 I'm a little surprised that the room isn't packed with this title. I thought that this would definitely bring everybody. It's not as daunting a title as it sounds. We all know what elliptical curve fitting is. Do we have a worst candle? He always has one. We all know what elliptical curve fitting is. We all remember probably what algebra is. And a generalized eigensystem is just a system where all the vectors are multiplied by things which do not change their direction. What I'm going to talk about is making elliptical models out of data from corneal samples and then applying those models to learn things about what's going on with surgical stuff. So, dsec, d-a-l-k, p-k-p, all of those things involve cutting corneal tissue with a microkeratome or with femtosecond laser. And these tissues, once cut, have particular shapes which are not necessarily round and do not necessarily have the exact edges that we might intend them to. And so these can be modeled mathematically in order to learn more about what those shapes are. Those shapes can then be evaluated with ray tracing or other methods to determine what might be the best surgical methods in cutting those tissues. In general, conic sections can be used to model things in the eye. And in particular, the ellipse is one of the more accurate models for the literature over the last 20 years. So if we take this case of the Sante OCT of corneal tissue, we could do a measurement, say from here to here and from here to here and generate data points based on cuts at, say, zero and 180, 45 and 225 and 90 and 270 and so on around and use these slices here to generate data points around a Cartesian coordinate system. Those data points then, so this is eight data points, can then be plotted on a graph and so if my laser pointer worked, you can generally see though. You can see the data points at each of the places where the measurements were made under the Sante. And I'm sorry, I'm actually gonna go through the math a little bit. The top is a generalized equation for a conic section. So if we take the 10 data points and plug them in, we get eight equations. And then using a paper by Fitzgibbons and Palau, who are mathematicians, there's a constraint, four times A times C minus B squared equals one from the previous equation that defines an ellipse. And so by minimizing the data points of those eight equations to that particular constraint, we force all outcome to be elliptical. And then we throw out the imaginary numbers and we can solve for the various important features of the ellipse, like its center, which are in this case defined as H and K, the angle of its major axis, and we can also generate an equation for the ellipse. And so ultimately this is a standard equation of the ellipse and if we re-center the ellipse so that H and K are zero and using that original data point, we get this equation for that data set. Then this equation can be used to calculate various things about what our original desect button or any other thing that we are modeling looks like. And so when we solve the eight equations plus our constraint for a least squares fit, we get this ellipse for that set of data points. And as you can see, it's pretty good in terms of fit. And we can also learn a variety of things about it. So in this case, this was a piece of corneal tissue cut with a microcaretome that went in at the epithelial surface, then cut across the stroma and back out at the epithelial surface. And so if we model the epithelial surface and the stromal surface with the epithelial surface as blue and the stromal surface as red, you can see that we can determine what the rotation of the cut is, how round the cut is, how the two cuts are rotated with respect to each other and where the center of the two cuts are with respect to each other. In particular, I'm just gonna mention one thing in terms of roundness. E-centricity may be a little hard to imagine. So a circle, by definition, has a eccentricity of zero and a line, by definition, has a eccentricity of infinity. And so in between, you have a nonlinear scale for eccentricity. So an eccentricity of 0.3 is significantly more circular than an eccentricity of 0.8. And that'll be important later. So these are just some pictures showing some curve fits of corneal sections that were cut by Youssef Khalifa. And so you can see that some of them are pretty round and pretty regular. Others are not so round or regular. This is what my wife thought of. She was born with a science boring interest trading. So that's pretty much what my wife thought of the talk. So now we're gonna actually get to things that involve the I instead of just math. So the reason I started this was Youssef came to me and said, I've got this problem with cutting desect buttons. And when we make the incision through the epithelium and then across and back out, it's not always completely round and we're concerned that maybe we're incorporating tissue from the stroma edges as we make these cuts. So the blade, it cuts through the epithelium transverses across the stroma for a portion of the depth before reaching its maximal cut depth and presumably making a relatively straight cut across the stroma. As you can see from the pictures I showed you previously which are data from this, the cut shapes may be different at the stroma level than they were at the epithelial level. And also shearing of the tissue as the blade goes through may result in elliptical rather than circular buttons. And so I modeled these. The methods were 12 donor human eyes which were unsuitable for transplant were cut with a 300 or 350 micron microcaratone. To save saying this later, there was no significant difference between using either of these two microcaratones. We imaged it with the Vesante OCT, took measurements at, took eight measurements around the circle and then modeled the ellipses. And so here's an example again of how the measurements were made. This isn't the actual tissue, I don't have his photographs so I'm sorry I can't show you that. But they were made on slices like this. And as a result the average cut diameter was at the epithelial surface was 10.69 millimeters which is larger as expected than the average cut diameter at the stromal surface. And that means that there is what we call an annulus between the epithelial cut surface and the stromal cut surface where the blade was coming in and then coming out on the other side. And so we were interested in what is the size of this annulus and is it uniform? And in fact it's not uniform. The average size of the annulus was 0.85 millimeters but it ranged from 0.3 millimeters to 1.8. I'll explain. Just wait. Just wait. So what I'm saying with that slide is this difference in distance from here to here ranges from 0.3 to 1.8 millimeters. In addition we looked at eccentricity and it's roughly the same. There's no statistically significant difference between the epithelial cut and the stromal cut. Although neither of them is completely circular. And so what that is looking at is essentially a ratio of the major axis to the minor axis. And so if major axis and minor axis are equal it says eccentricity is zero and it's a circle. In addition the location of the center of the stromal cut in respect to the epithelial cut is generally off center in the direction of the cutting of the blade. So the blade cuts from this side to this side and in general you have the center of the stromal cut smeared towards or smeared away from the blade. And so most of them are off, most of the stromal beds are off center from the epithelial bed. Again that was measuring this point versus that