Astigmatism and axis error
THEastigmatism is an optical defect" oriented": in its classic expression in ophthalmology, it has a magnitude (in diopter) and a direction (axis). The pressure (-3×90°) corresponds to a astigmatism myopic (negative sign) of magnitude 3 Dioptre, "oriented" at 90°. This characteristic allows astigmatism to be easily represented in a vectorial way, the norm of the vector (length of the arrow) being proportional to the magnitude of the astigmatism.
The correction of astigmatism requires the use of a device which, when viewed in a vectorial way, generates an astigmatism vector equal in magnitude to that to be corrected, but opposite in direction.
The orientation of the corrective device (lens, toric lens, toric implant) must be carefully carried out; it must be perfectly aligned according to the "axis" of ocular astigmatism: spectacle glass, Contact lenstoric lens, laser photoablation profile (LASIK, PKR) / or the axis of corneal astigmatism: toric implant of Crystalline lensartificial (surgery of the cataract).
(see also page): rotation of a toric implant in a patient with Keratoconus)
For example, an astigmatism of (+1 x 0°) is corrected if a device that generates an astigmatism of (-1 x 0°) is applied by aligning it with the same mark.
Sometimes an "axis error" occurs (see example : a toric implant axis error). Instead of being oriented along the axis of the astigmatism to be corrected (e. g. 90°), the corrector device is "shifted" by a few degrees (e. g. 95°). This can be caused by an untimely rotation of a toric implant after placement, cyclotorsion movements during laser photoablation surgery on the cornea, etc.
A legitimate question then arises: what is the result of this axis shift on the refractive result of the correction?
Several approaches can be used to calculate the effect of an error in axis for the correction of astigmatism: analytical approach (trigonometric calculations: astigmatism is expressed as a function A x cos (2T), or approach vector (astigmatism is treated as a vector of which the 'norm' is proportional to the magnitude of astigmatism, and orientation in line with the axis expressed in the form of astigmatism - ex) (: 90 °). Can also be used to representation by complex numbers.
The vector method is particularly suited to a 'Visual' understanding of the consequences of an error in axis. Here is the ' graphical representation ' about an example where to correct astigmatism is expressed by the formula + 1 x 90 °.
All astigmatism formulated as an optical prescription ophthalmic can be converted into formulation in positive cylinder (example:-1 x 0 ° plan is equivalent to + 1 x 90 ° with a sphere of + 1 D)
The method described below applies to any situation: if the initial astigmatism is different from +1 x 90°, add to the final result the difference with 90°; and multiply the magnitude of the cylinder obtained in this calculation by the value of the initial cylinder. In this representation, we use a classic angular representation"defined at 360°". This representation leads to the consideration of double angle values.
NB: the use of a "Double plot" graph for astigmatism Allows you to draw vectors of the same sign, whose angles are automatically doubled.
The Trigonometric representation of astigmatism Also allows you to see the effect of a correction axis alignment error.
Vector representation of astigmatism over 360°
With the 360° vector graphic method, the astigmatism to be corrected is represented by a vector (an arrow) of length +1 and axis 90°, oriented in a graduated reference frame in degrees (remember that this method requires converting the initial astigmatism to be corrected into a positive cylinder formula). Upward pointing arrows have by convention a positive standard (length), downward pointing arrows a negative standard.
We can represent the astigmatism (+ 1 x 90 °) like this:
Astigmatism correction without offset
The "opposite" astigmatism, which adding compensates exactly + 1 x 90 ° is:-1 x 90 °. He may be represented by an arrow "down", according to the 90 ° axis.
The 'sum' of these vectors of astigmatism is a vector, and a situation where the astigmatism + 1 x 90 ° is perfectly corrected by the addition of a device that induces - 1 x 90 °.
Astigmatism correction with offset
Imagine that a 30 ° axis error occurs in anti clockwise: the corrective device is more placed at 90 ° but (90 °-30 °). Compensatory astigmatism is a vector oriented at 120 °:-1 x (90 ° + 30 °) or - 1 x 120 °. The situation can be represented as follows:
Due to the modulation of the refractive astigmatism on 180 ° (non 360), it must be 'double' the angle corresponding to the axis error (30 °) to continue our graphic resolution of the problem: this angle becomes so 2 x 30 ° = 60 °. We then do an additional 30 ° rotation of the arrow of our vector corresponding to astigmatism induced by the correction device.
We can then achieve the vector sum and an arrow which length corresponds to that of residual astigmatism: in this example, this arrow also has a length equal to 1 (the triangle formed by the arrows is equilateral, every angle being equal to 60 °!). When the error of axis is 30 °, residual astigmatism has the same magnitude as the initial astigmatism! On the other hand, its axis is changed (we say colloquially that the axis of astigmatism has "turned"!)
The geometry of the figure suggests that this axis is (with the horizontal axis) 30 ° and 60 ° with the direction of astigmatism to be corrected (90 °). Again because of the double modulation of astigmatism on 360 °, divide the angle with the axis of initial astigmatism (here located according to 90 °) 2; 60 ° / 2 = 30 °.
An error of axis of 30 ° (-1 x 120 ° instead of-1 x 90 °) induces a residual of 1 diopter astigmatism oriented to 60 °: + 1 x 60 °. The magnitude of astigmatism has not changed, but its axis has changed!
The "double plot" graphical representation (where the angles are doubled) precisely avoids doubling the angles! (this is done directly since a full turn corresponds to 360/2 = 180°).
The value of the rotation suffered by the axis of astigmatism in case of error correcting device axis with respect to the 90 ° axis is always equal to (90 ° e) / 2 where E is the error in absolute terms (in degree). In the example above: (90 ° - 30 °) / 2 = 30 °. Well, the axis was diverted 30 ° (angular difference between 90 ° and 60 °).
Another "graphically" remarkable example is an error of 45 °. Instead of (-1 x 90 °) we 'deals' by mistake (-1 x 135 °). As we need to double the value of the angle before sommer arrows, and as 2 × 45 ° = 90 °, we get easily by looking at the geometry of the figure that the magnitude of residual astigmatism is equal to the square root of 2 (or 1.4 D about).
The axis of residual astigmatism is (90 ° - 45 °) / 2 = 22.5 °. The final formula is of (+ 1.4 x 67.5).
The vector method is particularly well suited to "visually" apprehend the consequences of an axis error for the correction of astigmatism. The consequences of this type of error are an astigmatism of different axis, and of residual magnitude depending on the axis error. This magnitude increases if the axis error is greater than 30°..
This method is practical if you do not have a computer or suitable software. It is important to remember that the formula of the refraction to be corrected must be converted into a positive cylinder; the correcting device is then a vector of the same axis but of opposite magnitude. The use of a "double plot" graph allows the vectors to be plotted directly (without doubling the axes, since they are already doubled in the circular reference frame whose complete rotation corresponds to 180° and not 360°).
This method is useful for understanding residual astigmatism observed after photoablation (LASIKPKR), toric implant placement (cataract surgery), or Contact lenstoric placement.
In clinical practice, axis errors generally do not exceed a few degrees. Residual astigmatism is therefore low in magnitude, and its orientation is often oblique with respect to the initial axis; a vector graphic representation of the consequences of an error of a few degrees explains this!
A small axis error represented in vectorial form induces a moderate residual astigmatism, whose axis is located in an oblique direction (about 45 °) vis-à-vis the initial direction.