Optics for photorefractive keratectomy

Optics for photorefractive keratectomy Jean-Philippe Colliac, M.D., H. John Sham mas, M.D.

ABSTRACT Matrix calculations are applied to Gaussian optics in the study of myopic correction by photorefractive keratectomy. The Colliac™ matrix formula determines the postoperative anterior curvature radius of the treated cornea. It also calculates the maximal depth of the removed corneal tissue and the ablation zone diameter needed to achieve emmetropia. Key Words: emmetropia, excimer laser, optics, photorefractive keratectomy

Photorefractive keratectomy (PRK) modifies the central cornea's refractive properties by removing concentric discs of superficial stromal tissues with an excimer laser. Since 1983, numerous researchers have used the excimer laser to correct myopia and have reported PRK's effect on animal and human eyes. I-I I Advances in PRK necessitate the improvement of the predictability and the stability of results. This article presents the Colliac™ formula based on matrix calculations to determine what anterior radius of corneal curvature is needed postoperatively for emmetropia. The formula also calculates the maximal depth of the removed corneal tissue and the ablation zone diameter.

neal curvature, expressed in millimeters, which is translated into diopters by considering the entire corneal power to be at the anterior surface. The relationship between the keratometric readings (K in diopters) and the value of the anterior corneal radius (r in millimeters) is: K

= (1.337: - 1) 1000 = 337,5/r

The Colliac formula uses the true refractive index of the cornea, which is 1.3771 (n 2). The relationship between the real value of the power of the cornea's anterior surface (DI in diopters) and the keratometric readings (K in diopters) is:

MATERIALS AND METHODS

Determination of the Ocular Parameters

We applied matrix calculations to Gaussian optics using the measured ocular parameters. Keratometry. The refractives indices we used were 1.3771 for the cornea, 1.3374 for the aqueous humor, 1.42 for the lens, and 1.336 for the vitreous. We prefer these indices, which Le Grand l2 measured with a sodium light (0.591l), instead of the Gullstrand's indices I 3 (1.376 for the cornea, 1.336 for the aqueous humor and the vitreous, 1.4085 for the lens). Original keratometers (Javal) measured total corneal power, not front surface power. That method incorrectly assumed that the back and front corneal radii were equal and therefore used a tear film index of 1.336. Later, to "standardize," an arbitrary index of refraction of 1.3375 was used so that a radius of 7.5 mm would yield 45.0 diopters (D): 45 = (l - 1.3375)/7.5 X 10-3 • 14 The keratometer measures the anterior radius of cor-

D\

= 1.1173 x

K

(1-2)

In our calculations the negative dioptric effect of the posterior corneal surface and the lens dioptric power is a constant; it is not necessary to know the posterior corneal curvature radius, lens design, or indices of refraction of the aqueous humor and lens. Clinical keratometers average the curvature radius over the central 3.5 mm of the cornea. They take two measurements from the corneal surface to determine its curvature and are unable to locate precisely the visual axis, the corneal vertex, or the corneal apex. The cornea is aspheric with individual variations and is best evaluated by keratoscopy. The photo keratoscope projects about 20 rings on the corneal surface and measures the corneal surface from a central area of less than 0.30 mm diameter to a peripheral area greater than 10 mm.15 Refraction. In young adults, a cycloplegic refraction is preferred. Spherical and cylindrical components are considered separately. The value of the spherical equiv-

From the Institut Arthur Vernes, Paris, France (Colliac), and the Department o/Ophthalmology, University School 0/ Medicine, Los Angeles, California (Shammas). Reprint requests to Jean-Philippe Colliac, M.D., 9 rue de Montalembert, 75007, Paris, France. 356

(1-1)

J CATARACT REFRACTSURG-VOL 19, MAY 1993

0/ Southern California

alent diopter, measured preoperatively, is used for the calculations. The refraction (R), or ametropia of the patient, is the inverse of the abscissa of the Remotum (R), which is measured from the corneal vertex VI, (XR = v.R): (1-3)

The relationship between the refraction and the value of correction (S) of a spectacle lens is: 1

R=

1

+S

-b

(1-4)

where b is the distance between the spectacle lens and the eye.

