Introduction

In cataract surgery, creating the capsulorhexis is a critical step1,2. A well-executed capsulorhexis ensures the secure and stable fixation of the IOL within the capsular bag3. Capsulorhexes can be created manually by a surgeon or automatically with a software-driven laser. Laser capsulotomy offers precise and consistent capsulorhexes with a predefined size, shape, and centration. This would in theory help to reduce IOL decentration and tilt. The use of femtosecond laser technology in cataract surgery has raised concerns, however, due to a higher incidence of anterior capsule tears, an important issue that merits careful consideration2,4,5,6. For manual capsulotomy, circular curvilinear capsulorhexis (CCC) is the most commonly performed capsulorhexis2.

Once the IOL performs the optical task of the removed crystalline lens, its position within the capsular bag is essential. IOL decentration or tilt may lead to some reduction of optical/visual quality7,8,9, of which the extent may depend on the IOL design10.

Multiple factors contribute to IOL decentration and tilt, including the ocular condition, capsulorhexis symmetry, and zonule behavior11,12. Zonular instability leads to instability of the IOL13. Besides, Takimoto et al.13 found that in eyes with zonular instability, a capsular tension ring can prevent marked IOL decentration and tilt.

Finite element analysis offers a quantitative approach to examine the impact of capsulorhexis size and centration on IOL alignment. The method allows comprehensive analyses of the IOL’s mechanical behavior under various circumstances, such as a highly decentered capsulorhexis or other capsulorhexis configurations. Many numerical studies related to IOL implantation have been reported, such as by Cabeza et al. who investigated how IOL stability is affected by lens design14 and were able to predict IOL’s stability in the capsular bag15. Meanwhile, Cornaggia et al. simulated different shapes and locations of an IOL by modeling the IOL with a set of tuned nonlinear rod connectors that produce the same force of a folding/unfolding haptic16, Wang et al. developed a model to assess the stability of various IOL types17, Rossi et al. built a capsular bag with capsulorhexes of different sizes and locations18, and Han et al. built a model of the crystalline lens to investigate the tear force during capsulotomy19. These models had their limitations, however, since Cabeza’s and Cornaggia’s models were based on a collapsed capsular bag, and Rossi et al.’s model presumably used the same zonular force for pseudophakia as for the crystalline lens in accommodation. Modelling the capsular bag with a collapsed model may alter the area and mechanical properties of capsular bag. While the forces of zonular fibers during accommodation and after cataract surgery may not be the same, so the actual zonular force in vivo after cataract surgery is largely unknown. Hence, there is currently no study that on an IOL implantation inside a capsular bag with its preoperative shape and zonular fibers based on elastic properties rather than external forces.

This study aims to determine numerically how the size and location of a capsulorhexis affect IOL decentration and tilt using a Finite Element (FE) model. It provides numerical results of IOL decentration and mechanical stability for different types of capsulorhexis and provides a reference of capsular stretching caused by the IOL.

Materials and methods

In this study, tilt is defined as the angle between the central axis of the IOL and the pupillary axis, while decentration is the distance between IOL and pupillary axis in the pupil plane10, as shown in Fig. 1.

Fig. 1
figure 1

The definition of the post-surgical decentration and tilt of the IOL with respect to the pupillary axis. Here, d is the decentration of IOL, the distance between IOL center to the pupillary axis, and α is the IOL tilt.

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Geometry of capsular bag and zonules in the FE model

The model included the structure of capsular bag, IOL, and zonular fibers. The initial shape of the capsular bag was based on a 45-year-old crystalline lens, with a diameter of 8.896 mm and a 1.882 mm gap between the capsule equator and ciliary body, following Burd et al.20, and Strenk et al.21 (Fig. 2). The anterior and posterior capsule curvatures were approximated using fifth-order polynomials as in the research of Brown et al.22. A 3D model was generated with the capsular bag as a thin profile and the ciliary body as a rigid torus, positioned 6.605 mm from the equator of the capsular bag (Fig. 2).

Fig. 2
figure 2

3D configuration of capsular bag with centered capsulorhexis. The capsular bag retained the shape of crystalline lens. The capsulorhexis in this figure has a capsulorhexis of 4.5 mm diameter.

