Vision correction by intraocular lenses

ISSN 8756-6990, Optoelectronics, Instrumentation and Data Processing, 2014, Vol. 50, No. 2, pp. 188–200. c© Allerton Press, Inc., 2014.

Original Russian Text c© G.A. Lenkova, 2014, published in Avtometriya, 2014, Vol. 50, No. 2, pp. 95–110.

OPTICAL INFORMATIONTECHNOLOGIES

Vision Correction by Intraocular Lenses

G. A. Lenkova

Institute of Automation and Electrometry,Siberian Branch of Russian Academy of Sciences,

pr. Akademika Koptyuga 1, Novosibirsk, 630090 Russia

E-mail: [email protected]

Received June 19, 2013

Abstract—The advantages of correcting the refractive errors of the eye (nearsightedness and farsight-edness) by introocular phakic (i.e., without removing the crystalline lens) lenses over the other typesof correction are considered. A relation between the optical power of spectacle glasses and contact andphakic lenses is obtained and analyzed. New, more accurate approximate formulas for calculating theoptical power of intraocular artificial lenses and phakic lenses are derived. It is shown that the deviationof calculations by the proposed formulas from the calculations by the formulas based on geometricaloptics are much less than the deviation of calculations by the regressive formulas used in ophthalmicpractice.

Keywords: correction of eye refraction, intraocular lens, contact lens, phakic lens.

DOI: 10.3103/S8756699014020113

INTRODUCTION

The most frequent lack of vision that leads to fuzzy visibility, i.e., to blurred images of objects on theretina, is the refractive errors of the eye: nearsightedness, farsightedness, astigmatism, and presbyopia.

There are several ways to correct one’s vision. The first is extraocular, which does not affect the eye andis carried out with spectacle glasses and contact lenses. The second involves changes in the refractive powerof the cornea. The third involves removal of the natural lens and implantation of an artificial one, called anintraocular lens (IOL). The fourth is implantation of an additional intraocular lens called phakic (PhIOL),which means “with a crystalline lens.” The last three ways relate to the field of refractive surgery.

Implanting conventional refractive IOLs improves the vision only in a single region (far or near, dependingon the patient’s wishes), and the patient needs spectacles to use the other region as the accommodationinterval is not fully restored. The problem of accommodation can be solved by using bifocal diffractive-refractive IOLs, on one of whose surfaces a microstructure in the form of a Fresnel phase zone plate isformed. Such an intraocular lens called MIOL Akkord [1] was developed at the Institute of Automationand Electrometry, Siberian Branch, Russian Academy of Sciences (Novosibirsk) with the participation ofthe Novosibirsk Branch of Fedorov Eye Microsurgery Complex and the Reper-NN Scientific ProductionEnterprise (Nizhny Novgorod). After the MIOL Accord lens is implanted, patients can see well into thedistance and up close without glasses.

An optimal combination of the refractive and diffractive components of the IOL and the eye elementsreduces the chromatic aberration characteristic of diffractive elements and expands the accommodationrange, i. e., creates pseudo-accommodation conditions. The use of diffractive elements in the IOL design notonly broadens the functionality of the lens, but also decreases its thickness, which is important to improveits flexibility. The problem of creating thin lenses is essential for PhIOLs, especially for high optical powers,as these lenses are inserted mainly in the small gap between the crystalline lens and the iris, whose size isapproximately 0.5 mm. Before analyzing the possibility of using diffractive elements in the construction ofphakic lenses, it is necessary to consider the characteristics of vision correction by conventional refractivephakic lenses.

188

VISION CORRECTION BY INTRAOCULAR LENSES 189

The purpose of this paper is to compare vision correction by phakic refractive lenses with other types ofcorrection; analyze the relationship between the optical powers of spectacle glass, contact lenses, and phakiclenses; derive and analyze new approximate formulas for calculating the optical power of IOLs and PhIOLs;compare the obtained formulas with the known regressive formulas.

REFRACTIVE ANOMALY DEFINITIONS

For clarity of further discussion, we define some of the concepts adopted in ophthalmology. Eye refrac-tion is the refractive power of the optical system of the eye expressed in diopters. As a physical phenomenon,it is due to anatomical features of the eye: radii of curvature, refractive indices, and mutual distances ofrefractive media. However, ophthalmologists are interested in the relation between the optical power of theeye and the length of the eye rather than its absolute power. Clinical refraction is characterized by theposition of the rear main focus of the eye in the state of accommodation rest, where the crystalline lensis in a relaxed state. The refraction is called emmetropia (proportionate) if the rear main focus coincideswith the retina, and it is called myopia (nearsightedness) or hyperopia (farsightedness) if the focus is locatedahead of or behind the retina. The latter two types are disproportionate and called ametropia. There is alsoanisometropia, which is a difference in refraction betwen the two eyes, mostly not exceeding 0.5 diopters. Inthe case of astigmatism, which is two different focuses in two mutually perpendicular planes, the concept ofspherical equivalent — average arithmetic value of refraction in two directions — is introduced.

An emmetropic eye sees well into the distance and up close, which is achieved by accommodation —the process of changing the refractive power of the eye. The mechanism of accommodation is based on theability of the lens to change shape under tension or relaxation of the ciliary muscle fibers and slight changesin the length of the eye. With age, the lens loses its elasticity and ability to focus on close objects, whichleads to age-related farsightedness known as presbyopia.

