Page 103 - Advances in Biomechanics and Tissue Regeneration

6.2 BASICS OF AORTIC WALL MECHANICS AND PASSIVE BIOMECHANICAL ROLE OF SMCS 99

FIG. 6.3 Schematic representation of the mechanical stresses in the aortic wall, and particularly in the media. The intima is neglected in the case of
aneurysms, but it cannot be the case for pathologies resulting in an intimal thickening; t is the thickness of the wall, r is the internal aortic radius, P is
the blood pressure, and t MLU is the mean thickness of an MLU.

6.2.3 Passive Mechanics of the Aortic Tissue
The passive behavior refers to the behavior of the aortic wall in the absence of vascular tone. It is mainly due to ECM
components, namely elastin and collagen fibers. If the elastin is responsible for the wall elasticity, the collagen fibers
are progressively tightened from their initial wavy configuration while the wall stress is increasing, and they tend to
protect the other components from overstress [10, 21, 57].
Given that the tissue contains about 70%–80% of water, it is often assumed as incompressible. As a heterogeneous
composite material comprising a fluid part (i.e., water) and a solid part (i.e., ECM and cells) [14, 27], divided into sev-
eral layers with different mechanical properties (see Section 6.2.1.2), the aortic wall has a complex anisotropic mechan-
ical behavior. To predict the rupture risk of ATAAs [14, 27, 57], the passive mechanical behavior of the ECM is relevant.
Numerous in vitro tests using the bulge inflation device [7, 8, 64–67] confirmed that elastin in the media is the weak
element of the wall toward rupture.

6.2.4 Multilayer Model of Stress Distribution Across the Wall

Single-layered homogenized models of arterial wall mechanics have provided important visions of arterial func-
tion. For example, Bellini et al. [68] proposed a bilayer model with different material properties for the media and
adventitia layers. They split the passive contributions of elastin, SMC, and collagen fibers (modeled with four different
families). Eventually, the strain-energy function (SEF) at every position may be written as [68, 69]:
n
X m m m
e
e
e
W ¼ ρ W ðI Þ + c i c i c i (6.3)
4
4
1 ρ W ðI Þ + ρ W ðI Þ
i¼1
where superscripts e, c i , and m represent, respectively, the elastin fiber constituent, the constituent made of each of the
n possible collagen fiber families, and the SMC constituent, with all these constituents making the mixture. In Eq. (6.3),
j
j
j
ρ refers to mass fraction, and W stands for the stored elastic energy of each constituent, depending on the first (I ) and
1
j
fourth (I ) invariants of the related constituents of the mixture (j 2{e, c i , m}). Let the mechanical behavior of the elastin
4
constituent be described by a neo-Hookean SEF as in Refs. [68, 70–72]
μ e
e e e (6.4)
1
1
W ðI Þ¼ ðI 3Þ
2
e
eT e
e
e
e
where I ¼ trðC Þ and μ is a material parameter with a stress-like dimension. C ¼F F denotes the right Cauchy-
1
e
e
Green tensor where F ¼ FG h is the deformation gradient of the elastin constituent. F is the corresponding deformation
e
gradient of the arterial wall mixture and G h is the deposition stretch of elastin with respect to the reference
I. BIOMECHANICS