Sensitivity of the corneal-plane refractive compensation to change in power and axial position of an intraocular lens

If an intraocular lens is displaced or if its power is changed what are the consequences for the refractive compensation of the eye? Gaussian optics is used to obtain explicit formulae for the sensitivity of the corneal-plane refractive compensation (also called the refraction, refractive state, etc) to change in power and axial displacement of a thin intraocular lens implanted in a simple eye. In particular, for a pseudophakic Gullstrand simplified eye with intraocular lens placed 5 mm behind the cornea the sensitivity to errors in the power of the intraocular lens is about 71 . 0 − 1 for an intraocular lens of power 20 D, that is, the refractive compensation decreases by about 0.71 dioptres per dioptre increase in the power of the intraocular lens. More generally the sensitivity is approximately ( ) I m 0037 . 0 63 . 0 F − − 3 ( ) I m 0037 . 0 63 . 0 F − )FI where FI is the power of the intraocular lens. Also for Gullstrand’s simplified eye the sensitivity of refractive compensation to axial displacement of the intraocular lens is approximately linear in FI , about (64D) FI, in fact. That is, for each dioptre of the power of the intraocular lens the refractive compensation increases by about 0.064 dioptres per millimetre of axial displacement towards the retina.


Abstract
If an intraocular lens is displaced or if its power is changed what are the consequences for the refractive compensation of the eye?Gaussian optics is used to obtain explicit formulae for the sensitivity of the corneal-plane refractive compensation (also called the refraction, refractive state, etc) to change in power and axial displacement of a thin intraocular lens implanted in a simple eye.In particular, for a pseudophakic Gullstrand simplified eye with intraocular lens placed 5 mm behind the cornea the sensitivity to errors in the power of the intraocular lens is about 7 1 .0 − 71 for an intraocular lens of power for an intraocular lens of power 20 D, that is, the refractive compensation decreases by about 0.71 dioptres per dioptre increase in the power of the intraocular lens.More generally the sensitivity is approximately where F I is the power of the intraocular lens.Also for Gullstrand's simplified eye the sensitivity of refractive compensation to axial displacement of the intraocular lens is approximately linear in F I , about (64D) F I , in fact.That is, for each dioptre of the power of the intraocular lens the refractive compensation increases by about 0.064 dioptres per millimetre of axial displacement towards the retina.

Introduction
How important is the positioning and the power of an intraocular lens?If the lens becomes displaced or the incorrect power is used what is the consequence for the refractive compensation of that eye?An accompanying paper 1 describes a general procedure for calculating the sensitivity of an optical property of an eye to change or error within the eye.The purpose of this paper is to apply the method to an eye containing an intraocular lens.More specifically the paper cal-culates the sensitivity of the corneal-plane refractive compensation of the eye to change in power and axial position of the intraocular lens.In order to display the ideas, minimally obscured by mathematics, use is made of the simplest reasonable optics (Gaussian optics) and the simplest pseudophakic eye (a single refracting surface for the cornea and a thin intraocular lens).For detail on the linear optics of the pseudophakic eye, not necessarily thin, and on the effect of axial position of the intraocular lens the reader is referred elsewhere 2, 3 .

Pseudophakic simplified eye
Consider a pseudophakic simplified eye.The eye can be regarded as consisting of two systems, the anterior part SA and the posterior part S P (Figure 1).The anterior part has entrance plane T 0 immediately in front of the cornea K and exit plane T I immediately in front of the intraocular lens I.It consists of a thin system (the cornea) of power F K followed by a homogeneous gap of reduced width A ζ .The posterior part has entrance plane T I and exit plane T immediately in front of the retina R. Its structure is optically similar: it also consists of a thin system (an intraocular lens I) of power F I and a homogeneous gap of reduced width P ζ .The index of refraction is n within the eye and 1 in front of the eye.Each part of the eye has a transference of the form given by Equation 19 of the accompanying paper 1 ; for the anterior part and for the posterior part It follows from Equation 11of that paper that the transference of the pseudophakic eye is The bottom row of the matrix is not needed below and is omitted.

Sensitivity of the refractive compensation to change in power of the intraocular lens
Applying Equation 4above to Equation 16 of the previous paper 1 one obtains the corneal-plane refractive compensation We are interested in the sensitivity of the cornealplane refractive compensation to change in power of the intraocular lens, that is This shows in particular that the sensitivity is independent of the corneal power F K .

Sensitivity of the refractive compensation to axial displacement of the intraocular lens
Suppose instead one is interested in the sensitivity of the corneal-plane refractive compensation to change in axial position of the intraocular lens.We need to work in terms of distances instead of reduced distances.Equation 5 can be expressed as z A z and z P z are the widths of the anterior and posterior parts of the eye respectively.
Axial displacement of the lens changes both z A z and z P z .One needs to find an independent variable that can be changed while all others remain fixed.An obvious procedure is to define z the length of the eye: In order to find the sensitivity of the corneal-plane refractive compensation to change in distance A z with K F , I F , z and n constant we need to differentiate with respect to A z .The result turns out to be ) a sensitivity which, as expected, is also independent of the corneal power.

Application to a pseudophakic simplified eye
Consider Gullstrand's simplified eye from which the lens has been removed.The index of the aqueous and vitreous humors is

Figure 1
Figure 1 A simplified eye with cornea K of power F K and an implanted thin intraocular lens I of power F I .The distance between the cornea and the intraocular lens is A z and between I and the

F
) (The denominator, with form reminiscent of the form of what is commonly called the equivalent power of a thick lens, might be regarded as an equivalent or effective reduced length of the eye.)Equation 5 defines independent variables F K , A ζ , F I and P ζ .It also implies four derivatives, that is, four sensitivities.One of them is obvious: the sensitivity of the compensation to change in corneal power, that is, agreement with what we would expect.

S
Afr Optom 2009 68(4) 176-179 WF Harris and RD van Gool -Sensitivity of the corneal-plane refractive ... position of an intraocular lens F , z and n and each of the five derivatives represent sensitivities.
22 mm.The corneal-plane refractive compensation is given by Equations 5, 7 or 9; for an im-Because the coefficient of F I in the denominator is in metres F I is necessarily in dioptres. .Thus for 1 D increase in power of D�D.Thus for 1 D increase in power of the intraocular lens the corneal-plane refractive compensation will decrease by about 0.71 D, thus changing the compensation to an estimated value of about 4 5 . 1 − 45 D. (This agrees with the true compensation D. (This agrees with the true compensation calculated via Equation 5 for 2 1 I = F 21 D.) A decrease of 1D results very nearly in emmetropia.The dependence of sensitivity on the power of the intraocular lens is roughly affine, the binomial expansion being we obtain the sensitivity of the refractive compensation to change in axial position.For 2 1 I = F 20 D the sensitivity is about 1280 D 2 or about 1.28 D�mm.An axial shift towards the retina of 1 mm will increase the refractive compensation by about 1.28 D to an estimated compensation of about D. The actual refractive compensation, calculated via Equation 5, is 0.52 D. Hence the error in the estimate is 0dioptre of power of the intraocular lens the corneal-plane refractive compensation increases