Fig. 10.1
Cornea in a natural scenario
Pathological scenario , that is, presence of anomalous alterations (see Fig. 10.2), in which the tissues that form part of the corneal architecture display structural weakness due to a pathological process, as in ectatic pathological diseases like keratoconus [13]. This weakened structure is manifested clinically by an alteration to the cornea’s surface morphology. This implies redistributing the corneal pachymetry , changes in the surface geometry and loss of optical power [14].
Fig. 10.2
Cornea in a pathological scenario
10.2.1 Geometry of the Cornea: Natural Scenario
In the natural scenario (see Fig. 10.1), the cornea presents corneal surface regularity, which makes it act as a lens and confers the cornea the ability to act as the most important refractive element of the eye scheme. Thus, the anterior corneal surface is responsible for two-thirds of the human eye’s total optical power at the highest point of the cornea, known as the corneal apex (see Fig. 10.3).
Fig. 10.3
Corneal apex
A healthy cornea presents the following morphological characteristics:
Between both surfaces, pachymetry or corneal thickness is defined with an average value in its geometrical centre of about 0.54 mm [14, 18, 19].
Several works in the scientific literature have demonstrated that the average curvature and corneal thickness values remain stable in the natural scenario [18].
Besides the aforementioned parameters, cornea characterisation implies knowing the morphology of the anterior and posterior cornea surfaces. From a geometrical point of view, surfaces are similar to a spheroid, which is a geometrical shape whose generatrix is obtained when an ellipse revolves around one of its main axes. This shape has two axes: the symmetry axis, located on the Cartesian coordinate axis, denoted with a letter ‘c’; the axis perpendicular to the symmetry axis, denoted with a letter ‘a’. Letters ‘c’ and ‘a’ are defined on the spheroid by international convention (see Fig. 10.4).
Fig. 10.4
Spheroid
The spheroid can simulate three different configurations according to the relation between its main axes ‘c’ and ‘a’:
Oblate spheroid: if a > c, the symmetry axis is the smallest (see Fig. 10.5).
Fig. 10.5
Oblate spheroid
Prolate spheroid: if a < c, the symmetry axis is the largest (see Fig. 10.6).
Fig. 10.6
Prolate spheroid
Sphere: if a = c; the symmetry axis is equal (see Fig. 10.7). This is a special case because the surface generated by the generatrix is a sphere; consequently, this curve is not an ellipse, but a circumference.
Fig. 10.7
Sphere
All this can be extrapolated to the ophthalmology field, where the cornea displays similar geometrical behaviour to an elliptic profile [6]:
This equation presents three spatial coordinates (X, Y, Z), on the basis of which the cornea can be interpreted as a three-dimensional refractive element by the following three parameters (see Fig. 10.8):
(10.1)
Fig. 10.8
Sections for different types of conical curves
Corneal asphericity (Q): a parameter that characterises the nature of the conical curve, represented by the mathematical expression.
The known revolution axis (Z).
Conoid radius (R): the radius of the highest point of the anterior corneal surface (corneal apex).
Corneal asphericity is defined as the degree of curvature, or slope, of the cornea from the central region to the peripheral region. This slope can present a plainer tendency (a geometrical scenario for a healthy cornea) or a more curved state (a geometrical scenario for a pathological cornea) (see Fig. 10.9).
Fig. 10.9
Asphericity. Concept
The corneal asphericity coefficient (Q) can be mathematically calculated using the relation between the central (b) and peripheral (a) hemi-axes of the corneal elliptic profile (see Fig. 10.9).
In light of the earlier, the elliptic profile of the corneal surface can be classified as:
Spherical (Q = 0). A singular case in which the horizontal or peripheral (a), and central or vertical (b) hemi-axes of the ellipse are equal, that is, the corneal surface has the same curvature in the central region as in the peripheral zone.
Prolate ellipse (0 > Q > −1). In this case the elliptic profile of the corneal surface flattens as it moves away from the highest point of the corneal surface (the apex). This behaviour occurs in a natural scenario which, from a refractive point of view, implies the cornea having a low spherical aberration.
Oblate ellipse (Q > 0). In this other case, the elliptic profile of the corneal surface becomes more curved as it moves away from the corneal apex. This behaviour occurs in a pathological scenario which, from a refractive point of view, implies the cornea having a spherical aberration, which is proportional to the Q factor.
