Corneal Topography





Introduction


Corneal topography provides an assessment of the anterior corneal surface shape, and corneal tomography includes an assessment of the posterior corneal surface shape, pachymetry, as well as biometry of the anterior segment. Elevation and curvature are mathematically related; both can be calculated with any topographic/tomographic technology. The apex of the cornea is defined relative to curvature, as the point with the greatest curvature, and the vertex is defined relative to elevation, as the highest point on the corneal surface. However, after myopic refractive surgery, the corneal apex no longer has the greatest curvature. In addition, the highest point is a function of the plane against which it is measured, making this indeterminate as well. Therefore for the purposes of this chapter, the center of any topographic map will be referred to as the central corneal topography (CT) axis .




Curvature


Curvature is mathematically defined as the rate of change of the tangent vector with respect to the arc length. Practically, this means that the curve that has the most bend over the shortest distance has the greatest curvature. To determine the curvature of a three-dimensional surface, such as a cornea, planes of intersection must be defined. The most common planes of intersection are termed the tangential planes , which all include the central CT axis. Each point on the corneal surface has two principal curvatures, which represent the maximum and minimum curvatures through the point. The best illustration of this concept is at the center of the cornea, where the difference in curvature between the flat and steep meridians through the center represents astigmatism of the cornea. Similarly, there are two principal curvatures at each noncentral point. For each point, a plane of intersection that is perpendicular to the tangential plane and includes the surface normal is termed a sagittal (or transverse ) plane. The tangential and sagittal planes are illustrated in Fig. 4.1 . For most topographic and tomographic systems, only the curvature in the tangential planes of intersection is calculated and displayed.




Fig. 4.1


Schematic illustration of the meridional (tangential) and transverse (sagittal) planes of an optical surface. R 1 is the radius of curvature in the tangential plane. R 2 is the radius of curvature in the sagittal plane. C is the center of curvature in the tangential plane.

(Adapted from Gao Y. Polynomial Modeling of the Corneal Surface to Improve the Measurements of Topography [Dissertation]. Columbus, OH: Ohio State University; 1994.)


By definition, curvature must be measured in units of inverse-millimeters. The radius of curvature at a point is defined as the inverse of curvature; thus it is measured in millimeters. In other words, the greater the curvature, the shorter is the radius of curvature. For purposes of corneal topography, curvature is converted to units of diopters (D) using the familiar keratometric formula given in Eq. (1) , which dates back to von Helmholtz and Gullstrand.


<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='(1−1.3375)/r×1000,’>(1−1.3375)/?×1000,(1−1.3375)/r×1000,
(1−1.3375)/r×1000,
where r is the radius of curvature in millimeters. Although this formula is appropriate for the central measurement of keratometry, it presents a few problems of interpretation for topography, which measures a much larger region, approximately 8 to 10 mm of the corneal surface. The keratometric formula is valid only in the center of the cornea, where curvature is directly proportional to power. This will be discussed further in the Functional Representations section. Unfortunately, corneal topography/tomography manufacturers often incorrectly label curvature maps as power maps. Even though common, this terminology should not be adopted, as it is very misleading. It is critical to differentiate curvature from power to avoid misinterpretation of postoperative maps and their impact on vision.


The most commonly used topographic curvature maps are axial maps and tangential curvatures maps. Synonyms for tangential curvature include meridional, instantaneous, or local curvature. Tangential curvature is converted to diopters using Eq. (1) . Axial curvature is the average of tangential curvature over an interval from the CT axis to the surface point along the corresponding tangential plane of intersection. Axial curvature is converted to diopters using Eq. (2) :


