80
Dynamics
of
the Vascular System
isometric (constant length) contraction. The regional tension and global
ventricular pressure relationship is seen to be parallel however, only
during this phase (Fig. 4.1.3, Li,
1987).
The anisotropic properties and
differential epicardial and endocardia1 segmental contraction due to their
fiber orientations complicate the direct translation of mechanics from the
muscle level to the global ventricular level.
Much of the biomechanics of muscle contraction can be traced back
to Hill, who was concerned about the mechanical efficiency, in terms of
work and speed, of human muscles. Although the concept of mechanical
spring as an energy storage element was introduced to model muscle
behavior before him, Hill accounted for the energy dissipation through
the introduction of a viscoelastic model. This leads to the expression:
dl
dt
k(Z
-
I,
)
-
Po
-
=
F
(4.1.2)
where
k
and
Po
are constants,
is
I
and
10
are the instantaneous and the
initial muscle lengths, respectively.
F
is the applied force. The velocity
of shortening is represented by the rate of change of fiber length
dZ
dt
v=-
(4.1.3)
This velocity term thus gives rise to the viscous effect. At any given
muscle length
I,
a larger load
F
is lifted with a lower velocity than a
smaller load.
Thus, the ability to generate force and the extent
of
velocity
of shortening of the contractile element has an inverse
relationship.
Hill's
two
element model, consisting of a passive series elastic
element and a contractile element has become popular and follows the
general expression:
(F
+
a)(v
+
b)
=
k,
(4.1.4)
where
k,
is a constant. The velocity of shortening
is
a function
of
initial
length of the muscle fiber.
Combined with the earlier force-length
