Change in refractive errors with changes in IOL parameters

This study considered two questions associated with intraocular lens (IOL) power and refraction: (1) Given a refraction with a particular IOL in the eye, what will be the refraction for the IOL or another IOL if located differently with regard to tilt or anterior–posterior position? (2) For a target refraction, what is the power of another IOL if located differently with regard to tilt or position? A thin lens technique was developed to address these questions. For the first question, light was traced through the initial correcting spectacle lens to the cornea, refracted at the cornea, transferred to the position of the initial IOL, refracted at this IOL, transferred to the position of a new IOL (which may be the same IOL but with a different position and/or tilt), refracted backwards through the new IOL, transferred to the cornea and refracted out of the eye to give a new correcting spectacle lens power. For the second question, light was traced through the initial correcting spectacle lens to the cornea, refracted at the cornea, transferred to the position of the initial IOL, refracted at the initial IOL and transferred to the position of a new IOL. Light was also traced through the second correcting spectacle lens, refracted at the cornea and transferred to the position of the second IOL. The difference between the reduced image vergence for the first raytrace and the reduced object vergence for the second raytrace gave the effective power of the second IOL, and from this, the power of the second IOL was determined. Examples are presented for different situations, including a case report.


INTRODUC TION
While intraocular lens (IOL) tilt is usually not high in most eyes, with a recent study finding a mean tilt of 4.0 ± 1.8°, it also reported values up to 10.7°. 1 In a previous paper, 2 we determined the effect of tilting an IOL on effective power and refraction.The IOL power necessary to compensate for the tilt was also determined.This was a raytracing approach using thin lenses and was highly accurate.
This work is extended here in more general terms.If the tilt of an implanted IOL is to be surgically corrected, then a surgeon needs to consider whether removing the tilt is possible and, if so, whether this is sufficient to eliminate the induced refraction, particularly unwanted astigmatism.The surgeon may consider it is better to implant a second IOL, and want to know the likely refraction for the particular position and tilt.Alternatively, the surgeon needs to know the IOL's power and cylindrical axis to achieve a particular refraction (e.g., emmetropia) for a particular combination of position and tilt.Two questions will be addressed: 1. Given a refraction with a particular IOL in the eye, what will be the refraction for the IOL or another IOL if located differently with regard to tilt or anteriorposterior position?2. For a target refraction, what is the power of another IOL if located differently with regard to tilt or position?
Here ';1' is used for parameters associated with forward raytracing into the eye, and ';2' is used for backward raytracing out of the eye.In detail, light is traced through a correcting spectacle lens of power Refraction;1 to the cornea, refracted at the cornea, transferred to IOL;1, refracted at IOL;1 and transferred to the position of IOL;2 (which may be the same IOL as IOL;1 but with a different position and/ or tilt).It is then refracted backwards through IOL;2, transferred to the cornea and refracted out of the eye to give the new correcting spectacle lens of power Refraction;2.See Figure 1 left.
Question 2 can be approached in the following way, given simplistically as Refraction;1 + IOL 1 ;1 = Refraction 2 ;1 + IOL 2 ;1 Again ";1" is used for parameters associated with forward raytracing into the eye, but there is no backward raytracing out of the eye.The subscript '2' is added for raytracing after the surgery.In detail, as before, the light is traced through a correcting spectacle lens of power Refraction;1 to the cornea, refracted at the cornea, transferred to IOL;1, refracted at IOL;1 and transferred to the position of IOL 2 ;1.Light is also traced through a correcting spectacle lens of Refraction 2 ;1, to the cornea, refracted at the cornea and transferred to IOL 2 ;1.The difference between the reduced image vergence for the first raytrace and the reduced object vergence for the second raytrace gives the effective power of IOL 2 ;1, and from this the power of IOL 2 ;1 is determined.See Figure 1 right.
Powers of refracting elements (refraction lens, cornea and IOL) are given in sphero-cylinder form as or in cross-cylinder form as where F is the sphere power, C is the cylinder power, is the cylinder axis in degrees, F is the power along the principal meridian and is equal to F, and F ±90 is the power along the principal meridian ± 90 • .The ± sign is used as appropriate to keep the latter meridian within the range 0-180°.
If a refracting element and the corresponding reduced object vergence do not have the same principal meridians, vector refraction is required for a forward raytrace.Similarly, if a refracting element and the corresponding reduced image vergence do not have the same principal meridians, vector refraction is required for a backward raytrace.Here, reduced object vergence is the inverse of the 'object' distance from a surface, multiplied by the refractive index of the object side media, and reduced image vergence is the inverse of the "image" distance from a surface, multiplied by the refractive index of the image side media.For refraction, mean sphere M, astigmatism J 180 and oblique astigmatism J 45 components are given by: These can be converted back into sphero-cylinder format by equations of the form 3,4 If a four-quadrant inverse tangent function is not available, the following steps may be needed to get the correct .If J 180 is 0 and J 45 < 0, = 135°.If J 180 is 0 and J 45 ≥ 0, = 45°.
To keep within the clinical conventional range of 0-180°, the following equations must be applied for a calculator giving results between −90° and 90°: Transfer from one element to another is determined by the calculations of the form: for a forward raytrace, and for a backward raytrace, where n is the refractive index of the medium, L is the reduced vergence at a particular position, L+ is the reduced vergence a distance d along the path and L− is the reduced vergence at a distance −d along the path.

