**Meet the editor**

Michael Goggin FRCSI (Ophth), FRCOphth, FRANZCO, MS, is a practicing ophthalmic surgeon with a particular interest in cataract, corneal disease and refractive surgery. He is a Senior Lecturer in Ophthalmology at the University of Adelaide, Australia and has researched, published, presented and taught widely on the subject of astigmatism as it relates to ophthalmic surgery.

## Contents

### **Preface XI**


#### **Part 2 Diagnosis and Imaging of Astigmatism 57**

	- **Part 3 Correction of Astigmatism 91**

X Contents


## Preface

This book presents a modern understanding of the underlying physiology of primary astigmatism and the pathology of astigmatism associated with disease and trauma. It explores many therapeutic approaches to amelioration of the effects of astigmatism in its many forms by surgical and optical means. It has been my pleasure to assist in the presentation of the evidence behind our present management of this significant refractive error, the treatment of which is seldom simple. The authors and I hope that the reader will find its content accessible and useful both in daily practice and in arriving at a deeper understanding of the condition.

> **Dr. Michael Goggin** The Queen Elizabeth Hospital, The University of Adelaide, Australia

## **Part 1**

## **Development, Physiology and Optics of Astigmatism**

## **Physiology of Astigmatism**

Seyed-Farzad Mohammadi, Maryam Tahvildari and Hadi Z-Mehrjardi

*Eye Research Centre, Farabi Eye Hospital, Tehran University of Medical Sciences Iran* 

## **1. Introduction**

We know that the expression emmetropia is a conventional one and that in fact all normal human eyes have mild degrees of spherocylindrical errors (Shilo, 1997) or consist of a bitoric optical system, i.e. have principal meridians of relatively higher and lower powers at right angles. It is generally accepted that genetic factors have a significant role in determining ocular refractive status as well as astigmatism (Hammond et al., 2001) but many conditions and procedures such as surgery, suturing, wound healing, and ocular comorbidities modify the cylindrical status of the eye. Induced astigmatism and surgical correction of astigmatism are extensively addressed in other chapters of this book.

Manifest astigmatism is the vectorial sum of anterior corneal toricity and internal astigmatism. A variety of factors change the magnitude and shift the meridians of these cylindrical components and the perceived subjective astigmatism throughout life. Astigmatism is an extremely dynamic phenomenon, and changes in the shape of optical interfaces, refractive index, optical aperture, eyeball-extraocular structures (eyelids and extraocular muscles) interaction, visual tasks, accommodation, binocularity, tear film status, and even body position induce and modify baseline ocular astigmatism. In this chapter we shall focus on factors that determine baseline, diurnal, functional, and dynamic aspects of 'physiological astigmatism'.

## **2. Natural course of astigmatism in normal eyes**

## **2.1 Age**

Age-related evolution of ocular astigmatism in terms of power and axis has been observed in epidemiologic studies (Abrahamsson et al., 1988; Atkinson et al., 1980; Attebo et al., 1999; Baldwin & Mills, 1981; Ehrlich et al., 1997; Gwiazda et al., 1984; Hirsch, 1963; Kame et al., 1993; Sawada et al., 2008; Stirling, 1920). It is well documented that a high degree of astigmatism is present in neonates and infants; however, the reported amounts show discrepancies (Abrahamsson, et al. 1988; Howland & Sayles, 1985; Isenberg et al., 2004; Kohl & Samek, 1988; Varughese et al., 2005; Wood et al., 1995). The degree of astigmatism is even higher in preterm newborns and has an inverse association with postconceptional age and birth weight (Friling et al., 2004; Varghese et al., 2009). In near retinoscopy without cycloplegia, Gwiazda and colleagues found astigmatism of at least 1 D in about 55% of infants younger than 5 months, 10% of whom displayed a cylinder power of 3 D or more (Gwiazda et al., 1984). In another study, photorefractive techniques showed that almost all infants at the age of 3 months had at least 1 D of astigmatism, which had decreased to adult levels by the age of 18 months (Atkinson et al., 1980). Likewise, a longitudinal study found astigmatism of at least 1 D in about 40% of infants at 3 months of age with a significant decrease to 4% by the age of 36 months. This reduction appears to be caused by the decrease in toricity of the cornea and the anterior lens (Mutti et al., 2004). Several studies have suggested that corneal shape changes throughout life. The linear reduction of the astigmatism to lower values with age is apparently a part of normal eye maturation (Friling et al., 2004) and emmetropisation. It has been suggested that the high astigmatism in early life induces and activates accommodation (Campbell & Westheimer, 1959; Howland, 1982).

Reports on the axis of astigmatism in infants are contradictory. According to Gullstrand, the natural form of the cornea is against the rule (Gullstrand 1962). Several studies have found a plus cylinder axis at 180±20 (i.e. against the rule) in the majority of infants (Abrahamsson et al., 1988; Baldwin & Mills, 1981; Dobson et al., 1984; Gwiazda et al., 1984; Saunders 1986, 1988). while recent studies have raised questions about the reliability of previously used techniques. These studies have shown that with-the-rule astigmatism is more frequent among infants (Ehrlich et al. 1997; Isenberg et al., 2004; Mutti et al., 2004; Varghese et al., 2009).

As the child grows older, much of the early astigmatism will gradually disappear and transform into with the rule owing to eyelid pressure (Gwiazda et al. 1984). Most of the changes occur at ages 1-3 years, when the vertical and horizontal diameters of the cornea and its elasticity attain adult size and amount (Karesh, 1994). This with-the-rule astigmatism in preschool children is stabilized towards adolescence (Goss, 1991; Huynh et al., 2007; Shankar & Bobier, 2004). However, this is not always true, and a role for myopia development has been attributed to ocular astigmatism, as it may degrade optical blur cues and disrupt emmetropisation, which can lead to axial myopia progression in school-aged children (Gwiazda et al., 2000).

In early adulthood, astigmatism of more than one diopter is infrequent and is still with the rule. Lin and colleagues found a slight increase in the amount of astigmatism in medical students after five years (Lin et al., 1996). Other cross-sectional studies have indicated that mean total astigmatism changes with age, varying from as much as 0.62 D with the rule during adolescence to as much as 0.37 D against the rule in older ages (Baldwin & Mills, 1981). Baldwin and Mills found that steepening of the cornea in the horizontal meridian accounts for a major proportion of the increase in against-the-rule total astigmatism among older patients (Baldwin & Mills 1981). In the Blue Mountains Eye Study (Attebo et al., 1999), mean total astigmatism increased with age from 0.6 D to 1.2 D in youngest (49-59) to oldest (80-97) age groups. From a related study, an average of 1.6 D rise in total corneal astigmatism is documented for each five years of increase in age (Ho et al., 2010). Nuclear sclerosis cataract and the change in refractive index of the crystalline lens at older ages may contribute to myopic astigmatism (Fotedar et al., 2008).

Anterior corneal (and total) astigmatism shows flattening in the vertical meridian with aging, in contrast to a trend towards with-the-rule astigmatism on the posterior corneal surface (Ho et al., 2010). Against-the-rule astigmatism is the most common type of astigmatism in adults over 40 years of age. Interestingly, men are significantly more likely to develop against-the-rule astigmatism (Goto et al., 2001). In general, corneal toricity accounts

(Gwiazda et al., 1984). In another study, photorefractive techniques showed that almost all infants at the age of 3 months had at least 1 D of astigmatism, which had decreased to adult levels by the age of 18 months (Atkinson et al., 1980). Likewise, a longitudinal study found astigmatism of at least 1 D in about 40% of infants at 3 months of age with a significant decrease to 4% by the age of 36 months. This reduction appears to be caused by the decrease in toricity of the cornea and the anterior lens (Mutti et al., 2004). Several studies have suggested that corneal shape changes throughout life. The linear reduction of the astigmatism to lower values with age is apparently a part of normal eye maturation (Friling et al., 2004) and emmetropisation. It has been suggested that the high astigmatism in early life induces and activates accommodation (Campbell & Westheimer, 1959;

Reports on the axis of astigmatism in infants are contradictory. According to Gullstrand, the natural form of the cornea is against the rule (Gullstrand 1962). Several studies have found a plus cylinder axis at 180±20 (i.e. against the rule) in the majority of infants (Abrahamsson et al., 1988; Baldwin & Mills, 1981; Dobson et al., 1984; Gwiazda et al., 1984; Saunders 1986, 1988). while recent studies have raised questions about the reliability of previously used techniques. These studies have shown that with-the-rule astigmatism is more frequent among infants (Ehrlich et al. 1997; Isenberg et al., 2004; Mutti et al., 2004; Varghese et al.,

As the child grows older, much of the early astigmatism will gradually disappear and transform into with the rule owing to eyelid pressure (Gwiazda et al. 1984). Most of the changes occur at ages 1-3 years, when the vertical and horizontal diameters of the cornea and its elasticity attain adult size and amount (Karesh, 1994). This with-the-rule astigmatism in preschool children is stabilized towards adolescence (Goss, 1991; Huynh et al., 2007; Shankar & Bobier, 2004). However, this is not always true, and a role for myopia development has been attributed to ocular astigmatism, as it may degrade optical blur cues and disrupt emmetropisation, which can lead to axial myopia progression in school-aged

In early adulthood, astigmatism of more than one diopter is infrequent and is still with the rule. Lin and colleagues found a slight increase in the amount of astigmatism in medical students after five years (Lin et al., 1996). Other cross-sectional studies have indicated that mean total astigmatism changes with age, varying from as much as 0.62 D with the rule during adolescence to as much as 0.37 D against the rule in older ages (Baldwin & Mills, 1981). Baldwin and Mills found that steepening of the cornea in the horizontal meridian accounts for a major proportion of the increase in against-the-rule total astigmatism among older patients (Baldwin & Mills 1981). In the Blue Mountains Eye Study (Attebo et al., 1999), mean total astigmatism increased with age from 0.6 D to 1.2 D in youngest (49-59) to oldest (80-97) age groups. From a related study, an average of 1.6 D rise in total corneal astigmatism is documented for each five years of increase in age (Ho et al., 2010). Nuclear sclerosis cataract and the change in refractive index of the crystalline lens at older ages may

Anterior corneal (and total) astigmatism shows flattening in the vertical meridian with aging, in contrast to a trend towards with-the-rule astigmatism on the posterior corneal surface (Ho et al., 2010). Against-the-rule astigmatism is the most common type of astigmatism in adults over 40 years of age. Interestingly, men are significantly more likely to develop against-the-rule astigmatism (Goto et al., 2001). In general, corneal toricity accounts

Howland, 1982).

children (Gwiazda et al., 2000).

contribute to myopic astigmatism (Fotedar et al., 2008).

2009).

for the major component of total astigmatism (Asano et al., 2005; Ho et al., 2010); it is suggested that, with aging, upper eyelid pressure on the cornea and the tone of orbicularis muscle decrease (Marin-Amat, 1956). It has also been demonstrated that with-the-rule astigmatism decreases when eyelids are retracted from the cornea (Wilson et al., 1982). When relative steepening in the vertical meridian is abated, the intrinsic lenticular againstthe-rule astigmatism will manifest. Decreases in action of extraocular muscles, especially the medial rectus (Marin-Amat, 1956), and vitreous syneresis and liquefaction may also contribute (Mehdizadeh, 2008).

The contribution of the lens to the ocular astigmatism is relatively constant throughout life (Hofstetter & Baldwin, 1957). Development of this lenticular astigmatism may be due to an emmetropisation phenomenon, as it effectively decreases manifest astigmatism in the early decades of life. But in older ages, lenticular astigmatism is manifested as an against-the-rule astigmatism when the corneal astigmatism is decreased (Artal et al., 2000, 2001; Ehrlich et al., 1997).

#### **2.2 Diurnal changes of astigmatism in the normal eye**

The magnitude and axis of astigmatism vary during the day; this variation can be described with regard to changes in eyelid pressure, extraocular muscle tension, pupil size and accommodation. From previous reports, it is postulated that generally the cornea has its flattest shape on awakening and steepens slightly until the evening (Manchester, 1970). Kiely et al. reported fluctuations in corneal asphericity during the day without recognizing a specific pattern (Kiely et al., 1982). Recently, diurnal variations in corneal topography have been studied by Read et al.: corneal wavefront error analysis revealed significant changes in astigmatism during the day (Read et al., 2005); see below.

#### **2.2.1 Lid pressure and muscle tension in near tasks**

Changes in corneal contour exerted through eyelid pressure have been widely discussed since the mid-1960s, and transient bilateral monocular diplopia after near work due to temporarily induced toricity in the cornea has been reported by a number of investigators (Bowman et al., 1978; Golnik & Eggenberger, 2001; Knoll, 1975; Mandell, 1966). It is agreed that visual tasks with significant downward gaze, such as reading, can alter corneal curvature owing to eyelid pressure (Collins et al., 2006; Read et al., 2007a). This will lead to horizontal bands on red reflex during retinoscopy (Ford et al., 1997) with concomitant topographic changes and corresponding distortions in Zernike wavefront analysis. Buehren et al. have reported changes towards against-the-rule astigmatism (Buehren et al., 2003).

In a recent study by Shaw et al., the average trend in the astigmatism axis due to near work was said to be against the rule, with approximately 0.25 D change within 15 minutes of 40 ̊ downward gaze, where both the upper and lower eyelids are in contact with the central 6 mm of the cornea. They also reported that eyelid tilt, curvature and position are important in the magnitude of corneal changes (Shaw et al., 2008).

Collins et al. demonstrated greater topographical changes in astigmatism during downward gaze with a larger angle (45° versus 25°) and with lateral eye movements (Collins et al., 2006). Studies on the time course of astigmatism regression have revealed slower recovery after longer periods of reading. Moreover, patterns of regression are similar among individuals, with a rapid recovery within the first 10 minutes after reading, and resolution takes between 30 to 60 minutes (Collins et al., 2005).

Regarding the role of extraocular muscles on corneal astigmatism, Lopping has mentioned that continuous use of the medial rectus muscle, especially during near tasks, imposes a force on the cornea which increases its radius of curvature in the horizontal meridian resulting in a shift towards against-the-rule astigmatism (Lopping & Weale, 1965).

These observations have implications for clinical testing, and it would be prudent that examinees avoid near tasks at least 30 minutes prior to refractive and topographic assessments.

## **2.2.2 Eyelid slant and tension**

Apart from temporary changes of corneal curvature due to eyelid pressure, the cumulative effect of the eyelids contributes to naturally occurring astigmatism in healthy adults (see 2.1 above).

Slanting of the palpebral fissure is an important factor affecting corneal toricity (Read et al., 2007b). The magnitude of astigmatism increases as the palpebral fissure diverges from the horizontal plane. Male subjects show more downward fissure slanting, whereas female subjects show more upward fissure slanting (Garcia et al., 2003). People with Down's syndrome (Akinci et al., 2009; Little et al., 2009) or Treacher Collins syndrome (Wang et al., 1990) will show oblique astigmatism partly due to upward or downward slanting of the palpebral fissure.

Thicker or tighter eyelids tend to correspond with higher degrees of astigmatism as well. Asians and Native Americans show higher degrees of corneal astigmatism than other races (Osuobeni & Al Mijalli, 1997).

Corneal rigidity can also contribute to the amount of astigmatism caused by eyelid pressure. For instance, nutritional deficiencies are presumed to decrease corneal rigidity and flatten the horizontal meridian while steepening the vertical one (Lyle et al., 1972).

#### **2.2.3 Pupil dynamics**

We know that the optical system of the eye is not coaxial and at least three important axes have been described: optical axis (corneal optical center to lens's optical center), visual axis (object of regard to fovea; line of sight), and pupillary axis. There is a mild physiological pupil decentration in the nasal direction. Such physiological asymmetries, which have long been described (Walsh, 1988), induce coma (Wilson et al., 1992).

The pupil is the aperture for light entrance into the eye; excluding pharmacologic changes, pupil size and its (centroid) lateral position around the optical axis of the eye change according to ambient light (Walsh, 1988; Wilson et al., 1992), accommodative effort, and emotional status (Wilson et al., 1992).

Pupil size correlates with both the magnitude and orientation of astigmatism. Larger mesopic pupil sizes are detected with higher cylinder powers and are also associated with with-the-rule astigmatism rather than against-the-rule and oblique astigmatism (Cakmak et al., 2010). Larger pupil sizes—in low lighting conditions— increase the amount of higher order aberrations such as coma and may intensify the cylinder power in subjective/manifest refraction. Coma has been shown to be correlated with greater amounts of astigmatism (Hu et al. 2004). On the contrary, pupillary accommodative constriction reduces higher order aberrations including lenticular astigmatism (Sakai et al., 2007).

About 0.4 mm temporal pupil centroid shift in darkness was first reported by Walsh (Walsh, 1988); Wilson and Campbell (Wilson et al., 1992) then found shifts of up to 0.6 mm with decreased illumination, in nasal or temporal directions.

