- Email: [email protected]
Intraocular lens power formula based on vergence calculation and lens design
Intraocular lens power formula based on vergence calculation and lens design Kristian Naeser, MD
ABSTRACT Purpose: To present a new method of intraocular lens (IOL) power calculation that takes into account the power-dependent variation in lens design. Setting: Department of Ophthalmology, Aalborg, Denmark. Methods: Information on exact IOL design is derived from the manufacturers' cutting cards. These data were built into an IOL calculation formula based on exact ray tracing and vergence calculation. All algorithms are demonstrated. Results: The method is optically correct as all refractive surfaces are characterized with respect to both position and refractive power. In simulation models, the Naeser and the Holladay formulas performed similarly, while the SRK(T formula predicted higher postoperative refractions for low-power IOLs. Conclusion: It is possible to incorporate the exact IOL design into an IOL power calculation formula. Theoretically, the Naeser formula should increase the accuracy of IOL power calculation; however, this has yet to be proved from empirical data. The formula provides an advantage in analysis of postoperative pseudophakia for experimental/scientific purposes because all intraocular distances and powers may be measured or calculated. J Cataract Refract Surg 1997; 23:1200-1207
I
n the past two decades, several empirical 1,2 and newer theoretical3-8 intraocular lens (lOL) power calculation formulas have been developed. However, both types of formulas erroneously assume that all 10Ls have an identical and constant thickness. In reality, 10L surface curvature and thickness vary with power. This power-dependent change in 10L architecture has a profound impact on postoperative anterior chamber depth (ACD) and the lens' effective power.
Previous studies have reported on calculation of lens thickness (LT),9 prediction of postoperative ACD (PRPO-ACD),10,11 and 10L power calculation using exact ray tracing and vergence calculation. 12 This paper presents a new theory of 10L power calculation that combines the methods in these previous studies while taking into account the power variation in 10L design.
Materials and Methods General Aspects and Descriptive Terms
From the Department o/Ophthalmology, Adlborg Sygehus Syd, Adlborg, Denmark. Reprint requests to Kristian Naeser, MD, Department 0/ Ophthalmology, Aalborg Sygehus Syd, 9100 Aalborg, Denmark. 1200
The distances, refractive indices, and powers of the schematic pseudophakic eyes are shown in Figure 1 and Tables 1 and 2. In Table 1, constants marked with an asterisk (*) are default values that may be changed, and
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
IOL POWER CALCULATION FORMULA
FC-
Powers:
nO
Ref. ind:
n2 <=
Distances:
c=
Figure 1. (Naeser) Refractive powers, refractive indices, and distances of the schematic pseudophakic eye .
V
4=
-
-
-
All distances involving the cornea are measured from the anterior corneal surface. The abbreviations are defined in Tables 1 and 2.
PO-DI & PRE-DI
LPCD
O-CAPAL
-
Table 1. Descriptive terms for the schematic pseudophakic eye. Definition
Constant/Formula
V*
Vertex distance
12
Abbreviation Distances (mm)
K1 & K2
Radii of principal anterior corneal planes
PRE-AL
Preop measured AL
PRPO-ALt
Predicted postop AL
276
PRPO-CApt
Predicted position of posterior lens capsule
3.33
PRPO-ACO
Predicted postop ACO
PRPO-CAP - LPCO - LT
LT
Central IOL thickness
R1 & R2
Radius of curvature anterior and posterior of IOL surfaces
LPCO
Lens-posterior capsule distance
PRE-OI
Preop predicted distance, posterior IOL surface-retina
MEPO-ACO
Postop measured ACO
MEPO-CAP
Measured postop capsule position
MEPO-ACO
PO-OI
Postop calculated distance, posterior IOL surface-retina
(N2 x 1000)/PO-LL4 See below.
OPT-AL
Optimized postop AL resulting in zero
MEPO-ACO
+ 0.89 x PRE-AL + 0.07 x PRE-AL
PRPO-AL - (PRPO-CAP - LPCO)
+ LT + LPCO
+ LT + PO-OI
predicting error
Refractive Indices (m x D) NO'
Air
1.0
N1 *
Cornea (fictitious)
1.3375
N2*
Aqueous and vitreous
1.336
N3*
IOL
PMMA = 1.491
'Constant default value that may be changed. tConstant in regression equation that may be updated using new empirical data.
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
1201
IOL POWER CALCULATION FORMULA
Table 2. Descriptive terms for the schematic pseudophakic
Table 3. lens manufacturer cutting card showing the physical
eye; powers of the various refractive planes.
characteristics of a biconvex IOl. Information from the cutting card is used to produce individual IOls.
Abbreviation Power
Definition Location
Constant/Formula Diopter
Powers (D) PC1, PC2
Principal anterior corneal planes
[(N1-NO) x 1000j/K1 [(N1-NO) x 1000j/K2
PC
Average corneal power
(PC1 + PC2)/2
P1
Anterior IOl surface
[(N3-N2) x 1000j/R1
P2
Posterior IOl surface
[(N2-N3) x 1000j/R2
P
Total IOl power
P1 + P2
constants in regression equations marked with a dagger (t) may be updated using new empirical data.
Calculation ofJOL Thickness Although it is possible to calculate 10L thickness using elementary trigonometric formulas,9 it is easier and more accurate to obtain cutting cards from lens manufacturers. Table 3 shows an example of a cutting card for a biconvex lens. For each total lens power (P), we must know the powers of the anterior (PI) and posterior (P2) surfaces and the LT. Therefore, a small database containing these relevant characteristics of all available 10L types in all available powers can be developed. Part of a sample database is shown below.
