Translation model for CW chord to angle Alpha derived from a Monte-Carlo simulation based on raytracing.

BACKGROUND: The Chang-Waring chord is provided by many ophthalmic instruments, but proper interpretation of this chord for use in centring refractive procedures at the cornea is not fully understood. The purpose of this study is to develop a strategy for translating the Chang-Waring chord (position of pupil centre relative to the Purkinje reflex PI) into angle Alpha using raytracing techniques.
METHODS: The retrospective analysis was based on a large dataset of 8959 measurements of 8959 eyes from 1 clinical centre, using the Casia2 anterior segment tomographer. An optical model based on: corneal front and back surface radius Ra and Rp, asphericities Qa and Qp, corneal thickness CCT, anterior chamber depth ACD, and pupil centre position (X-Y position: PupX and PupY), was defined for each measurement. Using raytracing rays with an incident angle IX and IY the CW chord (CWX and CWY) was calculated. Using these data, a multivariable linear model was built up in terms of a Monte-Carlo simulation for a simple translation of incident ray angle to CW chord.
RESULTS: Raytracing allows for calculation of the CW chord CWX/CWY from biometric measures and the incident ray angle IX/IY. In our dataset mean values of CWX = 0.32±0.30 mm and CWY = -0.10±0.26 mm were derived for a mean incident ray angle (angle Alpha) of IX = -5.02±1.77° and IY = 0.01±1.47°. The raytracing results could be modelled with a linear multivariable model, and the effect sizes for the prediction model for CWX are identified as Ra, Qa, Rp, CCT, ACD, PupX, PupY, IX, and for CWY they are Ra, Rp, PupY, and IY.
CONCLUSION: Today the CW chord can be directly measured with any biometer, topographer or tomographer. If biometric measures of Ra, Qa, Rp, CCT, ACD, PupX, PupY are available in addition to the CW chord components CWX and CWY, a prediction of angle Alpha is possible using a simple matrix operation.

Background

The angles Alpha and Kappa have previously been widely discussed for their potential impact on visual performance after cataract surgery and corneal refractive surgery [1]. Especially with premium lenses such as toric, enhanced depth of focus (EDOF) or multifocal lenses [2] or with enhanced excimer or femtosecond laser refractive surgery procedures [3-15] large angles Alpha or Kappa were identified as potential risk factors for an unsatisfactory outcome. In corneal refractive surgery, centration of the corneal ablation pattern can be directly performed based on the corneal centre (limbus centre [16]), pupil centre [17], or the Purkinje reflex PI originated from a coaxial light source in the surgical microscope [5, 12], and the benefits and drawbacks of the different centring strategies has been discussed controversially in many scientific papers. In contrast, in cataract surgery intraoperative centration of the lens implant could be performed with some restrictions [18], but the final lens position is mostly determined by wound healing effects and capsular shrinkage in the postoperative interval [8].

For indication or patient counselling prior to cataract surgery a determination of angles Alpha or Kappa is strictly recommended in the literature, especially when implanting lenses that are known to be sensitive to decentration and tilt. However, it is known that angles Alpha and Kappa cannot be directly determined in a clinical setting as the visual axis and the optical axis or pupillary axis are not properly defined or cannot be measured [12, 19]. In classical schematic model eyes such as the Gullstrand-Emsley or Kooijman eye, all refractive surfaces, either spherical or aspheric, are coaxially aligned and therefore the optical axis passing through all vertices of the surfaces is well defined. However, in those classical model eyes, the visual axis or line of sight coincides with the optical axis as the fovea is assumed to be located at the posterior pole of the eye. In contrast, in modern schematic model eyes such as the Liou-Brennan eye [20], while all refractive surfaces are still coaxially aligned, the fovea is shifted slightly temporally and therefore the incident ray angle is tilted with respect to the optical axis, and in addition the pupil centre is decentred half a millimetre in the nasal direction. While the optical axis can still be assumed as the connecting line between the vertices of all 4 refracting surfaces, the visual axis defined as a connecting line originating from an object point to the object-side nodal point of the eye, and again a connecting line from the image-side nodal point to the fovea, the line of sight defined by the axis connecting the object point and the centre of the entrance pupil (not necessarily hitting the fovea), or the fixation axis connecting the object with the centre of rotation of the eye cannot be determined with optical instruments such as tomographers or biometers. Only the pupillary axis can be easily determined from anterior segment tomography, but this axis intersects neither the object nor the fovea.

The angle Alpha is formed between the optical axis and the visual axis of the eye, and the angle Kappa defines the angle between the pupillary axis and the visual axis measured at the centre of the entrance pupil. As these axes cannot be determined with measurement devices, angles Alpha and Kappa cannot be measured by clinicians.

In 2014 Chang and Waring published a paper discussing this specific issue in detail [21]. In response to the many inconsistencies in the definitions and interpretation of these axes, the authors recommended the use of measures which can be derived directly from any biometer, topographer or tomographer. These instruments measure distances or surface geometries under patient fixation with a fixation target projected to infinity to avoid instrument myopia. Chang and Waring recommended extracting the outline of the entrance pupil, and the Purkinje reflex PI originated from the corneal front surface. From the relative position of the Purkinje reflex PI and the centre of the pupil they defined the Chang-Waring (CW) chord, replacing the confusing terminology of angles Alpha and Kappa by direct measures from the topographer, tomographer or biometer. However, in the literature there is little information on normative data for CW chord [22, 23], and even more importantly, there is no strategy for interpretation of CW chord or for translating CW chord to angles Alpha or Kappa.

The purpose of this study was

to find a way to translate CW chord to angle Alpha based on biometric data of the anterior segment of the eye,

to extract the lateral position of the pupil centre and the Purkinje reflex PI using raytracing to determine CW chord,

to apply this strategy to a large clinical dataset from a modern optical anterior segment tomographer in terms of a Monte-Carlo simulation, and

to generate a multivariable linear model for a simple application in a clinical setting which translates CW chord to angle Alpha and vice versa.

