Hill's Muscle Model - Three-element Model

Three-element Model

The three-element Hill muscle model is a representation of the muscle mechanical response. The model is constituted by a contractile element (CE) and two non-linear spring elements, one in series (SE) and another in parallel (PE). The active force of the contractile element comes from the force generated by the actin and myosin cross-bridges at the sarcomere level. It is fully extensible when inactive but capable of shortening when activated. The connective tissues (fascia, epimysium, perimysium and endomysium) that surround the contractile element influences the muscle's force-length curve. The parallel element represents the passive force of these connective tissues and has a soft tissue mechanical behavior. The parallel element is responsible for the muscle passive behavior when it is stretched, even when the contractile element is not activated. The series element represents the tendon and the intrinsic elasticity of the myofilaments. It also has a soft tissue response and provides energy storing mechanism.

The net force-length characteristics of a muscle is a combination of the force-length characteristics of both active and passive elements. The forces in the contractile element, in the series element and in the parallel element, and, respectively, satisfy

On the other hand, the muscle length and the lengths, and of those elements satisfy

During isometric contractions the series elastic component is under tension and therefore is stretched a finite amount. Because the overall length of the muscle is kept constant, the stretching of the series element can only occur if there is an equal shortening of the contractile element itself.