2.1.

Two-optics System

For a relaxed (un-accommodated) eye after the insertion of an AIOL, the emmetropic state is described by an equation based on Gaussian optics^{7, 8, 11}

In above equations, we have used the MKS units: X and S in mm, C and P in diopter (or 1/m). 1336 is from the refractive index of the aqueous (n = 1.336) converted to the MKS units. We have also defined dS > 0 (dS < 0) for axial movement toward (backward) to the cornea. The power reduction factor (Z) has a typical value of Z = 0.84 for S = 5.0 mm, C = 43 diopters.

From above equations, it may be realized that the accommodation rate function (M) is proportional to the AIOL power (P). However, it should be noted that M is governed by the combined effect of the 5 ocular parameters (C,P,X,S,L) and related by the emmetropic state Eq. 1. Therefore calculating the M value will require 4 of these 5 parameters and subject to Eq. 1. Examples are show as follows: (i) for L = 23.8, S = 5.0 mm, M = (1.1, 1.62) (D/mm) for and AIOL power of P = (17, 25.6)D and the associate corneal power of C = (45, 39)D, calculated from Eq. 1; (ii) for fixed C = 43D, L = 23.6 mm, we found M = (1.08, 1.36) (D/mm), for various S = (2.0, 6.0) mm. More details will be shown later.

2.2.

Three-optics System

We shall now extend the above two-optics formulas to a three-optics system as shown in Fig. 1. The dual-optics AIOL is much more complex than the single-optics AIOL due to the fact that either the front or the back optics can be mobile and the system overall power is influenced by multiple ocular parameters. In order to manipulate this complex system, we first define the AIOL effective total power (P) by its front and back optics power P ₁ and P ₂,

Using Eq. 3 and the revised X and S, the derivative of Eq. 1 with respect to the lens separation (s), we obtain, after some length but straight forward derivation, the accommodation rate due to the movement of the front (M ₁) and back (M ₂) optics as follows

Alternatively, Eq. 5 may be derived by the following simple argument for a deeper physics insight. In the revised X and S, the dual-optics AIOL reduced to one-optics. The net power change (P/net) due to the front optics forward-movement may be found by the power change due to both optics [TeX:] $(P_1 + P_2)$ $(P_1+P_2)$ forward-movement minus the power change due to the backward-movement of the back-optics ( [TeX:] $P_2$ $P_2$ ) and the interaction term (sB). Mathematically, above statement is given by [TeX:] $P/net = (P_1 + P_2)({\it sM}) - P_2 ({\it sM}) - {\it sB}$ $P/net=(P_1+P_2)\left(\mathit{sM}\right)-P_2\left(\mathit{sM}\right)-\mathit{sB}$ . Therefore, one may easily find M₁ given by M ₁ = (P/net)/s = [TeX:] $g^{\prime \prime} M - B$ $g^{\prime \prime }M-B$ , which is an approximate express of Eq. 4a having an error about 2%.

2.3.

Total Accommodation Amplitude (A)

Formula given by Eq. 5 is for the case that only one of the dual-optics is mobile. In general, both optics of the AIOL are allowed to move in response to the ciliary body contraction and could be in either forward or backward directions. The total accommodation amplitude (A) may be expressed by

2.4.

Exact Numerical Solution

The analytic equations of Mj(j = 1,2) shown by Eq. 5, are based on a linear theory assuming a linear accommodation rate which is true for a small movement of the optics. To study the nonlinear effects due to large movement, we calculate the corneal power changes, based on Eq. 1, for each of the 1.0 mm increase of the lenses separation (s), but with different initial values of s. We shall use Mj (j = 1,2) = C(s = 1.0 mm) – C (at s = 0) for linear regime; and Mj = C(s = 2.0 mm) – C(s = 1.0 mm) for nonlinear regime. These numerical data will also justify the accuracy of our analytic formulas for the M function.

3.1.

Single-optics AIOL

Figure 2 shows the effect of corneal power (C) on the accommodation rate (M) for various IOL power (P) from –30 to +30 D, where we have plot the absolute values of M. We note that higher M is found for positive-IOL (for hyperopia correction) than negative-IOL (for myopia correction). This feature may be easily realized by Eq. 2 that the (2C+ZP) term has a higher value for P > 0 than for P < 0 which has a cancellation over the 2C term. Our analytic formula, Eq. 2, also indicates that M is a deceasing function of the anterior chamber depth (S), but is an increasing function of the product of corneal power and IOL power (PC). This implies that patient with flat cornea or lens, or short axial length is less efficient in AIOL comparing to a long axis eye or more curved cornea or lens. Also shown in Fig. 2 is the asymmetric feature of M (with respect to AIOL power signs P > 0 or P < 0) and the nonlinear behavior of M versus the IOL power (P).We should note that the above features demonstrated in Fig. 2 can be easily realized by our analytic formulas, Eq. 2, but not by raytracing method. In producing curves in Fig. 2, we have fixed S = 3.5 mm and X is found from the emmetropic state condition. Eq. 1 for a set of (P,C,S) parameters.

