Vector Analyses of Astigmatism: The Good, the Bad and the Ugly
By Mathieu Gauvin, B. Eng., PhD and Avi Wallerstein, MD, FRCSC
Vectorial analysis of the change in preoperative to postoperative astigmatism is a fundamental and essential part of refractive surgery outcomes reporting. Unfortunately, many authors are deterred from performing such “advanced” analyses and omit to use this critical tool in their practice and manuscripts, despite the availability of free software.
In this article, we aim to explain the key issues requiring consensus to allow the universal adoption of an analysis and reporting standard. These include definitions of vectors, orientation conventions, preferred graphical representations, and terminology.
Is astigmatism a vector?
A vector is a mathematical quantity that has both a magnitude and a direction. Astigmatism certainly seems to fit these criteria as stated by Prof. Alpins in his 2004 paper:
“Astigmatism, with cylinder power and axis (refractive) or magnitude and meridian (corneal), fits this description” .
A critical property of vectors is that they are subject to mathematical operations. For example, the surgically-induced astigmatism (SIA) vector can be calculated from the preoperative and postoperative astigmatism.
Geometrically speaking, a vector can “rotate” 360º before returning to its initial value. In contrast, astigmatism returns to its origin after rotating 180 º. Therefore, we must convert astigmatism data to X and Y cartesian coordinates by using basic trigonometry before performing any vectorial calculations. We do so by multiplying the astigmatism magnitude by the cosine and sine of the double of the astigmatism axis. This doubling of the axis is precisely why we term the 360º plots “double-angle plots”. Only then can the astigmatism be “represented” as a mathematical vector in a 2D space where vectorial operations can be performed (black lines in Figure 1).
Following vectorial calculations, one may elect to keep and display the vectors in the double-angle “mathematical representation” or to use a single-angle plot to go back to the “clinical representation”, conventionally reported between 0 and 180º by optometrists and ophthalmologists.
Vectors may have a different meaning in mathematics, and physics, compared to ophthalmology. A mathematician will agree that astigmatism can be represented as a vector mathematically. Nevertheless, in our very specific context of clinical ophthalmology, it is essential to understand that astigmatism is not a “calculated vector”. Astigmatism is a “measured” quantity and not a vectorial analysis of change. Only “calculated vectors” such as the target-induced astigmatism (TIA) vector, the surgically-induced astigmatism (SIA) vector, or the difference (DV) vector are vectors quantifying vectorial changes. These concepts are best illustrated in Figure 1.
In our small ophthalmic world, we should try to simplify complexities of which there are undoubtedly many. Making the above distinction between astigmatism and “calculated vectors” makes the ophthalmic world a little simpler to comprehend. It might be worth spending some time to reach an agreement on these concepts in our scientific community.
Figure 1. Concept of vectors quantifying actual vectorial change of astigmatism (TIA, SIA, and DV) versus preoperative clinical astigmatism, target astigmatism, and postoperative astigmatism (black lines).
Axis orientation convention
When we started performing vectorial analyses, we naively and legitimately assumed that the American National Standards Institute (ANSI) paper  on vectorial analyses of astigmatism was the universally accepted method and terminology that authors had to adhere to when performing vectorial analyses. After all, the acronym “ANSI” commends a lot of credibility and remains the method requested by the FDA for excimer laser trials.
Utilizing the ANSI paper methodology as the gold standard approach ultimately led to a mix-up in the first version of our open source AstigMATIC software.
Let us explain. In Table 2 of the ANSI paper, one can see that the TIA axis of 80º, termed IRC or “intended refractive correction” by the authors, is equal to the preoperative refractive cylinder axis, also reported as 80.º In our paper describing AstigMATIC  we accordingly wrote that the target-induced astigmatism (TIA) axis was equal to the preoperative refractive astigmatism axis, in agreement with the ANSI paper .
Authors from around the world (Cuba, USA, Turkey, Spain) contacted us with confusion over this simple orientation convention. They did not understand why the TIA axis in our AstigMATIC graphs were not 90º away (perpendicular) from the preoperative refractive cylinder axis, as per the Alpins Method. We realized that this was a fundamental “convention” issue, rather than a “methodological” issue.
Simply put, the cylinder that we need to induce with an excimer laser has to be perpendicular to the preoperative refractive cylinder to be able to cancel it out. Therefore, if the refractive cylinder is 1.00 D X 90º, then the target cylinder (TIA) necessary to neutralize is indeed 1.00 D X 180 º. Put another way, the TIA axis is the same as the negative cylinder refractive axis.
Indeed, the combination of two perpendicular cylinder planes of equal magnitude cancel each other out. This is the rationale for establishing that the TIA should be perpendicular to the refractive cylinder by convention, as per the Alpins Method .
AstigMATIC has since been modified to follow the latter convention. Our first version used the ANSI orientation convention, and as such, the vector angles, and the Angle or Error parameter were not respecting the Alpins Method. We are grateful to all clinicians who contacted us to correct our AstigMATIC software to reflect the Alpins Method convention, hence avoiding unnecessary confusion.
