The Masculoskeletal System & Tissue Mechanics

•

Tissue M~....._,~.....

This book is for people who are interested in applying the principles of engineeringmechanics to analyze the mechanical and structural funcrions of the musculoskeletalsystem, including the design' of orthopaedic devices that are used to treat skeletaldiseases or repair damage to the skeleton: We can do this because the skeletal systemis ~ machine. That is, it is a coinbiil~tion of rigid or resistant bodies having definitemotions and capable of performing useful work. The links of this machine, are thebones along with the soft tissue structures that are associated with them. These linksare connected to each other at joints that enable the body to move quickly in a rela-tively agile manner and transmit large forcesfrom link to link.

In general, the relative motion between bones is not as precisely defined as themotions between the links of most engineered machines. The hip joint is an excep-tion. It is a ball and socket joint, and the ,motio~ between the femur and the pelvis atthe hip is well defined. The knee joint, 'however; allows at least five degrees offree-dom, and six if separation of the femur and the tibia is included. The motion at theknee is constrained by soft tissue structures: rheIigarnents that connect bone tobone, the joint capsule that encloses the joint space, the menisci between the femurand the tibia, and the tendons that connect the muscles to the bone and transmit themuscleforces across the joint. It is possible to describe the general characteristics ofthe motion of the knee joint under the loads associated with normal daily activitiesto a precision sufficient for purposes of analysis and design.' Under other loadingconditions, however, the precise relative motions of the joint and the contact forcesacross the joint depend upon the funcrionalloads applied to the skeleton.

In niost machines, it is possible to consider the kinematics of the mechanismwithout regard for the external loads acting on it and the internal forces transmittedfromlink tolink, Such is not, the case with biological joints. Because kinematic con-straint at the joint is provided in part by ela:stic structures, which are much less stiffthan the bony links, the loads across the joint and the motion of the joint are cou-pled. Changes in force will produce changes in joint motion, and changes in the rel-ative motion between the bones will result in changes in joint' forces.

x Preface

(Chapters 5 and 6). The material on loads and motion (Chapter 2) and on structur-al analysis (Chapters 5 and 6) are designed to provide an opportunity for studentsto develop confidence and competence in applying fundamental concepts ofmechanics to biomechanical systems. Furthermore, they also provide a commonapproach and notation for students who come from a variety of introducrory courses.The remaining chapters introduce applications of the fundamentals addressed inthe first part of the book. Chapter 7 provides a basic introduction to bone-implantsystems, which is followed by chapters on fracture' fixation devices, hipreplace-ments, 'knee replacements, and articulating surfaces, '

The material can be organized in a number of ways for courses with particularemphases. For example, although the chapters provide a logical progression, the top-ics in Chapters Slmd 6, can also be addressed in parallel with the applications pre'sented in the later chapters, 1)1e bookis generally broader than any of the courseswe have offered, which at one time or another have stressed dynamics, bone mechan-ics, or bone-implant systems. But while the text is quite broad, there are parts of thisfascinating field that we have only touched upon lightly or not even 'covered. Whatwe present represents our interests and areas of expertise and doe~n't attempt tocover the entire field of musculoskeletal biomechanics and orthopaedic design 'andengineering, A number of other works are referenced in thetext to help direcr the. stu:'dent to other sources. It should be noted also that ~e ~v.e srayedaway from subjectsthat are still controversial. Our philosophy was to focus on fundamental topics thathave stood and will stand the test of 'time.' As such, this' is riot a survey of currentresearch, but indeed much more of a traditional textbook, ' , , " '

Many people have contributed direcrly and indirectly to the development ofthis text and thecoursesfrom which it grew. We would liketo thank the 'sh.dentswho have taken our courses, ;a~k:edgood' questions, and :ferreted out 'errors as wedeveloped the manuscript. The graduate students who have servedas teaching assis~rants to us in teaching our coih.-ses (jver ilie~yea'~s"aiso ,desecY'e s~e~i~i mention.Indeed, all three of us have been influenced by colleagues and. graduate students fartoonumerous to'Ii~i:. We thailk them all for what they have'ta'u@it us as we ~orkedarid learned together. We would, however, like, to thank several individuals specificcally, First, we thank our colleagues Clare Rirtmac and Ken Mann for theirirnpor-rant contributions to the section on implant materials.'We also thank 'post-doctoraltrainee chris Hernandez; and graduate students J~nni Buckley and Eric Nauman fortheir contributions to die tissue mechanics' sections.' S~ial thank~ roGinnyGilesand Judy Thoroughman for 'their, valuable assistance arid theii''i,atlence lit rtealihgwith word processors ana professors as the book' took form. Finally, we' would liketo thank' our families for their support during the course' of this projecr. Perhaps thisbook will give them additional insights into what has occupied our time and energyas we have pursued various aspects of the fasC41~ting world of biomechanics. ' , '

