Patent Publication Number: US-8540370-B2

Title: Devices and methods for selecting intraocular lenses

Description:
This application claim priority to and is a continuation application of U.S. application Ser. No. 11/270,991, filed on Nov. 11, 2005, now U.S. Pat. No. 8,087,782, which claims priority to provisional application No. 60/627,430, filed on Nov. 12, 2004 and foreign application No. SE 0402769-4, filed on Nov. 12, 2004, each of which is hereby incorporated by reference in their entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to devices and methods for selecting an intraocular lens and more specifically to devices and methods of finding suitable powers and/or locations of intraocular lenses to be implanted into an eye in order to obtain a predetermined refractive outcome, taking into account such parameters as the asphericity of the cornea and/or the intraocular lens. 
     2. Description of the Related Art 
     U.S. Pat. No. 5,968,095, herein incorporated by reference, refers to a method of preoperatively selecting the power of an intraocular lens (IOL) to be implanted into an eye having a lens haptic plane. The method involves selecting eye parameters to construct an eye model for finding a correct representation of the intraocular lens as axially positioned in the eye following surgical implantation. However, this method is not designed to be applicable when any of the optical surfaces is aspheric. In particular this method is not applicable when using aspheric lenses designed to reduce or eliminate the spherical aberration of the cornea. Other commonly applied methods to determine IOL power, such as the widely used SRK/T formula, and other widely applied methods such as the Hoffer Q and Holladay 1 and Holladay 2 formulas, suffer the same shortcoming in being based on thin lens vergence calculations and/or spherical lens surfaces. Paul-Rolf Preussner et al. disclose an alternative method of predicting outcome of choice of IOL model and power in J Cataract Refract Surg, 2004, Vol. 30, pp. 2077-2083, which is herein incorporated by reference. 
     As aspheric IOLs capable of correcting spherical aberrations now are becoming available on the market (e.g., Tecnis® brand of IOL, available from AMO Inc., Santa Ana, Calif.), there is a demand to obtain reliable methods to select aspheric IOL powers in order to achieve the desired patient outcome in terms of spectacle correction and/or image quality. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments of the present invention may be better understood from the following detailed description when read in conjunction with the accompanying drawings. Such embodiments, which are for illustrative purposes only, depict the novel and non-obvious aspects of the invention. The drawings include the following figures, with like numerals indicating like parts: 
         FIG. 1  is a graphical representation of elements of an eye model used in various embodiments of the present invention. 
         FIGS. 2   a  and  2   b  are magnified views of the corneal region of the graphical representation shown in  FIG. 1   
         FIG. 3  is a flow chart showing a method of selecting an IOL according to one embodiment of the invention. 
         FIG. 4  is a flow chart showing a method of selecting an IOL according to another embodiment of the invention. 
         FIG. 5  is a graphical representation of the elements of computing system for selecting an IOL according to embodiments of the present invention. 
         FIG. 6  is printout of the formulas programmed into each cell of an Excel spreadsheet used to provide the ray tracing program in accordance with embodiments of the present invention. 
         FIG. 7  is illustrates the numerical result of the calculation in each cell of an Excel spreadsheet used to provide the ray tracing program in accordance with embodiments of the present invention. 
         FIG. 8  is a ray trace of the numerical results presented in  FIG. 7 . 
         FIG. 9A-9D  are through-focus MTF plots used to determine maximum MTF of an IOL. 
     
    
    