Methods

Dataset for the Monte-Carlo simulation

In total, a dataset with 11,277 measurements (measurements performed between October 2017 and April 2021) from one clinical centre (Augenklinik Castrop, Germany) taken using the Casia2 anterior segment tomographer (Tomey, Nagoya, Japan) was considered for this retrospective study. Duplicate measurements of eyes were already discarded at the time point of data export. Measurements from pseudophakic eyes or in mydriasis and data indexed with situation after refractive surgery, ectatic corneal diseases such as keratoconus or keratoglobus, or other corneal pathologies were omitted from the dataset. The data were transferred to a.csv data table using the data export module of the Casia2 software. Data tables were reduced to the relevant parameters required for our raytracing and analysis, consisting of: laterality (left or right eye), central corneal curvature of the corneal front (Ra) and back (Rp) surface in mm, asphericity of the corneal front (Qa) and back (Qp) surface, central corneal thickness (CCT in mm), external anterior chamber depth (ACD) measured from the corneal front apex to the lens front apex in mm, Pupil size (Pup) in mm, as well as the location of the pupil centre (PupX in horizontal and PupY in vertical direction), both in mm.

The data were transferred to Matlab (Matlab 2019b, MathWorks, Natick, USA) for further processing. A waiver was provided for this study by the local ethics committee (Ärztekammer des Saarlandes, 157/21).

Preprocessing of the data and raytracing

Custom software was written in Matlab using class libraries. By convention we defined a Cartesian coordinate system with an origin at the corneal front apex, X axis to the right, Y axis in the superior direction, and Z axis towards the retina. Fig 1 shows a schematic drawing of the situation of a left eye from above. The optical model is represented by a structure consisting of a cornea with 2 aspheric surfaces defined by Ra, Qa, Rp, Qp and CCT and an aperture stop with a diameter Pup (Fig 1). Accordingly, the coordinates of the elements in the optical model were X / Y / Z = 0 / 0 / 0 for the corneal front apex, 0 / 0 / CCT for the corneal back apex, and PupX / PupY / ACD for the pupil centre (decentration PupX in horizontal and PupY in vertical direction), respectively. To avoid unnecessary complexity the front and back surfaces of the cornea were defined to be simple rotationally symmetric aspherical surfaces coaxially aligned on an optical axis coincident with the Z axis. For air / cornea / aqueous humour we used refractive index values of 1.0 / 1.376 / 1.336 respectively, taken from the Liou-Brennan schematic model eye [20]. This schematic model eye is widely used for optical simulations and raytracing purposes, as it appears to be anatomically and physiologically correct and, importantly, it includes non-centred optical elements and a non-coaxial entrance beam thereby considering the decentred location of the fovea in the eye. Without loss of generality, to consider all samples as left eyes, the optical model was flipped horizontally for right eyes, which means that the sign for PupX was changed keeping PupY unchanged. Positive values for X and Y refer to the temporal and superior direction.

Fig 1

Schematic drawing of a left eye from above (the vertical coordinates are not shown in the plot).

The corneal front and back surface (with radius of curvature Ra and Rp and asphericity Qa and Qp respectively) are coaxially aligned with the Z axis of the coordinate system. The incident ray is tilted by angle IX (negative values for IX as the ray is tilted to nasally). The location of the pupil centre (PupX) is used to calculate the location of the pupil centre at the corneal apex plane (PupCX). PurkinjeCX refers to the projection of the Purkinje image PI at the corneal apex plane. Coordinates PurkinjeCX and PupCX are rotated to a plane perpendicular to the entrance beam ro read out the coordinates PurkinjeRX and PupRX as determined by the tomographer. The Chang Waring chord CWX is derived from the coordinate points of PurkinjeRX and PupRX. CCT / ACD / Pup refer to the central corneal thickness / the anterior chamber depth as the distance from the anterior corneal apex to the front apex of the crystalline lens / the pupil size as measured with the Casia 2 tomographer.

As the incident ray angle with components IX / IY cannot be measured, and therefore we do not have normative values, we used for the mean value the data from the Liou-Brennan schematic model eye with mean(IX) = -5° (from the nasal direction, all eyes are considered as left eyes) and mean(IY) = 0°. Additionally, we added without loss of generality a normal distribution with a standard deviation of 2° and finally discarded values outside the range [-9°; -1°] for IX and [-4°; 4°] for IY. With this strategy we considered the eccentric location of the fovea shifted in the temporal direction (positive values of X).

Calculating the CW chord coordinates

Pupil centre

For initialisation a parametric ray with an incident ray angle IX / IY was projected to the corneal front apex and traced through both corneal surfaces to the pupillary plane. The distance of the ray-pupil intersection was extracted, and using an iterative nonlinear optimisation strategy (Interior Point Method (IPM) [24]) the ray was shifted in X and Y to pass through the pupil centre. At the corneal apex plane the horizontal and vertical coordinates of this ray were PupCX and PupCY.

Purkinje reflex PI

For initialisation a parametric ray with an incident ray angle IX / IY was projected to the corneal front apex and refracted by the corneal front surface. From the normalised direction of the ray in front of and behind the corneal front surface, the dot product was analysed. This refers to the cosine of the angle (always in a range from -1 to 1) between the non-refracted and the refracted ray. With an iterative nonlinear optimisation strategy (Interior Point Method [24]), the ray was shifted in X and Y to maximise the dot product (minimise 1-dot product) to identify the ray which is collinear with the surface normal and which is therefore not refracted at the corneal front surface. At the corneal apex plane the horizontal and vertical coordinates of this ray were considered as PurkinjeCX and PurkinjeCY.

In a next step the positions of both points (PupCX / PupCY and PurkinjeCX / PurkinjeCY) were rotated by Euler angles–IX / -IY with respect to the corneal apex position (Fig 1), to project the pupil centre and Purkinje reflex PI from the Cartesian X / Y / Z to coordinates on a plane perpendicular to the ray (PupRX / PupRY and PurkinjeRX / PurkinjeRY). This rotation was performed using quaternion transformation [25].

In a next step we calculated the horizontal and vertical coordinates of CW chord from the offset of the Purkinje PI and the pupil centre by CWX = PupRX—PurkinjeRX and CWY = PupRY—PurkinjeRY (Fig 1).

