David Simoncini

Sebastien Verel

Manuel Clergue

Philippe Collard

A premilinary investigation of Anisotropic Selection in Cellular Genetic Algorithms

We introduce the two concepts of Fuzzy Neighborhood and Anisotropic Selection to control the exploration/exploitation tradeoff in cellular genetic algorithms.

Introduction

In this paper we deal with the exploration/exploitation tradeoff in cellular genetic algorithms (cGAs). This concept is closely related to the notions of selective pressure and diversity. We propose to use anisotropic selection (AS) to promote diversity and to control selective pressure in genetic search. Previous studies on cGAs selected the size and the shape of neighborhoods [START_REF] Sarma | An analysis of the effects of neighborhood size and shape on local selection algorithms[END_REF], or the shape of the grid [START_REF] Alba | Cellular evolutionary algorithms: Evaluating the influence of ratio[END_REF][START_REF] Giacobini | Modelling selection intensity for linear cellular evolutionary algorithms[END_REF][START_REF] Giacobini | Modeling selection intensity for toroidal cellular evolutionary algorithms[END_REF] as basic parameters to tune the search process. Altering these structural parameters entails a deep change in the way we deal with the problem. For instance, there is no built-in mean to swap from a rectangular grid to a square grid without misshaping the neighborhood relation. One important advantage of the anisotropic selection scheme is that it allows to control the exploration/exploitation tradeoff without effect, neither on the grid topology, nor on the neighborhood shape; so the cellular genetic algorithm we propose can work on a square grid and a simple von Neumann neighborhood shape. The paper is divided in 5 sections. In the first one we give a brief definition of cGA and an overview of existing techniques used to control the exploration/exploitation tradeoff. Section 2 introduces the anisotropic selection concept. Section 3 studies the influence of AS on the selective pressure. In section 4 we compare anisotropic selection and rectangular grids topologies' influence on the selective pressure. Finally, in section 5 we show how AS promotes the emergence of niches.

1 Cellular Genetic Algorithms 1.1 Characterizing the CGA Cellular Genetic Algorithms (cGAs) are a subclass of Genetic Algorithms (GAs) in which exploration and population diversity are enhanced thanks to the existence of small overlapped neighborhoods [START_REF] Spiessens | A massively parallel genetic algorithm: Implementation and first analysis[END_REF]. Such a kind of structured algorithms is specially well suited for complex problems [START_REF] Jong | On decentralizing selection algorithms[END_REF]. We assume a two-dimensional toroidal grid as a spatial population structure. Each grid cell contains one individual of the population. The overlapping neighborhoods provide an implicit mechanism for migration of genetic material throughout the grid. A genetic algorithm is assumed to be running simultaneously on each grid cell, continuously selecting parents from the neighborhood of that grid cell in order to produce an offspring which replaces the current individual according to a replacement method.

Takeover Time

A standard technique to study the induced selection pressure without introducing the perturbing effect of variation operators is to let selection be the only active operator, and then monitor the growth rate of the best individual in the initial population [START_REF] Goldberg | A comparative analysis of selection schemes used in genetic algorithms[END_REF]. The takeover time is the time it takes for the single best individual to conquer the whole population. The grid is initialised with one cell having the best fitness and all the other having a null fitness. Since no other evolution mechanism but selection takes place, we can observe the way the best individual spreads over the grid by counting generation after generation the number of copies of this one. A shorter takeover time thus means a higher selective pressure. It has been shown that when we move from a panmictic population, as in standard GA, to a spatially structured one of the same size with synchronous updating of the cells, the global selection pressure induced on the entire population is weaker [START_REF] Sarma | An analysis of the effects of neighborhood size and shape on local selection algorithms[END_REF].

The takeover time is influenced either by the neighborhood size and shape an dby the grid topology. Neighborhood size and shape in a cGA are parameters that have some influence on the takeover time. A larger overlap of local neighborhoods of the same shape speeds up the best individual's spreading over the grid. The influence of the shape is given by Sarma and De Jong through a measure on the neighborhood which represents the spatial dispersion of a cell pattern [START_REF] Sarma | An analysis of the effects of neighborhood size and shape on local selection algorithms[END_REF]. More than the size of the neighborhood in terms of individuals, the determinant particularity of a local neighborhood is its radius. The takeover time decreases with this radius in a spatially structured population.

To measure the relation between grid topology and selective pressure, we consider rectangular grids, so-called RG, where the population size is fixed to 4096. We use the following grid shapes: 2×2048; 4×1024; 8×512; 16×256; 32×128 and 64 × 64 individuals. The selection strategy is a binary tournament. For each cell we randomly choose two individuals in its neighborhood (Von Neumann neighborhood) and the best one replaces the individual of the cell on the grid if it has a better fitness or with the probability 0.5 if the fitnesses are equal. Figure 1 shows the average on 20 independant runs of growth of best individual copies against generations, the takeover time is reached when the number of copies is equal to the size of the grid (see table 1). The average of increase rate, that is the number of new best individuals copies at each time step, of these curves for three rectangular grid shapes (64 × 64, 32 × 128, 16 × 256) is plotted figure 2. This figure helps to understand the growth of the best individual copies. The increase rate of the best individual is the same for all grids in the first generations. Then, the spreading speeds down to reach a constant speed for the rectangular grids. This constant is 2lp 1 where p 1 is the probability of selecting the best individuals when there is 1 copies of it in the neighborhood 1 . More accuratly, the spreading starts speeding down when a smallest diameter of the torus (a diameter of the smallest dimension of the grid) is filled by copies of the best individual (see figure 3(b)). Then, the speed becomes constant until the best individual has spread over a diameter of the other dimension (see figure 3(c)); this is the reason why the 64 × 64 grid curve has no constant period: radii in the two dimensions are filled at the same time. Finally, the increase rate of the growth curves falls down to zero as the best individual finishes to conquer the grid. The results of the experiments we conducted are in agreement with E.Alba and J.Troya observations that narrow grid shapes induce low selective pressure [START_REF] Alba | Cellular evolutionary algorithms: Evaluating the influence of ratio[END_REF]. We will see in the next sections that quite the same behavior is observed with the anisotropic selection method.

Anisotropic selection

This section introduce the Anisotropic Selection (AS) method where not all the neighbors of a cell have the same probability to be selected.

Von Neumann Fuzzy Neighborhood

The von Neumann neighborhood is defined as all cells reachable in a fixed axial direction (N orthSouth or EastW est). In the von Neumann Fuzzy Neighborhood, we assign different probabilities to choose one cell in the neighborhood according to the direction. Let p ns be the probability to choose a cell in the N S direction and p ew = 1 -p ns the probability to choose a cell in the EW direction. More, we assume the probability to stay in place is 1 5 as for uniform distribution. So, the probability to reach N or S is 2 5 p ns , and the probability to reach E or W is 2 , and α = 1 is the limiting case for fuzzy neighborhood where p ns = 1 and p ew = 0. In this last case, there is a vertical neighborhood with three neighbors only.

