Corneal power after refractive surgery for myopia: contact lens method

Corneal power after refractive surgery for myopia: Contact lens method Wolfgang Haigis, PhD Purpose: To clarify the theoretical background of the rigid contact lens overrefraction (CLO) method to determine corneal power after corneal refractive surgery. Setting: University Eye Clinic, University of Wu¨rzburg, Wu¨rzburg, Germany. Methods: Using paraxial geometrical optics, the measurement situation for the contact lens method was analyzed and the definitions of corneal refractive power were reviewed. Based on the theoretical Gullstrand eye, model eyes were constructed, representing 1 emmetropic and 2 myopic eyes (primary refraction –5.21 diopters [D] and –10.25 D, respectively) before and after photorefractive keratectomy and laser in situ keratomileusis. In these eyes, the application of the CLO was mathematically simulated using Gaussian thick-lens optics and commercial ray-tracing software. Results: The CLO method measured neither the equivalent (total) power nor the vertex (back) power of the cornea but rather the quantity 336/R1C (R1C ⫽ anterior corneal radius). Based on these results and the Gullstrand eye, new formulas are proposed to derive the equivalent power and vertex power of the cornea by the CLO method. Conclusions: Depending on whether intraocular lens calculation formulas are based on equivalent (total) corneal power or vertex corneal power, the respective new formulas for the CLO method should be applied in patients after corneal refractive surgery. An increase in prediction accuracy of the refractive outcome is expected. J Cataract Refract Surg 2003; 29:1397–1411 © 2003 ASCRS and ESCRS

I

n 2000, approximately 550 000 cataract surgeries with subsequent intraocular lens (IOL) implantations were performed in Germany1; the estimate in the United States is approximately 2.3 million cataract procedures per year.2 Progress in microsurgery, sophisticated new measurement techniques, and improvements in IOL calculation algorithms offer excellent chances to restore good vision in today’s cataract patients. Whereas Accepted for publication October 28, 2002. From the University Eye Clinic, University of Wu¨rzburg, Wu¨rzburg, Germany. The author has no financial or proprietary interest in any material or method mentioned. Reprint requests to Dr. Wolfgang Haigis, Universita¨ts-Augenklinik, Josef-Schneider-Strasse 11, D-97080 Wu¨rzburg, Germany. E-mail: [email protected]. © 2003 ASCRS and ESCRS Published by Elsevier Inc.

most patients benefit from this progress without restrictions, a subgroup presents special problems in IOL calculation3– 6; that is, those who have had corneal refractive surgery. According to a study of approximately 60% of German ophthalmology surgical centers, more than 17 000 refractive procedures were performed in Germany in 2000. The numbers in the U.S. are much higher; for laser in situ keratomileusis (LASIK) alone, the estimate is approximately 931 000.2 Because more than two thirds of the eye’s total refractive power is provided by the cornea, its refractive power must be determined as accurately as possible for IOL calculation. Refractive surgery alters the corneal shape considerably; changes in corneal radii of 20% to 25% are fairly normal in formerly myopic eyes. As a consequence, classic keratometry in these eyes produces erroneous results,3,4,7 which in turn affect the accuracy 0886-3350/03/$–see front matter doi:10.1016/S0886-3350(02)02044-8

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

of IOL calculations. Thus, the future holds increasing problems in IOL calculation when patients with previous refractive surgery develop cataract. Hoffer8 and Holladay9 describe 5 ways to estimate corneal power in eyes after corneal refractive surgery. They are, in descending order of accuracy, refractive history, contact lens overrefraction (CLO), videokeratography, automated keratometry, and manual keratometry. The refractive history method is considered to be the procedure of choice 4 – 6,10 if all necessary information is available. If not, the CLO method is recommended. It appears as though the results of the contact lens measurement method have not been correctly interpreted. This paper, therefore, analyzes the CLO method to determine corneal power.

plano); and P2 is the contact lens base curve power in diopters (D). The Appendix lists all notations of physical quantities used in the text. A rigorous analysis of equation 1 reveals that the formula is not mathematically correct. It may give a clinically acceptable estimate of the corneal back vertex power in normal eyes without previous corneal surgery. In eyes with surgically reshaped corneas, however, equation 1 would theoretically lead to equivalent and vertex powers of the cornea that are too high by 1.00 to 2.00 D. The purpose of this paper is to demonstrate the anticipated errors of the CLO method in model calculations and to derive formulas to overcome them.

Materials and Methods Formulas

Hard Contact Lens Method Introduced by Holladay,9 the hard contact lens method is based on determining the difference between the manifest refraction with and without a rigid “plano” contact trial lens of a known base curve. An unchanged refraction indicates that the tear lens between the cornea and contact lens has zero power and that the effective anterior corneal radius is equal to the posterior radius (base curve) of the trial lens. If a myopic shift in refraction occurs with the contact lens, the base curve is steeper (ie, the tear lens forms a plus lens and vice versa). The idea is to determine corneal radius by finding the trial lens that does not change the refraction with and without contact lenses and then read the power from the contact lens base curve. For this method to work, visual acuity must be good enough. Holladay9 sets the lower limit of visual acuity at 20/80. Refraction should, of course, not be influenced by changes caused by cataract. Furthermore, contact lens decentration may cause problems. Hill recommends monitoring the lens position during the measurement (W. Hill, personal communication, 2002), which should be performed as a trial frame refraction and not as an autorefraction. Mathematically, the hard contact lens method reads K ⫽ Rx 1⬘ ⫺ Rx 0⬘ ⫹ P ⫹ P 2

(1)

in which K is the corneal power; Rx1⬘ and Rx0⬘ are the vertex-corrected refraction with and without a contact lens, respectively; P is the contact lens power (normally 1398

First, the power definitions underlying standard keratometry and the origin of keratometer indices were reviewed. For this purpose, theoretical eye 2 of Gullstrand11 (Table 1) was used. Elementary geometric optics was applied; some basic formulas are listed and described in the Appendix. Then, the measurement situation during application of the CLO method to determine corneal power was analyzed within the framework of geometric optics. An expression for the measured quantity resulting from the application of the CLO method was derived that was different from the formula used up to now.8,9 Based on these results, new formulas for the equivalent and back vertex powers of the cornea measured by the CLO method are proposed.

Model Eyes To check the validity of the new formulas, model calculations were performed. Starting from the Gullstrand eye, 2 myopic model eyes were simulated by adding 2.0 mm and 4.0 mm to the axial length provided by the Gullstrand eye (24.149 mm). The corneal curvature, anterior chamber, and lens were kept unchanged. (Only the vitreous length was increased by the respective distance.) Using custom computer programs based on thick-lens optics12 and, independently, commercial ray-tracing software (WinLens 4.1, Linos Photonics GmbH), the paraxial properties of the model eyes were determined: the myopic eye with axial length of 26.149 mm (named “myopic 5”) turned out to have a refraction of –5.21 D; the eye 4.0 mm longer (“myopic 10”) refracted to –10.25 D. In these eyes, LASIK procedures were simulated by varying the respective anterior corneal radii until both eyes were emmetropic. Thus, apart from the emmetropic Gullstrand eye, there were 4 model eyes (2 before and 2 after LASIK). These eyes were subsequently used for simulated mathematical applications of the CLO method. The model eyes and their characteristic parameters are shown in Table 2.