Key points
• This paper provides a simple method for determining a new refraction when an IOL is replaced by a second IOL at a different tilt and/or anterior-posterior position.• The method described provides a technique for determining the power of this second IOL for a target refraction.
The procedure that follows allows for change in corneal parameters between the first and second situations as appropriate.
It is necessary in the procedure to determine the effective power of a tilted IOL.The IOL has a refractive index and is tilted by angle about axis .The untilted power vector fo is given by with mean sphere, astigmatism and oblique astigmatism components being given by The tilted matrix M is given by The tilted power vector f is given by matrix multiplication again with M, J 180 and J 45 refraction components.The matrix multiplication is not shown here.These can be converted into sphero-cylinder format as described in Equations ( 2)-(4b).
The determination of the (untilted) power of an IOL that would account for an effective (tilted) IOL power follows from the above.The untilted power vector fo is obtained from the tilted power vector f and the matrix M −1 , the inverse to M, in Equation ( 8) by The determination of M −1 and the matrix multiplication to get fo are not shown here.
The steps of procedures for Questions 1 and 2 are described fully in the Appendix 1.

R ESULTS Question 1
Four examples are shown in Table 1 and full calculations are shown in the Data S1.Examples 1 and 2 are purely theoretical.Example 1 illustrates how much refraction change occurs by altering the tilt of an IOL.A +20.00/−2.79× 90 powered IOL was tilted incorrectly by 20° relative to the line of sight, about an axis of 35°, giving a refraction of −0.67/−2.00× 31.3.Removing the tilt gave a Plano refraction.
Example 2 has the same initial IOL of Example 1 (+20.00/−2.79× 90).A new IOL with the same tilt of power 17.94/−2.64× 66.3 gives a plano refraction.The determination of this power was given previously. 2 Examples 3 and 4 are part of a case report.A 67-year-old woman, with a history of bilateral LASIK for myopia in 2006, had cataract surgery in her right eye in November 2022 due to difficulty driving at night.A distance refraction was targeted using a +23 D, non-toric, Alcon AcrySof Vivity IOL (Model #DFT015) (alcon.com/ ).Five weeks post-operatively, her uncorrected visual acuity was 6/15 in the right eye.The manifest refraction was −1.00/−1.00× 155 and corrected visual acuity was 6/6.The patient complained of headaches and nausea with the right eye being 'completely blurry'.Manifest refraction 9 weeks post-operatively was −1.25/−0.75× 155.
On examination with a slit lamp and an anterior optical coherence tomography (OCT) (Heidelberg Anterion, busin ess-lounge.heide lberg engin eering.com/ us/ en/ produ cts/ anter ion/ ), the IOL was found to be tilted considerably (Figure 2 left).The superior haptic was in the sulcus and the inferior haptic was in the capsular bag.Superiorly, the anterior capsule appeared fused to the posterior capsule.Relative to the visual axis, the intraocular lens had a tilt of 12° about an axis of approximately 145°.The anterior chamber depth (cornea to IOL) was 4.39 mm (Lenstar, Haag-Streit, https:// haag-streit.com/ en), which was estimated to be approximately 0.3 mm too close to the cornea.The corneal power was determined as +42.82/+1.02× 95.
A further surgical procedure was performed to correct the unintended myopia and unexpected astigmatism, by reopening the capsular bag and replacing the IOL with a new lens.Adding half the intraocular lens thickness of 0.69 mm to the anterior chamber gives a new value of 4.74 mm.Thus it was estimated that if the original IOL was removed and replaced by a +22 D IOL without the tilt and at the correct position (5.03 mm), then the refraction would change from the 9-week post-operative value to +0.37/−0.25 × 20 (Table 1, Example 3).Four months post-operatively, a second IOL exchange was performed with a +22.0 D Alcon Clareon Vivity IOL #CNWET0 with an incision at 160°.The 'fused' anterior and posterior capsules were easily separated with viscodissection.At a 1-week post-operative visit, the patient had an uncorrected distance visual acuity of 6/6 −2 , Jaeger (J) 2 vision at an intermediate distance and J2 at near.Refraction was plano/−0.25 × 107 giving 6/6 vision.Five weeks postoperatively, the manifest refraction was plano (6/6).
The new IOL was well centred in the capsular bag, but the tilt was 7° about 132° (Table 1, Figure 2 right).Further, the corneal power changed little from +42.82/+1.02× 95 to +42.73/+1.12× 92, and the anterior chamber depth was 4.64 mm.Adding half of the lens thickness of 0.64 mm to the anterior chamber depth gave a new value of 4.96 mm.When these measurements replace the zero tilt, the predicted anterior chamber depth and the original corneal power, the refraction prediction was +0.07/-0.08× 15, that is, similar to the manifest refraction.(Table 1, Example 4).

