THE OPTICAL SYSTEM OF THE HUMAN EYE
The human eye has an optical system which is composed of: (1) four main noncoaxial optical elements (anterior and posterior corneal and lens surfaces); (2) the pupil; and (3) the retina, which is aplanatic to compensate for the native spherical aberration (SA) and coma through its nonplanar geometry. The optical surfaces are aligned almost coaxially; however, the deviations from a perfect optical alignment results in a range of axes and their interrelationships (Fig. 1.1). This guides us to the following definitions:
- The visual axis (VA): It is the line connecting the fixation point with the foveola, passing through the two nodal points of the eye, but not necessarily through the pupil center.
- The optical axis (OA): It is the axis connecting the center of curvatures of the optical surfaces of the eye. It can be recognized by the Purkinje images I, II, III, and IV, namely of the outer corneal surface (I), inner corneal surface (II), anterior surface of the lens (III), and the posterior surface of the lens (IV). If the optical surfaces of the eye were perfectly coaxial, these four images would be coaxial, which is seldom observed.
- The principle line of sight (LOS): It is the ray from the fixation point reaching the foveola via the center of the entrance pupil (EP).
- The pupillary axis (PA): It is the normal line to the corneal surface that passes through the center of the EP and the center of curvature of the anterior corneal surface.
- The achromatic axis: It is defined as the axis connecting the center of the EP with the nodal points.
- The vertex keratoscope (VK) normal: It is the axis that is perpendicular to the plane of the capturing machine (originally, the keratoscope) and intersecting with anterior corneal surface at corneal apex (corneal vertex).Fig. 1.1: Optical system of the eye (superior view of the right eye). Surfaces, angles, and axes. [EP: entrance pupil (the opening within the dotted line); F: foveola; FP: focal point; LOS: line of sight; N: nodular point; OA: optical axis; PA: pupillary axis; VA: visual axis; VK: video keratoscope axis]
- Angle kappa (measured in degrees): It is the angle between PA and VA. Measuring angle kappa is very important in refractive surgery in terms of laser ablation centration and multifocal intraocular lens (MFIOLs) implantation. Large angle kappa has an adverse clinical impact on the visual outcomes after laser vision correction (LVC), particularly hypermetropic treatment and astigmatic treatment when the magnitude of astigmatism is more than 1 diopter (D). Pupil-offset technique is recommended in such cases. In addition, large angle causes photic phenomenon and decreased effectiveness of the MFIOL. MFIOL implantation is contraindicated when angle kappa is larger than 400–500 µm.
Normal distribution in angle kappa was studied by using Orbscan II (Placido-based) and the Synoptophore. It was found that values of angle kappa measured by the Orbscan II were almost as twice as when measured by the Synoptophore. Based on Orbscan II, Hashemi and associates determined an average value of angle kappa of 5.46 ± 1.33° in Iranian adults with insignificant intergender difference. In another study, Gharaee and associates determined an average value of 4.96 ± 1.38° in total, an average horizontal angle kappa of –0.02 ± 0.49 mm, and an average vertical angle kappa of –0.09 ± 0.32 mm.
In addition, studies reporting normative angle kappa values in different conditions found that angle kappa was significantly larger in exotropes than in esotropes or controls, and tended to be larger in the left eye than in the right eye. Moreover, there was a positive correlation between angle kappa and positive refractive errors, which can be explained by the negative correlation with the axial length of the globe.
Unlike Placido-based topographers, Scheimpflug-based tomographers cannot measure angle kappa. This raises the need to find a way to estimate this angle in Scheimpflug-based topographers. However, the VA can roughly be considered as passing in between the center of the EP and the corneal apex, and might be half the distance. Therefore, in Scheimpflug-based devices, angle kappa can roughly be half values of X and Y coordinates of the EP center.
- Angle alpha (measured in degrees): The angle formed at the first nodal point by OA and VA.
- Angle lambda (measured in degrees): The angle between PA and LOS.
- Chord µ (measured in mm): It is the chord length of angle kappa in polar coordinates relative to the center of the EP.
