ACCOMMODATIVE IOL

Accommodating IOLs (AIOL) including single-optics and two-optics configurations have been designed and analyzed by many researchers.^1-11 The accommodation rate (M) defined by the accommodation amplitude per 1.0 mm of the IOL axial movement has been studied both

Fig. 15.2: The conversion function (CF) for plus and minus IOL in 3-optics system for 3 cases: (a) d’=1.0, (b) d’=2.0, (c) d’=3.0

Fig. 15.3: The power difference Delta=P-Po where 3-optics IOL-power (P) is higher than 2-optics (Po)

Fig. 15.4: Delta vs. the separation (d’) between piggyback and primary-IOL for (a) Po=+10.0 D and (b) Po=-10.0 D

Table 15.4: Typical values of ELP, A-constant and surgeon’s factor(s) for different IOL implants IOL Implant ELP (mm) A-constant s-factor Anterior chamber lens 2.8 – 3.1 115.0 – 115.3 -0.7 to -0.4 Iris-supported lens 3.3 – 3.5 115.5 – 115.7 -0.1 to +0.1 Posterior chamber lens

in the sulcus 3.7 – 4.1 115.9 – 117.2 +0.1 to +0.7 in the bag 4.3 – 5.1 117.5 – 118.8 +0.9 to +1.6

analytically by Gaussian optics and numerically by raytracing method.^2,3,5 Single-optics AIOL is much simpler than the dual-optics AIOL which has been analyzed recently by Ho et al using raytracing method. However, analytic formulas are not yet available (Fig. 15.5)

This Chapter presents and derives analytic formulas for dual-optics AIOL, in which either or both the front

Fig. 15.5: Accommodation rate M (per 1.0 mm of forward movement) versus IOL-power at various axial length L=(22, 24, 26) mm

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and back optics are allowed to move in both forward and backward directions. Calculated results are consistent with that of raytracing method used by Ho et al. However, analytic formulas provide more insight features which are not readily available in raytracing.

Two-optics Formulas (Lin, 2005) The accommodation amplitude (A)

A = M(dS), (8.a)

M = (Z²P)(2/X – P/1336), (8.b) Z = 1 – S(Dc/1336), (8.c)

where the rate function M (in diopter/mm) is governed by four parameters the corneal power Dc, the IOL power P, X and S as defined earlier. Note that A>0 (for near vision accommodation) when dS>0 defined by a movement toward the cornea. The values of M ranging (0.5 to 1.9) depends on ocular conditions. Examples are shown as follows.

Case (1): fixed X = 18.8 mm and S = 5.0 mm, (or L = X + S

= 23.8 mm).

M = (1.1, 1.29, 1.45, 1.62) (diopter/mm) for corneal power of Dc = (45, 43, 41, 39 ) diopter and the required IOL-power for emmetropia P = (17, 20, 22.4, 25.6) diopter. It shows that higher IOL power, in general, produces higher accommodation for a given amount of anterior movement (dS).

Case (2): fixed corneal power Dc = 43 diopter and axial length L = 23.8 mm.

M = (1.08, 1.14, 1.2, 1.29, 1.36) (diopter/mm), for S=(2.0, 3.0, 4.0, 5.0, 6.0) mm. It shows that M is an increasing function of the IOL position, where X decreases when S increases for a fixed axial length L = X+S.

Case (3): fixed IOL-power P = 20 diopter and L=23.8 mm.

M = (1.36, 1.34, 1.32, 1.29, 1.25) (diopter/mm), for S = (2, 3, 4, 5, 6) mm.

The decrease of M for larger S is the net result of the competing factors from the decrease of X = L – S, and the emmetropic condition which requires a higher corneal power for a given IOL power.

Case (4): fixed Dc = 43 diopter and S = 5.0 mm

M = (0.56, 0.82, 1.22, 1.87) (diopter/mm), for IOL-power P = (9.5, 12.4, 19.1, 27.3) diopter and axial length L = (27, 26, 24, 22) mm calculated from L = X + S. It shows that, for a given corneal power, longer eye with lower IOL power result in a smaller M, in consistent with raytracing method of Nawa.

Case (5): fixed P = 20 diopter, S = 5.0 mm.

M = (1.38, 1.26, 1.19) for L = (22, 24, 26) mm and X = (17, 19, 21) mm.

Case (6): fixed Dc = 43 diopter and S = 5.0 mm (with L varies for emmetropic state).

M = (0.6, 0.93, 1.29, 1.68), for P = (10, 15, 20, 25) diopters.

Three-optics Formulas (Lin. 2006) Mobile Front-optics Case

As shown in Figure 15.6, a doublet IOL (dual-optics) consists of 2 optics separated by p’, where the front optic separates from the corneal vertex surface by a distance p.

M = Z (D₁/1336)(2Dc+ZD₁)- Delta,(9.a)

Delta = Z²(D₁D₂/1336), (9.b)

Z = 1-S(Dc/1336), (9.c)

where S (in the unit of mm) being the effective anterior chamber depth (ACD) of the IOL; D₁, D₂ and Dc are, respectively, the front and back IOL power and corneal power.

The accommodation rate (M function) is almost linearly proportional to the moving front IOL-power (D₁).

M is positive for positive-power moving optics, that is an increase of p’ (or dp’>0 and dp<0) or a forward movement to the cornea is needed in order to have increasing accommodation (or myopic shift) for near vision of a presbyopia. These features are consistent with the numerical raytracing method of Rana et al and Ho et al.

Fig. 15.6: Dual optics AIOL

(showing separations of p and p’) (see text) EXAMPLE # 1

M is a decreasing function of S, or the corneal-IOL separation (p). For example, given Dc = 43 diopter, p’ = 1.0 mm, and the configuration with (D₁, D₂) = (+30, -10) diopter, g = –0.5, M = (2.39, 2.35, 2.29), for p = (2.5, 3.0, 3.5) mm or S = (2.0, 2.5, 3.0) mm, respectively.