In a last step we traced a bundle of collimated rays (10,000 equidistant rays) with a diameter of 7 mm through the cornea and the aqueous humour to the pupillary plane. The rays passing through the aperture stop with diameter Pup were marked, and the intersection of these rays with the corneal front apex plane was calculated. A constraining ellipse was derived using eigenvalue decomposition, and the relevant characteristics of this ellipse (centre EllipseCX and EllipseCY) together with the major and minor diameter (with angular orientation) were documented. Again, the coordinates of the constraint ellipse were rotated by–IX / -IY using quaternion operation to obtain projections to a plane perpendicular to the incident ray (centre EllipseRX and EllipseRY; major diameter Dlong@Along; minor diameter Dshort@Ashort).

Setup of the multilinear regression model

A stepwise linear regression [26] was implemented to analyse the relevant effect sizes for a multilinear prediction model for the target parameters CWX and CWY from the potential input parameters Ra, Qa, Rp, Qp, CCT, ACD, PupX, PupY, Pup, IX, and IY. The stepwise strategy begins with an initial constant model and takes forward and backward steps to add or remove variables, until a stopping criterion is satisfied. As stopping criteria we restricted the number of iterations to a maximum of 100, iteration steps smaller than 10e-9, or improvement of the root mean squared prediction error by less than 10e-12. The tolerance for adding terms was a significance value less than 0.05, and the tolerance for removing terms was a significance value larger equal 0.05.

With the effect sizes identified with this stepwise fit a multivariable linear model was set up to predict the coordinates of the incident ray angle (angle Alpha) to coordinates of CW chord. In addition, for prediction of angle Alpha from the CW chord coordinates, we reversed this linear model to obtain IX, and IY from the input variables Ra, Qa, Rp, Qp, CCT, ACD, PupX, PupY, Pup, CWX, and CWY.

Results

From the 11,277 measurements exported from the Casia 2 device, totals of 1188 / 789 / 1272 / 365 measurements were indexed as pseudophakic measurements / measurements in mydriasis / eyes with ectatic corneal diseases / incomplete measurements respectively. After quality approval of the dataset and filtering out measurements in pseudophakic eyes, eyes in mydriasis, eyes with ectatic corneal diseases and incomplete data, a final total of N = 8959 measurements (4223 right and 4736 left eyes from 5213 patients) were used for our Monte-Carlo simulation. The process time for extracting the location of the pupil centre, the Purkinje reflex PI, derivation of CW chord, and tracing 10,000 rays through the 8959 optical models took 17,315 seconds (4 hours 49 min) on a standard office PC. Table 1 shows the explorative data for Ra, Qa, Rp, Qp, CCT, ACD, Pup, and the pupil centre location PupX and PupY derived from the Casia2 anterior segment OCT, together with the random values defined for the incident ray angle IX and IY.

Table 1

Explorative data extracted from the dataset of the Casia2 anterior segment tomograph.

Ra, Qa, Rp, Qp, CCT, ACD, Pup, PupX, PupY, IX, IY refer to the corneal front surface curvature and asphericity, corneal back surface curvature and asphericity, central corneal thickness, anterior chamber depth measured from the corneal front apex, lateral position of the pupil centre in X (positive values in the nasal direction) and Y (positive valuesin the superior direction), and simulated incident ray angle in X and Y. Right eyes were flipped in X. MEAN, SD, MEDIAN, 5% CL, and 95% CL refer to mean value, standard deviation, median, and 90% confidence interval, respectively.

N = 8959	Ra in mm	Qa	Rp in mm	Qp	CCT in mm	ACD in mm	Pup in mm	PupX in mm	PupY in mm	IX in °	IY in °
MEAN	7.76	-0.22	6.56	-0.11	0.55	3.36	3.24	-0.30	-0.10	-5.03	0.01
SD	0.28	0.13	0.25	0.11	0.04	0.40	0.81	0.21	0.19	1.77	1.47
MEDIAN	7.74	-0.22	6.56	-0.10	0.55	3.37	3.31	-0.30	-0.10	-5.03	0.02
5% CL	7.33	-0.44	6.17	-0.33	0.49	2.67	2.45	-0.70	-0.40	-7.99	-2.45
95% CL	8.27	-0.01	6.99	0.06	0.60	3.99	4.66	0.00	0.20	-2.08	2.41

Table 2 shows the descriptive data for the coordinates of the pupil centre PupCX / PupCY and the Purkinje reflex PI (PurkinjeCX / Purkinje CY) after raytracing in a plane perpendicular to the incident ray with the origin at the corneal front apex. The total displacement of the pupil centre (sqrt(PupCX2+PupCY2)) was 0.43±0.19 mm (MEDIAN: 0.41 mm), and for the Purkinje reflex PI it was 0.71±0.23 mm (MEDIAN: 0.71 mm), respectively. These positions are observed from an ophthalmic instrument with coaxial illumination and patient fixation to a far target. The coordinates of CW chord are derived from the difference of PupCX / PupCY and PurkinjeCX, / Purkinje CY. The data of the constraint ellipse fitted to the rays passing through the pupil in a forward raytracing model characterised by the ellipse centre (EllipseRX, EllipseRY) and the long (Dlong) and short axis (Dshort) together with the aspect ratio Dlong / Dshort are also provided in Table 2. Comparing Dlong and Dshort to the pupil diameter measured by the Casia 2 (see also Pup at Table 1) we notice a mean pupil magnification for the entrance pupil of around 15%.

Table 2

Explorative data after raytracing and processing: PupCX, PupCY, PurkinjeCX, Purkinje CY, CWX, CWY refer to the horizontal and vertical coordinates of the pupil centre, Purkinje reflex PI, and Chang-Waring chord as would be noticed by an anterior segment analyser under patient fixation with a far target (projected from Cartesian coordinates X / Y / Z to a plane perpendicular to the incident ray).

EllipseRX, EllipseRY, Dlong, and Dshort refer to the horizontal and vertical coordinates of the centre and the long and short diameter of the constraint ellipse of the pupil outline derived from raytracing with a bundle of 10,000 rays. Right eyes were flipped in X. MEAN, SD, MEDIAN, 5% CL, and 95% CL refer to mean value, standard deviation, median, and 90% confidence interval, respectively.

