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Calculating the power of toric phakic intraocular lenses Achim Langenbucher1, No´ra Szentma´ry2 and Berthold Seitz3 1
Department of Medical Physics, University of Erlangen-Nu¨rnberg, Erlangen, Germany, 2Department of Ophthalmology, University of Budapest, Hungary, and 3Department of Ophthalmology, University of Saarland, Homburg (Saar), Germany
Abstract Background and purpose: A toric phakic intraocular lens (IOL) implanted in the anterior or posterior chamber of the eye has the potential to correct high or excessive ametropia and astigmatism with high predictability of the postoperative refraction and preservation of phakic accommodation. The calculation of spherical phakic lenses has been described previously, but a formalism for estimating the power of toric phakic lenses has not yet been published. The purpose of this study is to describe a mathematical strategy for calculating toric phakic IOLs. Methods: The method presented in this paper is based on vergence transformation in the paraxial Gaussian space. Parameters used for the calculations are the spherocylindrical spectacle refraction before implantation, corneal power (sphere and astigmatism) and (spherocylindrical) target refraction, together with the vertex distance and the predicted position of the phakic IOL. The lens power is determined as the difference in vergences between the spectacle-corrected eye and the uncorrected eye at the reference plane of the predicted lens position. The axes of the preoperative refraction, the target refraction and the corneal astigmatism are at random (not necessarily aligned). Results: The method was applied to two clinical examples. In example 1 we calculate the power of a phakic lens for the simple case, when the target refraction is plano and the axis of the preoperative refraction is aligned to the axis of the corneal astigmatism. In example 2, the cylindrical axis of the preoperative refraction is not aligned to the corneal astigmatism and the target refraction is spherocylindrical (and the axis of the target refraction is not aligned to the preoperative refractive cylinder or the corneal astigmatism). The calculations for both examples are described step-by-step and illustrated in a table. Conclusions: The calculation scheme can be generalized to an unlimited number of crossed cylinders in the optical pathway. Based on paraxial raytracing, the spherical and cylindrical power as well as the orientation of the cylinder are determined from the preoperative refraction (including vertex distance), the corneal power, the intended target refraction (including vertex distance) and the predicted position of the phakic lens implant provided by the lens manufacturer. This calculation scheme can be easily implemented in a simple computer program (i.e. in Microsoft EXCEL or MATLAB). Keywords: intraocular lens power calculation, paraxial raytracing, phakic intraocular lenses, toric lenses, vergence transformation
Introduction Received: 22 August 2006 Revised form: 8 January 2007, 24 January 2007 Accepted: 25 January 2007 Correspondence and reprint requests to: Achim Langenbucher. Tel.: 49 9131 8525557; Fax: 49 9131 8522824. E-mail address:
[email protected] The authors have no proprietary interest in the development or marketing of this or any competing instrument or piece of equipment. ª 2007 The Authors. Journal compilations ª 2007 The College of Optometrists
The difficulties of correcting high ametropia (especially in combination with astigmatism) due to the inherent optical aberrations and secondary psychological problems of spectacles, and frequent contact lens intolerance, justify the search for and investigation of new technologies in refractive correction. However, the surgical correction of high ametropia and astigmatism has been doi: 10.1111/j.1475-1313.2007.00487.x
