Biomedical Engineering Reference
In-Depth Information
An example of such a measure can be [8]
p
T
0 |
1
T
p d t
y
(
s
)=
y
(
s
,
t
) |
,
(7.22)
where p is a parameter and the dot denotes the derivative with respect to the cardiac
cycle time t . This quantity defines the influence of the time “microscale” t on the
time “macroscale” s , and it is sensitive to perturbations in the frequency as well as to
arbitrary variations of the wave form of the variable y
on the cardiac cycle. Once
the metric that bridges the cardiac cycle time scale to the G&R time scale has been
chosen, the general constitutive equations for the G&R process can be formulated.
We need to model the active contractile property endowed by the smooth muscle
cells. We chose to model the smooth muscle activation level with a scalar quantity
C denoting the net concentration of vasoconstrictors. Experimental evidence reveals
that the concentration of vasoactive molecules is modulated by molecular pathways
regulated by the endothelial cells according to the hemodynamic forces they experi-
ence. It is then reasonable to assume that C depends on the wall shear stress
(
s
,
t
)
τ w and
on its rate of variation on the cardiac cycle
τ w
) , τ w (
C
(
s
)=
C
( τ w (
s
s
)) .
(7.23)
This constitutive assumption asserts that the level of smooth muscle activation is
driven directly by hemodynamic processes on the cardiac cycle.
It has been shown that fibroblast activity is governed largely by the stress in the
tissue [34, 37], on the rate of variation of hemodynamic loads on the cardiac cycle
[8], and on the concentration of vasoactive molecules [29, 31, 32]. We thus assume
that mass production rates m k depend on a measure of the stress in the component
k ,
σ
on its rate of variation on the cardiac cycle σ
k , and on the constrictor concentration
in the tissue C , namely
m k
m k
k
k
(
s
)=
σ
(
s
) , σ
(
s
) ,
C
(
s
)
.
(7.24)
Because the stress and constrictor concentration depend on the hemodynamic loads,
also the mass production rate depends ultimately on the hemodynamics.
Cell apoptosis and matrix degradation are complex phenomena that are also in-
fluenced by the tensional state induced by the hemodynamic loads. Thus we assume
that the survival function q k
(
, τ )
s
(i.e., the fraction of the constituent k that was de-
posited at time
and survives at time s ) depends on the time course of the stress in
the constituent and of its rate over the cardiac cycle time scale, that is
τ
q k
q k
k
n
k
n
]) , σ
(
s
, τ )=
σ
( τ ) ([ τ ,
s
( τ ) ([ τ ,
s
])
.
(7.25)
This constitutive assumption reflects the importance of hemodynamics also on the
removal rate of structurally significant constituents.
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