Improving estimation of corneal refractive power by measuring the posterior curvature of the cornea

Ahmet Eryildirim, M.D., Tugrul Ozkan, Sureyya Eryildirim, M.D., Suleyman Kaynak, M.D., Guray Cingil, M.D.

ABSTRACT We developed a formula to measure the posterior curvature of the cornea. In our study, real corneal power in 51 eyes was calculated by using anterior and posterior curvatures and corneal thickness. The corneal refractive index was found to be a variable value between 1.3243 and 1.3339 (mean 1.3304 ± 0.0003).

Key Words: anterior curvature, corneal power, corneal thickness, posterior cur­vature, refractive index

Many parameters must be used to calculate intraoc­ular lens (IOL) power. Some can be measured for each individual but others are constant. Corneal power, an important parameter in calculating IOL power, is mea­sured by keratometry using anterior curvature of the cornea and a constant refractive index. Some kerato­meters use different refractive indexes, so the results can vary.

In conventional measurement of corneal power, the cornea has a single refractive surface. It is calculated as follows:

P = (N2- Nl)/R

where P = power of cornea, N 1 = refractive index of air, N2 = refractive index of cornea, R = radius of anterior curvature of cornea.

This refractive index (N2) does not represent the real corneal index. It is an arbitrary value found as a function of corneal and aqueous humor refractive indexes. We call it the anterior curvature dependent refractive index (nacd).

In fact, the cornea has two surfaces. It is easy to mea­sure the anterior surface curvature by keratometry, but there is no system that measures posterior curvature directly. We developed a formula to calculate the pos­terior curvature, thus ascertaining the relationship be­tween anterior curvature with posterior curvature and nacd and the effect of the posterior corneal curvature on corneal power (Figure 1 ).

MATERIALS AND METHODS

In this study, 51 eyes were measured for real corneal power. Anterior curvatures were measured using the

R anterior curvature posterior curvature central thickness 1 ,zon thickness

Yb

01 = center of anterior curvature 02 = center of postenor curvature

y

Xb

1.75 mm = Yb

X

' anterior corneal surface

posterior corneal surface

Fig. 1. (Eryildirim) Development of the posterior curvature calculation formula.

Haag-Strait and Sbisa 4186/85 Javal keratometers. Mean anterior curvature was calculated by averaging horizontal and vertical values. The Biophysic Paxiel sys­tem was used for pachymetry. Corneal thickness was measured at the center and at 1. 7 5 mm from the center. The peripheral cornea was measured on four meridians (0, 90, 180, 270), and the mean value calculated by averaging these measurements. For conventional calcu­lation, 1.3375 was used as the refractive index of the cornea.

The posterior corneal curvatures were measured us­ing the following formula (see Appendix for complete formula):

From the Dokuz Eylul University Medical Faculty Eye Department, Balcovajlzmir, Turkey.

Reprint requests to Ahmet Eryildirim, M.D., Cagdas cad. 2/22, 35330, Balcovajlzmir, Turkey.

J CATARACT REFRACT SURG-VOL 20, MARCH 1994 129

r=~~~~~(R~--b~)~x~(l~.7_5~/R~)~~---­Sin(ATAN((SQR(R 1\ 2- 1.75 1\ 2) x (R- b)/

R- R + a)/((R- b) X 1.75/R)) X - 2)

where a = central thickness of cornea, b = peripheral thickness ( 1. 7 5 rnm from center of cornea), R = anterior curvature, r = posterior curvature.

Calculated values were plotted according to anterior curvature.

Real corneal power was calculated using the thick lens formula 1:

p = N2 - Nl + N3 - N2 _ d x N3 R r R X N2 r x N3

N2 - Nl x N3 - N2

where P =total power, R =anterior curvature, r = pos­terior curvature, N1 = refractive index of air (1), N2 = refractive index of cornea ( 1.376), N3 = refractive index of aqueous ( 1.336).

Anterior curvature dependent refractive indexes (nacd) were calculated by multiplying the real corneal power and anterior curvature for each individual. These values were plotted according to anterior curvature. Re­lationships between anterior curvature with posterior curvature and refractive index (nacd) were evaluated by using a least square method.

RESULTS

Table 1 shows the measured and calculated values. There is a linear relationship between anterior curvature and posterior curvature (Figure 2):

r = -6.19553 + 1.658615 x R (r = 0.85)

Calculated individualized refractive indexes varied between 1.3339 and 1.3243. The relationship between anterior curvature and refractive index was linear (Figure 3):

nacd = 1.290197 + 0.005154 X R (r = 0.57)

The effect of the corneal thickness was ignored.

DISCUSSION

While measuring anterior curvature, keratometers use the Purkinje-1 image. This image is the reflection of the light from the cornea's anterior surface.2•3 Another image, Purkinje-11, is the reflection from the surface between cornea and aqueous humor (posterior surface of the cornea). Using the Purkinje-11 image, it is possible to measure the posterior curvature of the cornea, but in practice, observation of this image is extremely difficult. 2

We used ultrasonic pachymetry to calculate the pos­terior curvature of the cornea. Although we were com­puting corneal power, we used a trigonometric approach to calculate posterior curvature because the 3.50 mm part of the central cornea is like a sphere. The periphery zone was measured 1. 7 5 mm from the center of the cornea. Since the pachymeter probe has a diameter of 2.00 mm, we overlapped the areas under probe in the

8.5

!!! 8.8 :I - 7.5 ca > ... :I u 7.8 ... .-.. o E ·a: E 6.5 s._. Ill ... oO 6.8 c. a..

