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Residual astigmatism produced by toric intraocular lens rotation
LABORATORY SCIENCE
Residual astigmatism produced by toric intraocular lens rotation Adelina Felipe, PhD, Jose M. Artigas, PhD, Amparo Díez-Ajenjo, OD, MSc, Carmen García-Domene, OD, MSc, Pablo Alcocer, MD, PhD
PURPOSE: To analyze changes in the eye’s refractive properties when a toric intraocular lens (IOL) rotates. SETTING: Fundaci on Oftalmol ogica del Mediterraneo, Valencia, Spain. DESIGN: Experimental study. METHODS: The matrix definition of astigmatism was used in this theoretical study and compared with another vector representation. Two methods were compared: (1) The cylinder, C, resulting from the addition of 2 cylinders C1 and C2 whose axes form an angle a, is obtained by the addition of 2 vectors of values C1 and C2 forming an angle 2a; (2) the power matrix, F, of a thin astigmatic dioptric system that decomposes naturally into 3 orthogonal components: the purely spherical part Fnes, the ortho-astigmatism For, and oblique astigmatism Fob. RESULTS: The residual cylinder was one third of the corneal astigmatism when a toric IOL rotated G10 degrees when the cylinder values for the cornea (C1) and IOL (C2) were equal. Nevertheless, in most cases C1 is greater than C2; therefore, the residual astigmatism did not change noticeably with small rotations. The angle of rotation, b, which annuls the astigmatism correction, could be obtained from the following: cos(p C 2b) Z r/2, with r being the ratio between the IOL and cornea cylinders. CONCLUSIONS: The 2 methods gave equivalent results. When the IOL cylinder had a value different from that of the corneal astigmatism, a better choice would be a lower, rather than higher, cylinder value to reduce residual astigmatism. In general, toric IOL rotations less than 10 degrees changed the eye’s refraction less than 0.50 diopter. Thus, small axis rotations are not an obstacle for satisfactory astigmatism correction with toric IOLs. Financial Disclosure: No author has a financial or proprietary interest in any material or method mentioned. J Cataract Refract Surg 2011; 37:1895–1901 Q 2011 ASCRS and ESCRS
Toric intraocular lenses (IOLs) are a good solution for astigmatism correction for patients who desire spectacle independence.1–6 New designs of these IOLs were developed to provide better performance and safety. Astigmatism correction involves the position of the IOL in the eye and the axis lying in the correct direction; for this reason, many studies specifically analyze toric IOL rotation after surgery as well as the origin of the rotation to avoid this problem in new designs.7–14 The importance of controlling possible IOL rotation prompted studies of more suitable methods of assessing IOL alignment and toric IOL stability. Improvements introduced in haptics were fundamental in solving this problem.13,14 As a consequence, there have been several studies of the postoperative stability of the toric IOL.11,15–18 Nevertheless, despite the Q 2011 ASCRS and ESCRS Published by Elsevier Inc.
reported high proportion of stability, Chang15 found a small percentage (approximately 1.1%) of patients with significant postoperative rotation and proposed a method calculating the correct angle for repositioning the IOL. Others report that no eye had significant rotation (minor or !20 degrees) and 96.7% had a rotation of less than 10 degrees.4 In a more recent study,18 96% of eyes had rotation of less than 5 degrees and 99% had less than 10 degrees. In any event, there are many cases of eyes with off-axis toric IOLs due to postoperative IOL rotation or an initial error in implantation. Theoretic analyses shows a steady loss of astigmatism reduction as toric IOLs become more off axis.9 If the cornea and IOL cylinders are equal, the amount of corrected astigmatism is zero if the IOL is off axis by 30 degrees. 0886-3350/$ - see front matter doi:10.1016/j.jcrs.2011.04.036
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We performed a theoretical study analyzing in what way and to what extent the refractive properties of an eye, which is supposedly best corrected with a toric IOL, vary when small rotation of the IOL occurs. In a comprehensive study, Harris19 showed the different representations of astigmatism in the optometric and ophthalmologic literature. There are different mathematic expressions that represent the same phenomenon because some authors seem to look for useful mathematic expressions to accomplish a particular job rather than looking for the exact mathematic expression (there is no choice, as Harris said) for astigmatism. Harris presented a universally applicable representation of astigmatism for quantitative work. We tested the usefulness of the matrix definition of astigmatism proposed by Harris19 in this study and contrasted the results with those obtained by the classic addition of vectors20–22 which represent cylinders. Although in both cases we used vectors, Harris’ theory is formally more precise because his definition of astigmatism is invariant under spherocylindrical transposition and constitutes a vector space with all its properties. Both methods must lead to equivalent results; however, an advantage of matrix notation is the possibility of using calculation programs operating with the matrix. This makes it possible to apply the rotation matrix, R, over any power astigmatic matrix as follows: cosb sinb RZ sinb cosb This provides a complete, quick way to obtain a sure result. Moreover, the matrix representation is always valid, even for a thick system, and other methods are not.
equals the cylinder of the cornea and the IOL axis is positioned orthogonally with the corneal astigmatism axis. For example, expressing the power of the astigmatic cornea as sphere (S) and cylinder (C) axis (a), consider an eye with a corneal power of C40.00 C2.00 30 (in diopters [D]) and an IOL with a power of C20.00 C2.00 120 Z C22.00 2.00 30 in the corneal plane. It is easy to see that the addition of the corneal power and IOL power gives a sphere of 62.0 D because the addition of 2 cylinders having the same axis direction and equal absolute power, but opposite sign, is null. It is known that the effective lens position must be considered to obtain the relationship between the IOL cylinder in the corneal plane and the real toric IOL cylindrical power. Thus, to neutralize astigmatism, the actual toric IOL cylindrical power is always higher than the real corneal astigmatism.