The MusculoskeletalSystem

DONAlD L. BARTELCornell University

DWIGHT T. DAVYCase Western Reserve University

ToNY M. 'KEAVENYUniversity of California at Berkeley

CHAPTER

2 The Musculoskeletal System• Section 1.1 Anatomical Overview 3Chapter 1

One may be tempted to say, as a colleague once did, that when analyzing theskeletal machine instead of an engineered machine, only the nouns are different; theverbs are the same. This is not quite the case. Most engineering analyses of machinesare done for devices we design; and we design them so the' analysis tools we haveavailable are applicable. When analyzing the skeletal system, we are confrontedwith a machine we did not design and one that is, in many ways, more complex thanthe machines we produce. Consequently, we are often confronted with the need tofirst understand the basic functioning of the skeletal system by using fundamentalprinciples of mechanics.

Skeletal loads and motions vary from individual to individual and from time totime in a particular person. The structural characteristics of the skeleton also changewith time. There are changes with disease and aging that we might wish to avoid or in-hibit, but there is also the remarkable ability of the skeleton to heal damage. Even moreremarkably, the skeleton has the ability to adapt to changing demands-the geometryand mechanical properties of bones and soft tissues change when the loading changes.

It would be nice, at least in some cases, to precisely estimate the motion, stresses,and deformations in a particular individual for specific circumstances. Of course,this is not practical, arid our modeling of disease and damage must be done while re-alizingrhe uncertainties of intersubject and intrasubject behavior.' Often, precisemeasures of stresses or motions in particular situations are unnecessary. What isneeded, instead, are models that describe the fundamental behavior of the system.To this end we can use basic concepts from' dynamics and strengthof materials tounderstand many of the effects of damage and disease and to evaluate the methodsand techniques used to restore function. The choice of mOdel is critical in this endeavorand is one of the underlying themes of subsequent chapters. First.we need to look inmoredetail at several aspects of this wonderful machine ..

and the thorax consist of 80 bones. The spine (Figure 1.2) consists of 33 vertebrae inthree sections: 7 in the cervical spine (the neck)" 12 in the thoracic spine (surroundingthe chest and rib cage), 5 in the lumbar spine (the lower back), 5 in the sacral spine(which are fused), and 4 in the coccygeal region. The appendicular skeleton has 126bones. There are 4 bones in the pectoral girdle, 60 bones in the upper limbs, 2 bonesin the pelvic girdle, and 60 bones in the lower limbs.

Bones come in a wide variety of shapes and sizes, but most fall into threegroups. Long bones, such as the femur, the tibia, and the humerus, are long in onedirection and have cross sections in the central shaft that are tubular. Short bones,such as the bones of the wrist and the ankle, are bones or portions of bones thathave about the same dimensions in all directions; the other bones of the hand andfoot fall somewhere between long and short bones (Figures 1.3 and 1.4). Flat bones(sometimes called tabular bones), which make up portions of the skull, the scapula,the pelvis, and the transverse and spinous processes of vertebrae, are much smallerin onet dimension than in theothers. Irregular bones are those that do not fit neatly

1.1 I Anatomical OverviewEngineering analysis of the skeletal system is a relatively recent 'activity and involvescollaboration with, physicians and life scientists who understand in depth theanatomical and physiological details of thebody, To be effective collaborators, weeach.need tolearn something of the language of the other. But the anatomy of themusculoskeletal system is not a part of most engineering or H,omedlca( engineeringcurricula, although, increasingly, ail introduction to biology is either expected or re-quired. Consequently, in this section we provide an overview of the anatomy of themusculoskeletal system, along with the associated terminology, to give sufficientbackground for the topics introduced 'in subsequent chapters.' The scope is necessar-ily very general and emphasizes the systems and subsystems we have chosen to ad-dress in the rest of the book. There are any number of good reference books onanatomy and physiology available for. those who would like to see these topics andothers in greater depth. . '

Bones of the skeleton. There are 206 bones in the human skeleton (210 if wecount the two sesamoid bones that lie under the head of the first metatarsal in eachfoot in most people) (Figure 1.1). The axial skeleton, the skull, the vertebral column,

RGURE 1.1 A frontal viewof the adult skeleton.

4 Chapter 1 The Musculoskeletal System

FIGURE 1.2 A side andfrontal view of the spinal col-umn. C: cervical region, T:thoracic region, L: lumbarregion.