In the state of ametropia, insufficient accommodation does not allow good vision at different distances.A nearsighted person sees well up close, but cannot see well into the distance without glasses with negative,i. e., light-scattering, lenses, which make it possible to make the image of distant objects coincident with theretina. If one has hyperopia and presbyopia, on the contrary, seeing up close requires glasses with positive,i. e., collecting, lenses. Sometimes the amount of accommodation decreases so much that the person needsbifocal glasses to see well into the distance and up close.

The cause of nearsightedness is typically an increased refractive power of the cornea and/or extendedlength of the eyeball. The causes of farsightedness are opposite: low refractive power of the cornea and/orreduced length of the eyeball. Astigmatism is caused by an incorrect shape of the cornea and/or the lens.

REFRACTION CORRECTION METHODS

Vision improvement by extraocular spectacle correction had no competition over 300 years. In the secondhalf of the XX century, contact lenses were developed, which eliminated the problems caused by glasses, butrequire daily high-maintenance care. Particular discomfort is experienced by people with high degrees ofrefractive errors, who are constantly forced to use strong glasses and contact lenses. In the end of theXIX century, there were first unsuccessful attempts to change the refractive power of the optical systemof the eye surgically. But the basic development of refractive surgery as an alternative method of visioncorrection that frees people from the need to wear glasses or contact lenses, began in the 1980s–1990s.

Initially, high degrees of refractive anomalies were corrected by surgical methods involving changing therefractive power of the cornea using various technologies. The refractive power of the cornea can be changedby manipulations at its periphery (keratotomy, thermokeratoplasty, etc.) or in its center (keratophakia,keratomileusis, laser keratomileusis (LASIK), etc.) by a mechanical knife or a laser beam (excimer laser). Forexample, keratotomy involves placing radial or longitudinal incisions to a depth of 500 µm, more commonlyknown as the famous Fedorov incisions (named after Academician S. N. Fedorov) at the edges of the cornea,which led to flattening or protrusion of the central part of the cornea. The shape of the cornea can bemodeled directly in the center due to a decrease or increase in its thickness by making incisions or using adonor element as a lens. As a result of mechanical or radiation exposure, the radius of curvature of the frontsurface of the cornea and, consequently, the refractive power of the eye as a whole change.

However, the surgical methods related to changes in the corneal shape cannot be applied to all patients.The method of incisions allows removing a maximum of 6 to 7 diopters, whereas the correction effect weakensover time. Laser methods present some danger as the cornea becomes thinner and the process itself becomes

OPTOELECTRONICS, INSTRUMENTATION AND DATA PROCESSING Vol. 50 No. 2 2014

190 LENKOVA

irreversible. Moreover, nearsightedness can be corrected by no more than 12 diopters. Surgical visioncorrection methods are considered in detail in [2].

A unique and effective correction method for the cases of high myopia and hyperopia is to remove thecrystalline lens and implant an IOL, whose optical power is selected depending on the refraction of thepatient’s eye. But this makes sense if the crystalline lens does not function properly, i.e., it is damaged orclouded by a cataract. The crystalline lens is removed using the phacoemulsification procedure, in which itis emulsified by ultrasound and aspirated from the eye through a small hole (∼2.5 mm in diameter), which isself-sealed without sutures. If the crystalline lens is in good condition, the deviations of the refraction fromemmetropia can be compensated by placing a so-called phakic intraocular lens in front of the crystallinelens. This condition of the eye is defined as biphakic (with two lenses) in the diagnosis after surgery.

PHAKIC LENSES

The method of implanting phakic intraocular lenses has gained popularity in the U.S. and in manyEuropean countries in the last 20 years. PhIOL implantation has several advantages. The main differencesof refraction from the other types of correction lies in the fact that the cornea is saved in its entirety and thatthe process is reversible: if necessary, a phakic lens can be removed from the eye or replaced by another lens.Furthermore, the accommodative ability of the eye is saved because the natural lens remains in place. Manyleading ophthalmologists believe that the method of implantation of phakic lenses is safer, more predictable,and more efficient as compared with refractive surgery related to changes in the corneal shape.

The operation is as follows: a phakic lens is introduced through a small incision in the cornea and is setwithin the eye by one of three ways: it is fixed in the corner of the anterior chamber in front of the iris(anterior-chamber), mounted on the iris (iris-clips) or behind the iris in front of the crystalline lens supportedby the ciliary sulcus (posterior chamber).

Anterior chamber PhIOLs made of polymethylmethacrylate (PMMA) were the first to be implanted asearly as in the mid XX century in Italy, but the test was not entirely successful as the cornea was injured.Then, in various countries, there have been attempts to fix the lens to the iris, but also without success. Atthe turn of the 1980s–1990s, a posterior chamber PhIOL was developed, with all errors taken into account,in the Eye Microsurgery Complex under the supervision of S. N. Fedorov . It was made of silicon andsubsequently collagen copolymer. This model is the prototype of all modern posterior chamber lenses.

Fedorov’s patent was acquired by Swiss-American STAAR Surgical Company, which, in 1993, began toproduce these lenses and then improved ones, known in the world under the general name of ICL (ImplantableCollamer Lenses or Implantable Contact Lenses. The approximate cost of ICL implantation in one eye isfrom

60 to 65 thousand rubles, which is more expensive than refractive LASIK surgery.In Russia, three types of posterior chamber PhIOLs are produced: PCK-3 for correction of myopia from

4 to 26 diopters (pupil-supported Fedorov–Zuev model of a mushroom shape ); PCK-1 and PCK-1(3) forcorrection of hyperopia from 1 to 15 diopters and correction of myopia from 4 to 26 diopters, respectively(both ciliary sulcus-supported). The lenses are custom fabricated in the Eye Microsurgery Research andExperimental Production (Moscow).

