Modeling of Muscular Systems
summarized from The Mechanical Properties of Human Muscle
by Arthur E. Chapman
Exercise & Sport Sciences Reviews. 13:443-501, 1985.
Ever since A. V. Hill’s pioneering work in the early 20th Century, researchers have attempted to develop a mechanical model of the muscle. Most current models describe a muscle as consisting of a contractile component and an elastic component in series; the model may also include a passive elastic element in parallel, as shown in Figure 1. While these mechanical components will be discussed separately, they should probably not be thought of as representing separate anatomical features.
The Contractile Component
The contractile component is a theoretical construct that accounts for the many force-producing elements of a muscle. Most versions of the CC are based on the cross-bridge/sliding filament theory of force generation. There are several factors which modify muscle force, including voluntary activation level, muscle shortening velocity, muscle length, prior history of the state of contraction, and fiber type.
The first of these factors, muscle activation, is fairly intuitive. All skeletal muscles are activated by voluntary stimulation. Greater stimulation recruits more motor-neuron groups (bundles of muscle fibers), which in turn generates greater force. Activation can be measured indirectly through use of the electromyogram.
The relationship between force and shortening velocity has been studied extensively, but researchers must confront difficulties in making accurate measurements and in separating the contributions of different parameters. While there is no universally applicable F-V relationship, Hill’s equation is useful in many situations. This equation is given by (P + a)(V + b) = (Po + a)b, where P is force, V is velocity, Po is maximal tetanic force, and a and b are empirical constants. As can be seen, as contraction velocity increases, contraction force decreases and vice-versa. A plot of the force-velocity curve is given in Figure 2.
Another important and well-researched relationship, shown in Figure 3, is that between force and length. Studies of muscle force under isotonic conditions have shown a bell-curve shaped relation between force and muscle length, with small tension at extremes of length and maximal tensions in between. This phenomenon fits well with the sliding filament theory of muscle contraction; according to the theory, more cross-bridges can be formed, and therefore more force can be generated, when the overlap between the actin and myosin filaments is greatest. Force-length relationships and optimal muscle lengths must be determined for individual muscles. Also, changes in force due to length changes should be considered when studying force-velocity relationships.
Properties of the contractile component are further influenced by events preceding their measurement. For example, fibers forced to shorten rapidly produce subsequent isometric forces lower than would otherwise be expected, while isometric force is enhanced following muscle stretching.
Finally, dynamic properties of muscle vary with fiber type. Humans have both “fast-twitch” and “slow-twitch” muscle fibers, which activate at different rates. Other intrinsic fiber properties, such as maximum shortening velocity, also influence performance.
The Series Elastic Component
The series elastic component in the model is an attempt to account for the stiffness, or spring-like resistance to stretch, of muscle. At least three different methods to measure the properties of the SEC have been developed, but all have been somewhat criticized on their assumptions. Experimental results have shown that the series elastic component does not exhibit a fixed elastic modulus when different forces are applied across it–stiffness seems to increase with increasing force. Approximately half of muscle stiffness can be accounted for anatomically by the tendon, but the source of the rest of the series elastic component is still debated.
The Parallel Elastic Component
While it has been shown that the parallel elastic component of the muscle model is responsible for significant forces under certain conditions, most research shows that it is negligible under the normal range of human joint movement.
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