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The zonular fibers, forming a radial network connecting the ciliary body to the lens equator, were modeled with 90 spring dampers equally distributed among anterior, equatorial, and posterior zones, following the 6:1:3 ratio25. The fibers were tension-free in the initial configuration, representing the fully accommodated lens. This approach simulates the fibers’ geometry and elasticity, similar to linear connector elements15.

The IOL considered in this study has the generic shape of a modern 1-piece IOL with two open-loop haptics, designed for the placement into the capsular bag, with a tip-to-tip distance of 13 mm and an optic diameter of 6 mm, and three single-piece TECNIS ZCB00 IOLs (Johnson & Johnson Vision) with different powers of 5 D, 20 D, and 34 D to investigate the effect of IOL power on lens misalignment.

Material properties

The material properties of the capsular bag and zonular fibers were based on experimental data taken from the literature. Although the in vivo mechanical properties of the lens capsule are essential, this data is still incomplete23. Instead, in vitro data of stretching and deformation tests or pressure loading inflation tests with different geometries of capsule18 had to be used. Fisher reported a Young’s modulus of the capsular bag of 3–4 MPa for a 50-year-old capsule23,24, while Krag’s stretching test25 showed a considerably smaller elastic modulus of 1.5 Mpa26. Since Fisher’s result is related to the stiffness at strain larger than 10%, Krag’s elastic modulus was used in the current model.

A linear elastic model was used for the IOL, the values got from the dynamic mechanical thermal analysis (DMTA), with Young’s modulus of 1.911 Mpa, Poisson’s ratio was 0.47.

Fig. 3
figure 3

Discretized geometry in finite element models. The mesh was built with tetrahedral mesh and mapped mesh for capsular bag and IOL, IOL haptics and equatorial capsular bag were meshed with finer elements because of the simulation of the contact between those two parts. a Mesh of capsular bag and ciliary ring, b mesh of IOL.

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For the zonules, Fisher reported no appreciable change in stiffness with age27, while Burd et al. deduced their stiffness by modeling accommodation20. The spring constants of zonules were defined as 66 N/m, 11 N/m, and 33 N/m20, respectively for the anterior, equatorial, and posterior zonules group, for 90 spring dampers in total. Given that the zonules were represented in the model by three groups of 30 springs, the individual value assigned to each spring was derived by dividing the overall spring constant by 30.

Modeling IOL implantation

The model was built in COMSOL Multiphysics and consisted of 39,088 three-dimensional (3D) elements for the IOL, capsular bag and ciliary body (Fig. 3). A mesh refinement study was done by discretizing the capsule and IOL into a mesh with 44,253, 67,023, 238,036 elements, and assessing the IOL decentration and capsular bag equator deformation for each case. The difference in capsular bag deformation between meshes was less than 0.07 μm, and IOL decentration varied less than 0.3 μm, neither of which is detectable in clinic. To best resolve the geometry in the contact area and considering the computing time, the mesh with 67,023 elements was used for analysis.

The process of IOL implantation was simulated in two steps, including a stationary study and a time-dependent study. First, an edge load was applied on the haptic with a value of 0.75 mN. The IOL was compressed to a size that allowed it to be fully contained within the capsule at the equatorial region, ensuring no contact with the inner surface of the capsular bag. The force value applied on the haptics is comparable with the previous studies17,28. During this stage, the ciliary body, a portion of the optics, and the entire capsular bag were fixed, while no force was exerted on the zonules. Consequently, the compression led to an approximate 3.5 mm displacement of the haptic tips towards the center of the IOL optic, effectively encapsulating the entire lens within the capsular bag without contact.

Next, the force acting on the haptics was released in a stepwise fashion, slowly introducing the contact between the IOL haptics and capsular bag to facilitate model convergence. A spring foundation was used to fixate the shape of the capsular bag and the position of the IOL during the contact process to improve the convergence of the solution, and the spring foundations were ramped down to 0 by the auxiliary sweep at the end of the study.

Fig. 4
figure 4

Modeling different locations of capsulorhexis. (left: capsulorhexis shift on X axis, right: capsulorhexis shift on Y axis) used in this study from anterior view and how the lens haptics are aligned with respect to the capsulorhexis. The dashed line shows the centered capsulorhexis. The solid circles show the center of the capsulorhexis with displacement on X or Y axis. The longest approach line between two haptics were aligned with the cartesian coordination X axis.

