ACTA 0 P H TH A L M 0 L O G I C A 72 (1994) 715-720

Posterior corneal curvature and its influence on corneal dioptric power

Carsten Edmund

Department of Ophthalmology, Rigshospitalet, University of Copenhagen, Denmark

Abstract. An algorithm for estimation of the posterior corneal curvature is presented and applied on data from normal and keratoconic eyes. Radius of central posterior corneal curvature are demonstrated to be (mean f SD)

6.71 f 0.23 mm and 5.58 f 0.78 mm in normals and kera- toconic eyes, respectively. This corresponds to a ratio be- tween posterior and anterior corneal curvature at 0.85 and 0.83 in the groups mentioned. Both these ratios are significantly smaller than the corresponding ratio at 0.88 in Gullstrand’s schematic eye which on corneal dioptric power results in offset errors at 0.20D and 0.46D in nor- mal and keratoconic eyes. It is further demonstrated that the ratio is not constant over the corneal surface, result- ing in central peripheral dioptric offset errors between 0.2D and -0.31D in normals and between 0.46D and -0.38D in keratoconic eyes. On corneal dioptric power it is finally shown that a variation in the refractive index of aqueous humor has a 30 times larger influence than a similar variation in corneal refractive index.

Key words: posterior corneal curvature - corneal thickness profile - photokeratoscopy - keratometry - corneal diop- tric power - keratoconus - intraocular lens implant.

From an optical point of view cornea makes up a thick concavoconvex lens separating the surround- ing air and aqueous humor. The dioptric corneal power therefore depends on radius of anterior and posterior curvature, corneal thickness and the in- dices of refraction of air, cornea and aqueous humor. By keratometry and topographic keratos- copy, radius of the anterior corneal curvature is measured in various locations and converted to corneal dioptic power, assuming the other dioptric components mentioned bo be constant.

The purpose of the present paper is to evaluate the influence on corneal dioptric power of varia- tion in the assumed constant dioptric elements, with special attention to biological variation in radius of posterior corneal curvature.

Methods

Determination of posterior corneal curvature The anterior corneal surface may be measured by photokeratoscopy and described as a part of conic section (circle, ellipse, parabola, hyperbola) (Edmund (Sjontoft 1985). The corneal thickness profile may be measured by topographic pacho- metry (Edmund 1987) and described as

T(x,) = T X (1 +TV X x:) (1)

where T(x,) is the corneal thickness in the chord distance x from the visual axis, T is the central cor- neal thickness (x, = 0) and TV is the thickness vari- ation coefficient. The foot sign (a) denotes anterior surface.

In Fig. 1 the anterior corneal surface may be de- scribed as

where R is radius of the central corneal curvature and E = 1 - e2, where e2 is the squared eccentricity of the conic section.

By subtracting the corneal thickness profile from the anterior surface, the posterior surface in

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Fig. 1. Corneal sections (hyperbola, parabola, ellipse, circle) as a descriptive model of anterior and posterior corneal sur- faces in the horizontal meridian. Z and X axis denotes sa- gittal depth and chord distance, respectively. T(x) indi- cate corneal thickness in chord distance x

Fig. 1, may be approximately described as a conic section with the equation

Z: = R;/E; - l/Ep X xP2 (3)

where the foot sign (,) denotes posterior surface. For x, = 0, (z, - zp) = T and equation 1 to 3 gives

Rp= E, x (R,/E, - T) Ep= xp2/((R,/E,)2 - zb2) (5)

(4)

From Fig. 1 it appears

Tan@) = - dz,/dx, = x,/(z, x E,) zb= Z, - T(x,) x C O S ( ~ )

xb= X, - T(x,) x sin(@)

(6) (7) (8)

For x,= 3 mm (average chord distance), z, is determined from equation (1). Thus, for known values of R,, E,, T and Tv - p, z b and xb may be es- timated from equation (6) to (8) and employing these values the parameters of the posterior cor- neal surface R, and E, (= 1 - eb2) may be deter- mined from equation (4) and (5).

Using a conic section as a descriptive model of the corneal surface, the approximate radius of cor- neal curvature may be expressed as

where R(x) is radius of corneal curvature in chord distance x from the visual axis, R is radius of cen- tral corneal curvature (x = 0) and RV is the radius variation coefficient defined as

RV = 1.5 x e2/R2 (10)

where e2 is the squared eccentricity of the actual conic section (Edmund & Sj~ntoft 1985).

Thus, from equation (1) to (10) it appears that the back corneal curvature radius is a function of the parameters R,, RV,, T and TV.

