正常成人角膜数学模型的探讨
摘要
自从人们第一次真正意识到角膜在眼球屈光系统中的重要作用以后，已经有越来越多的研究者相继投身于该领域的研究，尤其在角膜形状方面。而且随着研究的逐渐深入，人们已逐渐认识到角膜的复杂性，正常角膜实际是一个非常复杂的光学及解剖结构。
各种计算机辅助的角膜形态及光学特征检测设备和仪器的广泛应用，极大地推动了该领域临床和基础研究的发展，可以说带来了质的飞跃。然而，它们固有的一些不可避免的缺陷，大大地影响了其测量准确性，如近轴和偏轴光学系统的一概而论，检测数据来源的单一性，个体应用的局限性等。
角膜是一个曲面，即具有某种性质的点的集合。从理论上讲，每张曲面都有它自己的数学模型。数学模型是描述物体表面形状的数学表达式，也称为物体的几何模型。它是对物体进行分析、计算和绘制的根据，是研究曲面性质的工具，可以说从一定程度上代表了曲面的本质特性，因而，对这类物体进行分析、处理的首要任务就是为其建立数学模型。
计算机辅助几何设计，尤其是曲面重建技术正是根据已有曲面去构建反映其形状的数学模型的过程。它的飞速发展和广泛应用，为角膜形状特征的研究开辟了崭新的途径。
目的 本研究拟从数学角度对角膜形状特点进行较为深入的探讨分析，同时结合曲面重建技术建立其数学模型，旨在了解其本质，明确其内在特征，为今后的科研、临床乃至检测设备仪器的制造完善等多方面的工作提供科学的理论基础和指
浙江大学博士生学性伶灰
导。同时，建立该模型的三维显示软件，使其更加直观形象，并可用于不同群体
角膜模型的构建。
方法
1.数据采集应用Orbscanll角膜地形图系统采集角膜顶点，0”、朋“、60“、
90”、1200、150”、180。、210。、2400、270“、__300“、330“子午线__}_几五牡
角膜顶点分别为1.smm、2.smm、3.smm、4.smm处点的角膜前、后表面曲率半径、
总角膜曲率半径、角膜厚度数值。
2.坐标建立建立笛卡儿空间三维直角坐标:以角膜顶点为原点，眼球光轴方向
为Z轴，水平方向为X轴，垂直方向为Y轴，并确定各点的空间坐标值。
3.数据处理应用SPSS10.0及office 2000 Excel统计软件包对所有数据行统
计学分析及计算处理。统计学方法包括:描述性统计分析、独立样木t检验、双
变量相关分析。
4.数学模型的建立
(1)角膜前后表面数学模型以Bennett’s锥形切面数学理论为基础，根据公
式yZ=a，z十aZzZ并结合二次曲面截痕法，明确角膜前后表面的形状，并建立其相应
数学表达式。
(2)形状系数数学模型根据锥形切面数学公式yZ=Zrl)z一(1一eZ)22推导出eZ=1-
(Zrl)z一yZ)/22，并以此建立角膜前后表面两正交主子午线的形状系数的数学表达
式。
(3)角膜曲率分布根据托力克面斜轴子午线曲率F’计算公式
F’二F汁(F卜一民)* SinZQ，确定角膜前、后表面斜轴子午线曲率及斜轴子午线总角膜
曲率的理论值，并通过分析该理论值与实测值之间的一致性，明确角膜曲率分布
特征。
5.数学模型三维重建及显示软件
础，采用Windows
中文Windows2000、
V 1 sua1Studio
应用PC/CE台式兼容机，以上述数学模型为基
(vC+十6.0)编程软件包自行设计而成的，可在
XP、
6.数学模型软件的应用
结果
NT等环境下运行。
构建正视状态正常成人的角膜数学模型。
浙协大嵘博士生含位论式
1.角膜前后表面数学模型前后表面均为近似椭球面，其相应数学模型分别为:
角膜前表面:xZ/aZ+yZ/bZ+(z一e)2/e，==1(a，b，c>o)
角膜后表面:x，/aZ+yZ/bZ+(z一e一d，，)2/eZ=一(a，b，e>o)
本研究中，正视状态成人群体的角膜前、后表面数学模型分别为:
角膜前表面:xZ/8.053，+yZ/7.9732+(z一8.226)2/8. 2262砚
角膜后表面:XZ/6.8362+yZ/6 .745斗(z一6.974)2/7 .5272=1
2.角膜一前后表面形状系数的数学模型
角膜前表面陡峭子午线:eZ二1一(2r。z一y勺/22
扁平子午线:eZ二l一(2r。z一x今/22
角膜后表面陡峭子午线:eZ二1一「Zr，，(z一山)一yZ」/(z一山)“
扁平子午线:eZ二1一〔Zrl、(z一山)一xZ〕/(z一山)2
本研究中，正视状态成人群体的角膜性状系数数学表达式分别为:
角膜前表面陡峭子午线:eZ二1一(15.612一yZ)/z，
扁平子午线:eZ二l一(15.612一x今/z，
角膜后表面陡峭子午线:eZ二1一[12.254(z一0.553)一y，]/(z一0.553)2
扁平子午线:e‘=1一[12，254(z一0.553)一x，]/(z一0.553)2
3.角膜曲率分布一除300 4.smm、3300 3.smm、3300 4.smm处总角膜曲率及30”
3.smm、210“4.smm处角膜后表面曲率的理论值与实测值之间有显著性差异
(只0.05)外，其余各点均呈高度一致性。由扁平子午线到陡峭子午线，角膜前、
后表面曲率、总角膜曲率呈正弦相关的规律性分布。
4.数学模型的三维重建及显示软件采用颜色曲率分布图方式描述角膜前后表面
的曲率?
