How To Analyze Surgically Induced Astigmatism

Transcription

1 Acta Ophthalmologica Scandinavica Thesis Assessment and Statistics of Surgically Induced Astigmatism Kristian Næser Faculty of Health Sciences, University of Aarhus, Denmark. Eye Clinic, Regional Hospital Randers and Department of Ophthalmology, Aarhus University Hospital and Department of Opthalmology, Aalborg Hospital, Aarhus University Hospital, Denmark Aarhus 2008

2 Acta Ophthalmologica Scandinavica 2008 Denne afhandling er i forbindelse med nedensta ende anførte tidligere offentliggjorte artikler af det Sundhedsvidenskabelige Fakultet ved Aarhus Universitet antaget til offentligt at forsvares for den medicinske doktorgrad. Aarhus Universitet, den 5 februar 2008 Søren Mogensen, Dekan Forsvaret finder sted fredag den 2 maj 2008, klokken præcis i Auditorium 424, Anatomisk Institut, Aarhus Universitet. The present thesis is based on the following publications, which are referred to in the text by their Roman numerals. None of these publications have previously been part of a doctoral or Ph.D. thesis. I Naeser K. Conversion of keratometer readings to polar values. J Cataract Refract Surg 1990; 16: II Naeser K, Behrens JK & Næser EV. Quantitative assessment of corneal astigmatic surgery: Expanding the polar values concept. J Cataract Refract Surg 1994; 20: III Naeser K & Behrens JK. Correlation between polar values and vector analysis. J Cataract Refract Surg 1997; 23: IV Naeser K. Assessment of surgically induced astigmatism; Call for an international standard. J Cataract Refract Surg 1997; 23: V Naeser K & Hjortdal J. Bivariate analysis of surgically induced regular astigmatism. Mathematical analysis and graphical display. Ophthal Physiol Opt; 1999, 19: VI Naeser K & Guo S. Precision of autokeratometry expressed as confidence ellipses in Euclidian 2-space. Ophthal Physiol Opt 2000, 20: VII Naeser K & Hjortdal J. Polar value analysis of refractive data. J Cataract Refract Surg 2001; 27: VIII Naeser K. Popperian falcification of methods of assessing surgically induced astigmatism. J Cataract Refract Surg 2001; 27: IX Naeser K & Hjortdal J. Multivariate analysis of refractive data. Mathematics and statistics of spherocylinders. J Cataract Refract Surg 2001; 27: X Naeser K, Knudsen EB & Kaas-Hansen M. Bivariate polar value analysis of surgically induced astigmatism. J Refract Surg 2002; 18: XI Naeser K & Hjortdal J. Concepts of regular astigmatism in first order optics and wave front analysis. S Afr Optom 2004; 63: XII Naeser K & Hjortdal J. The power of a cylinder in an oblique meridian: revisiting an old controversy. Ophthal Physiol Opt 2006; 26: This thesis is available online at: Copyright ª Acta Opthalmol. Scand. Printed in the UK by the Charlesworth Group 2

3 ACTA OPHTHALMOLOGICA SCANDINAVICA 2008 Contents Acknowledgements 4 Abstract 5 Part I. Introduction 5 I.1. Synopsis of part I 6 Part II. Basic concepts 6 II.1. Abbreviations, directions, and formats 6 II.2. First order optics 7 II.3. Definition of regular astigmatism 7 II.4. Optical and physiological effects of ocular astigmatism 8 II.5. Refraction and aberration 8 II.6. Description of astigmatic surfaces by differential geometry 9 II.7. The sine-squared correlation and its controversy 9 II.8. Measurement of regular astigmatism 10 II.9. Corneal refractive index 10 II.10. Corneal structure and its change by surgery 11 II.11. Synopsis of part II 11 Part III. Assessment methods for surgically induced astigmatism 11 III.1. Definitions 11 III.2. Early methods 11 III.3. Polar value method 12 III.3.1. Polar values as a general equation 12 III.3.2. Surgically induced astigmatism and error of refractive procedures 13 III.3.3. Clinical significance of polar values 13 III.3.4. Decomposition to polar values with a Fourier approach 14 III.3.5. Bivariate and trivariate polar value analysis 14 III.3.6. Reconversion from polar values to a spherocylindrical format 15 III.3.7. Statistical analysis of polar values 15 III.4. Contemporary methods 16 III.4.1. Methods based on Stokes s principle and the sine-squared correlation 16 III.4.2. Methods based on Long s power matrix 16 III.5. Advanced methods 16 III.5.1. Fourier analysis of irregular surfaces 16 III.5.2. Wavefront analysis 17 III.5.3. Other advanced methods 17 III.6. Synopsis of part III 17 Part IV. Polar value and other component analyses of refractive data 17 IV.1. Precision of autokeratometry examined by bivariate statistical methods 17 IV.2. Trivariate analysis of the accuracy of autorefraction 18 IV.3. Effect of misalignment of an astigmatic correction 18 IV.4. The flattening effect of various cataract incisions 19 IV.5. Bivariate analysis of surgically induced astigmatism after cataract extraction 19 IV.6. Javal s rule in pseudophakie eyes 19 IV.7. Aggregate analysis on corneal laser ablative refractive surgery 20 IV.8. Synopsis of part IV 20 Part V. Conclusions 20 Part VI. Perspectives 21 Part VII. Summary in Danish - Dansk Resumé 22 VII.1. Baggrund 22 VII.2. Polærværdi metoden 22 VII.3. Andre metoder 23 VII.4. Konklusion og perspektiver 23 References 23 Appendix Numerical Examples 27 3

4 Acknowledgements This thesis is based on 12 papers published from 1990 to 2006 during my employment as consultant eye surgeon at the Department of Ophthalmology, Ålborg Sygehus Syd, the University Department of Ophthalmology, A rhus Sygehus and the Eye Clinic, Randers Centralsygehus. I have truly enjoyed this research work, which has made the daily clinical activities much more interesting and rewarding. The first three papers in the thesis gained international attention. One study formed the basis of the computer program Astigmeter, which was distributed internationally by the Leo Medical Company. In the 1990s the issue of surgically induced astigmatism (SIA) was the topic of many oral and written discussions in Denmark and abroad. I would like to express my gratitude to the many persons involved in the project, and assure that I have enjoyed both the friendly words and the controversies. My interest in cataract surgery was aroused during my training period as a cataract surgeon in the Department of Ophthalmology, Vejle Sygehus, under Dr Leif Corydon s inspiring leadership. The younger ophthalmologists were given ample opportunities for studying and travelling abroad, and the original inspiration for this thesis was a lecture on SIA given in 1987 by Dr Thomas Cravy, in Uppsala, Sweden. It has been a privilege and much fun to cooperate with all co-authors. All authors have been full-time clinicians, and thus time has been scarce, but the studies have generally served as vehicles for new learning and social life in the departments. First of all I would like to acknowledge the invaluable support and commitment of Dr Jesper Hjortdal, who is co-author in five of the studies, and who performed the computer graphics in three additional papers. Jesper Hjortdal s vast knowledge in the field of clinical optics, his expertise on computer graphics and his efficacy has been crucial for the whole project. Jesper Hjortdal was also very helpful in giving the review paper an appropriate format. I am very thankful to my other co-authors, Drs Jens Christian Behrens, Erik Vincent Næser, Suping Guo, Ellen-Birthe Knudsen, and Mette Kaas-Hansen. Dr Suping Guo had to learn some Danish in order to perform the clinical examinations and was much missed by the patients, when she returned to Dalian, China. Professor Niels Ehlers gave invaluable advice on the final format and submission of the thesis. Professor Toke Bek was very helpful in organizing my sabbatical leave during the final stages of the writing process. I have had expert help from the following professionals from the University of A rhus: Associate Professor Svend Terp from the Institute of Economy, Professor Michael Væth from the Department of Biostatistics, Professor Eva Vedel Jensen from the Institute of Mathematics, and Associate Professor Jørgen Ellegaard Andersen from the Institute of Mathematics. I thank my former colleagues Drs Jørgen Andersen and Carl Uggerhøj Andersen from A lborg for many inspiring discussions. My present colleagues, Drs Peter Isager, Henrik Vorum, and Jan Kjær Pedersen, were very helpful in their constructive criticism and proof reading of the thesis. Thomas Næser, M.A., performed additional linguistic corrections. Former Managing Editor of Acta Ophthalmologica Mrs. Vibeke Allen performed the final proof reading. Professor William Harris from Johannesburg, South Africa, and Dr Noel Alpins from Melbourne, Australia, are thanked for constructive discussions on the topic of astigmatism. My secretaries Karin Fredriksen, Lene Røjkjær, and Charlotte Christensen helped type and edit the review paper. Helle Brandstrup Larsen and John Bra uner from the Department of Clinical Photography, Randers Centralsygehus, constructed some of the computer drawings in the final manuscript. All are gratefully acknowledged. Finally, I want to express my deepest thanks for the patience and support provided by my parents Johannes ( ) and Anna Elise Naeser, my beloved wife Ruth and our children Thomas, Esben, and Johan. The study was supported by Statens Sundhedsvidenskabeligt Forskningsra d, Cykelhandler P. Th. Rasmussen og hustrus Mindelegat, Hotelejer Carl Larsen og Hustru Nicoline Larsens Mindelegat, Landsforeningen til Værn om Synet, Forskningsinitiativet for A rhus Amt, Alcon Denmark, and Desire e and Niels Ydes Fond. Conflicts of interest I have no potential conflicts of interest to declare. 4

5 Thesis Assessment and Statistics of Surgically Induced Astigmatism Kristian Næser Faculty of Health Sciences, University of Aarhus, Denmark ABSTRACT. The aim of the thesis was to develop methods for assessment of surgically induced astigmatism (SIA) in individual eyes, and in groups of eyes. The thesis is based on 12 peer-reviewed publications, published over a period of 16 years. In these publications older and contemporary literature was reviewed 1. A new method (the polar system) for analysis of SIA was developed. Multivariate statistical analysis of refractive data was described 2 4. Clinical validation studies were performed. The description of a cylinder surface with polar values and differential geometry was compared. The main results were: refractive data in the form of sphere, cylinder and axis may define an individual patient or data set, but are unsuited for mathematical and statistical analyses 1. The polar value system converts net astigmatisms to orthonormal components in dioptric space. A polar value is the difference in meridional power between two orthogonal meridians 5,6. Any pair of polar values, separated by an arch of 45 degrees, characterizes a net astigmatism completely 7. The two polar values represent the net curvital and net torsional power over the chosen meridian 8. The spherical component is described by the spherical equivalent power. Several clinical studies demonstrated the efficiency of multivariate statistical analysis of refractive data 4,9 11. Polar values and formal differential geometry describe astigmatic surfaces with similar concepts and mathematical functions 8. Other contemporary methods, such as Long s power matrix, Holladay s and Alpins methods, Zernike 12 and Fourier analyses 8, are correlated to the polar value system. In conclusion, analysis of SIA should be performed with polar values or other contemporary component systems. The study was supported by Statens Sundhedsvidenskabeligt Forskningsra d, Cykelhandler P. Th. Rasmussen og Hustrus Mindelegat, Hotelejer Carl Larsen og Hustru Nicoline Larsens Mindelegat, Landsforeningen til Værn om Synet, Forskningsinitiativet for A rhus Amt, Alcon Denmark, and Desirée and Niels Ydes Fond. Key words: Astigmatism, refractive surgery, cataract surgery, polar values, statistics, mathematics, differential geometry, cornea. Acta Ophthalmol. Scand. 2008: 5 28 ª 2008 The Author Journal compilation ª 2008 Acta Ophthalmol Scand doi: /j x I. Introduction The aim of cataract and refractive surgery is to achieve a good visual function with the refraction tailored to each patient s needs. This goal may be reached through disposal of a number of methods with known refractive effect and subsequent election of a suitable method based on the patient s preoperative refraction I. Cataract surgery is increasingly developing into refractive surgery, and the annual number of cataract and refractive procedures amount to several millions worldwide. In Denmark, approximately cataract 13 and 2500 (year 2000 level) refractive 14 procedures are performed each year. Quality assurance of surgically induced astigmatism (SIA) and other refractive components has therefore become a public health issue, and analysis of SIA is as important as the technical part of the procedure 15. The most pronounced effect of cataract and refractive surgery is a change in sphero-cylindrical power. The spherical power is defined by a single variable 16. The astigmatic power is a complex entity, characterized by direction in degrees and magnitude in dioptres. These incompatible entities are not suitable for quantitative analysis IV,VI,VIII, The mathematical difficulties involved in management of spherocylinders have led to the publication of a number of more simple methods These methods are still used in scientific publications, but will often produce systematic errors or inconsistent results. More importantly, they may 5

