From vergence to thin lens IOL power formula
This page concerns the development of a theoretical formula for calculating the power of an intraocular lens (IOL) from an eye model made up of two thin lenses (the cornea, and the implant whose power we want to calculate to achieve the desired postoperative refraction). The formula that will be established is the core of many IOL power calculation formulas.
The establish the thin lens IOL power formula, the reader is expected to know the definition of vergence and to be relatively familiar with the method of calculating vergence step-by-step as the light emitted from a distant source propagates through the ocular media. In short, when a wavefront meets a refractive element (cornea or IOL), we add the value of the vergence of this element, under the vergence formula. When the wavefront propagates in a medium, we subtract or add a segment to the distance to the focus, which corresponds to the distance traveled by the wavefront in the medium.
Vergence after refraction by the IOL in an emmetropic or well-corrected eye
The vergence is equal to the ratio between the refractive index of the propagation medium, and the distance to the focus. In the case of an emmetropic eye or one corrected for its refractive error in glasses, the vergence of a light wavefront after refraction by the implant (VIF) is equal to the ratio between the refractive index of the medium (nv) and the distance between the implant and the fovea:
To obtain an expression making it possible to calculate the power of the implant intended to obtain the desired postoperative refraction, we will calculate the value of the vergence propagated from a source point located at infinity to the intraocular lens (more precisely at the output of it). Once this expression is established, we will then set equality with the VIF vergence established above.
Vergence of the incoming wavetrain
Let’s start by establishing the value of the vergence of a wave train coming from the distant source in contact with a spectacle lens correcting our pseudophakic eye. The power of this lens is equal to the target postoperative refraction (which is generally between emmetropia = plano lens = 0D, and low myopia for reading = -2.50 D).
Vergence after refraction by the spectacle glass
As soon as the wavefront emerges from the spectacle lens, its vergence is equal to that of this spectacle lens. This vergence corresponds to the inverse of the distance to the focus. In the case of a concave glass (correcting myopia), the focus is on the left, the wavefront is diverging and the vergence is negative.
Vergence at corneal plane
At the corneal plane, the vergence is easily calculated as the inverse of the sum of the distance to the focus with the distance between the spectacle lens and the cornea. This vergence has a negative sign (divergence).
Vergence after refraction by the cornea
After the cornea, a change in the refractive index takes place. The vergence is obtained by adding to the vergence at the corneal plane the vergence of the cornea. The distance to the new focus is obtained by dividing the refractive index of the aqueous humor by the vergence after the cornea. It is positive (converging to the right).
Vergence at IOL plane
It is now necessary to calculate the vergence in the plane of the artificial lens implant (vergence at IOL plane). It suffices to subtract the distance corresponding to the space between the cornea and the implant from the previously calculated focusing distance (which is equal to the ratio na/VAC). This space, the measure of which is not known after the operation, is an important variable for the implant calculation, usually referred to as “Effective Lens Position”(ELP). For certain numerical applications, such as that of calculating the power of the implant which interests us here, a value must be chosen for the ELP but it can only be anticipated.
Vergence after refraction by the IOL
The last step is to calculate the vergence after refraction by the IOL (VAI). It suffices, by virtue of the vergence formula, to add the power of the implant to the vergence calculated previously (vergence at IOL plane). The expression obtained is equal to VIF if the starting condition is verified: the power of the implant placed provides the eye with the desired refraction R. It is then sufficient to solve the equation for the value P to obtain an implant power calculation formula intended to induce the refraction R (null when targeting emmetropia), in an eye of axial length AL and of total corneal power VK. The value of the ELP must be chosen in advance: the modalities and determinants of this choice constitute what still differentiates most of the theoretical implant calculation formulas today.