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Optimal Astigmatism to Enhance Depth of Focus after Cataract Surgery
Optimal Astigmatism to Enhance Depth of Focus after Cataract Surgery MARK R. SAWUSCH, MD,l DAVID L. GUYTON, MD2
Abstract: A small amount of myopic astigmatism can enhance the depth of focus of the pseudophakic eye, optimally providing at least 20/30 visual acuity for both near and distance fixation. For given spherocylindrical refractive errors and fixation distances, the cross-sectional area of Sturm's conoid at the retina was calculated for a schematic eye. These data were used to determine the optimal astigmatic error needed to obtain maximum depth of focus and least theoretical blur for any given spherical equivalent refractive error. Optimal depth of focus was obtained when the plus cylindrical component equaled negative sphere - 0.25 diopters. The near and distance visual acuities of ten pseudophakic patients with induced refractive errors were highly correlated with this model. Low myopic astigmatism after cataract surgery may represent an alternative to multifocal intraocular lenses by providing spectacle independence. Ophthalmology 1991; 98:1025-1029
The calculation of intraocular lens (lOL) power, and intraoperative or postoperative modification of corneal astigmatism allow the surgeon significant control of the refraction after cataract surgery. This raises the question of how to select the optimal refraction in a monofocal, pseudophakic eye lacking accommodation. The optimal postoperative refraction for a given patient may depend on several factors, including occupation and lifestyle and the need to maintain isometropia. 1 Most surgeons advise low myopia (in sedentary patients) or emmetropia (in active patients) as an optimal refraction, thereby allowing relatively good acuity without glasses at the fixation distances most often used by the patient. 2 However, bifocal lenses are usually necessary to provide good acuity for both near and distance fixation. Multifocal IOLs are currently under development to enhance the depth of focus and to restore a degree of Originally received : September 28, 1990. Revision accepted: March 5, 1991. 1 2
The Doheny Eye Institute, USC School of Medicine. Los Angeles. The Wilmer Ophthalmological Institute, The Johns Hopkins University School of Medicine, Baltimore.
Presented at the American Academy of Ophthalmology Annual Meeting, Atlanta, Oct/Nov 1990. Correspondence to Mark R. Sawusch, MD, Doheny Eye Institute, 1355 San Pablo St, Los Angeles, CA 90033.
"accommodation" to pseudophakic patients. However, some patients with monofocal IOLs have been noted to have paradoxically good visual acuity at both near and distance (pseudoaccommodation) without glasses. 3-8 Pseudoaccommodation also has been noted to occur in aphakic patients wearing their distance correction only.9-11 The enhanced depth of focus in these patients may be explained by small pupil size8 (pinhole effect) or possibly by uncorrected myopic astigmatism. Peters l2 and Huber4 •5 have shown that the visual acuity in uncorrected myopic astigmatism can be nearly constant and 20/40 or better from near to far. This may be partly explained by consideration of the interval of Sturm. The simple myopic eye is in optimal focus only when the object of regard is at the far point. The blurred retinal image for a closer or farther point light source is a circle, which rapidly increases in size (increasing blur) as the light source is moved away from the far point. The eye with myopic astigmatism, however, is never in perfect focus, but has two fixation distances with minimal retinal blur, occurring when each focal line of Sturm's conoid fans on the retina. As a point light source is brought toward the eye, both focal lines move posteriorly in tandem, or anteriorly as it is taken away. With the point light source at near, the anterior focal line will approach the retina, and with the point light source at distance, the posterior focal line will approach the retina. In the ideal, aberration-free eye, the shape of the retinal image of a 1025
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all refraction is assumed to take place at the front surface of the cornea. The vergence relationship can be used: U+D=V
[2]
where U = vergence of object rays entering the cornea; D = dioptric power of the eye, acting at the cornea; and V = vergence of image rays leaving the cornea. Hence: 1 n --+D=u v Fig 1. Schematic representation oflight rays in an astigmatic eye forming the conoid of Sturm. The cross-section3I area of Sturm's conoid at the retina determines the blur area of the retinal image. The shape of the retinal image is a line, ellipse, or circle, depending on the location within the conoid of Sturm.
[3]
where u = object distance from the cornea; n = refractive index of the schematic eye = 1.33; and v = distance from the cornea to the focal line in question. Therefore: n
v=--D - 11u
[4]
Referring to the reduced schematic eye in Figure 2 and by the law of similar triangles: Bp r=A
[5]
where A = v - d; B = L - v; p = half pupillary diameter; r = radius of Sturm's conoid at the retina. Thus: r=
Fig 2. Reduced schematic eye for determination of retinal blur area.
p(L - v)
[6]
(v - d)
substituting 4 into 6: point light source is determined by the intersection of Sturm's conoid with the retina (Fig 1). This shape is either a circle, ellipse, or line, depending on the refraction and object distance. The area of the image on the retina may be used as a measure of image blur, which reduces visual acuity. '2 For given spherocylindrical refractive errors and object distances, the cross-sectional area of Sturm's conoid at the retina was calculated for a schematic eye. These data were used to determine the optimal astigmatic error needed to obtain maximum depth of focus with least theoretical blur for any given refractive error. The near and distance acuities of ten pseudophakic patients with induced refractive errors were obtained to compare with these theoretical calculations.
