Refractive, anterior corneal and internal astigmatism in the pseudophakic eye

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1 Refractive, anterior corneal and internal astigmatism in the pseudophakic eye Jesper F. Bregnhøj, 1,2 Pourang Mataji 1,2 and Kristian Næser 1,2 1 Department of Ophthalmology, Aarhus University Hospital, Aarhus, Denmark 2 Department of Ophthalmology, Randers Regional Hospital, Randers, Denmark ABSTRACT. Purpose: To evaluate the correlation between astigmatism (RA) and anterior corneal astigmatism (ACA), and determine the internal astigmatism (IA) in 184 pseudophakic eyes. Methods: The study was a prospective non-masked single-centre study. Patients were examined 8 weeks after phacoemulsification with implantation of aspheric one-piece monofocal IOLs. Examination included autokeratometry and subjective refraction. All data were converted to the corneal plane. The corneal index, taken to be 1.376, was used to estimate the ACA. All astigmatisms were converted to net curvital and net torsional powers with the steeper corneal plane as the reference meridian. Curvital power is the force acting along a given meridian, and torsion is the power twisting the astigmatic direction out of that plane. The internal astigmatism (IA) was calculated as the difference between RA and ACA. Results: For curvital powers, the astigmatism (KP(Φ) RA ) could be described as a function of anterior corneal astigmatic magnitude (KP(Φ) ACA ) and direction a by the multiple linear regression equation: KP (Φ) RA = *KP(Φ) ACA *cos2a, (r 2 = 0.59, p < ). The average internal astigmatism amounted to 0.47 D inclined 92 relative to the steeper anterior corneal meridian. The magnitude of internal astigmatism depended on the angle a of the steeper anterior corneal meridian, averaging 0.86 D at 91 for with-the-rule, 0.37 D at 95 for oblique and 0.17 D at 97 for against-the-rule corneal astigmatisms. Conclusions: The internal astigmatism varies as a function of the direction of the anterior steeper corneal meridian. In patient candidates to surgical correction of astigmatism, measuring only the curvature of the anterior corneal surface and neglecting that of the posterior corneal surface can lead to inaccurate evaluation of total corneal astigmatism. Key words: Astigmatism Javal s rule polar values posterior corneal curvature residual astigmatism Acta Ophthalmol. 2015: 93: ª 2014 Acta Ophthalmologica Scandinavica Foundation. Published by John Wiley & Sons Ltd doi: /aos Introduction The goal of current cataract surgery is an excellent uncorrected visual acuity (UCVA) in all fixation ranges, which requires a zero postoperative astigmatism (RA). This goal may be reached by implantation of toric lenses and by a variety of corneal procedures, such as on-axis phaco incisions, limbal relaxation incisions, anterior keratotomy and postoperative corneal laser touch-ups. Following cataract surgery, RA is predominantly determined by the total corneal astigmatism. It is therefore important to establish a numerical correlation between corneal and astigmatisms. Ocular astigmatism (RA) is measured with manifest refraction. Ocular RA is caused by the toricity, tilt or lack of axis alignment of the ocular surfaces (Næser 2008). Anterior corneal astigmatism (ACA) refers to the toricity of the anterior corneal surface, as measured with keratometry or videokeratography (Keller et al. 1996). The curvatures of the principal anterior corneal meridians are recorded, and their dioptric powers are calculated, employing the corneal index, taken to be Posterior corneal curvatures are usually not measured, and the total corneal astigmatism is therefore calculated with the use of effective or fictitious indices, in which the negative posterior corneal powers are included. In current clinical practice, effective corneal indices vary from to and are based on various assumptions over the relative distribution of anterior and posterior corneal powers (Olsen 1986; Holladay et al. 1998). Duke-Elder defines the residual astigmatism as the astigmatism due to the posterior surface of the cornea, the surfaces of the lens, decentration and the variability of the index of the lens (Duke-Elder 1970). In current ophthalmology, residual astigmatism 33

