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Corneal tissue ablation depth and the Munnerlyn formula
Corneal tissue ablation depth and the Munnerlyn formula Austin W. Chang, MD, Alexander C. Tsang, MD, Janice E. Contreras, MD, Paul D. Huynh, MD, PhD, Christopher J. Calvano, MD, PhD, Tracy C. Crnic-Rein, MD, Edmond H. Thall, MD
Purpose: To determine whether the error in ablation depth produced by approximations inherent in the Munnerlyn formula are clinically significant when estimating residual corneal stromal depth for the evaluation before refractive surgery. Setting: Department of Ophthalmology, University of Texas Medical Branch, Galveston, Texas, USA. Methods: Using identical geometric assumptions, the exact ablation depth was calculated and compared to the approximate ablation depth predicted by the Munnerlyn formula. An adjustment factor was then derived for large optical zones and corrections. Results: The exact ablation depth is always larger than the ablation depth predicted by Munnerlyn’s formula. Analysis found the error in ablation depth varied as the fourth power of the optical zone and linearly with correction. The initial corneal radius had little effect on the difference. The ablation depth could be reasonably approximated by adding an adjustment factor for large optical zones and refractive corrections. Conclusions: In patients with large optical zones, it may be preferable to calculate tissue ablation depth using the exact formula. Alternately, the Munnerlyn formula can be used to calculate ablation depth and then an adjustment factor can be added. J Cataract Refract Surg 2003; 29:1204 –1210 © 2003 ASCRS and ESCRS
P
rogressive corneal ectasia is an uncommon but serious postoperative complication of laser in situ keratomileusis (LASIK).1– 4 The etiology of it is not fully understood, but current thinking is that the problem can be minimized or avoided by maintaining a residual stromal depth (RSD) of 250 m.5,6 The RSD depends on many intraoperative variables. Flap thickness can differ from the microkeratome setting by 50 m.7,8 Current clinical ultrasonic pachymeters have an accuracy of 40 m.9,10 Ablation depth can vary with the ablation profile (multizone versus sinAccepted for publication September 25, 2002. Reprint requests to Austin W. Chang, MD, Department of Ophthalmology, University of Texas Medical Branch, Galveston, Texas 775550787, USA. © 2003 ASCRS and ESCRS Published by Elsevier Inc.
gle zone), atmospheric conditions, corneal hydration, corneal asphericity, and laser parameters.11 Intraoperative variables make it impossible to precisely predict the RSD. Nevertheless, clinicians must in some way estimate it preoperatively. The decision to perform LASIK or another refractive procedure or avoid refractive surgery is based partly on the estimated RSD. Currently, the RSD is calculated by subtracting flap thickness and ablation depth from the central pachymetry measurements. There are 2 ways to estimate ablation depth: using the preoperative parameters or using the Munnerlyn approximated formula.12 The Munnerlyn approximated formula is theoretical and based on the assumption that both the cornea and ablation profile are spherical. It is not exact but only an estimate of the ablation depth predicted by spherical 0886-3350/03/$–see front matter doi:10.1016/S0886-3350(02)01918-1
LABORATORY SCIENCE: CORNEAL ABLATION DEPTH AND MUNNERLYN FORMULA
geometry. The Munnerlyn approximated formula is based on the first 2 terms of a Taylor series expansion13 and systematically underestimates the theoretical ablation depth. Using the same geometric assumptions, we calculated the exact ablation depth and compared it to the approximate ablation depth predicted by the Munnerlyn approximated formula. We calculated the theoretical exact ablation depth for various optical zone sizes and various amounts of correction to determine whether the difference between the 2 formulas is clinically significant. We then analyzed the data and series expansion to derive an adjustment factor to better approximate the exact theoretical ablation depth predicted by spherical geometry (Appendix).
Materials and Methods The Appendix shows a detailed derivation of the formulas. Here, the issues are discussed qualitatively. Munnerlyn based the theoretical ablation depth for photorefractive keratectomy on a spherical corneal geometry. Figure 1 depicts a spherical myopic correction. Only myopic correction is considered because ablation depth is less critical in small hyperopic corrections. In myopic eyes, the ablation depth is greatest along the optical axis. The exact ablation depth for spherical geometry can be calculated using equation 1. Munnerlyn Theoretical Exact Ablation Depth ⫽ R1 ⫺ ⫺ ⫹
冑 冑冋
R 1 · 共n ⫺ 1兲 n ⫺ 1 ⫹ R1 · D
R 12 ⫺
OZ 2 4
R 1 · 共n ⫺ 1兲 n ⫺ 1 ⫹ R1 · D
册
2
⫺
OZ 2 4
(1)
where n is the refractive index of the cornea (1.376), D is the correction in diopters, and the other variables are illustrated in Figure 1. The Munnerlyn approximated formula12 used clinically is 2
Approximate Ablation Depth ⬵
OZ D 3
(2)
It is difficult to directly compare the 2 formulas because the Munnerlyn exact formula depends on 3 variables (optical zone, correction in diopters [D], and initial radius of curvature) whereas the Munnerlyn approximated formula depends
Figure 1. (Chang) Spherical myopic correction: R1 and R2 ⫽ initial and final radii of curvature; C ⫽ chord length or diameter of the optical zone; S1 and S2 ⫽ sagittal depth of the preoperative and postoperative cornea.
