An Applanation Resonator Sensor for Measuring Intraocular Pressure Using Combined Continuous Force and Area Measurement

Abstract

purpose. For diagnostic purposes and for follow-up after treatment, it is important to have simple and reliable methods for measuring intraocular pressure (IOP). The purpose of this study was to develop a new applanation method for IOP measurement that uses combined continuous force and area measurement and to develop and evaluate an applanation resonator sensor (ARS) tonometer based on that method.

methods. The tonometer was developed and evaluated in an in vitro porcine eye model, in which enucleated eyes were pressurized with a saline column. A model assuming that the applanation principle is valid over a certain interval of contact area was proposed. Continuous contact area was measured with a resonator sensor device, and contact force was measured with a force transducer, both mounted together in one probe. Reference IOP was measured in the vitreous chamber (IOP_VC) with a standard fluid pressure transducer.

results. An optimization algorithm determined the applanation interval that was optimal for calculating IOP_ARS. The corresponding time interval was 30 ± 3 to 77 ± 4 ms (mean ± SD, n = 418) after initial contact. The proposed model showed a degree of explanation of R ^{2 [supi]} = 0.991 (n = 410, six eyes), corresponding to a correlation of r = 0.995 (n = 410) between IOP_ARS and IOP_VC. The within-eyes precision (i.e., 95% confidence interval for the residuals between IOP_ARS and IOP_VC) was ± 1.8 mm Hg (n = 410, six eyes).

conclusions. In this study, the ARS method for measuring IOP was evaluated in an in vitro porcine eye model and showed high precision. The ARS method is, to the authors’ knowledge, the first to combine simultaneous, continuous sampling of both parameters included in the applanation principle: force and area. Consequently, there is a potential for reducing errors in clinical IOP tonometry.

Glaucoma is an eye disease that may be defined as a progressive optic neuropathy with characteristic changes of the optic nerve head and the visual field. The etiology is not completely understood, but one of the major risk factors is elevated intraocular pressure (IOP). ¹ All treatment so far has been intended to reduce IOP. Therefore, it is important for diagnostic purposes and for follow-up after treatment to have simple and reliable methods for measuring IOP, because tonometry is a standard procedure in all examinations of the eye. 

The applanation principle states that when a flat surface is pressed against a fluid-filled sphere with a flexible membrane, the internal pressure can be measured by the force exerted on the plane and the area of contact. ² For IOP measurement the applanation principle is generally described through the Imbert-Fick law ² by contact force (F _C) and area of contact (A) (equation 1) .  

\[\mathrm{IOP}\ {=}\ \frac{F_{\mathrm{C}}}{A}.\]

 

One of the most commonly used methods to measure IOP is Goldmann applanation tonometry (GAT). It is based on constant area applanation and operator adjustment of the force to reach that area. ² Thus, IOP is calculated from a one-point reading of area and force. 

We recently introduced a new sensor for tonometry, the applanation resonator sensor (ARS), and evaluated it in an in vitro porcine eye model. ³ The sensor is based on a piezoelectric element that is set to vibrate at its resonance frequency. It has been shown ³ that when the sensor is applied against the cornea, the frequency shift is proportional to the contact area. By using a constant force and frequency shift (i.e., contact area) measure, IOP can be calculated according to equation 1 . The study ³ demonstrated that the resonator sensor principle can be used to measure IOP in a physical model. As with GAT, the calculation of IOP is based on a one-point reading of area and force. That study, ³ as well as a later study, ⁴ showed an IOP measurement variation related to differences between eyes. In the latter study, normalization of frequency shift was shown to reduce between-eye variability, and a further development toward a method that incorporates a self-calibration, similar to the normalization, was suggested. 

The hypothesis in the present study was that a method based on a continuos force and ARS area recording during the initial applanating phase and an analysis of the differential relationship of these parameters would be less dependent on the absolute frequency shift, and therefore have the potential to reduce between-eye variability. The purposes of the present study were to develop further and evaluate the between-eye difference according to this new method for IOP measurement. The evaluation was performed in an in vitro porcine eye model. 