Question 2
Four examples are given in the Data S1, which reverse Examples 1-4 for Question 1. Rather than determining the refractions for particular IOL powers, the IOL powers for the particular refractions are determined.
For the case study given in Examples 3 and 4, the surgeon was unsure whether he would be able to open the superior capsule, which would mean that the 12° tilt would remain.It would then be necessary to determine the IOL power to give a desired refraction.Applying Question 2 with the original parameters in the example, and aiming for a plano refraction, the required IOL power was +21.32/−0.86× 156.Question 1 could be applied to estimate a new refraction of +0.23/−0.65 × 156.

DISCUSSION
A procedure has been developed to show how replacing one IOL by the same or a different one at a different tilt and/or location will affect the residual refraction (Question 1).The presented case study indicates that the procedure works well.In addition, for a particular target refraction, the power was determined for a replacement IOL located differently with regard to tilt or position (Question 2).The former was applied in the case study, but if it had not been possible to open the capsule, as mentioned above, then the latter question would have had to be addressed.
A supplementary file 'Tilt Problem OPO paper' includes the examples presented in this paper and the reader can replace these with new data.For Question 1, 'given a refraction with a particular IOL in the eye, what will be the refraction for this or another IOL located differently with regard to tilt or position?' in worksheet 'Tilted_IOLs Q1', rows 4-17 give the original set of results (highlighted orange), and any changes can be entered at rows 20-30 (highlighted green).The new refraction and changes to the correction are given in rows 52-58 (highlighted blue).Computations are given in the following rows.
For Question 2, 'For a target refraction, what is the power of another IOL located differently with regard to tilt or position?' in worksheet 'Tilted_IOLs Q2', rows 4-17 give the original set of results (highlighted orange) and any changes can be entered at rows 20-30 (highlighted green).The new IOL power and changes to the IOL power are given in rows 52-54 (highlighted blue).Computations are given in the following rows.

R E F E R E N C E S
Wang L, de Souza RG, Weikert MP, Koch DD.Evaluation of crystalline lens and intraocular lens tilt using a swept-source optical coherence tomography biometer.

S U P P O R T I N G I N FO R M AT I O N
Additional supporting information can be found online in the Supporting Information section at the end of this article.The correcting spectacle lens has power F l;1 ∕ C l;1 × l;1.

How to cite
The respective powers along l;1 and ± 90 l;1 are 2. Transfer to the cornea The reduced object vergence at the cornea is given along l;1 by and along l;1 ± 90 by

Power of the cornea
The cornea has power 4. Refraction at the cornea Using the procedure given in Equations ( 1)-( 4b), the reduced object vergences and corneal power are combined to give reduced image vergences for the cornea: and with modification to ′ c;1 as necessary according to the process described previously.

Transfer to the IOL
The reduced object vergence at the IOL is given along and along � c ± 90; 1 by where d c;1 is the distance between the cornea and IOL and n a is the refractive index of the aqueous medium following the cornea.

IOL power
The IOL has power 7. Tilted IOL power The IOL has refractive index ; 1, is in a medium of refractive index n a (the refractive index of the vitreous is assumed to be the same as that of the aqueous) and is tilted by angle ; 1 about axis ; 1.The untilted power vector fo 1 is Similar to Equation (1), mean sphere, astigmatism and oblique astigmatism components are given by The matrix M;1 for the tilted IOL is given by The tilted power vector f 1 is given by matrix multiplication where M IOL tilt;1 , J 180, IOL tilt;1 and J 45,IOL tilt;1 are its refraction components.The matrix multiplication is not shown here.The refraction components are converted tilted, or effective, IOL power in sphero-cylindrical format F IOL tilt;1 C IOL tilt;1 × IOL tilt;1 , where and with modification to IOL tilt;1 as necessary according to the process described previously.