on only 2 variables (correction and optical zone). To directly compare the 2 equations, a typical corneal radius of curvature of 7.5 mm (which corresponds to a K-reading of 45.0 D) was used. To determine the effect of the initial radius on the ablation depth, only the exact formula was used. Calculations were performed using commercial mathematics software (Mathcad 2001, Mathsoft, Inc.). To prepare graphs and tables, the data were exported to a spreadsheet program (Excel). Because of the nature of the approximations used, the exact ablation depth is always larger than the ablation depth predicted by the Munnerlyn approximated formula. The difference between the Munnerlyn exact formula and the Munnerlyn approximated formula is designated as delta (⌬). Table 1 shows the difference for optical zones between 3.0 and 9.0 mm in 1.0 mm increments and corrections from ⫺1.00 to ⫺12.00 D in 1.00 D increments. To illustrate the results, 2 graphs were created: the ablation depth versus the optical zone for a – 8.0 D correction with a typical initial corneal radius (7.5 mm) and a steep initial radius (7.0 mm, which corresponds to a K-reading of 48.0 D), and the ablation depth versus the diopters of correction for a 6.0 mm optical zone with typical and steep initial radii.
Results Table 1 is a complete tabulation of delta for various optical zones and corrections. Some typical values are worth attention. For an average cornea, the difference between the exact and approximate formulas for a 6.0 mm optical zone was 6.0 m for a ⫺5.0 D correction and 10.0 m for a ⫺10.0 D correction. For an 8.0 mm optical zone, the difference between the exact
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Table 1. Difference between predicted corneal depth derived from the exact formula and the Munnerlyn formula in microns, ⌬. Exact ablation given in parentheses. Diopters of Correction
Optical Zone Size (mm) 3
4
5
6
7
8
9
⫺1
*
(3)
*
(9)
1 (13)
3 (19)
4 (26)
8 (35)
⫺2
*
(6)
1 (11)
1 (18)
2 (26)
5 (38)
9 (51)
15 (69)
⫺3
*
(9)
1 (17)
2 (27)
4 (40)
7 (56)
13 (77)
22 (103)
⫺4
* (12)
1 (22)
2 (36)
5 (53)
9 (74)
17 (102)
28 (136)
⫺5
* (15)
1 (28)
3 (44)
6 (66)
11 (93)
20 (127)
34 (169)
⫺6
* (18)
1 (33)
3 (53)
7 (79)
13 (111)
23 (151)
39 (201)
⫺7
* (21)
2 (39)
4 (62)
8 (92)
15 (129)
26 (176)
45 (234)
⫺8
* (25)
2 (44)
4 (71)
9 (105)
17 (147)
30 (200)
50 (266)
(6)
1
⫺9
1 (28)
2 (50)
4 (79)
9 (117)
18 (165)
32 (224)
54 (297)
⫺10
1 (31)
2 (55)
5 (88)
10 (130)
20 (182)
35 (248)
59 (329)
⫺11
1 (34)
2 (61)
5 (97)
11 (142)
21 (248)
37 (272)
63 (360)
⫺12
1 (37)
2 (66)
5 (105)
12 (156)
22 (329)
39 (295)
66 (390)
*Less than 1.0 m. Values rounded to nearest micron.
and approximate formulas was 20.0 m for a ⫺5.0 D correction and 35.0 m for a ⫺10.0 D correction. The calculations show that optical zone size has the greatest effect on delta. The Munnerlyn approximated formula is based on the first 2 terms of a Taylor series expansion. The third term in the expansion varies as the fourth power of the optical zone size (equation A-3, Appendix). Consequently, the error in ablation depth varies as the fourth power of the optical zone (Figure 2). The effect of diopters of correction is less pronounced but clinically important. Delta is directly proportional to the diopters of correction, thus delta increases about linearly with correction (Figure 3). From Figures 2 and 3, it is clear that the initial corneal radius has little effect on delta. At the largest optical zones and diopters of correction, the difference in ablation depths between a steep cornea and a typical cornea is about 6.0 m. Ablation depth could be reasonably approximated by adding an adjustment factor given by equation 3 to the ablation depth predicted by the Munnerlyn approximated formula (Figure 4). Adjustment Factor ⬵ 0.0011 · OZ 4
(3)
Discussion Because of intraoperative variables, there are no accurate methods of predicting the exact RSD preopera1206
tively. Nevertheless, it is current clinical practice to estimate RSD to ensure that the postoperative cornea has sufficient mechanical integrity to prevent progressive corneal ectasia. Two methods are commonly used to estimate ablation depth preoperatively. One uses the software provided in some laser systems to calculate the ablation depth based on operative parameters (eg, optical zone size and desired correction). The other models the ablation geometrically and calculates the ablation depth based on the mathematical model. Each approach relies on a set of assumptions, and the validity of these assumptions is open to question. Neither approach has been shown to be more accurate. Laser manufacturers assume that each pulse removes a constant amount of tissue. Ablation depth is calculated by simply multiplying the number of pulses by the amount of tissue removed by each pulse. However, the actual amount of tissue removed per pulse depends on corneal hydration, the air in the operating room, plume evacuation method, and other patient- and surgeon-specific factors. Conversely, if the ablation depth is based on a mathematical model, the results will only be as accurate as the underlying geometric assumptions. Today, the Munnerlyn approximated formula is commonly used to estimate ablation depth and it is
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Figure 2. (Chang) Ablation depths for varying optical zones with a fixed correction of – 8.0 D comparing the Munnerlyn exact formula to the Munnerlyn approximated formula. The ablation depths for a normal cornea correspond to a K-reading of 45.0 D and for a steep cornea, to a K-reading of 50.0 D.