Example 1 This example shows various biometric variables for three lens powers in an eye with a preoperative axial length (PRE-AL) of 23.05 mm and average corneal power (PC) of 42.9 diopters (D). The first four items are rOL characteristics taken from the cutting card in Table 3. The other three-postoperative position of the posterior lens capsule (PRPO-CAP), PRPO-ACD, and postoperative refraction (PRPO-REF)-are calculated values from Examples 2 and 3, which are shown later. PRPOP
PI
P2
LT
CAP
PRPOACD
22.0
12
10.0
0.72
4.94
4.22
-0.44
22.5
12
10.5
0.73
4.94
4.21
-0.80
23.0
10
13.0
0.75
4.94
4.19
-1.09
1202
PRPO-
REF
8.0 9.0 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5 30.0
Name
Radius Front
Name
Radius Back
Central Thickness
38.860 31.081 51.880 51.880 38.860 33.860 31.081 31.081 25.902 25.902 22.189 22.189 19.409 19.409 25.902 25.902 22.189 22.189 19.409 19.409 17.245 17.245 15.515 15.515 14.099 14.099 12.918 12.918 15.515 14.099
4.0 4.0 7.0 7.5 7.0 7.5 7.0 7.5 7.0 7.5 7.0 7.5 7.0 7.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 13.0 12.5
-38.860 -38.860 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -11.918 -12.399
0.36 0.39 0.41 0.43 0.44 0.45 0.46 0.48 0.49 0.50 0.51 0.53 0.54 0.55 0.57 0.58 0.59 0.61 0.62 0.63 0.64 0.66 0.67 0.68 0.69 0.71 0.72 0.73 0.75 0.76
4
5 3 3 4
4 5 5 6 6 7
7 8 8 6 6 7 7
8 8 9 9 10 10 11 11 12 12 10 11 11
14.099
13.0
-11.918
0.77
12 12 13 13 14 14 13 15 14 14 15 15
12.918 12.918 11.918 11.918 11.062 11.062 11.918 10.319 11.062 11.062 10.319 10.319
12.5 13.0 12.5 13.0 12.5 13.0 14.5 13.0 14.5 15.0 14.5 15.0
-12.399 -11.918 -12.399 -11.918 -12.399 -11.918 -10.677 -11.918 -10.677 -10.319 -10.677 -10.319
0.78 0.80 0.81 0.82 0.83 0.85 0.86 0.87 0.89 0.90 0.91 0.92
Prediction ofPostoperative Chamber Depth First, the PRPO-CAP is predicted using a previously published method. 11 From this value, the LT for each specific lens power is subtracted to obtain a measure of the PRPO-ACD. In the original study, 11 a multiple linear regression formula was constructed in which PRPO-CAP is a
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
lOL POWER CALCULATION FORMULA
dependent variable and patient age, preoperative ACD, and PRE-AL are independent variables. Based on empirical observations, the equation was simplified as follows: PRPO-CAP == 3.33
+ 0.07
X PRE-Ai
(1)
The PRPO-ACD is calculated as PRPO-ACD == PRPO-CAP - LPCD - LT == 3.33 + 0.07 X PRE-Ai - LPCD - LT
(2)
where LPCD is the lens-posterior capsule distance,13 which is the centrally measured distance between the posterior lens surface and the posterior capsule. The LPCD varies with lens design and may be measured; however, the following values may be assumed: 13,14 posterior convex lenses, 0 mm; posterior plano lens, 0.14 mm; posterior concave lenses, 0.20 mm; lenses with laser space, same depth as that of the laser space. Thus, equation 2 generates a prediction of PRPOACD that is dependent on intraocular biometry but varies with lens power and thickness.
Example 2
In this example, the PRE-AL = 23.05. To calculate the PRPO-ACD for a 22.0 0, biconvex lens, we assume the LPCD is 0 mm. The LT for a 22.0 0 IOL is 0.72 mm (Table 3). Using formula 2, we get PRPO-ACD == 3.33 = 3.33 =
+ 0.07 + 1.61
X 23.05 - LPCD - LT
- 0 - 0.72
4.22 mm
Predicted values ofPRPO-ACD for this and other IOL powers are shown in Example 1.
vergence Calculation Vergence calculation formulas are based on exact paraxial ray tracing. 12 Table 4 defines and shows the vergence formulas used here; the locations of the calculated vergences in the schematic pseudophakic eye are shown in Figure 2. Prediction ofpostoperative refraction. This vergence calculation from right to left ("against the light") uses preoperative measurements. 12 The aim of these calculations is to predict the PRPO-REF for each lens power (Table 4). This allows the surgeon to choose a lens power with an appropriate assumed PRPO-REF.
Example 3 First, the PRPO-ACD for the eye and lens powers are calculated as in Example 1. The IOL has an assumed refractive index of 1.492. The following biometric variables are valid for all lens powers. PRPO-Ai == predicted axial length == 2.76 + 0.89 X PRE-Ai = 2.76 + 0.89 X 23.05 = 23.27 mm PRE-DI == predicted distance from poster IOL surface to retina == PRPO-Ai - PRPO-CAP == 23.27 - 4.94 = 18.33 mm
These values are inserted into the formulas shown in Table 4 (preoperative) and calculated for the three lens powers shown below. If a postoperative refraction of approximately -0.5 0 is desired, a +22.0 0 lens should be chosen.
PRPO-REF ... LCI LC2
LL2
-lL3
Figure 2. (Naeser) Predicted and measured spherically equivalent postoperative refractions and vergences of the schematic pseudophakic eye. The abbreviations for the vergences are defined in Table 4.