Material and Methods

ARS Probe

The probe for IOP_ARS measurement was based on a resonator sensor element for measuring the contact area between the sensor and cornea, and a force transducer for measuring the contact force (Fig. 1) . The area measuring device consisted of a rod-shaped (23 × 5 × 1 mm) piezoelectric element made of lead zirconate titanate (PZT). A PZT pick-up (5 × 5 × 1 mm) was glued on the center of the PZT element. A feedback circuit processed the signal from the pick-up and powered the PZT element to sustain the oscillations in the resonance frequency. A plastic piece (Duro; E Wiberger, Molndal, Sweden), used for contact against the cornea, was glued onto one end of the PZT element. The contact surface of the piece is convex, with a radius of curvature of ρ = 7 mm. The resonance frequency depends on the geometry and material properties of the PZT element and contact piece, the suspension of the element, and the frequency characteristics of the feedback circuit. ⁵ When the sensor is brought in contact with the cornea, the acoustic impedance of the cornea mechanically loads the sensor and a new oscillating system is formed with a new resonance frequency. Because the degree of load on the resonance system depends on the contact area, the resonance frequency of the system is related to the contact area between sensor and cornea. ³ The frequency of oscillation can be used for area measurement. The principle and theory of the resonator sensor system have been described earlier. ^{3 5 6} The force transducer (PS-05 kDa; Kyowa, Tokyo, Japan) and the resonator element were mounted in a sensor module consisting of a cylindrical aluminum case (Fig. 1) . 

Earlier reports on similar resonator probes have shown that the frequency shift has a linear relationship with area of contact. ^{3 5} In designing a model for the current ARS probe we therefore assumed that frequency (f) changes proportionally to contact area (A)  

\[A\ {=}\ \frac{1}{C_{\mathrm{ARS}}}\ (f{-}f_{0})\]

with f ₀ as the resonance frequency of unloaded sensor and C _ARS being a sensor-specific proportionality constant. Equations 1 and 2 yield  

\[F_{\mathrm{C}}\ {=}\ \frac{1}{C_{\mathrm{ARS}}}\ (f\ {-}\ f_{0})\ \mathrm{IOP}.\]

Differentiation leads to  

\[\mathrm{IOP}\ {=}\ C_{\mathrm{ARS}}\ \frac{dF_{\mathrm{C}}}{df}.\]

 

This was the basis for our model. To handle offset forces not related to IOP, a constant term was added to the model. The final model is described with general coefficients β _i  

\[\mathrm{IOP}\ {=}\ {\beta}_{1}\ \frac{dF_{\mathrm{C}}}{df}{+}{\beta}_{0}.\]

 

To include differences between eyes, the model was extended to a general linear model with eye number as a categorical factor (GLM analysis) when it was applied to data from six eyes grouped together. 

Higher order polynomial models were also tested. The models were defined as  

\[\mathrm{IOP}_{\mathrm{VC}}\ {=}\ {\beta}_{0}\ {+}\ {\beta}_{1}\ \frac{dF_{\mathrm{C}}}{df}{+}{\beta}_{2}\left(\frac{dF_{\mathrm{C}}}{df}\right)^{2}{+}{\beta}_{3}\left(\frac{dF_{\mathrm{C}}}{df}\right)^{3}.\]

These models were also tested on six eyes grouped together, and eye number was used as a categorical factor. 

Apparatus

The ARS probe was mounted on a servomotor-controlled lever and applied to the cornea of the eye with a controlled velocity of 6.4 mm/s. A frequency-to-DC voltage converter was used to measure the frequency. Contact force between the cornea and the ARS was measured with the force transducer. IOP in the vitreous chamber (IOP_VC) during applanation was measured with a standard monitoring kit (BD Biosciences, Lincoln Park, NJ). The vertical position of the sensor was measured with an inductive position transducer ⁶ placed at the lever. Indentation (L) was defined as the position change after contact (Fig. 2) . Maximum indentation was set so that the force at full indentation was between 40 and 50 mN at an IOP of 10 mm Hg, which resulted in a maximum indentation ranging from 1.20 to 1.63 mm. Frequency, pressure, force, and position data were sampled with a data acquisition card (DAQCard 1200; National Instrument, Inc., Austin, TX). The sampling rate was set to 1000 Hz, and recording time was set to 15 seconds. 