Determination of corneal dioptric power The dioptric corneal power in the chord distance x from the visual axis (Fap(x)) is given by

Fa,($ = Fa(3 + Fp(4 - d(x) X Fa(4 X Fp(4 (11)

where F,(x) = (n, - l)/Ra(x), F,(x) = (n, - nc)/Rp(x), d(x) = T(x)/n,

where F,(x), Fp(x), R,(x), d(x) and T(x) are dioptric powers of the anterior and posterior corneal sur- face, radius of the anterior and posterior corneal curvature, reduced corneal thickness and corneal thickness in chord distance (x) from the visual axis, respectively. nc and n, are the refractive indices of cornea and aqueous humor, respectively.

By numeric differentiation the partial derivates 6F,,/6Ra, 6Fa,/6Rb, 6FaP/6T, 6FaP/6n, and 6FaP/6n, may be estimated for normal values of R,, R,, T, n, and n,. In keratometers radius of anterior corneal cur-

vature is converted to dioptric corneal power using a constant converting factor. Setting nc = 1.376, na= 1.336, T=0.5 mm and Rp=kxR, where k = Rp/R, = 6.8/7.7 = 0.88 (mean values in Gull- strand's schematic eye (Olsen (1986) has demon- strated the optimal factor to be 331.5 so as

Fcurnea = F,,(x) = 331.5/Ra (12)

with R, measured in mm.

Analysis of variance The influence of variance in R,, R,, T, nc and na on the corneal dioptric power Fab(x) may be analyzed empoying the following general formula (Armit- age 1977)

i = k

i = l var(y) = c (6y/6pi)2 x var(p,) + . . .

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Normal (n = 37)

Mean SD SE

where y is a function of k variables and p i and pk denotes the i'th and k'th variable, respectively. It may be mentioned that the variance (var) is the square root of the standard deviation (SD). Equa- tion (13) assumes the covariances to be zero.

Further employing equation (13) SD(R,), SD(RV,), SE(R,) and SE(RV,) may be estimated from the partial derivates of R,, RV,, T and TV with respect to R, and RV, and the SD's and SE's mentioned in Table 1.

Keratoconus (n = 27)

Mean SD SE

Materials

To analyze the influence of variations in the par- ameters R,, RV,, T and TV on F,,(x) in normal as well as in abnormal corneas a previously published material of normal and kertaoconic corneas was employed (Edmund 1987). The material is shown in Table 1.

The SD expresses the biological variation of the descriptive parameters in the two groups. The SE expresses the uncertainty by which the parameters are estimated (Edmund 1989).

RP - Mean SD SE

Results

Table 2 demonstrates the estimated values of radius of the central posterior corneal curvature

RV,

Mean SD SE

(R,) and the posterior corneal curvature radius variation coefficient (RV,) in normal and kerato- conic eyes. As expected in normals and more pro- nounced in keratoconic eyes, R, are smaller and RV, are larger than the corresponding parameters of the anterior corneal surface (Table 1). This is be- cause the corneal thickness increases towards pe- riphery most pronounced in keratoconus. Further the uncertainty (SE) of the descriptive radius par- ameters is larger for the posterior curvature (Table 2) compared with the anterior curvature (Table 1). This is because the uncertainty of as well the ante- rior curvature radius parameters as the thickness is incorporated in the posterior curvature radius parameters.

Table 3 demonstrates the posterior corneal cur- vature influence on the dioptric power in normal and keratoconic eyes cdmpared with Gullstrands's schematic eye. The ratio (R,/R,) between the pos- terior and anterior corneal curvature at 0.85 and 0.83 in normal and keratoconic eyes, respectively, is statistically significantly different from the corresponding ratio in Gullstrand's schematic eye at 0.88 (t-test, p<O.OOl). In normal and keratoconic eyes this difference caused a dioptric difference of 0.20 and 0.46, respectively.

Fig. 2 illustrates the dioptric influence of the ac- tual central-peripheral variation in R,/R, in nor-

Table 2. Radius parameters of posterior corneal curvature (R,(x) = R, X (1 + RV, X x2)) in normal and keratoconic eyes.

