Model eyes for evaluation of intraocular lenses

0 downloads 0 Views 2MB Size Report
Sep 10, 2007 - With newer IOLs designed to compensate for the spherical aberration of the cornea ..... pupil will then vary depending on haptic design and.
Model eyes for evaluation of intraocular lenses Sverker Norrby,1,* Patricia Piers,1 Charles Campbell,2 and Marrie van der Mooren1 1

AMO Groningen BV, van Swietenlaan 5, NL-9728 NX, Groningen, Netherlands 22908 Elmwood Court, Berkeley, California 94705, USA *Corresponding author: [email protected] Received 16 October 2006; revised 22 May 2007; accepted 18 July 2007; posted 25 July 2007 (Doc. ID 76148); published 7 September 2007

In accordance with the present international standard for intraocular lenses (IOLs), their imaging performance should be measured in a model eye having an aberration-free cornea. This was an acceptable setup when IOLs had all surfaces spherical and hence the measured result reflected the spherical aberration of the IOL. With newer IOLs designed to compensate for the spherical aberration of the cornea there is a need for a model eye with a physiological level of spherical aberration in the cornea. A literature review of recent studies indicated a fairly high amount of spherical aberration in human corneas. Two model eyes are proposed. One is a modification of the present ISO standard, replacing the current achromat doublet with an aspheric singlet cut in poly(methyl methacrylate) (PMMA). The other also has an aspheric singlet cut in PMMA, but the dimensions of it and the entire model eye are close to the physiological dimensions of the eye. They give equivalent results when the object is at infinity, but for finite object distances only the latter is correct. The two models are analyzed by calculation assuming IOLs with different degrees of asphericity to elucidate their sensitivity to variation and propose tolerances. Measured results in a variant of the modified ISO model eye are presented. © 2007 Optical Society of America OCIS codes: 170.4470, 110.4100, 350.4800.

1. Introduction A.

Background

The international standard for evaluation of imaging quality of intraocular lenses [1] (ISO 11979-2, 1999) prescribes a model eye with an essentially aberrationfree artificial cornea. In practice, this is achieved with readily available achromat doublets. In this model eye, any aberrations are due to the intraocular lens (IOL) being measured. The metric by which the lens is evaluated is the modulation at 100 cycles!mm. The requirement is that this value be 0.43 or higher for a 3 mm aperture at the IOL using monochromatic light close to 546 nm (mercury green). The higher the value the better the IOL is for the patient—as long as the IOL has positive spherical aberration, which all IOLs had at the time of development of this standard. One of us (S. Norrby) was leading the development of the standard and performed an analysis of different 0003-6935/07/266595-11$15.00/0 © 2007 Optical Society of America

methods in use at that time for assessing imaging quality of intraocular lenses [2]. For IOLs with negative spherical aberration, designed to counter the positive spherical aberration of the cornea, lower modulation in the ISO model can no longer be interpreted as lower quality for the patient. An IOL exhibiting low modulation in the model could now be either better or worse than one exhibiting higher modulation. There is therefore a need for a model eye with a physiologic level of spherical aberration in the artificial cornea. The question is: what is the spherical aberration of normal corneas? B.

Eye Models

Gullstrand’s classic eye model [3] has all surfaces spherical. However, already Helmholtz [4] had measured the cornea and described it as a prolate conicoid of rotation, which was well known to Gullstrand. To take the effect of the spherical aberration of the cornea into account, the axial length of his model is such that paraxial focus falls on the retina 10 September 2007 ! Vol. 46, No. 26 ! APPLIED OPTICS

6595

under the assumption of !1 diopter (D) in the spectacle plane. Lotmar [5] was the first to introduce aspherical surfaces in an eye model. The anterior corneal surface was given a shape in accordance with measurements by Bonnet (referenced in [5]). The posterior lens surface was arbitrarily given a parabolic shape, while other surfaces remained spherical. The lens was treated as homogeneous, i.e., without a gradient index. Lotmar’s aspheric eye model is paraxially emmetropic. For principal rays at finite angles to the optical axis he calculated spherical aberration and found it to be largely in agreement with the experimental findings available to him. Several eye models have since been proposed. Analyzing them, Liou and Brennan [6] found that they all predicted more ocular spherical aberration than what was measured in living eyes. Following this analysis they proposed an eye model [7] with a cornea with aspheric surfaces and a lens with aspheric surfaces and a gradient index. Their model also takes chromatic aberration, pupil decentration, and angle alpha (angle between visual and optical axes) into account. The aspheric surfaces are described as conicoids of revolution by means of the Q value (Q ! "1 for a paraboloid and Q ! 0 for a sphere; values in between denote a prolate ellipsoid). They adopted anterior radius Ra ! 7.77 mm and anterior asphericity Qa ! "0.18 from measurements by Guillon and co-workers (referenced in [7]). The posterior radius was calculated to be Rp ! 6.40 mm from the ratio 0.823 (posterior!anterior) found by Dunne and co-workers (referenced in [7]), while the posterior asphericity Qp ! "0.60 was taken as the value best fitting literature experimental data of ocular spherical aberration at different ray heights. More recently Dubbelman and co-workers [8 –11] measured radii and asphericity of the corneal and lens surfaces, together with thicknesses of the cornea, aqueous, and lens, by means of corrected Scheimpflug imaging. Their data were used to establish the Dubbelman eye model [12], which describes the eye as a function of age and level of accommodation. It is a rotationally symmetric, monochromatic eye model with a homogeneous index lens. For the cornea Ra ! 7.87 mm, Qa ! "0.18, Rp ! 6.40 mm, and Qp ! "0.34, all independent of age and level of accommodation. C. Corneal Topography