MEPO-REF -
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
1203
IOL POWER CALCULATION FORMULA
Table 4. Vergence calculation in the schematic pseudophakic eye; preoperative and postoperative situations. VergenceJRefractlon at the Following Position
Abbreviation
Constant/Formula
Preoperative; agalnst-the-light (0) PRE-LL4
Posterior IOL surface after refraction by P2
(N2 x 1000)/PRE-DI
PRE-LL3
Posterior IOL surface before refraction by P2
PRE-LL4 - P2
PRE-LL2
Anterior IOL surface after refraction by P1
PRE-LL3/(1 + (LT x PRE-LL3)/(N3 x 1000»
PRE-LL 1
Anterior IOL surface before refraction by P1
PRE-LL2 - P1
PRE-LC2
Posterior corneal surface
PRE-LL 1/(1 + (PRPO-ACD x PRE-LL 1)/(N2 x 1000»
PRE-LC1
Anterior corneal surface before refraction by PC
PRE-LC2 - PC
PRPO-REF
Predicted postop SE refraction
PRE-LC1/(1 + (V x PRE-LC1)/(NO x 1000»
Postoperative; with-the-light (0) MEPO-REF
Measured postop SE refraction
PO-LC1
Anterior corneal surface before refraction by PC
MEPO-REF/[1 (V x MEPO-REF)/(NO x 1000)]
+ PC
PO-LC2
Posterior corneal surface after refraction by PC
PO-LC1
PO-LL1
Anterior IOL surface before refraction by P1
PO-LC2/[1 - (MEPO-ACD x PO-LC2)/(N2 x 1000)]
PO-LL2
Anterior IOL surface after refraction by P1
PO-LL 1
PO-LL3
Posterior IOL surface before refraction by P2
PO-LL2/(1 (LT x PO-LL2)/(N3 x 1000))
PO-LL4
Postop IOL surface after refraction by P2
PO-LL3
PO-DI
Postop calculated distance posterior IOL surface-retina
(N2 x 1000)/PO-LL4 (see Table 1 under "Distances")
Lens Power (D) Vergence (D) +22.0
+22.5
+23.0
PRE-LL4
72.89
72.89
72.89
PRE-LL3
62.89
62.39
59.89
PRE-LL2
61.04
60.54
58.14
PRE-LLl
49.04
48.54
48.14
PRE-LC2
42.46
42.10
41.82
PRE-LCI
-0.44
-0.80
-1.08
PRP02REF
-0.44
-0.80
-1.09
Calculation of vergence from left to right. This calculation is "with the light" in the postoperative situation. 12 All calculations are performed using postoperative refractive, ACD, and capsule position data but with the preoperative average corneal power. The aim 1204
+ P1
+ P2
of these calculations is to find the optimized postoperative axial length (OPT-AL), equivalent to the measured postoperative refraction (MEPO-REF). This optimizes the entire 10L power calculation formula for future cases.
Example 4 The following calculates the OPT-AL in the eye discussed in Examples 1 through 3 but with a 22.0 D IOL in place. Postoperative measurements are as follows: MEPO-REF, -0.6 D; MEPO-ACD, 4.0 mm. From Table 1, it follows that MEPO-CAP = MEPO-ACD + LT + LPCD = 4.0 + 0.72 + 0 = 4.72 mm
Inserting the values from Table 4 (postoperative) would yield the following:
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
IOL POWER CALCULATION FORMULA
Vergence
Lens Power 22.0 D
MEPO-REF
-0.6
PO-LCl
-0.6
PO-LC2
42.3
PO-LLl
48.4
PO-LL2
60.4
PO-LL3
62.2
PO-LL4
72.2
PO-DI = (N2 X 1000)/PO-LL4 = 1336/72.2 = 18.50 mm OPT-AL = MEPO-ACD + LT + PO-DI = 4.0 + 0.72 + 18.5 = 23.22 mm
Thus, the axial length that is consistent with the measured refraction, biometric data, and IOL characteristics is not 23.05 as measured preoperatively but 23.22 mm as calculated postoperatively.
Optimizing the Formula The formula may be optimized from empirical data in the prediction of postoperative capsule position 11 and correction of axial length. 12 These enhanced formulas, both based on linear regression analysis, may be used for future cases. Prediction of posterior lens capsule position. Using empirical data, we may find an improved correlation between PRE-AL and MEPO-CAP. Linear regression analysis with PRE-AL as an independent and MEPOCAP as a dependent variable yields a new expression for PRPO-CAP; PRPO-CAP may vary with lens design and should ideally be calculated for each IOL: MEPO-CAP = a
+b
X (PRE-AL)
The constants in the regression equation are given by the following: 15 a = Mean(MEPO-CAP) - b X Mean(PRE-AL) b = L [(PRE-AL - Mean(PRE-AL)) X (MEPO-CAP) - Mean (MEPO-CAP))]/ L (PRE-AL - Mean(PRE-AL))2
Correction ofaxial length. An expression for PRPOAL is obtained by performing linear regression analysis with PRE-AL as the independent and OPT-AL as the dependent variable: OPT-AL
= c+
d X (PRE-AL)
This analysis is independent of lens type and thus may be performed on the aggregate data. The values of c and d are found using the above calculation.
Discussion The effective power of an IOL depends on the exact intraocular locations of its refractive surfaces. The theoretical advantages of the refractive system described here is improved prediction of the positions of the refractive IOL surfaces in individual cases and more efficient correction of offset errors. All IOL power calculation formulas have an assumed postoperative ACD. In the empirical SRK formulas,1,2 both assumed postoperative ACD and lens design are expressed by the A-constant. However, the expression of these items is neither effective nor specific because the A-constant also includes all other offset and measurement errors. The A-constant is an average value that erroneously assumes identical ACDs in each eye. In reality, the A-constant should vary with lens power. The newer theoretical formulas4--8,16 assume an identical postoperative ACD for all powers of a given IOL type. These formulas estimate postoperative ACD by regression analysis of various preoperatively measured variables. Use of this approach yielded a great variation in significant predictors for various IOL types,IO,17 which led my colleagues and I to abandon the method. The variation probably results from the inconsistency of the method; the regression equation uses several biological factors as independent variables, while postoperative ACD depends on both biological variables and a nonbiological variable, namely IOL thickness. In the present study, biological and nonbiological factors were segregated to obtain a primary prediction of the position of the posterior lens capsule from biometric data; the lens thickness was then subtracted, yielding the postoperative ACD for each lens power. 11 The present system is optimized by regression analysis between the preoperatively estimated and the postoperatively measured values for ACD and axial length. The correction factors for postoperative ACD may vary for each lens type, while the correction factors for axial length are identical for all lens types. The use of regression constants rather than fixed values for optimizing axial length eliminates any correlation between prediction errors and short or long eyes.
J CATARACf REFRACT SURG-VOL 23, OCfOBER 1997
1205
IOL POWER CALCULATION FORMULA
Thus, it is possible to incorporate the exact IOL design into a power calculation formula. The question is whether this improves the prediction of postoperative refraction. All IOL power calculation formulas yield approximately identical results for "average" eyes (i.e., eyes with a biometry close to the mean thereby requiring an IOL of average power). As these eyes are by far the most numerous, detecting any significant difference between the formulas would require a large population. The value of sophisticated formulas is therefore greatest in eyes with unusual biometry or IOL power. This can be illustrated by calculating the postoperative refraction in the eye shown in Example 1 using the system described here and the SRKlT and Holladay formulas. The Naeser and Holladay formulas produce approximately similar results, while the SRKlT formula predicts higher postoperative refractions for low-power IOLs (Figure 3). The correlation between axial length and IOL power for emmetropia shows a similar increasing disparity for very long and very short eyes (Figure 4). These are only examples using highly optimized data. The correct assessment method is to optimize the formula on one data set and then test it on another, which will be attempted in a later study. The total average prediction error for any of these formulas first depends on the proportion of eyes with unusual biometry.
