BIOP2203 Biomechanics S1. BIOPHYSICS 2203: Biomechanics

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1 COURSE INTRODUCTION BIOPHYSICS 2203: Biomechanics MECHANICS -The study of motion of bodies. -Movement of all material bodies are subject to the laws of mechanics. Classical (or Newtonian) mechanics: -The size of the body is large compared to the size of the atom ( m). -The velocity of the body is small compared to the velocity of light ( m/s). Mechanics: - Kinematics - Describes the motion of one or more bodies (time, position, velocity and acceleration). - Dynamics - Cause of motion, relationships between motion, forces and properties of moving bodies (mass, force, momentum, energy and power). BIOMECHANICS - The mechanics of biological systems. - Interdisciplinary field involving Physics (mechanics) and zoology, botany, anatomy, physiology, medicine, human movement. Fields within biomechanics: - Muscle mechanics, ergonomics, orthopaedics, motor control, electromyography, locomotion, sports biomechanics. BIOMECHANICS OF LOCOMOTION Terrestrial locomotion (walking and running) - Mechanics. Aquatic locomotion (swimming) - Hydrodynamics. Aerial locomotion (flying) - Aerodynamics. Key Themes: - Effect of size on mode and speed of locomotion and the energy cost of locomotion. - Gait vs energy cost Dr Ralph James Room 5-2A ralph@physics.uwa.edu.au mobile:

2 1 ANIMALS as MACHINES 1.1 MECHANICAL MODEL of the BODY A Biophysicist views an animal as a machine: Definition of a machine: - A mechanism that transmits force from one place to another. (Usually also changes magnitude) ANIMAL CHARACTERISTICS: - Skeleton - Muscles - Tendons - Ligaments - Joints - Neural command - Reflexes MACHINE CHARACTERISTICS - Levers - Actuators - Joints - Pulleys - Springs - Dampers In animal locomotion: Skeleton - Is theframework of the animal. Muscles - Develop forces to move skeleton. Tendons - Transmit force generated by muscle to skeleton. - Act as spring (store some energy as strain energy, returned in elastic recoil). 2

3 Skeleton and muscle forms a lever system. - Muscle force,, acts with a moment arm,, on the bone to produce a joint torque. - may vary with the angle of the joint) In animal locomotion: Limb rotation whole body translation Classification of lever systems: - 1st order: The fulcrum lies between the effort and the load - 2nd order: The fulcrum is at one end, the effort at the other end and the load lies between the effort and the fulcrum - 3rd order: The fulcrum is at one end, the load at the other end and the effort lies between the load and the fulcrum 3

4 load moment arm of force Mechanical advantage: MA = =, force moment arm of load where 0 < MA < Joints in the human body: - Are most often 3rd order lever systems. - Operate at mechanical disadvantage (i.e. muscular force >> load ). - As result, achieve large range of motion (ROM) and speed of movement. Many muscles are multi-joint muscles (pull over 2 or 3 joints). - Complex (not well understood). - Produce economical movement. Functioning of the animal "machine": - Generation of forces by the machine. - Energy efficiency of the machine. - Effect of size on the structure and motion of the machine. Joints: Type Degrees of freedom example Hinge, pivot 1 Knee, ankle, elbow Ovoid, saddle 2 Wrist Ball and Socket 3 Hip, shoulder 4

5 Kinematic chain: - Bones of the extremities used for locomotion form an open-loop kinematic chain. - Large number of degrees of freedom are required in joints to achieve flexibility in the positioning of the extremity. Functioning of the animal machine : - Generation of forces by the machine. - Energy efficiency of the machine. - Effect of size on the structure and motion of the machine. 1.2 EXERTING FORCES with MUSCLES Striated muscles power the movement of most animals. Muscles exert forces to - Support body weight (overcome gravity). - Accelerate / decelerate the body and limbs (to overcome inertia). - Overcome resistance of fluid (air or water) through which the animal moves. Muscle Mechanics: Muscles operate by developing tension and shortening - The muscle pulls on bone (connected with a tendon) to move the joint. - Muscles cannot push, and consequently at least two muscles are required to work a joint. Muscles are comprised of: - Fascicles - Bundles of muscle fibres - Muscle fibres(cells) - Regular repeating pattern of filaments, diameter µm, length 2 40 mm. - Protein filaments - thick filaments (myosin) and thin filaments (actin & tropomyosin), myofibrils diameter µm 5