     DETAILED DESCRIPTION OF THE DRAWINGS 
     The present invention is directed to reliable methods and devices for selecting spherical and aspheric intraocular lenses (IOLs) that provide a predetermined refractive outcome for patients in need of cataract or refractive surgery. Embodiments of the invention may be understood by reference  FIG. 1 , which is a graphical representation of a model of an eye  20  comprising a cornea  22 , an iris  24 , a retina  26 , and an optical axis  28 . An IOL  30  is disposed within the eye  20 , the IOL  30  comprising an optic  32  and one or more haptics  34  having distal ends  38 . In general, the eye model may consist of the dimensional parameters illustrated by the geometry shown in  FIG. 1 , for example, the axial length of the eye (AL) and the anterior chamber depth (ACD). Other dimensional parameters that may be included in the eye model that are not shown in  FIG. 1  include, but are not limited to, the corneal radius (CR), the corneal power (K), and crystalline lens thickness (LT). The eye model may also include various other parameters such as, for example, the refractive indices of the various portions of the eye  20  and/or the IOL  30 . In certain embodiments, the distal ends  38  of the haptics  34  are disposed within a plane defined a lens haptic plane (LHP). In other embodiments, other information of the IOL  30  may be included in the eye model such as, for example, an effective principal plane of the optic  32  or other information regarding the optic  32  useful in determining the performance optic  32  and/or the location of the optic  32  within the eye  20 . 
     The graphical representation of the eye model illustrated in  FIG. 1  also has a coordinate system containing a horizontal axis  40  and a vertical axis  42 , which are labeled in units of millimeters. The graphical representation illustrated in  FIG. 1  also shows a plurality of rays  44  entering cornea  22  and the IOL  30  of the eye model. The plurality of rays  44  comprises a paraxial ray  50  that is disposed near the optical axis  28  and a marginal ray  52  that is disposed near edge of the opening formed by the iris  24 . The plurality of rays  44  additionally comprises an averaged ray  51  disposed between the paraxial ray  50  and the marginal ray  52 , for example, at a height at the pupil that is 1/√{square root over (2)} or ½ times the height of the entrance pupil height. In some embodiments, the eye model additionally contains information regarding an object or source represented by the plurality of rays  44  entering the eye  20 , for example, the distance of the source or object from the eye  20  and/or the extent of the source or object in units of length or arc length. 
     Referring to  FIG. 2   a , which is a magnified view of the region around retina  28 , the rays  50 - 52  may come to focus at different points along the optical axis  28 , which are labeled in the figure as marginal focus, best focus, and paraxial focus. As illustrated in the figure, the distance between the marginal focus and the marginal focus may be used to define a longitudinal spherical aberration (LSA). Such a result may be produced, for example, when the surfaces of the IOL  30  are spherical. Alternatively, one or more of the surfaces of the IOL  30  may comprise an aspheric profile that is configured to reduce or eliminate spherical aberrations produced either by an IOL made of spherical surfaces or by at least portions of the eye  20  (e.g., the cornea  22 ). In such embodiments, as illustrated in  FIG. 2   b , the rays  50 - 52  focus to a common or substantially common point along the optical axis  28 . 
     Embodiments of the invention may be used in conjunction with an eye model such as that illustrated in  FIG. 1  to select or determine a characteristic of an IOL to be implanted into the eye of a subject or a class of subjects, for examples subjects of a particular age group or condition (e.g., a class of subjects who have had a LASIK or similar procedure). In certain embodiments, measurements from a subjects eye (e.g., AL, ACD, CR, LT) may be used in conjunction with statistical data and/or an analytical tool (e.g., a ray trace program or algorithm) to determine the characteristic of the IOL. The characteristic of the IOL resulting from embodiments of the invention may include the thickness of the IOL, the power of the IOL, the asphericity of the IOL, and/or the location of the IOL within the eye of the subject or subjects. 
     Referring to  FIG. 3 , in certain embodiments, a method  100  of selecting an IOL comprises one or more of the following operational blocks  110 - 180 . The method  100  comprises an operational block  110 , which comprises determining physical characteristic of the eye such as an axial eye length or a pupil size at a desired light level. The method  100  further comprises an operational block  120 , which comprises determining a desired postoperative condition such as a postoperative refraction and/or spherical aberration. The method  100  further comprises an operational block  130 , which comprises determining an aspheric representation of the corneal curvature or curvatures. The method  100  also comprises an operational block  140 , which comprises selecting an IOL with one or more predetermined characteristics (e.g., with a predetermined power or asphericity) and determining the location of a plane of fixation of the IOL following implantation (e.g., the lens haptic plane or LHP). The method  100  additionally comprises an operational block  150 , which comprises employing the results of operational blocks  110 - 140  to establish an eye model. The method  100  also comprises an operational block  160 , which comprises computing, by means of an analytical tool (e.g., a ray tracing routine) with said eye model, an expected postoperative condition such as an expected postoperative refraction refraction and/or spherical aberration. The method  100  further comprises an operational block  170 , which comprises, in the case the expected postoperative condition is not within the desired postoperative condition, selecting another IOL with different power and/or asphericity and repeating operational blocks  150  and  160  until the expected postoperative condition is within the desired postoperative condition. The method  100  may also comprise an operational block  180 , which comprises selecting, for implantation, an implantable IOL of the nearest power and asphericity available or designing an implantable IOL that results in the desired postoperative refraction and spherical aberration. 
     The method  100  may also include transforming the measured axial length to a human population average scale by adding to the value a transformation constant. The axial length measured by ultrasound is not the same as the optical axial length, and as the axial length measured by one piece of equipment may differ from that measured by another one, there is a benefit to obtaining instrument independent measurements. Measurement of axial eye length for an individual patient may be obtained by ultrasound A-scan or the newer partial coherence interferometry (PCI) principle, available with the Zeiss IOLMaster. Regardless of the instrument and/or method used, the axial eye length may first be transformed to a human population average (HPA) scale. In certain embodiments, an underlying assumption for the HPA scale is that the mean axial length is about constant in any large group of adults. Transformation is discussed in more detail by Norrby et al. (J Cataract Refract Surg 2003; 29:100-105) and the HPA scale is introduced by Norrby et al. (J Cataract Refract Surg 2005; 31:1338-1344), both of which are herein incorporated by reference. Transformation amounts to the addition of a correction constant to the measured axial eye length. The correction constant is generally regarded as instrument specific, for example, as described in Norrby et al. (J Cataract Refract Surg 2005; 31:1338-1344). A general outline of a routine to obtain a common scale for axial lengths may include the following steps. First the postoperative anterior chamber depth consistent with axial length, corneal radius, postoperative refraction and IOL power implanted are calculated by thick lens ray tracing for a number of individual cases. The mean of the calculated anterior chamber depths may be calculated and compared with a previous study with the same lens (e.g., Pharmacia CeeOn® 809C brand of IOL published in Koranyi et al.: J Cataract Refract Surg, 2002; 28:243-247, herein incorporated by reference). The measured axial lengths may then be transformed by addition of a constant value, and the mean anterior chamber depth was calculated anew. This process may be iterated until the calculated mean anterior chamber depth coincided with that of the other. 
     The pupil size may be measured preoperatively at the desired light level, e.g. mesopic light (dusk). The pupil size at dusk is about 4 mm, but variations between at least about 2 mm to 6 mm or more can occur. 
     The aspheric representation of the anterior corneal is typically derived from corneal topography, most commonly based on the so-called Placido disk principle. Resulting height maps can be used to fit an aspherical description of the surface by a least squares optimization. Slit based methods such as implemented on the Orbscan® brand of topography systems (Bausch &amp; Lomb) may be used for the same purpose (Holladay et al. J Refract Surg 2002; 18:683-691). The Orbscan® can also be used to obtain an aspherical description of the posterior corneal surface in the same manner. Instruments based on the Scheimpflug principle, such as Nidek EAS-100, may also be used to obtain anterior and posterior curvatures of the corneal surfaces. By rotating the slit and taking multiple pictures the topography of both surfaces can be obtained. The newly presented Oculus Pentacam eye scanner, which is also based on the Scheimpflug principle, achieves this within a couple of arc seconds, and is suitable for use with the method  100 . 
     Independent of the measurement system used, the topography for the purpose of the method  100  is conveniently described as a conicoid surface of revolution, characterized by the aspheric constant k value (conic constant), optionally extended with additional polynomial terms. Preferably, k values are obtained for both anterior and posterior corneal surfaces, optionally in combination with additional polynomial terms. 
     The method  100  may be used to calculate an amount of at least one of a postoperative refraction and a postoperative spherical aberration for a lens that is implantable into the eye of a subject. Preferably the calculations are carried out using a ray tracing program or procedure, although other calculating means may also be used, such as an optics design program. One benefit of the method  100  is that it is capable of reducing the amount of computation necessary when using a ray tracing procedure and yet produces reliable information for lens power selection. Accordingly, only limited numbers of rays needs to be employed with the routine rather than the great number of rays normally traced for the purpose of optical design (several software packages are commercially available, e.g., those sold under the brand names of Zemax®, OSLO®, Code V®), which are cumbersome to employ, although they could be used for the purpose of the IOL power calculation and assessment of the resulting image quality. 
     In one aspect of the invention a ray entering the pupil at 1/√{square root over (2)} of the entrance pupil height is employed. This ray is here termed the focusing ray. Alternatively a ray at the full pupil diameter (marginal ray) and a ray with close to zero ray height (paraxial ray) are traced. Focus is in this case assumed to be at the midpoint of the foci of the marginal and the paraxial rays. The distance between the foci of the marginal and the paraxial rays, the longitudinal spherical aberration (LSA), can also be used as a simple metric for image quality. The smaller LSA is the better the image quality is. 
     According to one embodiment of the method  100 , one or more of the surfaces of an IOL such as the IOL  30  are described by the formula: 
                   x   =           (     1   R     )     ⁢     y   2         1   +       1   -         k   ⁡     (     1   R     )       2     ⁢     y   2               +       a   4     ⁢     y   4       +       a   6     ⁢     y   6       +   …             (   1   )               
wherein R is the radius of curvature at the apex, k the conic constant, y the radial distance from the optical axis and x the sag in the direction of light propagation. Depending on the value of k the first term is a conic section and describes a:
         hyperbola k&lt;0   parabola k=0   prolate ellips 0&lt;k&lt;1   circle k=1   oblate ellips k&gt;1       

     The coefficients for the additional polynomial terms a 4 , a 6 , etc. can either be set to zero, in which case the surface is a conicoid of revolution, or be given positive or negative non-zero values to modify the simple conic section rotational surface. Alternatively, the method  100  may be used with other forms of the above equation or other definitions of terms such as conic constant. 
     A method to design intraocular lenses for the purpose of correcting average corneal spherical aberration obtained from pooled corneal data of a an elected patient group is further explained in the U.S. Pat. No. 6,609,793, herein incorporated by reference. Corneas of the normal population are in the prolate range (0&lt;k&lt;1) however, the method  100  is applicable to all types of aspheric IOLs, such as IOLs with a hyperbolic (including parabolic) or oblate (including spherical) surfaces. 
     According to one aspect, patients having had corneal refractive surgery to correct myopia can have a hyperbolic anterior surface (k≦0), while those having had corneal refractive surgery to correct hyperopia can have an oblate anterior surface (k≧1) (Buehren et al., Scientific poster 144, AAO 2004, New Orleans). The method  100  demonstrates satisfying capacity in obtaining careful prediction of IOL powers also for such patients, including estimating the resulting retinal image quality in terms of LSA, although surfaces deviating considerably from prolate may be required. 
     The method  100  may further comprise obtaining the corneal apex radius, typically both anterior and posterior corneal apex radii, from the topography, or from corneal radius measured by conventional keratometry (at about 3 mm diameter) and corrected to the value at the apex by the method described by Dubbelman et al. (Vision Res 2005; 45:117-132), herein incorporated by reference. 
     There are both indirect and direct methods available to preoperatively determine the location of the lens haptic plane (LHP), i.e. the distance from the anterior cornea to the LHP. Direct methods include ultrasound biomicroscopy, optical coherence tomography and Scheimpflug photography as taught in U.S. Pat. No. 5,968,095, herein incorporated by reference. Newer, commercially available equipment having the capacity to conduct such direct measurements includes the following systems, which are available from the listed companies: Artemis (Ultralink LLC), Visante OCT (Zeiss), and Pentacam (Oculus). 
     Alternatively, the location of the lens haptic plane may be obtained with a prediction algorithm that includes preoperatively measured parameters such as axial eye length (AL), corneal radius (CR) or, alternatively, corneal power (K), anterior chamber depth (ACD), and crystalline lens thickness (LT). Norrby et al. (J Cataract Refract Surg 2005; 31:1338-1344) have studied prediction algorithms of the general type:
 
 LHP=a+b×AL+c×ACD+d×LT+e×CR+f×AL   2   +g×ACD   2   +h×LT   2   +i×CR   2   +j×AL×ACD+k×AL×LT+l×AL×CR+m×ACD×LT+n×ACD×CR+o×LT×CR   (2)
 
