Corneal retardation time as an ocular hypertension disease indicator

Corneal biomechanics (CB) is a branch of biophysical sciences that deals with deformation and equilibrium of corneal tissue when any external force is applied. In this sense, the mechanical properties of the corneal tissue depend on the specific organization of fibres, cells and ground substance within the structure. Collagen in Bowman's layer and stroma make a significant contribution to corneal elasticity, whereas the ground substance would give the viscous behaviour (Garcia-Porta et al 2014). The increasing interest in CB is due to, among others, its role in the detection and management of ectatic disease (Ortiz et al 2007, González-Méijome et al 2008, Roy and Dupps 2011, Ambrósio et al 2017, Padmanabhan and Elsheikh 2023) and an accurate estimation of IOP to manage pathological diseases such as glaucoma (Liu and Roberts 2005, Asejczk-Widlicka et al 2019, Consejo et al 2019, Susanna et al 2019, Chan et al 2021, Catania et al 2023).

There are different material models which describe with more or less accuracy the corneal biomechanics. In this respect, we can mention the visco-hyperelastic model, where a highly nonlinear elastic response is achieved when very large strains are applied (Ariza-Gracia et al 2015, Whitford et al 2018, Liu et al 2020), the viscoelastic model (i.e. the material's elastic stress-strain relationship depends on the strain rate) (Fraldi et al 2016, Maczynska et al 2019) or the finite element methods, where a complete 3D model of the cornea is designed to study its mechanical behaviour (Sánchez et al 2014, Simonini et al 2016). On the other hand, one-dimensional (1D) rheological models have been useful to describe the viscoelastic properties of the cornea (Glass et al 2008, Han et al 2014, Jannesari et al 2018), though they are not meant to study the 3D corneal deformation.

In this regard, 1D models consist of parallel and/or series combinations of springs and dashpots which mimic the elastic and/or viscous character of the cornea. Thus, a Kelvin-Voigt model with an additional spring can reproduce the instantaneous deformation of the cornea (Glass et al 2008), while a four-element viscoelastic model (i.e. the Burgers model) has also been selected for modeling the corneal biomechanics (Jannesari et al 2018). More recently, a more sophisticated rheological model that takes into account the elastic and viscous effects of cornea, crystalline lens and the whole eyeball has been reported (Jimenez-Villar et al 2022). Although more complex models with a greater number of elements (i.e. springs and/or dashpots) should be more accurate in corneal modeling (Kok et al 2014, Jannesari et al 2018), these approaches might not present a unique mathematical solution, due to the higher-order differential equations inherent in these models. So, as clearly stated by Jannesari et al (2018), rheological models combining simplicity with accuracy are desired.

Furthermore, it has been amply demonstrated that CB influences IOP measurements (Medeiros and Weinreb 2006, Grise-Dulac et al 2012, Brown et al 2018). For that matter, the corneal-compensated intraocular pressure provided by the non-contact tonometer ORA is less influenced by corneal biomechanics (Medeiros and Weinreb 2006, Hager et al 2008, Lee et al 2019), so it might be a reliable parameter to characterize OHT subjects. As it is well-known, an elevated IOP is the major risk factor for developing glaucoma (De Moraes et al 2012, Matlach et al 2019), however, this is not the unique factor. It has been reported glaucomatous damage at low IOP values (Anderson 2003), whereas no significant glaucoma progression has been found at IOPs greater than 22 mmHg (Kass et al 2002).

In this connection, the gold standard method widely used by ophthalmologists to evaluate structural changes in the optic nerve head (ONH) or the retinal nerve fiber layer (RNFL) and assist in the diagnosis of glaucoma has been the fundus photography (Chakrabarti et al 2016). The main advantage of this technique is its simplicity and cost-effectiveness, despite the clinical examination of ONH and RNFL is subjective and qualitative, leading to considerable intra- and interobserver variability in assessing the ONH among qualified specialists. Alternative methods such as optical coherence tomography (OCT) (Geevarghese et al 2021), scanning laser polarimetry (SLP) (Lemij and Reus 2008), and confocal scanning laser ophthalmoscopy (CSLO) (Yaghoubi et al 2015) have been developed to evaluate nerve fiber loss and optic disc changes in glaucoma. Nonetheless, these retinal imaging instruments are costly and present some drawbacks, among them, the susceptibility of CSLO to interobserver variabilities or the inability of SLP method to provide both RNFL and ONH data. Additionally, selective perimetry techniques such as short-wavelength automated perimetry (SWAP) and frequency-doubling technology (FDT) perimetry have been extensively studied as adjuncts to standard automated perimetry evaluation (Sharma et al 2008).