Definition

The Anisotropic Selection operator exploits the von Neumann Fuzzy Neighborhood; it works as follow: for each cell it selects k individuals in the cell neighborhood according to the p ns probability (k stands in the range [START_REF] Alba | Cellular evolutionary algorithms: Evaluating the influence of ratio[END_REF][START_REF] Spiessens | A massively parallel genetic algorithm: Implementation and first analysis[END_REF]). The k individuals participate to a tournament and the winner replaces the old individual if it has a better fitness or with probability 0.5 if the fitnesses are equal. Sarma and De Jong [START_REF] Sarma | An analysis of the effects of neighborhood size and shape on local selection algorithms[END_REF] established that increasing the neighborhood size creates a larger overlap and decreases the propagation time. Hence increasing local neighborhood sizes while keeping the grid size fixed should result in corresponding increases in selection intensity reflected by higher growth rates of the best individual. The control parameter α is a measure of anisotropy: α = 0 corresponds to standard selection, and α = 1 is the limiting case with the utmost anisotropy. We conjecture that the selection pressure decreases while increasing anisotropy.

Selective pressure and Anisotropic Selection

In this section we describe the relationship between selection pressure and anisotropic selection. First we measure the takeover time for different anisotropic degrees, then we go further into the analyse with the growth curve of the best individual.

We measure the effect of different anisotropy degrees on the takeover time. In our experiments, the anisotropic selection is based on tournament selection of size k = 2 on the square grid of side 64. All the 4096 cells are updated synchronously, they change their states simultaneously. For each cell, individual is replaced if the selected individual have a better fitness. For each value of control parameter α, we perform 20 independent runs. When α = 0.0, no direction is privileged and anisotropic selection is equivalent to the standard isotropic method. When α = 1.0, only one direction is exploited, the grid can not be filled and the takeover time is not defined. The figure 4 shows the influence of anisotropic selection on the takeover time. The takeover time increases when the parameter α increases. The correlation between takeover time and the anisotropic degree is not linear ; it fast increases after the value α = 0.9. These results are fairly consistent with our expectation that selection intensity decreases with the anisotropy degree. The figure 5 shows the average curves of the growth of best individual copies and its rate of increase as a function of time steps. The shape of the curve is decomposed in three parts: in the first one, the rate of increase is nearly proportional to the time steps; The growth curve is approximatively a parabola. In the second one, this rate becomes constant after a period of decrease; The growth curve becomes linear. In the last one, the rate decreases linearly until zero with a different slope from the first part. The higher the anisotropic degree α, the weaker the initial slope of the increase rate. In the same way in the second part, the slope of growth curve is smaller when α is higher. So, the selective pressure is lower when the anisotropic degree is higher.

The three part of the growth curve correspond to three periods in the spread of the best individual on the square grid (see figure 6). During the first one, the best individual spreads more in the privileged direction rather than the other one. This period is finished, as describe in section 1.2, when a diameter of the torus is filled by best individual copies in the privileged direction (see figure 6(b)). During the second period, the best individual fills the second direction of the grid until it has spread over a diameter in the less privileged direction (see figure 6(c)). The best individual front is sharp at the beginning, and becomes approximatively a horizontal line later. The third time finish to fill the grid. Taking into account those three times, it would be possible to have the equation of the growth curve as in [START_REF] Giacobini | Modeling selection intensity for toroidal cellular evolutionary algorithms[END_REF].

Anisotropic Selection vs. Rectangular Grid

The modification of the rectangular grid shape and the tuning of the anisotropic degree are two methods for varying the selective pressure. This section shows in which way they can be equivalent and the difference of those methods.

Takeover time analysis

Taking the experimental results from sections 1.2 and 3, we have computed the α parameter values for which we obtained the same takeover time as for each rectangular grid shapes. In figure 7 we plot the α values as a function of the l L ratio and exhibit the correlation between those parameters. We can see that α is proportional to the l L ratio. So, it is possible to have the same selective pressure using the two methods, according to the takeover time measure, with α = 1 -l L .

Growth curve analysis

Figure 8 shows the mean growth curves of the best individual spreading against time steps for all grid shapes and for the corresponding square grids using anisotropic selection. Although we found the same takeover time with rectangular grid shapes and with some values of anisotropic degree, the selective pressure is exerted in a different way for the two methods during the spread of the best individual. It is weaker during the first generations in the anisotropic case then it becomes slightly stronger, and finally reaches the same takeover time.

Anisotropic selection promotes niching

Many real optimization problems require the coexistence of diverse solutions during the search. In this section we show how the anisotropic selection is able to promote niching.

Niching methods

Niching methods have been proposed in the field of genetic algorithms to preserve population diversity and to allow the GA to investigate many peaks in parallel. As a side effect, niching prevents the GA from being trapped in local optima. Niching methods are inspired from nature where species specialize themselves to different ecological niches (for example food) in order to decrease the selective pressure they undergo. Niching GA's tend to achieve a natural emergence of niches in the search space. A niche is commonly referred to as an optimum of the domain, the fitness representing the resources of that niche [START_REF] Sareni | Fitness sharing and niching methods revisited[END_REF]. Niching methods are used to solve multimodal problems, and in dynamic optimization too [START_REF] Cedeno | On the use of niching for dynamic landscapes[END_REF]. For such problems a GA must maintain a diverse population that can adapt to the changing landscape and locate better solutions dynamically. There are different niching GA for panmictic population: sharing, crowding, etc. These methods are based on the concept of distance: sharing [START_REF] Golberg | Genetics algorithms with sharing for multimodal function optimiztion[END_REF] decreases the fitness according to the number of similar individuals in the population, and with crowding, replacement is performed considering the distance between solutions.

Experimental results

To show up to what extent anisotropic selection promotes niching, we have conducted experiments where two solutions with the best fitness (here 1) are placed on the grid at the initial generation. These solutions are farther from each other in the less favored direction (here oriented horizontally). We use a square grid with 64 × 64 individuals. Figure 9 shows some snapshots of the spreading of the two bests over generations for different anisotropic degrees. Cells in light grey (resp. dark grey) are copies of the first best (resp. the second best), and all white cells have a null fitness value. Generations grow from top to bottom, and the anisotropic parameter α increases from left to right. The lefthand row (α = 0) represents standard binary tournament schema ; we observe that isotropic selection is not able to maintain niches, after 1000 generations the grid is a mixture of the two optima. On the other hand, as α increases, frontiers between niches emerge and cells with the same lineage seems more and more closely linked.

Conclusions and perspectives

This paper presents a new selection scheme in cellular genetic algorithms. The main objective is to control in a flexible way the exploration/exploitation tradeoff. We propose to exploit the cellular GA characteristics to promote diversity during a genetic search process. Previous studies in cGA selected structural parameters, as neighborhood or grid shape, to tune the selective pressure. The main drawback of these techniques is that altering a structural parameter entails a deep change in the way we deal with the problem. The new selection scheme we propose is based on fuzzy neighborhood where a cell is chosen according to different probabilities. In order to favor one direction rather than the other one, anisotropic selection chooses individuals in fuzzy neighborhood. Experiments conducted to establish relation between the takeover time and the degree of anisotropy are consistent with our expectation that selection pressure decreases with the degree of anisotropy. Analysis of the growth curves allows to distinguish three different phases in the diffusion process. Then experimental results establish linear correlation in takeover between AS and cGA using rectangular grid. Finally we point out capabilities of AS to promote the emergence of niches.

This paper is a preliminary investigation and a more extensive analysis must be made to confirm that equilibrium between exploration and exploitation makes AS a good technique for complex problems in static or dynamic environments. Future works should address the following issues: compare AS and changes in the neighborhood shape and size, measure AS effects with cGA using mutation and crossover, change the balance of directions dynamically. This last point is an important feature: tuning the control parameter α, it would be possible to make the algorithm to self-adjust the selective pressure, depending on global or local measures of diversity. This adaptive ability have two important advantages: first, parameter α may vary in a continuous way, second, changes in this parameter have no effect, neither on the grid topology, nor on the neighborhood shape. In general, we have to continue investigation of Anisotropic Selection to assess its validity and generality.