N = 8959	PupCX in mm	PupCY in mm	PurkinjeCX in mm	PurkinjeCY in mm	CWX in mm	CWY in mm	EllipseRX in mm	EllipseRY in mm	Dlong in mm	Dshort in mm	Dlong / Dshort
MEAN	-0.35	-0.10	-0.77	0.00	0.33	-0.10	-0.36	-0.11	3.75	3.69	1.04
SD	0.22	0.19	0.24	0.20	0.30	0.26	0.22	0.20	0.84	0.84	0.05
MEDIAN	-0.34	-0.10	-0.68	0.00	0.33	-0.10	-0.35	-0.11	3.68	3.61	1.01
5% CL	-0.73	-0.41	-1.08	-0.33	-0.16	-0.54	-0.74	-0.40	2.56	2.49	1.00
95% CL	-0.02	0.20	-0.28	0.33	0.83	0.32	-0.04	0.19	4.88	4.85	1.14

The stepwise fit algorithm which qualifies the input parameters for our linear multivariable model shows that for modelling of the X component of CW chord (CWX), the relevant input values are: Ra (significance level: P<1e-9), Qa (P = 5.53e-56), Rp (P = 0.0016), CCT (P = 2.1e-7), ACD (P = 0.0394), PupX (P<1e-9), PupY (P = 0.0019) and IX (P<1e-9). Qp, Pup and IY did not qualify as input parameters for the model. In contrast, for the Y component of CW chord (CWY) the relevant input values are: Ra (P = 0.0900), Rp (P = 0.0213), PupY (P<1e-9) and IY (P<1e-9). Qa, Qp, CCT, ACD, PupX, Pup and IX did not qualify as input parameters for the model. Fig 2 shows the matrix of grouped scatterplots for the relevant input parameters identified with the stepwise fit algorithm together with the components of the CW chord (CWX in red, CWY in green). The respective histograms are plotted on the diagonal of the matrix. The relevant effect sizes for CWX are the anterior and posterior corneal curvature Ra and Rp, asphericity of corneal front surface Qa, central corneal thickness CCT, anterior chamber depth ACD, position of the pupil centre PupX and PupY, and the incident ray angle IX. The relevant effect sizes for CWY are identified for anterior and posterior corneal curvature Ra and Rp, position of the pupil centre PupY, and incident ray angle IY. The graph shows that there is a good correlation between Ra and Rp, whereas Qa and Qp are not correlated and show no dependency on Ra or Rp.

Fig 2

Matrix of grouped scatterplots for the relevant input parameters of the multivariable linear prediction model and the components of the CW chord (CWX in red, CWY in green).

The respective histograms are plotted on the diagonal of the matrix. The relevant effect sizes for CWX are identified with a stepwise fit algorithm to anterior and posterior corneal curvature Ra and Rp, asphericity of corneal front surface Qa, central corneal thickness CCT, anterior chamber depth ACD, position of the pupil centre PupX and PupY, and incident ray angle IX. The relevant effect sizes for CWY are identified to anterior and posterior corneal curvature Ra and Rp, position of the pupil centre PupY, and incident ray angle IY. The model is based on a dataset with N = 8959 eye measurements.

The regression model for predicting CW chord (CWXM and CWYM) with the regression coefficients composed to a matrix notation is given by the following estimation equation: where the non-relevant coefficients are set to zero. The difference in both estimation models (1st and 2nd row in the matrix) results from the differences between horizontal and vertical pupil offset (PupX and PupY) and differences between horizontal and vertical incident ray angles (IX and IY). The reversed model for predicting the coordinates of the incident ray angle (IXM and IYM) can be derived after some mathematical transformation as:

The performance of both models is shown in Fig 3. On the left side the model predictions CWXM and CWYM are plotted versus the output of the raytracing calculation, and on the right side the respective model predictions IXM and IYM are plotted versus the output of the raytracing calculation.

Fig 3

Performance of the linear models: In the left graph the predicted Chang-Waring chord from the linear multivariable model (X and Y components CWXM and CWYM) is plotted versus the respective components derived from raytracing calculations (X and Y components CWX and CWY).

In the right graph the reverse model is shown, where the predicted incident ray angle from the linear multivariable model (X and Y components IXM and IYM) is plotted versus the respective components derived from raytracing calculations (X and Y components IX and IY). The model is based on a dataset with N = 8959 eye measurements.

Discussion

As previously noted, while there has been much discussion regarding centration of corneal refractive laser procedures and the impact of angle Alpha or Kappa on the outcome of corneal refractive surgery, the impact of intraoperative centration or alignment of the intraocular lens in cataract surgery is less clear, as the final position of the lens is mostly determined by wound healing effects and capsular shrinkage.

As the angles Alpha and Kappa are not directly measurable in a clinical setting, we wished to explore the use of the CW chord (defined by Chang and Waring as the relative position of the Purkinje reflex PI and the centre of the pupil), as a parameter that can be measured using current instruments such as topographers, tomographers or biometers [27-30]. However, there is to date no calculation scheme which translates the CW chord to angle Alpha or vice versa.

Therefore, in the present paper we set up a raytracing strategy to derive the CW chord from biometric data and the incident ray angle. As discussed in the Methods section, this model was based on selected parameters from a large dataset of measurements from an anterior segment tomographer Casia2, together with additional parameters from the Liou-Brennan model eye. The raytracing model could easily be upgraded to astigmatic or free form surfaces or to non-coaxial arrangement of corneal front and back surface, but the number of effect sizes may increase with the consequence that interpretation of the results might become difficult and the required number of measurements for proper modelling may increase dramatically.