FIGURE 1.3 The bones ofthe foot and ankle joint.

Section 1.1

}-] Metacarpals

} Carpals

Ulna -

irito the first three categories and include the vertebral bodies arid the posterior ver-tebral elements (Figure 1.5).

Anatomical _Terms. When we discuss the bones _of the human skeleton, it is oftenuseful to distinguish between different regions ofrhe bone. Thefollowing terms arecommonly used when discussing the subjects considered in this book:

·0 Proximalaspect. Nearest.to the top of-the body. Usually only used in conjunc-tion with the bones of the appendicular skeleton. Thus, we talk of the proximalfemur, which is at the hip joint. 0

• Distal aspect: The opposite' of proximal-nearest the bottom of, the body.Again, this term is normally used in conjunction with the bones of the appen-dicular skeleton. Thedistalfemur, for example, is atthe knee joint.

• Inferior: Beneath or lower, Used to denote the bottom or underside of a tissueor strucnu:«;:.Especially.important when discussing bones of the axial skeleton.

• Superior: Opposite of inferior; same rules of usage.

Lateral (Side) View:Working Facet Joints

phalangealjoints

Anatomical Overview 5

FIGURE 1.4 The bones ofthe hand and wrist joint.

FIGURE 1.5 Spinal motionsegment. A motion segmentconsists of two vertebral bod-ies with posterior elementsand the intervening disc. Withforward flexion (left), thefacet joints tend to seperate,and with extension (right) theytend to compress.

6 Chapter 1 The Musculoskeletal System Section 1.2 The Functions of the Musculoskeletal System 7

• Lateral: The part closest to the outside of the body or farthest from the body'smidline. So the lateral aspect of the femur is on the outside of your (left orright) thigh.

• Medialr Opposite of lateral-the part closest to the inside or midline of thebody. Note that lateral and medial are referenced to the midline of the body,not to either the left or right sides.

• Anterior: Before or in front.

• Posterior: Behind or in back.Dorsal: Near or on the back.

• Ventral: Near or on the anterior or lower surface of an animal opposite theback. " ,

In addition to these terms, there are a number of other specialized terms for specificmotions of specific bones; books on functional anatomy can be consulted for a morecomprehensive list. '

1.2 I The, Functions of the Musculoskeletal System

To describe motions of the skeleton, we must first define the anatomic positionand the three anatomic planes. In the a~ai:Qmic position, the individual is stand-ing with head and palms facing forward{Figure 1.1). The frontal (coronal) planedivides the body front-back; the sagittal plane divides the body left-right, andthe transverse plane divides the body top-bottom. For example, a biceps curl is amotion in the sagittal plane; twisting one's head to the side is a motion in thetransverse plane. In order to describe motions in these planes, we use the follow-ing terms:

• Flexion: A folding movement in which the anterior angle between two bonesis decreased (except in the knee and toes, in which case the' angle ismeasuredposteriorly), It generally means that you are moving a bone closer to the bodywith respect to it's anatomical position; ,

• Extension: The opposite of flexion-an increaseinthe anterior-angle betweentwo bones (except in the knee and toes; in-which case-the angle is measuredposteriorly) .

• Abductiore- Movement away from the midline of the body, usually in the-frontalplane.

• Adduction: Movement toward the midline otthe body, usually in the frontalplane.

• Hyperextension: Coi'ttinuation of motion beyond the anat~mic position.• Lateral flexion: .Movemenr of the spine to the right or left, in the frontalplane,• Supination: A movement of the forearm to rotate the hand into the anatomic

position. For example, this would be a clockwise rotation of the right forearm(looking down the arm).

• Pronation: Opposite of supination-a movement of the forearm to rotate thehand so that the palm faces backwards.

• Dorsiflexion: Rotation of the ankle about a transverse axis so that the toes moveupwards (away from theground) in the sagittal plane. Used only for the ankle.

• Plantar flexion: Opposite of dorsiflexion-r~tation of the ankle so that thetoes move toward the ground. Used only for the ankle.

., Inversion: Rotation of the foot inward and upward (in the frontal plane).

• Eversion: Rotation of the foot outward and upward (in the frontal plane).

