# Asphericity

The Asphericity is a **geometric variable** That can be used to describe some variations in the curvature of a curve or surface. It confers optical properties peculiar to surfaces that possess this property of asphéricité: a reduced (or accentuated) rate of sphericity aberration (spherical aberration). The ocular surfaces are all Aspherical

Most manufacturers now offer aspheric implants for the replacement of the lens of the eye after cataract surgery. Laser ablation profiles used for the correction of myopia by LASIK or PKR are derived from models that take into account the asphericity of the cornea.

In presbyopia surgery, some laser (presby LASIK) profiles are based on a intentional modification of the corneal Asphericityto induce a high rate of negative spherical aberration. This aberration translated a gradient of focal power between the central area (area near the optical axis) and the edges of the pupil: the generated multifocalite allows to compensate for presbyopia, because it increases the depth of field. In this context, the modulation of the asphericity of the corneal profile is used to quickly vary the corneal power to allow an increase in the depth of field of the eye.

### Definition of the Asphericity

An optical surface is called aspherical when it does not marry the shape of a sphere (the profile of the meridians of such a surface differs from that of a circle): its curvature is not constant at all points, unlike that of a sphere. The curvature of a surface or a curve is a differential geometric notion. The next curve represents a section (profile) of the surface considered: at a particular point, the radius of curvature can be defined as that of the circle tangent to the surface at the point considered (this circle is called Circle osculating).

#### The corneal Asphericity

The human cornea has two convex surfaces (pre-and post-) naturally aspherical, that is, apart from the area immediately near the top (in the Central 3 mm) they don't match simply to a surface (spherical)**1**). The curvature of the cornea is slightly reduced the Summit to its edges (we're talking about in this case

If the cornea was perfectly spherical, the representation of the topography of corneal curvature of our patients would be uniformly monochrome!

The cornea is often also slightly o-ring. In this case, the area near its Summit not to marry the shape of a sphere but of a torus. The curvature of the corneal top (apical curvature) varies according to the profile considered between two extreme values. In this configuration, each Meridian cornea considered in isolation remains aspherical curvature ranging from the Center to the edges.

These principles do not only concern the cornea and can be applied to the description of a surface of glass bezel or a contact lens.

(No rings) purely spherical or aspherical surfaces present a symmetrical (you can rotate around their central axis without changing the geometric and optical properties). It is the case of the faces of the optics of implants phakes or pseudophakes no o-rings. The surfaces of the o-rings are obviously not symmetrical.

The asphericity of an optical diopter is not enough to predict its optical behaviour, which also depends on its index of refraction and distance in which is located the image object.

### Characterization of an aspheric surface

The relative simplicity of these definitions however hides the underlying complexity to the real notion of curvature. A spherical surface has a same local curvature in any point, regardless of his able guidance. An another convenient way to represent this property is to consider only two pieces of sphere of the same size but taken to production at random can be exchanged without any deformation of the spherical surface. This is not the case for an aspheric surface.

A simple experiment to illustrate the complexity of aspherical surfaces: consider a non o-ring aspheric surface. One end of a loaf of bread kneaded with care and well regularly provides a good example of aspheric surface no o-ring. If we cut vertically a range including the Summit of this wand (commonly called the "crouton") will be observed without penalty that this person has a symmetrical about an axis through its Summit and parallel to the main axis of the stick). If we now cut a slice of oblique way (using the other tip of the wand), we get this time a section of surface that we can't put it back without any deformation if we did turn to a certain angle. Student more finely the characteristics of this surface, observed that it is not only ring (since there is more symmetrical), but also asymmetrical (the curvature is not the same in two points of hand and side of the cut off piece)!

This simple experience foreshadows the optical consequences of the shift or tilt of aspherical devices. The significant shift of an aspheric lens implies the refraction of the incident rays through an optical surface to the neighbouring properties of the oblique section previously: source of regular induced astigmatism (related to the toricite induced and correctable by glasses glass) and irregular (optical aberrations of degree not correctable by glasses glass).

### Quantification of the Asphericity

The qualitative description of the cornea allows to describe briefly the form but does not offer the possibility to precisely study the optical properties. The mathematical formulation allows both the schematic representation (modeling) and the tool to explore the physico-optiques characteristics. Modern techniques of corneal refractive surgery, imaging, contactology have been designed through the use of theoretical models physico-mathematiques, developed from the collection of data from the study in vivo and in vitro of geometric and optical properties of the eye or the cornea. State of the art advances so over time by successive adjustments between models and measuring instruments.

To represent the aspheric reality of the profile of a lens or a cornea, it seems natural to orient the choice of a simple non-circular curve like an ellipse or a parabola. The latter, as well as the hyperbole and the circle, actually have something in common: they belong to a family of figures called **Conic Sections**.

#### Conic Sections

The conical sections are a family of mathematical curves that can be spawned (as their name implies) by the section of the tablecloth of a simple cone by a plane (**2,3**).