2 denotes astigmatism left over from procedures. In this study, we shall therefore use the term internal astigmatism (IA), defined as the difference between the and anterior corneal astigmatism. The IA in pseudophakic eyes predominantly reflects the astigmatism of the posterior corneal surface, as the astigmatic effect of IOL tilt and decentration usually is minimal (Rosales & Marcos 2007). A number of clinical studies comparing and corneal astigmatisms are detailed in Table 1. The resultant formulas may be regarded as variations over Javal s rule (Javal 1890). Early studies included with-the-rule (WTR) astigmatism and against-therule (ATR) astigmatism and excluded oblique astigmatisms. Some later studies transformed astigmatisms of all directions to vectors along fixed meridians of 0/90 and 45/135 degrees. Refractive indices of around were used in most studies. The aim of this study was to evaluate the correlation between the and anterior corneal astigmatism and to determine the internal astigmatism in 184 pseudophakic eyes. All data were converted to the corneal plane. The corneal index, taken to be 1.376, was used to estimate the ACA. All astigmatisms were transformed to polar values with the steeper corneal plane as reference meridian (Næser 2008). A new statistical correlation between and anterior corneal astigmatism was developed. Materials and Methods We conducted a prospective nonmasked single-centre study on cataract patients referred to the Department of Ophthalmology, Randers Regional Hospital in Denmark, for phacoemulcification with monofocal IOL implantation. Informed consent was obtained from all participants, and the study was conducted according to the tenets of the Declaration of Helsinki, including acceptance by the local ethics committee. Inclusion criteria were uncomplicated treatment with standard cataract surgery and postoperative corrected distance visual acuity (CDVA) 0.8 (Snellen visual scale). Exclusion criteria included previous corneal or intraocular surgery, corneal disease, iris abnor- Table 1. Review of clinical studies on correlation between and keratometric astigmatism. The table reports the first author, measurement methods and directions for astigmatism, number (n) of eyes and the resultant formulas representing variations in Javal s rule. In studies employing nominal values for corneal and astigmatism, only with-the-rule (WTR) and against-the-rule (ATR) astigmatisms were included, while oblique astigmatisms were excluded. Studies based on power vectors (J0, J45) included astigmatisms in all directions. Formula. Units in dioptres (D) Correlation coefficient Refractive Index Mean Age (Years) Phakic/Pseudophakic Study population (n) Astigmatisms included Study Methodology WTR and ATR 1058 N/A Phakic N/A RA = 1.00 * ACA N/A Keratometric, Grosvenor et al. (1988) WTR and ATR Phakic N/A RA = 0.97 * ACA r 2 = 0.70 Videokeratoscopy, Keller et al. (1996) r 2 = 0.71 r 2 = 0.48 All directions Phakic RJ0 = * CJ RJ45 = * CJ Autokeratometry, Tong et al. (2001) r 2 = 0.79 r 2 = 0.83 All directions Phakic RJ0 = 1.07 * CJ RJ45 = 1.46 * CJ Autokeratometry, Remon et al. (2009) WTR and ATR Pseudophakic N/A RA = 0.81 * ACA r 2 = 0.56 Autokeratometry, Bae et al. (2004) r 2 = 0.29 r 2 = 0.36 All directions Pseudophakic RJ0 = 0.52 * CJ RJ45 = 0.41 * CJ Autokeratometry, Teus et al. (2010) ACA = anterior corneal astigmatism, nominal value, RA = astigmatism, nominal value, RJ0 = astigmatism in the meridian of 0 /180, CJ0 = anterior corneal astigmatism in the meridian of 0 /180, RJ45 = astigmatism in the meridian of 45 /135, CJ45 = anterior corneal astigmatism in the meridian of 45 /135. Keller et al. (1996) reported formulas for data obtained with both 4 and 7 mm aperture for the applied videokeratoscopy. The formula for data obtained with the 4 mm aperture is listed. Correlation coefficient r 2 values calculated based on published r values. 34

3 malities, retinal disease, uveitis, glaucoma and neuro-ophthalmologic disease. Included subjects received standard preoperative examination with automated keratometry, measurement of CDVA, slitlamp biomicroscopy, pneumotonometry and indirect funduscopy under mydriasis. Biometric measurements were performed with IOL Master, version (Carl Zeiss Meditec AG, Oberkochen, Germany), and IOL power calculated with Naeser s vergence formula (Naeser 1997). Phacoemulsification in topical anaesthesia was performed in all cases. The procedure was performed under mydriasis through a 2.4-mm corneal incision temporally or in the steeper corneal meridian. After anterior capsulorhexis, hydrodissection and phacoemulsification, aspheric monofocal IOLs (one-piece acrylic aspheric TEC- NIS, ZCB00 IOL, Abbott Medical Optics) were implanted in the capsular bag. Finally, cefuroxime 1.0 mg was injected intracamerally. Healon GV â was used during capsulorhexis and later during IOL implantation. All