A New Method for Calculating Residual Astigmatism Produced by Toric Intraocular Lens Rotation

Aims: To report a new method for calculating residual astigmatism when exact and inexact cylinder power toric intraocular lenses (IOLs) are rotated. Methodology: Using Excel spreadsheets and sinθ values, we expressed the cylinder powers of toric IOLs in percentage terms compared to corneal astigmatic powers and calculated values for various corneal astigmatisms and toric IOLs. From these values, we created charts of residual astigmatic powers and axes demonstrating the results of IOLs ranging from 50% to 150% and rotated 0o to 30o. Results: Comparing residual astigmatic powers, the rank orders for each IOL were as follows: 100%<90%<80% at 5o of rotation, 90%<100%<80% at 10o of rotation, and 90%<80%<75%<100% at 15o of rotation. Thus, lower cylinder power IOLs perform better than higher power IOLs when rotated over 5o. Furthermore, the residual astigmatic powers of 125% IOLs were always higher than those of 100% IOLs; however, the residual astigmatic powers of 75% IOLs became lower than those of 100% IOLs when rotation exceeded 15o. Conclusion: Our method shows that lower cylinder power IOLs are advantageous in environments where IOL rotation is likely and when inexact IOLs are utilized. Original Research Article Yae and Kubota; OR, Article no. OR.2014.6.011 369


INTRODUCTION
Toric intraocular lenses (IOLs) are relatively effective in correcting corneal astigmatisms [1,2]. However, errors in measuring both the corneal astigmatic axis and postoperative IOL rotation may occur [3,4]. Moreover, inexact IOLs are sometimes utilized because the cylinder powers of toric IOLs are limited, while corneal astigmatic powers are infinite. Therefore, we need precise calculation methods for residual astigmatism when higher, equal, or lower cylinder power IOLs are rotated compared to the corneal astigmatic power.
In general, the astigmatic distribution curve of a cylinder lens of f diopters (Cylinder (C)+f Diopter (D)×0º) is represented by f sin 2 θ [5]. In the present study, we used sin 2 θ and Excel spreadsheets to calculate the residual astigmatism produced by toric IOL rotation. Furthermore, we expressed the cylinder powers of toric IOLs in percentage terms compared to corneal astigmatic powers. Therefore, our method can be applied to any corneal astigmatism and to any cylinder power IOL.
Several calculation methods have been previously reported. Two main types of methods have been discussed, namely vector [6,7] and matrix methods [8,9]. While these methods are detailed and precise, the calculations must be performed one-by-one. In contrast, our method allows us to perform many precise calculations simultaneously in Excel spreadsheets using percentage terms and sin 2 θ. Furthermore, our method also allows for visually comprehensible charts to be obtained. Consequently, we are able to decide whether a higher, equal, or lower cylinder power IOL would be most appropriate.
We would typically use the website "AcryS of Toric Calculator" when preoperatively deciding which IOL would be most optimal [10]; however, this does not explain why a particular IOL is selected. In contrast, our method allows us to better understand the concept as it sheds light on the theoretical background behind the calculations through its tables and figures.

METHODOLOGY
Firstly, we defined the corneal astigmatism as C+2.0 D×0° (f=2.0), which is represented by 2 sin 2 θ (Fig. 1). Furthermore, we expressed the cylinder powers of toric IOLsin percentage terms compared to the corneal astigmatic powers. Namely, the cylinder power of a 100% toric IOL is 2.0 D, and that of a 75% toric IOL is 1.5 D if the corneal astigmatic power is 2. Under these conditions, we created (Table 1) to explain our calculations. In the table, sin 2 θ is calculated using the actual formula (Microsoft Excel, Redmond, WA, USA) as follows: sin(θ×PI()/180)×sin(θ × PI()/180) Through this formula, we can calculate values using degrees rather than radians; moreover, any errors in the calculation results were less than 1×10 -10 D.

