Theoretical analysis of wavefront aberration from treatment decentration with oblique incidence after conventional laser refractive surgery

Fang Lihua; He Xingdao; Chen Fengying

doi:10.1364/OE.18.022418

1. Introduction

Excimer laser refractive surgery was designed to correct myopia, hyperopia, and astigmatism of the eyes and has become one of the most popular methods for correcting refraction in recent years. This is mainly due to recent advances in surgical techniques [1], which have enabled surgeons to achieve the desired refraction postoperatively, in most cases [2,3]. Refractive surgery can be implemented to change the curvature of the anterior corneal surface to achieve vision correction (the postoperative corneal surfaces being flatter for myopic spherical correction and steeper for hyperopic correction).

According to the data from clinical measurements, optical aberrations in human eyes include higher-order aberrations beyond defocusing and astigmatism. In general, lower-order aberrations have a large proportion of the aberration structure in an ametropic eye, so they play a more important role in the impact on visual function than do high-order aberrations. The goal of conventional vision correction is to correct these lower-order aberrations, namely defocusing and astigmatism. Based on a mathematical model of the anterior corneal surface, a treatment profile for conventional refractive surgery can be calculated from the clinical refractive errors and the k value of the cornea. Actually, fitting errors in the mathematical model of the cornea may affect the postoperative image quality in human eyes. Thus the treatment profile from the mathematical model, which describes the relationship between refractive errors and corneal ablation depth, is a key issue in refractive surgery. Finding a treatment profile for refractive surgery was first investigated by Munnerlyn et al. [4]. Chang et al. [5] showed the relationship between corneal ablation depth and the Munnerlyn formula. A mathematical model based on the corneal toric surface (using for simple myopia astigmatism, simple hypermetropia astigmatism, compound myopia astigmatism, compound hypermetropia astigmatism, and mixed astigmatism) was proposed by Shun et al. [6] However, there is a lack of theoretical analysis on the induced aberrations from these treatment profiles, so this issue deserves further study.

Moreover, patients are in a supine position in corneal refractive surgery, but they are in a seated position in the preoperative examination. A low to moderate amount of eye cyclotorsion has been observed in the transition from the seated to the supine position [7–9]. Additionally, the pupil size of a human eye changes due to a change of illumination or accommodation, and the pupil constriction is most likely not to be concentric [10]. The shift of the pupil center in refractive surgery will lead to treatment translation of the cornea because the illumination level for laser refractive surgery is inconsistent with that of the preoperative examination [11]. Clinical treatment decentration implies that the ideal ablation zone of the cornea is inconsistent with the practical ablation zone [12–14]. As a result, correction of lower-order aberrations such as defocusing and astigmatism is accompanied by an increase in postoperative higher-order aberrations, which have a highly significant correlation to visual function [15–19]. In theory, practical corrected aberrations of the cornea can be calculated by the translation and rotation transformation of ideally corrected aberrations [20]. Guirao et al. [21] have also theoretically investigated the impact of translation and rotation on individual Zernike terms by ocular wavefront transformation.

In addition, laser ablation is centered on the visual axis (defined as the broken line connecting the fixation point with the fovea, passing through the nodal points) of the eye in refractive surgery. In fact, because the model of the anterior corneal surface is an ideal toric surface, the laser beam is moved vertically parallel to the optical axis (defined as the line that most closely connects the four Purkinje–Sanson images) of the eye in theoretical analysis. The laser beam is approximately at normal incidence on the central area of pupil, and yet it is at oblique incidence in the peripheral area [22]. Jiménez et al. [23] have shown that the influence of reflection losses and non-normal incidence on the ablation depth of the cornea may be significant. The correction factor may be up to 9% for an incidence height of y = 2 mm and a corneal radius of R = 7.7 mm when these aspects are taken into account. Therefore, the laser-ablation algorithms that take reflection loss and non-normal incidence into account may be more accurate and more beneficial to the visual function after refractive surgery. The angle of incidence of the laser beam on the cornea can be calculated by a mathematical model of the anterior corneal surface for pure myopia and myopia astigmatism correction.

The ablation equation proposed by Shun et al. [6], as well as the Munnerlyn formula, is used to calculate the ablation depth of the cornea in the optical zone for corrected refraction. Based on the equation, what individual Zernike terms should be induced from these ablation profiles with treatment decentration, and what is the relationship between the amount of aberration and the degree of decentration? These questions deserve further study.

In this research, the relationship between induced aberrations with oblique incidence and corrected refraction was studied. Induced aberrations from treatment decentration were also researched. This was based on the ablation profile of the cornea for conventional laser refractive surgery. Theoretically, an effective strategy for designing a method of visual correction must reduce treatment decentration and take oblique incidence into account in order to minimize the impact of postoperative aberrations on optical quality in human eyes.