MATERIALS AND METHODS MATHEMATICAL MODEL,
The cross-sectional area of Sturm's conoid at the retina for a point light source was calculated in a reduced schematic eye as a function of object distance and ametropia. The cross-sectional area of Sturm's conoid is obtained using the formula for the area of an ellipse: [1]
where S is the area of the ellipse and r, and r2 are the half principal axes of the ellipse. In a reduced schematic eye, 1026
r = p
(L - n/(D -
l/u»
«n/(D - 1/u) - d)
[7]
where L = axial length = 22.26 mm; and d = anterior chamber depth = 3.5 mm. With the simplifying approximation that refractive error adds to or subtracts from the dioptric power of the eye at the front corneal surface, r, and r2 may be calculated using equation [7] for the two principal meridians, assuming a given size pupil (taken here as 3 mm in diameter). Then r, and r2 can be substituted into equation [1] to determine the cross-sectional area of Sturm's conoid at the retina (S). To compare the depth of focus of various ametropic eyes, the area of the retinal image for a point light source was calculated by a computer program for object distances between 0.5 and 6 m at 0.25-m intervals. Simulated spherocylindrical refractions ranged from +3.00 to -10.00 diopters (D) sphere combined with 0.00 to +8.00 D cylinder at 0.125-D increments each. To determine a composite amount of blur over the range of object distances, image area values were simply summed for each refraction. Refractions with the highest sums were assumed to produce the most significant overall reductions in visual acuity throughout the range of object distances. INDUCED REFRACTIVE ERRORS
The near and distance visual acuities for a series of induced refractive errors were determined for ten eyes of ten pseudophakic patients. All patients had identical style posterior chamber lenses implanted at the time of cataract
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surgery and had corrected visual acuities of 20/20 with less than 1.50 D of cylinder. The average pupil size was 3.4 mm. Each patient was refracted using a phoropter, and the refractive correction was placed in a trial frame. A series of refractive errors were induced by placing additional spherocylindrical lens combinations in the trial frame. All cylindrical lenses were placed with their axes at 90 0 • Visual acuity was tested at near using an illiterate E reading card placed at 0.5 m and at distance using an illiterate E Snellen chart at 6 m.
RESULTS MATHEMATICAL MODEL
The relationship of spherical equivalent, cylindrical component, and the summed image areas ("summated blur") is shown in three-dimensional representation in Figure 3. The local minimums of summated blur (the dips along the vertical axis) identify the ideal cylindrical component for a given spherical equivalent to enhance depth of focus; this minimum point set is graphed twodimensionally in Figure 4. Thus, Figure 4 indicates, for each spherical equivalent refraction, the optimal cylindrical component producing the least summated image area. For spherical equivalents of -0.25 D or less, the equation of this line is given by: plus cylinder
=
-2 (spherical equivalent) - 0.50
Because spherical equivalent
=
sphere
[9]
For spherical equivalents greater than -0.25 D, the equation of this line is given by: plus cylinder
=
2 (spherical equivalent)
6~----------------------------------~
[8]
+ 112 cylinder:
plus cylinder = - sphere - 0.25
Fig 3. Summed retinal image areas ("summated blur") plotted as a function of plus cylinder and spherical equivalent refractive error. Minimal blur occurs along the "trough" in the graph, which forms the line shown in the lower half of the graph. This line is plotted in two dimensions in Figure 4.
c
Plus Cylinder Refraction -2 (sph eq) -0.50
5
o
~
e
Ci
4
a:
+ 0.50
The refraction resulting in the least amount of summated blur throughout the range of object distances (0.5 to 6 m) was -1.00 sphere + 0.75 cylinder, followed closely by -0.75 sphere + 0.50 cylinder.
ii :!!
•
INDUCED REFRACTIVE ERRORS
The average visual acuities (in decimal Snellen fractions) for the series of refractive errors are shown in Table 1. The refractions for which equation [9] holds true (-0.75 + 0.50, -1.00 + 0.75, -1.75 + 1.50, -2.75 + 2.50) resulted in the best combined distance and near acuities in comparison with other refractions having the same spherical equivalent. This increase in depth of focus is at the expense of a small (0 to 2 line) decrease in visual acuity.
DISCUSSION It has been well demonstrated that low myopic astigmatism serves to enhance depth of focus in the pseudophakic eye. 3- 8 In this study, we supported this empiric observation by calculating the cross-sectional area of
o+-----~~--~~~~~~~~~~--~~
-4
-3 postoperative
-2
-1
Spherical
o
Equivalent
2
Refraction
Fig 4. Ideal postoperative cylinder refraction for a given spherical equivalent to provide optimal depth of focus. The equation of this line is shown for spheriCal equivalents of -0.25 D or less.