Accordingly, the aim of this work is to yield a reasonable indicator related to the viscoelastic corneal quality, which might be useful to discern which OHT subjects are more probable to develop ocular hypertensive disorders such as glaucoma. This parameter is the corneal retardation time τ, that is, the time in which about 63% of the final corneal strain is determined (Brinson and Brinson 2008, Jannesari et al 2018), and might serve as an indicator of how elastic or viscous a cornea should be. In other words, this metric would measure the cornea's ability to absorb IOP fluctuations. As we will show in this article, the τ parameter might explain why some OHT subjects (and even those with normal IOP values) should undergo glaucoma progression, while others not.

The paper is organized as follows. In section 2 we describe our 1D corneal viscoelastic model to derive a practical expression for the corneal retardation time τ, as a function of the corneal applanation pressures and their first derivatives. A detailed explanation of our methods to calculate the τ parameter is performed in section 3, and our corneal retardation results concerning a population of 100 healthy young subjects is presented in section 4. Finally, we discuss and summarize our results in section 5.

Let us first introduce the theoretical model for the corneal biomechanics, in order to derive a simple and useful expression for the corneal retardation time τ (i.e. our crucial parameter which might be used as a plausible OHT disease indicator).

When loaded, the cornea demonstrates some instantaneous deformation (purely elastic behavior) followed by a progressive viscoelastic deflection. This trend can be fairly described by the Kelvin-Voigt viscoelastic model of three elements (KVM) (please, see figure 1) where the dashpot η symbolizes the time-dependent viscous resistance to the applied force, and springs E₁ and E₂ mimic the purely elastic behavior (Glass et al 2008). When a stress σ is applied, a corneal strain is induced. This configuration allows an instantaneous deformation of the cornea through spring E₂. More precisely, the right-hand spring E₂ stretches immediately upon loading. Then, the dashpot η then takes up the stress, transferring the load to the second spring E₁ as it slowly varies over time. Upon unloading, E₂ contracts immediately and the left-hand spring slowly shortens, being held back by the dashpot.

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Though more sophisticated 1D rheological models have been recently studied (Jannesari et al 2018, Jimenez-Villar et al 2022), we have chosen the KVM over other viscoelastic approaches (such as the Zener or Burgers models) for two reasons: its ability to mimic the corneal response to an applied force (as above-mentioned, an instantaneous deformation followed by a progressive viscoelastic deflection) and the limited number of independent variables (thus, reducing its mathematical complexity). As currently explained by Torres et al (2022), the Kelvin-Voigt model is quite appropriate and straightforward to characterize the viscoelasticity of the cornea, as well as it has been experimentally validated with artificial phantom corneas (Glass et al 2008).

Our KVM relates the applied stress (σ) to the corneal strain () through the following set of equations (Kelly 2013)

$$\begin{eqnarray}\begin{array}{rcl}\sigma & = & {\sigma }_{1}+{\sigma }_{2}\qquad \epsilon ={\epsilon }_{1}+{\epsilon }_{2}\qquad {\sigma }_{1}={E}_{1}{\epsilon }_{1}\\ \sigma & = & {E}_{2}{\epsilon }_{2}\qquad \quad {\sigma }_{2}=\eta {\dot{\epsilon }}_{1},\end{array}\end{eqnarray} \tag{ 2.1 }$$

and ${\dot{\epsilon }}_{1}$ corresponds to the strain rate of the parallel elements. Assuming the cornea to be axisymmetric, a single elastic constant should govern corneal behavior (Glass et al 2008), so we can identify E₁ = E₂ = E. After Laplace transforming, the constitutive relation can be written as (Kelly 2013)

$$\begin{eqnarray}&&E\epsilon +\eta \dot{\epsilon }=2\sigma +\tau \dot{\sigma },\end{eqnarray} \tag{ 2.2 }$$

where $\dot{\sigma }$ is the stress rate and τ = η/E stands for the corneal retardation time. The latter parameter describes the time dependent response of the cornea with respect to the applied load.

