The purpose of the skeletal qrstem is to protectinternal organs, provide ritrd icinematic rinl6 andmurle attachnrent sites, ana facilitate muscle action'tnd t'*cdy movemenl Bone tras unique siruct.oal andnrtrhanical groperties that allow it io cary out theser'lt"s' Bone is among the bod/s turdest stnrct'res,'nlv dentin and enasrel in the teeth being harder. Itis ()ne of the most dynasdc and metaboEllly activetissues -.

*.. body and remains active throughoutlifc- A highlv vasgular tissue, it has -rn excetentt"lpacit)' for self-repair and ctn alter its properties andt''nfigu:ation in t"tpot t" to changes ';

mechanicaltlcmand- Fo1 *.o,pi.,,."h"og, in bone dersity are('()mmonly observed after Ft.ar of diilse and ofgrcatlv increased *: o*g, in bone shape :rrenote'd during fracturS hearini"r,a after certain or)er-att,ns' Thus, bon" adapts * a" mechanicat demandsplaced on it.This chapter des,qibes the composition and stmc-ture of bone dszue, the mechanical prcperties ofbone' and $,e--uetnvior of bone trnder differentloading conditio*- various hctors that affect themechanical behavio, of bone ir, ,.iuo and in vivo are.rlso di_<ctrssgd.

-r rzr''rlE rrl

BON.E CO,tfpOStTION ANDSTRUCTURE" VUITTVI '

Bone 6ssue is a specialized connective tissue"r'hose *,r.g ."*p"sitigir -rG

i, for its supportive-tn.i pro:ectire 'ollr- ut* o-,h.] connective tissues, it

BfOfl/IECFI,ANICSOF BONEMargarea NordinVictor H. Frankel

consists of cells and art orgarric extracellular matrix offibers and gronnd substance _produced by til ..ttr.

The distinFishing feattue of bone is ie t igh .oii.r,tof inoqganic materiah, in the forur of mineral salts,ftat combine intimately with the organic matrix. Theinorganic cornponent of bone maki ur. tissue hardTd-rrtrd, while the o{gardc component gives bone itsflelibility and resilience.

The ffi:=l portion of bone consists primaril y ofcalcitun and phosphate, mainly in the fo'rm of smattcryste ls res€mbli"g slEthetic h.yto*Jr"patite cystalswith the pmposition Gro(poj.(oH)r.'These miner-als, which account for at to /On of

-the bone,s dry

weight, F e bone its sotid coruistency. Bone seryes:ls a reservoir for essential minerah'in the body,partiailarlv calciun.

Bone mineral is embedded in rrariously orientedfibers of tg protein

5o!s9n, the fibrous portion ofthe octracellular matrix. CJUagen fibers are tough andpliable, I.t t"r resist sreiching

"r,l have rirtreatensibility. collagen cgmposes approximatery gsvo

of the ocracellular-matrix "t

a .*d,i"ts for abodt 25to 30% of the dry weight of bone. A universalbuilding block of the bodt, gollag.r, ir also the chiefftrous component of oti ei

'skeietal ,t ,r.t rr.r. (A

detailed dToiption of the misostmchrre and me_chanical behavior of collagen is provided in Chapters2 and 3.)The g"l".ti"gys tround substance surrounding themineralizecl.c.ollagen fibers coruists mainly of pr3t"ir,

polysaccharides, or Srcosaminoglycanr icairl, pi-maril)r in the form of iomprex macromorecules calred

BIOMECHANICS OF NSSUES AND STRUCTURES OF THE MUSCULOSKELETAL sYsTEM

proteoglycans (PGs). The GAGs ser:\re as a cementingsubstance between layers of mineralized collageifibers. These GAGs, along with various nonco[Jgu-nous glycoproteins, constitute about svo of the

"*iu-cellular matrix. (The structure of PGs, which are vitalcomponents of articular cartila ge, is described indetail in Chapter 2.)

Water is fairly abundant in live bone, account-ing for up to 25% of its total weight. About gsvoof the water is found in the organic matrix, aroundthe collagen fibers and ground substance, and in thehydration shells surrounding the bone crystals. Theother L57o is located in canals and cavities that housebone cells and carry nutrients to the bone tissue.

At the microscopic level, the fundamental struc-fural unit of bone is the osteon , ot haversian system(Fig. t-1). At the center of each osteon is a smallchannel, called a haversian canal, that contains bloodvessels and nen/e fibers. The osteon itself consists ofa concentric series of layers (lamellae) of mineralizedmatrix surrounding the central canal, d configurationsimilar to growth rings in a tree trunk.

Along the boundaries of each layer, or lamella, aresmall cavities known as lacunae, each containing one

CANALICULI

OSTEOCYTELACUNA

CIRCUMFERENTIALIAMELLAE

HAVERSIANSYSTEMS

CEMENT LINE

INTERSTITIALLAMELLAE

TRABECULAE

HAVERSIANCANALS

VOLKMANN'SCANALS

bone cell, or osteocyte (see Fig. l-tc). Numeroussmall channels, calea canali*-li, rad.iate from eachlacrrna/ connecting the lacunae of adjacent lamellaeand ultimately reaching the haversian canal. Cellprocesses extend from the osteorytes into the cana-liculi, allowing nufrients from the blood vessels in thehaversian canal to reach the osteocytes.

At the periphery of each osteon is a cement line, anarrow area of cementlike ground substance com-posed primarily of glycosaminoglycans. The canalic-uli of the osteon do not pass this cement Line. Like thecanaliculi, the collagen fibers in the bone matrixinterconnect from one lamella to another within anosteon but do not cross the cement line. Thisintertwining of collagen fibers within the osteonundoubtedly increases the bone's resistance to me-chanical stress and probably explains why the cementline is the weakest portion of the bone's microstruc-ture (Dempster and Colemdrr, L960; Evans and Bang,Le67).

A typical osteon is about 200 micrometers (pm) indiameter. Hence , every point in the osteon is no morethan 100 F.m from the centrally located blood supply.In the long bones, the osteons usually run longitudi

BLOOD VESSEL

F I G . I - IA. The fine structure of bone is iflustratedschematicalfy in a section of the shaft/of a longbone depicted without inner marrow. The os-teons, or haversian systems, are apparent as thestructuraf units of bone. fn the center of theosteons are the haversian canafs, which formthe main branches of the circulatory network inbone. Each osteon is bounded by a cement line.One osteon is shown exten ding from the bone(20x1. fAdapted from Bassen, 1965.l B, Eachosteon consists of f amef lae, concentric ringscomposed of mineral matrix surroun ding thehaversian canaf . (Adapted from Tortora andAnagnostakos, l9g4.l C. Along rhe boundariesof the lamelfae are small cavities known aslacunae, each of which contains a singfe bonecelf, or osteoqrte. Radiatin g from the lacunae aretiny canafs, or canaliculi, into which the ryto-pfasmic processes of the osteoq/tes extend.(Adapted from Tortora and AnagnostakoS,te8+.1

BRANCHES OFPERIOSTEALBLOOD VESSELS

LAMELLAE

nally, but they branch frequently and anastomoseextensively with each other.

Interstitial lamellae span the regions between com-plete osteons (see Fig. 1-1A). Th"y are continuouswith the osteons and are just the same material in adifferent geometric configuration. As in the osteons,no point in the interstitiat lamellae is farther than 100pm from its blood supply. The interfaces betweenthese lamellae contain an array of lacunae in whichosteorytes lie and from which canaliculi extend.

At the macroscopic level, all bones are composedof two types of osseous tissue: cortical, or compact,bone and cancellous, or trabecular, bone (Fig. l-Z).Cortical bone forms the outer shell, or cortex, of thebone and has a dense structure similar to that ofivory. Cancellous bone within this shell is composedof thin plates , ot trabeculae, in a loose mesh struc-ture; the interstices between the trabeculae are filledwith red marrow. Cancellous bone tissue is arrangedin concentric lacunae-containing lamellae, but it doesnot contain haversian canals. The osteorytes receivenutrients through canaliculi from blood vessels pass-irg through the red marrow. Cortical bone alwayssurrounds cancellous bone, but the relative quantityof each Vpe varies among bones and within individ-ual bones according to functional requirements.

FrG. 1 -2Frontal longitudinal section through thehead, neck, greater trochanter, and prox-imaf shaft of an aduft femur. Cancellousbone, with its trabecufae oriented in alattice, l ies within the shell of corticalbone. (Reprinted with permission fromGray, H.. Anatomy of the Human Body.l3th American Ed. Edired by C. D.Clemente. Philadelphia" Lea & Febiger,le8s. )

BIOMECHANICS OF BONE

Since the lamellar pattern and material composi-tion of cancellous and cortical bone appear identical,the basic distinction between the two is the degree ofporosify. Biomechanically, the two bone types can beconsidered as one material whose porosify anddensity vary over a wide range (Carter and Hayes,1977b). The difference in the porosif of cortical andcancellous bone can be seen in cross sections fromhuman tibiae (Fig. 1-3). The porosity ranges from 5to 30Vo in cortical bone and from 30 to over 90Vo incancellous bone. The distinction between porouscortical bone and dense cancellous bone is somewhatarbitrary.

All bones are surrounded by a dense fibrousmembrane called the periosteum (see Fig. 1-1A). Itsouter layer is permeated by blood vessels and nen/efibers that pass into the cortex via Volkmann's canals,connecting with the haversian canals and extendi*gto the cancellous bone. An inner, osteogenic layercontains bone cells responsible for generating newbone during growth and repair (osteoblasts). Theperiosteum covers the entire bone except for the jointsurfaces, which are covered with articular cartilage.In the long bones, a thinner membrane, the endos-teum, lines the central (medullary) cavity, which isfilled with yellow fut!y marrow. The endosteum

ff* ^&--*#*

{ f r

dt

(t BIOMECHANICS OF TISSUES AND STRUCTURES OF THE MUSCULOSKELETAL SYSTEM

contains osteoblasts and also giant multinucleatedbone cells called osteoclasts, which play a role in theresorytion of bone.

BIOMECF{ANICAL PROPERTIES OFBONE

Biomechanically, bone tissue may be regardedas a two-phase (biphasic) composite material, withthe mineral as one phase and the collagen andground substance as the other. In such materials (anonbiologic example is fiberglass)-in which astrong, brittle material is embedded in a weaker,more flexible one-the combined substances arestronger for their weight than either substance alone(Bassett, 7965).

Functionally, the most important mechanical prop-erties of bone are its strength and stiffness. These andother characteristics can best be understood for bone,or any other structure, by examining its behaviorunder loadinB, r.e., under the influence of externallyapplied forces. Loading causes a deformation, or achange in the dimensions, of the structure. When aload in a known direction is imposed on a structure,the deformation of that structure can be measuredand plotted on a load-deformation curve. Muchinformation about the strength, stiffness, and othermechanical properties of the structure can be gainedby examining this cun/e.