     One finding of the study mentioned is that AL and ACD measured with one piece of equipment can deviate systematically from that measured with another piece of equipment (Norrby et al. J Cataract Refract Surg 2003; 29:95-99; see also Koranyi et al. J Cataract Refract Surg 2002; 28:243-247, and Norrby, J Cataract Refract Surg 2001; 27:1656-1661, all of which are herein incorporated by reference). To correct measured AL and ACD the concept of a Human Population Average (HPA) scale was devised (Norrby et al. J Cataract Refract Surg 2005; 31:1338-1344). Algorithms containing LT and ACD in general were found to be unreliable when employing measurements obtained with different pieces of equipment, despite correction of ACD to the HPA scale (Norrby et al. J Cataract Refract Surg 2005; 31:1338-1344). Also the early attempts to model LHP in terms of and LT and ACD (Norrby and Koranyi, J Cataract Refract Surg 1997; 23:254-259, U.S. Pat. No. 5,968,095, both of which are herein incorporated by reference) were found unreliable. Regression formulas containing CR and AL in linear, quadratic and cross-terms, with or without the constant a, in accordance with the general formula above, gave consistent results independent of the measurement equipment used, when AL was transformed to the HPA scale. A preferred algorithm is
 
 LHP= 2.486+0.2174×( AL+ΔAL )−0.4213 ×CR   (3)
 
wherein AL is the measured axial eye length, ΔAL is the transformation constant (ranging from 0.2 mm to 1.0 mm depending on equipment used) and CR is the keratometric corneal radius (at about 3 mm diameter); (see also S Norrby et al. J Cataract Refract Surg 2005; 31:1338-1344). The position of the IOL  30  in the eye is determined by its vault height, i.e. the distance between the LHP and the anterior apex of the IOL  30 , where the LHP coincides with the plane of contact between the IOL haptics and ocular tissue (e.g. the capsular bag). The vault height is considered to be positive if the anterior IOL apex is posterior to LHP and negative if the anterior IOL apex is anterior to LHP.
 
     The present invention also relates to an improved eye model, which admits simple ray tracing procedures to evaluate suitable intraocular lenses for implantation and to select a lens available in terms of refractive power and/or asphericity. The eye model includes values of the axial eye length based on a measured axial eye length transformed to the human population average scale by addition of a transformation constant; the pupil size at a desired light level, an aspheric representation of the corneal curvature and a value of the lens haptic plane location (the plane of fixation of an implantable IOL following implantation). Routines of how to obtain the mentioned necessary values for the eye model from an individual are described above. Besides admitting a significant calculation simplicity, the invented eye model provides estimations that are substantially independent from what type of biometric instrumentation that are used for the eye axial length. 
     In certain embodiments, a method comprises determining the optical quality of an eye following the implantation of an implantable IOL. The method may be based upon using the above described eye model with an aspheric IOL and a ray tracing routine, for example, in which a marginal ray and a paraxial ray are used to calculate the longitudinal spherical aberration (LSA). If an undesired high value of LSA is obtained from the method, another lens with another power and/or asphericity is selected and the method is repeated until a lens is found that provides a predetermined optical quality, as represented by a low LSA. 
     Referring to  FIG. 4 , in certain embodiments, a method  200  of selecting an IOL comprises one or more of the following operational blocks  210 - 270 . Where appropriate, aspects of the method  100  discussed above herein may also be applied to embodiments of the method  200 . The method  200  comprises an operational block  210 , which comprises determining one or more ocular dimensions based on one or more measurements of at least one eye. The method  200  also comprises an operational block  220 , which comprises selecting a desired refractive outcome. The method  200  comprises an operational block  230 , which comprises selecting an IOL (e.g., the IOL  30 ) having at least one of a power, an aspheric profile, and a lens plane. The method  200  comprises an operational block  240 , which comprises establishing an eye model based on one or more characteristics of the at least one eye. The method  200  comprises an operational block  250 , which comprises determining a location of the lens plane. The method  200  comprises an operational block  260 , which comprises performing a calculation to determine a predicted refractive outcome based on the eye model and a ray tracing algorithm. The method  200  comprises an operational block  270 , which comprises comparing the predicted refractive outcome to the desired refractive outcome. The method  200  comprises an operational block  280 , which comprises, based on the comparison, repeating the calculation with an IOL having at least one of a different power, a different aspheric profile, and a different lens plane. The method  200  comprises an operational block  290 , which comprises selecting an implantable IOL configured for implantation into the eye of a subject. 
     Referring to operational block  210 , the method  200  incorporate one or more ocular dimensions based, for example, the eye model illustrated in  FIG. 1 . In certain embodiments, the method  200  may incorporate data from a database of eyes or from a plurality of eyes belonging to subject belonging to a particular population such as a population of cataract patients or subjects that have received a corneal treatment for vision correction. Such data is illustrated, for instance, in U.S. Pat. Nos. 6,609,793 and 6,830,332 and U.S. Patent Application Publication 2004/088050, all herein incorporated by reference. 
     Referring to operational block  220 , the desired refractive outcome may be, for example, providing a subject with distant vision and/or near vision. This may include providing the subject sufficient visual acuity that there is no need for external corrective spectacles or contact lenses for near and/or distant vision. Alternatively, the refractive outcome may be less stringent in terms of the degree of correction. For example the refractive outcome might to provide sufficient visual acuity such that normal vision is provided by the use of external corrective spectacles or contact lenses having a correction of less than about 3 Diopters, preferably less than 2 Diopters, and more preferably less than 1 Diopter. In some embodiments the desired refractive outcome is reduction of spherical aberrations or other higher order aberrations that would have been created by the use of, for example, and IOL having only spherical surfaces. Alternatively or additionally, the desired refractive outcome is reduction of spherical aberrations or other higher order aberrations induced by the cornea or some other part of the eye. Such criteria are discussed in U.S. Pat. No. 6,609,793. 
     Referring to operational block  250 , the lens plane may be lens haptic plane (LHP) illustrated, for example, in  FIG. 1 . Alternatively, the lens plane may be some other that is appropriate for determining, for example, the power, asphericity, and/or location of an IOL in within the eye of a subject. For example, the lens plane used in the method  200  may be an effective principal plane of the optic  32 . In such embodiments, a distinction may be made between lenses of various manufactures so that effect of different geometries on IOL performance may be taken into account. 
     Referring to operational block  260 , calculation of a predicted refractive outcome is based not simply on measurements and correlation databases, such as those used in currently existing formulas such as Holladay 1 and 2, Hoffer Q, and SRK/T and a ray tracing algorithm. Rather, the current method  200  calculates a predicted refractive outcome based on a ray tracing or wavefront analysis in addition to using measurement and correlation databases. This approach has been found by the inventor to provide a more reliable way of providing a patient a lens with the correction power to provide normal vision as well as provide the possibility of correcting for higher order ocular aberrations such as spherical aberrations. The one or more ocular dimensions may include, for example, any of the dimension of any of the elements of the eye  20  illustrated in  FIG. 1   
     In certain embodiments, a computer system  300  for selecting an IOL for placement into the eye of a subject comprises a processor  302  and a computer readable memory  304  coupled to the processor  302 . The computer readable memory  304  has stored therein an array of ordered values  308  and sequences of instructions  310  which, when executed by the processor  302 , cause the processor  302  to select an implantable IOL configured for implantation into the eye of a subject. The array of ordered values  308  may comprise data used or obtained from the methods  100 ,  200  or other methods consistent with embodiments of the invention. For example, the array of ordered values  308  may comprise one or more ocular dimensions of an eye or plurality of eyes from a database, a desired refractive outcome, parameters of an eye model based on one or more characteristics of at least one eye, and data related to an IOL or set of IOLs such as a power, an aspheric profile, and/or a lens plane. The sequence of instructions  310  may include one or more steps of the methods  100 ,  200  or other methods consistent with embodiments of the invention. In some embodiments, the sequence of instructions  310  includes determining a location of the lens plane of an IOL, performing one or more calculations to determine a predicted refractive outcome based on an eye model and a ray tracing algorithm, comparing a predicted refractive outcome to a desired refractive outcome, and based on the comparison, repeating the calculation with an IOL having at least one of a different power, a different aspheric profile, and a different lens plane. 
     The computer system  300  may be a general purpose desktop or laptop computer or may comprise hardware specifically configured performing the task of selecting an IOL for placement into the eye of a subject. In some embodiments, the computer system  300  is configured to be electronically coupled to another device such as a phacoemulsification console or one or more instruments for obtaining measurements of an eye or a plurality of eyes. In other embodiments, the computer system  300  is a handheld device that may be adapted to be electronically coupled to one of the devices just listed. 
     A number of examples will now be presented demonstrating how methods and devices according to embodiments of the invention may be used to determine a suitable lens for a patient in terms refractive power and/or reduced aberrations. These examples also demonstrate that these methods can be used to estimate the visual quality of the patient in terms of the longitudinal spherical aberration (LSA) of the retinal image. The examples given demonstrate that methods according to the invention are applicable for different k values of the cornea and the importance of considering pupil size and how consideration of k values for both anterior and posterior corneal surfaces effect the predictability of the methods. 
     Example 1 
     Demonstration of the Ray Tracing Technique for the Focusing Ray 
     In this and the following examples, a ray tracing procedure is used in determining various lens parameters such as, for example, IOL optic power and LHP. The ray tracing procedure utilized is in the form of a Microsoft Excel spreadsheet; however, any ray tracing program or routine may, in general, be utilized in various embodiments of the invention. In the ray tracing discussed here, a meridional ray impinges on a surface and follows a straight line, as expressed by
 