6 induce errors in scientific conclusions and clinical decision-making. The present thesis is based on 12 papers I-XII, published over a time span of 16 years. The aim of the studies has been to develop methods enabling analysis of past and optimization of future refractive procedures. The publications may collectively be termed polar value analysis of SIA. Polar values are mathematical transformations of astigmatic directions and magnitudes to orthonormal components in dioptric space III,VI,VII,IX,X. A dioptric space is a vector space with orthogonal axes and units of dioptres (D) VI,VII,IX,X,16,33. In this space a power vector represents the net cylinder or the spherocylinder 16. The components are coordinates of power vectors in dioptric space. In this space all types of mathematical and statistical analyses may be performed in the usual manner. Polar value analysis applies for both individual patients and for groups. The height of interest in SIA was reached in the early nineties, along with the transition from planned extracapsular cataract extraction to phakoemulsification. The major argument for this transition was the good control of postoperative astigmatism after phakoemulcification, and there was a great demand for relevant methods for assessment of SIA IV,15. The first paper I in the series of polar value analysis gained immediate international attention, and the described methodology was used in publications in the first special issue on SIA, published in the Journal of Cataract and Refractive Surgery in ,35. The methods I-V,IX have been gradually developed to match the increasing complexity and diversity of the refractive procedures. Multivariate statistical analysis V,IX of refractive surgical data was introduced in ophthalmology and presented in the second special issue on SIA published in the Journal of Cataract and Refractive Surgery in 2001 VII,IX. The polar value system was the first component-based method in ophthalmology. During the last 13 years several other component-based systems for assessment of SIA have been published Publication I in the thesis describes the surgically induced flattening along the vertical meridian following superior cataract incisions. Paper II reports on the flattening along a random meridian after phakoemulsification and refractive surgery. A complete description of SIA in the form of flattening and torque is given in publication III. A method for calculating the magnitude and direction of an average astigmatism is described in publication IV, while bivariate and multivariate statistical analyses of refractive data are reported in publications V and IX. Application of polar value analysis on various clinical problems is described in publications VI X. Concepts of regular astigmatism in first order optics and wavefront analysis are compared in publication XI. The validity of the sine-squared correlation is discussed in publication XII. In the thesis older methods are mentioned, contemporary methods reviewed, and the polar value method described in detail. It will be demonstrated that contemporary analysis of SIA requires conversion of refractive data to relevant orthonormal components in dioptric space followed by multivariate statistical analysis. I.1. Synopsis of part I Polar values are mathematical transformations of astigmatic directions and magnitudes to orthonormal components in dioptric space. II. Basic concepts II.1. Abbreviations, directions, and formats 1. Frequently used abbreviations:aca, Anterior corneal astigmatism; AKP, Astigmatic polar values ¼ polar values recorded along the preoperatively more powerful meridian; Dioptres, D; DC, Dioptre(s) of Cylinder(s); DS, Dioptre(s) of Sphere(s); KP(U), Polar value along the meridian U; LASIK, Laser in situ keratomileusis; M, Astigmatic magnitude M P 0; n, Refractive index; RA, Residual astigmatism; S +, Spherical power in plus cylinder transposition SEP ¼ The spheroequivalent power ¼ S þ þ 1=2M ð1þ TOA-Ab Cornea, Total ocular astigmatic aberration in the corneal plane; TOA-Ab Spectacle, Total ocular astigmatic aberration in the spectacle plane; TOA-Ref Cornea, Total ocular astigmatic refraction in the corneal plane; TOA-Ref Spectacle, Total ocular astigmatic refraction in the spectacle plane.. Vectors are shown in bold, for example as t, n or b 2. Directions (Fig. 1):, Symbol for degrees; a, Astigmatic meridian expressed in degrees ¼ direction of principal meridian of maximal positive (or least negative) power for a spherocylinder I-III,XII ; / ¼ (a + 90), Astigmatic axis in plus cylinder format; U, Direction of the plane under examination; b, Angle between axis of net cylinder and plane ðuþ under examination ¼ð/ UÞ o ¼ðða þ 90Þ UÞ o : ð2þ 3. Formats:K, a o ) ¼ net cylinder I-III,IX or net astigmatism in plus format, indicates the power along a. Spherocylindrical formats: Minus cylinder axis format ¼ ((S + +M)) M axis a o ) ¼ ((S + + M) ) M a o ) Plus cylinder power format ¼ (S + M along a o ) ¼ (S + M@a o ) Combined cylinder power format ¼ðS þ þ MÞ@a o ÞÞ and ðs o Þ A S T β 0 y T S A α Φ Ø x ð3þ Fig. 1. Three-dimensional representation of a planocylinder with its maximal power along SS and zero power along the axis in AA XII. An oblique section along TT is inclined b with respect to AA. The angles between the reference direction X and the various meridians are shown as symbols. Meridian SS along angle a. Meridian AA along angle Ø. Meridian TT along angle U. Any coordinate system and reference direction may be chosen and any meridian may be analyzed. The analysis is always based on the inclination angle b between the cylinder axis and the oblique meridian under investigation. 6

7 The magnitude (M) of the net astigmatism a o ) is the absolute difference in power between the two principal meridians. The direction (a) is the principal meridian of maximal positive (or least negative) power for a spherocylindrical surface. The direction is given in units of degrees ( ). See Example 1 in the Appendix. II.2. First order optics The propagation of electromagnetic radiation through space displays wave-like characteristics such as diffraction and interference, but may also be described as particles 46(pp.1)11and pp.545)554). Transmisssion of light is attenuated by absorption, reflection, and scatter. Refraction is the bending of light caused by the change in its velocity as it passes from one transparent media to another. The refractive index n for a medium is the ratio between the speed c of light in vacuum and the speed v m of light in that specific medium 46(p.8) n ¼ c ð4þ v m Snell s law of refraction over curved surfaces states that the angle of incidence I and the angle of refraction RF are related by the equation 47(p.43) : n I sin I ¼ n RF sin RF; ð5þ where n I and n RF represent the refractive indices of the two media separated by the refractive surface. Light propagating through the human eye is subjected to all the mentioned physical modifications, but only wavelengths ranging from about 380 to 780 nm are visible, i.e. capable of eliciting a retinal neural response. In physics exact Snell s law optics is applied in all calculations. In first order or paraxial optics only the most central and axis-parallel rays are considered. First order optics is exclusively used in optometry and ophthalmology. First order optics is a reminiscence from a time when computational complexity was a problem. First order optics assumes paraxiality, coaxiality and absence of tilt, and these assumptions are not met in complex optical systems, such as the human eye. Its continued use in ophthalmology is supported by the pupillar aperture, the asphericities of refractive surfaces, the Stiles-Crawford effect, the density of cones in the fovea, and the cerebral image processing, which all favour the narrow paraxial bundle of light. Calculations on the schematic eye XI and on videokeratographic models 48,49 indicate that the paraxial area is limited to the central 2 mm of the cornea. For small angles expressed in radians the sine function is approximated from 46(p.449) : sin a a a3 3! þ a5 5! a7 þ... ð6þ 7! For infinitely small paraxial angles the sine function may be expressed by the first of the terms on the right side of the equation sign, hereby transforming Snell s law of refraction to the first order approximation: n I I ¼ n RF RF ð7þ Unless otherwise stated, first order optics will be assumed in the following. In Snell s law and first order optics both the incident and refracted rays and the normal to the point of incidence are assumed to lie in one plane. The power of a lens or a refraction is defined as the reciprocal of the reduced focal length or far point, respectively 50(p.8),51. Some of the difficulties in calculating SIA are related to the continued use of the simplified model of first order optics XII. The power P of a curved surface (Fig. 2) with curvature C and radius r is derived as 50(p.8) : P ¼ C ðn RF n I Þ¼ 1 r ðn RF n I Þ ð8þ Power and curvature have the unit of D (m )1 ). Fig. 2. Each point of a plane curve is characterized by its curvature 52. For the point A the curvature C is the reciprocal of the radius of curvature of the best-fitting (osculating or kissing ) circle. The orthogonal unit tangent and normal vectors are denoted by t and n, respectively. A curvature is positive, if the curve turns in the same direction as the chosen normal of the surface. II.3. Definition of regular astigmatism In an optimal stigmatic (point-like) optical system, a point in the object space is focused as a point image (Fig. 3). The shape of the blurred 50(pp.62)77) image for out-of-focus point objects is always a circle 53,54. An astigmatic 55 optical system is nonpoint-like 47(pp.274)292). In a regular astigmatic optical system an object point is focused as two mutually perpendicular line segments delimiting an intermediate interval, termed Sturm s conoid 47(pp.62)64),56 (Fig. 4). No point focus is formed, but the blurred image attains different shapes and directions in Sturm s interval 47(pp.62)64), 50(pp.78)92). The focal lines define the orthogonal principal meridians of the conoid. Differential magnification in the principal meridians means that the focal line segments have different lengths 50(pp.78)92). Irregular astigmatism is the non-stigmatic and non-regular astigmatic part of the refractive Fig. 3. A stigmatic ocular optical system. In the absence of diffraction, aberration, and scatter a point in object space is focused as a point image. An object located in the far point is focused on the retina. Objects from any other position are defocused and blur circles are projected on the retina. Fig. 4. An astigmatic with-the-rule ocular optical system. The mutually perpendicular focal lines delineate Sturm s interval. The position of the circle of least confusion is the dioptric average between the two focal lines and is determined by the spherical equivalent power. No point focus is formed. The image projected on the retina is always blurred due to its variable shapes and directions. 7

8 spectrum 50(pp.78)92). Strictly speaking, only stigmatic and regular astigmatic refraction are defined within the concept of paraxial optics 57. Irregular astigmatic entities require broader optical bundles 57. Inclusion of the two terms on the right side of equation (6) produces the third order Seidel aberrations 46(pp.449)469). In wave-front analysis, stigmatic and regular astigmatic powers correctable by spherocylinders - are termed lower-order aberrations. Irregular astigmatic elements are called higher-order aberrations. Unless otherwise stated, the term astigmatism will refer to regular astigmatism in the following. Anterior corneal astigmatism (ACA) refers to the toricity of the anterior corneal surface, as measured by keratometry or videokeratography. Total ocular astigmatic refraction (TOA-Ref) is the astigmatism determined by manifest refraction or autorefraction. TOA-Ref is created by the toricity, tilt or lack of alignment of optical axes of the ocular refractive surfaces 47(pp.274)292),58. Astigmatism is essentially a 3-dimensional phenomenon, and astigmatic surfaces are usually torioidal or cylindrical. All astigmatic surfaces have one principal meridian of maximal curvature and an orthogonal principal meridian of minimal curvature. A withthe-rule corneal astigmatism is present when the most powerful (steeper) anterior corneal meridian is along 90 degrees I,47(pp.274)292). Conversely, a maximal curvature along the zero degree meridian indicates an againstthe-rule corneal astigmatism. The powers of the principal meridians may be calculated and expressed in D with equation (8). The astigmatic magnitude is the difference between these powers. II.4. Optical and physiological effects of ocular astigmatism The optical effects of ocular astigmatism are blur and distortion. Blur is the lack of point focus and altered shape of the retinal image caused by the astigmatism. Distortion is the altered shape of objects caused by unequal (differential) magnification of the retinal image in the various meridians. The distortion is proportional to the ametropia 59 and the distance from the entrance pupil of the eye to the sites of astigmatism, the cornea and the lens 50(pp.233)243). In an optically uncorrected astigmatic eye, the retinal image is blurred proportional to the astigmatic refractive error. As the cornea and lens are close to the entrance pupil, the distortion amounts to only approximately 0.3% per DC of uncorrected astigmatism 50(pp.233)243). In an optically corrected 59 astigmatic eye, blur may be eliminated. The distortion depends primarily on distance between the entrance pupil and the corrective refractive surfaces 59. The distortion is approximately 1.6% per DC in a spectacle correction and only 0.3% per DC in a contact lens correction 50(pp.233)243). The tilt of vertical lines is maximal for correcting cylinders in oblique directions 50,59, amounting to 0.4 degree of tilt per dioptre monocularly 50,59(pp.233)243). In a surgically corrected astigmatic eye both blur and distortion may be eliminated, as the correction is applied directly towards the sites of astigmatism in the cornea and lens. Eyes with an oblique axis of astigmatism often do not have a good corrected visual acuity. 60,61 In printed matter the vertical strokes of letters predominate. Therefore, when a focal line on the retina is oriented vertically, the letter visual acuity is improved. In with-the-rule uncorrected simple myopic astigmatism the vertical focal line falls on the retina for distant objects, but the horizontal focal line is closest to the retina for objects in the near distance 56,62,63 (Fig. 4). This eye would be expected to have better distant Snellen acuity than an eye with uncorrected against-the-rule astigmatism, which conversely would be superior for reading purposes. Physiological evidence also points to either a horizontal or vertical axis for optimal performance 64,65. Postoperative astigmatic directions along these meridians might therefore be desirable targets for some patients 56. II.5. Refraction and aberration The subjective refractive error indicates the optical (spectacle) correction allowing parallel incident light to focus on the fovea. The optical correction may actually be located in front of or in the eye. The optical correction modulates the incident wavefront. In the present context, aberration is conceived as a wavefront emanating from the eye. The correction and the total ocular aberration at any given plane should be identical, but of opposite signs: Aberration ¼ Correction ð9þ For example, in order to calculate the SIA of a toric intraocular lens (IOL), ray tracing for both the incident and emerging wave front to the plane of the IOL is required 66,67. However, for corneal refractive surgery the correction and aberration are calculated in the corneal plane, which requires conversion of the spectacle refraction (considering vertex distance) to the anterior corneal surface. Only the combined cylinder format shown in equation (3) may be used for vergence calculations 36,39,68. Isolated aberration of the anterior corneal surface is measured by conventional keratometry or video-keratography. Keratometric data are always in notation. A power format for the aberration will allow for direct comparison with keratometric data. A plus power format usually gives the best visualization of refractive data. Residual astigmatism (RA) 47(pp.274)292) is the component of total ocular astigmatism that is not attributed to the anterior corneal surface 69. RA is therefore the difference between the total ocular astigmatic aberration in the corneal plane (TOA-Ab Cornea ) and the anterior corneal astigmatism ACA, both expressed as net astigmatisms 69. RA ¼ TOA-Ab Cornea ACA ¼ -TOA-Ref Cornea ACA; ð10þ where TOA-Ref Cornea is the total ocular astigmatic refraction transformed from the spectacle to the corneal plane. RA amounts to approximately 0.5 D against-the-rule in younger subjects A14 tilt around the vertical axis of the crystalline lens would theoretically give this amount of RA 50(p.209). Actually, the tilt of the crystalline lens is much smaller 72,73, which points to the posterior corneal surface as the source of the RA. Both anterior and posterior corneal surfaces have the same principal meridians 74. The posterior corneal surface has twice the toricity and a third of the astigmatic power of the anterior surface 74. 8

