Biomedical Engineering Reference
In-Depth Information
is a length at which the muscle prefers to act, it being able to generate less active
force at lengths less than or greater than
L
0
. In contrast to striated muscle, however,
which develop their maximal active force at a length where passive tension is ab-
sent, a significant passive force is required to achieve
L
0
in smooth muscle prior to
activation. On the molecular level, intracellular free calcium Ca
2
+
is the primary
determinant of contractility. Increases in intracellular Ca
2
+
associate with contrac-
tion and decreases with relaxation. Finally, fibroblasts are important players in the
maintenance of the
adventitia
via the synthesis of type I collagen. In case of tissue
damage, the fibroblasts migrate quickly to the site of injury, proliferate and then syn-
thesize new collagen. Such activity is regulated in part by growth factors. It has been
observed that fibroblasts production of collagen filaments is strongly dependent on
the stress level in the tissue and on the concentration of vasoactive molecules. Thus
fibroblasts activity depends, ultimately, on the hemodynamic loads.
7.2 Models
7.2.1 Thin-wall models
Models proposed for the G&R of living tissues traditionally account for a single
time scale at which the biological processes take place [2, 12, 13]. Nevertheless the
influence of phenomena happening on different time scales might be relevant for
G&R. Some examples of phenomena that influence long-term adaptation, although
happening on shorter time scales, are the cardiac cycle, the respiratory cycle, and
the circadian cycles. For vascular growth and adaptation, the dynamics on the car-
diac cycle and the hemodynamic loads induced by the pumping action of the heart
are certainly non negligible [15, 36]. The characteristic G&R time (days to weeks) is
several orders of magnitude higher than the cardiac cycle time scale (seconds to frac-
tion of a second depending on species). This difference in characteristic evolution
times justifies the assumption of a multiple time scales approach to formulate the
G&R problem under hemodynamic loads. In particular let us denote by
s
the G&R
time variable and by
t
the cardiac cycle time and assume that negligible growth oc-
curs during the cardiac cycle [17].
In several cases [3, 5, 37] the artery can be modelled as a thin-walled, circular
cylinder of thickness
h
, internal radius
a
, and fixed length
L
subject to internal pres-
sure
P
and wall shear stress
τ
w
induced by the blood flow
Q
. Even in this simple case,
the description of blood flow rate, velocity and pressure result from the solution of a
fluid-structure interaction problem [11, 39]. For the sake of simplicity let us assume
that the blood flow rate and pressure are assigned as inputs into the model and that
they are prescribed by a Fourier series on the cardiac cycle time scale
t
A
i
cos
i
t
sin
i
t
2
π
2
π
)+
∑
i
(
,
)=
(
(
)
+
(
)
,
P
s
t
P
m
s
s
B
i
s
(7.1)
T
(
s
)
T
(
s
)
C
i
(
cos
i
t
sin
i
t
2
π
2
π
)+
∑
i
Q
(
s
,
t
)=
Q
m
(
s
s
)
+
D
i
(
s
)
,
(7.2)
T
(
s
)
T
(
s
)