Matrix Methods

The Colliac matrix formula applies matrix calculations to Gaussian optics. 16-20 The processes of refraction and light transfer may be performed mathematically by matrix operations because of the linear form of the refraction equation of a spherical diopter (interface). The matrix representation is an efficient way to study optical systems of two or more lenses. Furthermore microcomputers have made such calculations easier. Refraction M atrixfor a Spherical Diopter. A spherical diopter represents the len's front or back surface. Given a spherical diopter with D power, the origins are the principal points merged with the diopter's vertex; r is the radius of diopter curvature separating two mediums of n2 and n l , respective refractive indices. Therefore, the diopter's power is: n

-

n(

2 D=.......:;..-....:.

(1-5)

r

The refraction matrix is written: M =

[_~ ~]

Transfer Matrixfor a Lens. In a lens with thickness (t) and refractive index (n), for the first refraction, 0 1 origin is merged with the vertex of the first diopter. For the second refraction, O 2 origin is merged with the vertex of the second diopter. M =

[-~2 ~] [~ ~] [_~. ~] 0].

1 I . [OilS a trans atory matnx.

The transfer matrix of the lens is written:

0] [a b]

The transfer matrix of a system of lenses equals the matrix product formed by the product of translatory and refraction matrices. This matrix product is written from right to left, following the succession of diopters that refract light.

RESULTS Effects of PRK on the Cornea

In myopic PRK, the epithelium is removed mechanically. An excimer laser ablates a centrallenticule from Bowman's membrane and the underlying superficial stroma. The exposed area heals and forms a smooth pseudomembrane, which acts as a template for epithelial cells. This repro filing creates a new corneal surface with an increased curvature radius and a final flattening ofthe central cornea. Figure 1 shows the changes at the corneal surface. (The thickness of the ablated lenticule is greatly exaggerated.) Figure 2, the cornea before and after PRK, shows the following: V (: anterior vertex of the cornea before PRK V2: anterior vertex of the cornea after PRK V 3: posterior vertex of the cornea r.: radius of curvature of the initial surface of the cornea before PRK r2: radius of curvature of the final anterior surface of the cornea after PRK AB: d, diameter of the ablated area n2: refractive index of the cornea, 1.3771. D (: power of the initial anterior surface of the cornea before PRK D 2 : power of the final anterior surface of the cornea after PRK D( = (n2 - 1)/r(, D2 = (n 2 - 1)/r2 t2 = V (V3: initial thickness of the cornea before PRK t' 2 = V;V;-: final thickness of the cornea after PRK

to = t2 -

t' 2 = V (V2: maxim urn depth on the optical axis 02 = t 2/n 2, 0'2 = t' 2/n2'

2 1 - oD 2 = c d The formula to associate two diopters is: D = D. + D2 - oD.D2

(1-7)

We can calculate the following:

where 0 = tin is the reduced thickness of the single lens.

M= [ -D( - 1D2- +oo(oD(D

which are derived from the Gaussian coefficients. The cardinal points are properties of the lens and are computed from the matrix M. A reduced length is the quotient of a length by a refractive index (n) of the medium where this length is; x and x' are the reduced lengths of the projections of two conjugated points on the axis of a centered system. Those reduced lengths are counted, respectively, from any fixed origins 0 1 and O2. Let x' F be the reduced length of the image focus:

(1-6)

The a, b, c, and d are the Gaussian coefficients. They depend on the system and on the choice of the 0 1 and O2 origins. The six points-two focal, two principal, and two nodal-constitute the system's cardinal points,

to of ablated stroma

Transfer Matrix of the Ametropic and Noncorrected Eye

We assimilate the eye to a centered system having four surfaces (anterior and posterior corneal; anterior and posterior lens) separating areas of different indices

J CATARACT REFRACT SURG-VOL 19, MAY 1993

357

• •

Fig. 1.