9 Javal s rule 47(pp.274)292) is a crude correlation between the magnitudes of the refractive cylinder in the spectacle plane and the keratometer reading, which both are regarded as positive with-therule and negative against-the-rule: TOA Ab Spectacle ¼ 1:25 ACA 0:50 ð11þ A modified Javal rule with 1.0 as the regressions coefficient 75,76 apply to corneal apertures ranging from 2 to 7mm 75. Both corneal and total ocular astigmatisms increase with age and converts towards against-the-rule astigmatism with approximately 0.14 D in 5 years 77,78. The against-the-rule astigmatism may be attributed to changes in the cornea 77,78. Both vertical and horizontal corneal radii steepen with age, but the horizontal steepening is largest 50(pp.415)416),79,80, hereby causing an against-the-rule change 50,79,80. Some with-the-rule astigmatism should therefore be preserved in refractive procedures in younger patients. II.6. Description of astigmatic surfaces by differential geometry According to the theory of differential geometry 52, a space curve is uniquely determined except for its position in space - by its curvature and torsion XII,81. Curvature and torsion are measured in units of dioptres. For any point it is possible to find a plane with an optimal fit to the curve. This is the osculating plane 81. The mutually perpendicular unit tangent t and normal n vectors lie in the osculating plane, with the binormal vector b orthogonal to this plane (Fig. 5). The osculating Fig. 5. Each point of a space curve or surface is characterized by its curvature C and torsion T. The vectors (t, n, b) constitute a positively oriented orthonormal basis in a socalled Frenet frame or local coordinate system, where b ¼ t n 52(p.28). Axis Fig. 6. A cylinder with the radius r. A line parallel to the axis and located on the cylinder surface is called a ruling of a cylinder. A tangent line cutting the cylinder rulings with a constant angle b creates a circular helix along the cylinder surface 81. r plane contains a unique circle with contact of at least the second order 81(p.34) at a given point. This is the circle of curvature, characterized by a centre and radius r of curvature 81. The curvature (C) ¼ 1/r is a measure of the rate at which the curve is turning away from the tangent 81,or the change in slope by the tangent vector divided by the arch length 48,82. The torsion (T) is a measure of the rate at which the curve is twisting out of the osculating plane 81,83. At each point of a surface it is possible to identify the principal directions 84 ; one direction of maximal and one orthogonal direction of minimal curvature. Principal meridians of astigmatic surfaces display zero torsion, but in all other meridians both curvature and torsion are present 46(pp.233)264). Therefore, an oblique meridian of an astigmatic surface does not stay in one plane 83. Consider a cylinder with the radius r (Fig. 6). An oblique meridian with a constant angle b to a line parallel to the axis creates a circular helix on the cylinder surface, a segment of which is identical to the narrow band of light over the arch TT in Fig. 1. The function of the circular helix may be parameterized as F(t) ¼ (rcos(t),rsin(t), rcot(b)æt) 81(p.31). Curvature and torsion of a circular helix may be calculated with the following general equations XII,81(p.36)37) : Curvature C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðf 0 ðtþf 00 ðtþþ ðf 0 ðtþf 00 ðtþþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðf 0 ðtþf 0 ðtþþ 3 ¼ sin2 b r ð12þ Torsion T ðf 0 ðtþf 00 ðtþþ F 000 ðtþ ¼ ðf 0 ðtþf 00 ðtþþ ðf 0 ðtþf 00 ðtþþ sin b cos b ¼ ð13þ r In these equations F (t),f (t) and F (t) are the first, second and third derivative of a parameterized vector 81,(x y) the cross product, and (xæy) the scalar product of two random vectors 85. The sign of the torsion is ambigious 52,81,83,86, but is correct as stated, when the cylinder axis is chosen as reference direction. This mathematical description of a surface is in accordance with the general definition of an astigmatic surface, based on two orthogonal principal meridians. The following text will show that the curvature and torsion for a cylindrical helix and the dioptric power in an oblique meridian of a cylinder may be described by similar equations XII. II.7. The sine-squared correlation and its controversy The sine-squared correlation describes the power of a cylinder in an oblique meridian and is the backbone of both optical equations and most electronic measuring devices for astigmatism XII,51, However, its validity has repeatedly been challenged during the last century XII, as experiments have failed to demonstrate a specific power of an isolated oblique meridian 82, 83, A plano-cylinder has its maximal power, M, along its meridian a and zero power in the plane along its axis Ø. The sine-squared correlation 46(pp.233)264) states that the power P in an oblique meridian inclined b with respect to the axis of the cylinder is given by I,XII,16,26,46,82,97,98 : 9

10 P ¼ M sin 2 b ð14þ From equation (2) follows that this power alternatively may be expressed as XII : P ¼ M sin 2 ðða þ 90Þ UÞ ð15þ Similarly, for a toroidal surface (Fig. 7) with principal powers S + and (S + + M) the meridional or curvital power is XII,16,28,87 : P Curvital ¼ S þ þ M sin 2 b ¼ S þ þ M sin 2 ðða þ 90Þ UÞ ð16þ The net torsional power may be described as XII : P Torsional ¼ 2Msinb cos b ð17þ Multiplying (n RF ) n I ) with equation (12) yields expression (14) for curvital power. Multiplication with equation (13) gives equation (17) for net torsional power III, XII. Thus, the expressions for curvital and torsional power, derived from the sine-squared correlation, are similar to the formal mathematical description of astigmatic surfaces. In the paraxial simplification astigmatic power is defined solely by the curvature of principal meridians XII. The sine-squared paradox is explained by a model of two operating powers S+M 2.0 SEP 1.5 S P(Φ) α = 60 Ø = 150 Fig. 7. Meridional power plot of a spherocylinder as a function of U. Abscissa: the meridian U in degrees. Ordinate: the meridional power P(U) in dioptres. The spherocylinder (1.0 DS ) has its maximal power of +2.0 D along 60 and its minimal power of +1.0 D along 150. The meridional power is the sum of the sphere and the astigmatic magnitude M, attenuated by the sine-squared correlation: P(U) ¼ sin 2 (( ) ) U). Alternatively XII, meridional power may be expressed as a cross cylinder of magnitude 1/2M, oscillating around the spherical equivalent power, SEP: P(U) ¼ cos 2(60 ) U). ½M M Φ in first-order optics. Both powers are calculated by the sine-squared correlation. Curvital power is responsible for refraction in the meridional plane, while the torsional component twists the light out of that plane 83,91. In all meridians, the combined curvital and torsionsal powers direct incoming wavefronts into the two orthogonal focal lines. In conclusion, the sinesquared correlation is valid and may therefore be used in optical equations for assessment of SIA. II.8. Measurement of regular astigmatism In most autokeratometers and autorefractors the mean value of several measurements is simply the median of the accepted measurements for sphere, cylinder, and axis. This separate analysis of refractive components does not consider the multivariate nature of a spherocylinder (Fig. 8). In some corneal topographs, regular astigmatism is indicated as non-orthogonal entities and sometimes even in hemimeridians 99. This is clearly a misconception 18, as an astigmatic surface is defined by its orthogonal principal meridians and specific symmetry over a 360 degrees surface 50(pp.78)92). An expression for regular astigmatism for a refractive surface or wavefront XI,100 is always possible with proper data analysis XI,100. Fig. 8. Typical print-out from a commercial auto-keratorefractometer. The measurement with an asterix (*) is the median value for each separate component of a spherocylinder. Astigmatic direction and magnitude are separated and a few outliers may distort the measurements. For a scientific calculation see Figure 18. The average cornea may be described as a prolate ellipsoid with an asphericity Q-value of approximately ) ,79. Bennett 101,102 defined two planes to describe peripheral corneal curvature in an aspherical cornea: the tangential and sagittal. A tangential section at a specific point contains both the normal and the corneal axis. A sagittal section contains the normal and is perpendicular to the tangential plane. Radii of curvature may be calculated for both ellipsoids of revolution 48,102 and for ellipsoids in general 103. In corneal topography, tangential and axial curvature maps correctly reflect power in the paraxial area, but grossly underestimate the spherical aberration of the peripheral cornea 48,49,104. Both autorefractors and conventional keratometers measure over a diameter of approximately 3 mm 69. Using the sagittal 101,102 radius of curvature, Dunne 69 demonstrated minimal change in astigmatism from the apex to a distance of 3 mm from the apex in an aspherical corneae. In conclusion, corneal asphericity only marginally affects central corneal power, measured by keratometry. Following surface ablation procedures the cornea cannot always be described as an ellipsoid, and conventional keratometry of corneal power is generally unreliable. II.9. Corneal refractive index The average corneal index of refraction of (p.387) should be used for procedures such as photorefractive keratectomy and laser in situ keratomileusis (LASIK), where only the anterior corneal surface is changed 36. In the effective corneal refractive index the negative power of the posterior corneal surface is taken into account. In Guldstrand s schematic eye the ratio between the posterior and anterior corneal radii of curvature is 6.8/7.7 ¼ (p.387). More contemporary assumptions vary from to Most keratometers use an effective corneal refractive index of , but recommended values include , , , and An effective corneal refractive index should be used for astigmatitic keratometry, limbal relaxing incisions, and cataract surgery, in which an equal change in both anterior and posterior corneal surfaces is assumed. 10

11 II.10. Corneal structure and its change by surgery The stroma has preferred collagen orientation in vertical and horizontal directions, but at the limbus the fibrils assume a tangential direction and fuse to form a dense circum-corneal annulus 107. Incisional surgery such as cataract extraction and astigmatic keratotomy creates gaping of the wound, addition of tissue in the incision, and elongation of the radius of curvature with subsequent flattening of the central cornea along the surgical meridian 31,108. Surgically induced astigmatism following tangential and arcuate incisions is characterized by coupling, by which an astigmatic change in one meridian is followed by a similar change in the orthogonal meridian with no net change in SEP 23,56,109. Tight sutures will flatten the tissue around the incision with an initial steepening of the central cornea 108,110. Sutures have no long-time effect on astigmatism 20, and an initial with-the-rule astigmatism is followed by against-the-rule decay after superior sutured cataract incisions 108. The effect of surgery is therefore similar to placing a cross cylinder with its minus-power along the surgical meridian 56. Surgically induced astigmatism after incisional surgery is predominantly influenced by wound length 114 and placement 56, while patient age, preoperative astigmatism, and intraocular pressure play minor roles 115. Ablational laser surgery does not generate a coupling effect, and the spherical effect of ablation for astigmatism must be taken into account. II.11. Synopsis of part II In regular astigmatism an object point is focused as two mutually perpendicular line segments. No point focus is formed. The uncorrected astigmatic eye is blurred due to absence of point focus, and distorted because of differential magnification in the two principal meridians. A meridian of an astigmatic surface is fully described by its curvature and torsion, which may be derived by either first order optics and the sine-squared correlation or by differential geometry. Tangential corneal incisions are followed by flattening along the surgical meridian. Deep corneal incisional surgery is characterized by coupling, by which an astigmatic change in one meridian is followed by a similar change in the opposite direction in the orthogonal meridian. In ablational corneal surgery no coupling is induced. III. Assessment methods for surgically induced astigmatism III.1. Definitions Following a refractive procedure the postoperative spherocylinder may be conceived as the sum of the preoperative and the surgically induced spherocylinder. The surgically induced spherocylinder is therefore defined as the difference between the post- and the preoperative spherocylinders I,II,VII,X,31,32,116 : Surgically induced spherocylinder ¼ Postoperative spherocylinder Preoperative spherocylinder ð18þ Two spherocylinders placed in contact with their axes at random in general have the same fundamental properties as a single spherocylinder 50(pp.86)89). Equation (18) may be regarded as a specialized case for subtraction of spherocylinders, based on the theory of addition of spherocylinders in oblique directions Several reviews of SIA have been published VIII,25,26,28,56,100, A method for assessing SIA should be optically meaningful for the surgeon, correct for both single and aggregate data and allow for statistical analysis VIII. The traditional refractive format represents a general problem with analysis of SIA. The three components of a spherocylinder are unsuited for mathematical and statistical analyses, as they are dependent on each other 19, specified in polar form 16, incompatible in units VI, and yield systematic errors for aggregate data (IV, VIII). Furthermore, as a quantitative measure a spherocylinder is mathematically unorthodox: it is not a vector, matrix, or any other entity, whose rules for algebraic conversion are known 33. In an orthonormal coordinate system the basis vectors are mutually perpendicular and normalized to unit length. Following transformation of spherocylinders to appropriate components in orthonormal vector space, mathematical and statistical analyses may be performed with usual methods V,IX. A vector space V is characterized by closure under addition, so that any two vectors a and b determine a vector (a + b), which belongs to V. It is further determined by closure under multiplication, so that any vector a and some real number k determine the vector ka in V 85,122. Non-zero pair wise orthogonal vectors are always linearly independent. These vectors can be functions and for any two basis vectors f(x) and g(x) orthogonality is proved by the inner product (scalar or dot product) equating to zero 85 : Z p p gðxþ fðxþdx ¼ 0 ð19þ The definition of vector space and equation (19) will be used in the following discussion of various assessment methods for SIA. III.2. Early methods The theory of adding obliquely crossed cylinders was originally described by Stokes 123, who suggested a graphical vector analysis with all astigmatic angles drawn with twice their angles. The sine-squared correlation with subsequent differentiation 26,97,124 or other methods 87,125,126 have been used to elaborate equations for SIA. Gartner 127 first described the optical decomposition of a cylinder, which formed the working principle in Humphrey s Vision Analyser 128,129. In vector analysis with Bennet s astigmatic decomposition 50(pp.78)92), any spherocylinder may be decomposed into a SEP and two cross cylinders at (0/90 ) and (45/135 ) (Fig. 9). These magnitudes are mathematically (linearly) independent. All angles are doubled according to Stokes 123, and for a cylinder of magnitude M and axis /, the two vector components are III,129 : C 0 ¼ M cos 2/ ð20þ C 45 ¼ M sin 2/ ð21þ Calculating a combination or an average of spherocylinders is accomplished by compiling the elements C 0, C 45, and SEP, independently. 11