Vector Addition To show how a small rotation affects the refractive characteristics of the eye corrected with a toric IOL, this problem was analyzed in detail using the vector addition method. For this calculation, it was assumed that the IOL cylinder and cornea cylinder are equal but with orthogonal (a Z 90 degrees) axes and then, from this initial position, the IOL is off axis in angle b. It is known that the cylinder, C, resulting from the addition of 2 cylinders C1 and C2 with an angle a between the 2 axes is obtained by the addition of 2 vectors of values C1 and C2 forming an angle 2a (Figure 1), giving the following result: C2 Z C21 þ C22 þ 2C1 C2 cosð2aÞ
(1)
In this case, because the 2 cylinders (those of the cornea and the IOL, respectively) when added have the same C1 value, the previous equation can be written as C Z 2C1 cosðaÞ
(2)
in which it can seen that C Z 0 when axes of both cylinders are orthogonal. If from this position a Z 90 degrees, the lens axis is off axis in angle b the cylinder resultant would be
MATERIALS AND METHODS To correct a patient’s astigmatism, the toric IOL is calculated so that in the corneal plane the IOL cylinder value
Submitted: November 8, 2010. Final revision submitted: March 30, 2011. Accepted: April 7, 2011. From Fundacion Oftalmologica del Mediterraneo (Felipe, Artigas, Dıez-Ajenjo, Garcıa-Domene, Alcocer) and Departamento de Optica (Felipe, Artigas,. Dıez-Ajenjo) Facultad de Fısica, Universidad de Valencia, Burjassot (Valencia), Spain. Supported by Catedra Alcon-Universitat de Valencia, Spain. Corresponding author: Adelina Felipe, PhD, Departamento de Optica, Facultad de Fısica, Universidad de Valencia, C/Dr Moliner, 50, E46100-Burjassot (Valencia), Spain. E-mail: adelina.felipe@ uv.es.
Figure 1. The addition of 2 cylinders, with values C1 and C2 diopters, whose axis directions form an angle, a, is equivalent to the addition of 2 vectors having the same values and forming an angle 2a.
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C Z 2C1 cosða þ bÞ Z 2C1 sinðbÞ
(3)
and when b!0 / C Z 2C1 sinðbÞ
angle c. Given the direction of the resultant cylinder axis, which is obtained from the expression tanð2cÞ Z
40:5 0:866 1 0 Z 41 0:866 41:5 0 1 1 0 0 1 0:5 0:866 0 1 1 0
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Fcornea Z
(12)
C2 sinð2aÞ C1 þ C2 cosð2aÞ
(4)
it can be seen that if C1 Z C2, equation 4 is reduced to tanð2cÞ Z tanðaÞ 0 2c Z a and when a rotation occurs: tanð2cÞ Z tanða þ bÞ 0 2c Z a þ b (5) Notice that the c angle (Figure 1) gives the direction of the resulting cylinder axis from the cornea axis position. Thus, in this example, taking b Z 10 degrees and c Z (90 to 10 degrees)/2 Z 40 degrees means the resultant cylinder axis is in a direction 40 degrees from the corneal axis; that is, 40 degrees C30 degrees Z 70 degrees. To obtain the direction of the cylinder resulting from this method, the corneal axis angle (30 degrees in the example) must be added to the angle c (Figure 1). On the other hand, when b O 0, ie, the rotation is produced by distancing the axis of the IOL from the corneal axis (a C b O 90 degrees), the result is C ! 0 and c O 45 degrees while when b ! 0, the rotation goes in the opposite way ( a C b ! 90 degrees) and results in C O 0 and c ! 45 degrees.
Dioptric Power Matrix The symmetric dioptric power matrix introduced by Long in 197623 makes it possible to express the power (S C a) of a thin system as a 2 2 symmetric matrix: f f (6) F Z 11 12 f21 f22 where the components are calculated by f11 Z S þ C sin2 a
(7)
f 12 Z f 21 Z Csina cos a
(8)
f 22 Z S þ Ccos2 a
(9)
The example proposed above can also be calculated following the matrix method introduced by Harris19 as the most complete solution for defining and calculating astigmatism. According to this method, the power matrix, F, of an astigmatic dioptric system decomposes naturally into 3 orthogonal components as follows: F Z Fnes þ Fast Z Fnes þ For þ Fob
(10)
where Fnes is the purely spherical part of the general power and Fast the purely astigmatic part, which itself decomposes in orthoastigmatism For and oblique astigmatism Fob. Using this theory, the cornea and IOL power matrix of the example added by previously writing their power matrix as a linear combination of the 3 linearly independent basic astigmatic powers as follows: 1 0 1 0 0 1 þ For þ Fob (11) F Z Fnes 0 1 0 1 1 0
21:5 þ0:866 1 0 Z 21 þ0:866 20:5 0 1 1 0 0 1 þ 0:5 þ 0:866 0 1 1 0
(13)
1 0 Fcornea þ FIOL Z 62 þ0þ0 0 1
(14)
FIOL Z
It is easily shown (Appendix) that For Z C/2 cos(2a) and Fob Z C/2 sin(2a), which coincides with the values of vector components J0 and J45, respectively. The cylinder, axis direction, and sphere values of the spherocylindrical formula can be obtained from the matrix representation by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G For þ Fob Z GC=2 and S þ C=2 Z Fnes (15) with tan ð2aÞ Z Fob =For Equation 15 provides only the absolute value of the angle 2a; however, the sign of the 2 components For and Fob indicates the sign of cos(2a) and sin(2a), respectively; from this information, the quadrant where the angle is can be deduced. In fact, when equation 15 was applied to both the cornea and IOL in the example (C Z C2), the same result a Z 30 degrees was obtained in both cases. However, for the cornea both sin(2a) and cos(2a) are positive, which means the angle is in the first quadrant, while for IOL both sin(2a) and cos(2a) are negative, indicating that the angle is in the third quadrant. In conclusion, it is necessary to determine the sign of the 2 astigmatism components to ascertain the cylinder direction. If the transposed formula (C Z 2) is used, it will provide the angles corresponding to that case.
