**Purpose.**:
To compare the accuracy of different methods of calculating human lens power when lens thickness is not available.

**Methods.**:
Lens power was calculated by four methods. Three methods were used with previously published biometry and refraction data of 184 emmetropic and myopic eyes of 184 subjects (age range, 18–63 years; spherical equivalent range, −12.38 to +0.75 D). These three methods consist of the Bennett method, which uses lens thickness, a modification of the Stenström method and the Bennett-Rabbetts method, both of which do not require knowledge of lens thickness. These methods include *c* constants, which represent distances from lens surfaces to principal planes. Lens powers calculated with these methods were compared with those calculated using phakometry data available for a subgroup of 66 emmetropic eyes (66 subjects).

**Results.**:
Lens powers obtained from the Bennett method corresponded well with those obtained by phakometry for emmetropic eyes, although individual differences up to 3.5 D occurred. Lens powers obtained from the modified-Stenström and Bennett-Rabbetts methods deviated significantly from those obtained with either the Bennett method or phakometry. Customizing the *c* constants improved this agreement, but applying these constants to the entire group gave mean lens power differences of 0.71 ± 0.56 D compared with the Bennett method. By further optimizing the *c* constants, the agreement with the Bennett method was within ±1 D for 95% of the eyes.

**Conclusions.**:
With appropriate constants, the modified Stenström and Bennett-Rabbetts methods provide a good approximation of the Bennett lens power in emmetropic and myopic eyes.

^{ 1 –5 }they are currently too complicated to be used in large-scale studies or clinical practice.

^{ 6 }who used a thick-lens description that makes assumptions about the shape and refractive index distribution of the lens based on the Gullstrand-Emsley schematic eye.

^{ 7 }From this, he could calculate the equivalent lens power in a way that has been shown to be accurate in comparison with phakometry.

^{ 8 }However, his method requires knowledge of the lens thickness, which is sometimes not available.

^{ 9,10 }and by Bennett and Rabbetts.

^{ 11 }These approaches might be useful in a clinical practice using biometry devices that do not provide lens thickness (e.g., IOL Master; Carl Zeiss Meditec, Dublin, CA), or in analysis of historical biometry data.

^{ 8 }found between the Bennett method and phakometry; to (2) compare lens powers obtained with the Bennett method, our modification of the Stenström method, and the Bennett-Rabbetts method for previously published data of emmetropic and myopic eyes, and (3) to provide customized constants to optimize the performance of these three methods. The results allow improvement of our statistical eye model

^{ 12 }by including a more reliable method to estimate lens power when lens thickness is not available.

^{ 13 }for a group of 66 eyes of 66 emmetropic subjects (32 men, 34 women; 62 Caucasian, 4 non-Caucasian). The average spherical equivalent refraction of this group was +0.01 ± 0.38 D (range, –0.88 to +0.75), and the mean age was 42.4 ± 14.4 years (range 19–69).

^{ 14 }This dataset contained 118 eyes of 118 emmetropic and myopic subjects (43 men, 75 women; 74 Caucasian, 44 non-Caucasian) with a mean spherical equivalent refraction of −2.69 ± 2.79 D (range, −12.3 to +0.75 D) and an average subject age of 25.4 ± 5.1 years (range, 18–36 years). No phakometry data were available for this second dataset.

^{ 13 }similar to that described by Rosales and Marcos.

^{ 3 }Note that phakometry data were not available for the second dataset.

^{ 6 }calculates lens power

*P*

_{L}when lens thickness

*T*is available by keeping the distances from the surfaces to the principal planes of the lens in the same proportion as in the lens of the Gullstrand-Emsley eye model.

