 Engineering, 2013, 5, 477-481 http://dx.doi.org/10.4236/eng.2013.510B098 Published Online October 2013 (http://www.scirp.org/journal/eng) Copyright © 2013 SciRes. ENG The 3D Computer Image of the Anterior Corneal Surface Bo Wang1, Xueping Huang2, Jinglu Ying3, Mingguang Shi3 1School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, China 2Wenzhou Medical College, Wenzhou, China 3Department of Ophthalmology, The Second Affiliated Hospital of Wenzhou Medical College, Wenzhou, China Email: bowa ng@live.com, pshimg@hotmail.com Received 2013 ABSTRACT In this paper, we derive a nonlinear equation of corneal asphericity (Q) using the tangential radius of curvature (rt) on every semi-meridian. We transform the nonlinear equation into the linear equation and then obtain the Q-value of cor- neal semi-meridian by the linear regression method. We find the 360 semi-meridional v ariation rule of the Q-value us- ing polynomial function. Furthermore, we construct a new 3D corneal model and present a more realistic model of shape of the anterior corneal surface. Keywords: Cornea; Computer; Image 1. Introduction It is well established that the anterior surface of the cor- nea is the major refractive element of the human eye, being responsible for approximately 75% of the eye’s total unaccommodated refractive power . Guillon  and Bennett  assumed the human cornea to have a conic section which can be described by the Baker’s eq- uation: 2202yr xpx= − . Here, p describes the as- phericity of the corneal section. Cheung  calculated the corneal asphericity (p) using the sagittal radius of curvature (sr) from corneal axial power map according to Bennett’s equation: 22 20(1 )sr rpy= +− . Schwie- gerling  used corneal height data from corneal height map and Zernike polynomials to describe the shape of the cornea. Corneal topography is commonly presented as axial power map, tangential power map and height map. To our knowledge, there is no report of investigat- ing the corneal asphericity (Q, 1Qp= −) calculation by the tangential radius of curvature (rt) from tangential power map. In most previous studies, it has been reported the Q- values which is representative of all corneal meridians or the Q-values of two principal meridians. Dubbelman  measured k-values (where 1kQ= +) of six semimeri- dians (0˚, 30˚, 60˚, 90˚, 120˚, 150˚) using Scheimpflug photograph y and modeled the meridional variatio n of the k-value using the 2cos function. However, it indicated that the 2cos function is not an adequate model to de- scribe t he va riati on. Sagittal radius of curvature (rs) is spherically biased and is not a true radius of curvature [9-11] and it will lead to erroneous result for an asymmetric corneal sur- face. Tangential radius of curvature (rt) is a true radius of curvature which can better represent corneal shape and local curvature changes especially in the periphery . In this paper we derive a nonlinear equation of corneal asphericity (Q) using the tangential radius of curvature (rt) on every semi-meridian for the first time. We obtain the Q-value of corneal semi-meridian by the linear regres- sion method and find the 360 semi-meridional variation rule of the Q-value using polynomial function. Further- more, we construct a new 3D model of shape of the ante- rior corneal surface. 2. Derivation of the Corneal Model The Bausch & Lomb Orbscan II corneal topographer is used to acquire images of the topography of the right eye of 66 normal young subj ects. All subj ec ts have no history of ocular disease and ocular surgery with emmetropic eyes. A series of data point on a semi-meridian are ar- ranged at 0.1 mm intervals. The interval between two semi-meridians is 1˚. The tangential radius of curvature (rt) and perpendicular distance from the point to optical axis (y) of all data point on a se mi-meridian and vertex radius of curvature (0r-value) can be obtained from the raw data of tangential power map of anterior corneal surface. A three dimensional Car tesian coordinate system is set with its origin at vertex nor mal to the corn eal interse ction of the optic axis of the corneal topographer ). The Z-axis, Y-axis, X-axis of the coordinate represent the optical axis direction, the vertical direction and the hori- B. WANG