Kiely and co-workers [13] measured 176 corneas with the autocollimating keratoscope developed by Clark (referenced in [13]). This was a forerunner to presentday placido disk topographers. They found Ra ! 7.72 # 0.27 mm and Qa ! "0.26 # 0.18, a value long considered as the gold standard for corneal asphericity. They also found that asphericity could vary depending on meridian, but there was no tendency for any meridian to be more aspheric, while the horizontal meridian was flatter (with-the-rule astigmatism). Holladay and co-workers [14] measured corneal topography with the Orbscan I (Bausch & Lomb) scan6596

APPLIED OPTICS ! Vol. 46, No. 26 ! 10 September 2007

ning slit topographer in 71 eyes of 71 cataract patients. The topography height data were fitted by Zernike polynomials using the first 36 terms (up to the seventh order). The surface shape thus described was used to model a single surface cornea adopting the commonly used keratometric refractive index of 1.3375. By ray tracing, the aberration produced by this cornea was obtained in terms of Zernike coefficients. These were averaged and only astigmatism c"2, "2#, trefoil c"3, "3#, and spherical aberration c"4, 0# were found to be significantly different from zero. Both the surface shape fitting and the ray tracing were done for a pupil of 6 mm on the cornea. The average spherical aberration was found to be c"4, 0# ! 0.27 # 0.20 $m (standard deviation misprinted as 0.02 $m in the paper [14]). This value was used to design what later became known as the TECNIS IOL (Advanced Medical Optics). Wang and co-workers [15] in a mixed group of refractive and cataract patients measured 228 eyes of 134 subjects and found c"4, 0# ! 0.280 # 0.086 $m for a 6 mm pupil at the cornea. They used a Humphrey Atlas (Carl Zeiss) topographer and the CTVIEW program (Sarver and Associates) to calculate Zernike coefficients from the topography data. Bellucci and co-workers [16] found c"4, 0# ! 0.276 # 0.036 $m for a 6 mm pupil in 25 eyes of 25 patients measured postoperatively. Measurements were made with the Topcon KR-9000PW cornea topographer!aberrometer, which also provided the Zernike coefficients. Guirao and co-workers [17] in a group of 70 cataract patients found c"4, 0# ! 0.32 # 0.12 $m preoperatively and c"4, 0# ! 0.34 # 0.19 $m postoperatively for a 6 mm pupil. They used the EH-290 EyeMap (Alcon) topographer and their own computational procedure. It thus appears that for a 6 mm entrance pupil values around c"4, 0# ! 0.3 $m are obtained with a variety of equipment and different calculation schemes. There also seems to be little or no surgically induced change in spherical aberration on average. D.

Age Effects

Oshika and co-workers [18] found the spherical aberration of the cornea to be independent of age, while Guirao et al. [19] found a slight though statistically significant increase with age. Dubbelman and co-workers [10] found radii and asphericity to be independent of age. However, they measured only in the vertical meridian. In a more recent study [20] they measured the right eyes of 57 males and 57 females in six meridians and found more complex relations. There was a significant influence of gender on corneal radius: Ra ! 7.87 # 0.30 mm and Rp ! 6.60 # 0.23 mm for males; Ra ! 7.72 # 0.23 mm and Rp ! 6.456 # 0.23 mm for females (standard deviations were obtained by multiplying the standard errors in the paper by $57). There was no gender variation in any other variables. While the Q value of the anterior surface was independent of meridian, that of the posterior surface was signifi-

cantly more prolate in the vertical meridian. There was statistically significant age-dependence in Q value of both surfaces. On average Qa ! "0.24 % 0.003 & age and Qp ! "0.006 & age. In other words the posterior cornea is nearly spherical at a young age, while the anterior cornea approaches spherical with aging. E. Summary of Earlier Work

To facilitate comparison between the models and measurements presented in Subsections 1.B–1.D, ray tracing calculations (OSLO EDU, freeware from Lambda Research Corporation) were applied to find corresponding c"4, 0# values from Q values, and vice versa. The calculations were all done for an entrance pupil of 6 mm using the best focus criterion of OSLO EDU. Other focus criteria had limited influence on calculated c"4, 0# values. All calculations were done for monochromatic light with refractive indices representative for light of %546 nm wavelength. OSLO EDU outputs spherical aberration as the eighth term in millimeters, nonnormalized. Multiplication by 1000 and division by $5 yield the normalized value in micrometers. The cornea models in Table 1 are subdivided into 2-surface and 1-surface models. Among the 2-surface ones, only those due to Dubbelman et al. [10,20] are actually based on measurement of both corneal surfaces. Dubbelman 1 [10] with c"4, 0# ! 0.219 $m is close to that of the widely recognized Liou and Brennan [7] eye model &c"4, 0# ! 0.258 $m'. The newer Dubbelman 2 [20] results, averaged over all meridians, indicate a more aberrated cornea at 60 yr of age (c"4, 0# ! 0.304 $m for males and c"4, 0# ! 0.323 $m for females), close to the spherical one of Gullstrand [3] &c"4, 0# ! 0.335 $m'. The eye model of

