Alpins method of astigmatism analysis
The Alpins Method of astigmatism analysis is a method of astigmatism analysis developed by Australian ophthalmologist Noel Alpins. The seminal publication for the method, which employs vector mathematics, was in the Journal of Cataract and Refractive Surgery in 1993. Since its introduction, the method has been further advanced by Alpins, and has been used or cited by many other investigators involved in analyzing the results of refractive, corneal, and cataract/intraocular lens (IOL) surgical procedures. Although Alpins’ earlier work was sometimes overlooked or underacknowledged in some subsequent publications on the subject, the Alpins Method of astigmatism analysis eventually became a standard in the field, and the foundation for an approach to astigmatism analysis endorsed by the American National Standards Institute (ANSI) Astigmatism Project Group. Based on the use of the Alpins Method in a study of cataract surgical patients receiving phacoemulsification, one commentator lauded the Alpins Method for its “elegance and usefulness…”
Alpins has obtained a number of patents on the method; these patents are programmed into a commercially available ophthalmic surgical analysis system, called ASSORT (Alpins Statistical System for Ophthalmic Refractive Surgery Techniques), designed to help plan and analyze the results of refractive, corneal, and cataract surgical procedures.
In brief, the Alpins Method determines a goal for astigmatism correction and a treatment required to achieve that goal. The method also allows the calculation of the principal components by which an operation fails to achieve its goal, and other components that assist in comparing the results of astigmatism surgery for individuals and groups of individuals. The Alpins Method has become an accepted standard worldwide for reporting the results of studies that include refraction and corneal astigmatism measurements.
In the early 1990s, Alpins first began to examine astigmatism analysis and treatment as they applied to laser modalities. It became apparent to him that the approach to astigmatism at that time was inconsistent and confusing. He noted that many approaches simply compared pre- and postoperative astigmatism magnitude values with no consideration of the axis of astigmatism or the amount of attempted change. Other approaches calculated a mean of the axes. None of the methods assessed the success of the results nor the extent to which surgical goals had been achieved.
A description of surgically induced astigmatism vector (SIA), one of the central measures of the Alpins Method, dates back to the 19th century. Other authors have also contributed to a vector analysis approach to corneal incisions at the 90° and 180° meridia and polar values outside the 90° and 180° meridia. The Alpins Method enabled a corneal astigmatism analysis to be performed even where treatment parameters were based on refractive values. The advent of excimer laser technology (e.g., laser-assisted in situ keratomileusis, or LASIK), however, introduced a conundrum between incisional and ablation techniques; specifically, should treatment be planned according to refractive cylinder values as introduced with laser refractive surgery, or corneal astigmatism parameters as had been customary with incisional surgery.
The Alpins Method provided a consistent, logical approach to quantifying and comparing the success of various refractive surgical procedures, and refining/planning surgery to achieve even better results, both in individuals and groups of individuals receiving refractive surgery. The Alpins Method determines a treatment path and defined astigmatic target that in many instances is not zero, although prior to the Alpins Method zero had been a nearly unanimous, but inachievable, preference.
 Refractive error and astigmatism in the population
The ideal cornea of the eye is a perfect dome with a base that is a perfect circle. For people with astigmatism, the dome is not perfectly spherical and the base is elliptical to one degree or another (Figure 1).
Since the late 1970s, eye surgeons have been taking advantage of the refractive power of the cornea to correct nearsightedness (myopia) and farsightedness (hyperopia). In general, they flatten the dome of the cornea to correct myopia, and steepen the dome of the cornea to correct hyperopia. Either way, the goal is to produce a focused image on the retina, a light-sensitive tissue in the back of the eye that serves like the film in a camera.
Myopia and hyperopia in the absence of astigmatism are said to be spherical. The correction of spherical myopia and hyperopia by changing the shape of the cornea is a relatively straightforward process. When astigmatism is thrown into the equation, however, refractive surgery becomes logarithmically more complicated. It is estimated that as many as half of the patients who are candidates for refractive surgery, who may constitute half of the world's population, have a degree of astigmatism sufficient to be of concern to the refractive surgeon. For these patients, planning, implementing, and analyzing refractive surgery is of upmost importance.
The reader is referred elsewhere for basic descriptions of the cornea and available refractive surgical techniques. The American Academy of Ophthalmology offers a detailed description of current techniques and the respective characteristics of suitable patient candidates for each approach.
 Basics of the Alpins Method
 The golf analogy
The Alpins Method of astigmatism analysis has many parallels to the game of golf (Figure 2).