On the other hand, non-contact tonometers such as ORA applanate the cornea in two instants, when the strain is minimum. Consequently, the strain rate cancels at these applanation moments and $\dot{\epsilon }=0$. So, from equation (2.2) we can write for the first applanation time t_ap,1

$$\begin{eqnarray}&&\begin{array}{l}E\epsilon ({t}_{\mathrm{ap},1})=2\sigma ({t}_{\mathrm{ap},1})+\tau \dot{\sigma }({t}_{\mathrm{ap},1})\\ \quad =-2| \sigma ({t}_{\mathrm{ap},1})| +\tau | \dot{\sigma }({t}_{\mathrm{ap},1})| ,\end{array}\end{eqnarray} \tag{ 2.3 }$$

where it is assumed a compressive stress (σ(t_ap,1) < 0) during the load stage ($\dot{\sigma }({t}_{\mathrm{ap},1})\gt 0$). Moreover, for the second applanation time (also a compressive regime with σ(t_ap,2) < 0), we have

$$\begin{eqnarray}&&\begin{array}{l}E\epsilon ({t}_{\mathrm{ap},2})=2\sigma ({t}_{\mathrm{ap},2})+\tau \dot{\sigma }({t}_{\mathrm{ap},2})\\ \quad =-2| \sigma ({t}_{\mathrm{ap},2})| -\tau | \dot{\sigma }({t}_{\mathrm{ap},2})| ,\end{array}\end{eqnarray} \tag{ 2.4 }$$

and now, during the unload process, $\dot{\sigma }({t}_{\mathrm{ap},2})\lt 0$. Provided that the strain at applanation is the same for both load-unload processes (i.e. (t_ap,1) = (t_ap,2)), we obtain the following expression for the corneal retardation time τ from equations (2.3) and (2.4)

$$\begin{eqnarray}&&\tau =\displaystyle \frac{2\left(| \sigma ({t}_{\mathrm{ap},1})| -| \sigma ({t}_{\mathrm{ap},2})| \right)}{| \dot{\sigma }({t}_{\mathrm{ap},1})| +\dot{\sigma }({t}_{\mathrm{ap},2})}.\end{eqnarray} \tag{ 2.5 }$$

Let us now analyze how the different pressures act on the anterior and posterior corneal surfaces at applanation (please, see figure 2). The intraocular pressure IOP on the posterior surface of the cornea is subtracted to the sum of the tonometer applied pressure P_t(t_ap,i) and the tear film pressure s, so as to obtain the resultant intraocular pressure at applanation P_r(t_ap,i) (Liu and Roberts 2005, Glass et al 2008, Kotecha et al 2015)

$$\begin{eqnarray}&&{P}_{{\rm{r}}}({t}_{\mathrm{ap},{\rm{i}}})={P}_{{\rm{t}}}({t}_{\mathrm{ap},{\rm{i}}})+s-\mathrm{IOP},\quad \mathrm{for}\ i=1,2.\end{eqnarray} \tag{ 2.6 }$$

This radial stress P_r(t_ap,i) can be related to the membrane stress σ(t_ap,i) in the KVM via the Laplace law (Glass et al 2008)

$$\begin{eqnarray}&&\sigma ({t}_{\mathrm{ap},{\rm{i}}})=\displaystyle \frac{{R}_{{\rm{c}}}}{2e}{P}_{{\rm{r}}}({t}_{\mathrm{ap},{\rm{i}}}),\quad \mathrm{for}\ i=1,2,\end{eqnarray} \tag{ 2.7 }$$

where R_c and e state for the corneal radius of curvature and corneal thickness, respectively. Therefore, introducing equation (2.7) into (2.5) and performing some elementary calculations, we derive the final expression for the corneal retardation time