The refractive power of the human eye comes mainly from the cornea and the crystalline lens. In emmetropia, corneal power ranges in between 39 D and 48 D (average 43.05 D), while the power of the crystalline lens ranges from 15 D to 24 D (average 19.11 D). The refractive media in the human eye are: tear film (n = 1.336), cornea (n = 1.376), aqueous humor (n = 1.336), crystalline lens (n = 1.406), and vitreous humor (1.336), where “n” is the refractive index (RI) of the media measured relatively to air (n = 1.000). The dioptric power of these media is determined by the radius of curvature, the RI, and the distance between various interfaces.
CORNEAL GEOMETRY
The cornea has two surfaces separated by corneal substance. The anterior surface is coated with the tear film, and together forms one refractive surface separating air from corneal substance. The posterior surface separates corneal substance from aqueous humor. The cornea is not a part of a perfect sphere. The shape of both surfaces is defined as: an aspheric prolate, toric, and asymmetric conoidal shape (Figs. 1.2 and 1.3). Each of the previous expressions will be explained in detail in the following paragraphs.
Corneal Dimensions
Corneal dimensions include diameters, meridians, radii of curvature, corneal zones, corneal thickness, corneal shape, corneal power, and geometric landmarks.
- Diameters: The sclerocorneal junction (base of the cornea) is an ellipse. The vertical corneal diameter is 10.6 mm on average, whereas the average horizontal corneal diameter is 11.7 mm.
- Meridians: The normal adult cornea has two meridians that are 90° apart.Due to the elliptical base of the cornea at the sclerocorneal junction, the vertical diameter is generally shorter than the horizontal one, meaning that the vertical meridian is steeper (smaller radius of curvature) than the horizontal one (greater radius of curvature). Due to this difference, corneal shape is considered as toric. This toricity is responsible for corneal astigmatism. In younger eyes, this toricity is represented as with-the-rule (WTR) astigmatism, where the vertical meridian is steeper than the horizontal one. This reverses with age, causing against-the-rule (ATR) astigmatism.
- Radius of curvature: The cornea has two surfaces, anterior with an approximate radius of 7.8 mm, and posterior with an approximate radius of 6.5 mm. These two radii are for the central (axial) zone of the cornea. The radii increase while moving to the periphery, indicating a flatter corneal periphery. The normal cornea flattens progressively from center to periphery by 2–4 D, with the nasal area flattening more than the temporal area, and this is shown on the curvature map as the nasal side becoming blue (flat) more quickly (Fig. 1.4). The normal average anterior/posterior radii ratio is 1.21 in virgin nonoperated corneas. This ratio is altered by keratorefractive surgeries which is a major source of error in intraocular lens (IOL) measurements after these surgeries.
- Corneal thickness: Due to the difference in radius between the two corneal surfaces, the cornea is thinner in the central zone than at periphery. There are two important values in corneal thickness, the central corneal thickness (CCT) and thinnest corneal thickness (TCT). Both will be discussed later in this chapter.
Corneal Zones
Clinically, the cornea is divided into zones that surround fixation (corneal vertex or apex) and blend into one another:
- The central zone (central 3 mm): It overlies the pupil and is responsible for high definition vision. The central part is almost spherical and is also called the apical or axial zone.
- The paracentral zone (3–6 mm): It has a doughnut shape with an outer diameter of 6 mm. It represents an area of progressive flattening toward the third zone.
- The central and paracentral zones are responsible for the refractive power of the cornea, and are used for contact lens fitting.
- The peripheral zone (6–9 mm): It is also known as the transitional zone. This zone is asymmetrically flatter than the central zone. The nasal and superior segments are flatter than the temporal and inferior ones.
- The limbal zone (>9 mm): It is adjacent to the sclera and is the area where the cornea steepens prior to merging with the sclera at the limbal sulcus.
Being steeper in the center and flatter at periphery gives the cornea what is known as a “prolate” aspheric shape.
Corneal Shape
Corneal shape is “conoidal” (Fig. 1.3). It is a composition of toricity, asphericity, and asymmetry. From a meridional 6viewpoint, the cornea is “toric”, which is the source of corneal astigmatism. From the zonal viewpoint, the cornea is “aspheric” because the radius of curvature differs from the center toward the periphery. From a sectorial viewpoint, the cornea is asymmetric because the nasal sector is usually flatter than the temporal sector as shown in Figure 1.4.