The system described here allows measurement of all powers and positions of refractive elements in the postoperative eye. Analysis of these data may enable us to optimize values for refractive indices, ultrasound velocities, and "fictitious" corneal refractive indices, thereby further optimizing the entire formula. There is still room for improvement, however. All the mentioned formulas use rather primitive first-order optics in the calculations. In the future, it may be possible to use more advanced, computer-assisted models based on multiple ray tracing and Snell's law. Theoretically, the formula presented here should increase the accuracy of IOL power calculation; however, this has yet to be proved. The possibility exists that the current relatively inaccurate methods for ultrasonography will prevent detection of differences among the advanced formulas. However, these biometric methods undoubtedly will improve, increasing the demand for the most optically correct methods. The system described here represents an advantage in analysis of postoperative pseudophakia for experimental/scientific purposes because all intraocular distances and powers can be measured or calculated. This formulation is being incorporated into a database for all subsequent IOL calculations used in my department. This will allow analysis in large numbers of patients. Surgeons may use the definitions and descrip-
_10
e.z o
8 + ; 6 J
~~ : t ° tL
~ -2 ~ -4
~Q.
·t
••
11 · IJ ·1 · ··. · 11-.
:.
. . .. ..
i
-6 1 -8 J.;
.
- ii---ii -- + - - - + --+--
8 10 12 14 16 18 20 22 24 26 28 30 IOL POWER (0)
,_NAESER 1206
DHOLLAOAV] J CATARACT REFRACT SURG-VOL 23. OCTOBER 1997
Figure 3. (Naeser) The correlation between IOL power and predicted postoperative refraction using the Naeser, SRK/T, and Holladay formulas for an eye with the biometry and IOL characteristics given in Example 1. All three formulas were optimized to yield a postoperative refraction of 1.05 0 for a 20.00 0 IOL. The Naeser and the Holladay formulas performed almost identically for all lens powers, while the SRK[T formula predicted higher postoperative refractions for low-power IOLs.
IOL POWER CALCULATION FORMULA
-
8 .29
I
a: w 27
~ 25 023 a.. ..J 21
Q 19 Q 17
z
~
15 13
~
9
~ 11
~ 7
20
21
I_NAESER
22
23
24
25
AXIAL LENGTH (mm)
_SRK-T
26
26,5
I
Figure 4. (Naeser) The correlation between axial length and IOL power for emmetropia using the Naeser, SRK[T, and Holladay formulas for an eye with a constant average corneal radius of 7.5 mm with the IOL shown in Table 3. The three formulas were optimized to yield emmetropia for an axial length of 22.0 mm and an IOL power of 22.1 D. There was an increasing disparity between the formulas for very short and very long eyes. For a 20.0 mm axial length, the IOL power for emmetropia was 29.9, 29.0, and 29.4 0 using the Naeser, SRK{T, and Holladay formulas , respectively.
DHOLLADAVI
tive terms given here to enter the entire formula into their own spreadsheets.
References 1. Sanders DR, Kraff Me. Improvement of intraocular lens power calculation using empirical data. Am Intra-Ocular Implant Soc] 1980; 6:263-267; erratum 1981; 7:82 2. Retzlaff J. Posterior chamber implant power calculation: regression formulas. Am Intra-Ocular Implant Soc ] 1980; 6:268-270 3. Binkhorst RD. Intraocular Lens Power Calculation Manual. A Guide to the Author's TI 58/59 IOL Power Module, 2nd ed. New York, NY, RD Binkhorst, 1981 4. Olsen T. Theoretical approach to intraocular lens calculation using Gaussian optics. ] Cataract Refract Surg 1987; 13:141-145 5. Holladay JT, Musgrove KH, Praeger TC, et al. A threepart system for refining intraocular lens power calculations. J Cataract Refract Surg 1988; 14: 17-24 6. Retzlaff]A, Sanders DR, KraffMe. Development of the SRKlT intraocular lens implant power calculation formula. ] Cataract Refract Surg 1990; 16:333-340; correction p 528 7. Retzlaff]A, Sanders DR, KraffMC, et al. Comparison of the SRKlT formula and other theoretical and regression formulas. ] Cataract Refract Surg 1990; 16:341-346 8. Barrett GO. An improved universal theoretical formula for intraocular lens power prediction. ] Cataract Refract Surg 1993; 19:713-720
9. Naeser K, Naeser EY. Calculation of the thickness of an intraocular lens.] Cataract Refract Surg 1993; 19:40-42 10. Naeser K, Boberg-Ans ], Bargum R. Prediction of pseudo-phakic anterior chamber depth from pre-operative data. Acta Ophthalmol 1988; 66:433-437 11. Naeser K, Boberg-Ans ] , Bargum R. Biometry of the posterior lens capsule: a new method to predict pseudophakic anterior chamber depth. ] Cataract Refract Surg 1990; 16:202-206 12. Naeser K. The vergence-based, empirically modified intraocular lens equation. Eur] Implant Refract Surg 1991; 3:201-206 13. Naeser K, Rask KL, Hansen TE. Morphological changes after extracapsular cataract extraction with implantation of posterior chamber lenses; a prospective clinical study. Acta Ophthalmol 1986; 64:323-329 14. Boberg-Ans L], Naeser K, Bargum R. Vaulted posterior chamber lenses and the posterior capsule. J Cataract Refract Surg 1990; 16:325-328 15. Armitage P, Berry G. Statistical Methods in Medical Research, 2nd ed. Oxford, England, Blackwell Scientific Publications, 1987; 141- 159 16. Olsen T, Corydon L, Gimbel H. Intraocular lens power calculation with an improved anterior chamber depth prediction algorithm. ] Cataract Refract Surg 1995; 21:313-319 17. Haigis W, Waller W, Duzanec Z, Voeske W Postoperative biometry and keratometry after posterior chamber lens implantation. Eur ] Implant Refract Surg 1990; 2:191-202
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
1207
ABSTRACT Purpose: To present a new method of intraocular lens (IOL) power calculation that takes into account the power-dependent variation in lens design. Setting: Department of Ophthalmology, Aalborg, Denmark. Methods: Information on exact IOL design is derived from the manufacturers' cutting cards. These data were built into an IOL calculation formula based on exact ray tracing and vergence calculation. All algorithms are demonstrated. Results: The method is optically correct as all refractive surfaces are characterized with respect to both position and refractive power. In simulation models, the Naeser and the Holladay formulas performed similarly, while the SRK(T formula predicted higher postoperative refractions for low-power IOLs. Conclusion: It is possible to incorporate the exact IOL design into an IOL power calculation formula. Theoretically, the Naeser formula should increase the accuracy of IOL power calculation; however, this has yet to be proved from empirical data. The formula provides an advantage in analysis of postoperative pseudophakia for experimental/scientific purposes because all intraocular distances and powers may be measured or calculated. J Cataract Refract Surg 1997; 23:1200-1207
I
n the past two decades, several empirical 1,2 and newer theoretical3-8 intraocular lens (lOL) power calculation formulas have been developed. However, both types of formulas erroneously assume that all 10Ls have an identical and constant thickness. In reality, 10L surface curvature and thickness vary with power. This power-dependent change in 10L architecture has a profound impact on postoperative anterior chamber depth (ACD) and the lens' effective power.
