Biomedical Engineering Reference
In-Depth Information
fibres are incorporated in the tissue. Therefore, at G&R time s, the constituent
k
deposited at time
τ
experiences, according to [5] and [37], the deformation gradient
(
τ
)
−
1
G
k
F
n
(
τ
)
(
s
)=
F
(
s
)
F
,
(7.18)
while on the cardiac cycle time scale
t
, at a given G&R instant
s
, its deformation
gradient is given by
(
τ
)
−
1
G
k
F
n
(
τ
)
(
,
)=
(
)
(
)
,
s
t
F
s
t
F
s
F
(7.19)
were the further deformation gradient
F
s
(
t
)
is superimposed to configuration
β
s
(cor-
responding to the mean pressure
P
m
(
s
)
and flow
Q
m
(
s
)
) and brings to the current
s
. It is worth noticing that, due to
strong material non-linearity, the deformation gradient
F
n
(
τ
)
(
configuration on the cardiac cycle time scale
β
s
)
does not correspond
to the average of
F
n
(
τ
)
(
over
t
. We assume that the constituents are deposited at
the configuration of the vessel corresponding to the mean transmural pressure and
flow rate on the cardiac cycle
s
,
t
)
β
s
(see Eqs. 7.1 and 7.2).
With the assumption of time scales separation, the mass density of the mixture,
that is a function of the G&R time
s
only, can be computed as
M
)=
∑
k
M
k
(
s
(
s
)
where
M
k
is the mass density of the constituent
k
given by the convolution integral
[4, 37]
s
M
k
M
k
Q
k
m
k
q
k
(
s
)=
(
0
)
(
s
)+
(
τ
)
(
s
,
τ
)
d
τ
,
(7.20)
0
where
Q
k
0 (assumed
as the instant at which the perturbation of the system from the normal conditions oc-
curs) that survives at time
s
,
m
k
(
s
)
∈
[
0
,
1
]
accounts for the mass fraction present at time
s
0
=
≥
0 is the true mass density production rate function,
and
q
k
that
survives at time
s
. The mass fraction of the constituent
k
at G&R time
s
is therefore
given by
(
s
,
τ
)
∈
[
0
,
1
]
is the mass fraction of material produced at time
τ
∈
[
0
,
s
]
k
M
k
φ
(
s
)=
(
s
)
/
M
(
s
)
. Similarly the mixture strain energy density density
)=
∑
k
W
k
where
W
k
is given by the sum
W
(
s
,
t
(
F
(
s
,
t
))
is the contribution of the
constituent
k
computed via the convolution integral
W
k
(
(
,
))
F
s
t
s
)
W
k
F
n
(
0
)
(
,
τ
)
W
k
F
n
(
τ
)
(
d
M
k
m
k
(
0
)
(
τ
)
ρ
Q
k
q
k
=
(
s
s
,
t
)
+
(
s
s
,
t
)
τ
,
(7.21)
ρ
0
where
is the mixture mass density per unit volume, assumed as constant herein
(i.e., changes in mass parallel changes in volume), and
W
k
is the strain energy den-
sity per reference unit volume for the constituent
k
. Typical assumptions for
W
k
are
the neo-Hookean model for elastin and Fung-type exponentials for collagen fibres
and passive smooth muscle cells.
Given a quantity
y
ρ
(
s
,
t
)
, defined on the cardiac cycle time scale, we denote by
(
)
y
s
a suitable measure of its rate of variation on the cardiac cycle at G&R time
s
.