6 BIOP2203 Biomechanics Fascicles (bundle of muscles fibres) Cross-section of fascicles 6 S1

7 The thin filaments are held together by Z discs. The distance between each Z disc is defined as 1 sarcomere that contains 2 sets of thin filaments and 1 set of thick filaments. Muscular activation - Occurs from the electrical stimulation of muscle fibres by motor nerves. - An individual muscle fibre obeys an all-or-nothing law when stimulated (i.e., a single fibre has a given electrical stimulation threshold). - However, different muscular fibres have different electrical stimulation thresholds. - As a result, the overall tension in a muscle is fine-tuned by the variety of activation thresholds present in a group of muscle fibres. The sliding filament theory of muscular contraction involves cross-bridges in thick filament (myosin). - The thick filament attaches to a thin filament, bends (swings), detaches, swings back and then reattaches (i.e., pulls on the thin filament). - Thin filaments are pulled between thick filaments. - The degree of overlap increases and so the sarcomere shortens. - Energy for force generating cycle of cross bridges is obtained from stored chemical energy, ATP (Adenosine triphosphate). - Cycle time 50 microseconds. 7

8 Muscular contraction (Crossbridge formation) - Myosin binds with actin. Muscular loosening (Crossbridge release) - Tropomyosin blocks actin filament so myosin cannot bind. 8

9 Limited range of muscular contraction - Minumum length 70% maximum length. Force exerted by muscle number of fascicles activated number of attached crossbridges (depends on overlap of filaments). Maximum force is generated when all crossbridges are attached. For muscle-tendon unit: Tension Due to contractile elements in muscle. Tension Due to connective tissue (tendon). 9

10 Maximum force Cross-sectional area of muscle Maximum stress, F/m N/m N/mm 2 i.e., Human biceps A mm 2 F 1500 N (large force) But since the mechanical advantage of the biceps 1/10, Lift only 15 kg. Work done by muscles (force.distance) W 200 J/kg Maximum power (work per unit time) P 1.0 kw/kg (short-term power generation) 0.3 kw/kg (sustained power generation) Hill equation: Force velocity curve for muscle shortening. v ( F + a) = b ( F o F) where F Force (N) F o Isometric tension developed by muscle. v Shortening velocity a,b Muscle dependant constants ( a 0.2 F o ) Maximum velocity of shortening occurs when F = 0, b = a F o v max Force and power curves for muscle shortening velocity: - Faster contraction lower maximum force (less resistance faster contraction) - Maximum power generation ( P = F.v ) at intermediate velocity. The force and power that a muscle has depends on the rate of shortening (Alexander, 1982). 10

11 The sliding filament - cross-bridge theory: a critical evaluation Reference: GH Pollack "Muscles & molecules - uncovering the principles of biological motion" (1990) BIOL Q The sliding-filament theory was developed in the 1950s by HE and AF Huxley. It is a composite of two theories: the sliding-filament model in which filaments of constant length slide past each other, and the cross-bridge theory which describes the propelling mechanism. In the latter mechanism the cross-bridge passes through a specific cycle of attachment, rotation and detachment, which is powered by ATP hydrolysis and can be repeated many times during contraction. Although much experimental evidence supports these theories there remain a number of discrepancies and problems with it. Despite the weight of contradictory evidence there theory has remained popular in virtually all text books in the last decades. Sir Andrew Huxley (1957) pointed out that muscle scientists have a long history of getting stuck in dogmatic grooves, firmly believing models that subsequently turn out to be incorrect. In the small subdisclipline that is muscle, not only are the dissenters from currently orthodoxy forbidden to appear at meetings, but their work disappears from the record by never being quoted. Some of these difficulties are reviewed in Pollack's book: Thick filament length. In unactivated sarcomeres, filament lengths are confirmed to remain constant, as predicted by the theory. But activated thick filaments are frequently reported to shorten, sometimes substantially. Tension-length relationships. Some studies show little or no variation in tension as the sarcomere is extended from full overlap to almost half-maximal overlap - as though reducing the number of bridges has no effect on tension. Cross-bridge morphology. Electron micrographs of the filament lattice reveal cross-bridges as rung-like structures that interconnect adjacent thick filaments. Bridges apparently lack the freedom to swing. ATP usage. The theory's expectation that stages of the power stroke are coupled to stages of the ATP hydrolysis cycle is not supported. Input and output energy do not follow the same time course. In vitro models. Observation of reverse gliding leads to interpretational difficulties. Random action. Random action by numerous cross-bridges leads to the prediction smooth shortening; however, the shortening waveform is stepwise. In situations with few cross-bridges, random action predicts tension fluctuations; however, the tension waveform is fluctuation free. Pollack offers an alternate mechanism involving the helix-coil transition for the myosin rods in the presence of calcium. 11