2. Methods

2.1 Mathematical model of anterior corneal surface

The anterior corneal surface model provides a way in which to determine corneal ablation depth in refractive surgery. How to establish such a mathematical model may be summarized in the following way. Strictly, the astigmatism component is universally included in refractive errors of the human eye. Naturally occurring astigmatism is regular astigmatism, generally. Corneal radii of curvature of various meridians are not equal, one of which has a minimum radius of curvature (the steepest meridian) and represents the strongest refractive power. The flattest meridian (the largest curvature radius), which is perpendicular to the steepest meridian, depicts the weakest refractive power. The steepest and flattest meridians are defined as the principal meridians. Therefore, the anterior corneal surface can be modeled as a surface with two principal meridians with different radii of curvature [6]. The radii of curvature of the principal meridians of the corneal surface can be conveniently denoted by R_ix and R_iy, respectively. R_ix represents the radius of curvature of the flattest meridian, and R_iy conveys that of the steepest meridian. As for simple myopia or simple hyperopia, R_ix = R_iy. Figure 1 sketches the anterior corneal surface of myopia astigmatism.

Fig. 1 Principal meridians for anterior corneal surface with the curvature radii of the steepest and flattest meridians R_ix and R_iy.

Download Full Size | PDF

A mathematical model of the anterior corneal surface can be shown as

Z_{i} (x, y) = \sqrt{{(\sqrt{R_{i x}^{2} - x^{2}} + R_{i y} - R_{i x})}^{2} - y^{2}} .

2.2 Effect on laser-ablation algorithms of oblique incidence on the anterior cornea

In the laser technology of refractive surgery, one of the decisive factors of the ablation depth per pulse is the laser energy per illuminated area. Additionally, the laser beam may be at oblique incidence on the anterior cornea when the laser beam is moved vertically parallel to the optical axis of the cornea. Two aspects in the effects of oblique incidence on the effective corneal ablation depth are important. First, when the ablation is performed outside the optical axis, the illuminated area of the cornea changes due to the oblique incidence. Second, reflection loss of laser energy on the corneal surface changes with the incidence angle of the laser beam. Hence, the exposure received at one point of the cornea (F₀’) as a function of the incident exposure of the laser (F₀) is

F_{0}' = F_{0} (\cos (α) \cdot (1 - R)) .

Here, α represents the angle between the incident laser beam and the normal vector of the corneal surface. R conveys the reflectivity of the cornea. According to the normal vector of the anterior corneal surface, the cos(α) at one point (x, y, z) is deduced as

\cos (α) = \frac{1}{\sqrt{1 + {(\frac{d}{d x} z (x, y))}^{2} + {(\frac{d}{d y} z (x, y))}^{2}}} .

Here, z(x,y) conveys the anterior corneal surface.

The reflectivity of the cornea (R) is calculated using the Fresnel equations as follows:

R = \frac{1}{2} [{(\frac{n^{2} \cos α - \sqrt{n^{2} - \sin^{2} α}}{n^{2} \cos α - \sqrt{n^{2} - \sin^{2} α}})}^{2} + {(\frac{\cos α - \sqrt{n^{2} - \sin^{2} α}}{\cos α + \sqrt{n^{2} - \sin^{2} α}})}^{2}] .

In this study, n = 1.52 represents the refractive index of the stroma for laser wavelength (λ = 193nm).

The ablation depth per pulse (d) is approximated by Lambert–Beer’s law [24] when the oblique incidence is not taken into account.

d = m \cdot \ln (\frac{F_{0}}{F_{t h}}) .

Here, factor m depicts the ablation depth at a radiant exposure, F ₀ = e* F _th. F ₀ is the radiant exposure and F_th conveys the ablation threshold.

When the oblique incidence is taken into account, the effective ablation depth per pulse (d) can be approximated by the function using Eqs. (2) and (5).

d = m \cdot \ln (\frac{F_{0}}{F_{t h}}) \cdot (1 + c \cdot ln (\cos (α) \cdot (1 - R))) .

Here, the constant is c = 1/ln(F ₀/F_th), which depends on the specific laser used, and the value is 0.88 in this study.

2.3 Ablation profile for conventional laser refractive surgery

Assuming that the cornea is a geometric sphere, only pure myopia or hyperopia components are included in the refractive errors of the whole eye. Based on the Munnerlyn equation, the ablation depth of cornea for myopic correction is given by

l (r) = \sqrt{R_{1}^{2} - r^{2}} - \sqrt{R_{f}^{2} - r^{2}} + \sqrt{R_{f}^{2} - {(S / 2)}^{2}} - \sqrt{R_{1}^{2} - {(S / 2)}^{2}} .

Here, r is the distance of any arbitrary point in the pupil plane to the center of the pupil, and R ₁ conveys the radius of curvature of the anterior corneal surface before refractive surgery. R _f represents the radius of curvature after refractive surgery. S is the diameter of the optical zone. R _f can be calculated as follows:

R_{f} = \frac{1000 (n - 1) R_{1}}{(n - 1) + D_{s p h} R_{1}} .

Here, D_sph depicts the myopic (hence negative) refraction in diopters.

On the other hand, myopia and astigmatism components are generally included in the refractive errors in human eyes, namely myopia astigmatism. In the present paper, we study the ablation depth of the cornea for myopia astigmatism correction shown in Fig. 2 when the anterior corneal surface is a spherical surface after corneal refractive surgery.

Fig. 2 Corneal shape and tissue ablation depth after laser refractive surgery with the curvature radii of the anterior corneal surface. After refractive surgery, R_f; before surgery, R_ix and R_iy; and the diameter of the optical zone S.

