Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 304

numerical formulation to solve Eq. ( 1 ) can be found in [ 15 ]: the equilibrium dis-
placement field is solved by a finite element method coded in FORTRAN 77 [ 43 ].

2.1.1 Strain-Energy Functions for Heterogeneous Aneurysmal Tissue

The arterial wall is modelled as two layers. The inner layer models the mechanical
response of the media (and intima), with contributions from the elastinous constituents
(ground substance, elastin fibres and passive smooth muscle cells) and a double helical
pitch of collagen fibres with orientations c M p to the azimuthal axis: p ¼þ; p ¼
denote positively (c M þ [ 0) and negatively (c M \0) wound fibres, respectively. The
outer layer models the mechanical response of the adventitia, which is considered to
have a small elastinous contribution and a double helical pitch of collagen fibres with
orientations c A p (p ¼ ) to the azimuthal axis. The mechanical response of each layer
is modelled as the sum of a neo-Hookean strain energy function (SEF) [ 44 ] and a
highly nonlinear SEF which represents the mechanical response of the collagen [ 32 ].
Spatially and temporally dependent functions are introduced for the concentration of
the elastinous and collagenous constituents and the configuration in which the col-
lagen fibres begin to be recruited to load bearing.
Recruitment stretch variables define the factor the tissue must be stretched,
relative to the unloaded configuration, in the direction of a collagen fibre for it to
begin to bear load. Remodelling the recruitment stretches enables the remodelling
of the collagenous fabric during aneurysm evolution to be simulated. Constituent
concentration variables define the ratio of the mass density of a constituent at time
t to the mass density at time t ¼ 0 and enable the growth/atrophy of the constituent
to be simulated. For further details of the theoretical formulation to describe the
G&R the interested reader is referred to [ 33 ]. The SEFs for the elastinous con-
tributions in the medial and adventitia are multiplied by a normalised spatially and
temporally dependent concentration function, denoted m E ð h 1 ; h 2 ; t Þ: This is
employed to prescribe the degradation of the elastinous constituents, where
m E ð h 1 ; h 2 ; t ¼ 0 Þ¼ 1 : Fields of spatially and temporally dependent fibre recruit-
ment stretch k J p ð h 1 ; h 2 ; t Þ and concentration m J p ð h 1 ; h 2 ; t Þ variables are defined
throughout the midplane of the arterial wall, where the subindex J denotes the
media M or the adventitia A : The fibres within each layer are orientated at an angle
of c J p to the azimuthal axis, where p denotes the pitch c J relative to the azimuthal
axis in the unloaded reference configuration. Hence, the SEFs are

W J ¼ m E K J ð E 11 þ E 22 þ E 33 Þ
þ X
p ¼; E J p [ 0

n h i 1 o ;

ð 2 Þ

exp A C ð E J p Þ 2 m J p K J J ¼ M ; A ;

where the material parameters for the elastinous constituent are denoted by K J ;
whilst K J

and A C

are parameters that relate to the collagen fabric. The Green-