Regarding the role of extraocular muscles on corneal astigmatism, Lopping has mentioned that continuous use of the medial rectus muscle, especially during near tasks, imposes a force on the cornea which increases its radius of curvature in the horizontal meridian

These observations have implications for clinical testing, and it would be prudent that examinees avoid near tasks at least 30 minutes prior to refractive and topographic assessments.

Apart from temporary changes of corneal curvature due to eyelid pressure, the cumulative effect of the eyelids contributes to naturally occurring astigmatism in healthy adults (see 2.1

Slanting of the palpebral fissure is an important factor affecting corneal toricity (Read et al., 2007b). The magnitude of astigmatism increases as the palpebral fissure diverges from the horizontal plane. Male subjects show more downward fissure slanting, whereas female subjects show more upward fissure slanting (Garcia et al., 2003). People with Down's syndrome (Akinci et al., 2009; Little et al., 2009) or Treacher Collins syndrome (Wang et al., 1990) will show oblique astigmatism partly due to upward or downward slanting of the

Thicker or tighter eyelids tend to correspond with higher degrees of astigmatism as well. Asians and Native Americans show higher degrees of corneal astigmatism than other races

Corneal rigidity can also contribute to the amount of astigmatism caused by eyelid pressure. For instance, nutritional deficiencies are presumed to decrease corneal rigidity and flatten

We know that the optical system of the eye is not coaxial and at least three important axes have been described: optical axis (corneal optical center to lens's optical center), visual axis (object of regard to fovea; line of sight), and pupillary axis. There is a mild physiological pupil decentration in the nasal direction. Such physiological asymmetries, which have long

The pupil is the aperture for light entrance into the eye; excluding pharmacologic changes, pupil size and its (centroid) lateral position around the optical axis of the eye change according to ambient light (Walsh, 1988; Wilson et al., 1992), accommodative effort, and

Pupil size correlates with both the magnitude and orientation of astigmatism. Larger mesopic pupil sizes are detected with higher cylinder powers and are also associated with with-the-rule astigmatism rather than against-the-rule and oblique astigmatism (Cakmak et al., 2010). Larger pupil sizes—in low lighting conditions— increase the amount of higher order aberrations such as coma and may intensify the cylinder power in subjective/manifest refraction. Coma has been shown to be correlated with greater amounts of astigmatism (Hu et al. 2004). On the contrary, pupillary accommodative constriction reduces higher order

About 0.4 mm temporal pupil centroid shift in darkness was first reported by Walsh (Walsh, 1988); Wilson and Campbell (Wilson et al., 1992) then found shifts of up to 0.6 mm with

the horizontal meridian while steepening the vertical one (Lyle et al., 1972).

been described (Walsh, 1988), induce coma (Wilson et al., 1992).

aberrations including lenticular astigmatism (Sakai et al., 2007).

decreased illumination, in nasal or temporal directions.

resulting in a shift towards against-the-rule astigmatism (Lopping & Weale, 1965).

**2.2.2 Eyelid slant and tension** 

above).

palpebral fissure.

(Osuobeni & Al Mijalli, 1997).

emotional status (Wilson et al., 1992).

**2.2.3 Pupil dynamics** 

#### **2.2.4 Accommodation and convergence**

Three decades ago, Brzezinski introduced the expression 'accommodative astigmatism' and claimed that changes in lenticular astigmatism can neutralize corneal astigmatism and reduce the eye's overall toricity (Brzezinski, 1982). Other investigators have suggested that astigmatism increases as the accommodative response becomes larger (Denieul, 1982; Ukai & Ichihashi, 1991). According to Brzezinski, accommodative astigmatism is related to lens distortion due to inhomogeneous lens elasticity, variable constriction in ciliary muscles (which itself changes the lens power), and nonhomogeneous tension of the extraocular muscles during convergence (which causes corneal distortion). These may explain 'lag of accommodation', the phenomenon of less accommodative response than the accommodative stimulus in the horizontal meridian and the resultant with-the-rule astigmatism (Tsukamoto et al., 2001). Pupillary constriction may contribute to such changes as well (see above).

In a more recent study, Tsukamoto et al. found that all emmetropic subjects became astigmatic during accommodation, 93% with the rule (mean -1.96 D). Corneal astigmatism of with-the-rule orientation with mean values of 0.84 D and 0.91 D, respectively, for right and left eyes was detected without a direct association with the amount of accommodation. The eyes became emmetropic just after relaxation (Tsukamoto et al., 2000). Cheng et al. examined wavefront aberrations in a large adult population and found changes in astigmatism towards with the rule with an average of -0.1 D during maximum accommodation (Cheng et al., 2004). The mentioned pupillary and accommodative effects interact with the factors considered above during near tasks (see above).

Accommodation always accompanies convergence during near-vision tasks (Tait, 1933; Rosenfield & Gilmartin, 1988), and it is known that slight changes in cylinder power and axis (towards with the rule) occur during convergence alone (Beau Seigneur, 1946; Lopping & Weale, 1965; Tsukamoto, Nakajima et al., 2000). Seigneur has mentioned that this change is seen in a small percentage of eyes (Beau Seigneur, 1946), and many of the individuals do not experience any discomfort when using the same spectacles for far and near activities; nevertheless, for those who experience such an alteration, separate spectacle prescriptions for near and far distance vision might be beneficial.

#### **2.2.5 Cyclotorsion and binocularity**

Eye rotation around the Z axis (rolling or cyclotorsion) modifies the axis of ocular astigmatism in relation to the outside world. There is a complex interaction between accommodation, baseline astigmatism, and torsional alignment (Buehren et al., 2003; Read et al., 2007). These features contribute to eye fusional potential, depth perception, and depth of field (Regan & Spekreijse, 1970).

A number of reasons are implicated for physiological ocular torsion including unmasking of cyclophoria during monocular fixation and fusion loss (Tjon-Fo-Sang et al., 2002; Borish & Benjamin, 2006) and changing of body position from upright to supine (Park et al., 2009; Hori-Komaii et al., 2007; Fea et al., 2006; Chernyak, 2004; Swami et al., 2002); these changes gain remarkable clinical significance when an individual is examined in the seating position but undergoes laser ablation in the supine position. Binocularity is also disturbed during corneal topography and wavefront aberrometry; binocular viewing is not normal during laser ablation in the supine position either.

Although several studies have shown significant incyclotorsion or excyclotorsion of about 2- 4 degrees (maximum 9 to 14 degrees) as a result of changing the body position from seated to supine (Swami et al., 2002; Chernyak, 2004; Fea et al., 2006; Neuhann et al., 2010), a number of investigations have reported insignificant axis changes of less than 2 degrees, which can hardly affect astigmatic correction (Tjon-Fo-Sang et al., 2002; Becker et al., 2004). It has been suggested that axis misalignment of about 4 degrees will lead to 14% cylinder undercorrection during laser ablation (Swami et al., 2002; Neuhann et al., 2010).

As mentioned above, cyclotorsion is frequently seen when switching from binocular to monocular vision, especially in those who have significant cyclophoria (Borish & Benjamin, 2006). Although it has been believed that an occluded eye shows excyclophoria under monocular occlusion of several hours (Graf et al., 2002), it is indeterminate whether ocular torsion resulting from monocular occlusion for a short period during refractive surgery or retinoscopy and monocular subjective refraction is clockwise or counter-clockwise; Hori-Komai et al. and Chang et al. both demonstrated that the magnitude and direction of cyclotorsion is different for each individual (Chang, 2008; Hori-Komai et al., 2007). This has significance in subjective refraction refinement and binocular balancing as well; in fact, a novel position in the phoropter allows maintenance of binocularity (fusion) while clarity of the images of the eyes are independently assessed (Borish & Benjamin, 2006).

Apart from body position and monocularity, which account for static eye rotational alignment, dynamic cyclotorsion also occurs during laser ablation and may result in astigmatic undercorrection and/or induced astigmatism (Neuhann et al., 2010; Chang, 2008; Hori-Komai et al., 2007). Fea et al. showed that blurring of the fixation target happens during ablation (after epithelium removal in surface ablation and following flap lifting in LASIK) and is an important factor for dynamic cyclotorsion, the magnitude of which seems to be significantly higher in the supine position (Fea et al., 2006). Modern eye trackers now are designed to dynamically follow the eye during laser ablation.

#### **2.2.6 Tear film**

The tear layer has a refractive index near to that of the cornea (1.33 versus 1.376) and refraction at the air-tear film interface accounts for the majority of refractive power of the anterior ocular surface (Oldenburg et al., 1990). Use of hard contact lenses to correct refractive errors creates a 'tear lens' in the contact lens-cornea interface which resolves the keratometric cylinder (Astin, 1989). This decouples anterior corneal astigmatism from internal astigmatism and manifests as 'residual astigmatism'. The nature of this astigmatism is frequently against the rule and at times can cause eye strain (see above).

The superior eyelid exerts pressure on the cornea, and tear accumulates over the lower eyelid margin due to gravity; this combination induces a vertical coma which may manifest as a cylinder (Montés-Micó et al., 2004a). Localized aggregation of lacrimal fluid is also caused by peripheral corneal lesions such as pterygium (Oldenburg et al., 1990; Walland et al., 1994; Yasar et al., 2003), limbal conjunctival carcinoma (Leccisotti, 2005), or nodules (Das et al., 2005). Such changes cause corneal astigmatism and are largely resolved after excision of the lesion or drying of the tear pool (Leccisotti, 2005; Yasar et al., 2003).

The tear film effect can also be discussed with regard to ocular wavefront changes during blinking. It has been agreed that higher order corneal aberrations show micro-fluctuations during the inter-blink interval. These dynamic variations of ocular surface topography have been widely investigated using high speed videokeratoscopes (E. Goto et al., 2003; T. Goto et al., 2004; Koh et al., 2002; Kojima et al., 2004; Montés-Micó et al., 2004a; Németh et al., 2002). Zhu et al. found that the height of the ocular surface increased about 2 mm within 0.5 s after blinking at the upper edge of the topography map. They also declared that absolute values

to supine (Swami et al., 2002; Chernyak, 2004; Fea et al., 2006; Neuhann et al., 2010), a number of investigations have reported insignificant axis changes of less than 2 degrees, which can hardly affect astigmatic correction (Tjon-Fo-Sang et al., 2002; Becker et al., 2004). It has been suggested that axis misalignment of about 4 degrees will lead to 14% cylinder

As mentioned above, cyclotorsion is frequently seen when switching from binocular to monocular vision, especially in those who have significant cyclophoria (Borish & Benjamin, 2006). Although it has been believed that an occluded eye shows excyclophoria under monocular occlusion of several hours (Graf et al., 2002), it is indeterminate whether ocular torsion resulting from monocular occlusion for a short period during refractive surgery or retinoscopy and monocular subjective refraction is clockwise or counter-clockwise; Hori-Komai et al. and Chang et al. both demonstrated that the magnitude and direction of cyclotorsion is different for each individual (Chang, 2008; Hori-Komai et al., 2007). This has significance in subjective refraction refinement and binocular balancing as well; in fact, a novel position in the phoropter allows maintenance of binocularity (fusion) while clarity of

Apart from body position and monocularity, which account for static eye rotational alignment, dynamic cyclotorsion also occurs during laser ablation and may result in astigmatic undercorrection and/or induced astigmatism (Neuhann et al., 2010; Chang, 2008; Hori-Komai et al., 2007). Fea et al. showed that blurring of the fixation target happens during ablation (after epithelium removal in surface ablation and following flap lifting in LASIK) and is an important factor for dynamic cyclotorsion, the magnitude of which seems to be significantly higher in the supine position (Fea et al., 2006). Modern eye trackers now

The tear layer has a refractive index near to that of the cornea (1.33 versus 1.376) and refraction at the air-tear film interface accounts for the majority of refractive power of the anterior ocular surface (Oldenburg et al., 1990). Use of hard contact lenses to correct refractive errors creates a 'tear lens' in the contact lens-cornea interface which resolves the keratometric cylinder (Astin, 1989). This decouples anterior corneal astigmatism from internal astigmatism and manifests as 'residual astigmatism'. The nature of this astigmatism

The superior eyelid exerts pressure on the cornea, and tear accumulates over the lower eyelid margin due to gravity; this combination induces a vertical coma which may manifest as a cylinder (Montés-Micó et al., 2004a). Localized aggregation of lacrimal fluid is also caused by peripheral corneal lesions such as pterygium (Oldenburg et al., 1990; Walland et al., 1994; Yasar et al., 2003), limbal conjunctival carcinoma (Leccisotti, 2005), or nodules (Das et al., 2005). Such changes cause corneal astigmatism and are largely resolved after excision

The tear film effect can also be discussed with regard to ocular wavefront changes during blinking. It has been agreed that higher order corneal aberrations show micro-fluctuations during the inter-blink interval. These dynamic variations of ocular surface topography have been widely investigated using high speed videokeratoscopes (E. Goto et al., 2003; T. Goto et al., 2004; Koh et al., 2002; Kojima et al., 2004; Montés-Micó et al., 2004a; Németh et al., 2002). Zhu et al. found that the height of the ocular surface increased about 2 mm within 0.5 s after blinking at the upper edge of the topography map. They also declared that absolute values

undercorrection during laser ablation (Swami et al., 2002; Neuhann et al., 2010).

the images of the eyes are independently assessed (Borish & Benjamin, 2006).

are designed to dynamically follow the eye during laser ablation.

is frequently against the rule and at times can cause eye strain (see above).

of the lesion or drying of the tear pool (Leccisotti, 2005; Yasar et al., 2003).

**2.2.6 Tear film** 

in horizontal coma and secondary astigmatism at 45° significantly increased during the inter-blink interval, while secondary astigmatism at 0° decreased considerably (Zhu et al., 2006). In another study, irregular astigmatism induced by tear film breakup was measured and significant increases were observed in coma, spherical aberration and total higher order aberrations (Koh et al., 2002).

It is therefore suggested that measurement of corneal wavefront aberrations for refractive surgery purposes should be done at a fixed interval after each blink (Montés-Micó et al., 2004b). Based on a number of studies that evaluated the variability of topography maps (Buehren et al., 2001; Iskander et al., 2005; Montés-Micó et al., 2004b), an interval of 1 to 4 seconds after blinking is suggested as the optimal time (Zhu et al., 2006).

#### **2.2.7 Retinal astigmatism**

From a historical point of view, directional variability in photoreceptor arrangement was proposed as a source of astigmatism (Mitchell et al., 1967); in other words, functional retinal elements may be more abundant or thicker in one axis than the other (Shlaer, 1937). More recently, a 'tilted' retina was simulated and it was observed to manifest as some degree of cylindrical error (Flüeler & Guyton, 1995). This could be the result of unequal lengthening of the sclera in different meridians during axial growth.

#### **3. Conclusion**

Although most of the materials presented in this chapter are of investigational interest, there is a resurging interest in these physiological issues owing to refractive surgery. Variations in tear film status, torsional alignment, and pupil features are sources of error in ocular refractive assessment and laser photoablation. Our objective should be firstly not to spoil the innate versatility and optical quality of the virgin eye; and secondly, to avoid inconsistencies in the outcome. The available optical models do not simulate optical performance of the eye perfectly, and the refractive surgery technology —in terms of diagnosis and treatment does not fully follow our optical models either.

On the positive side, if we intend to make 'super vision' a reality (Applegate et al., 2004), we have to better understand the mentioned dynamics and interaction and 'personalize' treatments. Iris registration for example, can be used to avoid the negative effects of pupil centroid and astigmatism axis shifts during excimer laser ablation (Porter et al., 2006; Jing et al., 2008; Khalifa et al., 2009; Park et al., 2009). But this is just the beginning and we need dynamic optical models and advanced simulations to fulfil the mentioned objectives.

Additionally, an in-depth understanding of physiology of ocular astigmatism may throw light on the pathobiology of refractive errors and lead to new avenues for the prevention of clinically significant astigmatism.

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## **Etiology and Clinical Presentation of Astigmatism**

David Varssano

*Department of Ophthalmology, Tel Aviv Medical Center, Tel Aviv University Israel* 

## **1. Introduction**

14 Astigmatism – Optics, Physiology and Management

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Astigmatism exists to some extent in virtually every eye. Multiple terms include the word "astigmatism": regular astigmatism, fully correctable by a cylinder; irregular astigmatism (with dual use: both as the refractive error not correctable by sphere and cylinder, and as a component in Fourier analysis); and components in Zernike's polynomials (primary and secondary astigmatism).

