Mechanics of Muscle 9780814788776

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MECHANICS OF MUSCLE: SECOND EDITION

NEW YORK UNIVERSITY BIOMEDICAL ENGINEERING SERIES General Editor: Walter Welkowitz Mechanics of Muscle: 2nd Edition Daniel J. Schneck Cardiovascular Biomechanics K. B . Chandran Engineering Principles of Physiologic Function Daniel J. Schneck Neuroelectric Systems Sid Deutsch and Evangelia Micheli-Tzanako u Biological Materials: Structure, Mechanical Properties, and Modeling of Soft Tissues Frederick H. Silver Arterial System Dynamics John K-J. Li Engineering Hemodynamics: Application to Cardiac Assist Devices: 2nd Edition Walter Welkowitz

MECHANICS OF MUSCLE: SECOND EDITION

Daniel J . Schnec k Virginia Polytechni c Institut e an d Stat e Universit y

NEW YOR K UNIVERSITY PRES S New Yor k and Londo n

Copyright © 1992 by New York University All rightsreserved Manufactured in the United States of America Library of Congress Cataloging-in-Publication Data Schneck, Daniel J. Mechanics of muscle / Daniel J. Schneck.—2nd ed. p. cm.—(Ne w York University biomedical engineering series) Includes bibliographical references and index. ISBN 0-8147-7935-2 (alk. paper) 1. Muscle contraction. 2 . Striate d muscle—Mechanical properties. 3. Muscles—Physiology . I . Title . II . Series. [DNLM: 1. Biomechanics . 2 . Models , Biological. 3 . Muscle Contraction—physiology. 4 . Muscles—physiology . W E 103 S3575] QP321.S34 199 1 612.7'4—dc20 DNLM/DLC for Library of Congress 91-2760 3 CIP

New York University Press books are printed on acid-free paper, and their binding materials are chosen for strength and durability.

To Judi, Cyndi, and Patti, Because they were always there when I needed them ...

Table o f Content

s

List of Figures xi List of Tables xv List of Symbols xvii Foreword xxxi Preface and Acknowledgments xxxiii Chapter 1 Introduction , Anatom y and Physiology o f Striated Skeletal Muscl e .. . The Source of Human Powe r 1 Introduction 1 Anatomy o f Muscl e 4 Molecular Ultramicroscopic Structure of Contractile Elements 4 Micro structure of Motor Units 1 Macro structure of the Striated Skeletal Muscle 2 Physiology o f Muscula r Contractio n 2 Excitation and Establishment of an Active State 2 Excitation-Contraction Coupling 3 Relaxation and Recovery Following a Single Twitch 3 Summation of Contractions, the Staircase Effect, Tetanus 4 The Metabolic Equivalent, or "MET' Measure of Exertion 4 References fo r Chapte r 1 4

3 8

Chapter 2 A Nonlinear, Elastic , Continuum Constitutiv e Model for Striated Skeletal Muscl e 5 Introduction 5 Analysis o f Strai n 5 Analysis o f Stress 6 Nonlinear Passiv e Behavior o f Striated Skeleta l Muscl e 6 Active Nonlinea r Behavio r o f Striated Skeleta l Muscl e 6 References fo r Chapte r 2 7

1 1 2 0 6 9 8

Chapter 3 A Linear, Viscoelastic, Phenomenologica l

3 1 5 5 0 5 0

viii

TABLE OF CONTENTS

Constitutive Mode l for Striated Skeletal Muscl e 7 Introduction an d Mathematica l Formulatio n 7 Passive Rheologi c Behavio r o f Muscl e 8 Step-Strain Input : Stres s Relaxatio n 8 Step-Stress Input : Cree p 9 Sinusoidal Input : Phas e an d Amplitud e Relationship s 10 Active Rheologi c Behavio r o f Muscl e 11 The Relaxatio n Tim e Spectru m 12 References fo r Chapte r 3 13