Parabola or hyperbola (Q ≤ −1). This is an exceptional case in healthy eyes and is exclusive in central ectatic cornea pathologies. From the refractive surface viewpoint, the cornea has a negative spherical aberration index, which is also proportional to Q factor.
Apart from the aforementioned asphericity, the degree of cornea curvature can also be expressed according to other factors, for instance:
They are all interrelated (see Table 10.1 ).
Eccentricity factor (e):
(10.2)
Shape factor (p)
(10.3)
Table 10.1
Relationship among different types of geometric curves of the cornea in function of several parameters
Cornea profile | Eccentricity e 2 | Form factor   | Asphericity  |
|---|---|---|---|
Sphere |  |  |  |
Oblate ellipse |  | ||
Prolate ellipse |  |  |  |
Parabola |  |  |  |
Hyperbola |  |  |
In addition, the corneal elliptical cross-section profile is defined according to the central curvature radius (Rc) (see Fig. 10.10), thus:
The most widely used term in the scientific literature to characterise corneal morphology is asphericity (Q). In normal eyes, the numerical asphericity value for the anterior surface ranges from −0.29 to −0.13 [15, 16, 20], and from −0.34 to −0.38 for the posterior surface [6, 15]. This indicates that both surfaces have a prolate morphological characterisation.
Fig. 10.10
Corneal asphericity
(10.4)
(10.5)
It must be made clear that asphericity values change according to the corneal region under study, in such a way that the morphology of the same cornea can present different asphericity values.
10.2.2 Geometry of the Cornea: Pathological Scenario—Keratoconus
In this scenario (see Fig. 10.2), the bilateral ectatic pathology, named keratoconus, causes progressive loss of corneal thickness. This morphological change in the corneal architecture is manifested in the form of protrusion. It presents a conical-type geometry which causes the appearance of an irregular astigmatism or loss of visual acuity in patients [4].
This protrusion creates thinning, which affects the tissues that form part of the corneal architecture. This situation occurs progressively as the degree of disease severity advances, leading to morphological changes that follow a characteristic pattern which is related to a structural weakening of the tissues that make up the cornea (see Fig. 10.11). From the geometric perspective, keratoconus may give way to several keratoconus subtypes according to cone size and shape [21], these being [4]:
Nipple. The corneal topography analysis presents a cone-like morphology with a diameter of roughly 5 mm that affects at least 50 % of the corneal area. Its morphology becomes rounded and curved, but the rest of the corneal surface takes its natural form. The apex is located in the central or paracentral area, but can sometimes shift inferonasally.
Oval. The corneal topography analysis presents a larger sized cone than the nipple kind; it affects one quadrant or two, and is generally located in the inferotemporal area. Its morphology is ellipsoid-type.
Globe. This morphology practically covers the whole corneal area because it presents an extension of more than 6 mm in diameter. This morphology is characterised by generalised thinning.
Astigmatism. This characterisation presents a conical protrusion characterised by vertical bow tie-type astigmatism whose asymmetry is inferosuperior, and frequently more curved in the upper area, thus affects less than 50 % of the corneal area.
This cone-based classification serves in particular to describe cone extension and to classify keratoconus according to its shape. However, it offers no information about corneal pathology severity.
The scientific literature offers many publications that employ corneal topography techniques to represent the typical topographical patterns of the disease [4, 13, 22–25] (see Fig. 10.12).
Fig. 10.12
Keratoconus characteristic topographical patterns
One of the main morphological characteristics that the corneal architecture presents and is related with this pathology is the focal curvature that the anterior corneal surface undergoes. This protrusion is more frequently manifested, in fact in 72 % of cases, in the paracentral region, and in the central region in around 25 % of cases. Therefore, 97 % of cases with keratoconus are localised for a radius from r = 0 mm to r = 4 mm, which include the central and paracentral corneal regions [24]. Presence of a protrusion in the peripheral region is considered unusual [26, 27].
The increased curvature produced by keratoconus in the corneal morphology due to the weakening of the tissues that form part of the corneal architecture not only affects the anterior corneal surface, but also the posterior corneal surface, even in incipient or early stage keratoconus cases [4, 27]. Thus, knowledge of geometric behaviour of posterior corneal surface is most interesting in the ophthalmology field to early diagnose keratoconus [28–30] (see Fig. 10.2).