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controversial (Scarpatetti, 1983; Lackner et al., 2003; Saxena et al., 2003). The implantation of phakic lenses is a highly effective and predictable procedure to correct high ametropia such as myopia, hyperopia, or astigmatism (van der Heijde et al., 1988). In contrast to corneo-refractive surgery with the excimer laser or radial keratotomies, the potential range of correction is much greater, the optical zone is much wider (which enhances vision performance especially in dim light) and there is no risk of keratectasia (Gu¨ell et al., 2003). Especially in corneal astigmatism, the range of correction is limited with transverse or arcuate keratotomies or LASIK. Intraocular procedures capable of correcting ametropia include phakic lens implantation, which permits optical correction of the refractive error while maintaining accommodation, and clear lens extraction, which must be considered with caution especially in myopic eyes because of the potentially higher risk of retinal detachment and the loss of accommodation in young patients. Similar to corneo-refractive laser surgery, phakic intraocular lenses (IOLs) use a smaller optical zone to treat higher ametropia (Gu¨ell et al., 2003). The implantation of anterior chamber IOLs in phakic eyes has proved to be an effective and predictable technique. However, the risk of damage to anterior chamber structures and especially the corneal endothelium (Menezo et al., 1998) initiated the development of a new concept of posterior chamber lens implantation. In 1986, Fechner and Worst (1989) modified the iris claw lens for correcting aphakia introduced by Fyodorov into a negatively biconcave lens for correction of high myopia. To increase the safety of this lens, the design of the lens optics was changed in 1991 to a concave–convex shape. This lens design was reported to cause less alteration to the corneal endothelium. With the introduction of the toric phakic IOL (Gu¨ell et al., 2003) there are completely new options for correcting corneal or lenticular astigmatism whilst preserving the physiological accommodation of the eye. Especially in high or excessive astigmatism (where corneo-refractive laser surgery may fail due to potential complications such as flap striae, haze, reduced contrast sensitivity or glare) toric phakic IOLs may be an appropriate surgical option. In clinical practice, phakic IOLs are normally calculated using the so-called van der Heijde formula (van der Heijde et al., 1988). In this fundamental paper a calculation strategy for spherical phakic Worst-Fechner IOLs for correction of myopia using classical vergence transformation was described. With a prediction of the cardinal point of the phakic lens of 3 mm behind the corneal vertex, the refractive power of such a lens implant can be calculated from the pre-existing myopia,
the vertex distance of the spectacle correction and the corneal power. The purpose of the present paper is to generalize the van der Heijde formula for calculation of toric phakic IOLs with spherocylindrical target refraction using a vergencebased formalism by moving the position of the refractive correction from the spectacle plane to an IOL plane. The axes of the corneal astigmatism, the pre-existing refractive cylinder and the cylinder of the target refraction are randomly positioned. The applicability of this calculation scheme is demonstrated in two clinical examples. Methods Definition of a prescription in different notations Our calculation scheme makes use of the principle of astigmatic decomposition, which is commonly used in optical aberration theory. A prescription (vergence or refractive power of a surface) in the spherocylindrical form S/C · Q (S ¼ sphere, C ¼ plus or minus cylinder, Q ¼ axis) is converted into a component notation (Bennett, 1986a,b; Dunne et al., 1994, 1997): C SEQ ¼ S þ 2 C0 ¼ C cosð2HÞ ; C45 ¼ C sinð2HÞ where SEQ refers to the spherical equivalent refractive power, C0 refers to the projection of the cylinder to the 0/180 meridian and C45 to the projection of the cylinder to the 45/135 meridian. Converted to this form, the three elements of any number of astigmatic prescriptions can be algebraically summed without further processing. The re-conversion into the commonly used standard notation requires the following steps: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 C ¼ C02 þ C45 1 C C0 H ¼ tan : C45 C S ¼ SEQ 2 The above-mentioned method is suitable to compound two obliquely crossed spherocylinders and neither the direction of rotation in positive or negative angles (clockwise or counter clockwise) nor the sign (plus or minus) of the cylinder need to be considered. s
s
Vergence transformation and prediction of the power of the phakic lens implant For our calculations, the spectacle lens, the cornea and the phakic IOL are considered as thin refractive ª 2007 The Authors. Journal compilations ª 2007 The College of Optometrists