RESULTS In practice, both methods described lead to equivalent results as long as thin systems are being dealt with, as was the case. Figure 2 represents the residual cylinder as a function of rotation angle b, in which the rotation direction was considered negative to thus obtain a positive cylinder. The rotation direction did not change the resulting cylinder value, in absolute values, only its sign changes. The resulting cylinder value varied between 0 and double the value of the corneal cylinder (2C1). From equation 3, one obtains C Z C1 when b Z 30 degrees, which shows that the toric IOL has no correction effect when it rotates 30 degrees. Between 0 degree and 30 degrees, one can consider a linear relationship between the residual cylinder C and the angle of rotation. Thus, the residual cylinder when the IOL rotates 10 degrees is 0.33C1. Nevertheless, this result (Figure 2) has been obtained considering equal cylinder values for the cornea and IOL; however,
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Table 1. Effects of IOL rotation. Cylinder (D) Scenario/Rotation Angle b Corneal C1 IOL C2 Residual C
Figure 2. The value of the residual cylinder, C, as a function of the rotation angle of the IOL axis, when cornea and IOL cylinders have equal values (C1 Z C2). The ordinate axis ranges from zero to 2C1, C1 being the corneal astigmatism.
it is not always possible to have IOL cylinders with equal values because the corneal cylinders and the result would change if the 2 cylinders were different (equations 1 and 4). The angle of rotation, b, which annuls the astigmatism correction, could be obtained from cosðp þ 2bÞ Z r=2 with r being the ratio between IOL and cornea cylinders, respectively. Table 1 shows the value of the residual cylinder for different values of IOL cylinder (in the cornea plane) and rotation angle. This shows the result in astigmatism correction according to whether the IOL cylinder equals the cornea cylinder and for different rotations of the IOL axis. Another point is how IOL rotation changes the corrective power in the 2 lens astigmatism axes and the resulting residual refraction. Figure 3 shows that the powers in the axes of the lens are S and S C C (sphere and sphere C cylinder) initially and as the IOL rotates (angle b), the power in these 2 directions increases and decreases, respectively, so S C C/2 values (the socalled spherical equivalent) are obtained in both axes when b Z 45 degrees. As a consequence of these changes in power, the refraction of the eye initially considered (b Z 0 degree), which had been corrected by the IOL, will start to increase in accordance with the rotation angle b as shown in Figure 4; the refraction value will be C/2 and CC/2, respectively, when b Z 45 degrees. A 20-degree rotation only changes the eye refraction in the principal axis directions by approximately 0.25(C1/2) Z C1/8 D (Figures 3 and 4).
Effect of slight rotation with equal IOL and corneal cylinder 0 degree 2.00 1.50 0.50 0 degree 2.00 2.00 0.00 0 degree 2.00 2.50 0.50 G5 degrees 2.00 1.50 0.58 G5 degrees 2.00 2.00 0.35 G5 degrees 2.00 2.50 0.60 G10 degrees 2.00 1.50 0.78 G10 degrees 2.00 2.00 0.69 G10 degrees 2.00 2.50 0.92 Effect with corneal astigmatism partially corrected Case 1 0 degree 4.00 2.50 1.50 G5 degrees 4.00 2.50 1.60 G10 degrees 4.00 2.50 1.90 Case 2 0 degree 7.00 3.50 3.50 G5 degrees 7.00 3.50 3.60 G10 degrees 7.00 3.50 3.90 IOL Z intraocular lens
Figure 5 shows the value of For and Fob (J0 and J45 astigmatism components) of the IOL astigmatism as a function of rotation angle b. The J0 component, whose initial value was C/2 becomes null after a 45-degree rotation, while the initial value of J45 was zero and becomes C/2 after a 45-degree rotation. The 2 components become equal, with a value of 0.35 C, when the IOL is rotated 22.5 degrees.
Figure 3. The 2 IOL powers, in the 2 principal cornea directions, as a function of the rotation angle of the IOL axis, when cornea and IOL cylinders have equal values (C1 Z C2).
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Figure 4. Changes in eye refraction in the 2 principal cornea directions when the IOL rotates from an initial position in which astigmatism was perfectly corrected.
Continuing with the example proposed above, the matrix representing the IOL when rotated 5 degrees, that is 20 (C2) 115 Z 22 (2) 25, will now be 21:6428 þ0:7660 1 0 Z 21 FIOL ð5 Þ Z þ0:7660 20:3572 0 1 1 0 0 1 þ0:6428 þ 0:7660 0 1 1 0
21:6428 þ0:7660 1 Z 21 þ0:7660 20:3572 0 1 0 0 þ0:342020 þ 0:93969 0 1 1
FIOL ðþ5 Þ Z
0 1
1 0
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Figure 5. The variation of the 2 astigmatism components as the IOL axis rotates.
