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Abstract
The eye has specific optical and biomechanical properties that jointly regulate the eye's quality of vision, shape, and elasticity. These two characteristics are interdependent and correlated. Contrary to most currently available computational models of the human eye that only focus on biomechanical or optical aspects, the current study explores the inter-relationships between biomechanics, structure, and optical properties. Possible combinations of mechanical properties, boundary conditions, and biometrics were specified to ensure the opto-mechanical (OM) integrity to compensate for physiological changes in intraocular pressure (IOP) without compromising image acuity. This study evaluated the quality of the vision by analyzing the minimum spot diameters formed on the retina and drew how the self-adjustment mechanism affects the eye globe shape by adopting a finite element (FE) model of the eyeball. The model was verified by a water drinking test with biometric measurement (OCT Revo NX, Optopol) and tonometry (Corvis ST, Oculus).
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The human eyeball is repeatedly subjected to tensions altering its OM properties [1]. IOP fluctuations, e.g., physiological changes during the day ranging between 2-4 mmHg in the healthy eye, alter the focal length of the eye (in humans), but it does not affect visual acuity [2,3]. It is postulated that the opto-mechanical self-adjustment (OMSA) hypothesis compensates for the subtle IOP variations effect [1], as shown in Fig. 1.
Fig. 1. Opto-mechanical self-adjustment hypothesis. H and H’ are principal planes, and P indicates the IOP levels. Subscripts 1 and 2 illustrate different IOP levels. F and F’ refer to focal points on the retina. Assumption: Emmetropic eye without accommodation.
Computational models of the human eye fall into two main groups. The first group consists of biomechanical models utilized to advance the understanding of the mechanical behavior of the human eye under various physiological and pathological conditions [4]. The second group consists of optical models used to analyze and predict how changes in eye biometrics affect the refraction and aberrations of the eye's optical system [5].
In the case of the first group, biomechanical models, studies were conducted to investigate the biomechanical changes of the eye structure upon varying IOPs. A study found that the IOP increment caused increased stress in all ocular elements of the anterior segment of the human eye [6]. The authors reported that the elevated IOP moved the lens posteriorly and displaced the cornea anteriorly, showing that as IOP increases, accordingly the axial length of the eye increases. Another study simulated the mechanical behavior of eye components and the damage process under high IOP levels [7]. It showed that the biomechanical properties of the sclera determine the biomechanical changes of the optic nerve head. Additionally, the posterior part of the sclera has been observed to adopt a heterogeneous, anisotropic, and non-linear behavior when exposed to acute elevations of IOP (beyond 30 mm Hg), causing it to stiffen. In another study, distributions of stresses and strains in human elements were determined for different IOP values [8]. The research clarified that as IOP increases, so does the radius of curvature and the amount of von Mises stress on the cornea and lens. In contrast, the sclera indicated adverse behavior in terms of stress values. Another study found an approximately linear increasing trend of the maximum nodal displacement, stress, and strain characteristics as the IOP increased from 10 mmHg to 50 mmHg [9]. They found that the cornea moves anteriorly, and the anterior eye segment tissues experience greater displacement with IOP elevation. Another study showed that the human eyeball, except for vitreous humor and lens, experiences increased stress levels for increasing the value of IOP. Consequently, an increase in the displacements is also observed for IOP increment [10].
The optical analysis of the whole eye biometry has shown that there is a correlation between IOP fluctuations and critical parameters, including central corneal thickness (CCT), corneal curvature (1/R), anterior chamber depth (ACD), axial length (AL), and vitreous chamber depth (VCD). Kiely et al. [11] reported that the corneal curvature becomes steeper during the day and positively correlates with corneal thickness. Some studies have shown a mutual correlation between IOP fluctuations and the diurnal ocular axial length [12–14]. Leydolt at el. [15] proposed that increased IOP levels cause eyeball lengthening without affecting the thickness and position of the lens or ACD [15]. Another study found that drug-induced increases in ACD lowered IOP without affecting visual acuity or changing lens thickness [16]. The above literature emphasizes that IOP fluctuations, swift and considerable variations, change the relative position of the visual elements. Consequently, these alterations reposition the corneal apex and lens relative to the retina, yet vision quality does not change during IOP fluctuations.