12 Fig. 9. Net cylinder of the power M in the axis Ø. The figure shows the principle behind Bennet s astigmatic decomposition: The true axis of the cylinder is along Ø. After all axes are doubled, the cylinder is represented by the vector OR, which may be projected on the abscissa and ordinate as OA and OB, respectively. These methods were developed in optometry and were most frequently used to find the resultant of two obliquely crossed cylinders. However, in 1968 Naylor 32 suggested the use of vector analysis for the assessment of SIA. Jaffe and Clayman 31 later described the following three methods, based on Stokes double angle concept: 1. Graphical vector analysis. 2. Analysis by the law of sines and cosines. For an individual patient or observation this equation has the following general format VIII : M 2 SIA ¼ M2 preop þ M2 postop 2M preop M postop cosð2/ postop 2/ preop Þ ð22þ / SIA ¼1=2arctan½ðM postop sinð2/ postop Þ M preop sinð2/ preop ÞÞ =ðm postop cosð2/ postop Þ M preop cosð2/ preop ÞÞŠ ð23þ 3. The rectangular coordinate method, where the components are identical to Bennet s C 0 and C All methods yield identical results in analysis of a single data set. Only the rectangular coordinate method has the potential for giving correct results for aggregate data, but the precise methodology for aggregate analysis was not published. Many researchers have correctly calculated astigmatic direction and magnitude for each individual patient, but only averaged the astigmatic magnitude for aggregate analysis in the subsequent Jaffe vector analysis. The method, also termed Astigmatic magnitude not considering axis VIII, gives inconsistent results and systematic errors VIII,26,46. Due to its simplicity this method is still extensively used and has the following format IV : Average of astigmatic magnitudes P P q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M C 2 0 ¼ ¼ þ C2 45 ð24þ n n This entity is a scalar. According to the theory of vector spaces the average of several astigmatic vectors is not a scalar, but a new astigmatic vector with direction and magnitude IV,122,126. The correct calculation of magnitude and direction of the average astigmatism is IV : Magnitude of average astigmatism IV sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 P 2 C0 C45 ¼ M Mean ¼ þ n n ð25þ Axis of average astigmatism 50ðpp:78 92Þ P C45 / ¼ arctan M mean n P C0 n ð26þ Several later papers fall into the group of Jaffe vector analysis. They are variations over the law of sines and cosines and typically have small improvements in calculation of SIA for an individual patient, but identical systematic errors for aggregate data. The vector decomposition VIII,25,26 method calculates the with-the-rule and against-the-rule components from equations (22) and (23). However, for aggregate data the method yields a systematic error VIII,14, which may be quantified 130. Cravy 23 described a mathematical method for reporting the with-the-rule and against-the-rule change after surgery. The method is not based on optical principles. Naeser 22 later reported an identical method with a simplified mathematics. In the algebraic method 20 the astigmatic magnitudes with meridians between 45 and 135 degrees are multiplied with +1 and are considered with-the-rule; for all other values of a the magnitude is multiplied with )1 and is considered against-the-rule. For aggregate data the algebraic method yields results very similar to Cravy s and the polar value method, KP(90) VIII,26. In the simple subtraction method astigmatic magnitudes are subtracted without considering axis directions VIII,26. The algebraic and the simple subtraction methods are inaccurate for single data set VIII. Finally, some researchers have suggested separate analysis of astigmatic directions 21,24,42,131, which does not give meaningful results for aggregate data 25. In summary, with the exception of the Gartner 127 /Bennett 129 /Humphrey 128 methods and of the rectangular coordinate version of the Jaffe/ Claymann 31 method, all these early approaches are erroneous for either single or aggregate data. Many of these methods are still used in scientific publications, but may induce errors in scientific conclusions and clinical decision-making. The reasons for these flaws are isolated analysis of astigmatic direction and magnitude 20,21, incorrect components 22,23, and the use of the method Astigmatic magnitude not considering axis VIII, In the latter method relevant components are actually used for calculation of SIA in individual patients, but for that purpose the data are reconverted to the usual spherocylindrical format. It is necessary to remain with components in dioptric space in order to perform calculation of average values and statistical analysis. III.3. Polar value method The polar value method I-XII is based on the sine-squared correlation. All refractions and keratometries are converted to a plus power (@) net cylinder format. A net astigmatism with magnitude in dioptres and direction in degrees is transformed to orthonormal polar values with units in dioptres. Any refractive astigmatic surface may be described. The surgically induced polar value in the meridian U is the meridonal I,II,XII power causing a decrease or increase in power along that plane. The surgically induced polar value in the meridian (U + 45) is the torsional III,XII force twisting the astigmatic direction in a counterclockwise or clockwise direction. These components in the form of polar values characterize the SIA completely. III.3.1. Polar values as a general equation For a toroidal surface the polar value in a random meridian U is defined as the difference between the meridional or curvital power along U and its orthogonalmeridian(u + 90) I,II,XII (Fig.10). 12

13 Fig. 10. Net cylinder of the power M in the axis Ø. The polar value KP(U) in a random meridian U is the difference in meridional power between the power projected on the planes along U and (U + 90). The figure illustrates the polar value in 90 degrees, but any meridian may be analyzed. According to equation (16) the polar value in U degrees emerges as: KPðUÞ¼ðS þ þ M sin 2 ðða þ 90Þ UÞÞ ðs þ þ M cos 2 ðða þ 90Þ UÞÞ ¼ M ½sin 2 ðða þ 90Þ UÞ cos 2 ðða þ 90Þ UÞŠ ¼ M cos2ðða þ 90Þ UÞ ð27þ KP(U) is the net refractive power acting along the plane U and may be termed the net meridional, net on-axis or net curvital power along U. The polar value in (U + 45) degrees is the difference between the powers of the oblique meridians in (U + 45) and (U ) 45) degrees II,III,XII (Fig. 11): KPðU þ 45Þ ¼ S þ þ M sin 2 ðða þ 90Þ ðu þ 45ÞÞ ðs þ þ M cos 2 ðða þ 90Þ ðu þ 45ÞÞÞ ¼ M 2 sinðða þ 90Þ UÞ cosðða þ 90Þ UÞ ¼ M sin 2ðða þ 90Þ UÞ ð28þ Fig. 11. The polar value in (U + 45) degrees. In this example U ¼ 90 degrees, and the figure illustrates the polar value in 135 degrees, KP(135). KP(U + 45) is the power twisting the astigmatic direction towards the plane through either (U + 45) or (U ) 45) and may be termed the net oblique, net off-axis or net torsional power over U III. Naeser et al. III used the term rotation for the off-meridional change in astigmatism, but torque or torsion is probably a better description 44 for the tangential force producing a change in both astigmatic direction and magnitude X. The spherical equivalent power, SEP, is the average of any two orthogonal powers, as: 1=2½ðS þ þ M sin 2 ðða þ 90Þ UÞÞ þðs þ þ M cos 2 ðða þ 90Þ UÞÞŠ ¼ S þ þ 1=2M ð29þ The three components, net curvital power, net torsional power, and SEP characterize a spherocylinder. Net curvital and net torsional power may be regarded as cross cylinders in directions (U,U + 90) and (U + 45,U ) 45) degrees, respectively IX. Linear and mathematical independence is shown by equation (19). Any mathematical conversion such as addition, subtraction or averaging of spherocylinders may therefore be performed independently by these orthogonal components. See Example 2 in the Appendix. III.3.2. Surgically induced astigmatism and error of refractive procedures 1. SIA expressed as polar values is the difference between the postoperative and the preoperative polar values I,II,VII,X : ðkpðuþ SIA ; KPðU þ 45Þ SIA Þ ¼ðKPðUÞ Postop ; KPððU þ 45Þ Postop Þ ðkpðuþ Preop ; KPððU þ 45Þ Preop Þð30Þ Generally, a positive KP(U) SIA indicates a surgically induced increase in power, and a negative KP(U) SIA a surgically induced decrease in power of the meridian U. For the anterior corneal surface, a positive KP(U) SIA indicates a surgically induced steepening, and a negative KP(U) SIA a surgically induced flattening of the meridian U. For KP(U + 45) SIA a positive value means a counter-clockwise change, and a negative value a clockwise torque III. See Example 3 in the Appendix. 2. The error of the refractive procedure is the difference between the actually achieved and the planned postoperative refraction, expressed as polar values I-III,VII,X : ðkpðuþ Error ; KPðU þ 45Þ Error Þ ¼ðKPðUÞ Postop ; KPðU þ 45Þ Postop Þ ðkpðuþ Planned ; KPðU þ 45Þ Planned Þ ð31þ Positive and negative values for KP(U) Error means an undercorrection and an overcorrection, respectively. A positive KP(U + 45) Error indicates an unintended counter-clockwise torque, a negative KP(U + 45) Error an overly clockwise torque. See Example 4 in the Appendix. III.3.3. Clinical significance of polar values Polar value analysis may be applied on any type of refractive surgery and for any refractive plane. Aberrations and refractions may be reconverted to the corneal and spectacle plane with vergence formulas 36,67,68,132. However, most often surgery is performed on the cornea. The aim of incisional or ablational corneal astigmatic surgery is to flatten the steeper meridian, steepen the flatter meridian 90 away, or a combination II. KP(U) SIA, the surgically induced change in polar values along a surgical meridian U, always indicates the efficacy of the surgical procedure. A positive value signifies a surgically induced net steepening, a negative value a net flattening of the surgical meridian. KP(U + 45) SIA represents the net torsion along the surgical meridian III. A positive value indicates a counter-clockwise torque, a negative value a clockwise torque. Usually, no surgically induced torsion is desired and KP(U + 45) SIA is therefore an index of the accuracy of the refractive surgery. To target a spherical cornea the incision should be placed in the steeper preoperative meridian, and an incision with a known flattening effect equivalent to the preoperative steepening should be elected. No torsion should be produced III. By choosing different values for U in equations (27) and (28) various situations may be analyzed III : 1. For U ¼ direction of the surgical plane, a measure of the flattening and 13

