232
5 Contraction
Finally, as a check, vertical force equilibrium for the free-body diagram in
Fig. 5.11b demands
p i a =
b
a
σ θ dr.
(5.43)
Calculations confirm that equilibrium is indeed satisfied (see Problem 5.9).
5.4.4 Alternative Models for Contraction
Many authors simulate contraction using a variation of the present model. They do
this by simply increasing the active stress, but the concept of a changing ZSS is not
formally introduced.
Over the decades, several other models have been proposed for muscle contraction. Microstructural models based on cross-bridge attachment and detachment
rates capture many of the known behaviors of muscle (McMahon 1984). An early
model is Hill’s three-element model (McMahon 1984; Fung 1993), which includes
a contractile element in series with a passive spring, both of which are parallel to
another spring. In effect, the series components are equivalent to the present model
for a CE, and the parallel spring represents the passive part of the CF model.
5.5 Case Study: Cardiac Mechanics
The heart is essentially a mechanical pump composed of four muscular chambers—
the left and right atria and the left and right ventricles (Fig. 5.13a). Valves prevent
blood from flowing in the wrong direction as the heart beats. The mitral valve
is located between the left atrium and left ventricle, the tricuspid valve between
the right atrium and right ventricle, the aortic valve between the left ventricle and
aorta, and the pulmonary valve between the right ventricle and pulmonary artery
(Fig. 5.13a).
The muscle of the myocardium is fed by coronary arteries. If the blood supply
is reduced, such as by an arterial blockage, oxygen demand may exceed supply,
causing a condition known as myocardial ischemia, which can cause cardiac
dysfunction or damage. Severe and persistent ischemia can cause a myocardial
infarction (heart attack), which can lead to a region of dead muscle that disrupts
pumping and electrical activity. If the infarcted area is relatively extensive, the
dead muscle may bulge outward passively during systole as the surrounding muscle
contracts inward. During the first 10 days post-infarction, this region is susceptible
to rupture. If the patient survives this acute phase, the dead cells are replaced by
5 Contraction
Finally, as a check, vertical force equilibrium for the free-body diagram in
Fig. 5.11b demands
p i a =
b
a
σ θ dr.
(5.43)
Calculations confirm that equilibrium is indeed satisfied (see Problem 5.9).
5.4.4 Alternative Models for Contraction
Many authors simulate contraction using a variation of the present model. They do
this by simply increasing the active stress, but the concept of a changing ZSS is not
formally introduced.
Over the decades, several other models have been proposed for muscle contraction. Microstructural models based on cross-bridge attachment and detachment
rates capture many of the known behaviors of muscle (McMahon 1984). An early
model is Hill’s three-element model (McMahon 1984; Fung 1993), which includes
a contractile element in series with a passive spring, both of which are parallel to
another spring. In effect, the series components are equivalent to the present model
for a CE, and the parallel spring represents the passive part of the CF model.
5.5 Case Study: Cardiac Mechanics
The heart is essentially a mechanical pump composed of four muscular chambers—
the left and right atria and the left and right ventricles (Fig. 5.13a). Valves prevent
blood from flowing in the wrong direction as the heart beats. The mitral valve
is located between the left atrium and left ventricle, the tricuspid valve between
the right atrium and right ventricle, the aortic valve between the left ventricle and
aorta, and the pulmonary valve between the right ventricle and pulmonary artery
(Fig. 5.13a).
The muscle of the myocardium is fed by coronary arteries. If the blood supply
is reduced, such as by an arterial blockage, oxygen demand may exceed supply,
causing a condition known as myocardial ischemia, which can cause cardiac
dysfunction or damage. Severe and persistent ischemia can cause a myocardial
infarction (heart attack), which can lead to a region of dead muscle that disrupts
pumping and electrical activity. If the infarcted area is relatively extensive, the
dead muscle may bulge outward passively during systole as the surrounding muscle
contracts inward. During the first 10 days post-infarction, this region is susceptible
to rupture. If the patient survives this acute phase, the dead cells are replaced by