FrG. | -3r{. Reflected-f ight photomicrograph ofcortical bone from a human tibia(+}xl. (Courtesy of Dennis R. Carter,Ph.D.l B. Scanning electron photo-micrograph of cancelfous bone froma human tibia {30 x,. (Courtesy ofDennis R. Caner, Ph.D.|

A hypothetical load-deformation curve for a some-what pliable fibrous structure, such as a long bone, isshown in Figure "1,-4. The initial (straight line)portion of the curye, the elastic region, reveals theelasticify of the structure, i.e., its capacity,for return-

PI.ASTIC REGION C

tULTIMATEFAILUREPOINT

YIELD ,/ D

POINT ;ooJ

I

ENERGY

DEFORMATIONFIG. I _4Load-deformation curue far a structure composed of a some-what pf iable material. f f a load is applied within the elastic rangeof the structure (A to B on the curvef and is then refeased, nopermanent deformation occurs. lf loading is continued past theyield point (B) and into the structure's pfastic range (B to Con thecurvef and the load is then released, permanent defoimationresults. The amount of permanent deformation that occurs if thestructure is foaded to point D in the plastic region and thenunfoaded is represented by the distance between A and D,. lfloading continues within the pfastic range, an ultimate failurepoint (C) is reached.

fI

If

II

ff

Ir

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F

f f ii!:a

r*a,,!.rlt

*;+Ni,ii:,

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ir:

ing to its original shape after the load is remov€d. Asthe load is applied, deformation occurs but is notpermanen| the structure recovers its original shapewhen unloaded. As loading continues, the outermostfibers of the structure begin to yield at some point.This yield point signals the elastic limit of thestructure. As the load exceeds this limit, the structureexhibits plastic behavior, reflected in the second(curved) portion of the curve, the plastic region. Thestructure will no longer return to its original dimen-sions when the load has been released; some residualdeformation will be permanent. If loading is progres-sively increased, the structure will fail at some point(bone will fracture). This point is indicated by theultimate failure point on the curve.

Three parameters for determinirg the strength of astructure are reflected on the load-deformation curve:(1) the load that the structure can sustain beforefailing, (2) the deformation that it can sustain beforefailing, and (3) the energy that it can store beforefailing. The strength in terms of load and deforma-tion, or ultimate strength, is indicated on the curve bythe ultimate failure point. The strength in terms ofenergy storage is indicated by the size of the areaunder the entire curve. The larger the area is, thegreater the energy that builds up in the structure asthe load isr applied. The stiffness of the strucfure isindicated by the slope of the curve in the elasticregion. The steeper the slope is, the stiffer thematerial.

The load-deformation curye is useful for determin-irg the mechanical properties of whole structuressuch as a whole bone, an entire ligament or tendon,or a metal implant. This knowledge is helpful in thestudy of fracture behavior and repair, the response ofa strucfure to physical stress, or the effect of varioustreatment programs; however, characterizing a boneor other structure in terms of the material thatcomposes 7t, independent of its geome{r, requiresstandardization of the testing conditions and the sizeand shape of the test specimens. Such standardizedtesting is useful for comparing the mechanical prop-erties of two or more materials, such as the relativestrength of bone and tendon tissue or the relativestiffness of various materials used in prostheticimplants. More precise units of measure can be usedwhen standardized samples are tested, i.€., the loadper unit of area of the sarnple (stress) and the amountof deformation in terms of the percentage of changein the sample's dimensions (strain). Tha curve gen-erated is a stress-strain curve.

Stress is the load, or force, per unit area thatdevelops on a plane surface within a structure in

BIOMECHANICS OF BONE

response to externally applied loads. The three unitsmost commonly used for measuring stress in stan-dardtzed samples of bone are newtons Per centimetersquared (NI/cm2); newtons per meter squared, orpascals (N/m2, Pu); and meganewtons per metersquared, or me gapascals (MN/m2, MPa).

Strain is the deformation (change in dimension)that develops within a strucfure in resPonse toexternally applied loads. The two basic tyPes of strainare linear strain, which causes a change in the lengthof the specimen, and shear strain, which causes achange in the angular relationships within the struc-ture. Linear strain is measured as the amount oflinear deformation (lengthening or shorted^g) of thesample divided by the sample's original length. It is anondimensional parameter expressed as a percentage(for example, centimeter per centimeter). Shear strainis measured as the a*onttt of angular change (^y) in aright angle lying in the plane of interest in thesample. It is expressed in radians (one radian equalsapproximately 57.3 degrees) (International Society ofBiomechanics, 1988).

Stress and strain values can be obtained for boneby placing a stan dardtzed specimen of bone tissue ina testing jig and loading it to failure (Fig. 1-5). Thesevalues can then be plotted on a stress-strain curve

FfG. I -5

Standardized bone specimen in a testing machine. The strain inthe segment of bone between the two gauge arms is measuredwith a strain gauge. The stress is calculated from the totaf foadmeasured. (Courtesy of Dennis R. Carter, Ph.D.)

g BTOMECHA.N|CS OF n55UE5 AND STRUCTURES OF rHE MUSCULOSKEJTAL SYSTEM

(Fig. 1-6). The regions of this cr.rn'e are similar tothose of the load-deformation curve. Loads in theelastic region do not cause perrnanent cieformation,but once the yield point is exceeded, some deforma-tion is permanent. The strength of the material interms of enerry storage is represented by the areaunder the entire curve. The stiffness is representedby the slope of the curve in the elastic region. A valuefor stiffness is obtained by dividing the stress at anypoint in the elastic (straight line) portion of the curveby the strain at that point. This value is called themodulus of elasticity (Young's modulus). Stiffermaterials have higher moduli.

Mechanical properties differ in the two bone t1pes.Cortical bone is stiffer than cancellous bone, with-standing greater stress but less strain before failure.Cancellous bone in vitro does not fracture until thestrain exceeds 75Vo, but cortical bone fracttrres whenthe strain exceeds ZVa. Because of its porous stnrc-ture, cancellous bone has a large capacity for energystorage (Carter and Hayes, 1975).

Stress-strain curves for cortical bone, met-|, andglass illustrate the differences in mechanical behavioramong these materials (Fig. L-n The variations instiffness are reflected in the different slopes of thecurves in the elastic region. Metal has the steepestslope and is thus the stiffest material.

FlG. | -7

Sress-strain cun/€s for three materials. Meal has rhe steepest

sfope in rhe etasric region and is thus the sdffest material. The

elastic porcion of trre cune br menf is a suaight fine, indicatinglinearty etasuc befravior. The fact dut meaf has a long pfastic

region irrdicates fiat trris tlpical ducdle material deforms

exrensivcly before failure. Gla<s. a brinfe material, exhibis linearty

elastic befravior but faifs abrupdy wifr licle deformadon, as

indicared by me |act of a plasdc region on trte sUess-strain curve.

Conical bone. u/hicfi possesses both ductile and brinle qualities.

e,rhibirs nonlinear elasric behavior. This behavior is demon-

strared Ay a slight curue in trre elastic region, which it'dicates

some yefding durlng loading witrrin dris region. Conical bone

condnus to deform before failure but to a lesser extent than

does meal.

The elastic portion of the cturle for glass and metalis a straight line, indicating linearly elastic behavior;virtually no yielding takes place before the yield pointis reached. By comparisor, precise testing of corticalbone has shown that the elastic portion of the curye isnot straight but is slightly cruved, indicating tlratbone is not linearly elastic in its behavior but yieldssomerr"hat dudng loading in the elastic region @one-field and Li, I95n.

After the )""td point is reached, glass deforms verylittle before failing, ?s indicated by the absence of aplastic region on the stress-strain curve. By contrast,metal €rKhibits extensive deformation before failing,as indicated by a long plastic region on the curve.Bone also deforms before failing but to a much lesserextent than metal. The difference in the plasticbeharior of metal and bone is due to differences in

aU'UJEFo

aal,rl(rFU'

STRAIN c'

FlG. | -6 !Sress-strain curve for a conrcal bone sampfe tested in tersion(pufledf. Yield poinr (Bl pornt past whrch some permanentdeformarion of the bone sample occurred. Yield sress lB'l: foadper untt area susrarned by rhe bone sampfe before plasdcdeformation took place. Yield svain lB'l' amount of de!'onnauonwrthsrood by rhe sampte before plastic deformauon occuned.The strain at any point in the elasuc region of the cun€ isproponional to rhe srress at that pornl tJtomarc fatlure point (Cl:the pofnr pasr which faiture of fie sample occuned. Ultirnarcsrress lC'l: ioad per unir area sustained by the sarnple beforefarfure. Ultimate srran lCl: amount of deformauon susulinecl bythe sample before fallure.

B "

STRAIN

micromechanical et'ents at yield. Yield.ing in metal(tested in teruion,-or pulled) is caused Uy pLil flowand formation of prastic rlp lines; ,fup n ,., areiormed when the molecul"r oi the lattice srrrcture ofmetal dislocate' Yieldingin bone (tested in tension) iscaused by debonding of the osteons at the cementlines and microfriactrrre.

lvlaterials are classified as brittle or ductile depend-ing on the extent of deformation before failure.'Ct *is a t'"ical brittle material, and soft metal is a tlet-ductile material- The difference in the

"o,o#t of

deforrration is reflected in the fractrrre surfaces of thehvo materi*

-(Hg- l-g). when pieced together aftertrachue, the ductile material *iit not conform to itsorigrnal shaPe whereas the brittle material will. Bonee.xhibits more brittte or more ductile behavior de-pelding on its age

9-o,*ger bl"being more ducrile)and the rate at whidr it is loaded (b"i" being morebrittle at higher loading speeds).

I

Because the stmchrre of Uor,e is dissimilar in thetransverse and the longitudinal directions, it exhibitsdifferent mechariicat pioperties when loaded alongdifferent axes, a characteristic known as arrisotropy.Figure l-9 shows the variations in strength andsriffness for cortical bone samples from a humanfemoral shaft, tested in tension in four d.irections(Frankel and Burstein , rg70). The 'alues for bothparameters.T highest for the sampres loaded in thelongitudinal direcfion. Although the relatior,ship be_trveen loading_ patterns and tf," mechanical prop"r_ties of bone throughout the skereton is ..*t"*er),compl€X, it cat g".'-*rally be said that bone strengthand stiffness are gre-atest in the direction in whichloads are most co'uncnty imposed (Franker andBurstein, 1970).

BIOMECT{ANTCAL BEFI,AWOR OFBONE

The mechanical behavior of bone-its behaviorunder the influence of forces and moments-isaftected by its mechallcal properties, its geometriccharacteriiHcs, the roading ;-,ode applied, the rate ofluadi.g, and the fr"qr"ncy of loadirg.

