I’ve done 10th grade math 3 times: a couple of years ago with my daughter, recently with my son, and many years ago myself. And I can tell you that it certainly becomes easier the third time around. One of my favorite math topics is the study of vectors, which (as you recall), involve both a direction and a magnitude. The classic problem is to determine the effect of a crosswind on the path and speed of an airplane. It turns out that astigmatism is also a vector with a specific direction and a magnitude.
The refractive astigmatism of the eye is a combination of both the corneal astigmatism and the lenticular astigmatism. When looking to neutralize astigmatism at the time of cataract surgery, we are primarily looking at the corneal astigmatism, because the crystalline lens will be removed and replaced with an intraocular lens (IOL) implant. If the corneal astigmatism is not significant, we can choose a non-toric IOL which will only address the refractive spherical correction, assuming the IOL is well-positioned and not tilted within the eye. A toric IOL can be selected to offset the existing corneal astigmatism and address the refractive spherical correction, resulting in an excellent visual outcome.
The challenge, however, is that our incisions, particularly our main corneal or limbal phaco incisions, will change the astigmatism. Every incision in the cornea will induce a degree of flattening at that axis, and an overly tight suture will induce steepening instead. Small incisions, such as our paracentesis incision which tends to be 1 mm or less in width, tend to have an insignificant change to the corneal astigmatism. But our phaco incisions are approximately 2 to 3 mm in width, and they can induce anywhere between 0 and 2 diopters of astigmatism, with most having about 0.5 diopters of flattening. The exact astigmatic effect of our incision cannot be accurately predicted because there are so many other factors such as corneal pachymetry, corneal elasticity, patient age, and corneal diameter, which can give a variable tissue response to the exact same incision.
This is where going back to the airplane vector problem is helpful. Imagine that our plane encounters only a perfectly aligned headwind or tailwind, with no crosswinds. This will make the calculation much easier, since we know for certain that the plane will stay on course and the only difference will be the speed, which will be boosted by the tailwind or slowed by the headwind. The magnitude of the vector changes, but not its direction. Applying this to astigmatism means that if we always make our phaco incision on either the steep or flat axis, then we will not change the direction of the astigmatism, we will just change the amount of it. This means that we will be able to consistently align the toric IOL with the correct, steep axis, and if there is any variability in the effect of our phaco incision, it will only affect the magnitude of the astigmatism.
Putting this into practice means that a phaco incision on the steep axis would steer the surgeon toward a lower degree of toric correction, while an incision on the flat axis would mean using a higher degree of toric correction. Incisions that are askew to the steep and flat axes should be avoided because they will change both the magnitude and the direction of the astigmatism. This is particularly important in lower degrees of astigmatism, which is the more common case in our patient population. This concept has helped me to improve my toric IOL outcomes and achieve a higher level of accuracy as well as patient satisfaction. Who knew that 10th grade math would become so important in my daily life?