Geometric optics or ray optics assumes that the wavelength of the light is sufficiently small so that the light propagation can be described in terms of rays. The path of the rays is determined by reflection and refraction. The latter is defined by Snell's law of refraction. A ray, obeying Snell's law, is called exact ray or real ray. Analyzing optical systems by tracing many rays based on Snell's law is therefore known as exact-ray tracing, finite-ray tracing or real ray-tracing (RRT), a common approach in technical optics.1,2

Such a real ray is opposed to a paraxial ray obeying the paraxial approximation of Snell's law. This first-order approximation of Snell's law is a simplification of geometric optics, in that it assumes small angles of the rays with respect to one common optical axis. Every point in the object space can be one-to-one transformed into a point in the image space. Paraxial optics provides a quick overview about the optical properties of an optical system; however, it is no more than an idealized model.3 Moreover, lens optics is based on paraxial optics, where the essential quantity describing optical components is the diopter. In contrast, analysing optical systems with RRT leads to a more comprehensive description, especially including effects of irregular surface shapes. In terms of geometrical optics, every deviation from a perfect optical system can be quantified in terms of the so called optical aberrations.

The application of RRT in ophthalmology in general is not new. It has been used to address various issues, typically as soon as paraxial optics is not sufficient to describe optical properties accurately. One field of application is corneal refractive surgery, where the shape of the cornea is modified by means of some surgical procedure, for example by removing a certain amount of tissue—the so called ablation profile—with an excimer laser.

Klonos et al. did some early investigations of ablation profiles in a model eye.4 Ortiz et al. used RRT for choosing the optimal ablation parameters from individualized eye models.5-7 Most studies using RRT in the context of simulating and analyzing ablation profiles for refractive surgery are mainly focused on corneal wavefront aberration and they all include measured anterior corneal surface data, the so called corneal topography.8-16

Intraocular lenses (IOLs) are a second major field of investigation for RRT. IOLs are the artificial lenses that are used in cataract surgery to replace the cloudy human crystalline lens. There is literature that evaluates IOL designs (mainly spherical versus aspheric) with RRT, without making use of corneal topography data,17-23 while others explicitly use measured corneal topographies;24-29 some studies additionally included polychromatic effects.20,28 Franchini et al. used real ray- tracing to study different lens-edge designs by analyzing the reflected glare images.30 Ho et al. investigated accommodating IOLs using RRT.31 There is also some literature regarding the actual calculation—beyond a simple analysis—of optical components, such as IOLs, employing RRT.32-40

There are other published studies that are not directly IOL- or refractive-surgery-related; for instance, some studies used RRT for the analysis of schematic eyes.41-43 Fink et al. simulated human optics with real ray- tracing.44-46 Others carried out investigations using RRT based on measured corneal topography data.47-50 Sarver et al. also used corneal topography data and are principally capable of calculating optical components with their Visual Optics Lab* program.51 Only few authors did RRT calculations including a gradient-index model of the human crystalline lens.4,52-56

The individual virtual eye is a computer model of a human eye that accounts for its optical properties, and which is designed to contribute to all fields of research where ray tracing has been used so far. However, it does not include yet the human crystalline lens. Established schematic eye models, such as Gullstrand's model,57 are intended to represent properties of an average eye based on averaged population measurements. In contrast, the individual virtual eye is based on measurements of a single person. The geometry of the eye is important, including axial length and anterior corneal surface. While theoretical surfaces can also be used, the focus lies on the inclusion of measured corneal topography data.

All optical components of a pseudophakic eye –an eye with an artificial lens– are modeled by means of computer methods. Based on this virtual eye model, its geometrical optics properties, such as the wavefront aberration, can be simulated with RRT. Moreover, optical components can be calculated using computer optimization procedures; which leads to potential clinical application fields in ophthalmology. Most studies using ray tracing found in literature investigate specific issues in vision science and often rely on commercially available optical design programs like ASAP (Breault Research Organization, Inc., Tucson, AZ), CODE V (Optical Research Associates, Pasadena, CA), OSLO (Lambda Research Corporation, Littleton, MA), or ZEMAX (ZEMAX Development Corporation, Bellevue, WA). In contrast, this study focuses on the development of an autonomous computer eye model that may serve as basis for various applications in the clinical practice. The virtual eye is designed and implemented in a way that would allow for its integration in a single medical device, thus gaining in efficiency and usability.

