26-th ECMI Modelling Week Final Report Dresden, Germany

Transcription

1 26-th ECMI Modelling Week Final Report Dresden, Germany

2 Group 4 Mathematics of the eye: modelling corneal curvature. Federica Bani Dept. of Mathematical Sciences, University of Florence, Viale Morgagni, 67/a , Florence, Italy. Éric Lluch Dept. of Mathematical Sciences, Universitat Autónoma de Barcelona, Barcelona, Cerdanyola del Valles 08193, Barcelona, Spain. Olexandra Dmytrenko Dept. of Applied Math, National Technical University of Ukraine Kyiv Polytechnic Institute, 37, Prospect Peremohy, 03056, Kyiv-56, Ukraine. Juan Sánchez Ingenieria Termica y de Fluidos, Universidad Carlos III de Madrid, Avda. De la Universidad 30, Leganes, Madrid, Spain. Christoph Wanzke Dept. of Mathematics, Technische Universität Kaiserslautern Gottlieb-Daimler-Straße, Kaiserslautern, Germany. Alisa Zeleva Department of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Milchakova, 8a, , Rostov-on-Don, Russia. Instructor: Lukasz P lociniczak Faculty of the Fundamental Problems of Technology, Wroc law University of Technology, Institute of Mathematics and Computer Science ul. Janiszewskiego 14a Wroc law, Poland. 2

3 Abstract Keratoconus is one of the corneal diseases that influences the ability to see. This condition modifies the shape of the eye, which leads to the loss of vision. Therefore, this illness has to be diagnosed on its early stage. If a person feels a change of a vision in one eye, he needs a medical examination. Videokeratoscope is frequently used to measure eye parameters, with the help of which the diagnosis of eye condition can be done. For many diseases the corneal curvature is the main indicator that can give a conclusion about the problem. The aim of our work is to find the method to diagnose the Keratoconus with the help of automatic analysis on computer to eliminate a doctor s mistake. To reach the result, using MATLAB we processed cornea s curvature original data from the healthy eye and developed the polynomial formula for its surface. The model of the sick eye was based on the analysis of images of sick ones. The database with information about the other eye corneal diseases can be made by collecting the data. Using it gives the possibility to make a diagnosis and taking advantages from the built model.

4 2 Mathematics of the eye: modeling corneal curvature 4.1 Introduction Our eyes are very powerful instruments used for perception and cognition of the world. This is because the fact that with the ability to see a person gets 80% of the information about the environment. However, it is a very delicate and sensitive mechanism. That is why it is very important to keep them safe. Unfortunately, sometimes they are damaged by diseases or mechanical injury. According to statistics one in three has problems with eye health. The most common diseases are myopia, hyperopia, cataracts and astigmatism. Keratoconus occurs much less often, but in the last decade due to environmental degradation the incidence of keratoconus has increased. Keratoconus is a degenerative non-inflammatory disease of the eye in which the cornea thins and takes a conical shape. Keratoconus can cause serious vision loss. Most often patients complain of photophobia, diplopia, blurring the image. Until now, keratoconus remains a little studied disease, reasons for its occurrence are not clear, as it is not possible to predict the course of the disease and the diagnosis. Disease are treated better and easier in the early stages. When it is not possible to cure the disease often it can be held up. The first step is to examine the eye in order to put up the diagnosis. To perform it, the complex structure of eye has to be taken into consideration. First of all we have to decide which features healthy eyes posses, and what is a deviation from them. The difficulty lies in the fact that all eyes are individual, it is important for healthy eyes to show common characteristics by which laws of health can be established. If the patient s eye does not correspond to indicators of healthy eyes then the conclusion of the disease should be done. In our work we used a database from actual eye that includes indicators like axial curvature, elevation and thickness of its cornea. The model of an eye can be created in several ways, for instance, ellipsoid has a similar shape. Moreover, paraboloid and more complex functions can be also applied. In our view, the model of eye can be done from these database using splines. We chose this method because polynomial functions are convenient to operate and its coefficients can give us a lot of information. For all calculations and plotting we used MATLAB. The main goal of our work was to study the shape of the curvature of an eye affected by the keratoconus and compare it to the curvature of a healthy eye. As a result we have got that healthy eye has eight-shape while the illness eyes has a deformed shape. 4.2 Eye anatomy The eye is a very powerful instrument of vision which has relatively complicated structure. On the Fig. 4.1a you can see some of its parts.