1 0 þ 0:5 0 1 0 1 (19) 0 1 0:866 1 0
Fcornea þ FIOL ð30 Þ Z 62
1 0
Here, the astigmatism is the same as in the cornea before surgery, but now the axis lies in the other direction, just between the direction of the corneal axis and that of the IOL, because both cylinders have the same value. DISCUSSION
(16)
and the astigmatism after the addition of this IOL and the cornea (equation 2) is not null 1 0 Fcornea þ FIOL ð5 Þ Z 62 0 1 (17) 1 0 0 1 þ 0:1428 0:1 0 1 1 0 Again, we can be proved that when the IOL is rotated 30 degrees, its corrective effect is null. In fact, 22 0 1 0 Z 21 FIOL 30 Z 0 20 0 1 (18) 1 0 0 1 þ1 þ0 0 1 1 0 This IOL added to the cornea of equation 2 gives
We would like to address the following 2 ideas about IOL rotation: (1) A certain rotation of angle b can be achieved by rotating the IOL to the right or to the left. In both cases, one obtains the same amount of corrected astigmatism; however, there the sign and axis direction are different for the residual cylinder. (2) One might think that by using a larger cylinder than that of the cornea and slight rotation (which decreases its correcting effectiveness), it would be possible to wholly cancel out the astigmatism. However, when the cylinder value of an IOL is greater than that of the cornea cylinder, by slightly rotating the IOL, one can achieve a component in the orthogonal direction of the cornea. This cancels out the corneal astigmatism, even though slight residual astigmatism always remains in an intermediate direction, as shown in Figure 6, A. Indeed, Harris19 noticed the advantages of using the graphic representation; Figure 6, A, shows the vectors representing the corneal astigmatism and the IOL astigmatism viewed in the example. The scheme clearly shows that only if the vectors are
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than the corneal astigmatism because in this way the residual cylinder is lower, as the top of Table 1 shows. The bottom half of Table 1 shows the case in which corneal astigmatism is high and the cylinder of the available IOL is lower. In this case, we can see that IOL rotation less than 10 degrees hardly affects the residual astigmatism value because the variation is under 0.5 D. The matrix method has the advantage of being formally more correct, invariant under spherocylindrical transposition, and universally usable. Moreover, suitable software is available, which makes matrix calculation easier. We believe this method will be more widely used in the future when calculation by software is generally applied. We are also in agreement with Harris19 regarding the use of the graphic representation as a helpful tool for understanding what happens. Although both methods are absolutely equivalent, depending on the calculation, one method may be easier than the other. In any event, both present the same graphic representation, which is basic and fundamental to understanding what occurs without the need for calculations, as shown in Figure 6, A and B. In conclusion, small IOL rotations change the eye’s refraction by less than 0.50 D under general conditions. Hence, we can state that they are not an obstacle to the satisfactory correction of astigmatism with toric IOLs. APPENDIX Figure 6. The Fcornea and FIOL astigmatisms, when 2a Z 180 degrees, are 2 equal and opposite vectors that annul each other. It is possible to obtain a component opposite and equal to the cornea annulling the astigmatism with a rotated IOL; however, a small astigmatic component will always remain in an intermediate direction.
orthogonal (2a Z 180 degrees) can the astigmatism be completely canceled out (2 equal but opposite vectors). It also shows the information stated in the previous paragraph regarding the small residual cylinder after IOL rotation. The graph in Figure 6, B shows that the residual astigmatism is equal to the corneal astigmatism (ie, astigmatism correction is null) when the IOL rotation is 30 degrees. The information published about residual cylinder after rotation,9 as well as the results in the present study (Figures 2 to 5), were obtained for the case in which C1 Z C2. Nevertheless, sometimes it is not possible to equal a corneal astigmatism with an IOL cylinder. In this case, it does not matter whether the IOL cylinder with a higher or lower value than the corneal astigmatism is chosen as long as both cylinders are orthogonal. However, because small rotation of the implanted IOL is sometimes unavoidable, in such cases the best solution is to choose an IOL cylinder lower
f 11 f 22 S þ C sin2 a S Ccos2 a Z 2 2 C 2 C Z sin a cos2 a Z cos2a 2 2 C Fob Z f 21 Z C sin a cos a Z sin 2a 2
For Z
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function and patient experience after bilateral implantation toric intraocular lenses. J Cataract Refract Surg 2010; 36:609–616 Shimizu K, Misawa A, Suzuki Y. Toric intraocular lenses: correcting astigmatism while controlling axis shift. J Cataract Refract Surg 1994; 20:523–526 Leyland M, Zinicola P, Bloom P, Lee N. Prospective evaluation of a plate haptic toric intraocular lens. Eye 2001; 15:202–205 Till JS, Yoder PR Jr, Wilcox TK, Spielman JL. Toric intraocular lens implantation: 100 consecutive cases. J Cataract Refract Surg 2002; 28:295–301 Chang DF. Early rotational stability of the longer Staar toric intraocular lens; fifty consecutive cases. J Cataract Refract Surg 2003; 29:935–940 € tzner A, Becker R, Pavlovic S. Weinand F, Jung A, Stein A, Pfu Rotational stability of a single-piece hydrophobic acrylic intraocular lens: new method for high-precision rotation control. J Cataract Refract Surg 2007; 33:800–803 Jampaulo M, Olson MD, Miller KM. Long-term Staar toric intraocular lens rotational stability. Am J Ophthalmol 2008; 146:550–553 De Silva DJ, Ramkissoon YD, Bloom PA. Evaluation of a toric intraocular lens with a Z-haptic. J Cataract Refract Surg 2006; 32:1492–1498 Chang DF. Comparative rotational stability of single-piece openloop acrylic and plate-haptic silicone toric intraocular lenses. J Cataract Refract Surg 2008; 34:1842–1847 Chang DF. Repositioning technique and rate for toric intraocular lenses. J Cataract Refract Surg 2009; 35:1315–1316