Fig. 1 .

 1 Fig. 1. Growth curves of the best individual on different grid shapes.

2 5

 2 p ew . Let α = 2(p ns -0.5) be the control parameter in the range [0, 1]; the case α = 0 corresponds to standard von Neumann neighborhood with p ns = p ew = 1

Fig. 3 .

 3 Fig. 3. Spreading of the best individual over a 32x128 grid

Fig. 4 .

 4 Fig. 4. Average of the takeover time as a function of the anisotropic degree α.

Fig. 5 .

 5 Fig. 5. Growth curves of the best individual on a square grid (a) and the rate of increase of the corresponding curves (b) for different anisotropic degree α.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Spreading of the best individual with α=0.75

Fig. 8 .

 8 Fig. 8. Comparison between growth curves on different grid shapes and on a square grid with adapted anisotropy degree.

Fig. 9 .

 9 Fig. 9. Spreading of two copies of the best individual

Table 1 .

 1 Increase rate against time steps for three rectangular topologies. Takeover time for different rectangular grid topologies.

	1 As defined in [4]


Thibaut Gaboriau

Remi Dubois

Boris Foucque

Florence Malet

Cedric Schweitzer
email: cedric.schweitzer@chu-bordeaux.fr.

Glaucoma 24-Hour Monitoring of Intraocular Pressure Fluctuations Using a Contact Lens Sensor: Diagnostic Performance for Glaucoma Progression

Keywords: glaucoma, intraocular pressure (IOP), contact lens sensor (CLS), visual field, glaucoma progression, circadian rhythm

PURPOSE.

The purpose of this study was to compare 24-hour intraocular pressure (IOP) related fluctuations monitoring between 2 groups of visual field progression rates in patients with open angle glaucoma (OAG).

METHODS.

Cross-sectional study performed at Bordeaux University Hospital. Twenty-fourhour monitoring was performed using a contact lens sensor (CLS; Triggerfish; SENSIMED, Etagnières, Switzerland). Progression rate was calculated using a linear regression of the mean deviation (MD) parameter of the visual field test (Octopus; HAAG-STREIT, Switzerland). Patients were allocated into two groups: group 1 with an MD progression rate <-0.5 dB/year and group 2 with an MD progression rate ≥-0.5 dB/year. An automatic signal-processing program was developed and a frequency filtering of the monitoring by wavelet transform analysis was used to compare the output signal between the two groups. A multivariate classifier was performed for prediction of the faster progression group.

RESULTS.

Fifty-four eyes of 54 patients were included. The mean progression rate was -1.09 ± 0.60 dB/year in group 1 (n = 22) and -0.12 ± 0.13 dB/year in group 2 (n = 32). Twenty-four-hour magnitude and absolute area under the monitoring curve were significantly higher in group 1 than in group 2 (group 1: 343.1 ± 62.3 millivolts [mVs] and 8.28 ± 2.10 mVs, respectively, group 2: 274.0 ± 75.0 mV and 6.82 ± 2.70 mVs respectively, P < 0.05). Magnitude and area under the wavelet curve for short frequency periods ranging from 60 to 220 minutes were also significantly higher in group 1 (P < 0.05).

CONCLUSIONS.

The 24-hour IOP related fluctuations characteristics, as assessed by a CLS, may act as a risk factor for progression in OAG. In association with other predictive factors of glaucoma progression, the CLS may help adjust treatment strategy earlier.

G laucoma is a neurodegenerative disease defined by a progressive loss of optic nerve axons and retinal ganglion cells resulting in a characteristic enlargement of the optic nerve head cup and associated visual field defects. [START_REF] Weinreb | Primary open-angle glaucoma[END_REF] Several studies have demonstrated that the level of intraocular pressure (IOP) plays an important role in glaucoma onset or progression, even in glaucoma cases with an IOP measured in the normal range using the standard Goldman applanation tonometry. [START_REF] Normal | Comparison of glaucomatous progression between untreated patients with normal-tension glaucoma and patients with therapeutically reduced intraocular pressures[END_REF][START_REF] Normal | The effectiveness of intraocular pressure reduction in the treatment of normal-tension glaucoma[END_REF][START_REF] The | The Advanced Glaucoma Intervention Study (AGIS): 7. The relationship between control of intraocular pressure and visual field deterioration[END_REF][START_REF] Chauhan | Canadian Glaucoma Study: 3. Impact of risk factors and intraocular pressure reduction on the rates of visual field change[END_REF][START_REF] Garway-Heath | Latanoprost for open-angle glaucoma (UKGTS): A randomised, multicentre, placebo-controlled trial[END_REF][START_REF] Heijl | Reduction of intraocular pressure and glaucoma progression: Results from the Early Manifest Glaucoma Trial[END_REF][START_REF] Musch | Intraocular pressure control and long-term visual field loss in the Collaborative Initial Glaucoma Treatment Study[END_REF] Despite the importance of IOP measurements in glaucoma care to preserve visual function and related quality of life, the current method of measurement could miss IOP peaks or 24-hour fluctuations and delay treatment changes. Indeed, the majority of IOP peaks occurs at night or early in the morning and thus outside office-hour limits. [START_REF] Barkana | Clinical utility of intraocular pressure monitoring outside of normal office hours in patients with glaucoma[END_REF][START_REF] Jonas | Single intraocular pressure measurements and diurnal intraocular pressure profiles[END_REF][START_REF] Mottet | 24-hour intraocular pressure of young healthy humans in supine position: Rhythm and reproducibility[END_REF] Jonas et al. found that any single IOP measurement taken within office hours limits had a higher than 75% chance to miss the highest point of IOP over a 24-hour cycle. [START_REF] Jonas | Single intraocular pressure measurements and diurnal intraocular pressure profiles[END_REF] Moreover, 24-hour IOP monitoring by iterative IOP measurements during hospitalization is time-consuming and expensive for a frequent use in clinical practice. Furthermore, this method of measurements cannot be performed under physiologic conditions because of the awake and standing conditions required for IOP measurements during the night period. Additionally, although the role of 24-hour IOP fluctuations in glaucoma onset or progression still remains controversial, Grippo et al. found that untreated patients with ocular hypertension who converted to glaucoma had significantly different 24-hour IOP pattern from healthy controls. [START_REF] Bengtsson | Fluctuation of intraocular pressure and glaucoma progression in the early manifest glaucoma trial[END_REF][START_REF] Caprioli | Intraocular pressure fluctuation a risk factor for visual field progression at low intraocular pressures in the advanced glaucoma intervention study[END_REF][START_REF] Grippo | Twenty-four-hour pattern of intraocular pressure in untreated patients with ocular hypertension[END_REF][START_REF] Liu | Twenty-four-hour intraocular pressure pattern associated with early glaucomatous changes[END_REF] Liu et al. also found that the diurnal to nocturnal IOP profile was significantly different between early glaucoma and healthy patients. [START_REF] Liu | Twenty-four-hour intraocular pressure pattern associated with early glaucomatous changes[END_REF] Finally, in the Advanced Glaucoma Intervention Study, Caprioli and Coleman showed that long-term IOP fluctuations were associated with visual field progression in patients with low mean IOP but not in patients with high mean IOP. [START_REF] Caprioli | Intraocular pressure fluctuation a risk factor for visual field progression at low intraocular pressures in the advanced glaucoma intervention study[END_REF] Some intra-ocular and extra-ocular devices have recently been developed to estimate IOP fluctuations and provide a permanent 24-hour monitoring as a potential additional biomarker for glaucoma. [START_REF] Aptel | 24-h monitoring devices and nyctohemeral rhythms of intraocular pressure[END_REF] Triggerfish (Sensimed AG, Lausanne, Switzerland) is an extra-ocular contact lens sensor (CLS) device with a strain gauge assuming that changes in corneal curvature and circumference could be related to IOP fluctuations. This relationship was previously validated using an in vitro model of cannulated porcine eyes. [START_REF] Leonardi | Wireless contact lens sensor for intraocular pressure monitoring: Assessment on enucleated pig eyes[END_REF] The CLS assumes to record IOP related fluctuations in a reallife setting for up to 24 hours including during sleep time. Published studies also showed fair to good reproducibility of the CLS 24-hour monitoring patterns without any significant serious adverse events related to the device. [START_REF] Smedt | 24-hour intraocular pressure fluctuation monitoring using an ocular telemetry Sensor: Tolerability and functionality in healthy subjects[END_REF][START_REF] Mansouri | Continuous 24-hour monitoring of intraocular pressure patterns with a contact lens sensor: Safety, tolerability, and reproducibility in patients with glaucoma[END_REF] Agnifili et al. found that the CLS was able to differentiate output signal patterns of glaucomatous eyes from healthy subjects with glaucoma cases exhibiting more prolonged peaks and higher fluctuations than healthy subjects. [START_REF] Agnifili | Circadian intraocular pressure patterns in healthy subjects, primary open angle and normal tension glaucoma patients with a contact lens sensor[END_REF] Interestingly, the authors also showed different output signal patterns between healthy eyes and normal tension glaucoma eyes. Tojo et al. showed that patients with exfoliative glaucoma had a significantly larger range of 24-hour IOPrelated fluctuations and an earlier acrophase than healthy subjects: all healthy eyes had their maximum CLS value during the night period, compared to only 64% of glaucomatous eyes. [START_REF] Tojo | Fluctuations of the Intraocular Pressure in Pseudoexfoliation Syndrome and Normal Eyes Measured by a Contact Lens Sensor[END_REF] De Moraes et al. found that the CLS output signal could be associated with the rate of progression of treated glaucomatous eyes. [START_REF] Moraes | Visual Field Change and 24-Hour IOP-Related Profile with a Contact Lens Sensor in Treated Glaucoma Patients[END_REF] However, despite published evidence, there is currently no consensual method for the analysis and interpretation of the CLS output signal. Given that potential bias could affect the signal recorded by the CLS, definition of discriminant features within the 24-hour output signal pattern by signal processing methods is therefore needed. Additionally, although this device could be of interest to discriminate glaucomatous eyes from healthy eyes, an early detection of patients with glaucoma with a high risk of visual field progression is also an important issue for patients in order to adjust treatment earlier. Therefore, biomarkers of early detection of glaucoma progression are scarce, particularly when IOP seems controlled, and the CLS performance for glaucoma progression has to be analyzed.