•

•

Vl

•

(Co Iliac) Schematic of the cornea before and after PRK. The ablated corneallenticule (shadows) is not to scale.

of refraction. To this centered system we apply the paraxial approximation, in which the light rays stand at a small angle to the axis and the angles of incidence on the surfaces are small. If the 0, origin is the vertex V, of the anterior surface of the cornea, and the O2 origin is the vertex V7 of the posterior surface of the crystalline lens, the transfer matrix of the ametropic and non corrected eye between the corneal vertex and the posterior surface of the lens is obtained using the following formula:

Fig. 2.

(Colliac) Geometric drawing of the cornea before and after PRK.

The conjugate equation is: aX R

= CXRX'R -

b-

dX'R

=0

(2-1)

The reduced abscissa of the conjugate point of the Remotum is: , _ _ aX R X R-

CX R -

b

(2-2)

d

(2-3)

<--->

<--->

ametropic eye

<---->

corneal anterior corneal thickness surface

in which the transfer matrix of the system (refractive surfaces) between the posterior vertex of the cornea and the posterior surface of the lens is:

[azz dbz]z

This factor is a constant in our calculations and it is not affected by PRK. Given the transfer matrix of the system between the anterior corneal vertex before PRK, V, and the posterior corneal vertex, V3:

~:] = [~ ~z][ _~I ~] = [1 =ri~1 ~z]

[~ ~]=[~~ = [az(l -

c2 (1 -

°° ° ° 2

1) -

2

1) -

b2 0 1 d2 0 1

a2 02 + b2 ] C2 02

+ d2

Let XR be the reduced abscissa of the Remotum, let X'R be the reduced abscissa of its conjugate point, the retina (r). The transfer matrix between the Remotum (R) and theretina(r)is:

-ax R -

(2-4)

= l/xR

This refraction is measured from the corneal vertex (XR =

ViR).

Transfer Matrix of the Corrected Eye

After PRK, the radius of curvature of the final anterior surface of the cornea is r2> the final thickness ofthe cornea is t' 2 and the maximum depth of ablated stroma on the optical axis is to (Figure 1). If the 0, origin is the vertex V2 of the cornea's anterior surface and the O 2 origin is the vertex V7 of the crystalline lens posterior surface, the transfer matrix of the corrected eye between the corneal vertex and the posterior surface of the lens is obtained thus:

[~ ~] = [~~ ~~] [~~'2]

<--->

corrected eye

<--->

[-~z~]

<

>

corneal corrected anterior thickness corneal surface

Given the transfer matrix of the system between the anterior corneal vertex after PRK, V2 and the posterior cornealvertex, V3:

+ b + dX'R] +d

CXRX'R - CX R

358

R V"

C

[::

The refraction or ametropia of the patient is:

J CATARACT REFRACT SURG-VOL 19, MAY 1993

::J

[~ ~] [~~ ~~][:~ =

= [azm. + b

azmz + b Z m4] czm z + dZm4 Let x' F be the reduced abscissa of the image focus of the corrected eye. Z

m3

The diameter ofthe treatment zone is obtained by the formula of the cupola (spherical segment) height, written as follows:

czm. + d Zm 3

x'

A

azm.

=- - =-

+ b1 m 3

Fe d czm. + 2m3

(2-5)

Condition for Correction

The eye is corrected if the length of the image focus of the corrected eye, measured from the posterior surface of the crystalline lens, equals the distance between this posterior surface of the crystalline lens and the retina. The condition for correction is: X'R

= X'F

(2-7)

c2 (a.X R - b.)

d.

To simplify our calculation, we may write: k

a.xR - b. = -m. = -'-=-:..---'. m3

C.XR -

(2-8)

d.

Hence we obtain: nzr. - tzn z

+ t z - t1r.R

tz k = + - (2-9) n z(1 - n1 - r.R) 1 - n2 - r.R n z We know that there is only one circle passing through three points, which are not on a straight line. These three points are the corneal vertex and the two ends of the diameter of the ablation zone: V I or V2, A and B. There are three roots for the equation of correction: the radius of curvature of the final corneal surface (r 2), the diameter (d) of the treatment zone, and the maximum depth (to) of ablation stroma on the optical axis. We obtain a solution if either the maximum depth (to) or the diameter (d) of the treatment zone is fixed before the operation (Figure 1). r.

