Ray tracing error correction in ophthalmic optics

Ray tracing error eorreetion in ophthalmie opties Toyohiko Kashiwagi, M.D. , Patricia M. Khu, M.D.

ABSTRACT Ray tracing error correction (RTEC), a new method for various refractive calculations, is introduced. It consists of two procedures: (1) ray tracing that determines whether the ray originating from an infinite point focuses on the fovea and (2) error correction that varies the curvature of the different refractive surfaces according to an error correction formula when the ray is not focused at the fovea. These calculations are continued until the ray is focused at the fovea. The refraction of Gulistrand's schematic eye at relaxation and at accommodation, the desired postoperative corneal curvature for postoperative emmetropia after corneal ablation surgery, and the intraocular power of convex-plano, biconvex, and plano-convex intraocular lenses are presented.

Key Words: cataract surgery, corneal curvature, Gullstrand's schematic eye, intraocular lens power calculation, postoperative emmetropia, prediction, ray tracing error correction (RTEC), refracting surface, refraction

In the early 19th century, Gullstrand 1 derived the optical model of the eye including two refracting surfaces for the cornea and four refracting surfaces for the lens. But many optical calculations in ophthalmology were done on a reduced model; i.e., all theoretical intraocular lens (IOL) formulas have some simplifications or modifications of this optical model. 2 Because manual derivation of a formula based on Gaussian optics requires tremendous effort when the number of refracting surfaces is large, the original Gullstrand's model is usually not used. To eliminate the complicated manual procedure in deriving a formula for different refractive surfaces, ray tracing error correction (RTEC) was developed by one ofus (T.K.) in 1985 and applied to IOL power determination. 3 ,4 In this paper, we will explain the theory and procedure, showing its application in

determining the refraction of G ullstrand's schematic eye at relaxation and at accommodation in different axial lengths, the desired postoperative curvature after corneal ablation surgery, and the IOL power of convex-plano, biconvex, and plano-convex lenses.

MATERIALS AND METHODS Ray tracing error correction is an autofocusing process on computer. One or more curvatures ofthe optical system are varied according to the correction method; curvature of spectacles, cornea, or IOL is varied for correction by spectacles, corneal refractive surgery, or IOL implantation, respectively. In the calculations, the curvature of every refracting surface , the distance between adjacent surfaces, and every refractive index are considered. These data are based on Gullstrand's schematic eye.

From The Eye Clinic, Sakai , Osaka, Japan (Kashiwagi ), and the Division of Ophthalmology, Brigham & Womens Hospital, Boston, Massachusetts (Khu). Reprint requests to Patricia Khu , M.D., Brigham & Womens Hospital, Division of Ophthalmology, 75 Francis Street, Boston , Massachusetts 02115, or Toyohiko Kashiwagi, M.D., The Eye Clinic , 5-768-1 Nakamozu-cho, Sakai-City, Osaka, 591 Japan. 194

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Spectacles

R (m)

Retina

1.4-------0 (m)-------..I 1
Fig. 1.

(Kashiwagi) Principle of RTEC. A ray parallel to the optical axis originating from infinity is refracted on the surface of spectacles, cornea, and lens and crosses the optical axis at the second focal point (SFP). U sing ray tracing technique, D(m), the distance between anterior corneal surface and SFP is obtained. Ifthe optical system is properly corrected, SFP should be on the fovea and D(rn) is equal to the axiallength DO. If D(m) is different from DO, the anterior curvature of the spectacles is varied according to the error correction formula. As the calculations proceed, D(m) will eventually be equal to DO, and the corresponding curvature RO obtained is the curvature of the spectacles that corrects the refraction.

Ray tracing error correction consists of two procedures, a ray tracing and an error correction process (Figure 1). To illustrate, the refraction of Gullstrand's schematic eye using the optical parameters in Table 1 and varying the anterior curvature of spectacles is given. The initial curvature R(I) is arbitrarily determined. Corresponding second focal point (SFP) and its distance from the anterior corneal surface, D(I), is measured by ray tracing. R(2) is obtained from the formula R(2) = l/(lIR(l) + (1ID(1) -1/DO)/(n2 - n1) where nl and n2 are the refractive indices of both sides of the varied surface and DO is the distance from the anterior corneal surface to the fovea, which is equal to the axiallength ofthe eye. Corresponding

D(2) is measured by ray tracing. R(3) is determined from R(I), R(2), D(I), D(2) using the error correction formula 11R(m + 1) = 11R(m) + (llR(m) - 11R(m -1))/(1/D(m)-1I D(m -l))*(l/DO -l/D(m)), (m = 3,4,5 ... )

Corresponding D(3) is determined by ray tracing. Iterating error correction and ray tracing, two progressions R(I), R(2) ... R(m) ... and D(I), D(2) . .. D(m) ... are obtained. The progressions will eventually converge to RO' and DO, respectively, where RO is the curvature of the spectacles that corrects the refraction. The software program was written in Fortran. All calculations are done by double precision with effective digits of 16.

