eye ratio between the front and back curvature.1 Finally, the radius is measured in a ring area of approximately 3 mm, which may not be representative of the effective central pupillary zone.
The technological development of instruments for the analysis of the corneal shape has improved greatly since the development of the original concept of the K-reading. Today, scanning devices such as Scheimpflug cameras or optical coherence tomography may provide clinical information of both surfaces of the cornea, enabling the analysis of the corneal optics in more detail. The purpose of this study was to describe how exact ray tracing can be used to analyze the three-dimensional measurements of both the front and back surface of the cornea in terms of the focal length and consequently the effective power of the cornea.
Patients and Methods
Twenty patients referred for cataract or refractive lens surgery were included. Excluded from this study were cases with corneal astigmatism of greater than 4.00 D, keratoconus, previous trauma, or anterior segment surgery. All cases were subjected to a routine preoperative evaluation including standard keratometry, biometry, and intraocular lens power calculation. The clinical data appear in Table 1.
Clinical Data of 20 Patients Scheduled for Cataract or Refractive Lens Surgery
Conventional keratometry was performed using the Nidek ARK700 automated keratometer (Nidek Ltd., Gamagori, Japan). The axial length was measured using the Zeiss IOLMaster 500 (software version v.1; Carl Zeiss Meditec, Jena, Germany). Scheimpflug tomography was performed using the Oculus Pentacam HR software version 19r21 (Oculus Optikgeräte, Wetzlar, Germany). The Pentacam variables “Sim-K” of the front surface, representing the standard simulated keratometer reading in a 3-mm ring area; “True Net Power” (TNP), based on thick lens calculation of the corneal power in a 3-mm zone area; and “Total Corneal Refractive Power” (TCRP), based on proprietary ray tracing of the 3-mm zone area by the Pentacam software, were recorded. The matrix of the mapped elevation data containing approximately 4,200 points from the anterior and 3,800 points from the posterior surface was exported using the “CSV” file export option of the Pentacam software and analyzed as described below.
Ray-tracing Analysis of Corneal Power
The Pentacam measurements of the front and back surface of the cornea were available in the format shown in Figure 1. Each elevation point was represented in microns in an x–y coordinate system where the origin (0,0) represented the corneal vertex. Each increment in the x- and y-directions represented a 0.1-mm step in the horizontal and vertical direction of the cornea, respectively.
Pentacam (Oculus Optikgeräte, Wetzlar, Germany) matrix of elevation data in microns. Each point is represented in an x–y coordinate system where origin (0,0) represents the corneal vertex of zero (0) elevation (yellow field in insert). Each increment in the x- and y-direction represents a 0.1-mm step in the horizontal and vertical direction of the cornea, respectively.
Zemax (Zemax LLC, Redmond, WA) is a professional optical engineering software for the design and analysis of imaging systems such as telescopes or cameras. The advantage of such software is that it allows for the analysis of the optical properties of any surface provided the exact physical shape and refractive index of the material are known. To analyze the refraction of light, only Snell's law is necessary:
is the incident and θ2
is the refracted angle of a ray of light with respect to the normal and n1
are the respective refractive indices of the two media.
To import the elevation data into Zemax, it was necessary to convert the x-y-z matrix of the Pentacam elevation matrix into a series of three-dimensional triangles of a polygonal format interpretable by Zemax. In Zemax, the corneal model was constructed as a so-called “non-sequential component,” allowing the ray-tracing analysis to be performed on the real physical shape, in this case of a multi-faceted, triangulated type.
The refractive index of the cornea was assumed to be 1.376 and that of the aqueous 1.336.2 The central thickness of the cornea was measured by the Pentacam instrument. The illumination point source was placed 20 feet in front of the cornea, simulating the normal testing distance of a visual acuity chart. A high number of rays (> 500.000) were “fired” through the corneal model and analyzed for the resulting point spread function (PSF) at the image plane (Figures 2–3), which was varied to analyze for the best focus. The focal length was defined as the distance giving the sharpest PSF or the highest peak irradiance.
