Astigmatism and the analysis of its surgical correction

Thank you for the opportunity to reply to the letter from Alpins concerning our recent article in the BJO.^1,2

Noel Alpins is a widely respected contributor to many international meetings, having written comprehensively on the use of astigmatism vector analysis. His software program assort is widely used for the planning of refractive surgery and provides many derived indices (transformations) from the vector analysis of both refractive and topographic astigmatism. Although the derived indices are summary measures, we have argued that their usefulness for statistical analysis is limited. This is because the perception of astigmatism is a psychophysical phenomenon altered by the orientation of the axis of astigmatism (the power meridians of the cornea and crystalline lens). Unfortunately, the perceptual response means that the measurement of the axis of astigmatism (which is with an arbitrary 180° scale) is non-linear in outcome terms, as related to visual acuity outcome. Astigmatism obliquity is the least desirable outcome but this is separated into two on the scale “with the rule” astigmatism (WTR), which is generally the most desirable outcome. Oblique astigmatism also separates the two groups of “against the rule” astigmatism (ATR) from the WTR astigmatism. Developments of vector analysis so far have not resolved this issue of non-linearity of the axis of astigmatism compared with the visual outcome. Alpins recognised the relative value of WTR astigmatism and described how to plan refractive corrections using this principle (his Fig 10a) (reference 33 in our article.³ We suggested the WTR transformation would help eliminate the problem of divided oblique and WTR astigmatism, but this makes the use of vector analysis difficult and does result in data compression.

We agree that our understanding of astigmatism is incomplete. With over 4000 responses to a search for astigmatism on PubMed, there is much to know and yet more still unknown. The references cited in the article were simply representative or illustrative of the arguments discussed in the article. By way of apology, the correct reference for the “surgical error” (originally given as reference 34) in Figure 7, equation 20, and the relevant text page1131 should in fact be reference 70, Alpins's first article on vector analysis.⁴ The surgical error is the arithmetic result of the preoperative vector combined with the surgically induced vector (SIA), less the target induced vector (TIA), which is analogous to neutralising a lens with another of the opposite sign (hence the reverse direction arrow). This produces two outcome measures, the difference in magnitude and axis. However, as our article’s discussion on obliquely crossed cylinders described, misalignment due to rotation of the corrective cylinder produces not only an error in cylinder magnitude and axis, but also a spherical power error. Because of this interdependence all three should be analysed together, but this produces statistical difficulties. One way around this problem is to use an appropriate summary measure of the outcome instead.

The surgical error may be applied to treatments targeting non-zero goals despite not addressing changes in corneal shape. As an outcome measure of the surgical process it is equally applicable to the arithmetic result of the SIA with the TIA as it is with the SIA and the preoperative astigmatism vector. The transformation of the error between the SIA and the preoperative astigmatism vector into the “difference vector” (which is a mathematically precise and absolute measure of the surgical error) unfortunately does not address the problem of non-linearity (that is, the relative value in terms of visual acuity outcome) so is not useful as a summary measure of the outcome. However the difference may be useful in understanding the effects of the surgery (that is, as a process measure) and for deriving the “index of success.”

Alpins describes the SIA “torque” effect with reference to the preoperative axis of astigmatism (or the TIA) not cited in our article.⁵ Torque needs to be distinguished from the effect of rotation of the corrective cylinder that is derived from the postoperative astigmatism value. Unfortunately, our discussion on the optical decomposition did not clearly state that the 45° polar value is derived from the postoperative result, thus correctly describing the rotation effect (as discussed with the obliquely crossed cylinder effects). We apologise for creating some confusion with the “torque” effect.

We agree that the healing response is connected to the surgical process; however, healing is a very individual response. Vector analysis in terms of the SIA can only reflect the surgical process. Although a “vector” could be used to represent the measurement of the healing response at any point in time, it may not be representative of the healing responses at other times because the healing process is continuous. Furthermore an individual's response may not be well represented by the aggregate or mean vectorial response, which as discussed, is compounded by the non-linearity problem of the separation of the oblique and ATR astigmatism axis values (see reference 104 in our article).⁶

In his early article² (reference 33) Alpins discusses surgical treatment planning combining the topographic astigmatism values with the refractive values to produce an optimal corneal curvature. Alpins suggests that the surgical emphasis is best directed towards a WTR result when there is a disparity between the values requiring some residual astigmatism after surgery. Without recognising Javal's rule, Alpins, none the less, has ascribed a better relative value to ATR astigmatism suggesting that optimal treatment planning be based on this psychophysical phenomenon. As we stated “only using keratometric data for the planning of refractive surgery” would create a problem otherwise.

It is understandable that Alpins feels that the concepts presented in our article are in conflict with some of his own, but these do not diminish the value of vector analysis as a process measure, particularly for individual cases. It is the use of vector analysis as an outcome measure relative to the visual acuity that was critically evaluated by our article.