1. INTRODUCTION

Examination of the actual value of intraocular pressure (IOP) is a current challenge of ophthalmology. Existing, non-invasive methods of IOP measurement do not allow direct measurement. They are performed through the cornea or sclera, so the obtained results are disturbed by biomechanical properties of the anterior part of the eye and variability of IOP during measurement. This may hinder proper diagnosis of glaucoma or classification to the glaucoma risk group. On the other hand, modern, state-of-the-art devices not only allow indirect measurement of IOP, but also try to take into account the influence of biomechanical properties of the eye and correct the obtained result [1–4].

One such device is the non-contact Corvis ST tonometer, available on the market since 2011. To measure IOP with this instrument, an air pulse is directed from the tonometer nozzle towards the center of the cornea. As a result of the air blast, the central part of the cornea changes its shape from convex to concave, and then returns to the initial convex form. During the measurement, the cornea is illuminated with blue light (455 nm) through a horizontal, 9 mm aperture. This kind of illumination allows registration of images of horizontal cross section of the cornea, deforming under the air puff. Deformations of the horizontal corneal section are recorded using the built-in ultrafast camera. During a single measurement, a sequence of 140 images is captured with the acquisition time of 32 ms.

Corvis ST is accompanied by software designed for analysis of IOP, corneal profile thickness distribution, and dynamics of the corneal profile deformation. The basic, uncorrected IOP given by Corvis ST (CVS-IOP) is calibrated based on the time at which the central part of the cornea has been flattened (applanation time). This method is commonly used in non-contact tonometers [1]. Additionally, this software enables to determine a value of biomechanical corrected IOP (bIOP) as a parameter that takes into account the central corneal thickness and patient’s age, in order to predict the real value of pressure inside the eye, not affected by influence of the biomechanical properties of the cornea [4].

An air blast applied to the center of the cornea induces not only the corneal apex applanation and indentation (deflection), but also the whole eye displacement and rotation [5], as well as vibrations of deformed cornea [6–11]. It can be observed on the obtained image sequences that the right and left sides of the cornea deformed by the air puff do not displace symmetrically inward or outward of the eye center, but manifest distinct vibrations.

Vibrations of the cornea observed on video sequences from the Corvis ST tonometer can be classified into two groups:

The mentioned types of corneal vibrations are schematically shown in Fig. 1. They can also be observed in Visualization 1.

 

Fig. 1. Symmetric and asymmetric corneal vibrations; white arrows: direction of vibration at a given moment; gray, unfilled arrows: vibration direction in the next moment.

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The presence of corneal vibrations under the air blast was reported much earlier before the introduction of the Corvis ST on the market. Several attempts were made to use them in the measurement of IOP [12] or to determine the mechanical properties of the cornea [13]. The corneal vibrations from the image sequences captured by means of Corvis ST were described in [6–10]. The authors drew attention to the relationship between the observed vibrations and biomechanical properties of the eye. In these studies, the authors described corneal vibrations in terms of amplitudes or frequencies. However, neither the repeatability for a given eye nor the factors that may affect vibration properties were analyzed. Koprowski and Ambrosio presented an algorithm that involved corneal vibrations to classify the sequences of images from Corvis ST made for healthy and keratoconic eyes [11]. However, the authors used central corneal thickness, IOP, and length of the first applanation as main parameters to categorize the measured eyes to healthy or keratoconus groups.

The vibrations of the cornea caused by the air blast may result from both IOP [12,14,15] and mechanical properties of the cornea [13]. The ranges of corneal vibration frequencies, as well as their repeatability for individual eyes and different groups of eyes, are interesting not only from the biomechanical point of view, but also because of possible applications in medical diagnostics. The aim of this study was to determine characteristic frequencies of such vibrations and to investigate their relations with CVS-IOP and bIOP.

2. METHODS

A group of healthy volunteers consisted of 10 women and three men aged 22–31 years, and one man 65 years old. After acquainting the participants with the purpose of the test and related procedures, informed consent was obtained from all of them. The study was performed according to the Declaration of Helsinki. The participants were examined using the non-contact Corvis ST tonometer (Type 72100, Oculus Optikgerate GmbH). A complete series of eight measurements under constant conditions for one subject’s eye lasted about 30 min. A total number of 112 image sequences of deforming corneas, as well as pressure and deformation parameters were exported from the device using a new version of Corvis ST software (ver. 6.08r22, [16]).

