Astigmatism and axis error
THEastigmatism is an optical defect "oriented": it has a magnitude (in diopter) and a direction (axis). This characteristic allows astigmatism to be easily represented vectorally, the standard of the vector (length of the arrow) being proportional to the magnitude of astigmatism. The correction of astigmatism requires the use of a device which, when apprehended vectorally, generates a vector of astigmatism equal in magnitude, but opposite in direction. The orientation of the corrective device must be carried out with care; This must be aligned according to the "axis" of astigmatism: glass of glasses, Contact lens O-Ring, Crystalline lens Artificial cataract surgery ), Profile of Photoablation (LASIK(, PKR), etc.
(see also page): rotation of a toric implant in a patient with Keratoconus)
For example, an astigmatism of (+ 1 x 0 °) is corrected if we apply a device that generates an astigmatism of (-1 x 0 °).
It sometimes happens that a "axis error" occurs (see example): a toric implant axis error). Rather than be positioned according to the axis of astigmatism (ex: 90 °), the correction device is 'shifted' by a few degrees (ex: 95 °). This can be caused by inadvertent rotation of a toric implant after installation, movements of cyclotorsion during surgery on the cornea laser photoablation, etc.
A legitimate question arises: what is the result of this shift of axis on the refractive correction result?
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 °, the difference with 90 ° should be added to the final result; And multiply the magnitude of the cylinder obtained in this calculation by the value of the original cylinder. In this representation, a classical angular representation is used, "on 360 °". This representation leads in certain calculation steps to consider doubled 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.
With the vector graphics method on 360 °, one represents astigmatism to be corrected by a vector (one arrow) of length + 1 and axis 90 °, oriented in a graduated mark in degree (remember that this method requires converting the initial astigmatism to Correct in positive cylinder formula). The arrows pointing upward have by convention a positive standard (length), the arrows pointing downwards a negative norm.
We can represent the astigmatism (+ 1 x 90 °) like this:
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 °.
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 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 adapted to apprehend the consequences of an axis error for the correction of astigmatism in a "visual" way. The consequences of this type of error consist of 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 useful if you do not have a computer calculator or a suitable software. It is important to remember that the formula of refraction to be corrected should 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 you to directly plot the vectors to be summed (without doubling the axes, since they are already doubled in the circular coordinate system, a complete rotation of which corresponds to 180 ° and not 360 °).
This method is useful for the understanding of the observed residual astigmatismes after photoablation (LASIK, PKR), poses a toric implant (cataract surgery), or ask a toric contact lens.
In clinical practice, axis errors do not usually exceed a few degrees. The residual astigmatism is therefore low in magnitude, and its often oblique orientation vis-à-vis the initial axis; A vector graphic representation of the consequences of a few degree error explains this.