A Mathematical Approach for Studying Ca-Regulated Smooth Muscle Contraction - Computer Models in Biomechanics - page 46

the direct membrane depolarizing stimulation (e.g., potassium) pathway, which also
leads to an increase in intracellular Ca 2 + ]

. It should be noted that the length-tension
relationship in smooth muscle is obtained for both membrane depolarization and
agonist stimulation. A more plausible hypothesis is that the length-tension relation-
ship may originate from the structural rearrangements within the smooth muscle
contractile unit, when stretched. This would influence the filament overlap between
the actin and myosin filaments in a smooth muscle contractile unit which has a di-
rect relation to the number of attached cross-bridges, and hence the active tension
produced by the smooth muscle. A connection between the length of the contractile
unit (sarcomere) and the length-tension behavior in muscle has been hypothesized
for a long time (Gordon et al., 1966 ).

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4.2.3 Force-Velocity Relationship

The importance of the chemical and mechanical model combination is demonstrated
when it comes to modeling the characteristic force-(shortening) velocity relation-
ship of muscle. When the isotonic shortening velocity is measured for different
forces (after-loads) a hyperbolic relationship of the force and the shortening velocity
is obtained (Woledge et al., 1985 ). When extracting the force-velocity relationship,
two certain times are of importance: (i) the time at which the quick-release is per-
formed, i.e. the amount of time of isometric contraction before the quick-release,
and (ii) the time at which the velocity is measured during the isotonic contraction.
When the force-velocity relationship is extracted at different time of isotonic
quick-release, the relationship changes. The shortening velocity is higher when the
quick-release is performed at an early stage of the isometric contraction rather than
at a later stage, see Fig. 4.2 D. This behavior supports the hypothesis of non-cycling
latch cross-bridges which are dominant at a later stage of an isometric contraction.

4.2.4 Smooth Muscle Modeling

By assuming the well-established three-element Hill muscle characteristic, as de-
scribed by Fung ( 1970 ) for smooth muscle, the smooth muscle contractile unit is
represented by an elastic serial element and a contractile element. The active ten-
sion produced by the smooth muscle depends on two main principal parameters:
(i) the number of attached load-carrying cross-bridges, and (ii) the (average) elastic
elongation of the attached cross-bridges, both phosphorylated and dephosphorylated
(cf. Rachev and Hayashi, 1999 ; Yang et al., 2003 ; Stålhand et al., 2008 ; Murtada et
al., 2010a ).
The kinetics of the smooth muscle myosin phosphorylation, which regulates the
activation of smooth muscle contraction, can be used to define the number of at-
tached load-carrying cross-bridges. The kinetics of myosin phosphorylation and