Unlike surgical methods aimed at changing the refractive corneal power, the phakic lens implantation issuitable for people with myopia and hyperopia over a wide range (from −30 to +16 diopters). It is possiblethat soon phakic intraocular lenses become the leading method, especially for correction of myopia from 12to 18 diopters [3]. These are the cases where other methods are powerless or ineffective enough. Myopiaof such a high degree is very common today. All methods aimed at myopia correction are applicable tohyperopia. However, hyperopia is much more difficult to correct as positive lenses are thicker and the gapbetween the iris and the crystalline lens is not large enough.

The main difficulties in the application of phakic lenses are high requirements to the accuracy of lenscalculation and selection, detailed diagnostics, and the perfect quality of the ophthalmic surgeon’s workduring lens implantation.

ANALYTICAL CALCULATION OF THE OPTICAL POWEROF INTRAOCULAR LENSES

PhIOL calculation should be preceded by a calculation of a conventional IOL for emmetropic andametropic (with postoperative refraction correction) eyes. Formulas for IOL calculations have been derived



l2l1

l2l1

l1cor

fcor

l2cor

fs

(b)

l

L

L

d l

(a)

n

n

A

A

41

2

3

41

2

3

Fig. 1. Optical circuits of eye models: with a spectacle glass (a) and a phakic lens (b).

by many authors [4, 5]. The calculation paths differ, and sometimes mistakes are made, but, in general, thefinal results are the same. In our view, it is more convenient to calculate an IOL on the basis of opticalpowers and reduced convergences (reciprocals of distances divided by refractive index) [6]. Reduction [7]means bringing the distances to the same medium, such as air, so that the focal and other distances indifferent media were comparable between each other.

The optical power of the IOL is calculated on the basis of measurement of eye parameters. In addition,for the ametropic, the postoperative refraction is specified in the form of the optical power of the spectaclelens. IOL calculations are simplified if contact correction data are used. Therefore, we first determinethe relationship between the optical powers of the glasses and contact lenses. The calculation is carriedout assuming that an image of a distant object is projected on the retina (Fig. 1). In the figure 1 is thecornea, 2 IOL, 3 the retina, 4 spectacle glass (Fig. 1a) and PhIOL (Fig. 1b); L is the eye length, l theanterior chamber length, i.e., the distance from the cornea to the IOL, n the refractive index of the chamberhumidity and vitreous body, d is the distance between the spectacle glass and the cornea, and fs and fcor

are the focal lengths of the spectacle glass and cornea, respectively. For contact lens correction, the focallength for it fcon = fs− d, or, in another representation, fcon = 1/Ds− d = 1/Dcon, where Ds is the opticalpower of the spectacle lens and Dcon is the optical power of the contact lens equivalent to the action of the


192 LENKOVA

spectacle glass with optical power Ds or the optical power of the spectacle glass reduced to the cornea. Thelast formula for Dcon yields the relation

Dcon =1

1/Ds − d. (1)

On the basis of the reduced convergences of the objective D1cor = 1/l1cor and imaging D2cor = n/l2cor

beams, we can represent the optical power of the cornea Dcor in the form

Dcor = D2cor −D1cor = (n/l2cor)− (1/l1cor), (2)

where l1cor and l2cor are the distances from the cornea to the object and image, respectively. Next, we writean equation relating the object and image to the IOL:

DIOL = D2 −D1 = (n/l2)− (n/l1) (3)

Here DIOL is the optical power of the intraocular lens, D1 = n/l1 and D2 = n/l2 are the reduced convergencesof the objective and imaging beams for the IOL, and l1 and l2 are the distances from the IOL to the objectand to the image, respectively.

It is seen from the construction of Fig. 1a that l1cor = fs − d = fcon and, hence, D1cor = 1/l1cor =1/fcon = Dcon. Using these relations, from Eq. (2) we determine that l2cor = n/(Dcor + Dcon). Then, inEq. (3), we substitute l1 = l2cor − l = n/(Dcor + Dcon)− l, l2 = L− l (see l1 and l2 in Fig. 1b) and obtainexpressions for DIOL of the ametropic eye in terms of Dcon (for contact lens correction):

DIOL =n

L− l− n

n/(Dcon + Dcor)− l; DIOL =

n

L− l− n(Dcon + Dcor)

n− l(Dcon + Dcor). (4)

Substituting Dcon from Eq. (1) into Eq. (4), we write expressions for DIOL in terms of Ds (for spectacleglass correction)

DIOL =n

L− l− n

n/(1/(1/Ds − d) + Dcor)− l;

DIOL =n

L− l− nDs+ cor

n(1− dDs)− lDs + cor,

(5)

where Ds + cor = Ds + Dcor − dDsDcor is the total optical power of the spectacle glass and cornea. Theresult of calculating DIOL for the ametropic eye (5) coincides with the results shown in [8].