BON-E BEF{,AVIOR UNDER VARIOUSLOADING MODES

Forces and moments can b" "pplied

to a stmc-ture in various dite.dons, producing tension, com_pression, bendi f,g, shear, 'torsion,

and combinedloadits (Fig- 1- l0i. &rne in rivo is subjected to all of

EiCMECHAAJICS OF EONE

l i

DUCTILE FRACTURE

FlG. I _gFracure surftces of sarnples of a ducrile and a britde marenal.The broken frnes on fte dr,rcrife materiat indicate tfre origirullengfi of dre sampfe. before ft deformed. The bride marenatdeformed rcry lirde befure ftacnrre.

these loading modes- The folrowing descriptions ofthese *d- appb' to strrcturr in equilibri,.r* (atrest or moring at a T"rTt speed); toaaing prod,r..,an internal deforrning effect'on the structure.

Tersim

Dudry teruile .l*dTg, equal and oppositeIoads are applied outward Fo^ the strrface of thestmchrre' and tensrle shess and strain result insidethe sbnrchre- Tensrle stness can be thought of asErany small forces directed aw ay from the surface ofthe stnrchrre- ldaximal tensile stress occurs on a

pJT" perpsrdicular to the applied load (Fig. 1-11).under tens.le loading, the rt*.t ,," lengthens andnarrows- At the microscopic level, the far,rre mech-anism for bone tissue loaded in tension is mainlv

ctCNUTccFCn

STRATNFfc . t -9Anisotroplc be-avio r of c3nical bone specrmens ftoma humanfemoral s|Eft :esred in rension (puiledl in four direcrions:fongirudinal (L,- :lted 30 cegrees wirh respec to g1e neuraf axrsof dre bone. =:ed & decrees. and rransverse fTl. (Data fromFrankel and B,-iern . lg73 l

-\-.-/:>

-l;lLJ-

I

J">57

@I

f T I

' t i i lt t: 1

t o BtoMEcHANrcs oF TrssuEs AND srRUCfuREs oF THE MUSCULOSKELETAL SYSTEM

vUNLOADED TENSION

-.COMPRESSION BENDING

SHEAR

FrG. I - l0Schematic representation

FrG. | - 12Reflected light photomicrograph of a human corfical bone

specimen tested in tension {30 x }. Arrows indicate debonding at

the cement lines and pulling out of the osteons. (Courtesy of

Dennis R. Carter, Ph.D.)

F t G . I - t 3Tensile fracture through the calcaneus produced by suongcontraction of the triceps surae muscle during a tennis match.

{Courtesy of Robert A. Winquist, M.D.)

Compression

During compressive loading, equal and oPPo-

site loads are applied toward the surface of the

structure and comPressive stress and strain result

inside the strucfure. Compressive stress can be

thought of as many small forces directed into the

F t G . t - t lTensile loading.

a a vD-t:t

ryLt4->

TORSION COMBINEDLOADING

of various loading modes.

debonding at the cement lines and pulling out of the

osteons (Fig. L-Lz).Clinic ally, fractures produced by tensile loading

are usually seen in bones with a large ProPortion ofcancellous bone. Examples are fractures of the base ofthe fifth metatarsal adjacent to the attachment of theperoneus brevis tendon and fracfures of the calca-neus adjacent to the attachment of the Achillestendon. Figure 1- L3 shows a tensile fracture throughthe calcaneus; intense contraction of the triceps suraemuscle produced abnormally high tensile loads onthe bone.

nilUU

FlG. 1- t4ComPressive loading.

surface of the structure. Maximal compressive stress

occurs on a plane PerPendicular to the applied load

(Fig. 1,-1,4). Under comPressive loading the structure

shortens and widens. At the microscopic level, the

failure mechanism fof bone tissue loaded in comPres-

sion is mainly oblique cracking of the osteons (Fig.

1- 15) .Ctinically, compression fractures are commonly

found in the vertebrae, which are subjected to high

compressive loads. These fractures are most often

seen in the elderly, whose bones weaken as a

function of apng. Figure 1,-1,5 shows the shortetitg

and widening that took place in a human vertebra

subjected to a high comPressive load. In a joint,

compressivb loading to failure can be produced by

abnormally strong contraction of the surroundingmuscles. An example of this effect is Presented in

Figure L-L7; bilateral subcapital fractures of thefemoral neck were sustained by u patient undergoing

electroconvulsive therapy; strong confractions of themuscles around the hip joint comPressed the femoralhead against the acetabulum.

F f G . t - 1 5Scanning electron photomicro graph ofa human coftical bone specimentested in compression {30x). Arrowsindicate oblique cracking of the os-teons. (Courtesy of Dennis R. Carter,Ph.D.l

BIOMECHANICS OF BONE

FfG. t - 16Compression fracture of a human first lumbar vertebra. Thevertebra has shortened and widened.

l l

l2 B|OMECHAN|CS OF TTSSUES AND STRUCTURES OF THE MUSCULOSKELETAL SYSTEM

FrG. | - t7Bilateraf subcapital compression fractures of the femoral necks ina patient who underwent electroconvulsive therapy.

Shear

During shear loading, a load is applied parallelto the surface of the structure, and shear stress and

strain result inside the structure. Shear stress can be

thought of as many small forces acting on the surfaceof the structure on a plane Parallel to the applied load(Fig. 1-18). A structure subjected to a shear load

deforms internally in an angular manner; right angles

on a plane surface within the strucfure become

obtuse or acute (Fig. L-L9). Whenever a structure is

subjected to tensile or comPressive loading, shearstress is produced. Figure L-20 illustrates angulardeformation in structures subjected to these loadingmodes.

Clinic ally, shear fracfures are most often seen incancellous bone. Examples are fracfures of the femo-ral condyles and the tibial plateau. A shear fracture ofthe tibial plateau is shown in Figure L-zL.

FtG. I - t8Shear loading.

BEFORE LOADING UNDER SHEAR LOADINGFfG. | - 19When a structure is loaded in shear, lines originally at rightangles on a pfane surface within the structure change theirorientation, and the angle becomes obtuse or acute. Thisangular deformation indicates shear strain. (Adapted fromFrankel and Burstein , 11970.1

Human adult cortical bone exhibits different val-ues for ultimate stress under compressive, tensile,and shear loading (Fig. L-22). Cortical bone canwithstand greater stress in comPression than intension and greater stress in tension than in shear(Reilty and Burstein, 1975). The value for the stiffnessof a material under shear loading is known as theshear modulus rather than the modulus of elasticity.

Bending f.

In bendirrg, loads are applied to a structure in amanner that causes it to bend about an axis. When abone is loaded in bendin1, it is subjected to acombination of tension and comPression. Tensile

+t tUNLOADED UNDER

TENSILELOADING

UNDERCOMPRESSIVE

LOADING

FtG. | -20The presence of shear strain in a structure foaded in tension andin compression is indicated by angular deformation. (Adaptedfrom Frankel and Burstein, 1970.'l

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FfG. | -21

Shear and compression fracture oF the lateraf tibial plateau.

{Courtesy of Steven Lubin, M.D.)

FfG. | -22

Ultimate stress for human adult corticaf bone specimens testedin compression, tension, and shear {average of data from Reillyand Burstein, 1975l|. shaded area indicates ultimate stress forhuman aduft cancellous bone with an apparent densiry of 35o/otested in tension and compression {Carter, 1979).

FfG. | -23

Cross section of a bone subjected to bending, showingdistribution of stresses around tne neutral axis. Tensile

stresses act on the Superior side, and compressive Stressesact on the inferior side. The stresses are highest at theperiphery of the bone and lowest near the neutraf axis. The

tensile and compressive Stresses are Unequal because the

bone is asymmetrical.

BOMECHANICS OF BONE | 3

stresses and strains act on one side of the neutral axis,

and compressive stresses and strains act on the other

side (Fig. 1,-23); there are no stresses and strains

along the neutral axis. The magnitude of the stresses

is proportional to their distance from the neutral axis

of the bone. The farther the stresses are from the

neutral axis, the higher their magnitude. Because

bone is asymmetrical, the tensile and comPressive

stresses may not be equal.Bending may be produced by three forces (three-

point bending) or four forces (four-point bending)

(Fig. 1,-24). Fractures produced by both tyPes of

bending are commonly obsenred clinically, Particu-larly in long bones.

Three-point bending takes place when three forces

acting on a structure Produce two equal moments,

each being the product of one of the two peripheral

forces and its perpendicular distance from the axis of

rotation (the point at which the middle force is

applied) (see Fig. 1,-24A). If loading continues to the

yield point, the structure, if homogeneous and sym-

metrical, will break at the Point of application of the

middle force.A typical three-Point bending fracture is the "boot

top" fracture sustained by skiers. In the "boot toP"

fracture shown in Figure 1 -25, one bending moment

acted on the proximal tibia as the skier fell fonnrard

over the top of the ski boot. An equal mornent,

produced by the fixed foot and ski, acted on the distal

tibia. As the proximal tibia was bent fonvard, tensile

stresses and strains acted on the Posterior side of the

bone and compressive stresses and strains acted on

the anterior side. The tibia and fibula fractured at the

top of the boot. Since adult bone is weaker in tension

than in compression, failure begins on the side

subjected to tension. Immature bone may fail first in

compression, and a buckle fracfure may result on the

compressive side.Four-point bending takes place when two force

couples acting on a structure produce two equal

moments. A force couPle is formed when two parallel

forces of equal magnitude but oPPosite direction are

applied to a structure (see Fig. '1,-248).

Because the

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l 4 BIOMECHANICS OF TISSUES AND STRUCTURES OF THE MUSCULOSKELFTAL SYSTEM

rnagnitude of the bending moment is the samethroughout the area between the fwo force couples,the stmcture breaks at its weakest point. An exampleof a four-point bending fracture is shown in FigureI-26. A stiff knee joint was manipulated incorrectlyduring rehabilitation of a patient with a femoralfracture. Duting the manipulation, the posterior kneejoint capsule and tibia formed one force couple andthe femoral head and hip joint capsule formed theother. As a bending moment was applied to thefemur, the bone failed at its weakest point, theorigrnal fracture site.

FrG. | -25Lateral roentgenogram of a "boot top" fracture produced bythree-point bending. (Courtesy of Robert A. Winquist, M.D.)

FrG. | -2+Two types of bending. A.Three-point bending. B.Four-point bending.

Torsion

In torsion, a load is applied to a structure in amanner that causes it to twist about an axis, and atorque (or moment) is produced within the structure.When a strucfure is loaded in torsion, shear stressesare distributed over the entire structure. As inbendin1, the magnitude of these stresses is ProPor-tional to their distance from the neqtral axis (Fig.

L-27). The farther the stresses are from the neutral

axis, the higher their magnitude.Under torsional loading, maximal shear stresses

act on planes parallel and PerPendicular to theneutral axis of the structure. In addition, madmaltensile and compressive sfresses act on a planediagonal to the neutral axis of the structure. FigUre

1-28 illustrates these planes in a small segment of

bone loaded in torsion.The fracture pattern for bone loaded in torsion

suggests that the bone fails first in shear, with the

formation of an initial crack parallel to the neutral axis

of the bone. A second crack usually forms along the

plane of maximal tensile stress. Such a pattern can be

seen in the experimentally produced torsional frac-fure of a canine femur shown in Figure L -29.