 y=y   o   +ut   (4)
 
where y o  is the radial height at the origin, u the angle (in radians) between the ray and the optical axis and t the distance, along and parallel with the optical axis, between the origin and the intersection with the surface.
 
     The condition for intersection is that the radial height y at the surface and of the ray is the same. The calculation can be set up in an Excel spreadsheet and the Goal Seek (or Solver) utility can be used to find the value for t for which there is zero difference between the heights of the surface and of the ray. In this examples, the additional polynomial terms (a 4 y 4 ,a 6 y 6 ) are set equal to zero for simplicity, but the method is valid for non-zero values also. 
     The slope of the surface at the point of intersection is found by numeric differentiation. From the slope, the angle of the normal is found, and Snell&#39;s law of refraction is applied to find the angle of the refracted beam. The intersection of the refracted beam with next surface is sought as before, the slope at the point of intersection is found as before, Snell&#39;s law of refraction is again applied to find the angle of the beam leaving this surface, etc. until after refraction at the last surface the intersection between that beam and the optical axis (focus) is sought. This calculation can be set up as a macro program to perform these calculations. 
     A ray traced at 1/√{square root over (2)}≈0.7 of the height of the entrance pupil height is an average ray in the sense that it divides the pupil into two surfaces of equal area, one outer annular ring and a central circle. It is here termed the focusing ray and its intersection with the optical axis is adopted as one definition of best focus. 
     An alternative definition of best focus is the midpoint between a marginal ray (i.e. a ray entering at the margin of the pupil) and a paraxial ray (i.e. entering infinitesimally close to the optical axis at the pupil). The distance between the foci of the marginal and paraxial rays, the longitudinal spherical aberration (LSA) is a simple metric for optical quality of the image formed at the photoreceptor layer of the retina. The sign convention is here taken that if the paraxial ray focuses posterior to the marginal ray, the spherical aberration is termed positive. Conversely if the paraxial ray focuses anterior to the marginal ray, the spherical aberration is negative. The best image quality is when LSA is zero. The smaller the absolute value of LSA, the better the image quality. 
     The entrance pupil (on the first spectacle lens surface) is 5 mm in this example. 
     
       
         
           
               
            
               
                   
               
               
                 CORNEA 
               
            
           
           
               
               
               
               
            
               
                   
                 Surface 
                 Apex radius (mm) 
                 k 
               
               
                   
               
               
                   
                 anterior 
                 7.7 
                 0.82 
               
               
                   
                 posterior 
                 6.8 
                 0.66 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 LENS 
               
            
           
           
               
               
               
               
            
               
                   
                 Surface 
                 Apex radius (mm) 
                 k 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 anterior 
                 12.154 
                 −5 
               
               
                   
                 posterior 
                 −12.154 
                 −5 
               
               
                   
               
            
           
         
       
     
     Coefficients a 4 , a 6 , etc. are all set equal to zero in this example. 
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Object 
                 Spectacle 
                 Vertex 
                 Corneal 
                 Anterior 
                 Intraocular 
                 Axial 
                 Transf. 
               
               
                 distance 
                 lens thickness 
                 distance 
                 thickness 
                 chamber depth 
                 lens thickness 
                 length 
                 const. 
               
               
                   
               
               
                 6 m 
                 2 mm 
                 12 mm 
                 0.5 mm 
                 4.9 mm 
                 1.13 mm 
                 23.77 mm 
                 0.23 mm 
               
               
                   
               
            
           
         
       
     
     Anterior chamber depth is defined here and in subsequent examples as the distance from the anterior apex of the cornea to the anterior apex of the lens (whether the natural lens or an IOL). The transformation constant, here assumed to be 0.23, transforms the measured axial length to the human population average (HPA) scale. 
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle  
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous  
                 lens 
                 Vitreous 
               
               
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458  
                 1.336 
               
               
                   
               
            
           
         
       
     
     The macro program “Sub trace( )” is run to determine the ray path with the given input, followed by “Sub spectacle( )” to find the spectacle power giving zero ray height at the retina, i.e. the power to focus the image on the photoreceptor layer of the retina. Because changing the spectacle power changes the ray incidence on the cornea, “Sub trace( )” is run again followed by “Sub spectacle( )”. Repeating this sequence a few times results in sufficient accuracy in the final result. 
       FIG. 6  illustrates the formulas programmed into each cell of the Excel spreadsheet used to provide the ray tracing program, while  FIG. 7  illustrates the numerical result of the calculation in each cell. The Sub trace( ) and Sub spectacle( ) routines used in the spreadsheet model are as follows: 
     
       
         
           
               
             
               
                   
               
             
            
               
                 Sub trace( ) 
               
            
           
           
               
               
            
               
                   
                 Range(“D12”).GoalSeek Goal:=0, ChangingCell:=Range(“D10”) 
               
               
                   
                 Range(“E12”).GoalSeek Goal:=0, ChangingCell:=Range(“E10”) 
               
               
                   
                 Range(“F12”).GoalSeek Goal:=0, ChangingCell:=Range(“F10”) 
               
               
                   
                 Range(“G12”).GoalSeek Goal:=0, ChangingCell:=Range(“G10”) 
               
               
                   
                 Range(“H12”).GoalSeek Goal:=0, ChangingCell:=Range(“H10”) 
               
               
                   
                 Range(“I12”).GoalSeek Goal:=0, ChangingCell:=Range(“I10”) 
               
               
                   
                 Range(“J12”).GoalSeek Goal:=0, ChangingCell:=Range(“J10”) 
               
            
           
           
               
            
               
                 End Sub 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
             
               
                   
               
             
            
               
                   
                 Sub spectacle( ) 
               
            
           
           
               
               
            
               
                   
                 Range(“J8”).GoalSeek Goal:=0, ChangingCell:=Range(“D4”) 
               
            
           
           
               
               
            
               
                   
                 End Sub 
               
               
                   
               
            
           
         
       
     
     In the example, the spectacle power becomes +0.01 D.  FIG. 8 . 
     Example 2 
     Selecting Power of a Spherical IOL 
     The entrance pupil (on the first spectacle lens surface) is 5 mm in this example. 
     
       
         
           
               
            
               
                   
               
               
                 CORNEA 
               
            
           
           
               
               
               
               
            
               
                   
                 Surface 
                 Apex radius (mm) 
                 k 
               
               
                   
               
               
                   
                 anterior 
                 7.87 
                 0.82 
               
               
                   
                 posterior 
                 6.40 
                 0.66 
               
               
                   
               
            
           
         
       
     
     The radii apply at the center of the cornea. Corneal radius determined with a keratometer applies at a circle of about 3 mm diameter. With the k-values given, 7.90 mm and 6.42 mm, respectively, would have been measured. 
     