patients were re-examined in our clinic 4 23 weeks (average 8.3 weeks) after surgery. The examination included autorefraction and autokeratometry with the Humphrey, HARK 599, autorefractokeratometer (Carl Zeiss Meditec AG, Oberkochen, Germany), followed by meticulous subjective refraction by the authors with a Nidek RT-2100 phoropter (NIDEK Co., Ltd., Gamagori, Japan). Subjective refraction was performed before pupil dilatation with an accuracy of 0.25 D in sphere and cylinder magnitude, and 5 accuracy of astigmatic direction. Thorough clinical ophthalmological examination was performed in the same manner as preoperatively. Data management Any given surface may be recorded in cross-cylinder format, namely as a principal meridian of maximal power S Max along the meridian a and an orthogonal principal meridian of minimal power S Min along the axis (a + 90). The difference in power is the magnitude of astigmatism M. In the present context, the direction a of the astigmatism is the principal meridian of maximal power, S Max = S Min + M, where M 0. Fig. 1. Bivariate polar value analysis of the internal astigmatism (IA). Individual values in dioptres and the 95% confidence ellipse for observations. The combined mean value amounted to ( 0.47 D, 0.04 D) for the total population of eyes (n = 184). The Pearson correlation coefficient between these two variables amounted to r = Abscissa: IA expressed as the curvital polar value, KP(Ф) IA. Ordinate: IA expressed as the torsional polar value, KP(Ф + 45) IA. Net astigmatism is the general power format for describing any astigmatism and is given as (M at a) or M along a, where M is the astigmatic magnitude in dioptres (D), and a is the astigmatic direction in degrees (Næser 2008). This format can be used to characterize a single astigmatism, but cannot be used for calculations, which require transformation to components, such as the polar value system. Any net astigmatism is fully characterized by the net curvital and torsional powers, where the former is the power acting along a given meridian Φ, and the latter is the power twisting the astigmatic direction out of that plane (Keating 1986; Harris 1997; van Gool & Harris 1998), hereby giving rise to a cylinder rotation. In this study, the meridian Φ is always the direction a of the steeper anterior corneal meridian, different for each eye, measured from the horizontal meridian. See Fig. 1 in Næser (2008) for a description of these meridians. Deriving and reducing equations from a previous study by Næser (2008): curvital power along any random plane Φ may be described by the sine-squared correlation as: PðUÞ ¼S Min þ M cos 2 ða UÞ ð1þ Curvital power along the orthogonal plane (Φ + 90) is given as: PðU þ 90Þ ¼S Min þ M cos 2 ða ðu þ 90ÞÞ ¼ S Min þ M sin 2 ða UÞ ð2þ Curvital power along the oblique plane (Φ + 45) counter-clockwise to Φ is given as: PðU þ 45Þ ¼S Min þ M cos 2 ða ðu þ 45ÞÞ ¼ S Min þ M cos 2 ðða UÞ 45Þ ð3þ Curvital power along the oblique plane (Φ 45) clockwise to Φ is given as: PðU 45Þ ¼S Min þ M cos 2 ða ðu 45ÞÞ ¼ S Min þ M cos 2 ðða UÞ þ 45Þ ð4þ The net curvital power is the difference between the curvital powers along Φ and (Φ + 90): 35

4 Net curvital power ¼ KPðUÞ ¼ polar value along the meridian U ¼ M cos 2 ða UÞ Msin 2 ða UÞ ¼ M cosð2 ða UÞÞ ð5þ The net torsional power over Φ is the difference between the curvital powers along (Φ + 45) and (Φ 45). The net torsional power is also the difference between the counter-clockwise and the clockwise torsions over Φ. Net torsional power ¼ KPðU þ 45Þ ¼ polar value along the meridian ðu þ 45Þ ¼ M cos 2 ðða UÞ 45Þ M cos 2 ðða UÞþ45Þ ¼ M 2 sinða UÞcosða UÞ ¼ M sinð2 ða UÞÞ ð6þ These expressions may be further reduced when the reference meridian Φ is identical to the direction a of the net astigmatism. This is the case for the anterior corneal astigmatism in this study. Inserting Φ = a in equations (5) and (6), we obtain: KPðUÞ ¼Net curvital power ¼ M cosð2 ða aþþ ¼ M cosð0þ ¼M ð7þ KPðU þ 45Þ ¼Net torsional power ¼ M sinð2 ða aþþ ¼ M sinð0þ ¼0 ð8þ So, when the astigmatic direction and the reference meridian are coinciding, net curvital power is identical to the nominal astigmatic magnitude M, and the net torsion is zero. See Example 1 Keratometry. The corneal net astigmatisms were directly converted to polar values with equations (5) and (6). Refractive data were transformed from vertex (spectacle) to corneal plane as described by Holladay et al. (1998). Manifest refraction was converted to cross-cylinder power format. Each of these cylinders was transformed from the