Since it was first discovered that the corneal is the major refracting surface of the ocular refractive system, more and more investigators have devoted themselves to the research of this field, while to the research of the corneal shape especially. With the deepening of the investigation, the complexity of the cornea is being realized little by little. In fact, the normal cornea is an optical and anatomic structure which is sophisticated and precise extraordinarily.
The widespread applications of various computer-assisted corneal detecting instruments, accelerated the development of the clinical and scientific investigations in the corneal domain greatly. But some inevitable defects reduce their measuring accuracy, such as the confusion of paraxial and nonparaxial systems, the narrowness of datum information and the 1 imitated applicable coverage.
Cornea is the curved surface which has its own specific mathematical model theoretically. And this model is the mathematical formula
representing the surface shape of the object, namely geometric model. It' s the foundation to give an analysis, calculation or description of the object, and the key to detect the nature of the curved surface. It reflects the essential qualities of the curved surface to some degree, therefore, to make up the mathematical model of the body is the chief task of the accurate analysis.
Computer Aided Geometric Design, curved planar reformation chiefly, is just the way to construct the mathematical model according to the existing body. And its accelerated development and extensive application bring a brand-new approach to the research of the corneal shape. Objective To detect the inner characters of the corneal shape from a mathematical point of view and construct its correct geometric model, in order to provide the reliable theoretical guide and basis to the scientific and clinical investigations, the design and improvement of the detecting instruments and so on. To devise the three-dimension reconstruction and display software of the model so that it will be more stereoscopic, visual and vivid and can be used to display the corneal models of various human groups. Methods
1. Data collection Anterior corneal radius of curvature, posterior corneal radius of curvature, total corneal radius of curvature and corneal thickness of the points located 1. 5mm, 2. 5mm, 3. 5mm and 4. 5mm away from the corneal apex on certain meridians, including 0° , 30° , 60° , 90° , 120° , 150° , 180° , 210° , 240° , 270° , 300° and 330° meridians, were measured with OrbscanIIcorneal topography system,
2. Coordinate Cartesian coordinate was established: the origin 0 was placed at the corneal apex, and the optical, horizontal and vertical axises were defined as the Z , X and Y axises respectively. Then the locations
of all the points on the cornea were determined referred to the coordinate.
3. Datum processing SPSS 10. 0 for windows and Microsoft Excel of Office 2000 were used for statistical analysis. The statistical methods included Descriptive statistics, Independent-Sample T Test and Bivariate Correlations.
4. Mathematically modeling
(1) Mathematical model of anterior and posterior corneal surface Based on Bennett' s notation about conic sections and quadric theory, the corneal shape was analyzed and described with the quadric formula: y2=a1z +a2z2.
(2) Mathematical model of corneal shape factor(SF) The equation ez=l -(2rnz -y2)/ z2 which derives from Bennett' s notation about conic sections (y2=2r0Z - (1-e2) z2), was used to define the shape factors of the steepest and flattest meridians on anterior and posterior corneal surfaces.