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

Table 1. Corneal data of the theoretical eye of Gullstrand (see Figure 1). For the tear lens, a center thickness of 10 ␮m is assumed. Parameter

Gullstrand Eye

Symbol

Unit

Refractive index of air

n1

—

1.000

Refractive index of cornea

nc

—

1.376

Refractive index of aqueous

n2

—

1.336

Center thickness of cornea

dc

mm

0.50

Anterior radius of corneal curvature

R1c

mm

7.70

Posterior radius of corneal curvature

R2c

mm

6.80

Refractive index of tear lens

nT

—

1.336

Center thickness of tear lens

dT

mm

0.01

Radius of curvature of tear lens

RT

mm

7.70

Equivalent power of cornea

PCe

D

43.05

Vertex (back) power of cornea

PCv

D

43.83

Equivalent and vertex powers of corneal lens in water

—

D

⫺0.68

Surface power of anterior tear surface

—

D

43.64

Surface power of anterior corneal surface vs. tears

—

D

5.19

Surface power of anterior corneal surface vs. air

P1c

D

48.83

Surface power of posterior corneal surface

P2c

D

⫺5.88

Model Contact Lenses The CLO method involves a set of rigid afocal contact lenses with a broad base curve range of approximately 30.00 to 45.00 D (W. Hill, personal communication, 2002). Therefore, a set of 10 plano model contact lenses of poly(methyl methacrylate) (PMMA) with a refractive index nL of 1.490 was designed with back curves covering the full range in question. They had different front and back radii as shown in Table 3 and a constant center thickness dL of 0.5 mm. Equivalent and vertex powers were 0 D for all lenses.

lens, tear lens, cornea, crystalline lens). The crystalline lens data from the Gullstrand eye were kept constant. The data of the cornea, tear lens, and contact lens were defined by the respective model conditions. By thick-lens calculations and paraxial ray tracing (WinLens 4.1), the posterior radius of the spectacle glass was iteratively varied to match the specific vitreous length of the model eye (Table 2). By this process, the refraction with contact lens was determined, making it possible to calculate the corneal power according to the classic formula as well as to the proposed algorithms.

Results

Model Calculations For each model eye, the CLO measurement procedure using the 10 model contact lenses was mathematically simulated in terms of a thick-lens system comprising 8 (centered) spherical surfaces; that is, 5 lenses (spectacle glass, contact

Corneal Refractive Powers and Keratometer Indices Keratometry is performed to determine corneal power. Keratometers or ophthalmometers measure the

Table 2. Axial length, vitreous length, anterior corneal radius, and refraction in the study’s model eyes. The emmetropic eye is equivalent to Gullstrand eye 2. Myopic eyes were derived by increasing the original vitreous length by 2.0 mm (myopic 5) and 4.0 mm (myopic 10).

Model Eye

Axial Length (mm)

Vitreous Length (mm)

Anterior Corneal Radius (mm)

Emmetropic

24.149

16.944

7.7000

0.00

Myopic 5, pre-LASIK

26.149

18.944

7.7000

⫺5.21

Myopic 10, pre-LASIK

28.149

20.944

7.7000

⫺10.25

Myopic 5, post-LASIK

26.149

18.944

8.5603

0.00

Myopic 10, post-LASIK

28.149

20.944

9.4697

0.00

J CATARACT REFRACT SURG—VOL 29, JULY 2003

Refraction (D)

1399

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

Table 3. Construction details of plano model contact lenses of PMMA (nL ⫽ 1.490; dL ⫽ 0.5 mm). R1L (mm)

R2L [mm]

P2L (D)

7.5644

7.4000

45.41

7.9644

7.8000

43.08

8.3644

8.2000

40.98

8.7644

8.6000

39.07

9.1644

9.0000

37.33

9.5644

9.4000

35.74

9.9644

9.8000

34.29

10.3644

10.2000

32.94

10.7644

10.6000

31.70

11.1644

11.0000

30.55

Note: Equivalent and vertex powers ⫽ 0 for all model lenses. R1L ⫽ anterior radius; R2L ⫽ posterior radius; P2L ⫽ base curve of contact lens (L) (336/R2L)

anterior corneal radius of curvature by evaluating the size of the first Purkinje reflex for a given luminant object in the instrument. The translation from the measured curvature radius R1C (in millimeters) into an optical power, PC in diopters, is done by instrumentspecific keratometer formulas such as PC ⫽

331.5 R 1C

(2)

PC ⫽

337.5 R 1C

(3)

or

Equation 2 is used, for example, in Zeiss instruments (rounded to 332/R1C) and equation 3, in Javal-Schio¨ tz type keratometers.13 The different instrument calibrations result in a system-specific inherent difference of ⬇ 0.7 D in corneal power if a radius of curvature of 7.7 mm is measured with a Zeiss (PC ⫽ 43.12 D) or a Javal (PC ⫽ 43.83 D) instrument. Both formulas are based on the surface power Ds (according to equation A.1 in the Appendix) Ds ⫽

n⬘ ⫺ 1 R

of a spherical interface with radius R, separating air (refractive index ⫽ 1) from a medium with refractive index n⬘ given by 1.3315 (Zeiss) or 1.3375 (Javal). 1400

The Javal value of 1.3375, which is also used in all videokeratoscopes,14 is comparable to the refractive indices of tear film, aqueous, and vitreous (1.336). According to Helmholtz, a respective value of 1.336 should be applied to convert corneal radii into power.14 His measurements yielded 1.337. Javal also used this value14 and later arbitrarily changed it to 1.3375, only to produce a corneal power of 45.00 D from a radius of curvature of 7.5 mm. Apart from generating the pair 7.50 mm/45.00 D, the refractive index of 1.3375 has another important property: It gives—as discussed below—the vertex power of the cornea under certain conditions. The “true” refractive index of the cornea is 1.376, again dating back to Helmholtz and Gullstrand.14 The keratometer index 1.3315 may be easily derived15 from Gullstrand’s theoretical eye 2,16 as shown in the Appendix. A value of 1.3315 (ie, 1.332) was also obtained by Hartinger13 and Littmann.17 Olsen,18 by plausibility arguments, also adopted this value. Similarly, Gobbi and coauthors7 derived a value of 1.3315. A crucial assumption in the derivation of the Zeiss index is that a given cornea has the same ratio of its radii as the Gullstrand eye (R1C/R2C ⫽ 7.7/6.8). Only then is it possible to determine the equivalent corneal power using equation 2. The Javal index of 1.3375 in equation 3 can be deduced by calculating the vertex power of the Gullstrand eye, as demonstrated in the Appendix. Again, for a given cornea, it must be guaranteed that its radii follow the Gullstrand ratio. Only then will the Javal index (ie, the use of equation 3) give the corneal back vertex power. This was also noted by Norrby19 and Bennett and Rabbetts.20 Contact Lens Method Measurement Situation Figure 1 shows a schematic of the measurement situation when applying the CLO method. On the left, no contact lens is present. The actual refraction is measured; the spectacle glass S0 is positioned in the working (vertex) distance d in front of the eye. A tear lens T0 covers the anterior surface of the cornea C. On the right, a rigid contact lens L has been placed on the tear lens, which has changed its shape to become T1. The combination of contact lens L and tear lens T1 has altered the refractive power of the total system (contact lens, tear lens, and eye). To compensate for this change, the spec-

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

tacle glass must be changed accordingly. With the new glass S1 in the working distance d in front of the eye, a sharp image can be viewed again. To assess the effects of the different lenses’ powers, the spectacle lenses may be replaced by theoretical thin lenses positioned directly in front of the other lenses present. To allow for the change in position, these theoretical lenses receive their powers from the actual spectacle lenses S0 and S1 by vertex correction. With the abbreviations in the Appendix, the vertex-corrected equivalent powers PS0⬘ and PS1⬘ of glasses S0 and S1 are given by PS 0⬘ ⫽

PS 0 PS 1 and PS 1⬘ ⫽ 1 ⫺ d · PS 0 1 ⫺ d · PS 1

Figure 1. (Haigis) Schematic of the corneal lens without (left) and with (right) tear lens (notation according to Table 1).