Download Full Size | PDF

The corresponding ablation profile of the cornea is given by

l (x, y) = \sqrt{{(\sqrt{R_{i x}^{2} - x^{2}} + R_{i y} - R_{i x})}^{2} - y^{2}} - \sqrt{R_{f}^{2} - x^{2} - y^{2}} + \sqrt{R_{f}^{2} - {(S / 2)}^{2}} + R_{i x} - R_{i y} - \sqrt{R_{i x}^{2} - {(S / 2)}^{2}} .

Here, R_ix and R_iy convey the radii of curvature of two principal meridians of the anterior corneal surface before refractive surgery. R_f represents the radius of curvature after refractive surgery. S is the x-axis diameter of the optical zone.

In practical application, the values of R_ix, R_iy, and R_f for myopia astigmatism correction in individual human eyes are calculated as follows:

R_{i x} = \frac{1000 (n - 1)}{D_{k} + 0.5 D_{c y l}}, R_{i y} = \frac{1000 (n - 1)}{D_{k} - 0.5 D_{c y l}}, R_{f} = \frac{1000 (n - 1)}{(D_{k} + 0.5 D_{c y l} + D_{s p h})} .

Here, D_k represents the measured k value of a cornea from a corneal topographer. The refractive errors in the pupil plane are an equivalent sphere (denoted by D_sph) and cylinder (denoted by D_cyl). The n represents the refractive index value for the cornea, and the value is 1.376 for visible light. Particularly, D_sph and D_cyl are negative for myopia astigmatism correction.

2.4 Decentration of wavefront aberration

When the center of the ablation profile for refractive surgery is inconsistent with the center of the actual aberrations in human eyes, it indicates treatment decentration. Theoretically, treatment decentration can be simulated by ocular wavefront transformation in this section; that is, wavefront aberration coefficients are converted to express them with respect to a different reference frame. For the sake of simplicity, only linear conformal mappings between reference frames are explicitly considered in this study, namely wavefront aberration transformations consisting of lateral displacements and rotations.

Let XY and X’Y’ be two Cartesian reference frames defined in a plane. Figure 3 shows that X’Y’ is displaced and rotated with respect to XY. Let G_R be a circle centered at O, origin of XY, which represents the ideal corrected aberrations. Whereas, let G_R’ be a circle centered at O’, origin of X’Y’, which conveys the actual corrected aberrations. Therefore, the Zernike coefficients for the actual corrected aberrations in XY can be obtained from the transformation of the ideal corrected aberrations.

Fig. 3 Actual corrected wavefront aberrations (axis X’Y’) decentered with respect to the eye (axis XY). The aberrations are the rotated (angle α) version and the translated (Δx, Δy) version of the ideal corrected aberrations.

Download Full Size | PDF

The actual (W_d) and ideal (W_i) corrected wavefront aberrations are denoted as follows [25]:

\begin{array}{l} W_{d} = \sum_{i} C_{i}^{} Z_{i}^{} (x, y), \\ W_{i} = \sum_{k} a_{k}^{} Z_{k}^{} (x, y) . \end{array}

Here, Z(x,y) is the Zernike term defined following [25], whereas C_i and a_i represent the Zernike coefficient.

Figure 3 indicates that any arbitrary point of the plane is determined by its position (x, y, with units of length) in the XY frame and, equivalently, by its position (x’, y’) in X’Y’. The relationship between the coordinates of the point expressed in both reference frames is

\begin{array}{l} x' = (x - Δ x) \cos α + (y - Δ y) \sin α, \\ y' = (y - Δ y) \cos α - (x - Δ x) \sin α . \end{array}

Here, Δx depicts the displacement in the x axis, and Δy represents the displacement in the y axis. α is the angle formed by X’ and X measured counterclockwise from X.

The new Zernike coefficients (denoted by C_i) are calculated from the coefficients (a_k) of the ideal corrected aberrations by matrix multiplication as follows:

C_{i} = \sum_{j, k} T_{i j} R_{j k} a_{k} .

Here, R is the rotation matrix and, equivalently, T is the translation matrix. In addition, C_i conveys the Zernike coefficients for the whole eye.

2.5 Induced wavefront aberration in conventional laser refractive surgery from treatment decentration

The ablation depth (D(x,y)) of the cornea for myopia astigmatism correction in the optical zone can be calculated by the clinical refractive errors based on the mathematical model of the anterior corneal surface. Then the ablation depth is translated into the ideal corrected aberrations (W_i). The Zernike coefficients of the ideal corrected aberrations are obtained by surface fitting. The actual corrected aberrations with treatment decentration are achieved as follows.

The required number of laser pulses for correction in any arbitrary point (x,y) in the optical zone of cornea without treatment decentration is given by the following equation:

N (x, y) = \frac{D (x, y)}{d} = \frac{D (x, y)}{m \cdot \ln (\frac{F_{0}}{F_{t h}}) \cdot (1 + c \cdot ln (\cos (α) \cdot (1 - R)))} .

Here, d represents the effective ablation depth per pulse denoted by Eq. (6).