Sturm's conoid at the retina of a reduced schematic eye for given refractive errors and object distances. These data were used to determine the optimal astigmatic error necessary to minimize blur over a fixation range of 0.5 to 6.0 m. The visual acuity performance predicted by this model agrees closely with that found in a series of pseudophakic patients with induced refractive errors. Certain assumptions made in developing this model mayor may not agree with the clinical counterpart. Cal1027
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Table 1. Decimal Snellen Acuities for Induced Refractive Errors in Ten Pseudophakic Patients Grouped by Spherical Equivalent* Average Acuities Refraction
Spherical Equivalent
Near
Distance
Overall
-0.50 + 0.00 -0.75 + 0.50 -1.00 + 1.00 -1.25 + 1.50 -1.50 + 2.00 -1.00 + 0.75 -1.00 + 0.00 -1.25 + 0.50 -1.50 + 1.00 -1.75 + 1.50 -2.00 + 2.00 -2.25 + 2.50 -1.50 + 0.00 -2.00 + 1.00 -2.50 + 2.00 -2.75 + 2.50 -3.00 + 3.00
-0.50 -0.50 -0.50 -0.50 -0.50 -0.62 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.50 -1.50 -1.50 -1.50 -1.50
0.50 0.67 0.80 0.80 0.67 0.80 0.67 0.67 0.50 0.80 0.67 0.50 0.67 0.67 0.67 0.67 0.67
0.40 0.67 0.40 0.40 0.29 0.67 0.40 0.40 0.33 0.29 0.29 0.25 0.25 0.25 0.25 0.29 0.20
0.45 0.67 0.60 0.60 0.48 0.73 0.53 0.53 0.41 0.54 0.48 0.37 0.46 0.46 0.46 0.48 0.43
* Near and distance visual acuities were averaged to provide the estimate of "overall" visual acuity.
culation of the cross-sectional area of Sturm's conoid at the retina cannot be used directly to predict the visual acuity in ametropia, because diffraction effects, orientation biases of the test letters, the Stiles-Crawford effect, and image enhancement by the retina are all ignored. This method is useful, however, in comparing relative differences in blur patch areas in various ametropic eyes, bearing in mind that Gullstrand l4 has shown that Sturm's conoid is only an approximation of the actual envelope of light rays in the eye. The model examines object distances ranging between 0.5 and 6.0 m, at 0.25-m intervals. For negative spherical equivalents (-0.25 D or less), equation [9] shows that least summated blur is obtained when the spherical and cylindrical components of the refraction are of nearly equal magnitude and opposite sign. When the most distant object point is assumed to approach infinity rather than 6 m, this equation was noted to approach: plus cylinder = - sphere
[10]
for refractions with negative spherical equivalents. This formula defines the state of simple myopic astigmatism. When such an eye fixes an object point at distance, one focal line falls in the vitreous and the other on the retina. As the object moves closer to the eye, the interval of Sturm moves posteriorly, but straddles the retina for all object distances from infinity to the far point of the myopic meridian. With Sturm's interval straddling the retina, the shape of the blurred image, more than its area, will change for different object distances. 5 Depth of focus is thus enhanced at the expense of a mild loss in visual acuity. This model predicts that the best overall acuity throughout the range of object distances between 0.5 and 6 m is obtained with a refraction of -1.00 sphere + 0.75 1028
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cylinder. For this refraction, a focal line will be present at the retina for object distances of 1 and 4 m. Between 0.5 m and infinity, one focal line will always fall within 0.50 D of the retina. Excellent agreement was found between this theoretical model and the visual acuities for a series of patients with pseudophakia and induced refractive errors. The refractions for which equation [9] holds true (-0.75 sphere + 0.50 cylinder, -1.00 sphere + 0.75 cylinder, -1.75 sphere + 1.50 cylinder) resulted in the overall best combinations of distance and near acuities for given spherical equivalent, albeit with up to a two-line reduction in best acuity. The best depth offocus was noted in patients having a -1.00 + 0.75 refraction, in agreement with the theoretical calculations of this model. While this theoretical model assumes image blur is independent of the axis of astigmatism, clinical studies indicate that visual acuity is better for astigmatism at horizontal and vertical axes than at oblique axesY This is believed to be related to the horizontal and vertical orientation of Snellen optotypes. There is also some evidence to suggest that myopic patients have better distance Snellen acuities if they have with-the-rule instead of againstthe-rule astigmatism. 16 For a patient having with-the-rule simple myopic astigmatism, the vertical focal line is closest to the retina when fixating an object at distance. Because many Snellen letters are easier to read if their vertical strokes are in focus rather than their horizontal strokes, better distance acuity would be expected for with-the-rule myopic astigmatism. 16 However, the opposite is true when the same patient reads at near, since the horizontal focal line will then be closest to the retina. Because the eye is more often called on to discriminate fine detail at distance rather than near, myopic astigmatism with-the-rule appears to be preferable to myopic astigmatism against-therule. Our model may be useful in a clinical setting to guide selection ofIOL power and postoperative suture cutting. To provide the patient with postoperative myopic astigmatism approaching -1.00 + 0.75 X 90, calculate the IOL power (using any standard formula) to provide a postoperative refraction of approximately -0.62 sphere. Of course, if this refraction is likely to yield excessive anisometropia, an alternative refraction may be necessary. Huber4,5 has shown that change in corneal power after cataract surgery is practically always spherocylindrical with only a very weak change in spherical equivalent (essentially zero). An increase in corneal curvature in the 90 0 meridian is compensated by a flattening in the 180 0 meridian. Thus, if we induce a refraction of +0.75 D cylinder X 90 by wound closure, we add approximately -0.37 D sphere overall, for a net change in spherical equivalent of zero. With the target spherical equivalent of -0.62 D, the overall refraction, therefore, becomes -1.00 + 0.75 X 90. Intraoperative and postoperative modification of astigmatism should be guided by knowledge of the natural course of astigmatism following extracapsular cataract extraction. Richards et al 17 found that 1.50 to 2.00 D of astigmatism induced at the time of surgery had regressed