Fig. 2

Accommodation rate (M) versus IOL power (P) for various corneal powers (C).

3.2.

Dual-optics AIOL

Accommodation rate (in absolute value) for moving front-optics and moving back-optics are shown in Figs. 3 and 4, respectively, where the solid curves are the linear case based on Eq. 5 and dotted curves are the nonlinear case. We have used anterior chamber depth S ₀ = 3.5 mm and lens separation s = 2.0 mm and total power of P = 40D. These curves justify the accuracy of our analytic formulas for Mj (j = 1,2). It should be noted that the linear approximation, Eq. 5, is very accurate for positive-optics (with P1 > 0), whereas errors occur for negative-optics (with P1 < 0) particularly for high diopters. Both Figs. 3 and 4 show the asymmetric features of M versus the power signs. For front optics is mobile, M1 is higher for P ₁ > 0 than for P ₁ < 0; for example, M ₁ = 1.38 (D/mm) for P = +20 versus M ₁ = 0.83 (D/mm) for P = –20 as shown by Fig. 3. However, an opposite trend is shown in Fig. 4 for M ₂ when back-optics is mobile.

Fig. 3

Accommodation rate for mobile front-optics versus its power; solid curve is from linear Eq. 5 and dotted curve for nonlinear case.

Fig. 4

Same as Fig. 3 but for mobile back-optics.

Figures 5 and 6 show the accommodation rate for mobile front and back optics, respectively, for various IOL front-optics power and for fixed anterior chamber depth S ₀ = 3.5 mm and total AIOL power of [TeX:] $P = P_{1} + P_{2} = 20 $ $P=P_1+P_2=20$ D. We shall note that, as shown by Fig. 5, M ₁ with P ₁ = +10 D is slightly higher than P ₁ = −10 D. In addition, M ₁ = 4.5 (D/mm) for P ₁ = +30 D (with P ₂ = −10 D) with a 2.0 mm lens separation. In comparison, same M ₂ value will require back optics power of P ₂ = +40 D (with P ₁ = −20 D) when back optics is mobile, as shown in Fig. 6. M ₁ is about −2 (D/mm) for the back optics (with a negative-power of P ₂ = −10 D, when P ₁ = +30 D) moving 2 mm toward the cornea. Figure 6 also shows when P ₁ = +20 D (with P ₂ = 0, for total power P = 20 D), M ₂ = 0 as expected. It should be noted that M is defined by the change rate of accommodation amplitude (A) in the forward direction, toward the cornea. Therefore positive A may be achieved for a presbyopia to see near by either a forward movement of a plus-IOL, or a backward movement (toward the retina) of a minus-IOL. Greater detail for the general feature of dual-optics AIOL will be discussed in the next section.

Fig. 5

Accommodation rate for moving front-optics versus the lens separation (s) for various front-optics power and for S ₀ = 3.5 mm.

Fig. 6

Same as Figure 5 but for back optics is mobile.

3.3.

Important Features

Many of the important features readily available from the analytic formulas of Eqs. 5, 5 are further discussed as follows.

3.4.

Applications

The M function (for the single-optics AIOL) may be used to find the piggy-back IOL power (P') placed at a distance from the corneal plan [TeX:] $d=(S_{0}-s$ $d=(S_0-s$ ) by the following relation

Another application of the M function is to estimate the error caused by the uncertainty of the IOL position. It was known that it is difficult to implant an IOL at a preset position, in which the refractive error could be estimated by Eq. 1 for a set of ocular parameters. For example, for an IOL power P = 20 D, the refractive error caused by 1.0 mm mis-position is about 1.3 to 1.7 D depending on the corneal power of 38 to 48 D, as shown by Fig. 2, or by Eq. 2.

In conclusion, we have derived analytic formulas for the accommodation rate function (Mj) for both two-optics and three-optics systems. In the dual-optics AIOL, the total accommodation amplitude (A) has a maximum when the front positive-optics moving forward to the corneal plan and the back negative-optics moving backward. Our analytic formulas precisely predict that greater accommodative rate may be achieved by using a positive-powered front optics. This general feature is true for either front or back optics is mobile and is consistent with that of raytracing method. The complex nonlinear features of three-optics system is explicitly formulated by a geometric factor (g) and the lens “interaction” term (B). These features are difficult, if not possible, to be predicted by raytracing method. The M function may be used to find the piggy-back IOL power (P ^′) for customized design based on the individual ocular parameters.