Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 311

degradation of elastin occurs whilst lower values of WSS give rise to deg-
radation of elastin. Furthermore we suppose that there exists a value of WSS,
say s X : 0 s X \s crit ; at which maximum degradation occurs. We assume a
simple quadratic functional form for the degradation function F D that
describes the relation between the local WSS and the degree of the degra-
dation of elastin, i.e.

8
<

0 ; s s crit ;

2

F D ð s ð h 1 ; h 2 ; t ÞÞ ¼

s crit s
s crit s X

ð 16 Þ

;

s X \s\s crit ;

:

1 ; s s X :

In this study, D max ¼ 0 : 5 ; s crit ¼ 0 : 5 Pa and s X ¼ 0 Pa.

2.3.2 Collagen Adaption

The adaption of the collagen fabric during aneurysm evolution consists of two
distinct mechanisms: growth and remodelling.

2.3.3 Remodelling the Recruitment Configuration of Collagen

Evolving the reference configurations that fibres are recruited to load bearing
simulates the mechanical consequences of: (i) fibre deposition and degradation in
altered configurations as the aneurysm enlarges; (ii) fibroblasts configuring the
collagen to achieve a maximum strain during the cardiac cycle, i.e. the attachment
strain E AT ¼ðð k AT Þ 2 1 Þ= 2 : The recruitment stretches adapt so that the maxi-
mum strain in the collagen fibres remodels to E AT : We utilise linear differential
equations for the remodelling of the recruitment stretches, i.e.

!

dk J p
dt ¼ a

E J p j max E AT

E AT

; ð 17 Þ

where a ¼ 0 : 3 years 1 ;

E J p j max ¼ max E J p j sys ; E J p

j dias ð 18 Þ

and E J p j sys and E J p j dias denote the magnitude of the collagen strains evaluated in the
systolic and diastolic configurations, respectively. Note, [ 15 ] assumed that peak
strains occur at systolic pressure and thus collagen remodelling is simulated by
considering the steady deformations that occur in the systolic configuration. How-
ever, it was observed that for a more complex geometry, the fibres may not achieve
maximum strains at systolic pressure: towards the distal and proximal ends of the
AAA the axial strains increase as the pressure is reduced [ 15 ]. Hence, the subtle