Another differentiating element of the keratoconic cornea exists: a parameter that characterises the nature of the conical curve or asphericity. In this scenario, variation in asphericity occurs on the anterior/posterior corneal surfaces, and normally with negative values; that is, the healthy cornea displays a more prolated geometrical ellipse behaviour in relation to both surfaces [6, 31, 32]. Based on all this, the curvature of a pathological cornea’s morphology is greater in the central region than in the peripheral region, and its behaviour tends to be more parabola like than prolate ellipse like. Hence the asphericity value increases with more advanced degrees of keratoconus [4].
10.3 Corneal Topography: Stages
Nowadays, the geometrical characterisation of corneal morphology can be done by Corneal Topography. This is a non-invasive exploratory technique that allows cornea morphology to be analysed both quantitatively and qualitatively and is able to identify standard patterns and to rule out the potentially devastating alterations for eyesight that derive from ectatic pathological disorders [9, 33–36].
Topography is performed with equipment known as corneal topographers (see Fig. 10.13). These instruments are widely used by the ophthalmological medical community and allow anatomical and geometrical corneal characterisation based on its surface reconstruction [35].
Fig. 10.13
Phases of a corneal topographer
Today’s corneal topographers are based on two technologies : (1) systems based on light reflected on the cornea; (2) systems based on slit light projected on the cornea. In turn, they both present a standard photographic system or are based on Scheimpflug’s photography principle.
Corneal topographers are based on the above-described technologies (see Fig. 10.13) and provide different topographic maps of the anterior and posterior corneal surfaces, pachymetry maps, and summary indices to follow-up and clinically evaluate corneal topographies.
They also contemplate two clearly different stages to present the data obtained from corneal topography to doctors, who can use them as a basis to clinically diagnose corneal ectasis:
The first stage corresponds to the internal data collection and processing procedure.
The second stage corresponds to geometric corneal surface reconstruction and to corneal topography generation.
10.3.1 Stage I: Internal Data Collection and Processing
In this stage topographers obtain altimetry data, as a matrix of elevations, on a discrete and finite set of points that is representative of the corneal surface, known as raw data.
10.3.2 Stage II: Geometric Corneal Surface Reconstruction—Corneal Topography Generation
In this second stage, by using the data that form the matrix of elevations, the corneal surface is reconstructed to later compare it with the reference surface, which is usually a sphere, and to obtain the topographic map of the corneal surface from the comparison made of both surfaces.
The geometrical reconstruction algorithm can be based on two surface reconstruction methods: the so-called modal and zonal methods (see Fig. 10.14).
Fig. 10.14
Reconstruction methods for corneal surface
10.3.2.1 Modal Methods
Modal methods are based on approaching the surface by a combination of basic functions defined globally throughout the data domain. This combination may depend on a certain number of parameters and the quantity they require to retrieve relevant information about the surface in an attempt to avoid overfitting their measurement error [37].
In the ophthalmology field, most of the modal methods used for surface reconstruction from standard heights employ Zernike’s polynomials-based developments [38–40], where expansion coefficients are interpreted in terms of the basic aberrations or degradation in the image in optic systems [41, 42].
However, this procedure entails a series of problems that have been well discussed in the scientific literature [37, 43–45]. In particular, there is growing concern about this method being inaccurate in abnormal situations as it does not obtain a reliable reconstruction in complex topographies, which are of much more clinical interest.
They also present another problem, that of correctly estimating the number of Zernike polynomials to be used in the reconstruction process: given the general nature of their support, a relatively small number of them are needed for healthy corneas, but a much larger number is required for pathological corneas [43, 46]. To overcome this problem, there are techniques available that objectively estimate the number of expansion coefficients that need to be considered in the reconstruction polynomial. However, they are computationally very complex and difficult to implement [47, 48].
Regarding the earlier problems , some alternative techniques have been proposed in the literature:
However, they all entail the same problems as Zernike , and even with other ones, such as high computational complexity when dealing with residual errors, or obtaining controversial results.
10.3.2.2 Zonal Methods
In zonal methods the data domain is divided into more elemental subdomains, and the surface is approached in each subdomain, defined independently of the others [37, 40].
This reconstruction method is characterised as it presents a flexible and accurate fitting when considering data inside a zone delimited for the calculation of a surface point. This means that representation is local, and therefore local irregularities do not affect the global surface representation, unlike the global functions adopted in modal reconstruction as high-order quadratic or polynomial surfaces, which lack local information. Hence local surface irregularities or defects cannot be suitably represented by these approaches [3, 56].