(3) Distributions of corneal curvature With the formula F' =Fn+(Fh-F,n*. Sin2a which is about the curvatures of oblique meridians on toric surface, the theoretical values of anterior, posterior and total corneal radius of curvature of the oblique meridians were obtained. The concordance between the theoretical and the actual values were detected in order to determine the distributing rul
各种计算机辅助的角膜形态及光学特征检测设备和仪器的广泛应用，极大地推动了该领域临床和基础研究的发展，可以说带来了质的飞跃。然而，它们固有的一些不可避免的缺陷，大大地影响了其测量准确性，如近轴和偏轴光学系统的一概而论，检测数据来源的单一性，个体应用的局限性等。
角膜是一个曲面，即具有某种性质的点的集合。从理论上讲，每张曲面都有它自己的数学模型。数学模型是描述物体表面形状的数学表达式，也称为物体的几何模型。它是对物体进行分析、计算和绘制的根据，是研究曲面性质的工具，可以说从一定程度上代表了曲面的本质特性，因而，对这类物体进行分析、处理的首要任务就是为其建立数学模型。
计算机辅助几何设计，尤其是曲面重建技术正是根据已有曲面去构建反映其形状的数学模型的过程。它的飞速发展和广泛应用，为角膜形状特征的研究开辟了崭新的途径。
目的 本研究拟从数学角度对角膜形状特点进行较为深入的探讨分析，同时结合曲面重建技术建立其数学模型，旨在了解其本质，明确其内在特征，为今后的科研、临床乃至检测设备仪器的制造完善等多方面的工作提供科学的理论基础和指
浙江大学博士生学性伶灰
导。同时，建立该模型的三维显示软件，使其更加直观形象，并可用于不同群体
角膜模型的构建。
方法
1.数据采集应用Orbscanll角膜地形图系统采集角膜顶点，0”、朋“、60“、
90”、1200、150”、180。、210。、2400、270“、__300“、330“子午线__}_几五牡
角膜顶点分别为1.smm、2.smm、3.smm、4.smm处点的角膜前、后表面曲率半径、
总角膜曲率半径、角膜厚度数值。
2.坐标建立建立笛卡儿空间三维直角坐标:以角膜顶点为原点，眼球光轴方向
为Z轴，水平方向为X轴，垂直方向为Y轴，并确定各点的空间坐标值。
3.数据处理应用SPSS10.0及office 2000 Excel统计软件包对所有数据行统
计学分析及计算处理。统计学方法包括:描述性统计分析、独立样木t检验、双
变量相关分析。
4.数学模型的建立
(1)角膜前后表面数学模型以Bennett’s锥形切面数学理论为基础，根据公
式yZ=a，z十aZzZ并结合二次曲面截痕法，明确角膜前后表面的形状，并建立其相应
数学表达式。
(2)形状系数数学模型根据锥形切面数学公式yZ=Zrl)z一(1一eZ)22推导出eZ=1-
(Zrl)z一yZ)/22，并以此建立角膜前后表面两正交主子午线的形状系数的数学表达
式。
(3)角膜曲率分布根据托力克面斜轴子午线曲率F’计算公式
F’二F汁(F卜一民)* SinZQ，确定角膜前、后表面斜轴子午线曲率及斜轴子午线总角膜
曲率的理论值，并通过分析该理论值与实测值之间的一致性，明确角膜曲率分布
特征。
5.数学模型三维重建及显示软件
础，采用Windows
中文Windows2000、
V 1 sua1Studio
应用PC/CE台式兼容机，以上述数学模型为基
(vC+十6.0)编程软件包自行设计而成的，可在
XP、
6.数学模型软件的应用
结果
NT等环境下运行。
构建正视状态正常成人的角膜数学模型。
浙协大嵘博士生含位论式
1.角膜前后表面数学模型前后表面均为近似椭球面，其相应数学模型分别为:
角膜前表面:xZ/aZ+yZ/bZ+(z一e)2/e，==1(a，b，c>o)
角膜后表面:x，/aZ+yZ/bZ+(z一e一d，，)2/eZ=一(a，b，e>o)
本研究中，正视状态成人群体的角膜前、后表面数学模型分别为:
角膜前表面:xZ/8.053，+yZ/7.9732+(z一8.226)2/8. 2262砚
角膜后表面:XZ/6.8362+yZ/6 .745斗(z一6.974)2/7 .5272=1
2.角膜一前后表面形状系数的数学模型
角膜前表面陡峭子午线:eZ二1一(2r。z一y勺/22
扁平子午线:eZ二l一(2r。z一x今/22
角膜后表面陡峭子午线:eZ二1一「Zr，，(z一山)一yZ」/(z一山)“
扁平子午线:eZ二1一〔Zrl、(z一山)一xZ〕/(z一山)2
本研究中，正视状态成人群体的角膜性状系数数学表达式分别为:
角膜前表面陡峭子午线:eZ二1一(15.612一yZ)/z，
扁平子午线:eZ二l一(15.612一x今/z，
角膜后表面陡峭子午线:eZ二1一[12.254(z一0.553)一y，]/(z一0.553)2
扁平子午线:e‘=1一[12，254(z一0.553)一x，]/(z一0.553)2
3.角膜曲率分布一除300 4.smm、3300 3.smm、3300 4.smm处总角膜曲率及30”
3.smm、210“4.smm处角膜后表面曲率的理论值与实测值之间有显著性差异
(只0.05)外，其余各点均呈高度一致性。由扁平子午线到陡峭子午线，角膜前、
后表面曲率、总角膜曲率呈正弦相关的规律性分布。
4.数学模型的三维重建及显示软件采用颜色曲率分布图方式描述角膜前后表面
的曲率?