A golf putt is a vector, possessing magnitude (length) and axis (direction). The intended putt (the path from the ball to the hole) corresponds to Alpins' target induced astigmatism vector (TIA), which is the astigmatic change (by magnitude and axis) the surgeon intends to induce in order to correct the patient's preexisting astigmatism to the derived or calculated target. The actual putt (the path the ball follows when hit) corresponds to Alpins' surgical induced astigmatism vector (SIA), which is the amount and axis of astigmatic change the surgeon actually induces. If the golfer misses the cup, the difference vector (DV) corresponds to the second putt—that is, a putt (by magnitude and axis) that would allow the golfer to hit the cup (the surgeon to completely correct) on a second attempt.
 The double-angle vector diagram
Figure 3 is a double-angle vector diagram (DAVD) that allows calculations in a 360° sense and permits the use of rectangular (Cartesian) coordinates. It is important to note that vectors can only be calculated; they cannot be measured like astigmatism. The analytical technique here simplifies interpretation of differences among preoperative, desired, and achieved astigmatic values, and allows the calculation of the magnitude and direction of surgical vectors. The trigonometry is described in these references:
Line 1 in Figure 3 defines a patient's preoperative astigmatism by magnitude (length of the line) and axis (an angle from the x axis representing twice the patient's measured axis of preoperative astigmatism). Line 2 defines the target astigmatism—that is, the magnitude and axis of the correction the surgeon would like to achieve. Line 3 represents achieved astigmatism—that is, the magnitude and axis of the postoperative astigmatism. The dashed lines are the TIA, SIA, and DV, as described above. The TIA, SIA, and DV, and the description and calculation of their various relationships, comprise the essence of the Alpins Method.
- Correction index (CI)—The ratio of the SIA to the TIA—what the surgery actually induced versus what the surgery was meant to induce. The CI is preferably 1; it is greater than 1 if an overcorrection occurs and less than 1 if there is an undercorrection. The CI is calculated by dividing the SIA (actual effect) by the TIA (target effect).
- Coefficient of adjustment (CA)—The inverse of the CI, the CA quantifies the modification needed to the initial surgery plan to have achieved a CI of 1, the ideal correction. If the surgeon achieves an overcorrection, for example, the CA might by 0.9, indicating that the surgeon should have selected a correction 90% of what was actually selected. The CA can be used to refine nomograms for future procedures.
- Magnitude of error (MofE)—The intended correction minus the actual correction in diopters.
- Angle of error (AE)—The angle described by the vectors of the intended correction versus the achieved correction (SIA minus TIA). By convention, the AE is positive if the achieved correction is on an axis counterclockwise to where it was intended, and negative if the achieved correction is clockwise to its intended axis.
- Index of success (IOS)—The IOS is calculated by dividing the DV (how far the target is missed) by the TIA (the original target effect). The IOS is a relative measure of success; that is, if golfer John attempts a long putt and golfer Bob a shorter one, and each ends up the same distance from the cup, John's putt can be considered more successful because he had the longer initial putt and a lower IOS (zero being perfect). The IOS is a valuable measure of the relative effectiveness of various surgical procedures.
Unlike previous available approaches to astigmatism analysis, the indices Alpins describes can be subjected to conventional forms of statistical analysis, generating averages, means, standard deviations, etc., for each individual component of surgery.
 Regular and irregular astigmatism
There are 3 types of astigmatism: (1) naturally occurring regular astigmatism; (2) naturally occurring irregular astigmatism; and (3) irregular astigmatism associated with disease, trauma, or prior ocular procedures. The Alpins Method applies mainly to the first 2 types of astigmatism.
Although irregular astigmatism is commonly associated with prior ocular surgery and trauma, it is also naturally occurring and prevalent. Corneal topography or computer-assisted videokeratography (CAVK)—a technique that produces an image map based on the refractive power of the cornea at many discrete points on its surface—shows that irregular astigmatism comes in various configurations. The 2 steep hemimeridians, 180° apart in regular astigmatism, may be separated by less than 180° (a situation called nonorthogonal); and the 2 steep hemimeridians may be asymmetrically steep—that is, one may be significantly steeper than the other, as shown by a larger magnitude value. The continuation of corneal irregular astigmatism can be quantified in diopters (D) as the topographic disparity (the vectorial difference between the 2 opposite semimeridian values for magnitude and meridian) when displayed using a DAVD.
Unlike other astigmatism analysis approaches, the Alpins Method can independently analyze the 2 hemimeridians of irregular astigmatism. This capability assumes greater importance as refractive lasers gain the ability to treat discrete parts of the cornea.
