Page 73 - Advances in Biomechanics and Tissue Regeneration

68 4. MECHANICAL AND MICROSTRUCTURAL BEHAVIOR OF VASCULAR TISSUE

where α represents the amount of cross-links, being α 2 [0, 0.5]. For the extreme cases, α ¼ 0 means no cross-links on the
tissue while for α ¼ 0.5, the degree of links is high enough to consider a fully isotropic distribution of the fibers. The
physical meaning of this parameter α is the inclusion in the model of the attachments between the main fibers of col-
lagen, which provides a certain degree of stiffening in the transverse direction of these fibers.

4.2.2.3 Microstructural Model
Alastru e et al. [36] proposed a microsphere-based model to account for the dispersion of the collagen fibers around a
preferential direction, overcoming the one-dimensional (1D) limitation of the previous characterization of the collagen
fiber. 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 2 coll ŠÞ dA, (4.7)
j¼1 j¼1 j¼1 
where n is the chain density, N denotes the number of families of collagen fibers, N ¼ 2 according to the experimental
results of orientation of collagen fibers [9], and applying a discretization to the continuous orientation distribution on
2
j
the unit sphere  ,[Ψ coll ] corresponds to the expression
m
j X nρðr ;Z,QÞψ i ðλ i
i
½Ψ coll Š ¼ coll coll Þ, (4.8)
i¼1
2
i
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 the r direction, and ψ coll (λ coll ) the strain energy
i
density function associated with the r direction. Using Eqs. (4.7), (4.8) it results in
N m
XX i i j
ðw nρ½ψ ŠÞ ,
Ψ coll ¼ coll (4.9)
j¼1 i¼1
i
where w , i ¼ 1, …, m denote related weighting factors and ρ is the orientation density function (ODF) to take into
account the fibril dispersion [7].
In this work, we used the Bingham ODF [41] initially proposed by Alastru e et al. [36] for the incorporation of anisot-
ropy in a microsphere-based model with application to the modeling of the thoracic aorta. One of the main advantages
of the Bingham ODF is the possibility of considering three different concentration parameters in three orthogonal
directions of the space. These orientations can be easily correlated with the three main directions of a blood vessel:
circumferential, radial, and axial. That function is expressed as
dA 1 t t dA
4π 4π
ρðr;Z,QÞ ¼½F 000 ðZފ expðtrðZ Q r r QÞÞ , (4.10)
3
where Z is a diagonal matrix with eigenvalues κ 1, 2, 3 , Q 2  defines the orientation of the three principal orthogonal
directions with respect to the reference basis, and
1 Z t
4π  2
F 000 ðZÞ¼ expðtrðZ r r ÞÞdA: (4.11)
Thus, the probability of finding a find in a specific direction is controlled by the eigenvalues of Z, which might
be interpreted as concentration parameters. Specifically, the difference between pairs of κ 1, 2, 3 —that is, [κ 1 κ 2 ],
[κ 1 κ 3 ], and [κ 2 κ 3 ]—determines the shape of the distribution over the surface of the unit sphere. Therefore, the
value of one of these three parameters may be fixed to a constant value without reducing the versatility and different
distributions of a family of fibers achieved for a constant value of κ 2 and varying values of κ 1 and κ 3 .
The exponential-like SEF proposed by Holzapfel et al. [9] was used to approach the fiber response
i 2 2
nψ i ðλ i c 1coll e c 2coll ððλ coll Þ 1Þ 1 if λ i 1 otherwise ψ ðλ i Þ¼ 0,
coll coll Þ¼ f i (4.12)
2c 2coll
because it is usually considered that collagen fibers only affect the global mechanical behavior in tensile states; see
i
i
i
Holzapfel et al. [9]. The affine kinematics define the collagen fiber stretch λ coll ¼kt k in the fiber direction r . Finally,
c 1coll and c 2coll are stress dimensional and dimensionless material parameters, respectively.

I. BIOMECHANICS