The optical apparatus is created by an interaction of the cornea, the crystalline lens and the fovea. Any imperfection translates into a refractive error. The sphere component of the refractive error is influenced by the relationship between all three components. Any other refractive error, including any astigmatism, is influenced by the cornea and crystalline lens only. The aetiology of astigmatism is highly diverse. Its aetiology and its presentation are outlined in this chapter

## **2. Types of astigmatism**

To many, the term astigmatism is replaceable with cylinder. This certainly is not the case. Astigmatism is usually described as regular and as irregular astigmatism.

## **2.1 Regular astigmatism**

Regular astigmatism is the type or refractive error correctable by a cylinder lens. Such a lens can be used in spectacles, in a soft contact lens, and in an intraocular lens – replacing the crystalline lens or being added to it.

A perfect spherocylindrical apparatus is composed solely of spheres and cylinders. A positive sphere is a lens that converges parallel light rays to a single spot. The amount of conergence of the light is inversely proportional to the distance from the source. If the distance is expressed in meters, the vergence is in units of dioptres.

$$\text{Diopters} = 1/\text{Distance in meters} \tag{1}$$

Thus, a +5 diopter sphere lens converges parallel light rays to a single spot, 1/5 of a meter, or 20 cm, away from it.

A positive cylinder is a lens that converges parallel light rays to a straight line, parallel to the cylinder axis. Here again, the amount of convergence of the light is inversely proportional to the distance from the source. A +5 diopter cylinder lens converges parallel light rays to a straight line, as long as the length of the lens and parallel to its axis, 20 cm away from it.

In regular astigmatism, the optical power of the optical system can be perfectly described by a single sphere and a single cylinder. The actual combined lens is neither a sphere nor a cylinder. A convex (positive, convergent) pure sphere lens is part of a perfect sphere. The radius of curvature of the lens is identical in all meridians. A convex spherocylindrical lens is like a "bent" sphere. It may look more like a part of an american football, or a part of a donut: the radius of curvature in two perpendicular meridians is not the same.

While the sphere power of the optical system of the human eye is usually more than 60 diopters, the cylinder is much lower. Typically, the cylindrical power of an eye that has not undergone surgery is less than one diopter, seldom surpassing 2 or 3 diopters.

A refractive error caused by a cylinder is corrected with a negative cylinder placed on the same axis. When the correct lenses are placed in the correct orientation, the eventual image is sharp.

#### **2.1.1 With-the-rule and against-the-rule astigmatism**

The condition in which the meridian of greatest power is vertical, or within 30 degrees of the vertical, is most common and it is called with-the-rule astigmatism. It is corrected with a plus cylinder at the same vertical axis, or with a minus cylinder at an axis perpendicular to it. When the meridian of greatest power is horizontal, or within 30 degrees of the horizontal, it is called against-the-rule astigmatism. When the astigmatism is neither with the rule nor against the rule it is called oblique astigmatism.

## **2.2 Irregular astigmatism**

In the traditional representation of refractive errors, any refractive error not corrected by a sphere or a cylinder is an irregular astigmatism. While regular astigmatism, or a spherocylindrical refractive error, is a theoretical approximation, irregular astigmatism is what happens in real life. Any irregularity in the surfaces of the cornea and the crystalline lens and any local change in the refractive index of the lens or the cornea changes the optical power of the system in that location in a way that a spheroclylindrical lens can not fully correct.

### **2.3 Presentation by Zernike polynomials**

The refractive error of the eye can be presented in several methods. The measurement of optical aberrations is based on the principle of Tscherning's aberroscope (Mierdel et al., 1999), Hartmann-Shack's aberroscope (Moreno-Barriuso et al., 2001) or light ray tracing (Moreno-Barriuso et al., 2001). The joint representation of all the raw aberroscope data constitutes the spot diagram, which can be taken as a rough estimate of the shape of the retinal point spread.

The results of the measurement are often presented by the Zernike polynomials. These polynomials were invented by Frits Zernike, (1888-1966), a Dutch Nobel prize in physics laureate, as a research tool in optics. The Zernike polynomials are used in ophthalmology to describe and display the optical aberrations of the entire optical pathway or of any of its components.

Figure 1 presents all the optical aberrations and Zernike polynomial values (Zernike's coefficients) of the anterior corneal surface (corneal topography, Wavelight, Allegro Topolyzer) of a normal eye, computed to the 8th order.

the distance from the source. A +5 diopter cylinder lens converges parallel light rays to a straight line, as long as the length of the lens and parallel to its axis, 20 cm away from it. In regular astigmatism, the optical power of the optical system can be perfectly described by a single sphere and a single cylinder. The actual combined lens is neither a sphere nor a cylinder. A convex (positive, convergent) pure sphere lens is part of a perfect sphere. The radius of curvature of the lens is identical in all meridians. A convex spherocylindrical lens is like a "bent" sphere. It may look more like a part of an american football, or a part of a

While the sphere power of the optical system of the human eye is usually more than 60 diopters, the cylinder is much lower. Typically, the cylindrical power of an eye that has not

A refractive error caused by a cylinder is corrected with a negative cylinder placed on the same axis. When the correct lenses are placed in the correct orientation, the eventual image

The condition in which the meridian of greatest power is vertical, or within 30 degrees of the vertical, is most common and it is called with-the-rule astigmatism. It is corrected with a plus cylinder at the same vertical axis, or with a minus cylinder at an axis perpendicular to it. When the meridian of greatest power is horizontal, or within 30 degrees of the horizontal, it is called against-the-rule astigmatism. When the astigmatism is neither with the rule nor

In the traditional representation of refractive errors, any refractive error not corrected by a sphere or a cylinder is an irregular astigmatism. While regular astigmatism, or a spherocylindrical refractive error, is a theoretical approximation, irregular astigmatism is what happens in real life. Any irregularity in the surfaces of the cornea and the crystalline lens and any local change in the refractive index of the lens or the cornea changes the optical power of

The refractive error of the eye can be presented in several methods. The measurement of optical aberrations is based on the principle of Tscherning's aberroscope (Mierdel et al., 1999), Hartmann-Shack's aberroscope (Moreno-Barriuso et al., 2001) or light ray tracing (Moreno-Barriuso et al., 2001). The joint representation of all the raw aberroscope data constitutes the spot diagram, which can be taken as a rough estimate of the shape of the

The results of the measurement are often presented by the Zernike polynomials. These polynomials were invented by Frits Zernike, (1888-1966), a Dutch Nobel prize in physics laureate, as a research tool in optics. The Zernike polynomials are used in ophthalmology to describe and display the optical aberrations of the entire optical pathway or of any of its

Figure 1 presents all the optical aberrations and Zernike polynomial values (Zernike's coefficients) of the anterior corneal surface (corneal topography, Wavelight, Allegro

the system in that location in a way that a spheroclylindrical lens can not fully correct.

donut: the radius of curvature in two perpendicular meridians is not the same.

undergone surgery is less than one diopter, seldom surpassing 2 or 3 diopters.

**2.1.1 With-the-rule and against-the-rule astigmatism** 

against the rule it is called oblique astigmatism.

**2.3 Presentation by Zernike polynomials** 

Topolyzer) of a normal eye, computed to the 8th order.

**2.2 Irregular astigmatism** 

retinal point spread.

components.

is sharp.

Fig. 1. Zernike analysis of a normal corneal topography

The first orders of the Zernike polynomials, Z0 0 (named piston) and Z1 1, Z1 -1 (named tilt), have little direct meaning on refraction. The second orders of the Zernike polynomials are Z20, (sphere, Figure 2) and Z22, Z2-2, (cylinder, Figure 3, Figure 4). In traditional nomenclature, these would the sphere and cylinder of the refractive error. These can be corrected with spherocylindrical spectacle lenses, soft contact lenses or intraocular lenses.

Figure 5 represents the sum of all the optical aberrations of the higher orders (in the specific example, 3rd to 8th orders were calculated). These are named high order aberrations, or HOA's. HOA's can not be corrected with spherocylindrical spectacle lenses, soft contact lenses or intraocular lenses. Correction of HOA's can be attempted in several methods: Using rigid contact lenses to minimize the HOA's originating from the anterior corneal surface; using excimer laser to reshape the anterior corneal surface so that all optical aberrations are treated (wavefront guided ablation); using excimer laser to reshape the anterior corneal surface so that aberrations originating from the anterior corneal surface are treated (topography guided ablation); using an intraocular lens that, apart from correcting sphere and sometimes cylinder, can also address other aberrations, namely spherical aberrations (Z4 0).

#### **2.4 Presentation by Fourier analysis**

Fourier analysis is another way of representing the refractive qualities of a system (e.g. a whole eye) or of an optical component within the system (e.g. the anterior corneal surface). Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist, who showed that representing a function by a trigonometric series can greatly simplify its study. Fourier series harmonic analysis can be applied to

Fig. 2. Isolated Zernike coefficient Z20 (sphere) of a normal corneal topography

Fig. 3. Isolated Zernike coefficient Z22 (cylinder) of a normal corneal topography

Fig. 2. Isolated Zernike coefficient Z20 (sphere) of a normal corneal topography

Fig. 3. Isolated Zernike coefficient Z22 (cylinder) of a normal corneal topography

topographic analysis (Maeda, 2002). A Fourier series consists of trigonometric sine and cosine functions with increasing coefficients. A Fourier series can be used to transform any periodical functions into trigonometric components. Therefore, by applying the Fourier analysis to the polar data of the dioptric corneal power for each mire, one can break down the complex information given in corneal topography into spherical, regular astigmatism, decentration, and irregular astigmatism components. Figure 6 presents the Fourier analysis of the anterior corneal surface (corneal topography, Wavelight, Allegro Topolyzer) of the same normal eye in as in figures 1-5. Irregular astigmatism (termed "Irregularities") is of a much lower amplitude than regular astigmatism in this normal cornea.

It seems that for the presentation of the optical system of the eye, Zernike polynomials are better suited than Fourier analysis (Yoon et al., 2008). The Zernike method outperformed the Fourier method when representing simulated wavefront data from topography maps. Even 2nd through 5th order Zernike polynomials were enough to outperform the Fourier method in all populations. Up to 9th order Zernike modes may be required to describe accurately simulated wavefront in some abnormal eyes.

Fig. 4. Isolated Zernike coefficient Z2-2 (cylinder) of a normal corneal topography

### **3. Prevalence of astigmatism**

It appears that the prevalence of astigmatism in mainly population related. Age probably plays a smaller role.

#### **3.1 Populations**

Corneal astigmatism and total refractive astigmatism were measured in a presbyopic population in Germany (Hoffmann & Hutz, 2010). Mean corneal astigmatism was 0.98±0.78 diopters. Less than 1.00 D of corneal and refractive astigmatism was found in 63.96% and 67.97% respectively. Astigmatism value ≥1.00 D and <2.00 D in 27.95% and 22.55% respectively, ≥2.00 D and <3.00 D in 5.44% and 6.09% respectively, ≥3.00 D and < 4.00 D in 1.66% and 2.18% respectively, ≥4.00 D and <5.00 D in 0.56% and 0.80% respectively, ≥5.00 D and <6.00 D in 0.25% and 0.28% respectively and only 0.18% and 0.13% respectively had astigmatism of 6.00 D or more. Lekhanont et al., 2011, reported on corneal astigmatism in cataract candidates in Bangkok, Thailand. Less than 0.50 D was found in 19.71%, ≥0.50 D and <1.00 D in 42.49%, ≥1.00 D and <2.00 D in 29.92%, ≥2.00 D and <3.00 D in 6.30% and 3.00 D and above in 1.58%. KhabazKhoob et al. (2010) found the corneal astigmatism in residents of Tehran, Iran to be 0.98 D (C.I. 0.89-1.06), with no apparent age effect.

Fig. 5. Combined Zernike coefficient Z2 to Z8 (high order aberrations) of a normal corneal topography

Astigmatism was studied (Allison, 2010) in a Polish immigrant population in Chicago. Fifteen percent of the patients exhibited astigmatism ≥1.00 D.

Prevalence of astigmatism >1.00 D in indigenous Australians within central Australia (Landers et al., 2010) was found to be 6.2%.

Direct comparison between these different population groups is not easy, because of different cutoff points and different age groups. The impression from the above figures is that there seems to be a substantial population or location difference in prevalence and magnitude of astigmatism between these studies.

#### **3.2 Children**

The astigmatism in young children (6-72 month old) was studies in African American and Hispanic children (Fozailoff et al. 2011). Mean refractive astigmatism was 0.58±0.61 D

diopters. Less than 1.00 D of corneal and refractive astigmatism was found in 63.96% and 67.97% respectively. Astigmatism value ≥1.00 D and <2.00 D in 27.95% and 22.55% respectively, ≥2.00 D and <3.00 D in 5.44% and 6.09% respectively, ≥3.00 D and < 4.00 D in 1.66% and 2.18% respectively, ≥4.00 D and <5.00 D in 0.56% and 0.80% respectively, ≥5.00 D and <6.00 D in 0.25% and 0.28% respectively and only 0.18% and 0.13% respectively had astigmatism of 6.00 D or more. Lekhanont et al., 2011, reported on corneal astigmatism in cataract candidates in Bangkok, Thailand. Less than 0.50 D was found in 19.71%, ≥0.50 D and <1.00 D in 42.49%, ≥1.00 D and <2.00 D in 29.92%, ≥2.00 D and <3.00 D in 6.30% and 3.00 D and above in 1.58%. KhabazKhoob et al. (2010) found the corneal astigmatism in residents

Fig. 5. Combined Zernike coefficient Z2 to Z8 (high order aberrations) of a normal corneal

Fifteen percent of the patients exhibited astigmatism ≥1.00 D.

(Landers et al., 2010) was found to be 6.2%.

magnitude of astigmatism between these studies.

Astigmatism was studied (Allison, 2010) in a Polish immigrant population in Chicago.

Prevalence of astigmatism >1.00 D in indigenous Australians within central Australia

Direct comparison between these different population groups is not easy, because of different cutoff points and different age groups. The impression from the above figures is that there seems to be a substantial population or location difference in prevalence and

The astigmatism in young children (6-72 month old) was studies in African American and Hispanic children (Fozailoff et al. 2011). Mean refractive astigmatism was 0.58±0.61 D

topography

**3.2 Children** 

of Tehran, Iran to be 0.98 D (C.I. 0.89-1.06), with no apparent age effect.

Fig. 6. Fourier analysis of a normal corneal topography

(mean±SD) in the right eye, 0.58±0.60 D in the left eyes in the African American population, and 0,66±0.76 D in the right eye, 0.64±0.75 D in the left eye in the Hispanic population. The overall prevalence of astigmatism ≥1.50 D was 12.7% in African Americans and 16.8% in Hispanic children. Prevalence of astigmatism ≥1.50 D decreased with age group in both ethnicities (P<0.0001). The overall prevalence of astigmatism ≥3.00 D was 3.0% for Hispanic and 1.2% for African American children. Astigmatism ≥3.00 D showed no significant trend with age in either ethnicity (P≥0.50).

A population based (Stratified random clustering) method was used to examine refractive and corneal astigmatism in white school children in Northern Ireland (O'Donoghue, 2011). The prevalence of refractive astigmatism (≥1.00D) did not differ significantly between 6-7 year-old children (24%, 95% CI 19-30) and 12-13-year-old children (20%, 95% CI 14-25). The prevalence of corneal astigmatism (≥1D) also did not differ significantly between 6-7-yearold children (29%, 95% CI 24-34) and 12-13-year-old children (25%, 95% CI 21-28). Whilst levels of refractive astigmatism and corneal astigmatism were similar, refractive astigmatism was predominantly oblique (76%, 95% CI 67-85 of 6-7-year-olds; 59% 95% CI 48-70 of 12-13-year-olds), but corneal astigmatism was predominantly with-the-rule (80%, 95% CI 72-87 of 6-7-year-olds; 82% 95% CI 74-90 of 12-13-year-olds).

Prevalence of refractive astigmatism was studied in preschool children in Taiwan (Lai et al. 2010). In this group, 49.5% of the total had astigmatism ≥0.50 D, 25.4% had 0.75 D or more, and 13.3% had 1.00 D or more. The prevalence of astigmatism >1.50 D was 4.0% (95% CI, 2.8%-5.2%). The prevalence of astigmatism was unassociated with age or sex. Children with with-the-rule astigmatism had greater mean cylinder power than those with against-the-rule or oblique astigmatism. Refractive astigmatism correlated with its corneal component.

The age effect is not clear in these studies. Age did and did not play a role in the magnitude of astigmatism in different studies. This may be a result of the population differences, as seen in the previous paragraphs.

## **4. Location of astigmatism**

As stated above, astigmatism can originate from any refractive surface in the optical system. Each surface adds some astigmatism, regular and irregular, and the total astigmatism of the system is the product of all components. In some cases, the effect of one surface may negate the effect of another, thus improving the total result. This fact is utilized in refractive surgery, when changes of the anterior corneal surface are made in an attempt to lower the total optic aberrations of the entire eye, including sphere, cylinder and high order aberrations.