While currently available computational models of the human eyeball concentrate mainly on its biomechanical or optical aspects, the literature needs to include a computational model to address the OM self-regulation mechanism of the human eye comprehensively. The current study extends the validity of the self-adjustment mechanism [3] that takes advantage of the combination of ocular mechanics and human eye optics to find the most proper and demanded combination of the OM circumstances to keep the focal point in place on the retina. Accordingly, this study develops a FE model to explore the inter-correlations between biomechanics, structure, and optical properties.
2. Method
The present study is based on a two-step calculation, as shown in Fig. 2:
- 1) Developing a FE model in ABAQUS FEA Software. (Release 6.14-2, Dassault Systèmes).
- 2) Verifying the optical prerequisites in OpticStudio 18.7 (Zemax, Seattle, WA).
This research adopts the available geometries and various material models of the human eyeball (available literature) to assess the existence of optomechanical self-regulation. The ultimate goal of the hypothesis is to determine a non-invasive technique to find the human eye’s mechanical properties. As the development of such techniques largely relies on computational modeling of the eyeball, the study utilized the OMSA hypothesis as a tool to find out the most proper and demanded combination of the properties of the biological tissues. The research builds a foundation (first-generation model) for further studies.
Tracking error refers to the situation of following the refracted incoming optical rays (IOR). The image of the IORs should be a spot on the retina. Changing the geometry and the optical elements’ mutual position influence the image formation's position, causing a defocus.
The flow chart indicates the state where the focal point seeks to be inside its acceptable Opto-physiological deviations. Section (I) finds the certified geometry for the initial configuration (when the Nominal IOP level inside of the human eye globe is 16 mmHg (2133 Pa) to reflect the incoming optic rays on the retina. Section (II) strives to extend the validity of the verified geometry of sector (I) to reach the most proper combinations of the material properties meeting the optical prerequisites in the specific range of daily IOP variations. The combination of both sections determines the authenticated FE model in terms of biometry (geometry) and material properties, which satisfy the OMSA hypothesis.
2.1 Finite element model of an ocular globe
This study developed a 2D axisymmetric first-generation FE model of the ocular globe, including the sclera [17–19], cornea, limbus, zonules, extraocular tissue, and lens (Fig. 3). The cornea, limbus, and sclera were all considered separate parts due to their fibrous structure and mechanical response differences. Based on the literature and available material properties, the cornea, sclera, extraocular tissue, and limbus were considered hyperelastic materials [20–22].
The ocular eye globe profile was created based on the parameters of the Goncharov model for a healthy 30-year-old eye [23]. A conic section approximated the anterior and posterior corneal profiles, as shown in equation [Eq. (1)]:
where x and y are shown in Fig. 4. Term e represents the eccentricity of a conic section, and R is the mean axial radius of curvature of the cornea surface.The zonules peripheries were attached to the lens with an anterior intersection region of 1.5 mm and a posterior region of 0.3 mm. This study approximated the zonules fibers and ciliary body as one structure. The geometrical details are summarized in (Table 1 and Fig. 4) [20–22,24]. The geometry, location, and thickness variation of the extraocular tissue (orbital fat) profile and zonules fiber were taken from recent publications [20–22,24]. The calculations were assumed for the emmetropic status (without accommodation).
Fig. 4. Ocular Globe dimensions in cross-sectional view, CCT, LAT, LCT, LPT, SET, PPT, VCD, LD, LT, and ACD refer to central corneal thickness, anterior limbal thickness, central limbal thickness, posterior limbal thickness, scleral equatorial thickness, posterior pole thickness, vitreous chamber depth, lens diameter, lens thickness, and anterior chamber depth respectively.
Table 1. Nomenclature (geometrical parameters of the ocular components) Unit: [mm]
Human ocular tissues are generally viscoelastic and exhibit nonlinear material properties [25–27]. This nonlinear material property of the human eye ranges widely due to its complex nature. Quasi-incompressible hyperelastic models were applied to describe the biomechanical behavior of the eye components. This study assessed the OM response of the ocular globe in a range of daily IOP fluctuations for the different combinations of material properties. Ocular parameters of various structures and their material properties were obtained from previous literature (Table 2) [20–22]. It is worth mentioning that the available literature on the mechanical properties of human eye elements can be divided into two general categories. The first group indicated that the posterior part of the eye (sclera) has a stiffer behavior than the cornea, and the second group demonstrated that the sclera has lower coefficients and is thus more deformable than the cornea (lower Neo-Hookean coefficient, lower Ogden coefficient for the same alpha values). The study found that the latter category satisfies the self-regulation hypothesis for the desired geometry (Figure 2 & Table 1).