Figure 3. (Chang) Ablation depths for diopters of correction with a fixed optical zone of 6.0 mm comparing the Munnerlyn exact formula to the Munnerlyn approximated formula. The ablation depths for a normal cornea correspond to a K-reading of 45.0 D and for a steep cornea, to a K-reading of 50.0 D.
based on spherical geometry. Recently, mathematical models of other ablation profiles have been introduced.14 We did not investigate alternative geometric models or determine the validity of the assumptions underlying the Munnerlyn approximated formula. Empirical evidence has shown that this formula provides a reasonable estimate of ablation depth for small optical zones when the ablation consists of a single optical zone with a blend. Multizone ablations are not modeled well by the Munnerlyn approximated formula (although theoretically it may be possible to model multizone ablations by repeated application of the Munnerlyn approximated formula).
In this age of electronic computation, why not simply calculate the ablation depth using the exact formula? There are theoretical and practical reasons to use an approximation. Theoretically, the exact formula requires 3 variables (initial radius of curvature in addition to correction and optical zone size), whereas the Munnerlyn approximated formula involves only 2 variables (correction and optical zone size). As we have shown, the extra variable adds little to the accuracy of the calculation and is a potential source of error. It is immediately obvious from the Munnerlyn approximated formula that ablation depth increases as the square of the optical zone size, an important clinical relationship that is not apparent in the exact formula.
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Figure 4. (Chang) Ablation depths for varying optical zones with a fixed correction of – 8.0 D comparing the Munnerlyn exact formula to the Munnerlyn approximated formula with the correction factor added. The ablation depths for a normal cornea correspond to a K-reading of 45.0 D and for a steep cornea, to a K-reading of 50.0 D.
There are also practical reasons to use an approximation. It is possible to “program” a formula into a computer or calculator, but the more complicated the formula, the more likely an error will occur in the program. A programming error may be subtle and go undetected, giving erroneous results in many patients. Even if properly programmed, a computer may not be readily available, so it is helpful to have a formula that is easily memorized and calculated. Data entry errors are a potential source of error. The Munnerlyn approximated formula is so simple that it can be estimated mentally and used to check the results of any calculation. At the time the Munnerlyn formula was derived, it was anticipated that optical zones would be 5.0 mm or smaller. The Munnerlyn approximated formula is extremely accurate for small optical zones so there was no advantage to using a more complicated formula. However, for today’s larger optical zones and corrections, the Munnerlyn approximated formula underestimates the ablation depth. An accurate estimation of ablation depth is most critical in patients with large optical zones and large corrections since they are at greatest risk for postoperative ectasia. The correction factor we derived is slightly more complicated than the Munnerlyn approximated formula because it involves the fourth power of the optical zone instead of just the square of the optical zone size. Nevertheless, the correction factor is similar to the Munnerlyn approximated formula in that it involves only 2 variables, is easy to memorize, and is relatively easy to calculate. While it is always possible to use the exact 1208
formula to calculate ablation depth, the correction factor should provide sufficient accuracy for clinical purposes.
Appendix Munnerlyn and coworkers began by modeling the anterior corneal surface as a sphere and derived a formula for the entire ablation profile (ie, ablation depth as a function of distance from the optical axis) for photorefractive keratectomy. Since we are only interested in ablation depth, we present a similar but slightly simpler derivation. The sagittal depth of a circle, S, is the distance between a chord and the circle measured along the chord’s perpendicular bisector. Figure 1 illustrates a myopic ablation assuming spherical geometry. The ablation depth at the optical axis is the difference between the sagittal depths of the preoperative and postoperative cornea. In general, for any circle of radius, R, the exact sagittal depth is S⫽R⫺
冑
R2 ⫺
C2 4
(A-1)
where C is the chord length, which equals the optical zone diameter. Factoring out R and replacing C, with the optical zone, OZ, gives
冉 冑
S⫽R 1⫺
1⫺
OZ 2 4R 2
冊
(A-2)
The square root term can be expanded in a power series
冑 冉 冊 1⫺
OZ 4R
2
⬵ 1⫺
OZ 4 OZ 6 OZ 2 ⫺ ⫺ 8 · R 2 128 · R 4 1024 · R 6
⫺ Higher Order Terms
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(A-3)
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For optical zones that are small compared to the corneal radius, only the first 2 terms are required to reasonably approximate the sagittal depth. Thus,
冋 冉 冉 冊
Approximate Sagittal Depth ⬵ R 1 ⫺ 1 ⫺
OZ 2 8R 2
2
⫽R
OZ OZ ⫽ 8R 2 8R
冊册
冉
OZ 2 1 1 ⫺ 8 R1 R2
(A-4)
冊
R2 ⫽
⫹
so the amount of correction produced by the ablation in diopters, D, is
冉
冊
(A-7)
Thus,
冉
冊
1 1 D ⫺ ⫽ R2 R1 共n c ⫺ 1兲
(A-8)
Substituting this into equation A-5, the approximate ablation depth is Approximate Ablation Depth ⬵ ⫺
OZ 2 Approximate Ablation Depth ⬵ D 3
(A-10)
where it is understood that D is positive for myopic ablations. This equation is most often referred to in the literature as the Munnerlyn formula. Equation A-10 is an approximate
冑冋
冑
R 1 · 共n c ⫺ 1兲 ⫺ nc ⫺ 1 ⫹ R1 · D
R · 共n c ⫺ 1兲 nc ⫺ 1 ⫹ R1 · D
册
2
⫺
R 12 ⫺
OZ 2 4
OZ 2 4
(A-13)
Approximate Ablation Depth
冉 冊
This is the equation published by Munnerlyn and coauthors.12 For myopic ablations, D is negative. Today, most clinicians use a positive value for D even though the ablation is myopic so the minus sign can be dropped. A slightly different formula can be obtained by substituting a numerical value for nc, the corneal refractive index. Munnerlyn and coauthors used 1.377 for the value of nc, the refractive index of the cornea: 1.377 ⫺ 1 ⫽ 0.377, which is approximately 0.375 or 3/8. Thus,
(A-12)
gives the exact formula for the ablation depth in terms of the initial corneal radius, R1; correction, D; optical zone size, OZ; and corneal refractive index, nc. This is the formula used to calculate the exact ablation depth as shown in Munnerlyn’s paper.12 An approximate but more accurate formula may be obtained by using the first 3 terms in the power series expansion. Equation 4 would then be
OZ 2 D 8 nc ⫺ 1
(A-9)
R 1 · 共n c ⫺ 1兲 共n c ⫺ 1兲 ⫹ D · R 1
Thus, Exact Ablation Depth
(A-6)
1 1 ⫺ D ⫽ P f ⫺ P i ⫽ 共n c ⫺ 1兲 R2 R1
(A-11)
Alternately, equation 7 can be rearranged to solve for R2:
⫽ R1 ⫺
共n c ⫺ 1兲 R
OZ 2 4
⫺ R 2 ⫹ 冑R 22 ⫺ O.