^{ 7 }Using the parameters defined in Table 1, the steps in his method can be combined as the single equation: with

*n*= 4/3 the aqueous and vitreous index,

*c*

_{1}

*T*= 1000

*n*(

*n*−

*n*

_{L})

*T*/(

*n*

_{L}

*P*

_{L}

*r*

_{La}) the distance between the anterior lens surface and first lenticular principal plane, and

*c*

_{2}

*T*= 1000

*n*(

*n*−

*n*

_{L})

*T*/(

*n*

_{L}

*P*

_{L}

*r*

_{La}) the distance between the posterior lens surface and second lenticular principal plane. The latter is negative because the principal plane is in front of the back surface. Bennett estimated the

*c*

_{1}and

*c*

_{2}constants using the Gullstrand-Emsley eye model, for which the lens refractive index

*n*

_{L}= 1.416.

Parameter | Unit | Calculation | Description |
---|---|---|---|

S | D | Spherical refraction at spectacle back vertex plane | |

S _{CV} | D | S/(1 − 0.014 S) | Spherical refraction at corneal vertex |

S _{PP} | D | S/(1 − 0.0155 S) | Spherical refraction at first principal plane of the eye |

K | D | Corneal power | |

ACD | mm | Anterior chamber depth (corneal epithelium to anterior lens) | |

T | mm | Lens thickness | |

L | mm | Axial length | |

V | mm | L-ACD-T | Vitreous depth |

P _{L} | D | Lens power | |

n | — | Refractive index of aqueous and vitreous humors | |

n _{ L } | — | Refractive index of crystalline lens | |

P _{L,Bennett} | D | Equation 1 | Lens power using Bennett method |

r _{La} | mm | Anterior radius of curvature of lens | |

r _{Lp} | mm | Posterior radius of curvature of lens | |

P _{La} | D | (n _{L} − n)/r _{La} | Power of anterior lens surface |

P _{Lp} | D | (n − n _{L})/r _{Lp} | Power of posterior lens surface |

c _{1} T | mm | 1000 n(n − n _{L})T/(n _{L} P _{L} r _{Lp}) | Distance between anterior lens surface and first principal plane of lens |

c _{2} T | mm | 1000 n(n − n _{L})T/(n _{L} P _{L} r _{La}) | Distance between posterior lens surface and second principal plane of lens |

P _{L,Sten} | D | Equation 2 | Lens power using modified-Stenström method |

P _{eye} | D | Equation 3 | Equivalent power of combination of eye and a thin correcting lens placed at the cornea |

c _{ Sten } | mm | Equation 2 + 3 solved for c _{ Sten } | Distance between anterior lens surface and first principal plane of lens |

P _{L,BR} | D | Equation 4 | Lens power using Bennett-Rabbetts method |

c _{BR} | mm | Equation 4 solved for c _{BR} | Distance between thin lens position and anterior lens surface |

*T*is not available, one can estimate the lens power

*P*

_{L}using Stenström's method,

^{9,10}which provides the lens power referenced to its anterior vertex rather than to the principal planes. We modified the method by including the parameter

*c*

_{Sten}, which is the estimated distance between the anterior lens surface and the first lenticular principal plane. The modified-Stenström method is given by: using the parameters in Table 1, with

*n*= 1.336. This equation contains the equivalent power of the eye

*P*

_{eye}. Based on Stenström's derivation, we calculated this as: Here the ocular refraction at the first principal plane of the eye

*S*

_{PP}is used. Lens power

*P*

_{L}can be found by substituting the value for

*P*

_{eye}derived from equation 3 into the right side of equation 2.

^{ 15 }who used the approximation

*P*

_{eye}= 1392/

*L*−

*S*

_{PP}. However, since this simplification deviates considerably from values obtained from equation 3 for

*c*

_{Sten}> 0 mm, we did not include it in our analysis.

*P*

_{L}without knowing

*T*is to modify an equation proposed by Bennett and Rabbetts

^{ 11 }for the purpose of calculating the spherical refraction of an eye when its biometry is known. They replaced the lens with an equivalent thin lens located at the midpoint between the lenticular principal planes, using the Bennett-Rabbetts eye model.