vertex to the corneal plane with the distance equation: REF C ¼ REF V =ð1 ðref V V=1000ÞÞ; ð9þ where REF C is the corneal plane refraction, REF V the vertex plane refraction and V the vertex distance in mm. The resulting net astigmatism was transformed to polar values (Naeser 2012). All calculations are illustrated in Example 1 Manifest refraction. The internal astigmatism (IA) was calculated as the difference between and anterior corneal astigmatism: (KP(Φ) IA, KP(Φ + 45) IA ) = (KP (Φ) RA, KP(Φ + 45) RA ) (KP(Φ) ACA, KP(Φ + 45) ACA ) See Example 1 Internal astigmatism Subgroup analyses of IA were performed for WTR (60 a < 120 ), oblique (30 a < 60 and 120 a < 150 ) and ATR (0 a < 30 and 150 a 180 ) anterior corneal astigmatisms. Any single set of polar values and the result of any compilation of polar values may be reconverted to the usual net cylinder notation by the following general equations: M ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KPðUÞ 2 þ KPðU þ 45Þ 2 M KPðUÞ a ¼ arctan þ U KPðU þ 45Þ ð10þ ð11þ Example 1 Keratometry. The radii of curvature are measured as r 1 = (7.34 mm at 171 ) and r 2 = (8.07 mm at 81 ). The anterior corneal astigmatism, using the corneal index of 1.376, is given as: ( ) * (1000/ /8.07) = 4.63 D. The net anterior corneal astigmatism therefore is (4.63 D at 171 ). Converting to polar values, with the reference meridian Φ = the steeper anterior corneal meridian = 171 : KP(Φ) ACA = 4.63*cos(2 ( )) = 4.63 D; KP(Φ + 45) ACA = 4.63*sin(2( )) = 0D. Manifest refraction. These transformations were carried out as described on page of Naeser (2012). Manifest refraction (= spectacle correction) with a vertex distance of 15 mm: axis 75. Converting to cross-cylinder power format: (2.0 D at 75 ) and ( 1.75 D at 165 ). Inserting in the distance equation (9) to obtain the corneal plane powers: REF C = 2.0/(1-(2.0*15/ 1000)) = 2.06 D; REF C = 1.75/ (1-(( 1.75)*15/1000)) = 1.71 D. The corneal plane cross-cylinder format therefore is (2.06 D at 75 ) and ( 1.71 D at 165 ). Change signs to obtain ocular astigmatism hereby enabling a comparison with keratometric data: ( 2.06 D at 75 ) and (1.71 D at 165 ). The net astigmatism in plus power format is (3.77 D at 165 ). Converting the net astigmatism to polar values with equations (5) and (6), with the reference meridian Φ = the steeper anterior corneal meridian = 171 : KP(Φ) RA = 3.77*cos (2( )) = 3.69 D; KP(Φ + 45) RA = 3.77*sin(2( )) = 0.78 D. Internal astigmatism. The internal astigmatism (IA) is subsequently calculated as: (KP(Φ) IA, KP(Φ + 45) IA ) = (3.69 D, 0.78 D) (4.63 D, 0 D) = ( 0.94 D, 0.78 D). All data met the normality requirements as defined by the Kolmogorov Smirnov statistics. Relations between keratometric, and internal astigmatism data were performed with Pearson correlation coefficient and linear regression analyses. Comparison of data was performed with Student s t-test and analysis of variance test (ANOVA). A p- value of less than 0.05 was considered significant. Statistical analyses were performed with GRAPHPAD PRISM 6 for Windows, GraphPad Software, Inc., SIGMAPLOT for Windows Version 10.0, Systat Software, Inc., and SPSS version 18 for Windows (SPSS, Inc., Chicago, IL, USA). Graphical bivariate analysis of the internal astigmatism was performed with Systat version 13.0, Systat Software, Inc. Results Demographic data are shown in Table 2. The mean nominal astigmatism (RA) was 0.68 dioptres (D) (SD 0.54 D, range 3.77 to 0.00), which is approximately twothirds of the mean anterior corneal astigmatism (ACA) of 1.01 D (SD 0.70 D, range ). Table 3 summarizes the astigmatic data after transformation to polar values in the corneal plane. The mean curvital power KP(Φ) RA was 0.54 D, and the average anterior corneal curvital power KP(Φ) ACA was 1.01 D. Internal astigmatism net curvital (KP(Φ) IA ) and torsional (KP (Φ + 45) IA ) powers varied considerably and averaged 0.47 D and 0.04 D, respectively. The average IA amounted to 0.47 D inclined 92 relative to the 36