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16. Wolffsohn JS, Buckhurst PJ. Objective analysis of toric intraocular lens rotation and centration. J Cataract Refract Surg 2010; 36:778–782 17. Rozema JJ, Gobin L, Verbruggen K, Tassignon M-J. Changes in rotation after implantation of a bag-in-the-lens intraocular lens. J Cataract Refract Surg 2009; 35:1385–1388 18. Buckhurst PJ, Wolffsohn JS, Naroo SA, Davies LN. Rotational and centration stability of an aspheric intraocular lens with a simulated toric design. J Cataract Refract Surg 2010; 36:1523–1528 19. Harris WF. Astigmatism. Ophthalmic Physiol Opt 2000; 20: 11–30 20. Deal FC Jr, Toop J. Recommended coordinate systems for thin spherocylindrical lenses. Optom Vis Sci 1993; 70:409–413 21. Thibos LN, Wheeler W, Horner D. A vector method for the analysis of astigmatic refractive errors 14–17. Available at:. In: Visual Science and Its Applications. Technical Digest Series volume 2. Washington, DC, Optical Society of America, 1994. http://www. opt.indiana.edu/people/faculty/thibos/PV/pv.html. Accessed May 6, 2011 22. Thibos LN, Wheeler W, Horner D. Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error. Optom Vis Sci 1997; 74:367–375. Available at: http://journals.lww.com/optvissci/Abstract/1997/06000/Power_ Vectors__An_Application_of_Fourier_Analysis.19.aspx. Accessed May 6, 2011 23. Long WF. A matrix formalism for decentration problems. Am J Optom Physiol Opt 1976; 53:27–33
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Residual astigmatism produced by toric intraocular lens rotation Adelina Felipe, PhD, Jose M. Artigas, PhD, Amparo Díez-Ajenjo, OD, MSc, Carmen García-Domene, OD, MSc, Pablo Alcocer, MD, PhD
PURPOSE: To analyze changes in the eye’s refractive properties when a toric intraocular lens (IOL) rotates. SETTING: Fundaci on Oftalmol ogica del Mediterraneo, Valencia, Spain. DESIGN: Experimental study. METHODS: The matrix definition of astigmatism was used in this theoretical study and compared with another vector representation. Two methods were compared: (1) The cylinder, C, resulting from the addition of 2 cylinders C1 and C2 whose axes form an angle a, is obtained by the addition of 2 vectors of values C1 and C2 forming an angle 2a; (2) the power matrix, F, of a thin astigmatic dioptric system that decomposes naturally into 3 orthogonal components: the purely spherical part Fnes, the ortho-astigmatism For, and oblique astigmatism Fob. RESULTS: The residual cylinder was one third of the corneal astigmatism when a toric IOL rotated G10 degrees when the cylinder values for the cornea (C1) and IOL (C2) were equal. Nevertheless, in most cases C1 is greater than C2; therefore, the residual astigmatism did not change noticeably with small rotations. The angle of rotation, b, which annuls the astigmatism correction, could be obtained from the following: cos(p C 2b) Z r/2, with r being the ratio between the IOL and cornea cylinders. CONCLUSIONS: The 2 methods gave equivalent results. When the IOL cylinder had a value different from that of the corneal astigmatism, a better choice would be a lower, rather than higher, cylinder value to reduce residual astigmatism. In general, toric IOL rotations less than 10 degrees changed the eye’s refraction less than 0.50 diopter. Thus, small axis rotations are not an obstacle for satisfactory astigmatism correction with toric IOLs. Financial Disclosure: No author has a financial or proprietary interest in any material or method mentioned. J Cataract Refract Surg 2011; 37:1895–1901 Q 2011 ASCRS and ESCRS
Toric intraocular lenses (IOLs) are a good solution for astigmatism correction for patients who desire spectacle independence.1–6 New designs of these IOLs were developed to provide better performance and safety. Astigmatism correction involves the position of the IOL in the eye and the axis lying in the correct direction; for this reason, many studies specifically analyze toric IOL rotation after surgery as well as the origin of the rotation to avoid this problem in new designs.7–14 The importance of controlling possible IOL rotation prompted studies of more suitable methods of assessing IOL alignment and toric IOL stability. Improvements introduced in haptics were fundamental in solving this problem.13,14 As a consequence, there have been several studies of the postoperative stability of the toric IOL.11,15–18 Nevertheless, despite the Q 2011 ASCRS and ESCRS Published by Elsevier Inc.
reported high proportion of stability, Chang15 found a small percentage (approximately 1.1%) of patients with significant postoperative rotation and proposed a method calculating the correct angle for repositioning the IOL. Others report that no eye had significant rotation (minor or !20 degrees) and 96.7% had a rotation of less than 10 degrees.4 In a more recent study,18 96% of eyes had rotation of less than 5 degrees and 99% had less than 10 degrees. In any event, there are many cases of eyes with off-axis toric IOLs due to postoperative IOL rotation or an initial error in implantation. Theoretic analyses shows a steady loss of astigmatism reduction as toric IOLs become more off axis.9 If the cornea and IOL cylinders are equal, the amount of corrected astigmatism is zero if the IOL is off axis by 30 degrees. 0886-3350/$ - see front matter doi:10.1016/j.jcrs.2011.04.036
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We performed a theoretical study analyzing in what way and to what extent the refractive properties of an eye, which is supposedly best corrected with a toric IOL, vary when small rotation of the IOL occurs. In a comprehensive study, Harris19 showed the different representations of astigmatism in the optometric and ophthalmologic literature. There are different mathematic expressions that represent the same phenomenon because some authors seem to look for useful mathematic expressions to accomplish a particular job rather than looking for the exact mathematic expression (there is no choice, as Harris said) for astigmatism. Harris presented a universally applicable representation of astigmatism for quantitative work. We tested the usefulness of the matrix definition of astigmatism proposed by Harris19 in this study and contrasted the results with those obtained by the classic addition of vectors20–22 which represent cylinders. Although in both cases we used vectors, Harris’ theory is formally more precise because his definition of astigmatism is invariant under spherocylindrical transposition and constitutes a vector space with all its properties. Both methods must lead to equivalent results; however, an advantage of matrix notation is the possibility of using calculation programs operating with the matrix. This makes it possible to apply the rotation matrix, R, over any power astigmatic matrix as follows: cosb sinb RZ sinb cosb This provides a complete, quick way to obtain a sure result. Moreover, the matrix representation is always valid, even for a thick system, and other methods are not.