Previous studies have reported on calculation of lens thickness (LT),9 prediction of postoperative ACD (PRPO-ACD),10,11 and 10L power calculation using exact ray tracing and vergence calculation. 12 This paper presents a new theory of 10L power calculation that combines the methods in these previous studies while taking into account the power variation in 10L design.
Materials and Methods General Aspects and Descriptive Terms
From the Department o/Ophthalmology, Adlborg Sygehus Syd, Adlborg, Denmark. Reprint requests to Kristian Naeser, MD, Department 0/ Ophthalmology, Aalborg Sygehus Syd, 9100 Aalborg, Denmark. 1200
The distances, refractive indices, and powers of the schematic pseudophakic eyes are shown in Figure 1 and Tables 1 and 2. In Table 1, constants marked with an asterisk (*) are default values that may be changed, and
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
IOL POWER CALCULATION FORMULA
FC-
Powers:
nO
Ref. ind:
n2 <=
Distances:
c=
Figure 1. (Naeser) Refractive powers, refractive indices, and distances of the schematic pseudophakic eye .
V
4=
-
-
-
All distances involving the cornea are measured from the anterior corneal surface. The abbreviations are defined in Tables 1 and 2.
PO-DI & PRE-DI
LPCD
O-CAPAL
-
Table 1. Descriptive terms for the schematic pseudophakic eye. Definition
Constant/Formula
V*
Vertex distance
12
Abbreviation Distances (mm)
K1 & K2
Radii of principal anterior corneal planes
PRE-AL
Preop measured AL
PRPO-ALt
Predicted postop AL
276
PRPO-CApt
Predicted position of posterior lens capsule
3.33
PRPO-ACO
Predicted postop ACO
PRPO-CAP - LPCO - LT
LT
Central IOL thickness
R1 & R2
Radius of curvature anterior and posterior of IOL surfaces
LPCO
Lens-posterior capsule distance
PRE-OI
Preop predicted distance, posterior IOL surface-retina
MEPO-ACO
Postop measured ACO
MEPO-CAP
Measured postop capsule position
MEPO-ACO
PO-OI
Postop calculated distance, posterior IOL surface-retina
(N2 x 1000)/PO-LL4 See below.
OPT-AL
Optimized postop AL resulting in zero
MEPO-ACO
+ 0.89 x PRE-AL + 0.07 x PRE-AL
PRPO-AL - (PRPO-CAP - LPCO)
+ LT + LPCO
+ LT + PO-OI
predicting error
Refractive Indices (m x D) NO'
Air
1.0
N1 *
Cornea (fictitious)
1.3375
N2*
Aqueous and vitreous
1.336
N3*
IOL
PMMA = 1.491
'Constant default value that may be changed. tConstant in regression equation that may be updated using new empirical data.
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
1201
IOL POWER CALCULATION FORMULA
Table 2. Descriptive terms for the schematic pseudophakic
Table 3. lens manufacturer cutting card showing the physical
eye; powers of the various refractive planes.
characteristics of a biconvex IOl. Information from the cutting card is used to produce individual IOls.
Abbreviation Power
Definition Location
Constant/Formula Diopter
Powers (D) PC1, PC2
Principal anterior corneal planes
[(N1-NO) x 1000j/K1 [(N1-NO) x 1000j/K2
PC
Average corneal power
(PC1 + PC2)/2
P1
Anterior IOl surface
[(N3-N2) x 1000j/R1
P2
Posterior IOl surface
[(N2-N3) x 1000j/R2
P
Total IOl power
P1 + P2
constants in regression equations marked with a dagger (t) may be updated using new empirical data.
Calculation ofJOL Thickness Although it is possible to calculate 10L thickness using elementary trigonometric formulas,9 it is easier and more accurate to obtain cutting cards from lens manufacturers. Table 3 shows an example of a cutting card for a biconvex lens. For each total lens power (P), we must know the powers of the anterior (PI) and posterior (P2) surfaces and the LT. Therefore, a small database containing these relevant characteristics of all available 10L types in all available powers can be developed. Part of a sample database is shown below.