14 torque over this meridian III is obtained. Changes in the astigmatism projected on this incision are called with-the-wound and against-thewound, respectively 24. Most often the surgical meridian coincides with the preoperative steeper meridian. 2. For U ¼ steeper preoperative meridian, the astigmatic polar values, AKP and AKP(+45) III, are obtained. Changes in the astigmatism projected on the preoperatively steeper and flatter meridian are called with-thepower and against-the-power, respectively II. All changes in astigmatism are referred to the preoperative steeper meridian. For the preoperative net cylinder U) the preoperative values of AKP and AKP(+45) always amount to M and zero, respectively. 3. Inserting U ¼ 90 yields KP(90) and KP(135). KP(90) I, the polar value in 90 degrees, is defined as the difference between the with-the-rule and againstthe-rule components. KP(90) is positive for with-the-rule and negative for against-the-rule corneal astigmatism. See Fig. 10. KPð90Þ ¼M ðsin 2 a cos 2 aþ ¼ M ðcos 2aÞ ð32þ KP(135), the polar value in 135 degrees, calculates the balance between the components projected on the 135 degrees and 45 degrees meridian (Fig. 11). KP(135) is positive for net astigmatisms in the right quadrant and negative for net astigmatisms with meridians varying between zero and 90 degrees. KPð135Þ ¼M ðsin 2 ða 45Þ M cos 2 ða 45ÞÞ ¼ M ðsin 2aÞ ð33þ These polar values are used for analysis of surgical procedures along the 90 degrees meridian and for population statistics. For corneal surgery, a positive KP(90) SIA indicates a withthe-rule change and a negative KP(90) SIA an against-the-rule change with net flattening of the 90 degrees corneal meridian I. 4. U ¼ 0 yields the polar values in zero and 45 degrees: KPð0Þ ¼M ðcos 2aÞ ð34þ KPð45Þ ¼M ðsin 2aÞ ð35þ These polar values are used for analysis of SIA with temporal cataract Diopters +M -M M cos 2α M sin 2α Fig. 12. KP(0) and KP(45) as a function of the astigmatic meridian a in degrees. The two identical, but phase-shifted functions are related by their derived functions, as: dðkpðuþþ ¼ dð cosð2bþþ dðbþ dðuþ ¼ 2 sinð2bþ¼ 2 KPðU þ 45Þ,KP ðu þ 45Þ¼ 1=2 dðkpðuþþ dðuþ incisions and for population statistics. KP(0) and KP(45) as a function of a are shown in Fig. 12. Polar value analysis along fixed meridians for procedures in various surgical planes gives information on the result of surgery. Off-axis refractive treatments will invariably create torque in different preoperative corneal net cylinders 44. Polar value analysis of the surgical meridian yields information on the process in the form of surgically induced flattening and torque. Only polar value analysis of the surgically meridian is relevant for planning and optomizing future surgeries. When the surgical meridian is selected to coincide with the preoperative steeper meridian, maximal reduction in astigmatism may be achieved by smallest possible surgical means. III.3.4. Decomposition to polar values with a Fourier approach The expression for spherocylindrical meridional power may be transformed XII to the sum of a constant and two harmonically related sine and cosine waves 16,100, which is the definition of a Fourier transformation. PðUÞ ¼S þ þ M sin 2 ðða þ 90Þ UÞ ¼ SEP þ 1=2M cos 2ða UÞ ¼ SEP þ 1=2KPð0Þ cos 2U þ 1=2KPð45Þ sin 2U ð36þ This curve is shown in Fig. 13. For any point the meridional power is equal to the sum of the spherical equivalent and two cross cylinders, one cross cylinder with axes along (0,90 ) and another cross cylinder with axes along(45/135 ) XII. The coefficients in form of the polar values KP(0) and KP(45) determine the amplitude of the harmonic curves. P(Φ) S+M 2.0 SEP 1.5 S P(Φ) cos 2Φ 0.44 sin 2Φ Power of cross cylinder (D) Φ Fig. 13. Meridional power of a spherocylinder may be decomposed to the sum of the SEP and two cross cylinders, representing net curvital and net torsional power of the net cylinder. The spherocylinder shown in Figure 7 and with the traditional format ( ) attains the Fourier format: P(U) ¼ 1.5)0.25 cos 2U sin 2U. Axes as in Figure 7. Fourier components are per definition orthogonal 16 and therefore mathematically independent 133. III.3.5. Bivariate and trivariate polar value analysis 1. Bivariate analysis of cylinders V. A cylinder may be described as a single point in an orthonormal 2-dimensional vector space of two polar values separated by an angle of 45 degrees. This vector space is cylinderequivalent with unit bases of 1.0 DC. Fig. 14 illustrates the calculations from Example 3 in the Appendix. 2. Trivariate analysis of spherocylinders IX. A spherocylinder may be described as a single point in an orthonormal 3-dimensional vector space of SEP combined with any pair of polar values separated by an arch of 45 degrees. An example is shown in Fig. 15. The vector space is sphereequivalent with unit bases of 1.0 DS IX,4. The power vector has the following form: Fig. 14. Point O ¼ Origo. Vector OP ¼ Preoperative astigmatism ¼ (0.75,0). Vector OT ¼ Postoperative astigmatism ¼ ()0.98, )0.17). Vector OS ¼ Vector PT ¼ SIA ¼ ()1.73,)0.17)

15 III.3.6. Reconversion from polar values to a spherocylindrical format There is a correlation between polar values and vector analysis by the astigmatic decomposition method III, as KP(90) and KP(135) are mathematically identical to Bennet s C0 and C45 129, shown in equations (20) and (21). The average of several polar values may be retransformed to the average net cylinder with equations (25) and (26). Any single set of polar values and the result of any compilation of polar values may be reconverted to spherocylinder notation by the following general equations III,V,X : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ¼ KPðUÞ 2 þ KPðU þ 45Þ 2 ð39þ Fig. 15. The difference between autorefraction and manifest refraction IX. The various refractions are shown as points in a 3-dimensional space IX, determined by the power vector n ¼ [1/2KP(90),1/2KP(135),SEP]. O is the origio. Vector OA ¼ manifest refraction,¼ ()0.25 DS)1.0 DC 120 ) ¼ [0.25,0.44,)0.75] Vector OB ¼ autorefraction ¼ ()0.25 DS)0.75 DC 119 ) ¼ [0.20,0.32,)0.63] Autorefraction)manifest refraction ¼ vector AB ¼ vector OC ¼ [)0.05,)0.12,0.12] ¼ (0.25 DS)0.26 DC 34 ) =2 KPðUÞ n ¼ 4 1=2 KPðU þ 45Þ 5 ð37þ SEP The Pythagorian length, n IX, of the power vector n is a metric representing 16, the overall blurring strength of a given spherocylinder. The expression is independent of astigmatic direction: jnj ¼ This metric is a better predictor of visual acuity loss than the root-meansquare or SEP of a spherocylinder 135. In this system defoci produced by the pure sphere (1.0 DS) and the pure cross cylinder ()1.0 DS 2.0 DC) are identical and equal to 1.0. The scalar n may also indicate the distance in dioptric space between two spherocylinders. See example 5 in the Appendix. For a net cylinder the plus-root is consistently taken. a ¼ arctan M KPðUÞ þ p 180 þ U KPðU þ 45Þ ð40þ In equation (40) p is integer and determines the periodic function. Any value may be used, but by convention an astigmatic meridian between zero and 180 degrees is chosen. The spherical equivalent is calculated independently and the sphere is given by S ¼ SEP 1=2M: ð41þ III.3.7. Statistical analysis of polar values Statistical analysis requires conversion to components and cannot be performed with net astigmatisms V,16,19,33. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1=2 KPðUÞÞ 2 þð1=2 ðkpðu þ 45ÞÞ 2 þ SEP 2 ¼ ð1=2 MÞ 2 þ SEP 2 ð38þ Fig. 16. Bivariate analysis of SIA in 100 eyes following LASIK VII. Individual values and 95 % normal areas. In the upper figure, each point represents a single induced astigmatism uniquely characterized by a pair of polar values in the form of D AKP and DAKP(+45). Any point (¼combination of polar values) inside the ellipse is within the 95% distribution of observations. In the lower figure, the same phenomenon is displayed as net astigmatisms in a polar plot. For each pair of polar values, there are 2 solutions of net astigmatism: one in the interval from zero to 180 degrees and one in the interval from 180 to 360 degrees. The dashed line connecting the single point of polar values to the two equivalent net astigmatisms below illustrates this phenomenon. The confidence perimeter is constructed by a point-for-point retransformation from the ellipse in the upper figure. 1. Univariate analysis. Descriptive statistical analyses of univariate data are performed in the usual manner. 2. Multivariate analysis. As the refractive components are correlated, multivariate statistical analysis is usually necessary and always more correct than the univariate approach V. The following formats are relevant for paraxial optics: a. Bivariate analysis of cylinders V,VII,X. Statistical analysis may be performed with relatively simple mathematical and graphical methods V. Bivariate statistical analysis is performed with Hotelling s 137(pp.397)434) T 2. The confidence and normal region V is an ellipse centred at the combined mean of the two polar values, while its orientation is determined by the two variances and the correlation coefficient (Fig. 16). For a correlation coefficient of zero the axes of the ellipse are parallel to the axes of the coordinate system. For non-zero correlations coefficients the two sets of axes are non-parallel. b. Trivariate analysis of spherocylinders IX. Trivariate distributions require matrix algebra and are fully characterized by the average and covariance matrices. Multivariate analysis in any dimension may be performed with Hotelling s 137(pp.397)434) T 2 statistic. The confidence region for trivariate analysis is an ellipsoid in 3- dimensional space IX. The graphical analysis is most conveniently performed in a new equidistant coordi- 15

16 nate system with origio in the combined average and axes parallel to the axes of the ellipsoid, determined by the eigenvectors 137(pp.77)91) of the distribution. In this system the (1 ) c)æ100% confidence ellipsoid for the combined mean value is given by: ð1=2 KPðUÞÞ 2 k 1=2KPðUÞ n ð1=2 KPðU þ 45ÞÞ2 þ k 1 =2KPðUþ45Þ n þ SEP2 k SEP ¼ qðn 1ÞF ½c;q;n qš ð42þ n q n In equation (42) k symbolizes the eigenvalues IX of the various distributions, n the number of observations, c the chosen significance level, and q the number of dimensions. F is an F- test with (q, n)q) degrees of freedom. For a trivariate distribution q is 3. The tolerance or normal ellipsoid for the individual observations is derived by division by n in the numerators on the left side of the equation sign in (42). A minimal spread around the targeted refractions is always desirable in refractive surgery. In multivariate analysis of orthonormal components the total variance is the sum of the variances of the individual components V,IX. The total variance of the difference between achieved and planned postoperative refraction is therefore a robust measure of the quality of refractive surgery IX. III.4. Contemporary methods All methods convert spherocylinders to orthonormal components in dioptric space and are invariant (unchanged) during rotation of the reference coordinate system and transposition from plus to minus cylinder format. III.4.1. Methods based on Stokes s principle and the sine-squared correlation The polar value method and other approaches 138 belong to this general type of analysis. 1. Alpins method is based on Stokes double angles and double-angled plots Alpins introduced indices to ease evaluation of refractive surgery 42,45. An index may work for univariate data, such as isolated description of SEP. However, refractive data are multivariate and cannot properly be described by univariate indices 139. The indices Magnitude of error and Angle of error 45 may be appropriate for an individual patient but actually represent separate analyses of astigmatic direction and magnitude. Aggregate analysis will therefore yield systematic errors 139. Alpins has described methods to optimize corneal laser surgery by combining refractive and topographic data 43. Alpins astigmatic components are: (X, Y) ¼ (KP(0), KP(45)) 45. The system is based on net cylinders in plus power format, exactly as for polar values. 2. In Holladay et al. used a modification of Jaffe s 31 vector analysis. Later, however, they 36 recognized problems with this method s aggregate analysis and described a new method based on Stokes double-angled-principle. The combined mean in the double angle plot was given the term centroid. Holladay et al. 39,131 later reported a bivariate statistical method, which did not include the covariance between the components. This model is not in accordance with formal multivariate statistics 140. Holladay s astigmatic components are: (x,y) ¼ (MÆ cos (2ÆAxis),MÆ sin (2ÆAxis)) 36,39. In the first publications in the series both axis, power, and transposed cylinder formats were used 24,27. In the last paper 131 a plus power format was consistently advocated, and with this format Holladay s components are identical to (KP(0), KP(45)). 3. Thibos et al. 16,40 elegantly described the decomposition of a spherocylinder by a Fourier transformation. A similar approach was later used by Naeser XII and the principle is outlined in equation (36). The Fourier method allows for all types of mathematical transformations, calculation of scalars and multivariate statistical analysis. Thibos astigmatic components are: (J 0, J 45 ) ¼ (1/2KP(0), 1/2KP(45)). A general approach for decomposition of complex surfaces with a Fourier approach has previously been described by Hjortdal 100. III.4.2. Methods based on Long s power matrix Long 141 originally described an elegant matrix operation to transform a spherocylinder to a 2 2 power matrix of the form: F ¼ f 11 f 12 f 21 f 22 " # S þ M sin 2 / M sin / cos / ¼ ; M sin / cos / S þ M cos 2 / ð43þ where f 11 and f 22 are cylinders projected on the (0, 90) degrees meridian, while f 21 and f 12 are cross cylinders in the (45,135) degrees meridian. This power matrix has been modified and used by Keating 142,143 and Harris 17,18,134 to solve a number of optometric problems. The components f 21 and f 12 are identical for thin lenses, but usually disparate in thick optical systems. Harris has described a modified 3-dimensional power vector h of the same length, h, as F 17,134 : h 1 f pffiffiffi 7 h ¼ 4 h 2 5 ¼ 4 2 f21 5 h 3 f S þ M sin 2 / 6 p ¼ ffiffiffi M sin / cos / 5 S þ M cos 2 / qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jhj ¼ f 2 11 þ 2 f2 21 þ f2 22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ S 2 þðs þ MÞ 2 ð44þ The elements in the power matrix are cylinders and represent an orthonormal cylinder-equivalent space 134. Any cylinder of power ±1.0 DC is defocused by 1 D, while a sphere of power ±1.0 DS is defocused by Ö(2) 134. Similarly, the lengths of the power vectors are given by: h ¼ F ¼ Ö(2)Æ n SIA is the difference between the post- and preoperative matrices 37,38,116,144. Multivariate statistical analysis with stereoscopic graphics has been described 145,146. Harris components in the vector h are related to the polar values in the vector n as (h 1,h 2,h 3 ) ¼ (SEP) 1/2KP(90),)(Ö(1/2))ÆKP(135), 1/2KP(90)) V. See example 6 in the appendix. III.5. Advanced methods SEP+ III.5.1. Fourier analysis of irregular surfaces Hjortdal et al. 100 used Fourier series harmonic analysis to decompose power plots from topographic images. Polar data from each mire were separated into SEP, second harmonic regular astigmatic, and higher harmonic 16

