Determining axis misalignment and power errors of toric soft lenses

Clinical Article

Determining Axis Misalignment and Power Errors of Toric Soft Lenses Richard

G. Lindsay, BScOptom, MBA, FAAO, DipCL, Adrian S. Bruce, BScOptom, PhD, FAAO, Noel A. Brennan, PhD, MScOptom, FAAO, and Michael J. Pianta, MScOptom

This article describes a method of determining the misalign ment of a toric soft kns, by use of thepatient’s ocular refraction

as well a the refraction obtained over the mislocatinglens. Matrix optics are used to facilitate the calculation of the effective back vertex power of the toric lenson the eye. On the basisof this calculation, the degreeof axis mislocation can be identified along with any errors in the power of the manufactured lens. The appropriateformulae are outlined and incorporated into a spreadsheet program that can be easilyapplied in clinical practice. Additionally, a Java appkt for this purposeis accessible on the Internet at (http://www,optometry.unimelb,edu.auljawa/ ophthalmicak.html). 0 EIsevier ScienceInc. 1997 Keywords: Toric soft lenses; oblique crossed cylinders; overrefraction; matrix optics

Introduction Over the last decade, notable advances in toric soft lens technology have been made such that the correction of astigmatism with soft lenses is now a viable option for most contact lens patients with significant degrees of astigmatism. Innovations in lathing technology and lens design, especially with regard to providing a stable axis orientation on the eye, have greatly improved the reproducibility of these lenses and generally led to a much higher success rate in their fitting.’

Address reprint requests to Richard Lindsay, Victorian College of Optometry, Cnr Keppel and Cardigan Streets, Carlton VIC 3053, Australia. Accepted for publication June 19, 1997. ICLC, Vol. 24, May/June, 1997 0 Elsevier Science Inc. 1997 655 Avenue of the Americas, New York, NY 10010

A common problem that the contact lens practitioner confronts in prescribing toric soft contact lenses is determining the extent of lens misalignment if lens rotation is observed. Hanks and Weisbarth showed that, on average, toric soft lenses will tend to rotate nasally by about 5 to lo”, where nasal rotation is classed as rotation toward the nose with respect to the inferior aspect of the lens.2 They also showed, however, that there was significant variability between toric soft lens wearers in the actual amount and direction of lens rotation, due to such factors as the patient’s lid anatomy, the thickness profile of the lens, and the fitting relationship between the lens and the eye. Calculation of the effective back vertex power of the contact lens on the eye, by using the spherocylindrical refraction obtained over the mislocating toric soft lens, is one technique that can be used to calculate the degree of lens misalignment. Using the graphical method of resolving obliquely crossed cylinders (Stokes construction),3 Dain4 has previously shown that overrefraction can be used to deduce the degree of axis mislocation in toric soft contact lenses. This may be obtained by comparing either the cylinder axis in the overrefraction with the required cylinder axis or the overrefraction cylinder power with the contact lens cylinder power. Both of these techniques had the limitation of assuming that the mislocated lens was in error only in its axis, although a difference in results obtained by the two methods did indicate an incorrect power for the

prescribedlens. Thomson’s drical lenses’

formulae for obliquely crossed spherocylincan also be used to determine the degree of 0892-8967/97/$17.00 PII SO892-8967(97)00035-7

Chid

Article

toric lens misalignment.5 Application of these formulae is made easier by the use of matrix methods. Using the matrix formulations outlined by Long6 and Keating,7 we will reveal in this article a simple technique for determining the effective contact lens power of a mislocating toric soft lens by consideration of both the spherocylindrical refraction over this lens and the patient’s ocular refraction. This algorithm can easily be incorporated into a spreadsheet. A Java applet is also described to facilitate calculation via the Internet.