Toric phakic IOLs: A. Langenbucher et al. elements. The vergence in front of the spectacle lens is assumed to be zero, which means that the optical system eye is corrected for far distance before and after surgery: n refers to the refractive index of aqueous humour. To calculate the effective refractive change in vergence between successive surfaces, the orthogonal projections of the powers in each principal refracting meridian (S and S + C) may be summed to the respective orthogonal projections of the vergence. The distance between two successive surfaces has to be reduced by the refractive index of the medium between them (d/n). A vergence V is transformed through a homogeneous optical medium by V0 ¼
V ; 1 ðd=nÞV
where V refers to the vergence in front of the medium and V¢ refers to the vergence behind the medium. This transformation has to be made separately with the vergences in each meridian (S and S + C). The orientation of the cylinder remains unchanged. For example, a vergence V of )10 D at 12 mm spectacle distance (from an object at 10 cm) is transformed to a vergence of V0¼
10 D ¼ 8:928 D: m 1 0:012 1:000 : 10 D
Before implantation of the phakic IOL, the optical system eye consists of the spectacle correction PSpr, the interspace between the spectacle and the cornea (vertex distance, ds), the cornea PC, the interspace between the cornea and the crystalline lens (anterior chamber, dac), the crystalline lens PL, and the interspace between the crystalline lens and the retina (vitreous, dv). A bundle of rays entering the system at the spectacle plane with an object vergence V0pr ¼ 0 (object at infinity) is refracted (spectacle refraction, vergence V0pr¢) and transformed to the corneal plane (vergence V1pr). After being refracted by the cornea (vergence V1pr¢) the bundle of rays is passing the anterior chamber of the eye. The reference plane for our calculation scheme where we have to determine the vergence (V2pr) is the predicted lens position. After implantation of the phakic IOL, the optical system eye consists of the spectacle correction (¼target refraction) PSpo, the interspace between the spectacle and the cornea (vertex distance, ds), the cornea PC, the interspace between the cornea and the phakic lens implant dIOL, the phakic IOL, the interspace between the phakic lens and the crystalline lens dl, the crystalline lens PL, and the interspace between the crystalline lens and the retina (vitreous, dv). A bundle of rays entering the system at the spectacle plane with an object vergence V0po is refracted (spectacle refraction, vergence V0po¢) and transformed to the corneal plane (vergence V1po). After being refracted by the cornea (vergence V1po¢) and ª 2007 The Authors. Journal compilations ª 2007 The College of Optometrists
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passing the aqueous humour to the phakic IOL (vergence V2po) it is refracted by the crystalline lens (vergence V2po¢). If the vergence V2pr at the plane of the phakic lens implant and the vergence V2po¢ at the back surface of the phakic lens implant are equal, the entire optical system has the same focal plane preoperatively and postoperatively. Thus, the refractive power of the phakic IOL is calculated as the difference of the vergence V2pr and the vergence V2po. Results For both examples we assume a vertex distance of 14 mm and a predicted position of the phakic IOL of 3.0 mm behind the corneal vertex as provided by the lens manufacturer. The refractive index of air is 1.000 and the index of aqueous humour is 1.336. Corneal refraction is assumed to be 41.50/+6.50 · 15. Example 1 For the first example (Table 1) we assume a spectacle refraction )8.00/)5.00 · 15, where the refractive cylinder is aligned in axis to the corneal astigmatism. Target refraction is plano (0 D spherical). As we calculate the power of the phakic IOL for far distance, the object vergences V0pr and V0po in the standard and component notation (rows 1 and 10, columns 3–9) equal zero. In the first step, the corneal power PC (rows 6 and 15), the preoperative refraction PSpr (row 2) and the target refraction PSpo (row 11) are entered in the calculation scheme in Table 1 in standard notation (columns 3–6). The distances ds (row 4) and dIOL (row 8) are entered in column 1. In the second step, the corneal power, the preoperative refraction and the target refraction are transformed to component notation and the data are entered in columns 7–9. The distances d from column 1 are transformed to reduced distances d/n and the data are entered in column 2. In the third step, the vergences V0pr¢ and V0po¢ are calculated by adding the three respective components (columns 7–9) of V0pr and PSpr and V0po and PSpo. After re-conversion to standard notation (columns 3–6) the vergences S and S + C are transformed separately through the homogeneous optical medium (with reduced distance d/n) to get the vergences V1pr and V1po at corneal plane. Both vergences are converted to component notation (columns 7–9) and the respective components of the corneal surface are added up to get the vergences V1pr¢ and V1po¢ directly behind the corneal surface in component notation. After re-conversion to standard notation (columns 3–6), the vergences S and