## **4.1 Cornea**

The corneal surface (and mainly the anterior corneal surface) is probably the largest source of astigmatism in the eye. It is not the only source. A study in school children (O'Donoghue, 2011) found that levels of refractive astigmatism and corneal astigmatism were similar, but refractive astigmatism was predominantly oblique (76% of 6-7-year-olds; 59% of 12-13-year-olds), while corneal astigmatism was predominantly with-the-rule (80% of 6-7-year-olds; 82% of 12-13-year-olds). Another cylindrical vector being added to the corneal cylinder vector, causing a new direction of the compound vector explains this finding.

## **4.1.1 Anterior corneal surface**

In most measurement devices, the data on the corneal power including corneal astigmatism is derived from the anterior corneal surface, by use of reflection rather than refraction to compute curvature, and then to assess refraction by assuming the refractive index of the cornea, and the posterior corneal curvature. This assumption is close to reality in most normal eyes, but grossly incorrect in others. A large exception is the cornea after excimer laser refractive surgery, when the shape of the anterior corneal surface is markedly changed, while the posterior corneal surface is roughly unchanged.

Measuring the anterior corneal surface only, several studies have reported age-related changes to corneal astigmatism. A shift in the axis of corneal astigmatism from with the rule toward against the rule associated with increasing age was found.

Gudmundsdottir et al. (2000) did autokeratometry on adults in Reykjavik. Using linear regression the change to against the rule was 0.14 D (0.14 D in males and 0.13 D in females) in five years. The Distribution of corneal astigmatism 0.75 D or more measured by keratometry shows a marked shift towards against-the-rule astigmatism. Their younger age groups (50-59 year old) had ~5% against-the-rule astigmatism and 60%-70% with-the-rule astigmatism. The older age groups change gradually, and the 80+ year olds have ~5% with-

Prevalence of refractive astigmatism was studied in preschool children in Taiwan (Lai et al. 2010). In this group, 49.5% of the total had astigmatism ≥0.50 D, 25.4% had 0.75 D or more, and 13.3% had 1.00 D or more. The prevalence of astigmatism >1.50 D was 4.0% (95% CI, 2.8%-5.2%). The prevalence of astigmatism was unassociated with age or sex. Children with with-the-rule astigmatism had greater mean cylinder power than those with against-the-rule or oblique astigmatism. Refractive astigmatism correlated with its corneal component. The age effect is not clear in these studies. Age did and did not play a role in the magnitude of astigmatism in different studies. This may be a result of the population differences, as

As stated above, astigmatism can originate from any refractive surface in the optical system. Each surface adds some astigmatism, regular and irregular, and the total astigmatism of the system is the product of all components. In some cases, the effect of one surface may negate the effect of another, thus improving the total result. This fact is utilized in refractive surgery, when changes of the anterior corneal surface are made in an attempt to lower the total optic aberrations of the entire eye, including sphere, cylinder and high order

The corneal surface (and mainly the anterior corneal surface) is probably the largest source of astigmatism in the eye. It is not the only source. A study in school children (O'Donoghue, 2011) found that levels of refractive astigmatism and corneal astigmatism were similar, but refractive astigmatism was predominantly oblique (76% of 6-7-year-olds; 59% of 12-13-year-olds), while corneal astigmatism was predominantly with-the-rule (80% of 6-7-year-olds; 82% of 12-13-year-olds). Another cylindrical vector being added to the corneal cylinder vector, causing a new direction of the compound vector explains this

In most measurement devices, the data on the corneal power including corneal astigmatism is derived from the anterior corneal surface, by use of reflection rather than refraction to compute curvature, and then to assess refraction by assuming the refractive index of the cornea, and the posterior corneal curvature. This assumption is close to reality in most normal eyes, but grossly incorrect in others. A large exception is the cornea after excimer laser refractive surgery, when the shape of the anterior corneal surface is markedly changed,

Measuring the anterior corneal surface only, several studies have reported age-related changes to corneal astigmatism. A shift in the axis of corneal astigmatism from with the rule

Gudmundsdottir et al. (2000) did autokeratometry on adults in Reykjavik. Using linear regression the change to against the rule was 0.14 D (0.14 D in males and 0.13 D in females) in five years. The Distribution of corneal astigmatism 0.75 D or more measured by keratometry shows a marked shift towards against-the-rule astigmatism. Their younger age groups (50-59 year old) had ~5% against-the-rule astigmatism and 60%-70% with-the-rule astigmatism. The older age groups change gradually, and the 80+ year olds have ~5% with-

seen in the previous paragraphs.

**4. Location of astigmatism** 

**4.1.1 Anterior corneal surface** 

while the posterior corneal surface is roughly unchanged.

toward against the rule associated with increasing age was found.

aberrations.

**4.1 Cornea** 

finding.

the-rule astigmatism and ~60% against-the-rule astigmatism. Oblique astigmatism prevalence retained 25% to 35% levels in all age groups, 50 years and above.

Gudmundsdottir et al. (2005) found in a follow-up study an against-the-rule astigmatic shift in keratometry in adults in Reykjavik. The shift, for a 5 year period, was 0.03 D in people 50- 59 year old, 0.08 D in 60-69 year olds, and 0.17 on 70 or more year olds.

Goto et al. (2001) found that although corneal curvature in younger men and women was similar, people older than 50 had gender related differences. In older men, 81.1% had against-the-rule astigmatism and 18.9% had with-the-rule or no astigmatism. In contrast, in older women, 22.5% had against-the-rule astigmatism and 77.5% had with-the-rule or no astigmatism. The differences between genders in terms of the frequency of against-the-rule astigmatism were statistically significant (p < 0.001). With age, the pattern of astigmatism tends to change from with-the-rule to against-the-rule. Older men had a significantly higher potential for this change than women (p < 0.001).

Asano et al. (2005) examined the astigmatism in middle-aged and elderly Japanese. Corneal astigmatism (mean±SD) slightly increased over time: 0.83±0.57, 0.82±0.61, 0.85±0.63, 0.95±0.70 and 0.86±0.63 D in the 40–49 year old, 50–59 year old, 60–69 year old, 70–79 year old and all groups respectively. The axis gradually changed over time, from with-the-rule astigmatism to against-the-rule astigmatism. With-the- rule astigmatism was the most frequent in the 40s age group; its prevalence was over 60% in this group and decreased with age to ~25%. In contrast, the prevalence of against-the- rule astigmatism increased with age from ~15% to ~50% (P<0.0001)

#### **4.1.2 Posterior corneal surface**

In contrast to the abundance of research on the anterior corneal surface, much less was studied concerning the posterior corneal surface. However, instruments capable of capturing the posterior corneal curvature made such research possible.

In a report from Taipei, Taiwan, (Ho et al. 2009), a rotating Scheimpflug camera (Pentacam) was used. Subjects with healthy corneas were randomly selected from the Taipei City Hospital ophthalmology clinic visitors. Astigmatisms of the anterior and posterior corneal surfaces were determined. The total corneal astigmatism was derived using power vector summation and vergence tracing. Age-related changes to corneal astigmatism were evaluated using polar value analysis.

For the anterior and total cornea, the proportion of with-the- rule astigmatisms decreased and those of oblique and against-the-rule astigmatisms increased with age. For the posterior cornea, most eyes displayed against-the-rule astigmatisms in all age groups. There was a significant trend toward against-the-rule astigmatism associated with increasing age for both anterior and total corneal astigmatisms (mean changes of -0.18 and -0.16 diopters/5 years, respectively), and toward with the rule in posterior corneal astigmatism (a mean

change of 0.022 diopters/5 years). Regarding shape changes, a ''flat meridian toward a more vertical orientation'' trend with increasing age for both the anterior and posterior corneal surfaces was observed (mean changes of 0.0295 and 0.0224 mm/5 years, respectively). In the posterior cornea, proportions of with-the-rule, oblique, and against-the-rule astigmatisms are 0%, 1.7%, and 98.3% in the 21–30 age group and 9.1%, 2.3%, and 88.6% in the ≥71 age group. In contrast, in the anterior cornea, the proportions of with-the-rule, oblique, and against-the-rule astigmatisms are 91.4%, 5.2%, and 3.4% in the 21–30 age group and 31.8%, 29.5%, and 38.6% in the ≥71 age group. The posterior corneal surface compensated for the astigmatism arising from the anterior corneal surface in 91.4% and 47.7% of eyes in the 21– 30 and ≥71 year groups, respectively.

Using similar equipment, Mas et al. (2009) compared the optics of the anterior and posterior corneal surfaces. Their data shows that there was a low correlation between the astigmatic components of both surfaces: for the power vectors J0 of the first and second corneal surfaces, the regression line is: y = -0.13x + 0.17; r2 = 0.49; p < 0.01. For the power vectors J1 of the first and second corneal surfaces, the regression line is y = -0.10x -0.00; r2 = 0.23; p < 0.01.

## **4.2 Crystalline lens**

Shankar & Bobier (2004) examined preschool children (mean age ±SD 51.1±8.4 months). They calculated the lenticular astigmatism by substracting corneal astigmatism from the total refractive astigmatism. This method ignores the effect of the posterior corneal surface. However, the results of this paper are that the magnitude of total and corneal cylinder was significantly greater in high astigmats, but overall lenticular cylinder was similar in both groups. However, the Fourier transforms showed high astigmats to have significantly lower lenticular J0 (with-the-rule or against-the-rule astigmatism) and higher lenticular J45 (oblique astigmatism) than the normal astigmats. Both the high and the low astigmatism groups in the study had higer corneal astigmatism than total astigmatism, so that the lenticular component (and actually, the posterior corneal component as well) of the astigmatism served to lower the astigmatic effect of the anterior corneal surface.

### **4.2.1 Structural causes**

Physical changes influencing the crystalline lens can also cause change in the optical properties an doptical effect of the lens.

A 6-year-old boy developed lenticular astigmatism with a regular component of 5.5 D within 6 weeks of a penetrating scleral injury that included vitreous prolapse (Ludwig et al. 2002). Visible indentational folds in the posterior lens capsule, caused by anterior vitreous fibers and anterior hyaloid, were presumed to be the origin of the astigmatism. A pars plana vitrectomy partially helped, reducing the preoperative astigmatism to 4.0 D.

Another paper (Urrets-Zavalia, 1989) described lenticular astigmatism following subluxation of the crystalline lens.

## **4.2.2 Cataract**

The connection of cataract formation and appearance of lenticular astigmatism has been reported many years ago (Vaughn & Schepens, 1981). Rapidly progressive lenticular astigmatism related to cataract formation caused a cylinder of 12.00 D (Tint et al. 2007), and another case was reported (Tatham & Prydal, 2008), where the astigmatism changed from 0.25 to 5.00 during a period of 20 months without signs of lens opacity, returning to 0.25 D apter cataract surgery. As with myopic shift, changes in the aging lens can cause astigmatism.

## **5. Etiology**

Astigmatism may arise from many reasons. The following sections are an effort to sample the vast literature dealing with etiology of astigmatism.

### **5.1 Corneal thinning disorders**

Several noninflammatory conditions may lead to corneal thinning, including keratoconus, pellucid marginal degeneration, keratoglobus and Terrien's marginal degeneration. Other corneal thinning disorders are secondary to or associated with inflammation and necrosis, such as peripheral ulcerative keratitis and Mooren ulcer (Jain et al. 2011). To this list one must add laser in situ keratomileusis (LASIK) as an iatrogenic cause of a keratoectatic disorder. All these disorders are associated with regular and irregular astigmatism, ranging from low magnitude to many diopters.

#### **5.2 Post corneal surgery**

24 Astigmatism – Optics, Physiology and Management

Using similar equipment, Mas et al. (2009) compared the optics of the anterior and posterior corneal surfaces. Their data shows that there was a low correlation between the astigmatic components of both surfaces: for the power vectors J0 of the first and second corneal surfaces, the regression line is: y = -0.13x + 0.17; r2 = 0.49; p < 0.01. For the power vectors J1 of the first

Shankar & Bobier (2004) examined preschool children (mean age ±SD 51.1±8.4 months). They calculated the lenticular astigmatism by substracting corneal astigmatism from the total refractive astigmatism. This method ignores the effect of the posterior corneal surface. However, the results of this paper are that the magnitude of total and corneal cylinder was significantly greater in high astigmats, but overall lenticular cylinder was similar in both groups. However, the Fourier transforms showed high astigmats to have significantly lower lenticular J0 (with-the-rule or against-the-rule astigmatism) and higher lenticular J45 (oblique astigmatism) than the normal astigmats. Both the high and the low astigmatism groups in the study had higer corneal astigmatism than total astigmatism, so that the lenticular component (and actually, the posterior corneal component as well) of the astigmatism

Physical changes influencing the crystalline lens can also cause change in the optical

A 6-year-old boy developed lenticular astigmatism with a regular component of 5.5 D within 6 weeks of a penetrating scleral injury that included vitreous prolapse (Ludwig et al. 2002). Visible indentational folds in the posterior lens capsule, caused by anterior vitreous fibers and anterior hyaloid, were presumed to be the origin of the astigmatism. A pars plana

Another paper (Urrets-Zavalia, 1989) described lenticular astigmatism following

The connection of cataract formation and appearance of lenticular astigmatism has been reported many years ago (Vaughn & Schepens, 1981). Rapidly progressive lenticular astigmatism related to cataract formation caused a cylinder of 12.00 D (Tint et al. 2007), and another case was reported (Tatham & Prydal, 2008), where the astigmatism changed from 0.25 to 5.00 during a period of 20 months without signs of lens opacity, returning to 0.25 D apter cataract surgery. As with myopic shift, changes in the aging lens can cause astigmatism.

Astigmatism may arise from many reasons. The following sections are an effort to sample

Several noninflammatory conditions may lead to corneal thinning, including keratoconus, pellucid marginal degeneration, keratoglobus and Terrien's marginal degeneration. Other

and second corneal surfaces, the regression line is y = -0.10x -0.00; r2 = 0.23; p < 0.01.

served to lower the astigmatic effect of the anterior corneal surface.

vitrectomy partially helped, reducing the preoperative astigmatism to 4.0 D.

**4.2 Crystalline lens** 

**4.2.1 Structural causes** 

properties an doptical effect of the lens.

subluxation of the crystalline lens.

**5.1 Corneal thinning disorders** 

the vast literature dealing with etiology of astigmatism.

**4.2.2 Cataract** 

**5. Etiology** 

The cornea is a physical structure under internal pressure. Surgical intervention in the physical integrity of the cornea, intended or unintended, is a possible cause of change of shape and optical properties. The cornea is subject to many forms of surgery, and induced astigmatism evolves in many of the patients.

## **5.2.1 Penetrating keratoplasty**

Penetrationg keratoplasty is the ocular procedure with the largest wound length, and the donor tissue is held in place with multiple sutures. Astigmatism is the common result in most cases. A review (Price MO & Price FW, 2010) reports of average astigmatism of 4.2, 4.7 and 3.9 D, 2, 2 and 8 years after surgery respectively, in three articles on 297 patients. Much of the astigmatism is irregular, forcing patients to resort to rigid gas permeable contact lenses rather than spectacles or soft contact lenses.

## **5.2.2 Posterior lamellar keratoplasty**

A review (Price MO & Price FW, 2010) reports of a low average refractive cylinder of 1.2, 1.5 and 1.5 D achieved 6, 6 and 3 months following Descemet stripping automated endothelial keratoplasty (DSAEK). Similar results were reported following Descemet membrane endothelial keratoplasty (DMEK): 0.85 D, close to the average astigmatism of a normal cornea.

#### **5.2.3 Anterior lamellar keratoplasty**

Astigmatism following deep anterior lamellar keratoplasty (DALK) was reported in two groups (Kubaloglu et al. 2011): descemetic DALK (dDALK), or pre-descemetic DALK (pdDALK). The results were similar: 3.73±1.42 on the pdDALK group (mean±SD), and 3.52±1.53 in the dDALK group.

In a small group of patients, femtosecond laser-assisted sutureless anterior lamellar keratoplasty (FALK) was shown in 13 patients not to induce significant astigmatism (Shousha et al 2011). Preoperatine cylinder was 1.8±2.2 D (mean±SD), while 12 month cylinder measured in all patients was 2.2±2.3 D. However, adjunctive surgeries included phototherapeutic keratectomy, a procedure that may have improved surgically induced astigmatism.

#### **5.2.4 Refractive surgery**

Refractive surgery is basically aimed at reducing refractive errors, including astigmatism. Astigmatism is induced only when complications occur during or after surgery.

#### **5.2.4.1 Excimer laser**

Several complications of LASIK such as central islands, corneal ectasia and decentration can induce regular and irregular astigmatism (Johnson and Azar, 2001).