Calculating the Diameter of the Treatment Zone with a Fixed Depth of Ablation

The given values are the refraction of the noncorrected eye (R), the initial radius of curvature of the cornea (r l ), the maximum depth (to) of ablated stroma on the optical axis, and the refractive index of the cornea (n2)' We want to calculate the radius of curvature (r2) of the final corneal surface and the diameter (d) of the treatment zone. The radius of curvature of the final corneal surface (r2) is calculated by the equation: r2 = (n - 1) (_ z

to ___r.:...._ _)

(2-10)

nz 1 - nz - r.R Figure 2 shows a schematic drawing ofthe cornea before and after treatment. The measurement of the ablation diameter (d[AlJ]) is based on geometric calculations.

= r.

- ./r.z - dZ/4,

hz

= rz -

./r/ - dZ/4

(2-11)

in which hi is the height of the cupola (HV;") before PRK; h2 is the height of the cupola (HV;) after PRK, r I and r 2 are the initial and final radii of corneal curvature, and d is the diameter (AB) of the ablation zone. The maximum depth to of ablated stroma on the optical axis 1S:

to =

h. - h2

(2-12)

The diameter (d) of the treatment zone is calculated to be:

(2-6)

~""':""":":--':"'+dl

C.XR -

h.

/

+ 'V (r. 2 + r/)

d=

(r 2 - (I

r

-

Z)Z

f - (to to - r. + rz 2

r.

+ rz)l (2-13)

Figure 3 shows the diameter of the ablation zone needed to correct myopia, using a fixed central depth of 20, 30, 40, 50, or 60 Jlm. Matrix calculations have also shown that an approximation of 0.01 mm ablation zone diameter produces the same results whatever the value of the initial radii of corneal curvature. Table 1 shows the exact diameter needed to correct the preexisting myopia when the ablation depth ranges between 10 Jlm and 100 Jlm.

Calculating the Depth of the Ablation Zone with a Fixed Diameter

The given values are the refraction of the noncorrected eye (R), the initial radius of curvature the cornea (r I)' the diameter (d) of the treatment zone, and the refractive index of the cornea (n2)' We want to calculate the radius of curvature (r2) of the final corneal surface, and the depth (to) of the removed cornea at the vertex. The radius of curvature of the final corneal surface (r 2) is calculated using the equation:

12

E

11

§.

10

~

9

2 g

Fixed center depth _ 60 _

~

:a01

_

40

GI

_

30

___

20

= '0 .

(~m)

50

~

E 01

is •

1

•

2 . 3 - 4 . 5

_ 6

.

7 -

8

-109 - _11. 12 _ 13.14 - 15

Correction (Diopters)

Fig. 3.

(Colliac) Diameter of the ablation zone with a fixed depth of cut on the optical axis for myopic correction.

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359

Table I. Diameter of ablation zone needed to correct myopia. Depth of Ablation on the Optical Axis (!lm)

-I

-2

-3

-4

-5

-6

-7

-8

-9

-10

-II

-12

-13

-14

-15

10 20 30 40 50 60 70 80 90 100

5,3 7,2 8,5 9,5 10,4 11 11,8 12 12,6 12,9

3,8 5,3 6,4 7,2 7,9 8,6 9,3 9,6 10 10,4

3,1 4,4 5,3 6,1 6,7 7,2 7,9 8,2 8,6 8,9

2,7 3,8 4,6 5,3 5,9 6,4 7 7,3 7,6 7,9

2,4 3,4 4,1 4,8 5,3 5,8 6,4 6,6 6,9 7,2

2,3 3,1 3,8 4,4 4,8 5,3 5,9 6,1 6,4 6,7

2,1 2,9 3,5 4 4,5 4,9 5,5 5,7 6 6,2

1,9 2,7 3,3 3,8 4,2 4,6 5,1 5,3 5,6 5,9

1,8 2,5 3,1 3,6 4 4,4 4,8 5 5,3 5,6

1,7 2,4 3 3,4 3,8 4,2 4,6 4,8 5,1 5,3

1,7 2,3 2,8 3,3 3,6 4 4,4 4,6 4,8 5,1

1,6 2,2 2,7 3,1 3,5 3,8 4,2 4,4 4,6 4,9

1,5 2,1 2,6 3 3,3 3,7 4 4,2 4,5 4,7

1,5 2 2,5 2,9 3,2 3,6 3,9 4,1 4,3 4,5

1,4 2 2,4 2,8 3,1 3,4 3,8 3,9 4,2 4,4

Correction (D)

