Edwin J. Sarver, PhD
Introduction
Accommodation allows the phakic eye to change focus from distant objects to closer objects. During accommodation, an interaction between the ciliary muscle and the crystalline lens allows the required optical power change to take place. In a pseudophakic eye, which is an eye implanted with an IOL following cataract surgery, certain IOLs are designed to provide accommodation as well.
For the purposes of this tutorial, there are two primary types of accommodating IOLs-those comprising a single lens and those with two lenses. This accommodation is achieved by altering lens position. In the single lens case for distance vision, rays from a distant object focus at the retina (Slide 1). As the lens moves forward, accommodation is provided, and rays from a near object will now focus at the retina (Slide 2). The dual lens case for distance vision is illustrated in Slide 3. Just as for the single lens case, as the anterior lens moves forward, accommodation is provided, and rays from a near object are focused on the retina (Slide 4).
Lens Effectivity
The optical behavior mentioned above can be explained via the thin lens optics concept of effective power. For a fixed image plane, moving a lens along the optical axis alters its effective power in the optical system. Moving a lens away from the image plane effectively increases positive power or decreases negative power. For example, a positive lens that is too weak to focus distant objects onto the image plane is illustrated in Slide 5. To bring the distant rays into focus, either the lens can be moved to the left, as shown in Slide 6, or the power of the lens can be increased, as shown in Slide 7. The following two cases show the mathematical relationship between these two alternatives:
| (1) | |
| (2) |
The effective power relation in equation (2) demonstrates how accommodation can be achieved in the pseudophakic eye where the ciliary muscle contraction and relaxation translates the position of the IOL.
Definitions
Three related accommodation terms are defined as follows.
- Pseudophakic accommodation: the dynamic range, in diopters, of the refractive state of the pseudophakic eye due to interactions of the ciliary muscle and the IOL.
- Pseudophakic pseudoaccommodation: the range of focusing error, in diopters, which does not result in objectionable deterioration in retinal image quality for a static eye.
- Effective pseudophakic accommodation: the sum of pseudophakic accommodation and pseudophakic pseudoaccommodation.
Pseudophakic accommodation
In the following, the effective change in power if an IOL translates forward 1 mm is now investigated. The paraxial pseudophakic accommodation power equation for a single lens IOL is given by
| (3) |
where
n = index of the eye times 1000 (1336)
K = corneal power (D)
P = IOL power (D)
elp = expected lens position in unaccommodated state (mm)
d = distance IOL is translated, positive to the right (mm)
Equation (3) is similar to other equations entered in IOL power calculations. It is now possible to compare the accommodation values predicted by this equation to those given in recent publications and find that there is close agreement. In Table 1, values for the paraxial pseudophakic accommodation power equation and those given by Nawa et al1 and Langenbucher et al2 are presented.
Table 1. Comparison of the paraxial pseudophakic accommodation power equation and results published by Nawa et al and Langenbucher et al. The lens translation d is fixed at 1 mm.
| K | P | elp | al | A | Nawa | Langenbucher |
| 43.83 | 30 | 4.7 | 21 | 2.2 | 2.3 | 2.2 |
| 43.83 | 20 | 6.0 | 24 | 1.3 | 1.3 | 1.3 |
| 43.83 | 11 | 8.1 | 27 | 0.6 | 0.8 | 0.6 |
The small difference of 0.2 D between the value obtained by equation (3) and Langenbucher et al and that reported by Nawa et al in the last row and next to the last column, is likely due to a difference in the optical model assumed for the eye. In an effort to simplify the paraxial pseudophakic accommodation power equation to make it more usable, we can look at how sensitive the calculations are to a range of values for each parameter. If the IOL power and elp values are kept constant and the corneal power is varied, the values in Table 2 are obtained.
Table 2. Comparison of the paraxial pseudophakic accommodation power equation for a range of corneal powers from 40 D to 50 D while keeping P and elp fixed at 20 D and 5 mm, respectively. The lens translation d is fixed at 1 mm.
| K | P | elp | al | A |
| 40 | 20 | 5 | 25 | 1.2 |
| 45 | 20 | 5 | 23 | 1.3 |
| 50 | 20 | 5 | 21.4 | 1.4 |
If the corneal power and elp are fixed at 44 D and 5 mm, respectively, and P varies from 10 D to 30 D, we have the results indicated in Table 3.
Table 3. Comparison of the paraxial pseudophakic accommodation power equation for a range of P from 10 D to 30 D while keeping K and elp fixed at 44 D and 5 mm, respectively. The lens translation d is fixed at 1.0 mm.
| K | P | elp | al | A |
| 44 | 10 | 5 | 26.3 | 0.6 |
| 44 | 20 | 5 | 23.4 | 1.3 |
| 44 | 30 | 5 | 21.2 | 2.1 |
Finally, K and P are fixed at 44 D and 20 D, respectively, allowing elp to vary from 3 mm to 7 mm, and the results in Table 4 to be obtained.
