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Hill model of force production

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

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Hill model of force production

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  1. 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

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

  3. Temperature • Both development and recovery of tension are slower when cold

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

  5. 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

  6. 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

  7. 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

  8. 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)

  9. 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

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

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

  12. 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 tworest lengths • Time-dependent behavior from internal mechanics

  13. 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

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

  15. 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

  16. 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 Vmax -0.5 0 0.5 1 Shortening Velocity

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

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

  19. 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

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

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

  22. Model accuracy? 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

  23. 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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