## Introduction

Phakic intraocular lenses (PIOLs) are implanted in phakic eyes. Piggyback IOL implantation involves the placement of two IOLs in the eye. Piggyback IOL implantation can be primary, by putting in both lenses at the same stage, or secondary, by putting the second lens in at a second stage. Formulae for PIOLs and secondary piggyback IOLs are essentially the same, as they share the same principle for calculating IOL powers.

Several formulae have been developed for phakic IOL and secondary piggyback IOL calculations; the most established one is van der Heijde’s equation. Published in 1988, the equation determined the IOL power needed for phakic myopia and aphakia (hyperopia). Van der Heijde’s formula, which assumed postoperative emmetropia, was modified by Holladay, who took into account the desired postoperative refraction and published his equation in 1993. Thereafter, other formulae, such as the matrix formula and the Olsen–Feingold formula, were developed. Most are derived from the vergence formula; a few of the formulae use regression analysis. All of these formulae have good clinical predictability but most are complex, such that the IOL powers could be calculated only with the use of computers. Tables and graphs have been used to approximate the desired IOL powers.

The authors have designed a simplified formula that enables surgeons to double-check the desired powers of lenses without using computers and tables. In this chapter, we will review how the vergence formula was converted to formulae for PIOLs and secondary piggyback IOLs. We will also compare the parameters that require greatest accuracy to avoid IOL power calculation errors.

## Van der Heijde’s Equation and Holladay’s Equation

The two equations are based on the vergence formula. The (reduced) vergence of light ( *L* , in diopters) is defined by the refractive index of the medium ( *n* ) divided by the vergence distance (1, in meters):

L=n1.

*L*′) can be calculated by adding the object vergence and the power of the lens (

*F*):

L+F=L′.

*L*) of the refracting surface,

_{i }*i*, can be deduced from the previous image vergence (

*L*

_{i -1}′) and the distance (

*d*) between them:

_{i }Li=ni⋅Li−1′ni−di⋅Li−1′,

*n*is the refractive index of the corresponding refracting surface. This equation is usually applied for the prescription of a contact lens when

_{i }*n*is equal to the refractive index of air (

*n*= 1):

Ps′=PS1−t⋅PS,

*P*is the power of the spectacles or Brechkraft der Brille (in diopters),

_{S }*P*′ is the spectacle correction at the corneal vertex (in diopters), and

_{S }*t*is the back vertex distance (in meters).

The image vergence of the lens after operation (without spectacles) will be the same as the one of the corresponding site before operation (with spectacles; Fig. 28.1 ).

Preoperatively, the image vergence, *L* _{2} ′ (preop), at the imaginary principal plane of the PIOL is presented as follows ( Fig. 28.2 ):

FIOL=nnK+Ps′−d−nnK−d,

*n*is the refractive index of aqueous (

*n*= 1.336),

*d*is the effective lens position (ELP),

*K*is the corneal power, and

*P*′ is the spectacle correction at the corneal vertex and is equal to Eq. (4) :

_{S }Ps′=Ps/(1−t⋅Ps),

*t*is the back vertex distance and

*P*is the spectacle power.

_{S }