Biomedical Engineering Reference
In-Depth Information
viscoelasticity of the actin-myosin cross-bridges. The empirical values that describe this
component of the system can also be quantified by stimulating muscle to contract and
monitoring the relaxation of the system.
Since 1939, many experiments have been conducted in order to determine the empirical
constants that quantify this model. A full description of these experiments and the mathe-
matical derivation of these formulae can be found in Y.C. Fung's biomechanics texts (see
Further Reading material). To summarize the salient results, the total force acting within
the muscle (F) can be formulated as a summation of the parallel element force (P) and the
contractile/series element force (S):
F
5
P
1
S
ð
4
1
Þ
:
P is a function of the length of the muscle fiber. It is proportional to the number of sarco-
meres present and can be formulated as the length of one myosin fiber plus the length of
two actin molecules minus the overlap of the myosin and actin proteins. Assuming that all
sarcomeres have the same physical structure, the parallel element force (P) can then be
multiplied by the fiber length (or number of sarcomeres). If the sarcomere is under a
stressed state, there is an added constant to account for the viscoelastic extension (
). S has
been shown to be an exponential function that is dependent on the initial tension in the
fiber and the extension of the fibers. Hill's model neglects the kinetics of the biochemical
reactions that need to take place in order for actin and myosin to form cross-bridges
between each other. These (and other) necessary components of the heart model have
been incorporated into theoretical models by others, which represent the tension and
length curves reasonably well. However, Hill's model provides a fairly good prediction of
how cardiac tissue will respond to forces.
To understand heart motion, we will briefly discuss the solid mechanics associated with
the heart. As learned in previous solid biomechanics courses, biological tissue is non-
homogenous, is viscoelastic, is subjected to a very complex loading condition, and has a
complex three-dimensional geometry. This makes its analysis relatively difficult. However,
to first understand the loading conditions and how the elasticity can affect blood flow, let
us first consider the simplest case. If the left ventricle can be modeled as a homogenous
isotropic spherical shell, then the radial stress can be defined as
η
p
o
b
3
ð
r
3
2
a
3
p
i
a
3
ð
b
3
2
r
3
Þ
Þ
σ
r
ð
r
Þ
5
ð
4
2
Þ
Þ
1
:
r
3
ð
a
3
2
b
3
r
3
ð
a
3
2
b
3
Þ
The circumferential stress, which is greatest along the inner surface of the sphere, can
be defined as
p
o
b
3
2r
3
1
a
3
p
i
a
3
2r
3
1
b
3
ð
Þ
ð
Þ
σ
θ
ð
r
Þ
5
ð
4
3
Þ
Þ
2
:
2r
3
ð
a
3
2
b
3
2r
3
ð
a
3
2
b
3
Þ
The variables in
Equations 4.2 and 4.3
are the inner radius of the sphere (a), the outer
radius of the sphere (b), internal pressure (p
i
), and external pressure (p
o
)(
Figure 4.10
). As
the reader can see,
Equation 4.3
is maximized when r is equal to a (remember that b must
be greater than a). Looking at these two formulas, one can see that these stresses would
induce a stress on the fluid next to the ventricle wall. In fact, the internal pressure can be
equal
to the left ventricular hydrostatic pressure (which is a function of
time, see
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