Transfer to position of second IOL
As the IOL for the reverse raytrace (IOL;2) may be at a different position from the first IOL (IOL;1), compensation can be made.If the position of IOL;2 from the cornea is d c;2 , the distance travelled in the vitreous is d c;2 − d c;1 , and the refractive index of the vitreous is n a (the same as the aqueous), the image reduced vergence for IOL;2 is given along ′ tilt;1 by and along � IOL tilt ± 90; 1 by Note that ′ IOL tilt;2 is the same as ′ IOL tilt;1 , and � IOL tilt ± 90; 2 and the same is � IOL tilt ± 90; 1. Formally Question 1: Backward raytrace for second IOL

IOL power
The IOL has power 2. Tilted IOL power The procedure is the same as for the first IOL.Replacing ";1" by ";2", the tilted, or effective, IOL power in spherocylindrical format IOL IOL tilt;2 × IOL tilt;2 : and with modification to IOL tilt;2 as necessary according to the process described Refraction at IOL Using the procedure given in Equations ( 1)-(4b), the reduced image vergences and tilted IOL power combined to give reduced object vergences for the IOL: � IOL tilt ± 90; 2 = � IOL tilt ± 90; 1 and with modification to ′ c;2 as necessary according to the process described previously.Note here that ′ c;2 is used as it will be the axis when passing back to the cornea.

Transfer to the cornea
The reduced image vergence at the cornea is given along ′ c;2 by and along � c;2 ± 90 by where d c;2 is the distance between the cornea and IOL and n a is the refractive index of the aqueous medium.

Power of the cornea
The cornea has power 6.Refraction at the cornea Using the procedure given in Equations ( 1)-(4b), the reduced object vergences and corneal power are combined to give reduced image vergences for the cornea: and with modification to ′ l;2 as necessary according to the process described previously.Note here that ′ l;2 is used as it will be the axis when passing back to the spectacle lens.

Transfer to the intraocular lens
The reduced image vergence at the intraocular lens is given along ′ l;2 by and along � l±90;2 by 8. Power of the correcting spectacle lens (the final refraction) The reduced image vergences correspond to the power of the spectacle lens so that its sphero-cylinder representation is F l;2 ∕ C l;2 × l;2 , where and Question 1: Change from refraction 1 to refraction 2 The difference Δ S ∕ ΔC × between the two spectacle lenses can be found from the following: where d c2;1 is the distance between the cornea and IOL, and n a is the refractive index of the aqueous medium following the cornea.The reduced image vergence at IOL 2 ;1 follows as similar form as those for IOL;2 at Equations (30a)-(31b) for point 1.Along ′ IOL tilt2;1 , this is and along � IOL tilt ± 90; 1 it is The tilted (effective) power of IOL 2 ;1 is the difference between reduced image vergence and reduced object vergence: This has components as follows: The refraction components are converted to tilted IOL power is in sphero-cylindrical format as F IOL tilt2;1 ∕ C IOL tilt2;1 × IOL tilt2;1 , where and with modification to IOL tilt2;1 as necessary according to the process described previously.The final task is to determine the untilted power of IOL 2 ;1 for the specified tilt.The IOL has a refractive index of μ 2 ;1 and tilt ϕ 2 ;1 about axis θ 2 ;1.The tilted power vector f is A matrix M 2 ;1 tilted by angle 2 ; 1 about axis 2 ; 1 is given by The matrix that is the inverse to M 2 ;1 can be determined, and the untilted power vector fo is now with components IOL 2 ;1 has power in sphero-format given by where (66a)

1
Left: Raytracing schema for Question 1 showing (top) forward raytracing to the position of IOL;2, and (bottom) backward raytracing from IOL;2.Right: Raytracing for Question 2 showing (top) forward raytracing to the position of IOL 2 ;1, and (bottom) forward raytracing for IOL 2 ;1.R and T indicate refraction and transfer, respectively.IOL, intraocular lens.

F
Left: Optical coherence tomography (OCT) image of the original intraocular lens (IOL) along the 55° meridian; angle of tilt relative to the visual axis is 12°.Right: OCT image of the replacement IOL along the 42° meridian; angle of tilt relative to the visual axis is 7°.AC K N O W L E D G E M E N T S Heidelberg Engineering provided software modules for use with the Heidelberg Anterion.The authors thank Steve Thomson and Richard Cromwell from Heidelberg Engineering, USA.Open access publishing facilitated by Queensland University of Technology, as part of the Wiley -Queensland University of Technology agreement via the Council of Australian University Librarians.CO N F L I C T O F I N T E R E S T S TAT E M E N T David Atchison: Heidelberg Engineering provided software modules for use with the Heidelberg Anterion.David Cooke: None.O R C I D David A. Atchison https://orcid.org/0000-0002-3099-6545