Conformal geometry of the retinal nerve fiber layer

Abstract

The nerve fiber layer of the human retina is made up of the retinal segments of ganglion cell axons. Its geometry can be described mathematically as a fibration of a 2D domain: a partition of a certain region into smooth curves. Here, we present a simple family of curves that closely models the observed geometry of the nerve fiber layer. For each retina, the pattern depends on 2 parameters, A and B: A computer program determines A and B for a given retina and the theory matches the retina with a standard deviation of Ϸ6 -8°. These particular curves turn out to be the curves that would be generated if the growing ganglion cell axon tip moved down a gradient toward a source of diffusible neuroattractant at the disk and away from a weaker macular diffusible repellant. Thus, this model provides morphological evidence that diffusible substances provide positional information to the embryonic ganglion cell axons in finding their way to the optic nerve head.diffusion ͉ embryology ͉ optic nerve ͉ retina T he human optic nerve consists of bundles of axons originating in the ganglion cells of the retina. These axons first traverse the retina in a typically unmyelinated layer called the nerve fiber layer (NFL), before converging at the optic disk and, subsequently, emerging from the globe in the optic nerve. Thus, the detailed anatomy of the optic nerve head depends crucially upon the axon paths in the NFL. (Fig. 1A) The NFL was first described in 1914 by Vogt (1). Because of its transparency, the NFL is difficult to visualize and was not photographed for another fifty years (2). An improved method for wide angle, conformal (angle-preserving) photography was later devised (3).The fascicles of the NFL have the following characteristics:Y Macula-avoiding. The ganglion cells are displaced radially from the macula and their axons radiate from the macula, so that no axons cross over the macula. Y Horizontal raphe. The direction of nerve fiber varies continuously throughout the retina except along the horizontal raphe, the ray originating at the macula directed away from the nerve. No fibers cross the raphe, which is a sort of part or watershed in the retinal fibration. Axons just superior to the raphe are directed superiorly, while neighboring fiber just inferior abruptly take the opposite direction. Y Noncrossing. When axons from distal cells pass over proximal fibers, they follow the same path as the more proximal fibers, but on a more inner retinal layer. Thus, the fibers passing over any point determine a unique direction (excluding, of course, the macula and horizontal raphe-the two areas without overlying fibers). Vogt's article (1) overlooked this noncrossing feature.Any theory of retinal pathfinding by embryonic ganglion cell axons should account for the three features detailed above. Crick (4) made plausible the proposal of Ramon y Cajal (5) that positional information in embryology is specified by gradients of ''morphogens,'' diffusible substances acting as attractants or repellants-in this case, for the growing gangl...

Optimizing Structure–Function Relationship by Maximizing Correspondence Between Glaucomatous Visual Fields and Mathematical Retinal Nerve Fiber Models

“…Several of these mathematical RNFL models exist. 11,12,14 However, the complexity of these models may be very different. For example, an axial model is very simple but quickly becomes inaccurate with increasing distance from the optic nerve head (ONH), while a model with many parameters may be able to largely capture normal morphologic variation between different eyes.…”

“…Airaksinen et al 11 optimized the shape parameters A and B on observed nerve fiber bundle patterns in fundus photographs.…”

“…A more detailed description and interpretation of a, b, and the parameters A and B can be found elsewhere. 11 In Equation 1, the value of W selects a specific nerve fiber bundle. The angle a * at which this nerve fiber bundle meets the ONH can be derived from Equation 1 by setting y to 0 and b to p=2, yielding…”

“…The model used in this paper describes the path of the nerve fiber bundles by a single equation that contains two parameters 11 :…”

“…Previously, the parameter optimization was done by first tracing nerve fibers in fundus photographs and then selecting the shape parameters that resulted in the best match between observed and modeled nerve fibers. 11,12 This is a very timeconsuming approach and its accuracy may depend on the operator. 13 In this paper, we introduce a method to optimize mathematic RNFL models based on visual field data of moderately glaucomatous eyes, thereby establishing a relation between structural and functional measurements.…”