Animals

In this study seven cadaveric eyes from approximately 6-month-old Landrace pigs were enucleated immediately after the pigs were put to death at the abattoir (SQM, Skellefteå, Sweden). The eyes were kept in saline, and measurements were performed within 12 hours after enucleation. ⁷ Six eyes were used for evaluation of the sensor’s ability to measure IOP. One eye was used for evaluation of off-center positioning dependence of the sensor. After IOP measurement was completed, the equatorial diameter of the eyeball (∅_eye) and corneal diameter, (∅_cornea) were measured with a vernier caliper. The cornea was then removed from the eye and the corneal thickness, T _cornea, was measured with a micrometer. Anatomic measurements are found in Table 1 . 

Protocol

The in vitro method has been described in detail by Eklund et al. ⁵ Briefly, the eyes were mounted firmly in a Petri dish with agar solution (15 g/L) that covered the eye approximately 50%. A winged, thin-walled cannula of dimensions 0.8 × 19 mm (Terumo Corp., Tokyo, Japan) was introduced through the side of the eyeball into approximately the middle of the vitreous chamber. The interface between the cannula and the tissue was sealed with cyanoacrylate adhesive (True Bond; Pro-Gruppen AB, Stockholm, Sweden) to avoid leakage. ⁸ The cannula was connected to a saline column consisting of PVC tubing, a three-way stopcock, and, at the distal end, a partially saline-filled syringe open to air. The syringe was movable and mounted on a stable stand. The eye was pressurized for 10 seconds by opening it to the saline column. The pressure level was calculated from the measured height of the saline column. Just before measurement the stopcock was closed to create a closed system that approximated the normal state of the eye. ⁸ To account for the continuous pressure decline after the stopcock was closed, IOP_VC was defined as the mean monitored pressure in the interval 1.0 to 0.5 seconds before ARS contact. To avoid drying of the cornea, the eye was moistened before every pressurization with room temperature saline. To simulate blinking, the saline was applied to the eye with one sweep of a very soft goat hair brush (Kreatima 922; Schormdanner Pinsel, Nürnberg, Germany). 

IOP Dependence.

In each eye, IOP was increased in steps of 5 mm Hg, starting from 10 mm Hg and reaching 40 mm Hg. The measurements were repeated 10 times on each IOP level. 

Off-Center Position Dependence.

IOP was set to the 20-mm Hg level. Applying gauge blocks between the Petri dish and a firmly fixed border varied the center of contact between the contact surface of the sensor element and the cornea. The center position and four 1-mm off-center positions around the center position were evaluated. The measurements were repeated 10 times at each contact position. 

Frequency Interval Optimization Algorithm

The unloaded frequency (f ₀) for each measurement was determined as the mean frequency in the interval 1.0 to 0.5 seconds before contact. The force-frequency and position-frequency analyses focuses on an interval in the initial phase of applanation (Fig. 2) . The frequency interval is described by a starting offset f _start = f ₀ −f ₁ and an ending offset f _end = f ₀ − f ₂ (Figs. 2 3) . The interval length was defined as f _interval = f ₁ − f ₂. To optimize the frequency interval, an algorithm stepped through all combinations of start points and interval lengths (50 < f _start < 250 Hz; 50 < f _interval < 500 Hz), increasing with steps of 10 Hz. Using linear regression, we applied the model of equation 5 to the force and frequency data of the chosen frequency interval from all measurements. The precision of the model according to equation 5 was interpreted from the SD of the residuals. Optimal frequency interval was determined as the frequency interval that produced the lowest SD. 

Statistics

Data are expressed as the mean ± SD. The Kolmogorov-Smirnov statistic was used to test the hypothesis that the data were normally distributed. A one-way ANOVA with Bonferroni post hoc test was used to test for differences between groups. The Pearson correlation coefficient was used for correlation analysis. P < 0.05 was considered statistically significant. Within each eye, the replicate observations on each IOP level were based on independent measurements of both IOP_VC and dF _C/df and were regarded as independent measurements in the analysis. Terms used in the equations are listed and defined in ^{Appendix 1}. 

Results

Typical Measurement

Results from a typical measurement are presented in Figure 2 . For the estimation of IOP, a measurement interval in the initial phase when the sensor surface applanates, the cornea was used. The interval was determined by the selected frequencies f ₁ = f ₀ −f _start and f ₂ = f ₀ −f _end. The force, F _C, was plotted against the frequency and dF _C/df was calculated as the slope of a linear regression in the selected interval (Fig. 3) . 