(4)

in which PS0 and PS1 are the equivalent powers d and is vertex distance (0.012 m). With these corrections, the optical system at the bottom left of Figure 2 has the same total power as the optical system at the bottom right. Because the crystalline lens (which has not yet been mentioned) and the cornea are unaltered by the application of the CLO method, we will neglect them. Thus, the optical power of the system S0⬘–T0 is identical to that of the combination S1⬘–L–T1. Hence, treating all lenses as thin lenses with negligible separation, we may write PS 0⬘ ⫹ PT 0 ⫽ PS 1⬘ ⫹ PL ⫹ PT 1

(5)

It can easily be shown3,7 that the refractive effect of the tear lens T0 is negligible; that is, PT0 ⫽ 0, so that we obtain PT 1 ⫽ PS 0⬘ ⫺ PS 1⬘ ⫺ PL

(6)

Knowing the values for PL and for PS0⬘ and PS1⬘, we need to determine the power PT1 of the tear lens. T1 is affected by both the contact lens L anteriorly (by R2L) and the cornea C posteriorly (by R1C). Assuming, for ease of calculation, an infinitely thin air layer separating the different lenses, we may calculate the power PT1 of the tear lens T1 according to the thinlens formula (equation A.5 in Appendix): n T ⫺ n 1 n 1 ⫺ n T 336 336 ⫹ ⫽ ⫺ PT 1 ⫽ R 2L R 1C R 2L R 1C

Figure 2. (Haigis) Schematic of lenses in front of the eye including cornea when applying the contact lens method for the determination of corneal power (S0, S1 ⫽ spectacle glass; T0, T1 ⫽ tear lens; L ⫽ rigid contact lens; C ⫽ cornea; PS0, PT0, PC, PS1, PL, and PT1 ⫽ effective lens powers). Left: Without contact lens. Right: With contact lens. Top: Spectacle glass in working distance (d). Bottom: Spectacle glass corrected for vertex distance. Finite spaces between lenses are drawn for illustration purposes; dimensions are not to scale.

where we have used the refractive indices of air n1 ⫽ 1000 mm/m and water nT ⫽ 1336 mm/m. The first term on the right side of equation 7 is identified as the base curve power P2L of the contact lens L⬘; that is, 336 R 2L so that we obtain from equation 6 P 2L ⫽

(7)

J CATARACT REFRACT SURG—VOL 29, JULY 2003

(8)

1401

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

336 ⫽ PS 1⬘ ⫺ PS 0⬘ ⫹ PL ⫹ P 2L R 1C

(9)

The right side of equation 9 is the measurement result PCLO from applying the CLO method.With the substitution P CLO ⫽ PS 1⬘ ⫺ PS 0⬘ ⫹ PL ⫹ P 2L

(10)

we finally have from equation 9

PC e ⫽

376 ⫺ 5.78 R 1C

(11)

376 · 共PS 1⬘ ⫺ PS 0⬘ ⫹ P L ⫹ P 2L兲 ⫺ 5.78 336

PC e ⫽ 1.119 · P CLO ⫺ 5.78

Normally, equation 9 comes in the form of equation 1 K ⫽ Rx 1⬘ ⫺ Rx 0⬘ ⫹ P ⫹ P 2 Equations 1 and 9 are not identical for several reasons. Equation 9 contains the term PS1⬘ – PS0⬘, which is a difference of equivalent (vertex-corrected) lens (spectacle-glass) powers, whereas in equation 1, the term Rx1⬘ – Rx0⬘ is included, which is a difference of (vertex-corrected) refractions. Yet, refraction is a back vertex power by its nature. However, because we have assumed thin lenses during the derivation of equation 9, there is no distinction between the back vertex and equivalent powers as both are equal for a thin lens. In addition, in typical real lens glasses between –10.00 D and ⫹5.00 D, possible differences between the 2 lens powers are on the order of 0.10 D (Appendix) and thus irrelevant in clinical practice. Therefore, the right sides of equations 1 and 9 may be taken as identical. Yet the left sides of equations 1 and 9 are different. This is very important as it clearly shows that the commonly used interpretation of the CLO method is faulty Neither the corneal power nor the equivalent nor the vertex power is measured by the contact lens method; instead, the quantity 336/R1C is measured. To compute the equivalent or the back vertex power of the cornea, we must not rely on fictitious refractive indices but use the “true” ones and make allowance for the corneal back radius R2C. If R2C were known, and thus the posterior surface power P2C, the equivalent corneal power PCe can be calculated from equation A.11 in the Appendix, with R1C measured by the CLO method according to equa-

(12)

Insertion of equation 9 into equation 12 gives the final result for the equivalent corneal power PC e ⫽

336 ⫽ P CLO R 1C

1402

tion 9. If not, we may assume that P2C of a given eye is equal to that of the Gullstrand eye (–5.88 D) (Table 1). Hence, we have from equation A.11 in the Appendix

(13)

For the back vertex power, we start from equation A.16 in the Appendix. If P2C were known, we would obtain with equation 13 PC v ⫽ 1.119 · P CLO ⫹ 0.000455 · P CLO2 ⫹ P 2C (14) Assuming, as above, that the back surface power (P2C ⫽ –5.88 D) is given by the Gullstrand eye, we arrive at PC v ⫽ 1.119 · P CLO ⫹ 0.000455 · P CLO2 ⫺ 5.88 (15) or PC v ⫽ PC e ⫹ 0.000455 · P CLO2 ⫺ 0.1

(16)

Model Calculations Table 4 shows typical paraxial system parameters for the model eyes examined. The data relate to model myopic 5 before and after LASIK and to the CLO simulation on this model eye with 2 different contact lenses. All system parameters except for the spectacle glass were given by the respective eye model. The condition to be fulfilled was that the distance between the posterior lens vertex and the paraxial focus equals the vitreous length (18.944 mm in Table 4) of the respective model eye. This could be achieved by varying the spectacle glass radius R2 until the requested distance was reached (exact to 3 decimals). As a result of this procedure, the spectacle glass parameters were obtained. Refraction was calculated from these parameters as back vertex power using equation A.4 in the Appendix. Thick-lens optics and paraxial ray tracing produced identical numerical results.

