Combining refractive and topographic data in corneal refractive surgery for astigmatism

Transcription

1 Combining refrative and topographi data in orneal refrative surgery for astigmatism A new method based on polar value analysis and mathematial optimization Kristian Næser Department of Ophthalmology, Randers Regional Hospital, Denmark ABSTRACT. Purpose: To provide a theoretial approah for ombining refrative and topographi data in the planning of orneal laser refrative surgery for astigmatism. Methods: Refrative and topographi data for astigmatism were transformed to the orneal plane. Net astigmatisms were onverted to polar values. An optimization proess was performed with the use of differential alulus. Results: With this method, the magnitude of the orneal astigmatism is redued or unaltered, while its orientation is maintained. The method identifies the redution in orneal astigmatism, whih will yield the largest redution in refrative astigmati magnitude. Conlusions: The advantage of the optimization method desribed in this artile is a onsistent redution in orneal astigmatism towards spheriity. No new orneal astigmatism is arved on the ornea, and probably less tissue is ablated. The optimization method may also be used to ombine refrative and topographi data for higher order aberrations with sinusoidal omponents. However, ompared to the traditional purely refrative driven treatment, more refrative astigmatism will remain in the eye in most ases. A ontrolled linial trial is neessary for omparing these two treatment modalities. Key words: astigmatism orneal refrative surgery mathematis optimization polar values refration Ata Ophthalmol. 2012: 90: ª 2011 The Author Ata Ophthalmologia ª 2011 Ata Ophthalmologia Sandinavia Foundation doi: /j x There are two prinipal types of astigmatism with relevane for urrent orneal laser refrative surgery: (1) The total oular astigmatism (TOA), whih is the optial orretion neessary for neutralizing the astigmatism generated by the oular surfaes (Naeser 2008). This is the astigmati omponent of the manifest refration, usually as expressed in the spetale orretion. The value may also be objetively determined with auto-refration or wave-front analysis. (2) The anterior orneal astigmatism (ACA), whih is the topographi orneal astigmatism as measured with keratometry, orneal topography or orneal aberrometry (Naeser 2008). The greatest soure of refrative astigmatism is usually the toriity of the anterior orneal surfae. In suh eyes, there is often a good alignment between the axes of the refrative and the anterior orneal astigmatisms. Conversely, a onsiderable differene in alignment is often observed in eyes with large lentiular astigmatisms. But, in virtually all eyes, there will be some differene both in the orientation and the magnitude of the two types of astigmatism. Eximer laser photoastigmati refrative keratetomy (PARK) reshapes the anterior orneal surfae, but is usually based on preoperative refrative data only (Alpins 1997). The orretion of the refrative astigmatism is therefore loaded on the ornea. When the magnitude and orientation of the topographi and refrative astigmatisms differ, orneal toriity may be inreased with subsequent degradation of the retinal image and indution of higher order aberrations (HOAs), suh as oma and trefoil. In the extreme ase of a spherial ornea and a onsiderable lentiular astigmatism, the orneal astigmatism is inreased in full proportion to the refrative astigmatism. These optial problems may inrease over time, as the orneal astigmatism usually inreases in magnitude and hanges orientation with age (Gudmundsdottir et al. 2000). 768