Frequency Interval

From the optimization algorithm, the best frequency interval was determined to be between f _start = 150 and f _end = 470 Hz (Fig. 2) . This frequency interval was then used in the analysis throughout this study. The corresponding indentations for the frequency interval were between L ₁ = 0.19 ± 0.02 mm (n = 418) and L ₂ = 0.49 ± 0.02 mm (n = 418; Fig. 2 ). The number of measurements (n = 418) originated from 6 eyes × 7 IOP levels × 10 repetitions. Two measurements were omitted during experiments because of operator error. The corresponding time interval after initial contact was t ₁ = 30 ± 3 (n = 418) to t ₂ = 77 ± 4 ms (n = 418; Fig. 2 ). 

Indentation versus Frequency

Analysis of the residuals from a general linear model for frequency, with indentation as a covariant and IOP level and eye number as factors, using all measurement series (n = 418), put together, showed that the residuals were normally distributed (P > 0.20, n = 19,959) around zero. The number of samples (n = 19,959) originated from 418 measurement series, with interval length varying between 36 and 57 samples in each measurement series. This means that the relationship between frequency and indentation was linear in the frequency shift interval from 150 to 470 Hz below the unloaded frequency. The linear relationship in that interval can be described by  

\[f{-}f_{\mathrm{start}}\ {=}\ {\alpha}_{fL}\ (L{-}L_{\mathrm{start}})\]

 

The proportionality constant (α _fL ) was determined for each measurement series. There was a significant but low correlation between α _fL and IOP_VC (r = 0.47, n = 418). The linear relationship was described by α _fL = −2.30 IOP + 1152, resulting in a 6% variation over the 10- to 40-mm Hg interval. SD of the residuals was 44 Hz/mm (n = 418). 

Using equation A7 ^{(Appendix 2)}, the measured indentation interval (L = 0.19–0.49 mm) and radius of corneal curvature approximated with human corneal curvature (R = 7.5 mm) ⁹ dA/dL was nearly constant, with a range from 22.77 to 22.80 mm²/mm. 

Calculated from the estimated indentation interval and equation A6 ^{(Appendix 2)} the area interval used was approximately 4.3 to 11.2 mm². From these convex areas, the effective area normal to the axial direction can be calculated to 4.3 to 11.0 mm². 

IOP versus dF/df

The degree of explanation for the general linear model between dF _C/df and IOP_VC, based on the model of equation 5 and eye number as a categorical factor (GLM analysis), was R ² = 0.987 (n = 418, Table 2 ). Residual analysis from the linear fit showed a standard deviation of 1.14 mm Hg (n = 418) and that the deviations were not normally distributed (P < 0.001, n = 418). Exclusion of eight observations and a subsequent GLM analysis on the reduced data set yielded R ² = 0.991 (n = 410, Table 2 ) and showed that the residuals were normally distributed (P = 0.16, n = 410). The reduced set (n = 410) was used in the subsequent analysis. The residuals of the excluded observations all exceeded 3 SD of the normally distributed residuals of the reduced data set. ¹⁰ The mean of the residuals was 0.00 mm Hg (n = 410), with a standard deviation of 0.94 mm Hg (n = 410), indicating that the 95% confidence interval for the residuals between IOP_ARS and IOP_VC was ± 1.8 mm Hg. The GLM analysis also showed that the systematic contributions from the eyes were −0.71, −0.21, 0.61, −0.14, 0.34, and 0.11 mm Hg (n = 410) for the six eyes. Predicted IOP_ARS data are plotted against IOP_VC in Figure 4 . Table 3 shows that there was no clear trend regarding the standard deviation of the residuals in relation to IOP. 

Higher order polynomial models increased the R ² and reduced the residual standard deviation moderately (Table 2) . The analysis of individual eyes (Table 4) showed that the variation in slope (β₁) was less than ±6%, compared with the GLM analysis (Table 2) . The 95% confidence intervals for the individual slopes did not all overlap (Table 4) , indicating differences between eyes. 

Corneal Thickness

To investigate the relationship between corneal thickness and dF _C/df, a linear regression model including IOP_VC and corneal thickness was used:  

\[\frac{dF_{\mathrm{C}}}{df}{=}{\beta}_{\mathrm{Const}}{+}{\beta}_{\mathrm{VC}}IOP_{\mathrm{VC}}{+}{\beta}_{\mathrm{CT}}T_{\mathrm{cornea}}\]

where β represents the coefficients for each variable. The regression showed that all coefficients were significantly separated from zero (β_Const = −0.0161, β_VC = −0.00351, and β_CT = 0.0312, n = 410, six eyes) which means that dF _C/df was dependent on both IOP_VC and T _cornea. 