Lotmar [5] describes a cornea with considerably less aberration &c"4, 0# ! 0.038 $m'. Lotmar’s description, which was based on measured data, is a paraboloid with two modifying radial terms (see Table 1). Kiely et al. [13] in their paper approximated the equation given by Lotmar as a prolate ellipsoid &Qa ! "0.286' with one modifying radial term &"3.27 & 10"7 & r6'. For a 6 mm entrance pupil this approximation gives c"4, 0# ! 0.106 $m, three times as much as the original. Kiely et al. from their own corneal shape measurements found Qa ! "0.26 corresponding with c"4, 0# ! 0.179 $m. Earlier studies thus tend to indicate less spherical aberration of the cornea than more recent studies do. Possibly development in measurement techniques and calculation methods can explain some of the differences, but the age composition of the groups studied could also be of influence (see Table 1). At 20 yr of age the Dubbelman 2 model has c"4, 0# at %0.20 $m compared to %0.30 $m at 60 yr. From corneal topography only 1-surface models can be established. Wang et al. [15] found c"4, 0# ! 0.280 $m on average from corneal elevation data. The refractive index given by them (1.3771) indicates that they assumed the LeGrand eye model [21]. We have chosen the anterior corneal radius according to that model &7.8 mm' to calculate the corresponding Qa value &"0.1025'. We used the same assumptions for the conversion of reported c"4, 0# values to Qa values for Bellucci et al. [16] and for Guirao et al. [17]. Use of the high refractive index in combination with the absence of a second corneal surface explains the lower f numbers for these studies (see Table 1). Backcalculating from the Qa value found the Wang cornea with the common keratometric index 1.3375 results

Table 1. Selection of Theoretical Cornea Models and Cornea Models Based on Measurement of Shape or Topography

Type 2-surface corneas

1-surface corneas

Cornea Designation Thickness Rab Rpb (References) (mm) (mm) (mm)

b

b

Qa

Qp

0 Specialc #0.18 #0.18 #0.06

0 0 #0.60 #0.34 #0.36

Gullstrand [3] Lotmar [5] Liou [7] Dubbelman 1 [10] Dubbelman 2 male [20] Dubbelman 2 female [20]

0.50 0.55 0.50 0.574 0.579

7.7 7.8 7.77 7.87 7.87

6.8 6.5 6.40 6.40 6.60

0.579

7.72

6.456 #0.06

Kiely [13] Holladay [14] Wang [15] Bellucci post [16] Guirao pre [17] Guirao post [17]

3.6 3.6 $ $ $ $

7.72 Plano #0.26 7.575 Plano #0.14135 7.8d — #0.1025 7.8d — #0.1085 7.8d — #0.0455 7.8d — #0.0168

Corneal Aqueous c[4, 0]b f RIb RI ("m) Number 1.376 1.3771 1.376 1.376 1.376

#0.36 1.376 — — — — — —

1.3375 1.3375 1.3771d 1.3771d 1.3771d 1.3771d

Age (Years)

1.336 1.3374 1.336 1.336 1.336

0.335 0.038 0.258 0.219 0.304

3.87 3.93 3.94 4.00 3.98

— — — Mean 38 At 60

1.336

0.323

3.91

At 60

1.336 1.336 — — — —

0.179 0.270 0.280 0.276 0.32 0.34

3.81 3.74 3.45 3.45 3.45 3.45

16–80 Mean 74 Mean 50 63–84 Mean 70 Mean 70

a For those models for which Q values are known, the corresponding Zernike coefficient for spherical aberration, c[4, 0], was calculated for a 6 mm entrance pupil, and vice versa, to allow comparison. The f numbers also apply for this pupil size. See text for additional information. b Legend: Ra, anterior radius; Rp, posterior radius; Qa, anterior corneal Q value; Qp, posterior corneal Q value; RI, refractive index; c[4, 0], Zernike spherical aberration coefficient. c Qa % #1 for the conicoid with modifying radial terms !1.881 & 10#4 & r4 and #1.443 & 10#6 & r6. d Assumed in accordance with the LeGrand eye model [21] for the purpose of conversion from c[4, 0] to Qa.

10 September 2007 ! Vol. 46, No. 26 ! APPLIED OPTICS

6597

in c"4, 0# ! 0.270 $m. The keratometric index is thus of influence but not crucial. For practical use in IOL design calculations, Holladay et al. [14] truncated the cornea with a plano posterior surface behind which the refractive index was that of aqueous. Truncating the 1-surface Wang cornea in the same way results in c"4, 0# ! 0.283 $m, which justifies the method. Bellucci et al. [16] found almost the same level of spherical aberration, while the values of Guirao et al. [17] are at about the level of the spherical Gullstrand cornea. Coincidentally, elimination of the posterior asphericity of the Dubbelman 2 60 yr old female cornea reduces spherical aberration to c"4, 0# ! 0.280 $m. The prolate posterior surface asphericity thus increases the amount of positive spherical aberration of the cornea. 2. Methods A.