SAWUSCH AND GUYTON
to emmetropia by 3 years after surgery. Likewise, Jampel et al l8 found that an average of 2.2 D of with-the-rule astigmatism had regressed to 0.35 D of against-the-rule astigmatism after a mean follow-up of 5 months. On the basis of these studies, if a final postoperative astigmatism of +0.75 X 90 is desired, one should attempt to induce approximately +3.00 D of with-the-rule astigmatism at the time of surgery to allow for the natural regression in wound tension, and to allow for modification of the axis and magnitude of astigmatism by selective suture cutting. Wound closure with interrupted nylon sutures is preferable for this technique. Surgeons using small-incision phacoemulsification surgery should ideally induce less astigmatism at the time of surgery since the regression in postoperative astigmatism may be considerably smaller. 19 Postoperatively, the patient's actual spherical equivalent will remain constant. 4,5 If this spherical equivalent is determined, the ideal plus cylinder refraction to accompany it may then be calculated from equation [8]. Once most of the natural postoperative regression has occurred, if the actual plus cylinder is greater than the calculated ideal plus cylinder, then sutures may be cut in an attempt to approach the ideal plus cylinder. If the actual plus cylinder is less than the ideal plus cylinder, then further suture cutting would not tend to enhance depth of focus. While it is not possible to control postoperative astigmatism precisely with suture cutting, our calculations indicate that the optimal depth of focus is not significantly reduced as long as the actual cylinder is within ± 0.37 D of the calculated ideal value. It may be advisable to wait approximately 8 weeks before cutting sutures after extracapsular surgery, during which time approximately half of the expected regression in astigmatism has taken piacep,I8 Sutures may be cut earlier for small-incision surgery, since stabilization of the wound may occur as early as 4 weeks after surgery. 19 Multifocal IOLs are currently under development to enhance the depth of focus of pseudophakic patients. However, concerns regarding the use of these lenses include decreased contrast sensitivity,20 glare (from interfaces of refractive zones), and loss of effect from decentration. Simple myopic astigmatism may represent an alternative to multifocal IOLs by allowing spectacle independence after surgery. In addition, there is less risk of reduction of best correctable visual function. Although it may not be possible to provide an acceptable depth offocus for all patients, the principles set forth here may be useful in at least improving the balance between uncorrected near and distance acuity after cataract surgery.
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REFERENCES 1. Sanders DR. Intraocular lens power calculations: techniques and results. In: Stark WJ, Terry A, Maumenee AE, eds. Anterior Segment Surgery: IOLs, Lasers, and Refractive Keratoplasty. Baltimore: Williams & Wilkins, 1987; 37-47. 2. Drews RC. Determination of intraocular lens power. In: Peyman GA, Sanders DR, Goldberg MF, eds. PrinCiples and Practice of Ophthalmology. Vol. I. Philadelphia: WB Saunders, 1980; 665-70. 3. Terry CM. Selection and calculation of intraocular lens power. In: Engelstein JM, ed. Cataract Surgery: Current Opinions and Problems. Orlando: Grune & Stratton, 1984; 65-71. 4. Huber C. Myopic astigmatism: a substitute for accommodation in pseudophakia. Doc Ophthalmol1981; 52:123-78. 5. Huber C. Planned myopic astigmatism as a substitute for accommodation in pseudophakia. J Am Intraocul Implant Soc 1981; 7: 244-9. 6. Sugitani Y, Komori T, Kitoh R, Hayano S. Apparent accommodation (pseudoaccommodation) in pseudophakia. Folia Ophthalmol Jpn 1979; 30:326-30. 7. Hoshi H, Kamegasawa A, Chikuda M, et al. Visual functions of the intraocular lens implanted eye (pseudophakic eye) with special reference to pseudoaccommodation. Folia Ophthalmol Jpn 1980; 31: 1409-19. 8. Nakazawa M, Ohtsuki K. Apparent accommodation in pseudophakic eyes after implantation of posterior chamber intraocular lenses. Am J Ophthalmol1983; 96:435-8. 9. Zentmayer W. Apparent accommodation in aphakia. Am J Ophthalmol 1918; 1:570-1. 10. Horton JJ. Apparent accommodation in the aphakic eye. Am J Ophthalmol1929; 12:489-90. 11. Bettman JW. Apparent accommodation in aphakic eyes. Am J Ophthalmol1950; 33:921-8. 12. Peters HB. The relationship between refractive error and visual acuity at three age levels. Am J Optom 1961; 38:194-8. 13. Askovitz SI. The circle of least confusion on Sturm's conoid of astigmatism. Arch Ophthalmol1956; 56:691-7. 14. Gullstrand A. Beitrag zur Theorie des Astigmatismus. Skandin Arch Physiol 1891; 2:269-359. 15. Eggers H. Estimation of uncorrected visual acuity in malingerers. Arch Ophthalmol1945; 33:23-7. 16. Friedman B. Acceptance of weak cylinders at paradoxic axes. Arch Ophthalmol1940; 23:720-6. 17. Richards SC, Brodstein RS, Richards WL, et al. Long-term course of surgically induced astigmatism. J Cataract Refract Surg 1988; 14: 270-6. 18. Jampel HD, Thompson JR, Baker CC, Stark WJ. A computerized analysis of astigmatism after cataract surgery. Ophthalmic Surg 1986; 17:786-90. 19. Shepherd JR. Induced astigmatism in small incision cataract surgery. J Cataract Refract Surg 1989; 15:85-8. 20. Holladay JT, Van Dijk H, Lang A, et al. Optical performance of multifocal intraocular lenses. J Cataract Refract Surg 1990; 16:413-22.