9 9 3 7 4 1 8 8 0

Chapter 4 A Nonlinear, Viscoelastic, Stochastic Structura l Constitutive Mode l fo r Striated Skeletal Muscl e 13 Introduction 13 Formulation o f a Stochasti c Constitutiv e Mode l 13 The Separatio n o f Variables Formulatio n o f Hatz e 14 The Excitation-Dynamics Function 14 The Kinetics Functions 15 Closing Remark s 15 References fo r Chapter s 2 , 3 , and 4 15

1 1 2 3 5 1 5 8

Chapter 5 Mechanic s and Energetics of Muscular Contractio n 16 Introduction 16 Kinematics o f Muscula r Contractio n 16 Energetics o f Muscula r Contractio n 16 Strain Energy Due to Longitudinal Deformation 16 Strain Energy Due to Transverse Deformation 17 Total Energy of Contraction 17 The "Fundamental Equation of Muscle Contraction" 17 Kinetics o f Muscula r Contractio n 18 Efficiency o f Muscula r Contractio n 18 References fo r Chapte r 5 19

5 5 6 8 8 1 2 8 6 8 3

Chapter 6 Contro l of Muscular Contractio n 19 Introduction 19 Control o f Muscula r Contractio n Throug h Spina l Reflexes 20 The Myotatic (or Myotactic, or Proprioceptive or Stretch Reflex 20 ANATOMY O F MUSCL E SPINDLE S 20 PHYSIOLOGY O F TH E MONOSYNAP TIC REFLE X AR C 21 Golgi Tendon Organs and the Inverse Myotatic Reflex 21

7 7

THE CLASP KNIFE REFLEX 22

The Flexion or Nociceptive Reflex Arc 22 Control o f Muscula r Contractio n Throug h Highe r Centers 22 Thermoregulation 22 Proprioception 22

4 4 5 0 8

2 3

7 7 8

TABLE OF CONTENTS

Mines the sis23 THE SEMICIRCULA R CANAL S 23 THE OTOLIT H ORGAN S 23 LABYRINTHINE REFLEXE S 23 Subconscious Control — Dreaming 23 "Will" - The Ultimate Control - Or Is It? 23 Chemical an d Hormona l Contro l o f Muscula r Con traction 25 Control Through the Endocrine System 25 Control Through the Sympathetic Nervous System 26 Control Through Other Chemical Substances 26 SOME CHEMICA L FACTOR S REGU LATING BLOO D FLO W THROUG H MUSCLES 26 SOME CHEMICA L FACTOR S ASSOCIATE D WITH ELECTROLYT E DISTRIBUTIO N AND ACTIVIT Y I N STRIATE D SKELETAL MUSCLE S 26 SOME CHEMICA L FACTOR S THA T AFFECT SYNAPTI C TRANSMISSION 26 Concluding Remark s 27 References fo r Chapte r 6 27

ix

0 0 3 5 6 8 0 1 0 5 5

8 9 1 1

Chapter 7 Mathematica l Modellin g o f Neuromuscula r Control Systems 27 Introduction 27 The Canonica l For m o f a Feedbac k Contro l Syste m 27 The Alpha Moto r Neuron : Controllin g Elemen t o f the Syste m 28 Mathematical Analysis of the Stretch Reflex 28 TRANSFER FUNCTION S FO R TH E STRETCH REFLE X 29 Transfer Functions for the Inverse Myotatic Reflex 31 Transfer Functions for Some Other Reflex Pathways 31 The Moto r Unit : Controlle d Elemen t o f the Syste m 32 Excitation 32 THE SYNAPS E 32 THE SARCOLEMM A AN D T-SYSTE M 32 Excitation-Contraction Coupling 33 SARCOPLASMIC RETICULU M CISTERN S 33 FINALIZATION O F TH E ACTIV E STAT E 33 Contraction: The Development of a Power Stroke 33 References fo r Chapte r 7 34

