Some Simple Ophthalmology Mathematics

simple ophth math title

Ophthalmology has a large degree of math, particularly when it comes to refractive surgery.  Our most commonly performed and most powerful refractive surgery is cataract surgery, so knowing some basic ophthalmology math is a prerequisite for success.

For treatment of astigmatism with corneal incisions, we often look at the cylinder in plus form (positive values). This is because if we want to make a pair of limbal relaxing incisions (LRIs) in a pseudophakic eye, the axis of the LRIs is the same as the positive cylinder axis of the refraction, and also the same as the steeper corneal axis. A pseudophakic eye with a refraction of -0.50 +1.00 x 90 and K values of 44.00 x 90 / 43.00 x 180 will do well with paired LRIs made at the 90 degree axis.

If we are treating the astigmatism with an excimer laser, then we want to have the cylinder in minus form (negative values). This is because the excimer laser will be removing tissue via laser ablation. The negative cylinder axis of the refraction will correspond to the axis of the negative cylinder excimer laser ablation. An eye with a refraction of 0.00 – 1.00 x 180 will have the negative cylinder laser ablation oriented at 180 degrees.

To convert from + to – cylinder (or – to + cylinder) in the refraction: add the cylinder to the sphere, change the sign of the cylinder (+ to – or – to +), then change the axis by 90 degrees.

We have previously explored the effect of the refraction on the post-operative focal point achieved with cataract surgery. When determining this focal point, we are looking at the spherical equivalent of the refraction. This is calculated by taking half of the cylinder and adding it to the sphere.

convert cyl and SE


For IOL calculations, there is a factor which relates to the effective lens position and also incorporates other variations such as equipment calibration, surgeon technique, and even lens geometry. For most methods of IOL calculation this is referred to as the “A-constant” and it is used by the SRK/T formula, the Ladas Super Formula AI, and more. Other formulas will use different methods such as determining a surgeon factor (SF: used in Holladay 1 & 2) or personalized ACD (pACD: used in Hoffer Q). Most online IOL calculators, biometry machines, and lens calculation software packages will automatically convert between A-contant, pACD, and SF. This is akin to converting between inches, feet, and yards for my US readers (the rest of the world, of course, uses the metric system).

convert Aconst SF ACD


Which Astigmatism measurement? Refraction Cylinder or Keratometry

Which astigmatism value do we use for pre-op cataract patients versus post-op cataract patients? The issue is the human lens, which can help off-set the pre-existing corneal astigmatism. Let’s look at three scenarios:

  • 30 year old patient, wants LASIK: refraction -4.00 -0.25 x 180, keratometry 45.00 x 90 / 44.00 x 180.
    • We treat the -0.25 diopters of astigmatism in the refraction, not the 1.0 diopters of corneal astigmatism.
    • This is because the patient’s crystalline lens (no cataract) is offsetting most of the corneal astigmatism.
  • 70 year old patient, needs cataract surgery: refraction -4.00 -0.25 x 180, keratometry 45.00 x 90 / 44.00 x 180.
    • Here we treat the 1.0 D of corneal astigmatism measured in the keratometry, not the 0.25 diopters measured in the refraction.
    • This is because we will be removing the patient’s crystalline lens (the cataract) which is currently offsetting most of the corneal astigmatism, so we will need to make up for that.
    • a toric IOL would work well to treat this 1 diopter of corneal astigmatism
  • 70 year old patient, already had cataract surgery with a monofocal (non-toric) IOL but didn’t achieve a plano outcome as desired. Current refraction: 0.00 – 1.00 x 150, keratometry 45.00 x 90 / 44.00 x 180.
    • We treat the refraction because we will be keeping the existing IOL.
    • What makes the patient see well in trial frames in the office is the ablation that we should perform with the excimer laser


In future posts, we will start looking at pre-operative consultation data (biometry) and show examples of surgical planning.



  1. What if the location of the LRI’s post IOL surgery is determined to be where the original clear corneal incision was made ? Can it be done or must it be avoided and laser abalation offered.

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