Multi-Scale Modelling of Vascular Disease: Abdominal Aortic Aneurysm Evolution - Computational Modeling in Tissue Engineering - page 312

sophistication for the functional form for the remodelling of the recruitment stret-
ches, i.e. ( 17 ).

2.3.4 Growth/Atrophy of the Collagen Fabric

Fibroblasts deposit collagen fibres and secrete proteases to degrade the collage-
nous material. They adhere to the ECM via specialised cell surface receptors, in
particular integrins [ 58 ]. The integrins act as stretch sensors [ 59 ], transducing
mechanical signals to the fibroblast interior, enabling fibroblasts to sense their
mechanical environment. We assume that the rate of change of the concentration
of the collagenous constituents is dependent on the concentration of fibroblasts m J p
(ratio of the density of the fibroblast cells at time t to the density at time t ¼ 0) in
the arterial wall and rate of synthesis and degradation of collagen. Let F S and F D
be functions depicting how the rate of collagen synthesis and the secretion of
matrix metalloproteases are related to the deformation of the fibroblast cell during
a cardiac cycle, respectively. Rates of activity of vascular cells increase with the
magnitude of the cyclic deformation [ 60 , 61 ]. We assume F S and F D to be
functions of the maximum strains E J p j max and cyclic areal stretches A CS of the
cells during the cardiac cycle. Under these assumptions, the rate of change of the
collagen concentration is

dm J p
dt

¼ m J p F S ð E J p j max ; A CS Þ m J p F D ð E J p j max ; A CS Þ:

ð 19 Þ

In response to increased stretch, fibroblasts attempt to reduce their stretch and
reach a new equilibrium by restructuring their cytoskeleton and ECM contacts, i.e.
fibroblasts reconfigure their natural reference configuration. Hence, to simplify the
mathematical analysis, we assume that the local (natural) reference configuration
of a fibroblast cell is identical to that of the collagen fabric it is maintaining. Hence
the GL strain of the fibroblast cell, say E J p ; is assumed to be equal to the GL strain
of the collagen, i.e. E J p E J p : Furthermore, we assume that the concentration of
fibroblasts in the arterial tissue is proportional to the concentration of collagenous
constituents, i.e. m J p ¼ n 0 m J p : n 0 [ 0 : These assumptions imply that the rate of
evolution of the collagen fibre concentration m J p is

dm J p
dt

¼ m J p F G ð E J p j max ; A CS Þ ð 20 Þ

where F G ¼ n 0 ð F S F D Þ: If E J p j max ¼ E AT the collagen fabric is in homeo-
stasis, i.e. the secretion of ECM is balanced by the degradation and there is no
change in concentration. Hence, for E J p j max ¼ E AT it is required that F G ð E J p Þ¼ 0 :
The exact functional form of F G is unknown. However, if the substrate is stret-
ched, a net positive force acts on the cell and signalling to the nucleus results in an