The musculoskeletal system has four main functions; the first two are physiologicalin nature, the second two, mechanical:

• hematopoiesis• mineral storage• protection of the Vital organs• support and motion

HematopoiesisTrabecular bone, the spongy, highly porous bone found at the ends (epiphyses) of thelong bones;' the vertebrae, and several other locations (skull, pelvis, sternum) providessites for the formation of red blood cells, a process known as hematopoiesis. This.oc-curs only in the red bone marrow. Yellow bone marrow, which is found in the middle

, (or diaphysis) of most long bones, serve~ primaril~ as a storage area forfar cells:

Min~ralStorage 'The ~keletal system also has 'an important physiological function in that it acts as amineral bank; especially for calcium and phosphorous. Bone is made up primarily ofa mixture of collagen (a compliant and ductile protein polymer) and hydroxyapatite(a brittle calcium phosphate, ceramic). Approximately 99 percent of the calcium inthe human-body is stored in.rheskeleron. One of the ways that the body regulatesthe-level Of these minerals in the bloodstream is by a-continuous process ofremodel-ing (the resorption and formation of bone tissue). If the body falls short of its' dailycalcium intake via the gut, if will turn to the bones to get what it needs. Thus, indi-viduals with calcium-deficient diets' are at risk of losing bone mass', which in turnwould lead to weakening of their bones. Since calcium absorption .decreases withaging, 'elderly people are advisedto increase their daily intake of calcium to helpre-duce the risk of osteoporotic bone fractures.

Protection of the VitarOrgansIn order to protect the vital organs such as the brain, heart, spinal cord, and lungs,the.skeleton has developed various structures that allow it to absorb large amounts ofenergy, yet remain lightweight. For example, the cranial bones of the skull have asandwich construction consisting of a stiff cortical shell surrounding a relatively com-pliant trabecular bone core, called the diploe[Figure 1.6}. The outer cortical shell dis-tributesexternal forces evenly to the underlying trabecular bone, which absorbs mostof the 'energy on compression. The ribs and sternuni protect the lungs and heart in a'similar manner as do the spine and pelvis their respective soft tissue organs.


FIGURE 1.6 Sagittal cross-sectional view of the humanskull. (Atlas of the HumanBody. Harper Perennial,New York, 1989.)

right parietal bone

arterial sulci

right temporal bonenasal

left cheekoccipttal bone

left maxillaryleft external

~----- auditory foramen

'-------Ieft temporal bone

'--------~jaw bone'

SUPJX!rtand MotionAs' suggested previously, the' primary .f411Qion of 'the skeletal system addressed inthis book is its ability to provide forceand motion for mobility and to take care ofthe functions of daily living. The functions of daily living vary, of course, from indi-vidual to individual. For some, it is a matter of simple survival; for others, it 'may 1.0-elude strenuous activities such as high-performance athletics. The bones and jointsoperate together as levers.-with the muscles providing the active torque. about thejoint. When the muscles contract, theyproduceforces that cause a bone or an entirelimb to rotate about a joint, .thereby generating movement. For example, when thebiceps. contract, the forearm rotates about the elbowjoint, an action known as flex-ion (Figure .1.7). Since muscles are usually attached (or inserted) quite close to thejoint; there is a mechanical disadvantage at most joints-the muscle must pull witha force that is much larger than the externalload, Therefore, the joint loads (inter-nal forces) are much larger .than the functionalloads (the external forces on theskeletal system); .. .This is -eas~y visualized by imagining holding a weight in the hand (the func-tionalload) with the upper arm vertical and the forearm horizontal. The moment armfor the biceps muscle is a small fraction of the moment arm for the weight in the hand.

Even in activities such as normal gait, the. loads ;tcros~ joinrs are surprisinglylarge. For example, hip joint loads have been measured to be two to three timesbody weighdor level walking at constant speed. The largest loads on the foot.in thiscase are only about 1.2 times bodyweight.i'Ihe loads on joints of athletes may 'bemuch greater, For example, when a professional basketball player jumps to dunk theball, the load on the foot has been measured to be as high as 13 times body weight.It is no wonder that these individuals often end up with damaged joints.

The advantage of having- the tendons close to the joint is that small excursionsof the muscle (changes in length) can 'produce large. angular motions at the joint; theskeleton is designed both to provide. fast motion and to withstand large loads. We

r

(a)

Contractingtriceps

(b)

will see that muscles can only produce large loads when they are moving slowly.Speed is only possible when the loads are light. All in all, the skeletal muscles areamazingly efficient actuators that one would be hard pressed to duplicate with anyengineered device of similar size and weight. .

II.

.,'

)

1.3 I BonesIt is important to distinguish be~een bones.the structures, and bone tissue, the ma-terial from which bones are made. There are two basic types of bone tissue.Cortical, or compact bone, is the most dense bone in the skeleton. The diaphysis, orcentral shaft, of a long bone such as the femur or tibia is made of cortical bone..Cancellous bone (or trabecular or spongy bone) is much less dense than corticalbone and is found in. the epiphyseal regions at the ends of long bones" within the

.very thin cortices of vertebral bodies, and between the more dense outer layers ofbones such as the skull and pelvis.