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

Table 4. Paraxial system parameters for myopic model eye 5 derived from the Gullstrand eye by increasing the original axial length by 2.0 mm to 26.149 mm (ie, vitreous length ⫽ 18.944 mm). Pre-LASIK corneal powers: equivalent, PCe ⫽ 43.05 D; back vertex, PCv ⫽ 43.83 D. Post-LASIK powers: 38.14D and 38.75 D, respectively. Data in normal type are unchanged. Those in bold type are different for different eye models. Spectacle Glass

Air

Contact Lens

Tear Lens

Cornea

Aqueous

Lens

Vitreous

R1 (mm)

100.0000

—

—

7.7000

7.7000

—

10.000

—

R2 (mm)

48.8730

—

—

7.7000

6.8000

—

⫺6.000

—

n

1.500

1.000

1.490

1.336

1.376

1.336

1.413

1.336

d (mm)

1.00

12.00

0.50

0.01

0.50

3.10

3.60

18.944

—

—

8.5603

8.5603

—

10.000

—

⫺6.000

Parameter Pre-LASIK without model contact lens*

Post-LASIK without model contact lens† R1 (mm)

99.9999

R2 (mm)

99.9999

—

—

8.5603

6.8000

—

n

1.500

1.000

1.490

1.336

1.376

1.336

1.413

1.336

d (mm)

1.00

0.50

0.01

0.50

3.10

3.60

18.944

12.00

—

Pre-LASIK with model contact lens‡ R1 (mm)

100.0000

—

8.7644

8.6000

8.5603

—

10.000

—

R2 (mm)

103.2000

—

8.6000

8.5603

6.8000

—

⫺6.000

—

n

1.500

1.000

1.490

1.336

1.376

1.336

1.413

1.336

d (mm)

1.00

0.50

0.01

0.50

3.10

3.60

18.944

—

7.5644

7.4000

8.5603

—

10.000

—

⫺6.000

12.00

Post-LASIK with model contact lens§ R1 (mm)

100.0000

R2 (mm)

43.9750

—

7.4000

8.5603

6.8000

—

n

1.500

1.000

1.490

1.336

1.376

1.336

1.413

1.336

D (mm)

1.00

0.50

0.01

0.50

3.10

3.60

18.944

12.00

—

*Refraction RX0 ⫽ ⫺5.21 D; vertex-corrected refraction RX01 ⫽ ⫺4.91 D † Refraction Rx0 ⫽ 0.00 D ‡ P2L ⫽ 39.07 D; refraction RX1 ⫽ ⫹0.17 D; vertex-corrected refraction RX1⬘ ⫽ ⫹0.17 D; CLO 3 corneal power PCLO ⫽ ⫹0.17 D – 0.00 D ⫹ 0.00 D ⫹ 39.07 D ⫽ 39.24 D. § P2L ⫽ 45.41 D; refraction RX1 ⫽ ⫺6.35 D; vertex-corrected refraction RX1⬘ ⫽ 5.90 D; CLO 3 corneal power PCLO ⫽ ⫺5.90 D ⫺ 0.00 D ⫹ 0.00 D ⫹ 45.41 D ⫽ 39.51 D.

The first 2 sets of measurements in Table 4 list the surfaces of the myopic eye model before and after LASIK yet without a contact lens. The last 2 sets of measurements in Table 4 give the postoperative results with 2 different trial contact lenses (base curves 39.07 D and 45.41 D). Only the contact lens with base curve 39.07 D produces a refraction close to zero, whereas the other one causes a myopic shift. Using equation 10, the following corneal powers are obtained from this simulated CLO measurements:

PCLO ⫽ 39.24 D (base curve 39.07 D) and PCLO ⫽ 39.51 D (base curve 45.41 D). The results of the CLO simulation on all model eyes with the best trial lenses are shown in Table 5. The most suitable lenses were those that caused the smallest changes in refraction with and without contact lenses. The table also shows the true values for equivalent (PCe) and vertex (PCv) powers of the cornea (following directly from the model definitions). These data must be compared to the right side of Table 5, which shows the

J CATARACT REFRACT SURG—VOL 29, JULY 2003

1403

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

Table 5. Simulated application of the CLO method on model eyes to derive corneal power. Data from Model

Eye Model

Rx (D) Rx (D) w/o CL with CL

P2L (D)

Corneal Power (D) According to . . .

PCe PCv True (D) True (D) Rxⴕ (D) Rxⴕ (D) CLO Zeiss Javal Present 1 Present 2 Eq. A.3 Eq. A.4 w/o CL with CL Eq. 10 Eq. 2 Eq. 3 Eq. 13 Eq. 15

0.00

⫹0.53

43.08

43.05

43.83

0.00

⫹0.53

43.61

43.02 43.80

43.02

43.80

Myopic 5

⫺5.21

⫺4.40

43.08

43.05

43.83

⫺4.91

⫺4.18

43.81

43.22 44.00

43.24

44.02

Myopic 10

⫺10.25

⫺9.14

43.08

43.05

43.83

⫺9.12

⫺8.24

43.96

43.37 44.16

43.41

44.20

Myopic 5

0.00

⫹0.17

39.07

38.14

38.75

0.00

⫹0.17

39.24

38.72 39.42

38.13

38.75

Myopic 10

0.00

⫺0.26

35.74

33.91

34.40

0.00

⫺0.26

35.48

35.01 35.65

33.94

34.43

Emmetropic Pre-LASIK

Post-LASIK

Note: Numbers in bold (normal and italic) mark best agreements between model values and results from CLO simulation. w/o ⫽ without; CL ⫽ contact lens; Eq. ⫽ equation; CLO ⫽ contact lens method according according to equation 10; Zeiss ⫽ R1c calculated with equation 9, then inserted into equation 2; Javal ⫽ R1c calculated with equation 9, then inserted into equation 3; Present 1 ⫽ present calculation according to equation 13; Present 2 ⫽ present calculation according to equation 15

results of different processing methods of the CLO simulation: The CLO column gives the classic results (equation 10), and the Zeiss and Javal columns were obtained by calculating R1C from equation 9 with the respective calibrations in equations 2 and 3. On the far right side of Table 5, the results with the newly derived equations 13 and 15 are shown. Whereas Table 5 relates to the model CLO measurements with the closest contact lenses, Figures 3 and 4 show the results for all contact lenses applied to our emmetropic (Figure 3) and myopic 10 (Figure 4) model eyes. The upper parts of the figures show the differences between the true equivalent power and the calculated power according to the different calculation schemes. The lower parts show the differences relative to the true back vertex power of the cornea.

