Ординатура / Офтальмология / Английские материалы / Hyperopia and Presbyopia_Tsubota, Boxer Wachler, Azar_2003

Table 1 The Prescription of the Modified Navarro Eye for Modeling the Relationship Between Refilled Lens Volume and Ametropia Correction

tion to the anatomical and geometrical parameters of the crystalline lens. This lens model was based on combining two half-ellipsoids of revolution, as employed by past workers (27,32). The anterior and posterior radii of curvature of the lens were the same as those of the Navarro eye. By setting the length of the major axes (perpendicular to the optical axis) of the half-ellipsoids to be identical, the continuity of the lens surface at the equator was ensured. We chose 200 L as a reasonable nominal initial volume of the model lens to simulate the natural human lens (33).

Given the assumed curvatures, equatorial diameter, and lens volume, the asphericity and half-thickness of each half-ellipsoid were calculated employing equations relating to the apical radius of curvature and shape factors of conic sections (34).

Finally, the model eye was “emmetropized” by adjusting the vitreous chamber depth (distance between posterior lens surface and retina).

The resultant prescription of the modified Navarro model eye for analysis of controlling refilled lens volume is given in Table 1. The volume of this lens model is 208 L.

The equatorial diameter of 8.8 mm is slightly less while the thickness of 5.12 mm is slightly greater than the respective parameters for the typical adult lens (28). However, this was necessitated by a compromise in providing reasonable optical and geometrical properties to the model.

c. Refilling Model

No information is available in the literature about the quantitative relationship between lens curvatures and lens volume. Hence, a validated model of the change in lens curvature with refilling is not possible at this stage. In view of this paucity of information, we developed two simple but plausible mathematical models for lens refilling. These were:

1.Model 1: “spherization”

2.Model 2: proportional expansion

These two models (Fig. 5) provide contrasting relationships between lens thickness and curvature with increasing lens volume during refill. In general, Model 1 predicts that the anterior curvature and half-thickness of the lens will change more quickly than the posterior curvature and half-thickness as the lens refills during Phaco-Ersatz, while Model 2 predicts the converse. The intention of testing two such disparate models is to provide a “bracketing” of the results, such that the actual life situation might lie somewhere in between.

Model 1: “Spherization.”

This model assumes that as the lens is filled and then overfilled; it converges toward a sphere (i.e., the length of the major and minor axes at the endpoint of filling is the same). Hence, the posterior and anterior curvatures would converge with overfilling. Given that the anterior radius of curvature is greater at the “normal” volume, this model predicts that with overfilling, the anterior curvature would change more rapidly than the posterior curvature. We assumed that the lens equator expands slightly with overfilling.

Model 1 is represented mathematically as follows (Fig. 5). The anterior and posterior half-lenses are represented by half-ellipsoids of revolution such that their two-dimensional cross sections may be described by

where x is the distance along the optical axis

y is the distance across the optical axis

a is the half-length of the minor axis representing the half-thickness of the lens-half at normal volume

b is the half-length of the major axis representing the half-diameter of the lens at its equator at normal volume

where a′ is the half-thickness of the over-or underfilled half-lens.

b is the half-diameter of the over-or underfilled lens.

be is the radius of the endpoint sphere towards which the shape of an overfilled lens will converge.

s is a scaling factor defining the amount of over-or underfilling (s 0 is normal volume of filling, s 0 is overfilling, and s 0 is underfilling).

Model 2: “Proportional Expansion.”

Model 2 assumes that as the lens is filled and then overfilled, the posterior and anterior half-ellipsoids increase in axial dimensions (i.e., the length of the minor axis) in the same ratio. In contrast to Model 1, the posterior curvature in Model 2 would increase more rapidly than the anterior curvature as the lens overfills. As in Model 1, we assumed that the lens equator expands slightly with overfilling.

Model 2 is represented mathematically as follows (Fig. 5):

The anterior and posterior half-lenses are again represented by half-ellipsoids of revolution according to Eq (1). During lens refilling, Model 2 defines the following changes in lens shape:

where nomenclatures are as for the previous equations and subscript a values pertaining

to the anterior half-lens; subscript p values for the posterior half-lens. For both models, we assumed be to be 4.9 mm.