Eyes that had decentered LASIK ablation were compared to eyes that underwent uneventful surgery (Padmanabhan et al. 2009). There was a statistically significant (P<.05) linear correlation between the distance of decentration and the magnitude of induced tilt, coma and secondary astigmatism. The induced changes in tilt, oblique astigmatism, vertical coma, and spherical aberration were statistically significantly higher in eyes with decentered ablations than in eyes with well-centered ablations. A statistically significantly higher percentage of eyes (87%) with well-centered ablations than eyes with decentered ablations (70%) had a postoperative uncorrected visual acuity (UCVA) of 20/20 or better.

Proper placement of the treatment is crucial, and eye tracking devices are used to ensure that. In a study on the effect of different eye tracking methods during LASIK, some 400 eyes were operated in three groups (Prakash et al 2011). In the first group, no iris registration was used (no-iris-registration group). In a second group, preablation static iris registration was performed (static-iris-registration group). In the third group, preablation iris registration with dynamic rotational eye tracking was used (dynamic-iris-registration group). Alpins analysis showed that the indices for assessment of astigmatism outcomes were best in the dynamic-iris-registration group followed by the static-iris-registration group: better ability to treat in the right place yielded better ability to predict the outcome.

#### **5.2.4.2 Incisional**

Several incisional procedures are aimed to induce a cylinder effect, therefore undercorrection or over correction can be an expected result. Other procedures are aimed at correcting sphere (radial keratotomy, hexagonal keratotomy) and not affecting cylinder, but less than perfect construction of the incisions can still have an asymmetric effect.

Radial keratotomy erduced myopia by inducing instability to the peripheral cornea. The effect can be unpredictable. In the report of the PERK (prospective evaluation of radial keratotomy) study (Waring et al 1985), ten percent of patients increased astigmatism by more than 1.00 diopter. In a large scale survey of radial keratotomy complications (Marmer, 1987), irregular astigmatism was one of the reported complications.

Hexagonal keratotomy was used to treat hypermetropia. One article reports of 18 consecutive eyes of 12 patients that underwent hexagonal keratotomy (Werblin 1996). In addition to the primary procedures, 14 enhancements were required in seven eyes for both astigmatism and undercorrection. The author declared he no longer performed or recommended hexagonal keratotomy.

#### **5.3 Post cataract surgery**

Cataract surgery is often referred to as a refractive procedure. Cataract surgery is among the safest surgical procedures, but as the most performed ophthalmic procedure, every pro mil of complication is translated to thousands of suffering patients.

#### **5.3.1 Wound related**

Large incision cataract surgery was compared with phacoemulcification (Minassian et al 2001). The two planned treatments were: extracapsular cataract extraction (ECCE), and small incision surgery by phacoemulsification. In ECCE, a 12–14 mm corneoscleral section was made, while in phacoemulcification a self sealing 3.2 mm clear corneal incision was made on the steep axis of the corneal astigmatism. The post operative astigmatism ws markedly different in both groups. The phacoemulcification group kept the astigmatism just under 1 D, similar to the preoperative value. The ECCE group' on the other end, had a rise

Eyes that had decentered LASIK ablation were compared to eyes that underwent uneventful surgery (Padmanabhan et al. 2009). There was a statistically significant (P<.05) linear correlation between the distance of decentration and the magnitude of induced tilt, coma and secondary astigmatism. The induced changes in tilt, oblique astigmatism, vertical coma, and spherical aberration were statistically significantly higher in eyes with decentered ablations than in eyes with well-centered ablations. A statistically significantly higher percentage of eyes (87%) with well-centered ablations than eyes with decentered ablations

Proper placement of the treatment is crucial, and eye tracking devices are used to ensure that. In a study on the effect of different eye tracking methods during LASIK, some 400 eyes were operated in three groups (Prakash et al 2011). In the first group, no iris registration was used (no-iris-registration group). In a second group, preablation static iris registration was performed (static-iris-registration group). In the third group, preablation iris registration with dynamic rotational eye tracking was used (dynamic-iris-registration group). Alpins analysis showed that the indices for assessment of astigmatism outcomes were best in the dynamic-iris-registration group followed by the static-iris-registration group: better ability

Several incisional procedures are aimed to induce a cylinder effect, therefore undercorrection or over correction can be an expected result. Other procedures are aimed at correcting sphere (radial keratotomy, hexagonal keratotomy) and not affecting cylinder, but less than

Radial keratotomy erduced myopia by inducing instability to the peripheral cornea. The effect can be unpredictable. In the report of the PERK (prospective evaluation of radial keratotomy) study (Waring et al 1985), ten percent of patients increased astigmatism by more than 1.00 diopter. In a large scale survey of radial keratotomy complications (Marmer,

Hexagonal keratotomy was used to treat hypermetropia. One article reports of 18 consecutive eyes of 12 patients that underwent hexagonal keratotomy (Werblin 1996). In addition to the primary procedures, 14 enhancements were required in seven eyes for both astigmatism and undercorrection. The author declared he no longer performed or

Cataract surgery is often referred to as a refractive procedure. Cataract surgery is among the safest surgical procedures, but as the most performed ophthalmic procedure, every pro mil

Large incision cataract surgery was compared with phacoemulcification (Minassian et al 2001). The two planned treatments were: extracapsular cataract extraction (ECCE), and small incision surgery by phacoemulsification. In ECCE, a 12–14 mm corneoscleral section was made, while in phacoemulcification a self sealing 3.2 mm clear corneal incision was made on the steep axis of the corneal astigmatism. The post operative astigmatism ws markedly different in both groups. The phacoemulcification group kept the astigmatism just under 1 D, similar to the preoperative value. The ECCE group' on the other end, had a rise

(70%) had a postoperative uncorrected visual acuity (UCVA) of 20/20 or better.

to treat in the right place yielded better ability to predict the outcome.

perfect construction of the incisions can still have an asymmetric effect.

1987), irregular astigmatism was one of the reported complications.

of complication is translated to thousands of suffering patients.

recommended hexagonal keratotomy.

**5.3 Post cataract surgery** 

**5.3.1 Wound related** 

**5.2.4.2 Incisional** 

of astigmatism to more than 3 D, 3 weeks after surgery, declining and stabilizing 6 and 12 months after surgery at slightly less than 1.5 D.

Manual small-incision cataract surgery (SICS) was compared with phacoemulcification (Venkatesh et al 2010). SICS was performed through a 6.5 to 7.0 mm superior frown-shaped sclerocorneal tunnel, while phacoemulsification was performed through a temporal 3.0 mm scleral tunnel incision The mean surgically induced astigmatism (SIA) was 0.80±0.24 D in the phacoemulsification group and 1.20±0.36 D in the manual SICS group.

Smaller incison were compared in a prospective randomized study (Can et al 2010). Patients had standard coaxial (2.8 mm incisions), microcoaxial (2.2 mm incisions), or biaxial microincision (1.2 to 1.4 mm trapezoidal incisions) phacoemulsification. The mean SIA 90 days postoperatively was 0.46 diopter (D), 0.24 D, and 0.13 D, respectively (P<.01). Biaxial microincision surgery, with the smallest incisions, induced the least amount of astigmatism

As would intuitively be suggested, as the wound becomes smaller in size - from 12-14 mm down to 1.2-1.4 mm – the amount of surgically induced astigmatism is reduced.

#### **5.3.2 Subluxed intraocular lens**

The refractive results of displacement of an intraocular lens are known for many years (Lakshminarayanan, 1986). Using a modified Gullstrand schematic model eye, the authors have computed the amount of spherical and cylindrical errors that are induced due to the tilt and/or displacement of the intraocular lens. This refractive change can become a reason for repositioning and suturing the lens in place.

#### **5.4 Post trauma**

Trauma can cause refractive changes of the corneal and of the lens. In most but not all cases, the change is to the worst.

#### **5.4.1 Effect on cornea**

Akinci et al (2007) report of Trauma-induced astigmatism associated with regular astigmatic patterns in corneal topography in 14% of eyes suffering blunt ocular trauma. Induced astigmatism ranged from 1.75 D to 3.60 D.

Reddy et al (2007) report of a blunt trauma causing a large radial partial thickness corneal laceration at the vertical meridian, with several smaller lacerations in the periphery. Corneal topography revealed central flattening, and refraction changed from -3.50-1.50X175 to -1.50 only' reaching 20q20 vision with that correction. The corneal lacerations caused a spherocylindrical effect that luckily was consistent with good vision.

#### **5.4.2 Effect on lens**

Akinci et al (2007) report of Trauma-induced astigmatism associated with lens subluxation in 7% of eyes suffering blunt ocular trauma. Small and hard objects induced astigmatism significantly more frequently than others.

## **6. Presentation**

Astigmatism presentation is both subjective, based on the patient's description, and objective, based on instrument output.

## **6.1 Signs and symptoms**

An astigmatic eye produces blurred vision. When corrected with spectacles, the different refractive power in the two principal meridians may cause distortion of the image on the retina.

## **6.1.1 Visual acuity**

Visual acuity is lower with uncorrected astigmatism. The effect on vision depends both on amount of astigmatism and pupil size. In an experimental setting (Kamiya et al 2011), with astigmatism of 1, 2 and 3 D, logMAR UCVA was 0.04±0.08, 0.09±0.09 and 0.16±0.16 for 1 mm pupils, -0.01±0.09, 0.12±0.15 and 0.33±0.24 for 2 mm pupils, 0.02±0.09, 0.20±0.19 and 0.46±0.30 for 3 mm pupils, 0.02±0.08, 0.24±0.20 and 0.48±0.21 for 4 mm pupils, and 0.08±0.10, 0.33±0.18 and 0.53±0.22 for 5 mm pupils, respectively. The variance of the data was statistically significant (p=0.03 for 1 D, p<0.001 for 2 D, p<0.001 for 3 D, analysis of variance). With-the-rule and against-the-rule astigmatism had similar effect.

## **6.1.2 Visual disturbance / discomfort**

Visual discomfort from small amounts of astigmatism was examined (Wiggins et al 1992). The volunteers wore soft contact lenses, leaving between 0.50 and 1.00 D of residual astigmatism in each eye (mean = 0.68D). They were then examined using either full correction in a trial frame or a control lens (=0.12 D). Analysis of the data indicated greater reported visual comfort for the test lens pair over the control lens pair.

## **6.2 Visual quality**

The optical performance of the eye is related to a few interconnected terms: the point-spread function (PSF), Strehl ratio, and retinal-image spot radius (Miháltz et al 2011). The PSF of an optical system is the irradiance distribution of light from a point source projected onto the retina; it indicates the extent of blurring of the retinal image. The Strehl ratio is the ratio of the peak height of the PSF divided by maximum intensity of PSF in the diffraction-limited perfect eye. The Strehl ratio range is from 0 to 1; the greater the Strehl ratio, the better the quality of vision. Quality of vision can also be described by the minimum spot radius in the retina. Comparing groups of keratoconus eyes, subclinical keratoconus eyes and normal eyes, ocular aberrations were measured with a Hartmann-Shack sensor. The Strehl ratio significantly discriminated between the control group and the two ectatic groups, and the spot ratio separated each group from the other two.

## **6.3 Instrumentation**

Our understanding of phenomena is channeled by the tools we have to measure them: in cataract, loss of visual acuity is easier to quantify than the change in the quality of life caused by the cataract. Visual acuity is therefore the parameter we turn to when considering surgery, although improving quality of life should be our real goal. Through this human property we use our instruments to define our understanding of the term "astigmatism".

## **6.3.1 Keratometry**

The keratometer is used to approximate the refracting power of the cornea (BCSC 2008-2009). The central cornea can be thought of as a very powerful (about 250 D) convex spherical mirror. An illuminated object is placed in front of the cornea. A microscope is used to magnify the image reflected from the corneal surface, and the radius of curvature of the corneal surface is calculated. The final step is to convert radius of curvature into an estimate of the cornea's dioptric refractive power. This step is prone to error, since the anterior corneal surface is measured, but the posterior surface is only estimated. Another drawback of the keratometer is that it measures the central 3 mm of the cornea, and not the entire surface.

The keratometer is used to measure the two main meridians of the cornea, The difference between these two results is the keratometric astigmatism. If the astigmatism is regular, the two meridians perpendicular to each other.

## **6.3.2 Retinoscopy**

28 Astigmatism – Optics, Physiology and Management

An astigmatic eye produces blurred vision. When corrected with spectacles, the different refractive power in the two principal meridians may cause distortion of the image on the

Visual acuity is lower with uncorrected astigmatism. The effect on vision depends both on amount of astigmatism and pupil size. In an experimental setting (Kamiya et al 2011), with astigmatism of 1, 2 and 3 D, logMAR UCVA was 0.04±0.08, 0.09±0.09 and 0.16±0.16 for 1 mm pupils, -0.01±0.09, 0.12±0.15 and 0.33±0.24 for 2 mm pupils, 0.02±0.09, 0.20±0.19 and 0.46±0.30 for 3 mm pupils, 0.02±0.08, 0.24±0.20 and 0.48±0.21 for 4 mm pupils, and 0.08±0.10, 0.33±0.18 and 0.53±0.22 for 5 mm pupils, respectively. The variance of the data was statistically significant (p=0.03 for 1 D, p<0.001 for 2 D, p<0.001 for 3 D, analysis of

Visual discomfort from small amounts of astigmatism was examined (Wiggins et al 1992). The volunteers wore soft contact lenses, leaving between 0.50 and 1.00 D of residual astigmatism in each eye (mean = 0.68D). They were then examined using either full correction in a trial frame or a control lens (=0.12 D). Analysis of the data indicated greater

The optical performance of the eye is related to a few interconnected terms: the point-spread function (PSF), Strehl ratio, and retinal-image spot radius (Miháltz et al 2011). The PSF of an optical system is the irradiance distribution of light from a point source projected onto the retina; it indicates the extent of blurring of the retinal image. The Strehl ratio is the ratio of the peak height of the PSF divided by maximum intensity of PSF in the diffraction-limited perfect eye. The Strehl ratio range is from 0 to 1; the greater the Strehl ratio, the better the quality of vision. Quality of vision can also be described by the minimum spot radius in the retina. Comparing groups of keratoconus eyes, subclinical keratoconus eyes and normal eyes, ocular aberrations were measured with a Hartmann-Shack sensor. The Strehl ratio significantly discriminated between the control group and the two ectatic groups, and the

Our understanding of phenomena is channeled by the tools we have to measure them: in cataract, loss of visual acuity is easier to quantify than the change in the quality of life caused by the cataract. Visual acuity is therefore the parameter we turn to when considering surgery, although improving quality of life should be our real goal. Through this human property we use our instruments to define our understanding of the term "astigmatism".

The keratometer is used to approximate the refracting power of the cornea (BCSC 2008-2009). The central cornea can be thought of as a very powerful (about 250 D) convex spherical mirror. An illuminated object is placed in front of the cornea. A microscope is used to magnify the

variance). With-the-rule and against-the-rule astigmatism had similar effect.

reported visual comfort for the test lens pair over the control lens pair.

**6.1 Signs and symptoms** 

**6.1.2 Visual disturbance / discomfort** 

spot ratio separated each group from the other two.

**6.1.1 Visual acuity** 

**6.2 Visual quality** 

**6.3 Instrumentation** 

**6.3.1 Keratometry** 

retina.

The streak retinoscope is a tool to determine objectively the spherocylindrical refractive error, as well as determine whether astigmatism is regular or irregular, and to evaluate opacities and irregularities (BCSC 2008-2009). The examiner adds sphere and cylinder lenses until all spherocylindrical refractive error of the eye is neutralized. Whatever irregularity in the light reflex that remains is irregular astigmatism, or in other nomenclature – high order aberrations.

## **6.3.3 Corneal topography**

Corneal topography is similar in concept to conventional keratometry. However, unlike keratometry, that measures tow pairs of spots in the central 3 mm of the cornea, corneal topographers map the surface of the cornea, from close to the center out to 4 or 5 mm from the center (BCSC 2008-2009).

Most topographers are based on circular mires, similar to a Placido disc, consisting of many concentric lighted rings. The size and shape of the reflected images of the mires are the data, from which multiple calculations, similar to those behind the concept of the keratometer, are performed. The end result is a color map, graphically illustrating the corneal curvature in many thousands of spots on the corneal surface. Many topographers also calculate the SIM K (simulated keratometry) value, providing the power and location of the steepest and flattest meridians for the 3-mm optical zone

Different patterns of corneal topography have been described (Rabinowitz 1998): One (round) describes a spherical surface with no astigmatism. Two more (oval and symmetric bow tie) describe a spherocylindrical surface with no irregular astigmatism. All other patterns describe different amounts of irregularity: superior steepening, inferior steepening, irregular, symmetric bow tie with skewed radial axes, asymmetric bow tie with inferior steepening, asymmetric bow tie with superior steepening, asymmetric bow tie with skewed radial axes.

