6. Paraxial optical power of the cornea
This page is dedicated to the study of thick lens formulas that characterize the cornea’s paraxial properties. These formulas allow the calculation of the power of the cornea’s anterior and posterior surfaces and of the total cornea.
The determination of the location of the principal planes of the cornea will also be discussed. The specificities of choosing a reduced index to estimate corneal power based solely on anterior curvature data will be examined. At the end of this page, an interactive calculator allows you to calculate all these values.
What are we looking for?
We want to determine a paraxial model of the cornea, which involves determining the position of the principal planes and the focal distances for a thick lens. The paraxial optical power or vergence of the cornea depends on its front and back surfaces’ curvature and thickness.
In this context, it is necessary to know the values of the paraxial radii of curvature of the anterior face and posterior face of the cornea and its thickness. These values can be obtained thanks to measurements in OCT, corneal topography, and certain biometers. Some biometers measure only the curvature radius of the cornea’s anterior surface without accounting for the posterior surface. This leads to approximating the total paraxial power of the cornea using a reduced index value.
What do we need?
The cornea’s topography (Scheimpflug or OCT) and certain biometers allow the measurement of the apical curvatures of the anterior and posterior corneal surfaces (Rca and Rcp, respectively) and the corneal thickness.
The anterior surface of the cornea is in contact with air, while the posterior surface of the cornea is in contact with the aqueous humor. It is necessary to know the values of the refractive indices of these media (n0 and na) and the refractive index of the corneal stroma (ns).
Paraxial formulas
Using such information, we can determine the vergence (paraxial optical power) of the anterior and posterior surfaces and the entire cornea (combined effect of the anterior and posterior surfaces). Finally, we can determine the positions of the principal planes.
All the necessary formulas gathered for these calculations are shown in this illustration:
This figure makes it possible to calculate the anterior surface, the posterior surface and the cornea assimilated to a thick lens:
-the paraxial power,
-the object focal distance,
-the image focal length,
– the position of the principal planes of the cornea (thick lens).
Numerical example
We consider the value of the refractive index of the corneal stroma (ns = 1.376, and of the aqueous humor (na = 1.337)
The anterior curvature has a radius Rca = 7.8 mm.
The posterior curvature has a radius Rcp = 6.5 mm
The central corneal thickness is dc = S1S2 = 0.55 mm
Calculation of the power of the anterior surface of the cornea in air:
Dca = (nc – 1) / Rca = (1.376 -1) /0.0078 = 48.21 D
Focal distances of the anterior surface:
The object focal length is given by fca = -1 / 48.21 = -0.02074 m
The image focal length is given by f’ca = 1.376 / 48.21 = 0.02854 m
Calculation of the power of the posterior surface side of the cornea:
Dcp = (na – ns) / Rcp = (1.337 -1.376) /0.0065 = -6.0 D the steeper the posterior surface (Rcp decreases), the more negative (divergent) its optical power.
Focal distances of the posterior surface:
Object focal length fcp = -1.376 / -6.0 m = 0.2293 m
The image focal length f’cp = 1.337 / -6.0 m = -0.2228 m
Cornea: thick lens equivalent
The next step is to calculate the power of the centered system equivalent to the cornea (Dc), taking into account the thickness of the cornea at the center (distance between the anterior and the posterior surface).
Power of the centered system equivalent to the cornea (Dc)
For this, we use Gullstrand’s formula, which concerns « thick » systems: we subtract from the sum of the power terms the product of the thickness by the front and back powers divided by the refractive index of the stroma.
Dc = Da + Dp – dc (DaDp) /nc = 48.21 -6.15 –0.00055 x 48.21x-6.15 / 1.376 = 42.32D
Focal distances of the cornea:
The object focal length of the cornea is equal to fc = -1 / Dc = -1 / 42.32 m = -0.02362 m
The corneal image focal length is equal to f’c = 1.337 / Dc = 1.337 / 42.32 = 0.03158 m
Position of the principal planes and points (intersection of the planes with the optical axis) of the cornea.
(a page is devoted to the explanations and calculation of the position of the principal planes of a paraxial system).
The distance between the principal object plane Hc and the vertex of the cornea S1 is:
S1Hc = dc x fc / fcp = 0.55 x (-23.62) / (22.93) = – 0.0567 mm
The distance between the principal image plane H’co and the vertex of the cornea S2 is:
S2H’c = -dc x f’c / f’ca = -0.55 x (31.58) / (28.54) = – 0.6085 mm
The negative sign of the distances corresponds to the position of the principal object plane located in front of the cornea (58.3 microns before S1). The position of the principal image plane is also located at 0.55-0.6085 = 0.0585mm, approximately 60 microns in front of the front surface (S1).
The distance between the principal planes of the cornea (1.8 microns) is clinically negligible.
In a thick lens paraxial model of the « pseudophakic » eye, which would associate cornea and IOL, the effective position of the IOL corresponds to the distance between the principal image plane of the cornea and the principal object plane of the implant.