ET AL. Copyright © 2013 SciRes. ENG 478 zontal direction, respectively. θ is the angle between the corneal meridian section and the XOZ plane. The corneal meridian section is located on the YOZ plane when θ = 90˚. At this time, we assume that the equa- tion of corneal meridian can be correspondingly de- scribed by the conic equation: 2212 12,,yazaza aR=+∈. This conic equation is an improvement of the Baker’s equation . While located on the XOZ plane when θ = 0˚ and described by the conic equation: 2212xaz az= +. For any other angle θ except for 0˚, 90˚, 180˚, 270˚, the YOZ plane can be coincided with the corneal meridian section by rotating the coordinate system. Thus the corneal meridian section of any other angle θ can also be described by the conic equation 2212yaz az= + in the new coordinate system (see Sec- tion 5.2) . Here let us take corneal meridian section with θ = 90˚ for example, the formula of the curvature of a point on the section can be expressed as [14,15]: 32"11( ')tyKry= =+ (1) where K is curvature, 'y and ''y are the first and se- cond derivatives with respect to z which is z-axis coordinate value of the point. Differentiating both sides of the conic equation 2212yaz az= + with respect to z, we get 1222a azyy+′=, 2134ayy−′′ = Then by substituting y′ and y′′ into Equation (1), we obtain: 322122214((1) )4tar aya= ++ (2) The conic equation 2212yaz az= + can be rewritten as: 21222211222()2144azayaaaa++=− (3) Since 2Qe= −, then by Equation (3) we have 2(1 )aQ=−+. Finally by substituting 2(1 )aQ=−+ into Equation (2), we obtain 32220201[]trr Qyr= − (4) 3. Solution to Q-Value Calculation Problem Since rt is a nonlinear function of y in Equation (4), it is difficult to calculate Q-value. To transform the non- linear problem to the linear one, the Equation (4) is con- verted to another form which can be written as: 223tyb cr= + (5) where b and c are constants, a straight line graph of 2y (on the ordinate) vs 23tr (on the abscissa) is plotted. By the linear regression method, we get 20rbQ= and 430rcQ= −, that is, 23bQc= −. The straight line gives a coefficient of determination (2R). The Q-value of the given semi-meridian is calculated includ ing from the first point at 0.1 mm to 3.5 mm. Figure 1 illustrates a func- tion scatterplot of perpendicular distance squared versus tangential radius of curvature to the two-thirds power on the nasal horizontal principal semi-meridian of the right eye for subject number 1. 4. 360 Semi-Meridional Rule of the Q-Value The corneal zone analyzed is up to diameter 7.0 mm which is large enough to cover the pupillary area. The near vertical meridians will have a diameter limit im- posed by the eyelids and eyelashes. In our earlier study, we found that the semi-meridians which the peripheral points wer e up to 3.5 mm were mainly distributed within 50˚ of the horizontal including 0˚ - 50˚, 130˚ - 180˚, 181˚- 230˚, 310˚ - 359˚. The Q-value of each semi-meridian in these near horizontal regions was calculated. According to the Q-values of the near horizontal semi-meridians, we use regression analysis to model the 360 semi-meridional variation of the Q-value and fit the Q-value of Figure. 1. Scatterplot of perpendicular distance squared versus tangential radius of curvature to the two-thirds power. B. WANG ET AL. Copyright © 2013 SciRes. ENG 479 each semi-meridian in the near vertical regions including 51˚ - 129˚, 231˚ - 309˚. The form of a polynomial func- tion is: 23401 234( )...f xppxpxpxpx=+++++ where x is semi-meridian angle θ (degree) and f(x) is corresponding Q-value. Here, the degree must be con- verted to the radian when calculating the polynomial fitting. To determine which degree polynomial will provide an optimal fit to the 360 semi-meridional variation of the Q-value, we calculate the RMS fit error of the polynomi- al function from 5th degree to 9th degree. We find that the RMS fit error become relatively stable at approximately 0.02 for fits higher than 6th degree. The 360 semi-meridional variation of the Q-value is well fitted using the 7th degree polynomial function for all subjects. Figure 2 shows an example of the variation of the Q-value as a function of semi-meridian for subject number 22 with the following 7th degree polynomial