Since we will be dealing primarily with long bones throughout this book, theywarrant a closer look (Figure 1.8). The shaft, or diaphysis, of a long bone growsfrom its ends at growth plates. The growth plates eventually close when the individ-.ual has matured. The epiphyses, or large ends of the. bone, grow from separate ossi-fication centers. The metaphysis is the region between the diaphysis and theepiphysis. . .

The medullary, or marrow cavity, of a long borie is filled with yellow marrowthat consists of fat and a few primitive blood cells. This marrow has mass and con:tributes to the' weight of the bone, but is of no structural consequence. Red marrowexists in the proximal ends ·of the humerus and femur. (It can also be found in the

'Section 1.3 Bones 9

RGURE 1.7 Flexion (a) andextension (b) of the elbowjoint. In each case, the agonistmuscles contract and the an-tagonist muscles relax, all in ahighly coordinated fashion.

10 Chapter 1. The Musculoskeletal System Section 1.4 Joints of the Body 11

FIGURE 1.8 The generalform of a long bone.

Diaphysis

the mass of the marrow is included and appropriate constraints for strength or stiff-ness are applied. It can also be argued that the enlarged ends of 100igbones are opti-mal for transmitting large joint loads without high contact pressure. The enlargedends distribute the load over a large area, and the cancellous bone transmits the loadprogressively to the dense cortical bone of the diaphysis, which has a relatively smallcross-sectional area. The design of these remarkable structures, whose form is sowell suited to their function, is indeed impressive and a topic of continuing study.

[ Epiphysis ..

[ MetaphYSis'

1.4 I Joints of the Body

[[

There are two ways to classify joints: functionally and structurally. The functionalclassification is based on the amount of relative motion permitted by the joint. Onethat allows essentially no relative motion between the bones is called a synarthrosis.If the joint allows slight motion; it is called an amphiarthrosis. Finally, a joint thatallows large relative motions is called a diarthrosis or a diarthrodial joint.

When considering the mechanics _ofvarious .types of joints, it is often moreuseful to employ the structural classifications. Fibrous joints and cartilaginousjoints are held together by fibrous connective tissue or cartilage, as their namesimply. For analysis of most human motion and the design of total joint replace-ments, the joints of most interest are the synovial joints, a subset of diarthrodialjoints (Figure 1.9). The bones forming a synovial joint are stabilized and con-strained by a fibrous joint capsule, which may contain connective ligaments. Thearticulating surfaces of the bones are, covered with a thin layer of articular carti-lage, and ·the joint contains highly viscous synovial fluid, which is secreted by a thinlayer of synovial' cells (the synovium). that lines the inside _of the joint capsule and

short and flat bones and in the central portions of vertebral bodies.) Red marrow isassociated with cancellous bone tissue and is where red blood cells are made.

The outer; or periosteal, surface of the borie is covered by a fibrous membranecalled the periosteum. This membrane covers the bone, .except at the flared endswhere the bone is covered by articular cartilage. The inner surface of the bone iscalled the endosteal surface;

The bone gets-its rich blood supply through two sets.of arteries. The periosteal ar-teries are associated with a dense network of vessels in the periosteum, The .medullaryor nutrient artery usually enters the bone somewhere close to midshaft through a canalthat is oblique to the long axis of the bone. The epiphyseal regions of the bone-usuallyhave additional blood supply from other arteries.

Finally, it is interesting to note that.a case, can be made that the various bonesare optimally suited for the functions they must perform. For example, the shafts oflong bones have wall thicknesses optimal for a structure that has minimum mass if

RGURE 1.9 The structure ofsynovial joints.


TABLE1.l

Human kneeHuman hipBrake material on cast ironSteel on steelBrass on steelGraphite on steelTeflon on steelPolyethylene on steel

0.005-{).020.01-{).04

0.40.6.cO.8

0.350.1

0.04-{).20.2-{).4

J. Charnley (1960)A. Unsworth (1975)

tUnless noted, from Table 8.4 of "Introduction-to the Biomechanics of Joints and Joint Replacement"D Dowson and V Wright, editors._MEP, London, 1981.

helps provide joint lubrication. Human synovial joints have extremely low frictionbecause of the lubrication mechanisms that occur in natural joints, which includeboundary lubrication and fluid film' lubrication between deformable surfaces ..when these complex lubrication processes are characterized by a simple coefficientof friction, 'it is found to be lower than virtually' any coefficient of friction for engi-neered bearing surfaces (Table 1.1). . .