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Abstract: A small amount of myopic astigmatism can enhance the depth of focus of the pseudophakic eye, optimally providing at least 20/30 visual acuity for both near and distance fixation. For given spherocylindrical refractive errors and fixation distances, the cross-sectional area of Sturm's conoid at the retina was calculated for a schematic eye. These data were used to determine the optimal astigmatic error needed to obtain maximum depth of focus and least theoretical blur for any given spherical equivalent refractive error. Optimal depth of focus was obtained when the plus cylindrical component equaled negative sphere - 0.25 diopters. The near and distance visual acuities of ten pseudophakic patients with induced refractive errors were highly correlated with this model. Low myopic astigmatism after cataract surgery may represent an alternative to multifocal intraocular lenses by providing spectacle independence. Ophthalmology 1991; 98:1025-1029
The calculation of intraocular lens (lOL) power, and intraoperative or postoperative modification of corneal astigmatism allow the surgeon significant control of the refraction after cataract surgery. This raises the question of how to select the optimal refraction in a monofocal, pseudophakic eye lacking accommodation. The optimal postoperative refraction for a given patient may depend on several factors, including occupation and lifestyle and the need to maintain isometropia. 1 Most surgeons advise low myopia (in sedentary patients) or emmetropia (in active patients) as an optimal refraction, thereby allowing relatively good acuity without glasses at the fixation distances most often used by the patient. 2 However, bifocal lenses are usually necessary to provide good acuity for both near and distance fixation. Multifocal IOLs are currently under development to enhance the depth of focus and to restore a degree of Originally received : September 28, 1990. Revision accepted: March 5, 1991. 1 2
The Doheny Eye Institute, USC School of Medicine. Los Angeles. The Wilmer Ophthalmological Institute, The Johns Hopkins University School of Medicine, Baltimore.
Presented at the American Academy of Ophthalmology Annual Meeting, Atlanta, Oct/Nov 1990. Correspondence to Mark R. Sawusch, MD, Doheny Eye Institute, 1355 San Pablo St, Los Angeles, CA 90033.
"accommodation" to pseudophakic patients. However, some patients with monofocal IOLs have been noted to have paradoxically good visual acuity at both near and distance (pseudoaccommodation) without glasses. 3-8 Pseudoaccommodation also has been noted to occur in aphakic patients wearing their distance correction only.9-11 The enhanced depth of focus in these patients may be explained by small pupil size8 (pinhole effect) or possibly by uncorrected myopic astigmatism. Peters l2 and Huber4 •5 have shown that the visual acuity in uncorrected myopic astigmatism can be nearly constant and 20/40 or better from near to far. This may be partly explained by consideration of the interval of Sturm. The simple myopic eye is in optimal focus only when the object of regard is at the far point. The blurred retinal image for a closer or farther point light source is a circle, which rapidly increases in size (increasing blur) as the light source is moved away from the far point. The eye with myopic astigmatism, however, is never in perfect focus, but has two fixation distances with minimal retinal blur, occurring when each focal line of Sturm's conoid fans on the retina. As a point light source is brought toward the eye, both focal lines move posteriorly in tandem, or anteriorly as it is taken away. With the point light source at near, the anterior focal line will approach the retina, and with the point light source at distance, the posterior focal line will approach the retina. In the ideal, aberration-free eye, the shape of the retinal image of a 1025
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all refraction is assumed to take place at the front surface of the cornea. The vergence relationship can be used: U+D=V
[2]
where U = vergence of object rays entering the cornea; D = dioptric power of the eye, acting at the cornea; and V = vergence of image rays leaving the cornea. Hence: 1 n --+D=u v Fig 1. Schematic representation oflight rays in an astigmatic eye forming the conoid of Sturm. The cross-section3I area of Sturm's conoid at the retina determines the blur area of the retinal image. The shape of the retinal image is a line, ellipse, or circle, depending on the location within the conoid of Sturm.
[3]
where u = object distance from the cornea; n = refractive index of the schematic eye = 1.33; and v = distance from the cornea to the focal line in question. Therefore: n
v=--D - 11u
[4]
Referring to the reduced schematic eye in Figure 2 and by the law of similar triangles: Bp r=A
[5]
where A = v - d; B = L - v; p = half pupillary diameter; r = radius of Sturm's conoid at the retina. Thus: r=
Fig 2. Reduced schematic eye for determination of retinal blur area.
p(L - v)
[6]
(v - d)
substituting 4 into 6: point light source is determined by the intersection of Sturm's conoid with the retina (Fig 1). This shape is either a circle, ellipse, or line, depending on the refraction and object distance. The area of the image on the retina may be used as a measure of image blur, which reduces visual acuity. '2 For given spherocylindrical refractive errors and object distances, the cross-sectional area of Sturm's conoid at the retina was calculated for a schematic eye. These data were used to determine the optimal astigmatic error needed to obtain maximum depth of focus with least theoretical blur for any given refractive error. The near and distance acuities of ten pseudophakic patients with induced refractive errors were obtained to compare with these theoretical calculations.