5. COMPUTATION AND ANALYSES

a. Calculation of Volume of Over-Underfilling

For a lens constructed of two half-ellipsoids of revolution, the volume V can be calculated as

From Eqs. (1) through (4) and assigning various values for scaling factor s, we can calculate the volume of the de novo lens at various amounts of filling.

From Eqs. (1) through (3) and (5a) and (5b), the assigned refractive indices, and assigning various values for the scaling factor s, the power of the de novo lens can be calculated at various amounts of filling.

c. Modeling Correctable Ametropia

Armed with the calculated lens curvatures and thicknesses at different levels of lens refilling, it is now possible to calculate the associated amount of ametropia that is correctable within the two models using the modified Navarro eye model.

One final variable had to be fixed—that of the position of the lens at different levels of filling. Because of the enormous complexity of the interactions between capsular tension, lens volume, vitreous influence, zonular tension and iris influence as well as the lack of precise, quantitative information in many of these factors, we assumed a simplified model in which the equatorial plane of the lens remains fixed in the axial (x) direction at all levels of filling. Under this assumption, the anterior and posterior lens surfaces would bulge forward and backward, respectively, by an amount equal to the change in the length of the minor axes of the respective half-ellipsoids.

The amount of ametropia correctable was computed in two ways—assuming refractive ametropia and assuming axial ametropia.

Paraxial optical equations were used in all calculations. Dioptric values of ametropia were referred to the plane of the cornea on the model eye.

6. Results

Figure 6 shows the relationship between the volume of lens refilled (expressed as a percentage of a lens with normal volume) and equivalent refractive power and thickness of the resultant lens. The resultant lens thicknesses are not greatly different for Models 1 and 2.

The refractive power of the refilled lens varied slightly more with change in refilled volume for Model 1 than for Model 2. This suggests that should lens refilling follow the shape and thickness changes predicted by Model 1, a greater amount of ametropia may be correctable.

From the results shown in Fig. 6, the overall power of the eye with a refilled lens, and hence the amount of ametropia correctable, was computed. This is shown in Fig. 7, which plots the amount of ametropia that is correctable under Models 1 and 2 for both axial and refractive ametropia. The maximum ametropia that is correctable occurs using Model 1 with axial ametropia (approximately 4 D). The minimum ametropia that is correctable occurs using Model 2 with axial ametropia.

7. Discussion

a. Eye Model

As noted, the geometrical dimensions of the lens model deviated from typical adult human lenses in that the equatorial diameter was slightly less and the thickness slightly higher. This compromise was necessary to enable simple ellipsoids of revolutions to be used to construct a model lens while maintaining reasonable values for the optics and geometrical parameters. A more mathematically sophisticated approach (35) has been developed to provide an anatomically precise representation of the adult human lens. However, that model employed a pair of parametric functions within a system of polar coordinates to describe the lens. Given the parametric nature of the model description, there is no direct method by which changes in volume and curvature may be modeled without a high level of mathematical complexity. Hence for convenience, we adopted the half-ellipsoids of revolution (27,32).

The assumed endpoint diameter of the lens was 8.9 mm, which represents a 0.1- mm increase in the diameter of a lens when overfilled to 200% (double the volume). Should this prove to be an overestimation, there would be an increase in the amount of ametropia that is correctable with controlled lens refilling due to the expected increase in curvatures.

This model is able to estimate only the amount of ametropia that is correctable by controlling the refilled volume of a lens with Phaco-Ersatz. Due to a number of limitations relating to quantitative understanding of the influence of lens volume on lens shape during accommodation, it is not yet possible to study the effect of overor underfilling on the amplitude of accommodation, as has been done for the refractive index strategy.

Hence, there may be detrimental effects on the amplitude of accommodation in the application of this strategy, which are yet to be determined.

The critical assumption in our lens model is that the lens capsule has a negligible effect on the change in curvature and thickness during refilling. Intuitively, this would not be the case, given that there is a difference in thickness between the anterior and posterior capsules (21,36,37), which would influence lens shape and thickness during refilling. However, we have introduced two contrasting models for refilling with the intention that the actual lens response would lie somewhere within this “bracketing.” We believe, therefore, that the robustness of the predictions within our model should be reasonable, especially considering that the two contrasting refilling models returned similar predictions.