17 non-regular astigmatic components. The Fourier coefficients for the regular astigmatic component were identical to Thibos s values and to 1/ 2KP(0) and 1/2KP(45). As demonstrated in Fig. 1 from Hjortdal s paper 100 Fourier analysis can decompose any irregular surface into regular astigmatic and other optically meaningful quantities. Fourier analysis has been used for assessment of both regular and irregular astigmatism after small-incision cataract surgery 133,147,148. III.5.2. Wavefront analysis Light can be decomposed into distinct and pure optical elements by wavefront analysis. The actual wavefront may be decomposed by a variety of polynomial series, including Zernike polynomials IX, Regular astigmatism is a polynomial of the second order combined with a sinusoidal harmonic component of the second order XI,149, Regular astigmatism is represented by the following paired polynomials with fixed directions: Z 2 2 ¼ pffiffi c2 2 6 q 2 cosð2xþ; ð45þ a polynomial with axes along the (0/ 90) degrees meridians, and pffiffi Z 2 2 ¼ c q 2 sinð2xþ; ð46þ a polynomial with axes along the (45/ 135) degrees meridians. Defocus is presented by the following polynomial without angular component: Z 0 2 ¼ pffiffi c0 2 3 ð2q 2 1Þ ð47þ These metrics are based on leastsquare fitting. In the equations, q indicates the normalized radial distance and x the angular component XI,155. Each coefficient c m n characterizes the actual shape of the specific polynomial and has the units lm. Conversion of Zernike coefficients of elevation to polar values in D is accomplished by XI,156,157 : 1=2KPð90Þ ¼ 2 p ffiffiffi 6 c 2 2 ð48þ r 2 1=2KPð135Þ ¼ 2 p ffiffi 6 c 2 2 r 2 SEP ¼ 4 p ffiffiffi 3 c 0 2 r 2 ð49þ ð50þ In equations (48) (50) r signifies the radius in mm of the examined area in the entrance pupil. From equations (48) (50) follow that there is a strong functional and mathematical correlation between Zernike lower order aberrations and first order optics XI. In both approaches the components are orthonormal and mathematically independent. The pair of Zernike polynomials for regular astigmatism may be composed to a common polynomial with a direction in degrees 153. However, exactly as in first order optics, only the decomposed Zernike elements allow for mathematical and statistical analyses. SIA is the difference in Zernike coefficients between the post- and preoperative measurements. Multivariate statistical analysis in any number of dimensions can be performed with the orthogonal Zernike coefficients. Zernike lower order aberrations sampled over a large aperture of the entrance pupil may differ from subjective refraction XI. Recent evidence suggests that alternative aberrationderived metrics might be superior to least-square fitting in predicting manifest refraction 156. III.5.3. Other advanced methods Rays may be traced paraxially through complex optical systems with 4 4 refraction and translation matrices 66,116,158,159. These methods have been used for analysis of SIA following LASIK 41 and toric intraocular lenses 66. Analysis of refractive surfaces with 3-dimensional Snell s law techniques 160 and differential geometry 103 has potential for more exact clinical results. III.6 Synopsis of part III Surgically induced astigmatism is the difference between the postoperative and the preoperative astigmatism. Analysis of SIA for single and aggregate data cannot be performed with net cylinders, but requires conversion to orthonormal components in dioptric space. There are two main groups of component analyses, based on either Long s power matrix or Stoke s double-angled principle. The polar value method, based on Stoke s principle and the sine-squared correlation, allows for analysis of SIA in any meridian and of any type of refractive surgery. Curvital and torsional polar value analyses in the surgical meridian characterize the quality of the astigmatic surgery completely. The final results of refractive surgery may be retransformed from components to net astigmatism on a point-to-point basis. IV. Polar value and other component analyses of refractive data Component analysis of SIA has been performed with a variety of surgical procedures However, component analysis of refractive data has also been used for diverse purposes such as comparison of astigmatism in twins 166, examination of residual astigmatism 69,132, study of the effect of hyperbaric oxygen administration on refractive astigmatism 167, and examination of the against-the-rule astigmatic decay 77 with age. However, as shown in the following, component analysis has also been used for more complex investigational purposes and with multivariate statistical analysis. IV.1. Precision of autokeratometry examined by bivariate statistical methods The precision of autokeratomtery in 50 patients with cataract was examined in paper VI. Previous precision studies have performed univariate statistical analysis along specific meridians, but the simultaneous variation in astigmatic direction and magnitude has barely been explored. In paper VI autokeratometry on right eyes was performed on two different occasions. Each net astigmatism was converted to KP(90) and KP(135). The precision was defined as the paired difference (D) between these respective components. Mean, standard deviation, and 95% confidence interval (CI) of the mean amounted to: DKP(90): )0.05, 0.30, CI: ()0.14)00.03); DKP(135): )0.04, 0.31, CI: () ). The correlation coefficient between DKP(90) and DKP(135) was There was no statistically significant difference between the combined means of the two measurements (Hotelling s T 2 ¼ 0.007, 17

18 Fig. 17. The paired difference between two keratometries in 50 eyes shown as pairs of polar values. Units in DC. Individual values and 50% (inner ellipse) and 95 % normal ellipses for observations. This is analogous to the use of standard deviations in univariate analysis. Fig. 18. Twenty autokeratometeries in one eye. Individual values and 95% normal ellipse, expressed as polar values. Units in DC. The two autokeratometries outside the parameter were excluded in the final calculation of mean keratometry. F ¼ with (2,48) degrees of freedom; p ¼ 0.85). Bivariate analysis with the 50% and 95% normal ellipses is shown in Fig. 17 VI. Autokeratometry was performed 20 times on one eye. All keratometries were converted to polar values, as shown in Fig. 18. KP(90) Mean amounted to 0.01 D and KP(135) Mean to )1.22 D. The average net astigmatism was 45 ). After exclusion of the outliers beyond the 95% perimeter the corrected mean polar values were: KP(90) Mean ¼ )0.05 D and KP(135) Mean ¼ )1.22 D. The corrected average net astigmatism was 44 ). This is an accurate method for averaging net cylinders. IV.2. Trivariate analysis of the accuracy of autorefraction Accuracy and precision of autokeratometry have been examined in several studies, but inconsistent power vectors and statistical methods have been used 169, Autorefraction and subjective refraction in 50 eyes without pathology were compared in publication IX. The accuracy was defined as the paired difference between autorefraction and subjective refraction, using the power vector n; see Fig. 15 for an example of such calculations. Individual values and the 95% normal ellipsoid are shown in 3-dimensional space in Fig. 19. Aggregate univariate, bivariate, and trivariate statistical analyses did not demonstrate a significant average difference between the two refraction methods. For the trivariate data Hotelling s T 2 was and the p-value with (3,47) degrees of freedom. The study concluded that in groups of normal eyes autorefraction can be used as a substitute for manifest refraction. In the recent years description of accuracy and precision of keratometers and autorefractors with multivariate statistics has become more common 176,177. IV.3. Effect of misalignment of an astigmatic correction A net corneal astigmatism of the format a o ) is eliminated by a flattening of M D along the meridian a or a steepening along the orthogonal meridian (a + 90). This correction may be produced by a corrective cylinder or a refractive procedure. However, when the correction is misaligned the flattening effect is reduced, torsion is produced, and the result is a new astigmatism with magnitude and direction 44,59,178. Consider the net astigmatism 0 ). This astigmatism is eliminated by the net cylinder 90 ) in the corneal Fig. 19. Trivariate analysis of the difference between autorefraction and subjective refraction. Individual refractive data and the 95 % normal ellipsoid for observations. Units in DS. plane. The refractive effect as a function of the angle a of misalignment is:! KPð0Þ Postop KPð45Þ Postop ¼ 1 cos 0 1 cos 2ða þ 90Þ þ 1 sin 0 1 sin 2ða þ 90Þ ¼ 1 1 cos 2a þ ð51þ 0 1 sin 2a According to equations (39) and (40) the magnitude M Postop and the direction h of the resulting postoperative cylinder is given by: M Postop qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1 þð 1 cos 2aÞÞ 2 þð 1 sin 2aÞ 2 ¼j2 sin aj ð52þ ðj2 sin aj ð1 cos 2aÞÞ h ¼ arctan sin 2a ð53þ There is a dramatic decrease of refractive efficacy by even small degrees of misalignment (Figs 20 and 21). With a misalignment of 30 the magnitude of the resulting net cylinder is unaltered from the preoperative value. With a 45 o off-axis incision the Power of new cylinder in dioptres M Postop KP(0) Postop KP(45) Postop Angle α of misalignment in degrees Fig. 20. The effect of misalignment of a corrective incision or cylinder on the resulting astigmatism. Abscissa: Misalignment a in degrees of the corrective incision or cylinder. Ordinate: Magnitude M Postop, net curvital power KP(0) Postop and net torsional power of KP(45) Postop of the resulting astigmatism. All functions have the period 180 degrees. Meridian θ of new cylinder in degrees Angle α of misalignment in degrees Fig. 21. Effect of misalignment of a corrective incision or cylinder on the resulting astigmatism. Abscissa: As in Figure 20. Ordinate: Direction h of the resulting postoperative cylinder in degrees. 18

19 flattening effect is eliminated, and the resulting net cylinder is 158 ). The magnitude of the astigmatism is doubled for misalignments of 90. The direction for the postoperative astigmatism is counter-clockwise for positive and clockwise for negative values of KP(45) Postop ¼ ) sin 2a. Any combination of preoperative refractions and corrective surgeries or cylinders may be analyzed by this general method. As a corollary, the combined curvital and torsional effect from surgery in an off-axis meridian may be replaced by a pure flattening effect along a correctly chosen surgical meridian. See Example 7 in the Appendix. IV.4. The flattening effect of various cataract incisions The PUBMED facilities were used to search for studies reporting the change in corneal astigmatism after cataract surgery. Studies with at least 50 patients, at least 1 year of follow-up, and analyses of SIA by either the polar value I-II, Cravy s 23, the algebraic 20, or Olsen s astigmatic decomposition method 25,26 were included. In the latter method 25,26 subtraction of with-the-rule and against-the-rule components were done. All these methods express the net dioptric change along vertical and horizontal corneal meridians VIII,130,179. The flattening effect for equidistant incisions was highest for superior corneal, followed by superior scleral and temporal corneal incisions (Table 1). The flattening after one year amounted to approximately 1.0, 0.5, and 0.25 DC for incision lengths of 10, 6, and 4 mm, respectively. Only the 3 mm temporal incision approached astigmatic neutrality, which is confirmed by several recent papers with short Table 1. Change in corneal curvature at specific postoperative intervals following superior limbal/scleral, superior corneal, and temporal corneal incisions. The table shows the name of the first author, the incision site and length, the presence or absence of sutures, and the average values for SIA in dioptres. A positive value for SIA always indicates a steepening, and a negative value a flattening of the surgical corneal meridian. For superior incisions, calculated by KP(90) or similar methods, positive values for SIA indicate a with-the-rule change in corneal astigmatism, negative values an against-the-rule change. For temporal incisions, calculated by KP(0) or similar methods, positive values for SIA indicate an against-the-rule change, negative values a with-the-rule change in corneal astigmatism. Surgically induced astigmatism in dioptres measured at the following specific postoperative intervals First author Incision Site Incision length (mm) Suture 1 day 30 days 90 days 1 year 2 years Parker 113 Superior scleral 11 (+) )0.10 )0.79 Talamo 183 Superior scleral 10 (+) )0.69 )1.41 Zheng 111 Superior scleral 10 + ) 3.0 )0.41 )0.80 )0.92 Cravy 181 Superior scleral )0.2 )0.38 )0.95 )1.5 Singer 34 Superior scleral )0.46 )0.61 )0.64 Olsen 25 Superior scleral )0.44 )0.51 Dam-Johansen 121 Superior scleral )0.49 )0.55 Zheng 111 Superior scleral )0.34 )0.36 )0.66 Davison 184 Superior scleral ) )0.23 Pfleger 185 Superior scleral 4.5 ) )0.46 )0.68 )0.68 )0.66 Davison 184 Superior scleral )0.34 Rainer 186 Superior scleral 4 ) 0 )0.19 )0.25 )0.21 )0.55 Roman 187 Superior scleral 4 ) 0.96 )0.18 )0.35 )0.41 Pfleger 185 Superior scleral 3.5 ) )0.26 )0.39 )0.43 )0.5 Zheng 111 Superior scleral 3 (+) 0.49 )0.07 )0.28 )0.35 )0.46 Goes 188 Superior corneal 3.2 ) )0.41 )0.7 )0.58 )0.57 Pfleger 189 Temporal corneal 5.2 ) )1.13 )0.77 )0.74 )0.48 Pfleger 189 Temporal corneal 4 ) )0.60 )0.44 )0.30 )0.15 Roman 187 Temporal corneal 4 ) )0.78 )0.60 )0.43 )0.33 Pfleger 189 Temporal corneal 3 ) )0.30 )0.21 )0.09 )0.09 Zeng 111 Temporal corneal 3 ( ) ) )0.01 )0.09 Symbols for sutures: No sutures: ), sutures in some cases and/or sutures removed: (+), remaining sutures: +. follow-up time 180. Very extended follow-ups indicate a continued flattening of the surgical meridian after scleral incisions 111,112,181,182. IV.5. Bivariate analysis of surgically induced astigmatism after cataract extraction A simultaneous description of surgically induced flattening and torque is only possible with bivariate statistics V. In publication X keratometry was performed in 200 eyes before and serially up to 1 year after cataract extraction with various superior corneal incisions. SIA was analyzed with the polar values KP(90) and KP(135). After 1 year the surgically induced flattening and torsion along the 90 degree meridian for 9, 5.5, and 4 mm incisions amounted to (1.02, 0.46), (0.71, )0.04), and (0.64, 0.1) D, respectively. Bivariate analysis of the combined means is shown in Fig. 22. The 4 and 5.5 mm incision had similar refractive effect, while the 9 mm incision caused more flattening and torsion than the two smaller incisions. The flattening with these superior corneal incisions appears larger than both superior scleral and temporal corneal incisions of similar lengths. A rebound steepening from 3 months to 1 year after surgery was observed. The same phenomenon is observed for the corneal incisions in Table 1 and is in contrast to the continued late against-the-rule decay seen after scleral incisions in Table 1. This observation has not previously been published and may be caused by excessive cautery in scleral incisions. Another possible explanation might be that corneal fibrils heal head-tohead and hereby tighten up, whereas the tangentially oriented 107 limbal fibrils might continue to gape 190 after incisions. IV.6. Javal s rule in pseudophakic eyes Apparently, Javal s rule and residual astigmatism (RA) have not previously been reported in pseudophakic eyes. The RA in pseudophakic eyes may theoretically be explained by a toricity of the posterior corneal surface 74 or by a tilt and decentration of the implant. Autokeratometry and manifest refraction were performed in 85 pseudophakic eyes. The correlation 19