5 Table 2. Demographic data of the 184 pseudophakic eyes. Right eyes Left eyes Overall Male Female Total Age; mean (range) in years 71.2 (41 85) 71.0 (41 94) 71.1 (41 94) Multiple linear regression with RA as dependent, and ACA and cosine 2a as independent variables, emerges as: KPðUÞ RA ¼ 0:09 þ 0:61 KPðUÞ ACA þ 0:33 cos2a; ðr 2 ¼ 0:59; p \ 0:0001Þð12Þ steeper anterior corneal meridian. The IA is shown in a bivariate plot with 95% confidence limits in Fig. 1. The magnitude of internal astigmatism depended on the angle a of the steeper anterior corneal meridian, the net astigmatism averaging 0.86 D at 91 for with-the-rule, 0.37 D at 95 for oblique and 0.17 D at 97 for againstthe-rule anterior corneal astigmatisms. Mean IA curvital powers for WTR, oblique and ATR corneal astigmatisms amounted to 0.86 D, 0.36 D and 0.17 D, respectively. The torsional component was negligible in all subgroups. Analysis of variance (ANOVA) revealed a statistically significant difference between average IA curvital values, but no significant difference between average torsional values in the WTR, oblique and ATR subgroups (Table 3). A statistically significant positive correlation between the magnitudes for and anterior corneal curvital power is shown in Fig. 2. Linear regression between RA and ACA curvital power emerges as: KP(Φ) RA = 0.55*KP (Φ) ACA 0.02, (p < ). The determination coefficient r 2 amounted to The direction a of the anterior corneal astigmatism is a periodic function with the period 180 degrees and is therefore not suitable for linear analysis. Curvital power is a double-angle cosine function, as shown in Equation 2. A linear correlation between curvital astigmatism and cosine 2a is shown in Fig. 3. Linear regression between RA curvital power and this transformed value emerges as follows: KP(Φ) RA = 0.26*cos2a , (p < ). The determination coefficient r 2 amounted to The determination coefficient r 2 increased to 0.59 by including both magnitude and cos2a of the ACA as independent variables in the regression equation. The correlation is shown in a three-dimensional plot in Fig. 4. The corneal index, taken to be 1.376, is theoretically correct, but in practical ophthalmology, effective indices incorporating some compensation of the negative posterior corneal power are usually employed. The IA magnitudes and the multiple linear regression equation are recalculated with the commonly used effective index of Assessments of anterior corneal toricity with this methodology were termed keratometric astigmatism (KA). For the effective corneal index n = the IA curvital power averaged 0.37 D ( 0.54 D) for the Table 3. Polar values for the anterior corneal astigmatism (ACA), astigmatism (RA) and internal astigmatism (IA) in the total group and in the three subgroups of with-the-rule (WTR), oblique and against-the-rule (ATR) corneal astigmatism. All curvital and torsional polar values in units of dioptres (D). The IA is the paired difference between the RA and the ACA, with the steeper anterior corneal meridian Φ as reference. The directions of RA and IA are angular inclinations in relation to the reference plane Φ. Curvital power is acting along the reference meridian Φ, while torsion is the dioptric force twisting the astigmatic axis out of this plane. All four groups demonstrated a statistically significant curvital IA, but no significant torsional IA power (Student s paired t-test). The net astigmatisms in the right column were calculated with equations (10) and (11) from the average curvital and torsional powers. Group Polar values Curvital power, KP(Φ) Torsional power, KP(Φ + 45) Mean (SD) Range Mean (SD) Range Net astigmatism M (D) at a Overall, n = 184 Anterior corneal astigmatism (ACA) 1.01 (0.70) 0.06 to D at steeper meridian Ф Refractive astigmatism (RA) 0.54 (0.61) 0.98 to (0.32) 0.85 to D at 178 Internal astigmatism (IA) 0.47 (0.57)* 2.37 to (0.32)ns 0.85 to D at 92 WTR, n = 73 Anterior corneal astigmatism (ACA) 1.20 (0.76) 0.06 to D at steeper meridian Ф Refractive astigmatism (RA) 0.34 (0.57) 0.98 to (0.30) 0.85 to D at 177 Internal astigmatism (IA) 0.86 (0.53)* 2.37 to (0.30)ns 0.85 to D at 91 Oblique, n = 29 Anterior corneal astigmatism (ACA) 0.71 (0.49) 0.12 to D at steeper meridian Ф Refractive astigmatism (RA) 0.35 (0.48) 0.44 to (0.43) 0.74 to D at 174 Internal astigmatism (IA) 0.36 (0.33)* 1.38 to (0.43)ns 0.74 to D at 95 ATR, n = 82 Anterior corneal astigmatism (ACA) 0.94 (0.67) 0.06 to D at steeper meridian Ф Refractive astigmatism (RA) 0.78 (0.61) 0.00 to (0.29) 0.78 to D at 179 Internal astigmatism (IA) 0.17 (0.46)* 1.29 to (0.29)ns 0.78 to D at 97 ANOVA: p< ANOVA: ns ns: Student s paired t-test, not statistically significant. * Student s paired t-test, statistically significant with a p-value <