Expressed in the terminology of raytracing, the centre of the entrance pupil is given by the image of the pupil formed by the cornea. This means that we have to find the ray by variation of the lateral position within the ray bundle passing exactly through the centre of the pupil. The Purkinje reflex position PI refers to a location at the corneal front surface where coaxial rays from the fixation target at infinity meet the corneal front surface perpendicularly. In other words, the incident ray is back-reflected; it is collinear with the surface normal (or collinear with the refracted ray). To fulfil this condition we evaluated the dot product between the normalised incident ray and the normalised refracted ray, which equals 1 if it intersects the corneal front surface perpendicularly. Both rays (passing through the pupil centre and hitting the corneal front surface perpendicularly) could be selected from a large number of rays (e.g. 10,000 rays) projected to the cornea considering an incident ray angle by identifying that ray which passes closest the pupil centre or which is refracted the least (shows the largest dot product). In the present study, we wanted to find the ray which passes exactly through the pupil centre and hits the corneal surface exactly perpendicularly. Therefore we implemented a nonlinear optimisation strategy which iteratively searches for the ray passing through the pupil centre and passing the corneal front surface perpendicularly by varying the lateral position of the incident ray until our stopping criterion for the iteration was fulfilled. For that purpose we used the Interior Point Method (IPM) [24]. Finally, to obtain the lateral coordinates of the pupil centre and PI as seen by the ophthalmic instrument we projected our locations for pupil centre and Purkinje reflex PI given in Cartesian coordinates to a plane perpendicular to the incident ray (Fig 1). This could be easily performed using quaternion operations based on the classical Euler angles, as well established in computer graphics programming. Finally, we traced 10,000 coaxial rays through our optical model and fitted a constraint ellipse to determine the magnification of the entrance pupil by the cornea and to read out the distortion of the circular pupil to an ellipse resulting from the oblique incident ray angle. However we feel that with a mean aspect ratio of 1.03, distortion of the pupil (3% between long and short axis on average) can be ignored in a clinical setting if the pupil centre has to be identified.

As such raytracing is currently not available in the software tools of ophthalmic instruments, we attempted to model the raytracing results using a linear multivariable prediction algorithm for CW chord. First the relevant effect sizes were determined separately for the horizontal and vertical components CWX and CWY. In a second step we calculated the regression coefficients for the relevant effect sizes by minimising the root mean squared prediction error. We discovered that the models for CWX and CWY differ significantly, both in the number of relevant effect sizes as well as in the regression coefficients. This is mostly due to the fact that the pupil position and the incident ray angles are not identical in X and Y, and that our simple linear setup does not consider non-linear effects with different magnitudes of pupil decentrations and incident ray angles. However, as can be seen from Fig 3, in the parameter space used for our Monte-Carlo simulation the model shows a good performance for the horizontal and vertical component. To provide a calculation for the incident ray angle (angle Alpha) from the biometric measures and CW chord, we also included the reverse prediction model. Both prediction models could be easily implemented in a simple program code, an Excel sheet, or even using the software packages installed on ophthalmic instruments to translate CW chord to angle Alpha or vice versa.

There are some limitations in our study: we restricted the scope to a simple optical model for the cornea with rotationally symmetric aspherical surfaces aligned on a common axis. Further, we are aware that the location of the pupil centre is affected by the pupil size as shown by Erdem et al. [30], Wildenmann and Schaeffel [31], and Yang et al. [32]. For our raytracing setup we used the pupil centre and pupil location as provided from the Casia 2. We did not however perform a series of measurements for each eye with different pupil sizes to assess the respective pupil centre position and the CW chord as a function of pupil size. This was deemed unnecessary as the results from the stepwise fit algorithm confirmed that the pupil size did not act as a relevant parameter in the linear model for translating CW chord to incident ray angle and vice versa. In contrast, when modelling the CW chord from biometric data we feel that the pupil size could be a relevant parameter in the model. And last but not least a translation of CW chord to angle Kappa is in general possible, but requires a more sophisticated optical model as the lens data from the crystalline lens or the intraocular lens after cataract surgery are required to extract the nodal points of the eye. Nevertheless, with the linear regression model the CW chord could easily be translated to angle Alpha if, in addition to CWX and CWY, measurements of Ra, Qa, Rp, CCT, ACD, PupX, PupY are available. Furthermore, if such a conversion is to be applied in the future, the device is a further factor that has to be accounted for. While most biometers and topographers display the apparent CW chord, Scheimpflug tomographers and OCT devices usually display the actual chord μ. These two are not interchangeable, as the apparent CW chord is defined as the chord length between Purkinje-Sanson image I and the apparent pupil centre viewed coaxially from a light source through the cornea, while the actual chord μ displays the actual distance from the visual axis and the actual pupil centre [22].

In conclusion, many ophthalmic instruments measure corneal front and back surface curvature and asphericity, central corneal thickness, anterior chamber depth and the location of the pupil in horizontal and vertical direction together with the CW chord defined as the chord between Purkinje reflex PI and the location of the pupil centre. With these data raytracing could be performed to translate CW chord into angle Alpha, or a simplified multivariable regression model as shown in the present paper could be applied.

8 Feb 2022

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Reviewer #2: Yes

Reviewer #3: Yes

**********

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Reviewer #2: Yes

Reviewer #3: Yes

**********

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Reviewer #2: Yes

Reviewer #3: Yes

**********

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**********

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Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Line 1-2: please indicate that this is a retrospective study.

Line 59: please define EDOF

Line 104 - 108:

The purpose of this study is to find a way to translate CW chord to angle alpha based on biometric data of the anterior segment of the eye.

There are many issues with the measurement of angle alpha / kappa. Chang-Warring overcame these issues with the introduction of chord mu / CW-chord. The value of chord mu is readily available in a number of machines. Thus, making life much easier for everybody.

Why do the researchers want to complicate things again by translating CW chord back to angle alpha ???

Line 114 - 116: please indicate the range of dates that the measurements were taken.

Line 117: what were the inclusion criteria?

Line 117: why were measurements from pseudophakic eyes or in mydriasis omitted from the data set?

LIne 123 - 124: Anterior chamber depth (ACD) represents the distance between the corneal endothelium and the anterior capsule of the crystalline lens.

However, the researchers measured the ACD from the corneal front apex to the lens front apex; which is incorrect.

Line 129 - 136: Please provide a schematic drawing / diagram to aid the understanding of the readers.

