How to Ensure Accuracy With Toric Soft Lens Prescriptions

BY DALE M KOERS, O.D., M.S., & THOMAS G. QUINN, O.D., M.S., F.A.A.O.
JAN. 1997

Preprogrammed calculators are useful tools to help you determine toric soft contact lens prescriptions. Accounting for lens rotation will help improve your accuracy.

Although you didn't realize it at the time, you learned the basic theory for fitting toric soft contact lenses in your high school trigonometry class. The Law of Cosines and the Law of Sines are used in the pre-programmed calculators provided by Sunsoft and CooperVision. These manufacturers claim you can get an accurate toric soft contact lens prescription if you input the labeled power of the diagnostic lens and the overrefraction with the diagnostic lens.

Both these companies recommend that you not compensate for diagnostic lens rotation, while other toric soft lens manufacturers say that you should. Who is correct? And can these high-tech preprogrammed calculators really give you an accurate prescription? Call on your high school trigonometry for the answer.

Preprogrammed calculators use basic trigonometric equations, so it's easy to check their accuracy. The following steps will show that you must account for lens rotation before and after inputting the values for the diagnostic lens and the overrefraction to arrive at an accurate prescription.

STEP 1: CALCULATING OVERREFRACTIONS

A patient's eye is an optical system that can be represented by a single optical lens. The eye's optical power and the diagnostic lens power can be combined to form an equivalent single lens called the resultant. If the patient is a -3.00D myope with a -2.00D diagnostic lens, you would calculate the resultant: +3.00D sphere (optical power of eye) combined with -2.00D sphere (diagnostic lens) = +1.00D sphere (resultant). The overrefraction, the lens power that neutralizes the resultant, is a -1.00D sphere.

At first, this may seem like a roundabout method to obtain the value of an overrefraction. However, calculating overrefractions is the key to proving the need to compensate for rotation when using preprogrammed calculators.

Finding the resultant of two spherical lenses is easy, but finding the resultant of two spherocylinder lenses requires more than simple addition. Spherocylinder lenses are perfectly suited for a special type of addition called vector addition. Vectors represent quantities that have both magnitude and direction. The cylinder power of the lens is the magnitude and the cylinder axis of the lens is the direction. The Law of Cosines formula simply 'adds' or combines two spherocylinder lenses (vectors) to find the resultant cylinder power. The Law of Sines formula calculates the axis of the resultant cylinder power.

Using the Law of Cosines and the Law of Sines, we can calculate the resultant and hence the overrefraction for any eye/toric soft lens combination. For example, if we fit a simple myopic astigmat whose vertexed spectacle prescription is plano -3.00 x 180 (optical power equal to plano +3.00 x 180) with a rotating diagnostic lens that has a labeled power equal to plano -2.00 x 180, we can either clinically measure or calculate an overrefraction. To calculate the overrefraction, assume that the diagnostic lens is accurately labeled, there is no lens flexure or lacrimal lens effect, and the diagnostic lens consistently rotates 20 degrees left when on the patient's eye (i.e., the bottom of the lens rotates to the doctor's left). The resultant is: plano +3.00 x 180 (optical power of eye) combined with plano -2.00 x 160 (rotated diagnostic lens) = -0.48 +1.95 x 021. (resultant).

The calculated overrefraction with the rotated diagnostic lens is: +0.48 -1.95 x 021. Use this overrefraction to check the calculator's accuracy. Enter the diagnostic lens power and the calculated overrefraction into the calculator. Since the lens rotates left 20 degrees on the eye, we know that the ideal prescription (and the answer the calculator should give us) is plano -3.00 x 020.

STEP 2a: DON'T COMPENSATE FOR ROTATION

We'll start by not compensating for rotation, per the manufacturer's recommendations.

Patient's spectacle Rx plano -3.00 x 180

Diagnostic lens plano -2.00 x 180

Diagnostic lens after 20°

left rotation plano -2.00 x 160

Overrefraction with rotated

diagnostic lens (from step 1) +0.48 -1.95 x 021

Preprogrammed calculator input: plano -2.00 x 180 (diagnostic contact lens with no compensation for rotation) combined with +0.48 -1.95 x 021 (overrefraction) = +0.35 -3.69 x 011 (toric soft contact lens order).