.5D) toric IOL rotated 15º
The residual astigmatic power is represented by the amplitude of the sine wave, the axes of which are the peak and trough of the sine wave, respectively (cf. Fig. 2).
The approximate residual astigmatic axes at 45º and 135º are shown in square.
Next, we demonstrate how to calculate the precise residual astigmatic power and axes. The residual astigmatic power is represented by the amplitude of the sine wave, while the residual astigmatic axes are the peak and the trough of the sine wave, respectively. Therefore, in (Table 1 and Fig. 2), the approximate residual astigmatic power is 1.250 (=0.375-(-0.875)) D, and the approximate residual astigmatic axes are 45º and 135º.
However, more precise calculations can be performed by calculating in 1º increments. Therefore, we created (Table 2 and Fig. 3) from area a in (Fig. 2). (Table 2 and Fig. 3) reveal thata more precise value for the residual astigmatic power is 1.2608 (= 0.3804-(-0.8804)) D, and more precise residual astigmatic axes are 41º and 131º.
This graph was created from ( Table 3). The cylinder power of an n% IOL is n/100 × the corneal astigmatic power (n=75, 100, and 125). Point B is the intersection of a 100% IOL and a 75% IOL (between 14º and 15º). The corneal astigmatic power is equal to 100% residual astigmatic power.

RESULTS
In (Table 3), the residual astigmatic powers produced by each toric IOL rotation can be compared. Namely, the rank orders of each IOL are 100%<90%<80%<75%<125% at 5º of rotation, 90%<100%<80%<75%<125% at 10º of rotation, and 90%<80%<75%<100% <125% at 15º of rotation. Therefore, 100% IOLs are the best selection for rotations within 5°, while 90% to 100% IOLs are best for rotations from 5º to 10º, and 80% to 90% IOLs are best for rotations over 10º. Moreover, (Fig. 4) reveals that the residual astigmatic powers of 125% IOLs are always higher than those of 100% IOLs. In contrast, those of 75% IOLs become lower than those of 100% IOLs when the IOLs rotation exceeds the point of intersection (point B: between 14º and 15º). Accordingly, the curves of the residual astigmatic powers of lower cylinder power toric IOLs have gentler slopes than those of higher cylinder power toric IOLs. This means that low cylinder power IOLs are a better choice than high cylinder power IOLswhen rotated.

Fig. 4. The residual astigmatic powers of 75%, 100%, and 125% intraocular lens (IOL) rotations
In (Fig. 5), corrections with lower cylinder power IOLs (50%) result in the residual astigmatic axis being close to the corneal astigmatic axis, while corrections with higher cylinder power IOLs (150%) result in it being close to the axis of the IOLs. In addition, it is demonstrated that IOLs closer to 100% produce more oblique (OB) astigmatism. In general, OB astigmatism is not very significant if its power is small; however, the power of OB astigmatism should be minimized [11]. Therefore, low cylinder power toric IOLs are also better from this viewpoint, when exact IOLs are not available for the corneal astigmatism.
The cylinder power of an n% IOL is n/100× the corneal astigmatic power (n=50, 75, 100, 125, and 150). In a 100% IOL (a), there is no axis when there is no rotation; however, the axis is approximately 45º when rotated by even 0.1º.

DISCUSSION
The advantage of our method is that the table (Table 3) and chart (Fig. 4) prepared beforehand are useful references that allow us to select the most appropriate toric IOLs. This method was facilitated by expressing the cylinder power of toric IOLs in percentage terms compared to the corneal astigmatic power and by using Excel tables; therefore, the calculations are simple. Consequently, (Table 3 and Fig. 4) indicate that a low cylinder power IOL would be a better choice than a high cylinder power IOL when rotated. This opinion is also shared by Felipe et al. [9].On the other hand, some methods have recommended high cylinder power IOLs, contrary to our opinion [12,13]. However, we must remember that slightly higher cylinder power IOLs will cause OB astigmatisms instead of becoming against-the-rule (ATR) astigmatisms when correcting with-the-rule (WTR) astigmatism (Fig. 5). Furthermore, the residual astigmatic powers corrected by high cylinder power IOLs are larger than those corrected by low cylinder power IOLs (Fig. 4). Kobayashi et al. reported that eyes with OB astigmatism had significantly lower visual performance than eyes with WTR or ATR astigmatism [11]. Moreover, the influence of IOL rotation becomes more severe with greater corneal astigmatism and when higher cylinder power IOLs are selected. In fact, one report suggested that the error was particularly relevant in patients in whom higher cylinder power IOLs were implanted [2]. Consequently, we recommend low cylinder power IOLs when correcting corneal astigmatisms.
Clinically, ATR astigmatic change in corneal astigmatisms with advancing age is a documented fact [14,15]. Therefore, ATR astigmatism calculations are important. However, we have only calculated WTR astigmatisms in order to make the charts easy to understand.