Combined Loading

Although each loading mode has been consid-ered separately,living bone is seldom loaded in onemode only. Loading of bone in vivo is complex fortwo principal reasons: bones are constantly subjectedto multiple indeterminate loads, and their geometric

strucfure is irregular. Measurement in vivo of the

A

B

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A BFfG. | -26r{. During manipulation of a stiff knee during fracture rehabili-tation, four-point bending caused the femur to fracture at itsweakest point, the originaf fracture site. B. Lateral roentgeno-gram of the fractured femur. {Counesy of Kaj Lundborg, M.D.)

BIOMECHANICS OF BONE T5

SHEAR

COMPRESSION

TENSION

FfG. | -28Schematic representation o1'a small segment of bone foaded intorsion. Maximal shear stresses act on planes parallel andperpendicular to the neutral axis. Maximal tensife and compres-sive stresses act on planes diagonal to this axis.

FrG. | -29Experimentalfy produced torsional fracture of a canine femur.The short crack (arrow) parallef to the neutral axis representsshear failure; the fracture f ine at a 30-degree angle to the neutrafaxis represents the pfane of maximal tensife stress.

strains on the anteromedial surface of a human adult

tibia during walking and jogstg demonstrated the

complexity of the loading patterns during these

common physiologic activities (Lanyon et dl., 1975).

Stress values calculated from these strain measure-ments by Carter (1978) showed that during normalwalking the stresses were comPressive during heelstrike, tensile during the stance phase, and againcompressive during push-off (Fi g. 1-30A). Values forshear stress were relatively high in the later portion ofthe gait cycle, denoting significant torsional loaditg.This torsional loading was associated with externalrotation of the tibia during stance and push-off.

During jogpng the stress pattern was quite differ-ent (Fig. 1-308). The compressive stress predomineit-ing at toe strike was followed by high tensile stressduring push-off. The shear stress was low through-

FfG. | -27Cross section of a ryfinder loaded in torsion, showingdistribution of shear stresses around the neutral axis.magnitude of the stresses is highest at the periphery of

ryf inder and fowest near the neutral axis.

theThethe

WALKING (1.4 m/sec)

STRESS

_- TENSTLE -

-.- @UPRESSI\IE

- SIGAR (EXTEF

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EIOMECI{ANIC5 CF NSSUES AND SIRUCTURES OF T}JE MUSCULOSKELEI'AL SYSTEM

1 2

G.o - 13EoutF1CT'

RNAL ROTATION}

TS TS.TOB

FrG. | -3011. Calculated sucses on the ancromedial conex of a hurnan adult tibia durirg wdking.HS-heel strike: FF-foot llat; FlO-heel-ofr TO-toeofr S-virq. {After |.arDon et al..1975; counesy of Dennis R C-aner, Ph.D.l B. Calculated $resses on fte anteromedial cortex of anaduft human tibb during jqgirg.TS-toe srike; TO-toeotr hfter Lanlon et al- 1975;courtesy of Dennis R. Caner, Ph.D.l t

out the stride, denoting minimal torsional loadingproduced by slight external and internal rotation ofthe tibia in an alternating pattern. The increase inspeed from slorv walking to jogpng increased boththe stress and the strain on the tibia (I-anyon et al.,L975). This increa* in strain with greater speed wasconfirmed in studies of locomotion in sheep, whichdemonstrated a fivefold increase in strain values fromslow walking to fast trotting (Lanyon and Bourn,1979).

Clinical exarrrination of fracture patterns indicatesthat few fractures are produc€d by one loading modeor even by hvo modes. lndeed, most frachrres areproduced Ly

" cornbination of several loading modes.

INFLUENCE OF MUSCLE ACTIVIW ONSTRESS DISTRIBUTION IN BONE

When bone is loaded in vivo, contraction of themusdes attached to the bone alters the stress distri-bution in the bone. This musde contraction decreases

JOGGING (22 m/sec)

STRESS

. -IE}ISiLE

-.- @UFRESSIVE

- $€An GXTERI{AL ROTATIOiO

- slGAlR (INTERNAL ROTATION)

or eliminates tensile sress on the bone by producingcompressive stress that neutralizes it either partialll'or totally.

The effect of muscle contraction can be illustratedin a tibia zubjected to three-point bending. Figure1-31A represents the l"g of a skier who is faltingforward, zubjecting the tibia to a bending moment.High tensile stress is produced on dre posterioraspect of the Hbda, and high compressive stress actson the anterior aspect. Contraction of the tricepssurae muscle produces great comPressive stress onthe posterior aspect (Fig. 1-3lB), neutralizt4g thegreat tensile sfress and therebv protecting the tibiafrom failnre in tension. ThiS musde contraction mayresult in higher comprressive stress on the anteriorstrrface of the tibia. Adult bone can usually withstandthis stress, but imrnahue bone, which is weaker, rnayfail in compression.

Musde contraction produces a similar effect in thehip joint (Fig. 1-32). During locomotion, bendingmoments ane applied to the femoral neck and tensilestress is produced on the superior cortex- Contraction

e 6o-EE4UJEF(t,

2

HS FFHO

FIO STO

A

t

ABFfG. | -3 |A. Disribmion of cornpressir'e aN tersite $resses in a dbiasubleced to firee-point bending. 8. Conracion of trre ricepssurae muscle produces high cqnpressirrre $rcss on trte po*erioraspecL neuualang dre hgh tensile stress.

FtG. I _32stress crsrnbuuon in a fernoral rreck subjecred to bending. whenine -oluteus rnecrus muscre is refaxed fopl. tensile stress acE onine sucerlor corTex and compressive sress acE on dre inferiorcciex' conc'acton of dris muscle (bonomf neuralizes trle tensile' : l i eSS.

EfoMECFr N|CS OF BONE l7

of the gluteus medius musde produces compressivestress that neutralizes this tensile stress, with the netresult that neither compressive nor tensile stress actson the superior cortex. Thus, the uruscle contractionallows the femoral neck to zustain higt er loads thanwould otherwise be possible-

RATE DEPENDENCY IN BONE

Because bone is a viscoetasfic material, itsbiomechanical behavior varies with the rate at whichthe bone is loaded (Le., the rate at rryirictr the load isapplied and removed). Bone is stiffer and sustains ahigher load to failure when loads are applied athigher rates. Bone also stores morre energy beforefailure at higho loading rates, provided tnat theserates are within the physiologic range.

The loaddeformation curves in Figure l-33 showthe difference in the mechanical properties of pairedcanine tibiae tested in vitro at a high and a very lowloading_ rate, 0.01. second and 200 seconds, respec-tively (Sammarco et aI., l97L)- The amount of enirrystored before failure approximately doubled at tfiehigher loading rate. The load to failure almostdoubl€d, but the deformation to failure did notchange significantly. The bone was about sa% stifferat the high"r speed.

The loading rate is dinically significant because itinfluences both the fracture pattern and the aurountof soft tissue damage at fnacture. When a bonefracfures, the stored energy is released. At a lowloading rate, the ener6y can dissipate through theformation of a single crack; the bone and soft tissues

FfG. r -33Rate dependency of bone is demonsrrated in paireC canineubiae tested at a high arrt a low foad,ng rare- The foad ro failureand the energy stored ro faifure afrpst doubled at d're high rare.lAdapted ftom Sammarco er al., lg7l.l

ooJ

LOADING RATE O.O1 SEC

DEFORTATION

T8 BIOMECHANICS OF NSSUES AND STRUCIUREs OF THE MUSCULOSKELETAL SYSTEM

remain relatively intact, and there is little or nodisplacement of the bone fragments. At a highloading rate, however, the greater energy storedcannot dissipate rapidly enough through a singlecrack, and comminution of bone and extensive softtissue damage result. Figure 7-34 shows a humantibia tested in vitro in torsion at a high loadin g rate;numerous bone fragments were produced, and dis-placement of the fragments was pronounced.

FrG. I -35Tensile strain values from a human adult tibia during logging{Lanyon et al., 19751 have been pfotted on a stress-strain curuefor bone samples tested to failure in tension. A small proponionof the total energy storage capacity of the bone is utilize d duringthis normal physiologic activiiy.

FtG. | -34Human tibia experimentafly tested tofailure in torsion at a high loading rate.Displacement of the numerous frag-ments was pronounced.

Clinic ally, bone fractures fall into three generalcategories based on the amount of energy released atfracture: low-en ergf , high-energy, and very high-energy. A low-enerry fracture is exemplified by thesimple torsional ski fracture; ahigh-energy fracture isoften sustained during automobile accidents; and avery high- energy fracture is produded by veryhigh- muzzle velocity gunshot.

only a small proportion of the total energy storagecapacity of bone is utilized during normal activity.Figure 1-35 illustrates just how little of this capacityis used during the normal physiologic activity ofjogsrg.

FATIGUE OF BONE UNDERREPETITIVE LOADING

Bone fractures can be produced by u single loadthat exceeds the ultimate strength of the bone orby repeated applications of a load of lower magni-tude. A fracture caused by repeated applications ofa lower load is called a fatigue fracture and is Vpi-cally produced either by few repetitions of a highload or by many repetitions of a relatively normalload.

The interpluy of load and repetition for anymaterial can be plotted on a fatigue curye (Fig. l-g1).For some materials (some metals, for example), thefatigue curve is asymptotic, indicating that if the load

U)@utEFa

PHYSIOLOGIC

STRAIN

BIOMECFTANICS OF EONE l 9

fracture in the lower extremities is outlined in thefollowing schema:

ooJ

FtG. t-lE--The inrerplay of load dfr repetidm is represen*ct qt a faog*CUn/€.

STRENUOUS EXERCISE

IFATIGUED IIUSCLE

^/- \'

7

\

Loss oF sHo{ }1.RED GAfrAITSOREING.CIPACITY

/

\ /ABNORMAL LOADING

IALTERED STRESS

/

OISTRIBUTIONis kepl below a certain level theoretiolly, trematerial will remain intact, no matter how manyrepetitions. For bone tested in vito, the curve is notasymPtotic- When bone is s,tbiected to repetitive lowloads, it may sustain fatigue microfractires (C-arterand Hayes, l9T7a). Testing of bone in vitro alsoreveals that bone fatigues rapi<tly nrhen load ordeformation aPProadres the )aeld strength of thebone (Carter and Hayes , l9i7a); that is, tfre nnrrrberof rePetitions needed to produe a fractr:re dimin-ishes rapidly.

