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 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

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 Relaxation
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 two rest 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
-0.5
0.5 1
Vmax
Shortening Velocity

17. 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

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? 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

23. 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