46 Acta Ophthal. 72.6 717

R ~ ’ R a

Mean 1 SD

Gullstrand 0.88 - 33 1.5 - Normal 0.85 0.019 330.0 0.20 Keratoconus 0.83 0.035 328.4 0.46

Converting Difference factor dioptre

mal and keratoconic eyes compared with the con- stant ratio at 0.88 in Gullstrand’s schematic eye. It appears that the dioptric error is between 0.20 and - 0.31 in normal eyes and between 0.46 and - 0.38 in eyes with keratoconus. The figure demonstrates that the converting factor between radius of curva- ture in mm and the dioptric curvature power is not constant over the surface. Further, the figure shows that an abnormal corneal shape (keratoconus) may involve a larger central-peripheral ratio variation compared with a normal corneal shape. Finally, the figure indicates that the converting factor at 331.5 is approximately the averaged central-pe- ripheral converting factor in as well normal and keratoconus (error(D) = 0 for chord distance about 2.5 mm).

In normal and keratoconic eyes Table 4 shows the influence on the precision of corneal dioptric power (SD(F)) of variations in radius of anterior (SD(R,)) and posterior (SD(R,)) central corneal curvature and in the central corneal thickness (SD(T). The values of SD(R,) are the measurement precisions of R, (= SE(R,), Table 1) in normal and keratoconic eyes, respectively. The standard devia- tions (SD(R,), SDO) indicate the biologic varia- tions in the respective parameters in normal and keratoconic eyes.

% = ((6F/6p) X SD(P))~ x 100/(SD(F)2). From Table 4 it appears that the variations men-

tioned above influence the precision of central cor- neal dioptric power with 0.26D and 1.26D in nor- mal and keratoconic eyes, respectively. The table further demonstrates that the main error source in the corneal dioptric power precision is the biologic variations of R, in as well normal (58.9%) as kerato-

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Normal

6F/6p SD (p) %

Error (D)

__I--

Chord dlsIance I mm

Keratoconus

6F/6p SD (p) Yo

Fig. 2. Central peripheral dioptric error in normal and kerato- conic eyes when uing a constant (Gullstrand eye 331.5) factor converting radius of anterior corneal curvature to corneal dioptric power.

conic (60.3%) eyes. However, in normal eyes the bi- ologic variation in R, has only negligible (0.20D) influence on the central corneal dioptre precision. The larger influence of R, on the corneal dioptre precision in keratoconic eyes (0.98D) i s the result of an increased biologic variation SD(R,) in kerato- conic eyes compared with normal.

Table 5 demonstrates the influence of hypo- thetic biological variations in the refraction in- dices of cornea (n,) and aqueous humor (n,) on the precision of central corneal dioptric power (SD(F)) in normal and keratoconic eyes. Hypothetic varia- tions are employed because the real biologic varia- tions in the refraction indices are not available. It appears that a variation of about 0.37% in na and nc influences the central corneal dioptre precisions with 0.73D and 0.88D in normal and keratoconic eyes, respectively. This means that real, rather

Table 4. The influence of variation in anterior (R,) and posterior (Rb) central corneal curvature K and RV and in central corneal thickness (T) on the precision on corneal dioptric power (SD(F)) in normal and keratoconic eyes.

Table 5. The influence of hypothetic variations in the refraction indices of cornea (n,) and aqueous humor (n,) on the pre- cision on corneal dioptric power (SD(F)) in normal and kera toconus.

I Normal I Keratoconus

"c -4.7 0.005? 0.1 -6.9 0.005? 0.1 "a 146.0 0.005? 99.9 175.3 0.005? 99.9 SD (F) 0.73 D 100 0.88 D 100

small biologic variations in the refractive indices may be the main sources of error in the central cor- neaI dioptre precision. Further, the table demon- strates that a variation in na has a much more signi- ficant influence on corneal dioptric power than a similar variation in nc. Finally, the table shows that variation in n, has about the same influence on corneal dioptric power in normal and abnormal (keratoconic) eyes.

Discussion

Employing photographs of corneal sections in the vertical meridian Lowe & Clark (1973) determined radius of the central posterior corneal curvature in normal eyes (n = 92). They determined radius of anterior and posterior curvature to 7.65 mm (k 0.27 mm) and 6.46 ( f 0.26 mm) with SD in parenthesis. This corresponds to a ratio between posterior and anterior corneal curvature radius at 0.84. Based on photography of Purkinje images in three fixed meridians and measurement of the central corneal thickness Dunne et al. (1992) esti- mated in as well the vertical as the horizontal meri- dian radius of central corneal curvatures in young normal eyes (n=60). They estimated radius of anterior and posterior curvature in the vertical meridian to 7.96 mm (f 0.23 mm) and 6.45 (f 0.31 mm) and a ratio at 0.81. The corresponding mean results in the horizontal meridian were estimated to 8.15 mm and 6.82 mm with a ratio at 0.84, which is very close to the presently demonstrated ratio at 0.85 (Table 3), also for the horizontal meridian. Thus, compared to ratio in Gullstrand's schematic eye at 0.88 the significantly smaller ratio at 0.85 ac- tually demonstrated seems to be a more realistic value.