In repetitive loading of living bone, the fatigueProcess is affected not only by the arnount of load i"athe number of -t"P"titions b'trt afso by tt* nrrnber ofapplications of the load rrithin a gro, time (fte-qu-ency of loading). sin-ce living bon"L selt-r€pairing,a fatigue frachrre results onlir,nhen the ,e-bdelingProcess is outpaced by the f",is"e process, Le., rvhenloading is so freg.r"t i that it irerrudes the remodel-lng necessary to prrevent faillrrre.

Fatigue fractures :rre 'sually' zustained drrringcontinuous strenuo's physical

"cti"ity, which cluses

the muscles to become r"ugu"d "t

i reduces theirability to contract- As a r"tirt thev ane less able tostore enerty and thus to neutrilize the stressesimposed on the bone. The resulting alteration of thestress distribution in the bone cluses abnormallyL ' r inrgn loads to be Tposed, and a fatigue fract're mayresult- Bone

T"y f"il on the tensile sile, the comp'es-sive side, or uour sides. Failure on the tensile sideresults in a transverce sack, and,'the bone proceedsrapidly to complete fractrrre. Fatsue fract'res on thecomPressive Id:

a.ppear to_b" produced rnore srowly;the remodeling is less easily oltp"ced h. th" fatigue

[:H::: and the bone -"i; i**d-to complete

This theory of muscle iatigue as a cause of fatitue

HIGH COMpRESStOfTf + COMBTNED-6Gg

I taow -pRocESS- FAST

ITENSION

III

DEBONDING OF OSTEONSTRANSVERSE CRACKS

ITRANSVERSE FRACTURE

INRUENCE OF BONE GEOMETRYON BIOMECI{ANICAL BEFI,AVIOR

The teometry of a bone greatly influences itsmeclranical behavior. In tension a'd compression,the load to failure and the stiffiress are proportionalto the cnoss-sectional area of the bone. ft e largerthe :rnea is, the stronger and stiffer the bone. I"bending, both the cross-sectional :uea and thedistribution of bone tissue around a neutral axisaffect the bone's mechanical behavior. The quantilvthat takes into account these two factors in bendingis called the area moment of inertia. A laqger aneamoment of inertia results in a stronger and stiffurbone.

Figure t-37 shows the influence of the areamoment of inertia on the load to failure and thestiffness of three rectangutar stmcj1rres that have thesame :rriea but different shalres. tir uending bea' Itris the stiffest of the three and c:ln wi**tana thehighest load, because the greatest asrount of materialis distributed at a distance from the neutral anis. Forrectangular cnoss sectiors, the forrrrula for the anea

OBUQUE CRACKSIII

OBLIQUE FRACTURE

t

REPETITIO}.I

20 BIOMECI.IANIG OF I65UE5 AND STRUCTURES OF fiE MUSCT'LOSKELETAT SIsIEM

2x2ll

moment of inertia is the width (B) multiplied by thecube of the h.ight (H1 dividedby 12:

B . H 3t2

Because of its large area moment of inertia, beasr IIIcan wiftstand four times more load in bending thanbearn I-

A third facbr, the length of the bone, influenc€sthe strength and stiffness in bending. The longer thebone is, the greater the magnitude of the bendingmoment caused by the application of a force. In a

BEAM A

FfG. | -37Three beams of equal area brx differentshapes subjecred to bending. Fs recangularooss secdors. fie area moment of inerua is

cahrlarcd ry fie formura B: $ rahere B is

t 2tfn widtrt aN H. trte height The areamqnent of inenia for beam I is elZ: for beamtf, lUlZ: aN for beam ltl. 64112. [Adapredfiorn Frankef aN Bursrein l970-t

1x4til

rectangutar stnrcture, the magnitude of the stressesproduced at the point of application of the bendingmoment is proportional to the length of the stmcttrre.Figure l-38 dqplcts the forces acting on two beamswift the same width and height but different lengths:beam B is twice as long as bearn A. The bendingmoment for the longer beasr is twice ilrat for theshorter bearu coruiequently, the stress magnitudettuoughout the beam is twice as high.

Because of their leogth, the long bones of the skel-eton are subjected to high bending moments and, so,high teruile and compressive stresses. Their tubuiar

FlG. | -39Eearn B is rwice as fong as beam A andsr.rsEins twice fie berxling momenl Hence.fie $ress rnagniude firo4hour beam B istwice as high. fAdapred from Frankel andBur$ein. 1970.1

STRESSMAGNITUDE = S

BEAM B

-.

L-.-JI STRESS

MAGNTTUDE = 2S

shape gives them the ability to resist bending mo-ments in all directions. These bones have a large areamoment of inertia because much of the bone tissue isdistributed at a distance from the neutral a)ds.

The factors that affect bone strength and stiffnessin torsion are the same ones that operate in bendingthe cross-sectional area and the distribution of boneHssue around a neutrial axis. The quantity that takesinto account these two factors in torsiornl loading isthe polar moment of inertia. The larger the polarmoment of inertia is, the stronger and sfitrd thebone.

'

Figure 1-39 shows distal and proximal cnosssections of a tibia subjected to torsional loading.Although the proximal section has a slightly surallerbony area than does the distal section, it tras a muchhigher polar moment of inertia because much of thebone tissue is distributed at a distance from theneutral ilds. The distal section, while it has a largerbony area, is subjected to much high"r shear stressbecause much of the bone tissue is distributed closeto the neutral axis. The magnifude of the shear stressin the distal section is approximately double that inthe proximal section. Clinically, torsiornl fractrrres ofthe tibia conrmonly occrr distally.

when bone begins to heal after ftactnre, bloodvessels and connective tissue from the periosteusrmigrate inlo the region of the fracture, forrting a cuffof dense fibrous tissue, or callus, anound the fr"ct rrusite, which stabiti-es that area (Fig. l-40A). Thecallus significantly increases the ate and polarmoments of inertia, thereby increasing the strengthand stiffness of the bone in bending and torslonduring the healing period. As the fracture heals and,h: bone gradually regains its norrnal strength, thecallus cuff is progressively resorbed and th" bonereturns to as near its normal size and shape aspossible (Fig. l-4OB).

certain surgical procedures produce defects thatgreatly weaken the bone, partiarlarty in tonsion.These defects fall into two categories: those whoselength is less than the diameteiof the bone (stressraisers) and those whose length exceeds the bonediarneter (open section defects).

o' .t"'I,Til:T ::HllH: : :Hffi ffiH il*strength is reduced because the stresses imposedduring loading_ are prevented from beirg distributeds1'ehlt throufhout the bone and instead becomeconcentrated around the defect. This defect is anaL.Bous to a rock in a stream, which diverts the water,producing high water turburence around it (Fig.

EIOMECFdAT'.IICS OF BONE 2 l

FfG. t -39Disribtttion of sfiear stress in two cross secrions of a ribiasubjeced to tqsirnal loadlng. The proximaf section (Al fras ahigher morrfem d Frnia dran does fre disal secion (Bl.because more Wry nEterial is distributed away from trre netrralaxis. fAdapted fun frankel and Bursrein, l971.l

1-41). The wealcening effect of a stress raiser ispartictrlarly marked under torsional loading the totaldecrease in bone strength in this loading mode canreach 60Vo.

Bnrstein and associates (1972) showed the effect ofstress raisers pTdrced by screws and by emptyscrew holes on the eneqty storage capacity of rabbitbones tested in torsion at a high loading rate. Theimmediate eftct of drilling a hole and inserting ascnew in a rabbit femnr was a74vo decrease in energystorage capacity. After 8 weel$, the stress raiser effectproduced W the scrervrs and by the holes withoutscrrews had disappeared completely because the bonehad resrodele& bone had been laid down around thescrews to stabilize them, and the empty screw holeshad been fiUed in with bone. In femora from whichthe screws had been removed immediately beforetesting howetrer, the energy storage capacity of thebone decreased by 50%, mainly because the bonetissue arund the scnew sustained microdamageduring screw removal (Fig. !-42).

An open section defect is a discontinuity in thebone caused by s,rrgrcal remorral of a piece of bonelonger than the bone's diameter (for example, by thectrtting of a slot during a bone biopsy). Because theouter surface of the bone cross section is no longercontinuous, the bone's abitity to resist loads isdtered, partiorlarly in torsion.

V(-

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.-.F v

\-o'rl\\-/\B

22 BIOMECHANICS OF NSSUES AND STRUCTURES OF THE MUSCULOSKELF|AL SYSTEM

FfG. | -41

Stress concentration around a defecg such a defect is analogousto a rock in a stream.

In a normal bone subjected to torsion, the shearstress is distributed throughout the bone and acts toresist the torque. This stress pattern is illustrated inthe cross section of a long bone shown in Figure1-43A. (A cross section with a continuous outer

FtG. | -40A, Early callus formation in a femoraf fracture fixed with anintramedullary naif. B, Nine months after injury the fracture hasheafed and most of the callus cuff has been resorbed. {Counesyof Robert A. Winquist M.D.)

0 1 2 3 4 5 6 7 8WEEKS

FtG. | -42Effect of screws and of empty screw hof es on the energy storagecapacity of rabbit femora. The energy storage for experimentafanimals is expressed as a percentage of the total energy storagecapacity for controf animals. When screws were removedimmediately before testing, the energy storage capacity de-creased by 50o/o. (Adapted from Burstein et af., 1972.)

surface is called a closed section.) In a bone with anopen section defect, only the shear stress at theperiphery of the bone resists the applied torque. Asthe shear stress encounters the discontinuity , it isforced to change direction (Fig. L-438). Through-

EoE(f

(,tr[rJzI.IJ

o.o<-.oEMPWSCREWHOLE

}.'o- < SCREW IN PLACEo SCREW REMOVED AT TESTING

,nrN\t'Y'//

'rr/(

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FlG. | -45

A patient sustained a tibialproduced open section defectafter operation.

BIOMECHANICS OF BONE 23

fracture through a surgicallywhen she tripped a few weeks

BA

3

rf3

1

I

3

))

FtG. | -43

Stress pattern in an open and closed section under torsionalloading. A. In the cfosed section, all the shear stress resists theapplied torque. B. fn the open section, onfy the shear stress atthe periphery of the bone resists the applied torgue. {Adaptedfrom Frankel and Burstein, 1970.1

out the interior of the bone, the stress runs parallel toto the applied torqu€, and the amount of bone tissueresisting the load is greatly decreased.

In torsion tests in vitro of human adult tibiae, anopen section defect reduced the load to failure andenergy storage to failure by as much as 90%. Thedeformation to failure was diminished by about 70Vo(Frankel and Burstein, 1970) (FiS. 1-M).

Clinically, surgical removal of a piece of bone cangreatly wdaken the bone, particularly in torsion.Figure 7-45 is a roentgenogram of a tibia from whicha graft was removed for use in an arthrodesis of thehip. A few weeks after operation, the patient trippedwhile twisting in an attempt to rescue the Christmasham from toppling onto the floor, and the bonefractured through the defect.