In this section, the displacement values of the center of the ablation zone are Δx and Δy. The angle of rotation of the ablation zone is α. Consequently, the actual required number of laser pulses in any arbitrary point (x,y) in the optical zone with treatment decentration is denoted as follows:

\begin{array}{l} D' (x, y) = N (x', y') \cdot m \cdot \ln (\frac{F_{0}}{F_{t h}}) \cdot (1+ c \cdot ln (\cos (α) \cdot (1 - R)) \\ = \frac{D (x', y')}{m \cdot \ln (\frac{F_{0}}{F_{t h}}) \cdot (1+ c \cdot ln (\cos (α') \cdot (1 - R'))} \cdot m \cdot \ln (\frac{F_{0}}{F_{t h}}) \cdot (1+c \cdot ln (\cos (α) \cdot (1 - R)) \\ = D (x', y') \cdot \frac{(1+ c \cdot ln (\cos (α) \cdot (1 - R))}{(1+ c \cdot ln (\cos (α') \cdot (1 - R'))} . \end{array}

Here, (x, y) and (x’, y’) are as shown in Fig. 3. Also, α and R represent the parameters of the ideal vision correction, and α’ and R’ are used for the actual vision correction.

Therefore, the wavefront aberrations (W_d) with treatment decentration can be shown: $W_{d} (x, y) = D' (x, y) \cdot (n - 1) .$

In conclusion, induced wavefront aberrations in laser refractive surgery from treatment decentration are calculated as follows:

\begin{array}{l} W_{r} (x, y) = W_{i} (x, y) + W_{d} (x, y) \\ = \sum_{i = 1}^{M} C_{i} Z_{i} (x, y) - \sum_{i = 1}^{M} a_{i} Z_{i} (x, y) . \end{array}

Here, C_i is the Zernike coefficient for vision correction with treatment decentration, and a_i is the coefficient without it.

3. Results

3.1 Induced wavefront aberration for pure myopic and hyperopic correction

We calculated the theoretical exact ablation depth for an optical zone size based on the ablation profile for pure myopic and hyperopic correction. Then Zernike coefficients without treatment decentration were obtained by surface fitting. The transverse translation of the center of the pupil was simulated by ocular wavefront translation transformation. Similarly, eye cyclotorsion in refractive surgery was simulated by wavefront rotation transformation. In refractive surgery, the laser beam is moved vertically parallel to the optical axis of the cornea. The incidence angle (α) of the laser beam at any arbitrary point (x,y) on the cornea is determined by the mathematical model of the anterior corneal surface. Then an adjustment factor (κ) of the ablation depth of the cornea at any arbitrary point in the optical zone is deduced from a change of the illuminated area and a change of the reflection loss of laser energy on the anterior corneal surface with oblique incidence. The impact of the adjustment factor on the ablation depth of a cornea is relatively unknown. We evaluated it using the ratio between the adjustment factors in an ideal and an actual optical zone with treatment decentration. Differences between the ideal and actual corrected aberrations revealed the induced wavefront aberrations in conventional laser refractive surgery. These were multiplied by the ratio of the adjustment factors. Thus, we obtained the induced Zernike coefficients by wavefront surface fitting.

Considering defocus aberration with rotational symmetry, no higher-order aberrations were induced from eye cyclotorsion for pure myopia or hypermetropia correction. Therefore, only relationships among the induced higher-order aberrations, corrected refraction, and amount of decentration were studied. The results have shown that the induced aberrations are mainly coma (C₃₁) and spherical aberrations (C₄₀). The induced trefoil (C₃₃), tetrafoil (C₄₄), and secondary astigmatisms (C₄₂) have small rms values and range from 0.01 to 0.1 μm. Furthermore, the induced coma changes with the corrected refraction and decentration in the x axis. The induced spherical aberration alters with the corrected refraction and decentration as well. Contour maps of these relationships are shown in Fig. 4 . Here, the diameter of the optical zone on the cornea is 6 mm. Panel A shows the induced coma and Panel B shows the induced spherical aberration. In addition, the first row in this figure shows the effects of the corrected defocus or decentration alone on the induced coma as well as on the induced spherical aberration.

Fig. 4 Contour maps of the induced coma and spherical aberration for correcting pure myopia. Panel A corresponds to the coma and panel B corresponds to the spherical aberration. The diameter of the optical zone is 6 mm.

Download Full Size | PDF

The results indicate in Fig. 4(a) that the induced coma increases linearly with an increase in the corrected refraction, and the coma is proportional to the translation of the center of the pupil. The horizontal coma is induced from the translation in the x axis. Additionally, the induced coma for hyperopic correction is larger than for myopic correction in the same magnitudes of refraction. In Fig. 4(b), the postoperative spherical aberration increases theoretically with an increase in the corrected refraction. The sign of the induced spherical aberration for myopic correction is negative, which is opposite the sign for hyperopic correction. However, the spherical aberration maintains almost constant with the change of translation. This result implies that spherical aberration is hardly induced from the translation of the center of the pupil. The reason may be attributed to the fact that spherical aberration is an axial aberration and is defined for objects in the axis. In addition, the fifth- and sixth- order wavefront aberrations are not correlated significantly with the corrected refraction or the value of translation.