Hence, we developed an automatic method of CLS output signal processing and we conducted a study to compare the 24-hour IOP related fluctuations monitoring pattern recorded by the CLS between two groups of patients with open angle glaucoma (OAG) with different rates of visual field progression. Then, we developed a prediction model of progression based on the output signal features significantly associated with a faster rate of visual field progression obtained after the comparison of our two groups.

MATERIALS AND METHODS

This was a cross-sectional prospective study performed in the department of Ophthalmology at Bordeaux University Hospital (France). The signal processing method was developed in the electrophysiology and heart-modeling institute (IHU LIRYC) at Bordeaux University (France). The aim of the study was to analyze and compare the 24-hour monitoring output signal provided by a CLS between 2 groups of patients with OAG with different rates of visual field progression measured with standard automated perimetry.

All subjects provided informed written consent for enrollment in the study. The research followed the Declaration of Helsinki's tenets and the study was approved by the institutional review board and by the ethical committee of Bordeaux in 2013. This trial is registered with ClinicalTrials.gov, number NCT01849536.

Participants

Between May 2015 and May 2016, we consecutively included patients with OAG with at least a 2-year follow-up and at least 5 reliable visual field tests. The worse eye of each patient was included.

Glaucoma was defined as a chronic neuropathy (asymmetric cup/disc ratio >0.2, rim thinning, notching, excavation, or retinal nerve fiber layer defect) with progressive changes of the visual field and reliable abnormal results. [START_REF] Weinreb | Primary open-angle glaucoma[END_REF] All patients underwent visual field tests using standard automated perimetry (Octopus; HAAG-STREIT, Koeniz-Berne, Switzerland) and a 30-2 dynamic strategy. Reliability of visual field was defined with false positive and false negative rates lower than or equal to 20% and with fixation losses rate lower than 10%. Glaucomatous visual field damage was defined with at least 3 contiguous test points within the same hemifield on the pattern deviation plot at P < 0.01, with at least 1 point at P < 0.005, on at least 2 consecutive examinations. All patients underwent at least 5 visual field tests over a period of at least 2 years before inclusion. Visual field progression was defined using a trend-based analysis and a linear regression was performed on the mean deviation (MD) of the visual field test to calculate a rate of progression expressed in dB/year. Then, patients were allocated into 1 of the 2 groups of glaucoma progression: group 1 for eyes having a rate of visual field progression faster than -0.50 dB/year and group 2 for eyes having a rate of progression slower than or equal to -0.50 dB/year. The threshold of -0.5 dB/year was chosen based on the approximate median of visual field progression rates of our population sample in order to define a group of fast visual field progression and a group of slow visual field progression.

Central corneal thickness (CCT) was measured using anterior segment optical coherence tomography and IOP was measured using Goldman applanation tonometry between 8:00 AM and 8:30 AM and before the CLS was placed on the surface of the eye.

We excluded patients with angle closure or secondary glaucoma, except exfoliative and pigmentary glaucoma, eyes suffering from corneal or conjunctival disease, or severe dry eye syndrome defined as severe symptoms and a tear break-up time lower than 5 seconds or corneal fluorescein staining, and eyes with history of glaucoma surgery. Eyes with unreliable visual field tests or with severe visual field damage defined by an MD value worse than 20 dB were also excluded. If both eyes were eligible, only the eye with the worse MD value was included in order to minimize the impact of the CLS wearing on daily activities.

24-Hour IOP Related Fluctuations Monitoring Using a CLS

We used a CLS with a strain gauge (Triggerfish; Sensimed AG, Lausanne, Switzerland) to assess 24-hour IOP related fluctuations monitoring for each eye included. The technology of the CLS was published by Leonardi et al. [START_REF] Leonardi | Wireless contact lens sensor for intraocular pressure monitoring: Assessment on enucleated pig eyes[END_REF][START_REF] Leonardi | First steps toward noninvasive intraocular pressure monitoring with a sensing contact lens[END_REF] The Downloaded from iovs.arvojournals.org on 04/18/2023 strain gauge is a platinum-titanium sensing-resistive gauge that enables a recording of circumferential changes in the area of the corneoscleral junction by varying its voltage according to its deformation. A microprocessor connected to the gauge sends an output signal proportional to changes of the strain gauge's voltage. Wireless power and data transfer are achieved using a patched periorbital antenna, which is made possible by using an inductive coupling system. Two coils are coupled: the first one is a gold coin connected to the strain gauge into the contact lens, and the second one is a copper coil inserted into the patched periorbital antenna. A cable connects the periorbital antenna to the portable recorder, which contains the battery that powers the device. The contact lens is made in silicone because of its oxygen permeability and water absorption making it insensitive to the hydration level at the ocular surface. To render silicone hydrophilic and thus achieve proper fitting conditions of the lens on the eye, the contact lens surface is treated with oxygen plasma. The CLS exists in three different base curves to fit a large range of corneas. CLS adaptation was performed according to the manufacturer's recommendations.