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Table 1. Example 1: Calculation of a toric phakic lens implant for correction of spherocylindrical refraction. The calculation scheme is based on vergence transformation. The values d and d/n refer to the distance and reduced distance between refractive surfaces. ÔVÕ refers to vergences, ÔprÕ or ÔpoÕ to the preoperative and postoperative state, and ÔPÕ to the refractive power of a surface. In standard notation, S and S + C mean prescriptions (power of an astigmatic surface or vergence) in both principal meridians, C is the difference between both meridional powers and Q the axis of the prescription. In the component notation, SEQ is the power equivalent of a prescription. C0 and C45 are the standardized cylindrical components in an axis 0 and 45. The shaded rows describe refractive surfaces, whereas the non-shaded rows mean vergences in front of or behind those surfaces. The upper block of the matrix (rows 1–9) refers to the spectacle corrected eye before phakic lens implantation and the middle block (rows 10–18) to the eye with a phakic lens and target refraction at spectacle plane (in this example equal to zero). In row 19, the refraction of the phakic lens is characterized with the difference of the vergences given in rows 9 and 18 s
S + C are transformed again through the homogeneous medium (reduced distances d/n) to the predicted position of the phakic IOL. Both vergences V2pr and V2po are converted to component notation (columns 7–9). In the fourth step, the three components of the vergence V2po are subtracted from the respective components of the vergence V2pr to determine the refractive power of the phakic IOL in component notation. The components are entered in row 19. In the last step, the
s
component notation of the phakic lens power (row 19, columns 7–9) is re-converted to standard notation (row 19, columns 3–6) to get the refractive power of the lens. In our example, a toric phakic lens with a sphere of )13.44 D and a cylinder of 4.85 D (equivalent power )11.01 D) has to be implanted to correct the eye to emmetropia. The orientation of the plus cylinder of the toric phakic lens (105) is perpendicular to the axis of the corneal cylinder (15). ª 2007 The Authors. Journal compilations ª 2007 The College of Optometrists
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Table 2. Example 2: Calculation of a toric phakic lens implant for correction of spherocylindrical refraction. A description of the variables is provided in the legend to Table 1 and in the text
Example 2 In example 2 (Table 2) we present the calculation in a more complex case, where we intend a spherocylindrical target refraction after the surgical intervention to support near vision (astigmatism against the rule) and the axes of the corneal astigmatism and the preoperative refraction are not aligned (i.e. there is lenticular astigmatism). We assume a spectacle refraction )4.00/ )4.50 · 5 (row 2) and the intended target refraction to be )0.50/)1.00 · 90 (row 11). For our calculations, the same corneal refraction and the same distances as in example 1 are used. Analogous to example 1, the object vergences V0pr and V0po in the standard and component notation (rows 1 and 10, columns 3–9) equal zero. ª 2007 The Authors. Journal compilations ª 2007 The College of Optometrists
The procedure has to be carried out in an analogue way from step 1 to step 4 (Table 2). After subtraction of the three components of the vergence V2po from the respective components of the vergence V2pr the refractive power of the phakic IOL in component notation is given in row 19, columns 7–9. These components are reconverted to standard notation (row 19, columns 3–6) to get the refractive power of the lens. In our example, the toric phakic lens with a sphere of )8.72 D and a cylinder of 5.93 D (equivalent power )5.75 D) has to be implanted to achieve a target refraction of )0.50/ )1.00 · 90. The orientation of the plus cylinder of the toric phakic lens (94.3) is neither aligned to the axis of the corneal cylinder (15) nor to the axis of the preoperative refraction (5).