12 Experiments with isolated muscle: A.V. Hill ( ) - physiologist (University College, London) Types of muscle contraction: - activation of muscle fibres causes a tendency to shorten. - what happens to muscle length depends on the load. 1. Concentric contraction - muscle shortens - work performed by muscle 2. Eccentric contraction - muscle lengthens - work performed on muscle 3. Isometric contraction (degrades mechanical energy to heat) - muscle length constant - muscle does no work 4. Isotonic contraction - muscle contracts or lengthens under constant load 12

13 Experimental apparatus: isometric contraction isotonic contraction twitch tetanus - transient rise in muscle force due to stimulation. - response to train of stimulations (> 30 Hz). Force tetanus unfused tetanus 0 Twitch 1 2 Time (s) The strength of the contraction in a twitch depends on the strength of the stimulus. The maximum occurs when the stimulus is sufficient to excite all the fibres in a muscle. Each individual fibre gives an "all-or-nothing" response. If the muscle is repeatedly stimulated and with such rapidity that no complete recovery periods are allowed between stimulations, the condition of fatigue appears. The response of the muscle becomes weaker until they fail entirely. This condition of fatigue will last until the oxidation and removal of wastes has been completed. 13

14 The results: The muscle produces the highest tension when held in the apparatus at the length it normally has in the intact animal. If held at longer (or shorter) lengths, the active tension produced is less. You may conclude from this that nature knows best. And, in fact she does. If a muscle is surgically reattached to an animal so that its length is changed, the muscle gradually adapts to its new length and, after a few weeks, is able to exert its maximum isometric contractions at the new length. The interpretation: A muscle stretched beyond its normal length has less overlap between the thick and thin filaments. Thus fewer cross-bridges can form to slide the filaments against each other. If the muscle is stretched so far that the thin filaments are pulled entirely away from the thick filaments, the muscle exerts no tension at all. As for the effect of holding the muscle at shorter than normal length, the thin filaments extend so far across the sarcomere that they interact with crossbridges exerting force the opposite way - reducing the tension generated. Tetanised muscle fibre: force - depends on overlap of filaments. number of attached crossbridges (independent force generators) 14

15 Tension vs filament overlap Force - length curve (for musculo-tendinous unit): isometric tension total tension passive active length (% of resting length) Active tension Passive tension - due to contractile elements in muscle. - due to connective tissue. Series elastic component (tendon) Parallel elastic component (connective tissue surrounding fibre) (sarcolemma) A giant biopolymer titin connects the end of each actin filament to the transverse structure thereby preventing the filaments from being completely pulled apart. 15

16 Hill Equation Force - velocity curve: Empirical relation for muscle shortening (Hill, 1938). v (F + a) = b (F o - F) with F force F o isometric force developed by muscle v shortening velocity a, b constants, depend on muscle type (a 0.2 F o ) Experiment: Several force-velocity curves for skeletal muscle. Shortening is most rapid when the muscle is loaded by a weight equivalent to that normally imposed by body structures. 16

17 Maximum velocity of shortening when F = 0 (no load) v max = bf o a Hill equation valid for: - concentric contraction (shortening) - constant velocity of shortening. Eccentric contraction: - muscle lengthens (v is -ve) when F > F o - muscle "gives" when F > 1.8 F o Power output of muscle: P = Fv P = v (bf o - av) v + b - maximum power generation at v 0.3 v max - the Hill equation can also be written: F F o = 1 - v v max 1 + 4v v max - Weight lifters can lift greater weights in the clean-and-jerk compared to the snatch. - Typically: F o 1.5 pn v max 5 µm s -1 17

18 Muscle contraction kinetics: [Volkenshtein "Biophysics", 1983 p ] A. Huxley (1957) Descherevsky (1968) A: actin M: myosin A + M AM form crossbridge AM + ATP A.ATP + M break crossbridge A.ATP A.ADP + P splitting of ATP n: number of pulling bridges m: number of retarding bridges n o : total number of active heads in half a thick filament v: velocity of relative motion L: distance between active sites on actin n o - n - m: number of available myosin heads for crossbridge formation Kinetics:. n = k 1 ( n o - n - m ) - v L n (1). m = v L n - k 2 m (2) k 1 : k 2 : rate of crossbridge formation rate of crossbridge breaking Newton's 2nd Law: M. v = f o (n - m) - f n o where M: mass moved f: external load per bridge f o : active force of single pulling bridge Steady state solution:.. n = m =. v = 0 v (f + a) = b (f o - f) where a = k 1 f o k 1 + k 2 b = k 1 k 2 L k 1 + k 2 Theoretical agreement with Hill's empirical result (1938). 18

