The astigmatic cornea

The ideal corneal dome, viewed from above, is a circle. The cornea with regular astigmatism is an ellipse. The astigmatic cornea immensely complicates analysis because an ellipse has two primary meridians, one long and one short. The meridians are at right angles to one another and, superimposed on an X-Y axis, can assume various orientations in the human eye (Fig. 1).



Fig. 1: The base of a cornea with regular astigmatism is an ellipse. The long axis of the ellipse (A) at 10° is perpendicular to the short axis (B). In the otherwise normal astigmatic human eye, the long axis can be found anywhere from 1° to 180°. However, astigmatism with the long axis near 180°, called "with-the-rule" astigmatism, is considered "better"; that is, people having with-the-rule astigmatism can see better and report less handicap than people with similar degrees of against-the-rule astigmatism.

The slope of the astigmatic corneal dome on the short meridian is steeper than the slope of the corneal dome along the long meridian. One can think of the bowl of a dinner spoon, which also has short (steeper) and long (flatter) meridians. If we use the flat meridian as a reference, on an X-Y axis the flat meridian can be found, in the otherwise normal human eye, at any point between 1° and 180°.

When the flat meridian is at or near 180°, the astigmatism is said to be "with the rule." When the flat meridian is at or near 90°, the astigmatism is said to be "against the rule." With-the-rule astigmatism is more common and is said to provoke less disruptive visual effects than against-the-rule astigmatism of similar magnitude.(12,13,14)

Since regular astigmatism is symmetrical, it can be defined by its magnitude (steepness) and its meridian (axis). The range between 1° and 180° is sufficient to describe regular astigmatism because it is symmetrical; that is, 3 D of astigmatism at the 10° axis, for example, describes the same situation as 3 D of astigmatism at the 190° axis, so by convention practitioners define regular astigmatism as occupying axes between 1° and 180°.

To correct myopic astigmatism and achieve a perfect dome, the procedure must flatten the cornea as a whole to correct the myopia, but flatten the steeper axis of astigmatism a little more (or, seen another way, relatively steepen the flatter axis). To correct hyperopic astigmatism, a procedure must steepen the cornea as a whole, but relatively steepen the flatter axis of astigmatism a little more.

One can now imagine that various amounts of spherical flattening or steepening can be accompanied by various amounts of astigmatic flattening or steepening, which can be accompanied by axis shifts to any point between 1° and 180°. The situation presents an extremely complicated analytical challenge. Traditional methods of reporting refractive surgical correction of astigmatic patients fall short mainly in their inability to convey axis shifts and handle aggregate data. The Alpins method, as will be described, determines a goal for astigmatism correction, and a vector (steepening force) required to achieve that goal. From this, the method allows the calculation of the principal components by which an operation fails to achieve its goal, and other components that assist in the comparative analysis of the results of astigmatism surgery for individuals and groups of individuals.

Here is an additional confounding factor we will revisit: Astigmatism as measured by refraction (the well-known test where various lenses are placed in front of the eye while the doctor asks, "Which is better, this or this?") often differs from astigmatism as measured by keratometry and corneal topography, tests considered more objective and quantitative.

The golf analogy

Fundamental concepts of the Alpins method are demonstrated by a golf putt performed on a flat green with no outside forces such as wind (Fig. 2).



Fig. 2: Vector mapping of a golf putt demonstrates fundamentals of the Alpins approach to astigmatism analysis.

A golf putt is similar to a vector, possessing magnitude (length) and axis (direction). The intended putt (the path from the ball to the hole) corresponds to Alpins' target induced astigmatism (TIA), which is the astigmatic change (by magnitude and axis) the surgeon intends to induce in order to correct the patient's preexisting astigmatism. The actual putt (the path the ball follows when hit) corresponds to surgical induced astigmatism (SIA), which is the amount and axis of astigmatic change the surgeon actually induces. If the golfer misses the cup, the difference vector (DV) corresponds to the second putt; that is, a putt (by magnitude and axis) that would allow the golfer to hit the cup on the second attempt.

