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4.2 MICROSTRUCTURAL MODELING OF THE CAROTID ARTERY 67
For the elastin, we use the classical neo-Hookean SEF
Ψ elas ¼ μ ½I (4.2)
elas 1 3
with I 1 the first invariant of the symmetric Cauchy-Green tensor and μ elas is a stress dimensional material parameter.
Following Bellin et al. [32], the behavior of SMC is modeled using an exponential function
c 1vsmc 2 2
½expðc 2vsmc ½λ
Ψ vsmc ¼ vsmc 1 Þ 1 (4.3)
2c 2vsmc
with λ vsmc ¼kF mk the stretch experienced by the smooth muscle where m is the unit vector associated with the
SMC orientation, and c 1vsmc and c 2vsmc are stress dimensional and dimensionless material parameters, respectively.
The contribution of the smooth muscle to the SEF was only considered in the media of the vessel and the SMCs were
supposed to be circumferentially oriented [33].
Ψ coll defines the anisotropic contribution of the collagen fiber. Based on the previous literature and histological
results, four SEFs representing collagen fibers are chosen: a PM [34], a cross-linked phenomenological model
(CLPM) [35], an MM [36], and a cross-linked microstructural model (CLMM) [27].
4.2.2.1 Phenomenological Model
Holzapfel et al. [34] proposed a multilayer fiber-reinforced composite model for the arterial wall that considers
the histological structure of arteries with a preferential direction for the fiber orientation and with an exponential
form that represents the prominent stiffening characteristics of the arterial collagen
c 1coll 2 2
X
½expðc 2coll ½ρ½I i 1 + ½1 ρ½I 1 3 Þ 1:
Ψ coll ¼ (4.4)
2c 2coll
i¼4,6
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] while I 4 1 and I 6 1 characterize the mechanical response in the preferential
directions of the fibers/cells [34]. c 1coll > 0 is the stress-like parameter and c 2coll > 0 is the dimensionless parameter.
The ρ 2 [0, 1] parameter is also dimensionless and accounts for the fiber dispersion. The SEF represents the strain
energy stored in a composite material reinforced in two preferred directions represented by the invariants I 4 and
I 6 . Both invariants can also be expressed as a function of the main stretches, Eq. (4.5). Due to each family of fibers rep-
resents the main direction of collagen bundles (θ 1 and θ 2 ) that are orientated in a helicoidal manner and both families of
fibers were assumed to have the same mechanical response. The anisotropy directions were assumed to be oriented at
θ degrees with respect to the longitudinal axis [9], so θ 1 ¼ θ and θ 2 ¼ θ in Eq. (4.5), see Holzapfel et al. [9].
2
2
2
2
2
2
2
2
I 4 ¼ λ cos θ 1 + λ sin θ 1 , I 6 ¼ λ cos θ 2 + λ sin θ 2 : (4.5)
1 2 1 2
This model was used by García et al. [3] to fit the mechanical properties of the porcine carotid arteries by uniaxial
tensile tests where θ was treated as an unknown phenomenological variable included in the minimization algorithm.
Proximal and distal samples were fitted separately considering independent material parameters for each position. It is
emphasized that the model defined by Eq. (4.17) should be considered as a phenomenological approach and cannot be
regarded as a structural model. Note that the parameter ρ has no histological meaning.
4.2.2.2 Cross-Linked Phenomenological Model
O’Connell et al. [40] in rat aortas and Sáez et al. [27] in pig carotids have shown that collagen fibers are bundled
around the SMC with some collagen fibrils linking the main fiber in a predominant perpendicular direction. In the case
of the media layer of carotid arteries, previous studies reported that SMC keep a highly predominant circumferential
direction [27]. With this orientation obtained experimentally, the PM presented in Holzapfel et al. [34] was unable to fit
the uniaxial test. The CLPM was initially proposed by Sáez et al. [35] and used to fit the mechanical properties of the
porcine carotid arteries from experimental data presented in García et al. [3], where the θ parameter was directly fixed
to the circumferential direction [27, 40].
A linear interpolation of a well-known isotropic and anisotropic SEF was used by Sáez et al. [35] to recover this
behavior as
c 1coll 2 c 1coll 2
X
½expðc 2coll ½I i 1 Þ 1 + α ½expðc 2coll ½I 1 3 Þ 1,
Ψ coll ¼ ½1 2α (4.6)
2c 2coll 2c 2coll
i¼4,6
I. BIOMECHANICS