Table 1. Optical parameters in calculating the ret'raction of Gullstrand's schematic eye at relaxation and after refractive corneal surgery. Initial curvatures of spectacles are infinite.

Sl2ectacles

Refracting Surfaces Cornea 1 1 2

1

2

Curvature (R;mm)

1.D+100

I.D+100

7.7

6.8

Distance (D;mrn)

-13.0

-12.0

0.0

0.5

Refractive index*

1.498

1.376

Lens 2

3

4

10.0

7.911

-5.76

-6.0

3.6

4.146

6.565

7.2

cortex

nucleus

1.406

1.386

*Others: 1.0 (air), 1.336 (aqueous and vitreous)

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Spectacles

Spectacles

Retina Lens

Fig. 2.

Retina Cornea IOL

(Kashiwagi) Optical model for calculating the refraction ofGullstrand's schematic eye, the desired postoperative corneal curvature, and refraction after refractive corneal surgery. Optical parameters are in Tables 1 and 2. The anterior curvature of spectacles and anterior corneal curvature are varied in calculating the refraction and after refractive corneal surgery.

Fig. 3.

(Kashiwagi) Optical model for calculating the IOL power and postoperative refraction. Optical parameters are in Table 3. The curvatures of anterior, posterior, or hoth IOL surfaces are varied in determining the power of convex-plano, plano-convex, and biconvex lenses, respectively. The anterior curvature of spectacles is varied in calculating the postoperative refraction.

ent axiallengths. Table 6 shows the desired postoperative corneal curvature for postoperative emmetropia after corneal ablation surgery in different axial lengths. Table 7 presents the different IOL powers for postoperative emmetropia in the three types oflenses. The difference among the three IOL shapes is greater at shorter axiallength. Values given are for posterior chamber lenses.

Using Gullstrand's schematic eye with different axiallengths, the refraction at relaxation and at accommodation, the desired postoperative corneal curvature for postoperative emmetropia after corneal ablation surgery, and the IOL power of convexplano, biconvex, and plano-convex lenses were determined by RTEC. The thickness ofthe IOL was 1 mm. Figures 2 and 3, Tables 1, 2, and 3 give the optical models and parameters used in the calculations.

DISCUSSION

RESULTS Table 4 shows an example of the RTEC process determining the refraction ofGullstrand's schematic eye. The initial curvature was infinite and corresponding SFP was at 24.385 mm. In the process of iterations, SFP gradually approached the fovea and finally coincided with it. This procedure required 3 seconds with iteration measurements of 5. Table 5 shows the refraction ofGullstrand's schematic eye at relaxation and at maximum accommodation in differ-

Versatility of RTEC . As shown in the results, calculations of various refractive methods are possible using the same error correction formula while varying the curvature ofthe refracting surface under study. Complex problems such as a large number of refracting surfaces (i.e., 15 or more) can also be handled by RTEC. Such versatility is possible because RTEC uses a trial and error procedure with subsequent error correction, which is different from other methods using rigid formulas derived from

Table 2. Optical parameters in calculating the refraction of Gullstrand's schematic eye at accommodation.

Sl2ectacles

Refracting Surfaces Cornea 1 2

1

2

1

Curvature (R;mm)

1.D+l00

1.D+ 100

7.7

6.8

Distance (D ;mm)

-13.0

-12.0

0 .0

0.5

Refractive index*

1.498

1.376

Lens 2

3

4

5.33

2.655

-2 .655

-5.33

3.2

3.3725 cortex

nucleus

1.406

1.386

*Others: 1.0 (air), 1.336 (aqueous and vitreous) 196

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7.2

Table 3. Optical parameters in calculating an IOL implanted eye.

Spectacles

Refracting Surfaces Cornea 1 2

1

2

Curvature (R;mm)

l.D+100

l.D+100

7.7

Distance (D;mrn)

-13.0

-12.0

0.0

Refractive index*

1.498

IOL 1

2

6.8

1.D+ 100

1.D+ 100

0.5

3.6

4.6

1.376

1.490

*Others: 1.0 (air), 1.336 (aqueous and vitreous)

Gaussian optics. Although this trial and error procedure requires a longer period than calculations by a formula, the time frame is usually within several seconds. Accuracy of RTEC. Since every refracting surface and refractive index and the distance between refracting surfaces are taken into consideration in RTEC, no modification or approximation is made on the optical model. The calculations are continued until the correct measurement is obtained. Intraocular Lens Power Determination and Effect of IOL Shape. Most theoretical formulas determining IOL power have some modification of the Gullstrand model. For instance, Binkhorst's formula5 did not consider the posterior surface of the cornea and the corneal thickness, and it used different corneal refractive indices. Although the posterior curvature of the cornea is not a clinically available measurement, it is important to give an approximation since a 1 mm difference can result in a 1 diopter change in IOL power. 2 In RTEC, the posterior curvature of the cornea is assumed to be 0.9 mm less than the anterior curvature based on Gullstrand's model. In Table 7, the difference in IOL power between convex-plano and plano-convex

lenses at axial lengths of 22 mm and 30 mm are 2.11081 and 0.14152 diopters, respectively. This difference is dependent on axial length, with increasing differences among the three types of IOL at shorter axial lengths. Thus, simple addition of a number or factor to the value obtained by a regression formula will certainly not be accurate; i.e., in the SRK formula only the A constant is changed for different IOL types. Binkhorst6 also introduced a modified version that considered both IOL surfaces. It is valid for biconvex, plano-convex, and meniscus lenses, provided the anterior chamber depth is defined as the distance from the cornea to the first principal point of any lens. Barrett7 developed the method for calculating the power of different IOLs but failed to consider the posterior surface of the cornea. Most traditional vergence formulas, i.e., Binkhorst, Holladay, are applicable for spherical refract-