The mostly widely used tool is Splines [57], especially B-Splines [3, 45, 58], which are numerically stable and flexible functions that offer good fit accuracy [59, 60].
B-Spline functions have been widely used in engineering to solve geometrical and computational problems which appear when we wish to use entities that are highly complex in geometric terms [59, 60].
In the ophthalmology field, these B-Spline functions have been successfully employed in the following applications:
In short, depending on the descriptions of the models and the previously mentioned purposes, and given the singular nature of corneal surface morphology, creating a model that offers the advantages of zonal reconstruction models by B-Spline functions based on Computer-Aided Geometrical Design (CAGD) is interesting for real corneal morphology characterisation. This model could, in turn, also be used to diagnose certain pathologies that relate to corneal morphology modifications, such as keratoconus.
10.4 Geometrical Model of the Cornea
Generating a geometric corneal model using CAGD tools requires representative points of corneal morphology for the reconstruction of the whole corneal surface and the subsequent generation of a solid representative model of the human cornea.
Some corneal topographers can export these points, specifically from the aforementioned first data collection stage. Such data correspond to a matrix of elevations that is representative of the corneal surface, known in the scientific literature as raw data, which can be obtained by an internal vision algorithm of the corneal topographer [2, 3, 37, 67–69].
Some authors have conducted works using these raw data, e.g. geometrical cornea characterisation done by evaluating corneal symmetry with shape and position parameters [68]; developing a personalised biomechanical model by obtaining the patient’s geometry [2, 67, 69]. Both these works were performed using Cartesian coordinates (x, y, z), provided by a Pentacam topographer . However, in both cases, raw data were interpolated given the above-cited extrinsic examiner–team–patient trinomial problems, which affected the measurement process. In order to obtain the representative spatial points of the whole corneal area, such data interpolation would imply them being biased. This would imply that throughout the process of reconstructing the whole corneal surface, minor morphological alterations caused by an incipient degree of the corneal pathology, such as keratoconus, not to be reliably reproduced.
To date, only raw spatial data obtained from the EyeTop 2005 corneal topographer (CSO, Italy) without them being interpolated have been considered in anterior cornea surface reconstructions [37], and also in the reconstruction of the anterior and posterior corneal surfaces [3] conducted with the data supplied by the Sirius corneal topographer (CSO, Italy).
Therefore with these data and CAGD tools, the corneal surface is reconstructed and the solid model of a human cornea is generated.
10.4.1 Computer-Aided Geometrical Design
In recent years, major technological progress has been made thanks to both the generation of virtual models and new computation tools to acquire and process images [1]. These have enabled 3D shapes to be produced [70], and performance models to be formulated that more reliably reproduce the geometry of a solid structure [71].
One of these tools is CAGD, which emerged when attempting to meet the technological requirements of companies in the automobile and aeronautics sectors. CAGD allows us to study geometrical and computational aspects of any complex physical entity, e.g. surfaces and volumes, to produce virtual models [72], which differ from physical models as they do not imply a destructive process, and reduce both production and time costs [73, 74].
In the bioengineering field, the development of virtual models by CAGD allows the characterisation of biological structures by establishing new experimental procedures in the field of medicine [71] for different purposes:
In the optics field, some models have been found which propose using CAGD to generate only the corneal surface and to perform its posterior optical analysis with the spatial points obtained by corneal topographers to be used later by the aforementioned conical or biconical functions [83–85], or using the images taken of the incisions made ex vivo to chicken eyes [86, 87]. In both cases, however, the produced cornea model was incomplete because, in order to generate the cloud of the points that constitute the corneal surface, which will be subsequently reconstructed by CAGD, data were interpolated, and only the corneal surface of optic interest was reconstructed. The use of CAGD tools allowed us to define and adopt a new methodology [3], which guarantees that the data employed for geometric reconstruction did not undergo any type of preprocessing or interpolation, which that could avoid quite accurately detecting minor distortions formed on both corneal surfaces due to keratoconus.
10.5 Geometric Reconstruction of the Cornea
The geometric reconstruction process of the cornea defined by the authors proposes carrying out the following stages in sequence (Fig. 10.15): (1) Obtaining the representative point clouds of the corneal surfaces , (2) geometrically reconstructing the corneal surfaces and (3) generating a representative solid model of the cornea.