       
         
           
               
            
               
                   
               
               
                 INTRAOCULAR LENSES 
               
            
           
           
               
               
               
               
               
            
               
                 Power 
                 Front radius 
                 Back radius 
                 Thickness 
                 Vault height 
               
               
                 (D) 
                 (mm) 
                 (mm) 
                 (mm) 
                 (mm) 
               
               
                   
               
               
                 20.0 
                 12.154 
                 −12.154 
                 1.10 
                 0.03 
               
               
                 20.5 
                 11.856 
                 −11.856 
                 1.11 
                 0.03 
               
               
                 21.0 
                 11.572 
                 −11.572 
                 1.12 
                 0.02 
               
               
                   
               
            
           
         
       
     
     Vault height is the distance from LHP to the anterior surface of the lens (positive if the lens surface is posterior to LHP). 
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
               
            
               
                 Object 
                 Spectacle lens 
                 Vertex 
                 Corneal 
                   
                 Axial  
               
               
                 distance 
                 thickness 
                 distance 
                 thickness 
                 LHP 
                 length 
               
               
                   
               
               
                 6 m 
                 2 mm 
                 12 mm 
                 0.574 mm 
                 4.36 mm 
                 23.92 mm 
               
               
                   
               
            
           
         
       
     
     LHP was calculated by the formula
 
 LHP= 2.486+0.2174×( AL+ΔAL )−0.4213 ×CR  
 
     where CR is the measured corneal radius (7.90 mm), AL is the measured axial length (23.69 mm) and ΔAL is the transformation constant, here assumed to be 0.23 mm (AL+ΔAL) is the axial length transformed to the human population average (HPA) scale, which is the value given in the table. The anterior chamber depth is LHP plus the vault height for the specific IOL chosen. 
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle 
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous 
                 lens 
                 Vitreous 
               
               
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458 
                 1.336 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
               
            
               
                   
                 IOL (D) 
                 Spectacle (D) 
               
               
                   
               
               
                   
                 20.0 
                 +0.40 
               
               
                   
                 20.5 
                 +0.03 
               
               
                   
                 21.0 
                 −0.37 
               
               
                   
               
            
           
         
       
     
     A surgeon would probably choose to implant the 21.0 D lens. Slight myopia is often preferred. 
     Using the midpoint between marginal and paraxial ray foci as focusing criterion the following results are obtained. 
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
               
            
               
                   
                 IOL (D) 
                 Spectacle (D) 
               
               
                   
               
               
                   
                 20.0 
                 +0.39 
               
               
                   
                 20.5 
                 +0.01 
               
               
                   
                 21.0 
                 −0.38 
               
               
                   
               
            
           
         
       
     
     These results are for all practical purpose equal to those obtained with the focusing ray as focusing criterion. The axial defocus of the marginal and paraxial rays in relation to the focusing rays are given in the following table. 
     
       
         
           
               
            
               
                   
               
               
                 DEFOCUS IN RELATION TO FOCUSING RAY 
               
            
           
           
               
               
               
            
               
                 IOL (D) 
                 Marginal ray 
                 Paraxial ray 
               
               
                   
               
               
                 20.0 
                 −0.267 
                 +0.259 
               
               
                 20.5 
                 −0.278 
                 +0.268 
               
               
                 21.0 
                 −0.289 
                 +0.279 
               
               
                   
               
            
           
         
       
     
     The marginal ray thus focuses anterior and the paraxial ray posterior to the focusing ray, indicating that the optical system has overall positive spherical aberration. The near symmetry in relation to the focusing ray is another indication of the agreement between the two focusing criteria in this example. 
     Example 3 
     Selecting Power of an Aspherical IOL 
     A generalized aspheric surface may be characterized using Equation (1), discussed in greater detail above herein. The entrance pupil (on the first spectacle lens surface) is 5 mm in this example. 
     
       
         
           
               
            
               
                   
               
               
                 CORNEA 
               
            
           
           
               
               
               
               
            
               
                   
                 Surface 
                 Apex radius (mm) 
                 k 
               
               
                   
               
               
                   
                 anterior 
                 7.87 
                 0.82 
               
               
                   
                 posterior 
                 6.40 
                 0.66 
               
               
                   
               
            
           
         
       
     
     The radii apply at the center of the cornea. Corneal radius determined with a keratometer applies at a circle of about 3 mm diameter. With the k-values given, 7.90 mm and 6.42 mm, respectively, would have been measured. 
     
       
         
           
               
            
               
                   
               
               
                 INTRAOCULAR LENSES 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Anterior surface 
                 Posterior surface 
                 Thick- 
                 Vault 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Power 
                 Radius 
                   
                 Radius 
                   
                 ness 
                 height 
               
               
                 (D) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 20.0 
                 12.154 
                 −7 
                 −12.154 
                 −7 
                 1.13 
                 0.01 
               
               
                 20.5 
                 11.856 
                   
                 −11.856 
                   
                 1.13 
                 0.00 
               
               
                 21.0 
                 11.572 
                   
                 −11.572 
                   
                 1.14 
                 0.00 
               
               
                 21.5 
                 11.301 
                   
                 −11.301 
                   
                 1.15 
                 −0.01 
               
               
                 22.0 
                 11.043 
                   
                 −11.043 
                   
                 1.16 
                 −0.01 
               
               
                   
               
            
           
         
       
     
     Vault height is the distance from LHP to the anterior surface of the lens. 
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
               
            
               
                 Object 
                 Spectacle lens 
                 Vertex 
                 Corneal 
                   
                 Axial 
               
               
                 distance 
                 thickness 
                 distance 
                 thickness 
                 LHP  
                 length 
               
               
                   
               
               
                 6 m 
                 2 mm 
                 12 mm 
                 0.574 mm 
                 4.36 mm  
                 23.92 mm 
               
               
                   
               
            
           
         
       
     
     LHP was calculated by the formula,
 
 LHP= 2.486+0.2174×( AL+ΔAL )−0.4213 ×CR,  
 
where CR is the measured corneal radius (7.90 mm), AL is the measured axial length (23.69 mm) and ΔAL is the transformation constant, here assumed to be 0.23 mm (AL+ΔAL) is the axial length transformed to the human population average (HPA) scale, which is the value given in the table. The anterior chamber depth is LHP plus the vault height for the specific IOL chosen.
 
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle 
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous 
                 lens 
                 Vitreous 
               
               
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458 
                 1.336 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
               
            
               
                   
                 IOL (D) 
                 Spectacle (D) 
               
               
                   
               
               
                   
                 20.0 
                 +1.19 
               
               
                   
                 20.5 
                 +0.87 
               
               
                   
                 21.0 
                 +0.57 
               
               
                   
                 21.5 
                 +0.26 
               
               
                   
                 22.0 
                 −0.04 
               
               
                   
               
            
           
         
       
     
     Using the midpoint between marginal and paraxial ray foci as focusing criterion, the expected spectacle refraction is −0.11 D with the 22.0 D IOL. The focus of the marginal ray is +0.067 mm in relation to the focusing ray, i.e. focuses posterior to the focusing ray. The focus of the paraxial ray is −0.118 mm in relation to the focusing ray, i.e. focuses anterior to the focusing ray. This system thus exhibits negative spherical aberration, reversing the focusing order of the rays. 
     Example 4 
     Demonstrating the Influence of k-Value 
     The entrance pupil (on the first spectacle lens surface) is 5 mm in this example. 
     The average k-value in the human population is 0.82, with a standard deviation of 0.18 (Dubbelman, M., Weeber, H. A., van der Heijde, G. L. and Völker-Dieben, H. J. Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography. Acta Ophthalmol Scand 2002; 80: 379-383). 
     For illustration a 20.5 D spherical lens and a 21.5 D aspherical lens with the following designs are chosen. 
     
       
         
           
               
            
               
                   
               
               
                 INTRAOCULAR LENSES 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Anterior surface 
                 Posterior surface 
                 Thick- 
                 Vault 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Power 
                 Radius 
                   
                 a 4   
                 a 6   
                 Radius 
                   
                 a 4   
                 a 6   
                 ness 
                 height 
               
               
                 (D) 
                 (mm) 
                 k 
                 (mm −4 ) 
                 (mm −6 ) 
                 (mm) 
                 k 
                 (mm −4 ) 
                 (mm −6 ) 
                 (mm) 
                 (mm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 20.5 
                 11.856 
                 1 
                 0 
                 0 
                 −11.856 
                 1 
                 0 
                 0 
                 1.11 
                 0.03 
               
               
                 spherical 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 21.5 
                 11.301 
                 −4 
                 −1 · 10 −4   
                 −1 · 10 −6   
                 −11.301 
                 −4 
                 0 
                 0 
                 1.15 
                 −0.01 
               
               
                 aspherical 
               
               
                   
               
            
           
         
       
     
     Assume that the surgeon has come to these lens powers with a calculation method that does not take corneal asphericity into account. Which influence will variation of up to 3 standard deviations of corneal asphericity have on postoperative refraction? 
     The keratometrically (at 3 mm diameter) measured anterior corneal radius is assumed to be 7.90 mm. The posterior radius is unknown, but is as in previous examples assumed to be 6.42 mm (at 3 mm diameter) and have a k-value of 0.66. The shape of the posterior surface is further assumed to be independent of that of the anterior surface and remain unchanged when the k-value of the anterior surface is varied. 
     