Since it was first discovered that the corneal is the major refracting surface of the ocular refractive system, more and more investigators have devoted themselves to the research of this field, while to the research of the corneal shape especially. With the deepening of the investigation, the complexity of the cornea is being realized little by little. In fact, the normal cornea is an optical and anatomic structure which is sophisticated and precise extraordinarily.
The widespread applications of various computer-assisted corneal detecting instruments, accelerated the development of the clinical and scientific investigations in the corneal domain greatly. But some inevitable defects reduce their measuring accuracy, such as the confusion of paraxial and nonparaxial systems, the narrowness of datum information and the 1 imitated applicable coverage.
Cornea is the curved surface which has its own specific mathematical model theoretically. And this model is the mathematical formula
representing the surface shape of the object, namely geometric model. It' s the foundation to give an analysis, calculation or description of the object, and the key to detect the nature of the curved surface. It reflects the essential qualities of the curved surface to some degree, therefore, to make up the mathematical model of the body is the chief task of the accurate analysis.
Computer Aided Geometric Design, curved planar reformation chiefly, is just the way to construct the mathematical model according to the existing body. And its accelerated development and extensive application bring a brand-new approach to the research of the corneal shape. Objective To detect the inner characters of the corneal shape from a mathematical point of view and construct its correct geometric model, in order to provide the reliable theoretical guide and basis to the scientific and clinical investigations, the design and improvement of the detecting instruments and so on. To devise the three-dimension reconstruction and display software of the model so that it will be more stereoscopic, visual and vivid and can be used to display the corneal models of various human groups. Methods
1. Data collection Anterior corneal radius of curvature, posterior corneal radius of curvature, total corneal radius of curvature and corneal thickness of the points located 1. 5mm, 2. 5mm, 3. 5mm and 4. 5mm away from the corneal apex on certain meridians, including 0° , 30° , 60° , 90° , 120° , 150° , 180° , 210° , 240° , 270° , 300° and 330° meridians, were measured with OrbscanIIcorneal topography system,
2. Coordinate Cartesian coordinate was established: the origin 0 was placed at the corneal apex, and the optical, horizontal and vertical axises were defined as the Z , X and Y axises respectively. Then the locations
of all the points on the cornea were determined referred to the coordinate.
3. Datum processing SPSS 10. 0 for windows and Microsoft Excel of Office 2000 were used for statistical analysis. The statistical methods included Descriptive statistics, Independent-Sample T Test and Bivariate Correlations.
4. Mathematically modeling
(1) Mathematical model of anterior and posterior corneal surface Based on Bennett' s notation about conic sections and quadric theory, the corneal shape was analyzed and described with the quadric formula: y2=a1z +a2z2.
(2) Mathematical model of corneal shape factor(SF) The equation ez=l -(2rnz -y2)/ z2 which derives from Bennett' s notation about conic sections (y2=2r0Z - (1-e2) z2), was used to define the shape factors of the steepest and flattest meridians on anterior and posterior corneal surfaces.