EXAMPLE # 2

A positive-power would have a greater M (absolute value) than that of negative-power moving optics. For example,

Dc = 43 D, S = 3.5 mm, Z₂= 1.0 and Z = 0.887; the calculated M = +0.99 (D/mm) for D₁ = +15 D and M = –0.724 (D/mm) for D₁ = –15 D, which is about 27% smaller (comparing the absolute values). This novel feature is consistent with that of Ho et al using raytracing method.

EXAMPLE # 3

For a given IOL total power of Dt = D₁+Z₁D₂, with Z₁ = 1-p’(D₁/1336), the M value for front positive and back-negative IOL is always larger than that of both optics are positive. This feature may be readily observed by re-expressing the moving front-optics power D₁=Dt-ZD₂ has a higher value when D2 < 0.

Mobile Back-optics Case

By symmetry feature of the dual-optics IOL, Lin (2006) also derived the case for mobile-back (with immobile-front) to obtain

M’ = Z(D₂/1336)(2Dc+ZD₂) – Delta. (10) Similar to the mobile front-optics, the M’ value is proportional to the power of the mobile optics and slightly influenced by the immobile optics, via the Delta term.

Both Optics Mobile

In order to have a “positive” accommodation (for myopic shift), forward movement (toward the cornea) is needed if the moving optics has a positive power. In contrast, a backward movement is needed for a negative power moving optics. Therefore cancellation effect may occur when both optics having opposite-power are moving in the same directions or 2 optics having the same power sign but moving in the opposite directions. The actual amount of each components, M and M’, shall depend on

the structure and implant configuration of the IOL, where contraction of the ciliary body results the direction and amount of each movement. Furthermore, one may also expect a small contribution from the non-linear coupling effect of M and M’, when both optics are allowed to shift in either direction. Therefore, the total (net) amount of accommodation (A) may be expressed as follows, in general,

A = M(dS1) + M’(dS2), (11.a)

= A1 + A2 (11.b)

where dS1 and dS2 are the movement amount (including directions) of the front and back optics, respectively.

Cancellation occurs when A1 and A2 have opposite signs.

Figure 15.6 shows the possible combinations for various IOL configuration. Because the sign of M and M’ are defined as the same as the power of the moving optics, one also needs to defined dS>0 for forward movement (toward the cornea), and dS<0 for backward (toward the retina) movement.

Analytic formulas are presented for dual-optics AIOL which could have the front or back-optics being the mobile element or both. The important features of the accommodation rate function (M) include:

• M is proportional to the moving optics power (including the sign).

• Positive accommodation amplitude, A=A1+A2, requires combination of plus-power optics moving forward to the cornea and minus-power optics moving backward to the retina (Fig. 15.7A).

• Convex-concave or bi-convex configuration for both front and back optics having positive power offers maximal A when both are moving forward to the cornea (Fig. 15.7B).

Fig. 15.7: Dual-optics accommodating IOL, where (←, →) stands for movement direction of (forward, backward) w.r.t the cornea position; (+, -) stands for (positive, negative) accommodation amplitude

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Dual-optics AIOL provides higher accommodation rates (M) than the 2-optics via optical configurations.

However, the combined surface aberration (SA) to compensate the balanced SA of cornea and natural lens for minimal high-order SA should also be considered in the IOL design.

DESIGN OF ASPHERICAL IOLs

It was known that the spherical aberration (SA) increases (or decreases) after myopic (or hyperopic) LASIK due to the increase (or decrease) of the corneal front surface asphericity (Q) or the shape factor (P = Q + 1).

Similar to the LASIK procedure which induces a positive SA, larger population of cataracts surgery patients show an increase of positive SA. Therefore, optimized aspheric IOLs to minimize the cataract-induced SA and desired for best image quality. Most population have eyes with a positive SA (or Q < 0) and about 10% of the population has negative SA, therefore most of the customized IOL would require a negative SA to minimize the remaining SA after the compensation from the natural lens which has a negative SA in general.

In addition, the aged-reduced positive SA must be also included as one of the factors to be considered. It was known that the natural crystalline lens has a SA of -25 um, 0 and +25 um at age 20, 40 and 60, although cornea SA remains constant throughout the whole life. When the positive SA increases in the eye, the presence of glare and night halos also increase. It was proposed that aging causes a gradient refractive index in the lens and causes a change from negative to positive SA of the lens over time. Therefore, as the eye ages, the combined positive SA of cornea and lens increase.

The need of customized IOL is to cause a perfect compensation of the positive SA of a prolate-cornea (with Q < 0) by the implanted aspheric IOL having a negative SA of the cataract patients. However, an opposite strategy will be needed for the 10% abnormal patients having negative SA after cataracts. Furthermore, for IOL implanted to phakic eyes or piggyback-IOL, the strategy for minimum SA or coma should base on a 3-optics system, in which the system is rather complex and requires minimal calculations.

Table 15.5: Formulas for spherical aberration (W) and Coddington shape factor (S)

W(total) = W (cornea) + W”(IOL)

S(lens) = (1+X)/(1-X),

S’(cornea) = (1+aX’)/(1-aX’),

SM = 1(N² – 1)/(N + 2),

S* = –1/[1-PR/(n”’ – 1.226)].

where X = surface radius ratio (R’/R)

N = refractive indes ratio of IOL (n”’) and humours (1.336).

SM = condition of S for minimum SA

P = IOL power with front surface radius R and index (n”’)

The formula in aspherical IOL using Coddington shape factor for minimum spherical aberration (SA) and coma are shown in Table 15.5.