#### **6.3.4 Corneal tomography**

Scheimpflug photography and densitometric image analysis are very precise techniques for light scattering measurement and biometry in the anterior segment of the eye (Wegener & Laser-Junga, 2009). Commercial instruments based on the Scheimpflug photography princilple take multiple images of the cornea, all centered on the corneal apex. The front and back surface of the cornea are detected for each image, and 3 dimentional representation of the front and the back surfaces of the cornea are built. The total optical effect of the cornea, from both front and back surfaces, can be calculated.

#### **6.3.5 Wavefront analysis and retinal raytracing**

Wavefront analysis and retinal raytracing are used to measure the lower and the higherorder optical aberrations of the entire eye. Wavefront analysis is the study of the shape of light waves as they leave an object point and how they are affected by optical media (BCSC 2008-2009). An ideal optical system with no aberrations would produce a flat wavefront. Any aberration would distort the shape of the wavefront. There are different methods to measure the wavefront. In Hartmann-Shack aberrometry a single spot of light is lit on the retina, and the wavefront of the exiting light is calculated. This method is considered outgoing aberrometry. In Tscherning aberrometry the ingoing light passes through a mask of holes before entering the eye. The resultant array of spots on the retina is captured with a high-magnification camera, and the wavefront of the entering light is calculated. This method is considered ingoing aberrometry. Retinal raytracing is another example of ingoing aberrometry. Here a laser beam is used to scan across the pupil. At each laser beam position, the amount of deviation is measured, and the degree of aberration can thereby be calculated.

## **7. Conclusion**

Astigmatism affects a large portion of people. Much of it is regular, correctable with spectacles or soft contact lenses. Other refractive irregularities, or high order aberrations, are partially or fully correctable with rigid contact lenses or refractive surgery.

The understanding of irregular astigmatism grew with the development of the field of refractive surgery. With instruments capable of manipulating tissue in the sub micron level, there is motivation to research and to treat. The future will bring more diagnostic devices and more treatment modalities, improving our ability to better treat refractive errors, including different forms of astigmatism.

## **8. References**


light waves as they leave an object point and how they are affected by optical media (BCSC 2008-2009). An ideal optical system with no aberrations would produce a flat wavefront. Any aberration would distort the shape of the wavefront. There are different methods to measure the wavefront. In Hartmann-Shack aberrometry a single spot of light is lit on the retina, and the wavefront of the exiting light is calculated. This method is considered outgoing aberrometry. In Tscherning aberrometry the ingoing light passes through a mask of holes before entering the eye. The resultant array of spots on the retina is captured with a high-magnification camera, and the wavefront of the entering light is calculated. This method is considered ingoing aberrometry. Retinal raytracing is another example of ingoing aberrometry. Here a laser beam is used to scan across the pupil. At each laser beam position, the amount of deviation is measured, and the degree of aberration can thereby be calculated.

Astigmatism affects a large portion of people. Much of it is regular, correctable with spectacles or soft contact lenses. Other refractive irregularities, or high order aberrations, are

The understanding of irregular astigmatism grew with the development of the field of refractive surgery. With instruments capable of manipulating tissue in the sub micron level, there is motivation to research and to treat. The future will bring more diagnostic devices and more treatment modalities, improving our ability to better treat refractive errors,

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## **Optics of Astigmatism and Retinal Image Quality**

M. Vilaseca, F. Díaz-Doutón, S. O. Luque, M. Aldaba, M. Arjona and J. Pujol *Centre for Sensors, Instruments and Systems Development (CD6) Universitat Politècnica de Catalunya (UPC) Spain* 

## **1. Introduction**

32 Astigmatism – Optics, Physiology and Management

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reconstruction algorithms in representing corneal aberration of normal and

In the first part of this chapter, the optical condition of astigmatism is defined. The main causes and available classifications of ocular astigmatism are briefly described. The most relevant optical properties of image formation in an astigmatic eye are analysed and compared to that of an emmetropic eye and an eye with spherical ametropia. The spectacle prescription and axis notation for astigmatism are introduced, and the correction of astigmatism by means of lenses is briefly described.

The formation of the retinal image for extended objects and the related blurring are also analysed, and the real limits of tolerance of uncorrected astigmatism are provided. Simulations of retinal images in astigmatic eyes, obtained by means of commercial optical design software, are also presented.

Finally, the clinical assessment of retinal image quality by means of wavefront aberrometry and double-pass systems in eyes with astigmatism is presented, and current trends in research related to this topic are highlighted.

## **2. Optics of astigmatism**

## **2.1 Definition, causes and classification**

Astigmatism is a meridian-dependent type of refractive error that is present in most human eyes (Rabbets, 2007; Tunnacliffe, 2004; Atchison & Smith, 2000). Astigmatic (or toroidal) surfaces have two principal meridians, with the curvature of the surface ranging from a minimum on one of these meridians to a maximum on the other. Clinically, this refractive anomaly is described as a bivariate quantity consisting of an astigmatic modulus and axis (McKendrick & Brennan, 1996).

The main known causes of ocular astigmatism are hereditary and involve a lack of symmetry on the optical surfaces of the cornea and the crystalline lens. The main factors contributing to corneal and lenticular astigmatism are the following:


Other less frequent causes of astigmatism can also be cited: refractive index variation in some meridians of the eye due to a rare pathological condition; and irregular astigmatism, which notably occurs in corneal conditions such as keratoconus, where the principal meridians are not perpendicular to each other.

Astigmatism can be classified according to several different factors:

	- Compound hypermetropic astigmatism, in which both principal meridians have insufficient refractive power for the length of the eye.
	- Simple hypermetropic astigmatism, in which only one principal meridian has insufficient refractive power for the length of the eye, while the other is emmetropic.
	- Mixed astigmatism, in which one principal meridian has insufficient refractive power for the length of the eye while the other has too much refractive power.
	- Simple myopic astigmatism, in which only one principal meridian has too much refractive power for the length of the eye, while the other is emmetropic.
	- Compound myopic astigmatism, in which both principal meridians have too much refractive power for the length of the eye.
	- With-the-rule astigmatism, in which the flattest meridian is nearer the horizontal than the vertical (90±30º).
	- Against-the-rule astigmatism, in which the flattest meridian is nearer the vertical than the horizontal (0±30º).
	- Oblique astigmatism, in which the principal meridians are more than 30º from the horizontal and vertical meridians (45±15º).
	- Regular astigmatism, in which the principal meridians are perpendicular to each other and therefore correctable with conventional ophthalmic lenses.
	- Irregular astigmatism, in which the principal meridians are not perpendicular to each other or there are other rotational asymmetries that are not correctable with conventional ophthalmic lenses.

Many authors have measured the values and types of astigmatism exhibited by the human population (Baldwin & Mills, 1981; Kragha, 1986). There are various causes of change in eye astigmatism, including age and accommodation (Artal et al., 2002; Saunders, 1986, 1988; Atkinson, 1980; Gwiazda et al., 1984; Ukai & Ichihashi, 1991; Millodot & Thibault, 1985) and surgery (Bar-Sela et al., 2009; de Vries, 2009; Yao et al., 2006; Vilaseca et al., 2009a).

## **2.2 The retinal image of a point object**

In an emmetropic eye or in an eye with spherical ametropia, rays diverging from a point on the axis are converged to a conjugate image point provided that the paraxial approximation is taken into account. In an eye with regular astigmatism, the image of a point object is not a point because of the different refractive powers corresponding to each of the principal meridians. In this case, the image of a point object is generally an ellipse, as shown in Figure 1.

The figure shows the main features of the refracted pencil in an astigmatic eye. For convenience, the principal meridians denoted as y and z are presented in the vertical and horizontal directions, respectively. In this particular case, the vertical meridian (y) has the greatest optical power and a focal line F'y. This means that parallel rays contained in a vertical plane will be converged onto a point located on this focal line, while parallel rays

which notably occurs in corneal conditions such as keratoconus, where the principal

Compound hypermetropic astigmatism, in which both principal meridians have

 Simple hypermetropic astigmatism, in which only one principal meridian has insufficient refractive power for the length of the eye, while the other is

 Mixed astigmatism, in which one principal meridian has insufficient refractive power for the length of the eye while the other has too much refractive power. Simple myopic astigmatism, in which only one principal meridian has too much refractive power for the length of the eye, while the other is emmetropic. Compound myopic astigmatism, in which both principal meridians have too much

With-the-rule astigmatism, in which the flattest meridian is nearer the horizontal

Against-the-rule astigmatism, in which the flattest meridian is nearer the vertical

Oblique astigmatism, in which the principal meridians are more than 30º from the

Regular astigmatism, in which the principal meridians are perpendicular to each

 Irregular astigmatism, in which the principal meridians are not perpendicular to each other or there are other rotational asymmetries that are not correctable with

Many authors have measured the values and types of astigmatism exhibited by the human population (Baldwin & Mills, 1981; Kragha, 1986). There are various causes of change in eye astigmatism, including age and accommodation (Artal et al., 2002; Saunders, 1986, 1988; Atkinson, 1980; Gwiazda et al., 1984; Ukai & Ichihashi, 1991; Millodot & Thibault, 1985) and

In an emmetropic eye or in an eye with spherical ametropia, rays diverging from a point on the axis are converged to a conjugate image point provided that the paraxial approximation is taken into account. In an eye with regular astigmatism, the image of a point object is not a point because of the different refractive powers corresponding to each of the principal meridians. In this case, the image of a point object is generally an ellipse, as shown in Figure

The figure shows the main features of the refracted pencil in an astigmatic eye. For convenience, the principal meridians denoted as y and z are presented in the vertical and horizontal directions, respectively. In this particular case, the vertical meridian (y) has the greatest optical power and a focal line F'y. This means that parallel rays contained in a vertical plane will be converged onto a point located on this focal line, while parallel rays

other and therefore correctable with conventional ophthalmic lenses.

surgery (Bar-Sela et al., 2009; de Vries, 2009; Yao et al., 2006; Vilaseca et al., 2009a).

meridians are not perpendicular to each other.

The associated spherical refractive errors:

than the vertical (90±30º).

than the horizontal (0±30º).

conventional ophthalmic lenses.

**2.2 The retinal image of a point object** 

1.

emmetropic.

The axis direction:

The regularity of surfaces:

Astigmatism can be classified according to several different factors:

refractive power for the length of the eye.

horizontal and vertical meridians (45±15º).

insufficient refractive power for the length of the eye.

Fig. 1. Plots showing image formation in an emmetropic eye or an eye with spherical ametropia (left) and in an eye with a with-the-rule astigmatic refractive error (right). The principal meridians (y, z), the first and second focal lines (F'y, F'z), and the disc of least confusion (DLC) are shown.

contained in a horizontal plane will be converged onto a point located on the focal line F'z. At any other distance other than that of the two focal lines, the cross-section of the refracted pencil is generally an ellipse. Precisely at the dioptric midpoint between the two focal lines, the cross-section of the pencil is circular and is called the disc of least confusion (DLC). The region between these two focal lines is known as the conoid of Sturm or Sturm's interval. The characteristics of the blurred ellipse depend on the pupil diameter and on the type of astigmatism (Charman & Voisin, 1993a, 1993b; Keating & Carroll, 1976).

#### **2.3 Ocular refraction: notation and correction**

Refraction (defined as the vergence of the eye's far point [or *punctum remotum*], i.e. the point conjugate with the fovea of the unaccommodated eye) is generally used to quantify any spherical or astigmatic ametropia. In the case of astigmatism (A), the absolute value of the difference between the refraction of the most powerful meridian (Ry) and that of the flattest one (Rz) is commonly used. This is equivalent to computing the difference in terms of refractive power between the least powerful meridian (Pz) and the most powerful one (Py):

$$\mathbf{A} = \left| \mathbf{R}\_{\mathbf{y}} - \mathbf{R}\_{\mathbf{z}} \right| = \left| \mathbf{P}\_{\mathbf{z}} - \mathbf{P}\_{\mathbf{y}} \right| \tag{1}$$

The notation commonly employed for astigmatism is the one also typically used for the prescription of sphero-cylindrical lenses. Astigmatism can therefore be thought of as being formed by the following components: sphere (S); cylinder (C), which describes how the most different meridian differs from the sphere; and axis () (Figure 2). In the notation for astigmatism, the refraction corresponding to the most powerful plane is often given first (Ry), followed by the value of the astigmatism (A), and finally the axis of the most powerful meridian (y) (see Equation 2). However, there is also another possibility: the refraction corresponding to the least powerful meridian can be given first (Rz), followed by the value of the astigmatism but with the sign changed (-A), and finally the axis of the least powerful meridian (z). These two options—the "plus cylinder notation" and the "minus cylinder notation"—are the two conventions for indicating the amount of astigmatism in a spectacle prescription.

$$\begin{array}{ccccc}\text{S} & \text{C} & a\\\text{R}\_{\text{y}} & \text{A} & a\_{y}\\\text{R}\_{z} & -\text{A} & a\_{z}\end{array} \tag{2}$$

The following is an example of spectacle prescription. Consider an eye with compound hypermetropic astigmatism with refractions of +1.00 D in the vertical meridian (R90º = +1.00 D) and +2.00 D in the horizontal meridian (R0º = R180º = +2.00 D), that is, with-the-rule. Using Equation 1, the astigmatism of this eye can be quantified (A= R90º - R0º = -1.00 D). Therefore, the notation of the astigmatism will be +2.00 -1.00 0º (or equivalently 180º) or +1.00 +1.00 90º.

Fig. 2. Components of a sphero-cylindrical lens prescription (S: sphere, C: cylinder, : axis)

From this analysis, it is clear that people with astigmatism have blurred vision at all distances, although this may be worse for distant or near vision, depending on the type of astigmatism. The most common way to correct astigmatism is by means of an astigmatic ophthalmic lens, although contact lenses and refractive surgery (laser corneal treatments and intraocular lens implants) are also available. In the astigmatic eye, the patient needs a different correction power for each principal meridian of the eye. Ophthalmic lenses for astigmatism correction usually have a spherical surface as well as a toroidal one that is generally located on the back surface of the lens, and, as mentioned above, are often called sphero-cylindrical lenses. For proper correction, the principal meridians of the lens must be aligned with those of the astigmatic eye, and the principal refractive powers must be such that each principal focus of the lens coincides with the eye's far point (Figure 3).

#### **2.4 The retinal image of an extended object**

The formation of the retinal image of an extended object can be thought of as being composed of images of many individual points, each giving rise to its own astigmatic pencil. As shown above, the intersection of an astigmatic refracted pencil with the retina may form an ellipse (with specific orientation and elongation), a circle, or a line.

Figure 4 shows some examples of images of extended objects as a function of the position of the retina. The most favourable orientation, in which blurring is least apparent, is always perpendicular to the most emmetropic meridian.

notation"—are the two conventions for indicating the amount of astigmatism in a spectacle

*y z* (2)

y z

S C R A R A

The following is an example of spectacle prescription. Consider an eye with compound hypermetropic astigmatism with refractions of +1.00 D in the vertical meridian (R90º = +1.00 D) and +2.00 D in the horizontal meridian (R0º = R180º = +2.00 D), that is, with-the-rule. Using Equation 1, the astigmatism of this eye can be quantified (A= R90º - R0º = -1.00 D). Therefore, the notation of the astigmatism will be +2.00 -1.00 0º (or equivalently 180º) or +1.00 +1.00

Fig. 2. Components of a sphero-cylindrical lens prescription (S: sphere, C: cylinder, : axis) From this analysis, it is clear that people with astigmatism have blurred vision at all distances, although this may be worse for distant or near vision, depending on the type of astigmatism. The most common way to correct astigmatism is by means of an astigmatic ophthalmic lens, although contact lenses and refractive surgery (laser corneal treatments and intraocular lens implants) are also available. In the astigmatic eye, the patient needs a different correction power for each principal meridian of the eye. Ophthalmic lenses for astigmatism correction usually have a spherical surface as well as a toroidal one that is generally located on the back surface of the lens, and, as mentioned above, are often called sphero-cylindrical lenses. For proper correction, the principal meridians of the lens must be aligned with those of the astigmatic eye, and the principal refractive powers must be such

that each principal focus of the lens coincides with the eye's far point (Figure 3).

an ellipse (with specific orientation and elongation), a circle, or a line.

The formation of the retinal image of an extended object can be thought of as being composed of images of many individual points, each giving rise to its own astigmatic pencil. As shown above, the intersection of an astigmatic refracted pencil with the retina may form

Figure 4 shows some examples of images of extended objects as a function of the position of the retina. The most favourable orientation, in which blurring is least apparent, is always

**2.4 The retinal image of an extended object** 

perpendicular to the most emmetropic meridian.

prescription.

90º.