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Discussion Toric IOLs for correction of aphakia after cataract extraction have become generally accepted in clinical routine since the end of the 1990s for the correction of corneal astigmatism during cataract surgery to improve postoperative visual performance of the patient (Frohn et al., 1999; Nguyen and Miller, 2000; Novis, 2000; Ruhswurm et al., 2000; Sun et al., 2000; Gerten et al., 2001; Leyland et al., 2001; Amm and Halberstadt, 2002; Gills and van der Karr, 2002; Tehrani and Dick, 2002; Till et al., 2002; Gills, 2003). While the calculation of spherical IOLs is well established in clinical practice (Gernet et al., 1970; Hoffer, 1975, 1993; Retzlaff, 1980; Holladay et al., 1988; Retzlaff et al., 1990; Haigis, 1995; Olsen et al., 1995; Naeser, 1997), there are only a few schemes for calculating toric lenses (MacKenzie and Harris, 2002; Langenbucher and Seitz, 2003; Sarver and Sanders, 2004). Different approaches for calculating toric IOLs can be used. For example, Sarver and Sanders (2004) developed a set of equations to calculate an appropriate IOL power for phakic or aphakic astigmatic eyes, and to predict the postoperative spectacle correction for a selected toric IOL, and provided a parameter to be used to optimize predictability of the calculations based on clinical data. Another option is to derive the power of ÔthinÕ or ÔthickÕ toric IOLs using a compact matrix-based scheme (Langenbucher et al., 2004a,b). In cases of high ametropia the implantation of phakic anterior or posterior chamber IOLs is a useful strategy for correcting the refractive error of the eye. In contrast to clear lens extraction for correction of excessive myopia, phakic IOLs are beneficial especially in young patients due to maintenance of phakic accommodation. In the range of low or medium ametropia, other surgical interventions such as LASIK or LASEK are more popular, because the risk of endothelial damage caused by contact with the phakic lens is not accepted (Lackner et al., 2003). Recently a series of different phakic IOLs were launched to the market for correction of myopia, hyperopia (Saxena et al., 2003) and – especially in the last 5 years – for correction of astigmatism (toric lenses) (Gu¨ell et al., 2003; Langenbucher and Seitz, 2003). The first paper to describe the calculation of refractive power of phakic IOLs was by van der Heijde et al. (1988). They presented a vergence-based strategy to determine the power of a spherical phakic lens from preoperative spectacle refraction, vertex distance, corneal power and the predicted position of the lens implant within the eye. He transformed the vergence to the plane of the predicted phakic lens position both through the spectacle-corrected eye and the ametropic (uncorrected) eye. The power of the phakic lens implant was determined as the difference of the preoperative and postoperative vergence at the predicted position of the
phakic lens implant. Whereas van der Heijde focused on the calculation of phakic lenses for emmetropia, Holladay and Haigis provided explicit formulas for arbitrary (spherical) ametropia after implantation of phakic IOLs (Holladay, 1993; Haigis, 1996). The concept of calculating toric IOLs presented in this paper is a straightforward method for tracing an astigmatic pencil of rays through the spectacle correction, the cornea and the anterior chamber to the predicted position of the phakic IOL. Refractive surfaces were considered by adding the representative components, SEQ and both cylindrical components C0 and C45 , to the respective components of the vergence at this position. Transformation through a refractive medium (space between spectacle lens and cornea; or anterior chamber) was performed by considering the change of vergence calculated separately for both principal meridians. With the difference of both vergences in front of and behind the (thin) phakic lens implant the power of the lens can be calculated. The refractive correction at the corneal plane can be directly extracted from the residual vergence at the corneal plane (rows 5 and 14 of Tables 1 and 2 for the preoperative and postoperative state). With this mathematical concept the vergence of an optical system consisting of an unlimited number of astigmatic surfaces in random axes can be assessed. If the refraction of both corneal surfaces (front and back surface) are known, the optical