Alpins' "Correction Index" is determined by the ratio of the SIA to the TIA (what the surgery actually induces versus what the surgery was meant to induce), and is preferably 1 (it is greater than 1 if an overcorrection occurs and less than 1 if there is an undercorrection). It is calculated by dividing SIA (actual effect) by TIA (target effect).

Alpins' "Coefficient of Adjustment" is the inverse of the Correction Index and quantifies the modification needed to the initial surgery plan to have achieved a Correction Index of 1, the ideal correction. If the surgeon achieves an overcorrection, for example, the Correction Index might be 1.10, in which case the Coefficient of Adjustment would be .90, indicating that the surgeon should have selected a correction 90% of what was actually selected. The Coefficient of Adjustment can be used to refine future procedures.

The Alpins method also includes concepts such as the "Magnitude of Error" (intended correction minus actual correction) and "Angle of Error" (the angle described by the vectors of the intended correction versus the achieved correction). The Angle of Error is positive if the achieved correction is on an axis counterclockwise to where it was intended, and negative if the achieved correction is clockwise to its intended axis.

Alpins' "Index of Success" is calculated by dividing the DV (how far you miss the target effect) by the TIA (the original target effect). The Index of Success is a relative measure of success. Going back to our golf analogy: If golfer John attempts a long putt and golfer Bob a shorter one, and each golfer's ball lands the same distance from the cup, John's putt can be considered more successful because he had the longer initial putt. The Index of Success constitutes a valuable new measure of the relative effectiveness of various surgical procedures, and even of the surgeons themselves.

Unlike other available approaches to astigmatism, the indices Alpins describes can be subjected to conventional forms of statistical analysis, generating such familiar friends as averages, means, standard deviations, etc., for each individual component of surgery.

The double-angle vector diagram

Fig. 3 is a double-angle vector diagram (DAVD) used by Alpins to allow calculations in a 360° sense and permit the use of rectangular coordinates. It is an analytical technique that simplifies interpretation of differences among preoperative, desired and achieved astigmatic values, and allows the calculation of the magnitude and direction of surgical vectors. The mathematical methods are described in detail in his articles and will not be covered here.



Fig. 3: The target induced astigmatism (TIA), surgical induced astigmatism (SIA) and difference vector (DV) correspond to the golf putt analogy in Fig. 2, and are calculated from 1 the patient's preoperative astigmatism; 2 the targeted astigmatism the surgeon plans to achieve; and 3 the actual achieved effect of the surgery in this double-angle vector diagram (DAVD).

Line 1 defines a patient's preoperative astigmatism by magnitude (length of the line) and axis (an angle from the X axis representing twice the patient's measured axis of preoperative astigmatism). Line 2 defines the target astigmatism; that is, the magnitude and axis of the correction the surgeon determines to achieve. Line 3 represents achieved astigmatism; that is, the magnitude and axis of the postoperative astigmatism. The dashed lines are the vectors for TIA, SIA and DV as described above. A general rule worth remembering is that astigmatism values can be measured, but vector values can only be calculated.

As Alpins wrote(1), the TIA, SIA and DV, and the "description and calculation of [their] various relationships...comprise the essence of this method, which has not, I believe, been previously described."

In one of his more recent papers,(4) Alpins applies his analysis to irregular astigmatism. Although irregular astigmatism is commonly associated with prior ocular surgery, it is also naturally occurring(15) and widely prevalent.(4) Corneal topography, a technique that produces an image map based on the refractive power of the cornea at many discrete points on its surface, reveals that irregular astigmatism comes in various configurations: the two steep hemimeridians, 180° apart in regular astigmatism, may be separated by less than 180° (a situation called nonorthogonal); and the two steep hemimeridians may be asymmetrically steep; that is, one may be significantly steeper than the other as shown by a larger magnitude value.

Unlike other available astigmatism-analysis approaches, the Alpins method can independently analyze the two hemimeridians of irregular astigmatism and calculate the necessary laser treatment for the two separate halves of the cornea. This capability assumes huge importance as refractive lasers gain the ability to treat discrete parts of the cornea, technology that is only now on the cusp of clinical application.

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