The variation in corneal thickness, calculated from its effect on dF _C/df (equation 8) , translates to a contribution of 0.75 mm Hg to IOP_ARS for a 10% change in corneal thickness. 

Off-Center Alignment

The IOP during the off-center evaluation was IOP_VC = 20.4 ± 0.1 mm Hg (n = 50). IOP_ARS, calculated from measured dF _C/df and the linear model with coefficient according to Table 2 , second row, varied between 20.7 and 24.6 mm Hg (mean, n = 10). The coordinates (X,Y) describe the off-center position of the eye. IOP_ARS for two off-center positions, IOP_ARS (−1,0) = 21.3 (n = 10) and IOP_ARS (0,1) = 21.3 (n = 10), corresponded well to the center position IOP_ARS (0,0) = 20.7 (n = 10). Measurements from two positions, IOP_ARS (1,0) = 23.8 (n = 10) and IOP_ARS (0,−1) = 24.6 (n = 10) deviated significantly from the center measurement (ANOVA post hoc, n = 50). 

Discussion

In this study a new method for IOP measurement, based on a combined continuous force and area measurement during the initial phase of contact between sensor and cornea, was proposed. The ARS probe was further developed according to that method and evaluated in an in vitro porcine eye model. The new method effectively reduced the problem caused by variation due to intereye differences as well as the problem of decreased precision with increased pressure, both of which occurred in earlier versions of the ARS probe for IOP measurement. ^{3 4}  

The model represented by equation 5 assumes that the change in frequency linearly corresponds to a change in area, as described in equation 2 . To evaluate this assumption the indentation versus frequency was analyzed. It showed that the frequency change was linearly related to indentation within the frequency interval of (f ₀ – 150) to (f ₀ – 470) Hz. Furthermore, because the relationship between change in contact area and change in indentation, dA/dL, was shown to be nearly constant, we can conclude that, within the chosen interval, a change in indentation corresponds to a linear change in area. This supports the hypothesis that a change in contact area, within this interval, corresponds linearly to a change in resonance frequency. 

Previous evaluations ^{3 4} of ARS probes for IOP measurement have been based on a constant-force method and area measurement with the ARS probe. Those studies showed a high reproducibility within each eye, but clear differences between eyes. The Imbert-Fick law assumes that the cornea is infinitely thin, perfectly elastic, and perfectly flexible and that the only force acting against it is the pressure of the applanated surface. ⁷ None of these assumptions is true. The cornea is not a membrane without thickness, and it offers resistance to indentation, varying with its curvature and thickness and the presence or absence of corneal edema. The surface of the cornea is covered with a liquid film. Therefore, during the applanation of the cornea, capillary attraction or repulsion forces between the contact piece and the cornea interfere with the measurement. ² The force term depends on the width of the ring (i.e., the amount of fluid). ² Thus, there are forces unrelated to IOP that are present, and the magnitude of the forces is dependent on properties that differ from eye to eye and even from measurement to measurement. In Eklund et al. ³ it was shown that the correlation between reference IOP and IOP_ARS was r = 0.92 (n = 360, six eyes together) for the constant-force ARS method. By the same calculation, the present study with a model based on the continuous force and area measurement showed a corresponding correlation of r = 0.99 (n = 410, six eyes together). The eye-dependent variation in proportionality coefficient (β₁) was –25% to +16% (calculated from Table 1 in Ref. ³ ) for the constant-force study and less than ±6% in the present study. This shows that the intereye variation was much less with the new method. One explanation for the improvement is that the differential force–area method is not sensitive to constant-force terms, because it analyzes the change in force over an area interval and is therefore unaffected by any constant terms. 