A formula for calculating DIOL (hereinafter, DI1, 2, 3) for the emmetropic eye can be obtained from Eq. (5)setting Ds = 0:

DI1 = D =n

L− l− n

n/Dcor − l; DI1 =

(n− LDcor)n(L− l)(n− lDcor)

, (6)

CALCULATING THE OPTICAL POWER OF THE IOL FOR THE APHAKIC EYE

In all formulas for calculating the IOL, the optical power depends on the length of the eye L. However,in aphakia, when the crystalline lens is removed from the eye, the optical power can be calculated withoutmeasuring the length of the eye, but based on the optical power of the spectacle glass (Ds) correcting aphakia.For this, in formula (2) we take into account that l1cor = fs − d = 1/Ds − d, l2cor = L and obtain a relationfrom which expressions for Ds and L can be determined:

n/L− 1/(1/Ds − d) = Dcor, (7)

Ds =n− LDcor

L(1− dDcor) + nd, (8)



L =n(1− dDs)

Ds + Dcor − dDsDcor. (9)

Next, we substitute Eq. (9) into Eq. (6) and calculate DIOL by the formula

DIOL =Dsn

2

(1− dDs)(n− lDcor)2 − lDs(n− lDcor). (10)

It should be noted that, in Eq. (10), Ds is the optical power of the spectacle glass that corrects aphakia (eyewithout the crystalline lens) rather than the one for correcting the eye with an IOL (Ds in formula (5)). It isinteresting to compare the optical powers of the IOL and spectacle glass that correct aphakia. For example,for the Gullstrand eye model parameters (n = 1.336, l = 4.15 mm, Dcor = 43 diopters, and L = 24 mm)and for d = 12 mm, from formulas (10) and (8), we obtain DIOL = 17.7 diopters and Ds = 11 diopters.

DERIVING APPROXIMATE FORMULA FOR THE OPTICAL POWER OF THE IOL

Equations (6) show that there is a complex relationship between DIOL and the eye parameters. Themost common formulas in ophthalmic practice are approximate regressive formulas with a linear dependence,obtained on the basis of processing multiple implantation results. In this paper, we propose an analyticalderivation of an approximate formula. We write Eqs. (6) in the form of a total differential (it is best to usethe first formula from Eqs. (6))

∆DIOL = a∆Dcor + b∆L + c∆l, (11)

where a, b, and c are the coefficients of partial derivatives with respect to the eye parameters:

a =∂DIOL

∂Dcor=

−n2

(n− lDcor)2; b =

∂DIOL

∂L=

−n

(L− l)2;

c =∂DIOL

∂l=

n

(L− l)2− nD2

cor

(n− lDcor)2= −b +

D2cor

na.

(12)

Next, we represent Eq. (11) and DIOL in the form

∆DIOL = DIOL −DIOL av = a(Dcor −Dcor av) + b(L− Lav) + c(l − lav);

DIOL = aDcor + bL + cl − aDcor av − bLav − clav + DIOL av,

(13)

where Dcor av, Lav, and lav are the average parameters of the eye that correspond to the Gullstrand model.Furthermore, for n = 1.336, Dcor av = 43 diopters, Lav = 24 mm, and lav = 4.15 mm, we use formu-las (6) and (12) to determine the values of the optical power of DIOL av and the coefficients a, b, and c:DIOL av = 17.68 diopters, a = −1.33, b = −3.39 mm−1, and c = 1.55 mm−1. The next stage is substitutionof the calculated values into Eq. (13), and then, assuming l = lav = 4.15 mm and consistently combiningthe numerical components, we obtain DIOL in diopters in the form

DIOL = 132.24− 1.33Dcor − 3.39L + 1.55l + 17.68, (14a)

DI2 = 149.91− 1.33Dcor − 3.39L + 1.55l, (14b)

DI2 = 156.33− 1.33Dcor − 3.39L. (14c)

For comparison, here is the regressive formula of SRK II (by Sanders, Retzlaff, and Kraff) used byophthalmologists [9, 10]:

DIOL = A− 0.9Dcor − 2.5L,


194 LENKOVA

12

3

L, mm

DI1

, D

I2, D

I3, di

opte

rs20 22 24 26 28

10

15

20

25

30

5

Fig. 2. Dependences of DI1 (curve 1), DI2 (2) and DI3 (3) on L for l = 4.15 mm and Dcor = 43 D.

4

5

6

1,32

L, mm

DI2

_ D

I1, D

I3 _

DI1

, di

opte

rs

20 22 24 26 28

-4

-2

0

2

4

-6

Fig. 3. Dependence of the differences DI2 −DI1 (curve 1–3) and DI3 −DI1 (curve 4–6) on L forl = 4.15 mm, Dcor = 38 D (curve 1, 4), 43 D (2, 5), 48 D (3, 6).

where the coefficient A depends on the depth of the anterior chamber of the eye. For l = 4.15 mm, theSRK II formula has the form

DI3 = 116.7− 0.9Dcor − 2.5L. (15)

Figure 2 shows the dependences of the optical powers DI1, DI2, and DI3 on L calculated from theexact equation (6) and approximate formulas (14c) and (15) for n = 1.336, l = 4.15 mm, and Dcor =43 diopters. The graphs show that the deviation of the results of calculations by the proposed formula (14c)from formula (6) is considerably smaller than by the regressive formula (15). The results in the form ofthe differences DI2 − DI1 and DI3 − DI1 are more illustrative. Figure 3 shows them for three values ofthe optical power of the cornea Dcor = 38, 43, and 48 diopters. It can be seen from the figure that, forDcor = 43 diopters, the differences DI2−DI1 and DI3−DI1 corresponding to formulas (14c) and (15) are inthe intervals −1.8 . . . 0 and −4.2 . . . +1.7 diopters. For the other values of Dcor, the interval corresponding toformula (14c) is virtually unchanged (−1.8 . . . +0.1 diopters) and that for formula (15) increases substantially(to −6.2 . . . +3.9 diopters). The results for Dcor = 38 and 48 diopters are practically the same in the firstcase (formula (14c))and strongly disagree in the second case (formula (15)).