In order to detect shapes of corneal profiles during deformations, image sequences were processed using MATLAB software, assuming that one pixel corresponds to the size $0.016\times 0.016\mathrm{mm}$ [17].

Afterwards, IC—temporal indentations of the corneal profile—were extracted as a difference between corneal profile displacements and the whole eye retraction [5].

Horizontal vibrations of corneal profiles observed in image sequences (images $i=1\text{:}140$) were determined as the difference between the corneal indentation $I{C}_i(k)$ and its smoothing $IC{g}_i(k)$, calculated in columns $k(k=1\text{:}576)$:

Figure 2 shows contour plots of $I{C}_i(k)$ and $IC{g}_i(k)$ and their difference being the contour map of the corneal vibrations. Areas of the most intensive vibrations appear at a distance of about 1–3 mm on the right and left sides of the corneal apex, in the area of two local profile peaks. These areas are marked on the vibration map as vertical dotted lines.

 

Fig. 2. Corneal indentation map (a) and smoothed corneal indentation map (b), whose difference presents the corneal vibration map (c). Dotted vertical lines in (c) indicate the positions of the corneal points located 1.92 mm left and right of the apex.

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A chart presenting exemplary vibrations of the corneal profile points located 1.92 mm (120 pixels) on the left and right sides of the apex is shown in Fig. 3(a). The difference in vibration of these points is shown in Fig. 3(b). Figure 3(c) presents frequency distribution of vibrations presented in Fig. 3(b), calculated by use of the fast Fourier transform (FFT). In this measurement, the points on the right and left sides of the cornea vibrate in counterphase for almost the entire time of indentation. Such behavior was observed in most of the captured sequences.

 

Fig. 3. (a) Exemplary vibrations of the corneal profile points located 1.92 mm (120 pixels) on the left and right sides of the apex; (b) mutual vibrations of these two points, determined as a difference of their vibrations; (c) frequency distribution of mutual vibrations presented in (b).

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The presented periodogram shows a clear, single maximum, indicating the characteristic frequency of the signal: frequency with maximum amplitude (MAF). However, analysis of the vibrations described above have shown that the frequency distributions obtained for some eyes manifested several local maxima with similar amplitudes, as presented in Fig. 4. This effect is likely caused by the non-stationarity of vibrations, appearing when the corneal vibration frequency varies during registration. In such cases, MAF cannot be treated as a characteristic frequency of examined vibrations. To select the characteristic frequency, which describes more accurately the characteristic vibrations of corneal peaks, another parameter: center of mass (CM50) was proposed. CM50 indicates the center of mass of the area under periodogram distribution according to the formula

 

Fig. 4. Mutual vibrations of the corneal profile points located 1.92 mm (120 pixels) on the left and right sides of the apex (a), and its frequency distribution with characteristic vibration frequencies marked (b). MAF, maximum amplitude frequency; CM50, frequency related to the “mass” center of the shaded area.

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A. Statistical Analysis

The average values and standard deviations of corneal vibration frequencies (MAF and CM50) were calculated for each subject. Then, Pearson’s correlation coefficients $r$ were used to describe correlations between average vibration frequencies and IOP [18]. Frequencies MAF and CM50 were also analyzed in terms of repeatability, using: ${\zeta }_w$, within subject standard deviation; CV, coefficient of variation; CR, repeatability coefficient; and ICC, intraclass correlation coefficient. CV is calculated as $\frac{{\zeta }_w}{|\mathrm{mean}|}$, and the lower values of CV indicate the higher repeatability of characteristic frequency estimation. CR is calculated as $1.96\sqrt{2{\zeta }_w^2}$, and for 95% of pairs of observations for the same subject, their difference is expected to be less than CR [19]. Pearson’s $r$ and ICC were considered statistically significant if their $P\text{-}\text{values}<0.05$.

3. RESULTS

Exemplary indentations of the corneal apex, as well as indentations of points located 120 pixels (1.92 mm) towards the nasal and temporal sides of the cornea are shown in Fig. 5.

 

Fig. 5. Indentations of the corneal apex and two corneal points located 1.92 mm to the right and to the left from the apex presented for two exemplary measurements.