Our anatomic measurements (Table 1) of the porcine eyeball and cornea diameter corresponds approximately to the measurements of the human eye (∅_eye = 24 mm and ∅_cornea = 12 mm). ¹¹ This supports the use of an in vitro porcine eye model for a first evaluation of new tonometers. However, factors such as scleral rigidity, ocular curvature, and pressure–volume relationships, all vary between species, and tonometers must be calibrated for each species. ¹²  

Corneal thickness has been shown in numerous studies, reviewed by Doughty and Zaman, ¹³ to affect IOP measurement performed with the applanation method. The corneal thickness of the eyes of the Landrace pigs used in this study varied between 0.80 and 0.90 mm, thicker than the cornea of a normal human eye (T _{human =} 0.534 mm). ¹³ Ideally, for an applanation method, the cornea should be as thin as possible, indicating that the accuracy for an in vitro human eye should be better than the results of this study. Because of differences in corneal thickness, both the frequency interval and the coefficients of the model have to be optimized and reestimated for use in humans. Although only a small number of eyes were studied, there was a significant dependence of the ARS method on cornea thickness. This dependence, however, was less than the sensitivity to corneal thickness shown with GAT in healthy eyes (1.1 ± 0.6 mm Hg for a 10% change in corneal thickness). ¹³ Further studies are needed to determine fully the relationship between corneal thickness and accuracy of IOP_ARS. 

The maximum systematic difference between eyes was approximately 1 mm Hg, indicating that the intereye difference was at an acceptable level. The mean deviation of the residuals was zero, indicating a perfect accuracy within eyes. This, of course, is natural because the ARS is calibrated against the IOP_VC. We therefore cannot draw conclusions about overall accuracy in this type of study. Because the mean deviation is zero, the within-eye precision is described by the variation of the residuals. The overall SD of the residuals in this study was 0.93 mm Hg. This is in parity with the results of Schmidt ¹⁴ for GAT on four fresh enucleated human eyes (residual SD = 0.85 mm Hg, n = 20, calculated from Table 2 in Ref. ¹⁴ ). The quadratic and cubic models did not produce any substantial improvement in the precision of the method. For simplicity and robustness, we therefore choose and recommend the linear model (equation 5) . 

The problem with asymptotic behavior at higher pressure and corresponding loss of precision, associated with the ARS constant-force method, ³ was effectively reduced with the current method, which showed an approximately similar precision at all pressure levels (Table 3) . The main difficulty with the previous ARS method was that it used constant force and measured the resultant contact area. The contact area was therefore inversely proportional to IOP (equation 1) which resulted in small areas of contact and a decrease in resolution at high pressures. ³ This problem was solved with the new method, which uses the same contact area interval, independent of IOP. Similar to GAT, this leads to a measured force that is directly proportional to the IOP at all pressure levels. In addition, the area interval used in the new ARS method (4.3–11.0 mm²) is close to the interval for which Goldmann ² (4.9–12.6 mm², calculated from contact diameters of 2.5–4 mm) suggested that the Imbert-Fick law is valid. 

For indentation methods such as Schiötz, a relatively large volume is displaced, resulting in a pressure increase during measurement that depends on the scleral rigidity of the eye. ¹⁵ With the GAT, the standard contact area guarantees that the displaced volume is small, and the pressure in the eye will be elevated only slightly. ¹⁵ For indentation–applanation methods with constant area and guard ring (Tono-Pen; Mentor, Norwell, MA) there is no control of the indentation and volume displacement. The area measurement of the ARS method and choice of area interval for analysis based on that measurement ensures that the indentation, as in the Goldmann method, is standardized (L ₂ ≈ 0.49 mm). Thus, the scleral rigidity-related IOP-increase during measurement is controlled and should be approximately the same for all measurements. 

In a previous study, ⁴ off-center alignment was acknowledged as a source of error with the ARS. The study with an ARS mounted in a biomicroscope indicated that the maximum off-center alignment was approximately 1 mm. It also showed that this sensitivity could be reduced with a convex contact surface. The results of the present study showed that a 1-mm off-center alignment could result in a 4-mm Hg overestimation of the IOP_ARS at the 20-mm Hg level. Although the current tip was smaller and may be easier to apply at the center, this has to be taken into consideration in the future development of the ARS system. 

Taking into account that the total time of application against the cornea can be less than 1 second and that the crucial phase used in the analysis is less than 80 ms after the initial contact, clinical application without requirement for anesthetic may be possible. ¹⁶  

Eight measurements were excluded after initial analysis, because they clearly differed from the general distribution of the others. All eight came from two eyes, and five were at the 40-mm Hg level, which ended the protocol on each eye. This indicates damage to the cornea as a possible explanation. Another problem with the in vitro model is that the cornea was moistened with a brush, and this produced uneven wetting. The ARS needs a certain level of moisture on the eye to get good acoustic contact. Insufficient moistening is therefore another possible factor in the excluded measurements. We suspect that in an in vivo setting, normal blinking would moisten the cornea more evenly. 