^{ 11 }If the ocular refraction at the corneal vertex

*S*

_{CV}is known, their equation can be rewritten to give

*P*

_{L}: with

*n*= 1.336 and

*c*

_{BR}the distance between the anterior lens surface and the thin lens position. This parameter can be found by solving equation 4 for

*c*

_{BR}when

*P*

_{L}is known.

^{ 16 }: with

*P*

_{La}and

*P*

_{Lp}as defined in Table 1.

*c*constants

*c*

_{1},

*c*

_{2},

*c*

_{Sten}, and

*c*

_{BR}for both Gullstrand-Emsley and Bennett-Rabbetts eye models. As both eye models will differ from actual ocular biometry, we determined the optimal

*c*constants also for each eye individually. For the Bennett method, these constants were easily determined by filling in the available phakometry of the emmetropic dataset into the formulas for

*c*

_{1}and

*c*

_{2}in Table 1, using

*n*= 1.336. The optimal

*c*constants of the modified-Stenström and Bennett-Rabbetts methods were found by using the phakometry lens powers of the emmetropic dataset for

*P*

_{L}and solving equations 2 and 3 and equation 4 for the

*c*constants, also using

*n*= 1.336. The analytical solution for

*c*

_{Sten}in the modified-Stenström method was mathematically complicated and could not be used in Excel (Microsoft, Redmond, WA); Mathematica (Wolfram Research, Champaign, IL) was used instead, to estimate values numerically. Means and standard deviations of these optimal

*c*constants were called the customized

*c*constants and are given in Table 2.

Method | Symbol | Eye Model | c Constants | Average | Within ±1 D from P _{L} (%) | Pearson Correlation Coefficients with Phakometry |
---|---|---|---|---|---|---|

Phakometry | P _{L} | 22.87 ± 2.42 D | ||||

Bennett | P _{L,Bennett} | Gullstrand-Emsley | c _{1} = 0.596; c _{2} = −0.358 | 22.50 ± 2.02 D | 45.5 | 0.778 (P < 0.001) |

Bennett-Rabbetts | c _{1} = 0.599; c _{2} = −0.353 | 22.74 ± 2.03 D | 50.0 | 0.779 (P < 0.001) | ||

Customized | c _{1} = 0.571 ± 0.028 | 22.54 ± 2.00 D | 45.5 | 0.778 (P < 0.001) | ||

c _{2} = −0.378 ± 0.029 | ||||||

Modified Stenström | P _{L,Sten} | Gullstrand-Emsley | cSten = 2.145 mm | 21.04 ± 1.94 D | 19.7 | 0.720 (P < 0.001) |

Bennett-Rabbetts | cSten = 2.221 mm | 21.36 ± 1.97 D | 27.3 | 0.720 (P < 0.001) | ||

Customized | c _{Sten} = 2.875 ± 0.763 mm | 22.78 ± 2.12 D | 42.4 | 0.721 (P < 0.001) | ||

Bennett-Rabbetts | P _{L,BR} | Gullstrand-Emsley | c _{BR} = 2.230 mm | 21.21 ± 1.96 D | 24.2 | 0.720 (P < 0.001) |

Bennett-Rabbetts | c _{BR} = 2.306 mm | 21.54 ± 1.99 D | 36.4 | 0.720 (P < 0.001) | ||

Customized | c _{BR} = 2.891 ± 0.778 mm | 22.81 ± 2.13 D | 40.9 | 0.721 (P < 0.001) |

*P*< 0.05 (analyses by Excel; Microsoft; SPSS ver. 12; Chicago, IL).

