Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 308

Now, let

! !

1
k CS
1

1
k CS
2

n 1 ¼ max k CS n 2 ¼ max k CS ; 1 ; ; 1 ; ð 11 Þ

1 2

so that n 1 þ 1 ; n 2 þ 1 1 define the factor by which the principal stretches (with
orientations defined relative to the diastolic configuration) increase during the
cardiac cycle. Note the emphasis on cardiac cycle, i.e. we are not distinguishing
between increases that occur between diastole and systole and increases that occur
between systole and diastole. This is necessary because in certain locations,
principal stretches may decrease as the pressure acting on the arterial wall
increases, i.e. k C a \1(a ¼ 1 and/or a ¼ 2). In this scenario, the factor by which
the principal stretches increase during the cardiac cycle is ð 1 = k C a Þ [ 1 : Hence to
determine the factor by which the principal stretches increase during the cardiac
cycle we consider max k CS
a a :

We now define the biaxial stretch index v BSI

; 1 = k CS

to be:

v BSI ¼ min n 1 ; n 2

ð Þ

Þ :

ð 12 Þ

max n 1 ; n 2 ð

Suppose ð k C 1 1 Þ¼ j ð k C 2 1 Þ; i.e. the cyclic (linearised) strain in the direction
of the first principal stretch is a factor j greater than the cyclic (linearised) strain in
the direction of the second principal stretch. Then it is straightforward to deduce
that v BSI ¼ 1 = j : This measure quantifies the degree of the biaxial distortion of the
tissue (during the cardiac cycle) and is independent of the magnitude of the strains.

2.2 Haemodynamic Modelling

To achieve fully developed flow in the region the aneurysm develops, extensions are
attached to the structural model of the artery/aneurysm. The upstream extension is
taken to be a cylinder (radius r S and length 100 mm) whilst a patient-specific illiac
artery bifurcation (obtained from Magnetic Resonance Imaging) is utilised for the
downstream extension [ 47 ]. Given that the real artery is not perfectly cylindrical, a
connecting patch is incorporated to join the geometries together (see Fig. 2 ).
The methodological approach to solve the haemodynamics as the aneurysm
evolves proceeds as follows. The geometry of the aneurysmal section is exported
(see Fig. 1 ) from the structural solver to the meshing suite ANSYS ICEM (ANSYS
Inc, Canonsburg, PA). ANSYS ICEM automatically integrates the aneurysmal
section into the larger geometrical domain, i.e. attaches the upstream and down-
stream extensions, and generates an unstructured tetrahedral mesh with prism layers
lining the boundary in a scripted-automated manner for the fluid domain. After
meshing, appropriate boundary conditions are applied (see below) and the flow is
solved by ANSYS CFX (ANSYS Inc, Canonsburg, PA) which solves the