Table 2. Material properties used for the various ocular components, α is a dimensionless quantity (material constant), D is an incompressible parameter used to indicate volume change, k is the bulk modulus, µ is the shear modulus of the material (Alternative definition of C is sometimes used, notably in commercial finite element analysis software such as Abaqus), E and ν are Young's modulus and Poisson's ratio respectively
Unlike an elastic material, where stress varies linearly with respect to strain, a hyperelastic material uses the strain energy density function to derive the relationship between stress and strain. This allows them to model the relationship between stress and strain more accurately. Linearly elastic materials are described through two material constants (Young’s modulus and Poisson ratio). In contrast, the strain-energy density can be used to derive a nonlinear constitutive model in hyperelastic materials [28].
In the Ogden hyperelastic material model, the strain energy density is expressed in terms of the principal stretches ${\lambda _i}$ (i = 1) as:
N, ${\mu _p}\textrm{}$ and ${\alpha _p}\textrm{}$ are empirically determined material constants. The shear modulus results from, as shown in equation 3:
The strain energy density for Yeoh hyperelastic model is proposed as:
where ${C_i}$ are material constants. The quantity $2{C_1}$ can be interpreted as the initial shear modulus.The strain energy density function for an incompressible neo-Hookean material in a three-dimensional description is:
${C_1}\textrm{}$ is a material constant, and ${I_1}$ is the first invariant (trace) of the right Cauchy-Green deformation tensor.
2.2 FE analysis
Nodal coordinates and displacement vectors were obtained as the result of FE analysis. Displacement is the distance from which one object moves from its original location when an external force is applied [29].
The proprietary algorithm of the software discretized all components of the FEM. The predefined size of each element was set with an approximate element size of 0.27 mm. It consisted of 3744 elements, 3667 linear quadrilateral elements of type CAX4H (a 4-node bilinear axisymmetric quadrilateral, hybrid, constant pressure), and 77 linear triangular elements of type CAX3H (a 3-node linear, hybrid with constant pressure). Figure 5 shows the mesh-divided computational model.
Fig. 5. Mesh geometry (a) and boundary condition (b) of the axisymmetric eye globe, the outer edge of the ocular tissue was fully encastered (b, B.C. 1). The boundary conditions of symmetry to the y-axis (b, B.C. 2) were considered for the parts of the eye globe. B.C. stands for boundary condition.
The mesh quality was evaluated using software built-in quality assessment. Due to the actual conditions of the human eyeball, surrounded by the skull, the outer edge of the ocular tissue was fully constrained (b, B.C. 1). In addition, the boundary conditions of symmetry to the y-axis (b, B.C. 2) were considered for the parts of the eye globe, as shown in fig 6. The equivalent stress of the initial state (nominal IOP level) was applied to the elements as the predefined fields (initial conditions). An initial uniform volumetric predefined stress field of 0.1 MPa, an average stress value of human eye tissue, was considered for the sclera, limbus, and cornea [30]. All models were solved using an implicit solver in ABAQUS. A computer with a 2.90 GHz CPU, 8 Cores (16 Logical Processors), and 32 GB RAM was used for this study.
Fig. 6. Schematic of the geometric sensitivity. a) Case 13, limbus, b) Case 14, extraocular tissue. The infographic just shows the changes, and it is not exact.
2.3 Study parameters
This study investigates the OM behavior of the human eye globe for 16 different study cases in a FE analysis, as shown in Table 3.
Table 3. The Description of the study cases
The study cases started from the conditions of the first case (the verified geometry of Section (I) in Fig. 2). Different combinations of the material models were assigned to eye components in each step. These changes aim to get material model sets with a close biomechanical response to the experimental test results and an acceptable optical response (tracking error) for the range of daily IOP changes. In each study parameter, the OM behavior of the eyeball was evaluated and compared with the previous cases. The pre-defined material models of study case one was considered as a reference, and all the changes were applied to this combination of material sets. As there is no precise and standard geometry of the limbus and extraocular tissue, and in some cases, the researcher misrepresents the limbus in the human eye analysis [32–34], cases 13 and 14 investigated the sensitivity of the limbus and extraocular tissue geometries in the authenticated FE model.
Cases 15 and 16 depict the results of different loading conditions when the pressure in the anterior chamber (${P_2}$) and vitreous body (${P_1}$) are not the same.