Ray Tracing Calculations

was also used to calculate modulation transfer function (MTF) curves for the model eyes proposed in this paper. Maximum MTF at 100 cycles!mm was chosen as the focusing criterion in accordance with the present ISO standard. Furthermore, the entrance pupil was adjusted to give the desired pupil size at the IOL as defined by the height of the marginal ray, traced through the aspheric system, at the plane of the apex of the front IOL surface. To unequivocally distinguish from “entrance pupil” the term “iris pupil” is used. For the purpose of illustration 22 D power intraocular lenses of the models Tecnis Z9000 (Advanced Medical Optics), LI61U, and SofPort AO (both Bausch & Lomb) have been used. Full details of these lenses are given in Altmann et al. [22]. Tecnis Z9000 has a modified prolate anterior surface to provide negative spherical aberration to compensate the positive spherical aberration of the cornea. LI61U is a conventional spherical lens and thus exhibits positive spherical aberration. SofPort AO has both surfaces prolate and no spherical aberration. Calculations were made for 3.0 and 4.5 mm iris pupils, which can be considered representative for photopic and mesopic light conditions, respectively. OSLO EDU

Table 2. Proposed Modified ISO Model Eye

Surface

Radius (mm)

Q Value

Thickness (mm)

Refractive Index

Object Cornea front Cornea back Window front Window back Iris pupil Window front Window back Image

— 19.24 Flat Flat Flat Flat Flat Flat —

— 0.226 — — — — — — —

Infinity 10.00 3.00 6.00 6.25 10.00 6.00 11.65 —

1 1.493 1 1.5168 1.336 1.336 1.5168 1 —

a Except for the cornea itself this model is identical to the present ISO model, which assumes Schott BK7 for the windows. The choice of glass is unimportant.

called the physiological model eye [Table 3 and Fig. 1(b)]. Note that the former has an oblate front surface and the latter a prolate one, while they both produce the desired amount of wavefront aberration upon the IOL. The distance from the iris pupil to paraxial focus in a medium with refractive index 1.336 would be 27.65 mm for both models, which is in accordance with the present ISO model. Figure 2 illustrates how the three IOLs perform in the two proposed models in comparison with the present ISO model.

3. Results A.

Model Eyes for Evaluation of Intraocular Lenses

For the design of the Tecnis IOL [22,23] the value c"4, 0# ! 0.270 $m was used. Clinical results obtained with this lens model corroborate that value. Postoperative spherical aberration was not significantly different from zero [24], though slightly on the positive side. Therefore c"4, 0# ! 0.280 $m as found by Wang and co-workers [15] on a larger sample can be chosen as representative. It is also in the middle of values found by many recent studies (Table 1). Two model eyes are proposed. One resembles the present ISO model eye and will be referred to as the modified ISO model eye [Table 2 and Fig. 1(a)]. The other has dimensions close to the natural eye and is 6598

APPLIED OPTICS ! Vol. 46, No. 26 ! 10 September 2007

Fig. 1. Two proposed new eye models. The iris pupil plane is indicated and its width is set at 3 mm in the drawings.

Table 3. Proposed Physiological Model Eye

Surface

Radius (mm)

Q Value

Thickness (mm)

Refractive Index

Object Cornea front Cornea back Iris pupil Window front Window back Image

— 7.80 7.02 Flat Flat Flat —

— #0.0205 0 — — — —

Infinity 0.50 4.00 10.00 1.00 16.65 —

1 1.493 1.336 1.336 1.5168 1 —

a For the window Schott BK7 has been assumed, though making the thin rear window making out of a microscope object glass would be a practical choice.

Decentration of the IOL will negatively influence the performance in a model eye. In the presence of IOL decentration the tangential and sagittal MTF curves are no longer identical and focusing at maximum MTF becomes ambiguous. A focusing criterion often used is minimum on-axis root-mean-square (rms) optical path difference (OPD). In Fig. 3 the

calculated consequences of 0.5 mm of IOL decentration in the physiological model eye are shown for the two focusing alternatives. Very similar results are obtained with the modified ISO model eye (not shown). The focus position is dependent on both decentration and pupil size as can be seen in Fig. 4. The variation is larger for the minimum OPD focusing criterion than for the maximum MTF one. The axial position of the IOL is of minor importance as seen in Fig. 5 for the physiological model eye and IOL models Tecnis Z9000 and LI61U. For these calculations the iris pupil was 0.5 mm more forward than in Table 3, and the IOL anterior position was displaced #0.5 mm around the nominal. Only for the highly corrected system with Tecnis Z9000 is there discernible difference due to the axial displacement. Using water (refractive index 1.333) instead of aqueous (1.336) has very little influence on MTF as seen in Fig. 6. Figure 7 shows the correspondence between 0.43 modulation at 100 cycles!mm with a 3 mm iris pupil, the approval limit in the present ISO model eye, for

Fig. 2. Three IOL models Tecnis Z9000 (Œ), LI61U (□), and SofPort AO ("), together with the diffraction limit (thick curve), in the present ISO model eye (top), modified ISO model eye (middle), and physiological model eye (bottom) at 3.0 and 4.5 mm iris pupil sizes. In (a) the diffraction limit curve covers that for the SofPort AO IOL. For the Tecnis Z9000 IOL that is the case in (c), (d), and (e). 10 September 2007 ! Vol. 46, No. 26 ! APPLIED OPTICS

6599

Fig. 3. Three IOL models Tecnis Z9000 (Œ), LI61U (□), and SofPort AO ("), together with the diffraction limit (thick curve), in the physiological model eye at 3.0 and 4.5 mm iris pupil sizes. With decentration the tangential (continuous curves) and sagittal (dashed curves) separate, except for SofPort AO. Top: Decentered 0.5 mm with maximum MTF at 100 cycles!mm as focusing criterion (compare with Figs. 1(e)–1(f) for the centered case). Middle: Decentered 0.5 mm with minimum on-axis rms OPD as focusing criterion. Bottom: Centered with minimum on-axis rms OPD as focusing criterion. In (e) the diffraction limit curve covers that for the Tecnis Z9000 IOL.