The Pentacam (Oculus Optikgeräte, Wetzlar, Germany) measurements of the front and back surface of the cornea have been imported into the Zemax software (Zemax LLC, Redmond, WA). Each surface has been constructed as a meshwork of miniature triangles connected by the individual data points. An aperture has been inserted to block peripheral rays simulating the effect of the pupil. Rays originate from a distant point source 20 feet in front of the cornea.
Graphic diagram of ray tracing experiment in Zemax software (Zemax LLC, Redmond, WA). From a light point source to the far left, rays enter through the cornea made up of a front and back surface as measured by the Pentacam (Oculus Optikgeräte, Wetzlar, Germany). A pupil has been added behind the cornea. At the right is a small “detector” that functions as a photographic plate, allowing for the analysis of the point spread function (PSF) at the selected image distance. Inserts show how the PSF changed from a 1-mm unfocused (PSF 1) to a focused (PSF 2) image.
The relationship between peak irradiance and image distance is shown for a typical case in Figure 4. Repeated ray-tracing experiments found that the peak irradiance was well fitted (correlation coefficients > 0.95) using a polynomial function of the form:
where P is peak irradiance, z is the image distance, and a, b, and c are constants derived from the linear regression analysis. By deriving the differential function:
the value of z where the slope of Equation 3 was zero (or P at its maximum) could be determined as:
From the determined focal distance Z0
, the corneal power C was finally calculated as:
where C is corneal power in diopters, Z0
is the focal distance in meters, and n is the refractive index of the medium. In the paraxial domain, the distance divided by the refractive index is also known as the distance “reduced to air.”2
Therefore, the total reduced distance was further calculated as the sum of the reduced thickness of the cornea (= CCT / 1.376, where CCT is the thickness of the cornea in meters) and the reduced distance of the aqueous (= AQ / 1.336, where AQ is the image distance minus the CCT).
The peak irradiance has been recorded as a function of defocus in three double experiments. The curves were found to assume a polynomial shape, which made it possible to solve mathematically for the x-coordinate of the peak (the focal point) by differential analysis.
The above calculation of corneal power expressed the effective corneal power referenced from the front vertex. This might be a useful value. However, because the second principal plane of the cornea in the paraxial domain is approximately 0.05 mm in front of the cornea, this small distance was finally added to the observed reduced focal length to “normalize” the power estimation with the paraxial value. This resulted in a small correction (approximately 0.10 D reduction) to the front vertex power.
The effect of pupil size was studied and comparisons made with the other metrics of corneal power. The reproducibility of the ray-tracing procedure was evaluated from repeated experiments in 10 cases as the standard deviation of the difference between the paired measurements.
The mean ± standard deviation difference between the 10 double experiments (3-mm pupil) was 0.004 ± 0.009 mm and −0.005 ± 0.011 D for the determination of focal length and the corneal power, respectively (Table 2). The maximum difference between the repeated corneal power estimation was 0.023 D.
Results of Repeated Ray Tracing Experiments to Determine the Effective Focal Length and the Corresponding Corneal Power of 10 Normal Patients
Increasing the pupil size increased the estimated corneal power in all experiments (case shown in Figure 5). On average, the corneal power increased 0.30 D when the pupil size increased from 3 to 5 mm (Table 3).
Pupil experiment of a normal case. As the pupil enlarges, more light is detected at the image plane but the best focus (at the maximum of the curve) is shortened somewhat due to spherical aberration. Inserts show the appearance of the point spread function at the selected image plane.
Results of Varying the Pupil Size on the Estimated Corneal Power From Ray-tracing Analysis of 20 Normal Patients
The mean ray-traced corneal power of 42.34 D was found to be 1.02 ± 0.50 D lower (range: 0.40 to 2.80 D) than the standard keratometry reading, assuming a keratometric index of 1.3375 and a 3-mm pupil (P < .001 by paired t test). The mean ray-traced corneal power was 0.44 ± 0.23 D (range: 0.07 to 1.19 D) lower than the TNP reported by the Scheimpflug device (P < .01). The mean difference between the ray-traced corneal power and the TCRP was 0.03 ± 0.25 D (range: −0.64 to 0.43 D), which was not statistically significant from zero (P > .05). The different methods have been compared in Figure 6.
