This is why they say a 1 degree rotation of Toric IOL leads to 3.3% loss in toric correction

Have you ever wondered why they say that for every 1 degree of rotation, there is a loss of 3.3% in toric correction at the cornea plane? Why 3.3%, why not 2%, why not 4%. Where on earth does the 3.3 come from? Here is a breakdown of where the number comes from. Although there is some math involved, I will keep it simple and conceptual.

To start with, in ophthalmology, 0 degree and 180 degrees are considered as the same meridian. Aligning the Toric IOL at 0 degrees in the eye, is the same as placing in the 180 degree. Agree?

But in math, which involves vector analysis, 0 degree is the opposite of 180 degree, and they are not the same. A pilot asked by the air traffic control to follow the zero degree path at a direction of south-east, cannot travel towards 180 degree as it will reach an opposite destination than the intended course of travel.

Surgically induced astigmatism (SIA), is based on vector based math. Centroid SIA, which takes into consideration the effect of incision on both magnitude and axis, is based on vector analysis. For example, a two diopter of corneal astigmatism at 0 degree with an incision at 145 degrees, will involve vector analysis to arrive at the final corneal astigmatism post incision on the cornea.

Therefore, in Ophthalmology,

• 0° and 180° are identical

• 90° is maximally different

So, the system repeats every 180°, not 360°.

This is not acceptable in vector analysis. In the world outside your and my eye, 0⁰ is maximally different to 180⁰. That 180° periodicity mathematically forces a double-angle representation.

To marry the concepts of eye alignment with math, we do a double-angle representation. Every degree is doubled, so that 0 and 180 degrees are represented on the same side of the plot, while 90 degree is at the opposite of the plot (Image 1).

Image 1 Double angle plot. Notice 90 degree is made to locate opposite to 0 degree which is representative of math. To do this, every degree is double such that 90 degree sits opposite to 0 degree. quickguide.org

If this is clear, let us come to our original question, 'Where does the loss in astigmatism correction of 3.3% for every 1 degree of rotation come from? Hmm, if you heard 3.4% or even 3%, that’s OK. You understand my question, right?

Here comes the application of trigonometry. Don’t worry, I will keep it simple, in fact just a one line. The angle of rotation can be represented by Sin(θ). The only reference that I will make is to the trigonometric chart (image 2 - link to the trigonometry calculator) that will help us to arrive at the magic number.

Taking double angle plot into consideration one degree of rotation can be mathematically represented as Sin (2 x angle of rotation). That is, if the lens rotate by 1 degree we double the angle of rotation to match the double angle plot for vector analysis.

Image 2 Above values of Sin (2degrees) To find the Sin (θ) values : https://acesse.one/LSnAf

Therefore, every one degree of rotation of the Toric IOL, we will calculate as Sin(2 x 1 degree of rotation), or Sin (2θ) = .34m= 3.4 % (see image 2)

• Thus a 15° rotation, would mean Sin (2 x 15degrees) = Sin 30 = .50 or 50% loss.

• Thus a 30° rotation would mean Sin (2 x30° ) = Sin 600 = 86%

Short answer: the loss in effective toric correction for a rotation θ is sin(2θ). For θ = 1° this is sin(2°) ≈ 0.0349 → 3.49% (often rounded in practice to 3.3%–3.5%).

I know you are wondering now, that a 30 degree rotation was said to lead to a complete loss in astigmatism correction by the Toric IOL. Where did the 86% come from? Then why do they say that with 30 degrees of rotation there is a complete loss in correction?

Because, at this stage a large oblique component is induced. Imagine a Toric IOL that was supposed to seat at 0 degree, is now sitting at 30°. It will add a large amount of astigmatism at the oblique axis, completely defeating the purpose of implantation that was originally intended to correct the astigmatism at 0 degree. Mathematically it would still correct some negligible amount at 0 degree, but clinically useless as it induces more than it corrects at the oblique meridian. So, in surgical language, clinicians simplify and say 300 is total loss.

The effect of every degree of rotation of a Toric IOL on astigmatism correction quickguide.org