(3) Distributions of corneal curvature With the formula F' =Fn+(Fh-F,n*. Sin2a which is about the curvatures of oblique meridians on toric surface, the theoretical values of anterior, posterior and total corneal radius of curvature of the oblique meridians were obtained. The concordance between the theoretical and the actual values were detected in order to determine the distributing rul
引文
1. Duke-Elder S(ed): Systems of Ophthalmology. St Louis, CV Mosby Co, 1970, vol 5: Ophthalmic Optics and Refraction.
2. Gullstrand A: Appendix Ⅱ to von Helmholtz's Treatise on Physiological Optics, Vol 1: Southall JPC(ed). Rochester, NY, The Optical Society of America, 1924.
3. Aubert H, or Ericksen, cited by Mandell RB: Contact Lens Practice, ed 3. Springfield, Ⅲ, Charies C Thomas Publisher, 1981.
4. Clark BAJ: Topography of some individual cornea. Aust J Optom 1974; 57: 65-9.
5．刘祖国，谢玉环，张梅．正常人角膜前后表面地形即全角膜厚度的研究．中华
眼科杂志 2001，3：37(2)：125-8．
6. Roberts C. The accuracy of 'powers' maps to display curvature data in corneal topography systems. Invest Ophthalmol Vis Sci 1994 Aug, 35(9): 3525-31.
7. Barsky, Brian A. (2003) "Geometry for analysis of corneal shape," in lectures on Computational Geometry, Morningside Center of Mathematics lectures on Computational Geometry, joint publication series in Advance Mathematics. International Press(Providence, Rhode Island)/American Mathematical Society. (Somerville, Mass), 2003: 33-56.
8. Halstead, Mark A; Barsky, Brian A; Klein, Sranley A; and Mandell, Robert B. (1995) 'A spline surface algorithm for reconstruction of corneal. topography from a videokeratographic reflection pattern'. Optometry and Vision Science 1995 Nov: 72(11): 821-7.
9. Bennett, A.G. The calibration of keratometers. Optician 151: 317-22.
10. Bibby M. M. (1976) Computer assisted photokeratoscopy and contact lensdesign. Ⅰ. Optician, 171(4423), 37, 39, 41,43-44,171(4424), Ⅱ,14-15,17 and 171(4426), 15-17
11. Gatinel D, Haouat M, Hoang-Xuan T. Areview of mathematical descriptors of corneal asphericity. J Fr Ophthalmo 2002 Jan; 25(1): 81-90.
12. Lindsay R, Smith G, Atchison D. Descriptors of corneal shape. Optometry and Vision Science 1998 Feb; 75(2): 156-8.
13．同济大学应用数学系．微积分．高等教育出版社，2000 Jan：39-45．
14. von Helmhotz H. In: Southall JPC, ed. Helmhotz's Treatise on Physiological Optics. Vol. 1. New York: The Optical Society of America; 1924:5.
15. Mandell RB, St. Helen R. Mathematical model of corneal contour. Br J Physiol Opt. 1971; 26: 183-197.
16. Patel S, Marshall J, Fitzke FW. Shape and radius of theposterior corneal surface. Refract Corneal Surg 1993; 9:173-81.
17. Clinical Optics. Printed in The United States of America.
2. Gullstrand A: Appendix Ⅱ to von Helmholtz's Treatise on Physiological Optics, Vol 1: Southall JPC(ed). Rochester, NY, The Optical Society of America, 1924.
3. Aubert H, or Ericksen, cited by Mandell RB: Contact Lens Practice, ed 3. Springfield, Ⅲ, Charies C Thomas Publisher, 1981.
4. Clark BAJ: Topography of some individual cornea. Aust J Optom 1974; 57: 65-9.
5．刘祖国，谢玉环，张梅．正常人角膜前后表面地形即全角膜厚度的研究．中华
眼科杂志 2001，3：37(2)：125-8．
6. Roberts C. The accuracy of 'powers' maps to display curvature data in corneal topography systems. Invest Ophthalmol Vis Sci 1994 Aug, 35(9): 3525-31.
7. Barsky, Brian A. (2003) "Geometry for analysis of corneal shape," in lectures on Computational Geometry, Morningside Center of Mathematics lectures on Computational Geometry, joint publication series in Advance Mathematics. International Press(Providence, Rhode Island)/American Mathematical Society. (Somerville, Mass), 2003: 33-56.
8. Halstead, Mark A; Barsky, Brian A; Klein, Sranley A; and Mandell, Robert B. (1995) 'A spline surface algorithm for reconstruction of corneal. topography from a videokeratographic reflection pattern'. Optometry and Vision Science 1995 Nov: 72(11): 821-7.
9. Bennett, A.G. The calibration of keratometers. Optician 151: 317-22.
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