Discussion Corneal Refractive Powers and Keratometer Indices Corneal powers and keratometer indices have been treated by several authors3– 6,14,18,21–24 for normal eyes as well as for eyes after corneal refractive procedures. Apart from the axial formula in standard keratometry, on which our treatment is based, 2 more formulas to derive corneal power are in use: the instantaneous23 and the refractive.24 Hugger et al.23 compared all 3 formulas to relate corneal power changes with refraction after photorefractive keratectomy (PRK). The correlation 1404

was good except for the instantaneous formula. It was, however, not possible to assess the observed refractive changes on the basis of a keratometric index of 1.3375; the authors had to apply 1.408 instead. This value is different from 1.376, which is recommended for example by Mandell14 and Holladay and Waring.23 In a recent publication, Hamed and coauthors21 deduced a mean refractive index of 1.3300 for post-LASIK eyes. This value, however, decreased with increasing refractive correction. Thus, it may not seem possible to use a single refractive index for eyes that have had PRK or LASIK.21 It is beyond the scope of this study to discuss “effective” refractive indices derived to adjust post-LASIK results. On the contrary, it is my intention to clarify the origin of such indices in an attempt to better understand what is actually being measured in keratometry. Therefore, within the present context of analyzing the CLO method, it is important to be aware that for eyes with corneal radii following the Gullstrand ratio R1C/R2C ⫽ 7.7/6.8, the equivalent power is given by the Zeiss calibration 331.5/R1C and the back vertex power by the Javal calibration 337.5/R1C. If, for a given eye, this ratio is different, as in all eyes after refractive surgery, then neither the use of the Zeiss calibration nor the Javal calibration will lead to physically meaningful corneal powers. The K-readings based on these indices do not represent any refractive power—neither the equivalent nor the vertex— of the

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

Figure 3. (Haigis) Simulation determination of corneal power of an emmetropic model eye using CLO with different contact lenses (open circle ⫽ true anterior corneal radius; horizontal axis: ⫽ back radii R2L of different contact lenses; vertical axis ⫽ difference between true equivalent power (top) or true back vertex power (bottom) and calculated power according to 5 different methods (1, present calculation for equivalent power [equation 13]; 2, classic CLO according to equation 10; 3, Zeiss calibration R1C from equation 9, then inserted into equation 2; 4, Javal calibration R1C from equation 9, then inserted into equation 3; 5, present calculation for back vertex power; that is, equation 15).

cornea. This is essential for the application of the CLO method and is discussed below. However, even in the simple case of normal corneas, it is important to clarify which kind of corneal power enters the calculation algorithms for IOLs. Olsen18 and I25 favor the index 1.3315 and recommend using this value for corneal power determination in IOL calculations. Thijssen26 suggests in this context to subtract 1.00 D from the power measured with a Javal keratometer; that is, a calculation according to PC ⫽

337.5 ⫺1 R 1C

Figure 4. (Haigis) Simulated determination of corneal power of a formerly myopic (–10.25 D) model eye (myopic 10) after simulated LASIK using CLO with different contact lenses. (Legend is the same as in Figure 3.)

(PC in diopters, R1C in millimeters). His approach is based on the theoretical eye of LeGrand and El Hage.16 The correction term of –1.00 D is equivalent to the power of the corneal meniscus in a watery medium with refractive index 1.3374, separated anteriorly by a tear film (with an assumed thickness of 10 ␮m) and posteriorly by aqueous (Figure 1, right). For the tear lens, it is assumed that the anterior curvatures differ by no more than 10 ␮m from the anterior corneal radius. Yet the net effect of the tear film on the equivalent power of the cornea is negligible, affecting only the 4th decimal.3,7 In fact, all developers of formulas give strict rules within their IOL calculation frameworks as to how corneal power must be calculated. In general, the SRK II formula27 assumes K-values from a Javal-type keratometer with a system-specific index of 1.3375. In the SRK/T formula,28 the same authors used a keratometer index of 1.333. Holladay22 and Holladay et al.,29 like Binkhorst,30 advocate an index of 4/3 for the translation

J CATARACT REFRACT SURG—VOL 29, JULY 2003

1405

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

of measured radii into corneal power. For their IOL calculation formulas, Holladay and Hoffer31 took it a priori for granted that the K-reading was based on a Javal-type calibration (ie, n⬘ ⫽ 1.3375). Holladay, to have the recommended index of 4/3 in effect, gives a formula for the translation of keratometric power Kk (given in diopters on the basis of the Javal calibration) into optical power Ko22: K o ⫽ 0.987654 · K k Therefore, if a given IOL formula is based on Javal-type keratometric readings, the back vertex power of the cornea must be determined and entered into the formula for a patient after corneal refractive surgery. Vice versa, IOL algorithms with the Zeiss index for the corneal power call for the equivalent power. Hence, it is important to know what the CLO method actually measures. Contact Lens Method Measurement Situation Our analysis has shown that by the contact lens method, the term 336/R1C is determined, which in general is different from the equivalent or the vertex power of the cornea. Yet it is close to 337.5/R1C, which is the corneal vertex power for eyes with corneal radii following the Gullstrand ratio. Therefore, for normal eyes, K-readings do not differ significantly from the corneal power results from CLO. This explains the good results of Zeh and Koch32 in their comparison of keratometry and the CLO method in normal eyes. If we apply equation 11 to their data to derive the anterior corneal radii R1C, calculate the according vertex powers by equation 2, and compare them to their averaged Ks (Kave), our results are not statistically different from theirs. In general, especially in eyes after corneal refractive surgery with a ratio of radii that does not match the Gullstrand ratio anymore, I recommend deriving the anterior corneal radius R1C from the CLO result in equation 11 and to compute the equivalent or vertex corneal power from the paraxial equations A.2 and A.4 in the Appendix. For this purpose, the true posterior corneal radius R2C is necessary. Problems arising in this context are discussed below. Model Calculations The model calculations confirm the validity of the above analysis. As shown in Table 5 for the emmetropic 1406

eye model with radius R1C calculated from equation 9, the true equivalent power of the cornea is given by the Zeiss calibration (equation 2) and equation 13 and the true vertex power, by the Javal calibration (equation 3) and equation 15. The classic CLO result from equation 10 coincides with neither power although it is—as expected and already mentioned— closest (⬇ 0.2 D smaller) to the Javal value. The emmetropic eye model and models myopic 5 pre-LASIK and myopic 10 pre-LASIK match the Gullstrand corneal curvature ratio (7.7/6.8 ⫽ 1.132). Thus, the Zeiss (equation 2) and formula 13 for the equivalent power (equation 13) as well as the Javal (equation 3) and equation 15 for the back vertex power yield similar results, which, however, are slightly higher (by ⬇ 0.2 D for model myopic 5 and ⬇ 0.3 D for model myopic 10) than the respective true values. This is because the corneal radius R1C as derived from the CLO method using equation 9 is slightly smaller than the true one, leading to power values that are too high. In turn, the classic application of the CLO method interpretation of equation 9 as the corneal power, with its effective index of 1.336, just compensates for the difference in radius. This explains the observation in Table 5 that the true vertex powers of the pre-LASIK myopic 5 and myopic 10 eyes are best matched by the classic CLO method. Because these model eyes are pre-LASIK, it would not be necessary in practice to apply the CLO method; classic keratometry would yield the correct equivalent (via equation 2) or vertex (via equation 3) powers. The same argument holds for equations 13 and 15, which make use of CLO results. These equations are not meant for pre-LASIK eyes and therefore cannot be expected to give good results. The bottom section of Table 5 covers the postLASIK situation. Contact lens overrefraction produces power values too high for both model eyes. For the model myopic 5, the CLO value is ⬇ 1.1 D higher than the true equivalent power and ⬇ 0.5 D higher than the true vertex power of the cornea. For the model myopic 10, the differences are even more pronounced: The respective results are ⬃ 1.6 D for the equivalent power and ⬃ 1.1 D for the vertex power. The close match of the results of the Zeiss calibration (equation 2) with the true vertex power for model myopic 5 is coincidental. On the other hand, the newly proposed formulas, equation 13 and equation 15, yield results that virtually