We also compared the calculations of the differences DI2 − DI1 by formulas (14b) and (14c) for thesame values of the optical power of the cornea but for several values of the anterior chamber depth (l = 3,4, 4.5 mm). The results for formula (14b) with the dependence on l are slightly better (the interval ofthe difference DI2 − DI1 is from −2.8 to +0.7 diopters) than for formula (14c) (interval from −2.9 to+1.8 diopters). The scatter of reading depends on the eye length. Near L = 24 mm, it is approximatelyequal to ±0.5 diopters in the first case and ±1.5 diopters in the second. Therefore, the optical power of theIOL in a wide range of Dcor and l can be more accurately calculated by formula (14b).



Table 1

No. Parameters Maximum values Average values Minimum values

1 −a 1,46 +9.6% 1.33 1.16 −13 %

2 Dcor, L, l 48, —, 4.8 — 43, —, 4.15 38, —, 2.5 —

3 −b 5.09 +50% 3.39 2.23 −34 %

4 Dcor, L, l —, 21, 4.8 — —, 24, 4.15 —, 27, 2.5 —

5 c 3.64 +135% 1.55 0.14 −91 %

6 Dcor, L, l 38, 21, 4.8 — 43, 24, 4.15 48, 27, 2.5 —

The differences DI3 − DI1 for the regressive formula (15) for changing l were not calculated becauseeven for an average value of l = 4.15 mm, the deviations from the exact formula are very large (−6.2 . . .+3.9 diopters).

It should be noted that the coefficients a, b, and c are defined for the average parameters Dcor, L, and l.It is shown in [6] how the values of a, b, and c change if one of the parameters changes (l = 3 and 5 mm,Dcor = 36 and 50 diopters, L = 21 and 28 mm) and the rest remain equal to the average values givenabove. The obtained coefficient a depends slightly on Dcor (±5%) and more significantly on l (∓7%). Thecoefficient b varies considerably depending on L (±33%) and slightly less on l (∓15%). The coefficients aand b are independent of L and Dcor, respectively, which also follows from formula (12). The coefficient cdepends largely on L (±77%) and slightly on Dcor (±45%) and l (∓13%).

In this paper, we study the dependences of a, b, and c for all mutual combinations of Dcor, L, and l inthe intervals l = 2.5–4.8 mm, Dcor = 38–48 diopters, and L = 21–27 mm. Table 1 shows the average valuescorresponding to formula (14b), the maximum and minimum values and deviations of the last values (4thand 7th columns) from the averages in percentage. The 2nd, 4th, and 6th rows contain the values of Dcor, L, l(Dcor in diopters and L and l in millimeters) at which these deviations are observed. The parameters thatdo not affect the coefficients a, b, and c are marked in the rows of Dcor, L,, and l by dashes. The tableshows that, for some combinations of Dcor, L, and l, the coefficients vary considerably: b changes by 34–50%, i. e., by a factor of approximately 1.5, and c changes by 91–135%, i. e., by a factor of about 2. Thisimplies that the values of the coefficients in Eqs. (14b) and (14c) remain within limited ranges of variationof the eye parameters. This should be taken into account when calculating the optical power of the IOL byapproximate formulas. To improve the calculation accuracy for large deviations of the eye parameters relativeto the average values, we should calculate the new coefficients a, b, and c in accordance with formula (12).

ANALYTICAL CALCULATION OF THE PhIOL

The optical power of a PhIOL (DPhIOL can be represented as the difference of the optical powers of theIOL for the emmetropic eye (on the basis of formula (6))

Dem = DPhIOL + DIOL =n

L− l− n

n/Dcor − l, (16)

and ametropic DIOL with a postoperative refractive correction (formula (4) in the form)

DPhIOL = Dem −DIOL =n

n/(Dcon + Dcor)− l− n

n/Dcor − l;

DPhIOL =Dconn2

[n− l(Dcon + Dcor)](n− lDcor).

(17)

Substituting Eq. (1) into system (17), we obtain the following expressions of the optical power of the PhIOL(hereinafter DPh1, 2, 3) in terms of the optical power of the spectacle glass Ds:

DPh1 =n

n/(1/(1/Ds − d) + Dcor)− l− n

n/Dcor − l;

DPh1 =Dsn

2

(1− dDs)(n− lDcor)2 − lDs(n− lDcor).

