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4.2 MICROSTRUCTURAL MODELING OF THE CAROTID ARTERY 69

4.2.2.4 Cross-Linked Microstructural Model
In an attempt to bring together the advantages from the MM and the CPM, Sáez et al. [27] proposed an anisotropic
contribution of the collagen fibers to the SEF as
1 Z
f
f ½ρ + α^ ρŠψ dA, (4.13)
Ψ coll ¼h½ρ + α^ ρŠψ ðλÞi ¼ 4π  2
where ρ denotes the Bingham ODF; see Eq. (4.10). In addition, ^ ρ ¼½maxðρÞ ρŠ=maxðρÞ is an effective orientation
density that accounts for cross-links between the main collagen fibers. ^ ρ represents the physical space where the main
collagen fibers do not exist, and therefore collagen cross-links can be interpreted similarly to the one described in the
CLPM. Specifically, α 2 [0, 1] represents the relative amount of cross-links with α ¼ 0 and α ¼ 1 denoting no cross-links
and fully cross-linked collagen, respectively. The fully cross-linked state results in an isotropic response of the collagen.
2
Finally, the numerical integration of Eq. (4.13), using m integration points over the unit sphere  , gives
m
X i i
w i ½ρ + α^ ρ Šψ ðλ Þ, (4.14)
Ψ coll ¼ i i coll coll
i¼1
i
with w i denoting integration point weights and ψ coll was defined in Eq. (4.12).
This model was formerly used by Sáez et al. [27] for fitting the mechanical properties of the porcine carotid arteries
by uniaxial tensile tests where proximal and distal samples were adjusted separately considering independent material
parameters for each position. Again, the physical meaning of this parameter α is the inclusion of the attachments
between the main fibers, which provides an additional reinforcement to the collagen tissue in the transverse direction
of the main fibers [35] described previously in the MM.

4.2.3 Results on Modeling the Porcine Carotid Artery
The fitting of the experimental mechanical data (Fig. 4.3) was developed by using a Levenberg-Marquardt minimi-
zation algorithm [42], by defining the objective function represented in Eq. (4.15). In this function, σ θθ and σ zz are the
Ψ
Ψ

Cauchy stress data obtained from the tests, σ and σ are the Cauchy stresses for the ith point computed using the SEF
θθ
zz
(Eq. 4.21) under the hypothesis of incompressibility [43], and k is the number of data points.
Ψ 2
Ψ 2
χ ¼ Σ k ½ðσ θθ σ Þ + ðσ zz σ Þ Š, (4.15)
2
i¼1 θθ i zz i
where
Ψ ∂Ψ iso Ψ ∂Ψ iso
σ ¼ λ θ σ ¼ λ z : (4.16)
θθ
zz
∂λ θ ∂λ z
The coefficient of determination of the normalized mean square root error ε 2 [0, 1] was computed for each fitting
s ffiffiffiffiffiffiffiffiffiffi
χ 2
k q Σ k ðσÞ i
ε ¼ ϖ . In this equation, ϖ is the mean value of the measured stresses, ϖ ¼ i¼1 k , q is the number of parameters
of the SEF, so k q is the number of degrees of freedom, and ϖ the mean stress already defined earlier. ε 0.1 typically
represents a good fit to the experimental data.
Both data (proximal and distal, see Fig. 4.3) were fitted at the same time considering only one set of parameters. That
is, we considered that the mechanical properties of the elastin, VSMCs, and collagen do not change along the carotid.
The values of ϕ r , θ r , and κ 1, 2, 3 were taken from Sáez et al. [27] and are given in Table 4.1. The values of ϕ elas , ϕ vsmc , and
ϕ coll were taken from the mean values for distal and proximal positions of the porcine carotids from García et al. [3]. For
PM and CLPM approaches, a total of nine elastic parameters (μ elas , c 1vsmc , c 2vsmc , c 1coll , c 2coll , ρ prox , ρ dist , θ prox , θ dist ) and
seven elastic parameters (μ elas , c 1vsmc , c 2vsmc , c 1coll , c 2coll , α prox , α dist ), respectively, should be fitted. Furthermore,
the number of adjusting variables for the MM and the CLMM is five elastic parameters (μ elas , c 1vsmc , c 2vsmc , c 1coll , c 2coll )
and seven (μ elas , c 1vsmc , c 2vsmc , c 1coll , c 2coll , α prox , α dist ), respectively.
The results of the fitting of the different SEFs proposed to the experimental data for the distal and proximal curves
are shown in Fig. 4.4. The material parameters are summarized in Table 4.2. In terms of fitting errors, both phenom-
enological models, PM and CLPM, present relatively low values, ε 0.05 0.11 and ε 0.08 0.12, respectively. How-
ever, the values of θ prox ¼ 43.4 degrees for PM contradicts experimental findings [27]. Contrarily, the MM resulted in
very large error values (ε p ¼ 0.74 and ε d ¼ 0.37) for the proximal and distal zones, respectively, failing to reproduce

I. BIOMECHANICS