equals the cylinder of the cornea and the IOL axis is positioned orthogonally with the corneal astigmatism axis. For example, expressing the power of the astigmatic cornea as sphere (S) and cylinder (C) axis (a), consider an eye with a corneal power of C40.00 C2.00 30 (in diopters [D]) and an IOL with a power of C20.00 C2.00 120 Z C22.00 2.00 30 in the corneal plane. It is easy to see that the addition of the corneal power and IOL power gives a sphere of 62.0 D because the addition of 2 cylinders having the same axis direction and equal absolute power, but opposite sign, is null. It is known that the effective lens position must be considered to obtain the relationship between the IOL cylinder in the corneal plane and the real toric IOL cylindrical power. Thus, to neutralize astigmatism, the actual toric IOL cylindrical power is always higher than the real corneal astigmatism.
Vector Addition To show how a small rotation affects the refractive characteristics of the eye corrected with a toric IOL, this problem was analyzed in detail using the vector addition method. For this calculation, it was assumed that the IOL cylinder and cornea cylinder are equal but with orthogonal (a Z 90 degrees) axes and then, from this initial position, the IOL is off axis in angle b. It is known that the cylinder, C, resulting from the addition of 2 cylinders C1 and C2 with an angle a between the 2 axes is obtained by the addition of 2 vectors of values C1 and C2 forming an angle 2a (Figure 1), giving the following result: C2 Z C21 þ C22 þ 2C1 C2 cosð2aÞ
(1)
In this case, because the 2 cylinders (those of the cornea and the IOL, respectively) when added have the same C1 value, the previous equation can be written as C Z 2C1 cosðaÞ
(2)
in which it can seen that C Z 0 when axes of both cylinders are orthogonal. If from this position a Z 90 degrees, the lens axis is off axis in angle b the cylinder resultant would be
MATERIALS AND METHODS To correct a patient’s astigmatism, the toric IOL is calculated so that in the corneal plane the IOL cylinder value
Submitted: November 8, 2010. Final revision submitted: March 30, 2011. Accepted: April 7, 2011. From Fundacion Oftalmologica del Mediterraneo (Felipe, Artigas, Dıez-Ajenjo, Garcıa-Domene, Alcocer) and Departamento de Optica (Felipe, Artigas,. Dıez-Ajenjo) Facultad de Fısica, Universidad de Valencia, Burjassot (Valencia), Spain. Supported by Catedra Alcon-Universitat de Valencia, Spain. Corresponding author: Adelina Felipe, PhD, Departamento de Optica, Facultad de Fısica, Universidad de Valencia, C/Dr Moliner, 50, E46100-Burjassot (Valencia), Spain. E-mail: adelina.felipe@ uv.es.
Figure 1. The addition of 2 cylinders, with values C1 and C2 diopters, whose axis directions form an angle, a, is equivalent to the addition of 2 vectors having the same values and forming an angle 2a.
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C Z 2C1 cosða þ bÞ Z 2C1 sinðbÞ
(3)
and when b!0 / C Z 2C1 sinðbÞ
angle c. Given the direction of the resultant cylinder axis, which is obtained from the expression tanð2cÞ Z
40:5 0:866 1 0 Z 41 0:866 41:5 0 1 1 0 0 1 0:5 0:866 0 1 1 0
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Fcornea Z
(12)
C2 sinð2aÞ C1 þ C2 cosð2aÞ
(4)
it can be seen that if C1 Z C2, equation 4 is reduced to tanð2cÞ Z tanðaÞ 0 2c Z a and when a rotation occurs: tanð2cÞ Z tanða þ bÞ 0 2c Z a þ b (5) Notice that the c angle (Figure 1) gives the direction of the resulting cylinder axis from the cornea axis position. Thus, in this example, taking b Z 10 degrees and c Z (90 to 10 degrees)/2 Z 40 degrees means the resultant cylinder axis is in a direction 40 degrees from the corneal axis; that is, 40 degrees C30 degrees Z 70 degrees. To obtain the direction of the cylinder resulting from this method, the corneal axis angle (30 degrees in the example) must be added to the angle c (Figure 1). On the other hand, when b O 0, ie, the rotation is produced by distancing the axis of the IOL from the corneal axis (a C b O 90 degrees), the result is C ! 0 and c O 45 degrees while when b ! 0, the rotation goes in the opposite way ( a C b ! 90 degrees) and results in C O 0 and c ! 45 degrees.