The synovial joints fall roughly into three groups: ball and socket Joints; bi-condylar joints, and multiple bone Joints. The hipand th'e shoulderareball andsocket joints. In the hip, the ball (head of the femur),-arid the socket (acetabulum)are conforming, and 'the geometry of the joint surfacesprovidessubsranrial kine-matic constraint. The socket of the shoulder joint (glenoid) does not conform tothe head of the humerus, but is nearly flat. Consequeritly, the shoulder is particu-larly dependent upon muscle forces and soft tissue structures for stability andconstraint.

Bicondylar joints', such ~s the knee joint (see later section on the knee), haverwo pairs of articulating surfaces/Two curved condyles articulate against relativelyflat surfaces on the mating bone; 'Consequeptly, the surface geometry provides little,if any, kinematic constraint, and.,:the joints depend upon soft tissue structures and.muscle fotces Iorconstraint.andstability, Bicondylar joints have a greater range ofmotion in one plane. For example; the primarymotionof the knee joint is flexion orextension in a sagittal plane. Other angular rotations and displacements are small.Each condyle of a bicondylar joint can· transmit load. The loads can be unequal,which enables the joint to resist externally applied moments about the joint in aplane perpendicular to the primary-plane of motion. This provides a mechanism forresisting moments due to functional loads in that plane without the need for largeamounts of muscle or soft.tissue.

The joints berween the long bones of the hand and foot are harder to classify.They seem to fall somewhere berween ball and socket joints and bicondylar joints inmost cases-some are approximately cylindrical in extension and bicondylat in flex-ion. Cylindrical surfaces will have functional characteristics similar to a bicondylarjoint like the knee.

rSection 1.5 Soft Tissue Structures ,~. 13

The wrist and the ankle are multiple bone joints. The wrist (carpus) is made upof eight small carpal bones, and the ankle (tarsus) is made up of seven tarsal bones.'These bones are closely fitted and' bound together by ligaments. They do have syn-ovial cavities, and some movement occurs berween them. Although the movementbetween adjacent bones in these joints ·is small, the combined motion of these boneswith respect to each other and with the 'bones on either side of the joint permits sub-stantial overall motion of the joint.

1.5 I Soft Tissue StructuresMusclesThere are approximately 700 different muscles in the human body, divided mio threedifferent types: skeletal, cardiac, and smooth or, visceral muscles. Skeletal muscle isvoluntary and striated and makes up approximately 36 percent of the total bodyweight in women and42 percent-in men. The cardiac. muscle is.also striated but is aninvoluntary muscle. Smooth muscle tissue is involuntary and is not striated.

We will be concerned almost exclusively with skeletal muscles, and because manyC01111rlonmovements are coordinated by muscles acting in groups,-we will often con-sider. them as such. For instance, the quadriceps (or more precisely, the quadricepsfemoris on the anterior ofthe thigh) includes the rectus femoris, vastus medialis, vastuslareralis, and vastus intermedius, but they.all act together to extend the knee. The maincondition for lumping muscles; together in. this fashion is that they all have a commoninsertion point on.the.bone, thereby creating.no moment about that point. For exam-ple, the quadriceps muscles all come together at the patella, and both the long and shortheads ofthe biceps attach to the radius through a single tendon. A summary of themajor muscles or muscle groups is provided in Table 1.2, along with the action thatthey effect.

Hip

flexionExtensionflexionExtensionAbductionAdductionflexionExtensionflexionExtension

Hamstrings, gastrocnemeus"Quadriceps femoris"Iliacus, psoas majorHamstring»' , gluteus maximusAbductors (gluteus minimus, gluteus medius, erc.)Adductors (adductor magnus, adductor longus, etc.)Rectus abdominis, internal and external obliquesErector spinaeBrachialis, biceps brachiiTriceps bracliii .

Lumbar Spine

Elbow

"The hamstrings consist of three different muscles, the semitendinosus, semimembranosus, and bicepsfemoris.• "The quadriceps consists of four different muscles, the rectus femoris, vastus lateral is, vastus medialis,and vasrus intermedius:

14 Chapter 1 Thetvlusculoskeletal System

The muscles activate, position, and stabilize the skeleton. Active contraction ofthe muscles produces motion of one bone with respect to another. Because the mo-ment arms of the muscles with respect to the joints are small, large motions of thebones can be produced quickly for small amounts of contraction. Contraction with-out a change in length of the muscle creates (along with the associated tendon) aspring. These springs are an important part of the energy exchanges that take placein activities such as running. Co-contraction of agonist and antagonist musclesaround a joint provides joint stability. A good example is the contraction of thehamstrings and quadriceps to stiffen and stabilize the knee joint in flexion-extension.Thus, the muscles not only position and move the skeleton, but they proyide infinitelyvarying stiffnesses at the joints as well. ",'

