A three-element description for muscle with viscoelastic passive elements

Abstract

This paper explores the implications of expanding Hill's three-element model by replacing the series and parallel elastic elements with nonlinear viscoelastic elements. The resulting model includes a contractile element whose shortening relates not to a mechanical length, but to the number of active cross-bridges. Given this interpretation, the contractile element behaves in qualitatively agreement with known sarcomere biophysics. It also explains why one cannot vary maximum tetanic force and maximum shortening velocity independently in skeletal muscle, why cardiac muscle fails to exhibit hyperbolic force-velocity curves and why the curve's shape depends on how one obtains the curve. This analysis also shows that the classical formula for contractile element velocity, V CE (t) = P (KP + C) , actually gives initial muscle shortening velocity for a twitch that begins shortening at time t, not contractile element velocity. This model suggests a simple hypothesis to relate the inotropic agents action at a subcellular level to the mechanical events and permits indexing inotropic state at constant muscle length with one parameter. These results follow from replacing the two purely elastic elements with viscoelastic ones; they would still follow even if the exact formulation proposed for the viscoelastic element was wrong.