MATERIALS AND METHODS MATHEMATICAL MODEL,
The cross-sectional area of Sturm's conoid at the retina for a point light source was calculated in a reduced schematic eye as a function of object distance and ametropia. The cross-sectional area of Sturm's conoid is obtained using the formula for the area of an ellipse: [1]
where S is the area of the ellipse and r, and r2 are the half principal axes of the ellipse. In a reduced schematic eye, 1026
r = p
(L - n/(D -
l/u»
«n/(D - 1/u) - d)
[7]
where L = axial length = 22.26 mm; and d = anterior chamber depth = 3.5 mm. With the simplifying approximation that refractive error adds to or subtracts from the dioptric power of the eye at the front corneal surface, r, and r2 may be calculated using equation [7] for the two principal meridians, assuming a given size pupil (taken here as 3 mm in diameter). Then r, and r2 can be substituted into equation [1] to determine the cross-sectional area of Sturm's conoid at the retina (S). To compare the depth of focus of various ametropic eyes, the area of the retinal image for a point light source was calculated by a computer program for object distances between 0.5 and 6 m at 0.25-m intervals. Simulated spherocylindrical refractions ranged from +3.00 to -10.00 diopters (D) sphere combined with 0.00 to +8.00 D cylinder at 0.125-D increments each. To determine a composite amount of blur over the range of object distances, image area values were simply summed for each refraction. Refractions with the highest sums were assumed to produce the most significant overall reductions in visual acuity throughout the range of object distances. INDUCED REFRACTIVE ERRORS
The near and distance visual acuities for a series of induced refractive errors were determined for ten eyes of ten pseudophakic patients. All patients had identical style posterior chamber lenses implanted at the time of cataract
SAWUSCH AND GUYTON
•
OPTIMAL ASTIGMATISM
surgery and had corrected visual acuities of 20/20 with less than 1.50 D of cylinder. The average pupil size was 3.4 mm. Each patient was refracted using a phoropter, and the refractive correction was placed in a trial frame. A series of refractive errors were induced by placing additional spherocylindrical lens combinations in the trial frame. All cylindrical lenses were placed with their axes at 90 0 • Visual acuity was tested at near using an illiterate E reading card placed at 0.5 m and at distance using an illiterate E Snellen chart at 6 m.
RESULTS MATHEMATICAL MODEL
The relationship of spherical equivalent, cylindrical component, and the summed image areas ("summated blur") is shown in three-dimensional representation in Figure 3. The local minimums of summated blur (the dips along the vertical axis) identify the ideal cylindrical component for a given spherical equivalent to enhance depth of focus; this minimum point set is graphed twodimensionally in Figure 4. Thus, Figure 4 indicates, for each spherical equivalent refraction, the optimal cylindrical component producing the least summated image area. For spherical equivalents of -0.25 D or less, the equation of this line is given by: plus cylinder
=
-2 (spherical equivalent) - 0.50
Because spherical equivalent
=
sphere
[9]
For spherical equivalents greater than -0.25 D, the equation of this line is given by: plus cylinder
=
2 (spherical equivalent)
6~----------------------------------~
[8]
+ 112 cylinder:
plus cylinder = - sphere - 0.25
Fig 3. Summed retinal image areas ("summated blur") plotted as a function of plus cylinder and spherical equivalent refractive error. Minimal blur occurs along the "trough" in the graph, which forms the line shown in the lower half of the graph. This line is plotted in two dimensions in Figure 4.
c
Plus Cylinder Refraction -2 (sph eq) -0.50
5
o
~
e
Ci
4
a:
+ 0.50
The refraction resulting in the least amount of summated blur throughout the range of object distances (0.5 to 6 m) was -1.00 sphere + 0.75 cylinder, followed closely by -0.75 sphere + 0.50 cylinder.
ii :!!
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INDUCED REFRACTIVE ERRORS
The average visual acuities (in decimal Snellen fractions) for the series of refractive errors are shown in Table 1. The refractions for which equation [9] holds true (-0.75 + 0.50, -1.00 + 0.75, -1.75 + 1.50, -2.75 + 2.50) resulted in the best combined distance and near acuities in comparison with other refractions having the same spherical equivalent. This increase in depth of focus is at the expense of a small (0 to 2 line) decrease in visual acuity.
DISCUSSION It has been well demonstrated that low myopic astigmatism serves to enhance depth of focus in the pseudophakic eye. 3- 8 In this study, we supported this empiric observation by calculating the cross-sectional area of
o+-----~~--~~~~~~~~~~--~~
-4
-3 postoperative
-2
-1
Spherical
o
Equivalent
2
Refraction
Fig 4. Ideal postoperative cylinder refraction for a given spherical equivalent to provide optimal depth of focus. The equation of this line is shown for spheriCal equivalents of -0.25 D or less.
Sturm's conoid at the retina of a reduced schematic eye for given refractive errors and object distances. These data were used to determine the optimal astigmatic error necessary to minimize blur over a fixation range of 0.5 to 6.0 m. The visual acuity performance predicted by this model agrees closely with that found in a series of pseudophakic patients with induced refractive errors. Certain assumptions made in developing this model mayor may not agree with the clinical counterpart. Cal1027
OPHTHALMOLOGY
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JULY 1991
Table 1. Decimal Snellen Acuities for Induced Refractive Errors in Ten Pseudophakic Patients Grouped by Spherical Equivalent* Average Acuities Refraction
Spherical Equivalent
Near
Distance
Overall
-0.50 + 0.00 -0.75 + 0.50 -1.00 + 1.00 -1.25 + 1.50 -1.50 + 2.00 -1.00 + 0.75 -1.00 + 0.00 -1.25 + 0.50 -1.50 + 1.00 -1.75 + 1.50 -2.00 + 2.00 -2.25 + 2.50 -1.50 + 0.00 -2.00 + 1.00 -2.50 + 2.00 -2.75 + 2.50 -3.00 + 3.00
-0.50 -0.50 -0.50 -0.50 -0.50 -0.62 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.50 -1.50 -1.50 -1.50 -1.50
0.50 0.67 0.80 0.80 0.67 0.80 0.67 0.67 0.50 0.80 0.67 0.50 0.67 0.67 0.67 0.67 0.67
0.40 0.67 0.40 0.40 0.29 0.67 0.40 0.40 0.33 0.29 0.29 0.25 0.25 0.25 0.25 0.29 0.20
0.45 0.67 0.60 0.60 0.48 0.73 0.53 0.53 0.41 0.54 0.48 0.37 0.46 0.46 0.46 0.48 0.43
* Near and distance visual acuities were averaged to provide the estimate of "overall" visual acuity.