Dioptric Power Matrix The symmetric dioptric power matrix introduced by Long in 197623 makes it possible to express the power (S C a) of a thin system as a 2 2 symmetric matrix: f f (6) F Z 11 12 f21 f22 where the components are calculated by f11 Z S þ C sin2 a
(7)
f 12 Z f 21 Z Csina cos a
(8)
f 22 Z S þ Ccos2 a
(9)
The example proposed above can also be calculated following the matrix method introduced by Harris19 as the most complete solution for defining and calculating astigmatism. According to this method, the power matrix, F, of an astigmatic dioptric system decomposes naturally into 3 orthogonal components as follows: F Z Fnes þ Fast Z Fnes þ For þ Fob
(10)
where Fnes is the purely spherical part of the general power and Fast the purely astigmatic part, which itself decomposes in orthoastigmatism For and oblique astigmatism Fob. Using this theory, the cornea and IOL power matrix of the example added by previously writing their power matrix as a linear combination of the 3 linearly independent basic astigmatic powers as follows: 1 0 1 0 0 1 þ For þ Fob (11) F Z Fnes 0 1 0 1 1 0
21:5 þ0:866 1 0 Z 21 þ0:866 20:5 0 1 1 0 0 1 þ 0:5 þ 0:866 0 1 1 0
(13)
1 0 Fcornea þ FIOL Z 62 þ0þ0 0 1
(14)
FIOL Z
It is easily shown (Appendix) that For Z C/2 cos(2a) and Fob Z C/2 sin(2a), which coincides with the values of vector components J0 and J45, respectively. The cylinder, axis direction, and sphere values of the spherocylindrical formula can be obtained from the matrix representation by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G For þ Fob Z GC=2 and S þ C=2 Z Fnes (15) with tan ð2aÞ Z Fob =For Equation 15 provides only the absolute value of the angle 2a; however, the sign of the 2 components For and Fob indicates the sign of cos(2a) and sin(2a), respectively; from this information, the quadrant where the angle is can be deduced. In fact, when equation 15 was applied to both the cornea and IOL in the example (C Z C2), the same result a Z 30 degrees was obtained in both cases. However, for the cornea both sin(2a) and cos(2a) are positive, which means the angle is in the first quadrant, while for IOL both sin(2a) and cos(2a) are negative, indicating that the angle is in the third quadrant. In conclusion, it is necessary to determine the sign of the 2 astigmatism components to ascertain the cylinder direction. If the transposed formula (C Z 2) is used, it will provide the angles corresponding to that case.
RESULTS In practice, both methods described lead to equivalent results as long as thin systems are being dealt with, as was the case. Figure 2 represents the residual cylinder as a function of rotation angle b, in which the rotation direction was considered negative to thus obtain a positive cylinder. The rotation direction did not change the resulting cylinder value, in absolute values, only its sign changes. The resulting cylinder value varied between 0 and double the value of the corneal cylinder (2C1). From equation 3, one obtains C Z C1 when b Z 30 degrees, which shows that the toric IOL has no correction effect when it rotates 30 degrees. Between 0 degree and 30 degrees, one can consider a linear relationship between the residual cylinder C and the angle of rotation. Thus, the residual cylinder when the IOL rotates 10 degrees is 0.33C1. Nevertheless, this result (Figure 2) has been obtained considering equal cylinder values for the cornea and IOL; however,
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Table 1. Effects of IOL rotation. Cylinder (D) Scenario/Rotation Angle b Corneal C1 IOL C2 Residual C
Figure 2. The value of the residual cylinder, C, as a function of the rotation angle of the IOL axis, when cornea and IOL cylinders have equal values (C1 Z C2). The ordinate axis ranges from zero to 2C1, C1 being the corneal astigmatism.
it is not always possible to have IOL cylinders with equal values because the corneal cylinders and the result would change if the 2 cylinders were different (equations 1 and 4). The angle of rotation, b, which annuls the astigmatism correction, could be obtained from cosðp þ 2bÞ Z r=2 with r being the ratio between IOL and cornea cylinders, respectively. Table 1 shows the value of the residual cylinder for different values of IOL cylinder (in the cornea plane) and rotation angle. This shows the result in astigmatism correction according to whether the IOL cylinder equals the cornea cylinder and for different rotations of the IOL axis. Another point is how IOL rotation changes the corrective power in the 2 lens astigmatism axes and the resulting residual refraction. Figure 3 shows that the powers in the axes of the lens are S and S C C (sphere and sphere C cylinder) initially and as the IOL rotates (angle b), the power in these 2 directions increases and decreases, respectively, so S C C/2 values (the socalled spherical equivalent) are obtained in both axes when b Z 45 degrees. As a consequence of these changes in power, the refraction of the eye initially considered (b Z 0 degree), which had been corrected by the IOL, will start to increase in accordance with the rotation angle b as shown in Figure 4; the refraction value will be C/2 and CC/2, respectively, when b Z 45 degrees. A 20-degree rotation only changes the eye refraction in the principal axis directions by approximately 0.25(C1/2) Z C1/8 D (Figures 3 and 4).
Effect of slight rotation with equal IOL and corneal cylinder 0 degree 2.00 1.50 0.50 0 degree 2.00 2.00 0.00 0 degree 2.00 2.50 0.50 G5 degrees 2.00 1.50 0.58 G5 degrees 2.00 2.00 0.35 G5 degrees 2.00 2.50 0.60 G10 degrees 2.00 1.50 0.78 G10 degrees 2.00 2.00 0.69 G10 degrees 2.00 2.50 0.92 Effect with corneal astigmatism partially corrected Case 1 0 degree 4.00 2.50 1.50 G5 degrees 4.00 2.50 1.60 G10 degrees 4.00 2.50 1.90 Case 2 0 degree 7.00 3.50 3.50 G5 degrees 7.00 3.50 3.60 G10 degrees 7.00 3.50 3.90 IOL Z intraocular lens
Figure 5 shows the value of For and Fob (J0 and J45 astigmatism components) of the IOL astigmatism as a function of rotation angle b. The J0 component, whose initial value was C/2 becomes null after a 45-degree rotation, while the initial value of J45 was zero and becomes C/2 after a 45-degree rotation. The 2 components become equal, with a value of 0.35 C, when the IOL is rotated 22.5 degrees.