A first application of the individual virtual eye was the calculation of spherical IOLs as described in a previous work.58 In the present paper, the individual virtual eye and the RRT calculation methods are described in more detail, especially regarding the computer technical implementation. Based on measured patient data customized aspheric lenses are calculated, thus showing the capability of the method to deal with advanced lens geometry.

The diagrams in figure 8 show the Zernike analysis of the wavefronts. Even though Zernike polynomials up to the 6th order (28 terms) were fitted to the wavefront, only those Zernike modes up to the 4th order are shown for the sake of simplicity (thus omitting 13 terms, including the Z(6,0) coefficient) without hiding important information, because the 5th- and 6th-order coefficients are rather small. The zeroand first-order terms, piston and tilt, are not shown either; piston is only a constant offset without further relevance. Tilt causes a displacement of the image with respect to the center of the fovea, which can likely be compensated by the eye by adjusting the fixation angle.66 Thus, the individual Zernike coefficients start in the 5th column, ranging from astigmatism to quadrafoil. All Zernike coefficients refer to RMS in microns, as they are calculated according to the standards of the Optical society of America.67 The left four columns give RMS error values representing combined Zernike modes: total RMS gives the total RMS error (excluding piston and tilt), LOA RMS combines 2nd-order (lower order) aberrations (astigmatism and defocus), HOA RMS combines 3rd- to 6th-order (higher order) aberrations, SA RMS combines primary spherical aberration Z(4,0) and secondary spherical aberration Z(6,0). Note that, per definition, the individual Zernike coefficients are relative and do have a numerical sign, while the combined RMS values are absolute and are thus always positive. When interpreting the diagrams we bear the reader to keep also in mind that the combined RMS errors can be calculated directly from the individual Zernike coefficients as the square root of the sum of the squared coefficients, e.g.68

Figure 8.

Detailed wavefront aberration for different IOL types. The left four columns show the RMS error for the combined Zernike modes, while the other columns show the individual Zernike coefficients. IOLs are calculated by minimizing A. The RMS WFE, or B. The RMS spot size.

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Figure 8A shows the residual wavefront aberration when optimized for RMS WFE. The bars and numbers indicate mean values, while the error bars specify the standard deviation. The mean defocus is around zero for all types of IOLs and the standard deviation is small. The big standard deviation corresponding to astigmatism is almost completely eliminated by the toric IOLs; this can also be seen when looking at the LOA RMS. The SA RMS is reduced both with the aspheric and with the toric aspheric IOLs, but not completely to zero. Figure 8B shows the results of the optimization for the same eyes but when taking RMS spot size as criterion.

These are very similar for the aspheric and the toric aspheric IOLs, but there are some differences for the spherical and the toric IOLs: in the presence of spherical aberration, the RMS spot size seems to be smaller when there is some amount of defocus left.

Table 2 shows the geometries of the different calculated IOLs. When comparing spherical and aspheric IOLs, the following interrelation can be observed: when some asphericity is added to the posterior surface, the radius of curvature of the anterior surface needs to be changed, too. The asphericities of the posterior surface for the IOLs are mainly hyperbolic; a big standard deviation is observed.

The calculation time is highly dependent on the starting conditions, the number of rays traced, the number of refracting surfaces and the number of variables in the optimization process. Table 3 shows the typical number of iterations and calculation time for the optimization tasks in this study. The ray-tracing program was developed in the C++ programming language, using Microsoft Visual Studio .NET 2003 and Microsoft Windows XP (Microsoft Corporation, USA) as operating system. All calculations were performed with a standard personal computer equipped with an Intel Pentium 4 2.6 GHz processor (Intel Corporation, USA).

The individual virtual eye combines two scientific fields: computer science and medicine. Computer science provides the computer technical structures and methods for the realization of a virtual eye to allow for simulation regarding the optical properties and, furthermore, to allow for the calculation of its optical components. This leads to contributions to medicine, as there are clinically relevant ophthalmological applications of this computer model.