Modern biometric calculation formulas, such as the PEARL DGS formula, use this type of calculation to estimate corneal power. The PEARL DGS formula, designed to be usable on biometers that do not measure the posterior corneal surface, estimates the posterior curvature radius using artificial intelligence based on an algorithm trained to predict the posterior corneal surface from extensive biometric data. Older formulas use a less personalized method to estimate total corneal power based on the anterior corneal curvature:
Keratometric power of the cornea
Biometers are equipped with a device that estimates total corneal power from the measurement of the radii of curvature of the front and back corneal surfaces. Most of them do not measure the curvature of the posterior surface of the cornea and instead use an approximation to estimate total corneal power: the posterior surface of the cornea has a negative optical power, and to account for this attenuation, the calculation of anterior corneal power (based on a millimetric measurement of the radius of curvature from specular imaging techniques) is performed using a reduced refractive index value, known as the keratometric index. The value chosen for the keratometric index is most often 1.3375 for reasons unrelated to estimating the corneal power of the cornea; this value ensures that a curvature radius measurement of 7.5 mm corresponds to a keratometric power of 45 D. Implant calculation formulas were also developed at a time when biometers lacked direct measurement of the posterior surface of the cornea using a modified index value (which is not always identical to the value used by the biometer to convert corneal curvature into keratometric power, see below).
This approximation corresponds to the use of a « minored » corneal refractive index – or keratometric index (nk), used on keratometers and in topograph software to calculate the keratometric power map (axial or tangential) in diopters. The interest in this approximation was related to the inability to measure the posterior surface of the cornea easily in clinical practice. It can provide a sufficiently precise estimate of the keratometric power in diopters for some applications.
However, it is not even valid to estimate the true total paraxial power for corneas where the effect of posterior curvature relative to the anterior surface power does not differ too much from the « average. »
It is even less accurate when we measure a curvature radius value of 8.5 mm (flat cornea encountered after myopic LASIK or myopic PKR), with an unaltered posterior curvature value (6.5 mm), the power calculation paraxial using the previous equations gives the following value: Dc = 38.34 D. Using the commonly accepted value of a keratometric index of nk=1.3375, we obtain Dk = 39.71 D, almost a 1.5 D discrepancy.
This discrepancy can have significant clinical consequences. It is therefore not recommended to rely on this approximation (cornea = spherical dioptre of curvature Rca with reduced index nk) after refractive surgery to perform IOL power calculation, especially since the other inaccuracies in the measurement of keratometry tend after refractive surgery to overestimate the paraxial power of the cornea … which induces an underestimation of the power of the implant to be placed and risks inducing hyperopia after cataract surgery.
Relation between the keratometric index and the corneal ratio of anterior vs posterior curvature
We explored the theoretical relationship between the keratometric index, anterior-posterior corneal curvature ratio (APR), and estimated total corneal power to refine the accuracy of corneal power estimation (read the related publication). Conventional keratometry relies on a fixed keratometric index, typically 1.3375, to approximate corneal power based on the anterior corneal curvature. However, this standard value can lead to inaccuracies, particularly when the posterior corneal curvature is not considered, as it plays a significant role in the total refractive power of the cornea.
Our objective was to derive theoretical equations that predict the most suitable keratometric index, which better aligns with the total Gaussian corneal power, incorporating both the anterior and posterior corneal radii and central corneal thickness. This is especially important in eyes that have undergone laser refractive surgery, where the curvature ratio is altered.
By analyzing the impact of varying anterior and posterior corneal radii and corneal thickness, we demonstrated that the keratometric index is not a one-size-fits-all value:
nk ≈ ns + (na – ns) (Ra / Rp)
Where:
nk is the ideal keratometric index,
ns is the refractive index of the stroma (typically 1.376),
na is the refractive index of the aqueous humor (typically 1.336),
Ra is the anterior radius of curvature,
Rp is the posterior radius of curvature,
dc is the central corneal thickness,
Ra / Rp is the anterior-posterior corneal curvature ratio.
Numerically, we can use this approximation;
nk ≈ 1.376 + 0.04 (Ra / Rp)
The average anterior-posterior corneal curvature ratio (Ra/Rp) in the general population is approximately 1.22. This ratio can be used to estimate the ideal keratometric index (nk) using the following formula:
nk ≈ ns + (na - ns) X
Where X is the anterior-posterior corneal curvature ratio (Ra/Rp), which is 1.22.
Substituting the values into the equation:
nk ≈ 1.376 + (1.336 - 1.376) × 1.22 nk ≈ 1.376 - 0.0488 nk ≈ 1.3272
Therefore, for an anterior-posterior corneal curvature ratio (Ra/Rp) of 1.22, the estimated keratometric index (nk) is approximately 1.3272. This value is still lower than the lowest keratometric index used in most conventional IOL power formulas (Haigis, Holladay, Hoffer, Olsen, etc.). It is important to note that intraocular lens (IOL) power formulas do not use a uniform keratometric index. For instance, the Hoffer Q formula uses 1.3375, the Haigis formula uses 1.3315, the SRK/T formula uses 1.333, and the Holladay 1 formula adopts 4/3. When a power calculation is performed by the biometer software using one of its formulas, the corneal power is recalculated based on the measured curvature radius and the keratometric index value used by the relevant formula. These variations in keratometric index, when applied to IOL power calculations, highlight the need for tailored indices based on individual corneal parameters. Our results show that most third- and fourth-generation formulas overestimate the value of the keratometric index, and thus the paraxial corneal power. »
Post-refractive surgery, the increase in APR leads to further discrepancies in corneal power estimation when using the fixed index of 1.3375, which typically results in an overestimation of the total corneal power.
Interactive Calculator for the paraxial optical power of the total cornea
This spreadsheet is used to calculate the different variables mentioned on this page, which concern the thick cornea’s paraxial optical power.
One can also use this approach to characterize the paraxial properties of IOLs.
A fair deal of linear matrix calculations applied to the human eye can be found in the papers of Evans and Rubin (ex ; Linear optics of the eye and optical systems), in BMJ Open Ophthalmology:
This is the most complete analysis that I’ve found for the cornea treated as a thick lens.
Where can I find this analysis in a matrix optics form?; specifically for the cornea.