FtG. | -+4

Load-deformation curves for hum an adult tibiae tested in vitro

under torsional loading. The control curve represents a tibia with

no defecg the open section curye represents a tibia with an open

section defect. (Adapted from Frankel and Burstein, 1970.1

oo

BONE REMODELING

Bone has the ability to rernodel, by altering itssize, shape, and structure, to meet the mechanicaldemands placed on it. This phenomenon, in whichbone gains or loses cancellous and/or cortical bone inresponse to the level of stress sustained, is summa-rized as Wolff 's law, which states that bone is laiddown where needed and resorbed where not needed(Wolff , 7892).

If, because of partial or total imm obllization, boneis not subjected to the usual mechanical stresses,periosteal and subperiosteal bone is resorbed (Jenkinsand Cochrdn, 7969) and strength and stiffness de-crease. This decrease in bone strength and stiffnesswas shown by Kazarian and Von Gierke (1969), whoirnmobilized Rhesus monkeys in full-body casts for 60days. Subsequent compressive testing in vitro of thevertebrae from the immobilized monkeys and fromcontrols showed up to a threefold decrease in load to

CONTROL

DEFORMATION

24 BIOMECHANICS OF TISSUES AND STRUCTURES OF IHE MUSCULOSKELETAL SYSTEM

oo

fracture had heared. The bone under th" il#fiJ$,:Jff: l",: * 3:l^? "-ff]: :l r:- q-""i"dt re sorb#and the diameter of Fudiaphysis bu.u*L

ryi:ffiHit*:' ; *fj"::'^": : :*-'l :! " ":;i,,.u

ai um e tgreatly decrea s es bone s trength, parricular; ;f i:T:-#ilg#i. 1o,,1:o-1 :,^ rl#ol."J it" ;F; il ;;i;'_"moments of inertia. A 2AVo decrease in bone diamerurtt#

ilT#iH"',i"":":ffi1'1,1",H":JT";K*iI*F'.suggest that rigid plates should be removea shortly

'fi

{t:::^[::yl:,lf heared ulJ berore ii; bone has

J*markedly diminished in size. such a ,cecr";;;ffi:

'

:i:" *_i:l"l1m::Tl_"Tud by secondary osteoporo_ "Jsis, which further #eakens inu-;;;;i6rrnr:i[:;

*1980) 1vr'*Lrtr (tL dr', tt;

An imprant may cause bone hypurfrophy at itsattachment sites el exampre of bohe hypertrophyaround screws is illusfrated in Figure r-4g. A nailplate was applied to a femoral neck fracture, and thebone hyperirophied around the screws in response tothe increased road at these sites Hrpertrophy mayalso result if bone is repeatedry ,"u;".ted to highmechanical stresses within the normal physiologicrange. Hypertrophy ofnormal adult bone in responseto strenuous exerciie has been observed (Jones et ar.,7977; Dal6n and Olsson, I9T4; Huddleiton et aI.,7980)' as has an increase in bone density (Nilsson andWestlin, 1g7I). ---'d.'-

1-potitive correlation exists between bone massand body weight. A greater body weight has beenassociated with a largei bot "

mass (Exne"r et ar. ,i,g7g).conversely, a proloriged condition of weightlessness,such as that experienced ,curing space travel, has

FtG. I _47Anteroposterior IAt and lateral tBf roenrge nograms of an ulnaafter plate removal show a decreased bone diameter due toresorption of the bone under the plate. cancelf izatian of thecortex and the presence of screw hores arso weaken the bone.{Courtesy of Marc Martens, M.D.f

Ffc. | _+6Load-deformation curues for vertebral segmenE L5 to L7 fromnormal and immob ilized Rhesus monkeys. fAdapted fromKazarian and Von Gierke, ..969.)

failure uld energy sjolage capaciv in the vertebraethat had been immobilizEd, ,,iffr,um was also signif_icantly decreased (Fig. I_4'6).An implant that remains firmry attached to a boneafter a fracfure has healed may arso diminish thestrength and stiffness of the bone. In the case of aplate fixed to the bone with screws, the plate and thebone share the road

l'.proporuons det"r*i'ed by thegeometry and material p-roperties of each structure. Alarge plate, carryilg high t,ouJr, unroads the bone toa great extenu the bone then atrophies in response tothis diminished load. (The bone may hypertrophy atthe bone-screw interface in an attempt to reducemicromotion of the screws.)Bone resorption under a prate is ilrustrated inFigure 1-47 ' A compression plate made of a material

,i;BL

IMMOBILIZED

DEFORMATTON

BloMECHANlcs oF BONE 25

been found to resultweight-bearirg bonesGazenko et dl., 1981).

in a decreased bone mass in

(Rambaut and |ohnston, 1979;

DEGENERATIVE CI.I,ANGES IN BONEASSOCTATED WITH AGING

A progressive loss of bone density has been

observed aJ part of the normal aging Process. The

longitudinal trabeculae become thinner, and some of

the transverse trabeculae are resorbed (Siffert and

Levy, 1981) (Fig. I-49). The result is marked reduc-

tionin the amount of cancellous bone and thinning of

cortical bone. This decrease in the total amount of

bone tissue and the slight decrease in the size of the

bone reduce bone strength and stiffness.

Stress-strain curves for specimens from human

adult tibiae of two widely differing ages tested in

tension are shown in Figure 1-50. The ultimate stress

was approximately the same for the young and the

old bone. The old bone specimen could withstand

only half the strain that the young bone could,

indicatitg greater brittleness and a reduction in

energy storage caPacitY.

FfG. | -+8 ' '

Roentgenogram of a fractured femoral neck to which a nail plate

was applied. Loads are transmitted from the plate io the bone

via the screws. Bone has been laid down around the screws to

bear these loads.

FtG. t-49Vertebral cross sections from autopsy specimens of young fAf and old lBf bone show a markedreduction in cancellous bone in the latter. (Reprinted with permission from Nordin, B.E.C.:Metabolic Bone and Stone Disease. Edinburgh, Churchill Livingstone, 1973.1 C. Bone reductionwith aging is schematically depicted. As normal bone (top) is subjected to absorption {shadedareaf during the aging process, the longitudinal trabeculae become thinner and some transversetrabeculae disappear {bottom). (Adapted from Siffert and Levy, 1981.)

j'ffiqffiffin

lu4ltnnc

BA

26 BIOMECI{ANICS OF NSSUES AND STRUCruRE OF THE MTSCULOSKEI..ETAL SYsTEM

FtG. | -50stress-srain cun es for sampfg of aduft lrurnan dbbe of nnovudely differing ages tested in tersion. (Adapred fiom Br.rrsrein etal., 1976.1

oq,trJcEFo

SUMII/IARY

1. Bone is a two-phas€ composite material, inor-ganic mineral salts b.irg one phase and anorganic matrix of collagen and ground strbstancrethe other. The ino€anic component makes bonehard and rigtd, whereas the oqganic componentgives bone its flexibility and resilience.

2- Microscopic"fly, the fundarnental stmctural unitof bone is the ost@n, or haversian qlsteul,composed of conc€ntric layers of mineralizedmatrix srurounding a central canal containingblood vessels and nerve fibers.

3. Macroscopic"lly, the skeleton is composed ofcortical and cancellous bone. Bone of both t,"escan be considered as one material whose potoiityand density vary over a wide range. .

4. Bone is an anisotropic material, exhibitirg differ-ent mechanical properties when loaded in differ-ent directions. Mahre bone is strongest andstiffest in .9-pression.

5. Bone is subjected to complex loading patternsdtrring courmon physiologic activities luch :rswalking and jogging. Most bone fracttrres ane

produced by a combination of seneral loadingmodes.

6. Musde contraction affects stress patErns in boneby producing compressive stress that partially ortotally nzutralizes the tensile stress acting on thebone-

7. Bone is stiffer, sustains higher loads beforefailing, and stores morre energ:r wherr loaded athigh"" rates.

E. Lirirg bone fatigues wher. the fnequencv ofloading precludes the remodeling necessarv toprevent f,ailure.

9. The mechanical behavior of a bone is influencedby its geometry Oength, cross+ectional area, anddistribution of bone tissue around the neutralaxis).

10. Bone remodels in response to the mechanicaldemands placed on it; it is laid down whereneeded and resorbed where not needed.

11. with agng there is a marked reduction in theamount of cancellous bone and a decrease in thethicrcness of cortical bone. These .h*ges dimin-ish bone strength and stiffness.

REFERENCES

Bassett, c. A. L.: Electrical effects in bone. sd. An., 213:lE,1965.

Bonefield, w., and u, c. H.: Anisotropy of nonel,astic flow inbone. I. Appl. Physics, fi:Z4fl, lgGZ.

damage- A

1977a.Carter, D. R, and Hayes, W. C: The compressive behavior of

bone as a trro-phase Porous stnrcture. I. Bone foint Suqg., 59A#\r9T7b.

Carter, D. R, Schwab, G. H., and Spengl,er, D. tv{-: The efuof apparent density on the tenslh and compressive prroperties ofcancellous bone. Transactions of the 5tr Anrrual Meeting ffiro-paedic Research Society, 4:t7, ly79-

Dd€n, N., and Olsson, K E: Bone minerat content andphysical aAivity. Acta ffirop. Scard -, 15:1V0, 1y74.

Dempster, W. T., and Colemaru R F.: Tensile strery$ of bqrealong and dcnoss the grain. I. AppL Physiol., 16355, tgff,.

Evans, F. G.: Bone and bones. f. BiomecL Eng., 704:1,9AEvans, F. G., and Bang, S.: Differences and relations betrreen

the phvsical properties and the micccopic strudurc of humanfemord, 6bial, and fibularcorticalbone- Arl. I- AnaL,720:79,l%7.

Exner, G. [J., et al.: Bone densibmetry using comp'utedtomognphy. Part I: Selective deternination of trabecular bonedensity and other bone minsal parasreters. Normal values inchildren and adults. Br. I. Radiol, 52:14, ly79-

Frankel, V. H., and Burstein, A FL: ffiropaedic Biomechan-ics. Philadelphia, Lea & Febiger, 197O.

Gazenko, O. G., C-€nin, A trL, and Yegorotr, A- D.: Majormedical results of the Salyut{@ru2 l8Sday space flight- NASANDB A47. Proceedings of the )OOCI C-ongress of the InternadonalAstronaudcal Federation, Rome, It"lJn, Septenrber 6-17, 1981.

GoJr, H.: Anatomy of the Human Body. 13Or Asrerican edition.Edited by c. D. Clemente. Philaddphia, [.ea & Rbiger, 19E5.

Huddleston, A. L., Rockwell, D., Kulund, D. N., and Flarrison,R. B.: Bone mass in lifetime ternis athlaes. I-q,MA,244:ll(n,lg8/J,.

International Society of Biomechanb: Quantities and UniS ofMeasurement in Biomechanics, lW7 (unpubli3hed).