2.4 Optics
The eye parameters were adopted from the Goncharov eye model for an emmetropic eye with an axial length of 23.9 mm and an optical system power of 60.13 D [23]. The Goncharov wide-field schematic eye model with gradient-index lens includes four aspherical refractive surfaces representing the cornea, lens, and aspherical retina (Table 4). Calculations were made for a pupil size of 3 mm and a monochromatic beam (λ = 589 nm). The effective focal length was 16.63 mm for the above-mentioned optical parameters. Figure 8 shows the schematic model of the human eye in OpticStudio.
Fig. 7. Schematic of the different loading conditions, P indicates the IOP levels. Subscripts 1 and 2 illustrate different IOP levels.
Table 4. The Parameters of the Goncharov eye model for a healthy 30-year-old eye
The simulation results for different study cases (section 2.3) were evaluated. The anterior and posterior corneal curvatures, CCT, ACD, LT, and VCD, were imported to Zemax; it was taken from FEM. The detailed procedure of optical calculations was presented in the previous work, and the numerical calculations were done by OpticStudio 18.7 (Zemax, Seattle, WA) [1]. The retinal image analysis was based on the spot diameter and the defocus size. It was presumed that the optical parameters of the eye vary within the range for which the induced refractive error is not greater than ±0.25D. This threshold was considered tolerable and not clinically significant regarding visual acuity. It is a necessary condition for the self-adjustment hypothesis. It corresponds to the range ±0.07 mm of defocus and the spot diagram less than 40 µm.
2.5 Test procedure and validation process
The Water Drinking Test (WDT) is a clinically feasible, reliable, and safe test that has attracted attention as an indirect tool for evaluating IOP fluctuations. The exact mechanism behind WDT has yet to be discovered [35–37].
Four healthy young volunteers (two females and two males) participated in this study. The mean age ± standard deviation was 26 ± 1 in the WDT groups. All participants underwent a complete ophthalmological examination before participation to determine their refractive and ocular health status. Participants were fully informed about the study objective and potential risks. The project was approved by the Ethics Committee of the Wroclaw University of Science and Technology (O-22-27) and adhered to the tenets of the Declaration of Helsinki. To perform the WDT, the patients refrained from food and fluid intake for two hours before the test. Participants were instructed to drink 10–15 mL/kg of water in 5 minutes. All WDTs were performed between 10 to 12 A.M. Each instrument was calibrated at the beginning of the study. The same experienced technician using the criteria provided by the manufacturer of each device, performed all measurements.
Biometry assessment took place every 15 minutes after the WDT. Two groups of results were measured during the test separately.
- 1) IOP and CCT parameters obtained using Oculus Corvis ST.
- 2) ACD, CCT, AL, and LT parameters obtained using, B-OCT optical biometry module, in Posterior segment OCT REVO NX 130, Software version 11.0.5 (Optopol Technology) [38]. For the axial length of individual ocular structures (AL, ACD, LT, and CCT), a series of 10 vertical and horizontal measurements were taken. The device rejected the outliers, and the mean was computed. The measurement and boundary identification were fully automated.
The starting point of the datasets was considered according to the initial FE modeling conditions (IOP = 16 mmHg, CCT = 550µm, LT = 3.69 mm, and ACD = 3.06 mm) rather than their exact initial values. Hence, the shifting technique along the y-axis, compared to the specified initial numbers, was done for parameters that will not affect the trend analysis process.
3. Results
The optical coherence tomography (OCT) biometry assessment results provided information on AL, CCT, ACD, and LT parameters, describing the behavior of the eye globe during IOP increments. The results of figures 9–13 showed that IOP increment (Fig. 9) has the following consequences:
Fig. 9. Intraocular pressure (IOP) trend during WDT. The information was recorded in three points, initially, after 15 and 30 minutes by Corvis ST. The straight line is a connector with no relation to the parameter trend between the two following points. R and L refer to the right and left eye, respectively.
Fig. 10. Central corneal thickness (CCT) trend during WDT. The information was recorded in three points, initially, after 15 and 30 minutes by Corvis. The straight line is a connector with no relation to the parameter trend between the two following points. R and L refer to the right and left eye, respectively.
Fig. 11. Axial length (AL) trend during WDT. The information was recorded in three points, initially, after 15 and 30 minutes, by Revo. The straight line is a connector with no relation to the parameter trend between the two following points. R and L refer to the right and left eye, respectively.