spherical IOLs of different shapes (plano– convex, convex–plano, and bi-convex) made of high (1.55) or low (1.43) refractive index material and having such a high power that they are on the limit, and that attained by those IOLs in the physiological model eye with a 3 mm iris pupil. Through-focus MTF profiles at 25 cycles!mm with a 3.0 mm iris pupil for SofPort AO in the modified ISO and physiological model eyes are shown in Fig. 8. The OSLO software has an option to calculate throughfocus profiles in image space. On an optical bench that would be comparable to measuring MTF by moving the detector through-focus. If the detector is in a fixed position one can measure through-focus curves by changing vergence in object space. A practical way to do that is putting spectacle correction lenses in front of the model eye. A more sophisticated setup would be to use a Badal lens for the purpose. Calculation shows that the MTF with spectacle lens cor6600

APPLIED OPTICS ! Vol. 46, No. 26 ! 10 September 2007

rection is negligibly different for that calculated for the corresponding object vergence distance. 4. Discussion A.

Eye and Cornea Models

Even though recent measurements of both corneal surfaces indicate higher corneal spherical aberration than inferred from the front surface topography alone, we have chosen to adopt the level found by Wang et al. to propose two model eyes for the purpose of evaluating IOLs. A slightly lower level was used to develop the Tecnis IOL, and clinical studies [24] show that postoperative spherical aberration is not significantly different from zero. In the context of the pseudophakic eye this level of corneal asphericity thus seems appropriate. It can be noted in passing that while the means in measured c[4, 0] are about the same in recent studies,

Fig. 4. Focus shift due to decentration and pupil size in the physiological model eye at different focus criteria. The focus position is the distance from the back window surface to focus. The Tecnis Z9000 IOL is not sensitive to pupil size but to decentration for the maximum MTF criterion, but not for the minimum OPD one. The SofPort AO and LI61U IOLs are insensitive to both pupil size and decentration at maximum MTF but quite sensitive to pupil size at minimum OPD.

the standard deviations differ (Table 1). The standard deviation comprises variation in the group studied and the random error of the measurement. There could be differences between the groups due to age effects and in the measurement due to equipment used. Figure 2(a) shows that the 22 D power IOLs of Tecnis Z9000, LI61U, and SofPort AO all are well above the approval limit of 0.43 modulation at 100 cycles!mm for monochromatic &546 nm' light with a 3.0 mm iris pupil in the present ISO model eye with its aberration-free cornea. With the SofPort AO the system is diffraction limited. For a 4.5 mm pupil, Fig. 2(b), the MTF curves drop for Tecnis Z9000 due to negative spherical aberration and for LI61U due to positive spherical aberration. The absolute magnitude happens to be about the same for this particular IOL power. However, the negative spherical aberration of Tecnis Z9000 is independent of power, while the positive spherical aberration increases with power for LI61U, as for any IOL with all surfaces spherical. With SofPort AO MTF drops slightly below the diffraction limit, because the combination of the aberration-free cornea and the aberration free IOL is not a totally aberration free system, which shows at larger pupils. The SofPort AO IOL is designed to be aberration-free at zero vergence, while the rays behind the cornea are converging.

Fig. 5. Influence of axial position of the IOL in the physiological model eye at different pupil sizes with the Tecnis Z9000 IOL (Œ) and the LI61U IOL (□) at the nominal center position (continuous curve), 0.5 mm anterior (short dashes) and 0.5 mm posterior (long dashes). The influence of axial position of the IOL is of some influence for the well-corrected system with the Tecnis Z9000 IOL but negligible for the aberrated system with the LI61U IOL.

The modified ISO and physiological eye models show practically identical results for the three IOLs at both 3.0 and 4.5 mm iris pupils [Figs. 2(c)–2(f)]. The curves are diffraction limited with Tecnis Z9000 for both pupil sizes, because the IOL fully compensates the aberration of the cornea. With LI61U the system suffers positive spherical aberration from both the cornea and the IOL. SofPort AO neither adds nor subtracts spherical aberration and the curves are representative for the aberration of the cornea alone. The difference between LI61U and SofPort AO is

Fig. 6. Influence of refractive index of the aqueous in the physiological eye model with the Tecnis Z9000 (Œ) and LI61U (□) IOLs. Nominal refractive index of the aqueous is 1.336 (dashed curve) compared with water 1.333 (continuous curve). This difference in refractive index appears to have a negligible effect on the MTF. 10 September 2007 ! Vol. 46, No. 26 ! APPLIED OPTICS

6601

B.

Fig. 7. Performance in the physiological model eye of spherical lenses made of high refractive index material (1.55) or low refractive index material (1.43) of different shapes (PC, plano– convex, plano anterior; CP, convex–plano, convex anterior; BC, equi-biconvex), in each combination for a power that is just passing the approval criterion, a modulation of 0.43 at 100 cycles!mm, in the present ISO model eye with 3 mm iris pupil. The powers of the lenses are: 1.55 PC: 26.10 D; 1.55 CP: 52.20 D; 1.55 BC: 47.59 D; 1.43 PC: 16.21 D; 1.43 CP: 26.48 D; 1.43 BC: 32.05 D. The mean modulation at 100 cycles!mm in the physiological model eye with a 3 mm iris pupil is 0.27.

thus the effect of the difference in spherical aberration of these IOLs. Altmann in a patent [25] discloses a physiological eye model but without giving any quantitative detail, which prevents comparison with our proposed models.