(A-5)
Now, the (paraxial) power of the anterior corneal surface is generally P⫽
冑
Exact Ablation Depth ⫽ S 1 ⫺ S 2 ⫽ R 1⫺ R 12 ⫺
2
Thus, the difference in sagittal depth between the preablation and postablation cornea is approximately Approximate Ablation Depth ⫽
but accurate formula for small optical zones. However, as the optical zone increases relative to the corneal radius, the approximation becomes inaccurate. The difference in sagittal depth between the preoperative and postoperative cornea can be calculated exactly using
冊
(A-14)
⬵
1· 1 OZ 2 ⫺ · 8 R1 R2
⫹
OZ 4 1 共R 22 ⫹ R 1R 2 ⫹ 1兲 1 ⫺ · · 128 R1 R2 R 12 · R 22
(A-15)
⬵
冉
冊
冉
OZ 2 1 OZ 4 1 1 1 ⫺ ⫹ · 3⫺ 8 R1 R2 128 R 1 R 23
This simplifies to Approximate Ablation Depth
冉 冉
冊 冊
Thus, Approximate Ablation Depth ⬵
·
OZ 2 OZ 4 D D⫹ · 3 128 共n c ⫺ 1兲
共R 22 ⫹ R 1R 2 ⫹ R 12兲 R 12 · R 22
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(A-16)
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where again the clinical convention is followed so D is positive for myopic ablations. The first term is simply the ablation depth predicted by the Munnerlyn formula. The second term is therefore a correction factor. We found that for several typical values of preoperative and postoperative corneal radii, the coefficient of the second term was about 0.0011. Thus, Adjustment Factor ⬵ .0011 · OZ 4 · D
(A-17)
For large optical zones and corrections, the ablation depth can be calculated by adding this adjustment factor to the depth calculated by the approximated Munnerlyn formula.
References 1. Joo C-K, Kim T-G. Corneal ectasia detected after laser in situ keratomileusis for correction of less than –12 diopters of myopia. J Cataract Refract Surg 2000; 26:292–295 2. Seiler T, Quurke AW. Iatrogenic keratectasia after LASIK in a case of forme fruste keratoconus. J Cataract Refract Surg 1998; 24:1007–1009 3. McLeod SD, Kisla TA, Caro NC, McMahon TT. Iatrogenic keratoconus: corneal ectasia following laser in situ keratomileusis for myopia. Arch Ophthalmol 2000; 118: 282–284 4. Amoils SP, Deist MB, Gous P, Amoils PM. Iatrogenic keratectasia after laser in situ keratomileusis for less than – 4.0 to –7.0 diopters of myopia. J Cataract Refract Surg 2000; 26:967–977 5. Seiler T, Koufala K, Richter G. Iatrogenic keratectasia after laser in situ keratomileusis. J Refract Surg 1998; 14:312–317
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6. Geggel HS, Talley AR. Delayed onset keratectasia following laser in situ keratomileusis. J Cataract Refract Surg 1999; 25:582–586 7. Durairaj VD, Balentine J, Kouyoumdjian G, et al. The predictability of corneal flap thickness and tissue laser ablation in laser in situ keratomileusis. Ophthalmology 2000; 107:2140 –2143 8. Liu K-Y, Lam DSC. Direct measurement of microkeratome gap width by electron microscopy. J Cataract Refract Surg 2001; 27:924 –927 9. Thornton SP. A guide to ultrasonic pachymeters. J Cataract Refract Surg 1986; 12:416 –419 10. Thornton SP. A guide to pachymeters. Ophthalmic Surg 1984; 15:993–995 11. Kim W-S, Jo J-M. Corneal hydration affects ablation during laser in situ keratomileusis surgery. Cornea 2001; 20:394 –397 12. Munnerlyn CR, Koons SJ, Marshall J. Photorefractive keratectomy: a technique for laser refractive surgery. J Cataract Refract Surg 1988; 14:46 –52 13. Stewart J. Calculus. Early Transcendals, 4th ed. Pacific Grove, CA, Brooks/Cole, 1999 14. Gatinel D, Hoang-Xuan T, Azar D. Determination of corneal asphericity after myopia surgery with the excimer laser: a mathematical model. Invest Ophthalmol Vis Sci 2001; 42:1736 –1742 From the Department of Ophthalmology, University of Texas Medical Branch at Galveston, Galveston, Texas, USA. None of the authors has a financial interest in any product mentioned.