 Topography versus refraction
The Alpins Method offers insight into a common situation where astigmatism as measured by refraction (the well-known test where various lenses are placed in front of the eye while the doctor asks, "Which is better, this or this?) differs from the astigmatism as measured by keratometry and corneal topography, tests considered more objective and quantitative. A refraction identifies the myopic or hyperopic correction, as well as the magnitude and axis of total astigmatic correction needed for clear vision. However, as with the patient shown in Figure 4, most people with astigmatism demonstrate differences in magnitude and axis between corneal topographic astigmatism (T) and refractive astigmatism (R). This difference can be quantified by calculating the vectorial difference between refractive and corneal astigmatism, and is known as the ocular residual astigmatism (ORA). Figure 4 demonstrates 3 different approaches (Figures 4A, 4B, and 4C) to handling a patient who has such a discrepancy; that is, the surgeon can treat based 100% on the corneal topographic astigmatism (Figure 4A); 100% on the refractive astigmatism (Figure 4B); or at some point in between (Figure 4C).
(Abbreviations used in Figure 4: DAVD, double-angle vector diagram; ORA, ocular residual astigmatism; R, refractive astigmatism; T, corneal topographic astigmatism; TIA, target induced astigmatism.)
Faced with a discrepancy between T and R as shown in the patient in Figure 4, most refractive surgeons would treat the patient's refractive (spectacle) astigmatism in the belief that reshaping the cornea to the patient's refractive preference will produce better visual results. However, Alpins has always contended that treating R may do nothing to alleviate T, and actually can result in increased corneal topographic astigmatism, violating fundamental principles of corneal surgery. Alpins therefore described an "optimal treatment," known as vector planning, where greater surgical emphasis is put on topographic astigmatism the more unfavorably the target astigmatism falls on the cornea—that is, toward an against-the-rule or even oblique orientation.
Additionally, in a small prospective study, Alpins and Stamatelatos recently showed that combining wavefront with vector planning provided better visual outcomes than using wavefront planning alone.
 Alpins Method combined with wavefront technology
A description of wavefront technology and its use in refractive surgery can be found in the listing for LASIK, a type of refractive surgery. In the early 2000s, wavefront technology was seen as a possible "holy grail" that may provide "super vision" for refractive surgery patients. In an invited 2002 editorial titled "Wavefront Technology: A New Advance That Fails to Answer Old Questions on Corneal vs. Refractive Astigmatism Correction," Alpins was one of the first ophthalmologists to raise a cautionary note. In that publication and others he pointed out the importance of combining vector analysis, often overlooked at the time, with the then-standard use of corneal topography and optical/refractive measurements.
Alpins' observation was confirmed in a small, prospective, masked study of patients receiving LASIK (21 eyes in 14 patients), published in 2008. Alpins and Stamatelatos found a greater reduction in corneal astigmatism and better visual outcomes under mesopic conditions using wavefront technology combined with vector analysis (the Alpins Method) than using wavefront technology alone, and also found equivalent higher-order aberrations. Noting this study, other investigators have acknowledged that a combined approach will be "the treatment approach of the future."
Alpins' contention is that the purely refraction-based approach represented by wavefront analysis may hold true for scientific instruments such as the Hubble space telescope, but may not be true for the living eye-brain system, and may contradict corneal surgical experience developed over many years. Refractive surgeons have long known that corneal regularity is the foundation of a superior visual outcome. If all corrections for internal optical errors are surgically sculpted onto the cornea, corneal irregularity can only increase. Additionally, wavefront analysis does not take into account the cerebral integration of visual images. A surgical approach that includes the patients' conscious perception of their astigmatism is likely to enhance patient satisfaction.
Alpins believes that the pathway to "supernormal vision" requires a greater customized reduction of corneal astigmatism than is usually attempted, and that any remaining astigmatism ought to be regular (as opposed to irregular), both fundamental principles of vector planning that are overlooked by a purely wavefront-guided treatment plan.
No good data can be found that compare the percentage of LASIK procedures that employ wavefront guidance versus the percentage that do not, nor the percentage of refractive surgeons who have a preference one way or the other. Wavefront technology continues to be positioned as an "advance" in LASIK with putative advantages; however, it is clear that not all LASIK procedures are performed with wavefront guidance.
 The ASSORT program
Alpins founded the company ASSORT Pty. Ltd., of Cheltenham, Victoria, Australia, to commercialize the Alpins astigmatism analysis methodology. The company offers:
- ASSORT—An ophthalmic surgical management system capable of analyzing all measurable ophthalmic parameters, such as intraocular pressures and medications, visual acuities, and personalized A constants for cataract surgery. The ASSORT program also incorporates the Alpins Method for astigmatism surgery planning and analysis. Surgical techniques can be compared in differing patient groups, and pre- and postoperative events can be documented and analyzed.
- iASSORT—Performs astigmatic analyses using the topography and/or wavefront values provided by the diagnostic instrument into which the software has been installed. By selecting the review visits required, iASSORT® will import the required parameters (Sim Ks from topography and second-order astigmatism values from aberrometry) and display the analyses. The iASSORT program is used in most available corneal topographers, and discussions are underway to enhance iASSORT by including a new corneal measurement called CorT, which is described in the next section.