~

C r2

=

n2

1

-

(n

C2 -

1 2 _

d

2

1) 4" (2-14)

1)2 -

----1 (n2 - 1)2

where C is a constant to make the calculations easy: C

= r l (1 +

n 1 - n2

2 -

rlR

.J

) -

2

rl2

_

d

4

(2-15)

The maximum depth to of ablated stroma is found by the equation: L '0 -

n

_

2

(~+ n - 1 2

rI

1 - n2

-

r IR

(2

)

-

16)

Figure 4 shows the maximum cut depth needed to correct myopia, using a fixed diameter of ablation zone of3, 4,5, or 6 mm. Matrix calculations have also shown that if we made an approximation of corneal ablation

depth of 0.2 J.1m, it produces the same results whatever the value of the initial radii of corneal curvature. Table 2 shows the exact depth required to correct the preexisting myopia when the diameter ranges from 3 mm to 7 mm. The accuracy of our formula was evaluated. In a previous report, Zabel et al.I I and Maguire and coauthors21 presented the results of five patients who had excimer laser refractive keratectomy to correct myopia three months after surgery. We tested their published data. Photoablations were performed with a 193 nm excimer laser (Taunton Technologies, Monroe, CT) pulsed at 5 Hz with a fluence, on average, of 100 mJ/cm 2. The ablated area, with a diameter of 5.0 mm, contained 15 concentric steps. The data and the results of the calculated corneal power to correct myopia are shown in Table 3. In Table 4 we compare the deviation of the postoperative and estimated corneal power from the postoperative and real corneal power.

200

DISCUSSION

E .=,

...

-

Fixed diameter (mm)

150

:::J

--0-

U

_5 __ 4

o

.c

ii ~ E :::J E

~

~

6

100

---<>--

3

501-----+__L-~r-------~

• 5

- 10

- 15

. 20

Correction (Oiopters)

Fig. 4. 360

(Colliac) Maximum depth of cut on the optical axis with a fixed diameter of ablation zone for myopic correction.

The Colliac matrix formula is assumed to be an exact mathematical model in the study of Gaussian optics. The only needed data to calculate the final anterior radius of corneal curvature are refraction, initial anterior radius of corneal curvature, and the ablation zone diameter or the depth. Unlike intraocular lens power calculations in cataract surgery, it is not necessary to know the axial length and the design of the lens to calculate the postoperative radius of curvature in PRK.20 Preoperatively, if we fix the ablation zone diameter, we can calculate the ablation depth on the optical axis to correct a given refraction. Reciprocally, if we fix the ablation depth on the optical axis, we can calculate the ablation zone diameter to correct a given refraction. Figures 3 and 4 show that to correct myopia, a deeper

J CATARACT REFRACT SURG-VOL 19, MAY 1993

Table 2. Depth ().Lm) of cut on the optical axis needed to correct myopia. Correction (D)

Diameter of Ablation (mm)

-I

-2

-3

-4

-5

-6

-7

-8

-9

-10

-ll

-12

-13

-14

-15

3

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

3,5

4

8

12

16

20

24

29

33

37

41

45

49

53

58

62

4

5

11

16

22

27

33

38

44

49

55

60

66

71

77

82

4,5

7

14

21

28

35

42

49

55

62

69

76

83

90

97

104

5

8

17

26

35

44

52

60

69

78

86

95

103

112

121

129

5,5

II

21

33

43

54

63

74

84

95

106

116

126

137

147

158

6

13

25

38

51

64

78

89

102

115

127

140

152

165

177

190

6,5

15

30

45

61

76

92

107

123

138

153

168

183

199

213

227

7

18

37

55

73

90

108

126

144

162

180

197

213

230

247

266

Table 3. Patient data and calculation of the estimated corneal power to correct myopia. Preoperative Patient Data