Fig. 12. Anterior chamber depth (ACD) trend during WDT. The information was recorded in three points, initially, after 15 and 30 minutes, by Revo. The straight line is a connector with no relation to the parameter trend between the two following points. R and L refer to the right and left eye, respectively.
Fig. 13. Lens thickness (LT) trend during WDT. The information was recorded in three points, initially, after 15 and 30 minutes, by Revo. The straight line is a connector with no relation to the parameter trend between the two following points. R and L refer to the right and left eye, respectively.
The CCT depicts the opposite and decreasing behavior (Fig. 10), the AL parameter showed an increasing trend (Fig. 11), the ACD illustrated an increase in values (Fig. 12), and LT parameter did not show a definite behavior (Fig. 13).
3.1 Simulation results
Each study set was simulated for four IOP levels (17, 18, 19, and 20 mmHg). The last column shows the variation of the focal point location. Anterior corneal curvature (${R_A}$), posterior corneal curvature (${R_P}$), CCT, ACD, LT, and VCD were imported to Zemax; it was taken from FEM, and Zemax obtained focal point location (Δf).
Table 6 depicts the results of study case 1 for the datasets of Table 5.
Table 5. Assumed combinations of the material properties for the study case 1, ${\alpha }$ is a dimensionless quantity (material constant), D is an incompressible parameter used to indicate volume change, ${\boldsymbol \mu }$ is the shear modulus of the material (Alternative definition of C is sometimes used), E and ν are Young's modulus and Poisson's ratio respectively
Table 6. The simulation results of study case 1, the assumed combinations of the material properties for study case 1, shown in Table 5. Anterior corneal curvature (${{\boldsymbol R}_{\boldsymbol A}}$), posterior corneal curvature (${{\boldsymbol R}_{\boldsymbol P}}$), CCT, ACD, LT, and VCD were imported to Zemax; it was taken from FEM, and Zemax obtained focal point location (Δf). The red highlighted value means it exceeds the range ±0.07 mm of defocus and cannot fulfill the OM self-adjusting hypothesis.
The results of Table 6 show that the mentioned possible datasets of biomechanical parameters of the eyeball, as shown in Table 5, cannot fulfill the OM self-adjusting mechanism conditions at the IOP level of 20mmHg.
Study cases 2-4, considered the material property of the human lens and zonules as a Neo Hookean hyperelastic model (1st order), while the material properties of the other components were not changed.
Changing the mechanical properties of the sclera, (Study cases 5-7), and the human limbus, (Study case 8), did not satisfy the OMSA conditions in its daily changes (19 and 20 mmHg), as shown in Table 8.
Table 7. The simulation results of study cases 2-4, considering the material property of the human lens, zonules as a Neo Hookean hyperelastic model. Zemax obtained the focal point location (Δf). The red highlighted value means it exceeds the range ±0.07 mm of defocus and cannot fulfill the OMSA hypothesis
Table 8. The simulation results of study cases 5-8, changing the mechanical properties of the sclera (Study cases 5-7) and the human limbus (case 8). Zemax obtained the focal point location (Δf). The red highlighted value means it exceeds the range ±0.07 mm of defocus and cannot fulfill the OM self-adjusting hypothesis
The results of study case 9, considering the material property of the human cornea as a Neo Hookean hyperelastic model, satisfied the OM self-adjusting mechanism conditions in its daily changes (16-20 mmHg), as shown in Table 9.
Table 9. The simulation result of study case 9, considering the material property of the human cornea as a Neo Hookean hyperelastic model. Zemax obtained the focal point location (Δf)
Table 10 shows the results of study cases 10-12 (as shown in Table 3), which satisfied the OM self-adjusting mechanism conditions in its daily changes (16-20 mmHg).
Table 10. The results of study cases 10-12, considering the material property of the human cornea, limbus, and sclera as a Neo Hookean hyperelastic and Yeoh model. Zemax obtained the focal point location (Δf)
In addition, the ratio of the $\left( {\frac{{{R_\textrm{P}}}}{{{R_\textrm{A}}}}} \right)$ was constant for the results of cases 9-12. ${R_\textrm{P}}\,\textrm{}and\textrm{}\,{R_\textrm{A}}$ $\textrm{}$ refer to the radius of the anterior and posterior central cornea's curvature. The average value of $\left( {\frac{{{R_\textrm{P}}}}{{{R_\textrm{A}}}}} \right)$ is 0.841, which is another prerequisite to the optical self-adjustment hypothesis [1].