Graphical representation issues
A Journal of Refractive Surgery (JRS) editorial  by Reinstein, Archer, and Randleman stipulates:
“The Journal would prefer authors to use single-angle polar plots rather than double-angle plots”
In contrast, the recently updated instructions for authors in the Journal of Cataract and Refractive Surgery (JCRS) now mentions (link accessed on May 20th, 2020):
“For clarity and uniformity, manuscripts about astigmatism should adhere to terminology and graphical representations described by Abulafia et al.” 
Those JCRS instructions also state: “For details of the Alpins methodology and graphical reporting, please consult the following resources”, pointing to the JRS editorial by Reinstein , which recommends the use of Single-Angle plots.
(1) “Single-angle polar plots do not require any further learning or understanding than what is taught to all ophthalmologists”
(2) “Data plotted on a single-angle plot are directly transferrable to the clinical situation of a topography, treatment, or eye”
(3) “Single-angle plots require less space on the page.”
On the other hand, the terminology and graphical representations described by Abulafia et al.  mentions three key advantages of double-angle plots:
(1) “The x and y scatterplot of the data on a double-angle plot maintains the spatial relationship of each astigmatic value.”
(2) “The double-angle plot allows the display of the magnitude and axis or meridian of the average astigmatism (centroid) and the confidence ellipse”
(3) “Because double-angle plots group data appropriately, qualitative assessment of group data is facilitated. One can easily visualize trends in group data and compare them to other datasets.”
Of note, the axes in the Abulafia double-angle plots are labeled as 45º, 90º, 135º and 180º. While we understand the rationale for reporting them as such, it can create potential confusion in the community. It would be “technically” more correct to report the labels as 90º, 180º, 270º and 360º since those vector angles are doubled.
We recognize all the valid advantages and disadvantages of both graphical representations and that both provide useful information. Regrettably, the lack of consensus and uniformity in graphical reporting standards between key opinion leaders, and now between our two leading subspecialty journals, is encouraging confusion and debate that does not benefit the scientific community.
We were recently asked why we termed our vectors “target-induced astigmatism” (TIA) and “surgically-induced astigmatism” (SIA), as per the Alpins Method, in our AstigMATIC software  instead of “intended refractive correction” (IRC) and “Surgically induced refractive correction” (SIRC), as per the ANSI method . We were even asked if it was possible to change the labels at the top of our graphs.
The fact that we were asked these questions vividly illustrates the confusion that still exists over terminology. Having different terms to characterize the same vectors used for metrics is a real recipe for confusion. The reasons why we should adhere to the Alpins terminology is thoroughly discussed in the excellent Editorial by Reinstein, Archer, and Randleman , as well as in the book Practical Astigmatism: Planning and Analysis by Prof. Alpins .
The topic of vector analyses continues to create confusion as well as highly-polarized debates. This is due to the lack of universal standardization and agreement of definitions of vectors, orientation conventions, preferred graphical representations, and terminology.
The lack of standardization in the refractive surgery community complicates study comparisons, limits research reproducibility, and promotes personal biases and confusion. The latter may lead to additional manuscript revisions and experimental calculations, thereby increasing time to publication, or can even lead to rejection (or acceptance) that is not entirely “objective”.
We would strongly recommend that editors who commonly publish refractive surgery papers adopt an international standard for vectorial analyses of astigmatism across journals.
Until very recently, this was already in place by JRS, JCRS, and Ophthalmology journals, which all agreed to adhering to the Alpins Method as the standard. Why was this consensus recently changed?
The Alpins method works, is scientifically sound and proven and was already chosen as the standard in refractive surgery. Let us all commit to it for the sake of clinical standardization, research reproducibility, and to ensure we are all speaking the same language.
1. Alpins NA, Goggin M. Practical astigmatism analysis for refractive outcomes in cataract and refractive surgery. Surv Ophthalmol 2004; 49: 109-122
2. Eydelman MB, Drum B, Holladay J, Hilmantel G, Kezirian G, Durrie D, Stulting RD,
Sanders D, Wong B. Standardized analyses of correction of astigmatism by laser systems that reshape the cornea. Journal of Refractive Surgery 2006; 22: 81-95
3. Gauvin M, Wallerstein A. AstigMATIC: an automatic tool for standard astigmatism vector analysis. BMC Ophthalmol 2018; 18: 255
4. Alpins N. Practical Astigmatism Planning and Analysis. SLACK Incorporated, 2017
5. Reinstein DZ, Archer TJ, Randleman JB. JRS standard for reporting astigmatism outcomes of refractive surgery. J Refract Surg 2014; 30: 654-659
6. Abulafia A, Koch DD, Holladay JT, Wang L, Hill W. Pursuing perfection in intraocular lens calculations: IV. Rethinking astigmatism analysis for intraocular lens-based surgery: Suggested terminology, analysis, and standards for outcome reports. J Cataract Refract Surg 2018; 44: 1169-1174