Age Patient (Years)

Refraction (Correction and Spherical Equivalent)

Keratometric Reading (Range and Mean)

Postoperative Patient Data

Spectacle Corrected Visual Attempted Acuity Correction

Refraction (Correction and Spherical Equivalent)

Corneal Power (Range and Mean)

Estimated Corneal Power

+0.25 + 0.75 x 110 +0.63

35.50/39.00 37.25

38.70

AC

47

-8.50 + 0.75 x 140 -8.125

45.00/45.50 45.25

20/25

8.00

CK

25

-8.750 -8.750

43.75/43.75 43.75

20/20

8.75

-2.25 + 0.75 x 165 -1.88

37.50/39.50 38.50

36.74

CW

44

-6 + 0.50 x 115 -5.750

44.50/45.00 44.75

20/20

6.00

0.00 0.00

37.90/42.20 40.05

40.00

CA

51

-5.000 -5.000

44.50/45.00 44.75

20/15

5.00

-0.25 -0.25

40.50/44.80 42.65

40.58

DT

45

-10.25 + 2.00 x 90 -9.250

41.75/43.25 42.5

20/20

9.25

-1.25 + 0.75 x 90 -0.88

34.4/41.4 37.9

35.12

Table 4. Real and estimated changes in corneal power.

Patient

Preoperative Refraction

Postoperati ve Refraction

Change in Refraction

Estimated Change in Corneal Power

Real Change in Corneal Power

Deviation of the Estimated from the Real Postoperative Corneal Power

AC

-8.13

+0.63

8.75

6.55

8.00

+1.45

CK

-8.75

-1.88

6.88

7.01

5.25

-1.76

CW

-5.75

0.00

5.75

4.75

4.70

-0.05

CA

-5.00

-0.25

4.75

4.17

2.10

-2.07

DT

-9.25

-0.88

8.38

7.38

4.60

-2.77

keratectomy is needed if the ablation zone diameter is enlarged. For a 4 mm ablation zone diameter, the maximum depth of removed cornea is about 5 /.lm per diopter, and for a 5.5 mm ablation zone diameter, the maximum depth of removed cornea is about 11 /.lm per diopter (Table 2). To avoid going beyond 50 /.lm in depth, a5.5 mm diameter has to be used for corrections

up to 4 D, a 5 mm diameter for a correction of 5 D, a 4.5 mm diameter for corrections of6 Dand 7 D,a4 mm diameter for corrections of 8 D and 9 D and a 3.5 mm diameter for corrections between 10 D and 12 D. Tables 1 and 2 show the exact relationship between the diameter and the depth of the treated area to achieve the necessary myopic correction.