The corneal power determined by the ray-tracing technique compared to the standard K-reading, the True Net Power (TNP), and the Total Corneal Refractive Power (TCRP) reported by the Oculus Pentacam instrument. The K-reading was approximately 1.00 diopter (D) higher than the ray-traced power, which was higher than the TNP. No significant difference was found between the TCRP and the ray-traced power, assuming a 3-mm pupil.
Equivalent Refractive Index of the Cornea
From the estimated corneal power derived from the ray-tracing experiments, it is possible to derive the corresponding thin lens equivalent keratometric index of the cornea (Equation 1). Rearranging Equation 1 and solving for the refractive index in each of the 20 cases shown in Table 3, the average values for the effective keratometric index ranged from 1.3207 to 1.3320 for pupil sizes 3 to 5 mm (Table 4).
Interpretation of the Effective Corneal Power
Exact ray tracing is a well-known technique used in physical optics for the design of imaging and illumination systems and, for obvious reasons, is of interest in the field of ocular optics. With the advent of newer scanning techniques that measure both surfaces of the cornea in multiple points, we now have better options to study the optics of the cornea by ray tracing, which may have advantages over classic methods based on the anterior surface of the cornea only. In this study, we wanted to show the steps involved in the calculation of the corneal power by ray tracing on the raw elevation data and how this compared with conventional methods.
The ray-traced corneal power was found to be more than 1.00 D lower than the conventional simulated keratometry reading using the standard keratometer index of 1.3375. This is in good agreement with other recent studies showing that the ray-tracing technique based on tomography gives lower values than the conventional keratometer values based on anterior topography.3–5 Our findings indicated the equivalent simulated keratometer index to be in the range of 1.3207 to 1.3320 for pupil sizes between 3 and 5 mm, respectively. This finding is in fact close to the old value of 1.3315 proposed by Olsen, assuming paraxial imagery and a ratio of 0.88 between the front and back surface of the cornea as in the exact schematic eye of Gullstrand.1 One may wonder how this can be, when numerous Scheimpflug studies have shown the true ratio to be only 0.83, which is lower than the original 0.88. The difference in calculated corneal power between the two ratios amounts to approximately 0.40 D in the paraxial domain. The reason for the apparent discrepancy is explained by the difference between paraxial and exact ray tracing. In paraxial ray tracing, the refraction is calculated using the approximation sin(i) = i, where i is the incident angle. In exact ray tracing, there is no such approximation. All rays are refracted according to Snell's law and for a spherical cornea this means peripheral rays are refracted stronger than central rays due to spherical aberration. The effect of the spherical aberration will depend on the pupil, so the effective power of the cornea will increase as the pupil size increases. The magnitude of the effect in the current study was a 0.30 D increase in corneal power, when the pupil widened from 3 to 5 mm. Hence, spherical aberration is why a ray-traced corneal power assuming a Gullstrand ratio of 0.83 does not differ much from the paraxial corneal power, assuming a Gull-strand ratio of 0.88.
We found good agreement between the mean value of our ray-traced corneal power and the TCRP reported by the Pentacam software (for a 3-mm pupil), although some spread between the individual measurements was noted. According to the manufacturer, the TCRP is also based on ray tracing and therefore in principle should be similar to our approach, although the details of the ray tracing are not described in the Pentacam manual. The Pentacam TNP (equivalent to the thick lens or paraxial power) was found to be 0.44 D lower than the ray-traced corneal power. As explained above, the ray-traced power is expected to be higher than the paraxial power for a normal cornea, depending on the pupil. In fact, if our finding of a 0.15 D increase in corneal power per 1 mm increase in pupil size is extrapolated backward to zero (the paraxial domain), the expected difference would be 3 × 0.15 D = 0.45 D. Thus, taking pupil size and higher order aberrations into account, there is remarkably good agreement between the paraxial and exact ray-traced power.