In summary, this article presents a new methodology for measuring IOP. The applanation principle states that IOP can be calculated as the ratio between force and contact area. Previous tonometry methods allow one or a few readings of force with constant area for estimating the IOP. The ARS method is, to our knowledge, the first to combine continuous sampling of both parameters during the application of the sensor onto the cornea, resulting in a linear curve for the force and area relationship. The IOP is then deduced from the slope of that curve, which is based on many points and is independent of constant forces. The ARS was evaluated in an in vitro porcine eye model and had a short measurement time and precision that was in parity with in vitro results produced with GAT. Consequently, there is a potential for reducing errors in the clinical routine with the use of a device based on this method. For further development toward clinical application, off-center dependence must be considered, and the sensor must be evaluated and recalibrated for in vivo human eyes. These investigations will be performed in a future study in which GAT will be used as a reference for the calibration. 

Appendix 1

Appendix 1

Nomenclature

ρ, radius of curvature for contact surface 

β, coefficients in polynomial models 

∅_cornea, corneal diameter 

∅_eye, equatorial diameter of the eyeball 

A, contact area between sensor and cornea 

α _fL , proportionality constant between frequency shift and indentation 

C _ARS proportionality constant between frequency shift and area 

f, frequency 

f ₀, frequency of unloaded sensor 

f ₁, frequency of starting offset for interval used in analysis 

f ₂, frequency of end offset for interval used in analysis 

f _start , f ₀ – f ₁  

f _end , f ₀ – f ₂  

f _interval, f _start – f _end  

F _C , contact force 

IOP_ARS, IOP measured with the ARS 

IOP_VC, IOP measured in vitreous chamber 

L, indentation of the cornea 

L ₁, indentation at beginning of the interval used in the analysis 

L ₂, indentation at the end of the interval used in the analysis 

r, correlation coefficient 

R, corneal radius of curvature 

R ², degree of explanation for a model 

t, time references to initial contact 

t ₁, starting time for interval used in analysis 

t ₂, end time for interval used in the analysis 

T _cornea, corneal thickness 

T _human, corneal thickness of human eye 

Appendix 2

Appendix 2

If perfect flexibility of the cornea is assumed, the contact surface (A) can be calculated from a the penetration depth (a) ¹⁷ :  

\[A\ {=}\ 2{\pi}{\rho}a\]

where ρ is the radius of curvature for the sensor contact surface. According to Figure A1 , the indentation after initial contact is  

\[L{=}a\ {+}\ b\]

where the distances b and c can be expressed as  

\[b\ {=}\ R\ {-}\ \sqrt{R^{2}\ {-}\ c^{2}}\ and\ c^{2}\ {=}\ {\rho}^{2}\ {-}\ ({\rho}\ {-}\ a)^{2}\]

inserted in equation A2  

\[L\ {=}\ a\ {+}\ R{-}\ \sqrt{R^{2}\ {-}\ 2{\rho}a\ {+}\ a^{2}}.\]

 

Rearranging and taking the square on both sides yields  

\[2a(R\ {-}\ L\ {+}\ {\rho})\ {=}\ 2LR\ {-}\ L^{2}.\]

 

With equation A5 , together with equation A1 , the contact area as a function of indentation can be written as  

\[\mathrm{A}\ {=}\ {\pi}{\rho}\mathrm{L}\ \frac{2R\ {-}\ L}{R\ {+}\ {\rho}\ {-}\ L}.\]

 

Thus, a geometric derivation, assuming perfect flexibility of the cornea, gives an expression of the contact area (A) as a function of indentation (L). The differentiation in equation A6 produces the relationship between change in indentation and change in area  

\[\frac{dA}{dL}\ {=}\ {\pi}{\rho}\ \frac{2R(R\ {+}\ {\rho}\ {-}\ L)\ {+}\ L\ (L\ {-}\ 2{\rho})}{(R\ {+}\ {\rho}\ {-}\ L)^{2}}.\]

 

 Supported by Grant 2001-00870 from the Swedish Agency for Innovation Systems and grants from the Objective 1 Norra Norrland European Union structural fund.