       
         
           
               
            
               
                   
               
               
                 ANTERIOR CORNEAL SURFACE 
               
            
           
           
               
               
               
            
               
                   
                 Surface characteristics 
                 Apex 
               
            
           
           
               
               
               
               
               
            
               
                   
                 ±SD 
                 k 
                 type 
                 radius (mm) 
               
               
                   
               
               
                   
                 −3 
                 0.28 
                 prolate 
                 7.79 
               
               
                   
                 −2 
                 0.46 
                 prolate 
                 7.82 
               
               
                   
                 −1 
                 0.64 
                 prolate 
                 7.84 
               
               
                   
                 ±0 
                 0.82 
                 prolate 
                 7.87 
               
               
                   
                 +1 
                 1.00 
                 sphere 
                 7.90 
               
               
                   
                 +2 
                 1.18 
                 oblate 
                 7.92 
               
               
                   
                 +3 
                 1.36 
                 oblate 
                 7.95 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                   
                   
                   
                   
                 Axial 
               
               
                   
                 Spectacle 
                   
                   
                   
                 length 
               
               
                 Object 
                 lens 
                 Vertex 
                 Corneal 
                   
                 (trans- 
               
               
                 distance 
                 thickness 
                 distance 
                 thickness 
                 LHP 
                 formed) 
               
               
                   
               
               
                 6 m 
                 2 mm 
                 12 mm 
                 0.574 mm 
                 4.36 mm 
                 23.92 mm 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle 
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous 
                 lens 
                 Vitreous 
               
               
                   
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458 
                 1.336 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
               
               
            
               
                   
                   
                   
                 Spectacle correction  
               
               
                   
                   
                   
                 (D) with 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                   
                 Spherical 
                 Aspherical 
               
               
                   
                   
                   
                 IOL 
                 IOL 
               
               
                   
                 SD 
                 k 
                 20.5D 
                 21.5D 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 −3 
                 −0.72 
                 0.21 
                 0.23 
               
               
                   
                 −2 
                 −0.54 
                 0.15 
                 0.15 
               
               
                   
                 −1 
                 −0.36 
                 0.09 
                 0.07 
               
               
                   
                 ±0 
                 −0.18 
                 0.03 
                 −0.01 
               
               
                   
                 +1 
                 0.00 
                 −0.04 
                 −0.10 
               
               
                   
                 +2 
                 0.18 
                 −0.10 
                 −0.18 
               
               
                   
                 +3 
                 0.36 
                 −0.17 
                 −0.27 
               
               
                   
                   
               
            
           
         
       
     
     This example shows that the effect of neglecting corneal asphericity in IOL power calculation has effect on the postoperative refraction for spherical as well as for aspherical IOLs. 
     Example 5 
     Finding the Influence of Pupil Size 
     The entrance pupil is defined on the first spectacle lens surface and is varied in this example. 
     For this example a 20.5 D spherical lens and a 21.5 D aspherical lens with the following designs are chosen. 
     
       
         
           
               
            
               
                   
               
               
                 INTRAOCULAR LENSES 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Anterior surface 
                 Posterior surface 
                 Thick- 
                 Vault 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Power 
                 Radius 
                   
                 a 4   
                 a 6   
                 Radius 
                   
                 a 4   
                 a 6   
                 ness 
                 height 
               
               
                 (D) 
                 (mm) 
                 k 
                 (mm −4 ) 
                 (mm −6 ) 
                 (mm) 
                 k 
                 (mm −4 ) 
                 (mm −6 ) 
                 (mm) 
                 (mm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 20.5 
                 11.856 
                 1 
                 0 
                 0 
                 −11.856 
                 1 
                 0 
                 0 
                 1.11 
                 0.03 
               
               
                 spherical 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 21.5 
                 11.301 
                 0 
                 −1 · 10 −3   
                 1 · 10 −6   
                 −11.301 
                 1 
                 0 
                 0 
                 1.15 
                 −0.01 
               
               
                 aspherical 
               
               
                   
               
            
           
         
       
     
     Normally pupil size is not considered in IOL power calculation. About 4 mm is common at mesopic light conditions (dusk), but individual variations from 2 mm up to 6 mm or even wider are known. What could the consequences be for patients depending on pupil size? 
     The following additional parameters are assumed. 
     
       
         
           
               
            
               
                   
               
               
                 CORNEA 
               
            
           
           
               
               
               
            
               
                   
                 Apex  
                   
               
               
                   
                 radius 
                   
               
               
                 Surface 
                 (mm) 
                 k 
               
               
                   
               
               
                 anterior 
                 7.87 
                 0.82 
               
               
                 posterior 
                 6.40 
                 0.66 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Spectacle 
                   
                   
                   
                 Axial  
               
               
                 Object 
                 lens 
                 Vertex 
                 Corneal 
                   
                 length (trans- 
               
               
                 distance 
                 thickness 
                 distance 
                 thickness 
                 LHP 
                 formed) 
               
               
                   
               
               
                 6 m 
                 2 mm 
                 12 mm 
                 0.574 mm 
                 4.36 mm 
                 23.92 mm 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle 
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous 
                 lens 
                 Vitreous 
               
               
                   
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458 
                 1.336 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
            
               
                   
                 Postoperative  
               
               
                   
                 refraction (D) with 
               
            
           
           
               
               
               
            
               
                   
                 Spherical  
                 Aspherical  
               
               
                 Pupil  
                 IOL 
                 IOL 
               
               
                 (mm) 
                 20.5D 
                 21.5D 
               
               
                   
               
            
           
           
               
               
               
            
               
                 2 
                 +0.62 
                 +0.02 
               
               
                 3 
                 +0.49 
                 +0.06 
               
               
                 4 
                 +0.29 
                 +0.10 
               
               
                 5 
                 +0.03 
                 +0.15 
               
               
                 6 
                 −0.32 
                 +0.20 
               
               
                   
               
            
           
         
       
     
     This example shows that the pupil size can have large effects on postoperative refraction, in particular in an eye with much spherical aberration, i.e. in the normal case an eye with a spherical IOL. The aspherical IOL in this example corrects for most of the spherical aberration of the cornea, but not all, hence there is some effect of pupil size on postoperative refraction. However, if the corneal aberrations were perfectly corrected by the IOL, there would be no effect of pupil size on postoperative refraction. 
     Example 6 
     Consequence of not Knowing the Posterior Corneal Curvature 
     The entrance pupil (on the first spectacle lens surface) is 5 mm in this example. 
     For this example a 20.5 D spherical lens and a 22.0 D aspherical lens with the following designs are chosen. 
     
       
         
           
               
            
               
                   
               
               
                 INTRAOCULAR LENSES 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Anterior surface 
                 Posterior surface 
                 Thick- 
                 Vault 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Power 
                 radius 
                   
                 radius 
                   
                 ness 
                 height 
               
               
                 (D) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 20.5 
                 11.856 
                 1 
                 −11.856 
                 1 
                 1.11 
                 0.03 
               
               
                 spherical 
                   
                   
                   
                   
                   
                   
               
               
                 22.0 
                 11.043 
                 −7 
                 −11.043 
                 −7 
                 1.16 
                 −0.01 
               
               
                 aspherical 
               
               
                   
               
            
           
         
       
     
     Coefficients a 4 , a 6 , etc. are all set equal to zero in this example. 
     In the normal case only the anterior radius of the cornea is measured, known and used in IOL power calculation. Corneal thickness, posterior radius and posterior asphericity is generally not known. What are the consequences of making assumptions about these unknown quantities? 
     Assume as before that the corneal curvature (at 3 mm) measured by keratometry was found to be 7.90 mm. 
     