The optical components of the virtual eye are modeled by means of mathematical surfaces that are feasible in terms of physiology and optical properties, as well as efficiently usable for computational ray tracing. The virtual cornea is modeled by applying a spline-based interpolation schema to the corneal elevation data of individual patients obtained experimentally in polar coordinate form, preferably from a Placido videotopometer. The interpolation schema enables the calculation of corneal elevation data and surface normal at any generic position. This leads to the possibility to do a ray tracing of uniformly distributed rays through the virtual eye. The implementation of this interpolation schema requires some time-consuming operations, such as the functional spline interpolation, to be performed only once in a preprocessing step, while the ray-surface intersection calculations don’t use expensive (time-consuming) routines. This efficient implementation makes the virtual cornea usable in conjunction with numerical methods for finding ray-surface intersections. The virtual lens is modeled by means of two biconic surfaces. Surface elevation and normals can be evaluated efficiently at any position algebraically. This modeling represents only a rough approximation to a human crystalline lens, which has a non-homogeneous gradient index69 – however most kinds of artificial lenses can be specified this way, including toric and aspheric lenses.

The optical properties of the virtual eye are simulated with RRT. Since the components were modeled particularly with regard to ray-surface intersection and surface normal calculations, many real rays can be traced through the virtual eye efficiently using numerical methods. Snell's law is used at every refracting surface and geometric optic criteria for the assessment of optical quality can be calculated: RMS spot size and wavefront aberration, both common metrics in technical optics. The wavefront aberration can be specified either as a combined RMS wavefront error or, alternatively, Zernike polynomials can be used for approximation to allow for a more detailed analysis of the kind of residual optical error.

Beyond only simulating optics, optical components can also be calculated by formulating and solving an optimization problem. The input of such an optimization problem can be the geometry of an IOL, as shown in this study. Depending on the type of lens, one or more input parameters are possible. The output criterion to be minimized can be either the RMS spot size of the RMS wavefront error. This optimization process is solved by means of numerical methods, resulting in an iterative minimization procedure either in one or in multiple dimensions.

One application of the individual virtual eye is IOL calculation for cataract surgery. In a previous work it was shown that RRT – used to calculate basic spherical IOLs – is able to compete with current IOL calculation formulae for normal eyes. Unlike those IOL calculation formulae currently used that rely on paraxial optics and corneal keratometry readings, the calculation of IOLs with RRT accounts for a potentially irregular corneal topography (including HOAs) and, therefore, has the potential to overcome certain limitations of current calculation methods, like lens calculation after refractive surgery.58 The ray-tracing and optimization methods used in the previous study are identical to the ones described in more detail in the present paper. The optimization necessary for the spherical IOLs consisted of a one-dimensional minimization procedure. The present paper moves one step further and shows that even more advanced IOL geometries can be calculated with the individual virtual eye, resulting in a multi-dimensional optimization. Such a calculation of IOLs of complex shape for an individual patient cannot be attained with the formulae that are currently used. The residual wavefront aberration for the different kinds of IOLs was finally analyzed by means of Zernike polynomials. Furthermore, as optimizing criterion both the RMS spot size and the RMS WFE was used. A difference in the resulting IOL geometry was found depending on the criterion of choice– so minimizing RMS WFE or minimizing RMS spot size is generally not the same. When choosing RMS spot size the results of all calculations show that there is some interaction between spherical aberration and defocus (Figure 8): when there is some positive SA RMS left, optimizing spot size leads to negative defocus, while optimizing WFE guarantees almost zero defocus. However, when aspheric IOLs are calculated, there tend to be only minor differences between optimizing RMS WFE and optimizing RMS spot size. Apart from the effect of choosing one optimization criterion or the other, the impact of the type of IOL on the residual wavefront aberration can be nicely seen by analyzing the Zernike polynomials. Figure 9 summarizes the most relevant combined Zernike modes from figure 8A: The more complex the IOL calculated is, the lower is the total wavefront error (Figure 9, left bars). The spherical IOL is only able to correct for the defocus, while toric IOLs also eliminate astigmatism (Figure 9, middle bars). The spherical aberration is additionally reduced with the aspheric and the toric aspheric IOLs (Figure 9, right bars).

Figure 9.