$$\begin{eqnarray}&&\tau =\displaystyle \frac{2\left(| {P}_{1}| -| {P}_{2}| \right)}{| {\dot{P}}_{1}| +| {\dot{P}}_{2}| }=\displaystyle \frac{2\ \mathrm{CH}}{| {\dot{P}}_{1}| +| {\dot{P}}_{2}| },\end{eqnarray} \tag{ 2.8 }$$

where P_i = P_t(t_ap,i) are the tonometer applanation pressures, CH = ∣P₁∣ − ∣P₂∣ the corneal hysteresis and ${\dot{P}}_{i}$ the tonometer applanation pressures rates. One observes that the corneal retardation time τ is directly related to CH, but with a clear different behaviour and physical meaning, as will be examined in the next sections.

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To our purpose, a total number of 200 eyes from 100 healthy European Caucasian subjects (mean age 24 ± 5 years old) were involved in the study. Within this group, about 10% had an elevated IOP with more than 21 mmHg, and only patient number #278 had been undergoing medical treatment for elevated IOP and diagnosed glaucoma disease during the measurements. The inclusion criterion was to be aged between 18 to 30 years old, whereas subjects with history of ocular pathologies, corneal injuries or surgery, contact lens wearers or irregular astigmatism were excluded. This study was reviewed by an ethical review board and conforms to the tenets of the Declaration of Helsinki (Ethical Committee of Research of the Health Sciences Institute of Aragón, Spain) approved with reference C.P.-C.I.PI20/377. All participants were informed about the nature of the project and signed an informed consent document.

Hence, participants involved in this study were divided into two groups: control (with intraocular pressure values less than 21 mmHg) and ocular hypertensive (where IOP_cc is greater or equal than 21 mmHg), all of them (as previously mentioned) healthy subjects without ophthalmological clinical manifestations, except for patient number #278 with diagnosed glaucoma disease (please, see table 1). This differentiation will be more necessary and evident when we study the corneal retardation time as a function of the intraocular pressure, as described in figure 7 of the next section.

The applanation pressure data were collected with the non-contact tonometer Ocular Response Analyzer (ORA^®; Reichert Ophthalmic Instruments, Depew, NY) which measures, apart from the Goldmann-correlated IOP (IOP_g) and the corneal-compensated IOP (IOP_cc), some biomechanical properties such as the corneal hysteresis (CH) (related to the capacity of the cornea to absorb and dissipate energy) or the corneal resistance factor (CRF). This last metric is thought to be a better indicator of the corneal viscoelasticity than CH (Gatinel 2007).

The ORA device generates a 25 ms collimated air jet to deform the cornea and uses an infrared (IR) detection system in which the IR emitter is aligned on one side of the cornea with an IR detector (Roberts 2014). As the cornea deforms under the applied air pressure, it rapidly traverses a state of applanation, causing the reflected IR light to align with the detector. As a result, the captured light increases significatively and a spike in the IR signal is recorded. Hereafter, the cornea takes on a slight concave shape, to then move outward in another applanation state. Finally, the cornea recovers its normal configuration state. In our study, four measures were carried out for each subject's eye in order to get averaged values.

In this sense, the accurate method to determine the corneal retardation time τ was performed via the two ORA's characteristic curves: the signal amplitude (corresponding to the IR light which is reflected off the surface of the cornea during perturbation) and the pressure amplitude (i.e. the external applied pressure P_t(t) as depicted in figure 2). The last curve can be fairly fitted by the following gaussian profile (0.985 < R² < 0.997 in all cases)

$$\begin{eqnarray}&&{P}_{{\rm{t}}}(t)={P}_{{\rm{t}},0}+\exp \left[-0.5{\left(\displaystyle \frac{t-{t}_{{\rm{c}}}}{{\rm{\Delta }}t}\right)}^{2}\right],\end{eqnarray} \tag{ 3.1 }$$

where Δt stands for the pressure amplitude width, and the gaussian center t_c is located near the corneal concave state. Hence, we have represented in figure 3 both the pressure and signal amplitudes versus time for subject #136 in our population, where the upper (bottom) panel shows the results for the left (right) eye. As easily noticed, the pressure curve conforms a clear gaussian shape in accordance with equation (3.1), where the gaussian center t_c is also depicted.