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In the case presented in Fig. 5(a), points located outside the apex oscillate in counterphase for most of the indentation time, which indicates the presence of asymmetric vibrations. Moreover, the smooth shape of the apex indentation curve indicates that symmetrical vibrations at high frequencies were not present. Such behavior of the cornea was most frequently observed.

Another example is shown in Fig. 5(b). In this case, the indentation curves present clear vibrating movement of the apex and the relatively smoother indentations of the corneal points located outside the apex. In this situation, symmetrical vibrations of the cornea prevail over asymmetric ones. Such behavior of the cornea was characteristic and repeatable for two out of 14 measured eyes. In the measurements of the remaining eyes, symmetrical corneal vibrations were sporadically observed. Therefore, only asymmetrical vibrations were analyzed in terms of quantity.

As presented in Fig. 5, the whole process of deforming the central part of the cornea and its return to original shape lasts about 15–16 ms. The indentation of points located about 2 mm from the corneal apex takes less time, but not less than 10 ms. Therefore, vibrations with frequencies below 60 Hz in the apex and 100 Hz outside the apex result from the duration of the air pulse.

The summary statistics of frequencies MAF and CM50 are included in Table 1. CM50 is characterized by a smaller range of values in subsequent measurements of individual eyes, and therefore exhibits greater repeatability than MAF. It also presents a smaller spread of the achieved values.

Table 1. Summary Statistics of Frequencies MAF and CM50^a

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Correlations between the means of MAF and CM50 (calculated for each particular subject) and the parameters obtained directly from the Corvis ST software are presented in Table 2. For each presented parameter from the tonometer software, the correlation with CM50 was higher than correlations with MAF. Parameters from tonometer software not included in this table have low and insignificant correlations with MAF and CM50.

Table 2. Pearson Correlation Coefficients $r$ for Parameters Given by Corvis ST Software and Average Vibration Frequencies MAF and CM50^a

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Reproducibility of the CM50 for individual eyes was mediocre (Table 1), what results from significant dispersions of obtained values for an individual subject. However, some correlations of the average frequency CM50 with average parameters from the Corvis ST software for examined subjects turn out to be high. Very high correlations $r\ge 0.90$ were determined between CM50 and: HC deflection area (area between maximum deflected corneal profile and corneal profile in its initial state), HC deflection amplitude (amplitude of the corneal apex indentation at the moment of the highest concavity of the cornea), deflection amplitude max (the highest indentation of the corneal apex reached during measurement), peak distance (distance between the two highest points of the corneal profile at the moment of the highest concavity of the cornea), and bIOP. All the parameters listed above are related to the value of IOP, which confirms its decisive influence on the frequency of vibrations.

Plots for three dependencies—CVS-IOP, CVS-IOP divided by central corneal thickness (CCT), and bIOP versus vibration frequency CM50—are presented in Fig. 6. These graphs clearly show high correlations between dependence of the mean vibration frequencies and differently expressed IOP. A presented CVS-IOP/CCT has higher correlation with the corneal vibration frequency than the unadjusted CVS-IOP. The lowest correlation occurs in Fig. 6(a) for CM50 and non-corrected IOP.

 

Fig. 6. Correlation plots for CM50 versus CVS-IOP (a), bIOP (b), and CVS-IOP divided by CCT (c). Subject means and one standard deviation above and below the means of data for individual subjects are represented by dots and whiskers, respectively. $r$, Pearson correlation coefficient for subject means.

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3D plot of dependencies between CCT, unadjusted CVS-IOP, and vibration frequency CM50 are presented in Fig. 7 and Visualization 2. This graph shows that presented points can be well approximated by the plane. The graph’s orientation was selected to show the best fitted plane, perpendicular to the paper page. The equation of the best fitted plane is as follows:

 

Fig. 7. 3D plot of dependencies between corneal thickness CCT, unadjusted CVS-IOP, and vibration frequency CM50 (see Visualization 2). Subject means and one standard deviation above and below the means of data for individual subjects are represented by dots and whiskers, respectively.

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4. CONCLUSION

In this paper, the rapid vibrations appearing on the surface of the cornea deformed during air-puff tonometry measurement were analyzed. Vibrations of the observed horizontal corneal section were divided into two types—symmetrical and asymmetrical. To our knowledge, this is the first time that such a division of corneal vibrations has been proposed.