Line 138- There are a few schematic / model eyes. However the researchers chose to use Liou-Brennan schematic eye. The researchers should provide justification for the choice of their schematic/model eye.

Line 138 - 140: In order to consider all samples as left eyes, the optical model was flipped horizontally for right eyes.

The pupil centre is significantly decentered relative to the corneal centre in the nasal and superior direction.

[Wildenmann U, Schaeffel F. Variations of pupil centration and their effects on video eye tracking. Ophthalmic Physiol Opt. 2013 Nov;33(6):634-41. doi: 10.1111/opo.12086. Epub 2013 Sep 17. Erratum in: Ophthalmic Physiol Opt. 2014 Jan;34(1):123. PMID: 24102513.]

Will flipping the right eye horizontally affect the results of this study?

Why don’t the researchers analyze the data for the right eyes and left eyes separately?

Line 145: Why standard deviation of 2 degree? not 1degree or 3 degree??

line 148 - 178: Please provide a schematic drawing / diagram to aid the understanding of the readers.

line 194: report the number of eyes at each stage of study, eg numbers potentially eligible, examined for eligibility, confirmed eligible, included in this study, and analysed. Give reasons for exclusion at each stage.

Line 270-271: To avoid unnecessary complexity the researchers simplified the front and back surface of the cornea to simple rotationally symmetric aspherical surfaces which are coaxially aligned. How will this simplification affect the outcome of this study?

Line 321: The location of the pupil centre is affected by the pupil size. Yet, the researchers chose to ignore this very very important fact.

How will this affect the validity of the results?

Reviewer #2: The authors describe the prediction of both the Chang Waring chord and angle Alpha from a using both a Monte-Carlo simulation and a multivariate regression model applied to a dataset of 8959 eyes. Data was extracted from from a Casia 2 anterior segment tomographer. They provide a comparison between the CW chord/angle alpha measurements predicted from both models.

Dataset size is sufficient for the described analysis. The models are well described and methodology outlined in adequate detail. The data generated (table 2) appears to have reasonable parameters and the CW chord measurement is consistent with published values from the original authors [1].

References

1) Chang DH, Waring GO 4th. The subject-fixated coaxially sighted corneal light reflex: a clinical marker for centration of refractive treatments and devices. Am J Ophthalmol. 2014 Nov;158(5):863-74. doi: 10.1016/j.ajo.2014.06.028. Epub 2014 Aug 12. PMID: 25127696.

Reviewer #3: Thank you very much for this demanding original work. The potential usefulness of the CW-chord in the planning of refractive procedures and cataract surgery has been presented very well. Your statistical applications are well thought out and implemented with foresight. For the interested reader, who previously had no contact with the topic, the work remains somewhat dry. I would encourage you to create an additional illustration of your design of experiments.

Your discussion is initially redundant and takes up the methodology too intensively again. Here you should avoid unnecessary repetitions and limit yourself to the actual discussion of your methodological/statistical approach and results.

**********

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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1 Mar 2022

Dear Professor Atzberger,

thank you for considering our manuscript entitled ‘Translation model for CW chord to angle Alpha derived from a Monte-Carlo simulation based on Raytracing’ for publication for PlosONE! We found the comments and recommendations of the co-editor and the reviewer very helpful and valuable and we have addressed the issues in the revised version of our manuscript. Our responses to the comments and recommendations of the co-editor and the reviewer are in blue.

Reviewer #1:

Line 1-2: please indicate that this is a retrospective study.

This study is an evaluation of CW chord using raytracing techniques and a modelling of the results with a linear prediction model, to be used in a clinical environment where raytracing is not available or not practical. Such Monte-Carlo studies are based either on a large dataset (as in this study) or on synthetic data. The term retrospective study might be misleading, as the data are primarily used as a basis for calculating the CW chord using raytracing techniques. We feel that Monte-Carlo simulation as used in the title is the better term. However, for clarity we have added the term ‘retrospective study’ both in the Methods section of the Abstract and in the Methods section of the text.

Line 59: please define EDOF

The reviewer is right! In the revised version we have added ’…enhanced depth of focus (EDOF)…

Line 104 - 108:

The purpose of this study is to find a way to translate CW chord to angle alpha based on biometric data of the anterior segment of the eye.

We thank the reviewer for this idea and we have reformulated the ‘purpose of the study’ accordingly.

There are many issues with the measurement of angle alpha / kappa. Chang-Warring overcame these issues with the introduction of chord mu / CW-chord. The value of chord mu is readily available in a number of machines. Thus, making life much easier for everybody.

The reviewer is right. However, the classical textbooks all describe the angles of the eye are described rather than the CW chord, and also the meaning of CW chord is somewhat different to the angle Alpha. CW chord refers to a projection of the Purkinje reflex PI and the pupil centre, and therefore we require biometric data such as corneal curvature and others to translate CW chord to angle alpha and vice versa.

Why do the researchers want to complicate things again by translating CW chord back to angle alpha ???

The angles of the eye (Alpha, Kappa etc.) are described in all the classical textbooks on ophthalmic optics. The purpose was to develop a concept for a translation of the ‘new’ CW chord and vice versa. What we discovered is that from biometric data of the eye CW chord could be easily predicted from the angle Alpha and vice versa using a simple linear model. As the model performance is surprisingly good such a model could be implemented in several software tools as a good alternative to a raytracing setup.

Line 114 - 116: please indicate the range of dates that the measurements were taken.

We have included the date range where the measurements were taken.

Line 117: what were the inclusion criteria?

Primary measurements on all phakic eyes were included

Line 117: why were measurements from pseudophakic eyes or in mydriasis omitted from the data set?

Pseudophakic eyes were excluded from the study as the CW chord might change in pseudophakic eye due to a potential backward shift of the iris, or a decentration or tilt of the intraocular lens implant. Measurements in mydriasis were excluded as it might be difficult to identify the pupil centre in a pharmacologically dilated pupil. In addition, as the data of the incident ray angle are derived from the Liou Brennan schematic model eye (which describes the situation of the eye without pharmacological stimulation of the pupil size) the mean IX/IY = -5°/0° used as average preset value might be incorrect. This could introduce a systematic shift of the reference point for the respective prediction model for translation of angle Alpha to CW chord and vice versa..

