Download lecture 4 muscle

KIN 840
Muscle Mechanics
© 2006 by Stephen Robinovitch, All Rights Reserved
2006-1
Outline
• anatomical structure
• eccentric and concentric contractions
• determinants of muscle force (stimulation frequency,
muscle fibre type, muscle length and velocity, crosssectional area)
• size principle
• Hill's equation for muscle shortening
• Hill’s active state model of muscle contraction
• quick-release experiments
• Huxley's sliding filament model
• Loeb's "virtual muscle" model
Muscle anatomy
Eccentric and concentric contractions
• Concentric (energy
generating, positive work)
contractions tend to
increase joint angular
velocity, and increase the
total energy of the system
• Eccentric (energy
absorbing, negative work)
contractions tend to
decrease (or brake) joint
angular velocity, and
reduce the total energy of
the system
Concentric
(accelerating)
contraction
of m2
Eccentric
(braking)
contraction
of m2
Positive and negative joint work
Joint
moment
(Nm)
B3
extensor
A4
B4
A3
B1
A2
A1
B2
flexion
extension
A6
B7
A7
flexor
• direction of joint
work depends on
polarity of joint torque
and direction of joint
rotation
• movements 2, 3, 4,
and 5 involve positive
work or concentric
contractions
• movements 1, 6, and
7 involve negative
work or eccentric
contractions
B6
A5
B5
Joint
rotation
(radians)
Muscle force results from
interaction between
contractile proteins
• sarcomere: the smallest anatomical
unit that contracts like a muscle
• sliding filament model proposes that
muscle force arises from cyclic
binding between thick and thin
filaments of the sarcomere
• thin filaments contain actin, troponin
C, and tropomyosin
• thick filaments contain myosin
• in the absence of calcium,
tropomyosin prevents myosin from
attaching to actin
Calcium is needed for muscle contraction
• at the onset of an action potential,
the sarcoplasmic reticulum (SR; a
membrane that surrounds the
myofibrils) releases calcium
• calcium binds to troponin, causing
a conformal change in tropomyosin
which reveals myosin binding sites
on the actin
• simultaneously, adenosine
triphosphate (ATP) is hydrolyzed by
ATPase in the myosin head,
providing the energy for cross-bridge
attachment
• the SR re-sequesters calcium at the
end of the action potential, thereby
inducing muscle relaxation
Factors affecting muscle force development (a
partial list)
• muscle fibre type
• number of activated motor neurons, frequency of
discharge
• muscle length
• velocity of shortening/ lengthening
• muscle geometry (physiological cross-sectional area
(PCSA), angle of pennation)
Muscle-nerve interaction
• a motor nerve enters muscle
and splits into numerous
axons; each axon contacts 102000 muscle fibres
• each muscle fibre is innervated
by only one motor nerve axon,
and contracts in response to an
action potential in that axon
• motor unit: a single motor
nerve axon and all the muscle
fibres it contacts
muscle
# muscle
fibers
# motor
units
platysma
27,100
1,100
av. fibers
per motor
u.
25
Brachioradialis
130,000
330
410
Tibialis anterior
250,000
450
600
gastrocnemius
1,120,000
580
2,000
Stimulation frequency affects muscle force:
twitch and tetanus
• muscle force can be modulated by
varying: (1) the number of recruited
motor neurons, and (2) the frequency
of discharge (i.e., stimulation rate) in
motor neurons
• a single action potential (S1) produces
a twitch contraction, a quick rise and
slow fall in force
• a tetanus occurs when a new action
potential (S2) arrives before the
previous twitch has dissipated, and
there is force summation
• at stimulation frequencies >30/s, there
are no twitch transients (fused tetanus)
Three types of muscle fibres and motor units, defined
by contraction speed, peak force, fatigue resistance
Size Principle
• When a stimulus is applied to
the ventral aspect of the spinal
cord, the smallest and most
excitable motor units are
activated first. These tend to
be slow (S) motor units which
innervate slow oxidative (SO)
muscle fibres. Larger FR and
FF motor units that innervate
FOG and FG fibres are
recruited only at high levels of
force.
• Sequence is reversed when
force level falls, with largest
motor units dropping out first.