Model Eye Tolerances and Pass Criterion

A practical bench model realization of the physiological model eye would have a groove or recess where the IOL haptics are placed. The design should be such that the IOL optic does not touch the artificial pupil. The axial position of the IOL in relation to the iris pupil will then vary depending on haptic design and tolerances. As shown in Fig. 5 that should not be a problem because the influence of axial displacement of the IOL within #0.5 mm is practically negligible on MTF. For the calculations the iris pupil was moved 0.5 mm anteriorly and its size modified to obtain the desired iris pupil size in the nominal position. The axial shifts are accompanied by shifts in effective pupil size: 3.0 # 0.025 mm and 4.5 # 0.08 mm. This can be compared to the tolerance of #0.1 mm in the ISO standard [1]. Incidentally, the standard deviation in anterior chamber depth is '0.5 mm in both phakic and pseudophakic eyes [26]. For the cornea reasonable tolerances are #0.025 mm in radii and #0.001 in Q value of the anterior surface. These tolerances give just discernable differences in MTF while they are achievable with precision lathes. The refractive index of pure water, 1.333, is sufficiently close to that of aqueous, 1.336 (Fig. 6) to have negligible influence on MTF. However, we know by our own experience that this is not true for diffractive optics, where the refractive index difference between the material and the aqueous should be matched closely to that in situ, i.e., the IOL material is in equilibrium with aqueous at 35 °C (the eye is cooler

Fig. 8. Through-focus MTF profiles at 25 cycles!mm with a 3.0 mm iris pupil for the SofPort AO IOL in the modified ISO and the physiological model eyes. Zero is at paraxial focus. Top: Through-focus profiles in image space as obtained by the OSLO software from "0.6 to %0.6 mm in 0.03 mm steps. Bottom: Through-focus profiles obtained by calculating the MTF for spectacle corrections from "2 D to %2 D in 0.1 D increments. For through-focus response in image space the two models are equivalent, but clearly only the physiological model eye is adequate when generating the through-focus response in object space. 6602

APPLIED OPTICS ! Vol. 46, No. 26 ! 10 September 2007

than the body). Solutions of sucrose or other inert molecules in water can serve the purpose of aqueous emulators. The pass criterion in the ISO standard [1] is a modulation of at least 0.43 at 100 cycles!mm in the present ISO model eye with a 3.0 mm iris pupil. Figure 7 shows how that translates to the physiological model eye for a 3.0 mm iris pupil for IOLs of different shapes and refractive indices. Modulation at 100 cycles per degree for these IOLs in the physiological model eye with a 3 mm iris pupil ranged from 0.23 to 0.31, with a mean of 0.27. The shapes and powers assessed are quite extreme, yet there is not very much difference in how 0.43 in the present ISO model eye translates to modulation in the physiological model eye. An approval limit of 0.27 modulation in the latter therefore appears reasonable. It can be noted in Fig. 2(c) and 2(e) that all three IOL models are well above this limit in both the modified ISO and the physiological model eyes. C. Decentration

A highly corrected optical system is more sensitive to disturbances than a less corrected one. With IOL decentration the tangential and sagittal MTF curves split, especially for Tecnis Z9000, which for a 3.0 mm iris pupil and 0.5 mm decentration drop to about the level of LI61U [Fig. 3(a)]. If the pupil is widened to 4.5 mm there is a further drop due to spherical aberration, and all three IOL models are roughly at the same level [Fig. 3(b)]. Similar results were found by Altmann et al. [22] (their Figs. 2(b) and 3(b), representing 4.0 mm and 5.0 mm pupils, respectively). However, in that paper the curves for Tecnis Z9000 drop even more when the IOL was decentered. This can be explained by the fact that they did not refocus the system for the defocus caused by decentration. In clinical practice the patient is refracted to find the best spectacle correction whether the IOL is decentered or not. To best mimic the clinical situation calculations should be made at best focus. That the result is very sensitive to focusing is shown in Figs. 3(c)–3(d) where minimum on-axis rms OPD was used as the focus criterion. The curves for LI61U and SofPort AO drop dramatically for the 4.5 mm iris pupil. The minimum OPD criterion was tried because with the tangential and sagittal MTF curves separated, finding the maximum MTF by calculation is ambiguous. The focus position shifts due to decentration, pupil size, and focus criterion are shown in Fig. 4 and are appreciable. In summary, calculations could be misleading if refocusing to optimum performance is not done. Therefore one should evaluate model eye calculations at optimum conditions as we have done in this paper. It could perhaps be argued that refocusing for defocus caused by different pupil sizes, as we have done in this paper, should not be applied, because the refraction given to a patient is obtained with the pupil size induced by the illumination in the examination room. The focusing criterion used should at any rate be clearly stated. However, it remains an

open question which focusing criterion best represents human visual preference. D.

Object and Image Space

The modified ISO model eye and the physiological model eye are equivalent for objects at infinity and can both be equally well used with this constraint. For depth of focus measurement they are also equivalent if the test bench has the provision to move the detector through-focus [Figs. 8(a)– 8(b)]. If, however, depth of focus is tested as done clinically by placing spectacle correction lenses in front of the eye, only the physiological model eye gives relevant results [Figs. 8(c)– 8(d)]. The reason is the difference in effective focal length. With the object at infinity they are designed to have the same vergence incident upon the IOL, but with the object at finite distances, in reality or by means of correction lenses in front of the eye, this is no longer the case. Therefore only the physiological model eye can be used for studies of situations in object space. For the same reason the present ISO model eye is also unsuitable for measurement of through-focus response by variation of object vergence. E.