       
         
           
               
            
               
                   
               
               
                 CORNEAL CASES 
               
            
           
           
               
               
               
               
            
               
                   
                 Anterior surface 
                 Posterior surface 
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 Apex 
                   
                 Apex 
                   
                 Thick- 
                 Ratio of radii 
               
               
                   
                 radius 
                   
                 radius 
                   
                 ness 
                 (posterior/ 
               
               
                   
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 anterior 
               
               
                   
               
               
                 Case 1 
                 7.87 
                 0.82 
                 6.40 
                 0.66 
                 0.574 
                 0.81 
               
               
                 Case 2 
                 7.87 
                 0.82 
                 6.40 
                 1.00 
                 0.574 
                 0.81 
               
               
                 Case 3 
                 7.87 
                 0.82 
                 7.30 
                 0.82 
                 0.574 
                 0.93 
               
               
                 Case 4 
                 7.87 
                 0.82 
                 6.40 
                 0.66 
                 0.000 
                 0.81 
               
               
                 Case 5 
                 7.87 
                 0.82 
                 6.95 
                 1.00 
                 0.574 
                 0.88 
               
               
                 Case 6 
                 7.87 
                 1.00 
                 6.40 
                 1.00 
                 0.574 
                 0.81 
               
               
                   
               
            
           
         
       
     
     Case  1  is considered to have the proper values for all variables. The ratio of radii is taken from Dubbelman et al. Acta Ophthalmol Scand 2002; 80:379-383. In Case  2  the posterior surface is assumed to be spherical. In Case  3  the posterior surface is assumed to be concentric with the anterior surface and having the same asphericity, which leads to the ratio of radii given. In Case  4  the corneal thickness is neglected. In Case  5  the ratio of radii is assumed to follow the classic Gullstrand model, i.e. 6.8/7.7. In Case  6  both surfaces are assumed spherical. 
     The following additional parameters are assumed. 
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
            
               
                 Object 
                 Spectacle lens 
                 Vertex  
                 Corneal 
                 Axial length 
               
               
                 distance 
                 thickness 
                 distance 
                 thickness 
                 (transformed) 
               
               
                   
               
               
                 6 m 
                 2 mm 
                 12 mm 
                 0.574 mm 
                 23.92 mm 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle 
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous 
                 lens 
                 Vitreous 
               
               
                   
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458 
                 1.336 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
            
               
                   
                 Postoperative  
               
               
                   
                 refraction (D) with 
               
            
           
           
               
               
               
            
               
                   
                 Spherical  
                 Aspherical  
               
               
                   
                 IOL 
                 IOL 
               
               
                   
                 20.5D 
                 22.0D 
               
               
                   
               
               
                 Case 1 
                 +0.03 
                 −0.04 
               
               
                 Case 2 
                 +0.11 
                 +0.04 
               
               
                 Case 3 
                 −0.74 
                 −0.78 
               
               
                 Case 4 
                 +0.25 
                 +0.18 
               
               
                 Case 5 
                 −0.42 
                 −0.47 
               
               
                 Case 6 
                 +0.04 
                 +0.03 
               
               
                   
               
            
           
         
       
     
     Whether the IOL is spherical or aspherical this example shows that the posterior corneal radius, i.e. the assumed ratio of radii, has the largest influence (Cases 3 and 5). Putting corneal thickness equal to zero (Case  4 ) causes less than a quarter of dioptre increase in refraction. Neglecting posterior corneal asphericity (Case  2 ) has little influence, and simultaneously disregarding asphericity of both surfaces (Case  6 ) has close to negligible influence. This result is coincidental though. Other initial asphericities would give different results as can be inferred from Example 4. 
     Example 7 
     Alternative Calculations Using Optical Design Programs 
     The entrance pupil (on the first spectacle lens surface) is 5 mm in this example. 
     For illustration a 20.5 D spherical lens and a 22.0 D aspherical lens with the following designs are chosen. 
     
       
         
           
               
            
               
                   
               
               
                 INTRAOCULAR LENSES 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                 Anterior  
                 Posterior  
                   
               
               
                   
                   
                 surface 
                 surface 
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 Power 
                 radius 
                   
                 radius 
                   
                 Thickness 
               
               
                   
                 (D) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 20.5 
                 11.856 
                 1 
                 −11.856 
                 1 
                 1.11 
               
               
                   
                 spherical 
                   
                   
                   
                   
                   
               
               
                   
                 22.0 
                 11.043 
                 −7 
                 −11.043 
                 −7 
                 1.16 
               
               
                   
                 aspherical 
               
               
                   
                   
               
            
           
         
       
     
     Using the optical design software OSLO alternative focusing criteria were evaluated
         Minimum on-axis spot size   Minimum RMS OPD on axis   Maximum MTF at 20 cycles/mm   Maximum MTF at 50 cycles/mm       

     Calculations are monochromatic assuming the following refractive indices. 
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Spectacle 
                   
                   
                 Intraocular 
                   
               
               
                   
                 Air 
                 lens 
                 Cornea 
                 Aqueous 
                 lens 
                 Vitreous 
               
               
                   
                   
               
               
                   
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.458 
                 1.336 
               
               
                   
                   
               
            
           
         
       
     
     The keratometrically (at 3 mm diameter) measured anterior corneal radius is assumed to be 7.90 mm. The posterior radius is unknown, but is as in previous examples assumed to be 6.42 mm (at 3 mm diameter) and have a k-value of 0.66. The apex radii are slightly steeper due to the asphericity. 
     
       
         
           
               
            
               
                   
               
               
                 CORNEA 
               
            
           
           
               
               
               
            
               
                   
                 Apex  
                   
               
               
                   
                 radius  
                   
               
               
                 Surface 
                 (mm) 
                 k 
               
               
                   
               
               
                 anterior 
                 7.87 
                 0.82 
               
               
                 posterior 
                 6.40 
                 0.66 
               
               
                   
               
            
           
         
       
     
     Other parameters are as follows. 
     
       
         
           
               
            
               
                   
               
               
                 AXIAL DISTANCES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 Object 
                 Spectacle 
                 Vertex 
                 Corneal 
                 Aqueous 
                 Vitreous 
               
               
                   
                 distance 
                 lens thickness 
                 distance 
                 thickness 
                 thickness 
                 thickness 
               
               
                 IOL type 
                 (m) 
                 (mm) 
                 (mm) 
                 (mm) 
                 (mm) 
                 (mm) 
               
               
                   
               
               
                 Spherical 
                 6 
                 2 
                 12 
                 0.574 
                 3.811 
                 18.425 
               
               
                 Aspherical 
                   
                   
                   
                   
                 3.771 
                 18.415 
               
               
                   
               
            
           
         
       
     
     Vitreous thickness includes an assumed 0.25 mm retinal thickness. 
     
       
         
           
               
            
               
                   
               
               
                 RESULTS 
               
            
           
           
               
               
            
               
                   
                 Spectacle power (D) with 
               
            
           
           
               
               
               
            
               
                   
                 Spherical IOL 
                 Aspherical IOL 
               
               
                 FOCUSING 
                 of 20.5D 
                 of 22.0D 
               
               
                 CRITERION 
                 power 
                 power 
               
               
                   
               
               
                 Minimum spot size 
                 −0.27 
                 +0.03 
               
               
                 Minimum RMS OPD 
                 −0.01 
                 −0.05 
               
               
                 Max MTF @ 20 c/mm 
                 −0.16 
                 +0.01 
               
               
                 Max MTF @ 50 c/mm 
                 +0.26 
                 −0.01 
               
               
                 Focusing ray 
                 +0.03 
                 −0.04 
               
               
                   
               
            
           
         
       
     
     It can be seen that the Minimum RMS OPD criterion, which is a commonly accepted definition of best focus, agrees well with the focusing ray for both the spherical and the aspherical IOLs. The considerable amount of spherical aberration in case of the spherical IOL causes the various focusing criteria to disagree. 
     The through-focus MTF plots (output by the OSLO program) at 20 and 50 cyc/mm used to determine maximum MTF are shown in  FIGS. 9A-9D . The horizontal line at the top is the diffraction limited MTF of the system at the spatial frequency given. 
     Example 8 
     Correcting Extreme Corneal Aberrations by Adjusting the Shape of the IOL 
     The entrance pupil (on the first spectacle lens surface) is 4 mm in this example. 
     The k-value can vary considerably outside the normal range (see Example 4) in persons who have undergone corneal refractive surgery. Correction of myopia tends to make the corneal spherical aberration more positive (towards oblate), while correction of hyperopia tends to make the corneal spherical aberration more negative (towards hyperopic) (Buehren et al., Scientific poster 144, AAO 2004, New Orleans). 
     Consider two eyes, one originally −5 D axially myopic and the other +5 D axially hyperopic. They thus differ in axial length and proportionally in anterior chamber depth. However, their corneas and lenses are originally assumed to be identical. Their refractive state is characterized by the spectacle spherical equivalent (SE) and longitudinal spherical aberration (LSA). The anterior chamber depth was estimated from clinical data for eyes of corresponding lengths. 
     For this example the following refractive indices are assumed. 
     