No attempt was made in the current study to correct for the existence of the retinal directional sensitivity (ie, the Stiles–Crawford effect6), which will tend to reduce the effect of the spherical aberration, especially for larger pupil sizes.7 In case of a positive spherical aberration (most corneas), the effect of the Stiles–Crawford effect will therefore be to decrease the effective corneal power, thus increasing the difference between the effective corneal power and the conventional K-reading.
Although ray tracing may have a theoretical advantage, this does not necessarily indicate a clinical benefit, which remains to be shown by clinical studies. An obvious application is intraocular lens power calculation. However, only a few studies seem to have addressed the potential benefit of tomography data for the accuracy of intraocular lens power calculation.8,9 One study could not find a significant benefit in normal eyes.8 However, care must be exercised to replace the K-reading with the ray-traced corneal power in conventional intraocular lens power formulas because most formulas have been calibrated against the (wrong) K-reading. However, a recent study found that the Pentacam TNP corneal power in combination with a modified Shammas formula gave better results than conventional methods after LASIK.9 Several corneal refractive surgery studies have found that the corneal power derived by ray tracing agrees more closely with the refractive changes than Sim-K readings, thus supporting the view that the ray-traced corneal power gives clinically relevant information.3,5,10,11
The potential advantage of ray tracing is that it allows for the detailed description of the refraction, taking all aberrations into account. Exact ray tracing does not assume the cornea to be of any defined shape, whether it is a fitted spheroid, ellipsoid, or of polynomial function. Snell's law will accurately predict the refractive properties of the optical component once the physical shape has been determined. Because of the assumption-free physical principle, it has been suggested that ray tracing may be used to build more sophisticated eye models and open the door for customized IOL power selection.12–14 Such applications might be of special interest in corneas with regular and irregular astigmatism, keratoconus, post-LASIK cases, post-keratoplasty cases, and other unusual corneas. Further studies are needed to investigate the clinical benefit of ray tracing of the corneal optics and ocular refraction in general.
- Olsen T. On the calculation of power from curvature of the cornea. Br J Ophthalmol. 1986;70:152–154. doi:10.1136/bjo.70.2.152 [CrossRef]
- Bennett A, Rabbetts R. Clinical Visual Optics, 2nd ed. Oxford: Butterworth-Heinemann; 1989.
- Wang L, Mahmoud AM, Anderson BL, Koch DD, Roberts CJ. Total corneal power estimation: ray tracing method versus gaussian optics formula. Invest Ophthalmol Vis Sci. 2011;52:1716–1722. doi:10.1167/iovs.09-4982 [CrossRef]
- Tonn B, Klaproth OK, Kohnen T. Anterior surface-based keratometry compared with Scheimpflug tomography-based total corneal astigmatism. Invest Ophthalmol Vis Sci. 2014;56;291–298. doi:10.1167/iovs.14-15659 [CrossRef]
- Pan C, Hua Y, Huang J, Tan W, Lu W, Wang Q. Corneal power measurement with the dual Scheimpflug-Placido topographer after myopic excimer laser surgery. J Refract Surg. 2016;32:182–186. doi:10.3928/1081597X-20160106-02 [CrossRef]