Example 1 This example shows various biometric variables for three lens powers in an eye with a preoperative axial length (PRE-AL) of 23.05 mm and average corneal power (PC) of 42.9 diopters (D). The first four items are rOL characteristics taken from the cutting card in Table 3. The other three-postoperative position of the posterior lens capsule (PRPO-CAP), PRPO-ACD, and postoperative refraction (PRPO-REF)-are calculated values from Examples 2 and 3, which are shown later. PRPOP
PI
P2
LT
CAP
PRPOACD
22.0
12
10.0
0.72
4.94
4.22
-0.44
22.5
12
10.5
0.73
4.94
4.21
-0.80
23.0
10
13.0
0.75
4.94
4.19
-1.09
1202
PRPO-
REF
8.0 9.0 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5 30.0
Name
Radius Front
Name
Radius Back
Central Thickness
38.860 31.081 51.880 51.880 38.860 33.860 31.081 31.081 25.902 25.902 22.189 22.189 19.409 19.409 25.902 25.902 22.189 22.189 19.409 19.409 17.245 17.245 15.515 15.515 14.099 14.099 12.918 12.918 15.515 14.099
4.0 4.0 7.0 7.5 7.0 7.5 7.0 7.5 7.0 7.5 7.0 7.5 7.0 7.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 10.0 10.5 13.0 12.5
-38.860 -38.860 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -22.189 -20.707 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -15.515 -14.776 -11.918 -12.399
0.36 0.39 0.41 0.43 0.44 0.45 0.46 0.48 0.49 0.50 0.51 0.53 0.54 0.55 0.57 0.58 0.59 0.61 0.62 0.63 0.64 0.66 0.67 0.68 0.69 0.71 0.72 0.73 0.75 0.76
4
5 3 3 4
4 5 5 6 6 7
7 8 8 6 6 7 7
8 8 9 9 10 10 11 11 12 12 10 11 11
14.099
13.0
-11.918
0.77
12 12 13 13 14 14 13 15 14 14 15 15
12.918 12.918 11.918 11.918 11.062 11.062 11.918 10.319 11.062 11.062 10.319 10.319
12.5 13.0 12.5 13.0 12.5 13.0 14.5 13.0 14.5 15.0 14.5 15.0
-12.399 -11.918 -12.399 -11.918 -12.399 -11.918 -10.677 -11.918 -10.677 -10.319 -10.677 -10.319
0.78 0.80 0.81 0.82 0.83 0.85 0.86 0.87 0.89 0.90 0.91 0.92
Prediction ofPostoperative Chamber Depth First, the PRPO-CAP is predicted using a previously published method. 11 From this value, the LT for each specific lens power is subtracted to obtain a measure of the PRPO-ACD. In the original study, 11 a multiple linear regression formula was constructed in which PRPO-CAP is a
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
lOL POWER CALCULATION FORMULA
dependent variable and patient age, preoperative ACD, and PRE-AL are independent variables. Based on empirical observations, the equation was simplified as follows: PRPO-CAP == 3.33
+ 0.07
X PRE-Ai
(1)
The PRPO-ACD is calculated as PRPO-ACD == PRPO-CAP - LPCD - LT == 3.33 + 0.07 X PRE-Ai - LPCD - LT
(2)
where LPCD is the lens-posterior capsule distance,13 which is the centrally measured distance between the posterior lens surface and the posterior capsule. The LPCD varies with lens design and may be measured; however, the following values may be assumed: 13,14 posterior convex lenses, 0 mm; posterior plano lens, 0.14 mm; posterior concave lenses, 0.20 mm; lenses with laser space, same depth as that of the laser space. Thus, equation 2 generates a prediction of PRPOACD that is dependent on intraocular biometry but varies with lens power and thickness.
Example 2
In this example, the PRE-AL = 23.05. To calculate the PRPO-ACD for a 22.0 0, biconvex lens, we assume the LPCD is 0 mm. The LT for a 22.0 0 IOL is 0.72 mm (Table 3). Using formula 2, we get PRPO-ACD == 3.33 = 3.33 =
+ 0.07 + 1.61
X 23.05 - LPCD - LT
- 0 - 0.72
4.22 mm
Predicted values ofPRPO-ACD for this and other IOL powers are shown in Example 1.
vergence Calculation Vergence calculation formulas are based on exact paraxial ray tracing. 12 Table 4 defines and shows the vergence formulas used here; the locations of the calculated vergences in the schematic pseudophakic eye are shown in Figure 2. Prediction ofpostoperative refraction. This vergence calculation from right to left ("against the light") uses preoperative measurements. 12 The aim of these calculations is to predict the PRPO-REF for each lens power (Table 4). This allows the surgeon to choose a lens power with an appropriate assumed PRPO-REF.
Example 3 First, the PRPO-ACD for the eye and lens powers are calculated as in Example 1. The IOL has an assumed refractive index of 1.492. The following biometric variables are valid for all lens powers. PRPO-Ai == predicted axial length == 2.76 + 0.89 X PRE-Ai = 2.76 + 0.89 X 23.05 = 23.27 mm PRE-DI == predicted distance from poster IOL surface to retina == PRPO-Ai - PRPO-CAP == 23.27 - 4.94 = 18.33 mm
These values are inserted into the formulas shown in Table 4 (preoperative) and calculated for the three lens powers shown below. If a postoperative refraction of approximately -0.5 0 is desired, a +22.0 0 lens should be chosen.
PRPO-REF ... LCI LC2
LL2
-lL3
Figure 2. (Naeser) Predicted and measured spherically equivalent postoperative refractions and vergences of the schematic pseudophakic eye. The abbreviations for the vergences are defined in Table 4.
