Hill model of force production Three element model Contractile Component Series Elastic Component Parallel Elastic Component Viscoelastic behavior Describe the three element model of force production Describe the behavior of each component during dynamic force production Implement a Hill-style model to predict force production

Release experiments Two-phase response Elastic decline in tension Monotonic recovery Increasing length of release

Temperature Both development and recovery of tension are slower when cold

Activation increases damping Set muscle vibrating on a spring Activate (b) Amplitude of vibration decreases

Viscoelasticity Elasticity Viscosity Voigt-Kelvin (parallel) Force depends on length (F = k x) Viscosity Force depends on velocity (F= b v = b dx/dt) Voigt-Kelvin (parallel) Equal displacement; forces sum Maxwell (series) Equal forces; displacements sum

Instantaneous response Length step dx/dt∞ viscous force ∞ Voigt (parallel) model fails Maxwell (series) model looks elastic Force step Voigt model looks viscous Maxwell model looks elastic

Adaptation Creep Relaxation Under persistent force, viscous element lengthens Voigt: countered by rising elastic tension Relaxation Voigt model fails Maxwell spring pulls damper until force  0

Length Step Maxwell Model Instantly elastic Relaxation Voigt Model Instantly immobile Steady-state elasticity dLd/dt = k(x-Ld)/b; F=k(x-Ld) F = kx + b(dx/dt)

Force step Maxwell Model Instantly elastic Creep Voigt Model Instantly immobile Finite creep dL/dt = (dF/dt)/k+F/b dL/dt = (F-kL)/b

Dynamic Response Maxwell Model: Length control Voigt Model: Force control First one is different Does not return to initial condition Out of phase

Standard Linear Solid Series spring isolates the Voigt construct from incompatible length changes “Best of both worlds” Viscous creep/relaxation Persistent force

Three element model A.V. Hill (1922) H.S. Gasser & Hill (1924) Fibers as elastic tube Elastic myosin gel Viscous cytoplasm Elastic cell membrane/ECM Active state Contractile “stuff” with two rest lengths Time-dependent behavior from internal mechanics

Hill’s activation & release Release resets CE balance Active state starts, CE reference length changes Instantaneous CE force resisted by damper Tension recovers to a lower level: force-length relationship Time course of tension rise and recovery don’t actually match in real muscle

Cyclic stretches Viscoelastic model has short-range stiffness ie, matches Rack & Westbury’s nonlinear result

Conceptual revisions There’s no actual viscous structure Phenomenological contractile element i.e.: curve fitting F = FL(x) * FV(v) Series elasticity: tendon (?) Parallel elasticity Epi-/peri-mysium? Titin? You can’t really match physical structures with a phenomenological model

Application of Hill model Series & Parallel elastic elements Contractile element Activation, force-length, force-velocity F = a(t) * FL(x) * FV(v) 1.8 1.6 1.4 1.2 Force 1 Po 0.8 0.6 0.4 0.2 -0.5 0.5 1 Vmax Shortening Velocity

Modeling Simulink Matlab Mathematica Excel 1 Force f(u) Sarcomere F-L Product SL*u/ML ML->SL F-V 3 Activation 2 Velocity Length Simulink Matlab Mathematica Excel

Experimental measures Raw, isokinetic data Force-velocity/length curve Sandercock & Heckman 1997

What is a modern “Hill model”? Phenomenological: curve fitting Extrapolation from Isometric force-length Isotonic force-velocity Extra features Activation dynamics (ECC) Short-range stiffness Nonlinearities

Hill model + architecture Muscle is one big sarcomere Scaling LfVmax, L0 PCSAP0

Complex simulation platforms SIMM (Musculographics) SimTK (NIH) Animatlab (GSU) Neuromechanic DADS (LMS) SimMechanics (Matlab)

Model accuracy? One big sarcomere assumption Simulation of continuously changing velocity not so good One big sarcomere assumption Steady-state to dynamic assumption Estimation of force-length pretty good Winters et al., 2011 Perreault & al., 2003

Summary 3-Element model Descriptive but practical Contractile element (active forces) Isometric force-length Isotonic force-velocity Series elastic element (transient dynamics) Parallel elastic element (passive forces) Descriptive but practical