Theory For a patient to be corrected appropriately with a toric soft contact lens, the back vertex power of the contact lens on the eye (BVP, siru) should be equal to the patient’s refraction at the ocular plane (Oc Rx). If a toric soft lens does not exhibit any rotation when placed on the eye, then the specified back vertex power will be the same as the BVP,, situ, provided that there is no significant tear lens formed under the soft lens and that the degree of lens flexure is negligible.6 This is also assuming that the lens has been made to specification. Where lens rotation is observed, the cylinder axis of the BVP, sinr will differ from the cylinder axis specified in the contact lens prescription by an amount equal to the degree of lens rotation. The specified cylinder axis should incorporate an allowance for this rotation, to ensure that the cylinder axis of the lens on the eye will coincide with the cylinder axis of the ocular refraction. When allowing for nasal rotation in the right eye, the amount of rotation should be subtracted from the required cylinder axis and vice versa for the left eye. When allowing for temporal rotation in the right eye, the amount of rotation should be added to the required cylinder axis and vice wersa for the left eye. The acronym “LARS” (left add, right subtract) for nasal rotation can be quite useful. Alternatively, clockwise rotation necessitates adding the allowance for rotation to the required cylinder axis and counterclockwise rotation requires subtracting the allowance for rotation to determine the final cylinder axis.1 For example, consider a toric soft lens being fitted to a left eye. On the basis of the ocular refraction, the required back vertex power of the contact lens is - 2.00/- 1 .OO X 70. No allowance is made for any lens rotation. When this lens is placed on the left eye, it is observed to rotate 10” nasally (i.e., clockwise). As a result, the cylinder axis of the BVP,, situ will be 60. Consequently, if we wish to make allowance for this 10” nasal rotation, the specified back vertex power must be - 2.00/- 1 .OO X 80. The expected nasal rotation of 10” will then move the cylinder axis of the lens around to axis 70, hence resulting in a BVP, siru of -2.00/-1.00 X 70. Conversely, if the same toric soft lens (BVP of -2.00/ - 1 .OO X 70) is fitted to a right eye with an ocular refraction of -2.00/1.00 X 70, nasal (i.e. counterclockwise) rotation will result in the cylinder axis of the lens now being along 80. In this case, if we wish to allow for this 10” nasal

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rotation, the specified back vertex power must now be -2.00/1.00 X 60. The expected nasal rotation of 10” will then move the cylinder axis of the lens around to axis 70, hence resulting in the required back vertex power of -Z.OO/ -1.00 x 70. When a toric soft lens axis mislocates on the eye, the BVP, siru will be different than that required. Significant cylindrical error can be induced as a result. For example, with a toric soft lens incorporating a cylindrical correction of -2.OODC X 180, a mislocation of the axis by 10” results in a cylindrical error of -0.69DC X 40. A 20” movement gives a cylindrical error of -1.37DC X 35, and a 25” rotation gives a cylindrical error of - 1.65DC X 32.5.8 The combination of the BVP, siru and the spherocylindrical overrefraction with this lens (OR) will be equal to the patient’s ocular refraction (Oc Rx).

RX

BVP,, ,i,,+OR=Oc This formula

can be rearranged

to solve for BVP,

siru

BVP,” situ=O~ RX-OR The BVP, situ will not only indicate if the lens is misalign ing, but also if the lens has been made to the correct specifications.

Method Given the overrefraction with the lens and the patient’s ocular refraction, we can resolve these obliquely crossed cylinders and calculate BVP,, sinr using matrix optics and the following method: 1. Express both the spherocylindrical ocular refraction and the spherocylindrical overrefraction in dioptric power matrix form. The matrix formulation for obliquely crossed spherocylindrical lenses uses the dioptric power matrix F, in a Cartesian coordinate system, in which a spherocylindrical lens is expressed using Long’s formula6: S+C sin28 -C sin0 cos0 F=J -C sin0 cos0 S+C cos281 where S is the sphere power, C is the cylinder and 8 is the axis (in radians) of the cylinder.

power,

2. Subtract the dioptric power matrix for the overrefraction from the dioptric power matrix for the ocular refraction, to obtain the dioptric power matrix, F,, for the BVP,, sit,,. F,=l

S,+C, sin28, -C, sine, cos0,

3. Convert the matrix form spherocylindrical notation lined by Keating7:

-C, sin& cos0, s,+c, cos2flr 1 of the BVP, sinr back to using the formulae out-

Assessing Toric Soft Lenses: Lindsay et al. If the lens power

matrix

is 1:::

Exam@

zinl

then trace (t)=al,+azz determinant

(d) = (allazz) - (alzazl)

To convert the matrix form of the BVP, sinr back to spherocylindrical notation, S,, C,, and 0, (the sphere power, cylinder power, and cylinder axis, respectively, using the trace of the BVP,, situ) can be determined and determinant properties of this dioptric power matrix. s =(t-c,) T

8 =a tan (.%-all) I

al2

180

X7

2

(where

8, is in degrees)

c,= - Jtz-4d (The minus sign before the radical symbol simply means that the final solution will be in minus cylinder form.) Application of this method use of two examples.