Line 123 - 124: Anterior chamber depth (ACD) represents the distance between the corneal endothelium and the anterior capsule of the crystalline lens. However, the researchers measured the ACD from the corneal front apex to the lens front apex; which is incorrect.

We have used the term "anterior chamber depth (ACD)" to refer to the measurement from the corneal front apex to the lens front apex, as this is the definition used in most biometers. This meaning of ACD is also used in all intraocular lens power calculation concepts. In tomographers, the manufacturers sometimes differentiate between ‘internal’ and ‘external’ anterior chamber depth as measured from the corneal endothelium or the corneal epithelium (Tomey Casia 2, Tomey TMS5); or they use the term ACD (for the measurement from the epithelium) and aqueous depth (AQD, measured from the endothelium, e.g. Haag Streit LenStar; Zeiss IOLMaster 700; Heidelberg engineering Anterion). To clarify our use of the term, we have added 'external' to the definition of ACD in the Methods section.

Line 129 - 136: Please provide a schematic drawing / diagram to aid the understanding of the readers.

We thank the reviewer for this advice! In the revised version of the manuscript we have included a schematic drawing (Figure 1) showing a view of a left eye from above. This drawing explains the situation with the incident ray angle IX, the coordinate system used for raytracing (X, Z), and the coordinates of the projections of the Purkinje PI and pupil centre before(PurkinjeCX and PupCX) and after rotation (PurkinjeRX and PupRX) to a plane perpendicular to the entrance beam using Euler angles.

Line 138- There are a few schematic / model eyes. However the researchers chose to use Liou-Brennan schematic eye. The researchers should provide justification for the choice of their schematic/model eye.

The Liou-Brennan schematic model eye is the most widely used model eye for optical simulations, e.g. in the context of developing new intraocular lens implant or assessing the optical performance. The benefit of this model eye is that that it considers an entrance beam which is not aligned to the ‘optical axis’. With a fully symmetrical model eye and a coaxial entrance beam the Purkinje reflex PI and the projection of the pupil will always be on axis, and the CW chord will always be zero. From the Liou-Brennan schematic model eye we did not use data on the corneal front or back surface curvature, corneal thickness, or the position of the pupil centre in the axial or lateral directions, restricting the use of the data to the refractive index of the cornea and aqueous humour, together with the ‘typical’ angle for the incident ray. We have added some text as justification why this subset of data from the Liou-Brennan model eye was used.

Line 138 - 140: In order to consider all samples as left eyes, the optical model was flipped horizontally for right eyes.

This is correct!

The pupil centre is significantly decentered relative to the corneal centre in the nasal and superior direction.

[Wildenmann U, Schaeffel F. Variations of pupil centration and their effects on video eye tracking. Ophthalmic Physiol Opt. 2013 Nov;33(6):634-41. doi: 10.1111/opo.12086. Epub 2013 Sep 17. Erratum in: Ophthalmic Physiol Opt. 2014 Jan;34(1):123. PMID: 24102513.]

We agree with the reviewer! We have added some text to the paragraph already included in the Discussion section of the original manuscript. In fact, using our stepwise fit algorithm the pupil diameter was found not to have any statistically significant impact in our prediction model for transalating the incident ray angle to CW chord or vice versa. We fully agree with the reviewer that the CW chord itself or the incident ray angle itself would be affected by the pupil size and the respective pupil centre dislocation. However, the pupil size is not relevant for translating the incident ray angle to CW chord or vice versa, as the pupil centre itself was considered in our model.

Will flipping the right eye horizontally affect the results of this study?

When we checked our data prior to analysis we observed that Ra, Qa, Rp, Qp, CCT, ACD and PupY showed no noticeable differences between left and right eyes. Additionally, we determined that the PupX values of left and right eyes were symmetrical about the y-axis, meaning that reversing the sign for all right eyes would produce a distribution identical to that for the left eyes. Together with the identical distributions that we used for IY and the symmetrical distributions that we used for IY for our left and right eyes (with symmetry about the y-axis), we feel that it is justified to flip either all left or all right eyes to get a common description of CW chord from the biometric parameters. Flipping all left eyes to right eyes will not affect our model, as we reversed the sign of all relevant parameters symmetrical with respect to left and right eyes (input parameters IX, PupX, and output parameter CWX).

Why don’t the researchers analyze the data for the right eyes and left eyes separately?

We did consider analyzing left and right eyes separately. However, as we noted that Ra, Qa, Rp, Qp, CCT ACD and PupY showed no noticeable differences and PupX showed symmetrical distributions between left and right eyes, we decided to condense our results into a single model (with description of the model performance and the respective plots) instead of 2 separate models.

Line 145: Why standard deviation of 2 degree? not 1degree or 3 degree??

The standard deviation of 2 degrees was used as an assumption. In general, in a Monte-Carlo simulation as outlined before, either all parameters are synthesized by assumptions on their distributions, or we have a large dataset where most of the parameters with their distributions and interactions can be used and only the missing parameters are synthesized. In this study, we used a large dataset from a clinically established tomographer and made use of all data which could be directly used for our raytracing setup. However, as the incident ray angle cannot be measured by any device, we assumed that on average the incident ray angle matches that of the Liou-Brennan schematic model eye.

As we expect some variation in the incident ray angle (IX / IY) in the population we added a normally distributed random value with some constrains defined for the lower and upper boundary. One of the most important issues in a Monte-Carlo simulation is that the parameter space is adequately addressed. In fact, the exact distribution and/or range of the parameters used for IX / IY as input parameters are not the most important issues for predicting the corresponding CW chord as response parameter. Ultimately the preset distribution of the incident ray angle (with mean and standard deviation) mostly determines the reference point of our linear model. As we do not see a large scatter in the model performance plot (Figure 3 in the revised version of the manuscript) for small/large incident ray angles (or for small or large CW chord values in the inverse model) we feel that the translation model works correctly for the entire range of incident ray angles (between -1° and -9° in the horizontal and -4° to 4° in the vertical direction). However, the reviewer is correct that we did not find any literature about the variance of incident ray angle, probably mostly due to the fact that this angle of the incident ray cannot be measured. With the boundary conditions for the horizontal incident ray angle [-1° to -9°] we wanted to ensure that the fovea is in all cases located temporally to the symmetry axis. Using smaller or larger standard deviations for the incident angle would most probably result in a slightly higher / lower performance of our linear prediction model of CW chord (slightly less or more scatter in Figure 3 (previously Figure 2)), but less / more predictive value for larger deviations of the incident ray angle from the mean value. However, in general, we would not expect any structural change in our prediction model.

line 148 - 178: Please provide a schematic drawing / diagram to aid the understanding of the readers.

