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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

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

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Temperature Both development and recovery of tension are slower when cold

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

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Viscoelasticity Elasticity – 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

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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

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

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

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

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

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

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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

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

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

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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

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Application of Hill model Series & Parallel elastic elements Contractile element – Activation, force-length, force-velocity – F = a(t) * FL(x) * FV(v) -0.500.51 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Po Vmax Shortening Velocity Force

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

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

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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

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Hill model + architecture Muscle is one big sarcomere Scaling – L f V max, L 0 – PCSA P 0

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

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

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

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