will now be demonstrated

by

Example 1: Toric Lens Misalignment Consider a toric soft lens being fitted to a patient’s kft eye. The ocular refraction is -3.00/-2.00 X 180 (spectacle refraction for the eye taken back to the ocular plane). Consequently, the BVP,, situ of the contact lens should equal -3.00/-2.00 X 180. A decision is made to allow for 10” nasal rotation, so the specified contact lens back vertex power is -3.00/-2.00 X 10. This lens is placed on the eye. Refraction with this lens in situ gives +OSO/1.00 X 37.5. Using the formula

2: Toric Lens Misalignmentand Incorrect Power

This method also has the advantage of being able to identify and quantify the error when a toric soft lens has not been manufactured with the correct power. Consider a toric soft lens being fitted to a patient’s right eye. The ocular refraction is - l.OO/--2.00 X 180 (spectacle refraction for the eye taken back. to the ocular plane). The BVP,, situ of the contact lens should therefore equal - 1 .OO/- 2.00 X 180. A decision is made to allow for 5” nasal rotation, so the specified contact lens back vertex power is - l.OO/ -2.00 X 175. This lens is placed on the eye. Refraction with this lens in situ gives +0.25/1.00 X 155. Solving for BVP, s,fu gives

(It actually gives -0.97/1.56 X 14.7”, but this is rounded off to the above.) The cylinder axis ordered was 175; therefore, the lens is exhibiting 20” nasal rotation on the eye (instead of the expected 5”). In addition, the effective back vertex power reveals that the cylinder power is only 1.50 D and not 2.00 D, as was originally specified. A spreadsheet design (with appropriate formula displayed) for determining the axis misalignment and power error of toric soft lenses is shown in Table 1. The application of this spreadsheet for Examples 1 and 2 is presented in T&es 2 and 3, respectively. Alternatively, these calculations can be performed simply via a Java applet that we have written and made available on the Internet. The following address may be accessed via a Java compatible browser, such as Netscape www.optometry.unimelb.edu.au/java/ version 3.0: (http:// ophthalmicalc.html). The ocular refraction, specified contact lens back vertex power, and overrefraction are entered into the program to give the BVP,, situ, as well as details regarding what back vertex power should actually be ordered on the basis of the calculated lens rotation. The interface of this applet is shown in Figure 1. One feature of the program is that rotation of the lens is described in terms of nasal/temporal according to whether right or left eye is selected.

BVP,, Sitl = Oc Rx-OR

Discussion

BVP,,,i,,=(-3.00/-2.00X180)-(+0.50/-1.00 x37.5) Resolving

by matrix

methods

gives

BVPinsrtu=-3.00/-2.00X

165

The specified cylinder axis was 10”; therefore, the lens is exhibiting 25” nasal rotation on the eye (instead of the expected 10’). To allow for this 25” nasal rotation, the contact lens would now have to be ordered with a cylinder axis of 25.

This article has demonstrated how the effective back vertex power of a toric soft lens on the eye can be determined by subtracting the refraction obtained over this lens from the ocular refraction of the patient. Any , degree - of axis mislocation can then be identified along with any errors in the power of the manufactured lens by comparing the back vertex power in situ with the specified back vertex power. Although this method of determining toric lens mislocation has obvious benefits for the clinician, it does require that some assumptions are made. These include no significant tear lens under the lens, a negligible degree of lens

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Table 1. Spreadsheet (with Formulae Displayed) for Determining A 1 2 3 4 5 6 7 8 9 10 11 12

Oc Rx MATRIX OR MATRIX SUM TRACE DET

13 BVPin situ 14 15

C

B SPHERE

CYLINDER

:;3+C3*(sIN(E3)~2) = -C3*SIN(E3)*COS(E3) 0.5 =B~+C~*(SIN(E~)A~) = -C6*SIN(E6)*COS(E6) =B4-B7 =B5-B8 =B9+ClO =(B9*ClO)-(BlO*C9)

z2C3*SIN(E3)*COS(E3) =B~+C~*(COS(E~)A~) -1 = -C6*SIN(E6)*COS(E6) =B~+C~*(COS(E~)A~) =c4-c7 =C5-C8

=-SQRT((Bllh2)-4*B12)

