Biomedical Engineering Reference
In-Depth Information
prescribed material properties and distribution of constituents prestretch and mass
fraction.
7.2.3 Growth & remodelling: a multiple-time-scales approach
The main novelty of the proposed model is in the formulation of tissue growth and re-
modelling and in the inclusion of muscle tone guided by hemodynamics. According
to the theory of constrained mixtures proposed in [23], we assume the arterial wall to
be composed of several structurally significant constituents that are deposited in the
current configuration, on the G&R time scale, with a given prestretch
G
k
relative to
their own natural (i.e., stress-free) configuration and that they follow the mixture de-
formation afterwards, with no relative motion. We usually assume that the deposition
prestretches
G
k
do not depend on time. One can use different approximations for the
collagen orientations, including continuous distributions about two diagonal direc-
tions [18]. Following [37], we assume a four collagen fibre families model (axial (
z
),
circumferential (
h
degrees from the longitudinal
axes) and denote elastin with
e
, collagen fibres with
c
, and smooth muscle with
m
.
The structural constituents turnover continuously on the G&R time scale. Let us de-
note with
F
the deformation gradient from a given reference configuration
θ
), and helical (
h
) inclined of
±
α
β
0
to the
current configuration on the G&R time
β
s
, and with
F
s
the deformation gradient from
β
s
to the current configuration on the cardiac cycle time
s
(see Fig. 7.3). Moreover
β
we denote with
τ
a generic instant on the G&R time scale at which new constituent
Fig. 7.3.
Relevant configurations in the multiple-time scales G&R framework. At the current G&R
time
τ
, the constituents are deposited in the current configuration on the G&R time scale
β
τ
at a
given prestretch
G
k
independent of
τ
. The deformation gradient from a common reference config-
uration
β
0
to the current configuration on the G&R time
β
τ
is denoted with
F
(
τ
)
. The deformation
s
corresponding to the
gradient from the configuration
β
s
at G&R time
s
to the configuration
β
cardiac cycle time
t
(at G&R time
s
) is denoted by
F
s
. Grey, dashed arrows show the intercon-
nections between the two time scales: the current wall geometries and material properties influence
the solution on the cardiac cycle, in turn the mass production and removal rates as well as the
constituents material properties depend on suitable rate measures on the cardiac cycle
(
t
)