culation of the cross-sectional area of Sturm's conoid at the retina cannot be used directly to predict the visual acuity in ametropia, because diffraction effects, orientation biases of the test letters, the Stiles-Crawford effect, and image enhancement by the retina are all ignored. This method is useful, however, in comparing relative differences in blur patch areas in various ametropic eyes, bearing in mind that Gullstrand l4 has shown that Sturm's conoid is only an approximation of the actual envelope of light rays in the eye. The model examines object distances ranging between 0.5 and 6.0 m, at 0.25-m intervals. For negative spherical equivalents (-0.25 D or less), equation [9] shows that least summated blur is obtained when the spherical and cylindrical components of the refraction are of nearly equal magnitude and opposite sign. When the most distant object point is assumed to approach infinity rather than 6 m, this equation was noted to approach: plus cylinder = - sphere
[10]
for refractions with negative spherical equivalents. This formula defines the state of simple myopic astigmatism. When such an eye fixes an object point at distance, one focal line falls in the vitreous and the other on the retina. As the object moves closer to the eye, the interval of Sturm moves posteriorly, but straddles the retina for all object distances from infinity to the far point of the myopic meridian. With Sturm's interval straddling the retina, the shape of the blurred image, more than its area, will change for different object distances. 5 Depth of focus is thus enhanced at the expense of a mild loss in visual acuity. This model predicts that the best overall acuity throughout the range of object distances between 0.5 and 6 m is obtained with a refraction of -1.00 sphere + 0.75 1028
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VOLUME 98
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NUMBER 7
cylinder. For this refraction, a focal line will be present at the retina for object distances of 1 and 4 m. Between 0.5 m and infinity, one focal line will always fall within 0.50 D of the retina. Excellent agreement was found between this theoretical model and the visual acuities for a series of patients with pseudophakia and induced refractive errors. The refractions for which equation [9] holds true (-0.75 sphere + 0.50 cylinder, -1.00 sphere + 0.75 cylinder, -1.75 sphere + 1.50 cylinder) resulted in the overall best combinations of distance and near acuities for given spherical equivalent, albeit with up to a two-line reduction in best acuity. The best depth offocus was noted in patients having a -1.00 + 0.75 refraction, in agreement with the theoretical calculations of this model. While this theoretical model assumes image blur is independent of the axis of astigmatism, clinical studies indicate that visual acuity is better for astigmatism at horizontal and vertical axes than at oblique axesY This is believed to be related to the horizontal and vertical orientation of Snellen optotypes. There is also some evidence to suggest that myopic patients have better distance Snellen acuities if they have with-the-rule instead of againstthe-rule astigmatism. 16 For a patient having with-the-rule simple myopic astigmatism, the vertical focal line is closest to the retina when fixating an object at distance. Because many Snellen letters are easier to read if their vertical strokes are in focus rather than their horizontal strokes, better distance acuity would be expected for with-the-rule myopic astigmatism. 16 However, the opposite is true when the same patient reads at near, since the horizontal focal line will then be closest to the retina. Because the eye is more often called on to discriminate fine detail at distance rather than near, myopic astigmatism with-the-rule appears to be preferable to myopic astigmatism against-therule. Our model may be useful in a clinical setting to guide selection ofIOL power and postoperative suture cutting. To provide the patient with postoperative myopic astigmatism approaching -1.00 + 0.75 X 90, calculate the IOL power (using any standard formula) to provide a postoperative refraction of approximately -0.62 sphere. Of course, if this refraction is likely to yield excessive anisometropia, an alternative refraction may be necessary. Huber4,5 has shown that change in corneal power after cataract surgery is practically always spherocylindrical with only a very weak change in spherical equivalent (essentially zero). An increase in corneal curvature in the 90 0 meridian is compensated by a flattening in the 180 0 meridian. Thus, if we induce a refraction of +0.75 D cylinder X 90 by wound closure, we add approximately -0.37 D sphere overall, for a net change in spherical equivalent of zero. With the target spherical equivalent of -0.62 D, the overall refraction, therefore, becomes -1.00 + 0.75 X 90. Intraoperative and postoperative modification of astigmatism should be guided by knowledge of the natural course of astigmatism following extracapsular cataract extraction. Richards et al 17 found that 1.50 to 2.00 D of astigmatism induced at the time of surgery had regressed
SAWUSCH AND GUYTON