Tables 11 and 12 show the results of geometric sensitivity for the limbus and extraocular tissues, respectively. The calculation is based on the material sets of case 9, as shown in Table 3.
Table 11. The results of the geometric sensitivity, study case 13 (Limbus, as shown in Fig. 6 - a)
Table 12. The results of the geometric sensitivity, study case 14 (Extraocular tissue, as shown in Fig. 6 - b)
Comparing the results of cases 13-14 (Tables 11 and 12) showed that the geometry of the human limbus and extraocular tissue influenced the simulation results and the variation of the focal point location. Attenuating the border and periphery of both elements, limbus and extraocular tissue, satisfy the OMSA hypothesis.
Table 13. The results of the different loading conditions, study cases 15 and 16 (As shown in Fig. 7). ${{\boldsymbol P}_1}$ and ${{\boldsymbol P}_2}$ refer to IOP levels of the vitreous chamber and anterior chamber, respectively. Zemax obtained the focal point location (Δf)
Figures 14 and 15 represent the displacement results for two study cases, considering the Neo-Hookean and Ogden model for human cornea at the IOP level of 20 mmHg. The results for different loading conditions for two cases 15-16 are presented in the Table 13. Change of intraocular pressure in presented ranges meet the OMSA assumptions.
Fig. 14. a) Displacement results of the Ogden material model for human cornea (mm). b) Displacement vector, ROI refers to the region of interest.
Fig. 15. a) Displacement results of the Neo-Hookean material model for human cornea (mm). b) Displacement vector, ROI refers to the region of interest.
3.2 Validation process
Validation occurs by comparing FE-predicted results and obtained experimental results for ACD and CCT. A quasi-quantitative validation of the FE model was done for two main reasons:
- 1) Inability to synchronize the output data from two devices (it is impossible to measure the output data from two devices simultaneously).
- 2) Each participant's IOP baseline (the initial IOP level) was different, and the increment of the IOP level was different for each case.
While a significant increase in IOP level from baseline was observed for all participants during the test, it was not sufficient to make a definite and reliable behavior for quantitative evaluations.
Figures 16–20 show the behavior of the eye globe during IOP increments; it was taken from the FE simulation. Parameters CCT, ACD, VCD, and the radius of the anterior and posterior central cornea's curvature are shown for IOP levels of 17-20 mmHg.
Fig. 16. Central corneal thickness (CCT) trend for study cases 9-12, it was taken from FEM. The straight line is a connector with no relation to the parameter trend between the two following points.
Fig. 17. Radius of the anterior central cornea’s curvature trend for study cases 9-12. The straight line is a connector with no relation to the parameter trend between the two following points.
Fig. 18. Radius of the posterior central cornea’s curvature trend for study cases 9-12. The straight line is a connector with no relation to the parameter trend between the two following points.
Fig. 19. Anterior chamber depth (ACD) trend for study cases 9-12. The straight line is a connector with no relation to the parameter trend between the two following points.
Fig. 20. Vitreous chamber depth (VCD) trend for study cases 9-12. The straight line is a connector with no relation to the parameter trend between the two following points.
Figures 16–20 depict the following behavior of the eye globe during IOP increments:
- ▪ CCT parameter illustrated an opposite decreasing behavior for all study cases 9-12, correlated with the result of Figure 2, Central corneal thickness (CCT) trend during WDT.
- ▪ The central cornea’s radius of the curvature for both anterior and posterior surfaces showed the same trend for all study cases 9-12.
- ▪ The VCD illustrated increased values for all study cases 9-12.
- ▪ The parameter ACD depicted the opposite behavior for study cases 9-11, and study case 12 showed an increasing trend, which is in agreement with the result of Figure 4, Anterior chamber depth (ACD) trend during WDT.
3.3 Convergence and mesh independence study
The simulation results for various element sizes were explored to ensure that the results are independent of the number and size of elements and meshes [31]. The maximum strain and displacement of FE results were observed in the human corneal part. Hence, the mesh convergence was done for this part, and in each step, the number of elements for this part increased, as shown in Table 14. The results are nearly consistent from 3654 elements, with less than 2% changes for the maximum displacement and maximum strain parameter onward, indicating that the simulation findings are reliable.