(18)


196 LENKOVA

DERIVING AN APPROXIMATE FORMULA OF THE OPTICAL POWER OF PhIOLs

As noted above, IOL calculations are simplified if we use contact correction data, i. e., the optical powerof the spectacle glass reduced to the cornea. The same applies to PhIOLs. In the present study, the opticalpower of PhIOLs is calculated by an approximate formula obtained from Eqs. (17) by dividing the numeratorby the denominator:

DPhIOL = Dcon + 2DconDcorl/n. (19)

Substituting n = 1.336 into Eq. (19), we obtain

DPh2 = Dcon + 1.5DconDcorl. (20)

It was recommended [1] to calculate anterior chamber PhIOLs using the regressive approximate formulaproposed by D. T. Azar and A. C. M. Wong, which, as for conventional IOLs, was been derived on the basisof processing the results of multiple implantations:

DPh3 = 1.06Dcon + DconDcorl. (21)

For posterior chamber negative PhIOLs, Zuev [1] proposed the formula DPhIOL = Dcon − P , where P is acorrection that depends on the eye length L and the anterior chamber depth l. The value of P is determinedby the nomogram. Introduction of the correction for L is unclear, as in the formula for calculating the opticalpower of the PhIOL (18), obtained on the basis of geometrical optics, L is absent.

To determine the range of possible values of DPhIOL and compare formulas (20) and (21), we analyze therelation between the optical powers of the spectacle glass Ds (subjective refraction), the contact lens Dcon,and the phakic lens DPh1 (18). Figure 4 shows the dependences of Dcon and DPh1 on Ds calculated byformulas (1) and (18) for n = 1.336, l = 4.15 mm, and d = 12 mm (d is the distance from the surface of thespectacle lens facing the eye to the cornea top, although the main plane can be at a distance of 2 to 3 mmfrom the lens) and Dcor = 43 diopters. The figure shows that DPh1 is slightly smaller than Dcon and close toDs in the negative range (Ds < 0), and it substantially exceeds Dcon and Ds in the positive range (Ds > 0).Starting from Ds of order of +15 diopters, DPh1 is approximately two times greater than Ds. Becausefabricating high-diopter (DPh1 ∼ 30 diopters) intraocular lenses is problematic, it does not make sense toconsider the range Ds > +15 diopters. In domestic and foreign catalogs, the optical powers of refractivephakic lenses are in the range DPh1 = −25 . . . +15 diopters. In view of Eq. (18) with the above-mentionedparameters and Fig. 4 (curve sl1), this interval corresponds to Ds = −26.5 . . . +9 diopters. It should benoted that the marking on PhIOL is unclear as it differs from the values of Ds, which is especially noticeablefor hyperopia correction (in the range Ds > 0).

Figure 5 shows the deviations of the optical powers DPh2 and DPh3 calculated according to the approx-imate formulas (20) and (21) from the values DPh1 determined by the exact formula (18) for n = 1.336,

1

2

Ds, diopters

Dco

n, D

ph1,

dio

pter

s

-30 -20 0-10 2010 30

-15

15

0

30

45

60

-30

Fig. 4. Dependences of the optical power of the phakic DPh1 (curve 1) and contact Dcon (curve 2)lenses on the optical power of the spectacle glasses Ds.



1

2

Ds, diopters

Dph

2 _

Dph

1, D

ph3 _

Dph

1, d

iopt

ers

-30 20-20 -10 0 10

-3

-2

-1

0

1

-4

Fig. 5. Dependences of the deviations DPh2 −DPh1 (curve 1) and DPh3 −DPh1 (curve 2) on theoptical power of the spectacle glass Ds.

Table 2

Ds, D l, mm DPh2 −DPh1, diopters(min(Ds); max(Ds))

DPh3 −DPh1, diopters(min(Ds); max(Ds))

Data from the graphs of Fig. 5

−25 . . . +15 4.15 −2.9(+15); +0.2(−8) −3.4(+15); +0.5(−12)

−25 . . . +9 4.15 −1.2(+9); +0.2(−8) −1.4(+9); +0.5(−12)

Myopia

−25 . . . −52.5 . . . 3.5 −0.6(−25); +0.2(−9) −0.8(−25); +0.4(−13)

4 . . . 4.8 −0.6(−25); +0.5(−11) −0.3(−24); +1.2(−19)

Hypermetropia

+5 . . . +152.5 . . . 3.5 −2.4(+15); −0.2(+6) −2.9(+15); −0.1(+6)

4 . . . 4.8 −4.5(+15); −0.4(+5) −5.4(+15); −0.4(+5)

+5 . . . +92.5 . . . 3.5 −1.0(+9); −0.2(+6) −1.2(+9); −0.1(+6)

4 . . . 4.8 −1.8(+9); −0.4(+6) −2.4(+9); −0.4(+6)

l = 4.15 mm and Dcor = 43 diopters. The graphs show that the calculations by the proposed formula (20)are less different from those by the exact formula (18) than the results of calculation by formula (21). Thedeviations DPh2 −DPh1 and DPh3 −DPh1 corresponding to formulas (20) and (21) lie in the interval from−2.9 to +0.2 diopters and from −3.4 to +0.5 diopters in the range Ds = −25 . . . +15 diopters and in theinterval of −1.2 2 to +0.2 diopters and from −1.4 to +0.5 diopters for Ds = −25 . . . +9 diopters.