point, the centers. And most of the major axis is roughly in the perpendicular direction. Now the direction of the major axis the more circular it gets the less relevant that becomes because if it's an exact circle then you don't even have a direction of the major axis because the major and minor axis are the same no matter where you put them. But in general we found that the major axis was perpendicular to the blade cut. And I think what happens is as the blade, as the micro keratone blade cuts to the tissue it causes the tissue to deform and smear somewhat away from the blade and so as the blade comes across the tissue deforms this way as you're cutting it and that's why most of them have a major axis in this direction rather than this direction. And again, so almost to where you asked. So in order to obtain a consistent donor lenticule after you've prepared your stromal bed for desec as Dr. Ambadi is saying, what you wanna do is you wanna make a cut that does not involve the annulus or the epithelium. If you include epithelium then you are essentially putting epithelial tissue in on your desec button. And if you include the annulus then you're going to have this transition zone where you've made your micro keratone cut and it's not going to have the same optical properties as the flat stromal bed because you've got essentially a lens at an angle there. If you rely on purely the epithelium to epithelium diameter to determine the size of the punch you would overestimate the size of the stroma where you would get a uniform cut depth. And I'll show a picture of this in just a second. In addition, if the circle is not concentric which I've shown that none of them were completely circular, you have to ask yourself, where exactly do I need to measure from in order to guarantee that I'm going in the direct center of the stromal bed? Particularly if you measure from the epithelium surface I've shown that the stromal bed is not centered with the epithelium but is in fact shifted a small amount up to about 0.15 millimeters on average from the epithelial cut back and forth all the way to the stromal bed because I can show some figures and in particular that the Santé I've shown already shows a wedge of stromal tissue at the side which demonstrates that it was not. Yeah, and Yusuf did the measurements but he measured that half. So in addition to the teacher. So it's not round. It's not centered. The two surfaces are not centered with each other. The incision annulus varies both in width and in angle as Dr. Olson pointed out and all of these errors would be increased if your punch as you're punching it is not exactly centered as well. And so what you could end up doing is having this bed and this is just a schematic and punching it such that you catch these edges and this is essentially one of our models or one of the ellipses. And so you can see that here the annulus is actually much thinner than here and so much steeper. So where exactly do you center this and can you see the edge of it here like you would likely here? So that was the first application. And so as we're talking about DSEC, it often does not result in 2020 best spectacle corrected vision. And DMEK has been reported to have a higher rate of that which may be related to having a interface which is thinner and doesn't incorporate these wedges of tissue at the edges. And so as I was saying, those wedges may affect the optical quality of your DSEC and one cannot use the epithelial distance or decrease the epithelial distance by a constant factor unless it's a large constant factor to reliably size DSEC buttons. And you don't wanna decrease it by a large factor because you wanna get a maximum number of endothelial size. So did I address here clear on slit length? In theory yes, because you don't have the deformation of the tissue by the blade and so on. And I started some of that in a way too which was the second idea which is what does this do to the optics of the eye? And I mean, you're talking about this and it's known that there's a bit of a hyperopic shift as well. And so I started the process of modeling and this is another Vesante. And by taking measurements, I converted the lenticules into a Cartesian coordinate system and can generate essentially lenses out of each of them. And I haven't gone through this but you can calculate the power of each of these lenses from the epithelial surface, the interface surface and the endothelial surface and presumably do ray tracing. In addition, you can also calculate the power of the edges seen here. And you can calculate the change in the power. And I think it would be interesting to look at the lens say from here to here of the DSEC button and then out here as well and compare the two because even here, just going from here to here, it's not uniform thickness and you would expect that to change the optical properties as well. And so then one last thing I applied this to just to show that you can apply this modeling system to something else. Dr. Mosfar asked me, when we cut corneal grass for Boston Capros, he thought if you cut the inner circle first and then the outer rather than the outer circle first and then the inner that you would have a more centered cut. So he asked me to do it and I cut 10 cornea using the standard equipment that they provide. Five I cut the inner first and then the outer and five I cut the outer and then the inner. I photographed the graphs, made measurements and then modeled them as ellipses and calculated where the centers are. And so this is how I did the measurements. So I have each axis and I get eight data points for the outer and inner and that's the first one I cut and you can see I didn't get it that centered. And here's an example of the model with the fifth cut. This was inner than outer, this was outer than inner. And this one is a bit more centered than that one. And so if we then graph how off center they are as the vector, as the scalar quantity of the vector between the two centers, we see that at the beginning I was roughly the same for the first three cuts, whether I cut inner first and then outer or outer first and then inner. But then for the last two, maybe I got better or maybe this is just random variation. But there was no statistical difference in all five points with a P of about 0.3. If you compare the last three points, it comes closer and it's a P of about 0.056, still not statistically significant by our definition of what is usually significant. But I wonder if you took somebody who was more experienced or if I kept doing them if it might actually start to make a difference. So the conclusion there was mainly that it didn't make a difference at least at first, whether you cut the inside first or the outer. But practice did seem to make a difference. And I improved over time and neither the inner first or the outer first nor practice allowed me to get one dead center. And so I wondered if you might be able to make a device that would allow you to cut both at the same time and give you a much more centered cut when you're making the corneal graph portion of your K-pro. So overall, what I was trying to demonstrate here was that you could use a curve fitting model to model various aspects of punching corneas or desec buttons or other things in the eye. And that those equations could then be used to generate important data to describe the physical properties of the systems. And using those mathematical models, you could begin to attempt to solve different optical and op-doms such as what is the contribution of shape or centration in a desec button to the overall final vision of the patient.