- VECTrAK—A comprehensive, simple-to-use astigmatic vector calculator developed for ophthalmic surgeons for implanting/exporting multiple eyes after surgery. VECTrAK can determine astigmatic changes occurring following cataract, incisional, and laser surgical procedures.
- Online calculators—Free Web-based applications for toric IOL calculations and femto LRI (limbal relaxing incision) nomograms. The toric IOL calculator helps determine the most effective (or closest to plano) toric power to use among all available toric IOLs. The femto LRI calculator helps plan LRIs and determine nomogram adjustments.
 Use of CorT in Refractive and Cataract/IOL Surgery
 The CorT concept
Alpins et al have developed a vector analysis algorithm, which they call CorT, to quantify corneal topographic astigmatism. The authors compared CorT with manual keratometry, simulated keratometry (SimK), corneal wavefront, and paraxial curvature matching. CorT proved to be significantly more reliable (less variable with smaller standard deviation) than the other approaches (Figure 5). The magnitude of Ocular Residual Astigmatism (ORA) using CorT was the least and its magnitude was closest to refractive cylinder than all other parameters examined.
The variability of SimK values has been a point of frustration in the field, most recently in regard to its meridian for the alignment of toric IOLs. Alpins expects that CorT will replace the SimK value, which has been the standard measure since the inception of Placido ring topography technology.
Unlike SimK, the CorT value uses measurements from the entire cornea and takes a vectorial average of Placido ring powers across the whole cornea—a novel approach and one that holds much promise. To obtain a CorT value, rather than using one Placido ring for calculating corneal astigmatism as is the case with SimK, the CorT measurement uses all Placido rings and takes a vectorial average across the entire cornea (and each of its hemidivisions) after determining the most effective set of complete rings. The authors are programming CorT into the iASSORT software, which is already used in many types of corneal topographers located in ophthalmology offices around the world.
 CorT in refractive surgery
The authors’ original description of CorT was based on a retrospective review of 971 eyes of 498 patients aged 19 to 64 years who were seen for refractive surgery assessment. CorT was compared with manual keratometry, SimK, corneal wavefront, and paraxial curvature matching. The standard deviation for CorT was significantly less than for the other measures of astigmatism (P<.001).
iASSORT imports topographic data (SimK values, at present) from topographers or second-order astigmatism values from aberrometers. The surgeon manually enters the sphero-cylinder as measured by manifest refraction or from wavefront aberrometry. From these parameters, the program calculates the ORA, which is a calculation of astigmatism due to noncorneal surface causes as noted above. The software also calculates how much total astigmatism potentially can be corrected by reshaping the cornea, as well as the power and axis of surgical correction required to achieve the maximum potential astigmatic correction.
This information can be used before surgery to help manage patient expectations. For 25% to 30% of patients with ORA values of 1.0 D or more, it may not be possible to achieve the degree of astigmatism correction they might expect, and patients can be counseled accordingly. Preoperative measures also are used to set a target for corneal astigmatism that minimizes total astigmatism after surgery. A target—which can be non-zero—is essential to analyze outcomes, as it provides a baseline for comparison in individuals and groups of individuals, and thus can be used to calibrate future adjustments.
 CorT in cataract/IOL surgery
CorT can provide a reliable meridian (and magnitude) for the alignment (and power) of a toric IOL. The current standard practice amounts to estimating the meridian from multiple differing inputs (e.g., SimK, manual keratometry, IOL master keratometry, and Lenstar), which often differ in the meridian identified as “steepest.” CorT provides the most reliable orientation of the steep meridian and the most effective magnitude of the corneal astigmatism.
CorT is also an accurate and reliable way of measuring irregular corneal astigmatism. CorT can provide measurements of the 2 semimeridians of the cornea, and a separate calculation for each half of the cornea, which generates a magnitude value and a meridian value for each half of the cornea. When these calculations are added vectorially, the result is a CorT for the entire cornea. When these values are subtracted vectorially, the surgeon can determine the dioptric difference of one half of the cornea from the other half (called topographic disparity, or TD). TD provides a common measurement tool, where all topographers can have a common measuring gauge for corneal irregularity.
 iASSORT availability
The iASSORT software is now interfaced with more than 10 brands of topographers. The first 4 that may contain CorT capability are expected to be the Humphrey Atlas (Carl Zeiss Meditec, Jena, Germany), Pentacam (Oculus, Wetzlar, Germany), the Sirius Corneal Topographer (CSO, Scandicci, Italy), and the Galilei (Ziemer Ophthalmic Systems, Port, Switzerland).
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