6 whole group. The IA averaged 0.74 D ( 0.48 D), 0.29 D ( 0.32 D), and 0.07 D ( 0.43 D) in the WTR, oblique, and ATR subgroups. Mean torsional values were close to zero. The corresponding multiple linear regression equation for the whole group was: KP(Φ) RA = *KP(Φ) KA *cos2a (r 2 = 0.59, p < ). Fig. 2. Refractive astigmatism (RA) in dioptres as function of anterior corneal astigmatism (ACA). Linear regression with 95% confidence bands (inner dotted lines), and 95% prediction bands (outer dotted lines) for the total population of eyes (n = 184). A significant positive correlation was found between the curvital power KP(Φ) RA and the anterior corneal astigmatism curvital power KP(Ф) ACA. The astigmatism is approximately half of the anterior corneal astigmatism for any given magnitude of the anterior corneal astigmatism. Fig. 3. The magnitude of the astigmatism (RA) as function of converted values (cosine 2a) of the direction a of the steeper anterior corneal meridian (ACA). Linear regression with 95% confidence bands (inner dotted lines), and 95% prediction bands (outer dotted lines) for the total population of eyes (n = 184). Abscissa: cosine 2a. Ordinate: RA expressed as the curvital polar value, KP(Ф) RA. Fig. 4. Multiple linear regression equation with astigmatism (RA) as dependent variable of the anterior corneal astigmatism (ACA), and the cosine 2a of the direction a of the steeper anterior corneal meridian for the total population of eyes (n = 184). The plane in the threedimensional plot displays the combined correlation of the magnitude of ACA, the cosine of 2a and the magnitude of RA. Discussion All data were converted to the corneal plane to facilitate comparison between and corneal astigmatism, as previous. The corneal index of was applied. The internal astigmatism was calculated as the difference between and anterior corneal astigmatism. As the astigmatic effect of IOL tilt and decentration is negligible in pseudophakic eyes, the internal astigmatism probably largely reflects the posterior corneal astigmatism in this pseudophakic cornea model (Rosales & Marcos 2007). Some previous studies listed in Table 1 employed vector analysis along fixed meridians to evaluate the correlation between and anterior corneal astigmatism, as measured by keratometry. For a unique description of internal astigmatic direction and magnitude in preparation of surgery, we thought that a reference meridian had to be individualized. We chose the anterior steeper corneal meridian in each eye as reference; see page of the paper by Næser (2008) for a comprehensive description of these methods. Results of steep-meridian analysis applied on this field have not previously been published. The advantage of using the steeper anterior corneal meridian as a reference for vector calculations is that surgical procedures, such as toric IOLs and limbal relaxing incisions, are placed according to this meridian, and surgeons need to know the influence of posterior astigmatism along it. As the average torsional power was negligible, the results suggest this choice of reference meridian to be advantageous. The average meridians are therefore coinciding as between RA and ACA, or orthogonal as between ACA and IA. There was no average rotation of axes between the different groups of astigmatism. If only anterior corneal astigmatism can be recorded, the presently described statistical evaluations of posterior corneal astigmatism 38

7 Table 4. Review of clinical studies with measurements of posterior corneal astigmatism. The table reports the first author, measurement methods for astigmatism, study population and magnitude of mean posterior corneal astigmatism. Study Astigmatism assessment methodology Study population (n) Mean age (years) Posterior corneal astigmatism (dioptres) Dunne et al. (1991) Purkinje images in 3 fixed meridians (Polaroid camera, autokeratometer) Dunne et al. (1996) Purkinje images in 4 fixed meridians (Zeiss photographic 132 N/A 0.22 slit lamp, videokeratoscopy) Dubbelman et al. (2006) Scheimpflug photography in 6 fixed meridians (Topcon SL-45 camera) Ho et al. (2009) Rotating Scheimpflug photography (Oculus Pentacam) Pin ero et al. (2012) Rotating Scheimpflug photography (Oculus Pentacam) Koch et al. (2012) Combined rotating Scheimpflug photography and placido images (Ziemer Galilei DSA) Prisant et al. (2002) Scanning-slit topography (Bausch & Lomb Orbscan) 40 N/A Mo dis et al. (2004) Scanning-slit topography (Bausch & Lomb Orbscan) should be considered. For example, Equation (12) predicts 0.19 D astigmatism for a WTR anterior corneal astigmatism of 1.0 D at 90. Conversely, an ATR keratometry of 1.0 D at 0 amounts to 0.85 D astigmatism. A procedure removing 0.85 D of astigmatism might be relevant for the ATR eye, but surgery is not indicated for the WTR eye with similar anterior corneal astigmatism. In this study, internal astigmatism curvital and torsional powers averaged 0.47 D and 0.04 