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by Mariarita de Luca
PhD student at MOX, Math. Dep, Politecnico di Milano
Visiting student at MeMS, University of Pittsburgh
The purpose of this work is to use mathematical and numerical tools to study
the nonlinear, inelastic behavior of the arterial wall, in healthy and diseased
states. Particular emphasis is given to cerebral arteries and their pathologies
called aneurysms. Saccular Aneurysms are abnormal dilations of cerebral arte-
ries, primarily found at apices of arterial bifurcations and on the outer wall of
curved arterial segments in or near the Circle of Willis. These pathologies are
highly outstanding since the rupture of the aneurysm wall causes a subarachnoid
hemorrhage, which can lead to death or to severe disability.
The two central passive load bearing components of healthy arterial walls
are elastin and collagen. However, the wall of aneurysms has little or no elastin,
in stark contrast to the surrounding arteries. Thus, in order to model the
development of an aneurysm from a segment of arterial wall, it is necessary to
include the breakdown of the elastin separately from the bulk failure of the wall.
In the dual mechanism model, proposed by Wulandana and Robertson, elastin
and collagen are modeled as separate mechanisms, making it possible to do so.
At low loads, only the elastin mechanism is active. As loading is increased the
collagenmechanism is recruited. The ultimate failure of the elastin mechanism is
determined by an activation criterion which can simply depend on the current
state of strain or be based on accumulated damage (arising from mechanical
loading or biomechanical response) [Li, Robertson].
In this work, we study the arterial wall models using a non-commercial code
called LIFEV, which is a finite element (FE) library providing implementations
of state of the art mathematical and numerical methods. The structural solver
which has been developed through this work was originally realized in the fluid-
structure interaction perspective. Two nonlinear, slightly compressible, hypere-
lastic materials are considered. The first is a single mechanism Venant-Kirchhoff
model, in which there is a linear relationship between the stress and displace-
ment gradient. This model has been shown not to be polyconvex, so there is no
guarantee of existence of a solution and well-posedness of the problem in high
deformations range. The second model is a dual mechanism model with two
exponential, isotropic mechanisms. The exponential strain energy function is
polyconvex, so the existence of a solution and the well-posedness of the problem
are guaranteed. The main contribution of this work is to develop a new me-
chanical model that is able to capture the complex behavior of the arterial wall,
and in particular to understand the onset and increase of cerebral aneurysms.
Simplified geometry as well as a real geometrical model obtained by CT scans
have been used to realize the numerical simulations.
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