5.4 Mechanical Properties of Contractile Fibers
223
in Fig. 5.7b. In essence, these curves represent stress-stretch curves for a CE that is
“frozen” at various degrees of contraction, i.e., the CE is stretched while keeping
the contraction ratio K(t) fixed.
Although such an experiment may be difficult to achieve in practice, it serves
as a useful thought experiment. The procedure would be to first allow the CE to
contract and shorten unimpeded along the line P a0 = 0 from λ = 1 to the active
ZSS. According to Eq. (5.3), this occurs when λ = K (λ ∗ = 1). Then, the specimen
is stretched, and active stress P a0 is measured as a function of λ. This process is
repeated for selected values of K until peak contraction (K = K min ) is reached,
thereby providing a series of stress-stretch curves that can be used to determine a
time-varying constitutive relation (Fig. 5.8b). Note that ˆ
λ represents the overstretch
ratio, beyond which the active stress is zero.
During a single twitch, the value of K decreases from 1 (passive state) to K min
(peak contraction) and then returns to one. Each curve in Fig. 5.8b reflects active
material properties at selected instants in time as the CE contracts and relaxes.
To look at this problem another way, consider two paths for getting from Point A
(the passive unloaded state) to point C on the stress-stretch curve for maximum
isometric contraction (Fig. 5.8c). Path 1 corresponds to the one followed in our
thought experiment, i.e., the unconstrained element first contracts stress-free to point
B, then stretching takes us up the active stress-stretch curve to point C. For Path 2,
we first stretch the passive CE (without stress) to point B . Then, contraction occurs
with λ held fixed, and the active stress increases until point C is reached at peak
contraction.
Such path independence characterizes an elastic material, consistent with our
assumption of pseudoelasticity. Although CEs are not perfectly elastic, experiments
suggest that pseudoelasticity is a reasonable approximation for beating heart muscle.
Viscoelastic effects may be important for other types of CEs, with the behavior
depending on the history of contraction and loading. Such effects are ignored here,
however.
While these active stress-stretch curves are similar for all types of muscle, the
passive response sometimes differs considerably. These differences are illustrated
in Fig. 5.8d, where the passive stress P p , active stress P a0 , and total stress P =
P p + P a0 are plotted for skeletal and cardiac muscle. In the normal operating range,
which falls just to the left of the peak in the active stress-stretch curve for both types
of muscle, passive stress is significant for cardiac muscle but not skeletal muscle
(Fig. 5.8d). In the heart, passive stress helps prevent overstretch of the sarcomeres
during passive filling (diastole), keeping the force that ejects blood during systolic
contraction near the optimal value. Attachment to bones limits stretching of skeletal
muscles, so passive stress is not as crucial for preventing overstretch. Moreover,
organisms usually can survive overstretched skeletal muscle.
Finally, we note that the curve for total stress in skeletal muscle has a negative
slope after the peak, i.e., the stiffness becomes negative. Theoretically, this could
lead to local instabilities, such as some sarcomeres becoming longer than others
during contraction. However, the effects of this behavior on force generation are not
observed experimentally, leading researchers to propose stabilizing mechanisms for
sarcomeres (Vilfan and Duke 2003; Herzog 2017).
223
in Fig. 5.7b. In essence, these curves represent stress-stretch curves for a CE that is
“frozen” at various degrees of contraction, i.e., the CE is stretched while keeping
the contraction ratio K(t) fixed.
Although such an experiment may be difficult to achieve in practice, it serves
as a useful thought experiment. The procedure would be to first allow the CE to
contract and shorten unimpeded along the line P a0 = 0 from λ = 1 to the active
ZSS. According to Eq. (5.3), this occurs when λ = K (λ ∗ = 1). Then, the specimen
is stretched, and active stress P a0 is measured as a function of λ. This process is
repeated for selected values of K until peak contraction (K = K min ) is reached,
thereby providing a series of stress-stretch curves that can be used to determine a
time-varying constitutive relation (Fig. 5.8b). Note that ˆ
λ represents the overstretch
ratio, beyond which the active stress is zero.
During a single twitch, the value of K decreases from 1 (passive state) to K min
(peak contraction) and then returns to one. Each curve in Fig. 5.8b reflects active
material properties at selected instants in time as the CE contracts and relaxes.
To look at this problem another way, consider two paths for getting from Point A
(the passive unloaded state) to point C on the stress-stretch curve for maximum
isometric contraction (Fig. 5.8c). Path 1 corresponds to the one followed in our
thought experiment, i.e., the unconstrained element first contracts stress-free to point
B, then stretching takes us up the active stress-stretch curve to point C. For Path 2,
we first stretch the passive CE (without stress) to point B . Then, contraction occurs
with λ held fixed, and the active stress increases until point C is reached at peak
contraction.
Such path independence characterizes an elastic material, consistent with our
assumption of pseudoelasticity. Although CEs are not perfectly elastic, experiments
suggest that pseudoelasticity is a reasonable approximation for beating heart muscle.
Viscoelastic effects may be important for other types of CEs, with the behavior
depending on the history of contraction and loading. Such effects are ignored here,
however.
While these active stress-stretch curves are similar for all types of muscle, the
passive response sometimes differs considerably. These differences are illustrated
in Fig. 5.8d, where the passive stress P p , active stress P a0 , and total stress P =
P p + P a0 are plotted for skeletal and cardiac muscle. In the normal operating range,
which falls just to the left of the peak in the active stress-stretch curve for both types
of muscle, passive stress is significant for cardiac muscle but not skeletal muscle
(Fig. 5.8d). In the heart, passive stress helps prevent overstretch of the sarcomeres
during passive filling (diastole), keeping the force that ejects blood during systolic
contraction near the optimal value. Attachment to bones limits stretching of skeletal
muscles, so passive stress is not as crucial for preventing overstretch. Moreover,
organisms usually can survive overstretched skeletal muscle.
Finally, we note that the curve for total stress in skeletal muscle has a negative
slope after the peak, i.e., the stiffness becomes negative. Theoretically, this could
lead to local instabilities, such as some sarcomeres becoming longer than others
during contraction. However, the effects of this behavior on force generation are not
observed experimentally, leading researchers to propose stabilizing mechanisms for
sarcomeres (Vilfan and Duke 2003; Herzog 2017).