2 Furthermore, after atarat surgery, the orneal toriity will be fully refleted in the refrative astigmatism. Conversely, topography driven treatments do not onsider the refrative astigmatism. This modality is used in highly irregular orneas, suh as in keratoonus, and following previously deentred ablations. Topography-guided treatment ombined with orneal ollagen ross-linking has reently been desribed in keratooni eyes (Kanellopoulus 2009; Stojanovi et al. 2010). Suh treatments will usually leave onsiderable amounts of refrative astigmatism in the eye. A solution to these problems is to onsider both the refrative and the topographi astigmatisms in the planning of laser ablative orneal proedures. Noel Alpins (1997) suggested a method, where the astigmatism is distributed on both the refrative and topographi segments. By varying the emphasis between these two omponents, the surgeon may elet a suitable diretion for the postoperative refrative astigmatism. In the present study, the theory for a new ombination of refrative and topographi astigmati data is developed. The method seeks to redue the astigmati magnitude, rather than to alter the astigmati diretion. The method is based on transformation of net astigmatisms to polar values (Naeser & Hjortdal 2001a,b; Naeser 2008) with subsequent mathematial optimization of the treatment plan. With this method, the magnitude of the orneal astigmatism is redued or unaltered, while its orientation is maintained. The method identifies the redution in orneal astigmatism, whih will yield the largest redution in refrative astigmati magnitude. Basi Aspets Transformation of astigmati data to polar values Consider the prinipal meridional power F 1 along the meridian a degrees ( ) and F 2 along (a +90), where F 1 > F 2. The net astigmatism is represented by (M at a), where M=F 1 )F 2, and where both F 1 and F 2 are expressed in dioptres (D). All mathematial operations may be performed following transformation of the net astigmatism to the following two polar values, all expressed in dioptres (D) (Naeser 2008): The polar value along zero degrees: KPðÞM 0 osð2aþ ð1þ The polar value along 45 degrees: KPð45Þ M sinð2aþ ð2þ These polar values may be reonverted to the usual net ylinder notation by the following general equations: M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KPð0Þ 2 þ KPð45Þ 2 ð3þ a artan M KPðÞ 0 ð4þ KPð45Þ Suh transformations may be performed with any omponent system (Harris 1991; Thibos et al. 1997; Kaye & Harris 2002). The orrelation between the polar value system and the omponents proposed by Thibos et al. (1997) is: KPðÞ2 0 J0 and KPð45Þ 2 J45 ð5þ Astigmati formats The subjetive refrative error indiates the optial (spetale) orretion allowing parallel inident light to fous on the fovea. The optial orretion balanes the total oular refrative power, generated by the oular refrative surfaes. This proess emmetropizes the entire spetale-eye system. The orretion and the oular refrative power at any given optial plane should be idential, but of opposite signs (Naeser 2008): Corretion Oular refrative power ð6þ To allow for omparison, both refrative and topographi astigmatisms must be transformed to the orneal plane and to the same oular refrative power format. The transformation of data is different for the two types of astigmatism: Corneal data The anterior orneal astigmatism, ACA, is based on an oular refrative surfae. Keratometri and orneal topographi data are always given in power at notation (Naeser 2008). Data transformation of orneal data is therefore easy, as the orneal net astigmatisms simply are onverted to polar values with eqns (1 and 2). Example 1 The anterior orneal net astigmatism ACA = (1.5 D at 75); KP (0) = 1.5 os(150 ) =)1.30 D KP (45) = 1.5 sin(150 ) = 0.75 D Refrative data Data are transformed in the following six steps: (1) Perform a metiulous subjetive refration (= orretion) and indiate in the usual spheroylindrial format. (2) Transform to rossed ylinder power format. (3) Transform data from the spetale to the orneal plane (Holladay et al. 1998; Naeser 2008). (4) Change sign to obtain oular refrative astigmatism. (5) Convert to net astigmatism. (6) Transform to polar values. Example 2 (1) Total oular astigmatism as manifest spetale plane refration (=orretion) in minus ylinder axis format = ()3.90 D 1.44 D axis 155 ). (2) Total oular astigmatism as manifest spetale plane refration in rossed ylinder power format = ()3.90 D at 155 ) and ()5.34 D at 65 ). (3) Total oular astigmatism as manifest orneal plane refration in rossed ylinder power format, assuming a vertex distane of 14 mm = ()3.70 D at 155 ) and ()4.97 D at 65 ). The data in points 1 3 were originally presented as a more detailed alulated example on page in Holladay et al. (1998). (4) Total oular astigmatism as orneal plane oular refrative power in rossed ylinder power format = (3.70 D at 155 ) and (4.97 D at 65 ). (5) Total oular astigmatism as orneal plane oular refrative power, expressed as net astigmatism = (1.27 Dat65 ). (6) Total oular astigmatism as orneal plane oular refrative power, expressed as polar values: KP (0) = )0.82 D; KP (45) = 0.97 D. The spherial equivalent power (SEP) = 4.34 D. Data used in the following are onsistently transformed to the orneal 769