 Submitted for publication November 1, 2002; accepted December 9, 2002.

 Disclosure: A. Eklund, (P); P. Hallberg, None; C. Lindén, None; O.A. Lindahl, (P)

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Experimental setup, with the sensor module consisting of a resonator sensor that measures contact area, mounted on a force sensor that measures the contact force. To avoid radial forces, a flexible washer at the lower end supports the resonator sensor. For contact against the cornea a plastic piece with a convex contact surface was glued to the end of the PZT element. The sensor module was mounted on a servocontrolled lever. Indentation was measured with an inductive transducer attached to the lever.

Figure 2.

 

Typical measurement taken at IOP of 20 mm Hg with indentation (L), force (F _C), and frequency (f) as a function of time. Time (t) is referenced to initial contact with the eye, based on a 10-Hz frequency change from f ₀. The zero point of indentation is set when the force has increased to 1 mN. It is the initial applanation phase that is used in the analysis for estimating the pressure. Vertical dotted lines: the chosen interval at the f ₁ = (f ₀ – 150) and f ₂ = (f ₀ – 470) frequencies. Because the liquid on the eye creates contact with the resonator sensor before the actual contact with the cornea, the frequency change is initiated before force is detected.

Table 1.

 

Diameter of the Eyeball, Corneal Diameter and Corneal Thickness for the 7 Eyes.

Eye	⊘eye	⊘cornea	T cornea
1	24.0	14.5	0.81
2	23.5	14.6	0.86
3	24.8	15.5	0.84
4	24.6	15.5	0.85
5	23.8	15.8	0.90
6	24.4	15.2	0.85
7	23.3	14.3	0.90

Figure 3.

 

Force verses frequency from four measurements taken at the 10-, 20-, 30-, and 40-mm Hg levels in the same eye. The f ₀ – 150 and f ₀ – 470 interval is marked with vertical dotted lines. The slope dF _C/df is estimated in that interval, and the IOP is linearly related to that slope, according to the model of equation 5 .

Table 2.

 

Regression Coefficients β, Degree of Explanation R ², and Standard Deviation of the Residuals for a Fit of Three Polynomial Models

Model	β0	β1	β2	β3	R 2	SD Residual	n
Linear*	3.36	−284			0.987	1.14	418
Linear	3.18	−283			0.991	0.94	410
Quadratic	4.56	−242	267		0.992	0.89	410
Cubic	5.53	−187	1080	3520	0.992	0.88	410

Figure 4.

 

IOP_ARS from the applanation resonator method plotted against IOP_VC (n = 418). To obtain normally distributed residuals, eight measurements included in this plot were regarded as outliers in the evaluation of the model.

Table 3.

 

Residuals from the GLM Analysis According to Equation 5 for Different IOP Levels.

IOP Level (mm Hg)	Mean Residuals (mm Hg)	SD (mmHg)	n
10	0.33	1.00	58
15	−0.08	0.69	59
20	−0.37	0.69	59
25	−0.15	0.74	60
30	−0.41	0.86	60
35	0.00	1.04	59
40	0.74	0.98	55

Table 4.

 

Regression Coefficients β, degree of Explanation R ², and Standard Deviation of the Residuals for a Linear Regression Fit of the Model of Equation 5 against Data from the Individual Eyes

Eye	β0	β1	β1 95% Lower*	β1 95% Upper*	R 2	SD Residuals	n
1	2.71	−280	−286	−274	0.99	0.88	69
2	3.30	−279	−282	−275	1.00	0.57	69
3	2.61	−298	−303	−293	1.00	0.72	70
4	2.93	−284	−289	−279	1.00	0.73	70
5	4.86	−265	−273	−257	0.99	1.18	65
6	2.49	−293	−301	−286	0.99	1.04	67

Figure 5.

 

Drawing of the geometric relationships for the contact between the sensor tip and the cornea. The large circle is the eyeball and the small circle represents the curvature of the cornea with radius R. The upper part of the drawing symbolizes the sensor tip with a radius of curvature ρ. The sensor tip indents the cornea with an indentation depth L. For mathematical purposes the indentation depth, L, was divided by the contact point between cornea and sensor into two distances a and b.

The authors thank Tomas Bäcklund, Department of Biomedical Engineering, for skillful technical assistance and useful discussions.