Page 16 - Advances in Biomechanics and Tissue Regeneration

10 1. PERSONALIZED CORNEAL BIOMECHANICS

81 combinations were simulated to mimic the inflation experiments. The in silico inflation curves were then compared
with experiments [48, 55] and the range of material parameters leading to curves within the experimental window was
determined. The identified range of parameters was set to D 1 [kPa] 2 (0.0492, 0.492), D 2 [ ] 2 (70, 144), k 1 [kPa] 2 (15,
130), and k 2 [ ] 2 (10, 1000).
Afterward, the MC analysis was used to generate the dataset. A uniform distribution of the material parameters was
assumed because there are no a priori data on the dispersion of the mechanical parameters in the human cornea and,
therefore, a total ignorance about the population is assumed. Otherwise, a bias could be introduced on the outcome of
the system. Additionally, to account for the physiological diurnal variations in the IOP [59], variations in the IOP rang-
ing from 8 to 30 mmHg along with the patient’s IOP at the moment of the examination were also considered in the MC
simulation. Hence, for each available geometry in the clinical database, 72 different samples of the material parameters
and the IOP, uniformly distributed in their respective ranges, were used to conduct 72 simulations of the noncontact
tonometry test. Consequently, a total of 9360 computations (i.e., 72 combinations 130 geometries) was scheduled.
The generated dataset consisted of the following variables: classification (healthy, KC, and LASIK), computation exit
status (failed or successful), material parameters (D 1 , D 2 , k 1 , and k 2 ), IOP, CCT, nasal-temporal curvature (R h ), superior-
inferior curvature (R v ), and the computed maximum displacement of the cornea (U num ).
After the dataset was generated, an ANOVA analysis was done to identify the most influential model parameters
(geometry, pressure, and material) on the numerical displacement, U num , obtained with the noncontact tonometry sim-
ulation. The results from this analysis were used to identify the geometric parameters to be included in the construction
of the predictor functions for the material parameters. ANOVA was conducted on the global dataset without differ-
entiation between the populations and for each of the populations (healthy, Keratoconus or KC, and LASIK). Because
the dataset is randomly generated, ANOVA cannot be conducted directly on the data. Instead, a quadratic response
surface was first fitted to U num (e.g., U num ¼ f(geometry, pressure, material)). Then, a Pareto analysis (i.e., it states
the most influential parameters on an objective variable, arranging them in decreasing order by taking into account
the cumulative sum of the influence until reaching a 95% variation on the objective variable) was used to determine the
most influential parameters on the dependent variable, U num .
The simulations show that the proposed material model is adequate to reproduce both the inflation and the bending
response of the cornea when subjected to an air puff for different levels of the IOP (see Fig. 1.5A). In particular, the range
of parameters used for the MC simulation is able to accommodate the experimental response to corneal inflation tests
reported in the literature (see Fig. 1.5B). Note that traditional model development of corneal mechanics has mainly con-
sidered inflation tests to identify the model parameters. However, when the response to an air puff is considered, we
found that there are a number of combinations for which the inflation response is within the experimental range but
the corneal displacement due to the air puff is not. An example of this situation is given by the red and blue lines in
Fig. 1.5A. In both cases, the response to the inflation test is identical, but the response to the air puff is not physiological
for the red line. Therefore, from the total number of samples in the MC simulation, only those samples that reconcile the
response to an inflation and to an air-puff test to be within the experimental ranges were considered [9, 13, 60]. After
including this exclusion criterion, only 29% (1127 of 3855) of the healthy cases, 30.5% (1327 of 4344) of the KC cases,
and 21.5% (219 of 1017) of the LASIK cases were included in the training dataset (see Fig. 1.5C for a healthy population).
The empirical distribution of the material parameters related to the matrix (D 1 and D 2 ) did not follow a uniform
distribution, whereas those related to the fibers (k 1 and k 2 ) were found to be uniformly distributed (the results are
shown in Ariza-Gracia et al. [17]). A Kolmogorov-Smirnov test showed nonsignificant differences between the mate-
rial parameters of the healthy LASIK and the KC LASIK populations (see Table 1.1). By contrast, significant differences
were found for D 1 and D 2 between the healthy KC populations.
When the cornea is under the action of the IOP (i.e., its physiological stress state), the cornea is in a membrane stress
state where the full cornea works in tension (i.e., both extracellular matrices and both families of collagen fibers), and
therefore, no bending effects exist. However, during an air puff, the cornea experiences bending. Whereas the anterior
surface goes from a traction state of stress to a compression state of stress, the posterior surface works in tension.
Hence, in the anterior corneal stroma, the collagen fibers are not contributing to load bearing because they do not sup-
port buckling and the stiffness of the cornea mainly relies on the extracellular matrix. At the same time, the collagen
fibers on the posterior stroma suffer from a higher elongation, resulting in an overall nonphysiological state of stress. In
this regard, due to the action of the IOP, no significant differences in the maximum principal stress and in the max-
imum principal stretch were observed between the different populations for both the anterior and posterior corneal
surfaces. In contrast, when the maximum principal stress and stretch are compared at the instant of the maximum
corneal displacement, significant statistical differences between all populations were found on the posterior surface
(see Table 1.2). However, at the anterior surface, significant differences were found only for the maximum principal
stretch, whereas for the maximum principal stress, differences were found only between the healthy and KC popula-
tions (see Table 1.2).

I. BIOMECHANICS