5 26th ECMI modelling week 3 This is what happens before we see something: light waves enter the eye through the cornea, go to the pupil and with the help of crystalline lens focuses on the retina. The received image is in inverse. It becomes right with the help of our brain. This process is shown on schematic Fig. 4.1b. From this description it s clear that the curves of eye elements, their density and clearness influence out vision. Let s consider them. (a) Eye structure (b) Formation of an image of a tree on the retina of an eye Figure 4.1: Eye and image formation schemes Cornea The main purpose of the cornea is to keep ultraviolet rays away from retina and protect the eye from infections and elementary damages. Speaking about its characteristics it has to be clear (transparent) and have the incorrect curvature, so that the angles that participate in the formation of the images are accurate. The human s and primate s cornea consists of 5 parts: corneal epithelium, Bowman s layer, corneal stroma, Descemet s membrane and corneal endothelium. The epithelium consists of fast-growing and easily regenerated cells that need to be moist with tears. The most significant component of the total refractive power of the eye is the air-tear film interface which is situated on this layer. The Bowman s layer consists of a similar irregularly arranged collagen fibers. Its main purpose is to protect the stroma. In contrast to this layer the cornea stroma consists of regularly arranged collagen fibers along with sparsely distributed interconnected keratocytes, which are the cells for general repair and maintenance. This layer takes up to 90% of cornea thickness. Descemet s membrane is so to say modified Bowman s membrane which also consists of collagen. It derives other cells, especially helps in endothelium regeneration. Its thickness depends on the age and sex of a person and varies from 3 to 20m. The corneal endothelium consists of mitochondria-rich cells which are responsible for regulating fluid and solute transport between the aqueous and

6 4 Mathematics of the eye: modeling corneal curvature corneal stromal compartments. The thickness of this layer is approximately 5m Pupil The pupil is a hole in the center of an iris through which light passes. Its main function is to control the amount of rays that move across the eye. For example when its dark the pupil is wide in order to let the maximum amount of light in, but when an eye looks at something bright it is narrow Crystalline lens Crystalline lens is responsible for keeping images in focus on the retina. Its shape is adjusted depending on the distance to the object Retina The retina is a film of an eye which converts light rays into electrical signals. These signals are sent to the brain through the optic nerve. It is responsible for peripheral, central and color vision. An interesting fact: the reason for the cat s eyes glow in darkness after some light falls on them is that there is a layer of silvery guanine crystals called tapetum which reflects the light. These crystals are also on the fish scale. As far as having the corresponding equipment it is relatively easy to get the parameters of cornea curvature and it has severe impact on the vision we examined it in the aspect of keratoconus. 4.3 Methods for Measuring Corneal Topography There are 3 different ways to specify the corneal topography: by the surface elevation having chosen a certain reference surface, by the local slopes with respect to a reference sphere or by the local curvature. There are lots of methods to measure these properties that allow to specify the corneal topography. Some of these methods can be classified according to the physical phenomenon used in the measurement The Specular reflection technique The Specular reflection technique considers the anterior corneal surface as a convex mirror. The corneal is treated as a mirror and with these techniques the reflection of light (described by the law of reflection) from a surface can be obtained. With these techniques the slope of the reflected rays with respect to a reference sphere can be measured. Then, using an algorithm and the slope data the curvature and the elevation properties of the cornea can be obtained.

7 26th ECMI modelling week 5 There are some different specular reflection techniques. One of them is called Placido Disk System. It consists of a series of concentric illuminated rings that are reflected off of the cornea and viewed with a video camera. Other techniques are the interferometric methods. They are useful to measure small separations between 2 wavefronts being able to know where large changes of curvature occur. Finally there is another important technique called Moire Deflectometry in which the corneal is mounted in the course of a beam followed by a couple of transmission gratings that are superimposed and placed at a distance from each other The diffuse reflection technique The diffuse reflection technique changes the specular reflection of the anterior surface of the cornea in order to make a diffuse reflector. Using this modification it s easier to obtain a fringe projection. With this fringe pattern on the reflecting corneal surface the elevation can be obtained. With the elevation data and using the first and the second derivatives of the elevation function some types of curvature can be obtained. There are 3 relevant diffuse reflection techniques: Moire Fringes, Rastertereography and the Fourier Transform Profilometry. All of them use similar optical systems. The optical axis of the fringe projector and the optical axis of the video camera stay in the same plane and intersect at a point close to the eye The scattered light technique The scattered light technique uses the scattering property of the light when the light beams are transmitted in the corneal tissue. Some of this scattered light appears in the anterior surface of the cornea and is obtained by an optical video system. Then the anterior and posterior corneal surface can be obtained by this optical system and by triangulation. In this way the elevation of both anterior and posterior corneal surfaces are measured with respect to a reference plane. There is only one technique that use the scattered light method. This technique is called Slit-Based systems. The slitlamp is an instrument consisting of a high-intensity light source that can be focused to shine a thin sheet of light into the eye with adjustable width over any desired location. 4.4 Curvature properties As we said in the previous part of the report, the curvature is the main tool to understand and investigate which kind of disease affects the eye. We need

8 6 Mathematics of the eye: modeling corneal curvature to define the notion of curvature that we will use from a mathematical point of view. First of all we recall the principal curvatures definition. For each point x S, S R 3, where S is a given surface, the principal curvatures are the maximum and the minimum of the curvature of a curve, γ(x) S, x γ. We can now give the following definitions. The Main curvature is the arithmetic mean of the two principal curvatures while the Gaussian curvature is obtained multiplying the two principal curvatures. However, in the study of the corneal curvature, we will refer to the Axial curvature. This has an unambiguous definition if the cornea shows axial symmetry. If the corneal normal is not in the meridional plane, we have at least three definitions. In the first case we can define in unambiguous way the axial distance which is the distance from the corneal point to the videokeratograph axis in the direction of the corneal normal. Axial curvature can be defined as 1 d in mm 1 or as A = d (diopters) where is the factor that, by convention, converts from inverse millimeters to diopters. The distance d is given by: d = r sin(α) where r is the distance from the corneal point to the axis (in a direction normal to the axis) and α is the slope angle of the corneal normal in the meridional plane as shown in (4.2). Figure 4.2: Axial curvture definition

