Up to now we have seen how the patients were before and after refractive surgery and related the spherical equivalents to each other but at the moment we have no information about the astigmatism, we haven't made any description graphic and we have no idea what is happening to the axis. One way to include somehow the cylinder of the astigmatism is to take as postop value in the graphic the defocus equivalent (DEQ). The DEQ correlates much better with the visus that the SE and is calculated with the following formula:

DEQ = absolute value of SE + ½ absolute value of the cylinder

The DEQ is always positive and if we relate it to the SE preop we wil get a graphic like this one:

The points in that graphic are jittered by 0.025 to avoid overplotting, that means they are moved randomly 0.025 D horizontally and vertically. In this graphic we still have no idea about the axis or the magnitude of the cilinder. I will show now a graphic to describe the cylinder and the axis of the astigmatism:

In this graphic the axis of the astigmatism is doubled. The reason to do this is that the 180º and 0º axis are in the opposite side of the graphic and doubling the angles we see them at the same point. There are other reasons for doubling the angle (vector analysis) but I won't go through them in this post. You can find further information about double angle plots in this paper by Holladay.

In the graphic each circle is -0.50 D of cylinder and the outter circle is -5 D, the number in parenthesis is the original axis of the astigmatism. The points that fall in the green area have astigmatism with the rule (WTR), in the blue area are against the rule (ATR) and in the read area the oblique astigmatism. The red cross is the mean cylinder considering the axis, an example to explain it: a cylinder of -1 x 90º and a cylinder of -1 x 180º the mean would be 0 since to correct a cylinder of -1.00 x 90º we need a +1.00 x 90º or a +1.00 -1.00 x 180º. The shaded ellipse around the red cross is the standard deviation of the x and y values in cartesian coordinates and we can interpretate it as the ratio between the ATR + WTR astigmatisms and oblique astigmatisms, when the horizontal axis is larger than the vertical there are more ATR + WTR than obliques astigmatisms, if the vertical is larger then there are more obliques astigmatisms.

In the next graphic I will plot the astigmatism postop related to how it was preop (ATR, WTR or oblique):

Now we see that the astigmatism is much less, as expected, but we can also see if we under or over corrected or if we treated another axis. The blue points were ATR preop, so if they fall in the blue area we undercorrected, if they are in the green area we overcorrected and if they are in the red area we treated the wrong axis, equivalently for the green points (WTR preop) if they fall in the green area we undercorrected, if they are in the blue area we overcorrected and if they are in the red area we treated the wrong axis. For the oblique (red points) we cannot see if we over or under corrected but we can see if we treated the right axis. We could separate the oblique astigmatism points in 30º-60º and 120º-150º giving them other colors and we could see if we over or undercorrected.

We can divide the graphic as with the SE by any factor we want, as an example the following graphic divides them by the amount of cilinder preop, more or less than 2 D:

Here we can see that for the cylinders preop under 2 D the points are closer to 0 and it seems we got more oblique astigmatisms, there is quite a lot of overplotting so we should change the scale of the plot making the outer ring -1.00 D for example and we will see much more detail.

This was the last post of the series to get a better insight of refractive surgery results. Remember, if you want any of this graphics but don't know how or don't have time, I can do them for you.