*P*

_{L}= 22.87 ± 2.42 D, which may be considered the target value that the calculation methods must approximate (Table 2). Using both the Gullstrand-Emsley and Bennett-Rabbetts eye models, the lens powers with the Bennett method were not significantly different from the phakometry powers. Using the customized

*c*constants did not improve the agreement. A Bland-Altman plot shows that the differences between Bennett and phakometry lens power remained between ±3 D (Fig. 1a) and for 45% and 50% of the eyes were less than ±1 D (Table 2). These differences did not correlate with subject age (Pearson < 0.01,

*P*> 0.05), which excludes accommodation as a possible source of these differences.

*t*-tests,

*P*< 0.01). By customizing the

*c*constants, the differences with phakometry reduced remarkably to nonsignificance (paired

*t*tests,

*P*> 0.05), and for ∼40% of the eyes the differences were less than ±1 D (Table 2).

*c*constants calculated for the combination of the two datasets (184 eyes) as a function of axial length

*L*. The lens power has a negative correlation with axial length for

*L*< 24 mm (

*r*= –0.624;

*P*< 0.001), with a slope that matches that of the measured lens power data. Above

*L*= 24 mm, approximately corresponding with the onset of myopia, the lens power plateaus to become constant (

*r*= –0.036;

*P*> 0.05). Because phakometry was not available for the second dataset, this plateauing could not be confirmed experimentally. However, a similar trend was found in the raw data published by Sorsby et al.

^{ 17 }Thus, in the absence of phakometry data for the entire dataset, the Bennett power with customized

*c*constants was used as a benchmark. This choice is based on the observation of Dunne et al.

^{ 8 }that the Bennett power corresponds well with phakometry in myopic refractions up to –9.37 D, including the long eyes for which the plateauing is shown in Figure 2a.

*c*constants (Table 2). These differences were statistically significant (paired

*t*-test,

*P*< 0.001). Using the customized

*c*constants, the modified-Stenström and Bennett-Rabbetts methods each yielded lens powers that were 0.71 ± 0.56 D greater than those with the Bennett method (Table 3), and this difference was also statistically significant (

*P*< 0.001).

Method | Symbol | Eye Model | c Constants | Average | Within ±1 D from P _{L,Bennett} (%) | Pearson Correlation Coefficient with Phakometry |
---|---|---|---|---|---|---|

Bennett | P _{L,Bennett} | Customized | c _{1} = 0.571 ± 0.028 | 22.31 ± 1.72 D | ||

c _{2} = −0.378 ± 0.029 | ||||||

Modified Stenström | P _{L,Sten} | Gullstrand-Emsley | c _{Sten} = 2.145 mm | 21.30 ± 1.61 D | 61.4 | 0.942 (P < 0.001) |

Bennett-Rabbetts | c _{Sten} = 2.221 mm | 21.62 ± 1.63 D | 71.7 | 0.943 (P < 0.001) | ||

Customized* | c _{Sten} = 2.875 ± 0.763 mm | 23.01 ± 1.76 D | 64.7 | 0.947 (P < 0.001) | ||

Customized 2 | c _{Sten} = 2.550 mm | 22.30 ± 1.69 D | 95.1 | 0.945 (P < 0.001) | ||

Bennett-Rabbetts | P _{L,BR} | Gullstrand-Emsley | c _{BR} = 2.230 mm | 21.45 ± 1.62 D | 67.9 | 0.946 (P < 0.001) |

Bennett-Rabbetts | c _{BR} = 2.306 mm | 21.77 ± 1.64 D | 78.8 | 0.947 (P < 0.001) | ||

Customized* | c _{BR} = 2.891 ± 0.778 mm | 23.02 ± 1.76 D | 66.8 | 0.950 (P < 0.001) | ||

Customized 2 | cBR = 2.564 mm | 22.31 ± 1.69 D | 95.1 | 0.948 (P < 0.001) |

*c*constant (customized 2) was determined for the modified-Stenström and Bennett-Rabbetts methods that minimized the mean lens power difference with the Bennett method over the entire population. When these customized 2 constants were used, the lens power differences with the Bennett method were no longer statistically significant (

*P*> 0.05) and were within ±1 D for about 95% of eyes (Table 3). For both methods, the power differences with the Bennett method were correlated significantly with axial length

*L*(

*r*= 0.390,

*P*< 0.001 and

*r*= 0.329,

*P*< 0.001 for the modified-Stenström and the Bennett-Rabbetts methods, respectively; Fig. 2b).

