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. 2001 Sep;85(9):1127–1138. doi: 10.1136/bjo.85.9.1127

Astigmatism and the analysis of its surgical correction

N MORLET 1, D MINASSIAN 1, J DART 1
PMCID: PMC1724117  PMID: 11520769

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Figure 1  .

Figure 1  

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Diagram showing the principal rays in the formation of an astigmatic image. Falpha is the anterior focal point, convergent light rays from the steeper meridian produce a horizontal line focus. Fbeta is the posterior focal point from the flatter meridian producing a vertical line focus. In this example, when Fbeta is coincident with the retina, simple myopic astigmatism with the rule is produced.

Figure 2  .

Figure 2  

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The ideal cylinder (C) for a given spherical equivalent (SEQ): C = −2SEQ − 0.50 (ie, C = −sphere − 0.25), where SEQ is −0.25 D or less (ie, a myopic correction) and C is a plus cylinder, and C = 2SEQ + 0.50, where SEQ is greater than −0.25 D (ie, a hypermetropic correction).26

Figure 3  .

Figure 3  

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Residual astigmatism produced when a cylinder axis is shifted away from its correct position: (Csinϕ) DS/−2Csinϕ) DC, axis (θ + 45 + ϕ/2), where C cylinder is set at ϕ angle from the true axis θ. Graphs illustrate how reducing the cylinder power to an "optimal value" can minimise the residual astigmatism. The optimal cylinder power in these circumstances is: Ccos2ϕ, where C is the original full dioptric power of the correcting cylinder, and ϕ is the angle the cylinder is rotated away from the correct position. So if the "optimal value" for the correct cylinder power is used the residual astigmatism is equal to Csin2ϕ, where C is the original full dioptric power of the correcting cylinder and ϕ is the angle the cylinder is rotated away from the correct position.16

Figure 4  .

Figure 4  

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Diagram demonstrating the principle of vector analysis of the change in the astigmatic refraction following surgery. The arrow direction represents the axis of astigmatism and the length the magnitude. The principle assumes that a theoretical spherocylinder, the surgical induced astigmatism (SIA or CSURG × β°) is "crossed" with the preoperative refraction to produce the postoperative refraction69: SI/CI × θ° + SSURG/CSURG × β° = SR/CR × (θ + ε)°, CSURG = (CI2 + CR2 − 2 CI CR cos2ε), SCYL = (CI + CSURG −C R)/2, sin2β = (CR/CSURG) sin2ε, SSURG = SR − SI − SCYL; CI is the initial or preoperative astigmatism vector at θ° axis (in "plus" cylinder notation), CR is the resultant or postoperative astigmatism vector at θ + ε° axis ("plus" cylinder notation), CSURG is the surgically induced astigmatism vector (SIA, a theoretical construct) at β° axis (in "minus" cylinder notation), SCYL is the spherical equivalent of all the cylindrical components.

Figure 5  .

Figure 5  

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Diagram demonstrating the principle of decomposition of the astigmatic refraction into x and y values. Cravy56 suggested that cylinder (or astigmatism) of M magnitude at θ° axis maybe characterised by a "y" magnitude at 90° axis (y = M sin θ) and an "x" magnitude at the 0-180° axis (x = M cos θ). Naeser described a single summary value for the astigmatism which he called the polar value of net astigmatism (KP).82 83 85 89-91 This was originally calculated as89: KP = M × (|sin θ| −|cos θ|), but modified later to90: KP = M × (sin2 θ − cos2 θ), where M is the magnitude and θ is the axis of astigmatism, and KP is the polar value referable to the 90° meridian (ie, encompassing the WTR and ATR concept). When sin θ > cos θ, (ie, y > x), the more WTR the astigmatism and the more positive the polar value. Conversely when sin θ < cos θ (ie, y < x) the more ATR the astigmatism and the more negative the polar value.

Figure 6  .

Figure 6  

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Diagram of the Fourier decomposition of a spherocylindrical lens (S = plano/C = −3.25 × θ = 70). The 360° lens surface power (ie, for all meridians) is represented by the thick line. The three Fourier component terms are: a spherical equivalent (SEQ = S + C/2), a cosine astigmatism term (J × cos2θ), and sine astigmatism term (J × sin2θ) where J = C/2.93

Figure 7  .

Figure 7  

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Diagram demonstrating the use of the vector representation of the astigmatic refraction to determine the quantities of surgical error. The magnitude error (CERROR) is the simple subtraction of the absolute CSURG magnitude from the absolute CI magnitude in dioptres cylinder. The direction or axis error (ϕ) is the CSURG axis (β) in minus cylinder notation from the CI axis (θ) in plus cylinder notation.34

Figure 8  .

Figure 8  

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Methods of presenting astigmatism data. The same base data set is used for all the methods of presentation shown in Figures 8, 9, and 10. The data are from two groups of patients who had extracapsular cataract surgery. (A) The change in mean magnitude of astigmatism. (B) The change in the mean Naeser polar value (positive values are with the rule, negative values against the rule). (C) The change in the "expectancy" of astigmatism from 1 week postoperative (no 1) to 12 months postoperatively (no 8) using the "by the rule" or mirror equivalent axis conversion. The "expectancy" is a statistically correct bivariate (or simultaneous) calculation of the magnitude and axis for the grouped astigmatism data (as opposed to the incorrect method of calculating the simple mean of the magnitude or axis independently). This was calculated using the contingency table method of Toulemont.78 The axis values were first converted to mirror equivalent values as shown in Table 2.

Figure 9  .

Figure 9  

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Methods of presenting astigmatism data. The same base data set is used for all the methods of presentation shown in figures 8, 9, and 10. The data are from two groups of patients who had extracapsular cataract surgery. The change in the "expectancy" of astigmatism regraphed as a balloon plot.

Figure 10  .

Figure 10  

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Methods of presenting astigmatism data. The same base data set is used for all the methods of presentation shown in figures 8, 9, and 10. The data are from two groups of patients who had extracapsular cataract surgery. The change in magnitude and axis (converted to mirror equivalent values) of astigmatism and Naeser polar values plotted as frequency histograms at 1 week and 3 months postoperatively.

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Selected References

These references are in PubMed. This may not be the complete list of references from this article.

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