A CONSTITUTIVE EQUATION FOR THE PASSIVE PROPERTIES OF MUSCLE *

4 constitutive equation for unstimulated muscle based on nonlinear elastic and linear damping elements is proposed. This equation is solved in closed form to predict creep and stress relaxation in passive muscle and. if combined with an active contractile element in a three-element model. initial response to a sudden change in length or force during an isometric twitch. A stress relaxation experiment on unstimulated muscle is used to explicitly compute the three constants. The values of the constants associated with the muscle’s elastic properties obtained this way are similar to those derived from sxperiments which measure only elastic properties. Inotropic agents do not affect the values of these constants which indicates that they truly reflect passive propertics of the muscle. In order to succinctly describe the passive properties of muscle, one must have a theoretical construction for the interpretation of experimental data. This paper uses published experimental data to develop sucha construct in the form ofanequation which relates Isngth, force and their rates of change (a constitutive equation) in unstimulated muscle. This equation permits one to describe the passive properties of muscle with merely three constants and to unify the published results of a variety of experiments performed on both unstimulated and stimulated muscle. There is considerable published evidence that viscous effects are important in unstimulated muscle. (The authors do not always identify these effects as such.) For example, Bozler (1936) reported that smooth muscle creeps when subjected to a constant load; Alexander (1962) showed creep-like phenomena in whole hearts; Hefner and Bowen (1967) showed creep in isolated cat papillary muscle (Fig. 1): Walker (1960) demonstrated hysteresis in isolated dog papillary muscle, as did Fung (1970) in isolated rabbit papillary muscle, and Alexander and Johnson (1965) with frog sartorius. There is also widespread evidence that muscle exhibits stress relaxation (Fig. 2). It has been observed in isolated papillary muscles of the cat (Hoffman et al.. 1965: Sonnenblick, 1962a). the rabbit (Brady, 1965), and the rat (Grimm and Whitehorn. 1966). as well as many types of skeletal muscle such as * Receised 10 September 1973. + Present address: Division of Cardiology. Stanford University Medical Center, Stanford, California 91305. U.S.A. This investigation was supported in part by Health Services Demonstration Grant No. HS 00116-03 from the Health Services and Mental Health Administration. frog gastrocnemius (Alexander. 1959; Little. 1969). frog sartorius (Jewel1 and Wilkie. 1960; Husley and Simmons. 1970), rat tricepts surae (Walker and Thomas, 1960). and other skeletal muscles (Hill. 1926: Levin and Wyman. 1927; Abbott and Lowry. 19571. puppy bladder (Alexander, 1959). cat intestine (Burnstock and Prosser. 1960). rabbit taeniae coli (Gordon and Siegman. 19713. b). and other smooth muscles (Bozler. 1936; Yin and Fung, 1971; Weiss er a[., 1972). In every case one can divide the response to a step change in length into two phases: a phase in which the force rapidly peaks and drops, followed by one in which it slowly approaches steady-state. Hill (1926). Abbott and Lowry (1957). and Burnstock and Prosser (1960) also noted that the larger the peak force. the faster it decays to steady-state. LMany investigators have analyzed these data mathematically. Some present mechanica models based on the chemical kinetics of muscular contraction (e.g. Apter and Graessley. 1970) to account for the viscoelastic properties of passive (i.e. unstimulated) muscle. Fig. I. An example of creep in isolated cat papillary muscle. Note that (x, .u_) : 2(.x, x_). (Data from Hefner and Bowen (1967). Fig. 3.) 137