to emmetropia by 3 years after surgery. Likewise, Jampel et al l8 found that an average of 2.2 D of with-the-rule astigmatism had regressed to 0.35 D of against-the-rule astigmatism after a mean follow-up of 5 months. On the basis of these studies, if a final postoperative astigmatism of +0.75 X 90 is desired, one should attempt to induce approximately +3.00 D of with-the-rule astigmatism at the time of surgery to allow for the natural regression in wound tension, and to allow for modification of the axis and magnitude of astigmatism by selective suture cutting. Wound closure with interrupted nylon sutures is preferable for this technique. Surgeons using small-incision phacoemulsification surgery should ideally induce less astigmatism at the time of surgery since the regression in postoperative astigmatism may be considerably smaller. 19 Postoperatively, the patient's actual spherical equivalent will remain constant. 4,5 If this spherical equivalent is determined, the ideal plus cylinder refraction to accompany it may then be calculated from equation [8]. Once most of the natural postoperative regression has occurred, if the actual plus cylinder is greater than the calculated ideal plus cylinder, then sutures may be cut in an attempt to approach the ideal plus cylinder. If the actual plus cylinder is less than the ideal plus cylinder, then further suture cutting would not tend to enhance depth of focus. While it is not possible to control postoperative astigmatism precisely with suture cutting, our calculations indicate that the optimal depth of focus is not significantly reduced as long as the actual cylinder is within ± 0.37 D of the calculated ideal value. It may be advisable to wait approximately 8 weeks before cutting sutures after extracapsular surgery, during which time approximately half of the expected regression in astigmatism has taken piacep,I8 Sutures may be cut earlier for small-incision surgery, since stabilization of the wound may occur as early as 4 weeks after surgery. 19 Multifocal IOLs are currently under development to enhance the depth of focus of pseudophakic patients. However, concerns regarding the use of these lenses include decreased contrast sensitivity,20 glare (from interfaces of refractive zones), and loss of effect from decentration. Simple myopic astigmatism may represent an alternative to multifocal IOLs by allowing spectacle independence after surgery. In addition, there is less risk of reduction of best correctable visual function. Although it may not be possible to provide an acceptable depth offocus for all patients, the principles set forth here may be useful in at least improving the balance between uncorrected near and distance acuity after cataract surgery.
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OPTIMAL ASTIGMATISM
REFERENCES 1. Sanders DR. Intraocular lens power calculations: techniques and results. In: Stark WJ, Terry A, Maumenee AE, eds. Anterior Segment Surgery: IOLs, Lasers, and Refractive Keratoplasty. Baltimore: Williams & Wilkins, 1987; 37-47. 2. Drews RC. Determination of intraocular lens power. In: Peyman GA, Sanders DR, Goldberg MF, eds. PrinCiples and Practice of Ophthalmology. Vol. I. Philadelphia: WB Saunders, 1980; 665-70. 3. Terry CM. Selection and calculation of intraocular lens power. In: Engelstein JM, ed. Cataract Surgery: Current Opinions and Problems. Orlando: Grune & Stratton, 1984; 65-71. 4. Huber C. Myopic astigmatism: a substitute for accommodation in pseudophakia. Doc Ophthalmol1981; 52:123-78. 5. Huber C. Planned myopic astigmatism as a substitute for accommodation in pseudophakia. J Am Intraocul Implant Soc 1981; 7: 244-9. 6. Sugitani Y, Komori T, Kitoh R, Hayano S. Apparent accommodation (pseudoaccommodation) in pseudophakia. Folia Ophthalmol Jpn 1979; 30:326-30. 7. Hoshi H, Kamegasawa A, Chikuda M, et al. Visual functions of the intraocular lens implanted eye (pseudophakic eye) with special reference to pseudoaccommodation. Folia Ophthalmol Jpn 1980; 31: 1409-19. 8. Nakazawa M, Ohtsuki K. Apparent accommodation in pseudophakic eyes after implantation of posterior chamber intraocular lenses. Am J Ophthalmol1983; 96:435-8. 9. Zentmayer W. Apparent accommodation in aphakia. Am J Ophthalmol 1918; 1:570-1. 10. Horton JJ. Apparent accommodation in the aphakic eye. Am J Ophthalmol1929; 12:489-90. 11. Bettman JW. Apparent accommodation in aphakic eyes. Am J Ophthalmol1950; 33:921-8. 12. Peters HB. The relationship between refractive error and visual acuity at three age levels. Am J Optom 1961; 38:194-8. 13. Askovitz SI. The circle of least confusion on Sturm's conoid of astigmatism. Arch Ophthalmol1956; 56:691-7. 14. Gullstrand A. Beitrag zur Theorie des Astigmatismus. Skandin Arch Physiol 1891; 2:269-359. 15. Eggers H. Estimation of uncorrected visual acuity in malingerers. Arch Ophthalmol1945; 33:23-7. 16. Friedman B. Acceptance of weak cylinders at paradoxic axes. Arch Ophthalmol1940; 23:720-6. 17. Richards SC, Brodstein RS, Richards WL, et al. Long-term course of surgically induced astigmatism. J Cataract Refract Surg 1988; 14: 270-6. 18. Jampel HD, Thompson JR, Baker CC, Stark WJ. A computerized analysis of astigmatism after cataract surgery. Ophthalmic Surg 1986; 17:786-90. 19. Shepherd JR. Induced astigmatism in small incision cataract surgery. J Cataract Refract Surg 1989; 15:85-8. 20. Holladay JT, Van Dijk H, Lang A, et al. Optical performance of multifocal intraocular lenses. J Cataract Refract Surg 1990; 16:413-22.
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