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Once this gaussian fit is performed, the signal amplitude provides the two applanation pressures P₁ and P₂ (via the sharp peaks in both panels) and the corresponding corneal hysteresis CH (which resulted to be 10.9 mmHg for the left eye and 10.7 mmHg for the right eye, respectively). The pressure rates ${\dot{P}}_{i}$ can be analytically evaluated from the first derivatives of the tonometer pressures. Ergo, the corneal retardation time τ is calculated via equation (2.8) (for subject #136, this parameter was 1.13 (1.06) ms for the left (right) eye, respectively). These corneal retardation time results are fairly close to the average mean of the total population, as explained in detail in the next section.

In this section we deal with the fundamental results concerning the corneal retardation time of our young population, and its important relationship with the intraocular pressure IOP.

To this aim, the histogram illustrated in figure 4 shows the corneal retardation time τ (calculated via equation (2.8)) of the 100 subjects that participated in the study. It can be observed an explicit gaussian profile centered at 1.10 ms with a full width half maximum (FWHM) of 0.39 ms. The R-squared parameter for this gaussian fit was 0.97. In view of these results it can be assumed that for a young and healthy population, the corneal retardation time should be ranged between 0.90 and 1.30 ms, where more elastic corneas are associated with low τ values (about 13.5% in our case). In addition, elevated corneal retardation times are related to viscoelastic corneas (roughly a 13% of our population), not necessarily being pathological cases those subjects with upper or lower τ values (though, in the later scenario, a clear connection with higher intraocular pressures is found, as briefly discussed).

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On the other hand, it is expected that our biomechanical parameter should be directly correlated to the time interval between the two applanation times Δt_ap. That is, more elastic (viscoelastic) corneas, which entail lower (higher) τ values, might take less (more) time during the applanation interval. In such a case (not shown in this work), the linear coefficient of determination resulted to be R² = 0.46 for Δt_ap = t_ap,2 − t_ap,1. Subsequent data analysis (please, see figure 5) demonstrated that the optimized time interval corresponded to Δt_ap,opt = 1.5t_ap,2 − 0.5t_ap,1, where now R² = 0.72. This time lapse is depicted in the inset of figure 5, however different applanation time intervals might also be considered for our study (with R² > 0.45, in all cases). As a matter of fact, the optimized time interval for subject #OS231 was 18.46 ms, fairly shorter than patient #OS174 with a time lapse of 20.49 ms. This may be interpreted assuming that the cornea of the former subject is more elastic (that is, it takes less time between both applanation times) than subject #OS174, with a more viscoelastic cornea. Moreover, these findings should be affected by the intraocular pressure, because elastic corneas with low IOP values might take longer to recover its original shape than viscoelastic corneas of OHT subjects. This relationship between the τ parameter and the IOP will be treated in detail shortly.

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But before embarking on this study, let us first analyze an important biomechanical parameter like the corneal hysteresis CH and its dependence on the corneal-compensated IOP_cc in our population (please, see figure 6). Assuming that normal IOP ranges from 10 to 21 mmHg (Badakere et al 2021), one notices that both parameters are not correlated, in consistency with previous published work (Luce 2005), where no statistical significance was found. For this reason, we have not differentiated between control and OHT populations. Nevertheless, it can be observed that low CH values (such as subjects #OD278 or #OS231) also possess high intraocular pressures and an possible risk of glaucoma progression. This result agrees with prior reported research, where low corneal hysteresis is thought to be related to the risk and development of glaucoma (Prata et al 2012, Deol et al 2015), though there is no consensus on this topic. As stated by Roberts (2014), low CH should not be interpreted as a damaged cornea, and further work is required to determine what component contributing to this viscoelastic parameter correlates to damage at the optic nerve. In our study, when the corneal hysteresis CH is divided by the sum of the first derivatives of the applanation pressures ${\dot{P}}_{i}$ (please, see again equation (2.8)), the uncorrelated scheme illustrated in figure 6 turns to a well-defined linear dependence, as immediately discussed.