Practical Realization of the Physiological Model Eye

Figure 9 shows a realization of the physiological model eye. This model eye can be put on an optical bench for measurement of MTF through it. It could also, with suitable relay optics, be used as lens on a

Fig. 9. Physiological model eye modified to be mounted on a headrest before a wavefront eye refractor where it can be measured just as a human eye (bottom). The pieces are shown disassembled (top left) and assembled (top right). Instead of a rear window it has a rear section designed to act as an artificial retina. The retina is painted with Weathered Black Floquil model train paint to act as a diffuse reflector. By rotation on a thread, the retina can be moved in or out to obtain focus on it depending on the IOL power. 10 September 2007 ! Vol. 46, No. 26 ! APPLIED OPTICS

6603

CCD camera to simulate what a patient sees. The imaging would be realistic because the effective focal length is in the same range as that of real eyes. By slight modification it can also be mounted on a headrest in front of a wavefront eye refractor, where it can be measured just as a human eye is measured. The physiological model eye is thus very versatile for multiple purposes, while the modified ISO model can only be used for bench measurement of MTF or other optical quality metrics with the object at infinity. F.

Measurements of Intraocular Lenses in a Model Eye

Figure 10 shows MTF curves measured for several IOL models in a variant of the modified ISO model eye. It uses a custom made glass lens as the cornea instead of the poly(methyl methacrylate) (PMMA) cornea of Table 2. It has slightly more spherical aberration, corresponding to c"4, 0# ! "0.30 $m [27]. As can be seen the AMO Tecnis Z9000 IOL corrects almost all of the spherical aberration of the cornea and the system is close to diffraction limited. The Alcon SN60WF and the Canon KS-3Ai IOLs both correct part and about the same amount of the spherical aberration of the cornea. The Bausch & Lomb L161AO IOL neither corrects nor adds spherical aberration and all aberration is due to the cornea. The Alcon SN60AT IOL and AMO CeeOn 911A IOL, having all surfaces spherical, add their own spherical aberration to that of the cornea.

Fig. 10. Several IOL models measured in a variant (see text) of the modified ISO model at two pupil sizes. Legend: AMO Tecnis Z9000 18.5 D (Œ), Alcon SN60WF 20.0 D (&), Canon KS-3Ai 20.0 D (‚), B&L L161AO 17.0 D ("), Alcon SN60AT 18.0 D (!), and AMO CeeOn 911A 19.5 D (□) IOLs. 6604

APPLIED OPTICS ! Vol. 46, No. 26 ! 10 September 2007

G.

Model Eyes and Real Eyes

Real eyes are considerably more complex than model eyes. Results with model eyes should therefore be taken with a grain of salt. In this paper decentration of the IOL by 0.5 mm caused the MTF curves for the spherical aberration correcting Tecnis Z9000 IOL to drop to the levels of noncorrecting IOLs (Fig. 3). Holladay et al. [14] calculated that the Tecnis Z9000 IOL should perform better than conventional spherical lenses for up to 0.4 mm of decentration and 7° of tilt. Altmann et al. [22] investigated the influence of tilt and decentration and found that the Tecnis Z9000 IOL performed better than the SofPort AO IOL for decentrations up to 0.15, 0.30, and 0.38 mm for, respectively, 3, 4, and 5 mm iris pupil sizes. However, they did not refocus for the defocus caused by decentration, which we consider a flaw in their paper. Moreover, Altman et al. introduced an apodizing filter to emulate the Stiles–Crawford effect. However, we do not feel that is a correct method for dealing with light entering the periphery of the pupil. The effect of this portion of the wavefront on image formation is completely accounted for when the point spread function is calculated and this point spread is mode matched with the retinal photoreceptor waveguide modes [28]. Real pseudophakic eyes have corneas with the whole spectrum of wavefront aberrations. After cataract surgery there is usually some degree of refractive error, which is at best corrected to the nearest 0.25 D. There is therefore some defocus. There is also chromatic aberration. These circumstances have a moderating effect on the influence of any single disturbance. Taking them into account Piers and co-workers [29] found that a better corrected pseudophakic eye is superior to a less corrected one for decentration up to 0.8 mm and tilt up to 10°. Nevertheless model eyes can provide very useful information and are indispensable tools in the development and characterization of intraocular lenses. However, judgment must be exercised when drawing conclusions from results obtained by model eye calculations or bench measurements. The proposed modified ISO and physiological model eyes are equivalent for objects at infinity, with some reservation for chromatic aberration, which has not been investigated. However, the longitudinal chromatic aberration between 500 and 640 nm is calculated to be 0.64 D in the physiological model eye without IOL, which agrees well with the 0.63 D that can be calculated for the cornea of the LeGrand eye model [21]. The modified ISO model eye has considerably less longitudinal chromatic aberration as a consequence of its longer effective focal length. Hence, it is not suitable for measurement of chromatic properties. The modified ISO model eye can be realized by exchanging the model cornea on existing present ISO model eye setups. The modified cornea is easily made and provides a physiologic level of spherical aberration. The physiological model eye is more versatile in