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

match the true values. Yet, this perfect agreement should not be overinterpreted as both formulas are based on the Gullstrand eye and compared to model eyes that are derived from the same theoretical eye. In particular, the posterior curvatures of our model eyes are identical to that of the Gullstrand eye, which might well be different in real eyes. Corresponding findings as to the performance of the respective formulas can be obtained from Figures 3 and 4, which illustrate the simulated use of trial contact lenses with different base curves on emmetropic and post-LASIK myopic eyes. It has been assumed in the derivation of the equations for the contact lens method that the vertex distances of all lenses gathered at the cornea were negligible so that the use of the thin-lens approximation was justified. This implies that the center thicknesses of all the lenses involved are small. Best results are to be expected when the refractions involved are close to zero and when both the contact lens and tear lens are afocal so that there is no refraction difference with and without a contact lens. For thin lenses, this is the case when the respective anterior and posterior radii are identical. In real plano lenses, however, the anterior and posterior curvatures are not exactly identical because of the mixed term in the Gullstrand formula (equation A.2, Appendix). This why the anterior and posterior radii of our model contact lenses differ by 0.1644 mm (Table 3). For the emmetropic eye, Figure 3 confirms that the results of the Zeiss calibration (equation 2) and equation 13 coincide just as those of the Javal calibration (equation 3) and equation 15. The error ⌬PCe between the true equivalent corneal power and the power values determined with the formulas given becomes zero when the posterior radius of the trial contact lens coincides with the anterior corneal radius. The same holds for the error ⌬PCv with respect to the vertex power. At this point, the error between the CLO value and the equivalent (vertex) power is ⬇ – 0.6 D (⬇ 0.2 D). It again becomes clear that although the Zeiss calibration (equation 2) describes the equivalent and the Javal calibration (equation 3) the vertex power of the cornea, the classic CLO result in normal eyes is a close match to the vertex power (ie, to the usual Javal-based K-values). From Figures 3 and 4, we can conclude for eyes that have had LASIK or PRK with initial refractions between

0 and –10.0 D, there is a constant difference of approximately 0.2 D between the classic CLO value and the power value calculated with the Javal calibration from the CLO result in equation 10. The Zeiss calibration is a better estimate for the equivalent power and the vertex power than the Javal calibration. The CLO value and CLO-derived powers both overestimate the equivalent and vertex powers. For eyes with an initial refraction of –10.0 D, the CLO error is approximately 1.6 D with respect to the true equivalent and approximately 1.0 D relative to the true vertex power of the cornea. All CLO and CLO-derived results are closer to the corneal vertex power than to the equivalent power. Role of Posterior Corneal Curvature In our derivation of the proposed equations 13 and 15, we made the assumption that the corneal back surface power is given by the Gullstrand value (–5.88 D, corresponding to R2C ⫽ 6.8 mm), although there are reports7,33–38 that in real eyes the posterior corneal radius may be as low as 5.80 mm (corresponding to – 6.9 D). Similarly, the theoretical eye of LeGrand and El Hage16 includes a corneal back surface power of – 6.11 D. In addition, although it is commonly accepted that the posterior corneal radius changes in incisional keratectomy,5,7 there are indications that this may39,40 or may not 5,33,41 happen in PRK and LASIK. I chose the Gullstrand eye because it is a widely used reference for model calculations and theoretical considerations. Our principal analysis of the CLO method is not affected by the above additional changes in posterior corneal curvature. These changes can be easily addressed by modifying accordingly equation A.10 or A.16 in the Appendix. In practice, R2C can be measured by slit-scanning topography37,39,40 with the Orbscan (Bausch & Lomb) or determined by a combination of keratometry and pachymetry34 or keratoscopy and pachymetry.33

Conclusion In dealing with corneal power in IOL calculation or in measurement procedures for power determination, the equivalent powers and (back) vertex powers must be clearly distinguished. Whereas for normal eyes, the keratometer index of 1331.5 is linked to the equivalent and

J CATARACT REFRACT SURG—VOL 29, JULY 2003

1407

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

1.3375 to the vertex power, this is no longer true in eyes after corneal refractive surgery. Therefore, K-readings based on these indices do not provide meaningful corneal refractive powers in these eyes. Similarly, the quantity measured when applying the CLO method is neither the equivalent nor the vertex power of the cornea but rather the expression 336/R1C (R1C, the anterior corneal radius). Treating this term without further differentiation as “corneal power” is not justified and will, in eyes that have had refractive surgery for myopia of –5.0 to –10.0 D, lead to an overestimation of approximately 1.0 to 1.5 D relative to the true equivalent power. With respect to the true vertex power, the resulting overestimation is approximately 0.5 to 1.0 D. Tentative formulas based on the Gullstrand eye to overcome these drawbacks are shown in equations 13 and 15.42– 45

References 1. Wenzel M, Reuscher A, Aral H. Derzeitiger Stand der Katarakt- und refraktiven Chirurgie. Ophthalmo-Chirurgie 2001; 13:213–218 2. Leaming DV. Practice styles and preferences of ASCRS members—2000 survey. J Cataract Refract Surg 2001; 27:948 –955 3. Haigis W. Keratometrie und IOL-Berechnung. In: Duncker G, Ohrloff C, Wilhelm F, eds, 12. Kongress der Deutschsprachigen Gesellschaft fu¨ r IntraokularlinsenImplantation und refraktive Chirurgie, 1998. Berlin, Springer, 1999; 461–470 4. Seitz B, Langenbucher A, Nguyen NX, et al. Underestimation of intraocular lens power for cataract surgery after myopic photorefractive keratectomy. Ophthalmology 1999; 106:693–702 5. Seitz B, Langenbucher A. Intraocular lens calculations status after corneal refractive surgery. Curr Opin Ophthalmol 2000; 11:35–46 6. Gimbel HV, Sun R, Furlong MT, et al. Accuracy and predictability of intraocular lens power calculation after photorefractive keratectomy. J Cataract Refract Surg 2000; 26:1147–1151 7. Gobbi PG, Carones F, Brancato R. Keratometric index, videokeratography, and refractive surgery. J Cataract Refract Surg 1998; 24:202–211 8. Hoffer KJ. Intraocular lens power calculation for eyes after refractive keratotomy. J Refract Surg 1995; 11:490 – 493 9. Holladay JT. Consultations in refractive surgery. Refract Corneal Surg 1989; 5:203 1408