Figure 3. The 2 IOL powers, in the 2 principal cornea directions, as a function of the rotation angle of the IOL axis, when cornea and IOL cylinders have equal values (C1 Z C2).
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Figure 4. Changes in eye refraction in the 2 principal cornea directions when the IOL rotates from an initial position in which astigmatism was perfectly corrected.
Continuing with the example proposed above, the matrix representing the IOL when rotated 5 degrees, that is 20 (C2) 115 Z 22 (2) 25, will now be 21:6428 þ0:7660 1 0 Z 21 FIOL ð5 Þ Z þ0:7660 20:3572 0 1 1 0 0 1 þ0:6428 þ 0:7660 0 1 1 0
21:6428 þ0:7660 1 Z 21 þ0:7660 20:3572 0 1 0 0 þ0:342020 þ 0:93969 0 1 1
FIOL ðþ5 Þ Z
0 1
1 0
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Figure 5. The variation of the 2 astigmatism components as the IOL axis rotates.
1 0 þ 0:5 0 1 0 1 (19) 0 1 0:866 1 0
Fcornea þ FIOL ð30 Þ Z 62
1 0
Here, the astigmatism is the same as in the cornea before surgery, but now the axis lies in the other direction, just between the direction of the corneal axis and that of the IOL, because both cylinders have the same value. DISCUSSION
(16)
and the astigmatism after the addition of this IOL and the cornea (equation 2) is not null 1 0 Fcornea þ FIOL ð5 Þ Z 62 0 1 (17) 1 0 0 1 þ 0:1428 0:1 0 1 1 0 Again, we can be proved that when the IOL is rotated 30 degrees, its corrective effect is null. In fact, 22 0 1 0 Z 21 FIOL 30 Z 0 20 0 1 (18) 1 0 0 1 þ1 þ0 0 1 1 0 This IOL added to the cornea of equation 2 gives
We would like to address the following 2 ideas about IOL rotation: (1) A certain rotation of angle b can be achieved by rotating the IOL to the right or to the left. In both cases, one obtains the same amount of corrected astigmatism; however, there the sign and axis direction are different for the residual cylinder. (2) One might think that by using a larger cylinder than that of the cornea and slight rotation (which decreases its correcting effectiveness), it would be possible to wholly cancel out the astigmatism. However, when the cylinder value of an IOL is greater than that of the cornea cylinder, by slightly rotating the IOL, one can achieve a component in the orthogonal direction of the cornea. This cancels out the corneal astigmatism, even though slight residual astigmatism always remains in an intermediate direction, as shown in Figure 6, A. Indeed, Harris19 noticed the advantages of using the graphic representation; Figure 6, A, shows the vectors representing the corneal astigmatism and the IOL astigmatism viewed in the example. The scheme clearly shows that only if the vectors are
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than the corneal astigmatism because in this way the residual cylinder is lower, as the top of Table 1 shows. The bottom half of Table 1 shows the case in which corneal astigmatism is high and the cylinder of the available IOL is lower. In this case, we can see that IOL rotation less than 10 degrees hardly affects the residual astigmatism value because the variation is under 0.5 D. The matrix method has the advantage of being formally more correct, invariant under spherocylindrical transposition, and universally usable. Moreover, suitable software is available, which makes matrix calculation easier. We believe this method will be more widely used in the future when calculation by software is generally applied. We are also in agreement with Harris19 regarding the use of the graphic representation as a helpful tool for understanding what happens. Although both methods are absolutely equivalent, depending on the calculation, one method may be easier than the other. In any event, both present the same graphic representation, which is basic and fundamental to understanding what occurs without the need for calculations, as shown in Figure 6, A and B. In conclusion, small IOL rotations change the eye’s refraction by less than 0.50 D under general conditions. Hence, we can state that they are not an obstacle to the satisfactory correction of astigmatism with toric IOLs. APPENDIX Figure 6. The Fcornea and FIOL astigmatisms, when 2a Z 180 degrees, are 2 equal and opposite vectors that annul each other. It is possible to obtain a component opposite and equal to the cornea annulling the astigmatism with a rotated IOL; however, a small astigmatic component will always remain in an intermediate direction.
orthogonal (2a Z 180 degrees) can the astigmatism be completely canceled out (2 equal but opposite vectors). It also shows the information stated in the previous paragraph regarding the small residual cylinder after IOL rotation. The graph in Figure 6, B shows that the residual astigmatism is equal to the corneal astigmatism (ie, astigmatism correction is null) when the IOL rotation is 30 degrees. The information published about residual cylinder after rotation,9 as well as the results in the present study (Figures 2 to 5), were obtained for the case in which C1 Z C2. Nevertheless, sometimes it is not possible to equal a corneal astigmatism with an IOL cylinder. In this case, it does not matter whether the IOL cylinder with a higher or lower value than the corneal astigmatism is chosen as long as both cylinders are orthogonal. However, because small rotation of the implanted IOL is sometimes unavoidable, in such cases the best solution is to choose an IOL cylinder lower
f 11 f 22 S þ C sin2 a S Ccos2 a Z 2 2 C 2 C Z sin a cos2 a Z cos2a 2 2 C Fob Z f 21 Z C sin a cos a Z sin 2a 2
For Z
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