Fig. 3. In the correction of astigmatism by means of ophthalmic lenses, the principal meridians of the lens are aligned with those of the astigmatic eye (y, z), and the principal refractive powers are such that each principal focus of the lens (F'y, F'z) coincides with the eye's far point (pry, prz). The figure shows the correction of an eye with compound myopic astigmatism with the far point in front of each eye.

#### **2.5 Simulations of the retinal image**

The retinal image in an astigmatic eye can be simulated using commercially available optical design software. For most purposes, ocular astigmatism can be studied using schematic eyes, which have specific constructional data, Gaussian constants, cardinal points and aberrations (Atchison & Smith, 2000). From these elements, relevant information about the optical performance of the eye can be obtained. Moreover, software of this sort makes it possible to analyse the influence of several factors, such as pupil size and extra-axial field, on the retinal image. Theoretical models are now being widely used to gain more insight into the performance of various optical systems, such as contact and intraocular lenses, together with the eye.

For an eye with astigmatism of 1 DC (dioptres cylinder), Figure 5 shows simulations of images obtained at different planes using the OSLO® commercial software. Simulations were carried out with artificial eyes using a paraxial model (Le Grand eye model) and a finite model with aspherical surfaces (Navarro eye model) (Atchison & Smith, 2000; Navarro et al., 1985). A 4-mm pupil diameter and an angular field of 25º were considered.

## **3. Vision and tolerances to uncorrected astigmatism**

#### **3.1 Vision in uncorrected astigmatism**

This section discusses vision in uncorrected astigmatism, taking into account the paraxial approximation. The size and blur of the retinal image for an uncorrected astigmatic eye are presented, taking into account the corresponding mean ocular refraction. These factors may have a relevant impact on current ophthalmologic practice.

Fig. 4. Examples of image formation for extended objects as a function of the position of the retina (DLC: disc of least confusion; F'y and F'z: first and second focal lines).

Firstly, let us introduce the concept of mean ocular refraction (Rmean), also commonly known as spherical equivalent, which is the mean of the refractive errors in the two principal meridians of an astigmatic eye. The spherical equivalent gives the position of the DLC in terms of ametropia. Continuing with the example presented in a previous section, an eye with compound hypermetropic astigmatism of +2.00 -1.00 180º would have a mean ocular refraction of +1.50 D. In this example, it is likely that the patient can put the DLC, which is

Fig. 4. Examples of image formation for extended objects as a function of the position of the

Firstly, let us introduce the concept of mean ocular refraction (Rmean), also commonly known as spherical equivalent, which is the mean of the refractive errors in the two principal meridians of an astigmatic eye. The spherical equivalent gives the position of the DLC in terms of ametropia. Continuing with the example presented in a previous section, an eye with compound hypermetropic astigmatism of +2.00 -1.00 180º would have a mean ocular refraction of +1.50 D. In this example, it is likely that the patient can put the DLC, which is

retina (DLC: disc of least confusion; F'y and F'z: first and second focal lines).

the most favourable cross-section of the astigmatic pencil, into focus, provided that sufficient accommodation is available (i.e. the patient is healthy and young). Taking all this into account, the diameter of the DLC (*DLC*) can be computed as a function of the astigmatism as follows:

$$\phi\_{\rm DL,C} = \left| \phi\_{\rm PE} \cdot \frac{A}{2 \cdot (R\_{\rm mean} + P\_{\rm mean})} \right| \tag{3}$$

where *PE* is the eye's entrance pupil diameter and Pmean is the average refractive power of the astigmatic eye, that is, *Py + Pz* 2 <sup>1</sup> .

As an example, Figure 6 shows the size of the DLC corresponding to eyes with uncorrected compound hypermetropic astigmatisms ranging from 0.25 to 3.00 DC. In accordance with the example given above (with-the-rule astigmatism), the following values are considered in the calculations: Py = +59.00D (Ry = +1.00D, if the reduced eye model is considered) and +56.00 ≤ Pz ≤ +58.75 D.

If the DLC has been put into focus, the principal meridians of the astigmatic eye form the corresponding images of an extended object located at any distance in front of and behind the retina, respectively, thus obtaining a blurred retinal image. As stated above, this retinal image can be thought of as being composed of many DLCs, each corresponding to one point of the object, and the size of the blurred retinal image (y') can be calculated as follows (Figure 7):

$$\mathbf{y}' = \mathbf{b} + \phi\_{\text{DLC}} = \left( \left| \frac{u}{R\_y + P\_y} \right| + \left| \phi\_{\text{PE}} \cdot \frac{A}{2 \cdot (R\_{\text{mean}} + P\_{\text{mean}})} \right| \right) \tag{4}$$

where b is the size of the basic (sharp) retinal image that would be formed for a distant object subtending the same angle u.

The degree of blurring (DB) is then computed as the ratio between the diameter of the DLC and the size of the basic (sharp) retinal image as follows:

$$\text{DB} = \left| \frac{\phi\_{\text{DLC}}}{\text{b}} \right| \tag{5}$$

Figure 8 shows two retinal images with different degrees of blurring but with basic (sharp) retinal images of the same size.

Figure 9 shows the degree of blurring corresponding to a far object (3 m) and a near object (40 cm) in eyes with uncorrected compound hypermetropic astigmatisms ranging from 0.25 to 3.00 DC.

In the case of an unaccommodated eye with spherical ametropia, the size of the DLC for a distant object can be computed as follows:

$$
\phi\_{DL,C} = \left| \phi\_{PE} \cdot \frac{R}{R+P} \right| \tag{6}
$$

where R is the spherical refraction and P is the refractive power of the ametropic eye.

Fig. 5. Spot diagrams showing images at different planes corresponding to a point object of an eye with astigmatism of 1 DC, located on and off (8.82 and 12.5 degrees) the optical axis, simulated using the Le Grand artificial eye model (top) and the Navarro artificial eye model (bottom). A pupil diameter of 4 mm and an angular field of 25º were considered. Spot size is in millimetres and focus shift is in dioptres.

Fig. 5. Spot diagrams showing images at different planes corresponding to a point object of an eye with astigmatism of 1 DC, located on and off (8.82 and 12.5 degrees) the optical axis, simulated using the Le Grand artificial eye model (top) and the Navarro artificial eye model (bottom). A pupil diameter of 4 mm and an angular field of 25º were considered. Spot size is

in millimetres and focus shift is in dioptres.

Fig. 6. Diameter of the disc of least confusion (DLC) as a function of the astigmatism (A) (D: dioptres).

Fig. 7. Formation of the image of an extended object located at any distance for an eye with compound hypermetropic astigmatism; for convenience, only the image corresponding to the vertical meridian, located behind the retina, is shown (b: size of the basic [sharp] retinal image that would be formed for a distant object subtending the same angle u; DLC: diameter of the disc of least confusion).

Fig. 8. Two examples of retinal images having basic (sharp) retinal images of the same size (b) but different degrees of blurring (DB) due to the different sizes of the discs of least confusion.

Fig. 9. Degree of blurring (DB) for far (3 m) and near (40 cm) objects as a function of the astigmatism (A) (D: dioptres).

Equations 3 and 6 show that, for a given entrance pupil size, the size of the DLC corresponding to a certain amount of astigmatism is only approximately half the size of the DLC generated by the same amount of spherical ametropia. Therefore, patients with astigmatism and the DLC focused on the retina theoretically have better vision than those with the same amount of spherical ametropia. This paraxial approximation supports the fact that a visual acuity of 6/18 was traditionally thought to indicate spherical ametropia of about 1.00 D or astigmatism of approximately 2.00 DC, provided that the DLC is focused on the retina (Rabbets, 2007). However, Section 3.3 will show that when all aberrations of the eye are considered—rather than the paraxial approximation—the difference in vision between spherical ametropia and astigmatism of the same amount is in fact much lower.

Finally, in eyes with myopic astigmatism, far vision cannot be improved by accommodation, so the patient would be expected to have vision similar to that of an eye with spherical ametropia equal to the corresponding spherical equivalent.

#### **3.2 Lens rotation and mismatches in the cylinder**

This section analyses the effects of lens rotation or mismatches in the cylindrical power of the lens used to correct astigmatism. It should be noted that the rotation of a cylindrical lens with respect to the eye results in a residual refraction consisting of a cylinder (C) and a sphere (S) due to the combination of two obliquely crossed cylinders.

In general, when two toroidal surfaces (with cylindrical powers of C1 and C2) are combined, they result in a residual error, which consists of the following components expressed in terms of sphero-cylindrical lens notation (Rabbets, 2007; Harris, 1988):

$$\begin{aligned} \mathbf{C} &= \pm \sqrt{(\mathbf{C}\_1 + \mathbf{C}\_2)^2 - 4\mathbf{C}\_1 \mathbf{C}\_2 \sin^2 \alpha \\ \mathbf{S} &= \frac{1}{2} \left( \mathbf{C}\_1 + \mathbf{C}\_2 - \mathbf{C} \right) \\ \theta &= \arctan \left( \frac{-\mathbf{C}\_1 + \mathbf{C}\_2 + \mathbf{C}}{\mathbf{C}\_1 + \mathbf{C}\_2 + \mathbf{C}} \right) \cdot \tan \alpha \end{aligned} \tag{7}$$

where is the angle between cylindrical powers C1 and C2, and is the angle measured from C1 to C (see Figure 10).

Fig. 9. Degree of blurring (DB) for far (3 m) and near (40 cm) objects as a function of the

Equations 3 and 6 show that, for a given entrance pupil size, the size of the DLC corresponding to a certain amount of astigmatism is only approximately half the size of the DLC generated by the same amount of spherical ametropia. Therefore, patients with astigmatism and the DLC focused on the retina theoretically have better vision than those with the same amount of spherical ametropia. This paraxial approximation supports the fact that a visual acuity of 6/18 was traditionally thought to indicate spherical ametropia of about 1.00 D or astigmatism of approximately 2.00 DC, provided that the DLC is focused on the retina (Rabbets, 2007). However, Section 3.3 will show that when all aberrations of the eye are considered—rather than the paraxial approximation—the difference in vision between spherical ametropia and astigmatism of the same amount is in fact much lower. Finally, in eyes with myopic astigmatism, far vision cannot be improved by accommodation, so the patient would be expected to have vision similar to that of an eye with spherical

This section analyses the effects of lens rotation or mismatches in the cylindrical power of the lens used to correct astigmatism. It should be noted that the rotation of a cylindrical lens with respect to the eye results in a residual refraction consisting of a cylinder (C) and a

In general, when two toroidal surfaces (with cylindrical powers of C1 and C2) are combined, they result in a residual error, which consists of the following components expressed in

1 2 12

 

1 2 1 2

where is the angle between cylindrical powers C1 and C2, and is the angle measured

C +C +C arctan tan C +C +C

2 2

(7)

C (C +C ) 4C C sin

1 2

S C +C C 1 2

astigmatism (A) (D: dioptres).

from C1 to C (see Figure 10).

ametropia equal to the corresponding spherical equivalent.

sphere (S) due to the combination of two obliquely crossed cylinders.

terms of sphero-cylindrical lens notation (Rabbets, 2007; Harris, 1988):

*= ± =*

*θ =*

**3.2 Lens rotation and mismatches in the cylinder** 

Fig. 10. Composition of two obliquely crossed cylinders with cylindrical powers of C1 and C2 (C: resultant cylinder).

Figure 11 shows an example of residual refraction in terms of cylinder and sphere as a function of the rotation angle, when the ophthalmic lens and the astigmatism of the eye have the same cylindrical powers and when they differ by 0.50 DC. Results are given for eyes with astigmatisms of 1 and 3 D, respectively.

Fig. 11. Residual refraction as a function of the rotation angle between the cylindrical lens and the astigmatism axis () when the cylindrical power of the lens and the astigmatism of the eye have the same value (left) and when the cylindrical power of the lens and the astigmatism of the eye differ by 0.50 DC (right). Results are given for astigmatic eyes with cylinders of 1 and 3 DC, respectively (C: cylinder; S: sphere; D: dioptres; º: degrees).

#### **3.3 Tolerances to uncorrected astigmatism**

The question of the extent to which the retinal image may be degraded by defocus or astigmatism before it starts to be noticeably blurred has already been analysed taking into account the paraxial approximation. This question may play an important role in corrective lens design and refractive surgery outcomes, among other issues. Some studies including experiments involving just-noticeable decreases in image clarity have suggested that the limits for astigmatism are about 1.4 times greater than those for defocus blur (0.32 DC as compared to 0.23 D) (Burton & Haig, 1984; Haig & Burton, 1987). Using a similar approach, Legras et al. found that the cross-cylinder blur limit was about 1.25 times higher than that found for defocus (Legras et al., 2004).

Sloan (1951) determined the relative effects of spherical and cylindrical errors on visual acuity and found that cylindrical errors reduce visual acuity at 0.8 the rate of spherical errors, while Raasch (1995) found that pure cylindrical errors reduced visual acuity at about 0.7 the rate of spherical errors. By carrying out visual acuity simulations, other authors have found that aberrations with low orientation dependence had greater effects than those with higher orientation dependence. In dioptric terms, the ratios of visual acuity loss with astigmatism as compared to defocus were 1.1 and 0.8 for high- and low-contrast letters, respectively (Applegate et al., 2003).

By using an experimental setup based on adaptive optics, Atchison et al. (2009) determined the level of additional aberration at which an individual with normal inherent levels of higher-order ocular aberration becomes aware of blur due to extra defocus or cross-cylinder astigmatism. Cross-cylinder astigmatic blur limits were found to be approximately 90% of those for defocus.

Charman & Voisin (1993b) approached the question of visual tolerance to uncorrected astigmatism by exploring the changes in the modulation transfer function (MTF), which represents the loss of contrast produced by the eye's optics as a function of spatial frequency. The authors found a reduction in the MTF and orientation dependence with the magnitude of the astigmatism. These effects tended to increase with the spatial frequency, but also with the pupil diameter and inversely with the wavelength, because all of these parameters affected the wavefront aberration. Tolerance to astigmatism deduced from purely optical considerations (based on the analysis of the wavefront aberration or changes in the ocular MTF) was found to be approximately 0.25 DC. Nevertheless, tolerance to refractive error depends upon both the changes in the optical image on the retina and the ability of the neural stages of the visual system to detect those changes. Some authors have suggested that, since in real life the peak of the neural contrast sensitivity function at spatial frequencies is lower than 10 cycles per degree (Campbell, 1965), the contrast of natural scenes is low, and the roll-off in the amplitude spectrum is approximately reciprocal to the spatial frequency (Tolhurst et al., 1992), visual tolerances to astigmatic error are more heavily weighted toward the lower end of the spatial frequency spectrum, which is less influenced by astigmatic errors. Hence, although the optical deficits of 0.50 and 0.75 DC appear substantial, some authors have argued that they might be well tolerated in everyday use by less critical observers, although visual discomfort might arise during more exacting tasks such as work at visual display terminals (Wiggins & Daum, 1991).

Although the paraxial approximation is a very useful tool for easily analysing the visual implications of spherical ametropia and astigmatism, the aforementioned results show that this optical approach may be misleading in analysing real tolerances to uncorrected astigmatism in the aberrated human eye (see Section 3.1). The experimental results described above support the notion that the vision differences reported between spherical and astigmatic refractive errors are considerably smaller than those suggested by the paraxial approximation.

lens design and refractive surgery outcomes, among other issues. Some studies including experiments involving just-noticeable decreases in image clarity have suggested that the limits for astigmatism are about 1.4 times greater than those for defocus blur (0.32 DC as compared to 0.23 D) (Burton & Haig, 1984; Haig & Burton, 1987). Using a similar approach, Legras et al. found that the cross-cylinder blur limit was about 1.25 times higher than that

Sloan (1951) determined the relative effects of spherical and cylindrical errors on visual acuity and found that cylindrical errors reduce visual acuity at 0.8 the rate of spherical errors, while Raasch (1995) found that pure cylindrical errors reduced visual acuity at about 0.7 the rate of spherical errors. By carrying out visual acuity simulations, other authors have found that aberrations with low orientation dependence had greater effects than those with higher orientation dependence. In dioptric terms, the ratios of visual acuity loss with astigmatism as compared to defocus were 1.1 and 0.8 for high- and low-contrast letters,

By using an experimental setup based on adaptive optics, Atchison et al. (2009) determined the level of additional aberration at which an individual with normal inherent levels of higher-order ocular aberration becomes aware of blur due to extra defocus or cross-cylinder astigmatism. Cross-cylinder astigmatic blur limits were found to be approximately 90% of

Charman & Voisin (1993b) approached the question of visual tolerance to uncorrected astigmatism by exploring the changes in the modulation transfer function (MTF), which represents the loss of contrast produced by the eye's optics as a function of spatial frequency. The authors found a reduction in the MTF and orientation dependence with the magnitude of the astigmatism. These effects tended to increase with the spatial frequency, but also with the pupil diameter and inversely with the wavelength, because all of these parameters affected the wavefront aberration. Tolerance to astigmatism deduced from purely optical considerations (based on the analysis of the wavefront aberration or changes in the ocular MTF) was found to be approximately 0.25 DC. Nevertheless, tolerance to refractive error depends upon both the changes in the optical image on the retina and the ability of the neural stages of the visual system to detect those changes. Some authors have suggested that, since in real life the peak of the neural contrast sensitivity function at spatial frequencies is lower than 10 cycles per degree (Campbell, 1965), the contrast of natural scenes is low, and the roll-off in the amplitude spectrum is approximately reciprocal to the spatial frequency (Tolhurst et al., 1992), visual tolerances to astigmatic error are more heavily weighted toward the lower end of the spatial frequency spectrum, which is less influenced by astigmatic errors. Hence, although the optical deficits of 0.50 and 0.75 DC appear substantial, some authors have argued that they might be well tolerated in everyday use by less critical observers, although visual discomfort might arise during more exacting

tasks such as work at visual display terminals (Wiggins & Daum, 1991).