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Accordingly, our fundamental result is exhibited in figure 7(a) where the corneal retardation time τ is represented as a function of IOP_cc. Now, a clear linear dependence is found (with a coefficient of determination R² = 0.70), where lower τ values are mostly associated with higher intraocular pressures. This can be easily understood since more pressurized corneas will behave more elastically than those with lower IOPs. So, for instance, subjects #OD170 or #OS174 have the highest τ values in our study (which should be connected with small IOPs) but it cannot be assured that such corneas are the most viscoelastic of our population. This fact should be corroborated with a relevant number of ocular hypotony patients, though it does not constitute a subject of study in our current research.

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Nonetheless, this is not a fundamental rule. Indeed, after inspection of figure 7(a), one notices that for similar IOP_cc values (such as for subjects #231, #193 or #278), the corneal viscoelastic behavior is different. Whereas subject #193 possess good viscoelastic corneas for both eyes (greater than the average of 1.10 ms, as illustrated in figure 4), other OHT patients like #231 or #278 have more elastic corneas (that is, with lower τ values). This means that the corneas of subject #193 would be more prepared to absorb IOP fluctuations (and avoid possible glaucoma progression) than the other OHT patients. As a consequence, hypertensive subjects with low corneal retardation times should be periodically monitored, in order to prevent possible optical nerve damage. Moreover, even normal IOP subjects present significant differences between their τ values, although these patients are not considered as a "risky population". In this sense, at a normal IOP of 18.4 mmHg (please note the vertical dashed line in figure 7), subject #OD209 exhibits a more elastic cornea (0.79 ms) than patient #OS224, where the corneal retardation time resulted to be 1.24 ms (a 57% higher than the latter). Additionally, the corneal elasticity of subject #OD209 (quantified by our τ parameter) is similar to some OHT patients in our study, so such normal IOP subjects should also be controlled, despite they do not belong to a risk group.

For the sake of clarity, we have also illustrated the corneal retardation time τ for the control population (Figure 7(b)) and OHT subjects (Figure 7(c)). Clearly, the linear correlation between the τ parameter and the intraocular pressure IOP_cc for the control group is even increased (as compared to the whole population), while no significant correlation for OHT subjects is found. This fact reflects the difficulty in predicting the corneal retardation time for our ocular hypertensive population, probably due to the small number of OHT participants in our study (please, see again table 1).

Summarizing, a detailed analysis of the corneal retardation time τ of a young population (200 eyes from 100 healthy subjects) has been carried out. Our results show that this parameter is highly correlated with the corneal-compensated intraocular pressure IOP_cc supplied by ORA tonometer, underlying the risk for OHT subjects with lower τ values to develop hypertension diseases (due to the inability of the poorly viscoelastic cornea to absorb IOP fluctuations). Indeed, viscous damping of the cornea should be crucial since increased damping capacity of the eye may actually buffer hazardous IOP fluctuations, diminishing the stress/strain on the optic nerve and peripapillary scleral tissues (Kaushik and Pandav 2012).

Furthermore, some authors argue that IOP_cc is overestimated (Martinez et al 2006) in comparison with the gold standard technique in measuring IOP, that is, the Goldmann applanation tonometry (Lee et al 2018). Thus, a possible discrepancy between our results for the τ parameter and the corneal-compensated intraocular pressure should be expected. However, given that all IOP_cc values in our population might be affected by the same (or similar) scale factor, the linear dependence depicted in figure 7 should remain the same, with comparable R-squared parameters.

As previously stated, the fundamental aim of our work is to yield an useful tool (i.e. the corneal retardation time τ) to systematically discern which ocular hypertensive patients are more likely to develop OHT diseases and ensure an early diagnosis. Among them, glaucoma plays a leading role since this eye illness is the most common cause of irreversible blindness and affects about 80 million people worldwide, with many more undiagnosed (Tribble et al 2023).