3.2 Induced wavefront aberration for myopia astigmatism correction

Clinically, most myopic eyes suffer from not only the myopic component of refractive errors but also the astigmatism component. The corneal ablation depth can be calculated with the ablation profile for myopia astigmatism correction. Similarly, eye cyclotorsion and transverse translation of the center of the pupil were simulated by ocular wavefront transformation. The induced wavefront aberrations in refractive surgery were computed as the difference between the ideal and actual corrected aberration multiplied by the ratio of the adjustment factor. The corresponding Zernike coefficients from treatment decentration were obtained by surface fitting. The results have shown that induced high-order aberrations do not change with the eye cyclotorsion. For this reason, Fig. 5 only shows the curves of the induced third- and fourth-order aberrations versus the translation of aberration in the x axis. In like manner, the diameter of the optical zone is 6 mm. Furthermore, Fig. 6 shows the relationship between the induced astigmatism and the translation of aberration in the x axis. In addition, the first row in this figure shows the effects of rotation or decentration alone on the induced astigmatism.

Fig. 5 Induced third- and fourth-order aberrations for correcting myopia astigmatism from treatment decentration. Panel A corresponds to the refractive errors (–5.0DS –2.0DC × 60°) and Panel B corresponds to another refractive errors (–7.5DS –1.5DC × 180°). The diameter of the optical zone is 6 mm.

Download Full Size | PDF

Fig. 6 Contour maps of the induced astigmatism for correcting myopia astigmatism from treatment decentration. Panel A corresponds to the refractive errors (–5.0DS –2.0DC × 60°) and Panel B corresponds to another refractive errors (–7.5DS –1.5DC × 180°). The diameter of the optical zone is 6 mm.

Download Full Size | PDF

The results indicate that induced coma increases linearly with an increase of translation [Fig. 5(a)]. Additionally, the induced spherical aberration is determined only by the corrected refraction, and it is hardly related to the treatment translation. In the third- and fourth-order wavefront aberrations, the induced Zernike aberration terms, except coma and spherical aberration, do not correlate with the translation, and their rms values are all less than 0.1 μm.

Figure 6 indicates that induced astigmatism correlates with the treatment translation and eye cyclotorsion. As there is an increase of translation, the induced astigmatism may be larger [Fig. 6(a)] but also may be smaller [Fig. 6(b)]. In contrast, the induced astigmatism increases with the increase of eye cyclotorsion. In addition, the induced astigmatism is related to the corrected astigmatism component; that is, it may be larger with the greater corrected astigmatism.

4. Discussion

4.1 Comparison with previous studies

Several factors, such as a change of pupil center location under different conditions, the surgeon manually locking the eye-tracker, or the surgeon’s visual alignment of the ablation center may contribute to treatment decentration. Additionally, conventional laser refractive surgery is based upon manifest refraction, and pupil size and location are not taken into account when obtaining manifest refractions. In fact, candidate ablation centers include the corneal intercept of the visual axis (CVA), the entrance pupil center (EPC), and so on [26,27]. The CVA would be an ideal ablation center, but it is difficult to locate accurately. In clinical practices, EPC is usually used as the ablation center, regardless of its deviation to the visual axis. That is why subclinical treatment decentration is inevitable in refractive surgery. In the present study, our mathematical model assumes that the CVA and EPC are the same. When EPC deviates from CVA in a real human eye, this deviation may be treated as part of the treatment decentration, and our results are useful theoretically and clinically. Decentration of the ablation profile during surgery may account for a portion of the postoperative increase in higher-order aberrations [28–30]. Our results are in agreement with previous studies that postoperative coma-like aberrations tend to increase [14]. It has been reported that subclinical treatment decentration is a major factor in increased coma-like aberrations after corneal laser surgery [12,31–34]. Thus, our results are sufficient to support the conclusion drawn from Wang et al. [35] that treatment decentration produces significantly greater higher-order aberrations of rms values than cyclotorsion misalignment in residual wavefront aberrations.

In this study, the induced spherical aberration was underestimated because the anterior corneal surface after laser refractive surgery is a geometric sphere. Additionally, our results, shown in Figs. 5(a) and 5(b), indicate that the induced spherical aberration is hardly related to treatment decentration. This result is consistent with the previous studies [33,34]. However, Mrochen et al. have found subclinical decentration has caused an increase in postoperative spherical-like aberrations [31]. This variety might result, in part, from differences in variations of methodology. Our analytical method was based on the ablation profile from the mathematical model of an anterior corneal surface. As a contrast, Mrochen used an eye model to calculate the higher-order aberrations after surgery and the wavefront aberrations located at the retinal level.

4.2 Without effect on laser-ablation algorithms of oblique incidence

In the optical zone, the ablation depth of a cornea was calculated on the basis of the ablation profile obtained from myopia astigmatism correction without the effect of oblique incidence. Similarly, the actual corrected aberrations were computed by ocular wavefront transformation. In this section, we investigated the relationship between induced higher-order aberrations and treatment decentration. The results are shown in Fig. 7 .

Fig. 7 Induced third- and fourth-order aberrations for correcting myopia astigmatism from treatment decentration without oblique incidence. Panel A corresponds to the refractive errors (–5.0DS –2.0DC × 60°) and Panel B corresponds to another refractive errors (–7.5DS –1.5DC × 180°). The diameter of the optical zone is 6 mm.