The CLS takes a measurement every 5 minutes (i.e. a total of 288 measurements for a 24-hour recording). Each measurement is composed of 300 samples of the voltage corresponding to the deformation of the corneoscleral junction during 30 seconds (10 Hz sampling rate). The CLS standard analysis reports the median value over each 30 second measurement, the output values are voltages expressed in millivolt (mV).

Each monitoring started between 8:00 AM and 8:30 AM following Goldman applanation tonometry IOP measurement and the CLS was removed 24 hours later.

Signal Processing

A three-step dedicated signal processing method was developed and implemented under MATLAB software (Mathworks, Natick, MA, USA). Sensimed (Lausanne, Switzerland) provided the raw outputs of the CLS data for each patient: 300 samples during 30 seconds every 5 minutes during the 24 hours (N = 86,400 samples). The company was masked to clinical data.

First, blinking artifacts were removed from the signal. During the day period, the 30-second periods are regularly distorted with abnormal high values that last for about 10 milliseconds. These outlier areas were identified as being associated with blinking movements (Fig. 1B during the day period, and Fig. 1C during the night period). An adaptive threshold-based derivative method was developed to automatically detect these areas and linear interpolation was performed to smooth the signal.

A positive voltage difference between the first values of the recording and the last values 24 hours later was observed in almost all recordings. Assuming that circadian biorhythms are endogenous, 24-hour oscillations of the physiological systems, including sleep/awake cycles, metabolic, or cardiovascular 24-hour activities, cell cycles, and 24-hour supranucleus brain activities, the positive trends were compensated by a linear interpolation between the first 5 minutes of the recording and the last 5 minutes 24 hours later [START_REF] Asgari-Targhi | Mathematical modeling of circadian rhythms[END_REF] (Fig. 2A).

Finally, a wavelet-based filtering approach was applied. This method, commonly used for biosignal analysis, decomposes the original signal into N sub-signals using a "mother" wavelet W at N scale levels (w 1 to w N ), each within a specific frequency range. [START_REF] Leise | Wavelet-based analysis of circadian behavioral rhythms[END_REF] Table 1 shows the correspondence between each wavelet w i and the range of frequency when using a decomposition over N = 10 scale levels. The decomposition of the signal on each wavelet w i allows the analysis of the corresponding frequencies contained in the signal (Fig. 2B).

Output Signal Features

We defined two sets of features to assess IOP fluctuations.

The first set was related to long-term fluctuations in IOP and aimed to differentiate between day and night periods; the second set was related to more rapid fluctuations in IOP (peaks of IOP). Features belongings to the first set were calculated on the sub-signal that lied in the frequency range of diurnal fluctuations (cumulative decomposition from W 4 to W 10 ). Two features were computed (Fig. 3A): IOP related to more rapid fluctuations (ultradian rhythm) were computed on the two sub-signals obtained from the decomposition on w 4 and w 5 corresponding to fluctuation of period 60 to 220 minutes, and 120 to 500 minutes, respectively. Two features per decomposition were computed (Fig. 3B):

-The magnitude of the sub-signal from w 4 decomposition: w4Magn and w5Magn, expressed in V. -The absolute value of the area under the curve of sub-signal from w 4 decomposition (w4Area) and from w 5 decomposition (w5Area) expressed in Vs.

Classification Between the Two Groups

The objective being to design a classifier that allows the automatic classification of the patients group 1 versus group 2: classification accuracy of each feature presented above used individually (univariate classification) was tested and reported as the area under the curve of the corresponding receiver operating curve (AUROC). Then, a multidimensional classifier (multivariate classification) based on linear discriminant analysis was optimized to account for several features simultaneously: a Downloaded from iovs.arvojournals.org on 04/18/2023 20-fold cross-validation strategy was used to divide the database into training sets and validation sets. The training sets were used to perform (i) feature selection and (ii) classifier parameter fitting. The results are presented on the validation sets. Feature selection was performed using the statistics-based method of machine learning presented by Stoppiglia et al. that orders the available features according to their contribution to a multivariate classifier. [START_REF] Stoppiglia | Ranking a random feature for variable and feature selection[END_REF] All available demographic, ophthalmological, and wavelets features of the output signal were included in the model.

Outcomes

The primary outcome was to compare the mean values of the features we selected from the 24-hour monitoring raw output signal recorded by the CLS between group 1 and group 2.

The secondary outcome was to analyze the diagnostic performances of our features to discriminate the two groups of patients with OAG. Finally, we performed an internal validation of our results and defined a score based on the performance of all the features to discriminate the output signal patterns of the two groups.

Statistical Analysis

Data are expressed as mean ± standard deviation (SD) or median (upper quartile-lower quartile). Qualitative variables were described using number of occurrences. Significance of differences between group 1 and group 2 was performed using a Student t-test, or a Mann-Whitney test for non-normal distribution. Frequencies were compared using chi-square tests.

The performance of each parameter was assessed using the receiver operating characteristic (ROC) curve method for each parameter.

Statistical analysis was performed using GraphPad Prism software version 5 (GraphPad Software Inc., La Jolla, CA, USA). Statistical significance was defined as P < 0.05.

RESULTS

Fifty-four eyes of 54 patients were included in the analysis. All raw data of the 24-hour IOP related fluctuations by the CLS were complete and available. Twenty-two eyes were allocated into group 1 and 32 eyes were allocated into group 2.

Demographic and ophthalmological characteristics of the population sample are shown in Table 2. Of the 54 patients, 27 patients were women and 27 were men. The mean age was 68.5 ± 6.0 years and was not significantly different between the 2 groups. Mean IOP before the CLS placement was 15.9 ± 3.1 mm Hg and was not significantly different between the 2 groups. The mean number of visual field tests available before the inclusion was 7.7 ± 2.3 and was also not significantly different between the 2 groups. The mean rate of progression was significantly faster in group 1 than in group 2 (-1.09 ± 0.60 dB/year and -0.12 ± 0.13 dB/year, respectively; P < 0.0001). The severity grade of glaucoma assessed by the visual field was higher in group 1 than in group 2 with MD value of -10.57 ± 6.97 dB and -5.43 ± 4.32 dB, respectively (P < 0.01). The mean CCT was significantly thinner in group 1 than in group 2 (511.9 ± 20.9 μm and 533.8 ± 22.7 μm, respectively; P < 0.05). No serious adverse events were reported in this study.

Table 3 shows the comparison of output signal features between the two groups. During the 24-hour period, the magnitude of monitoring curve (24hMagn) was significantly higher in group 1 (343.1 ± 62.3 mV) than in group 2 (274.0 ± 75.0 mV; P = 0.0027), as well as the absolute value of the area under the monitoring curve (24hArea; P = 0.0251). When analyzing short-term periods, the sub-signal decomposed on the wavelet with a frequency content ranging from 60 to 220 minutes exhibited a significantly higher magnitude (w4Magn) in group 1 (110.1 ± 33.4 mV) than in group 2 (86.1 ± 21.7 mV; P = 0.0316), as well as the absolute value of the area under the curve (w4Area; P = 0.0188). There was no significant difference between the two groups for the other wavelet features generated by the frequency filtering analysis.