10. Ladas JG, Boxer Wachler BS, Hunkeler JD, Durrie DS. Intraocular lens power calculations using corneal topography after photorefractive keratectomy. Am J Ophthalmol 2001; 132:254 –255 11. Bennett AG, Francis JL. Visual optics. In: Davson H, ed, The Eye, Vol. 4: Visual Optics and the Optical Space Sense. New York, NY, London, Academic Press, 1962; 3–7 12. Haigis W. Strahldurchrechnung in Gau␤scher Optik zur Beschreibung des Linsensystems Brille-KontaktlinseHornhaut-Augenlinse (IOL). In: Schott K, Jacobi KW, Freyler H, eds, 4. Kongress der Deutschsprachigen Gesellschaft fu¨ r Intraokularlinsen Implantation, Essen 1990. Berlin, Springer, 1991; 233–246 13. Zeiss C. Handbuch fu¨ r Augenoptik. Oberkochen, Carl Zeiss, 1993 14. Mandell RB. Corneal power correction factor for photorefractive keratectomy. J Refract Corneal Surg 1994; 10:125–128 15. Haigis W. Biometrie. In: Kampik A, ed, Jahrbuch der Augenheilkunde 1995: Optik und Refraktion. Zu¨ lpich, Biermann, 1995; 123–140 16. LeGrand Y, El Hage SG. Physiological Optics. New York, NY, Springer, 1980 17. Siebeck R. Optik des menschlichen Auges; Theorie und Praxis der Refraktionbest-immung. Berlin, Springer, 1960 18. Olsen T. On the calculation of power from curvature of the cornea. Br J Ophthalmol 1986; 70:152–154 19. Norrby NES. Unfortunate discrepancies [letter]. J Cataract Refract Surg 1998; 24:433 20. Bennett AG, Rabbetts RB. Clinical Visual Optics, 2nd ed. London, Butterworths, 1989; 468 21. Hamed AM, Wang L, Misra M, Koch DD. A comparative analysis of five methods of determining corneal refractive power in eyes that have undergone myopic laser in situ keratomileusis. Ophthalmology 2002; 109:651– 658 22. Holladay JT. Standardizing constants for ultrasonic biometry, keratometry, and intraocular lens power calculations. J Cataract Refract Surg 1997; 23:1356 –1370 23. Hugger P, Kohnen T, La Rosa FA, et al. Comparison of changes in manifest refraction and corneal power after photorefractive keratectomy. Am J Ophthalmol 2000; 129:68 –75 24. Mandell RB. The enigma of the corneal contour. CLAO J 1992; 18:267–273 25. Haigis W. Einflu␤der Optikform auf die individuelle Anpassung von Linsenkonstanten zur IOL-Berechnung. In: Rochels R, Duncker GW, Hartmann C, eds, 9. Kongress der Deutschsprachigen Gesellschaft fu¨ r Intraokularlinsen Implantation, Kiel 1995. Berlin, Springer, 1996; 183–189

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

26. Thijssen JM. The emmetropic and the iseikonic implant lens: computer calculation of the refractive power and its accuracy. Ophthalmologica 1975; 171:467–486 27. Retzlaff J, Sanders DR, Kraff M. Lens Implant Power Calculation; a Manual for Ophthalmologists & Biometrists, 3rd ed. Thorofare, NJ, Slack, 1990 28. Retzlaff JA, Sanders DR, Kraff MC. Development of the SRK/T intraocular lens implant power calculation formula. J Cataract Refract Surg 1990; 16:333–340; correction, 528 29. Holladay JT, Prager TC, Chandler TY, et al. A three-part system for refining intraocular lens power calculations. J Cataract Refract Surg 1988; 14:17–24 30. Binkhorst RD. The optical design of intraocular lens implants. Ophthalmic Surg 1975; 6:17–31 31. Hoffer KJ. The Hoffer Q formula: a comparison of theoretic and regression formulas. J Cataract Refract Surg 1993; 19:700 –712 32. Zeh WG, Koch DD. Comparison of contact lens overrefraction and standard keratometry for measuring corneal curvature in eyes with lenticular opacity. J Cataract Refract Surg 1999; 25:898 –903 33. Patel S, Marshall J, Fitzke FW. Shape and radius of posterior corneal surface. Refract Corneal Surg 1993; 9:173– 181 34. Camellin M. Proposed formula for the dioptric power evaluation of the posterior corneal surface. Refract Corneal Surg 1990; 6:261–264 35. Lowe RF, Clark BAJ. Posterior corneal curvature; correlations in normal eyes and in eyes involved with primary angle-closure glaucoma. Br J Ophthalmol 1973; 57: 464 –470 36. Royston JM, Dunne MCM, Barnes DA. Measurement of the posterior corneal radius using slit lamp and Purkinje image techniques. Ophthalmic Physiol Opt 1990; 10:385–388 37. Seitz B, Langenbucher A, Hofmann B, et al. Refractive power of the human posterior corneal surface in vivo in relation to gender and age [abstract]. Ophthalmologe 1998; 95(suppl 1):50 38. Dunne MCM, Royston JM, Barnes DA. Normal variations of the posterior corneal surface. Acta Ophthalmol 1992; 70:255–261 39. Naroo SA, Charman WN. Changes in posterior corneal curvature after photorefractive keratectomy. J Cataract Refract Surg 2000; 26:872–878 40. Kamiya K, Oshika T, Amano S, et al. Influence of excimer laser photorefractive keratectomy on the posterior corneal surface. J Cataract Refract Surg 2000; 26:867– 871 41. Herna´ndez-Quintela E, Samapunphong S, Khan BF, et al. Posterior corneal surface changes after refractive surgery. Ophthalmology 2001; 108:1415–1422

Appendix Notation of Physical Quantities in Text d ⫽ vertex distance (m) spectacle glass – cornea ⫽ 0.012 dT ⫽ center thickness of tear lens dC ⫽ center thickness of cornea De ⫽ equivalent power of a lens Dv ⫽ back vertex power of a lens Ds ⫽ surface power of a lens D ⫽ general refractive power of a lens K ⫽ corneal power (K-reading) Kk ⫽ keratometric power of the cornea Ko ⫽ optical power of the cornea n1 ⫽ refractive index of air (1.000) n2 ⫽ refractive index of aqueous (1.336) nC ⫽ refractive index of cornea (1.376) n⬘ ⫽ fictitious refractive index of cornea nT ⫽ refractive index of tear lens (1.336) PS0 ⫽ equivalent power (D) of spectacle glass without contact lens PS1 ⫽ equivalent power (D) of spectacle glass with contact lens PS0⬘ ⫽ equivalent power (D) of spectacle glass without contact lens, vertex corrected PS1⬘ ⫽ equivalent power (D) of spectacle glass with contact lens, vertex corrected PT0 ⫽ equivalent power (D) of tear lens without contact lens PT1 ⫽ equivalent power (D) of tear lens with contact lens PL ⫽ equivalent power (D) of contact lens PC ⫽ general refractive power (D) of cornea PCe ⫽ equivalent power (D) of cornea PCv ⫽ (back) vertex power (D) of cornea PCLO ⫽ corneal power obtained from contact lens overrefraction P1C ⫽ surface power (D) of anterior corneal surface P2C ⫽ surface power (D) of posterior corneal surface P2L ⫽ base curve power (D) of contact lens (336/R2L) P ⫽ general contact lens power P2 ⫽ contact lens base curve power R ⫽ general radius of curvature of a lens R1C ⫽ anterior radius (mm) of cornea R2C ⫽ posterior radius (mm) of cornea RT ⫽ radius of curvature of tear lens R2L ⫽ posterior radius (mm) of contact lens Rx0 ⫽ refraction (D) without contact lens Rx1 ⫽ refraction (D) with contact lens Rx0⬘ ⫽ refraction (D) without contact lens, vertex corrected Rx1⬘ ⫽ refraction (D) with contact lens, vertex corrected