1 1 7 2 3 3 7 3 3 5 6 0

Index 34

7

About the Author 35

5

7 7 8 4 4

List o f Figure s 1.1 Microstructur e o f Myosi n Molecules , illustratin g Ligh t Meromyosin (LMM ) intertwine d "spines " devoi d o f cross bridges — formin g th e I I-Zone o f th e structur e - - an d Heav y Meromyosin (HMM ) "appendages " formin g th e cross-bridg e region o f th e structure . Th e HM M regio n consist s o f a shor t side-chain segment , IIMM-S-2 , a t th e en d o f whic h lie s a globular cross-bridg e segment , HMM-S-1 . Th e latte r ha s binding site s fo r Adenosin e Triphosphat e (ATP) , and , durin g the contractil e process , Actin; 7 1.2 Schemati c illustratio n o f on e mechanis m b y whic h th e double helical configuratio n o f thi n Acti n molecules may intertwin e with th e thicke r Myosi n molecules , formin g cross-bridg e at tachments a t th e HMM-S- 1 site s o f th e Heav y Meromyosi n side chain s — eac h 30 ° o f circumferentia l increment , approxi mately ever y 6 0 t o 7 0 Angstrom s alon g th e longitudina l axi s of th e thicke r Myosi n polymer s — thu s establishin g th e "Active-State" Actomyosi n Comple x o f th e contractil e muscu lar Myofilaments ; 8 1.3 Schemati c Representatio n o f th e Anatomica l Cross-Striate d (Side Views) , Double-Hexagona l (Cross-Sectiona l En d View ) Microstructure o f Striate d Skeleta l Muscle ; 1 2 1.4 Polarized (insid e abou t 9 0 millivolt s negativ e relativ e t o out side) Striate d Skeleta l Muscl e Fibers . Bac h i s a singl e multinucleated cell , innervate d a t th e Myoneura l Junctio n b y an alpha-motoneuro n originatin g i n th e Ventra l Roo t o f th e Spinal Cord . Fo r detail s o f Cross-Section, se e Figur e 1.3 ; 1 5 1.5 Transvers e Tubul e (3 ) sandwiche d i n betwee n a pai r o f calcium-containing Termina l Cistern s (1,2 ) i n th e fuse d regio n of th e Sarcoplasmi c Reticulu m — the element s 1 , 2 , an d 3 forming wha t i s know n a s a Triad. Not e tha t th e tria d run s transverse t o th e muscl e filaments an d fibrils, which , them selves, ru n essentiall y paralle l t o th e polarize d sarcolemma . The T-Syste m originate s fro m invagination s o f the Sarcolemm a at th e Z-Lines ; 1 8

LIST OF FIGURES

xii

1.6 Som

e o f th e majo r muscle s an d muscl e group s o f th e huma n body, a s illustrate d fo r th e huma n mal e species . Thi s Figur e is reprinted b y permissio n fro m Keller , R. , Human Anatomy, Maplewood, Ne w Jersey, Hammond , Incorporated , 1968 ; 24

1.7 Th

e Adenosin e Triphosphat e (ATP ) Comple x Compose d o f Inorganic Phosphate , th e Suga r Pentose , an d th e Bas e Adenin e — the latte r tw o constitutin g Adenosine . Thi s compoun d (ATP) contain s th e high-energ y bond s tha t provid e potentia l energy (Gibb s Fre e Energy ) t o driv e th e biochemica l reaction s that ar e involve d i n muscula r contraction ; 2 7

1.8 Majo

r Biochemica l Pathway s an d Primar y Source s o f Energ y Responsible fo r Generatin g a Muscula r Contraction ; 3 1

1.9 Th

e Variou s Phase s o f th e Contractil e Process , Togethe r Wit h the Correspondin g Physiologi c Event s Takin g Plac e a t th e Time: (i) Cross-Bridge Attaches Itsel f Via an ATP-Driven Endergoni c Reaction t o Acti n Molecul e a t Poin t A ; (ii) ATP-Drive n Electromagneti c and/o r Therma l Agitatio n Provides th e Energ y o f Activation whic h allow s th e Cross Bridge t o Reorien t Itsel f t o a Mor e Stabl e Energ y Leve l During Whic h Configurationa l Chang e i t Move s Forward , Pulling th e Acti n Molecul e t o Whic h i t i s Attache d For ward t o Ne w Poin t A' ; (iii) Cross-Bridg e Release s fro m Acti n Molecul e a t Poin t A ' and Oscillate s Backwar d Agai n t o Recycl e th e Entir e Process; 32