We thank the reviewer for this advice! In the revised version of the manuscript we have included a schematic drawing (Figure 1) depicting a view of a left eye from above. This drawing explains the situation with the incident ray angle IX, and defines the coordinate system used for raytracing (X, Z), as well as the coordinates of the projections of the Purkinje PI and pupil centre both before (PurkinjeCX and PupCX) and after rotation (PurkinjeRX and PupRX) onto a plane perpendicular to the entrance beam using Euler angles.

line 194: report the number of eyes at each stage of study, eg numbers potentially eligible, examined for eligibility, confirmed eligible, included in this study, and analysed. Give reasons for exclusion at each stage.

In total, 11,277 primary eye measurements (one measurement per eye) were exported from the Casia 2 device. At the beginning of the Results section we have now specified as follows: ‘From the 11,277 measurements exported from the Casia 2 device 1188 / 789 / 1272 / 365 measurements were indexed as pseudophakic measurements / measurements in mydriasis / eyes with ectatic corneal diseases / incomplete measurements. After quality approval of the dataset and filtering out measurements in pseudophakic eyes, eyes in mydriasis, eyes with ectatic corneal diseases and incomplete incomplete data, a final total of N=8959 measurements…’

Line 270-271: To avoid unnecessary complexity the researchers simplified the front and back surface of the cornea to simple rotationally symmetric aspherical surfaces which are coaxially aligned. How will this simplification affect the outcome of this study?

In all tomographers currently on the market, the data for corneal front and back surface, corneal thickness, axial and lateral position of the pupil, and pupil size (or the outline data) are referenced to the measurement axis. There are no measurement data that enable us to read out the relative positioning and orientation of the refractive surfaces and the pupil with respect to any independent reference axis (which is not affected by the instrument axis during measurement). We therefore decided to use refractive surfaces aligned to a common axis (ignoring decentration and tilt). In contrast, we allowed for a decentration and size of the pupil (taken from the tomographer data) and used an incident ray angle with respect to the axis (defined by the refractive surfaces) to consider the peripheral location of the fovea. In general, our raytracing setup has the full flexibility to consider any decentration or tilt of refractive surfaces if respective data are available.

Line 321: The location of the pupil centre is affected by the pupil size. Yet, the researchers chose to ignore this very very important fact.

How will this affect the validity of the results?

This might be a misunderstanding! First, we excluded measurements on eyes in mydriasis as we are aware that pupil centre is typically dislocated with pharmacologically dilated pupil (as mentioned above). Second, we used the pupil centre (in axial and lateral position) as well as the pupil size for our raytracing simulation (please see line 124 and lines 132 and 133 of the original version of our manuscript). This means that finally we have for each processed measurement in our dataset also the respective pupil size Pup available. We fitted an ellipse to the projection of the pupil outline (entrance pupil) where the data are shown in Table 2. However, our final analysis showed that the pupil size Pup did not act as a relevant input parameter in our linear model for prediction of CWX or CWY (analysed with the stepwise fit algorithm, Results section lines 213-226). To clarify this we have added in the revised version a sentence for both models which lists the input parameters that did not contribute to the model.

Reviewer #2:

The authors describe the prediction of both the Chang Waring chord and angle Alpha from a using both a Monte-Carlo simulation and a multivariate regression model applied to a dataset of 8959 eyes. Data was extracted from from a Casia 2 anterior segment tomographer. They provide a comparison between the CW chord/angle alpha measurements predicted from both models.

Dataset size is sufficient for the described analysis.

We thank the reviewer for this comment!

The models are well described and methodology outlined in adequate detail. The data generated (table 2) appears to have reasonable parameters and the CW chord measurement is consistent with published values from the original authors [1].

[1] Chang DH, Waring GO 4th. The subject-fixated coaxially sighted corneal light reflex: a clinical marker for centration of refractive treatments and devices. Am J Ophthalmol. 2014 Nov;158(5):863-74. doi: 10.1016/j.ajo.2014.06.028. Epub 2014 Aug 12. PMID: 25127696.

We thank the reviewer for this favorable comment on our manuscript!

Reviewer #3:

Thank you very much for this demanding original work. The potential usefulness of the CW-chord in the planning of refractive procedures and cataract surgery has been presented very well.

We thank the reviewer for this positive comment!

Your statistical applications are well thought out and implemented with foresight. For the interested reader, who previously had no contact with the topic, the work remains somewhat dry. I would encourage you to create an additional illustration of your design of experiments.

For a better illustration we have added a drawing defining the coordinate system and showing the different coordinates used to identify the position of the Purkinje PI and the projection of the pupil centre.

Your discussion is initially redundant and takes up the methodology too intensively again. Here you should avoid unnecessary repetitions and limit yourself to the actual discussion of your methodological/statistical approach and results.

Thank you for this advice! We have shortened the Discussion section to avoid unneccesary repetitions in the text, referring back where appropriate to the Methods section.

Thank you for re-considering this manuscript for PlosONE

Achim Langenbucher

Submitted filename: CWchordRaytracingMC_response_ATC.docx

Click here for additional data file.

1 Apr 2022

Translation model for CW chord to angle Alpha derived from a Monte-Carlo simulation based on Raytracing

PONE-D-21-34567R1

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Reviewers' comments:

6 May 2022

PONE-D-21-34567R1

Translation model for CW chord to angle Alpha derived from a Monte-Carlo simulation based on Raytracing

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