Skeletal Muscle Mechanical Descriptions BME 615 Ray Vanderby

Crossbridge Mechanics Force measured is 3-4 pN per crossbridge Power stoke is only 11 nm!!!

Length-tension relationship (sarcomeres) (assumes isometric contraction) A.Optimum overlap B.Few available binding sites C.No available binding sites D.Fewer binding sites due to overlap W Herzog, Muscle Mechanics Not continuous F-L curve Isometric forces at max stimulation at various lengths

“Optimal sarcomere length” Length when there is most filament overlap & sarcomere produces maximum force Varies across mammals: –2.2  m in frogs –2.8  m in humans Considered constant across skeletal muscles in the human body…

First consider “isometric” force-length properties… At any fixed length, force transducer can be used to measure “passive” force (no stimulation) “active” pluse “passive” force (w/ stimulation)

Force-Length Properties of Whole Muscle Passive Adapted from McMahon, Muscles, Reflexes, and Locomotion Active part looks like a sarcomere force-length curve, active length range goes roughly from 0.5 to 1.5 times an “optimal fiber length” Where did the passive part come from? Active

Titin

What governs the passive F-L shape? Adapted from McMahon, Muscles, Reflexes, and Locomotion titin is coiled titin is straight

“Optimal fiber length” Muscle fiber length where muscle produces maximum active force … What would be the corresponding sarcomere length?

Isometric length-tension curve (muscles) W Herzog, Muscle Mechanics Not continuous F-L curve Isometric forces at max stim at various lengths Unique for each muscle due to: Fiber types Pennation PCSA Fiber length

Passive behavior of papillary muscle (a parallel fibered muscle) YC Fung, Biomech, 1993

Passive model of papillary muscle YC Fung, Biomech, 1993

Idealized contactile force versus length W Herzog, Muscle Mechanics Determined by numerous isometric F measurements

Normalized force versus length For a parabolic representation of muscle length behavior, normalize F by F 0 and L by L 0 This assumes a parabolic maximum isometric force versus muscle length relationship and a working range R = 0.8 L 0

Physiologic CSA  proportional to force production  muscle mass does not reflect force, as the arrangement must be considered

PCSA and Strength Physiological Cross-Sectional Area  Maximum Isometric Stress of Active Muscle  Maximum Isometric Force

Force-length properties Maximal isometric force, F 0 is primarily a function of physiological cross-sectional area (PCSA) PCSA=muscle volume/fiber length F 0 =k  PCSA where k is proportionality factor (20-40 N/cm 2 ) Force production is length-dependent –F-L properties have been derived for: Sarcomeres Isolated fibers Entire muscles –Depends on # of cross-bridges

Force versus cross section for a parallel fibered muscle W Herzog, Muscle Mechanics

Fiber length and the F-L curve You have two muscles  Muscle A has “optimal fiber length” = 4cm  Muscle B has “optimal fiber length = 10cm  How many sarcomeres in series does each fiber have? Muscle Length (cm)

Pennation W Herzog, Muscle Mechanics Pennation produces different: Working range R Optimal length L 0 Maximal force F 0

Parallel v Pennation  = 0°  = 30°  x x Force’ Force’ = xcos  Force’ = xForce’ = 0.87x

Parallel v Pennation 30° is very large pennation angle  increase in fiber packing more than offsets 13% reduction in force transmission  = 0°  = 30° 5 fibers7 fibers fiber packing

Pennation Angle (  ) Change Passive muscle 20° 45° Active muscle Fig Lieber (2002). LWW

Muscle Fiber Length  Indicates amount of muscle excursion   length =  excursion  Muscle velocity is proportional to muscle fiber length  longer muscle fibers have greater shortening velocities  sarcomeres are in series so shortening velocity is cumulative across fiber

FL/ML Ratio Low number indicates greater force (0.2) High number indicates greater velocity (0.6) Fig Lieber (2002) LWW

Force-length relation for muscle group W Herzog, Muscle Mechanics

Force-velocity relationship (Hill data) Force versus isotonic shortening from quick release of frog skeletal muscle at tetany. Obtained at optimal length. Max stress  200kPa YC Fung, Biomech, 1993 F is steady state force for shortening at velocity v