Figure 4A shows the correlation matrix of 24-hour and wave 4 output signal features, CCT and IOP. We observed a very low positive correlation between IOP and the output signal features ranging from 0.02 to 0.13 and a very low negative correlation between CCT and the output signal features ranging from -0.22 to -0.14.

The univariate discriminative power of each feature was evaluated using the AUROC. The highest AUROC value was observed for the 24-hour amplitude parameter 24hMagn (0.739; 95% confidence interval [CI] = 0.605 to 0.872) followed by 24hArea (0.680; 95% CI = 0.534 to 0.827), W4Magn (0.673; 95% CI = 0.525 to 0.821) and W4Area (0.689; 95% CI = 0.543 to 0.835) features. The AUROC for CCT was 0.694 (95% CI = 0.554 to 0.833).

Among all available demographic, ophthalmological, and wavelets parameters, the feature selection process Downloaded from iovs.arvojournals.org on 04/18/2023 for the multivariate classifier led to the selection of 4 features (24hMagn, 24hArea, W4Magn, and W4Area) from the wavelet analysis, as well as the CCT parameter. In this 5-dimensionnal input space, the multivariate classifier performances over the testing sets were as follows: accuracy of 77.7% (n = 42 eyes), sensitivity of 81.3%, and specificity of 72.7%. The AUROC value of the classifier was 0.730 (95% CI = 0.583 to 0.874; Fig. 4B).

DISCUSSION

We demonstrated that the 24-hour output signal pattern recorded by the CLS was different between patients with OAG with an MD rate of progression slower than or equal to -0.5 dB/year and patients with OAG with an MD rate of progression faster than -0.5 dB/year. Although IOP measurements performed just before CLS measurements were similar between the two groups of visual field progression, we found differences on the shape of wavelets with short frequency periods as well as on the shape of the curve over the 24-hour period. Whereas the amplitude and the absolute area under the monitoring curve were significantly higher in the group of faster rate of progression over the 24-hour period, particularly at night, we also found that the amplitude and the absolute area under the monitoring curve on shorter frequency periods ranging from 60 to 220 minutes were significantly higher in this group of patients with OAG.

Although the exact interpretation of the output signal pattern is still controversial, we speculate that the higher mean values of amplitude and absolute area under the curve we observed during the 24-hour period for the group of faster rate of visual field progression could be related to a higher and more prolonged night increase in corneal curvature and circumference as measured with the CLS. Additionally, our results are consistent with previously published results. [START_REF] Agnifili | Circadian intraocular pressure patterns in healthy subjects, primary open angle and normal tension glaucoma patients with a contact lens sensor[END_REF][START_REF] Tojo | Fluctuations of the Intraocular Pressure in Pseudoexfoliation Syndrome and Normal Eyes Measured by a Contact Lens Sensor[END_REF][START_REF] Moraes | Visual Field Change and 24-Hour IOP-Related Profile with a Contact Lens Sensor in Treated Glaucoma Patients[END_REF][START_REF] Moraes | Association Between 24-Hour Intraocular Pressure Monitored With Contact Lens Sensor and Visual Field Progression in Older Adults With Glaucoma[END_REF][START_REF] Martin | Use of Machine Learning on Contact Lens Sensor-Derived Parameters for the Diagnosis of Primary Open-angle Glaucoma[END_REF][START_REF] Tojo | Influence of Ocular Dimensional Change on 24-Hour Intraocular Pressure Measurement With Contact Lens Sensor[END_REF][START_REF] Tojo | Correlation between 24h continuous intraocular pressure measurement with a contact lens sensor and visual field progression[END_REF] Indeed, De Moraes et al. also analyzed the output signal provided by the CLS on a cohort of 40 patients with glaucoma with a progression of the visual field. [START_REF] Moraes | Visual Field Change and 24-Hour IOP-Related Profile with a Contact Lens Sensor in Treated Glaucoma Patients[END_REF] By using a different methodology of signal processing, they also observed that the output signal had a different shape for patients with a faster rate of visual field progression. They reported a higher number of long peaks and higher mean peak ratio when patients were awake. Tojo et al. also showed that high maximum amplitudes over the 24-hour period as well as during the night period were associated with a higher risk of visual field progression in glaucoma patients. [START_REF] Tojo | Correlation between 24h continuous intraocular pressure measurement with a contact lens sensor and visual field progression[END_REF] Finally, Martin et al. showed higher values of several features of the output signal features using machine learning methods in a large cohort of patients with OAG compared to control patients. [START_REF] Martin | Use of Machine Learning on Contact Lens Sensor-Derived Parameters for the Diagnosis of Primary Open-angle Glaucoma[END_REF] To evaluate short-term IOP related fluctuations (IOP peaks), we developed an original method of automatic signal processing based on a frequency filtering of the monitoring by wavelet transform. This method enables a detection of diurnal changes within the 24-hour period (ultradian biorhythm) using a wide range of frequencies. Interestingly, we found that signal amplitude and area under the curve of wavelet at periods ranging from 60 to 220 minutes were significantly higher in the group of patients with OAG with a faster rate of progression. These higher in amplitude and longer changes in corneal curvature or circumference observed in this group of patients with OAG could be related to short term IOP peaks, which may be associated with a higher risk of visual field progression. Although they used a different methodology of output signal analysis, De Moraes et al. also showed short-term changes in the output signal of patients with glaucoma. [START_REF] Moraes | Visual Field Change and 24-Hour IOP-Related Profile with a Contact Lens Sensor in Treated Glaucoma Patients[END_REF] Indeed, they observed that the amplitude and the length of short-term peaks were associated with a faster rate of visual field progression. As shortterm IOP peaks characteristics are not a priori known, our method of signal processing using a wavelet decomposition on a wide range of frequency periods could enable a more reproducible and accurate analysis of the raw output signal provided by the CLS. Hence, daily short-term IOP peaks, particularly peaks ranging from 60 to 220 minutes may play a role in glaucoma progression and the analysis of such IOP peaks could act as a biomarker to help predict the risk of visual field progression. However, our findings would need to be confirmed in other clinical and experimental studies.

