Understanding Angle Alpha & Angle Kappa- the finer aspects

There has been a lot of talk about angle kappa and angle alpha in the past two decades. Therefore I do not want to turn this article into another write up of what is angle kappa and angle alpha. It is well known that while angle kappa is the difference between pupillary axis and visual axis, angle alpha is the difference between the optical axis and the visual axis. Thus in both, visual axis is common.

Visual axis is the line that is drawn from the fixation target to the first nodal point of eye, and thereafter from the second nodal point of eye to the fovea. However, the angles of the eye are actually theoretical. There is no instrument in ophthalmology that can accurately determine the following:

Visual axis: visual axis cannot be determined by any clinical instrument, except the synaptophore. Thus angle kappa could be measured only with the synaptophore. All other machines are approximations.

Fig 1 Optical axis of a lens is an imaginary line joining the center of curvature of two refracting surfaces

Optical axis of eye and optical axis of cornea: By definition optical axis of an artificial lens is an imaginary line that passes through the center of the two refracting surfaces(Fig1). It is also called the Principal axis. A ray of light passing through this surface remains undeviated as it always passes perpendicular to surfaces.

However, in the eye the geometric center of the cornea (center of limbus) may not be the optical center of cornea. Even if we assume that geometric center is the optical center, it will still be difficult to exactly locate the geometric center of the cornea, as cornea is not a perfect sphere. Thus the optical axis of cornea is often substituted by a line perpendicular to the cornea plane and joining the center of the pupil (Fig2). The center of the pupil in itself is not constant, right? It varies with different lighting condition, as much as with age. So yes, there is a lot of assumption and simplification.

Fig 2 quickguide.org

That was about optical axis of cornea. The optical axis of eye will be the line that is perpendicular to the plane of cornea and intersects the center of pupil (optical axis of cornea) that extends and goes through the nodal point of eye and reach the retina. There are again several assumptions here. Even if we theoretically locate the optical axis of the human eye in this way, such a line does not reach the fovea. Thus this theoretical line that travels through the optical axis of cornea, nodal points, reach the retina slightly nasal to the fovea. In other words, the fovea lies temporal to the assumed optical axis of eye.

All cardinal points like the Principal Point, nodal points, centre of rotation of eye, primary and secondary focal points will then fall on this line. However, in reality such points do not coincide in the eye.

For multifocal IOLs centration plays a critical part. The surgeon may ask the patient to look at the coaxial light of the microscope in the operation theatre, and align the purkinje reflex from lens (P3) to that of cornea(P1). However, if major decentrations are there, the lens may take its natural position with capsular contraction in the capsular bag.

Thus if optical axis of eye is a theoretical concept, angle alpha, which is the difference between the optical axis and visual axis is difficult to measure in a clinical set up. Angle alpha is determined at the nodal point of eye. The primary nodal point is 7.039 mm posterior to corneal vertex and the secondary is 7.321 (1). Angle alpha as well as angle kappa could be measured by a Synoptophore which most practices do not have, today.

Fig 3: Optical biometry machines like IOL Master and Lenstar provide chord distances as XY coordinates

Chord mu and CW-Chord (Fig 3 highlighted in yellow) are alternative terminologies that have been proposed to work around the difficulty of obtaining angles alpha and kappa in a practice. The optical biometers report CHORD MU, as defined by Dan Chang and George Waring IV, which is the chord (straight line that joins end of two points) from the Vertex Normal Light Reflex (Purkinje 1) and the center of the pupil. The angle is actually Kappa. Chord mu is a distance between Purkinje 1 (light that reflex back from anterior cornea) and the pupil center, and are images at the pupillary plane, 4.0 mm posterior to the corneal vertex (and not at the corneal plane). Chord mu is reported on the IOL Master and LenStar as Px, Pcx and Py, Pcy. They are chords/distances and not angles. One must know all of the dimensions of the eye to determine the equivalent angle and the same chord gives different angles for different eye. In text books the nominal values of 1 mm = 7.5 degrees = 15 Prism diopters, but these are just averages, so Chord Mu is used directly(2).

Thus chords are in millimeters (because they signify distances) and not in angles, and is referred to as pupil barycenter (in Lenstar) for an equivalent of angle kappa and Iris barycenter for an equivalent of angle alpha. In the below the Ix and Iy signify the distance of between the coaxially sighted light reflex and the limbal centre. The CW chord below is the value of the displacement calculated as per Pythagoras theorem as the barycentric difference of the coaxially sighted light reflex at the pupillay plane (regarded as visual axis) from the centre of the pupil (4). Thus both parameters are showing decentrations on the horizontal and vertical plane, but not the lateral tilt in angles.

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