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 -
REVIEW OF RELATED LITERATURE
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
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.
As an example of calculation with the model
described, Figure 1 shows the surface of static deformation of a model fiber in
co-ordinates "activation extension - force."
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.
It is significant that the discussed model
distinguishes between isotonic mode at the level of EMU and at the level of muscle fiber.
At the same time, isometric mode is identical for either level.
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.
During this process each fiber is considered as a
non-linear visco-elastic controlled structural element. This allows studying external
force dynamics with relevant internal control.
Figure 1: Static deformation of a model fiber.
- Elliot, G.F. et al. Nature 226:417-420,
- Hatze, H. Myocybernetic Control Models of Skeletal
Muscle, Univ of South Africa, 1981.
- Hill, A.V. Proc. R. Soc. B. 159:297-318,
- Huxley, A.F. 243:1-43, 1974. J. Physiol.