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Models of the corneal contour
Models of the Corneal Contour Sheldon Wechsler, OD, FAAO and Douglas D. Miller, PhD
Formerly in private practice for twenty years in California, Sheldon Wechsler is now Associate Professor of Optometry at the University of Houston. He is also Vice-Chairman of the Contact Lens section of the American Academy of Optometry.
Introduction The corneal contour may be one of the parameters of the body that has been most widely investigated. Barry A. J. Clark 1 in his paper, Systems for Describing Corneal Topography, lists 39 references; in his paper, Less Common Methods of Measuring Corneal Topography 2, he lists 83 references. Most of the previously published work deals with the more central areas of the cornea and/or with exercises in trying to determine mathematical descriptions of the contour of the cornea. Interest in corneal topography waxes and wanes. A n d , perhaps because of the types of contact lenses that were in use, most of the descriptions of corneal contour are the result of investigations of the central 8mm of the corneal area. Today, however, with the large number of contact lenses of 12mm or more diameter, the contour of the whole corneal surface has assumed greater importance. In the United States today the description of the corneal surface as a segment of the prolate end of an ellipsoid of 0.5 eccentricity appears to be the most prevalent. Poster 3 4 and Bibby and Townsley 5 have used that description for their explanations of the rationale for the fitting of several of the lenses that are today available for use in the United States. The principles would be equally applicable to lenses used for patient care in other parts of the world.
Present Concept of Corneal Contour Contact lens practitioners talk as if the cornea could be described as a segment of a sphere. The keratometer makes that assumption, and in conversation we speak about a 45.00 D. cornea as having a radius of curvature of 7.50mm as if we were describing a segment of a sphere. It is always understood, however, that the cornea does indeed flatten toward the periphery.
18
The prolate side of an ellipse does flatten as the distance from the major axis increases. Conversely, the oblate side steepens as the the distance from the minor axis increases. Thus the prolate side of an ellipse, rather than the oblate side, appears to fit with the description of the cornea that we know to be true clinically. As the eccentricity of the ellipse increases the peripheral flattening increases and conversely as the eccentricity of the ellipse decreases the peripheral flattening is not as great. Finally, an increase in the radius of curvature at the point where the major axis intersects the prolate side of an ellipse makes the whole ellipse larger and a decrease in radius of curvature at that point makes the ellipse smaller. The keratometer that we use to measure the corneal curvature in the clinical setting reflects light from two points on the corneal surface. The measurement is actually a distance measurement which is averaged and converted to a figure that would indicate what the curvature of the central part of the cornea would be if it were a segment of a circle. The keratometer actually measures two points on the corneal surface separated by about 3mm.
Objective The purpose of this study is to investigate the contour of the verticial meridian of the cornea and to match that contour to a circle, ellipses of varying eccentricity, parabola and catenary curve.
Method The curve models for the study were generated on an Hewlett-Packard desk-top calculator (Model 9825) attached to an Hewlett-Packard plotter (Model 9862A). The following equations were used for the generation of the curves: Circle X 2 + y2 = R O 2
Journal of the British Contact Lens Association
Ellipse
(X/a)2+ (Y/b) 2 = 1where e q(a~ - b2)/a and R O = a ( 1 - e a) Parabola Y2 = 4cX where c = - R O / 2 . Catenary X = a*cosh (Y/a) where a = - R O . The curves covered a distance of 12mm on the Y axis. Calculations were made of the distances between the curves at several points along the curves. Four subjects' right eyes were photographed from a point 90 ° from the visual axis. Subjects were asked to look through an optical system at a spot of light. The light became invisible if the subject's head moved more than a couple of millimeters. A camera was mounted to point 90 ° from that visual axis and photographs were taken with Ektachrome film with the aid of a flash unit. Computer drawn curves were properly sized against the resultant profile photographs with the aid of a millimeter scale positioned just adjacent to the visual axis in the photographs. A properly sized computer curve-corneal profile photograph overlay was then produced for final evaluation. The curves were matched to the photographs of the eyes by two investigators. Computer drawings of the various curves for each subject eye were randomly numbered and the investigators were asked to rank the accuracy of the curve fits to the corneal contour. The radius of curvature of the several curves, RO, was computed taking into account the separation between the mires of the Bausch and Lomb keratometer. Thus, the curvature at the apex of the various curves, with the exception of the circles, is different from the K reading and more accurately reflects the apical curvature of the arcs whose K reading would be those of the subjects. The subjects' K readings were: 1. 45.00 (7.50) 2. 44.12 (7.65) 3. 43.25 (7.80) 4. 42.25 (7.99)