3 plane and expressed in oular refrative power format. Oular residual astigmatism The oular residual astigmatism (ORA) is the omponent of the TOA, whih is not attributed to the anterior orneal surfae (Dunne et al. 1994; Alpins 1997; Naeser 2008). The ORA is the ombined effet of the lentiular and the posterior orneal astigmatism. Following anterior orneal laser ablative surgery, both the lentiular and the posterior orneal urvatures are assumed to be unaffeted and the ORA is therefore unhanged. The ORA is defined as the vetor differene between the TOA and the ACA. In pratise, all net ylinders are onverted to polar values with eqns (1 and 2) and subtration is performed. In the following, vetors are symbolized in bold (Fig. 1). ORA TOA ACA KPð0Þ TOA KPð0Þ ACA KPð45Þ TOA KPð45Þ ACA ð7þ T A P R KP(45) 1.0 O S 1.0 KP(0) Fig. 1. Symbols: The origin: O; Vetor OT for the preoperative total oular astigmatism, TOA. Vetor OA = RT for the preoperative anterior orneal astigmatism, ACA. Vetor OR for the oular residual astigmatism, ORA. Vetor OS = TP for the surgially indued hange in the anterior orneal astigmatism, ACA SIA. Vetor OP for the optimal postoperative TOA. The optimal surgially indued orneal astigmatism ACA SIA, indiated as vetor OS = vetor TP, has the opposite diretion and 74% of the length of the preoperative anterior orneal astigmatism, vetor OA. The vetor sum of OT and OS yields the optimal postoperative TOA, vetor OP. Geometrially, the point P is the projetion of the origin O orthogonally on vetor RT. Optimizing by ombining topographi and refrative data In mathematis, optimization refers to methods for identifying minimal or maximal values of funtions. The extrema are found with the use of differential alulus. In the present ontext, the postoperative TOA is defined as the vetor sum of the preoperative TOA and the ACA SIA : Postoperative TOA ð8þ Preoperative TOA þ ACA SIA The minimal value for the postoperative TOA is found by differentiation of this funtion and subsequent equating to zero. For onveniene, the omponents of the preoperative astigmati vetors are abbreviated as: KPð0Þ TOA a KPð45Þ TOA b and ð9þ KPð0Þ ACA KPð45Þ ACA d The aim of the optimization proedure is to seure the largest possible redution in refrative astigmatism ombined with the least possible hange in orneal astigmatism in eah ase. Following surgery, the planned magnitude of the ACA shall vary between zero and the preoperative value and its diretion shall not be altered. The surgial objetive is to reate a more spherial ornea in eah ase. These limitations may be formalized mathematially as: ACA SIA t d ; ð10þ where t is a salar with permissible values restrited to: )1.0 t 0. The possible values for t may be interpreted in the following manner: (1) t < )1.0 indiates an overorretion with a 90 degree shift of astigmati meridian. This value is not permissible. The value t = )1.0 should be used in the treatment plan. (2) t = )1.0. The ACA is eliminated and the ornea beomes spherial. (3) )1.0 < t < 0. The postoperative ACA is redued in magnitude, but its diretion is unaltered. This is the most frequent treatment value. (4) t = 0. A zero surgially indued astigmatism. The ACA is not treated, and both orneal and refrative astigmati diretions and magnitudes remain unhanged. (5) t > 0. The astigmati magnitude is inreased, but the diretion is unaltered. This value is not permissible. The value t = 0 should be employed in the treatment plan. The postoperative TOA as a vetor parameterized with the variable t is abbreviated as OP(t). By ombining eqns (8 and 10), this vetor is given by: OPðtÞ Preoperative TOA þ ACA SIA a þ t b d ð11þ The length OP(t) of this vetor is determined by