B. WANG ET AL.

Copyright © 2013 SciRes. ENG

ter clockwise direction around the

-axis. Thus, the

plane can be coincided with the corneal meridian

section in the new coordinate system (

). The equa-

tions of the corneal meridian section in the new coordi-

nate system (

).are as follows:

222

12 0

0

2(1 )

x

yaz azrzQz

=

= +=−+

(7)

where

are the coordinates of the new coordinate

system (

).

Then by substituting

into

given in the

formula (6), we obtain the following coordinate rotation

equations of our corneal model:

sin cos

sin cos

xx y

yy x

θθ

θθ

= −

= +

(8)

We substitute the

given in Equation (7) into the

Equation (8). The equations of the corneal meridian sec-

tion on the original coordinate system (XOY) are as fol-

lows

22

0

sincos 0

( sincos)2(1)

xy

yxrz Qz

θθ

θθ

−=

+= −+

(9)

Finally, we transform the Equation (9) into the fol-

lowing f orm:

2

0

2

0

2(1)cos

2(1)sin

x rzQz

y rzQz

θ

θ

= −+

= −+

(10)

5.3. Generation of a 3D Corneal Model

360 semi-meridians are all chosen. Every point has an

coordinate. The

is a parameter and the

values of a semi-meridian are selected from 0 mm to 3.5

mm at 0.1 mm intervals. The

coordinate values of

every point are calculated by substituting the corres-

ponding

value into the Equation (11), 3D corneal sur-

face plot is generated with the Visual C++ 6.0 program-

ming [16]. Figure 5 shows a colorized 3D surface plo t of

anterior corneal surface from two different perspectives

for the same subject as in Figure 2. Variation of color

shows semi-meridional variation of the Q-value with

0.02 color steps. From the top to bottom of color scale,

the Q-value becomes more negative gradually. Figure 2

shows that the Q-value of each semi-meridian is negative

value (−1 < Q < 0) corresponding to the most common

corneal shape (prolate ellipse) ([17]). Thus, the 3D sur-

face plot of anterior corneal surface approximates a pro-

late ellipsoid shown in Figu re 5.

6. Conclusion

In contrast to the sagittal radius of curvature (rs), the

Figure 5. 3D surface plot of anterior corneal surface for the

same subject as in Figure 2.

tangential radius of curvature (rt) is a true radius of cur-

vature which can better represent corneal shape and local

curvature changes especially in the periphery.

In this paper, we proposed a nonlinear equation of

corneal asphericity (Q) using the tangential radius of

curvature (rt) on every semi-meridian. The 360 semi-

meridional variation of the Q-value was well fitted using

the 7th degree polynomial function for all subjects. We

constructed a new 3D corneal model and present a more

realistic model of shape of the anterior corneal surface.

Our mathematical model could be helpful in the contact

lens design and detection of corneal shape abnormalities,

such as kerat oconus or previous la s er surger y.

7. Acknowledgements

This study was supported by grant No. 30872816 from

the National Natural Scientific Found a tion of China.

REFERENCES

[1] K. Scholz, A. Messner, T. Eppig, H. Bruenner and A.

Langenbucher, “Topography-Based Assessment of Ante-

rior Corneal Curvature and Asphericity as a Function of

Age, Sex, and Refractive Status,” Journal of Cataract &

Refractive Surgery, Vol. 35, 2009, pp. 1046-1054.

http://dx.doi.org/10.1016/j.jcrs.2009.01.019

[2] M. Guillon, D. P. M. Lydon and C. Wilson, “Corneal

Topography: A Clinical Model,” Ophthalmic and Physi-

ological Optics, Vol. 6, 1986, pp. 47-56.

http://dx.doi.org/10.1111/j.1475-1313.1986.tb00699.x

[3] A. G. Bennett, “Aspherical and Continuous Curve Con-

tact Lenses,” Optometry Today, Vol. 28 ,1988, pp. 11-14,

140-142, 238-242, 433-444.

[4] T. Y. Baker, “Ray Tracing through Non-Spherical Sur-

faces,” Proceedings of the Physical Society, Vol. 55, 1943,

pp. 361-364.

http://dx.doi.org/10.1088/0959-5309/55/5/302

[5] S. W. Cheung, P. Cho and W. A. Douthwaite, “Corneal

Shape of Hong Kong-Chinese,” Ophthalmic and Physi-

ological Optics, Vol. 2, 2000, pp. 119-125.

http://dx.doi.org/10.1016/S0275-5408(99)00045-9

[6] A. G. Bennett and R. B. Rabbetts, “What Radi us Does the

Conventional Keratometer Measure?” Ophthalmic and

Physiological Optics, Vol. 11, 1991, pp. 239-247.

http://dx.doi.org/10.1111/j.1475-1313.1991.tb00539.x