The finding of +0.35 -3.69 x 011 has little resemblance to the plano -3.00 x 020 we know the final contact lens power should be.

STEP 2b: COMPENSATE FOR ROTATION

Let's try again, but this time we'll compensate for rotation of the diagnostic lens.

Patient's spectacle Rx plano -3.00 x 180

Diagnostic lens plano -2.00 x 180

Diagnostic lens after 20°

left rotation plano -2.00 x 160

Overrefraction with rotated

diagnostic lens (from Step 1) +0.48 -1.95 x 021

Preprogrammed calculator input: plano -2.00 x 160 (diagnostic lens with compensation for rotation) combined with +0.48 -1.95 x 021 (overrefraction) = plano
-3.00 x 180 (toric soft contact lens order).

The finding of plano -3.00 x 180 gives us the expected power, but the axis is incorrect.

In both Steps 2a and 2b, the calculator gives an incorrect prescription. Some might argue that the prescriptions are 'close enough' and that the patient will see clearly with either lens. To determine if this is true, calculate the overrefractions with the lenses ordered from Steps 2a and 2b.

STEP 3: CALCULATE THE OVERREFRACTIONS

The overrefraction using the prescription from Step 2a will be:

Patient's spectacle Rx: plano -3.00 x 180

Preprogrammed calculator

contact lens Rx (Step 2a) +0.35 -3.69 x 011

Contact lens Rx after 20°

left rotation +0.35 -3.69 x 171

Plano +3.00 x 180 (optical power of eye) combined with +0.35 -3.69 x 171 (rotated contact lens Rx) = -0.62 +1.25 x 057 (resultant).

The overrefraction for the prescription obtained with a preprogrammed calculator after 20 degrees left rotation is: +0.62 -1.25 x 057 (overrefraction).

The overrefraction using the toric soft contact lens prescription from Step 2b will be:

Patient's spectacle Rx: plano -3.00 x 180

Preprogrammed calculator

contact lens Rx (Step 2b) plano -3.00 x 180

Contact lens Rx after 20°

left rotation plano -3.00 x 160

Plano +3.00 x 180 (optical power of eye) combined with plano -3.00 x 160 (rotated contact lens Rx) = -1.03 +2.05 x 035 (resultant).

The overrefraction for the prescription obtained with the calculator after 20° left rotation is: +1.03 -2.05 x 035 (overrefraction).

The calculated overrefractions resulting from the prescriptions we obtained in both Steps 2a and 2b are unacceptable. As you can see, it doesn't matter if you compensate for rotation of the diagnostic lens before entering the lens power into the calculator because the calculator doesn't compensate for rotation after it performs the calculation. The Sunsoft and CooperVision calculators cannot determine the ideal lens order when used according to the manufacturers' recommendations.

Table 1 lists toric soft contact lens prescriptions and corresponding overrefractions for other amounts of diagnostic lens rotation. In most cases, the overrefraction is unacceptable.

THE FINAL STEP: LARS

In each case, Step 2b gives the correct power, but the incorrect axis. However, you can determine the ideal prescription by simply applying LARS to the calculator's resultant from Step 2b. If the initial diagnostic lens rotates five degrees left, add five degrees to the resultant in Step 2b. Similarly, if the initial diagnostic lens rotates 20 degrees left, add 20 degrees to the resultant in Step 2b.

DISCUSSION

The ideal toric soft lens prescription is one that completely neutralizes the eye's optical power. If there is no lens rotation, lens flexure or lacrimal lens effect, the contact lens prescription should equal the vertexed spectacle prescription.

Preprogrammed calculators have become increasingly popular because there is no method to directly measure the amount of lens flexure, or the lacrimal lens effects. The manufacturers of these calculators claim that the diagnostic lens overrefraction completely compensates for lens rotation, lens flexure and lacrimal effect. This is true, but there are two very important considerations that must be addressed.

