Each corneal topography image holds a vast array of data measurement points that need to be broken down into a more user-friendly form. By example, the Medmont E300 captures data at around 6,000 locations and displays each location in four different formats. I think it’s fair to say that any lens design requiring manual entry for each of these locations would be unlikely to attract many users. To overcome this problem the fit parameters for most lens designs are typically calculated to fit either a spherical or elliptical model of the corneal shape that is built from the captured topography data.
Spherical model (simulated keratometry)
Many of the earlier lens designs like Paragon CRT and Contex base first lens selection on keratometry values, which are easily obtained from topography as simulated K readings. They are called simulated because they haven’t captured the information in the same way as a traditional keratometer, but the value should be the same as if you had captured them in the traditional manner. Keratometry values can be described in mm or dioptres, and easily converted between the two as shown in the equations below. The degree of corneal astigmatism is calculated by subtracting the keratometry value in the steep meridian measured in dioptres from the flat keratometry value also measured in dioptres. E.g. a cornea that is 44.50D in the flat meridian and 48.00D in the steep meridian will have 44.50 - 48.00 = -3.50D corneal astigmatism - way too high to consider fitting with a spherical OrthoK lens!
A problem with modeling the cornea as a sphere is that most corneas flatten towards the periphery, which for K readings is calculated across a 2 to 3mm chord. This means that the spherical modal is fitted to the central part of the cornea while the landing zone of a modern OrthoK lens aligns across a larger chord of around 9mm. In a different post I covered how OrthoK lenses are fit using a sag based fitting philosophy, which means that to establish a good fit the sag of the lens at the peripheral bearing point needs to be the same as the sag of the cornea at the same location, and for this keratometry values do not provide sufficient information leading to potential for errors.
The OrthoK lens manufacturers that base first lens fit on keratometry readings are aware of this limitation and compensate accordingly - typically eyes with steeper K’s flatten towards the periphery at a faster rate than eyes with flatter K readings. So, they will calculate the sag of the modeled spherical keratometry at the lens bearing chord and then apply a compensation factor. This may sound like a fudge, but the respective manufacturers must have done a lot of work to establish the relationship between best fit lens and keratometry readings because from my experience, despite the limitations I have just described, the first fit selection for these lens designs when based on flat K tend to fit well in most cases.
Squashing the circle model along the central axis creates a prolate ellipse that by flattening towards the periphery more closely follows the corneal profile and hence creates a more accurate model of typical corneal shape (see above image). This, however, comes at the cost of requiring more variables to define the fitted model. Instead of a single K value, the cornea is instead described as an elliptical model of central (apical) curvature and eccentricity. The extra variable might not seem much, but it creates extra complexity when it comes to using a corneal model from which to base lens selection. You also need to be aware that different descriptors can be used to define elliptical shapes.
Designs that use K values are typically supported either by a slide rule or lookup table that cross-references refraction target against K value to establish the first lens to try. This becomes a lot more complex when working from the more accurate elliptical model because a 3-dimension slide rule or look-up table to reference Ro, eccentricity and refraction target is required. While this is achievable, it makes more sense for a lens manufacturer to create a computer program to calculate lens selection.
There has been a steady increase in software supporting OrthoK lens designs over recent years, either as an alternative and potentially more accurate alternative to paper-based selection calculators for older lens designs or in some cases as the only option for newer designs. There is nothing really to be gained over paper-based tools if you are only going to be feeding keratometry values and refraction target into a computer program. Where computers excel is in handling elliptical or more complex models of the eye, so if a software option is available for your OrthoK lens design of choice I suggest you use it in favor of less sophisticated paper-based tools.