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Capsulorhexis sizes and locations

To establish a baseline, first a centered capsulorhexis with a diameter of 4.5 mm was considered. The orientation of the intraocular lens (IOL) was specifically set with both tips of the haptics along the X axis (Fig. 4). To analyze the impact of capsulorhexis size, a model with a diameter of 5.5 mm was considered next. Finally, to assess the influence of capsulorhexis location, the center of the capsulorhexis was decentered by 0.5 mm and 1 mm in both the X- and Y- directions. This is repeated for IOLs of three different powers (5D, 20D, and 34D) to determine whether the decentration and tilt induced by the capsulorhexis size and position may be affected by IOL thickness and curvature.

Results

Capsular bag changes

Figure 5 shows the IOL in the capsular bag after the implantation, the FE model reached an equilibrium condition after the haptics stretched and deformed the capsular bag on the equatorial zone. The capsular bag was elongated on the long axis where the capsular bag had the largest diameter, and the short axis where the capsular bag had the shortest diameter was shortened. Long axis and short axis were not necessarily perpendicular in this study.

Fig. 5
figure 5

Intersectional view of ZCB 00 5D IOL in the capsular bag with capsulorhexis shifted 1 mm on Y axis.

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IOL has been implanted into the capsular bag and reached an equilibrium condition.

Figure 6 shows the deformation of the capsular bag along the long and short axis after the implantation of the IOL. The long axis was on average 9.25 mm after IOL implantation, compared to the initial diameter of 8.896 mm, corresponding with a change of 0.35 mm caused by the stretching of IOL haptics. For the short axis the changes were less than 0.1 mm from 0.0218 mm to 0.029 mm. Figure 6 shows that the deviation between the maximum and minimum value of capsular bag deformation was less than 0.01 mm.

Fig. 6
figure 6

Capsular bag deformation after the implantation of IOL. a Long axis change of capsular bag (mm) with different sizes and locations for the capsulorhexis. b Short axis change of capsular bag (mm) with different sizes and locations for the capsulorhexis.

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IOL alignment

The overall IOL decentrations were rather small and increased for larger decentrations along the X- or Y-directions (Fig. 7). Generally, these decentrations were larger along the X-axis than along the Y-axis in both the 4.5 mm and 5.5 mm capsulorhexis models. For the 4.5 mm capsulorhexis models, the decentration ranged from 0.5 μm to 2.4 μm, while for the 5.5 mm models, it varied from 0.43 μm to 8.3 μm. These values are considered too small to have noticeable optical consequences.

The IOL tilt ranged from 0.08° to 0.9° for the 4.5 mm capsulorhexis and from 0.02° to 0.64° for the 5.5 mm capsulorhexis, with larger tilt values for larger capsulorhexis decentration (Fig. 7). Especially the larger tilt values might produce minor amounts of coma aberrations that are unlikely to affect the visual quality by much, especially for monofocal IOLs.

Fig. 7
figure 7

IOL decentration in any direction (d) and tilt (α) with for different sizes and locations of the capsulorhexis. a Decentration of IOL for different capsulorhexes, b Tilt of IOL for different capsulorhexes.

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Looking at the X- and Y-components of the IOL decentration, the X-component was generally largest for capsulorhexis decentrations along the X-axis (Fig. 8). For IOL decentrations along the Y-axis the situation was less clear, as capsulorhexis decentrations in either direction would produce similar levels of IOL decentration. These findings imply that a displacement of the capsulorhexis along axis of the haptic contributes more significantly to IOL decentration.

Fig. 8
figure 8

Decentration of IOL along the X axis (dx) and Y axis (dy) for capsulorhexes with different sizes and locations. a IOL decentration along the X axis. b IOL decentration along the Y axis.

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Comparing different IOLs

No obvious differences in IOL decentration were seen between different lens powers. When the capsulorhexis shifted for 1 mm on X or Y axis, the magnitude of decentration increased with the IOL power; the decentration changed from 1.38 to 21.85 μm for 4.5 mm capsulorhexis, and 3.80 –11.55 μm for 5.5 mm capsulorhexis, as shown in Fig. 9a. Figure 9a shows the tilt of IOLs with different powers. The 34 D IOL had the largest tilt among the three IOLs, and in all cases of the simulation, the tilt was within 0.7 °.