Force-Velocity Relationship shortening velocity % force 0 V max % V max % power % MVC Power =

Contraction Types Isometric – force production with no length change Concentric – force production while muscle length decreases Eccentric – force production while muscle length increases

Force-Velocity Relationship shortening velocity % maximum tension 0 concentric (+)eccentric (-)

Length-Tension-Velocity Fig 2-8. Lieber (2002) LWW

Extended force-velocity curve W Herzog, Muscle Mechanics Decreased force potential with increased velocity of shortening Increased force potential with increased velocity of lengthening Note: max F max  2F 0 1 st quadrant is Hill- type data Higher eccentric force means injury more likely

Force-velocity relationship (Hill data) Force versus isotonic shortening from quick release of frog skeletal muscle at tetany. Obtained at optimal length. Max stress  200kPa YC Fung, Biomech, 1993 F is steady state force for shortening at velocity v

Hill’s Model - 1 Hill model includes 2 parts: 1.A maximal F versus v relationship for quick release from isometric contraction with isotonic force. 2.Lumped parameter model to phenomenologically represent elastic stiffness component of muscle Hill model was developed from energy balance concepts

Hill’s Model - 2 Hill showed: Muscle produced heat in isometric contraction When isometrically contracted, muscle was released under an isotonic load that allowed muscle shortening where H is shortening heat and x is distance shortened where a is constant of proportionality and is function of level of action, PCSA, etc. Hill also showed: where F 0 is max isometric force

Hill’s Model - 3 During shortening contacting muscle produces extra heat (greater than isometric) and mechanical work Total energy in excess of isometric contraction then is Therefore, the rate of extra energy liberation is: where v is the speed of shortening

Hill showed that rate of extra energy liberation was inversely proportional to load F applied to muscles in shorting experiments For isometric experiments Hill’s Model - 4 and For contracting muscle experiments where b is a constant associated with the rate of energy liberation (Equation 1)

If v 0 is the maximum velocity of contraction at F = 0 Then Hill showed that parameter was related to v 0 by Hill’s Model - 5

Hill’s Model - 6 Rewriting equation 1 yields Rearranging terms produces the Hill equation

Force-velocity relationship (Hill data) Force P (g weight) versus isotonic shortening V (cm/s) from quick release of frog skeletal muscle at tetany YC Fung, Biomech, 1993 Frog sartorius in Ringer’s solution

Sartorius muscle for Hill experiments Force P (g weight) versus isotonic shortening V (cm/s) from quick release of frog sartorius at tetany

Hill Equation Limitations Only defines max stimulation Only defines shortening No eccentric contraction (although this may be the most important mode of loading) All data at or near L 0 Applies only to quick release F-v relationship not unique, different for other muscles, species, etc. (only empirical data)

Will fiber length affect force-velocity? A Assume each muscle has the same volume Label the force-velocity curves B

Force-velocity property assuming fast- twitch or slow-twitch fiber composition W Herzog, Muscle Mechanics Data from human vastus lateralis. Slow twitch type 1, fast twitch type 2 fibers

Lumped parameter Hill model YC Fung, Biomech, 1993 Alternative arrangements of springs can be used Dashpots can be used to capture viscous behavior

Alternative lumped parameter models CE SE PE CE SE PE SE Each series or parallel element can be a viscoelastic element AB C Researchers often prefer model B (Hill model) because of the way the series and parallel elements can be experimentally isolated

Musculotendinous Unit PEC SEC CE

Isometric length-tension curve for Hill lumped parameter model parameters W Herzog, Muscle Mechanics Passive tension-length behavior can be used as a non-linear stiffness in a parallel element. Stiffness of contracted muscle at lengths below L 0 provide series element behavior.