9 26th ECMI modelling week 7 In the second case we can t calculate the distance d and the previous definition become ambiguous. That s why we provide an algorithm to model this curvature. 4.5 Polynomial fitting Data obtained from the corneal measurements consist of a huge amount of point coordinates. This data is not useful to make a corneal database because it needs a lot of memory. In order to reduce the data size, a fitting surface can be used. Many functions can be used for this purpose([3]) but in this project polynomial functions will be chosen. Polynomial functions are easy to manage and can provide a lot of information with only a few coefficients. Due the corneal shape, no more than 6th degree polynomial functions is needed. Also, the subroutine for the coefficients calculations are given. In this project the MATLAB function polyfit is used Fitting procedure Two independent polynomial functions, one for each direction x and y are used for the fitted surface, so this surface has the expression: F (x, y) = a n x n + a n 1 x n a 1 x + b m y m + b m 1 y m b 0, (4.1) where n is the polynomial degree in x direction and m the polynomial degree in the y direction. The zero-order term only needs to be calculated in one direction. The surface is a translation surface that uses the y curve as guideline curve. The x curve is the generatrix curve; this curve goes along the guideline to generate the surface To obtain the coefficients, a cross-section in the corresponding direction is used. In order to provide a noise reduction, the mean of three consecutive cross sections is used. The position of the section is determined by the maximum of the surface to fit (either elevation or curvature). Figure 4.3 shows this sections. The polynomial fit is made with the MATLAB function polyfit for each direction. The structure of of the process is showed in figure 4.4. Due the fitting, for a later representation of the surface, the grid data is not needed (Figure 4.5). Figures 4.6 and 4.7 show the real and the fitted surfaces of elevation and curvature. Figure 4.8 shows the error for two different fitting of a curvature surface. For a 6th degree polynomial fit, a 3% error is obtained in the main part of the eye.

10 8 Mathematics of the eye: modeling corneal curvature Figure 4.3: Position of the cross sections used for the fitting. determined by the maximum position They are SURFACE DATA - Mesh: x, y - Data: Z(x,y) ANALISYS DATA - Polinomial degree: n x, n y - Number of cuts: l 1. Surface cuts 2. Mean of cuts 3. Remove useless data 4. Fit POLYNOMIAL COEFFICIENTS X coef & Y coef Figure 4.4: Fitting process Disease parameter To provide an easy way to diagnostic eye diseases like Keratoconus, a simple parameter can be defined. Due to the eight shaped curvature of an healthy eye, a relation between the maximums and their positions can be defined: d 2 p := 1 + D 1 2 d D2 2 1 (4.2) the distances d 1, d 2, D 1 y D 2 are defined in the figure 4.9. For a healthy eye this parameter is near to 1. With the fitting provided in this work, this calculation only needs half of the calculated coefficients 4.6 Conclusion In this project a simple model of fitting corneal data surfaces is provided based on translation surfaces. This fitting can be used to save memory in databases. The polynomial fitting is a simple way to do this and a lot of

11 26th ECMI modelling week 9 SURFACE DATA Mesh: x, y POLYNOMIAL COEFFICIENTS SURFACE CONSTRUCTION -polynomial evaluation APPROXIMATE SURFACE -Z a (x,y) -Plot X coef & Y coef INDEX CALCULATION -Max & Min calculations SURFACE INDEX Figure 4.5: Latter representation of the fitted surface Figure 4.6: Elevation fitting subroutines are generated and optimised yet. Also, a parameter to evaluate Keratoconus is defined. This parameter gives a quick way to diagnose diseases and to sort and compare data in a eye data-base. The fitted surface also can be used in the measurement equipment. A polynomial surface is an easy way to integrate or derive properties(elevation or curvatures) from the measurements(elevation or slopes).

12 10 Mathematics of the eye: modeling corneal curvature Figure 4.7: Curvature fitting Figure 4.8: Curvature fitting error for a 2nd degree in x and 4th degree in y (left) and 6th degree for both directions (right). With only 14 coefficients the error is reduced to the 2% in main region of the eye.

13 26th ECMI modelling week 11 Figure 4.9: Dissease parameter definition. The graph represents a transversal cut in the direction of the two curvature maximums in a healthy eye. For a healthy eye this curve must be quite simetrical.

14 Bibliography [1] Klein, SA. Axial curvature and the skew ray error in cornela topography. Optometry and Vision Science 74(11), , ISSN [2] Mejía-Barbosa, Y and Malacara-Hernández, D. A review of methods for measuring corneal topography. Optometry and Vision Science 78(4), , ISSN [3] Okrasiński, W and P lociniczak, L. A nonlinear mathematical model of the corneal shape. Nonlinear Analysis: Real World Applications 13(3), , ISSN doi: /j.nonrwa URL S