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For example, for a -5 D correction, a 40!lm of depth of corneal removal, and a 7.8 mm of initial curvature radius, the ablation zone diameter will be 4.78 mm and the final curvature radius will be 8.69 mm. For the same -5 D correction, a 4 mm optical zone, and a 7.8 mm of initial curvature radius, the depth of central removal of corneal stroma will be 27.55!lm and the final curvature radius will be 8.69 mm. For a -10 D correction and a 4 mm optical zone, the depth of central removal of corneal stroma will be 54.93!lm and the final curvature radius will be 9.82 mm for a 7.8 mm of initial curvature radius, and respectively 55.45 !lm and 8.58 mm for 7 mm of initial radius of curvature, and 54.59 !lm and 10.96 mm for 8.5 mm of initial radius of curvature. These results compare favorably to those of Munnerlyn and coauthors,2 who found a central removal of 53 !lm for a - 10 D correction and a diameter of 4 mm; but Munnerlyn and coauthors did not indicate the values of the initial and final radii of curvature. Patients AC, CK, and CW (Tables 3 and 4) show good correspondence between the final change in central corneal power and the postoperative and estimated corneal power needed to obtain emmetropia and the attempted correction. The Colliac formula worked well for these three eyes. But in patients DT and CW there is poor correspondence between the final change in central corneal power and the final change in refraction. Additional investigation is needed in both eyes to resolve uncertainties about the inaccuracy of the corneal topography measurement technique. The calculations of the Colliac formula are limited by data error. For the anterior radius of corneal curvature, an error of 0.1 mm (keratometry reading = 0.55 D) affects the postoperative refraction by about 0.6 D. The deeper the wound, the more intense the cornea's healing response. Therefore, to correct larger amounts of myopia without making too deep a keratectomy, smaller diameter ablation zones will have to be accepted. Clinically, significant scarring is rare when keratectomies are shallower than 40 !lm. To prevent keratectomies deeper than 40 !lm, Seiler and coauthors6 •7 use an ablation zone diameter of 4.5 mm for corrections up to -6.0 D, gradually decreasing to 3.5 !lm diameter for a correction of -6.0 to -10.0 D. More pulses are needed to correct high myopia but increase the likelihood of thermal or mechanical effects that may stimulate collagen formation. The precision of ablation is limited by the accuracy of the measurement of the amount of tissue ablated per pulse at the fluence used. The relationship between the corneal ablation rate (!lm per pulse) and the logarithm of fluence (mJ/cm2) is not linear. The mechanism of ablation depends on the fluence used. 22 The ablation rate in Bowman's layer is two thirds that in the stroma. 23 We applied to the eye a paraxial or first order approximation known as Gaussian optics. But the real path of the light rays is not paraxial because the pupil is open wide. To understand the images and the optical aberrations created by the aspheric cornea, it will be 362

necessary to develop ray tracing programs, which is impractical. The aim of the refractive surgery is to obtain an optimum configuration of the cornea at its apex, which is why we believe that Gaussian optics were appropriate for our study. With matrix representation of Gaussian optics, it is easy to find the focal length and the principal points for a compound lens system composed of3 D, 4 D, or more. The Colliac matrix formula allows ophthalmologists to anticipate the value of the postoperative anterior radius of corneal curvature. There is a great interest to control and confirm the effects of the operating parameters of the used laser. The long-term effect of PRK on the cornea is not yet well established. Clinical studies need to compare the preoperative and postoperative refractions, as well as the initial and final corneal curvature radii. This will provide an objective assessment of the PRK's effect on the cornea and will allow for future refinement of the formula used. REFERENCES 1. Marshall J, Trokel S, Rothery S, Krueger RR. Photoablative reprofiling of the cornea using an excimer laser: photo refractive keratectomy. Lasers Ophthalmol 1986; 1:21-48 2. Munnerlyn CR, Koons SJ, Marshall J. Photorefractive keratectomy: A technique for laser refractive surgery. J Cataract Refract Surg 1988; 14:46-52 3. Taylor OM, L'Esperance FA Jr, Del Pero RA, et a1. Human excimer laser lamellar keratectomy: A clinical study. Ophthalmology 1989; 96:654-664 4. Hanna KO, Pouliquen Y, Waring GO III, et a1. Corneal stromal wound healing in rabbits after 193-nm excimer laser surface ablation. Arch Ophthalmol 1989; 107:895901 5. Fantes FE, Hanna KO, Waring GO III, et a1. Wound healing after excimer laser keratomileusis (photorefractive keratectomy) in monkeys. Arch Ophthalmol 1990; 108: 665-675 6. Seiler T, Kahle G, Kriegerowski M. Excimer laser (193 nm) myopic keratomileusis in sighted and blind human eyes. Refract Corneal Surg 1990; 6: 165-173 7. Seiler T, Wollensak J. Myopic photorefractive keratectomy with the excimer laser: One-year follow-up. Ophthalmology 1991; 98: 1156-1163 8. McDonald MB, Liu JC, Byrd TJ, et a1. Central photorefractive keratectomy for myopia: Partially sighted and normally sighted eyes. Ophthalmology 1991; 98: 13271337 9. Wilson SE, Klyce SO, McDonald MB, et a1. Changes in corneal topography after excimer laser photorefractive keratectomy for myopia. Ophthalmology 1991; 98: 13381347 10. Liu JC, McDonald MB, Varnell R, Andrade HA. Myopic excimer laser photorefractive keratectomy: An analysis of clinical correlations. Refract Corneal Surg 1990; 6:321328 11. Zabel RW, Sher NA, Ostrov CS, et al. Myopic excimer laser keratectomy: a preliminary report. Refract Corneal Surg 1990; 6:329-334 12. Le Grand Y. Optique physiologique, Tome 1. La diop-