Fig. 9
figure 9

IOL decentration decentrationa and tilt with for different sizes and locations of the capsulorhexis. a Decentration of IOL with capsulorhexis shifted 1 mm for different IOLs. b Tilt of IOLs with capsulorhexis shifted 1 mm for different powers.

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Figure 10 shows the axial movement of IOL anterior center. According to the results, the IOL moved 0.24 –0.37 mm to the posterior capsule, which reflects a hyperopic effect on the refraction of about + 0.37 D. Of the three IOLs considered, the capsulorhexis size and location least affected the position of the 20 D IOL.

Fig. 10
figure 10

Axial movement of IOL anterior center for different sizes and locations of the capsulorhexis. Axial movement of IOL with capsulorhexis shifted 1 mm for different IOLs. The axial movement values below zero means the movement of IOL is towards the posterior capsule direction.

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Discussion

The present study considered a finite element model of a 45-year-old capsular bag with a capsulorhexis to assess the shape changes of the capsular bag by varying the capsulorhexis’ size and location. This altered the shape of the capsular bag and capsulorhexis, as well as the IOL decentration and tilt, all of which were considered for further analysis.

This study modeled the effect of capsulorhexis with a crystalline lens shape capsular bag, and three families discretized zonular fibers, which provides a reference to model the IOL implantation process with a more practical simulation. Besides, this paper first compared the effect of the capsulorhexis on IOLs with different IOL power. After IOL implantation, the capsular bag diameter changed by about 0.35 mm along the long axis and 0.022 mm along the short axis, regardless of capsulorhexis diameter or position (Fig. 6). This is about 1.2 mm smaller than the clinical data by Wormstone et al.29 or the simulation by Cabeza et al.16 (Table 1), which may be due to different choices for the elastic modulus of the capsular bag. Compared with study of Cabeza et al.16, this study modeled the capsular bag with the crystalline lens shape instead of a collapsed capsular bag, and a linear elastic material was applied in this model instead of the Ogden model of Cabeza’s study. Besides, different boundary conditions can also lead to different results. Individual differences of capsule elastic modulus in clinical studies can be a possible reason contributing to the discrepancy of capsular bag deformation.

Table 1 Long axis and short axis of the capsulorhexis after implantation.
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The simulations indicate that the IOL remains well-centered in the capsular bag regardless of capsulorhexis size and position, aligning with clinical observations that IOL decentration and tilt typically match those of the original crystalline lens. Mester et al. mentioned that crystalline lenses and IOLs’ upward tilt were 2.2 degrees and 2.5 degrees, respectively30. In the simulation of this study, a minor shift of up to 8.3 μm was observed with decentered capsulorhexes along the x-axis, accompanied by a tilt of up to 1°. These values are unlikely to noticeably affect visual quality. Clinical data further support this, as Mester et al. reported average IOL decentrations of 0.06 ± 0.10 mm (horizontal) and 0.02 ± 0.18 mm (vertical), with tilt angles around 2.6 ± 1.5° horizontally and 2.5 ± 1.4° vertically30. In addition, study of Zhu et al. reported that the myopic group showed significantly inferior decentration in the capsular bag than the control group (0.21 ± 0.29 mm vs. 0.03 ± 0.22 mm)31, which the authors mentioned was becasue of the increasing size of capsular bag and the axial length. This study simulated the effect with one size of capsule, therefore the impact of myopia was unknown. Tabernero et al.32 similarly reported decentrations between 0.06 and 0.2 mm (horizontal) and − 0.07–0.17 mm (vertical), with tilts of approximately 2.2° horizontally and vertically. In further studies, Rosales et al.33 used Purkinje and Scheimpflug imaging, noting a horizontal decentration of 0.34 ± 0.19 mm (Purkinje) versus 0.23 ± 0.19 mm (Scheimpflug), along with a vertical decentration of 0.17 ± 0.23 mm (Purkinje) versus 0.19 ± 0.20 mm (Scheimpflug). Tilt values were consistent across methods, with averages of 1.89° horizontally and 2.34° vertically. Findl et al.34 observed that eccentric capsulorhexes resulted in a slight increase in decentration and tilt, whereas smaller capsulorhexes (< 4.5 mm) reduced decentration and tilt by approximately 55% and 30%, respectively.

