Thick lens paraxial eye model
Until recently, paraxial biometric formulas were based on a thin lens eye model, where the corneal power is estimated from the curvature of its anterior face, and the implant also likened to a thin lens. In this type of model, the axial length is equal to the distance between the cornea and the retinal photoreceptors, and the position of the implant is defined as the distance between the implant and the cornea. Intraocular implants (IOLs) have a non-negligible thickness on the scale of the axial length of the human eye. In addition, the distribution of the curvatures between their anterior and posterior faces can vary according to the models and the powers, which implies that at an equal distance from the cornea, two implants of the same nominal power do not induce exactly the same correction.
To overcome this problem, it is necessary to take into account the thickness of the elements involved in ocular refraction: the cornea and the lens. This page is devoted to the development of a paraxial thick lens pseudophakic eye model.
In thin lens models, it is intrinsically assumed that the anatomical position of the implant is equal to its « optical »position. But what is most important to consider, is that in a more realistic paraxial eye model called « thick lens », the anatomical position of the implant (defined by the distance between the vertex of the cornea and the vertex of the implant) is distinct from the « optical position » of the implant, which depends on the position of its principal planes.
The purpose of this page is to establish a relationship between the axial length of an emmetropic eye and the position of the implant using a paraxial thick lens model. It details the steps leading to the following relation, where ALT and ELPT are respectively a modified axial length and an effective position of the modified IOL within the framework of a thick lens model:
This expression also depends on the value of the corneal power (Dc), the power of the implant (Di), and those of the refractive indices considered (na, nv). Why is it fundamental to establish such an equation? Because it offers the possibility of being resolved for any of the variables it contains and in particular ELPT (effective position of the IOL in a thick lens eye model).
The following aims to justify this equation. You can ignore the derivation of this formula but it would be a shame because it highlights many elements relating to the properties of models in thick lenses.
Thick lens paraxial eye model: Gullstrand’s formula
Consider a thick lens eye model, consisting of a cornea and an implant. The cornea + implant system focuses incident light from a source located at infinity at a focal length F’e.
Using Gullstrand’s formula we can show that De, the optical power resulting from the cornea + IOL pair, is equal to:
ELPT is the optical distance between the cornea and the implant. In a thick lens model, this distance corresponds to the distance between the principal image plane of the cornea, and the principal object plane of the IOL.
The position of the focal point of the optical system corresponding to the cornea + IOL (F’e) with respect to the plane of the photoreceptors of the eye determines the refraction of the pseudophakic eye. If the focal point F’e is located in this plane, the pseudophakic eye is emmetropic. If it is located in the front, the eye is myopic, and if it is located behind, the eye is hyperopic.
Thin vs Thick lens eye paraxial eye model
For the same total powers of the cornea (Dc) and of the IOL (Di), for a thin lens eye model to be equivalent (to predict the same position for the focus F’e), it is necessary that the plane of the (thin) cornea coincide with the principal plane image of the (thick) cornea, and that the plane of the (thin) IOL coincides with the main object plane of the (thick) IOL. If and only if this condition is met, then the thin lens model is equivalent (same position of the image focus of the couple cornea + IOL) to the thick lens model.
The following figure is a schematic representation of this condition:
In clinical practice, thanks to ocular biometry, anatomical data (axial length, depth of the anterior chamber, etc.) are acquired. Let’s take a look at how this data fits into a thick eye model.
Effective lens position in a thick lens eye model (ELPT)
Let’s look at the thick lens model represented in the following figure, and where some important dimensions appear, including the formula that links the ELPT to the anatomical position of the implant (S1S3 = ALP). Recall that the distances are algebraic (their sign is positive or negative depending on whether the ends of the segment are stated from left to right, or from right to left under the upper line).
ALP is the anatomic lens position, from anterior corneal vertex to anterior IOL vertex
ILP is the internal lens position, from posterior corneal vertex to anterior IOL vertex.
The relations between the ELPT and ALP can be written as:
it is interesting to note that H’cS1 and S3Hi depend only on the cornea and the IOL to be inserted, respectively. They can therefore be calculated in advance if the information is available on the corneal topography as well as the type of IOL to be used.
It is only the ALP that cannot be known in advance with certainty but can simply be predicted. This is the main issue of IOL power calculation formulas.
We will now establish an expression that contains the main anatomical elements that play a role in calculating the potency of the intraocular implant. It is, therefore, necessary to establish relationships between the segments that, placed end to end, make up the anatomical axial length of an emmetropic pseudophakic eye.
Anatomical vs axial length in a thick lens paraxial eye model
The anatomical axial length (ALA) here corresponds to the value which connects the anterior corneal vertex to the photoreceptors’ plane at the fovea. Optical biometers provide the distance between the anterior corneal vertex and the retinal pigment epithelium using the technique of partial coherence interferometry.
In the case of an emmetropic eye, it is equal to the distance S1F’e, where F’e is the back focal point of the paraxial schematic pseudophakic eye.
We can split the anatomical axial length into two algebraic segments:
H’eF’e is the image focal length distance which is equal to the following ratio: nv/De
H’cHi is the ELPT, and H’iH’e corresponds to the position of the principal image plane of the thick dual-lens paraxial eye model. It is given by the following expression:
Finally, after rearranging the terms in the preceding equations, we obtain:
Let us define :
ALT is equal to the anatomical axial length of the emmetropic eye reduced by the distance between the principal planes of the implant HiH’i and the distance between the anterior surface of the cornea and the secondary principal plane of the cornea S1H’c (which is negative). The calculation of the value of the ALT can be carried out from data specific to the cornea of the eye to be operated on and from the characteristics of the intraocular lens which will have to be placed to replace the lens.
We finally obtain the expression:
This equation is valuable:
It provides the value of the axial length of an emmetropic eye provided with a cornea and an IOL considered as thick lenses.
It can be resolved for any of the variables it contains. This makes it possible to determine the value of certain variables in circumstances such as:
-calculating the value of the ELPT from the postoperative data. In this case, we must not lose sight of the fact that this expression concerns an emmetropic eye. In the case of a non-emmetropic eye, it is possible to modify the total vergence of the cornea to reduce in the case of an emmetropic eye.
-determining the total corneal power (Dc): this plays an important role in calculating the theoretical refraction (SE: spherical equivalent). Once this value is obtained, the refraction in the corneal plane is obtained by subtracting this value from that of real power measured using keratometry.
-last but not least, if we solve this equation for the power of the implant Di, we naturally obtain a biometric formula for calculating the power of IOL in a thick lens paraxial eye model.