=(Bll-C13)/2

Table 2. Application

B SPHERE

Oc Rx MATRIX

-3.00 -3.00 0.00 0.50 0.13 0.48 -3.13 -0.48 -8.00 15.01 -3.00

OR MATRIX SUM TRACE DET BVPin situ

flexure, performance of accurate refractions (ocular and that over the contact lens), and the cylinder axis of the lens being made as specified. The presumption of the cylinder axis being made to specification is extremely important. In Example 1, the lens was assumed to be exhibiting 25” nasal rotation based on a specified cylinder axis of 10 and a cylinder axis in situ of 165. Obviously, if the specified cylinder axis is incorrect, then the estimated degree of lens rotation will be wrong. Consider if the lens had been made with the cylinder axis at 5, instead of 10. In this case, the lens would really only be rotating nasally by 20” and so allowance for 25” nasal rotation would lead to an inappropriate back vertex power on the eye. The clinician may attempt to overcome this problem by assessing lens rotation according to the position of the reference marker that the laboratory usually places

104

and Power Error of a Toric Soft Lens E

D AXIS 180

=D3/57.2958

37.5

=D6/57.2958

=IF(57.2958*ATAN((Bl3-B9)/C9)>0, 57.2958*ATAN((B13-B9)/C9), 180+57,2958*ATAN((B13-B9)/C9))

of the Spreadsheet in Example 1 A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

the Axis Misalignment

ICLC, Vol. 24, May/June

1997

C CYLINDER

D

E

AXIS

-2.00 0.00 -5.00 -1.00 0.48 -0.13 -0.48 -4.87

180.0

3.14

37.5

0.65

-1.99

165.5

on the lens. However, this assumes that the reference marker has been placed in the correct position on the lens and that the practitioner is able to estimate, with some degree of precision, the amount of lens rotation. Although the technique outlined in this article cannot determine for certain if the lens has been manufactured with the appropriate cylinder axis, it does allow the practitioner to recognize when a toric soft lens has been made up in a power other than that specified. In Example 2, application of this method revealed the lens to have a cylinder power of 1.50 D and not 2.00 D, as was originally specified. A useful rule-of-thumb is that a lens made to specification but mislocating on the eye will produce an overrefraction with a spherical equivalent equal to zero. Where the sphere or cylinder power is also incorrect, the spherical

Assessing Toric Soft Lenses: Lindsay et al. Table

3. Application

of the Spreadsheet in Example 2 A

1

2 3 4 5 6 7

8 9 10 11

12 13 14 15

B SPHERE

Oc Rx MATRIX OR MATRIX

-1.00 -1.00

0.00 0.25 0.07

-0.38 SUM

-1.07

TRACE DET BVPin situ

0.38 -3.50 2.46 -0.97

1. Interface of the Java applet for calculating soft toric lens misalignment (showing the calculated BVP,, cIiu for Example 2). The applet can be easily accessed at the web-site (http://www. o~tometry.unimelb.edu.au/java/ophthalmicalc.html) Figure

equivalent of the overrefraction will not equal zero. This rule was noted by Long.’ It arises because the equivalent spherical powers of the ocular refraction and toric contact lens are equal and opposite, irrespective of lens axis. In Example 1, the spherical equivalent of the overrefraction (+OSO/1 .OO X 37.5) was zero, indicating that the misaligned lens had otherwise been made to the correct specifications. In Example 2, the spherical equivalent of the

C CYLINDER

-2.00

D

E

AXIS

180.0

3.14

155.0

2.71

0.00

-3.00 -1.00

-0.38 -0.57 0.38 -2.43 -1.56

14.7

overrefraction (+0.25/1.00 X 155) was not equal to zero, hence suggesting (and confirmed by calculation of the BVP,” J that the toric lens did not incorporate the specified powers. A second rule-of-thumb is that the direction of lens misalignment is always opposite to the axis of the overrefraction, relative to the prescribed cylinder axis. Consider Example 1. The overrefraction gave a cylinder axis of 37.5, which when compared with the prescribed cylinder axis (lo), was in the opposite direction to the effective cylinder axis on the eye (165). Intuitively, one might make the mistake of udding the overrefraction to the specified contact kns back vertex power to determine the required lens prescription. Although this method is appropriate for a spherical soft lens, it will give a totally spurious finding for a toric soft lens if there is an error in lens alignment. Once again, consider Example 1. Adding the overrefraction (+OSO/1.00 X 37.5) to the specified contact lens prescription (-3.00/-2.00 X 10) gives a totally meaningless answer of -2.65/-2.70 X 19. Clinicians should note that the overrefraction may not give an exact indication of the amount of lens rotation, because refractions are generally only performed to an accuracy of 0.25 D. For example, a toric soft lens incorporating a 1.00 D cylinder that has been made to the correct specifications but mislocates by 20” will give an overrefraction of +0.34/-0.68 X 35. However, in a clinical situation, the practitioner will probably obtain an overrefraction of +0.25/-0.50 X 35, indicating a lens misalignment of about 15”. One possible way to minimize this error would be to perform the overrefraction to an accuracy of 0.12 D. In this case, the overrefraction achieved would be +0.37/ -0.75 x 35, giving a more accurate figure (22”) for the degree of lens misalignment. For cylinder powers of lesser magnitude (~1.00 D), this error is probably not that significant. For higher cylinder powers, where axis alignment

