School of Optometry
Indiana University
Bloomington, IN 47405
email: thibos
indiana.edufax: 812-855-7045
Address for correspondence:
Larry N. Thibos
School of Optometry
Indiana University
Bloomington, IN 47405
email: thibosindiana.edu
fax: 812-855-7045
No proprietary interests.
Supported by National Institutes of Health grant R01 EY05109 to
LNT.
Purpose: To demonstrate the power vector method of representing and analyzing sphero-cylindrical refractive errors.
Setting: School of Optometry, Indiana University, Bloomington, Indiana, USA
Methods: Manifest and keratometric refractive errors were expressed as power vectors suitable for plotting as points in a 3-dimensional dioptric space. The three Cartesian (x, y, z) coordinates of each power vector correspond to the powers of three lenses which, in combination, fulfill a refractive prescription: a spherical lens of power M, a Jackson crossed-cylinder of power J0 with axes at 90º and 180º, and a Jackson crossed-cylinder of power J45 with axes at 45º and at 135º. The Pythagorean length of the power vector, B, is a measure of overall blurring strength of a sphero-cylindrical lens or refractive error. Changes in refractive error due to surgery were computed by the ordinary rules of vector subtraction.
Results: Frequency distributions of blur strength (B) clearly demonstrate the effectiveness of refractive surgery at reducing the overall blurring effect of uncorrected refractive error. Power vector analysis also revealed a reduction in the astigmatic component of these refractive errors. Paired comparisons revealed that the change in manifest astigmatism due to surgery was well correlated with the change in keratometric astigmatism.
Conclusions: Power vectors aid the visualization of complex changes in refractive error by tracing a trajectory in a uniform dioptric space. The Cartesian components of a power vector are mutually independent, which simplifies mathematical and statistical analysis of refractive errors. Power vectors also provide a natural link to a more comprehensive optical description of ocular refractive imperfections in terms of wavefront aberration functions and their description by Zernike polynomials.
The mathematical representation and statistical manipulation of sphero-cylindrical refractive errors is a recurring topic in the ophthalmic literature. 1-12 Much of this debate has revolved around the difficulties which arise when astigmatism is represented in the traditional polar form of magnitude and axis, rather than the more mathematically tractable Cartesian form. In general, statistical analysis of angular data (e.g. astigmatism axes, compass bearings, angle of disappearance of migratory birds over the horizon, etc.) is fundamentally different from the analysis of non-directional data.13 Consequently, the inappropriate application of conventional statistical methods to directional data can give very misleading results. If, however, astigmatism is represented in rectangular vector form, then conventional scalar methods may be applied separately to each vector component. Furthermore, standard multivariate statistics can be used to compute population means and variances, define confidence intervals, and test hypotheses.
Power vectors are a geometrical representation of
sphero-cylindrical refractive errors in terms of three fundamental
dioptric components.12 The first
component is a spherical lens with power M equal to the
spherical equivalent of the given refractive error. If this spherical
power is removed from the prescription, the result is a Jackson
crossed-cylinder (JCC) equivalent to a conventional cylinder of
positive power J at axis 
+90º
crossed with a cylinder of negative power -J at axis
.
By convention, we describe this astigmatic component as a JCC of
power J at axis (the
meridian of maximum positive power). This JCC can be further resolved
into the sum of two other JCC lenses, one with power J0 at
axis =0º=180º
and the other with power J45 at axis =45º.
Therefore, by this decomposition method we are able to express any
sphero-cylindrical refractive error by the three dioptric powers
(M, J0 , J45 ). It is convenient to
interpret these three numbers geometrically as the (x, y, z)
coordinates of a point in a three-dimensional dioptric space. Thus a
power vector is that vector drawn from the coordinate origin of this
space to the point (M, J0 , J45 ). The
length of this vector is a measure of the overall blurring strength
B of a sphero-cylindrical lens or refractive error.14,
15
The primary advantage of representing refractive errors by power vectors is that each of the three fundamental components of a power vector are mathematically independent of each other. In other words, a spherical lens cannot be produced by any combination of JCC lenses, a JCC lens with axis 0º cannot be produced by any combination of spherical lenses with JCCs at axis 45º, and a JCC lens with axis 45º cannot be produced by any combination of spherical lenses with JCCs at axis 0º. This notion of independence, which is formalized in the mathematical concept of orthogonality, greatly simplifies practical problems involving the combination, comparison, and the statistical analysis of sphero-cylindrical lenses or refractive errors. We demonstrate these advantages in this report by analyzing refractive data for 100 eyes treated with refractive surgery to correct myopia and astigmatism.
