A Mathematical Model of the Dynamic Force–length Relationship of Skeletal Muscle Operating on Ascending and Descending Limbs

Abstract

A new Hill-type model of skeletal muscle contraction referred to as the SL/ST model is proposed based on recent physiological ndings: intact human skeletal muscles operate on the ascending limb (hereinafter referred to as the as-limb) and the descending limb (ds-limb) of the isometric force-length relationship; and stretch-evoked force enhancement (ST-enhancement) is found on the ds-limb. Dynamic behaviors differ remarkably between the two limbs. The model has two modes: a sliding lament mode (SL mode) and a stretch-evoked force enhancement mode (ST mode). The SL mode operates on the as-limb, and on the ds-limb when the muscle is shortening, while the ST mode operates when ST-enhancement occurs in the ds-limb. Transient force responses of the model to length perturbations were similar to those of frog semitendinosus muscles in vitro. Length responses to step change in load of the model suggest that the muscle is unstable and not static in SL mode on the ds-limb, while it is stable and static in SL mode on the as-limb and in ST mode on the ds-limb.

A Novel Hill-type Two-mode Model of Skeletal Muscle to Simulate Mechanisms Underlying Position Control on the Descending Limb of the Force–Length Relation

“…The previous report [11] 7] and compared with physiological data of rabbit skeletal muscle [17]: the initial length is 1.17Lo (point S in Fig. 5) while the isotonic load is 0.2P 0 or 0.4P 0 .…”

“…A model in SL-mode is shown in Fig. 2B [11]. For mathematical expression, the CC is represented by a force generator (FG) and a viscous-like component (VC).…”

“…A model in ST-mode is shown in Fig. 2C [11]. A parallel elastic-like component (PEC ST ) is added to the CC.…”

“…In a computer simulation, force P(t) after the onset of stretch is calculated with Eqs. (1), (2), (4) and (8)- (11). L(t) is given as a ramp-hold function ( Fig.…”

“…While many models have been proposed [8][9][10], few reports have focused on properties on the ds-limb. In a previous paper, the author proposed a new Hill-type two-mode model comprising a sliding lament mode (SL-mode) and a stretch-evoked force enhancement mode (ST-mode) [11]. Note that ST-enhancement occurs in ST-mode.…”