First, the Law of Cosines and the Law of Sines can mathematically calculate the resultant of any two obliquely crossed spherocylinder lenses, but only if the correct values for the cylinder power and axis are used. If a diagnostic lens rotates on the eye, the lens has a new axis value, and this value must be used to calculate the resultant, i.e., you must compensate for lens rotation. In our example, the diagnostic lens has a labeled axis of 180 and rotates 20 degrees left. On the eye, this lens is now a spherocylinder lens with axis 160, and must be entered into the calculator as such. The new axis for the diagnostic lens on the eye can be found by using RALS (Right Add Left Subtract), the reverse of LARS.

Second, the resultant obtained by a preprogrammed calculator is not the power of the lens prescription. It is a new value of the eye's optical power that includes any lens flexure and/or lacrimal lens effect. You must use LARS on the calculator's resultant to compensate for the expected lens rotation.

In our example, the patient's spectacle prescription is plano -3.00 x 180. If the contact lens rotates 20 degrees left, add 20 to 180 for a prescription of plano -3.00 x 020. After rotation, the major optical meridians of the LARS calculated lens will align with the eye's major optical meridians.

In each case, when we compensate for rotation of the diagnostic lens (RALS), the formulae give a resultant equal to the optical power of the eye. You must use LARS on the calculator's resultant to compensate for the expected rotation of the lens on the eye.

TRIGONOMETRY REFRESHER The Law of Cosines states: The square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides times the cosine of the angle between them. c2 = a2 + b2 - 2ab cos * Other forms of the Law of Cosines are: a2 = b2 + c2 - 2bc cos * b2 = a2 + c2 - 2ac cos ß If *, * or ß = 90°, the Law of Cosines becomes the Pythagorean theorem since cos 90° = 0. If a and b represent the powers of the two obliquely crossed cylinder lenses, we can use the Law of Cosines to find the resultant power c and the Law of Sines to find the axis of the resultant cylinder power. To calculate the resultant of two obliquely crossed spherocylinder lenses, we must rearrange the triangle and modify the Law of Cosines formula. To find the resultant c, we must know the angle *. However, we only know the value of (180 ­ *), which is the angle between the two obliquely crossed cylinders. Since cos * = ­ cos (180 ­ *), the expression c2 = a2 + b2 ­ 2ab cos * becomes c2 = a2 + b2 + 2ab cos (180 ­ *). Let * = (180 ­ *), the angle between the axis of the two obliquely crossed cylinders. Second, multiply the angle * by two since we only use 180 degrees. Combining both modifications, the formula used in the Sunsoft and the CooperVision preprogrammed calculators to determine the resultant power value of two obliquely crossed cylinders becomes: c2 = a2 + b2+ 2ab cos 2* where a and b are the power values of the obliquely crossed cylinders, and * is the angle between their axes. The calculators then use the Law of Sines to calculate the axis of the resultant cylinder power: sin 2* = b/c(sin2*) where * is the angle between the first cylinder axis and the resultant cylinder axis. The additional spherical component, S, resulting from the combination of two obliquely crossed cylinders is: S = (a+b-c)/2 You can combine spherocylinder lenses in your lensometer to obtain a resultant that is equal to the result obtained from the preprogrammed calculators.

To obtain the exact toric soft lens prescription using a preprogrammed calculator:

1. Compensate for the rotation of the diagnostic lens using RALS.

2. Input the compensated axis and overrefraction into the calculator.

3. Use the calculator's output and compensate for lens rotation using LARS.

NEW METHOD ENSURES SUCCESS

You may have had some success with preprogrammed calculators because Sunsoft and CooperVision Toric lenses rotate very little. The calculator's failure to compensate for rotation is what introduces the error. If there is minimal rotation and the cylinder power is not large, the preprogrammed calculators work fine. In fact, an advantage is that they take into account lens flexure and lacrimal lens effects that may alter the lens power needed to precisely correct the eye's refractive error.

We now have a method that allows us to take advantage of the calculator's strengths and to correct for its weaknesses. CLS

Dr. Koers is an assistant professor at the Southern California College of Optometry and is in private practice in San Diego. Dr. Quinn, a contributing editor with Contact Lens Spectrum, has served as a faculty member and research associate at The Ohio State University College of Optometry. He is in group practice in Athens, Ohio.