the Pythagorean theorem as: jopðtþj 2 t 2 2 þ d 2 þ 2tða þ bdþþ a 2 þ b 2 ð12þ This funtion is a positively oriented parable with one minimal value. The t-value, t min, for whih OP(t) 2 is minimal is found by differentiation and subsequent equation to zero: dðjopðtþj 2 Þ 2tð 2 þ d 2 Þþ2ða þ bdþ dt ð13þ Determining t min by equating (13) to zero yields: ða þ bdþ t min ð14þ 2 þ d 2 Determination of t min identifies the minimal value for OP(t) 2 and hereby for OP(t). The signifiane of the t min may be easier to interpret geometrially. The minimal length of the OP(t) vetor is always the projetion P of the origin O orthogonally on vetor RT, so that vetor OP is normal to vetor RT (Fig. 1). Clinial Example Example 3 demonstrates the use of the optimization method on a single ase. In Table 1, the result is ompared to the refrative driven treatment, the topography driven treatment and to Alpins method. 770

4 Table 1. (Næser) Astigmati data from example 3 Method Preoperative astigmatisms ACA TOA ORA (1.70 D at 120 ) (1.40 D 107 ) (0.76 D at 57 ) Postoperative astigmatisms Example 3. This example was originally desribed in fig. 8A 11B in Alpins study from The preoperative total oular astigmatism, TOA, is expressed as the net ylinder (1.40 D at 107 ). In Fig. 1, this net ylinder is depited as vetor OT = the polar values (KP(0), KP(45)) = (a,b) = ()1.16 D, )0.78 D). The preoperative anterior orneal astigmatism, ACA, is given by the net ylinder (1.70 D at 120 ) = vetor OA =(,d) =()0.85 D, )1.47 D) in Fig. 1. The oular residual astigmatism, ORA, is alulated in aordane with eqn (7) as ()1.16 D, )0.78 D) ()0.85 D, )1.47 D) = ()0.31 D, 0.68 D) = vetor OR in Fig. 1. These polar values may be reonverted to the net ylinder (0.76 Dat57 ) with the use of eqns (3 4). t min is alulated from eqn (14) as: )[()1.16)*()0.85)+()0.78)*()1.47)] [()0.85) 2 +()1.47) 2 ]=)0.74. By hoosing this value for t min, the ACA SIA = vetor OS in Fig. 1 = )0.74Æ()0.85 D, )1.47 D) = (0.63 D, 1.09 D). These polar values may be reonverted to the net ylinder (1.26 Dat30 ). The postoperative ACA is given by ()0.85 D, )1.47 D) + (0.63 D, 1.09 D) = ()0.22 D, )0.38 D) = vetor RP in Fig. 1. These polar values may be reonverted to the net ylinder (0.44 D at 120 ) with eqns (3 4). The optimal postoperative TOA is alulated aording to eqn (8) as ()1.16 D, )0.78 D) + (0.63 D, 1.09 D) =()0.53 D, 0.31 D) = vetor OP in Fig. 1. These polar values may be Targeted ACA Targeted TOA ACA SIA Refrative (0.76 D at 147 ) (0 D at 0 ) (1.40 D at 17 ) Topographi (0 D at 0 ) (0.76 D at 57 ) (1.70 D at 30 ) Alpins (0.28 D at 147 ) (0.48 D at 57 ) (1.56 D at 26 ) Optimization (0.44 D at 120 ) (0.61 D at 75 ) (1.26 D at 30 ) The target postoperative topographi anterior orneal astigmatism (ACA), refrative total oular astigmatism (TOA) and orneal surgially indued astigmatisms (ACA SIA ) in ahieving these targets, as alulated with four different methods. The postoperative astigmatisms by Alpins method are alulated as indiated in fig. 10B on page 71 in Alpins paper from ORA indiates the oular residual astigmatism. reonverted to the net ylinder (0.61 Dat75 ). In this ase, a 74% redution in the preoperative ACA produed the optimal TOA. Table 1 illustrates that either refrative or topographi astigmatism will remain in the oular system following any treatment modality. Alpins method and the optimization method distribute the postoperative astigmatism on the refration and the anterior orneal surfae. The surgially indued anterior orneal astigmatism is lowest for the optimization method. Disussion Eximer refrative laser surgery is a young proedure, and eah year has brought improvements in measuring tehniques, energy delivery, ablation profiles and flap onstrution. The present study deals with a speifi surgial segment, namely Eximer laser PARK. The aim of the study is to provide a theoretial model for ombining topographi and refrative data in the surgial planning, hereby rendering PARK safer and less invasive. Both refration and topography driven treatment algorithms have their merits, but none of them an ompletely eradiate the astigmatism from the oular system: Photoastigmati refrative keratetomy surgery based on preoperative