Refractive data were supplied by Dr. Koch as manifest refractions
and corneal keratometry readings, before and after LASIK surgery.
Manifest refractions in conventional script notation (S,
C x )
were converted to power vector coordinates and overall blurring
strength by the following formulas:
These formulas apply regardless of whether the script is written in positive-cylinder form or negative-cylinder form, provided that C is a signed number (i.e. C > 0 for +cyl form, C < 0 for -cyl form)12.
Keratometry data were recorded as the power and meridian of maximum power, and the power and meridian of minimum power. These primary meridia were not always perpendicular, which suggested the presence of small amounts of measurement error and/or irregular astigmatism. For present purposes we dealt with this uncertainty by averaging the two given meridia and assuming the actual primary meridia were oriented ± 45° from this average meridian.
The change in refraction caused by some treatment (refractive surgery in the present case) is computed by the usual rules of vector subtraction. Since the power vectors P are given in rectangular coordinates, the difference vector is easily computed by subtracting corresponding values along each of the coordinate axes separately.
Table 1. Statistical summary of the distribution of manifest refractive errors before and after refractive surgery, referenced to spectacle plane.
We sought answers to the following pair of questions. First, was the refractive surgery successful in correcting the patient's refractive error? Second, was the change in manifest refraction produced by the surgery fully accounted for by change in corneal refraction? To answer the first question we examined the statistical distribution of manifest refractive errors before and after treatment. This was done by expressing each pre-surgical and each post-surgical refractive error as a power vector and separately computing the mean and standard deviation of each vector component. We also computed the length of each power vector and a statistical summary of the results is shown in Table 1. To get an overall sense of the effect of surgery on manifest refractive error, we collected the values of blur strength B into the frequency histograms of Fig. 1. The results clearly demonstrate that refractive surgery compressed a wide range of pre-surgical refractive errors into a narrow distribution near emmetropia. This visual impression is substantiated quantitatively by the numerical data in Table 1. For comparison, we also collected histograms of the spherical equivalent M of each refractive error as shown in Fig. 2. A comparison of Figs. 1 and 2 show that the post-surgical distribution of blur strength is narrower than the distribution of spherical equivalent. This difference is due to the fact that blur strength does not distinguish between myopic and hyperopic blur.
Figure 1. Reduction in overall blurring strength B of manifest refractive error as a result of refractive surgery. Values referenced to the spectacle plane.
Figure 2. Reduction in spherical equivalent M of the manifest refractive error as a result of refractive surgery. Values referenced to the spectacle plane.
To visualize the change in astigmatism caused by refractive surgery, we show in Fig. 3 the astigmatic component of the power vector as represented by the two-dimensional vector (J0, J45), which is the projection of the power vector into the astigmatism plane formed by the coordinate axes (J0, J45). For clarity, we do not show the entire vector extending from the origin but instead show only the endpoint of the vector. Since the origin in this graph represents the an eye free of astigmatism, we expected to see the cluster of points to collapse around the origin following surgery and this was indeed the result obtained.
A formal statistical test of the null hypothesis that the
population mean of post-surgical power vectors is equal to zero can
be constructed two ways. First, we may conduct a series of three
monovariate t-tests of the hypotheses that the mean power
vector component in each of the three dimensions is zero. The results
of these tests, conducted at the 0.01 level, indicated that neither
astigmatism component is significantly different from zero, but the
spherical component does differ significantly from zero. The second
approach is to use Hotelling's T2 test of
multivariate statistics to ask if the mean power vector is
significantly different from a vector of zero length.16
This test generates an F-statistic which may be compared against
tabulated values of Fisher's F-distribution. The computed F-statistic
for the post-surgical, three dimensional power vectors was 18.47.
This value is highly significant compared to the tabulated F-value
for 3 and 97 degrees of freedom of 3.99 at the =0.01
significance level, which indicates that the overall refractive error
was not fully corrected in this population of eyes. However, when the
analysis was repeated on the two-dimensional astigmatism component of
refractive error, the computed F-statistic was 2.35, which is not
significant at the 0.01 level. This result leads us to conclude that
the astigmatism component of refractive error was well corrected by
surgery in this population of eyes.
Figure 3. Manifest astigmatism before surgery (A) is reduced on average following surgery (B). Each data point represents the astigmatism component of a power vector for one eye, referenced to the spectacle plane.