function: Red: Fitted curve of 360 semi-meridional variation of the Q-value. Figure 3 shows that the majority of right eyes display the goodness of fit (r2) of polynomial function for all subjects for the asphericity above 0.9 and the median value is 0.94. The mean RMS fit error of polynomial fit is 0.02 ± 0.008. Figure 4 shows the variation in asphericity with semi- meridian region of anterior corneal surface for all sub- jects. It can be seen that the Q-value distribution of ante- rior corneal surface presents bimodal variation. These two peak va l ues repres ent the l e a s t negative Q-values. Figure 2. Typical example of the variation of the Q-value as a function of semi-meridian. Figure 3. Box and whisker plot for the goodness of fit (r2) of the polynomial function for all subjects for the asphericity (Q). Figure 4. Variation in asphericity as a function of semi- meridian region of anterio r corneal surface for all subjects. 5. Construction of a 3d Model of Corneal Shape 5.1. Rotation of the Coordinate System A new coordinate system (''X OY) is obtained by rotat- ing the original coordinate system (XOY)θ degree in the counter clockwise direction. P is an arbitrary point in the coordinate system with (, )Pxy in the original coordinate system and ( ',')Px y in the new coordinate system. We can obtain the following coordinate rotation formula: ' sincos' cossinxy xyy xθθθθ= += − (6) 5.2. The Parametric Representation of the Equations of the Corneal Meridian Section We set the angle between a given corneal meridian sec- tion and the XOZ plane is θ degree. A new coordi- nate system (XOY) is obtained by rotating the original coordinate system (XOY)0-(90 -)θ degree in the coun- B. WANG ET AL. Copyright © 2013 SciRes. ENG 480 ter clockwise direction around the Z-axis. Thus, the YOZ plane can be coincided with the corneal meridian section in the new coordinate system (XOY). The equa- tions of the corneal meridian section in the new coordi- nate system (XOY).are as follows: 22212 002(1 )xyaz azrzQz== +=−+ (7) where (, )xy are the coordinates of the new coordinate system (XOY). Then by substituting 0-(90 -)θ into θ given in the formula (6), we obtain the following coordinate rotation equations of our corneal model: sin cossin cosxx yyy xθθθθ= −= + (8) We substitute the ,xy given in Equation (7) into the Equation (8). The equations of the corneal meridian sec- tion on the original coordinate system (XOY) are as fol- lows 220sincos 0( sincos)2(1)xyyxrz Qzθθθθ−=+= −+ (9) Finally, we transform the Equation (9) into the fol- lowing f orm: 20202(1)cos2(1)sinx rzQzy rzQzθθ= −+= −+ (10) 5.3. Generation of a 3D Corneal Model 360 semi-meridians are all chosen. Every point has an (,,)xyz coordinate. The z is a parameter and the zvalues of a semi-meridian are selected from 0 mm to 3.5 mm at 0.1 mm intervals. The ,xy coordinate values of every point are calculated by substituting the corres- ponding zvalue into the Equation (11), 3D corneal sur- face plot is generated with the Visual C++ 6.0 program- ming . Figure 5 shows a colorized 3D surface plo t of anterior corneal surface from two different perspectives for the same subject as in Figure 2. Variation of color shows semi-meridional variation of the Q-value with 0.02 color steps. From the top to bottom of color scale, the Q-value becomes more negative gradually. Figure 2 shows that the Q-value of each semi-meridian is negative value (−1 < Q < 0) corresponding to the most common corneal shape (prolate ellipse) (). Thus, the 3D sur- face plot of anterior corneal surface approximates a pro- late ellipsoid shown in Figu re 5. 6. Conclusion In contrast to the sagittal radius of curvature (rs), the Figure 5. 3D surface plot of anterior corneal surface for the same subject as in Figure 2. tangential radius of curvature (rt) is a true radius of cur- vature which can better represent corneal shape and local curvature changes especially in the periphery. In this paper, we proposed a nonlinear equation of corneal asphericity (Q) using the tangential radius of curvature (rt) on every semi-meridian. The 360 semi- meridional variation of the Q-value was well fitted using the 7th degree polynomial function for all subjects. 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