Hill Muscle Model

Keywords— Flexion Extension, Hill Muscle Model, Gait cycle. 

The modeling and control of movements in this chapter relates to external control of muscles via so-called functional electrical stimulation. Macroscopic viscoelastic models started from the observation that the process of electrical stimulation transforms the viscoelastic material from a compliant, fluent state into the stiff, viscous state. Levin and Wyman [35] proposed a three-element model— damped and undamped elastic element in series. Hill s work [36] demonstrated that the heat transfer depends upon the type of contraction (isometric, slow contracting, etc). The model includes the force generator, damping and elastic elements. Winters [37] generalized Hill s model in a simple enhancement of the original, which... 

Winters, J.M., Hill-based muscle models a systems engineering prospective. In Winters, J.M. and Woo, S.L.-Y. (Eds.), Multiple Muscle Systems Biomechanics and Movement Organization. Springer-Verlag, New York, pp. 66-93,1990. 

In a Hill-type model, muscle s force-producing properties are described by four parameters... 

Some of the applicable muscle models include the Maxwell, Voigt, Hill and Carlson models (Figure 1). In particular, the Carlson (1957) equation is used in much of this work to describe the stress-velocity relationship of cardiac muscle over the entire cardiac cycle. Min et al. (1978) found very little difference in analyzing ventricular dynamics when he alternately used Carlson s equation only during isotonic contraction and Hill s equation during isovolumic contraction. 

The musculoskeletal model requires the muscle force simulation and the musculoskeletal geometry. For the muscle force simulation, Hill-based muscle models have been used for... 

FIGURE 24.6 Structure of the musculotendonal model (based on the Hill viscoelastic model). Abbreviations CE, contractile element SE, series elastic element PE, parallel viscoelastic element TE, tendonal elastic element , length M, muscle T, tendon. 

The muscle model identification problem can be categorized by the following factors (1) time domain continuous-time or discrete-time models (2) input types stimulus period (SP), that is, pulse frequency modulation, pulse width (PW) modulation, or a combination of the two (3) model outputs for example, muscle torque or force and muscle length or position (4) loading conditions isometric or nonisometric loads and load transitions and (5) model type linear models, nonlinear Hill-type models, and other nonlinear models. 

Just a year after Stephenson s classical paper of 1956, J. del Castillo and B. Katz published an electrophysiological study of the interactions that occurred when pairs of agonists with related structures were applied simultaneously to the nicotinic receptors at the endplate region of skeletal muscle. Their findings could be best explained in terms of a model for receptor activation that has already been briefly introduced in Section 1.2.3 (see particularly Eq. (1.7)). In this scheme, the occupied receptor can isomerize between an active and an inactive state. This is very different from the classical model of Hill, Clark, and Gaddum in which no clear distinction was made between the occupation and activation of a receptor by an agonist. 

Somlyo We did some work with David Trentham in which we looked for this in terms of modelling the lag phases, and at best we could come up with a Hill coefficient of 2. This was in smooth muscle, not Jurkat cells. My belief is that it is closer to 1. David wanted to be more conservative and said that it could be 2, but never 4 (Somlyo et al 1992). 

Fig. 8. Temperature dependence of tension. Curve 1 model fiber (A. Weber, 1951) la, actomyosin thread (Portzehl, 1950b), both in 3 X 10 M ATP curve 2 model fiber in 4 X 10 M ATP curve 3, model fiber without ATP (A. Weber, 1951) curve 4 muscle in tetanus (from A. V. Hill, 1951) curve 5 resting muscle (from Josenhans, 1949).

The difference between the contractile protein of the models and that of muscle becomes even greater if one attributes the elastic resistance of resting living muscle entirely to the sarcolemma and the connective tissue (Ramsey and Street, 1940, 1941 A. V. Hill, 1949a, 1950a). The con-... 

Some of the contractile properties of the models, however, differ markedly from those of the particular muscle from which they have been prepared (Table II, columns 1, 5 and 7). One of the most important of these differences, perhaps, is the slowness with which the models redevelop their tension after a release the time in the case of living muscle is short and characteristic of the speed of contraction (A. V. Hill, 1926 Gasser and Hill, 1924). It is hardly surprising that an actomyosin thread from a striated muscle no longer has the same short recovery time as that of the muscle (Table II, column 5), for the actomyosin from both slow and fast muscles appears to be much the same (Hamoir 1949). Actin and I.-myosin from quite different animals combine to give acto-myosins with the characteristic properties of the natural ones (Cigada... 

Eisenberg, E., Hill, T.L., and Chen, Y. 1980. Cross-bridge model of muscle contraction. Biophys. J. 29 195-227. 

FIGURE 48.3 The Hill model of muscle separates the artive properties of muscle into a contractile element, in series with a purely elastic element. The properties of the passive muscle are represented by the parallel elastic element. 

Hill TL, Eisenberg E, Chen YD, Podolsky RJ (1975) Some self-consistent two-state sliding filament models of muscle ctmtraction. Biophys J 15 335-372... 

One of the first attempts to examine the consequences of an electric-field-induced cooperative response was by Hill/ This effort was the basis for the later work of Blumenthal, Changeux, and Lefever and subsequently the work of Hill and Chen/ Most of these efforts, however, were directed toward understanding electric-field-induced excitability in those classes of membranes commonly referred to as excitable membranes, for example as in nerve and muscle. They were not attempts to model the interaction of external time-varying electric fields with biological membranes. Most all of these formulations were based on some type of mean-field theory and the use of lattice statistics. More recently Grodsky " and Denner and Kaiser performed somewhat analogous calculations with reference to a specific dipole model of an excitable membrane. Both analyses used a type of mean field theory to generate the thermodynamic expressions used to describe the behavior of the systems. 

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. 6.21). 

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