The reliability of corneal dioptric power estima- tion based on measurement of radius of anterior corneal curvature may include discussions of accu- racy and precision. Accuracy means agreement with the true value often denoted as the offset error. Precision means agreement between re- peated determinations on the same subject and is expressed as a standard deviation (SD) and often denoted as the standard error (SE).

In common clinical use of keratometers preci- sion is of most interest, but with e.g. intraocular lens (IOL) calculation or in evaluation of variations in corneal surface as demonstrated in various cor- neal mapping system, also of interest is the accu- racy of corneal dioptric power.

Many keratometers employ a factor at 337.5 (equation 12) at converting radius of anterior cor- neal curvature to dioptric corneal power. This re- sults in an accuracy (offset) error at 0.89D com- pared with the more correct factor at 330.0 esti- mated in the present study. In the SRK formula this error may be corrected by a suitable A-con- stant, but in general the manufacturers of IOL did not denote the suitable converting factor for a given A-constant. Thus, empirically as well as the- oretically, IOL calculation formulas should be based on radius of anterior corneal curvature rather than corneal dioptric power.

In the various corneal mapping systems the cen- tral-peripheral corneal curvature may be presented in terms of dioptric corneal power. In normals, applying a constant converting factor at 331.5 results in a variable offset error between +0.2D and -0.3D (Fig. 2). The offset error is as well larger as more variable in abnormal corneas (e.g. keratoconus). Thus, corneal mapping is more reliable when presented in terms of radius of ante- rior corneal curvature than in terms of corneal dioptric power where the maximum solution does not exceed 0.5D.

The measurement precision of R, in normal eyes denoted in Table 4 at 0.03 mm corresponding to 0.16D is determined by photokeratoscopic examinations. The measurement precision of e.g. an autokeratometer is in our department deter- mined to 0.25D (SE). Thus in the normal clinical determination of corneal dioptric power the kera- tometric measurement precision seems to be a more significant factor than the biologic variation in posterior corneal curvature and corneal thick- ness at least in normal eyes.

719 46 *

This statement implies constancy of corneal and aqueous humor refractive indices. It may be a little surprising to notice that a variation in na has about 30 times larger influence on corneal dioptric power than a similar variation in nc. The reason is that an increase inn, influences the dioptric power of as well the anterior as the posterior corneal sur- face in opposite directions resulting in a small de- crease in total corneal dioptric power. Conversely an increase in n, only influences the dioptric power of the posterior corneal surface resulting in a larger increase in the total corneal dioptric power. The refractive index of aqueous humor may be influenced by its content of especially large molecules. Thus, pathological conditions which in- crease blood vessel permeability of large molecules (inflammations, diabetes) may significantly in- fluence the corneal dioptric power and thus the re- fractive state of the eye.

Edmund C (1987): Determination of the corneal thick- ness profile by optical pachometry. Acta Ophthalmol (Copenh) 65: 147-152.

Edmund C (1987): Assessment of an elastic model in the pathogenesis of keratoconus. Acta Ophthalmol (Co- penh) 65: 545-550.

Edmund C (1989): Corneal topography and elasticity in normal and keratoconic eyes. A methodological study concerning the pathogenesis of keratoconus. Acta Ophthalmol (Copenh) 67 (Supplement 193).

Edmund C & Sjentoft E (1985): The central-peripheral radius of the normal corneal curvature. A photokera- toscopic study. Acta Ophthalmol (Copenh) 63:

Lowe R F & Clark B A J (1973): Posterior corneal curva- ture. Correlations in normal eyes and in eyes involved with primary angle-closure glaucoma. Br J Ophthal- mol57: 464-470.

Olsen T (1986): On the calculation of power from curva- ture of the cornea. Br J Ophthalmol70 152-154.

670-677.

References

Armitage P (1977): Statistical Methods in Medical Re- search, pp 96-98. Blackwell Scientific Publication, Ox- ford.

Dunne M C M, Royston J M & Barnes D A (1992): Normal variations of the posterior corneal surface. Acta Oph- thalmol (Copenh) 70: 255-261.

Received on January 6th, 1994.

Corresponding author: Carsten Edmund Rigshospitalet, Aviation Medicine Eye Clinic 20, Tagensvej DK-2200 Copenhagen N Danmark.

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