Download Full Size | PDF

After detailed analysis, the results show that induced aberrations from treatment decentration are mainly coma and spherical aberrations. However, induced higher-order aberrations are significantly larger than that of oblique incidence. To be specific, a coma rms with oblique incidence is about one-half of that without it.

4.3 Considering the angle between position vector of translation and astigmatism axis

A different location of the center of the optical zone may induce different amounts of wavefront aberrations for myopia astigmatism correction when the translation maintains a constant. Figure 8 sketches the angle between the position vector of the translation and the astigmatism axis. The center of the optical zone without decentration is located at the origin of the XY frame, and that of the optical zone with decantation is located at the origin of the X’Y’ frame. The position vector of translation (with units of length) is defined as a vector from O to O’. The axis represents the astigmatism axis. The angle (α) is defined as the angle difference, which is formed by the position vector and astigmatism axis measured counterclockwise from the astigmatism axis. In this section, the constant amount of translation is 1 mm. Figure 9 shows the curves of the induced wavefront aberrations versus the angle difference, and the orientation angle covers different ranges (from 0 to 180).

Fig. 8 Diagrammatic sketch of position vector of translation and astigmatism axis.

Download Full Size | PDF

Fig. 9 Relationship of induced third- and fourth-order aberrations for correcting myopia astigmatism versus the angle difference. Panel A corresponds to the refractive errors (–5.0DS –2.0DC × 60°) and Panel B corresponds to another refractive errors (–7.5DS –1.5DC × 180°). The diameter of the optical zone is 6 mm.

Download Full Size | PDF

The results show that the induced aberrations are in bilateral symmetry. The rms values of the induced aberration are at a maximum when the angle difference is 90°; that is, the position vector of the translation is perpendicular to the astigmatism axis. Among them, the induced astigmatism most obviously changes with the angle difference. On the contrary, the spherical aberration and the trefoil are hardly correlated with the angle difference.

4.4 Analysis of the impact of decentration on the wavefront aberration

From theoretical and clinical data analysis, the results have shown that a significant amount of coma may be induced from treatment decentration in refractive surgery, which interprets that a cornea with treatment decentration is an off-axis refractive system. Consequently, a larger amount of higher-order aberrations, especially coma, are inevitably induced from treatment decentration for a refractive system of the whole eye. When the oblique incidence is considered, the induced coma is obviously reduced because the center of the ablation zone deviates from the optical axis of the human eye, and the actual ablation depth in the center of the ablation zone is smaller than its theoretical depth. On the other hand, spherical aberration increases most significantly after laser refractive surgery from the clinical wavefront aberration data [36]. However, the predicted spherical aberration from the mathematical model is significantly different from the clinical results. This result implies that the increased spherical aberration after refractive surgery is not induced mainly from the ablation profile. Previous research has shown that an increase in postoperative spherical aberration may correlate to the change of corneal shape, the effect of a biological response of the cornea [37], or the effect of an oblique incidence of the laser spot on the cornea that leads the ablation depth of the periphery of the cornea less than the depth of the theoretical prediction.

Higher-order aberration terms besides coma and spherical aberration after refractive surgery are different from corresponding ones from theoretical prediction. The main reason for this result may be attributed to the following aspects. First, wavefront aberrations of human eyes are dynamic, and the results are different for each measurement. Second, the corneal flap is created and reposited in refractive surgery. Finally, the laser eye surgical systems for refractive errors include fitting errors, registration errors, tracking errors, errors caused by laser beam variability [38], and so forth. The transition zone may cause an increase in higher-order aberrations after refractive surgery for a dilated pupil.

Based on the above results, some critical factors in refractive surgery should be taken into account to achieve better postoperative visual performance. Clinicians should have extensive experience in refractive surgery to minimize treatment decentration. Then, more advanced devices should be developed to align with the visual axis [26] and to improve eye-tracking technology in order to prevent treatment decentration. In addition, the response of healing of the cornea also induces a considerable amount of spherical aberrations, and new surgical methods should be developed to minimize corneal trauma and to reduce induced higher-order aberrations as the cornea heals.

5. Conclusion

Based on a mathematical model of the anterior corneal surface, theoretical results indicated that significant coma was induced from treatment decentration, and it increased linearly with an increase in translation. Additionally, the amount of induced coma was also determined by refractive errors for myopia or myopia astigmatism. Furthermore, the induced coma from treatment decentration without oblique incidence was obviously larger than that with it. The induced spherical aberration was determined only by corrected refraction and was not related to the lateral translation. Except for the amount of coma and spherical aberration in higher-order aberrations, the ablation profile of a cornea for pure myopia or myopia astigmatism correction did not significantly influence the increase of Zernike aberration terms. The remarkable differences between the clinical and predicted data may be attributed to many factors besides treatment decentration and fitting errors.

Acknowledgements

This research is supported by the Natural Science Foundation of Tianjin City of China grants 033606011 and 07JCYBJC09500, the National Natural Science Foundation of China (NNSFC) grant 60777011, and the Science and Technology Supporting Plan of Jiangxi Province of China grant 2009BGB02700.