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Purpose: To determine whether the error in ablation depth produced by approximations inherent in the Munnerlyn formula are clinically significant when estimating residual corneal stromal depth for the evaluation before refractive surgery. Setting: Department of Ophthalmology, University of Texas Medical Branch, Galveston, Texas, USA. Methods: Using identical geometric assumptions, the exact ablation depth was calculated and compared to the approximate ablation depth predicted by the Munnerlyn formula. An adjustment factor was then derived for large optical zones and corrections. Results: The exact ablation depth is always larger than the ablation depth predicted by Munnerlyn’s formula. Analysis found the error in ablation depth varied as the fourth power of the optical zone and linearly with correction. The initial corneal radius had little effect on the difference. The ablation depth could be reasonably approximated by adding an adjustment factor for large optical zones and refractive corrections. Conclusions: In patients with large optical zones, it may be preferable to calculate tissue ablation depth using the exact formula. Alternately, the Munnerlyn formula can be used to calculate ablation depth and then an adjustment factor can be added. J Cataract Refract Surg 2003; 29:1204 –1210 © 2003 ASCRS and ESCRS
P
rogressive corneal ectasia is an uncommon but serious postoperative complication of laser in situ keratomileusis (LASIK).1– 4 The etiology of it is not fully understood, but current thinking is that the problem can be minimized or avoided by maintaining a residual stromal depth (RSD) of 250 m.5,6 The RSD depends on many intraoperative variables. Flap thickness can differ from the microkeratome setting by 50 m.7,8 Current clinical ultrasonic pachymeters have an accuracy of 40 m.9,10 Ablation depth can vary with the ablation profile (multizone versus sinAccepted for publication September 25, 2002. Reprint requests to Austin W. Chang, MD, Department of Ophthalmology, University of Texas Medical Branch, Galveston, Texas 775550787, USA. © 2003 ASCRS and ESCRS Published by Elsevier Inc.
gle zone), atmospheric conditions, corneal hydration, corneal asphericity, and laser parameters.11 Intraoperative variables make it impossible to precisely predict the RSD. Nevertheless, clinicians must in some way estimate it preoperatively. The decision to perform LASIK or another refractive procedure or avoid refractive surgery is based partly on the estimated RSD. Currently, the RSD is calculated by subtracting flap thickness and ablation depth from the central pachymetry measurements. There are 2 ways to estimate ablation depth: using the preoperative parameters or using the Munnerlyn approximated formula.12 The Munnerlyn approximated formula is theoretical and based on the assumption that both the cornea and ablation profile are spherical. It is not exact but only an estimate of the ablation depth predicted by spherical 0886-3350/03/$–see front matter doi:10.1016/S0886-3350(02)01918-1
LABORATORY SCIENCE: CORNEAL ABLATION DEPTH AND MUNNERLYN FORMULA
geometry. The Munnerlyn approximated formula is based on the first 2 terms of a Taylor series expansion13 and systematically underestimates the theoretical ablation depth. Using the same geometric assumptions, we calculated the exact ablation depth and compared it to the approximate ablation depth predicted by the Munnerlyn approximated formula. We calculated the theoretical exact ablation depth for various optical zone sizes and various amounts of correction to determine whether the difference between the 2 formulas is clinically significant. We then analyzed the data and series expansion to derive an adjustment factor to better approximate the exact theoretical ablation depth predicted by spherical geometry (Appendix).
Materials and Methods The Appendix shows a detailed derivation of the formulas. Here, the issues are discussed qualitatively. Munnerlyn based the theoretical ablation depth for photorefractive keratectomy on a spherical corneal geometry. Figure 1 depicts a spherical myopic correction. Only myopic correction is considered because ablation depth is less critical in small hyperopic corrections. In myopic eyes, the ablation depth is greatest along the optical axis. The exact ablation depth for spherical geometry can be calculated using equation 1. Munnerlyn Theoretical Exact Ablation Depth ⫽ R1 ⫺ ⫺ ⫹
冑 冑冋
R 1 · 共n ⫺ 1兲 n ⫺ 1 ⫹ R1 · D
R 12 ⫺
OZ 2 4
R 1 · 共n ⫺ 1兲 n ⫺ 1 ⫹ R1 · D
册
2
⫺
OZ 2 4
(1)
where n is the refractive index of the cornea (1.376), D is the correction in diopters, and the other variables are illustrated in Figure 1. The Munnerlyn approximated formula12 used clinically is 2
Approximate Ablation Depth ⬵
OZ D 3
(2)
It is difficult to directly compare the 2 formulas because the Munnerlyn exact formula depends on 3 variables (optical zone, correction in diopters [D], and initial radius of curvature) whereas the Munnerlyn approximated formula depends
Figure 1. (Chang) Spherical myopic correction: R1 and R2 ⫽ initial and final radii of curvature; C ⫽ chord length or diameter of the optical zone; S1 and S2 ⫽ sagittal depth of the preoperative and postoperative cornea.
on only 2 variables (correction and optical zone). To directly compare the 2 equations, a typical corneal radius of curvature of 7.5 mm (which corresponds to a K-reading of 45.0 D) was used. To determine the effect of the initial radius on the ablation depth, only the exact formula was used. Calculations were performed using commercial mathematics software (Mathcad 2001, Mathsoft, Inc.). To prepare graphs and tables, the data were exported to a spreadsheet program (Excel). Because of the nature of the approximations used, the exact ablation depth is always larger than the ablation depth predicted by the Munnerlyn approximated formula. The difference between the Munnerlyn exact formula and the Munnerlyn approximated formula is designated as delta (⌬). Table 1 shows the difference for optical zones between 3.0 and 9.0 mm in 1.0 mm increments and corrections from ⫺1.00 to ⫺12.00 D in 1.00 D increments. To illustrate the results, 2 graphs were created: the ablation depth versus the optical zone for a – 8.0 D correction with a typical initial corneal radius (7.5 mm) and a steep initial radius (7.0 mm, which corresponds to a K-reading of 48.0 D), and the ablation depth versus the diopters of correction for a 6.0 mm optical zone with typical and steep initial radii.