Finally, there are effects on aberrations that have not been considered. First, the gradient index of the natural lens has been eliminated by Phaco-Ersatz. This would alter

the aberration state of the eye at both distance and, as discussed, more critically at near. Further, as the lens shape changes, it is highly likely that its surface asphericity would also change, thereby altering the aberrations at different refill volumes. These considerations are beyond the scope of the current model but merit further investigation.

b. Correction of Ametropia

If we accept the above assumptions and exceptions discussed, then our theoretical analysis with the eye model predicts the following:

Even at the highest estimate predicted, the amount of ametropia that is correctable is too low to be practical (Model 1 and axial ametropia). Therefore controlling the refilled volume alone during Phaco-Ersatz for correcting ametropia is insufficient to correct any other than low degrees of ametropia.

The reciprocal of the average slope of the four curves in Fig. 7 indicates that in order to correct ametropia with an accuracy of 0.125 D, percentage refilling would have to be controlled to an accuracy of 2.3%. This equates to an approximate accuracy in volume of 5 L. This may pose a technical and surgical challenge.

c. Issues of Implementation

There are also issues of implementation relating to this strategy.

Figure 8 shows the predicted positions of the lens surfaces and retina with changes in refilled lens volume assuming a fixed position for the equatorial plane. Over the range of under-to overfilling analyzed (50 to 150%), the lens thickness changes from approximately 3.5 mm to approximately 8 mm. Even ignoring the effect of the iris and vitreous, it is certain that such a range would impose impractical values to the resultant anterior chamber depth. In particular, correction of medium to high hypermetropia would result in dangerously shallow anterior chambers.

In addition to considerations of anterior chamber depth, there are other practical limits to the amount of over-and underfilling that can be achieved. The minimum amount of underfilling is set by the lowest volume that can still produce an undistorted, optically useful de novo lens. Below this limit, prism due to sagging, and distortions due to rippling, warping, or waviness as a result of a lack of sufficient capsule tension may degrade vision below acceptable limits.

There is also an upper limit on the amount of overfilling. Beyond this limit, given the finite breaking strain of the capsule (38,39), rupture of the capsule would occur.

While reliable data on the limits of optical imperfection with excessive underfilling and capsule rupture with excessive overfilling do not exist, a reasonable estimation may be assumed on these lower and upper limits. Experience in our laboratories, which has been conducting surgical trials of Phaco-Ersatz, suggests that a “safe” limit for overand underfilling may be 20% of the normal volume. Referring back to Fig. 7, introduction of this limit to the volume of refilling predicts that the range of ametropia that is correctable is significantly reduced to approximately 2 D. This is probably not a viable range for useful correction of ametropia.

In addition to the above, other implementation issues relevant to the first strategy as listed in the previous section also apply. These include a limitation in the range of correction to spherical (nonastigmatic) refractive errors.

8. Summary

Due to the complexity of the interplay between physiology, mechanics, and optics, we have been able to test only a simplified model of controlled refilling. However, even this first approximation suggests that the range of corrections achievable within this strategy would be small because of the limitations of lens thickness as well as implementation difficulties.

9. Conclusions

In this chapter, we have analyzed, using theoretical modeling, the feasibility of two strategies that are intrinsic to Phaco-Ersatz for simultaneous correction of ametropia. It appears from the results and consideration of implementation issues that neither strategy on its own is sufficient or feasible for simultaneous correction of ametropia within Phaco-Ersatz.

We have not investigated the feasibility of a combination of the two strategies. However, in view of the above discussions, it may be expected that those combinations would also lack sufficient range and accuracy for applicability.

We therefore conclude, even in the absence of physical experimental results, that simultaneous correction using these two intrinsic strategies would be unattractive and probably not feasible.

However, predictions from models are only as reliable as the assumptions made and the values for parameters assumed. We recognize that there are potential shortcomings in our theoretical analyses, which warrant further research. In particular, the lack of reliable, quantitative knowledge of relationships such as lens shape and thickness with different level of refilling—as well as the effect of the capsule thickness, vitreous, and iris on lens shape and position—merits study as well in order to refine our models. A number of studies are under way in our laboratories seeking to address such issues (13,30,31). We believe that researchers in this area should persist in their efforts to understand those relationships.

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