20 KP(135) KP(135) KP(135) KP(135) KP(135) SIA Day SIA Day SIA Day 90 SIA Day SIA Day KP(90) KP(90) KP(90) KP(90) 4 mm 5.5 mm 9 mm KP(90) between TOA-Ab Spectacle and ACA had a determination coefficient of 0.78 and amounted to: TOA-Ab Spectacle ¼ 0:77 ACA 0:25 ð54þ Fig. 22. Bivariate analysis of SIA one year after three superior corneal cataract incisions. For each incision the mean induced astigmatism is the centre of the ellipse, which delineates the 95% confidence region. Confidence regions containing the point (0,0) signify mean values not differing significantly from zero. Common areas between two confidence regions reveal a lack of significant difference between the two involved combined means at the 5 % level. All mean induced astigmatisms deviated from astigmatic neutrality. SIA for the 9 mm incision differed significantly from the two smaller incisions. The correlation between total ocular aberration in the corneal plane and ACA was: TOA-Ab Cornea ¼ 0:76 ACA 0:25 ð55þ The average RA, expressed as a net astigmatism, amounted to ( ). This equation is similar to Grosvenor s findings in some phakic eye models 76. Javal s rule seems to apply for pseudophakic eyes. Therefore, some with-the-rule corneal astigmatism should be preserved by surgery in order to eliminate the refractive astigmatism. Fig. 23. The subjective refraction in the spectacle plane following LASIK in 100 eyes. Mean values and 95 % confidence areas of the combined means, expressed as polar values in the preoperatively steeper meridian. Units in DC. The mean and the confidence area are shown as follows: preoperative values ¼ line segment to the right; postoperative values (¼the dioptric error of the precedure) ¼ellipse in the middle; SIA ¼ ellipse to the left. The confidence area for the preoperative refractions emerged as a line segment along the abscissa, as the mean and the standard deviation for AKP(+45) Preop both were zero. All mean values differed significantly from zero and from one another. IV.7. Aggregate analysis on corneal laser ablative refractive surgery Polar value analysis was performed before and after LASIK in 100 eyes VIII. Subjective refraction in the spectacle plane and simulated keratometry were consistently converted to astigmatic polar values, where AKP and AKP(+45) indicate the flattening and torsion along the preoperatively steeper meridian. Hotelling s T 2 disclosed a statistically significant difference from zero for post-operative values for both refraction and keratometry. Analysis of the flattening effect (¼AKP SIA ) revealed an undercorrection by 12% for refractive data and by 19 % for keratometric data. No significant torsion was induced. The bivariate analysis of SIA for refractive data is shown in Fig. 23. This analysis VIII was published together with papers from five other invited groups in the Special Issue on SIA in the Journal of Cataract and Refractive Surgery from January, Comparison of these studies underlines the importance of selecting appropriate meridians for component analysis. Laser surgery was actually applied in various corneal meridians. Several studies calculated SIA along the (0/90) degrees meridian. Only component analysis along the surgical meridian will reveal the surgically induced flattening and torque with relevance for assessment of the present and optimization of future surgeries. IV.8. Synopsis of part IV Following conversion of spherocylinders to orthonormal components in dioptric space, calculations may be performed in a univariate, bivariate or trivariate manner. Formal multivariate analysis should be performed on aggregate data. V. Conclusions The surgically induced change in refraction is the difference between the postoperative and the preoperative refractions. Refractive data in the form of sphere, cylinder, and axis may define an individual patient or data set, but are unsuited for mathematical and statistical analyses. The three components of a spherocylinder do not represent 20

21 entities, whose rules for algebraic conversion are known IV,VI,VIII, Analysis of the spherical component is uncomplicated. The problems reside with the incommensurable astigmatic data. In a large number of assessment methods separate analyses of astigmatic magnitude and direction have been advocated. These methods are still used in scientific publications, but will invariably produce systematic errors and inconsistent results. More importantly, they may induce errors in scientific conclusions and clinical decision-making. Calculation of SIA for both single and aggregate data requires conversion of net astigmatisms to orthonormal components in dioptric space. A dioptric space is defined as a vector space with orthogonal axes and units of dioptres VI,VII,IX,X,16,33. In this space a power vector 16 represents the net cylinder or the spherocylinder. The components are coordinates of power vectors in dioptric space. In this vector space the components are mathematically independent, and all mathematical and statistical calculations may be performed with the wellknown methods. There are two main groups of component analyses, based on either Long s formalism or Stoke s doubleangled principle. The sine-squared correlation may be recognized in both these basal systems. The components in Long s 4-dimensional power matrix are two orthogonal cylinders and two cross cylinders. The system is cylinderequivalent. The components in Stoke s double-angled principle are the spherical equivalent power combined with two cross cylinders, separated by an angle of 45 degrees. This system is sphere-equivalent. Any of these component methods may be used for analysis of SIA. Systems based on Stoke s double-angled principles and the sinesquared correlation are probably more familiar to refractive surgeons 191. Systems developed in optometry usually analyze refractions along fixed horizontal and vertical meridians and are primarily used for description of refractions, prescription, etc. Systems developed for surgical purposes necessarily must have the capacity for analyzing SIA along a variety of surgical meridians. The polar value system was the first component-based method in ophthalmology. The polar value system is directly based on the sinesquared correlation, and any surgical or refractive meridian may be analyzed by varying the directions of the two cross cylinders, expressed by the polar values. The polar value system has gradually been developed to enable analysis of SIA for any refractive procedure, based on either keratometry, videokeratography, refraction, or wavefront analysis. Any surgical induced change may be conceived and calculated as net curvital and net torsional power along the specific meridian. The polar value system has changed the concept of SIA. In the polar value system a spherocylinder may be described in two different spaces. Sphere, cylinder, and axis for an individual patient are described in traditional format in one space. This space is conjugate with an orthonormal space of rectangular coordinate components, all described in dioptres. The conversion and reconversion of data formats between these spaces are determined by unambiguous equations. The final results of refractive surgery may be retransformed to net astigmatism, on a point-to-point basis. Statistical analysis of spherocylinders may be performed in a univariate, bivariate or trivariate manner. Multivariate statistical analysis is always more precise than univariate approaches, and formal multivariate analysis should be performed on aggregate data. It is a major advantage for both surgeons and researchers that multivariate statistics is available on all major statistical software packages. The graphical multivariate analysis is very illustrative; the mean and spread are directly related to the position and extension of the confidence ellipsoid, and the statistical significance may be read directly from the graphics. Informal statistical methods should be avoided 39,131. Original, orthonormal refractive components should be used for all assessments of SIA. Indices or semiquantitative approaches 42,45,131 should be used with caution, because multivariate refractive data cannot be properly described by univariate indices 139. Indices such as Magnitude of error and Angle of error 45,131 represent separate analysis of astigmatic direction and magnitude, and aggregate analysis will therefore yield systematic errors. Double-angled plots 36,39,42,45 can be used for description of regular astigmatism, but they may give the spurious impression of a polar plot rather than a rectangular coordinates system 192. Complex description of refractive surfaces can be performed with Fourier and Zernicke wavefront analyses. The components in these systems may be decomposed to paired sine and cosine elements, exactly as described for polar value analysis of regular astigmatism. Multivariate statistical analysis is also performed in a similar manner. Any desired combination of these orthonormal components may be analyzed. A unified approach is therefore available for decomposition and statistical analysis for the total refractive spectrum of spheres, regular astigmatisms, and irregular astigmatisms. This description is valid for both individual patients and for groups. First order optics is a simplification, defined by the powers of the principal meridians. The sine-squared correlation is a valid method to calculate meridional and torsional power. All meridians of an astigmatic surface participate in the refraction. In each oblique meridian, the combined net curvital and torsional powers direct incident wavefronts into the two orthogonal focal lines. It is reassuring that first order optics and formal differential geometry describe astigmatic surfaces with the same concept and with similar mathematical functions XII. In both cases the change of surface shape is determined by a calculation of a boundary value. Employment of Snell s law ray tracing could improve the description of refractive surfaces, but until now such exact descriptions have generally not been required in ophthalmology. VI. Perspectives Consistent transformation of refractive data to polar values or other relevant orthonormal components in dioptric space will allow for precise evaluation of refractive procedures. Multivariate statistics should be used for calculation of both means and spread of any refractive procedure. All refractive results, including subjective refraction, autorefraction, wavefront analysis, etc. should be 21