MEPO-REF -
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
1203
IOL POWER CALCULATION FORMULA
Table 4. Vergence calculation in the schematic pseudophakic eye; preoperative and postoperative situations. VergenceJRefractlon at the Following Position
Abbreviation
Constant/Formula
Preoperative; agalnst-the-light (0) PRE-LL4
Posterior IOL surface after refraction by P2
(N2 x 1000)/PRE-DI
PRE-LL3
Posterior IOL surface before refraction by P2
PRE-LL4 - P2
PRE-LL2
Anterior IOL surface after refraction by P1
PRE-LL3/(1 + (LT x PRE-LL3)/(N3 x 1000»
PRE-LL 1
Anterior IOL surface before refraction by P1
PRE-LL2 - P1
PRE-LC2
Posterior corneal surface
PRE-LL 1/(1 + (PRPO-ACD x PRE-LL 1)/(N2 x 1000»
PRE-LC1
Anterior corneal surface before refraction by PC
PRE-LC2 - PC
PRPO-REF
Predicted postop SE refraction
PRE-LC1/(1 + (V x PRE-LC1)/(NO x 1000»
Postoperative; with-the-light (0) MEPO-REF
Measured postop SE refraction
PO-LC1
Anterior corneal surface before refraction by PC
MEPO-REF/[1 (V x MEPO-REF)/(NO x 1000)]
+ PC
PO-LC2
Posterior corneal surface after refraction by PC
PO-LC1
PO-LL1
Anterior IOL surface before refraction by P1
PO-LC2/[1 - (MEPO-ACD x PO-LC2)/(N2 x 1000)]
PO-LL2
Anterior IOL surface after refraction by P1
PO-LL 1
PO-LL3
Posterior IOL surface before refraction by P2
PO-LL2/(1 (LT x PO-LL2)/(N3 x 1000))
PO-LL4
Postop IOL surface after refraction by P2
PO-LL3
PO-DI
Postop calculated distance posterior IOL surface-retina
(N2 x 1000)/PO-LL4 (see Table 1 under "Distances")
Lens Power (D) Vergence (D) +22.0
+22.5
+23.0
PRE-LL4
72.89
72.89
72.89
PRE-LL3
62.89
62.39
59.89
PRE-LL2
61.04
60.54
58.14
PRE-LLl
49.04
48.54
48.14
PRE-LC2
42.46
42.10
41.82
PRE-LCI
-0.44
-0.80
-1.08
PRP02REF
-0.44
-0.80
-1.09
Calculation of vergence from left to right. This calculation is "with the light" in the postoperative situation. 12 All calculations are performed using postoperative refractive, ACD, and capsule position data but with the preoperative average corneal power. The aim 1204
+ P1
+ P2
of these calculations is to find the optimized postoperative axial length (OPT-AL), equivalent to the measured postoperative refraction (MEPO-REF). This optimizes the entire 10L power calculation formula for future cases.
Example 4 The following calculates the OPT-AL in the eye discussed in Examples 1 through 3 but with a 22.0 D IOL in place. Postoperative measurements are as follows: MEPO-REF, -0.6 D; MEPO-ACD, 4.0 mm. From Table 1, it follows that MEPO-CAP = MEPO-ACD + LT + LPCD = 4.0 + 0.72 + 0 = 4.72 mm
Inserting the values from Table 4 (postoperative) would yield the following:
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
IOL POWER CALCULATION FORMULA
Vergence
Lens Power 22.0 D
MEPO-REF
-0.6
PO-LCl
-0.6
PO-LC2
42.3
PO-LLl
48.4
PO-LL2
60.4
PO-LL3
62.2
PO-LL4
72.2
PO-DI = (N2 X 1000)/PO-LL4 = 1336/72.2 = 18.50 mm OPT-AL = MEPO-ACD + LT + PO-DI = 4.0 + 0.72 + 18.5 = 23.22 mm
Thus, the axial length that is consistent with the measured refraction, biometric data, and IOL characteristics is not 23.05 as measured preoperatively but 23.22 mm as calculated postoperatively.
Optimizing the Formula The formula may be optimized from empirical data in the prediction of postoperative capsule position 11 and correction of axial length. 12 These enhanced formulas, both based on linear regression analysis, may be used for future cases. Prediction of posterior lens capsule position. Using empirical data, we may find an improved correlation between PRE-AL and MEPO-CAP. Linear regression analysis with PRE-AL as an independent and MEPOCAP as a dependent variable yields a new expression for PRPO-CAP; PRPO-CAP may vary with lens design and should ideally be calculated for each IOL: MEPO-CAP = a
+b
X (PRE-AL)
The constants in the regression equation are given by the following: 15 a = Mean(MEPO-CAP) - b X Mean(PRE-AL) b = L [(PRE-AL - Mean(PRE-AL)) X (MEPO-CAP) - Mean (MEPO-CAP))]/ L (PRE-AL - Mean(PRE-AL))2
Correction ofaxial length. An expression for PRPOAL is obtained by performing linear regression analysis with PRE-AL as the independent and OPT-AL as the dependent variable: OPT-AL
= c+
d X (PRE-AL)
This analysis is independent of lens type and thus may be performed on the aggregate data. The values of c and d are found using the above calculation.
Discussion The effective power of an IOL depends on the exact intraocular locations of its refractive surfaces. The theoretical advantages of the refractive system described here is improved prediction of the positions of the refractive IOL surfaces in individual cases and more efficient correction of offset errors. All IOL power calculation formulas have an assumed postoperative ACD. In the empirical SRK formulas,1,2 both assumed postoperative ACD and lens design are expressed by the A-constant. However, the expression of these items is neither effective nor specific because the A-constant also includes all other offset and measurement errors. The A-constant is an average value that erroneously assumes identical ACDs in each eye. In reality, the A-constant should vary with lens power. The newer theoretical formulas4--8,16 assume an identical postoperative ACD for all powers of a given IOL type. These formulas estimate postoperative ACD by regression analysis of various preoperatively measured variables. Use of this approach yielded a great variation in significant predictors for various IOL types,IO,17 which led my colleagues and I to abandon the method. The variation probably results from the inconsistency of the method; the regression equation uses several biological factors as independent variables, while postoperative ACD depends on both biological variables and a nonbiological variable, namely IOL thickness. In the present study, biological and nonbiological factors were segregated to obtain a primary prediction of the position of the posterior lens capsule from biometric data; the lens thickness was then subtracted, yielding the postoperative ACD for each lens power. 11 The present system is optimized by regression analysis between the preoperatively estimated and the postoperatively measured values for ACD and axial length. The correction factors for postoperative ACD may vary for each lens type, while the correction factors for axial length are identical for all lens types. The use of regression constants rather than fixed values for optimizing axial length eliminates any correlation between prediction errors and short or long eyes.
J CATARACf REFRACT SURG-VOL 23, OCfOBER 1997
1205
IOL POWER CALCULATION FORMULA
Thus, it is possible to incorporate the exact IOL design into a power calculation formula. The question is whether this improves the prediction of postoperative refraction. All IOL power calculation formulas yield approximately identical results for "average" eyes (i.e., eyes with a biometry close to the mean thereby requiring an IOL of average power). As these eyes are by far the most numerous, detecting any significant difference between the formulas would require a large population. The value of sophisticated formulas is therefore greatest in eyes with unusual biometry or IOL power. This can be illustrated by calculating the postoperative refraction in the eye shown in Example 1 using the system described here and the SRKlT and Holladay formulas. The Naeser and Holladay formulas produce approximately similar results, while the SRKlT formula predicts higher postoperative refractions for low-power IOLs (Figure 3). The correlation between axial length and IOL power for emmetropia shows a similar increasing disparity for very long and very short eyes (Figure 4). These are only examples using highly optimized data. The correct assessment method is to optimize the formula on one data set and then test it on another, which will be attempted in a later study. The total average prediction error for any of these formulas first depends on the proportion of eyes with unusual biometry.
