Biomedical Engineering Reference
In-Depth Information
allow for active regulation of geometrical or mechanical properties. For example,
arterial smooth muscle is known to contract in connection with acutely lowered
blood pressure. This response is believed to work as a first line of defense restoring
the transmural strain distribution and flow-induced shear stress back towards their
homeostatic values (Rachev and Hayashi,
1999
). Further, smooth muscle maintain
a constant contraction in arteries. This basal tone together with the residual stress
has been hypothesized to reduce the transmural stress gradient leading to a more
uniform stress distribution (Humphrey and Wilson,
2003
).
The muscle contraction is controlled by a complex chain of electrochemical
events involving depolarization of the cell membrane, binding of calcium ions to
calmodulin, phosphorylation of light myosin chains and ultimately formation of
cross-bridges between actin and myosin filaments. Most of the myosin undergoes
a continuous cross-bridge cycling where phosphorylated myosin heads attach actin,
perform a force-related power stroke, and finally detach actin. It is the combined
effect of all these cross-bridge cycles that generates active force and contraction.
In addition to these electrochemical processes, mechanics also impact on the force
generation, e.g., stretching the cell increases the phosphorylation rate and causes
a sensitization with respect to the calcium ion concentration. This clearly shows
that a reasonably complete smooth muscle model must be multi-scale and include
properties from both the electrochemical and the mechanical scales.
Relatively few studies have modeled smooth muscle. Among the exceptions are
studies by Gestrelius and Borgström (
1986
), Rachev and Hayashi (
1999
), Humphrey
and Wilson (
2003
), and Yang et al. (
2003
). The models in these studies are based on
experimental evidence and many constitutive equations are stated intuitively. For ex-
ample, in Rachev and Hayashi (
1999
) and Humphrey and Wilson (
2003
), the effect
of smooth muscle contraction is modeled by adding an active stress to the constitu-
tive equation. This active stress has the form
t
0
f
λ
f
Ca
2
+
, where
t
0
is the maximum
isometric stress,
f
λ
∈[
]
describes the bell-shaped stretch dependence associated
with the filament overlap, and
f
Ca
2
+
∈[
0
,
1
]
is the calcium ion dependent activa-
tion level. Although this additive technique is both simple and flexible, it has some
issues associated with it. First, the absence of a detailed kinematic description for
smooth muscle contraction may lead us to believe that the stretch dependence ob-
served in experiments is associated with the total stretch. A more detailed kinematic
analysis, see, e.g., Stålhand et al. (
2008
) and Murtada et al. (
2010
), suggests that
the total stretch should be decomposed into two pars: a filament translation and a
stretching of myosin heads. With this decomposition it makes more sense to take
the stretch dependence to be a function of the filament translation rather than the
total stretch. Second, the coupling of the electrochemical and mechanical subprob-
lems is made
ad hoc
and non-intuitive terms are easily overlooked, see Stålhand
et al. (
2008
). By deriving the model in a continuum thermodynamic framework,
this risk is minimized since couplings are implicitly given in the model. In addi-
tion, the continuum thermodynamic framework also give constraints on the consti-
tutive equations which guarantee smooth muscle contraction to be dissipative. This
is not guaranteed when adding extra terms to the constitutive equation but must be
checked for each case. Third, to guarantee a physically reasonable behavior such as
0
,
1