22 converted to components. This allows for examination and possible correction of the precision and accuracy of each measuring device. The consistent use of component analysis will therefore generally increase the quality of cataract and refractive surgery. The general demand for higher quality in cataract and refractive surgery will probably favour the use of component analysis of SIA. Only these methods can give the surgeon the feedback information needed to optimize future procedures. Implantation of aspherical, presbyopic, and multifocal intraocular implants is irrelevant unless accompanied by a minimal postoperative refractive astigmatism 193. In order to avoid enhancement surgery relevant assessment methods for SIA should consistently be used. Component analysis may be used to integrate refractive and corneal topographic data in the planning of corneal refractive surgery. It is possible to integrate not only regular astigmatism, but also higher order aberrations with component analysis. Unfortunately, in spite of multiple attempts for standardization IV,15,131,194, the ophthalmological community has still not reached a consensus on the correct description of SIA. When a consensus has been reached the resulting equations should be transformed to relevant computer programs, enabling entry of refractive data in any desired format. It is the present author s hope that the thesis may contribute in achieving consensus on methods for description and treatment of regular astigmatism with subsequent even higher quality in cataract and refractive surgery. VII. Summary in Danish - Dansk Resume VII.1. Baggrund Forma let med katarakt og refraktiv kirurgi er at bibringe patienten en god synsfunktion med en passende postoperativ refraktion (brydningsfejl). Den klassiske refraktion kan deles op i sfære og astigmatisme (bygningsfejl). Den sfæriske komponent er den del af brydningskraften, hvorved et punkt i objektrummet fokuseres skarpt som et punkt i billedrummet. Den sfæriske brydningsanomali kan karakteriseres ved brydningsstyrken, der er et enkelt tal i enheden dioptrier. Den regelmæssigt astigmatiske komponent er karakteriseret ved, at et punkt afbildes som to forskudte og vinkelrette pa hinanden sta ende liniestykker. Ved uregelmæssig astigmatisme forsta s mere uregelmæssige billeddannelser. Den astigmatiske komponent giver anledning til uskarp og forvredet billedannelse pa nethinden for objekter i alle fiksations afstande. I det følgende vil astigmatisme angive den regelmæssige astigmatisme. Astigmatisme korrigeres normalt af toroidale refraktive overflader. Brydningskraften udregnes separat for hver af de to orthogonale hovedmeridianer. Astigmatismens brydningsstyrke er den absolutte forskel i brydningsstyrke mellem de to hovedmeridianer. En netto astigmatisme er karakteriseret ved ba de en retning i grader og en brydningsstyrke, angivet i dioptrier. Den kirurgisk inducerede astigmatisme defineres som differencen mellem den postoperative og den præoperative astigmatisme. Refraktive indgreb udføres i forbindelse med katarakt kirurgi og ved primær refraktionskirurgi pa hornhinde eller linse. Alene i Danmark udføres a rligt ca kataraktoperationer og 2500 (antallet for a ret 2000) primært refraktive indgreb. Ved disse indgreb er det vigtigt at eliminere eller reducere astigmatismen hos den enkelte patient. Indenfor fysikken anvendes Snell s lov til beregning af stra legangen gennem optiske systemer. Indenfor oftalmologien og optometrien anvendes en approksimation, nemlig den sa kaldte første ordens optik, hvor kun den paraksiale stra legang beregnes. Anvendelse af første ordens optik giver en række praktiske fordele ved ha ndtering af linser og briller, men giver ogsa anledning til problemer ved beregninger. Netto astigmatismen eller sfærocylinderen karakteriserer eentydigt en enkelt patient eller en enkelt refraktion, men matematiske beregninger er ikke umiddelbart mulige med disse formater. Der er gennem de sidste 30 år udviklet en række analysemetoder, der ikke tager hensyn til disse særlige matematiske forhold. Metoderne anvender enten separat analyse af astigmatismens retning og størrelse, komponenter uden tilknytning til optikken eller korrekt analyse af kirurgisk induceret astigmatisme for den enkelte patient med efterfølgende separat analyse af astigmatismens retning og størrelse. Disse metoder vil sædvanligvis give anledning til inkonsistente resultater og systematiske fejl ved analyse af kirurgisk induceret astigmatisme for enkeltsta ende og/eller aggregerede refraktive data. På grund af deres enkle opbygning bliver disse tidlige metoder stadig brugt og nyudviklet, men vil normalt medføre fejl i videnskabelige konklusioner og klinisk beslutningstagen. VII.2. Polærværdi metoden Denne disputats er baseret pa 12 publicerede arbejder indenfor området kirurgisk induceret astigmatisme. Forma let med de samlede arbejder har været at udvikle metoder til analyse af allerede udførte operationer og optimering af fremtidige refraktive indgreb. De beskrevne metoder kan kollektivt benævnes polærværdi analyse af kirurgisk induceret astigmatisme. Polærværdier er matematiske transformationer af netto astigmatismer til orthonormale komponenter i et dioptrisk rum. Et dioptrisk rum er et vektorrum med orthogonale akser og basisvektorer i enheden dioptrier. Polærværdierne er komponenter eller koordinater til vektorer i dette rum. I det dioptriske rum kan matematiske udregninger og statistiske beregninger foretages pa sædvanlig vis. Om ønsket, kan slutresultatet af beregninger tilbageføres til velkendt format i form af netto astigmatismer eller sfærocylindre. Polærværdier er baseret pa den sinus-kvadrerede korrelation, hvorved den meridionale brydningskraft af enhver brydende toroidal overflade kan beskrives. Enhver kirurgisk induceret astigmatisme kan karakteriseres som en 3-dimensionel vektor i form af en sfærisk ækvivalent værdi og to polærværdier adskilt af en buelængde pa 45 grader; e n polærværdi til beskrivelse af brydningskraften langs den kirurgiske meridian og e n polærværdi til beskrivelse af torsionen over denne meridian. Disse tre komponenter er indbyrdes matematisk uafhængige, og enhver analyse kan foretages uafhæn- 22

23 gigt. Der pa vises en simpel korrelation mellem polærværdier og vektor analyse af kirurgisk induceret astigmatisme. Ved hjælp af denne generelle metode er udviklet almene formelsæt til beskrivelse af kirurgisk induceret astigmatisme efter katarakt eller refraktive kirurgi i en vilkårlig meridian. Der er udviklet specifikke formelsæt til beskrivelse af kirurgisk induceret astigmatisme efter superiore og temporale incisioner, og disse metoder kan samtidigt benyttes til generel beskrivelse af refraktive data som for eksempel refraktioner og brillerecepter. Polærværdier kan benyttes for ba de enkeltstående data samt for grupper af øjne. Statistisk analyse kan foretages univariat for hver enkelt komponent. Der er beskrevet metoder til bivariat og trivariat statistisk analyse af henholdsvis netto astigmatismer og sfærocylindre. Brugen af polærværdier med efterfølgende multivariat statistisk analyse er demonstreret for en række kliniske problemstillinger i 5 publikationer. Der er pa vist betydelig strukturel og matematisk lighed mellem polærværdi og bølgefronts analyse af refraktive data. Der er en lineær sammenhæng mellem polærværdier og Zernike koefficienterne for lavere ordens aberrationer. Den sinus-kvadrerede korrelation er grundstenen i de fleste ma leapparater af refraktive data og i en række formler til beskrivelse af kirurgisk induceret astigmatisme, herunder polærværdi metoden. Den sinus-kvadrerede korrelation har været omdiskuteret, da et isoleret stra lebundt i en skæv meridian af en sfærocylinder uvægerligt afbilledes svarende til hovedmeridianernes fokallinier. Der pa vises en tæt funktionel og matematisk relation mellem polærværdi begrebet og den fra differential geometerien kendte formelle beskrivelse af vilka rlige overfladers kurvatur og torsion. Det konkluderes, at den sinus-kvadrerede korrrelation er valid og kan anvendes til beregning af ba de den meridionale og torsionale brydningskraft langs en given merdian. For en given skæv meridian er den samlede virkning af disse to brydningskræfter, at parallelle indfaldne stra ler afbilledes i de to fokallinier. VII.3. Andre metoder Alle relevante metoder transformerer sfærocylindre til ortonormale komponeter i dioptrisk rum. Der er to hovedgrupper, baseret på enten Long s power matrix eller Stokes dobbelt-vinklede princip: 1. Long s power matrix er en 2 2 matrix besta ende af to ortogonale cylindre og 2 krydscylindre. Denne matrix kan omsættes til en 3-dimensionel vektor. Metoden er videreudviklet indenfor optometrien af Keating og Harris, og en række kliniske problemstillinger er belyst hermed. Der er beskrevet multivariat statistiske analyse. 2. Indenfor oftalmologien har Holladay og Alpins beskrevet metoder, baseret pa Stokes princip. Disse metoder ligger meget tæt pa polærværdi metoden, udregnet langs nul grader. Begge metoder har været vidt anvendt til beskrivelse af kirurgisk induceret astigmatisme efter refraktiv kirurgi. Alpins har forsøgt at beskrive kvaliteten af refraktive indgreb gennem et enkelt tal, et index. Da refraktive data er multivariate, er et sa dant monovariat index normalt ikke hensigtsmæssigt. Ingen af disse forfattere har anvendt eller beskrevet formel multivariat analyse af refraktive data. 3. Avancerede metoder. a. Fourier analyse. Det er muligt at finde udtryk for regelmæssig astigmatisme ved Fourier analyse af ba de uregelmæssige overflader (Hjortdal) og sfærocylindre (Thibos). Fourier koefficienterne er identiske med den meridonale og torsionale brydningskraft, udtrykt som polærværdier i retningen nul grader. b. Bølgefrontsanalyse af refraktive data giver et udtryk for regelmæssig astigmatisme. Ba de Fourier og Zernike koefficienter er matematisk uafhængige og kan analyseres ved hjælp af multivariate statistiske metoder. c. Tre-dimensionel ray tracing ved hjælp af Snells lov, differential geomtri eller 4 4 refraktions og translations matricer. Disse komplekse metoder er endnu ikke udbredte i klinisk praksis. VII.4. Konklusion og perspektiver Refraktive data i form af sfære, cylinder og akse karakteriserer en enkelt patient eller data sæt, men dette format kan ikke anvendes til matematisk eller statistisk analyse. Alle relevante metoder til beregning af kirurgisk induceret astigmatismer transformerer sfærocylindre til orthonormale komponeter i dioptrisk rum. Matematiske beregninger og statistiske analyser kan foretages pa normal vis i dette rum. Enhver af disse relevante metoder kan anvendes. Oftalmologer er normalt ikke fortrolige med formatet i Long s power matrix, men vil oftest anvende metoder bygget pa Stokes princip eller den sinus-kvadrerede korrelation. Metoder udviklet indenfor optometrien beskriver refraktive data ud fra faste reference-meridianer. Metoder udviklet til kirurgisk induceret astigmatisme ma nødvendigvis have muligheden for analyse langs vilkårlige meridianer. Polærværdi metoden er bygget pa den sinus-kvadrerede korrelation og giver mulighed for analyse af vilka rlige refraktive overflader samt vilka rlige meridianer. Efter beregning af polærværdierne langs den kirurgiske meridian kan forudga ende refraktive operationer analyseres og fremtidige refraktive indgreb planlægges og optimeres. Koefficienterne for regelmæssig astigmatisme i Fourier og bølgefronts analyse kan matematisk og statistisk behandles som polærværdier eller som andre komponent metoder. Samtlige komponenter kan analyseres med multivariate statistiske metoder. Pa trods af mange forsøg pa konsensus er der i det internationale oftalmologiske samarbejde ikke opnået enighed om metoder og udtryk til beskrivelse af kirurgisk induceret astigmatisme. Konsekvent indførelse af polærværdi eller andre komponent metoder med efterfølgene multivariat statistisk analyse ville give mulighed for mere nøjagtig refraktive resultater ved refraktive indgreb. 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J Cataract Refract Surg 2006;32: Hayashi K, Hayashi H, Nakao F, Hayashi F Influence of astigmatism on multifocal and monofocal intraocular lenses. Am J Ophthalmol 2000;130: Goggin M, Pesudovs K Assessment of surgically induced astigmatism: toward an international standard. J Cataract Refract Surg 1998;24: Appendix Numerical Examples Example 1. Derivation of net astigmatisms. a. Keratometer readings: 45 D along 75 degrees and 43 D along 165 degrees. M ¼ 45)43 ¼ 2D. a ¼ 75 degrees. Net astigmatism K ¼ (2@75 ). b. Manifest refraction (+1.0 DS )1.0 DC 0 ) ¼ ( ). M ¼ )1.0 ¼ 1.0. Net astigmatism K ¼ (+1.0@0 ). Example 2. Conversion of the net cylinders from Example 1 to polar values: By choosing U ¼ 90, the polar values in 90 and 135 degrees are obtained. a. Keratometry. Net cylinder 75 ). KP(90) ¼ )2Æ( cos 2(( ) ) 90)) ¼ )2Æ( cos 150) ¼ 1,73; KP(135) ¼ )2Æ( sin 2(( ) ) 90)) ¼ )2Æ( sin 150) ¼ )1 b. Refraction. Net cylinder ¼ 0 ). KP(90) ¼ )1Æ( cos 0) ¼ )1.0; KP(135) ¼ )1Æ( sin 0) ¼ 0 Example 3. Calculation of surgically induced astigmatism, expressed as polar values in 90 degrees. 27

28 Preoperative net corneal astigmatism ¼ 90 ) Cataract extraction through an incision in 90 degrees Postoperative net corneal astigmatism ¼ 5 ) (KP(90) SIA, KP(135) SIA ) ¼ ()1Æ cos 10,)1Æ sin 10) ) ()0.75Æ cos 180,)0.75Æ sin 180) ¼ ()0.98, )0.17) ) (0.75,0) ¼ ()1.73,)0.17) Surgery induced a flattening of 1.73 D along and a clockwise torque of 0.17 D over the surgical meridian in 90 degrees. Example 4. Calculation of the error of a surgical procedure, expressed as polar values. In the patient from example 3 a spherical cornea was planned (KP(U) Error,KP(U + 45) Error ) ¼ ()1Æ cos 10,)1Æ sin 10) ) (0Æ cos 0,)0Æ - sin 0) ¼¼()0.98,)0.17). The preoperative cylinder was overcorrected or flattened too much by 0.98 D. An unintended clockwise torque of 0.17 D was induced. This example is demonstrated graphically in Fig. 14. Example 5. The distance in dioptric space between the two refractions in Fig. 15 is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 0:05Þ 2 þð 0:12Þ 2 þ 0:12 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1=2 ð 0:26ÞÞ 2 þ 0:12 2 ¼ 0:18 D Example 6. The length of a spherocylinder in dioptric space, calculated with Long s power matrix. Harris 2 2 power matrix F and the length F of the spherocylinder (6.0DS )8.0 DC 135 ) 122. The power vector n and its length n is shown for comparison. In this example the polar values in (90,135) degrees were employed F ¼ 2 4 p jfj ¼2 ffiffiffiffiffi =2 KPð90Þ n ¼ 4 1=2 KPð135Þ 5 ¼ SEP 2 p jnj ¼2 ffiffiffi 5 Example 7. Selection of a surgical procedure in order to obtain a specific postoperative refraction. Consider the case from Example 3 with a preoperative net astigmatism ¼ 90 ). We now want to plan a surgical procedure to obtain the same postoperative net cylinder of 5 ). The required surgically induced astigmatism is calculated in the same manner as (KP(90) SIA, KP(135) SIA ) ¼ ()1.73,)0.17). Equations (39) and (40) translate these polar values into the net cylinder 3 ). The desired postoperative refraction may therefore be achieved by procedures aiming at either a steepening of 1.74 D along 3 degrees or a 1.74 D flattening along 93 degrees. Accepted on February 5, Correspondence: Kristian Næser, Eye Clinic, Regional Hospital Randers, 8900 Randers, Denmark Tel: Fax:: [email protected] 28