In this sense, our work might help in glaucoma diagnosis (as compared to previous existing methods already mentioned in the Introduction) due to the easiness and robustness of our method. More specifically, it is straightforward to measure the corneal retardation time τ with non-contact tonometers (i.e. via our equation (2.8) and the applanation pressures provided by such instruments). Besides, these values are not affected by subjective or qualitative factors, so the τ parameter can be considered a strong biomechanical indicator not subject to intra- or interobserver variabilities. Also, it has been suggested in the literature that increased viscoelasticity of ocular tissues may have a protective role in glaucoma (Murphy et al 2017, Del Buey-Sayas et al 2021), so the τ parameter could be an important metric to diagnose this disease, playing a leading role in explaining normotensive glaucoma. Instead, our method does not provide information about structural changes in the optic nerve head or the retinal nerve fiber layer, so a proper glaucoma diagnosis might not be guaranteed (and other techniques such as the fundus photography or optical coherence tomography are needed).

It has been widely reported in the literature that the major risk factors for glaucoma are genetics, age and an elevated IOP, so, intraocular pressure should be adequately controlled to avoid visual field deterioration. Beyond the diagnosis, the prediction of the future glaucoma progression of an individual patient is often extremely difficult for clinicians, due to the mix of the abovementioned risk factors. In this regard, corneal thickness, corneal hysteresis or horizontal and vertical cup-disc ratio constitute additional OHT risk factors (apart from an elevated IOP). As reported by Murphy et al (2017), about 3050% of glaucoma patients have normal IOP values, becoming evident that other elements should be taken into account. Provided the large number of independent risk factors, it might not seem plausible that a unique parameter as the corneal retardation time would effectively serve as a OHT indicator.

Nevertheless, the validity of our theoretical approach (as given by the fundamental equation (2.8) for the τ parameter) is based on the strength of the Kelvin-Voigt model to imitate the corneal viscoelastic behaviour. In other words, the τ metric constitutes a valuable indicator of the corneal viscoelastic quality. However, its validity is subject to obtaining well-defined signals via non-contact tonometers (please, see again figure 3): irregular signals with no evident applanation peaks will not give reliable corneal retardation time values.

In fact, our proposal is consistent with previous reported glaucoma research (Matsuura et al 2017) where it is suggested that careful consideration should be given to patients whose eyes are applanated fast in the first and second applanations (please, see again our fundamental results concerning the applanation time interval in figure 5, which are directly related to the τ parameter shown in figure 7(a)). Additionally, the loss of corneal viscoelasticity (which is correlated with lower τ values in our model) is a risk factor that can lead an ocular hypertensive subject to develop glaucoma disease (Roberts et al 2023). For that matter, the corneal retardation time might represent an early detector of those complications associated with ocular hypertension before clinical signs manifest. Nonetheless, such suspected glaucoma patients should be periodically monitored to confirm this fact.

Among the potential limitations encountered during our study, we can mention the reduced number of OHT subjects (in comparison with the normal IOP patients) and the difficulty in finding some OHT clinical cases. It is worth mentioning that our young study population consisted of 100 healthy subjects and, within this group, about 10% had an elevated IOP with more than 21 mmHg (only patient #278 was medically monitored due to its high IOP). Certainly, an increased number of OHT subjects is needed for a conclusive statement about the utility of the corneal retardation time as an ocular hypertension disease indicator. Moreover, it could be interesting to include participants with pathological conditions such as diabetic retinopathy or glaucoma for a more comprehensive analysis, as well as considering another age group. In addition, it should be studied the effects of physical parameters such as central corneal thickness (CCT) or corneal morphology on corneal biomechanics (in line with the proposals by Marcellán et al 2022) which is a subject of ongoing research by our group.

Oscar del Barco gratefully thanks Alfonso Jimenez Villar for helpful discussions on 1D corneal rheological models. The authors acknowledge the funding grant from Departamento de Ciencia, Universidad y Sociedad del Conocimiento del Gobierno de Aragón (research group E44−23R).

The data cannot be made publicly available upon publication because they contain sensitive personal information. The data that support the findings of this study are available upon reasonable request from the authors.

The authors declare no conflict of interest.