2.1 Schemati c Continuu m Representatio n o f a Striate d Skeleta l Muscle: (a) Before bein g subjecte d t o a Contractio n Unde r Load , F ; and, (b) After bein g Stressed Unde r Thi s Loading ; 5 4 2.2 Nonlinea r Passiv e Stress-Strai n Characteristic s o f Striate d Skeletal Muscle ; 68 2.3 Nondimensionalize d Activ e Tensio n Develope d b y Striate d Skeletal Muscl e a s a Functio n o f it s Lengt h — Activ e Stress Strain, o r Length-Tensio n Behavior ; 7 2 3.1 Four-Elemen 80

t Viscoelasti c Mode l fo r Striate d Skeleta l Muscle ;

3.2 Respons e o f Four-Elemen t Viscoelasti c Mode l fo r Striate d Skeletal Muscl e to : (A) Step-Strai n Inpu t (illustratin g th e phenomeno n o f Stres s Relaxation); and , (B) Step-Stres s Inpu t (illustratin g th e phenomeno n o f Creep).

xiii

LIST OF FIGURES

Both ar e for th e Passiv e Cas e Corresponding t o P = 0 in Figur e 3.1; 92 3.3 Passiv

e Hysteresi s Loops ; 99

3.4 Dynami c Viscoelasti c Behavio r o f Unstimulate d (Passive ) Striated Skeleta l Muscl e Subjecte d t o Cycli c (Sinusoidal ) Loading. Show n ar e the Phas e Angle and Amplitude Respons e of the Tissu e a s a Functio n o f the Disturbanc e Frequency ; 10 4 3.5 Activ

e Rheologi c Response s o f Striate d Skeleta l Muscl e t o both a n Isometri c (showin g Stres s Retardation ) an d a n Isotonic (showin g Strai n Retardation ) Contraction ; 12 1

3.6 Mass-Spring-Dashpot 5.1 Th

, Second-Orde r Viscoelasti c System ; 12 5

e "Fundamenta l Equatio n o f Muscl e Contraction, " Describ ing th e Kinetic s (Forc e vs . Velocity) o f an Isotonic , Concentri c (Positive Velocity ) Contraction ; 18 1

6.1 Anatom

y o f the Striate d Skeleta l Muscl e Spindle ; 207

6.2 Schemati c Representatio n o f th e Myotati c Refle x Arc , a s a Graph o f Sensory Nerv e Firin g Frequenc y vs . Spindl e Equato r Length; 21 1 6.3 Schemati c Representatio n o f th e Architectur e o f th e Semi Circular Canals ; 231 6.4 Sagitta

l Plan e (Two-Dimensional ) Representatio n o f a Five Linkage Mode l o f the Huma n Body ; 243

6.5 Schemati c Representatio n o f Som e o f th e Mechanism s Through Whic h Hormone s Ca n Contro l Muscula r Con traction; 25 2 7.1 Typica

l Schemati c Configuratio n o f a Closed-Loo p Feedbac k Control System ; 28 0

7.2 Viscoelasti c Mode l o f th e Mammalia n Muscl e Spindle , Com pared Wit h a Standar d Mechanica l Lea d Networ k an d a Standard Mechanica l La g Network ; 29 7 7.3 Nondimensionalize d Gain , i n Decibels , vs . th e Lo g o f Linea r Frequency, i n cycle s pe r second , fo r th e Viscoelasti c Mode l o f the Mammalia n Muscl e Spindl e Illustrate d i n Figur e 7.2 . Thi s Graph i s fo r Passiv e Behavio r Only , an d i s based o n Single input-single-output Mode l Characteristics ; 30 2 7.4 Nondimensionalize d Modulus , D\, an d Phas e Angle , \, vs . Linear Frequenc y illustrat e th e Passiv e Viscoelasti c Behavior of the Phenomenological Mode l Illustrate d i n Figur e 7.2 for th e Mammalian Muscl e Spindle ; 304

