A Physical Model Demonstrating Schachar's Hypothesis of Accommodation

RONALD A . SCHACHER , MD, PHD, DONALD P. CUDMORE , MS, RUSSEL TORTI , DSC , TRUMAN D. BLACK , PHD, AND TSENG HUANG , PHD (Printed in Annals of Ophthalmology January-February 1994;26:4-9) (compressed (gzip) postscript)

From the Departments of Physics and Civil Engineering, University of Texas at Arlington, Texas The authors thank G. W. H. M. van Alphen for supplying the original photographs of the profile of the unstretched and stretched human lens, which are published with permission from Permagom Press Ltd. Address for reprints: Ronald A. Schachar, MD, PhD, PO Box 796728, Dallas, TX 75379.

Scachar's hypothesis of accommodation states that there is increased zonular tension during accommodation and the observed in vivo changes in lenticular curvature that occur during accommodation are the result of zonular forces. To demonstrate that there can be steepening of the central curvature of the lens with increased zonular tension, profile photographs of an equatorially unstretched and stretched gelatin filled balloon and profile photographs from the literature of an unstretched and equatorially stretched human lens were digitized. Their radii and curvature were determined. We found that equatorial stretching of both the gelatin-filled ballon and the human lens produced central curvature steepening that was consistent with Schachar's hypothesis of accommodation.

Introduction

Schachar's 1 hypothesis of accommodation states that, during accommodation, there is increased zonular tension and that the changes in lenticular curvature can be attributed to zonular forces. This is not dependent on the topographic variation in thickness of the elastic capsule of the lens, 2 the vitreous, 3 or pressure changes between the anterior and posterior chambers.4 Presbyopia, according to Schachar's hypothesis, occurs as a result of the normal growth of the lens. The equatorial diameter of the lens increases at approximately 0.02mm/year. 5 However, the dimensions of the scleral shell in the normally emmetropic, hypermetropic, or myopic eye does not change significantly after 13 years of age. 6 The distance between the ciliary muscle and the equator of the lens decreases throughout life. 7,8 Therefore, the effective force that the ciliary muscle can apply to the lenticular equator is reduced in a linear fashion with age. 9,10 Consequently, the amplitude of accommodation decreases linearly with age, resulting in presbyopia.

Intuitively, we would expect that, by increasing zonular tension on the lenticular equator, the central radius of curvature would increase, i.e., the central surface curvature would become flatter. To demonstrate Schachar's counterintuitive hypothesis, we constructed a physical model.

Materials and Methods

Two rings of commercial grade aluminum were made with an internal diameter of 30.00cm and an outer diameter of 30.50cm. One of the rings had a thickness of 5.75mm and the other, a thickness of 13.15mm. The rings were fastened together with 16 number 2-56 stainless-steel screws. Eight evenly spaced 4.95mm holes were drilled at the junction of the two rings perpendicular to their central axis. Eight Pyrex (Corning Glass Works, Corning, NY) glass rods, each having a length of 15cm and a diameter of 5mm, were pressure fitted into these holes. A Mylar balloon (Pioneer Balloon, Wichita, KS) with a 4 inch diameter and uniform wall thickness was filled with gelatin (Knox, Englewood, NJ). The gelatin-filled balloon was clamped to the eight glass rods in the center of the aluminum ring by using small stainless-steel clamps (Figure 1). The horizontal and vertical diameters of the gelatin-filled balloon measured 8.90 and 9.00cm, respectively. All eight glass rods were manually moved outwardly so that the equatorial diameter of the balloon was stretched. The horizontal diameter increased to 9.05cm, and the vertical diameter increased to 9.18cm. A 35mm single lens reflex camera (Leica R5, Leitz, Lisbon, Portugal) with a macrolens (MacroElmarit-R 1:2.8/60, Leltz, Weltzlar, Germany) was positioned to take vertical profile pictures of the balloon before (Figure 2) and after equatorial stretching (Figure 3). The profile pictures were uniformly enlarged to ensure equal magnification to 8 by 10 inches.