Table 14. Sensitivity analysis, FE output values for various element numbers
4. Discussion
This research assessed the results of numerical simulations for some factors such as geometry, different loading conditions, and material properties and fine-tuned the OM characteristics of the above-mentioned primary components to maintain the visual image quality (within the specified range of IOP levels). This study showed that elevated IOP caused a slight dilation of the eyeball and the following sequence of actions. The corneal apex begins to move away from the retina. As a result, the eye experiences a slight change in the curvature of the cornea, as shown in Figures 17 and 18, which is consistent with studies conducted to investigate the effect of artificially elevated IOP on the central curvature of the cornea [39]. Douthwaite et al. found that in IOP increments, the mean (standard deviation) before and during the experiment was 15.6 (2.4) mmHg and 22.1 (2.3) mmHg, the mean changes in the radius of corneal curvature were 5%, which strongly correlated with the FE simulation results, showing this pressure rise could not distort the corneal surface centrally. The results presented in our study showed a slight axial movement of the limbus and an increase in the axial distance between the cornea and the crystalline lens (ACD), similar to the literature report [9]. Moreover, the OM response of the CCT in this work is consistent with the previous literature [11].
The results of study cases 1- 4 showed that although the choice of the mechanical properties of the human eye components is purpose-dependent, the material property of the human zonulas and lens do not significantly influence the OMSA hypothesis and are less important during a finite element analysis. This enables guiding the numerical simulation toward cost-effective results and increases their applicability in clinical practice. Comparing the results of study cases 9-12 showed that the material property of the cornea highly influences the simulation results and the variation of the focal point location. In addition, figures 14 and 15 show that by changing the material properties of the human cornea to Neo-Hookean, the maximum displacement is shifted towards the periphery of the limbus, keeping the focal point in place on the retina. The results of study cases 13-14 (Tables 7 and 8) showed that the geometry of the human limbus and extraocular tissue influenced the simulation results and the variation of the focal point location. Attenuating the border and periphery of both elements, limbus and extraocular tissue, satisfied the OMSA hypothesis. This step would be a base for further research to perform optimization to find the most acceptable and standard periphery of these elements. The study emphasized that neither all the possible combinations of material properties nor most of the statistical models (in terms of geometry) would be a possible choice for opto-mechanical studies. The results showed that by considering the verified geometry of the eye globe (Section I, Fig. 2) and applying the Neo-Hookean material set on the cornea, the mechanical behavior of the eye globe satisfies the self-regulation hypothesis in the range of daily IOP changes. The simulation results indicated that the opto-mechanical parameters of the cornea are significant and among the available material models in the literature, considering the Neo-Hookean (Cornea) and Yeoh model (Sclera) material models showed a close and similar behavior to the experimental tests. It is proposed that increasing the pressure inside the eye cavity (IOP) causes softer behavior and small changes in the cornea for the Neo-Hookean material model. Consequently, the first and second derivatives and the radius of curvature cannot have significant and notable changes.
Several limitations to our study need to be acknowledged, including using a 2D axisymmetric rotating body for the whole eye globe components and isotropic mechanical behaviors; the geometry of a real human eye is not symmetrical and isotropic. Second, the quantitative validation of the ocular FE simulation should be carried out in the follow-on work by more participants. Lastly, the model is two-dimensional both in Abaqus (in terms of the biomechanical side) and OpticStudio (in terms of optical aspects). As a result, the optical and geometrical axes were assumed identical at this stage.
The proposed OM model can be upgraded with new functions, which can be a solid basis for future research on ocular biomechanics. The results of the current study may have implications in both the research and clinical areas for better perspectives and invaluable information for determining the mechanical properties of the eye. This non-invasive technique enhances the understanding of individualized diagnostic solutions, certain ocular pathologies, predicting surgical outcomes, and developing personalized surgical procedures, including refractive surgery. All these research areas are of interest to support and develop ophthalmology diagnostics.
Future development of the described study should focus on improving the geometry of the current model (primarily detailing the profile and material properties of the eye tissues), which greatly impacts the simulation results (especially the anterior parts of the human eyeball). In addition, extensive model verification will allow for accurate numerical analysis (including statistical). In this way, it will be possible to find the general trends of changes in ocular biometric parameters.
Funding
Horizon Europe Marie Sklodowska-Curie ITN-ETN Action, 956720 OBERON project.
Disclosures
The authors clarify that they have NO affiliations with any institutions or entities with any financial or non-financial interest in the subject matter or materials discussed in this manuscript.
Data availability
Data underlying the results presented in this paper are publicly available and may be obtained from the authors upon reasonable request.
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