Page 79 - Advances in Biomechanics and Tissue Regeneration

74 4. MECHANICAL AND MICROSTRUCTURAL BEHAVIOR OF VASCULAR TISSUE

In this equation, I 1 represents the first invariant of the Cauchy-Green tensor [37] characterizing the isotropic mechan-
ical response of the elastin [38, 39]. μ > 0, k 1 > 0, and k 3 > 0 are stress-like parameters and k 2 > 0 and k 4 > 0 are dimen-
sionless. The SEF represents the strain energy stored in a composite material reinforced in two preferred directions
represented by the invariants I 4 > 1 and I 6 > 1, where it has been assumed that the strain energy corresponding to
the anisotropic terms only contributes to the global mechanical response of the tissue when stretched. A total of four
elastic parameters (μ, k 1 , k 2 , θ) should be fitted.

4.3.4 Structural Model
The Gasser, Ogden, and Holzapfel (GOH) model [8] extended the model of Holzapfel et al. [9] by the application of
generalized structure tensor H ¼ κ1+ (1 3κ)M 0 (where 1 is the identity tensor and M 0 ¼m 0
m 0 is a structure tensor
defined using unit vector m 0 specifying the mean orientation of fibers) and proposed a new constitutive model

k 1
X
^
Ψ ¼ μðI 1 3Þ + ðexpfk 2 E i g 1Þ , (4.19)
2k 2
i¼4,6
where
^
E i ¼ κI 1 + ð1 3κÞI i 1 i ¼ 4,6 (4.20)
and κ 2 [0, 1/3] is a dispersion parameter (the same for each collagen fiber family); when κ ¼ 0, the model is the same as
the one published in Holzapfel et al. [9], and when κ ¼ 1/3, it recovers an isotropic potential similar to that used in
1 R π 3
4 0
Demiray [45]. Note that the parameter κ ¼ ρsin θdθ could have histological meaning due to the fully characterized
distribution [8]. A total of five elastic parameters (μ, k 1 , k 2 , κ, θ) should be fitted.
4.3.5 Microfiber Model
As commented on in the carotid section, Alastru e et al. [36] proposed a microfiber model (microsphere-based
model) to account for the dispersion of the collagen fibers around a preferential direction, overcoming the 1D limitation
of previous characterizations of the collagen fiber. Consistent with the constrained mixture approach [31]
Ψ ¼ μðI 1 3Þ + Ψ coll , (4.21)
where the subscript coll refers to collagen fiber contribution. Ψ coll is defined as the sum of the contributions of each
collagen family of fibrils as
N N N Z
j hnρψ j ðnρ½ψ j
X X X 1
Ψ coll ¼ ½Ψ coll Š ¼ coll i ¼ 4π coll ŠÞ dA, (4.22)
j¼1 j¼1 j¼1  2
where N denotes the number of families of collagen fibers, N ¼ 2 according to the experimental results of the orien-
tation of collagen fibers [9], and applying a discretization to the continuous orientation distribution on the unit sphere
j
2
 ,[Ψ coll ] corresponds to the expression
m
j X nρðr Þψ i ðλ i
i
½Ψ coll Š ¼ coll coll Þ, (4.23)
i¼1
i
2
where r are the unit vectors associated with the discretization on the microsphere over the unit sphere  , m is the
i
i
i
i
i
number of discrete orientation vectors [7], λ coll ¼kF r k the stretch in r direction, and ψ coll (λ coll ) the SEF associated
i
with r direction. Using Eqs. (4.22), (4.23), this results in
N m
XX i i j
ðw nρ½ψ ŠÞ ,
Ψ coll ¼ coll (4.24)
j¼1 i¼1
where w i denotes related weighting factors and ρ is the ODF to take into account the fibril dispersion [7].
i¼1,…,m
The exponential-like SEF proposed by Holzapfel et al. [9] was used to deal with the fiber response
i 2 2
c 1coll
nψ i ðλ i e c 2coll ððλ coll Þ 1Þ 1 if λ i 1 otherwise ψ ðλ i Þ¼ 0, (4.25)
coll coll Þ¼ f i
2c 2coll
because it is usually considered that collagen fibers only affect global mechanical behavior in tensile states [9]. The
i
i
affine kinematics define the collagen fiber stretch λ coll ¼kt k in the fiber direction r . i
I. BIOMECHANICS