Although the paraxial approximation is a very useful tool for easily analysing the visual implications of spherical ametropia and astigmatism, the aforementioned results show that this optical approach may be misleading in analysing real tolerances to uncorrected astigmatism in the aberrated human eye (see Section 3.1). The experimental results described above support the notion that the vision differences reported between spherical and astigmatic refractive errors are considerably smaller than those suggested by the

found for defocus (Legras et al., 2004).

respectively (Applegate et al., 2003).

those for defocus.

paraxial approximation.

## **4. Clinical assessment of retinal image quality in eyes with astigmatism**

Retinal image quality can be clinically analysed using several different tools, such as wavefront aberrometers and double-pass systems. From them, relevant information on the optical quality of the retinal image, in particular of an eye with astigmatism, can be obtained.

Over the past decade, wavefront aberrometers have been widely used to determine retinal image quality in connection with customized wavefront-guided LASIK (Schallhorn et al., 2008; Dougherty & Bains, 2008). Most of these instruments are based on the Hartmann-Shack sensor (Prieto et al., 2000; Liang et al., 1994). They generally consist of a microlens array conjugated with the eye's pupil and a camera placed at its focal plane. If a plane wavefront reaches the microlens array, the image recorded with the camera is a perfectly regular mosaic of spots. If a distorted (that is, aberrated) wavefront reaches the sensor, the pattern of spots is irregular. From the displacement of the spots, the wavefront aberration can be computed by fitting to the Zernike polynomials and the MTF can also be calculated.

Some studies have analysed the wavefront aberrations of the eye as a function of age. Measurements of the total aberrations of the eye and of the aberrations of the anterior corneal surface suggest that astigmatism of the cornea is more widespread than astigmatism of the full eye in younger subjects (Artal et al., 2002). Other recent studies have focused on changes in refraction and peripheral aberrations as a function of accommodation (Mathur et al., 2009; Radhakrishnan & Charman, 2007). The authors of these studies generally found a small change in the astigmatic components of refraction or the higher-order Zernike coefficients, apart from fourth-order spherical aberration, which became more negative at all field locations. Researchers have recently demonstrated that certain combinations of nonrotationally symmetric aberrations, such as coma and astigmatism, can result in better retinal image quality as compared to the condition with the same amount of astigmatism alone (de Gracia et al., 2010). Other authors have studied the effect of cataract surgery on the optical aberrations of the eye, and astigmatism in particular (Montés-Micó et al., 2008; Guirao et al. 2004; Marcos et al., 2007). Most of these studies found that astigmatism increased significantly after surgery.

The double-pass technique has also been shown to be a useful tool for comprehensively evaluating retinal image quality in eyes affected by several optical conditions, such as defocus, astigmatism and higher-order aberrations. Double-pass systems are based on recording images from a point-source object after reflection on the retina and a double pass through the ocular media. Figure 12 shows a conventional layout of a double-pass system, which consists of a laser coupled to an optical fibre as a light source (LD). A motorized optometer consisting of two lenses (L3, L4) with a focal length of 100 mm and two mirrors (M2, M3) is used to measure the subject's defocus correction. A video camera (CCD1) records the double-pass images after the light is reflected on the retina and on a beam splitter (BS2). Pupil alignment is controlled with an additional camera (CCD2). A fixation test (FT) helps the subject during the measurements. The instrument has an artificial and variable exit pupil (ExP), controlled by a diaphragm wheel, whose image is formed on the subject's natural pupil plane.

Unlike standard wavefront aberrometry, double-pass systems directly compute the MTF from the acquired double-pass retinal image by Fourier transform, making possible the complete characterization of the optical quality of the eye (Santamaría et al., 1987). Because of the differences between the two technologies, recent studies have suggested that

Fig. 12. Double-pass experimental setup (LD: laser; L1-L5: lenses; EP: entrance pupil; ExP: exit pupil; BS1, BS2: beam splitters; FT: fixation test; CCD1, CCD2: CCD cameras; M1-M4: mirrors; DF: dichroic filter; IL: infrared LED).

wavefront aberrometers may overestimate retinal image quality in eyes where higher-order aberrations and intraocular scattered light are prominent (Díaz-Doutón et al., 2006).

Figure 13 shows the double-pass images and the corresponding intensity profiles as a function of the angle (averaged section of the double-pass image), MTFs, and simulated vision using a standard visual acuity chart for an emmetropic eye, an eye with uncorrected astigmatism and an eye with spherical ametropia. The measurements were taken with the Optical Quality Analysis System (OQAS, Visiometrics, S.L., Spain) (Güell et al., 2006; Saad et al., 2010; Vilaseca et al., 2010a), which is a double-pass system currently available for use in daily clinical practice that includes this application.

The double-pass technique has been used extensively to determine the optical quality of the eye, mainly by means of the ocular MTF. Studies have revealed the potential of this technique in basic research (Artal et al., 1995a; Williams et al., 1994, 1996) and in its application to ophthalmology, optometry, and ophthalmic optics testing. In particular, it has been used to assess retinal image quality in patients with keratitis (Jiménez et al., 2009) and patients undergoing refractive surgery, such as LASIK (Vilaseca et al., 2009a; Vilaseca et al., 2010b) and intraocular lens implants (Vilaseca et al., 2009a; Alió et al., 2005; Fernández-Vega et al., 2009; Artal et al., 1995b). This technique has also been used to evaluate presbyopia after photorefractive keratectomy (Artola et al., 2006), to study retinal image quality in contact lens wearers (Torrents at al., 1997), and to analyse in vitro optical quality of foldable monofocal intraocular lenses (Vilaseca et al., 2009b).

Pujol et al. (1998) used the double-pass technique to study retinal image quality in eyes with uncorrected astigmatism. Performing direct optical measurements to characterize retinal

Fig. 12. Double-pass experimental setup (LD: laser; L1-L5: lenses; EP: entrance pupil; ExP: exit pupil; BS1, BS2: beam splitters; FT: fixation test; CCD1, CCD2: CCD cameras; M1-M4:

wavefront aberrometers may overestimate retinal image quality in eyes where higher-order

Figure 13 shows the double-pass images and the corresponding intensity profiles as a function of the angle (averaged section of the double-pass image), MTFs, and simulated vision using a standard visual acuity chart for an emmetropic eye, an eye with uncorrected astigmatism and an eye with spherical ametropia. The measurements were taken with the Optical Quality Analysis System (OQAS, Visiometrics, S.L., Spain) (Güell et al., 2006; Saad et al., 2010; Vilaseca et al., 2010a), which is a double-pass system currently available for use in

The double-pass technique has been used extensively to determine the optical quality of the eye, mainly by means of the ocular MTF. Studies have revealed the potential of this technique in basic research (Artal et al., 1995a; Williams et al., 1994, 1996) and in its application to ophthalmology, optometry, and ophthalmic optics testing. In particular, it has been used to assess retinal image quality in patients with keratitis (Jiménez et al., 2009) and patients undergoing refractive surgery, such as LASIK (Vilaseca et al., 2009a; Vilaseca et al., 2010b) and intraocular lens implants (Vilaseca et al., 2009a; Alió et al., 2005; Fernández-Vega et al., 2009; Artal et al., 1995b). This technique has also been used to evaluate presbyopia after photorefractive keratectomy (Artola et al., 2006), to study retinal image quality in contact lens wearers (Torrents at al., 1997), and to analyse in vitro optical quality of foldable

Pujol et al. (1998) used the double-pass technique to study retinal image quality in eyes with uncorrected astigmatism. Performing direct optical measurements to characterize retinal

aberrations and intraocular scattered light are prominent (Díaz-Doutón et al., 2006).

mirrors; DF: dichroic filter; IL: infrared LED).

daily clinical practice that includes this application.

monofocal intraocular lenses (Vilaseca et al., 2009b).

Fig. 13. Double-pass images corresponding to an emmetropic eye (left), an astigmatic eye (middle) and an eye with spherical ametropia (right). The corresponding intensity profiles as a function of the angle (minutes of arc [arc min]), MTF curves and simulated vision using a standard visual acuity chart are also provided.

image quality in astigmatic eyes can be advantageous since the standard clinical evaluation using visual acuity tests may be affected by non-optical problems in the subjects' visual system that cannot be separated from the optical ones. Pujol et al. studied the influence of the amount of astigmatism and changes in axis of astigmatism on the eye's optical performance by means of numerical simulation using an emmetropic eye model (Navarro et al., 1985), an artificial eye, and three real subjects (JG, JP, VB) for a 4-mm artificial pupil. Different amounts of astigmatism were obtained by varying the cylindrical power of a lens situated in front of the eye, from 0.25 DC overcorrection to 1 DC undercorrection at intervals of 0.25 DC. Changes in the axis of astigmatism were obtained by rotating the lens, which neutralized the astigmatism in an angle of ±10° at 5° intervals.

The results showed a decrease in retinal image quality and an increase in the degree of image astigmatism as the amount of astigmatism increased (Figure 14) or when the angle between the lens and the eye axis was other than zero. In general, the largest variations were found when the astigmatism changed from 0 to 0.25 DC or when the axis changed from 0° to ±5°. Astigmatism reduced optical performance in the eye model, the artificial eye, and the living eyes, but in different proportions. The images obtained by simulation had better optical quality than those of the artificial eye, probably because exact focusing of the image was more difficult for the artificial eye than for the simulated eye. When these images were compared with those for the living eyes, considerable differences in shape and size due to the eye's optical performance were observed. The aberrations and intraocular scattering in living eyes introduced an additional blur into the retinal image that tended to reduce the loss of retinal image quality introduced by astigmatism.

Fig. 14. Double-pass images numerically simulated in an eye model (a), measured in an artificial eye (b) and measured in a living eye (c) for different amounts of astigmatism. Negative values of astigmatism mean overcorrection and positive values mean undercorrection (source: Pujol et al., 1998).

performance by means of numerical simulation using an emmetropic eye model (Navarro et al., 1985), an artificial eye, and three real subjects (JG, JP, VB) for a 4-mm artificial pupil. Different amounts of astigmatism were obtained by varying the cylindrical power of a lens situated in front of the eye, from 0.25 DC overcorrection to 1 DC undercorrection at intervals of 0.25 DC. Changes in the axis of astigmatism were obtained by rotating the lens, which

The results showed a decrease in retinal image quality and an increase in the degree of image astigmatism as the amount of astigmatism increased (Figure 14) or when the angle between the lens and the eye axis was other than zero. In general, the largest variations were found when the astigmatism changed from 0 to 0.25 DC or when the axis changed from 0° to ±5°. Astigmatism reduced optical performance in the eye model, the artificial eye, and the living eyes, but in different proportions. The images obtained by simulation had better optical quality than those of the artificial eye, probably because exact focusing of the image was more difficult for the artificial eye than for the simulated eye. When these images were compared with those for the living eyes, considerable differences in shape and size due to the eye's optical performance were observed. The aberrations and intraocular scattering in living eyes introduced an additional blur into the retinal image that tended to reduce the

Fig. 14. Double-pass images numerically simulated in an eye model (a), measured in an artificial eye (b) and measured in a living eye (c) for different amounts of astigmatism. Negative values of astigmatism mean overcorrection and positive values mean

neutralized the astigmatism in an angle of ±10° at 5° intervals.

loss of retinal image quality introduced by astigmatism.

undercorrection (source: Pujol et al., 1998).

The study also reported a reduction in the MTF with the presence of astigmatism. Figure 15 shows the MTF profiles in the direction of the low-power (solid curve) and high-power (dotted curve) principal meridians corresponding to the three subjects when astigmatism was 0 and 1 DC. Measurements were taken in the plane of the focal line corresponding to the low-power meridian. These profiles showed the minimum and maximum effects of

Fig. 15. MTF profiles in the direction of the low-power (solid curve) and the high-power (dotted curve) principal meridians for the three subjects and for astigmatisms of 0 and 1 DC (source: Pujol et al., 1998).

Fig. 16. Average MTF profiles for a simulated eye, an artificial eye, and a living eye (subject JG) at astigmatism values of 0 and 1 DC (source: Pujol et al., 1998).

astigmatism: the lower curve corresponds to the orientation showing the maximum elongation of the double-pass image, while the upper curve corresponds to the minimum elongation. In this context, Figure 16 shows the mean MTF obtained by averaging this function over all orientations for the simulated eye, the artificial eye, and the living eye (subject JG) when the astigmatism was 0 and 1 DC. For astigmatism of 0 DC, the highest optical performance was obtained for the eye model and the lowest for the living eye, which means that, when there is no dioptric blur in the retinal image, the aberrations present in the living eye are greater than those in the eye model. The optical performances of the living, artificial and simulated eyes were more similar for astigmatism of 1 DC than for 0 DC. At 0 DC, the associated dioptric blur was the greatest aberration.



Fig. 17. Refractive error of the three subjects (Table 1) and residual sphero-cylindrical power when the lens and eye axes are different (Table 2) (source: Pujol et al., 1998).

astigmatism: the lower curve corresponds to the orientation showing the maximum elongation of the double-pass image, while the upper curve corresponds to the minimum elongation. In this context, Figure 16 shows the mean MTF obtained by averaging this function over all orientations for the simulated eye, the artificial eye, and the living eye (subject JG) when the astigmatism was 0 and 1 DC. For astigmatism of 0 DC, the highest optical performance was obtained for the eye model and the lowest for the living eye, which means that, when there is no dioptric blur in the retinal image, the aberrations present in the living eye are greater than those in the eye model. The optical performances of the living, artificial and simulated eyes were more similar for astigmatism of 1 DC than for 0 DC. At 0

Fig. 17. Refractive error of the three subjects (Table 1) and residual sphero-cylindrical power

when the lens and eye axes are different (Table 2) (source: Pujol et al., 1998).

DC, the associated dioptric blur was the greatest aberration.

To evaluate the influence of variation in the axis of astigmatism on retinal image quality, the authors placed in front of the eye a lens whose cylindrical power was suitable for correcting the astigmatism and whose axis formed a particular angle with the axis of the astigmatic eye. This situation can occur in clinical practice when the correction axis and the eye axis are not coincident. Figure 17 shows the refractive errors and the residual sphero-cylindrical power for the three subjects and for the values of the angle formed by the lens and eye axes. According to Table 2, the retinal image quality decreased as the angular change in axis of astigmatism increased. For a particular angular value, the greatest variation was expected for subjects with a higher residual refractive error (JP, JG). However, the results did not show the proportional dependence on the residual refractive error that had been expected on the basis of the geometrical approximation. Again in this case, the aberrations (besides astigmatism) present in the living eye reduced the contribution of the residual spherocylindrical power to a decrease in retinal image quality.

Torrents et al. published a study in which they applied the double-pass technique to determine optical image quality in monofocal contact lens wearers and showed the

Fig. 18. Double-pass images for an eye with no lens, a soft contact lens and a rigid gaspermeable contact lens (a), and the corresponding intensity profiles as a function of the angle (b) (source: Torrents et al., 1997).

influence of ocular astigmatism on retinal image quality in this case (Torrents et al., 1997). In eyes with corneal astigmatism, the best results were obtained with rigid gas-permeable contact lenses because the lens offset the corneal astigmatism. The MTF was considerably smaller when no lenses or soft lenses were worn, even for small amounts of astigmatism (0.5 D) (Figure 18).

Finally, Vilaseca et al. (2009a) analysed the retinal image quality of eyes that had undergone kerato-refractive and phakic intraocular lens surgery by means of a double-pass system. In this study, residual astigmatism, and therefore retinal image quality, was found to be slightly different depending on the surgical procedure used. Although in general there was a high correction of astigmatism as well as good safety and efficacy indexes, some surgically induced astigmatism remained one month after surgery, especially in patients with intraocular lens implants.

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