Results Table 1 is a complete tabulation of delta for various optical zones and corrections. Some typical values are worth attention. For an average cornea, the difference between the exact and approximate formulas for a 6.0 mm optical zone was 6.0 m for a ⫺5.0 D correction and 10.0 m for a ⫺10.0 D correction. For an 8.0 mm optical zone, the difference between the exact
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Table 1. Difference between predicted corneal depth derived from the exact formula and the Munnerlyn formula in microns, ⌬. Exact ablation given in parentheses. Diopters of Correction
Optical Zone Size (mm) 3
4
5
6
7
8
9
⫺1
*
(3)
*
(9)
1 (13)
3 (19)
4 (26)
8 (35)
⫺2
*
(6)
1 (11)
1 (18)
2 (26)
5 (38)
9 (51)
15 (69)
⫺3
*
(9)
1 (17)
2 (27)
4 (40)
7 (56)
13 (77)
22 (103)
⫺4
* (12)
1 (22)
2 (36)
5 (53)
9 (74)
17 (102)
28 (136)
⫺5
* (15)
1 (28)
3 (44)
6 (66)
11 (93)
20 (127)
34 (169)
⫺6
* (18)
1 (33)
3 (53)
7 (79)
13 (111)
23 (151)
39 (201)
⫺7
* (21)
2 (39)
4 (62)
8 (92)
15 (129)
26 (176)
45 (234)
⫺8
* (25)
2 (44)
4 (71)
9 (105)
17 (147)
30 (200)
50 (266)
(6)
1
⫺9
1 (28)
2 (50)
4 (79)
9 (117)
18 (165)
32 (224)
54 (297)
⫺10
1 (31)
2 (55)
5 (88)
10 (130)
20 (182)
35 (248)
59 (329)
⫺11
1 (34)
2 (61)
5 (97)
11 (142)
21 (248)
37 (272)
63 (360)
⫺12
1 (37)
2 (66)
5 (105)
12 (156)
22 (329)
39 (295)
66 (390)
*Less than 1.0 m. Values rounded to nearest micron.
and approximate formulas was 20.0 m for a ⫺5.0 D correction and 35.0 m for a ⫺10.0 D correction. The calculations show that optical zone size has the greatest effect on delta. The Munnerlyn approximated formula is based on the first 2 terms of a Taylor series expansion. The third term in the expansion varies as the fourth power of the optical zone size (equation A-3, Appendix). Consequently, the error in ablation depth varies as the fourth power of the optical zone (Figure 2). The effect of diopters of correction is less pronounced but clinically important. Delta is directly proportional to the diopters of correction, thus delta increases about linearly with correction (Figure 3). From Figures 2 and 3, it is clear that the initial corneal radius has little effect on delta. At the largest optical zones and diopters of correction, the difference in ablation depths between a steep cornea and a typical cornea is about 6.0 m. Ablation depth could be reasonably approximated by adding an adjustment factor given by equation 3 to the ablation depth predicted by the Munnerlyn approximated formula (Figure 4). Adjustment Factor ⬵ 0.0011 · OZ 4
(3)
Discussion Because of intraoperative variables, there are no accurate methods of predicting the exact RSD preopera1206
tively. Nevertheless, it is current clinical practice to estimate RSD to ensure that the postoperative cornea has sufficient mechanical integrity to prevent progressive corneal ectasia. Two methods are commonly used to estimate ablation depth preoperatively. One uses the software provided in some laser systems to calculate the ablation depth based on operative parameters (eg, optical zone size and desired correction). The other models the ablation geometrically and calculates the ablation depth based on the mathematical model. Each approach relies on a set of assumptions, and the validity of these assumptions is open to question. Neither approach has been shown to be more accurate. Laser manufacturers assume that each pulse removes a constant amount of tissue. Ablation depth is calculated by simply multiplying the number of pulses by the amount of tissue removed by each pulse. However, the actual amount of tissue removed per pulse depends on corneal hydration, the air in the operating room, plume evacuation method, and other patient- and surgeon-specific factors. Conversely, if the ablation depth is based on a mathematical model, the results will only be as accurate as the underlying geometric assumptions. Today, the Munnerlyn approximated formula is commonly used to estimate ablation depth and it is
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Figure 2. (Chang) Ablation depths for varying optical zones with a fixed correction of – 8.0 D comparing the Munnerlyn exact formula to the Munnerlyn approximated formula. The ablation depths for a normal cornea correspond to a K-reading of 45.0 D and for a steep cornea, to a K-reading of 50.0 D.
