Page 198 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIOMECHANICS OF THE MUSCULOSKELETAL SYSTEM 175

7.5.2 Muscle Contraction Dynamics
Mechanism of Muscle Contraction. Our understanding of how a muscle develops force is
based on the sliding-filament theory of contraction. This theory, formulated by A. F. Huxley in
1957, proposes that a muscle shortens or lengthens because the thick and thin filaments slide
past each other without the filaments themselves changing length (Huxley, 1957). The central
tenet of the theory is that adenosine triphosphate (ATP) dependent interactions between thick
filaments (myosin proteins) and thin filaments (actin proteins) generate a force that causes the
thin filaments to slide past the thick filaments. The force is generated by the myosin heads of
thick filaments, which form cross-bridges to actin thin filaments in the AI zone, where the two
filaments overlap (see Fig. 7.2e). Subsequent structural changes in these cross-bridges cause the
myosin heads to walk along an actin filament. The sliding-filament theory correctly predicts that
the force of contraction is proportional to the amount of overlap between the thick and thin filaments
(see Fig. 7.3).
To understand the mechanism of muscle contraction, consider Fig. 7.20, which shows the vari-
ous steps involved in the interaction between one myosin head and a thin filament, also often referred
to as the cross-bridge cycle. In the absence of ATP, a myosin head binds tightly to an actin filament
in a “rigor” state. When ATP binds (Step 1), it opens the cleft in the myosin head, which weakens
the interaction with the actin filament. The myosin head then reacts with ATP (Step 2), causing a
structural change in the head that moves it to a new position, closer to the end of the actin filament
or Z disk, where it then rebinds to the filament. Phosphate is then released from the ATP-binding
pocket on the myosin head (Step 3), and the head then undergoes a second structural change, called
the power stroke, which restores myosin to its rigor state. Because the myosin head is bound to the
actin filament, this second structural change exerts a force that causes the myosin head to move the
actin filament (Fig. 7.20).

Modeling Contraction Dynamics. A. F. Huxley developed a mechanistic model to explain the
structural changes at the sarcomere level that were seen under the electron microscope in the late
1940s and early 1950s. Because of its complexity, however, this (cross-bridge) model is rarely, if
ever, used in studies of coordination. Instead, an empirical model, proposed by A. V. Hill, is used in
virtually all models of movement to account for the force-length and force-velocity properties of
muscle (Hill, 1938) (Fig. 7.21).
In a Hill-type model, muscle’s force producing properties are described by four parameters
(Zajac and Gordon, 1989): muscle’s peak isometric force (F o m ) and its corresponding fiber length
m
( ) and pennation angle (α), and the intrinsic shortening velocity of muscle (v max ). F o m is usually
l
o
obtained by multiplying muscle’s physiological cross-sectional area by a generic value of specific
α
tension. Values of optimal muscle fiber length, l o m , and , the angle at which muscle fibers insert
on tendon when the fibers are at their optimal length, are almost always based on data obtained
from cadaver dissections (Freiderich and Brand, 1990). v max is often assumed to be muscle
independent; for example, simulations of jumping (Pandy et al., 1990), pedaling (Raasch et al.,
1997), and walking (Anderson and Pandy, in press) assume a value of v max = 10 s −1 for all muscles,
which models the summed effect of slow, intermediate, and fast fibers (Zajac and Gordon, 1989).
Very few studies have examined the sensitivity of model simulations to changes in v max , even though
a change in the value of this parameter has been found to affect performance nearly as much as a
change in the value of F o m (Pandy et al., 1990).
Tendon is usually represented as elastic (Pandy et al., 1990; Anderson and Pandy, 1993).
Even though force varies nonlinearly with a change in length as tendon is stretched from its
resting length, l T s (see Fig. 7.5), a linear force-length curve is sometimes used (Anderson and
Pandy, 1993). This simplification will overestimate the amount of strain energy stored in ten-
don, but the effect on actuator performance is not likely to be significant, because tendon
force is small in the region where the force-length curve is nonlinear. Values of the four mus-
cle parameters plus tendon rest length for a number of musculotendinous actuators in the
human arm and leg can be found in Garner and Pandy (2001) and Anderson and Pandy (1999),
respectively.