The explanation of prevalence of one type of corneal vibrations over the other (asymmetric over symmetric) is neither obvious nor trivial at all. It is likely that the phenomenon of propagation of the corneal surface waving, induced during the air-puff deformation, can play a role. It can be supposed that the propagation of corneal waving may be subject to both asymmetry of the alignment of cornea in relation to the air pump of the device (perhaps even small misalignment may induce the asymmetric vibrations) and/or asymmetry of the corneal geometry (astigmatism with and against the rule) [20].

Due to the attachment of the muscles, biomechanical properties of the eye structures beyond the cornea also manifest differences between the vertical and horizontal cross sections, which can also affect propagation waving in the anterior segment of the eye. Examination of these vertically horizontal asymmetries of the anterior segment of the eye can be very interesting.

The asymmetrical corneal vibrations were observed in each single measurement. Average values of MAF and CM50 for all subjects and measurements are similar to each other and amount, respectively, 506 Hz and 515 Hz. A similar mean frequency value, $515\pm 99\mathrm{Hz}$, was obtained for horizontal vibrations of the most deformed area of the corneal profile in 10 image sequences captured by Corvis ST and analyzed by Koprowski and Wilczynski [9]. The obtained values also agree with the vibrational modes obtained in the numerical model of the eye presented by Aloy et al. [21].

For each parameter from the Corvis ST software, correlations with CM50 were higher than with MAF. This is due to the higher repeatability of CM50 for a given eye and may be an additional argument for choosing this parameter for describing corneal vibration frequencies. Interestingly, the correlation of average values CM50 and bIOP is very high ($r=0.91$), clearly higher than with unadjusted CVS-IOP ($r=0.80$). Average values CM50 also showed very high correlations ($r\ge 0.90$) among others with averages of deflection amplitude and peak distance (Table 2). All these values are related to IOP, which confirms its significant impact on the frequency of corneal vibrations.

Figure 6 shows three correlation plots between CM50 and different values related directly to CVS-IOP. According to the description given by the Corvis ST developer (and based on [4]), bIOP is calculated using an algorithm that takes into account CVS-IOP, patient’s age, and corneal thickness. The large similarity between Figs. 6(b) and 6(c) confirms that the main factor modifying bIOP in relation to CVS-IOP is corneal thickness.

The correlation coefficient for averages of only CM50 and CCT is low ($r=0.19$), and for this size of a group of subjects (14 people), statistically insignificant ($P\text{-}\text{value}=0.51$). It can be claimed that CM50—the frequency of corneal vibrations during deformation—is related to CVS-IOP and CCT in a specific way. A 3D plot of these three measured variables forms a plane with the high determination coefficient $r^2=0.78$ (see Fig. 7 and Visualization 2). It can be considered as an analogue of string vibrations, the frequency of which depends on two independent parameters—stress of the string and its length.

According to the best knowledge of the authors, the presented paper describes the first study demonstrating the high correlations of both CVS-IOP and CCT with the corneal vibration frequencies obtained from analysis of corneal deflection sequences captured by means of Corvis ST device.

Correlation coefficients of averages CM50 with the corneal stiffness parameters DA ratio max (2 mm) and SP-A1 were much lower ($r=-0.60$ and $r=0.55$, respectively). It seems that other biomechanical properties of corneas were not as important as CCT and CVS-IOP for these subjects. It was probably caused by the similar age of people from the measured group.

The symmetrical vibrations of the horizontal corneal profile were observed less frequently than asymmetric ones. For eyes in which the symmetrical vibrations of the cornea appeared in one or two measurements in eight, such vibrations may be treated as a result of environment factors or the impact of the nozzle setting in front of the eye in a given measurement. However, in some eyes, corneas manifested symmetrical vibrations in most of the measurements, which was probably caused by either anatomical or physiological factors.

As shown in Figs. 3 and 4, both the vibration frequency and amplitude of the corneal profile points undergo large changes during deformations (gradually increase until the maximum indentation, and then decrease). It does not resemble the vibrations presented in the numerical model presented by Han et al. [8], in which the vibrations of the cornea were modeled with a finite difference numerical approach. They obtained both vibration frequency and amplitude almost constant during deformation.

More detailed analysis of the corneal vibrations during air-puff deformations could be useful for better understanding of biomechanical behavior of the cornea during deformation, development of corneal modeling, and possible applications in future ophthalmic diagnosis.