D, respectively (Table 3). These values may be backcalculated to the net cylinder 0.47 D at 92. Hence, the present study estimated the average magnitude of the internal astigmatism to approximately 0.5 D and its direction as perpendicular to the steeper ACA. In average, the steeper anterior corneal and the flatter internal astigmatic meridians are therefore orthogonal, and the steeper meridians are aligned. The bivariate analysis in Fig. 1 illustrates the great individual variation in curvital and torsional IA. A proportion of corneas even had positive values for IA curvital power; in such eyes, the steeper anterior and the flatter IA meridians are aligned. The findings in this study are similar to results from previous studies of posterior corneal astigmatism, which reported average values from 0.21 D to 0.77 D, depending on the applied methodology (Table 4). Subgroup analysis revealed the influence of anterior meridional direction on the IA: mean IA curvital power averaged 0.86 D for WTR anterior corneal toricity, 0.17 D for ATR astigmatism and 0.36 D for intermediate oblique astigmatisms. Similar observations were carried out by Koch et al. based on dual Scheimpflug analyser measurements (Koch et al. 2012). The regression equation, KP(Φ) RA = 0.55*KP(Φ) ACA 0.02 (Fig. 2), predicts the astigmatism as approximately 55% of the corneal astigmatism. Similar variations in Javal s rule (Javal 1890) are summarized in table 1. However, the listed equations are quite different, both from each other and from the present equation. These differences may probably be attributed to the choice of corneal index, but also to variability in age of the investigated populations, and to various distributions of ACA magnitude and direction. The latter two variables were incorporated into the multiple linear regression equation, shown in Fig. 4 and in equation (12). To the best of our knowledge, the incorporation of both ACA magnitude and direction in a single equation to predict RA has not previously been reported. Equation (12) therefore represents a new method beyond Javal s rule. The determination coefficient of r 2 = 0.59 indicates that 59% of the variation in astigmatism may be attributed to the joint variations in astigmatic magnitude and direction. The present study has limitations. A larger population of pseudophakic eyes could potentially have strengthened the accuracy of the estimates. Of the total 184 eyes included, 134 were eyes from bilaterally pseudophakic patients. Supplementary statistical analyses on data from the 117 pseudophakic patients with one operated eye revealed similar results as the overall results presented. The HARK 599 autokeratometer, used in the present study, measures the anterior corneal curvatures in the central 3.0 mm part of the cornea. Hence, measurements may not reflect the true power of a larger part of the central cornea. The potential influence of the methods technical limitations was addressed by Keller et al. (1996) who assessed corneal astigmatism by videokeratoscopic measurements of corneal topography and found comparable estimates of average corneal astigmatism with apertures ranging from 2 to 7 mm. The study results were similar to Grosvenors keratometry data (Grosvenor et al. 1988). Hence, the appliance of keratometric measurements in the present study is warranted. The influence of the internal astigmatism may be particularly important to consider in IOL calculation in previously high-myopia LASIK-operated eyes with cataract, as posterior corneal abnormality can be present (Kamiya & Oshika 2003; Miyata et al. 2004). Several formulas have been suggested to accommodate for the altered properties of the anterior corneal curvatures in IOL power calculation in LASIKtreated eyes (Saiki et al. 2013), but these do not address the influence of internal astigmatism. Miyata et al. (2013) investigated the use of combined placido and Scheimpflug imaging of the corneal anterior and posterior surface in ray-tracing IOL power calculation in normal cataractous eyes, and found no association between the spherical prediction error and posterior corneal curvature. Development of methods for IOL power calculation that accommodate for individual internal astigmatism would certainly be advantageous for toric IOLs. 39