From this follows that the simulated IOL positions and tilts produced by all simulations in the literature are considerably smaller than the clinical values. This disparity between the clinical observations and simulations may in part be attributed to the different reference points used as the baseline center. Clinical studies typically measure decentration and tilt with respect to pupillary center, while this and other simulations consider the center of the crystalline lens as the reference point instead as the iris and pupil are typically not included in the model. Since the lens and pupil are not necessarily aligned, this can lead significant variations between the results obtained from simulations and clinical measurements.

The simulations in this study can also be compared with the work of Cornaggia et al., who found IOL decentrations between 0.29 and 24 μm and tilts between 0.057–12.7˚ for capsulorhexis sizes between 3.5 and 6.5 mm and decentrations varying from 0.5 mm to 3 mm using an IOL modeled as a linear elastic material16 (Table 2). Cabeza et al. simulated multiple lenses, and found tilts of 0.1˚ for a 4.5 mm capsulorhexis, 0.4˚ for a 6 mm capsulorhexis15. Finally, Rossi et al.18 modeled the influence of capsulorhexis sizes between 4 and 8 mm in diameter, reporting no notable changes in IOL decentration and small IOL decentrations along the z and x axis of 1.67–12.7 μm for different capsulorhexis positions, in their study, the IOL was modeled with a prescribed force which was applied on a flattened capsular bag. In this study, the IOL decentration caused by different capsulorhexis locations ranged between 0.04 and 7.9 μm with the reference of centered capsulorhexis, similar to the results by Cornaggia et al. and Rossi et al. The IOL decentration due to changes in capsulorhexis size ranged between 0.04 and 5.25 μm, which fell in the same range of the results of the other simulations. The tilt change caused by the change in capsulorhexis size ranged between 0.0016–0.25˚, comparable to the other simulations. Hence, there is a good overall agreement between the different models, and the remaining differences are likely due to different modelling approaches, such as differences in capsular bag geometry, IOL material, material properties of capsule, and IOL implantation into the capsular bag.

Table 2 Previous research results about simulation results of different sizes and locations of capsulorhexis and its effect on IOL decentration and tilt.
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IOL power seems to affect the decentration and tilt caused by the capsulorhexis size and position, with the overall largest decentration and tilt found for the 34D lens, but these effects are still small and would be barely noticeable to patients. Chen et al. investigated the influence of intraocular lens (IOL) weight on long-term IOL stability in highly myopic eyes. Their results indicated that higher-powered IOLs (≥ 14 D) exhibited greater overall decentration, with the MC X11 ASP and 920 H IOLs showing decentrations of 0.34 mm and 0.38 mm, respectively, compared to lower-powered IOLs (± 5 D), where the same IOLs showed decentrations of 0.17 mm and 0.31 mm, respectively. These results were larger than this study, and suggested that IOLs are more unstable in myopic eyes35. The axial position of the IOL is affected by the lens power, however, as implantation would lead to a shift of 0.24–0.37 mm towards the retina. This is of little clinical consequence, however, as this shift would be accounted for by the IOL-specific constants of the lens power calculation formulas.

In this study, the posterior capsular bag was not fully collapsed towards the IOL, which differs from the clinical situation. Since the removal of the crystalline lens material during surgery is difficult to simulate, the initial force on the capsular bag was considered as zero The omission of initial force on the empty capsular bag can lead to bias.

Another limitation of this study is that only one type of the lens was modeled in this study, so the influence of IOL design and material properties is not considered. The capsulorhexis showed different deformation along long axis and short axis, which may depend on the IOL type and the location of IOL implantation, which alter the stress and strain on anterior capsular bag. More configurations of the capsular bag will be considered in subsequent analyses to see how that affects its mechanical behavior. Similarly, the material properties of the capsular bag varies between studies23, which likely affects the IOL alignment results in the simulations. In addition, the adhesion after surgery may differ following the capsulorhexis, and the potential effect of the adhesion on the IOL stability is not known.

In conclusion, this study showed that the size and decentration of the capsulorhexis have small effects on IOL decentration and tilt after the implantation, given the calculation results from the simulations. A smaller capsulorhexis provides better stability of the IOL in the capsular bag. The capsulorhexis decentration in the direction of the haptics (X direction) contributes more decentration and tilt to the IOL.