Figure 3. (Chang) Ablation depths for diopters of correction with a fixed optical zone of 6.0 mm comparing the Munnerlyn exact formula to the Munnerlyn approximated formula. The ablation depths for a normal cornea correspond to a K-reading of 45.0 D and for a steep cornea, to a K-reading of 50.0 D.
based on spherical geometry. Recently, mathematical models of other ablation profiles have been introduced.14 We did not investigate alternative geometric models or determine the validity of the assumptions underlying the Munnerlyn approximated formula. Empirical evidence has shown that this formula provides a reasonable estimate of ablation depth for small optical zones when the ablation consists of a single optical zone with a blend. Multizone ablations are not modeled well by the Munnerlyn approximated formula (although theoretically it may be possible to model multizone ablations by repeated application of the Munnerlyn approximated formula).
In this age of electronic computation, why not simply calculate the ablation depth using the exact formula? There are theoretical and practical reasons to use an approximation. Theoretically, the exact formula requires 3 variables (initial radius of curvature in addition to correction and optical zone size), whereas the Munnerlyn approximated formula involves only 2 variables (correction and optical zone size). As we have shown, the extra variable adds little to the accuracy of the calculation and is a potential source of error. It is immediately obvious from the Munnerlyn approximated formula that ablation depth increases as the square of the optical zone size, an important clinical relationship that is not apparent in the exact formula.
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Figure 4. (Chang) Ablation depths for varying optical zones with a fixed correction of – 8.0 D comparing the Munnerlyn exact formula to the Munnerlyn approximated formula with the correction factor added. The ablation depths for a normal cornea correspond to a K-reading of 45.0 D and for a steep cornea, to a K-reading of 50.0 D.
There are also practical reasons to use an approximation. It is possible to “program” a formula into a computer or calculator, but the more complicated the formula, the more likely an error will occur in the program. A programming error may be subtle and go undetected, giving erroneous results in many patients. Even if properly programmed, a computer may not be readily available, so it is helpful to have a formula that is easily memorized and calculated. Data entry errors are a potential source of error. The Munnerlyn approximated formula is so simple that it can be estimated mentally and used to check the results of any calculation. At the time the Munnerlyn formula was derived, it was anticipated that optical zones would be 5.0 mm or smaller. The Munnerlyn approximated formula is extremely accurate for small optical zones so there was no advantage to using a more complicated formula. However, for today’s larger optical zones and corrections, the Munnerlyn approximated formula underestimates the ablation depth. An accurate estimation of ablation depth is most critical in patients with large optical zones and large corrections since they are at greatest risk for postoperative ectasia. The correction factor we derived is slightly more complicated than the Munnerlyn approximated formula because it involves the fourth power of the optical zone instead of just the square of the optical zone size. Nevertheless, the correction factor is similar to the Munnerlyn approximated formula in that it involves only 2 variables, is easy to memorize, and is relatively easy to calculate. While it is always possible to use the exact 1208
formula to calculate ablation depth, the correction factor should provide sufficient accuracy for clinical purposes.
Appendix Munnerlyn and coworkers began by modeling the anterior corneal surface as a sphere and derived a formula for the entire ablation profile (ie, ablation depth as a function of distance from the optical axis) for photorefractive keratectomy. Since we are only interested in ablation depth, we present a similar but slightly simpler derivation. The sagittal depth of a circle, S, is the distance between a chord and the circle measured along the chord’s perpendicular bisector. Figure 1 illustrates a myopic ablation assuming spherical geometry. The ablation depth at the optical axis is the difference between the sagittal depths of the preoperative and postoperative cornea. In general, for any circle of radius, R, the exact sagittal depth is S⫽R⫺
冑
R2 ⫺
C2 4
(A-1)
where C is the chord length, which equals the optical zone diameter. Factoring out R and replacing C, with the optical zone, OZ, gives
冉 冑
S⫽R 1⫺
1⫺
OZ 2 4R 2
冊
(A-2)
The square root term can be expanded in a power series
冑 冉 冊 1⫺
OZ 4R
2
⬵ 1⫺
OZ 4 OZ 6 OZ 2 ⫺ ⫺ 8 · R 2 128 · R 4 1024 · R 6
⫺ Higher Order Terms
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(A-3)
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For optical zones that are small compared to the corneal radius, only the first 2 terms are required to reasonably approximate the sagittal depth. Thus,
冋 冉 冉 冊
Approximate Sagittal Depth ⬵ R 1 ⫺ 1 ⫺
OZ 2 8R 2
2
⫽R
OZ OZ ⫽ 8R 2 8R
冊册
冉
OZ 2 1 1 ⫺ 8 R1 R2
(A-4)
冊
R2 ⫽
⫹
so the amount of correction produced by the ablation in diopters, D, is
冉
冊
(A-7)
Thus,
冉
冊
1 1 D ⫺ ⫽ R2 R1 共n c ⫺ 1兲
(A-8)
Substituting this into equation A-5, the approximate ablation depth is Approximate Ablation Depth ⬵ ⫺
OZ 2 Approximate Ablation Depth ⬵ D 3
(A-10)
where it is understood that D is positive for myopic ablations. This equation is most often referred to in the literature as the Munnerlyn formula. Equation A-10 is an approximate
冑冋
冑
R 1 · 共n c ⫺ 1兲 ⫺ nc ⫺ 1 ⫹ R1 · D
R · 共n c ⫺ 1兲 nc ⫺ 1 ⫹ R1 · D
册
2
⫺
R 12 ⫺
OZ 2 4
OZ 2 4
(A-13)
Approximate Ablation Depth
冉 冊
This is the equation published by Munnerlyn and coauthors.12 For myopic ablations, D is negative. Today, most clinicians use a positive value for D even though the ablation is myopic so the minus sign can be dropped. A slightly different formula can be obtained by substituting a numerical value for nc, the corneal refractive index. Munnerlyn and coauthors used 1.377 for the value of nc, the refractive index of the cornea: 1.377 ⫺ 1 ⫽ 0.377, which is approximately 0.375 or 3/8. Thus,
(A-12)
gives the exact formula for the ablation depth in terms of the initial corneal radius, R1; correction, D; optical zone size, OZ; and corneal refractive index, nc. This is the formula used to calculate the exact ablation depth as shown in Munnerlyn’s paper.12 An approximate but more accurate formula may be obtained by using the first 3 terms in the power series expansion. Equation 4 would then be
OZ 2 D 8 nc ⫺ 1
(A-9)
R 1 · 共n c ⫺ 1兲 共n c ⫺ 1兲 ⫹ D · R 1
Thus, Exact Ablation Depth
(A-6)
1 1 ⫺ D ⫽ P f ⫺ P i ⫽ 共n c ⫺ 1兲 R2 R1
(A-11)
Alternately, equation 7 can be rearranged to solve for R2:
⫽ R1 ⫺
共n c ⫺ 1兲 R
OZ 2 4
⫺ R 2 ⫹ 冑R 22 ⫺ O.