A strength of our study was to analyze raw data of the output signal in order to limit measurement bias and improve the accuracy of the 24-hour monitoring provided by the CLS in patients with OAG. First, by removing recurrent outliers of the output signal only observed during the day period that are very likely related to blinking, we hypothesize our method should have limited measurement bias and the comparison between the day and night periods. Indeed, these data points had very high values for a very short period of time. Instead of the removal of these outliers, the Triggerfish software uses the median of these measurements in the final analysis. Thus, we speculated that this processing method of data points could bias the measurement of change in corneal curvature or circumference during the 24-hour monitoring by overestimating the amplitude and length of peaks particularly during the diurnal period. Furthermore, these outliers could also bias the comparison of the output signal between the day and night periods. Second, as expected during a 24-hour monitoring, diurnal changes in corneal curvature or circumference should follow a circadian biorhythm. [START_REF] Asgari-Targhi | Mathematical modeling of circadian rhythms[END_REF] Thus, as the difference between the beginning and the end of the monitoring may overestimate the area and the amplitude of the output signal pattern recorded during the night period, we chose to compensate this difference by applying a linear regression to the entire monitoring. Although the origin of this difference is controversial, we assumed that this difference may be due to a measurement bias. Indeed, Hubanova et al. showed a significant increase in CCT following the 24-hour wearing of the CLS on the studied eye as compared with the fellow control eye of the same patient. [START_REF] Hubanova | Effect of overnight wear of the Triggerfish((R)) sensor on corneal thickness measured by Visante((R)) anterior segment optical coherence tomography[END_REF] However, the authors could not conclude whether changes in CCT observed during CLS wearing could significantly modify the corneoscleral junction angulation and influence the shape of the output signal at the end of the monitoring. Finally, to our knowledge, we were the first to apply a method of wavelet transform on the raw output signal to optimize information recorded during the 24-hour period. Indeed, this method is usually carried out in diurnal rhythm analysis to detect changes within the 24-hour period at progressing frequency periods from shortest periods to the 24-hour period. [START_REF] Leise | Wavelet-based analysis of circadian behavioral rhythms[END_REF] This method of signal processing also provides a smoothing of the signal to overcome the noise commonly observed in such monitoring. Interestingly, we also observed a very weak correlation between analyzed output signal features and IOP or CCT parameters. Thus, we speculate the influence of IOP or CCT on the amplitude and area of the 24-hour period and wavelet 4 features is likely weak. Hence, we believe the method of signal processing we used in our study can have standardized and optimized the analysis of short-term changes in corneal curvature and IOP-related fluctuations in our cohort of patients with OAG.

Although we found a significant difference of output signal patterns between the group of patients with OAG with a faster visual field progression and the patients with OAG group with a slower visual field progression, the diagnostic performance of the CLS to discriminate our two groups was moderate. However, our findings are in accordance with published results on diagnostic performance of the CLS to discriminate glaucoma from healthy eyes. Indeed, in a cohort of 435 subjects, Martin et al. observed a mean AUROC value of 0.611 with a best value at 0.759 using machine learning methods to analyze the CLS output signal pattern. [START_REF] Martin | Use of Machine Learning on Contact Lens Sensor-Derived Parameters for the Diagnosis of Primary Open-angle Glaucoma[END_REF] First, the moderate diagnostic performance of CLS could be related to the role played by IOP fluctuations in glaucoma onset or progression, which is not fully understood. Indeed, pathophysiology of glaucoma is multifactorial and besides IOP, some other factors could play an independent or an additive role to the progressive death of retinal ganglion cells, as blood supply around the optic nerve head, Downloaded from iovs.arvojournals.org on 04/18/2023 biomechanical properties of lamina cribrosa, or altered neuronal functionality of these cells. [START_REF] Weinreb | Primary open-angle glaucoma[END_REF] Then, the diagnostic performance of the CLS we observed could also be related to the threshold of -0.5 dB/year we chose to define our 2 groups of visual field progression. Although there is no consensus in the literature to define a fast rate of visual field progression, we chose the threshold of -0.5 dB/year because it was approximately the median of rate of progression of our population sample. In comparison, Heijl et al. reported a mean progression rate of -0.8 dB/year and a median progression of -0.62 dB/year. [START_REF] Heijl | Reduction of intraocular pressure and glaucoma progression: Results from the Early Manifest Glaucoma Trial[END_REF] Chauhan et al. [START_REF] Chauhan | Rates of glaucomatous visual field change in a large clinical population[END_REF] reported on a cohort of 2324 patients in which approximately a quarter of the eyes had a rate of progression of -0.30 dB/year or higher and they selected the threshold of -1 dB/year to define fast progressors. Boodhna et al. reported on a cohort of 18,926 eyes in which approximately a quarter of the eyes had a visual field progression higher than -0.51 dB/year. [START_REF] Boodhna | Are rates of vision loss in patients in English glaucoma clinics slowing down over time? Trends from a decade of data[END_REF] Hence, as there is no described specific biomarker to help predict visual field progression below -0.5 dB/year, we speculated that the monitoring of IOP-related fluctuations could be of interest in current practice for this category of patients with OAG to help adjust treatment strategy earlier. Additionally, despite moderate diagnostic performance of the CLS to discriminate the patients with OAG with a progression rate slower than or equal to -0.5 dB/year from patients with OAG with a rate of progression fast than -0.5 dB/year, the score we developed and tested using a sensitivity analysis strengthened our findings. Indeed, the performance of the score was fair in our population sample and this score could help define a predictive model of glaucoma progression. Hence, in association with other risk factors of glaucoma progression, the signal provided by the CLS may help predict the risk of glaucoma progression in current practice.

Our study may have some limitations that need to be considered. First, we chose to define glaucoma progression using a standard automated perimetry and a trend-based analysis with a linear regression of the mean deviation of visual field tests. [START_REF] Aref | Detecting Visual Field Progression[END_REF] Retinal sensitivity values measured with standard automated perimetry could fluctuate for the same eye with increasing test-retest variability with decreasing retinal sensitivity values across the retina. [START_REF] Gardiner | Differences in the Relation Between Perimetric Sensitivity and Variability Between Locations Across the Visual Field[END_REF] Furthermore, glaucoma progression can also be assessed using an eventbased analysis of the visual field. [START_REF] Aref | Detecting Visual Field Progression[END_REF] However, the trend-based analysis approach is commonly used in published studies analyzing glaucoma progression and also provides a rate of progression useful to analyze characteristics of glaucoma on a long-term period. [START_REF] Normal | Comparison of glaucomatous progression between untreated patients with normal-tension glaucoma and patients with therapeutically reduced intraocular pressures[END_REF][START_REF] Musch | Intraocular pressure control and long-term visual field loss in the Collaborative Initial Glaucoma Treatment Study[END_REF][START_REF] Chauhan | Practical recommendations for measuring rates of visual field change in glaucoma[END_REF] Furthermore, eyes were eligible when at least 5 reliable visual fields were performed over at least a period of 2 years and the mean number of visual field was 7.7 ± 2.3 in our population sample. At the beginning of standard of care follow-up visits, patients were classified as suspect of glaucoma progression on the visual field and progression was later confirmed with time and additional visual fields. Thus, we hypothesize that the rate of progression of our population sample was reliable and that our findings could be applied to such population groups. Another potential limitation could be related to the device we used to evaluate IOP related fluctuations and the potential bias that could be associated with this technology. Indeed, the 24-hour monitoring provided by the CLS measures the changes in corneal curvature and circumference expressed in voltage and thus does not provide real IOP measurements. Voltage measured with the strain gauge is supposed to be modified by changes in corneal circumference at the corneoscleral junction and the correlation between volumetric changes and IOP is not fully established. Hence, the correlation between voltage and IOP related fluctuations still remains unclear. [START_REF] Leonardi | Response to the letter to the editor entitled reverse 'Assessment of the Triggerfish contact lens sensor for measurement of intra-ocular pressure variations[END_REF][START_REF] Sunaric-Megevand | Assessment of the Triggerfish contact lens sensor for measurement of intraocular pressure variations[END_REF] Whereas Mansouri et al. found that the coefficient of correlation between CLS and pneumatonometer was R 2 = 0.914, Vitish-Sharma et al. found that the mean correlation coefficient between CLS output signal measurements and IOP measurements was r = 0.291. [START_REF] Mansouri | Efficacy of a contact lens sensor for monitoring 24-h intraocular pressure related patterns[END_REF][START_REF] Vitish-Sharma | Can the SENSIMED Triggerfish((R)) lens data be used as an accurate measure of intraocular pressure?[END_REF] Additionally, the association of corneal parameters as corneal thickness at the corneal junction and CLS measurements or the influence of a 24-hour wearing of the CLS on the cornea could also influence the recorded CLS output signal particularly at the end of the monitoring. [START_REF] Tojo | Influence of Ocular Dimensional Change on 24-Hour Intraocular Pressure Measurement With Contact Lens Sensor[END_REF][START_REF] Hubanova | Effect of overnight wear of the Triggerfish((R)) sensor on corneal thickness measured by Visante((R)) anterior segment optical coherence tomography[END_REF] In our population sample, we observed a 20μm difference in mean CCT measurements between the two groups at baseline. Although the CLS measures the changes in corneal curvature or circumference at the corneal junction and not at the apex of the cornea, the influence of this difference in central corneal thickness on our results remains unclear. However, the score we calculated took into account central corneal thickness parameter and was still able to discriminate the two groups of progression. Finally, although our multivariate classifier model showed good diagnostic performances to diagnose OAG with a faster rate of progression, this classifier would also need to be tested in an independent and larger population sample to confirm or refine its diagnostic performances (external validity analysis) and enable generalization of findings.