The calculations by formulas (20) and (21) were performed in broadened ranges of the parameters: Dcor =38 . . . 48 diopters, l1 = 2.5 . . . 3.5 mm (anterior chamber PhIOLs ) and l2 = 4 . . . 4.8 mm (posterior chamberPhIOLs) for myopia (Ds = −25 . . . −5 diopters) and hyperopia (Ds = +5 . . . +15 diopters) correction. Theselected ranges of l were the same as in [2]. The results of the calculations are shown in Table 2 as theminimum (min) and maximum (max) values of the differences DPh2 −DPh1 and DPh3 −DPh1, which arethe deviations between the optical powers of the PhIOLs calculated by Eqs. (20) and (21) and the exactformula (18). Next to the deviation values in parentheses are the values of Ds for which these deviations areobserved. The maximum deviations correspond mainly to the largest values of Ds in the interval studied.The table shows that almost always the results of calculations by the proposed formula (20) differ from thecalculations by the exact formula (18) far less than the results of calculations by formula (21). Nevertheless,in the positive range, the deviations DPh2−DPh1 reach −4.5 diopters and, even if the range of Ds is limitedto the value of +9 diopters, they are −1.8 diopters.

We attempted to specify the approximate formula (20) separately for anterior chamber (l1 = 2.5–3.5 mm)and posterior chamber (l2 = 4–4.8 mm ) PhIOLs. The formula coefficients were chosen so that, forDcor = 43 diopters and average l1 = 3 and l2 = 4.4 mm, the values of DPh2 were minimally differentfrom DPh1 in the entire range of Ds. The obtained formulas (20a), (20b) and (20c), (20d) for calculatingthe negative and positive PhIOLs, respectively, and the results of calculating the deviations DPh2 −DPh1


198 LENKOVA

Table 3

Formula for calculating DPh2 l, mm DPh2 −DPh1, diopters(min; max)

DPh3 −DPh1, diopters(min; max)

Ds = −25 . . . −5 D (myopia)

DPh2 = −0.4 + 0.96Dcon + 1.5DconDcorl (20a)(Fig. 6)

2.5 . . . 3.5 −0.2; +0.2 −0.8; +0.4

4 . . . 4.8 −0.2; +0.6 −0.3; +1.2

DPh2 = −0.5 + 0.96Dcon + 1.5DconDcorl (20b) 4 . . . 4.8 −0.3; +0.5 −0.3; +1.2

Ds = +5 . . . +15 D (hyperopia)

DPh2 = −0.28 + 1.11Dcon + 1.5DconDcorl (20c)(Fig. 7)

2.5 . . . 3.5 −0.7; +0.6 −2.9; −0.1

DPh2 = −0.5 + 1.21Dcon + 1.5DconDcorl (20d) 4 . . . 4.8 −1.1; +1 −5.4; −0.4

Ds = +5 . . . +9 D (hyperopia)

DPh2 = −0.28 + 1.11Dcon + 1.5DconDcorl (20c) 2.5 . . . 3.5 −0.2; +0.4 −1.2; −0.1

DPh2 = −0.5 + 1.21Dcon + 1.5DconDcorl (20d) 4 . . . 4.8 −0.2; +0.7 −2.4; −0.4

Ds, diopters

Dph

2 _

Dph

1, d

iopt

ers

-27 -3-23 -19 -15 -11 -7

-0.2

-0.1

0

0.2

0.3

-0.3

Dph

3 _

Dph

1, d

iopt

ers

-0.4

-0.6

-0.2

0

0.4

0.10.2

0.6

-0.8

(a) (b)

Ds, diopters-27 -3-23 -19 -15 -11 -7

Fig. 6. Deviations of calculations by approximate and exact formulas for a negative anterior chamberPhIOL (Ds = −5 . . . −25 D) versus Ds for l = 2.5 mm (three solid lines), l = 3 mm (three solidlines with a marker), l = 3.5 mm (three dashed lines) and for Dcor = 38, 43 and 48 D (the values ofDcor correspond successively to the three lines of the same form in the direction from top to bottom):(a) DPh2−DPh1 (difference of calculations by formulas (20a) and (18)), (b) DPh3−DPh1 (by formulas(21) and (18)).

Ds, diopters

Dph

2 _

Dph

1, d

iopt

ers

4 166 8 10 12 14

-0.4

0

0.4

0.8

-0.8

Dph

3 _

Dph

1, d

iopt

ers

-2.5

-2.0

-1.5

-0.5

-1.0

0

-3.0

(a) (b)

Ds, diopters4 166 8 10 12 14

Fig. 7. Deviations of calculations by the approximate and exact formulas for a positive anterior chamberPhIOL (the curve notationis the same as in Fig. 6): (a) DPh2 − DPh1 (difference of calculations byformulas (20c) and (18)), (b) DPh3 −DPh1 (by formulas (21) and (18)).



(between (20a)–(20d) and (18)) and DPh3−DPh1 (between (21) and (18)) are shown in Table 3. The choiceof the starting value of the range Ds = ±5 diopters for all cases was due to the fact that DPhIOL practicallyequals Ds at |Ds| < 5 diopters.

Table 3 shows that formula (20b) obtained for posterior chamber negative PhIOLs (l2 = 4–4.8 mm) givespractically the same differences DPh2 −DPh1 as those predicted by formula (20a) for the anterior chamberPhIOLs (l1 = 2.5–3.5 mm). Therefore, calculating PhIOLs for myopia correction can be carried out inalmost the entire range l = 2.5–4.8 mm by formula (20a). The results of calculations by formulas (20a) and( 20b) deviate from those by the exact formula by not more than 0.6 diopters (see column 3), which is twotimes smaller than by the regressive formula (21) (see column 4). The specified formulas (20c) and (20d)for hyperopia correction (Ds = +5 . . . +15 diopters) do not give a discrepancy with the exact formula lessthan 0.7 and 1.1 diopters in the absolute value (see column 3) near the values Ds = +13 and +15 diopters.In the same range, for formula (21), the discrepancy is −2.9 . . .− 5.4 diopters (see column 4). The results ofcalculation of the differences DPh2 −DPh1 and DPh3 −DPh1 corresponding to the proposed formulas (20a)and (20b) and the regressive formula (21) (see Table 3) are shown in Figs. 6 and 7.