Active force development in the sarcomere
depends on actin-myosin overlap
• (A): no overlap between actin
and myosin, zero developed
tension
• between (A) and (B): tension
increases linearly as overlap
increases
• between (B) and (C): maximum
overlap & maximum tension
• left of (C): interference between
actin filaments reduces ability of
crossbridges to develop tension
• left of (D): myosin filaments
collide with Z-lines and fold, and
force declines rapidly
Muscle length affects force development in
whole muscle
• the tension developed in a whole
muscle is the sum of active force
due to muscle contraction and
passive force due the passive
stiffness of tendon and muscle
• the passive force is negligible for
lengths less that the normal resting
length (l0)
• the active force follows the
tension-length behaviour of the
sarcomere, and scales with muscle
activation
Muscle velocity affects force development in
whole muscle
• force (T) is greater during
lengthening than shortening
contractions
• the greater the shortening
velocity (v), the smaller the
force (explains why we
cannot lift heavy objects
quickly)
• in the shortening regime,
mechanical power output is
maximum when T and v are
around one-third their
maximum values
Muscle force-velocity behaviour is described by the
Hill Equation
An empirical relation that
describes the force-velocity
behaviour of muscle during
shortening is the Hill Equation.
Greatest force is developed when lengthening
near resting length
Hill’s active state model of muscle contraction
• Hill assumed:
(1) for a given length, muscle always
develops the same peak force T0(x1,t);
(2) if the muscle is shortening, some
force is dissipated in overcoming
inherent viscous resistance
• B: muscle damping constant, which
must be a nonlinear function of
shortening velocity and temperature
• KSE: stiffness of the series elastic
component; represents forcedeflection properties of tendon
• KPE: stiffness of the parallel elastic
component; represents forcedeflection properties of sarcolemma,
epimysium, perimysium, and
endomysium
Quick-release experiments for determining the
Hill model parameters
• hold muscle length fixed with
the catch
• stimulate muscle to produce
peak (isometric) force T0
• instantly release catch
• at the instant of release,
muscle force is reduced to a
value T (where T < T0) that
depends on weight in pan
Quick-release experiments (cont)
• there is an instant change (Δ
x2) in the length of KSE
following release
• this is followed by a more
gradual change (Δx1) in the
length of the muscle
• as T increases, there is a
decrease in v (slope of dashed
line), reflecting that muscle
cannot shorten quickly under
high loads
• combinations of T and v
reflect the force-velocity
properties of a given muscle
Shortcomings of the active state model
• the model predicts a negative T0
during the end phase of the twitch
(marked by asterisk in diagram at
right)
• the liquid in muscle (water) does
not have the required nonlinear
damping characteristics
• the model cannot accurately predict
muscle force during lengthening
(the slope of the T-v curve is about
6-fold greater for lengthening than
shortening, and muscle length
increases rapidly when the load
exceeds 1.8T0)
Huxley’s sliding
filament model
• Force development is due to
stretch of elastic myosin
crossbridges, which can form
bonds with actin for x<h
• bonds can be maintained for x>h
(tensile) and x<0 (compressive)
• for (0<x<h), rate of attachment (f)
exceeds rate of detachment (g)
• rate of detachment (g) is slower
during lengthening than
shortening, thus accounting for
greater force under eccentric
conditions
Gerry Loeb’s virtual muscle (J Neuroscience
Methods, 2000)
• each muscle consists of (1) a
contractile element, (2) a
series elastic element, and
(3) a muscle mass
• instabilities in the model are
prevented by the muscle
mass, and a small viscosity
in the parallel elastic
element
• contractile element consists
of several motor units (3-5
for each fiber type), each
defined by its order of
recruitment, firing
frequency, force-lengthvelocity relationships,
passive parallel elastic
element, and total forceproducing capacity
(proportional to total crosssectional area)
•Matlab/ Simulink-based model for muscle and
tendon
•distributed freely over the Internet at
http://ami.usc.edu/projects/ami/projects/bion/muscu
loskeletal/virtual_muscle.html
Virtual Muscle (cont)
• the motor nucleus of the
whole muscle receives a
single, time-varying
neural activation
command signal, which
activates each motor unit
according to the userdefined recruitment order
• within each motor unit,
the frequency of
motoneuronal firing is
modulated in a realistic
manner
• the activation signal can
be based on pre-recorded
EMG data, simulated
feedback-driven reflex
dynamics, or cortical
commands
Virtual muscle (cont)
• user defines the properties of
each fiber type and the
morphometry of each
musculotendon element,
including the elastic
properties of tendon and
aponeurosis
• user then integrates these
elements with their defined
skeletal dynamics and control
system (developed in
Working Model)