Computer monitoring retina curvatures

Age-related macular degeneration is a leading cause of vision loss among the elderly. Retina tissue inflammation and detachment are manifestations of the eye disease. These morphological alterations result in changes of surface curvatures. The purpose of this paper is to provide mathematical requirements for software that determines retina curvatures. An emerging algorithm will evaluate the curvatures of tissue surface segmented by optical coherence tomography. Examinations may detect pathogenesis and enable computer monitoring of progress made in treatment. Inflammation of tissue is positioned within a Riemannian manifold of dimension two which is embedded in a Euclidean space of dimension three. A net of nonorthogonal coordinate curved lines covers the manifold, while location of points within the surrounding Euclidean space reference an orthogonal Cartesian system. Magnitudes of and the angles formed by covariant and contravariant base vectors are found. Scalar products of covariant bases result in the metric of the first fundamental quadratic form related to element of arc length. The scalar product of a differential normal vector into the differential surface vector gives the second fundamental quadratic form whose metric coefficients are triple scalar products. Normal curvature of the space curve is inferred from perpendicularity of normal and surface unit vectors. At every point of the tissue surface there exist two mutually orthogonal directions in which the normal curvature assumes extreme values. These numerical quantities are characterized by roots of a quadratic equation. Their sum yields the mean curvature, while the product implies the Gaussian curvature, at a point. A necessary and sufficient condition that a surface be isometric with the Euclidean plane is that the Gaussian curvature be identically zero. What is known of the biochemical processes on the molecular level and implicated in the affliction are reviewed first. Mathematical results are the- derived. Current and future capabilities of optical coherence tomography are discussed last.