*c*constants all three lens power calculation methods are in reasonable agreement with the phakometry lens power for emmetropic eyes. This answers the first purpose of this article, which was to confirm the agreement that Dunne et al.

^{ 8 }found between the Bennett method and phakometry. However, for individual eyes, differences between calculated and phakometric power of up to 3.5 D occurred (Fig. 1, Table 2), which is considerably larger than the differences of up to 0.77 D that Dunne et al. reported for the Bennett method. These differences could result from biometric errors and Bennett's assumption that the lens shapes of the eye models are representative for all eyes (the ratio

*P*

_{Lp}/

*P*

_{La}of posterior to anterior lens powers was 1.52 ± 0.19 for phakometry, but 1.67 and 1.70 for the Gullstrand-Emsley and Bennett-Rabbetts eye models, respectively). Using the argument of Bennett

^{ 6 }and Dunne et al.

^{ 8 }that lens power provided by the Bennett method is likely to be more accurate than phacometry, because of the inherent difficulties in performing the latter accurately, we considered that the Bennett method derived power as a reasonable approximation of the real equivalent lens power and used it as a reference to compare the modified-Stenström and Bennett-Rabbetts methods.

*c*

_{1}and

*c*

_{2}constants of the two eye models or the customized constants derived in this work.

*c*constants made little difference in the results, but in emmetropic eyes the customized

*c*of 2.875 and 2.891 mm for modified-Stenström and Bennett Rabbetts methods, respectively, gave nonsignificant lens power differences with phakometry and produced more accurate results than the constants of the eye models. When comparing lens powers for combined emmetropic and myopic eyes, the customized

*c*constants for emmetropic eyes produced systematic lens power differences between the Bennett method and the modified-Stenström and Bennett-Rabbetts methods. This was improved by new customized 2 constants for the latter two methods (

*c*= 2.550 and 2.564 mm for modified-Stenström and Bennett Rabbetts methods, respectively), which brought the lens power differences to within ±1 D for ∼95% of the eyes. If lens thickness is not available, both methods with the customized 2 constants may be considered to be good approximations of the Bennett method.

*c*

_{1}

*T = c*

_{Sten}and c

_{2}

*T = T − c*

_{Sten}). This can be confirmed mathematically by comparing equation 1 with equations 2 and 3 for the special case when

*S*

_{pp}=

*S*

_{CV}= 0. The more general case, when

*S*

_{pp}and

*S*

_{CV}are different from 0, could only be confirmed numerically because of the mathematically complicated equation 3. Although this seems to point at some common origin of both formulas, the meaning of this observation remains unclear.

*c*constants and lens refractive index values

*n*

_{L}determined from phakometry. For this reason, the results of the lens power calculations were given for each eye model separately. However, a significant correlation with

*n*

_{L}was seen only for

*c*

_{1}of the Bennett method; the other

*c*constants were either constant or randomly distributed.

^{ 18 }or the SRK/T formula

^{ 19,20 }to calculate the lens power, provided appropriate values for the IOL constants are used. Here, one has to deal with the added difficulty of estimating the final postoperative position of the lens,

^{ 21,22 }which may explain the large variety in IOL calculation formulas in the literature.

*c*constants. The modified-Stenström and Bennett-Rabbetts methods, with appropriate

*c*constants, provide reasonable approximations of equivalent lens power when lens thickness is not known. These methods allow the application of the concept of our statistical eye model

^{ 12 }to datasets without lens thickness or can be included in the software of a biometry device alongside IOL calculation formulas, thus providing physicians with access to the important parameter of lens power.

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