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Spline Mathematics Promotes Better Eye Surgery

Brian Barsky has spent twenty-five years deep in the field of theoretical mathematics. But in the last seven, he has taken that research to a place into which few mathematicians have ever wandered: contact lenses and optical surgery. In a person with normal vision, light enters the eye and is refracted a precise amount to focus on the retina. But for a patient with a misshapen cornea, the image will be out of focus. Eye doctors have been prescribing glasses for centuries and contact lenses for decades, but it is only recently that new surgical procedures such as radial keratotomy (RK) and various laser refractive surgeries have been developed.

In RK, the eye surgeon makes eight tiny cuts to the cornea, altering its shape. When the cornea heals, its new curvature allows the light rays to focus properly and voila! the patient should have 20/20 vision. Barsky, himself a victim of an unusual corneal condition, keratoconus, became interested in optometry in his own search for contact lenses to correct his vision. Every week for over a year he would try a different brand or style of contacts, trying to find an acceptable fit. "Due to the irregular shape of my cornea, my vision cannot be corrected with spectacles; contact lenses can correct my vision but fit so poorly that they can harm my cornea."

But as he researched further, he found that the knowledge of higher mathematics was absent from the field of practical optometry, and the ability to measure a patient's eye accurately was largely trial and error. Even more frighteningly, the procedures used to measure the eye for surgery were highly suspect and the results varied greatly from patient to patient. "There is mathematical theory about curves and surfaces, and geometry, and curvature, but this seemed rather absent in the practical level of eye surgery."

Barsky's research has been in the field of splines, finding appropriate mathematical expressions to describe real world curves and surfaces. Applying his own knowledge of splines and continuity ("smoothness" of a curve), he has developed a method to more accurately map the contours of the cornea. "Imagine holding a ringed fluorescent light to a mirror -- it reflects a circle. Now hold a series of concentric ringed lights and shine them onto the eye's surface and take a video image. The cornea of the eye will reflect these rings (like a mirror) with a fairly symmetrical reflection corresponding to relatively symmetrical regions of the cornea and distorted reflections resulting from more irregularly shaped areas of the cornea."

Barsky can use this data to provide eye doctors with a more accurate version of the shape of the cornea. And by using this scientific visualization, a multi-color image can be generated showing how the curvature varies locally over different locations on the cornea.



It's just a coincidence that I had a personal interest in corneas and a doctorate in spline research. Otherwise these two fields might have never come together.


"Brian Barsky develops new equipment for measuring curvatures of the eye", (c) Barsky 1996

"The data used can help eye surgeons get a more precise image of distortions", (c) Barsky 1996

 

These advances are especially useful now that refractive surgery is being performed with lasers to reshape the cornea, especially since the lasers can be finely calibrated to very exacting removal of tissue. "It's just a coincidence that I had a personal interest in corneas and a doctorate in spline research. Otherwise these two fields might have never come together."

 


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