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
– 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
– 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
dLd/dt = k(x-Ld)/b; F=k(x-Ld)
Voigt Model
•Instantly immobile
•Steady-state elasticity
F = kx + b(dx/dt)
Force step
Maxwell Model
•Instantly elastic
•Creep
dL/dt = (dF/dt)/k+F/b
Voigt Model
•Instantly immobile
•Finite creep
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
“Best of both worlds”
Viscous creep/relaxation
Persistent force
Series spring isolates
the Voigt construct from
incompatible length
changes
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
Time course of tension rise and recovery
don’t actually match in real muscle
Tension recovers to a
lower level: force-length
relationship
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
Force
– Activation, force-length, force-velocity
– F = a(t) * FL(x) * FV(v)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.5
Po
0
0.5
Shortening Velocity
1 Vmax
Modeling
Sarcomere F-L
•
•
•
•
1
Simulink
Matlab
Mathematica
Excel
SL*u/ML
f(u)
Length
ML->SL
1
Force
3
2
Activation
Velocity
Product
f(u)
F-V
0.1
6.2
Constant
Mux
Lengt h
du/dt
Velocit y Force
Act iv at ion
Simple
Stretch Ref ex
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?
Simulation of continuously
changing velocity not so
good
• One big sarcomere assumption
• Steady-state to dynamic assumption
Winters et al., 2011
Estimation of force-length
pretty good
Perreault & al., 2003
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