BTOMECF{,AAJ|CS oF BONE 27

Ienkins, D. P., and Cochnr, T. H.: Osteoporuis: The dramaticeffect of disuse of an stremity. Clin. Orthop., *7'rE, 1969.

lcres, H-, hiest, I., Hayes, W., and Nagel D.: Husreralhypenrophy in respcrse to exercise. f. Bone loint Surg., 59A2M,tyn-

IGzarian, L L, and Von Gierke, H. E.: Bone lr.ss as a result ofimnobilization and dreladon. Pretininary resuls m tvlnaa trnt-/rrtu- Cliru Orthop., 6267, 1969.

l.uryoru L E, and Bourn, S.: The influerre of mechanicalfunrrion (trr the dwdopnent and remodeling of the ftia. Anocp€limental study in sheEp. !. Bone loint Sulg,, 67A253, lgn.

bttyott, L E, Hanpsut, ll\f. G. f., Goodsltip, A E, and Shah,I. 5-: BqE dcfrrmation recorded in virc ftr strain gaug6attadred b the human dbial shaft. Acta Ortrop- Scand., 45:256,tgls-

lEssqu B. E, and Wesdin, N. E: Bqre dcnsit}r in athlebs.Clin- ffirop ., 77zlT), lqn.

t{mdiru B. E C: Meabolic Bone ard Stoa Disease. Edin-brrdr Ctnrrdrill Livingsone, lfin.

Rambaut, P. C, and lotrnsto& R S.: holonged weightlessnessand caiciurr lcs in man. Acta Astrronau6@, 5:1lIl, 1y79.

Reillp D., and Burstein, A.: The el-"tic and lriErate propertiesof urpact bon€ tissue. l. Biomech., ESy3, 1975-

Sammarco,1., Burstein, A., Davis, W., and Franket, V.: Thebimedunics a( tusional fracttrres: The etrect of loading onultimate Foperdes. f. Biomectr., 4:113,lyn.

$fu, R S., and lguy, R. N.: TrabecuLar lntterns and theinEnat adriEcturc of bone. Mt. Sinai I. Med-, gfll, 19E1.

SEffs, P., Paavoliainen, P., IGraturiu, E, ard Holcrstrom, T.:Sffiurat and biomectranical changes in bone after tigd platefintbn. C-an- f. Surg.,23247,1980.

Tortora, G. t., and Anagnostalcos, N. P.: Prirriples of Anatomyand Physiologr. 4th Ed. Nerv Yorlc, tlarp€r & Row, lg8d..

lVolff, f.: Das C'esetz der Transformafion der Knochen. B€rlin,Flirsdrwald, L$YL.

SUGGESTED READING

Whole Bones

_Albritht, f . A.: Bone: Physicat properties. Iz The Scientific Basis

:1 or,hopaedics. 2nd Ed. Edited by I. A. AIhUht and R A- Brand.New York, Appleton{entury<rofts, l9EZ, W- 213-240.

Albright, J. A., and Skirurer, H. c. W.t Bone Stnrctunlorganiretion and remodeling dynanics. ln The Scientific Basis oforthopaedics. 2nd Ed. Edita by I. a AlbrEht and R A. Brand,.New York, Appleton{entury6fu, lgg7, pp. tOf-lS..

Asang, E-: Biomechanics of the human teg itt alpinE'skiing. InBiomectranics rv. Edit"d by R c Nelson and C. A. Morehouse.Baltimore, university park pres s, 1974, pp. %-zttiL

.Asang, E-: Applied biomechanics of the human leg. A basis fortndividual protection from skiing iniuries. ffirop. Clin. NorttrAm., jgj-103, 1976.

._-lT"q E-: E*peri-ental biomectranics of the human reg. ADasrs for interpreting t'?ical skiing iniury mechanisurs. ortf,op.Clin. Nonh A.rr., 7:63-73, lg15.

Bassett, c- A- L-: Eectrical effecG in bone- scl Arl., 213:lg-2s,I 965.

Burstein, A- H-, and Frankel, v. H.:Theviscoelasdcpropertiesofsome biological materials. Ann. N.y. Acad- sd., I4d:158-165, ll)58.

Burstein, A H., et al.: Bone strength. llte effect of screw holes.

f. Bone loint Surg., SA:lliLl- 1156, lgT2.C:rter, D. R: Anisotlopic anatpis of strain ruette inforsradon

hon cortical bone. I. Biomech., 77:199-Tn, lil;lE.Cuney, l.: The Mectranical Adapatiqrs of Bones. Princeton,

Prilaton Univ€rsity Pness, 19E4, pp. 3-l5i/.Ilal€& N., and Olsson, IC-E.: Bone mirsal cpntent and

phystcal activit". Acta fthop. Scand., 45:l7O-44, 1y74.Deurpster, lAf. T., and Coleman, R F.: Tensile strength of

bone along and acnuis the grain. f. Appl. Phsiol., 15355-m,tt""*,

F. G.: Bone and bones. I. Biomech. Eng., 1o4:1-i , y9gz

Evans, R D. (ed.): Studies on the Anatomy and Fnnction ofBone and loints. New York, SpringenVerlag, l$5.

Enrer, G. IJ., et al.: Bone deruitosreEry using omptrtedtonography. Part k Selective detersrination d trabecrrlar boned€nsity and odrer bone mineral parartetes. Normal values indrildren and adults. Br. I. Radiol., 52:14-23, t9rTt.

Frankc{, V. H.: The Femoral Nedc Frrnction, Fracture Medra-nisuts, tnternal Fixation. Sptingfield, Charles C Thomas, tgff..

Fnnlc€L V. H., and Burstein, A. H.: fthopaedic Biomechan-ics. Ptriladelphia,l-ea & Febiger, l!70.

28 BIOMEGTAAIICS OF NSSUES AAJD STRUCTURES OF THE MUSCULOSKELETA SISTEM

Frankef, V. H-, and Burstein, A. H-: Biomechanics of thelocomotor srstem. In l\ledicl E"g"eering. Ed.itea bv c. D. Rav.chicago, Year Book Medical publisher, l!i4, pp. 505-51d.

Frost, H- M-: An Lntroduction to Bromeclnnics. Sprit g6eld,Charles C Thomas, 1976-

Hakim- N- 5-, and IGng; A. I.: hogranned replicadon of in situ(whole Uody) badiry conditions during in tii- (nrbst'ct're)testing of a vertebnl column s€tmenl I. Biomectr-, gfi='-6g2,rn6.

IrunarL V- T.: Functional aspects of the abductor murles of thehip. I. Bane loint Sqrg., ?f,A.;ffi2-6!9, L%7.

I€nkins, D. P., d Codrran, T. H.: Osteoporosis: The dramaticeffect of disuse of a qtremity. clin. ortho,p., &{:v1-l'+ .'969.

fensen, J. S., Ifanseru F. W., and loturrse&- I.: Tibial shaftftactures- A ourpAison of consernative beatm€nt and internalfixation wiffr onventbnal plates or AO compression plates- ActaOrthop. Scand- , 4,E 4#212, lgn.

lones, H-, Priesg I-, Hayes, w., and Nage{, D.: H'sreral!fp*"pLy in resPons€ to exercise. I. Bone Joint Suag. , SgAiln-w, lyn-

IGzari:rn,L, and Graves, G. A-: compressive strength charac-teristics of the hrrman vstebral centnln. spine , 2:l-ti, lgn.

IGzarfurn, L E., and Von Gierlc, H- E.: Bot "

loss as a result ofimmobilization and drdation. Preliminary resuls n lvInas trst-Ietu. Clin- Orthop -, 5SfiT-75, l%9.

Lakes, R, and saha, s.: cement rine motion in bone. science,2mflt-503, tgn.

[anyan,LE.,I{anFor" W. G. J., Goodship, A. E-, and Shah,L S-: Btf,|e deformation recorded in vivo from strain taugesattadred tD th€ hurnan bbiat shaft. Acta orthop. scand., {6-2fr-2fi, l9Tt-

Murphy, E F., and Burstein, A. H.: physical properties ofmaterials induding solid mechanics. In Atlas of Orthbtics. Biomechanical kinciPles and Application. St- ["ouis, C. V. Mosbry Co.,\*Tt, pp. 3-30.

l.letz, P., Eril$son, K, and sffimberg, L.: ultimate farllure ofdi"phys€al bqre. An ergerimenal studt on dogs. Acta orthop.Scand., 5I583-58E, f9fr).

Piziatl R L, and Nagel, D. a: Modelintof the human l"t ir,ski injuries. &thop- Gin. North A.trr., 7:LT--139, 1976.

Robinson, R. A-: Bone tissue: Composition and ftrnction- fohnsHopkins Med. J., ltIS:!0-Z4, lyn.

Rosse' C., and Clawson, D. K: Introduction to the tvtnsctr-loskeleal systen. l$ew York, Flarper & Row, ly,o.

Rybicki, E F., sinonen, F. A., and weis, E. B.: on themathemadcal andysis ot stress in the hnman fennr. f. Biomecl.,526-215, tyn.

Samsrarco, G. f-, Burstein, A H., Davis, W. L., and Frankel,V- H-: The biomechanics of torsional fracttrres: The etrect ofloading cr ultimate properties. I. Biomech. , 4:ll3- ll7, lyn.

Sandler' R 8., at'rd Herbert, D. L.: Qr,rantitative bone ass€ss-ments: Applbtions and eTpecta6ons. l. Alrr. Geriatr. soc.,29:9-1G3, f!)8l.

Sldnn€r, FL C. tV.: Bone minerali--tion. In The Scientific Basislf ot hopaedics. zrd Ed- Edited b.u I. a Alb,right and R. A- Brand.New York, Appletur-Centuqyr{rofts, 19g7, pp. f99- ZlZ.

Sleds. P., PaavoNainen, P., Karahariu, E., and Holms6m, T.:Stntchral and biocnechanical changes in bone after ritrd ptate6xation. Can- I. Su{g -, 23247-Lfl, igSO.

Sffimbery L., and Dal€n, N.: Experisrental measunement ofmaxirnusr tonque capacitv of long bones. Acta Orthop. Scand.,47:87-2fr, 1y75.

sffimberg, L., and Dal€n, N.: The irtfluence of freezing on themaximum torqre capaciv of long bones. An experimenal studvon dogs. Acta Ortltop. Scand -, 47:?5l-2fi, 1976.

Toridis, T. G-: Stress andysis of the femur. I. Biomech.,2:163-174, 1969.

Corticaf Bone

Bonefield, w-. and u, c H.: Anisotropy of nondastk flow inbone. I. Appl. Phssics, 3824fl-2/155, 1967.

Burstein, A- H-, ReilIy, D. T., and Marteru, M.: Agiry of bonetissue Mechanical properties. I. Bone Joint Surg., 58A82-E6, IyZ6.