Results
Figures 1 - 3 are the composite photos of subject #1. Figure 1 has an overlay of the optimal curve for this subject's eye which is a parabola. Figure 2 has a circle of 7.5mm radius overlayed. This circle impinges upon the corneal tissue beginning at a point just before 5mm from the apex of the cornea, and is well on to the corneal tissue at the timbus. Figure 3 has an overlay of a circle of 8.0mm radius. This circle with a radius approximating the base curve of a hydrogel lens that might be prescribed for this eye clearly contours the cornea much better throughout the distance from the superior limbus to the inferior one. Figures 4 and 5 are composite photos of subject #2. Figure 4 has an overlay of a parabola which in this case is too flat in the periphery to properly contour the subject's cornea. Figure 5 has an overlay of an
Figures 1 - 3
Composite photos of subject 1 with computer drawn curves overlayed on the subject's corneal profile. Figure 1 presents a parabolic curve that would produce a K reading of 45.00 D (7.50mm). Note the good apposition to the corneal profile near the superior and inferior limbus. Figure 2 presents a circle of 7.50mm radius which manifests a poor match peripherally. Figure 3 is a circle of 8.00mm radius. This may give some indication of why hydrogel lenses are usually fitted flatter than K.
(continued on page 24)
Journal of the British Contact Lens Association
19
(continued from page 19) in the periphery and have shorter "Z" values (the distance between the line that is normal to the apex of the curve and the curve itself at a point peripheral to the apex) appear to be better contour matches for the corneas photographed.
Figure 6 This composite photo overlays a catenary curve on the cornea of subject #3. Note the difference between the superior and inferior hemi-meridians.
Figures 4 - 5 Composite photos of subject #2. Figure 4, a parabolic overlay, is flatter than the corneal contour at the periphery. Figure 5, an ellipse of eccentricity 0.7 overlay, closely approximates the corneal contour.
ellipse of 0.7 eccentricity which more closely contours the subject's cornea. Figure 6 has an overlay of a catenary curve On subject #3. This cornea is unique among the four subjects in that the superior hemi-meridian of the cornea is flatter than the inferior hemi-meridian. The catenary curve does appear to contour the inferior hemi-meridian. Figures 7 and 8 have a catenary and ellipse of 0.7 eccentricity respectively overlayed on the cornea of subject #4. In these cases both curves contour the cornea rather well and it is difficult to choose the best corneal contour between the two photos. Discussion and Conclusions It is obvious from this study that the circle is a poor descriptor of the corneal contour; a circle whose radius of curvature is equal to the subject's K reading is a poor match for the corneal contour over a distance of 10 to 12mm. It seems equally apparent that an ellipse of 0.5 eccentricity is also a poor match. The parabola and catenary curves which are flatter
24
Figures 7 - 8 In these photos both the parabola and ellipse of eccentricity 0.7 closely approximate the corneal contour.
Journal of the British Contact Lens Association
It is evident, however, that the differences in spatial position of the curves do not become clinically significant within an 8mm region centered on the apex. On the other hand, the curvature separation between the circle and the parabola is greatest of the K Constant K: 43.25 ( 7 . 8 0 3 )
r
curves investigated (figure 9). Using the incorrect assumption that the K reading measures the corneal apical curvature, these separations are even greater (figure 10).
~
m i r e l e P l r S t l o n 3.078
Re
lepitr Itlon
(m 7.803 7.688 7.482
circle
elltplle (o .5) c41tonllr y
,.,.bo,.
47 120
47 83
\
.RO C o n l l e n t RO: 7.803 mire l e p | r l l i o n :
3.078
K circle
43.25 (7' 30)
cetenllry parabola
41.62 ( T . 9 2 ) 40.87 (8.2i5)
seperllton
197
llgl
I F i g u r e s 9 - 10
Computer drawings of curves with a K reading value of 43.25 D (7.803mm). Note the difference in
Journal of the British Contact Lens Association
separation between the peripheral parts of the curves •when the K reading is kept constant (Figure 9) and when the R O is kept constant (Figure 10).