       
         
           
               
            
               
                   
               
               
                 REFRACTIVE INDICES 
               
            
           
           
               
               
               
               
               
               
            
               
                   
                 Spectacle 
                   
                 Aqueous, 
                 Crystalline 
                 Intraocular 
               
               
                 Air 
                 lens 
                 Cornea 
                 vitreous 
                 lens 
                 lens 
               
               
                   
               
               
                 1 
                 1.5 
                 1.376 
                 1.336 
                 1.4274 
                 1.458 
               
               
                   
               
            
           
         
       
     
     The eyes can now be summarized as follows. 
     
       
         
           
               
            
               
                   
               
               
                 ORIGINAL STATUS OF THE EYES 
               
            
           
           
               
               
               
               
            
               
                   
                 Cornea 
                 Crystalline lens 
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                   
                 Anterior 
                 Posterior 
                   
                 Anterior 
                 Posterior 
                   
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Refractive state 
                 Apex 
                   
                 Apex 
                   
                 Thick- 
                 Apex 
                   
                 Apex 
                   
                 Thick- 
                 Ocular distances 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Original 
                 SE 
                 LSA 
                 radius 
                   
                 radius 
                   
                 ness 
                 radius 
                   
                 radius 
                   
                 ness 
                 AL 
                 ACD 
                 LHP 
               
               
                 ametropia 
                 (D) 
                 (mm) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
                 (mm) 
                 (mm) 
               
               
                   
               
               
                 Myopic 
                 −5.0 
                 0.084 
                 7.870 
                 0.82 
                 6.400 
                 0.66 
                 0.574 
                 10.670 
                 −3 
                 −5.848 
                 −2 
                 3.76 
                 25.43 
                 3.47 
                 4.74 
               
               
                 Hyperopic 
                 +5.0 
                 0.023 
                 7.870 
                 0.82 
                 6.400 
                 0.66 
                 0.574 
                 10.670 
                 −3 
                 −5.848 
                 −2 
                 3.76 
                 21.62 
                 2.96 
                 3.91 
               
               
                   
               
            
           
         
       
     
     LHP was calculated from the formula
 
 LHP= 2.486+0.2174×( AL+ΔAL )−0.4213 ×CR  
 
     in which the transformation constant ΔAL was set to 0.25 mm and the corneal radius at 3 mm CR is 7.896 mm with the apex radius and k-value as given in the table. 
     Assume that these eyes undergo corneal refractive surgery to make them emmetropic. Besides correcting the spherical equivalent the myopic eye is assumed to become one unit of k-value towards oblate, and the hyperopic eye is assumed to become one unit of k-value towards hyperopic. The myopic correction further results in decrease of the central thickness of the cornea amounting to 0.060 mm, while the hyperopic correction does not cause any change of the central thickness of the cornea. The decrease in corneal thickness causes a corresponding decrease in AL, ACD and LHP in the myopic case. The following situation ensues. 
     
       
         
           
               
            
               
                   
               
               
                 STATUS OF THE EYES AFTER CORNEAL REFRACTIVE SURGERY 
               
            
           
           
               
               
               
               
            
               
                   
                 Cornea 
                 Crystalline lens 
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Refractive state 
                 Apex 
                   
                 Apex 
                   
                 Thick- 
                 Apex 
                   
                 Apex 
                   
                 Thick- 
                 Ocular distances 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Original 
                 SE 
                 LSA 
                 radius 
                   
                 radius 
                   
                 ness 
                 radius 
                   
                 radius 
                   
                 ness 
                 AL 
                 ACD 
                 LHP 
               
               
                 ametropia 
                 (D) 
                 (mm) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
                 (mm) 
                 (mm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Myopic 
                 0.00 
                 0.434 
                 8.794 
                 1.82 
                 6.400 
                 0.66 
                 0.514 
                 10.670 
                 −3 
                 −5.848 
                 −2 
                 3.76 
                 25.37 
                 3.41 
                 4.68 
               
               
                 Hyperopic 
                 0.00 
                 −0.522 
                 6.967 
                 −0.18 
                 6.400 
                 0.66 
                 0.574 
                 10.670 
                 −3 
                 −5.848 
                 −2 
                 3.76 
                 21.62 
                 2.96 
                 3.91 
               
               
                   
               
            
           
         
       
     
     Note that the myopic eye now has considerable positive spherical aberration (LSA) and that the surgery of the hyperopic has even reversed the sign and resulted in considerable negative spherical aberration (LSA) of the entire eye. 
     Assume that these eyes several years later are eligible for cataract surgery. The aim of the surgery is emmetropia (with the target at 6 m) and elimination of spherical aberration. The following lenses are designed for this purpose. 
     
       
         
           
               
               
               
               
            
               
                   
               
               
                   
                 IOL 
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Vault 
                 Anterior surface 
                 Posterior surface 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Original 
                 Power 
                 Thickness 
                 height 
                 radius 
                   
                 a4 
                 A6 
                 radius 
                   
                 a4 
                 a6 
               
               
                 ametropia 
                 (D) 
                 (mm) 
                 (mm) 
                 (mm) 
                 k 
                 (mm −4 ) 
                 (mm −6 ) 
                 (mm) 
                 k 
                 (mm −4 ) 
                 (mm −6 ) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Myopic 
                 23.20 
                 1.19 
                 −0.02 
                 10.468 
                 −5.45 
                 −1.00 · 10 −3   
                 −4.85 · 10 −5   
                 −10.468 
                 1 
                 0 
                 0 
               
               
                 Hyperopic 
                 20.23 
                 1.13 
                 0.00 
                 12.012 
                 2.45 
                  8.80 · 10 −4   
                 −1.40 · 10 −5   
                 −12.012 
                 1 
                 0 
                 0 
               
               
                   
               
            
           
         
       
     
     The situation is now characterized as follows. 
     
       
         
           
               
            
               
                   
               
               
                 STATUS OF THE EYES AFTER CATARACT SURGERY 
               
            
           
           
               
               
               
            
               
                   
                 Cornea 
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Anterior 
                 Posterior 
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Refractive state 
                 Apex 
                   
                 Apex 
                   
                 Thick- 
                 Ocular distances 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Original 
                 SE 
                 LSA 
                 radius 
                   
                 radius 
                   
                 ness 
                 AL 
                 ACD 
                 LHP 
               
               
                 ametropia 
                 (D) 
                 (mm) 
                 (mm) 
                 k 
                 (mm) 
                 k 
                 (mm) 
                 (mm) 
                 (mm) 
                 (mm) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
            
               
                 Myopic 
                 0.00 
                 0.00 
                 8.794 
                 1.82 
                 6.400 
                 0.66 
                 0.514 
                 25.37 
                 4.66 
                 4.68 
               
               
                 Hyperopic 
                 0.00 
                 0.00 
                 6.967 
                 −0.18 
                 6.400 
                 0.66 
                 0.574 
                 21.62 
                 3.91 
                 3.91 
               
               
                   
               
            
           
         
       
     
     This example shows that intraocular lenses can be designed to correct rotationally symmetrical aberrations, i.e. sphere and spherical aberration, for eyes having extreme corneal spherical aberration. 
     The above presents a description of the best mode contemplated of carrying out the present invention, and of the manner and process of making and using it, in such full, clear, concise, and exact terms as to enable any person skilled in the art to which it pertains to make and use this invention. This invention is, however, susceptible to modifications and alternate constructions from that discussed above which are fully equivalent. Consequently, it is not the intention to limit this invention to the particular embodiments disclosed. On the contrary, the intention is to cover modifications and alternate constructions coming within the spirit and scope of the invention as generally expressed by the following claims, which particularly point out and distinctly claim the subject matter of the invention.