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trique de I'reil et sa correction, 2nd ed. Paris, Revue d'Optique, 1965; 147-159 Gullstrand A. Die Dioptrik des Auges. In: Helmholtz, HLF von, ed, Handbuch der Physiologischen Optik, 3rd ed. Hamburg/Leipzig, L. Voss, 1909; 41-375 Holladay JT, Waring GO III. Optics and topography of radial keratotomy. In: Waring GO III, ed, Refractive Keratotomy for Myopia and Astigmatism. St Louis, Mosby, 1992; 37-139 Wilson SE, Klyce SD. Advances in the analysis of corneal topography. Surv Ophthalmol 1991 ; 35:269-277 Brouwer W. Matrix Methods in Optical Instrument Design. New York, WA Benjamin, 1964 Bourdy C. Calcul matriciel et optique paraxiale: application al'optique ophtalmique. Rev Optique 1962; 41 :295308 Hecht E. Optics, 2nd ed. Reading, MA, Addison-Wesley, 1987; 128-220

13. 14.

15. 16. 17. 18.

19. Perez J-Ph. Optique Geometrique et Ondulatoire, 2nd ed. Paris, Masson, 1988; 40:74 20. Colliac J-P. Matrix formula for intraocular lens power calculation. Invest Ophthalmol Vis Sci 1990; 31 :374-381 21. Maguire U , Zabel RW, Parker P, Lindstrom RL. Topography and raytracing analysis of patients with excellent visual acuity 3 months after excimer laser photorefractive keratectomy for myopia. Refract Corneal Surg 1991; 7: 122-128 22. Van Saarloos PP, Constable IJ. Bovine corneal stroma ablation rate with 193-nm excimer laser radiation; quantitative measurement. Refract Corneal Surg 1990; 6:424429 23. Seiler T, Kriegerowski M, Schnoy N, Bende T. Ablation rate of human corneal epithelium and Bowman's layer with the excimer laser (193 nm). Refract Corneal Surg 1990; 6:99-102

APPENDIX Standard Gaussian Equations Constants n e, na , nv

= refractive indices of cornea, aqueous, and vitreous

Preoperative measured values

K rI Rs

=

keratometric reading

= anterior radius of curvature of the cornea = spheroequivalent spectacles correction

Other variables

VR

=

abscissa of the Remotum

= refraction from the primary principle plane

= corneal thickness on the optical axis before and after PRK = depth of the ablated cornea on the optical axis = power of the eye before and after PRK

= corneal power before and after PRK

= power of the front corneal surface before and after PRK

= power of the back corneal surface

= lens power

= anterior radius of curvature of the cornea after PRK = posterior radius of curvature of the cornea

r3

fml ' fl fm2' f2 ~, ~

~

1m , l'

object focal length before and after PRK image focal length before and after PRK = abscissa of the secondary principal plane of the cornea before and after PRK = abscissa of the primary principal plane of the lens = distance from the secondary principal plane of the eye to the retina before and after PRK = =

Equations

Dm = Dme + DL - DmPLom D = Dc + DL - DCDLO Dml = (ne - I)/r l Om = (-V1f.:; + VHLl)fna = (-~ + VHLl)/na fml = -I/D m

° Rh

C

= nv ~m -

fm2

= n)Dm

Dme = Dml + D2 - DmlD20me Dc = DI + D2 - D ID 20e DI =: (ne - 1)/r2 ome = tmdnc De = te/nc fl = -l/D

1\) - (Dm - D)

Depth of the ablated corneallenticule at a distance X from the optical axis tx = ~ - r l + ../r12 - X2 + r 2 - ../rl - X2

r l , f2 = anterior radius of curvature, before and after PRK ~ = maximum depth of the ablated cornea on the optical axis X =: distance from the optical axis tx = depth of the removed corneal lenticule at a given distance X from the optical axis.

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