Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 305

Lagrange (GL) strains of the elastin, i.e. E 11 ; E 22 and E 33 ; are defined relative to the
unloaded configuration; in the initial cylindrical configuration, these represent
strains in the axial, azimuthal and radial directions, respectively. The GL strains in
the collagen fibres are denoted by E J p ð h 1 ; h 2 ; t Þ and are defined relative to the con-
figuration in which the collagen fibres are recruited to load bearing. More specifi-
cally, the GL strains E J p of the collagen fibres are a function of the GL strains of the
elastin resolved in the directions of the collagen fibres, denoted E J p : Thus,

E J p ¼ E J p E J p ; ð 3 Þ

1 þ 2E J p

E J p ¼½ð k J p Þ 2 1 = 2 ; k J p ð h 1 ; h 2 ; t Þ are

where the recruitment stretches and

E J p ¼ E 11 sin 2 c J p þ E 22 cos 2 c J p þ 2E 12 sin c J p cos c J p :

2.1.2 Geometry, Physiological Data and Material Parameters

The material parameters for the media and the adventitia as well as all other values
which serve the basis for our subsequent computation are taken from [ 16 ] and are
summarized in Table 1 . In the systolic configuration, the artery has a radius r s ; an
axial pre-stretch of 1.3, a circumferential stretch of 1.25 and we assume that the
elastinous constituents bear 80 % of the load. We assume that the elastinous
response of the adventitia is an order of magnitude weaker than that of the media,
i.e. K A ¼ K M = 10 [ 32 ] and specify the ratio of the medial and adventitial collagen
material parameters to be K A ¼ K M = 4 : The remaining three independent material
parameters, namely K M ; K M ; A C are determined so that the SEFs adequately model
the mechanical behaviour of the artery (see [ 16 ]).
The Attachment Stretch k AT
Collagen fibres are in a continual state of deposition and degradation [ 30 , 45 ]
and they are attached to the artery in a state of stretch. Watton et al. [ 15 ]
hypothesised that fibroblasts configure the collagen fibres to achieve a maximum
stretch k AT during the cardiac cycle and introduced the terminology attachment
stretch to denote k AT :
Magnitude of Recruitment Stretches at t ¼ 0
For a cylindrical artery at fixed axial stretch and subject to radial inflation,
maximum collagen fibre stretches will occur in the systolic configuration. Hence,
the initial values of the recruitment stretches are determined so that the stretch of
the medial and adventitial collagen fibres at systole at t ¼ 0 equal the attachment
stretch, thus

q

k s cos 2 c J þ þ k z sin 2 c J þ k J þ j t ¼ 0 ¼ k J ¼ = k AT ð 4 Þ