In conclusion, we found significant difference of 24-hour as well as short-term IOP related fluctuations characteristics as assessed by a CLS and an automatic signal processing of the output signal between patients with OAG with rates of visual field progression slower than or equal to -0.5 dB/year and patients with OAG with rates of visual field progression faster than -0.5 dB/year. In association with other predictive factors of glaucoma progression, these CLS features could act as additive risk factors in clinical practice and help adjust treatment strategy earlier. However, further studies are still needed to confirm our findings -particularly the role of short-term IOP peaks in glaucoma progression -and strengthen the association between IOP fluctuations and CLS measurements as well as the influence of corneal parameters on CLS measurements.

FIGURE 1 .

 1 FIGURE 1. CLS 24-hour output signal raw data (A); voltage record during the day period showing recurrent high magnitude peaks associated with blinking movements (B); voltage during the night period without recurrent high magnitude peaks (C).

FIGURE 2 .

 2 FIGURE 2. Signal processing steps with MATLAB R2017b (Mathworks, Natick, MA, USA): main curve of the monitoring made by the median of each 30 seconds period after removing of blinks outliers (black line) (A); linear regression applied to the entire monitoring (dashed line), and frequency filtering of the monitoring by wavelets decomposition (bold black line) (B).

-

  The diurnal amplitude, which is the highest voltage magnitude of the curve over the 24-hour period of the monitoring (called h24Magn, expressed in Volt [V]). -The absolute area under the monitoring curve (h24Area, expressed in Volt.second [Vs]).

FIGURE 3 .

 3 FIGURE 3. Output signal after signal processing and features generated for analysis. (A) Maximum magnitude (24hMagn) is indicated by the arrow, absolute area under the monitoring curve (24hArea) for 24-hour fluctuations appears in grey; (B) Maximum magnitude (w4Magn) is indicated by the arrow, absolute area under the subsignal from the decomposition over the wavelet w 4 (w4Area) for fluctuations of frequency period ranging from 60 to 220 minutes appears in grey.

FIGURE 4 .

 4 FIGURE 4. Correlation matrix showing the correlations between 24-hour maximum magnitude (24hMagn), 24-hour absolute area under the monitoring curve (24hArea) and wavelet w 4 maximum magnitude (w4Magn), wavelet w 4 absolute area under the monitoring curve (w4Area) features of the output signal with central corneal thickness (CCT) and intraocular pressure (IOP) (A); area under the receiver operating characteristic curves (AUC) of the multivariate classifier (B).

TABLE 1 .

 1 Characteristics of the Frequency Content of Wavelet Transform Used for the Frequency Filtering of the Contact Lens Sensor Output Signal

		Frequency Content (Min)
	Wavelet	Minimum	Maximum
	W 1	10	30
	W 2	20	70
	W 3	30	125
	W 4	60	220
	W 5	120	500
	W 6	220	1000
	W 7	500	2000
	W 8 , W 9 , and W 10	≥800	≥4000

TABLE 2 .

 2 Baseline Demographic and Ophthalmological Characteristics Between the Group of Visual Field Mean Deviation Progression Rate ≤0.5 dB/Year and the Group of Visual Field Mean Deviation Progression Rate >0.5 dB/Year

			Group 1 MD	Group 2 MD
			Progression Rate	Progression Rate
	Characteristics	Total (n = 54)	<-0.5 dB (n = 22)	≥-0.5 dB (n = 32)	P Value *
	Mean age, y (SD)	68.5 (6.0)	69.6 (6.2)	67.7 (5.8)	0.24
	Gender -male subjects (%)	27 (50)	14 (63.6)	13 (40.6)	0.17
	Mean IOP (mm Hg) (SD)	15.9 (3.1)	15.7 (2.74)	15.1 (2.7)	0.56
	Mean CCT (μm) (SD)	525.3 (24.2)	511.9 (20.9)	533.8 (22.7)	0.015
	MD (dB) (SD)	-7.47 (5.85)	-10.57 (6.97)	-5.43 (4.32)	0.0072
	Mean rate of visual field progression (dB/y) (SD)	-0.52 (0.47)	-1.09 (0.60)	-0.12 (0.13)	<0.0001
	Number of visual field tests (SD)	7.7 (2.3)	8.3 (2.3)	7.2 (2.2)	0.08
	CCT, central corneal thickness; dB, decibel; IOP, intraocular pressure; MD, mean deviation; SD, standard deviation.

*

Results in bold are statistically significant.

TABLE 3 .

 3 Comparison of 24-Hour Contact Lens Sensor Output Signal Characteristics Between the Group of Visual Field Mean Deviation Progression Rate ≤0.5 dB/Year and the Group of Visual Field Mean Deviation Progression Rate >0.5 dB/Year Using Frequency Filtering and a Wavelet Transform Analysis

			Group 1 MD	Group 2 MD
			Progression Rate	Progression Rate
	CLS Output Signal Characteristics	Total (n = 54)	<-0.5 dB (n = 22)	≥-0.5 dB (n = 32)	P Value *
	24-h characteristics
	24hMagn (mV) [mean (SD)]	302.2 (79.1)	343.1 (62.3)	274.0 (75.0)	0.0027
	24hArea (10 3 Vs [mVs]) [mean (SD)]	7.42 (2.55)	8.28 (2.10)	6.82 (2.70)	0.0251
	Ultradian characteristics
	w4Magn (mV) [mean (SD)]	95.8 (27.7)	110.1 (33.4)	86.1 (21.7)	0.0316
	w4Area (10 3 Vs [mVs]) [mean (SD)]	1.10 (0.28)	1.24 (0.30)	1.01 (0.23)	0.0188
	w5Magn (mV) [mean (SD)]	100.9 (38.2)	108.7 (46.2)	95.6 (31.9)	0.57
	w5Area (10 3 Vs [mVs]) [mean (SD)]	1.38 (0.44)	1.40 (0.45)	1.37 (0.44)	0.85

24hArea, absolute area under the monitoring curve; w5Area, absolute area under the wave; CLS, contact lens sensor; mV, millivolt; SD, standard deviation; Vs, Volt.second.

* Results in bold are statistically significant.

Acknowledgments

Disclosure: T. Gaboriau, None; R. Dubois, None; B. Foucque, None; F. Malet, None; C. Schweitzer, None