Tendons and LigamentsTendons connect muscle to bone; ligaments' connect bone to bone, For example, thequadriceps tendon connects the quadriceps muscle to the patella, and the 'patellar liga-ment connects the patella to the proximal tibia at the tibial tuberosity. At one level, ten-dons and ligaments can be considered to be simple cables. Such simple models are oftenuseful in describing theoverall behavior of the musculoskeletal system. To describe thebehavior of 'these structural elements in more detail, one must consider their con-stituents. A closer look reveals that these cables are themselves structures consisting offibers: In 'some cases-for example, long tendons-the tendon consists ofabundle ofparallel fibers. In others-for example, the anterior cruciate ligament in the knee-thefibers-are of substantially different lengths and have a complex twisting orientationwith respect to the "axis" of the "cable." Ata microscopic level.we find thatthe fibersthemselves are made of fibrils and more than one, type of collagen, but primarily type I;from which tendons and ligaments (and bone) get their strength.

Articular CartilageArticular cartilage is also a complex structure consisting of an elastic matrix (itself astructure consisting of fibers), water, large molecules, and other elements. It is typically2-4 mm thick in hip and knee joints and has no blood or nerve supply, which makesit difficult to detect and repair damage. It can be modeled in various ways, dependingupon the questions being asked. For structural analyses of joints, it is sometimes ap-propriate to think 'of it as a single-phase elastic continuum. If its time-dependent bee -:havior is to be studied, it may be considered to: be biphasic or triphasic material.Others have establislied the overall behavior of cartilage by using micro mechanics tomodel the detailed structural interactions of fibers, proteoglycan molecules, and otherconstituents.

1.6 I The Hip, Knee, and SpineThree major joint systems of interest to us in later chapters are the hip, the knee,and the spinal motion segment. They are of primary importance to normal activi-ties of daily living and are also sources of some of the major clinical problems, in

Section 1.6 The Hip, Knee, and Spine 15

musculoskeletal medicine. Their failure to function properly has major consequences.They are also prototypical of three distinct kinds of joints that bring unique consid-erations in analysis and design of devices for treatment of associated disease or injury.Thus, we give them special consideration in. the sections that follow.

The HipBoth the hip and knee joints are synovial joints. The hip joint is a relatively simpleball and socket joint in which the head of the femur rotates relative to the acetabu-lum.in the pelvis (Figure 1.10). Because the hip joint is a ball-socket joint, contactcan be modeled by a single force that acts through the center of the joint. Obviously,this is not quite right, because there is a small, but finite, friction force and possiblysome forces generated by the joint capsular tissues that keep it from being an ideallyfrictionless joint. We are, however, often able to analyze the mechanics of the hipjoint and the femur by using relatively simple models to 'determine the forces andtheir, effects. For example, the femoral diaphysis can often be approximated by a

1,

FIGURE 1.10 The rightfemur: (a) anterior view;(b) posterior view.

Greatertrochanter

.~IlI~.

l;

iI1IIII

(a)


FIGURE 1.11 (a) Side and(b) anterior views of the kneejoint. Note that the fibularcollateral ligament is alsoknown as the lateral collateralligament and mils along thelateral aspect of the joint.(Tortora, G.J., Principles ofHuman Anatomy. Harper andRow, New York, 1983;)

hollow circular beam. The metaphysis (the region where the shaft starts to expand-remember, "meta" means "change") is more difficult to analyze since the load trans-fer to the surrounding bone is more complicated, and we must often resort to finiteelement studies. Also important to the load distribution and overall mechanicalanalysis are the ephiphysis and the linea aspera. The ephiphyses (one at each end ofthe bone) contain spongy trabecular bone and are two of the principal sites forhematopoiesis. During growth, bones lengthen by adding bone at the epiphysealplate, a soft cartilagirious tissue that eventually fuses and turns into bone. Fracturesacross the epiphyseal plate in children are therefore dangerous since they can inter-rupt or even terminate bone growth. The linea aspera is a raised bony ridge on theposterior side of the femur to which many of the muscle groups attach. The greaterand lesser trochanters on the proximal femur are additional examples of specializedmuscle attachment points (Figure 1.10). Biomechanically, ridges, trochanters; andother raised structures on bones create an increased moment arm for the muscleswith respect to the joint center.