Hill lumped parameter model limitations Includes no damping (viscous behavior) Separation between SE and PE are quite arbitrary Modifications are necessary to account for history dependent behavior Valid only for forces and velocities below the F-v relationships previously defined (only empirical data)

General lumped parameter models Composed of series or parallel elements that are usually linear. Elements include: 1.Contractile or active element representing actin- myosin complex which produces specified force (within physiologic limits) rapidly after stimulation 2.Spring or elastic element representing elasticity of the connective tissue within the muscle 3.Dashpot representing the frictional dissipation which occurs with relative sliding of components and with shearing of fluids within muscle during deformation Similarities to viscoelastic models of passive tissues are clear with unique active element whose behavior is voluntary

Viscoelastic behavior: The model would need damping elements to model these behaviors Data from Taylor et al (1990). Fig 8.8a - Enoka (2002). Human Kinetics Increased length at same applied load (78N) - Creep Decreased force needed to achieve constant displacement (10%) - Relaxation

A four parameter model - example Measured tension T(t) Force generator T 0 Contractile component  This is a 4 parameter lumped model to fit data of a frog gastrocnemius  TA McMahon in Muscles, Reflexes and Locomotion.  The contractile component uses a dashpot in series because data showed a significant difference (lag) between T 0 and measured tension T(t)

Skeletal Muscle Modeling Phenomenological Model (Hill-type) Based on 3 parameter elastic “standard solid” Describes structural behavior w/o structural representation using springs Provides limited insight to contraction and contractile proteins behavior Easy to use Used in whole tissue applications & for fibers Structural Models (Huxley- type) Uses tissue structure (cross-bridge theory) to provide detailed description of muscular contraction Harder to use Used by biophysicists and biochemists, but typically too complex for whole tissue mechanics

Muscle architecture Gluteus maximus Rectus femoris Semitendinosus Soleus Images used with permission from University of Washington online Musculoskeletal Atlas Describe differences you see between these skeletal muscles

Factors that Affect Musculotendon Force Production  Activation level  Muscle length  Physiological cross-sectional area  Pennation angle  Velocity  Tendon length and elasticity

Complications in muscle mechanics Force & velocity are voluntary, they can be whatever we decide they need to be within limits Limits of force/velocity can be measured, but there are many confounding factors and they can be acutely altered: –Warm-up induced changes –Changes due to previous loads –Stretch induced changes –Fatigue induced changes

Warm-up: Increase Core Temperature Physiological effects: –Increase O 2 dissociation from hemoglobin & myoglobin –Enhance metabolic reactions –Facilitate muscle blood flow –“Reduce” muscle viscosity – “pseudoelastic” –Increase connective tissue extensibility –Improve conduction velocity

Warm-up: Enhanced Muscle Function  No change in max force   peak velocity (V max )   peak power Fig Enoka (2002). Human Kinetics

Impulse Force Production Concentric preceded by Eccentric Greater concentric impulse with prior eccentric contraction

Stretching - Flexibility  Greater joint ROM  Dependent on ability to relax muscles  Not necessarily reflection of muscle length Fig Enoka (2002). Human Kinetics Data from McHugh, Fox and Gleim (1998).

Stretching & Flexibility: Theories of Stretching Effects Improve relaxation (PNF) –Reduced muscle activation occurs during and immediately after stretch –Greater tolerance to stretch discomfort repeatedly observed and correlated decreased sensory feedback or attenuated interpretation of feedback Increase passive compliance –only temporarily (few hours) Viscoelastic effect phenomenologically

Viscoelastic changes: Temporarily Reduced Passive Stiffness Data from Taylor et al (1990). Fig 8.8a - Enoka (2002). Human Kinetics Increased length at same applied load (78N) - Creep Decreased force needed to achieve constant displacement (10%) - Relaxation

Fatigue  Decreased force for repeated, sustained isometric contractions Fig Neumann (2002) Mosby

Fatigue: Endurance Time Fig Lieber (2002) LWW maintained indefinitely

Fatigue Mechanisms Muscle Fiber Neuromuscular Junction Peripheral Nerve Motor Command Psychological Factors peripheral central

Fatigue: Muscular Mechanisms  Reduced conduction velocity and amplitude of MUAP  elevated excitation threshold of muscle fibers  Slowing of relaxation phase of twitch contraction fatigued nonfatigued

Complications in muscle mechanics Chronic factors can also alter muscle mechanics –Exercise/training induced changes –Disuse induced changes –Metabolic/diet changes –Hormonal induced changes