25
The clinical impfications of this study have yet to be determined. Some indication of the explanation for fitting hydrogel lenses flatter than the K reading is present in figure 3 where it is evident that a circle that is flatter than the K reading by 0.5mm fits the corneal contour better than the "on K" circle. This being the case, it may be wise for contact lens manufacturers to consider making lenses with parabolar base curves. Macromolecular ftuorescein photographs of hydrogel lenses appear to support that possibility (figure 11). It must be emphasized that this study is a qualitative one and no attempt was made in this work to quantify the curve relationships. Further studies in quantification are warranted. Although the contour of the cornea has been
studied over a long period of time, virtually no work has been done on the contour of the limbal area. With the widespread use of hydrogel lenses that go beyond the limbus there is increasing need for investigation of the limbal contour and the far periphery of the cornea. It is our hope that future corneal contour research will expand its scope and include both these areas.
References 1. 2. 3.
4.
5.
Clark, Barry A. J., Systems for describing corneal topography, Australian Journal of Optometry, p. 48-56, February 1973. Clark, Barry A. J., Less common methods of measuring corneal topography, Australian Journal of Optometry, p. 182-192, May 1973. Poster, Maurice G., Equivalent fitting radius of the Softens and equivalent fitting radius of spherical posterior surface hydrophilic lenses by a three-point touch criterion, International Contact Lens Clinic, Vol. 4 (1), p. 60-67, January/February 1977. Poster, M. G., A rationale for fitting the Bausch and Lomb Softens, Journal of the American Optometric Association, 46 (3), p. 223-227, March 1975. Bibby, Malcom M. and Townsley, Malcom G., Analysis and description of corneal shape, Contact Lens Forum, p. 27-35, December 1976.
Figure 11 Macromolecular fluorescein photograph showing the peripheral clearance of a hydrogel lens on an eye. The clearance in part results from fitting a spherical base curve lens of longer radius of curvature than the K reading. The lens in this photo is 1.5ram flatter than K.
26
Address for further correspondence: University of Houston, College of Optometry, 4901 Calhoun, Houston, Texas 77004 USA.
Journal of the British Contact Lens Association
Formerly in private practice for twenty years in California, Sheldon Wechsler is now Associate Professor of Optometry at the University of Houston. He is also Vice-Chairman of the Contact Lens section of the American Academy of Optometry.
Introduction The corneal contour may be one of the parameters of the body that has been most widely investigated. Barry A. J. Clark 1 in his paper, Systems for Describing Corneal Topography, lists 39 references; in his paper, Less Common Methods of Measuring Corneal Topography 2, he lists 83 references. Most of the previously published work deals with the more central areas of the cornea and/or with exercises in trying to determine mathematical descriptions of the contour of the cornea. Interest in corneal topography waxes and wanes. A n d , perhaps because of the types of contact lenses that were in use, most of the descriptions of corneal contour are the result of investigations of the central 8mm of the corneal area. Today, however, with the large number of contact lenses of 12mm or more diameter, the contour of the whole corneal surface has assumed greater importance. In the United States today the description of the corneal surface as a segment of the prolate end of an ellipsoid of 0.5 eccentricity appears to be the most prevalent. Poster 3 4 and Bibby and Townsley 5 have used that description for their explanations of the rationale for the fitting of several of the lenses that are today available for use in the United States. The principles would be equally applicable to lenses used for patient care in other parts of the world.
Present Concept of Corneal Contour Contact lens practitioners talk as if the cornea could be described as a segment of a sphere. The keratometer makes that assumption, and in conversation we speak about a 45.00 D. cornea as having a radius of curvature of 7.50mm as if we were describing a segment of a sphere. It is always understood, however, that the cornea does indeed flatten toward the periphery.
18
The prolate side of an ellipse does flatten as the distance from the major axis increases. Conversely, the oblate side steepens as the the distance from the minor axis increases. Thus the prolate side of an ellipse, rather than the oblate side, appears to fit with the description of the cornea that we know to be true clinically. As the eccentricity of the ellipse increases the peripheral flattening increases and conversely as the eccentricity of the ellipse decreases the peripheral flattening is not as great. Finally, an increase in the radius of curvature at the point where the major axis intersects the prolate side of an ellipse makes the whole ellipse larger and a decrease in radius of curvature at that point makes the ellipse smaller. The keratometer that we use to measure the corneal curvature in the clinical setting reflects light from two points on the corneal surface. The measurement is actually a distance measurement which is averaged and converted to a figure that would indicate what the curvature of the central part of the cornea would be if it were a segment of a circle. The keratometer actually measures two points on the corneal surface separated by about 3mm.