8 In conclusion, the internal astigmatism varies as a function of the direction of the anterior steeper corneal meridian. In patient candidates to surgical correction of astigmatism, measuring only the curvature of the anterior corneal surface and neglecting that of the posterior corneal surface can lead to inaccurate evaluation of total corneal astigmatism. Acknowledgements Kristian Næser has received funding from the public health authorities through the Health Research Fund of Central Denmark Region. The authors have no conflict of interests to declare. References Bae JG, Kim SJ & Choi YI (2004): Pseudophakic residual astigmatism. Korean J Ophthalmol 18: Dubbelman M, Sicam VA & Van der Heijde GL (2006): The shape of the anterior and posterior surface of the aging human cornea. Vision Res 46: Duke-Elder S, ed (1970): System of Ophthalmology. Volume 5: Ophthalmic Optics and Refraction. St. Louis, MO: Mosby Dunne MC, Royston JM & Barnes DA (1991): Posterior corneal surface toricity and total corneal astigmatism. Optom Vis Sci 68: Dunne MC, Elawad ME & Barnes DA (1996): Measurement of astigmatism arising from the internal ocular surfaces. Acta Ophthalmol Scand 74: van Gool RD & Harris WF (1998): Curvital and torsional dioptric power and their polar profiles across meridians. S Afr Optom 57: Grosvenor T, Quintero S & Perrigin DM (1988): Predicting astigmatism: a suggested simplification of Javal s rule. Am J Optom Physiol Opt 65: Harris WF (1997): Dioptric power: its nature and its representation in three- and fourdimensional space. Optom Vis Sci 74: Ho JD, Tsai CY & Liou SW (2009): Accuracy of corneal astigmatism estimation by neglecting the posterior corneal surface measurement. Am J Ophthalmol 147: Holladay JT, Dudeja DR & Koch DD (1998): Evaluating and reporting astigmatism for individual and aggregate data. J Cataract Refract Surg 24: Javal E (1890): Memoires D 0 Ophthalmometrie. Masson G (ed.) Paris: Librarie de L 0 Academie de Medecine Kamiya K & Oshika T (2003): Corneal forward shift after excimer laser kerato surgery. Semin Ophthalmol 18: Keating MP (1986): Dioptric power in an offaxis meridian: the torsional component. Am J Optom Physiol Opt 63: Keller PR, Collins MJ, Carney LG, Davis BA & van Saarloos PP (1996): The relation between corneal and total astigmatism. Optom Vis Sci 73: Koch DD, Ali SF, Weikert MP, Shirayama M, Jenkins R & Wang L (2012): Contribution of posterior corneal astigmatism to total corneal astigmatism. J Cataract Refract Surg 38: Miyata K, Tokunaga T, Nakahara M et al. (2004): Residual bed thickness and corneal forward shift after laser in situ keratomileusis. J Cataract Refract Surg 30: Miyata K, Otani S, Honbou N & Minami K (2013): Use of Scheimpflug corneal anteriorposterior imaging in ray-tracing intraocular lens power calculation. Acta Ophthalmol 91: e546 e549. Modis L Jr, Langenbucher A & Seitz B (2004): Evaluation of normal corneas using the scanning-slit topography/pachymetry system. Cornea 23: Naeser K (1997): Intraocular lens power formula based on vergence calculation and lens design. J Cataract Refract Surg 23: Næser K (2008): Assessment and statistics of surgically induced astigmatism. ActaOphthalmologica 86: (Issue thesis1)1 28. Naeser K (2012): Combining and topographic data in corneal surgery for astigmatism: a new method based on polar value analysis and mathematical optimization. Acta Ophthalmol 90: Olsen T (1986): On the calculation of power from curvature of the cornea. Br J Ophthalmol 70: Pi~nero DP, Camps VJ, Mateo V & Ruiz-Fortes P (2012): Clinical validation of an algorithm to correct the error in the keratometric estimation of corneal power in normal eyes. J Cataract Refract Surg 38: Prisant O, Hoang-Xuan T, Proano C, Hernandez E, Awwad ST & Azar DT (2002): Vector summation of anterior and posterior corneal topographical astigmatism. J Cataract Refract Surg 28: Remon L, Benlloch J & Furlan WD (2009): Corneal and astigmatism in adults: a power vectors analysis. Optom Vis Sci 86: Rosales P & Marcos S (2007): Customized computer models of eyes with intraocular lenses. Opt Express 5: 15: Saiki M, Negishi K, Kato N, Arai H, Toda I, Torii H, Dogru M & Tsubota K (2013): A new central-peripheral corneal curvature method for intraocular lens power calculation after excimer laser surgery. Acta Ophthalmol 91: e133 e139. Teus MA, Arruabarrena C, Hernandez-Verdejo JL, Sales-Sanz A & Sales-Sanz M (2010): Correlation between keratometric and astigmatism in pseudophakic eyes. J Cataract Refract Surg 36: Tong L, Carkeet A, Saw SM & Tan DT (2001): Corneal and error astigmatism in Singaporean schoolchildren: a vector-based Javal s rule. Optom Vis Sci 78: Received on November 8th, Accepted on March 16th, Correspondence: Jesper F. Bregnhøj, MD Department of Ophthalmology Aarhus Hospital University of Aarhus Nørrebrogade Aarhus C Denmark Tel: Fax: jespbreg@rm.dk 40