In addition, van Alphen and Graebel 11 stretched a human lens equatorially in onc meridian and took profile pictures of the unstretched and stretched lens (Figure 4). We enlarged their photographs to 3.75 by 7 inches. The profile images of the gelatin filled balloon and human lens were scanned into a computer using a 1200-dot/inch scanner (Silverscan, La Cle, Beverton, OR). The profile images of the gelatin-filled balloon and the human lens were then magnified 175% and 300%, respectively. The contrast of the computer images was adjusted (Adobe Photoshop version 2.5, Mountain View, CA). The profile surfaces were converted to curved lines (Aldus Free Hand version 3. 1, Seattle, WA) and digitized (Digimatic, FEB Software, Richmond, VA). The digitized curves were found to best fit a fifth-degree polynomial (Figures 5 and 6). Using the computer (Mathematica version 2. 1, Wolfram Research, Champaign, IL), we determined the equations for each of the curves to be:

f(x) 1  = -0.914954 - 0.013686x + 0.000164145x 2  + 4.13774 * 10 -7 x 4  + 6.94863 * 10 -10 X 5  for the front profile of the unstretched balloon

f(x) 2  = -0.611803 - 0.0499991x + 0.00148316x 2  - 0.0000248132X 3  + 2.03334 * 10 -7 x 4 - 4.7457

* 10 -10 x 5  for the front profile of the equatorially stretched balloon

f(x) 3  = 21.0461 + 0.271607x - 0.00309546x 2   - 0.0000191544x 3  + 4.62617 * 10 -7 x 4  - 2.56143 * 10 -9 x 5  for the back profile of the unstretched balloon

f(x) 4  = 20.88 + 0.307185x - 0.00473147x 2  + 3.40799 * 10 -6 x 3  + 4.45935 * 10 -7 x 4  - 3.49152 * 10 -

9 x 5  for the back profile of the equatorially stretched balloon

f(x) 5  = 14.4407 + 0.163197x - 0.00380538x 2  + 0.0000932946x 3  - 1.51645 * 10 -6 x 4  + 6.83752 * 10 -9 x 5  for the anterior profile of the unstretched human lens

f(x) 6 = 12.7434 + 0.163719x - 0.000867987x 2  - 0.0000675304x 3  + 7.26234 * 10 -7 x 4  - 2.07047 * 10 -9 x 5  for the anterior profile of the equatorially stretched human lens

f(x) 7  = -13.8437 - 0.335635x + 0.000691177x 2  + 0.000126986x 3  - 1.68873 * 10 -6 x 4  + 7.69612 * 10 -10 x 5  for the posterior profile of the unstretched human lens

f(x) 8  = -14.8756 - 0.35376x + 0.00359075x 2  + 0.0000433018x 3  - 3.50395 * 10 -7 x 4  - 4.91635 * 10 -10 x 5  for the posterior profile of the equatorially stretched human lens

The  radius  of  curvature,  p,  of  each  surface  was  determined  by  using  the  Serret-Frenet formula. 12,13

p = [1 + {dx/dy} 2  ]  3/2   /  | d 2  x/dy 2  |

Figures

Results

The gelatin-filled balloon visually and mathematically demonstrated that, during equatorial stretching, the central radius of curvature decreased, i.e., became steeper (Figure 7). The analysis of the stretched human lens revealed the following.
  1. The anterior central radius of curvature decreased, i.e., became steeper.
  2. The peripheral anterior radius of curvature increased, i.e., became flatter.
  3. There was essentially no change in the posterior radius of curvature (Figure 8).
Equatorial stretching produced a dramatic change in the anterior radius of curvature of the human lens; there was essentially no change in the posterior radius of curvature. Similar changes in lenticular curvature have been observed in vivo in patients accommodation. 14.

Conclusions

Helmholtz's hypothesis of accommodation and its modifications, 2-4,15-19 which currently is accepted universally, states that, during accommodation, zonular tension and the equatorial diameter of the lens decrease. This reduced zonular tension allows the elastic capsule of the lens to contract, causing an increase in the anteroposterior diameter of the lens (the lens becomes more spheric) and resulting in an increase in the optical power of the lens. The central anterior radius of curvature of the lens decreases more than does the central posterior radius of curvature. This difference in change between the anterior and posterior curvatures is explained by the topographic difference in the thickness of the lens capsule. Helmholtz's hypothesis and its modifications state that presbyopia occurs as a result of the loss of elasticity of the lens capsule and/or sclerosis of the lens with age; when the zonutes are relaxed, the lens does not change its shape. According to Helmholtz's hypothesis and its modifications, presbyopia could only be reversed by changing the elasticity of the lens or its capsule.

Tscherning 20 postulated that there was increased zonular tension during accommodation; however, by contrast with Schachar's hypothesis, Tscherning attributed the changes in the shape of the lens to the vitreous and presbyopia to enlargement of the lenticular nucleus. Therefore, according to Tscherning, presbyopia could only be reversed by reducing the size of the nucleus of the lens.