J CATARACT REFRACT SURG—VOL 29, JULY 2003

1409

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

Equivalent Power and Vertex Power Lens powers may be expressed in terms of equivalent (total) powers or vertex powers; surfaces are characterized by their surface powers.1 For a spherical lens with radii R1 and R2, center thickness d, index of refraction n, separating 2 media with refractive indices n1 and n2 (the respective notations for the corneal lens in Figure 1, left [in text] are R1C, R2C, dC, nC), the surface powers D1 and D2 are calculated from the radii D1 ⫽

n ⫺ n1 n2 ⫺ n and D 2 ⫽ R1 R2

(A.1)

The lens total or equivalent power De is given by the Gullstrand formula De ⫽ D1 ⫹ D2 ⫺

d 䡠 D1 䡠 D2 n

(A.2)

Expressed in terms of R1 and R2, De ⫽

n ⫺ n1 n2 ⫺ n d n ⫺ n1 n2 ⫺ n ⫹ ⫺ 䡠 䡠 R1 R2 n R1 R2

De D1 ⫽ D2 ⫹ d d 1 ⫺ 䡠 D1 1 ⫺ 䡠 D1 n n

(A.4)

If the center thickness d is smaller than the lens diameter and radii of curvature, the thick-lens formula A.3 turns into the thin-lens formula n ⫺ n1 n2 ⫺ n ⫹ D ⫽ D1 ⫹ D2 ⫽ R1 R2

(A.5)

In this trivial case, equivalent and vertex powers are identical, as can be seen from equation A.4. To make allowance for that, the subscripts e and v have been omitted in the power descriptor D on the left side of equation A.5. The thin-lens approximation can usually be applied for minus glasses and not, in general, for plus glasses. In these cases, vertex powers are greater than effective powers. Typical plus spectacle glasses (50 mm diameter, n ⫽ 1.525) differ in equivalent and vertex powers by about 0.1 D for a nominal (vertex) power of 5.0 D and about 0.5 D for nominally 10.0 D (Rodenstock Corp., personal communication, 1989).

Equivalent Power of Cornea and Keratometer Index 1.3315 The keratometer index 1.3315 may be derived2 from Gullstrand’s theoretical eye 2,3 treating the cornea (schemat1410

PC e ⫽ P 1C ⫹ P 2C ⫺ ⫹

dC nC ⫺ n1 䡠 P 1C 䡠 P 2C ⫽ nC R 1C

n2 ⫺ nC dC nC ⫺ n1 n2 ⫺ nC ⫺ 䡠 䡠 R 2C nC R 1C R 2C

(A.6)

Keratometers restricted determining the anterior radius R1C. For the equivalent power PCe to be calculated correctly according to equation A.6, knowledge of the posterior radius and the center thickness is necessary. Without these data, it is not possible to compute corneal power with a formula such as

(A.3)

the Gullstrand formula is known as the lens-makers’ formula. The (back) vertex power Dv is defined by Dv ⫽

ically depicted in Figure 1 in the text) as a simple spherical lens with radii R1C and R2C, center thickness dC, index of refraction nC, separating air (refractive index n1) from aqueous (refractive index n2), its equivalent power PCe is, according to equation A.2,

PC ⫽

n⬘ ⫺ n 1 R 1C

(A.7)

where n⬘ is a fictitious index of refraction. If, however, the assumption is made that a given eye behaves like the theoretical Gullstrand eye,1 the following equation for n⬘ can be derived from equations A.6 and A.7: n⬘ ⫽ n C ⫹ 共n 2 ⫺ n C兲 䡠

冋

R 1C d C n C ⫺ n 1 ⫺ 䡠 R 2C n C R 2C

册

(A.8)

With the data from the Gullstrand eye in Table 1 (in the text), it follows directly from equation A.8 that n⬘ ⫽ 1.3315. Inserting this result into equation A.7, one obtains (for the radius R1C entered in millimeters) PC e ⫽

331.5 R 1C

(A.9)

The dominant term in equation A.8 is the Gullstrand ratio R1C/R2C (the second term in the right pair of brackets is but 1.8% of R1C/R2C). Only if a given cornea has the same ratio of its radii is it possible to measure its equivalent power according to equation A.9. If not, equation A.9 will lead to erroneous results. If, by one way or another, the posterior corneal radius R2C and thus the posterior surface power [P2C ⫽ (1336 – 1376)/R2C] were known, the true equivalent power can be calculated. With radii (in millimeters) and refractive powers (in diopters), the equivalent corneal power is (with equations A.1 and A.2 and data from Table 1 in the text)

J CATARACT REFRACT SURG—VOL 29, JULY 2003

LABORATORY SCIENCE: CONTACT LENS OVERREFRACTION TO DETERMINE CORNEAL POWER

PC e ⫽

·

376 共1376 ⫺ 1000兲 䡠 共1336 ⫺ 1376兲 ⫹ P 2C ⫺ R 1C 1376 dC R 1C · R 2C

(A.10)

The mixed term in equation A.10 is 0.10 D for the Gullstrand eye and does not change much even for different radii. Hence, we have from equation A.10 376 PC e ⫽ ⫹ P 2C ⫹ 0.1 (A.11) R 1C

Back Vertex Power of Cornea and Keratometer Index 1.3375 To derive the keratometer index 1.3375, we start from the back vertex power PCv of the cornea as defined by equation A.4. With equation A.1 we have PC v ⫽

PC e PC e ⫽ dC dC nC ⫺ n1 1⫺ 䡠 P 1C 1 ⫺ 䡠 nC nC R 1C

(A.12)

(A.13)

The equivalent power PCe is given—for eyes with the Gullstrand ratio R1C/R2C ⫽ 7.7/6.8 — by equation A.9. Therefore PC v ⫽ 1.01806 䡠

331.5 337.5 ⫽ R 1C R 1C

PC v ⫽ P 2C ⫹

P 1C dC 1⫺ 䡠 P 1C nc

(A.15)

Note that the second term in the denominator of equation A.15 is small compared to 1, irrespective of R1C. Therefore, we may carry out a series expansion [(1 – x) – 1 ⬇ 1 ⫹ x],4 yielding PC v ⫽ P 2C ⫹

376 䡠 R 1C

冉

1⫹

0.13663 R 1C

冊

(A.16)

where the respective data of the Gullstrand eye (Table 1 in text) have been inserted.

References

Inserting the values for the Gullstrand eye (Table 1 in text) into the denominator, equation A.12 becomes PC v ⫽ 1.01806 䡠 PC e

it possible to measure its vertex power according to equation A.14. If not, equation A.14 will lead to erroneous results. For the known posterior corneal radius R2C and surface power P2C, equation A.12 can be rewritten (with equation A.4) to give

(A.14)

Only if a given cornea has the same ratio of its radii is

1. Bennett AG, Francis JL. Visual optics. In: Davson H, ed, The Eye, Vol. 4: Visual Optics and the Optical Space Sense. New York, NY, London, Academic Press, 1962; 3–7 2. Haigis W. Biometrie. In: Kampik A, ed, Jahrbuch der Augenheilkunde 1995: Optik und Refraktion. Zu¨ lpich, Biermann, 1995; 123–140 3. LeGrand Y, El Hage SG. Physiological Optics. New York, NY, Springer, 1980 4. Bronstein IN, Semendjajew KA. Taschenbuch der Mathematik. Zu¨ rich, Frankfurt, MH Deutsch, 1969

J CATARACT REFRACT SURG—VOL 29, JULY 2003

1411