19 1.3 ENERGY COST of MOVEMENT Metabolic energy: - Muscle fuelled by metabolic energy (chemical energy of foodstuffs). - Oxidation of foodstuffs: carbohydrate + O 2 CO 2 + H 2 O + Energy fat, or protein - Consume 1 cm 3 of O 2 to release 20 J of energy (independent of food source). - Determine energy production by measuring oxygen consumption of animal. Convenient view of energy use in muscle Types of muscle contraction: - Activation of muscle fibres causes a tendency to shorten. - What happens to muscle length depends on the load. 1. Concentric: The muscle shortens while exerting a force (Muscle does +ve work). 2. Eccentric: Muscle extends while exerting force (Muscle does ve work). 3. Isometric: Muscle length does not change while exerting force (Muscle does no work). Muscle efficiency: - Metabolic energy work + heat - Conversion efficiency of metabolic energy work efficiency = metabolic power output metabolic power consumption - For muscles in-situ: Efficiency of +ve work 25% Efficiency of ve work -1.25% - Maximum efficiency (for a given amount of work) at 1/3 of maximum shortening velocity. cos t of exerting force = force! fascicle length! time economy of muscle type 19

20 There are 3 types of muscle fibre types: Type I: Slow-twitch - Aerobic metabolism requires oxygen. Slowly oxidative (SO) and features low contraction strength, small in size, high capillary density and highly resistant to fatigue. Muscles used to maintain posture with slow, repetitive movements feature high proportions of this tissue type. Type II: Fast-twitch Anaerobic metabolism requires no oxygen. a) - Oxidative and glyocolytic (FOG), feature large fibres, large capillary density and result in large contraction strength. However, they are more susceptible to fatigue than SO fibres. Muscles such as the eyelids with fast, repetitive movements feature high proportions of this tissue type. b) Glycolytic (FG) feature a high anaerobic capacity for short-term, intermittent, high force production such as sprinting. With small capillary supply, most susceptible to fatigue. Anaerobic metabolism: - Energy obtained by: Glucose Lactic acid + energy - Yields less energy than aerobic metabolism. - Short term energy generation process (muscle cannot tolerate high concentration of lactic acid) - Oxidize products and use energy produced to convert the rest back into carbohydrates (oxygen debt). Summary Locomotion is required to be used by animals to: - Find food. - Find mates. - Escape predators. Most animals use more energy for locomotion than any other purpose. - Humming bird 90% - Migrating animal 80% - Human 20% - Hibernating bear 5% Therefore, it is an advantage to keep energy costs as low as possible. 20

21 1.4 SCALING and SIMILARITY Scaling Scaling are important parameters in animal locomotion. Parameters that are proportional to: Length - Height, leg length. - aerodynamic drag on small animal at low speed. Area, i.e. (length) 2 - Max force exerted by muscle, max force on bone, tendon, or muscle. - Generation of hydrodynamic thrust by tail, or aerodynamic lift by wing. - Aerodynamic or hydrodynamic drag on large animal at high speed. Volume, i.e. (length) 3 - Body mass, buoyant force, momentum (at given speed). Geometrical Similarity Animals are geometrically similar if made identical by - Uniform changes in scale of length. In geometrically similar animals: surface area (length) 2 volume (length) 3 mass (length) 3 (assuming same density) Geometric similarity is widely observed in: - Ontogeny (Development of an individual). - Evolution of phyleticlines (if moderate size differences). Maximum work (max force) (length) (cross sectional area). (length) (length) 2 (length) (length) 3 mass - Muscles constitute about 40-45% of body mass irrespective of body size. 21

22 Allometric equation: y = b (length) a = b (mass) a/3 log y = a log (length) + log b Where a = 0 1 (size independent). = 1 length. = 2 area. = 3 volume ( mass). Allometry is a term used to describe relationships between dimension, size and proportions of organisms (Alexander, 1971). Disproportionate Scaling (elastic similarity): - For large size differences, not geometrically similar, mass (length) 3 - Usually elastically similar, i.e. mass (length) 4 not (length) 3 Bone diameter (length) 3/2 not (length) 1 Max muscle force (length) 7/2 not (length) 2 Bone diameter (terrestrial mammals) - Let us assume that bones are made of the same material. Bone strength cross sectional area (diameter) 2 - But the bones must support the animal's weight, so Bone strength mass Diameter (mass) 1/2 (length) 3/2 not (length) 1 i.e. Bones of elastically similar animals are not geometrically similar. Larger animals have relatively thicker bones. This is experimentally confirmed for leg bones of terrestrial mammals. 22