References and links

1. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wavefront sensor,” J. Opt. Soc. Am. A 11(7), 1949–1957 (1994). [CrossRef]

2. S. Mutyala, M. B. McDonald, K. A. Scheinblum, M. D. Ostrick, S. F. Brint, and H. Thompson, “Contrast sensitivity evaluation after laser in situ keratomileusis,” Ophthalmology 107(10), 1864–1867 (2000). [CrossRef] [PubMed]

3. T. Hiraoka, C. Okamoto, Y. Ishii, T. Kakita, and T. Oshika, “Contrast sensitivity function and ocular higher-order aberrations following overnight orthokeratology,” Invest. Ophthalmol. Vis. Sci. 48(2), 550–556 (2007). [CrossRef] [PubMed]

4. C. R. Munnerlyn, S. J. Koons, and J. Marshall, “Photorefractive keratectomy: a technique for laser refractive surgery,” J. Cataract Refract. Surg. 14(1), 46–52 (1988). [PubMed]

5. A. W. Chang, A. C. Tsang, J. E. Contreras, P. D. Huynh, C. J. Calvano, T. C. Crnic-Rein, and E. H. Thall, “Corneal tissue ablation depth and the Munnerlyn formula,” J. Cataract Refract. Surg. 29(6), 1204–1210 (2003). [CrossRef] [PubMed]

6. J. Shen, Y. Zhang, and W. Liao, “Mathematical model based corneal toric surface for excimer laser refractive surgery,” J. Southeast Univ. 36, 531–536 (2006) (Natural Science Edition).

7. D. A. Chernyak, “Cyclotorsional eye motion occurring between wavefront measurement and refractive surgery,” J. Cataract Refract. Surg. 30(3), 633–638 (2004). [CrossRef] [PubMed]

8. J. Schwiegerling and R. W. Snyder, “Eye movement during laser in situ keratomileusis,” J. Cataract Refract. Surg. 26(3), 345–351 (2000). [CrossRef] [PubMed]

9. H. Kim and C. K. Joo, “Ocular cyclotorsion according to body position and flap creation before laser in situ keratomileusis,” J. Cataract Refract. Surg. 34(4), 557–561 (2008). [CrossRef] [PubMed]

10. S. K. Webber, C. N. McGhee, and I. G. Bryce, “Decentration of photorefractive keratectomy ablation zones after excimer laser surgery for myopia,” J. Cataract Refract. Surg. 22(3), 299–303 (1996). [PubMed]

11. N. Asano-Kato, I. Toda, C. Sakai, Y. Hori-Komai, Y. Takano, M. Dogru, and K. Tsubota, “Pupil decentration and iris tilting detected by Orbscan: anatomic variations among healthy subjects and influence on outcomes of laser refractive surgeries,” J. Cataract Refract. Surg. 31(10), 1938–1942 (2005). [CrossRef] [PubMed]

12. J. Bühren, G. Yoon, S. Kenner, S. MacRae, and K. Huxlin, “The effect of optical zone decentration on lower- and higher-order aberrations after photorefractive keratectomy in a cat model,” Invest. Ophthalmol. Vis. Sci. 48(12), 5806–5814 (2007). [CrossRef] [PubMed]

13. J. Porter, G. Yoon, S. MacRae, G. Pan, T. Twietmeyer, I. G. Cox, and D. R. Williams, “Surgeon offsets and dynamic eye movements in laser refractive surgery,” J. Cataract Refract. Surg. 31(11), 2058–2066 (2005). [CrossRef]

14. P. Padmanabhan, M. Mrochen, D. Viswanathan, and S. Basuthkar, “Wavefront aberrations in eyes with decentered ablations,” J. Cataract Refract. Surg. 35(4), 695–702 (2009). [CrossRef] [PubMed]

15. L.-H. Fang, Z.-Q. Wang, W. Wang, and M. Liu, “The influence of wavefront aberration of single Zernike modes on optical quality of human eyes,” Acta Opt. Sin. 26, 1721–1726 (2006).

16. A. Guirao, J. Porter, D. R. Williams, and I. G. Cox, “Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes,” J. Opt. Soc. Am. A 19(3), 620–628 (2002). [CrossRef]

17. L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vis. Sci. 82(5), 358–369 (2005). [CrossRef] [PubMed]

18. R. A. Applegate, J. D. Marsack, R. Ramos, and E. J. Sarver, “Interaction between aberrations to improve or reduce visual performance,” J. Cataract Refract. Surg. 29(8), 1487–1495 (2003). [CrossRef] [PubMed]

19. X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vis. 4(4), 310–321 (2004). [CrossRef] [PubMed]

20. H. Zhao and B. Xu, “Position tolerance analysis for wavefront aberrations correction of human eyes,” Acta Opt. Sin. 28(5), 949–954 (2008). [CrossRef]

21. A. Guirao, D. R. Williams, and I. G. Cox, “Effect of rotation and translation on the expected benefit of an ideal method to correct the eye’s higher-order aberrations,” J. Opt. Soc. Am. A 18(5), 1003–1015 (2001). [CrossRef]

22. H. S. Ginis, V. J. Katsanevaki, and I. G. Pallikaris, “Influence of ablation parameters on refractive changes after phototherapeutic keratectomy,” J. Refract. Surg. 19(4), 443–448 (2003). [PubMed]

23. J. R. Jiménez, R. G. Anera, L. Jiménez del Barco, and E. Hita, “Effect on laser-ablation algorithms of reflection losses and nonnormal incidence on the anterior cornea,” Appl. Phys. Lett. 81(8), 1521–1523 (2002). [CrossRef]

24. M. Mrochen and T. Seiler, “Influence of corneal curvature on calculation of ablation patterns used in photorefractive laser surgery,” J. Refract. Surg. 17(5), S584–S587 (2001). [PubMed]

25. Organization for Standardization International (ISO), Ophthalmic Optics and Instruments – Reporting Aberrations of the Human Eye (ISO 24157, 2008).