The system described here allows measurement of all powers and positions of refractive elements in the postoperative eye. Analysis of these data may enable us to optimize values for refractive indices, ultrasound velocities, and "fictitious" corneal refractive indices, thereby further optimizing the entire formula. There is still room for improvement, however. All the mentioned formulas use rather primitive first-order optics in the calculations. In the future, it may be possible to use more advanced, computer-assisted models based on multiple ray tracing and Snell's law. Theoretically, the formula presented here should increase the accuracy of IOL power calculation; however, this has yet to be proved. The possibility exists that the current relatively inaccurate methods for ultrasonography will prevent detection of differences among the advanced formulas. However, these biometric methods undoubtedly will improve, increasing the demand for the most optically correct methods. The system described here represents an advantage in analysis of postoperative pseudophakia for experimental/scientific purposes because all intraocular distances and powers can be measured or calculated. This formulation is being incorporated into a database for all subsequent IOL calculations used in my department. This will allow analysis in large numbers of patients. Surgeons may use the definitions and descrip-
_10
e.z o
8 + ; 6 J
~~ : t ° tL
~ -2 ~ -4
~Q.
·t
••
11 · IJ ·1 · ··. · 11-.
:.
. . .. ..
i
-6 1 -8 J.;
.
- ii---ii -- + - - - + --+--
8 10 12 14 16 18 20 22 24 26 28 30 IOL POWER (0)
,_NAESER 1206
DHOLLAOAV] J CATARACT REFRACT SURG-VOL 23. OCTOBER 1997
Figure 3. (Naeser) The correlation between IOL power and predicted postoperative refraction using the Naeser, SRK/T, and Holladay formulas for an eye with the biometry and IOL characteristics given in Example 1. All three formulas were optimized to yield a postoperative refraction of 1.05 0 for a 20.00 0 IOL. The Naeser and the Holladay formulas performed almost identically for all lens powers, while the SRK[T formula predicted higher postoperative refractions for low-power IOLs.
IOL POWER CALCULATION FORMULA
-
8 .29
I
a: w 27
~ 25 023 a.. ..J 21
Q 19 Q 17
z
~
15 13
~
9
~ 11
~ 7
20
21
I_NAESER
22
23
24
25
AXIAL LENGTH (mm)
_SRK-T
26
26,5
I
Figure 4. (Naeser) The correlation between axial length and IOL power for emmetropia using the Naeser, SRK[T, and Holladay formulas for an eye with a constant average corneal radius of 7.5 mm with the IOL shown in Table 3. The three formulas were optimized to yield emmetropia for an axial length of 22.0 mm and an IOL power of 22.1 D. There was an increasing disparity between the formulas for very short and very long eyes. For a 20.0 mm axial length, the IOL power for emmetropia was 29.9, 29.0, and 29.4 0 using the Naeser, SRK{T, and Holladay formulas , respectively.
DHOLLADAVI
tive terms given here to enter the entire formula into their own spreadsheets.
References 1. Sanders DR, Kraff Me. Improvement of intraocular lens power calculation using empirical data. Am Intra-Ocular Implant Soc] 1980; 6:263-267; erratum 1981; 7:82 2. Retzlaff J. Posterior chamber implant power calculation: regression formulas. Am Intra-Ocular Implant Soc ] 1980; 6:268-270 3. Binkhorst RD. Intraocular Lens Power Calculation Manual. A Guide to the Author's TI 58/59 IOL Power Module, 2nd ed. New York, NY, RD Binkhorst, 1981 4. Olsen T. Theoretical approach to intraocular lens calculation using Gaussian optics. ] Cataract Refract Surg 1987; 13:141-145 5. Holladay JT, Musgrove KH, Praeger TC, et al. A threepart system for refining intraocular lens power calculations. J Cataract Refract Surg 1988; 14: 17-24 6. Retzlaff]A, Sanders DR, KraffMe. Development of the SRKlT intraocular lens implant power calculation formula. ] Cataract Refract Surg 1990; 16:333-340; correction p 528 7. Retzlaff]A, Sanders DR, KraffMC, et al. Comparison of the SRKlT formula and other theoretical and regression formulas. ] Cataract Refract Surg 1990; 16:341-346 8. Barrett GO. An improved universal theoretical formula for intraocular lens power prediction. ] Cataract Refract Surg 1993; 19:713-720
9. Naeser K, Naeser EY. Calculation of the thickness of an intraocular lens.] Cataract Refract Surg 1993; 19:40-42 10. Naeser K, Boberg-Ans ], Bargum R. Prediction of pseudo-phakic anterior chamber depth from pre-operative data. Acta Ophthalmol 1988; 66:433-437 11. Naeser K, Boberg-Ans ] , Bargum R. Biometry of the posterior lens capsule: a new method to predict pseudophakic anterior chamber depth. ] Cataract Refract Surg 1990; 16:202-206 12. Naeser K. The vergence-based, empirically modified intraocular lens equation. Eur] Implant Refract Surg 1991; 3:201-206 13. Naeser K, Rask KL, Hansen TE. Morphological changes after extracapsular cataract extraction with implantation of posterior chamber lenses; a prospective clinical study. Acta Ophthalmol 1986; 64:323-329 14. Boberg-Ans L], Naeser K, Bargum R. Vaulted posterior chamber lenses and the posterior capsule. J Cataract Refract Surg 1990; 16:325-328 15. Armitage P, Berry G. Statistical Methods in Medical Research, 2nd ed. Oxford, England, Blackwell Scientific Publications, 1987; 141- 159 16. Olsen T, Corydon L, Gimbel H. Intraocular lens power calculation with an improved anterior chamber depth prediction algorithm. ] Cataract Refract Surg 1995; 21:313-319 17. Haigis W, Waller W, Duzanec Z, Voeske W Postoperative biometry and keratometry after posterior chamber lens implantation. Eur ] Implant Refract Surg 1990; 2:191-202
J CATARACT REFRACT SURG-VOL 23, OCTOBER 1997
1207