|ARIEL DYNAMICS WORLDWIDE
by Guskov, A. Eliutin, A.
Vorobievl and G. Ariel
For numerous applications, including real-time analysis in computer-aided systems there is a need for a computationally simple but comprehensive model of muscle behavior. Using a simulation approach a finite muscle fiber model is formed, based upon a formalized rule of recruiting of elementary motor units (EMU). The scheme of EMU was supplemented by an absolutely rigid shell, which introduces a new characteristic - controlled self-tension.
Based on the constant volume of the fiber during contraction, Elliot, Rome and Spencer developed a hypothesis which implied changing of the axial distance between the filaments on shortening. That leads logically to the idea of variable axial density of EMUs and the existence of transverse processes. The fundamental monograph by Hatze (2) most adequately describes the dynamics of the muscular force output in response to control signals. Still, due to basically unidimensional nature of the models introduced they are unable to describe the dynamics of the muscle fiber behavior in the co-ordinates "length - force activation".
In the proposed model the muscle fiber is composed of EMUS, the behavior of which depends on two generalized co-ordinates: Z, which characterizes the deformation of the passive elastic element of the EMU; and Q which corresponds to the deformation of the active contractile element, encapsulated in a rigid shell. For each set Q,Q',Z,Z' and for a given activation rate T there exist unique extension level L and exerted force N. Note, that vice versa is not true, i.e. a given combination of external parameters L and N might be achieved through different combination of internal parameters Q,Z, and T. Activation rate T determines the level of self-tension and acts as a control signal for the fiber. Different values of T correspond to different values of the fiber "free length." For T=O free length coincides with the "natural length." For other values of T free length is always less than natural and implies off-axis recruiting of EMUS. This corresponds to increasing the cross-section of the fiber on shortening. With increasing level of extension L the number of EWs along the axis grows, and the linear density of EMU decreases in the process of on-axis recruitment.
Free parameters present in the model, determine the fraction of activated EMUS, the degree of recruitment, non-linearity of the elastic and contractile elements and may be identified by means of straightforward experiments.
Running the proposed model with different parameter settings has demonstrated that most of the known partial models and experimental data (24) might be described by the model. Particularly, the on-axis recruiting mechanism accounts for successful simulation of the declining portion of the force - extension diagram for an activated muscle fiber. The proposed model allows description of the contractile process in the muscle fiber corresponding to Hill's equation (3) with respect to the excitation curve and initial conditions applied.
Another merit of the model is its feasibility for modeling a real tendon-muscle complex as a set of fibers with individual physical properties, and functioning collectively, with the condition of deformation compatibility at the attachment points.