(A-5)
Now, the (paraxial) power of the anterior corneal surface is generally P⫽
冑
Exact Ablation Depth ⫽ S 1 ⫺ S 2 ⫽ R 1⫺ R 12 ⫺
2
Thus, the difference in sagittal depth between the preablation and postablation cornea is approximately Approximate Ablation Depth ⫽
but accurate formula for small optical zones. However, as the optical zone increases relative to the corneal radius, the approximation becomes inaccurate. The difference in sagittal depth between the preoperative and postoperative cornea can be calculated exactly using
冊
(A-14)
⬵
1· 1 OZ 2 ⫺ · 8 R1 R2
⫹
OZ 4 1 共R 22 ⫹ R 1R 2 ⫹ 1兲 1 ⫺ · · 128 R1 R2 R 12 · R 22
(A-15)
⬵
冉
冊
冉
OZ 2 1 OZ 4 1 1 1 ⫺ ⫹ · 3⫺ 8 R1 R2 128 R 1 R 23
This simplifies to Approximate Ablation Depth
冉 冉
冊 冊
Thus, Approximate Ablation Depth ⬵
·
OZ 2 OZ 4 D D⫹ · 3 128 共n c ⫺ 1兲
共R 22 ⫹ R 1R 2 ⫹ R 12兲 R 12 · R 22
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(A-16)
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where again the clinical convention is followed so D is positive for myopic ablations. The first term is simply the ablation depth predicted by the Munnerlyn formula. The second term is therefore a correction factor. We found that for several typical values of preoperative and postoperative corneal radii, the coefficient of the second term was about 0.0011. Thus, Adjustment Factor ⬵ .0011 · OZ 4 · D
(A-17)
For large optical zones and corrections, the ablation depth can be calculated by adding this adjustment factor to the depth calculated by the approximated Munnerlyn formula.
References 1. Joo C-K, Kim T-G. Corneal ectasia detected after laser in situ keratomileusis for correction of less than –12 diopters of myopia. J Cataract Refract Surg 2000; 26:292–295 2. Seiler T, Quurke AW. Iatrogenic keratectasia after LASIK in a case of forme fruste keratoconus. J Cataract Refract Surg 1998; 24:1007–1009 3. McLeod SD, Kisla TA, Caro NC, McMahon TT. Iatrogenic keratoconus: corneal ectasia following laser in situ keratomileusis for myopia. Arch Ophthalmol 2000; 118: 282–284 4. Amoils SP, Deist MB, Gous P, Amoils PM. Iatrogenic keratectasia after laser in situ keratomileusis for less than – 4.0 to –7.0 diopters of myopia. J Cataract Refract Surg 2000; 26:967–977 5. Seiler T, Koufala K, Richter G. Iatrogenic keratectasia after laser in situ keratomileusis. J Refract Surg 1998; 14:312–317
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6. Geggel HS, Talley AR. Delayed onset keratectasia following laser in situ keratomileusis. J Cataract Refract Surg 1999; 25:582–586 7. Durairaj VD, Balentine J, Kouyoumdjian G, et al. The predictability of corneal flap thickness and tissue laser ablation in laser in situ keratomileusis. Ophthalmology 2000; 107:2140 –2143 8. Liu K-Y, Lam DSC. Direct measurement of microkeratome gap width by electron microscopy. J Cataract Refract Surg 2001; 27:924 –927 9. Thornton SP. A guide to ultrasonic pachymeters. J Cataract Refract Surg 1986; 12:416 –419 10. Thornton SP. A guide to pachymeters. Ophthalmic Surg 1984; 15:993–995 11. Kim W-S, Jo J-M. Corneal hydration affects ablation during laser in situ keratomileusis surgery. Cornea 2001; 20:394 –397 12. Munnerlyn CR, Koons SJ, Marshall J. Photorefractive keratectomy: a technique for laser refractive surgery. J Cataract Refract Surg 1988; 14:46 –52 13. Stewart J. Calculus. Early Transcendals, 4th ed. Pacific Grove, CA, Brooks/Cole, 1999 14. Gatinel D, Hoang-Xuan T, Azar D. Determination of corneal asphericity after myopia surgery with the excimer laser: a mathematical model. Invest Ophthalmol Vis Sci 2001; 42:1736 –1742 From the Department of Ophthalmology, University of Texas Medical Branch at Galveston, Galveston, Texas, USA. None of the authors has a financial interest in any product mentioned.
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