xiv

LIST OF FIGURES

7.5 Nondimensionalize d Modulus , Z>9 , an d Phas e Angle , (/> 2, vs . Linear Frequenc y fo r a Linearize d Mode l o f the Golg i Tendo n Organ, illustratin g som e ke y breakpoin t Frequencie s an d cor responding Amplitudes ; 31 3 7.6 Schemati c Flo w Diagram , Illustratin g Som e o f th e Man y Extrinsic Input s t o th e Alpha-Motor-Neuro n tha t eithe r excit e the latter , o r inhibi t i t fro m dischargin g a t a firin g frequenc y of sufficient threshol d s o as to elici t a contractile respons e fro m the moto r uni t tha t th e neuro n controls ; 32 0

List o f Table s 1.1 Metaboli c Cost s o f a Variet y o f Physica l Activities , measure d in " M E T Units : 1 MET = 3. 5 m€ 0 2 (V0 2 ) pe r minute , pe r kilogram o f bod y weight . Tabl e reproduce d b y permissio n from Erb , B . D., "Applyin g Wor k Physiolog y t o Occupationa l Medicine," Occupational Health and Safety, Volum e 50 , Num ber 6 , Page s 20-24 , June, 1981 ; 45 3.1 Viscosit 4.1 Hatze'

y o f Som e Familia r Materials ; 95 s (1981 ) Value s fo r th e Muscle-Specifi c Constant s Ap -

pearing i n Equation s [4-31 ] to [4-35] ; 15 0 6.1 Intrinsi 6.2 Pathologi

c Contro l o f Muscula r Contraction ; 198-19 9 c Contro l o f Muscula r Contraction ; 201-20 3

6.3 Axo n Types ; 225 6.4 Extrinsi c Neura l Contro l o f Striate d Skeleta l Muscl e Con traction; 23 7 6.5 Hormona 6.6 Som

l Contro l o f Muscula r Contraction ; 254-25 6

e Endocrin e Disorder s Tha t Affec t Muscula r Contraction ; 261-263

7.1 Compariso n o f Significan t Maxima , Minima , an d Break-Poin t Frequencies amon g th e Muscl e Tissue , Itself , th e Muscl e Spindle, an d th e Tendo n Organs ; 31 6

List o f Symbol a

s

No-load, cross-sectional area of muscle fiber a t rest =

do a\

B+F

a

Series of constants in the expansion [4-26 ] for/2

J a*

Constant in the exponential expression [4-27 ] for/i

b

Ordinate-axis intercept of straight line segment (see Problem 2-3a )

b\ bt b*

Series of polynomial-expansion coefficient s (se e Problem 2-3b )

c

Dashpot constant in viscoelastic model of muscle

Interfilamentary-Overlap coefficien t (se e Problem 4-5); 0 < b* < 1

c\ c* di

Constant o f proportionality in the relaxation-time function [3-53 ] Location o f center s o f gravit y o f lim b segment s i n Nubar' s Tota l Body Model; /= 1,2,3,4, 5

eh

Efficiency associate d wit h the proces s whereby wor k appear s a s the result of the breakdown of high-energy phosphates

ep

Efficiency associate d wit h th e oxidativ e synthesi s o f High-Energ y Phosphates

f

Tensile force in muscle fiber

f(l*l)

Tensile force in group of fibers having given length /*/

/l/2

Alpha-motoneuron firin g frequency a t which

A

Tensile forc e i n dashpot-contractile-elemcnt-series-spring-elemen t loop of viscoelastic model of muscle

h

Tensile force i n parallel-spring-element loo p of viscoelastic model of muscle

LIST OF SYMBOLS

xviii

/VC) fx Tensil /(sub) Firin

e forc e i n annulu s (ring ) o f muscl e fiber s al l locate d a radial distance x from the longitudinal axis of the muscle g frequencies (impulse s per second) of afferent an d efferent nerve fibers innervatin g eithe r th e muscl e spindl e o r th e moto r en d plate ; (sub) = la , II , « , y , i (i't h fiber) , b , I b, A-II-/? , A-III-