23 Skeletal mass (terrestrial mammals) - Bone mass (cross sectional area) (length) (diameter) 2 (length) (length) 3/2 (length) 3/2 (length) (mass) (length) (length) 3 (length) (length) 4 not (length) 3 - Experimentally (terrestrial mammals): Bone mass (length) 3.25 Large mammals have smaller safety margin, i.e. they are more fragile. 23

24 Dynamical Similarity Motions are dynamically similar if made identical by - Uniform changes in scales of the dynamic parameters: length, time, mass, and force. - Scale factors length L λ L 2 = λ L 1 time t τ t 2 = τ t 1 mass m ξ m 2 = ξ m 1 force F χ F 2 = χ F 1 " and v 2 = v1 (velocity v = dl/dt)! " a 2 = a 2 1 (acceleration a = d 2 L/dt 2 )! From Newton's Law (inertial force): F = m a m # " =! 2 a 2 m 2 1 a1 # " $ = 2! 1 = $ # "! 2 24

25 m F L t 2 = $ # "! 2 m t1 F L = m t1 F L ml Ft 2 v = cons tan t = (since a L 2 L v = ) t Motions are dynamically similar if the dimensionless parameter is constant. - Define parameters length (L), velocity (v) and acceleration (a) to be typical of the motion. - a is the acceleration due to the dominant force (gravity, viscosity, pressure, surface tension). - Exact definitions of L, v, and a are not important, but use consistent definitions. (a) For motion in which gravity is important: Dynamically similar motion if same value of Froude number, where 2 v Fr = (dimensionless number) g L v velocity (m/s) L length (m) g gravitational acceleration (m/s 2 ) Fr 2 2 v v Fgrav = = Here, a = = g a L g L m = m v 2 / L! m g inertial force gravitational force (William Froude , engineer and naval architect) Example: Terrestrial legged locomotion depends on the interaction between inertia and gravity. Animals of different size walk / run in dynamically similar fashion (i.e. equal relative stride length) if speed is such as to give equal Froude number. where v velocity of locomotion L leg length In swimming and flying the effect of gravity is greatly reduced by buoyancy and lift, respectively. 25

26 (b) For motion in which viscosity is important: Dynamically similar motion if same value of Reynolds number,! v L Re = (dimensionless number) µ where v velocity (m/s). L length (m). ρ density of fluid (kg/m 3 ). µ coefficient of viscosity (N.s/m 2 ). (Osborne Reynolds , physicist / engineer) Re 2 2 v v = = Here, a L µ v L L 3! L = 3 2 v (! L )( v / L)! L µ = µ vl F a = viscous m inertial force! viscous force µ v A / L = 3! L Example: Swimming and flying depends on the interaction between inertia and viscosity of water or air, respectively. Animals of different size swim / fly in dynamically similar fashion if speed is such to give equal Reynolds number, where v velocity of locomotion (m/s). L length (m). 26

27 Reference data: Air Water (20 o C at sea level) (20 o C) viscosity, µ (N.s/m 2 ) density, ρ (kg/m 3 ) kinematic viscosity µ/ρ (m 2 /s) Review questions: 1) When balancing on the ball of one foot, the entire body weight is supported by the force in the Achilles tendon. (a) (b) Draw a simple mechanical diagram representing the moment arms, pivot, load and force corresponding to the foot, ankle joint, ground reaction force, and force exerted by the calf muscle through the Achilles tendon. (Assume that the foot is horizontal and the Achilles tendon is vertical.) Derive an expression for the mechanical advantage of the system, and for the force exerted by the Achilles tendon. What is the force in the Achilles tendon of a 60 kg person balancing on the ball of one foot (d 1 = 120 mm, d 2 = 47 mm)? (c) A human Achilles tendon has a cross-sectional area of about 90 mm 2. Calculate the stress in the tendon of a 60 kg person, and compare this to atmospheric pressure (p a x 10 5 Pa). 2) Individuals within a species are usually geometrically similar. (a) If a fishmonger sells a 20 cm fish for $2.00, how much do you expect a 40 cm fish to cost? (b) (c) (d) "And there went out a champion out of the camp of the Philistines, named Goliath, of Gath, whose height was six cubits and a span." 1 Samuel; 17:4. (Six cubits and a span is equivalent to 2.90 m, or 9' 6") How heavy was Goliath? How much did the tailor charge Goliath for his tunic? (The tailor charged the other men in his tribe 5 copper coins.) Compare the strength-to-weight ratio of Goliath's leg bones to those of the others in his tribe. Goliath was more worried than everyone else about tripping and falling on his face. Determine the relation between a person's height and the velocity at which the head hits the ground. (Assume the body is a uniform beam of length, l, and mass, m, that is pivoted at the feet.) 27