<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='(1−1.3375)/a×1000,’>(1−1.3375)/?×1000,(1−1.3375)/a×1000,
(1−1.3375)/a×1000,
where a is the axial distance in millimeters and represents the distance along the surface normal from the surface point to the intersection with the CT axis, as illustrated in Fig. 4.2 . In the center of the map, the axial and tangential curvatures are equivalent. However, because the axial curvature is essentially a running average of the tangential curvature, it smooths any extreme values of curvature. Therefore noncentral areas of flattening on the tangential map are steeper on the axial map and noncentral areas of high curvature on the tangential map are flatter on the axial map. For the special case of surfaces of rotational symmetry, the axial distance is equivalent to the radius of curvature and the axial curvature is equivalent to the tangential curvature. Examples of common clinical topographies are shown in Figs. 4.3 through 4.6 , demonstrating the difference between axial and tangential curvature maps. Once again, the tangential and axial topographic maps represent diopters of curvature, not power.


Fig. 4.2


The center of curvature of surface point S 1 is C 1 , and the radius of curvature is the length of the segment between the two points. The axial curvature is relative to the reference axis, and axial distance of S 1 is the length of the segment between S 1 and A 1 , demonstrating how axial curvature underestimates areas of high curvature. Analogously, axial curvature overestimates areas of lower curvature, as illustrated by surface point, S 2 with center of curvature, C 2 and axial distance S 2 – A 2 .



Fig. 4.3


An example of the axial (right) and meridional (tangential) maps (left) of a normal cornea.



Fig. 4.4


An example of the axial (left) and meridional (tangential) maps (right) of postoperative myopic laser in situ keratomileusis (LASIK). Note that the paracentral steepening is shown only by the meridional map. The extremes in curvature are reduced by the averaging nature of the axial map.



Fig. 4.5


An example of the axial (left) and meridional (tangential) maps (right) of an astigmatic cornea.



Fig. 4.6


An example of the axial (left) and meridional (tangential) maps (right) of a keratoconic cornea.


Elevation Topography


As previously described, curvature is the spatial rate of change of slope, which itself is the spatial rate of change of surface height, or elevation. Unlike curvature, however, elevation (height) requires a reference from which to measure. This is analogous to determining the height of a mountain, which can be measured relative to the ground at its base, or relative to sea level, or even relative to the center of the earth. The choice depends on the purpose. A mountain climber needs to know the height of the mountain above its base to determine how far to climb. However, the same climber also needs to know the height of the mountain above sea level to determine whether supplemental oxygen will be necessary in the higher altitudes of the climb. Both references are ultimately necessary. Similarly, corneal elevation can be measured relative to different references, including a plane, a sphere, an asphere, or a toric surface. Which is the best choice for the corneal reference? Just like the mountain, the answer depends on the application. First, it is important to remember that the choice of a reference is not an attempt to model the corneal surface. Rather, it simply provides a baseline against which to measure. A plane reference provides a surface of constant elevation as the baseline. However, since the major shape of the cornea is its overall curvature, use of a plane reference does not allow details to be appreciated. This is analogous to looking at the earth from space. The curvature of the earth can easily be observed, but the details of the surface topography, including the tallest mountains, are difficult to see. This is illustrated for two corneas in Figs. 4.7A and 4.7B . Despite the dramatic differences in surface shape for a keratoconic cornea and a postrefractive surgery cornea, it is not possible to differentiate the two corneas using a reference plane.




Fig. 4.7


The surface height relative to a plane for a keratoconic cornea (A) and a postmyopic refractive surgery cornea (B). Note that using a plane reference is not useful for determining characteristic patterns since the overall curvature of the cornea dominates. The corresponding elevation maps relative to a best-fit sphere are given for the keratoconic cornea (C) and postmyopic refractive surgery case (D). Note the best-fit sphere (BFS) ; thus the overall curvature is greater for the keratoconic cornea.