Objective The purpose of this study is to investigate the contour of the verticial meridian of the cornea and to match that contour to a circle, ellipses of varying eccentricity, parabola and catenary curve.
Method The curve models for the study were generated on an Hewlett-Packard desk-top calculator (Model 9825) attached to an Hewlett-Packard plotter (Model 9862A). The following equations were used for the generation of the curves: Circle X 2 + y2 = R O 2
Journal of the British Contact Lens Association
Ellipse
(X/a)2+ (Y/b) 2 = 1where e q(a~ - b2)/a and R O = a ( 1 - e a) Parabola Y2 = 4cX where c = - R O / 2 . Catenary X = a*cosh (Y/a) where a = - R O . The curves covered a distance of 12mm on the Y axis. Calculations were made of the distances between the curves at several points along the curves. Four subjects' right eyes were photographed from a point 90 ° from the visual axis. Subjects were asked to look through an optical system at a spot of light. The light became invisible if the subject's head moved more than a couple of millimeters. A camera was mounted to point 90 ° from that visual axis and photographs were taken with Ektachrome film with the aid of a flash unit. Computer drawn curves were properly sized against the resultant profile photographs with the aid of a millimeter scale positioned just adjacent to the visual axis in the photographs. A properly sized computer curve-corneal profile photograph overlay was then produced for final evaluation. The curves were matched to the photographs of the eyes by two investigators. Computer drawings of the various curves for each subject eye were randomly numbered and the investigators were asked to rank the accuracy of the curve fits to the corneal contour. The radius of curvature of the several curves, RO, was computed taking into account the separation between the mires of the Bausch and Lomb keratometer. Thus, the curvature at the apex of the various curves, with the exception of the circles, is different from the K reading and more accurately reflects the apical curvature of the arcs whose K reading would be those of the subjects. The subjects' K readings were: 1. 45.00 (7.50) 2. 44.12 (7.65) 3. 43.25 (7.80) 4. 42.25 (7.99)
Results
Figures 1 - 3 are the composite photos of subject #1. Figure 1 has an overlay of the optimal curve for this subject's eye which is a parabola. Figure 2 has a circle of 7.5mm radius overlayed. This circle impinges upon the corneal tissue beginning at a point just before 5mm from the apex of the cornea, and is well on to the corneal tissue at the timbus. Figure 3 has an overlay of a circle of 8.0mm radius. This circle with a radius approximating the base curve of a hydrogel lens that might be prescribed for this eye clearly contours the cornea much better throughout the distance from the superior limbus to the inferior one. Figures 4 and 5 are composite photos of subject #2. Figure 4 has an overlay of a parabola which in this case is too flat in the periphery to properly contour the subject's cornea. Figure 5 has an overlay of an
Figures 1 - 3
Composite photos of subject 1 with computer drawn curves overlayed on the subject's corneal profile. Figure 1 presents a parabolic curve that would produce a K reading of 45.00 D (7.50mm). Note the good apposition to the corneal profile near the superior and inferior limbus. Figure 2 presents a circle of 7.50mm radius which manifests a poor match peripherally. Figure 3 is a circle of 8.00mm radius. This may give some indication of why hydrogel lenses are usually fitted flatter than K.
(continued on page 24)
Journal of the British Contact Lens Association
19
(continued from page 19) in the periphery and have shorter "Z" values (the distance between the line that is normal to the apex of the curve and the curve itself at a point peripheral to the apex) appear to be better contour matches for the corneas photographed.
Figure 6 This composite photo overlays a catenary curve on the cornea of subject #3. Note the difference between the superior and inferior hemi-meridians.