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15 critical, the practitioner should be aware of the need to perform ah accurate an overrefraction as possible.

References 1. Lindsay KG, Westerhout IN. Toric contact lens fitting. In: Phillips AJ, Speedwell L (Eds.): Contact Lenses. 4th Ed. London, Butterworth-Heinemann, 1997. 2. Hanks AJ, Weisbarth RE. Troubleshooting soft toric contact lenses. ICLC 1983;10:305-317. 3. Jalie M. The Princi@s of Ophthalmic Lenses. 4th Ed. London, Assoc Disp Opticians, 1984.

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4. Dain SJ. Over-refraction and axis mislocation of toric lenses. ICLC 1979;6:57-61. 5. Bergenske PD. A guide to prescribing toric lenses (part 2). Contact Lens Spect 1996;11:3@-36. 6. Long WF. A matrix formalism for decentration problems. Am J Optom Physiol Opt 1976;53:?7-33. 7. Keating MI’. An easier method to obtain the sphere, cylinder, and axis from an off-axis dioptric power matrix. Am J Optom Physiol Opt 1980;57:734-737. 8. Holden BA. The principles and practice of correcting astigmatism with soft contact lenses. AtLst J Optom 1975;58:279299. 9. Long WF. Lens power matrices and the sum of equivalent spheres. Ootom Vis Sci 1991;68:821-822.

Assessing Toric Soft Lenses: Lindsay et al.

Richard Lindsay, BScOptom, MBA, graduated from the University of Melbourne in 1984 and is head of Contact Lens Clinics at the Victorian College of Optometry, where he also runs his own specialty contact lens practice. He is also an Academic Associate in the Department of Optometry and Vision Sciences at the University of Melbourne, where his major responsibilities include didactic and clinical teaching in the contact lens course. Mr. Lindsay is a member of various optometric and scientific groups including the Victorian College of Optometry and the National Vision Research Institute. He is a Fellow of the American Academy of Optometry and a Diplomate of its Cornea and Contact Lens Section. He is currently Treasurer and on the Executive Board of the International Association of Contact Lens Educators. He is a Past-President and a Founding Fellow of the Contact Lens Society of Australia. Dr. Adrian Bruce is a senior optometrist at the Victorian College of Optometry, Melbourne, Australia. He obtained a BScOptom degree from the University of Melbourne in 1985 and a PhD in 1991 and was a research fellow at Queensland University of Technology, in Brisbane, Australia from 1991 to 1995. The second edition of the handbook A Guide To Clinical Contact Lens Management, published by CIBA Vision International, was published in 1996 and is being widely distributed. He is a member of Melbourne Ocular Science & Technology (MOST) Enterprises, the Contact Lens Society of Australia, and the International Society for Contact Lens Research and is a Fellow of the American Academy of Optometry. Dr. Brennan is a former Associate Professor and Reader at the University of Melbourne. During his time there, he was awarded the Australian-American Education Foundation Senior Fulbright scholarship. In 1995, he established a private research company, Brennan Consultants Pty Ltd, which provides services to the contact lens and associated industries. More recently, he has been a founding partner of the joint venture company, Melbourne Ocular Science & Technology (MOST) Enterprises, which conducts large-scale clinical trials. His refereed scientific and clinical publications number in excess of 100. He is a Fellow of the American Academy of Optometry, Councillor of the International Society for Contact Lens Research, and a Founding Fellow of the Contact Lens Society of Australia. Mr. Michael Pianta obtained a BScOptom degree from the University of Melbourne in 1991 and an MSc in 1994 and was a Senior Tutor at the University of Melbourne from 1995 to 1996. He is a member of the Association for Research into Vision and Ophthalmology and is a Fellow of the Victorian College of Optometry.

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