Corneal asphericity is expressed by “Q-value”. The cornea is prolate (steeper in the center), oblate (flatter in the center) or spheric when Q-value is negative, positive, or zero, respectively. The average Q-value in the normal population is approximately –0.27 (–0.10 to –0.30). An abnormal Q-value means abnormal corneal asphericity, the origin of corneal SA. The Q-value at which no SA is found is –0.53 on average.
Corneal shape is discussed in detail in Chapter 9.
Corneal Power
The anterior corneal surface with its associated tear film layer plays a role of a convex refractive surface. Due to both its convexity and separation between two different media: (1) air (smaller RI; n = 1.000) and (2) corneal substance (larger RI; n = 1.376), it encounters the most powerful refractive surface in the optical system of the eye. The refractive power of the central (apical or axial) zone of the anterior corneal surface is approximately 49 D.
On the other hand, although the posterior surface of the cornea is convex, it acts as a negative concave surface because it separates corneal substance (higher RI; n = 1.376) from aqueous humor (lower RI; n = 1.336). The refractive power of the posterior corneal surface is approximately –6 D.
Moreover, corneal epithelium has an impact on corneal power. The shape of the epithelial layer is responsible for about 0.40 D of astigmatism. The mean Q-value is –0.20 ± 0.13 (0.06 to –0.60) with epithelium and –0.26 ± 0.23 (0.07 to –1.51) without epithelium. In other words, the cornea is more prolate without the epithelium, which means that the epithelial layer forms a negative lens (thinner in the center) as shown in Figures 1.5A and B. This fact has a clinical impact on LVC procedures, especially in surface ablation techniques. This fact is more important in cases with irregular corneal surface because the epithelium has a remodeling (filling) feature, which masks the real corneal power and a significant portion of the underlying corneal irregularities as shown in Figure 1.6. Moreover, the remodeling feature of the epithelium affects refractive results after surface ablation, characterized by undercorrection after both myopic and hypermetropic corrections. The epithelium forms a positive convex lens or a negative concave lens after myopic or hypermetropic correction, respectively.
Figs. 1.5A and B: The effect of corneal epithelium on corneal shape. The cornea is more prolate without the epithelium.
There are different methods to measure corneal power. They are discussed in Chapter 5.
Geometrical Landmarks
There are virtual landmarks of clinical importance in the cornea. They are corneal apex, thinnest location (TL), mean K-reading (Km), and maximum K-reading (Kmax) (Fig. 1.7).
- Corneal apex: As mentioned before, it is the point at which VK normal intersects with the anterior corneal surface; therefore, it is also known as vertex normal. Assumably, it represents Purkinje-Sanson reflex I. However, since the Pentacam does not include a Placido disk, the device considers the point which is confronting the fixation target as corneal apex, which is not true in 7most cases. The computer considers this point as the origin of coordinates, X for the horizontal and Y for the vertical axes (Chapter 4). The direction of X is from patient's right to their left, and the direction of Y is from the bottom up. All other landmarks are measured from the corneal apex. Therefore, the X and Y coordinates of this point have a value of 0.00. Corneal thickness at this point is usually referred to as CCT. Depending on the technology used for measuring corneal thickness, the average CCT ranges from 534 µm to 575 µm.
- Thinnest location: It is the location of the thinnest point in the measured cornea. Corneal thickness at this point is usually referred to as TCT. In an international multicenter study based on the Pentacam HR (OCULUS Optikgeräte GmbH, Wetzlar, Germany), Feng and associates reported an average TCT of 536 μm overall. Values less than 469 µm or 435 μm (–2 or –3 SD, respectively) would be expected in less than 2.5% or 0.15% of normal corneas, respectively. The X-coordinate averaged 0.44 mm temporally, and the Y-coordinate averaged 0.29 mm inferiorly in relation with corneal apex. Y-coordinates more than 1.00 mm inferiorly were found in less than 0.5% of normal corneas.
- Mean K-reading: It is the arithmetic average of the central Sim-Ks on the anterior corneal surface. Based on Placido-disk devices, normal Km is less than 47.2 D, while based on Holladay report (Pentacam), normal Km is less than 48 D.
- Maximum K-reading: It is the Kmax on the anterior corneal surface measured by the anterior sagittal curvature map. Interestingly, at this point in time, there is no normative data for Kmax.