Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 307

the artery is a cylinder of length k z L that has radius r s in the systolic configuration and
radius r d in the diastolic configuration. As the material constituents evolve, the
systolic/diastolic deformation fields to satisfy ( 1 ) are updated with a Newton-
Raphson method using the systolic/diastolic deformation fields from the previous
time-step as initial guesses; the positions of the boundaries of the domain (h 1 ¼ 0 ; L 1 )
in the systolic/diastolic configuration are held fixed as the aneurysm evolves.
Cyclic Areal Stretch
Let E 11 ; E 22 ; E 12 and E 11 ; E 22 ; E 12 denote the GL strains, in the diastolic (D) and
systolic (S) configurations, respectively. The cyclic areal stretch, here denoted A CS ;
is the ratio of the areal stretch at systole to the areal stretch at diastole, which
expressed in terms of the Green-Lagrange strains is:

s

ð 1 þ 2E 11 Þð 1 þ 2E 22 Þ 4E 12
ð 1 þ 2E 11 Þð 1 þ 2E 22 Þ 4E 12

A CS ¼ : ð 6 Þ

Biaxial Stretch Index v BSI
In the initial geometrical configuration, the artery is subject to cyclic stretch that
is one dimensional, i.e. the cyclic stretch is only in the circumferential direction.
As the aneurysm evolves regions of the artery experience biaxial stretching. To
characterise the evolution of the cyclic stretch environment, we consider a novel
Biaxial Stretch Index (BSI), denoted v BSI where 0 v BSI 1 : If v BSI ¼ 0 ; the
tissue is subject to 1D cyclic stretching whereas if v BSI ¼ 1 ; the tissue is subject to
equi-biaxial cyclic stretching. The index v BSI is calculated as follows. The prin-
cipal stretches k 1 ; k 2 can be expressed in terms of the GL strains (see [ 46 ]), i.e.

q

cos 2 u ð 2E 11 þ 1 Þþ sin 2 u ð 2E 11 þ 1 Þþ 2 sin u cos uE 12

k 1 ¼ ; ð 7 Þ

q

sin 2 u ð 2E 11 þ 1 Þþ cos 2 u ð 2E 11 þ 1 Þ 2 sin u cos uE 12 k 2 ¼ ; ð 8 Þ

where

u ¼ 1

2E 12
E 11 E 22 : 2 arctan

ð 9 Þ

We then denote the principal stretches in the diastolic configuration as
k a ¼ k a ð E 11 ; E 22 ; E 12 ; u D Þ (a ¼ 1 ; 2) where u D ¼ u D ð E 11 ; E 22 ; E 12 Þ is calculated
using ( 9 ). We then determine the magnitude of these principal stretches (with ori-
entations defined relative to the diastolic configuration) in the systolic configuration,
i.e. k D a ¼ k D a ð E 11 ; E 22 ; E 12 ; u D Þ: The cyclic variation of the (diastolic) principal
stretches, denoted k C a ; are thus:

¼ k DS k CS

a

a
k a

: ð 10 Þ