Remarkably, the discovery and characterization of these aspheric curves by a Greek scientist (Appolonius of Perga) took place several decades before our era. This discovery was prompted by a pure intellectual curiosity and occurred outside of any obvious application context (is that although later that you will discover that these curves are used to describe the movement of planets and some stars, the shape of some light interference fringes, some State of polarization of the light, etc.)

One can easily observe Conic sections in daily life: simply observe the light figures projected on the walls by the extra lamps or lights. As the walls are usually straight, formed curves are hyperbolic nature.

The Conic sections is the model most frequently used to describe the profile of the anterior face of the cornea, because less far from reality than the spherical model, particularly at the level of the 8 mm Central. It was also used to describe the asphericity of the front and back of the Crystal faces, and is more generally used for the profiles of the surfaces used for theoretical eyes patterns or aspheric implants.

#### Equation of the aspheric curves

Conic sections (ellipse, circle, parabola, and Hyperbola) all except the circle have a variable curvature. Two parameters are sufficient to their description: the apical curvature RADIUS and Asphericity factor. All Conic sections have a common mathematical equation where are these two parameters (equation of Baker): Y = (2 x Ro - (1 + Q) X ^ 2) ^ 0.5

Ro corresponds to the radius of curvature measured at the apex of the conical: it is the radius of the circle tangent to the top of the conic section, also called osculating circle.

Q is the factor of asphericity: he characterizes changes in the radius of curvature as you move away from this Summit.

#### Value of the factor Q of asphericity

The value of the variable Q determines the type of the conic section:

Q <-1: curve is a Hyperbola

Q = - 1: curve is a parabola

-1

Q = 0: the curve is a circle

Q > 0: curve is an oblate ellipse

Other descriptors of the Asphericity, known as p, e, are found in the literature. They can all be calculated using one of them as Q = p - 1 and p = 1 - e^{2} (**2**). The value of the eccentricity (e) isn't the most convenient to quantify the asphericity of a curve oblate (e^{2}<1), à moins de considérer cette excentricité comme un nombre alors « imaginaire ».

#### Clinical translation of the factor Q of asphericity

Ro determines the curvature of the surface in the immediate vicinity of the optical axis. In the case of the cornea, it allows to determine the power k Central, and in the case of an implant the focal power.

Q governs the optical properties of the surface seen at distance from the optical axis (no paraxiales conditions). Indeed, the sign of Q determines the way in which the radius of curvature varies from the optical axis to the edges (positive: the radius of curvature decreases compared to Ro, negative: the radius of curvature increases compared to Ro).

##### Q < 0: curve prolate

The curvature decreases from the Center (Summit) to the edges (edge). It is a physiological characteristics for the human cornea. The corneal curvature prolate character grows in case of Keratoconus, or after surgery to the hyperopia.

##### Q > 0: curve oblate

Believes the curvature Center (top) to the edges (edge). After conventional refractive surgery for the correction of myopia, corneal profile is generally oblate.

##### Q = 0: Asphericity zero

When Q is zero, the profile of the considered surface is spherical; its radius of curvature is constant at any point of it.

Conic sections as well as a model both simple and "intelligible" clinically. If we represent changes in the profile of a lens induced by different values of Q for a same Ro (for a constant apical RADIUS), we note that the variation of asphericity-induced changes are the size of micron, but are even more pronounced as you move away from the top of the conical section)**2**).

In optical terms,' the different eye dioptres (horny, crystalline) Asphericity greatly controls the rate of the aberrations of sphericity. The Asphericity is even more important that the pupil of the considered system entry will be broad. In other words, the sounding of the Asphericity on optical quality will be even more marked that the diameter of the entrance pupil is important (letting more rays refracted by a peripheral portion where the curvature so the vergence is changed according to the Asphericity). An increase in the character of the Asphericity prolate reduces the rate of positive spherical aberration, then increase the rate of negative spherical aberration.

The rise in the rate of negative spherical aberration is the mechanism that allows the multifocal contact lens Central addition, and the laser type presby LASIK profiles (supracor, laser blended vision, optimized to prolate ablation, etc.) to induce a multifocalite to compensate for presbyopia. This concept is more complicated to explain that changing the asphericity of the corneal profile (it is dependent on pupil), these techniques for correction of presbyopia are positioned as varying the corneal Asphericity rather that modulating the rate of spherical aberration.

Learn more about the parameters to describe the corneal Asphericity: JFO Asphericity article 2002 (JFO)

**Physiological variations of the Asphericity **

#### The corneal Asphericity

In mathematical terms, the profile of an aspherical surface like that of an anterior corneal section resembles that of the pointed vertex (prolate) of an ellipse. The average value of the asphéricité factor Q for the anterior face of the cornea is close to-0.2.