Our observed changes in curvature, which occur during equatorial stretching of the gelatin-filled balloon and the human lens, clearly demonstrate Schachar's hypothesis of accommodation. Interestingly, in 1896, Stradfeltll equatorially stretched a human lens in one meridian and observed that the central anterior radius of curvature decreased along that meridian with the Javal ophthalmometer. Equatorial stretching can account for the observed in vivo changes of curvature that occur during accommodation without implicating topographic variation in the thickness of the lenticular capsule, the vitreous, or pressure changes in the anterior or posterior chambers of the eye.

Our earlier findings included

  1. the small displacement qualitative mathematic model of increased zonular tension that predicts the curvature changes in the lens that occur during accommodation, 22
  2. the increase in optical power of the bovine lens that occurs with equatorial stretching, 23 and
  3. the clinical finding that expansion of the sclera in the region of the ciliary muscle can reverse presbyopia. 1
Coupled with our present observations, these seriously undermine Helmholtz's hypothesis of accommodation and its modifications.

References

1. Schachar RA: Cause and treatment of presbyopia with a method for increasing the amplitude of accommodation.  Ann Ophthalmol 1992;24:445-452.

2. Fincham EF: The mechanism of accommodation.  Br J. Ophthalmol  1937:8(suppl):5-80.

3. Coleman DJ: Unified model for accommodation mechanism. Am J Ophthalmol 1970:69:1063-1079.

4.  Hill L: Blood vessels and pressure.  Lancet 1920:1:359-366.

5. Rafferty NS: Structure, function and pathology, in Maisel H (ed): The 0cular Lens.  New York, Marcel Dekker, 1985: 2-5.

6. Duke-Eider S, Waybar KC: The anatomy of the visual system, in  Duke-Eider  S  (ed):  System  of Ophthalmology.   London, Henry Kimpton, 1961; vol.2:80-81.

7. Famsworth  PN,  Shyne  SE:  Anterior zonular shifts with  age. Exp Eye Res   1979;28:291-297.

8. Koretz JF, Kaufman PL, Neider MW, et al: Accommodation and presbyopia in the human eye-Aging of the anterior segment.  Vision Res 1989;29:1685-1692.

9. Gordon AR, Siegman MJ: Mechanical properties of smooth muscle: 1. Length-tensions and force-velocity relations.  Ami J Physiol 1971;15:1243-1249.

10. van Alphen GWHM, Robinette BS, Macri FJ: Drug effects on ciliary muscle and choroid preparations in vitro.  Arch Ophthalmol 1962;68:111-123.

11 I.vanAlphenGWHM,GraebelWP:Elasticity of tissues involved in accommodation. Vision Res 1991;31:1417-1438.

12. Hay GE: Vector and Tensor Analysis.  New York, Dover, 1953: 53-57.

13. Leithold L: The Calculus With Analytic Geometrv, 2nd ed.  New York, Harper and Row, 1972; 788-795.

14, Koretz JF: Accommodation and presbyopia, in Albert DM, JakobiecFA(eds):Principles and Practice of Ophthalmology, Basic Sciences.  Philadelphia, WB Saunders, 1994; 270-282

15. von Fielmholtz: Physiological Optics.  New York, Dover, 1962; vol. 1: 143-172, 375-415.

16. Fisher RFJ: Presbyopia and the changes with age in the human crystalline lens, J Physiol (Lond) 1973;228:765-779.

17. Stuhlnian O: An Introduction to Biophysics.   New York, John Wiley and Sons, 1948: 106-107.

18. Weale RA: New light on old eyes.  Nature 1963; 198:944-946.

19. Rohen JW: Scanning electron microscopic studies of the zonular apparatus in human and monkey eyes. Invest Ophthalmol Vis Sci. 1979; 18:133-144.

20. Tscherning M: Physiological Optics. Philadelphia, Keystone, 1904; 160-189.

21. Stratfeld AE: Die veranderung der linse bei traction der zonula.Kiin Moiiatshl Aiigenheilkd 1896@34:429-43 1.

22. Schachar RA, Huang T, Huang X: Mathematic proof of Schachar's hypothesis of accommodation.  Ann Ophthalmol 1993;25:5-9.

23 Schachar RA, Cudmore DP, Black TD: Experimental support for Schachar's hypothesis of accommodation.  Ann Ophthalmol 1993;25:404-409.