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Thank you for the opportunity to reply to the letter from Dr Alpins concerning our recent article "Astigmatism and the analysis of its surgical correction". 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 TM is widely used for the planning of refractive surgery a...
Thank you for the opportunity to reply to the letter from Dr Alpins concerning our recent article "Astigmatism and the analysis of its surgical correction". 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 TM 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 psycho-physical 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 by with-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. Dr Alpins recognised the relative value of WTR astigmatism and described how to plan refractive corrections using this principle in his January 1997 article (Figure 10a, J Cataract Refract Surg 1997;23:65-75, reference 33 in our article). We suggested the "by-the-rule" 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.
I agree that my 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 34) in Figure 7, equation 20, and the relevant text page 1131 should in fact be reference 70, Noel Alpins' first article on vector analysis. 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 (i.e. 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 (i.e. as a process measure) and for deriving the "index of success".
Dr Alpins describes the SIA "torque" effect with reference to the preoperative axis of astigmatism (or the TIA) in his December 1997 article not cited in our Perspective article (J Cataract Rafract Surg 1997;23:1503-14). Torque needs to be distinguished from the effect of rotation of the corrective cylinder that is derived from the post-operative 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 describes 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 from our article).
In his early 1997 article (reference 33) Noel Alpins discusses surgical treatment planning combining the topographic astigmatism values with the refractive values to produce an optimal corneal curvature. Dr 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, Dr Alpins nonetheless has ascribed a better relative value to ATR astigmatism suggesting that optimal treatment planning be based on this psycho-physical phenomenon. As we stated "only using keratometric data for the planning of refractive surgery" would create a problem otherwise.
It is understandable that Dr 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.
(1) Morlet N, Minassian D, Dart J. Astigmatism and the analysis of its surgical correction. Brit J Ophthalmol 2001;85:1127-38.
In a recent "Perspective" article by Morlet et al titled "Astigmatism and the analysis of its surgical correction"  there are a number of omissions and fundamental errors of content that lead to erroneous conclusions. These significant inaccuracies overlooked in the review process compromise the article's broad contribution.
In Dr Morlet's attempt to detail "the use and limitations of vector...
In Dr Morlet's attempt to detail "the use and limitations of vectors. . . for the analysis of change in astigmatism" he displays an incomplete understanding of the subject. He has made a valiant attempt to assemble an abundance of historical and contemporary references on a subject of significant interest, but key material has been omitted or misquoted. This has resulted in leading statements of the article, in both the body of the text and even the conclusion that require re-evaluation and substantial revision.
The most obvious omission, is the paper's absence of any discussion of the difference vector, a precise absolute measure of surgical error described in reference 70 . When the difference vector is related to the treatment (ie: TIA or target induced astigmatism vector) one has an extremely useful relative value of success of astigmatism treatment. Dr Morlet has overlooked this key vectorial entity and struggles to find any useful alternative. In sharp contrast Dr Doug Koch, Editor of the Journal of Cataract and Refractive Surgery, in his editorial introduction to the Analyzing Astigmatism issue of January 2001  described the difference vector and the index of success as "remarkably useful and intuitive means of understanding the effects of the surgery".
The authors state more than once for their principle foundation of the article that "Vector analysis alone does not provide any indication of the relative value of the surgical procedure and that it (vector analysis) does not assign a value to the outcome". These statements are erroneous, and the author's failure to discuss or dispute the value of the difference vector and index of success leaves the assertion unsupported and lacking credibility. If the surgical induced astigmatism vector (SIA) (and its further translation) was the only product of vector analysis, indeed vector analysis would be a limited tool. This seems to be Dr Morlet's contention. This is far from the truth and as a result the restatement in the conclusion that "vector analysis does not give a measure of outcome" is factually inaccurate.
In addition, Dr Morlet's interpretation that the off-axis effects of treatment at 45 degrees to the surgical plane are deemed to be rotation, would more accurately be termed "torque" the component of the SIA that has been ineffective in reducing astigmatism. The relevant reference  describing flattening, steepening, torque and effect of off-axis treatments has been omitted from the attempt at a comprehensive list of relevant published material. The phenomena of rotation and torque are fundamentally different physical processes. The polar value at 45 degrees to the "surgical plane" quantifies the torque which causes an increase in the existing astigmatism associated with its change in orientation. It does not properly gauge the cylinder rotation where no concurrent change in the amount of existing astigmatism occurs. Rotation includes some associated flattening (or steepening) effect occurring as a result of the SIA.
The article's conclusion that "a better evaluation of the effect of astigmatism axis requires the use of the 'by the rule' or mirror equivalent axis notation, or by a manual scoring method to produce an outcome summary measure" is convoluted and unworkable. If implemented this would adversely affect the comprehension of astigmatism outcome analysis by the average general ophthalmic or refractive surgeon.
It is unfortunate the reviewers of this paper did not direct the author to other significant fallacies that merited revision. The statement "vector analysis is only valid in the early post-operative period" because "the healing response has modified the initial result of the surgery" shows the authors' failure to understand that the healing response cannot be divorced from the surgical process. It is part of it. The amount of astigmatism correction (SIA/TIA) achieved shows consistent trends over time when examining aggregate data, and this phenomenon requires surgeons to examine outcomes facilitating adjustment of nomograms based on longterm (at least 6 months) and not immediate outcomes. The later statement "the use of vector analysis over time is conceptually invalid, because unlike the initial surgical event, the wound healing process is continuous" is seriously flawed. Vector analysis is an essential component of this refinement process. In fact, vector analysis could be used to determine the astigmatic effect of the healing process itself by comparison of data at various stages in the post-operative period.
The recommendations promoted by Dr Morlet introduce greater complexity to an already complicated subject. For example, mixing negative and positive cylinder notation is unnecessary. The technique put forward does not address the changes that occur in corneal shape as measured by keratometry and topography, and cannot be readily applied when targeting non-zero goals associated with incomplete or off-axis refractive astigmatism treatments.
It is probable that the authors are careless in raising phantom "problems" for planning techniques based on incorrect quoting of information (such as reference 33).  The merits of this customized treatment technique are that refractive as well as keratometric data are employed (contrary to its misrepresentation that the technique "only uses keratometric data for the planning of refractive surgery").
Dr Morlet's unfortunate statement of opinion that "a lack of critical evaluation" has resulted in "the surgical vector's adoption as the de facto standard used in most reports concerning the surgical management of astigmatism" is not shared by many experienced investigative surgeons in the field. This has been shown by its admitted prevalence by the authors, and the usefulness of vectorial analysis in understanding the surgical process . Indeed, many of the erroneous statements and omissions in the Perspective article might lead one to ask where the "lack of critical evaluation" actually lies.
(1) Morlet N, Minassian D, Dart J. Astigmatism and the analysis of its surgical
correction. Br J Ophthalmol 2001; 85:1127-1138
(2) Alpins NA. A new method of analyzing vectors for changes in astigmatism.
J Cataract Refract Surg 1993;19:524-533
(3) Koch DD. How should we analyze astigmatic data? J Cataract Refract Surg
(4) Alpins NA. Vector Analysis of astigmatism changes by flattening, steepening
and torque. J Cataract Refract Surg 1997;23:1503-1514
(5) Alpins NA. Astigmatism analysis by the Alpins Method. J Cataract Refract
(6) Alpins NA. New method of targeting vectors to treat astigmatism. J Cataract
Refract Surg 1997;23:63-75