- Stiles WS, Crawford BH. The luminous efficiency of rays entering the eye pupil at different points. Proc R Soc (Lond) B. 1933;112:428–450. doi:10.1098/rspb.1933.0020 [CrossRef]
- Olsen T. On the Stiles-Crawford effect and ocular imagery. Acta Ophthalmol. 1993;71:85–88. doi:10.1111/j.1755-3768.1993.tb04966.x [CrossRef]
- Shirayama M, Wang L, Koch DD, Weikert MP. Comparison of accuracy of intraocular lens calculations using automated keratometry, a Placido-based corneal topographer, and a combined Placido-based and dual Scheimpflug corneal topographer. Cornea. 2010;29:1136–1138. doi:10.1097/ICO.0b013e3181d3d689 [CrossRef]
- Potvin R, Hill W. New algorithm for intraocular lens power calculations after myopic laser in situ keratomileusis based on rotating Scheimpflug camera data. J Cataract Refract Surg. 2015;41:339–347. doi:10.1016/j.jcrs.2014.05.040 [CrossRef]
- Savini G, Calossi A, Camellin M, Carones F, Fantozzi M, Hoffer KJ. Corneal ray tracing versus simulated keratometry for estimating corneal power changes after excimer laser surgery. J Cataract Refract Surg. 2014;40:1109–1115. doi:10.1016/j.jcrs.2013.11.032 [CrossRef]
- Tang M, Li Y, Avila M, Huang D. Measuring total corneal power before and after laser in situ keratomileusis with high-speed optical coherence tomography. J Cataract Refract Surg. 2006;32:1843–1850. doi:10.1016/j.jcrs.2006.04.046 [CrossRef]
- Tabernero J, Piers P, Benito A, Redondo M, Artal P. Predicting the optical performance of eyes implanted with IOLs to correct spherical aberration. Invest Ophthalmol Vis Sci. 2006;47:4651–4658. doi:10.1167/iovs.06-0444 [CrossRef]
- Canovas C, Artal P. Customized eye models for determining optimized intraocular lenses power. Biomed Opt Express. 2011;2:1649–1662. doi:10.1364/BOE.2.001649 [CrossRef]
- Zhu Z, Janunts E, Eppig T, Sauer T, Langenbucher A. Tomography-based customized IOL calculation model. Curr Eye Res. 2011;36:579–589. doi:10.3109/02713683.2011.566978 [CrossRef]
Clinical Data of 20 Patients Scheduled for Cataract or Refractive Lens Surgerya
|Age (y)||Axial Length (mm)||IOL Power (D)||Pentacam Rfront (mm)||Pentacam Rback (mm)||Gullstrand Ratiob|
|56.21 ± 10.79 (44 to 79)||23.69 ± 2.79 (19.65 to 31.29)||22.63 ± 9.84 (0.00 to 39.00)||7.80 ± 0.27 (7.38 to 8.32)||6.48 ± 0.25 (6.08 to 6.91)||0.831 ± 0.018 (0.79 to 0.88)|
Results of Repeated Ray Tracing Experiments to Determine the Effective Focal Length and the Corresponding Corneal Power of 10 Normal Patients
|Reproducibility (n = 10)||Experiment 1||Experiment 2||Difference|
|Focal length (mm)||31.04 ± 0.87 (29.84 to 32.89)||31.05 ± 0.87 (29.84 to 32.91)||0.004 ± 0.009 (−0.010 to 0.019)|
|Corneal power (D)||42.99 ± 1.18 (40.56 to 44.69)||42.99 ± 1.18 (40.53 to 44.69)||−0.005 ± 0.011 (−0.023 to 0.014)|
Results of Varying the Pupil Size on the Estimated Corneal Power From Ray-tracing Analysis of 20 Normal Patientsa
|Zemax Derived Corneal Power (D)/Pupil Size||Pentacam Variables (D)b||Keratometer (D)|
|3 mm||4 mm||5 mm||TNP||TCRP||‘K-reading’|
|42.34 ± 1.33 (39.79 to 44.69)||42.52 ± 1.38 (39.86 to 45.19)||42.64 ± 1.41 (39.96 to 46.46)||41.91 ± 1.29 (39.50 to 43.65)||42.38 ± 1.28 (39.90 to 44.05)||43.36 ± 1.53 (40.74 to 45.95)|
Interpretation of the Effective Corneal Powera
|Pupil 3 mm||Pupil 4 mm||Pupil 5 mm|
|1.3207 ± 0.0037 (1.3165 to 1.3329)||1.3310 ± 0.0041 (1.3165 to 1.3378)||1.3320 ± 0.0043 (1.3170 to 1.3403)|