The curvature of the cornea is slightly reduced from the top (apex) to its periphery. Apex which corresponds generally to the arched part of the corneal surface, the average radius of curvature is 7.8 mm (about 43 D). The curvature decreases then (the radius of curvature increases) to the periphery in cornea prolate case. This variation, which has important optical effects, is significant on the macroscopic plan and is undetectable to the biomicroscopique review

There is however a great interpersonal variability and about 20% of normal subjects have a so-called cornea oblate (Q > 0: increase the curvature of the Center to the periphery) according to some studies)**3**). Oblate profile is however found in nearly 100% of the corneas after keratotomii radiaire (KR), procedures (PKR) Refractive and LASIK for myopia)**4**).

#### Asphericity of the natural crystalline lens

The asphericity of cristalliniennes refractive surfaces is more difficult to assess because of changes in geometry related to accommodation and the age, and the relative imprecision of the instruments of measurement of the anterior segment (the appreciation of the Asphericity requires an accuracy of the order of a micron). Different Q values between-6 and + 1 have proposed for the front and back of the natural lens faces. The insertion of an implant to spherical surfaces does not return of the optical quality of a young natural crystalline not accommodating. The increase in the volume of the lens with age and some structural changes (variation of the index gradient refractive and spherize surfaces) help to explain the increase of the spherical aberration over time.

#### Relationship between corneal Asphericity, keratometry, ametropia

There is no statistical link between the value of the corneal Asphericity and the Central keratometry in the general population.

It does not appear to be either of relationship between Asphericity and ametropia, apart from the high myopia, where a study found a higher than average rate of corneas Oblate)**5**). We did not find difference of corneal Asphericity between myopic and normalsighted subjects)**6**).

### Optical role of the Asphericity

It is important to make a clear distinction between curvature and optical power. The optical power depends of the curvature and the change of index of refraction and the angle of the rays refracted by the considered surface.

Given a spherical lens of index (ex: n = 1.49) has a constant curvature, but greater optical power at its periphery (increase the angle of the rays not close of axis) to the Center. The peripheral rays are so focused in front of those refracted at the centre (positive spherical aberration). This variation of optical power would translate to the level of a wave front emitted plan by a phase shift between the centre and its edges.

An aspheric lens has a non constant curvature but may have a constant optical power, if the reduction of power optics related to peripheral flattening exactly compensates for the effect of the increase of the angle of incidence of the peripheral rays (or even further the case of the oval of Descartes). This optical principle is similar to that of the so-called aspherical implants "aberration free" where the optical power is constant over the entire optical surface.

Peripheral corneal flattening (profile prolate) allows to reduce aberrations of sphericity or aberrations spherical that one would observe if the cornea was truly spherical, but it is not enough to completely cancel the eye aberrations of sphericity. Classically considered the lens alleviates some of the remaining positive spherical aberrations, thanks to its aspheric geometry and its refractive index gradient (the refractive index of the lens is higher in the center than towards its Equator))**7,8)**.

It is futile to attempt to define an "optimal" Asphericity for the cornea to improve night vision or, on the contrary, increase the multifocalite (correction of presbyopia). The Central corneal curvature, the diameter of the pupil IRIS, the depth of the anterior Chamber of the eye are all factors that alter the optical consequences of a same Asphericity.

The induction of a hyper Asphericity prolate is used in corneal surgery for presbyopia compensation: it allows, in case of myopisation of the eye (in the center of the optical zone), quickly reduce the ocular vergence to the most peripheral region of the optical box to get a by far sufficient Visual acuity.

**References**

*(1) Gatinel D. principles and interest of the corneal modeling in refractive surgery. In: "refractive surgery". Saragoussi JJ, Arne JL, Colin J, Montard M, pp 84-95, French society of Ophthalmology and Masson, 2001.*

*2) Gatinel D, Haouat M, Hoang Xuan T. Review of mathematical descriptors of corneal asphericity. J Fr Ophtalmol, 2002; 25 (1): 81-90*

*(3) f., Yeung KK., Maloney RK Eghbali. Determination of topographic corneal asphericity and its lack of effect on the refractive outcome of radial keratotomy. Am J invest 1995; 119: 275-280*

*(4) Hersh PS, Fry K, Blaker JW Spherical aberration after laser in-situ keratomileusis and also keratectomy. Clinical results and theoretical models of etiology. J Cataract Refract Surg. 2003; 29 (11): 2096-104.*

*(5) LG, Mainstone JC, Henderson BA Carney. Corneal topography and myopia. A cross-sectionnal study. Invest invest Vis Sci 1997; 38: 311-320*

*(6) Haouat M, Gatinel D, Duong MH, Faraj H, that O, F, Hoang Xuan T. Corneal asphericity in myopic Reyal. J Fr Ophtalmol 2002, 25; 488-492*

*(7) Barbero S, Marcos S, Merayo Lloves J. Corneal and total optical aberrations in a unilateral aphakic patient. J Cataract Refract Surg, 2002; 28:1594 - 1600*

*(8) Gatinel D. monochromatic Aberrations of high degree: definitions and implications for Visual function. In Gatinel D, Hoang Xuan T: "the LASIK: from theory to practice", Elsevier, 2003; pp151-159*

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