5.5 7 7.4 7.8 8.2 8.6 9

Anterior curvature R (mm)

Fig. 2. (Eryildirim) Relationship between anterior and pos­terior curvatures: r = -6.19553 + 1.658615 x R (r = 0.85).

1.348 Cl) 1.338 >

1.336 Q);:: ... u :I ca 1.334 -... ~Gi 1.332 ~ !:~ 1.338 u c " ... G) r=l'G

1.328 0 'tl ._. ·;: c >C

1.326 Cl) 8. Cl)

'ECP"g 1.324 ca 'tl ·-1.322 1.328

7.2 7.6 8 8.4 8.8

Anterior curvature R (mm)

Fig. 3. (Eryildirim) Relationship between anterior curva­ture (R) and refractive index (nacct): nacct = 1.290197 + 0.005154 X R {r = 0.57).

Table 1. Measured and calculated values.

Mini-Value mum

Anterior curvature 7.44 (rnm)

Posterior curvature 5.81 (rnm)

Central thickness 434 (~rn)

Peripheral thickness 445 (1.75 mm) (~m)

Corneal power, 38.79 conventional (D)

Corneal power, 38.37 calculated (D)

Refractive index 1.3243 (nacd)

* Standard error

Maxi-mum Main

8.70 7.80

8.10 6.73

596 523

604 539

45.36 43.34

44.24 42.42

1.3339 1.3304

SE*

0.04

0.07

4.83

4.59

0.20

0.17

0.0003

130 J CATARACT REFRACT SURG-VOL 20, MARCH 1994

center and periphery while positioning it. Peripheral probe positioning errors were minimized by averaging four measurements on four different axes.

In IOL power calculation formulas, different corneal refractive indexes are used. All of these values are con­stant. Early theoretical formulas used 1.336. Binkhorst used 4/3 (1.33 ... ) to compensate for a postoperative change in corneal curvature.4 Olsen used Gullstrand's mean anterior and posterior curvature values5 and by dividing them, found a ratio. He also found a refractive index of 1.3315 to calculate corneal power.6 In the SRK/T formula, 1.33 ... is used as the corneal refractive index.4

We found that the nacd is not a constant value. It varies depending on anterior curvature changes. This variable index value will provide more precise corneal power calculations.

REFERENCES

l. Jenkins FA, White HE. Geometrical optics. In: Jenkins FA, ed, Fundamentals of Optics. Singapore, McGraw­Hill, 1981; 1-212

2. Moses RA. Accommodation. In: Moses RA, Hart WM eds, Adler's Physiology of the Eye. St Louis, CV Mosby Co, 1981; 291-310

3. Mohrman R. The keratometer. In: Duane TD, ed, Clinical Ophthalmology. Philadelphia, Harper & Row, 1986; (1)60: 1-12

4. Retzlaff JA, Sanders DR, K.ratTMC. Development of the SRK/T intraocular lens implant power calculation for­mula. J Cataract Refract Surg 1990; 16:333-340

5. Katz M. The human eye as an optical system. In: Duane

TD, ed, Clinical Ophthalmology. Philadelphia, Harper & Row, 1986; ( 1 )33: 1-52

6. Olsen T. On the calculation of power from curvature of the cornea. Br J Ophthalmology 1986; 70:152-154

APPENDIX The Posterior Curvature Calculation Formula

Sin 0 = Yb/R Cos 0 = Xb/R n+r+a=R XB' = (R - b) x Cos 0 YB' = (R - b) X Sin 0 (R - b) x Cos 0 - n = r x Cos 0 1

(R - b) X Sin 0 = r X Sin 0 1

from 1 and 2:

(R - b) X Cos 0 - R + r + a = r x Cos 0 1 = (R- b) x Cos 0- R +a= r x (Cos 0 1 - 1) =

since as a trigonometric conversion Cos A = Cos2 A/2 - Sin2 A/2

r x (Cos2 0 1/2- Sin2 01/2- 1) =

since as a trigonometric conversion Cos2 A + Sin2 A = 1 and Cos2 A - 1 = -Sin2 A

(1)

(2) (3)

(R- b) X Cos 0- R +a= r X- 2 Sin2 01/2 (4)

division 4 by 3: .

(R- b) X Cos 0- R +a= r X -2 Sin2 01/2 = (R - b) X Sin 0 r X Sin 0 1

since as a trigonometric conversion, Sin A = 2 Sin A/2 X Cos A/2

r x -2 Sin2 01/2 r x 2 Sin 0 1/2 x Cos 0 1/2

(R-b)XCosO-R+a=- 0'/2 (R - b) x Sin 0 tg

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