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# Calculation of the optical power (vergence) of the cornea

Vergence of the cornea or paraxial optical power depends on the curvature of its faces front and rear, as well as its thickness.

What is being?

A determine the vergence (optical power) of the entire cornea (a combination of the refraction of the front and back sides). We wish to determine a model paraxial of the cornea, which implies the determination of the position of the principal planes, and of the focal lengths, as for a thick lens (see the page on the justification of) paraxial optics and main plans)

Experienced - we?

The topography of the cornea to measure the apical curvature of the cornea, and it is possible to obtain the thickness by tomographic measurement.

We know the value of the index of refraction of the corneal stroma (nco = 1,376, and aqueous humor (nHaq=1.336)

The anterior curvature has a RADIUS R1 = 7.8 mm

Subsequent curvature has a RADIUS R2 = 6.5 mm

E thicknessco = S1S2 = 0.55 mm Representation of the data which we know the numerical values for the calculation of the power of the cornea paraxial.

## Calculation of the power of the front of the cornea

D1 (n =co (1) /R1 = (1,376 - 1) / 0.0078 = 48.21 D

The focus distance object is given by f1 = 1/48.20 m = - 0,02074 m

The focal image is given by f'1= 1.376/48.21 m = 0.02854 m

## Calculation of the power of the face rear of the cornea

D2 (n =Haq -nco) /R2 = (1336-1.376)/0.0065 =-6.15 D over the cornea is cambered (R)2 decreases), more the back made "diverge" light rays!

The focal length f object2 =-1.376/-6.15 m = 0,2237 m

The focal length f image'2 = 1.336/-6.15 m =-0,2172 m

The next step is to calculate the power of the equivalent to the cornea (D centered systemco), taking into account the thickness of the cornea at the centre (distance between the anterior and the posterior).

## Equivalent to the cornea (D centered system powerco)

Used for this form of Gullstrand, referring to systems 'thick': we subtract the sum of the terms of power the product of thickness by the powers front and rear divided by the index of refraction of the stroma.

Dco = D1 + D2 -eco(D1D2) / nco = 48.21 - 6.15 - 0.00055 x 48.21x-6.15/1.376 = 48.21 - 6.15 + 0.12 = 42.17D

The focal object of the cornea (front and rear) is equal to fco = - 1 / Dco = 1/42.17 m = - 0,02371 m

The focal image of the cornea is equal to f'co = 1,337 / Dco 1.337/42.17 = 0.03170 = m

It remains to determine the position of points (intersection of the planes with the optical axis) and plans major of the cornea.

## Position of the main plans of the cornea

The distance between (point) plan main purpose:co  and the top of the cornea Sis:

S1Hco  e =co fco / f2 = 0.55 x (-23.71) / (223.7) = - 0.0583 mm

The distance between (point) plan main image: 'co  and the top of the cornea S2  is:

S2H'co  = EIco f'co / f'1 = - 0.55 x (31.70) / (28.54) = - 0.6101 mm

(pages are devoted to the paraxial a modeling thick lens and to the  explanation and calculation of the position of main points of a paraxial system and plans)

The negative sign of the distances is a position of the plan main object located in front of the cornea (58.3 microns before S)1). The position of the plan main image is also located at 0.55 - 0.6101 = 0.0601 mm = 60.1 microns in front of the front S1.

## Conclusion

The gap between the main plans of the cornea is very low (1.8 microns), and the distance from the top of the cornea (vertex in S1) is negligible.

We can assimilate the behavior perspective the cornea of a paraxial spherical dioptre. A such diopter would have a curvature such that:

R = (1.336-1)/42.17 = 7.97 mm.)

The same calculation can be done for the lens (see the paraxial of the lens power calculation). The lens is not comparable to a spherical diopter, and the calculation of its paraxiales properties matches made for a thick lens.

## Consequences: approximation of the optical power of the cornea

The RADIUS calculated with the formula for the spherical dioptre is relatively close to the radius of curvature previous 'real' of the cornea (7.8 mm). The index is considered should be for this calculation is aqueous (n = 1336) mood. If one chooses a somewhat smaller value for this index (ex: n = 1.333), it gets even closer to the initial value of the anterior radius of curvature.

This approximation is the use of an index says "minus" or k, used on the keratometres and the software of the topographers to calculate the maps of power (should say curvature) axial and tangential in diopter. The interest of this approximation was impossible to measure the posterior face of the cornea in a easy way in clinical practice-related.  It also provides a precise enough estimate of power k in dioptres for most common applications.

However, it is valid only for corneas with the effect of posterior curvature on the earlier power is not too different from the 'average '.

For example, if you measure a value of radius of 8.5 mm (flat cornea, after) LASIK or PKR myopic), but it keeps the same value of posterior curvature (6.5 mm), the power calculation paraxial using the previous equations provide the following value: Dco = 38.2 D.

Using the approximation with n = 1.333, obtain a power equal to 39.2 paraxial D: an error of 1 d (overestimate) may have significant clinical consequences. It is thus recommended to use this approximation (cornea = spherical with index minus dioptre) after refractive surgery to perform a calculation biometric, and this all the more that the other inaccuracies in measurement of the keratometry tend after refractive surgery to overestimate the Central corneal power... which leads to an underestimation of the power of the implant to ask and may induce a hyperopia after cataract surgery.

The position of the implant (after insertion in the capsular bag that is carried out during the cataract surgery) can vary for reasons relating to the geometry of the operated eye (depth of the anterior Chamber, behavior of the implant in the capsular bag, etc.). In a paraxial of the eye 'pseudophake' model, which would combine corneal and implant, the actual position of the implant corresponds to the main image of the cornea, the distance between the plan and the plan main purpose of the implant.

Finally, in a paraxial calculation (calculation biometric paraxial for example), the "optical" eye length is not quite equal to the anatomical length: do indeed start the length of the eye at the level of the plan main object, which is located about 50 microns in front of the cornea. This explains the term "+ 0.050" (in millimeters) which is often add to the axial (L) length, measured in biometric formulas (ex: formula Colebrander).

Thanks to the calculation of the focal powers and position of the cardinal elements of the cornea, one can calculate the Optical power of the entire eye.