26. L. Wu, X. Zhou, R. Chu, and Q. Wang, “Photoablation centration on the corneal optical center in myopic LASIK using AOV excimer laser,” Eur. J. Ophthalmol. 19(6), 923–929 (2009). [PubMed]

27. H. Uozato and D. L. Guyton, “Centering corneal surgical procedures,” Am. J. Ophthalmol. 103(3 Pt 1), 264–275 (1987). [PubMed]

28. N. Sakata, T. Tokunaga, K. Miyata, and T. Oshika, “Changes in contrast sensitivity function and ocular higher order aberration by conventional myopic photorefractive keratectomy,” Jpn. J. Ophthalmol. 51(5), 347–352 (2007). [CrossRef] [PubMed]

29. N. Yamane, K. Miyata, T. Samejima, T. Hiraoka, T. Kiuchi, F. Okamoto, Y. Hirohara, T. Mihashi, and T. Oshika, “Ocular higher-order aberrations and contrast sensitivity after conventional laser in situ keratomileusis,” Invest. Ophthalmol. Vis. Sci. 45(11), 3986–3990 (2004). [CrossRef] [PubMed]

30. K. Pesudovs, “Wavefront aberration outcomes of LASIK for high myopia and high hyperopia,” J. Refract. Surg. 21(5), S508–S512 (2005). [PubMed]

31. M. Mrochen, M. Kaemmerer, P. Mierdel, and T. Seiler, “Increased higher-order optical aberrations after laser refractive surgery: a problem of subclinical decentration,” J. Cataract Refract. Surg. 27(3), 362–369 (2001). [CrossRef] [PubMed]

32. T. Mihashi, “Higher-order wavefront aberrations induced by small ablation area and sub-clinical decentration in simulated corneal refractive surgery using a perturbed schematic eye model,” Semin. Ophthalmol. 18(1), 41–47 (2003). [CrossRef] [PubMed]

33. T. Oshika, G. Sugita, K. Miyata, T. Tokunaga, T. Samejima, C. Okamoto, and Y. Ishii, “Influence of tilt and decentration of scleral-sutured intraocular lens on ocular higher-order wavefront aberration,” Br. J. Ophthalmol. 91(2), 185–188 (2007). [CrossRef]

34. F. Taketani, T. Matuura, E. Yukawa, and Y. Hara, “Influence of intraocular lens tilt and decentration on wavefront aberrations,” J. Cataract Refract. Surg. 30(10), 2158–2162 (2004). [CrossRef] [PubMed]

35. L. Wang and D. D. Koch, “Residual higher-order aberrations caused by clinically measured cyclotorsional misalignment or decentration during wavefront-guided excimer laser corneal ablation,” J. Cataract Refract. Surg. 34(12), 2057–2062 (2008). [CrossRef] [PubMed]

36. Y. Wang, K. X. Zhao, J. C. He, Y. Jin, and T. Zuo, “Ocular higher-order aberrations features analysis after corneal refractive surgery,” Chin. Med. J. (Engl.) 120(4), 269–273 (2007).

37. G. Yoon, S. Macrae, D. R. Williams, and I. G. Cox, “Causes of spherical aberration induced by laser refractive surgery,” J. Cataract Refract. Surg. 31(1), 127–135 (2005). [CrossRef] [PubMed]

38. G. M. Dai, E. Gross, and J. Liang, “System performance evaluation of refractive surgical lasers: a mathematical approach,” Appl. Opt. 45(9), 2124–2134 (2006). [CrossRef] [PubMed]

Theoretical analysis of wavefront aberration from treatment decentration with oblique incidence after conventional laser refractive surgery

Abstract

1. Introduction

2. Methods

2.1 Mathematical model of anterior corneal surface

2.2 Effect on laser-ablation algorithms of oblique incidence on the anterior cornea

2.3 Ablation profile for conventional laser refractive surgery

2.4 Decentration of wavefront aberration

2.5 Induced wavefront aberration in conventional laser refractive surgery from treatment decentration

3. Results

3.1 Induced wavefront aberration for pure myopic and hyperopic correction

3.2 Induced wavefront aberration for myopia astigmatism correction

4. Discussion

4.1 Comparison with previous studies

4.2 Without effect on laser-ablation algorithms of oblique incidence

4.3 Considering the angle between position vector of translation and astigmatism axis

4.4 Analysis of the impact of decentration on the wavefront aberration

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (9)

Equations (16)

Optics Express