George H. Handelman

Analysis of human crystalline lens curvature as a function of accommodative state and age

Slit-lamp photographs from four human subjects, aged 11, 19, 29, and 45 were reanalyzed using computer-based digitization and curve-fitting methods in order to obtain more complete information on internal lens curvature changes during accommodation. All discernible curves (N = 742) could be fit to parabolas with chi 2 less than or equal to 0.001 irrespective of lens age, accommodative state, or curve location within the lens. For each lens, the coefficients of the parabolas, when displayed in graphic form, exhibit a linear relationship between location within the lens and the coefficient of the chi 2 term. The slope of this line remains unchanged over accommodation for a given lens, but is shifted in position. The slope changes as a function of age. The age 45 lens exhibits these characteristics to a limited extent only, the differences possibly related to the development of presbyopia. The further a given curve is located from the lens surface, the smaller the region of its arc that can be considered approximately circular. A roughly hourglass figure is generated by these circular bounds; the waist of the hourglass decreases with increasing accommodation, since changes in radius of curvature with accommodation are more pronounced internally. Calculations of arc lengths as a function of increasing accommodation indicate that these lengths change very little over the entire accommodative range.

Model of the accommodative mechanism in the human eye

The crystalline lens of the age 11 human eye has been modelled mathematically, using simplified assumptions about lens curvature, internal organization and elasticity. From this representation, expressions for description of strain and stress during accommodation have been obtained. Solution of these equations indicates that the lens capsule acts as a force distributor, spreading tension applied by the suspensory apparatus evenly over the surface of the underlying lens material. It also becomes clear that the vitreous body provides an essential support function during the accommodative process. Finally, the relative contribution of lens-associated structures has been determined for five different values of the Poisson ratio. In order for accommodation to occur by relaxation of zonular tension, this value must be greater than 0.38; with an additional constraint of the net axial force equalling zero during a small accommodative change, the Poisson ratio equals 0.46.

A model for accommodation in the young human eye: The effects of lens elastic anisotropy on the mechanism

An explanation of the mechanism of visual accommodation depends heavily on understanding the mechanical properties of the lens, as well as the way in which its shape is altered in small accommodative changes. An initial attempt to relate these properties to a mechanism has already been performed (Koretz and Handelman, 1982) for the young (age 11 yr) human lens, using certain simplifying assumptions (spherical curvature on the anterior lens surface and elastic isotropy). However, since it has been shown that the lens behaves as an anisotropic body, the previous treatment has been extended to include the variation of lens elastic properties in the polar and radial directions. With this modified representation, it is found that only one combination of elastic constants is consistent with the generally accepted qualitative theory of accommodation and with clinical data on the accommodative range of the emmetropic age 11 human eye. For this unique solution of the equations, however, the general mechanism already suggested by us, which includes support by the vitreous and alteration of the magnitude and angle of application of zonular force with accommodation, remains little changed.

Modeling age-related accomodative loss in the human eye

Abstract A simplified mathematical representation of accommodation—the process by which the eye focuses on near objects—is described. This model is based upon assumptions which limit its applicability to the young human eye, but it nevertheless leads to new insights about the physiological mechanism. A hypothesis about the underlying cause(s) of presbyopia—the loss of accommodative amplitude with age—is also presented here, which combines aspects of the expanded accommodative mechanism with the aging characteristics of the human lens. Using a similar mathematical strategy which incorporates new experimental data about lens shape and aging, a more general model of the accommodative mechanism can be developed and used to test the presbyopia hypothesis.

Vibrations of rotating beams with tip mass

ZusammenfassungEs werden die zur Rotationsebene normalen Biegeschwingungen eines rotierenden Stabes mit Endmasse studiert; insbesondere wird im Anschluss an eine frühere Arbeit, die sich auf die Grundschwingung beschränkte, der Einfluss der Endmasse bei grösseren Winkelgeschwindigkeiten auf die höheren Eigenschwingungen untersucht. Dabei werden Störungsmethoden und asymptotische Näherungen in Verbindung mit den klassischen Verfahren vonRayleigh-Ritz undSouthwell angewendet. Für Stäbe veränderlichen Querschnittes wird nachgewiesen, dass für Eigenschwingungen genügend hoher Ordnung und hinreichend grosse Winkelgeschwindigkeiten die Eigenfrequenzen durch die Endmasse erhöht werden. Für Stäbe konstanten Querschnittes werden numerische Schranken angegeben, welche dieses Resultat bereits für die erste Oberschwingung beweisen. Dieses Verhalten steht im Gegensatz zu demjenigen der Grundschwingung, deren Eigenfrequenz durch eine Endmasse stets erniedrigt wird.

Sturm—Liouville Inverse Eigenvalue Problems

Summary Starting with a second-order Sturm–Liouville problem in canonical form with separated boundary conditions, the inverse question which arises when the lowest n eigenvalues and corresponding normalization constants are replaced by different values is considered. It is shown that using the known eigenfunctions and the solutions to an initial value problem, the new first n eigenfunctions and the new potential are found by solving a system of non-singular, linear, algebraic equations. The new separated, homogeneous boundary conditions and the remaining eigenfunctions are found. These eigenfunctions are shown to form a complete system.