To answer the second question posed above, we transferred manifest
refractive errors from the spectacle plane to the corneal plane
(assuming a vertex distance of 14 mm) so that they could be directly
compared with keratometry data.17 These
refractive errors were then converted to power vectors to facilitate
calculation of the change in manifest refractive error and the change
in corneal refractive error produced by the surgical treatment. Thus
for each eye we have a pair of power vectors representing the change
in manifest refraction and the change in corneal refraction, as
illustrated schematically in Fig. 4. If the change
in manifest refraction was due solely to the change in corneal power,
then the two power vectors should be parallel and of equal length, so
their difference should be zero. To test this prediction we computed
the difference between pairs of change vectors for every eye and the
results are shown in Fig. 4. As expected, the
population of difference vectors is tightly clustered about the
origin. For this sample of eyes, 40% of all points are within 0.25 D
of the origin and 80% of all points are within 0.50 D of the origin.
Such small discrepancies are within normal clinical tolerances for
measurement of refractive error, so we conclude that the change in
manifest refraction following surgery can be accounted for by the
change in corneal refraction. This conclusion was substantiated by
the results of a statistical T2 test which indicated that
the mean of the data in Fig. 4 is not significantly
different from zero at the =
0.05 level.
Figure 4. Power vector test of the hypothesis that the change in manifest astigmatism was due to the change in corneal astigmatism produced by surgery. For this hypothesis to be true, the two astigmatism vectors with large arrow heads shown in the upper diagram should be parallel and the same length. Filled and open symbols represent astigmatism before and after surgery, respectively. The difference between the two change vectors is shown by the vector with a small arrowhead. The distribution of this latter quantity is displayed for a sample of 100 eyes in the scatter graph below.
Manifest Change Corneal Change Difference M J0 J45 M J0 J45 M J0 J45 Mean 5.444 -0.352 0.016 4.118 -0.343 0.029 1.326 -0.008 -0.013 Standard Deviation 2.049 0.42 0.3011 1.582 0.359 0.302 0.776 0.283 0.261 Table 2. Statistical summary of the change in astigmatism produced by refractive surgery, referenced to corneal plane. Manifest refractive errors were transferred from spectacle plane to corneal plane prior to this statistical analysis.
The statistical summary of the change in astigmatism produced by refractive surgery reported in Table 2 indicates that the manifest change in spherical equivalent M was 1.3D greater than the corneal change in M on average. In other words, the manifest degree of myopia was reduced 1.3D more than could be accounted for by the change in corneal curvature on average. This discrepancy may indicate an inappropriate choice of refractive index used by the keratometer for converting corneal curvatures to refractive power.17
One conceptual advantage of power vectors is that they represent a complicated entity (a sphero-cylindrical lens or refractive error) as a simple point in a 3-dimensional space. If the optical characteristics of an eye change over time due to surgery, contact lens wear, or other forms of treatment, injury or disease then the trajectory of this point graphically depicts the resulting changes in refractive error. In the present study we followed the trajectory of 100 such points to gain insight into the optical outcome of refractive surgery. We found that the change in manifest refractive error was well correlated with the change in corneal refractive error. Although the difference between these two changes was small, it was greater than could be accounted for by statistically variability. Perhaps the post-surgical corneas were more aberrated than the pre-surgical corneas,18 which would have rendered keratometry less reliable. To pursue this suggestion would require a more comprehensive optical assessment before and after surgery of the higher-order aberrations of the eye, like that obtained with a Shack-Hartmann aberrometer.19
Power vectors were conceived as a way of transforming a conventional sphero-cylindrical refractive error into independent, orthogonal components better suited to mathematical and statistical analysis. In a broader sense, all of the classical optical aberrations of eyes (e.g. coma, spherical aberration, etc.), as well as the higher-order aberrations which are loosely grouped together by the term "irregular astigmatism", may be similarly transformed into orthogonal components which are mutually independent. The traditional basis functions used for such a decomposition are the Zernike polynomials.20 This series of analytical functions includes three polynomials of the second degree which describe (1) the spherical wavefront aberrations produced by defocus, (2) the toric wavefront aberrations produced by with-the-rule or against-the-rule astigmatism, and (3) the toric wavefront aberrations produced by astigmatism with axes at 45º or 135º. The weighting coefficients applied to each of these three basic forms of wavefront aberration required to describe any particular eye correspond to the three components of a power vector. Thus, another advantage of the power vector approach is that it creates a natural link from current clinical practice to an optical theory powerful enough to provide a comprehensive description of all of the refractive imperfections an eye. This link will become increasingly important as clinicians aim to go far beyond the correction of defocus and astigmatism to achieve total vision correction and a perfect retinal image.21