Table 5. Refractions ofGullstrand's schematic eye at relaxation and at accommodation in different axiallengths.

Axial Lengths (rnrn) Table 4. Example of RTEC procedures in calculating the refraction of Gullstrand's schematic eye. Spectacle thickness is 1 mrn.

Iteration

Spectacles Curvature Power (mrn) (diopter)

SFP (rnrn)

20 21

Refraction (diopter) Relaxation Accommodation 11.68666 8.949273

3.776869 0.0516194

22

6.255536

-3.681204

23

3.604395

-7.421622

24 25

0.9948458

-11.69660

-1.574081

-14.92534

1

1.E+100

0.0000000

24.38541

26

-4.103328

-18.68869

2

756.2216

0.6585371

24.13003

27

-6.593807

-22.45972

3

498.3905

0.9992165

23.99831

28

-9.04640

-26.23847

4

500.5898

0.9948266

24.00001

29

-11.46197

-30.02495

5

500.5801

0.9948458

24.00000

30

-13.84135

-33.81920

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Table 6. Desired postoperative corneal curvature at emmetropia after corneal ablation surgery of Gullstrand's schematic eye in different axiallengths. Axial Length (mm)

Corneal Curvature (mm)

Table 7. Intraocular lens power of three lens types-for postoperative emmetropia of Gullstrand's schematic eye at different axiallengths. Intraocular lens thickness is 1 mm. Axial Length (mm)

Convex-Plano

IOL (diopter) Biconvex* Plano-Convex

20

6.011331

21

6.381025

20

33.18669

34.674645

36.30792

22

6.759542

21

28.44704

29.67536

31.01809

23

7.147201

22

24.22547

25.2358169

26.33628

24

7.544336

23

20.44146

21.267012

22.16337

25

7.951301

24

17.03031

17.697865

18.42066

26

8.368464

25

13.93951

14.470916

15.04487

27

8.796213

26

11.12595

11.539214

11.98458

28

9.234957

27

29

9.685125

28

6.193638

6.4130840

6.648693

10.147170

29

4.01995

4.1593482

4.308774

30

2.011582

2.0799362

2.153101

30

8.553936

8.8640192

9.197533

*Symmetrical

ing surfaces. For larger pupils (greater than 3 mm), however, the asphericity of the cornea may result in a significant error in IOL power determination. Since RTEC uses ray tracing, it can calculate not only para-axial rays but any meridional ray that passes through the cornea. In our software program, each refracting surface can be aspheric-asphericity being expressed as a function of the distance from the optical axis. Hence, corneal asphericity is considered in RTEC calculations. Though several factors still affect the predictability of postoperative refraction such as the accuracy of A-mode ultrasonography in determining axial length, the postoperative anterior chamber depth, the change between preoperative and postoperative corneal curvature, and the spherical aberration of the IOL implanted eye, RTEC remains the most versatile and promising method in determining the IOL power for different lens types.

Ray Tracing Error Correction in Refractive Corneal Surgery. The values given in Table 6 are for

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emmetropia. By varying the refraction, RTEC can also determine the corneal curvature to achieve the desired postoperative refraction in excimer corneal laser surgery. It can also be used in the refractive calculations for keratomileusis, epikeratophakia, corneal inlay surgery, and contact lenses. REFERENCES 1. Fjelstad EO. Book review. Helmholtz's Treatise on Physiologieal Optics. Northwest J Optom 1924; 1:213 2. Kashiwagi T. Various methods of IOL power determination and their problems. J Ophthalmie Surg (Japan) 1988; 1:7-19 3. Kashiwagi T, Manabe R. New method onOL power calculation. J Ophthalmol Opt Soc Jpn 1986; 7:93-98 4. Kashiwagi T, Ohji M, Kiridoshi A, et al. Determination of intraocular lens power by ray tracing error correcting method (RTEC). Jpn J Clin Ophthalmol 1987; 41:133-135 5. Binkhorst RD. The optieal design of intraocular lens implants. Ophthalmie Surg 1975; 6(3):17-31 6. Binkhorst CD, Loones LH. Intraocular lens power. Trans Am Acad Ophthalmol Otolaryngol1976; 81:0P-70-0P-79 7. Barrett GD. Intraocular lens calculation formulas for new intraocular lens implants. J Cataract Refract Surg 1987; 13:389-396

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