Table 4. Comparison of the paraxial pseudophakic accommodation power equation for a range of elp from 3 mm to 7 mm while keeping K and P fixed at 44 D and 20 D, respectively. The lens translation d is fixed at 1.0 mm.
| K | P | elp | al | A |
| 44 | 20 | 3 | 22.4 | 1.4 |
| 44 | 20 | 5 | 23.4 | 1.3 |
| 44 | 20 | 7 | 24.3 | 1.2 |
By comparing Tables 2 through 4, one can see that the paraxial pseudophakic accommodation power equation is not sensitive to K or elp values, but is comparatively sensitive to P. These observations allow the derivation of the approximate paraxial pseudophakic accommodation power equation given by
| (4) |
where
P = IOL power (D)
d = distance IOL is translated, positive to the right (mm)
If the distance translated is restricted to 1 mm, then the accommodation power change is about 7% of the IOL power.
Pseudophakic pseudoaccommodation
Effective pseudophakic accommodation is the sum of pseudophakic accommodation and pseudophakic pseudoaccommodation. Up to this point, the author has provided basic analysis to calculate pseudophakic accommodation. Attention is now turned to pseudophakic pseudoaccommodation. Atchison and Smith3 define the depth of field as the vergence range of focusing error, which does not result in objectionable deterioration in retina image quality. To illustrate this definition, suppose we have some tolerable blur defined as a circle at the retina (Slide 8). Given this tolerable blur circle, the near object point and far object point that just intersect the circle are found. The distance between these two points is the depth of field. The depth of field is dependent upon pupil size as shown in Slide 9. The depth of field is also due to aberrations of all orders. In Slide 10, a through focus point spread function (PSF) for an 8-mm pupil is shown. The PSF planes are located at the image plane and ± 1 mm.
For a well-corrected eye, the PSF at the image plane is compact, indicating clear vision. At the ± 1-mm planes, the PSF for the well-corrected eye has a lot of energy in the off-axis regions, indicating poor vision. For an eye with typical amounts of spherical aberration, the PSF at the image plane is not as well focused as the well-corrected eye, but at the ± 1-mm planes, the eye with spherical aberration is much better focused than the well-corrected eye. The trade off is aberrations increase depth of field at the expense of decreased best focus performance. The depth of field has experimentally been determined for a large number of subjects. The mean depth of field after age 50 is about 1.5 D.4,5 Using this value, the approximate paraxial pseudophakic accommodation power equation can be updated to include a component for pseudophakic pseudoaccommodation giving:
| (5) |
Samalonis6 reported accommodation values in the range of 2 D to 3 D using an accommodation IOL. Our approximate effective pseudophakic accommodation equation gives 2.6 D for typical lens movement and typical IOL power, indicating the equation is consistent with clinical observations.
For higher accuracy prediction of effective pseudophakic accommodation, one would include the individual eye's actual measurements of, for example, corneal surface aberrations, the exact location of the lens, and the eye's axial length. The optical model of the accommodating IOL and its expected movement during accommodation should also be considered along with pupil size dependence.
Conclusion
There are two primary types of accommodating IOLs-the single and the dual lens. Accommodation in these lenses is achieved by lens translation. This translation modifies the effective power in the optical system. The depth of field describes the range of acceptable objects for a static optical system. The total effective pseudophakic accommodation amplitude is a combination of pseudophakic accommodation (also called paraxial accommodation power, the dynamic part) and pseudophakic pseudoaccommodation (also called depth of field, the static part).
References
- Nawa Y, Ueda T, Nakatsuka M, et al. Accommodation obtained per 1.0 mm forward movement of a posterior chamber intraocular lens. J Cataract Refract Surg. 2003; 29:2069-2072.
- Langenbucher A, Reese S, Jakob C, Seitz B. Pseudophakic accommodation with translation lenses-dual optic vs. mono optic. Ophthal Physiol Opt. 2004; 24:450-457.
- Atchison DA, Smith G. Optics of the Human Eye. Oxford, U.K.: Butterworth-Heinemann; 2000:213.
- Hamasaki D, Ong J, Marg E. The amplitude of accommodation in presbyopia. Am J Optom Arch Am Acad Optom. 1956; 33:3-14.
- Sun F, Stark L, Nguyen A, et al. Changes in accommodation with age: static and dynamic. Am J Optom Physiol Opt. 1988, 65, 492-498.
- Samalonis LB. Accommodative IOLs coming a long way. EyeWorld. 2002; 3:18.