Evaluating refractive change from ortho-k: which map to use?

Corneal topography is a powerful tool in modern myopia control and orthokeratology (ortho-k) practice. But beyond simply interpreting colored maps, understanding what those maps mean geometrically is key to better clinical decision making.

The following schematic illustrates the geometry of a spherical surface compared to a typical corneal surface at an off-axis point, indicating:

• Spherical surface, with its optical axis (horizontal reference line) and Peripheral locations on the surface, where we draw the tangent (slope) to the surface.
• From each point, the normal (perpendicular to the tangent) back to the optical axis, meeting it at O’.
• The distance along that normal line is the sagittal radius (often called axial radius) at that point, labeled O for each of the displayed peripheral locations.

By definition, o’ is a common center of curvature for all tangents on a perfect sphere — thus the sagittal radius measured from any surface location through the normal returns to o’.

However, real corneas are not perfect spheres and instead become more prolate (flatten in the periphery). At each surface location progressively further from the center:

• The local curvature at surface locations becomes flatter.
• The common center shifts progressively further from the surface, now labeled s’ instead of o’.
• The sagittal radius s is increasingly greater.

Tangential (instantaneous) curvature considers not “how a given location relates to a central reference,” but “how the surface bends light at that location”.

In this model:

• A small circle (or arc) is fit to align with the curvature immediately around the measured location.
• The radius of that circle is t, the tangential radius.
• Because t references the local curvature, it typically changes more quickly as you move away from the optical axis, roughly threefold faster in a conicoid model compared to axial radius.

Because it is acutely sensitive to local shape fluctuations, the tangential measure reveals subtle peaks, inflections, or decentrations more readily than the smoother, more averaged sagittal metric.

Our previous article Understanding and interpreting corneal topography maps for orthokeratology displays in detail how the theory we’ve discussed maps onto everyday clinical imaging. In summary:

Sagittal (axial) maps provide a smoothed, global picture of power distribution. You’ll notice that peripheral irregularities are dampened because the axial representation “averages over neighbors.”

Tangential (instantaneous) maps highlight steep rings or focal distortions. Because these maps respond rapidly to small shape changes, they vividly display:

• The steep ring around an ortho-k treatment zone
• Any localized steepening or flattening
• Decentrations or asymmetries

Even relatively small deviations are much more apparent in tangential maps than in sagittal maps.

In effect: axial gives you the “bigger picture,” tangential gives you the “fine detail”, but which map best indicates the refractive changes induced from ortho-k lens wear? A practical way to think about this is:

For axial curvature, if incoming light is restricted to pass through a narrow annulus encompassing the measurement point of interest, the best focus converges at S’, the sagittal center. A focus corresponding to T’ (from the tangential construction) is not observed.

The only way to observe a focus corresponding to the tangential curvature is to restrict incoming light to a small circular aperture centered around that location.

While neither map represents what a person sees as light comes in over the whole pupil, it could be argued that the axial map is more meaningful than the tangential map, by aligning more closely with how light is focused across the pupil. Indeed, when converting wavefront aberration measurement maps across the pupil into power maps, it is the axial maps that are usually used.

However, tangential curvature will be steeper than axial curvature at a given point within the peripheral plus ring induced by ortho-k lens wear and will consequently be associated with a numerically greater ‘refractive power’. This may explain why tangential curvature is used in many research publications investigating ortho-k for myopia management. Particularly when indicating peripheral plus power to have a ‘dose dependent’ effect on myopia control efficacy with a higher ‘plus’ power being more aligned with this narrative.

Ultimately, sagittal and tangential maps are complementary representations of the same corneal surface, each valuable, neither complete. Axial curvature aligns more closely with how light is focused across the pupil and is therefore arguably more meaningful when estimating refractive change. Tangential curvature, with its sensitivity to local shape variation, remains indispensable for assessing treatment zone centration, steep ring integrity, and fine structural detail. The key is to understand what each map is showing you, and to use both.