Figures 4 - 5 Composite photos of subject #2. Figure 4, a parabolic overlay, is flatter than the corneal contour at the periphery. Figure 5, an ellipse of eccentricity 0.7 overlay, closely approximates the corneal contour.
ellipse of 0.7 eccentricity which more closely contours the subject's cornea. Figure 6 has an overlay of a catenary curve On subject #3. This cornea is unique among the four subjects in that the superior hemi-meridian of the cornea is flatter than the inferior hemi-meridian. The catenary curve does appear to contour the inferior hemi-meridian. Figures 7 and 8 have a catenary and ellipse of 0.7 eccentricity respectively overlayed on the cornea of subject #4. In these cases both curves contour the cornea rather well and it is difficult to choose the best corneal contour between the two photos. Discussion and Conclusions It is obvious from this study that the circle is a poor descriptor of the corneal contour; a circle whose radius of curvature is equal to the subject's K reading is a poor match for the corneal contour over a distance of 10 to 12mm. It seems equally apparent that an ellipse of 0.5 eccentricity is also a poor match. The parabola and catenary curves which are flatter
24
Figures 7 - 8 In these photos both the parabola and ellipse of eccentricity 0.7 closely approximate the corneal contour.
Journal of the British Contact Lens Association
It is evident, however, that the differences in spatial position of the curves do not become clinically significant within an 8mm region centered on the apex. On the other hand, the curvature separation between the circle and the parabola is greatest of the K Constant K: 43.25 ( 7 . 8 0 3 )
r
curves investigated (figure 9). Using the incorrect assumption that the K reading measures the corneal apical curvature, these separations are even greater (figure 10).
~
m i r e l e P l r S t l o n 3.078
Re
lepitr Itlon
(m 7.803 7.688 7.482
circle
elltplle (o .5) c41tonllr y
,.,.bo,.
47 120
47 83
\
.RO C o n l l e n t RO: 7.803 mire l e p | r l l i o n :
3.078
K circle
43.25 (7' 30)
cetenllry parabola
41.62 ( T . 9 2 ) 40.87 (8.2i5)
seperllton
197
llgl
I F i g u r e s 9 - 10
Computer drawings of curves with a K reading value of 43.25 D (7.803mm). Note the difference in
Journal of the British Contact Lens Association
separation between the peripheral parts of the curves •when the K reading is kept constant (Figure 9) and when the R O is kept constant (Figure 10).
25
The clinical impfications of this study have yet to be determined. Some indication of the explanation for fitting hydrogel lenses flatter than the K reading is present in figure 3 where it is evident that a circle that is flatter than the K reading by 0.5mm fits the corneal contour better than the "on K" circle. This being the case, it may be wise for contact lens manufacturers to consider making lenses with parabolar base curves. Macromolecular ftuorescein photographs of hydrogel lenses appear to support that possibility (figure 11). It must be emphasized that this study is a qualitative one and no attempt was made in this work to quantify the curve relationships. Further studies in quantification are warranted. Although the contour of the cornea has been
studied over a long period of time, virtually no work has been done on the contour of the limbal area. With the widespread use of hydrogel lenses that go beyond the limbus there is increasing need for investigation of the limbal contour and the far periphery of the cornea. It is our hope that future corneal contour research will expand its scope and include both these areas.
References 1. 2. 3.
4.
5.
Clark, Barry A. J., Systems for describing corneal topography, Australian Journal of Optometry, p. 48-56, February 1973. Clark, Barry A. J., Less common methods of measuring corneal topography, Australian Journal of Optometry, p. 182-192, May 1973. Poster, Maurice G., Equivalent fitting radius of the Softens and equivalent fitting radius of spherical posterior surface hydrophilic lenses by a three-point touch criterion, International Contact Lens Clinic, Vol. 4 (1), p. 60-67, January/February 1977. Poster, M. G., A rationale for fitting the Bausch and Lomb Softens, Journal of the American Optometric Association, 46 (3), p. 223-227, March 1975. Bibby, Malcom M. and Townsley, Malcom G., Analysis and description of corneal shape, Contact Lens Forum, p. 27-35, December 1976.
Figure 11 Macromolecular fluorescein photograph showing the peripheral clearance of a hydrogel lens on an eye. The clearance in part results from fitting a spherical base curve lens of longer radius of curvature than the K reading. The lens in this photo is 1.5ram flatter than K.
26
Address for further correspondence: University of Houston, College of Optometry, 4901 Calhoun, Houston, Texas 77004 USA.
Journal of the British Contact Lens Association