system can be enhanced using the thick lens model of the cornea. This modification may have the potential to offer a suitable concept for IOL calculation after refractive surgery (Naeser, 1997; Norrby and Koranyi, 1997), i.e. in cases where only one refractive surface of the cornea has been selectively changed in geometric shape by excimer laser ablation (Seitz and Langenbucher, 2000). In addition, if the shape of the phakic IOL (central thickness, refractive index, front and back surface geometry) is known, the optical system can be improved by including these data in the calculation scheme. The surgically induced astigmatism may be considered with a cylindrical component in the target refraction. For example, if from the surgeon’s experience the induced astigmatism is estimated to be )0.5 D at 10 and the (spherical) target refraction is intended to be )0.75 D, the surgeon may enter a target refraction of )0.25/)0.5 · 100 or )0.75/+0.50 · 10. Alternatively, if the surgeon has experience with the change in keratometry during surgery, this change in corneal shape can be directly considered by adding the respective values in the component notation to the corneal refraction in row 15 of the calculation scheme (Tables 1 and 2). A persisting problem is that the estimation of the postoperative Ôeffective lens positionÕ cannot be solved with this new calculation scheme. For this purpose, we s
s
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Toric phakic IOLs: A. Langenbucher et al. have to rely on the experience of the manufacturer of the phakic IOL. For the cataract surgery with posterior chamber lens implantation, the inventors of standard IOL formulae such as the SRK/T (Retzlaff, 1980; Sanders et al., 1988; Retzlaff et al., 1990), the HofferQ (Hoffer, 1975, 1993), the Holladay (Holladay et al., 1988), or Haigis (1995) or the calculation scheme presented by Naeser (1997) present a more or less sophisticated way of estimating the Ôeffective lens positionÕ. All these theoretical optical formulae provide an estimation of the postoperative lens position using multiple regression analysis using the lens specific constant (A-, ACD-, pACD-constant, surgeon factor…) together with the keratometric readings, the axial length, or the phakic anterior chamber depth. Naeser et al. (1990) developed a new methodology for prediction of the pseudophakic anterior chamber depth based on biometric measurements of the posterior lens capsule. Norrby and Koranyi (1997) developed a new concept (lens haptic plane concept) for estimation of the effective postoperative position of a thick IOL. Another problem with (phakic) toric IOLs, which is still unresolved, is the rotation of the axis. Especially in high cylindrical corrections, the axis of the keratometry should be validated by a topographer and should be marked directly before surgery using the slit-lamp to avoid a misplacement of the IOL axis by more than 5. Theoretically, a toric lens (calculated for a spherical target refraction) rotated 5/10 away from the ideal axis induces a postoperative astigmatism of 17%/35% of the initial astigmatism before surgery. In addition, the rotation of the IOL axis during the follow-up period may reduce the effectivity of this procedure (Langenbucher and Seitz, 2003; Langenbucher et al., 2005). In conclusion, we have presented a vergence-based strategy for calculation of toric phakic IOL power with the flexibility of crossing an unlimited number of cylinders. With that concept based on paraxial raytracing, the spherical and cylindrical power as well as the orientation of the cylinder is determined from the preoperative refraction (including vertex distance), the corneal power, the intended target refraction (including vertex distance) and the predicted position of the phakic lens implant provided by the lens manufacturer. This calculation scheme can be easily implemented in a simple computer program (i.e. in Microsoft EXCEL or MATLAB; The MathWorks Deutchland Offices, Munich, Germany). References Amm, M. and Halberstadt, M. (2002) Implantation einer torischen Linse zur Korrektur eines hohen postkeratoplastischen Astigmatismus. Ophthalmologe 99, 464–469. Bennett, A. G. (1986a) Two simple calculation schemes for use in ophthalmic optics. – I: Tracing oblique rays through ª 2007 The Authors. Journal compilations ª 2007 The College of Optometrists
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