Burstein, A- fa, Currey, I. D., Frankel, v. H., and Reillv, D. T-:The ultimate p'rqerdes of bone tisstre: The effects of ygdi" & I.Biomedr., S'35 'L I*TZ

C-arter, D. R, arrd Hayes, w. C: Compact bone fatigue dam-age. A micrwcog* exaninatbn. Clin. Orthop., 127{65-24, l%n .

C-artrr, D. R, ad Hayes, w. C: The compressiw belravior ofbone :u; a hrc?tras€ porcus stnrcture. f. Bone Ioint Suqg.,59A9il-%2, rgn-

Currey, I. D.: The medranical properties of bone. crn.Ortho'p ., 73210-Z1 Lyn.

DaI€n, N., FHlsffim, L€., and facobson, B.: Bone mineralcontent and mecbanical strength of the femoral neck- Acta Orthop.Scand., 47'ffi-lE, 1y76.

Enneking, hl-: Principles of Musculoskeletal Patholory.Cninesyille, FL, Storter Printing Company, lg7f,.

Errans, F. G: Mechanical Properties of Bone- Spngfield,Charles C Thom=, 1973.

Evans, F. G-, and Bang s.: Differences and relations betrpeenthe physical properdes and the microscopic strucuue of humanfemoral, ftial, :'yl fibular cortical bone. Asr. f. Anat , rz(hzg-gg,t967.

Fredensborg N., and Mlsson, B. E.: The bone Eineral contentand cortical thidm€ss in young women with femoral neck fracttrre.Clin. Clrthop ., 7J24:.161- 16,4., lEn.

Frost, H. M-: The l-aws of Bone struchrre. Sprin$eld, CharlesC Thomas, lgfl|.

[.akes, R., and Saha, S.: long-term torsional cr€€p in compactbone- f. Biomech- Eng., 7U2:178-180, 1980.

Mueller, IC I.L, Trias, A., and R.y, R D.: Borr density andcomposition. AgFrelated and pathological changes in water andmineral contenl f- Bone Ioint Surg., aISA:140-148, 1966.

Nilsson, B. E, and wesdin, N. E.: Bone densitv in athletes.Clin. Orthop., V:179-le, lYn.

PoF, M. FL, and Outwater, I. O.: The fracture draracteristicsof bone nrbsara- f. Biomech. , 5:457-ffi, lyn.

Reilly, D. T., and BursEin, A. H.: The mechaniol p-perties ofcortical bone. I- Bone loint Surg., 55A:1fi)l-1m2, lln4.

Reillv, D. T-, and Burstein, A. H.: The elastic and ultimateproperties of compact bone tissue. f. Biomech., 8:3S-4(E, ly71.

Reilly, D. T-, Bnrsteiru a H., and Frankel, V. H.: The elasticmodulus for borE- I. Biomech., 7.Zl-275, 1y74.

S€dlin, E. D-: A rheologc"l model for cortical bone. A study ofthe physical prryerties of hunan fiemoral samples- Acta Orthop.--Scand., Suppl. 63:1 -87, f965. r

Viano, D., Fldferuteiru U., Anliker, M., ird Rfregsegger, p.:Elastic properties of cortical bone in female husran- fesrurs. I.Biomectr ., 9:78-710, Ly76.

weaver, I. lC. The mi<roscopic hardness of bone- I. Bone IointSrtB-, 484:273-:38, 19ff..

Cancelfous Bone

Behreru, J. c, walker, P. s., and shoii, H.: Variations instrength and stlrcture of cancellorrs bone at the kne- f. Biomech.,7:2Ol-297, 1974-

C-a,r+er, D. R., and Haves, w. c.: Bone compressive strerrgth:The influence of densitv and strain rate. Science , Igl:1174-1176,1975.

Carter, D. R., and Hayes, w. C.: The compressive behavbr of'r'one as a two-phase porous struchrre. l. Bone lctnt Srrt.,59.{:9il -952, 1977 -

Carter, D. R., Schwab, G. H., and Spengter, D- M-: The ftof appa*tt-t density on the tensile and compressive properties ofcancellous bone. Transactions of the 25th Annual Meedn& ffiro-paedic Research Societ-v, 4:t7, 1979.

chung; 5. M- IC, Batterrran, s. c., and Brighbn, c r-: strearstrength of the husran femoral capital epiphyseat plate- f- BureIoint Surg., fiA:91-1(8, 1Y76.

Enneking, W.: Prirrciples of Musculoskeletal Fatrolo6il.Caine*'ille, FL, Storter Printing Compar/, lgTO.

Errans, F. G.: Medranical Properties of Bone. Springfieid,Charles C Thomas, lgn-

Frankel, v. H., and Burstein, A. H.: [.oad capacity of tubqlarbone- In Biomechanics and Related Bio-engineerint Topics- Effibl' R. M. Kenedi. New York, Perganon Press, 1965, pp. 3gl -ffi.

Frost, H. M.: The laws of Bone Stmctrrre. Spri"gfield, CharlesC Thomas, 196/..

C,alante, f., Rostoker, W., and Rav, R. D.: Ph)rsical properties oftrabecnlar bone. Calcif. Trssue Res., 5:236-245, lg7o.

Haves, w. c., and C-arter, D. R.: Postyietd behavior ofsutrhondral trabectrlar bone. f. Biomed. Mater. Res-, 7-S3Z-::p4,1e76.

Haves, w. c., Boyle, D. 1., and velez, A.: Ftrnctional adaF-tion in the trabecular architecttrre of the human patdla. Transac-tions of the 23rd 4ry1'r:l Meeting, Orthopaedic Research Sciety,J: I l{, 1977.

Undail, O.: Mechanical properties of dded defatted spongytrrnc. Acta Orthop. Scand., 47-.tl-19, 1976.

PoP., M- H., and Outrrater, f. O.: The fracture ctraracteristicstrl txrne substance. I. Biomech., 5:457-16s, 1972.

Pop", M. H., and outrnrater, I. o.: Mechanical properties ofhtnt' as a fnnction of position and orientation. l. BiomecL.,l :61-66, ly74-

siffert, R. s., and L"ny, R. N.: Trabectrlar patterns and theinternal architecture cf bone. Mt. Sinai I. Med. , ig-J!]t-ZEt, f$l.

lveaver' I- K-: The microscopic hardness of bone. I. Bone loint5rrg., 484:273-288, 1966.lveaver, I. K., and chalmers, I.: cancellous bone: Is strerrgth

rnd changes with agng and an evaluation of some mettrods forntcasuring its mineral content. I. Age changes in cancellous bone.| &rne.foint Surg. , #A.2gg-Zgg, l-gffi.

Fatigue Fracture

Baker, I-, Frankel, v. H., and Burstein, A. H.: Fatigue fractures:Bt'rmechanical considerations. f. Bone foint s*g., 5{r{:1345-r}rf,I e;1.

Burrorvs, H- I-: Fatigue infraction of the middle of the tibia inb'-rflet dances. f. Bone ioint Srrg., 3gB:g3 _g4,195d.Dt'vas, M-: stress io.trr"r. Edinburgh, London, and New\ tI rrrh. Lhurchill Livingstone, tgTS.

EToMECF|AN|CS OF BONE Zg

Frankel, v. H., and tlang, Y.-S-: Recent advances in thebiomedranics of sport iniudes. Acta Orthop. Scand., 4G-.48+-497,1975.

Friedenber& Z B.: Fa6gue fractnres of the tibia. CIin- Orthop.,76:lll-115, 1971.

McBryde, A. M.: stress fractures in athletes. t. Spora \fed.,32t2-2t7,1976.

Wdter, N. F-, and Wolf, M. D-: Stness fractgres in Fungathletes- Arl. I. Sports Med-, 5:165-lI , lgn-

Bone Remodellng

Abramsm, A-:8ate rncturbances in iniuries to the spinal cordand cauda eqdna (panplegh). Their prcven6on by ambuhbn. f-Bone lcint Surg ., XDMUiL-gEf7, 194E.

Bartley, lf,- fL, Anold, f. S., Fladam, R trC, and tee, W. S. S.:The reladonstrip of bone strcngth and bone qgantitf in health,disease, and aglry- I. Gerqrol,2l:5l7-Sll, !gff'.

Bassett, C A- L: Bblogic significance of piezoelectridt-v. Caldf.Tissue Res., 725i2-Zn, 196E.

Ctramaf, A., and TschanE, P.: Mectranical influences in boneremodeling. Experimental research on Wolff's Law. J. Biomectl.,5:lT3-180, lyTZ

Dal€n, N., and Otssort K-E.: Bone mineral curtent and,phrical activity. Acta ffirop. scand-, 4szl70-174, lg74-

Franlce, 1., et aL: Ph1nnd properties of fluorosis bone- ActaOrttrop. Scand .,. 4l/.A-27, 1976-

Gietsvik, A.: Bone respdeling and piezoelectricit)'-L |. Bio-mech., 6:69-77,1973.

Horal I-, Nadresrsoru A., and scheller, s.: cliniel andradiologitzl long Eur follow-up of rrertebral fractures in ctulldren.Acta Orthop. Scand -, 43:491-5(8, lytz

Huddleston, A L, Ro*rreII" D., Kulund, D. N., and tlarrison,R. B.: Bone m;rss in lifetime teruris athletes. fAtvIA , 2ti4:!lg7-ll0g,1980.

fawoski, Z. F- G.: Ph''sblogy and patholoy of bqre renrod-eling, Ceilular bads of bone stmcture in health and in oseoporo-sis. ffirop. Clin. Nordr An., 12:485- SXL, 1981.

Ienkins, D. P., and Coduan, T. FL: Osteoporosis: The dnsraticeffect of disuse of an octrenity. Gin- orthop., 64:128-13+ 1969.

fones, H. H., et al.: Hunerat hlpertnophy in respqrs€ toexercise- I. Bone tcint Surg-, SIA2CF.-208, lgTZ.

I-arryon, L. E, and Bcrrn, 5.: The influence of medranicalfunction on the derelcpment and resrodeling of the &ia. Anexperimental study in sheep. f. Bone loint S*g., 61A26"-2TZ,t979.

I\rlartin, R 8., Plckett, I- C., and Znaich, S.: Studies of skeletalremodding in aging men. CIin. Orthop., 149:269-282, 19fl1.

Rambaut, P. C, and fotrnston, R. S.: Prolonged weighdessnessand calciusr loss in man. Acta Astronautica, 6:1113-ll??, lyn.

*hock, C. C, Noyes, F. R, Madr€ws, C. H. 8., and C-rcuch,M. M.: The effect of activitv on surface remodetling in the Rhesqsmonkey rib and fennrr. Transactions of the 23rd Annuat Meeting;Orthopaedic Researdr Sod,ety, 2:36, 1977.

sharpe, w. D.: Age changes in hurnan bone: An o.erview.BuIl. N.Y. Acad. Med., 55tr37-773, 1929.

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