that it provides relevant responses for objects at finite distances. It is not very complicated to make either. References 1. ISO 11979-2, Ophthalmic implants—Intraocular lenses—Part 2: optical properties and test methods (International Organization for Standardization, 1999). 2. N. E. S. Norrby, “Standardized methods for assessing the imaging quality of intraocular lenses,” Appl. Opt. 34, 7327–7333 (1995). 3. A. Gullstrand, “The dioptrics of the eye,” in Helmholtz’s Treatise on Physiological Optics, J. P. C. Southall, ed. (Optical Society of America, 1924), Vol. 1, pp. 351–352. 4. H. Helmholtz, Handbuch der Physiologischen Optik (Leopold Voss, 1867), pp. 8 and 142. 5. W. Lotmar, “Theoretical eye model with aspherics,” J. Opt. Soc. Am. 61, 1522–1529 (1971). 6. H. L. Liou and N. A. Brennan, “The prediction of spherical aberration with schematic eyes,” Ophthalmic Physiol. Opt. 16, 348 –354 (1996). 7. H. L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A 14, 1684 – 1695 (1997). 8. M. Dubbelman and G. L. van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index, and the lens paradox,” Vision Res. 41, 1867–1877 (2001). 9. M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “The thickness of the aging human lens obtained from corrected Scheimpflug images,” Optom. Vision Sci. 78, 411– 416 (2001). 10. M. Dubbelman, H. A. Weeber, G. L. van der Heijde, and H. J. Völker–Dieben, “Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography,” Acta Ophthalmol. Scand. 80, 379 –383 (2002). 11. M. Dubbelman, G. L. van der Heijde, and H. A. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vision Res. 45, 117–132 (2005). 12. S. Norrby, “The Dubbelman eye model analyzed by ray tracing through aspheric surfaces,” Ophthalmic Physiol. Opt. 25, 153– 161 (2005). 13. P. M. Kiely, G. Smith, and L. G. Carney, “The mean shape of the human cornea,” Opt. Acta 29, 1027–1040 (1982). 14. J. T. Holladay, P. A. Piers, G. Koranyi, M. van der Mooren, and N. E. S. Norrby, “A new intraocular lens design to reduce spherical aberration of pseudophakic eyes,” J. Refract. Surg. 18, 683– 691 (2002). 15. L. Wang, E. Dai, D. D. Koch, and A. Nathoo, “Optical aberrations of the human anterior cornea,” J. Cataract Refractive Surg. 29, 1514 –1521 (2003).

16. R. Bellucci, S. Morselli, and P. Piers, “Comparison of wavefront aberrations and optical quality of eyes implanted with five different intraocular lenses,” J. Refract. Surg. 20, 297–306 (2004). 17. A. Guirao, J. Tejedor, and P. Artal, “Corneal aberrations before and after small-incision cataract surgery,” Invest. Ophthalmol. Visual Sci. 45, 4312– 4319 (2004). 18. T. Oshika, S. D. Klyce, R. A. Applegate, and H. C. Howland, “Changes in corneal wavefront aberrations with aging,” Invest. Ophthalmol. Visual Sci. 40, 1351–1355 (1999). 19. A. Guirao, M. Redondo, and P. Artal, “Optical aberrations of the human cornea as a function of age,” J. Opt. Soc. Am. A 17, 1697–1702 (2000). 20. M. Dubbelman, V. A. D. P. Sicam, and G. L. Van der Heijde, “The shape of the anterior and posterior surface of the aging human cornea,” Vision Res. 46, 993–1001 (2006). 21. Y. LeGrand, Form and Space, translated by M. Millodot and G. G. Heath (Indiana U. Press, 1967). 22. G. E. Altmann, L. D. Nichamin, S. S. Lane, and J. S. Pepose, “Optical performance of 3 intraocular lens designs in the presence of decentration,” J. Cataract Refractive Surg. 31, 574 –585 (2005). 23. S. Norrby, P. Artal, P. A. Piers, and M. van der Mooren, inventors, Pharmacia Groningen BV, assignee, “Methods of obtaining ophthalmic lenses providing the eye with reduced aberrations,” U.S. patent 6,609,793 (26 August 2003). 24. U. Mester, P. Dillinger, and N. Anterist, “Impact of a modified optic design on visual function: clinical comparative study,” J. Cataract Refractive Surg. 29, 652– 660 (2003). 25. G. E. Altmann, inventor: Bausch & Lomb, Inc., assignee, “Lens-eye model and method for predicting in vivo lens performance,” U.S. patent 6,626,535 (30 September 2003). 26. T. Olsen, H. Olesen, K. Thim, and L. Corydon, “Prediction of pseudophakic anterior chamber depth with the newer IOL calculation formulas,” J. Cataract Refractive Surg. 18, 280 – 285 (1992). 27. M. H. van der Mooren, H. A. Weeber, and P. A. Piers, “Verification of the average cornea eye ACE model,” poster 309 presented at the Association for Research in Vision and Ophthalmology (ARVO), Fort Lauderdale, Florida, USA, 30 April– 4 May 2006. 28. B. Vohnsen, I. Iglesias, and P. Artal, “Guided light and diffraction model of human-eye photoreceptors,” J. Opt. Soc. Am. A 22, 2318 –2328 (2005). 29. P. A. Piers, H. A. Weeber, P. Artal, and S. Norrby, “Design and performance of customized IOLs,” J. Refract. Surg. 23, 374 – 384 (2007).

10 September 2007 ! Vol. 46, No. 26 ! APPLIED OPTICS

6605