The most common reference for corneal elevation is the best-fit sphere (BFS). The term “best-fit” means that an error minimization approach is used to determine the radius and center of the sphere. The BFS reference is quite convenient for a number of reasons. First, it provides a constant radius of curvature reference, which produces characteristic patterns of shape for various conditions, such as astigmatism, postrefractive surgery, or keratoconus. Second, due to the nature of the mathematical fit, the BFS can be considered a measure of the average curvature of the region over which the sphere is fit, with about half of the corneal surface fitting above the sphere and half fitting below the sphere. These two concepts are illustrated in Figs. 4.7C and 4.7D . The topographies are not readily identified as keratoconus and postoperative myopic refractive surgery. The characteristic pattern of keratoconus has a high zone in elevation, corresponding to the region of high curvature at the location of the conus, surrounded by low areas of elevation. The characteristic pattern of postoperative myopic refractive surgery has a low zone in central elevation, corresponding to the central flattened zone, surrounded by a high zone corresponding the paracentral steepening.


The relationship of the highs and lows in an elevation map to the highs and lows of curvature depends on the pattern of elevation. A central area that fits below the sphere and is circumscribed by higher zones is flatter than the reference sphere, similar to the myopic refractive surgery case already described. This is not a concavity or a depression; it is simply lower than the sphere, as illustrated schematically in Fig. 4.8 . A central area that fits higher than the sphere has greater curvature than the sphere. In the case of astigmatism, however, the relationship between high versus low and steep versus flat is opposite, since these corneas usually fit with one meridian above the sphere and one meridian below, with a close match to the reference sphere in the center. In the periphery, the half of the cornea that fits above the BFS is the flat meridian and the half that fits below is the steep meridian. This is illustrated in Fig. 4.9 , which shows both anterior and posterior surfaces having with-the-rule astigmatism. The steeper meridian is low (shown by the blue colors), since it has greater curvature than the reference sphere and therefore falls away from the center faster than the reference sphere.




Fig. 4.8


Schematic illustration of a best-fit sphere to a centrally flat cornea (top) , such as a postoperative myopic laser in situ keratomileusis (LASIK) patient, and the best-fit sphere to a centrally steep cornea (bottom) , such as in keratoconus. Blue areas show negative values on an elevation map and red areas show positive values on an elevation map. Note that the cornea remains convex in the center on the top map even with negative elevation values. It is important to also note that flat areas are not always low on an elevation map. Peripheral low areas, such as in astigmatism, represent the meridian with the greatest curvature, such as in Fig. 4.9 .



Fig. 4.9


Normal with-the-rule astigmatism, demonstrating the relationship between surface representations. Top left is the anterior elevation map and top right is the posterior elevation map. Both elevation maps demonstrate the same with-the-rule astigmatism, with the steepest axis below the sphere and the flat axis above the sphere. The bottom left is the tangential curvature map, also showing with-the-rule astigmatism. The bottom right is the pachymetry map.


Other references that are used include a user-defined sphere, an asphere, and a toric, usually in an attempt to fit as close as possible to the shape of the cornea. The disadvantage of all of these surfaces as a baseline reference is that characteristic patterns are not as readily apparent since part of the shape of the cornea is embedded in the reference. The average curvature of the corneal surface is no longer represented by the radius of the BFS. The importance of the choice of reference is given in Figs. 4.9 and 4.10 , which show both the anterior and posterior elevation maps of an astigmatic cornea. Both the anterior and posterior elevation maps show with-the-rule astigmatism, with the steep meridian at 90° fitting lower than the reference sphere. The color scales are the same for both anterior and posterior maps, giving the impression that the surfaces are similar, except that the astigmatism looks more prominent on the posterior surface. However, the BFS is quite different between the two surfaces. The posterior has much greater curvature than the anterior surface, which is evident in the 6.88-mm posterior reference sphere and the 8.33-mm anterior reference sphere. The lesson to be learned from this analysis is that the magnitude of the reference sphere is critical for interpretation of the elevation maps.


Tags:
Oct 10, 2019 | Posted by in OPHTHALMOLOGY | Comments Off on Corneal Topography

Full access? Get Clinical Tree
Get Clinical Tree app for offline access