Page 15 - Advances in Biomechanics and Tissue Regeneration

1.4 PATIENT-SPECIFIC MATERIAL BEHAVIOR 9

based on combinations of patients of a real clinical database (the patient-specific corneal geometry and the Goldmann
IOP [12]) and of corneal material properties of the numerical model to predict the corneal apical displacement.
In brief, the FE model is used to perform a Monte Carlo (MC) simulation in which the material parameters and the
IOP uniformly vary within an established range. The range of the material parameters was determined by considering
the experimental results from an inflation test reported in the literature [48, 55] and the physiological response of the
cornea to an air-puff device (i.e., displacement of the cornea using a CorVis device).
First, the inflation tests were used to initially screen the model parameters, to constrain the search space of the opti-
mization, and to avoid an ill-posed solution [56]. Second, the range of each material parameter was then determined
such that the in silico inflation curve was within the experimental window. In this way, both physiological behaviors of
the cornea are simultaneously fulfilled: the response to an inflation test (biaxial stress) and the response to an air-puff
test (bending stress). Subsequently, the generated dataset was used to implement different predictors for the mechan-
ical properties of the patient’s corneal model in terms of variables that are identified in a standard noncontact
tonometry test.

1.4.1 Material Model
One feasible form of the strain energy function for modeling the cornea corresponds to a modified version of that
proposed by Gasser et al. [57] for arterial tissue, where the neo-Hookean term has been substituted by an exponential
term

N
~
2
X

ψðC,n α Þ¼ D 1 fexp½D 2 ðI 1 3ފ 1g + k 1 fexp½k 2 hE α i Š 1g
2 k 2
α¼1
J 1
2
el (1.2)
2
+ K 0 lnðJ el Þ ,
~
~
def κ ðI 1 3Þ + ð1 3κÞ ðI 4ðααÞ 1Þ,

with E α ¼
p ffiffiffiffiffiffiffiffiffiffiffiffi
detC is the elastic volume ratio; D 1 , D 2 , k 1 , and k 2 are material param-
where C is the right Cauchy-Green tensor; J el ¼
~
eters; K 0 is the bulk modulus; N is the number of families of fibers; I 1 is the first invariant of the modified right Cauchy-
2=3 ~
Green Tensor C¼ J C; and I 4ðααÞ ¼ n α C n α is the square of the stretch along the fiber’s direction n α . The parameter
el
κ describes the level of dispersion in the fiber’s direction and has been assumed to be zero because it has been reported
that a dispersion in the fibers of 10 degrees about the main direction results in a maximum variation of 0.03% on the
maximum corneal displacement [12].

Thestrain-liketerm E α in Eq. (1.2) characterizes the deformation of the family of fibers with preferred direction n α .
The model assumes that collagen fibers bear load only in tension while they buckle under compressive loading.

Hence, only when the strain of the fibers is positive, that is, E α > 0, do the fibers contribute to the strain energy func-

tion. This condition is enforced by the term hE α i,where theoperator h i stands for the Macauley bracket defined as
1
hxi¼ ðjxj + xÞ. The model has been implemented in a UANISOHYPER user subroutine (Abaqus, Dassault Systèmes).
2
Due to the random distribution of the fibers, far from the optic nerve insertion, the sclera has been assumed to be an
isotropic hyperelastic material [58](Eq. 1.3).
3 3
2 i ~ i
X X
Y
ψ ¼ K i ðJ el 1Þ + C i0 ðI 1 3Þ , (1.3)
i¼1 i¼1
where C 10 ¼ 810 [kPa], C 20 ¼ 56, 050 [kPa], C 30 ¼ 2, 332, 260 [kPa], and K i [kPa] is automatically set by the FE solver
during execution.
1.4.2 Monte Carlo Simulation
In order to obtain the personalized corneal material parameters for a given patient, it is necessary to build a reliable
dataset on which to fit or train a predictive model. In the present case, we chose to construct our dataset using an MC
analysis. First, the upper and lower boundaries of the material parameters were searched to restrict the number of
combinations. This prescreening experiment used ex vivo inflation experiments [48, 55] to establish a reliable range
of material parameters that made our simulations behave physiologically under membrane tension. A total of
I. BIOMECHANICS