If we confine ourselves to the interval DPhIOL = −25 . . . +15 diopters, set for domestic and foreignphakic refractive lenses, which corresponds to Ds = −26.5 . . . +9 diopters, the discrepancies with formula(18) reduce. They do not exceed −0.2 . . . +0.7 diopters for the proposed formulas (20c) and (20d) and liewithin −2.4 . . . −0.1 diopter for the regressive formula (21) (see Table 3 and Figs. 6 and 7).

CONCLUSIONS

In this paper, the advantage of correcting eye refraction anomalies (nearsightedness and farsightedness)by intraocular (phakic, i.e., without removal of the crystalline lens) refractive lenses over other kinds ofcorrection was considered. The relations between the optical powers of the spectacle glassDs, contact Dcon

and phakic DPh lenses was analyzed, which showed that the value of DPh1 is close to Ds in the negative(myopia) range (Ds < 0) and considerably greater than Ds in the positive (hyperopia) range. The intervalof Ds in the positive range was limited to the value of +15 diopters, at which DPh1 is approximately twotimes greater than Ds.

New, more accurate approximate formulas for calculation of the optical power of IOLs and PhIOLs with alinear dependence on the eye parameters were derived. The results of calculations by the proposed formulasand known regressive formulas used in the ophthalmic practice were compared. They were represented asdeviations from the calculations by exact formulas derived on the basis of geometrical optics.

It is shown that, for the average values of the eye parameters, the deviation of the optical power of theIOL calculated by the proposed formula (14b), is approximately two times smaller than that calculated fromthe regression equation (15). In the broadened range of the eye parameter variation (Dcor = 38–48 dioptersand l = 3–4.5 mm), the discrepancy with the exact formula increases. The accuracy of determining theoptical power of the IOL can be improved by introducing corrections using graphs or by calculating thecoefficients of the formula for a more limited range of the eye parameters.

The results of calculations of the optical power of the PhIOL by the proposed (20) and regressive (21)formulas were close to each other. The maximum deviation from the exact formula was ∼3 diopters. After theintroduction of corrections to the proposed formula, the accuracy of calculations increased significantly (bya factor of 2–4). Three formulas were obtained: (20a) to calculate negative (correction of myopia ) PhIOLsin the entire range l = 2.5–4.8 mm; (20c) and (20d) to calculate positive (hyperopia) anterior chamber(l = 2.5–3.5 mm) and posterior chamber (l = 4–4.8 mm), PhIOLs, respectively. The refined formulas forthe correction of hyperopia, despite improvements, do not allow one to obtain a discrepancy with the exactformula less than 0.7–1.1 diopters in the absolute value for large values of Ds (close to 13–15 diopters). If weconfine ourselves to the interval Ds = −26.5 . . . + 9 diopters corresponding to the nomenclature of domesticand foreign refractive PhIOLs, the discrepancy with formula (18) is reduced: it is reduced to 0.2–0.7 dioptersfor the proposed formulas (20c) and (20d) and to 0.1–2.4 diopters for the regression formula (21).

REFERENCES

1. V. P. Koronkevitch, G. A. Lenkova, V. P. Korol’kov, and I. A. Iskakov, “Bifocal Diffraction-Refraction IntraocularLenses,” J. Opt. Technol. 74 (12), 818-822 (2007).

2. L. I. Balashevich, Surgical Correction of Refractive and Accommodation Errors (Chelovek, Saint-Petersburg,2009) [in Russian].


200 LENKOVA

3. Ya. Grabar’, Implanted Lens, Interview by I. S. Fedorova // Women’s Health, January, 2004, http://www.wh-lady.ru/archive/?SECTION ID=284&ELEMENT ID=2741.

4. N. M. Sergienko, Intraocular Correction (Zdorov’e, Kiev, 1990) [in Russian].

5. S. N. Fedorov, A. I. Kolinko, and A. I. Kolinko, “Methods of Calculating the Optical Power of an IntraocularLens,” Vestn. Oftal’mologii, No. 4, 27–31 (1967).

6. G. A. Lenkova, “Effect of Optical Parameters of the Eye on the Choice of Refraction of Monofocal and BifocalIntraocular Lenses,” Avtometriya, No. 5, 96–102 (2001).

7. S. V. Kravkov, The Eye and Its Work (USSR Academy of Science, Moscow–Leningrad, 1950) [in Russian].

8. J. T. Holladay, “Refractive Power Calculations for Intraocular Lenses in the Phakic Eye,” Amer. J. Ophthalmol.116 (6), 63–66 (1993).

9. D. R. Sanders, J. Retzlaff, and M. C. Kraff, “Comparision of the SRK II Formula and Other Second-GenerationFormulas,” J. Cataract Refract. Surg. 14 (2), 136–141 (1988).

10. T. Olsen, K. Thim, and L. Corydon, “Theoretical Versus SRK I and SRK II Calculation of Intraocular LensPower,” J. Cataract Refract. Surg. 16 (2), 217–225 (1990).


Documents

Vision correction by intraocular lenses