# Biomechanical Study of the Eye with Keratoconus-Type
## 2 Materials and Methods
### Abaqus(r): Undformed Geometry and Finite Elements Analysis  
To perform finite element analysis, it is necessary to know the mechanical properties of the material used. In this case of the cornea, it is a challenging task to complete because it is a live tissue that is not likely to be uniform [21]. However, because it is a matter of great importance for the development of new applications, some studies have attempted to model corneas' material behavior using different approaches [22]. By conducting in vivo tests, some studies have attempted to model the cornea numerically, including linear, as in  
Figure 1: Description of the process of construction of the solid model (right cornea): (**a**) point clouds of the anterior and posterior surfaces of the cornea. (**b**) square-like surfaces obtained from the point clouds. (**c**) cylinder that sliced the surfaces to match its circular form, and (**d**) final stage of the 3D model. elastic models, nonlinear, as in viscoelastic models [23], or hyperelastic [3; 24; 25] materials. Others have described the cornea as anisotropic due to the collagen fibers' particular distribution [26; 27]. For this model of the cornea, we considered isotropic hyperelastic materials for nonpathological and keratoconus eyes described in [24] as second-order Ogden models, with specific constants due to their computational simplicity and material data that were more readily accessible [24]. The incorporation of anisotropic materials is beyond the scope of this study. Boundary conditions for this model were placed at the edges of the cornea with respect to the horizontal axis; it is of interest to mention that the model was constrained against displacement and rotation (Figure 3) [28]. In addition, the mesh used for these models was hexahedral and had a lineal geometrical order. The seed size was 250 \(\upmu\)m. This meshing resulted in the normal model having 4524 elements and 5910 nodes, the keratoconus cornea with 3285 elements and 4584 nodes, and the model for keratoglobus with 3504 elements and 4876 nodes (Figure 4). Figure 3: Boundary condition of displacement and rotation applied to the edge of the cornea model. Figure 2: Side view of the keratoconus cornea. Concerning the loads that were taken into consideration for the simulation, this study contemplated that since the model was based on in vivo data collection, it was already deformed by forces such as normal IOP or atmospheric pressure. An important part of simulating a patient-specific cornea is retrieving the undeformed geometry related to the unstressed state. The data obtained from the Pentacam correspond to a physiological state in which there are already forces acting upon the model. The most significant would be IOP, which varies among patients. In this work, the recovery of undeformed geometry was preferred over the pre-stress state. In this matter, the cornea models were subjected to successive analysis under physiological IOP (normal = \(-1.8\) mmHg; keratocnus = \(7.1\) mmHg; keratoglobus = \(13.9\) mmHg), as described in [26], in which the first set of coordinates were the physiological corneas:  
1. The IOP was applied to the internal surface of the cornea. 2. The resulting deformation was subtracted from the physiological state, thus creating a new set of coordinates. 3. The new set of coordinates was compared to its predecessor by the magnitude of the difference in their sequential coordinates. 4. The first three steps were iterated until the magnitude was smaller than the defined tolerance (5% of the minimum thickness of that cornea). 5. The obtained undeformed geometry was used to conduct further simulations. The physiological IOPs were applied as a load to the obtained models. Simultaneously, this simulation worked with the IOP variation in the eye experiences throughout the day of approximately \(1.5\) mmHg or \(199.984\) Pa [28] to compare how different corneas would behave in the same increment of IOP.