refrative data only enables treatment of large refrative astigmatisms. The aim of the treatment is a spherial refration, and the target ACA is therefore )ORA (Table 1). The aim of a topography-guided treatment is a spherial ornea, whih will leave a postoperative TOA idential to the ORA, as illustrated in Table 1. Alpins (1997) developed a system for ombining topographi and refrative data in PARK surgery and devised a method for apportioning the ORA between the topographi and the refrative astigmatisms. The emphasis between the two astigmatisms is deided by the individual surgeon. In two later linial series, the treatment was set to leave 60% of the ORA orreted on the ornea and 40% in the refration (Alpins & Stamatelatos 2007, 2008). The present study has a different surgial planning strategy. The ACA is redued or eliminated in eah eye. The orneal astigmati magnitude is never inreased, and its diretion is never hanged. The subsequent optimization proess will identify the smallest possible refrative astigmatism within these restraints. The method is ustomized, and there is only one mathematially optimized solution to eah eye. The advantage of the present optimization method is a onsistent redution in orneal astigmatism towards spheriity. No new orneal astigmatism is arved on the ornea and probably less tissue is ablated. The optimization method may also be used to ombine refrative and topographi data for HOAs with sinusoidal omponents. However, ompared to a purely refrative driven treatment, more refrative astigmatism will remain in the eye in most ases. A ontrolled linial trial is neessary for omparing these two treatment modalities. Aknowledgement Supported by a grant from Region Midtjyllands Forskningsfond. Referenes Alpins NA (1997): New method of targeting vetors to treat astigmatism. J Catarat Refrat Surg 23: Alpins N & Stamatelatos G (2007): Customized photoastigmati refrative keratetomy using ombined topographi and refrative data for myopia and astigmatism in eyes with forme fruste and mild keratoonus. J Catarat Refrat Surg 33:

5 Alpins N & Stamatelatos G (2008): Clinial outomes of laser in situ keratomileusis using ombined topography and refrative wavefront treatments for myopi astigmatism. J Catarat Refrat Surg 34: Dunne MC, Elawad ME & Barnes DA (1994): A study of the axis of orientation of residual astigmatism. Ata Ophthalmol (Copenh) 72: Gudmundsdottir E, Jonasson F, Jonsson V, Stefansson E, Sasaki H & Sasaki K (2000): With the rule astigmatism is not the rule in the elderly. Reykjavik Eye Study: a population based study of refration and visual auity in itizens of Reykjavik 50 years and older. Ieland-Japan Co-Working Study Groups. Ata Ophthalmol Sand 78: Harris WF (1991): Representation of dioptri power in Eulidian 3-spae. Ophthalmi Physiol Opt 11: Holladay JT, Dudeja DR & Koh DD (1998): Evaluating and reporting astigmatism for individual and aggregate data. J Catarat Refrat Surg 24: Kanellopoulus AJ (2009): Comparison of sequential vs. same day simultaneous ollagen ross-linking and topography-guided PRK for treatment of keratoonus. J Refrat Surg 25: S812 S818. Kaye SB & Harris WF (2002): Analyzing refrative data. J Catarat Refrat Surg 28: Naeser K (2008): Assessment and statistis of surgially indued astigmatism. Ata Ophthalmol 86(Suppl. 1): Naeser K & Hjortdal J (2001a): Polar value analysis of refrative data. J Catarat Refrat Surg 27: Naeser K & Hjortdal J (2001b): Multivariate analysis of refrative data. Mathematis and statistis of spheroylinders. J Catarat Refrat Surg 27: Stojanovi A, Zhang J, Chen X, Nitter TA, Chen S & Wang Q (2010): Topographyguided transepithelial surfae ablation followed by orneal ollagen ross-linking performed in a single ombined proedure for the treatment of keratoonus and pelluid marginal degeneration. J Refrat Surg 26: Thibos LN, Wheeler W & Horner D (1997): Power vetors: an appliation of Fourier analysis to the desription and statistial analysis of refrative error. Optom Vis Si 74: Reeived on August 4th, aepted on May 11th Correspondene: Kristian Næser, MD, Dr.Si. Department of Ophthalmology Randers Regional Hospital Skovlyvej Randers Ø Denmark Tel: (+45) KRISNAES@rm.dk; kristian.naeser@dadlnet.dk 772