Summarized wavefront aberration with different IOL types for minimized RMS WFE. As the complexity of the IOL increases, the residual total wavefront error decreases. This is the result of the toric component reducing astigmatism (Ast RMS) and the aspherical component (SA RMS) reducing spherical aberration.

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Although these simulated results of the residual wavefront aberration using individual patient data are theoretical so far, relevant implications may already be derived for clinical practice. The astigmatism of a pseudophakic eye can be corrected by either manipulating the shape of the cornea itself with incisional corneal relaxing techniques or with laser ablation or –without changing the cornea– by means of spectacles, contact lenses or toric IOLs.70 In this study the astigmatism was corrected with the toric surface of the IOL – but can principally be corrected just as well by means of spectacles without being affected by the asphericity of the lens and its resulting impact on the spherical aberration. The use of aspheric IOLs is being discussed extensively in literature. A conventional IOL with spherical surfaces has positive spherical aberration, which amplifies the average positive corneal spherical aberration of a patient. To avoid this, clinical practice is to use aspheric IOLs with a fixed amount of zero or negative spherical aberration, intended to compensate to a certain degree the average positive corneal spherical aberration.28 In contrast to this approach and as a more advanced one, the aspheric IOLs calculated in this study are customized, meaning that the asphericity of the IOL is calculated for each patient individually. The resulting wavefront analysis shows that the spherical aberration of the eye is reduced, even though total HOAs are still large according to the theoretical prediction, so the potential benefit in terms of visual performance may be limited, even with customized aspheric IOLs. The wavefront aberration calculated in this study can be compared to actual measurements of pseudophakic patients, for example those obtained with a Hartmann-Shack wavefront sensor. As a contribution to the controversial discussion about the benefits of aspheric IOLs,71 such a comparison and a discussion of the clinical implications will be presented in a separate investigation since it is beyond the more technical scope of this paper.

Which criterion the human eye actually uses for focusing is unknown.59 [p. 152] Both, the RMS WFE and RMS spot size are metrics used in technical optics. Visual performance metrics, such as visual acuity or contrast sensitivity, can not be generally assessed from geometrical optical quantities alone, since there are also other physiological factors involved, from the retina and the brain. Even though Applegate et al. found a significant correlation between RMS WFE and visual performance,72 – in a later study they confined their findings to relatively high levels of aberration, and concluded that the RMS WFE is not a good predictor of visual acuity for low levels of aberration.73 Other studies have also shown that different kinds of aberrations interact with each other74 and that they are not all equally important regarding their impact on visual performance.75,76 For a future extension of the virtual eye model, regarding issues beyond geometrical optics, it might be very interesting to incorporate the results of recent investigations studying the relation between wavefront aberration and visual performance.77-81

The accuracy of RRT is directly affected by the input data from measurements of the patient eyes – so it will directly benefit from improvements in the measurement devices, especially regarding corneal elevation. It will also benefit from the inclusion of measured geometry that is currently not assessed in most cases in clinical practice, such as the posterior corneal surface. Currently, measurements from different devices are used to construct the individual virtual eye (like those from a videokeratometer and a those from a PCI device to measure axial length). Once possible – maybe thanks to the combination of different measuring methods – it would be desirable to have one single measurement device that completely measured the geometry of the whole eye including cornea, lens and segmental lengths. Since the calculations performed in this study might also be done with other ray-tracing programs, the focus of this computer model lies in the development of a specifically designed standalone program. High value was placed on obtaining an efficient implementation of the time-critical numerical ray-tracing and optimization procedures, leading to short calculation times. This eye model could be integrated in such a combined measuring device, capturing the whole eye and leading to practicable methods, especially for IOL calculation. Moreover, the effects of misalignment of IOLs could be assessed, like tilt and decentration. Further application fields also include phakic IOLs, ablation profiles for corneal refractive laser surgery or combined surgical procedures. One major remaining challenge, in order to be able to extend the capabilities and, therefore, the potential applications of the individual virtual eye, is the proper simulation of the human crystalline lens. Another challenge is the further development trying to attain a more realistic model, by incorporating polychromatic issues or additional physical properties beyond geometrical optics, like diffraction and scattering. Finally, there is a wide scope when extending the individual virtual eye by addressing those issues beyond optics regarding physiology of the retina and brain.