The KneeThe knee joint (Figure 1.11) is a bicondylar joint that allows the femur and tibia torotate, twist, and slide relative to one another. Each type of motion is important tothe stability of the joint and must be reproduced in an artificial knee replacement.Otherwise, abnormal forces can develop on the cartilage or in the ligaments,which can lead to their deterioration. This intimate relationship between the kine-matics and loads is a characteristic of most synovial joints. Among the most im-portant structures in the knee are the medial collateralligatnent (MCL), the lateralcollateral.ligament (LCL);" quadriceps tendon, patellar ligament, anterior cruciateligament (ACL), and posterior cruciate ligament (PCL). Athlete's, especially foot-ball players and gymnasts, are well acquainted with the MCL, LCL, ACL, and thepain associated with injuries to these structures. The ACL and PCL lie within thejoint capsule; the other ligaments are outside the capsule (LCL) or are part of the cap-sule (MCL). The cruciate ligaments pass through the notch between the femoralcondyles (Figure 11.1b); The ACL attaches at the anterior side of the tibial plateau

(a)

Section 1.6 The Hip, Knee, and Spine 17

and runs posteriorly and somewhat laterally to an attachment on the femur. ThePCL artaches at the posterior side of the proximal tibia and runs anteriorly to anattachment on the distal femur (Figure 1.l1b). The menisci are also importantstructures in the knee. These are two crescent-shaped pads that help distribute theloads from the femoral condyles to the tibial plateaus. Without the menisci, thenonconJormity of the condyles with respect to the plateaus would result in increasedcontact stresses.

The SpineOne of the few cartilaginous joints that we will study is the anterior part of the in-tervertebral joint (Figure 1.12a). The' connective tissue is a complex fibrocartilagestructure known as the intervertebral disc. The healthy disc is analogous to an in-flated tire and is the largest avascular tissue in the body (Figure 1.12b). The outsideis aset of concentric rings of collagen sheets (the annulus fibrosis), while the centeris filled with a highly viscous gel (the nucleus pulposis). With aging, this gel losesmuch of its hydration and solidifies, which can have significant biomechanical con-sequences. This particular joint allows small motions between the bones it compris-es, but the combinatiori of multiple intervertebral joints along the spine provides thegreat flexibility to the trunk. Unfortunately, the intervertebral joint is often a site ofinjury, hence its significance to the orthopaedic biomechanics community.

Figure 1.13 provides two. views JJf a lumbar vertebra, the largest and strongestin the vertebral column, The principal load-bearing region is the vertebral body. Thecortex is only about 350 JLm thick, andtheinterior is composed of some of the low-est density trabecular bone in the body, particularly in the' elderly. The posteriorstructures of the vertebral bodies provide several functions. As seen in Figure 1.13,the pedicles and laminae forman arch" called the vertebral foramen (derived fromthe Latin for "hole"), through which the spinal cord passes. Adjacent vertebrae arealso connected throughsmall articulating joints (known as facet joints) that con-strain the motion' between' the venebrae, and' limit twisting and' extension of thespinal column. The articular processes are on the laminae (Figure 1.13). The entirestructure, then, serves to support the upper body; provide for the motion of thetrunk, neck, and head; and protect the spinal cord from trauma.

FIGURE 1.12 The interver-tebral joints (a) and disc (b).(Tortora, G.J., Principles ofHuman Anatomy. Harper andRow, New York, 1983.)


FIGURE 1.13 Lumbar ver-tebra: superior (a) and rightlateral (b) views. (Tortora,G.]., Principles of HumanAnatomy. Harper and Row,New York, 1983.)

When we analyze the mechanics of.the lower spine, we often assume that theback muscles (the erector spinae, among others) act at a distance of 5 em posteriorto the center of the vertebral body, although this tends to be an oversimplification,The center of rotation of the joint for sagittal bending is within the disc, althoughthis can vary with aging. Clearly, the ,motion of thespine and even the motion of~single motion segment is a complex problem that requires more sophisticated analy-ses to study the details of its inechanics. However, as' we shall see, much 'can belearned by looking at relatively simple ideas, such as axes 'of rotation and stressescalculated by using straightforward strength ofmaterials models:

1.7 I Damage and RepairBecause the skeletal system functions as a machine, it is not surprising that it can bedamaged, given the length of time it must function and the unusual loads to which itis often subjected. Bones break; joints wear out; and tendons, ligaments, and othersoft-tissue structures rupture. In addition, diseases may independently cause skeletalelements to fail or may. contribute to the occurrence of failures under normal use.

FracturesFracture of bones may be the result 'of abnormalloadingord~creased strength undernormal loads due to some underlying pathology. Fracture is' most often the result oftrauma that produces one-time loads that exceed the strength of the bone. But frac-ture can also result from cyclic loading of the bone during which the magnitudes ofthe loads never exceed the ultimate strength of the bone. In particulat, bones that

Section 1.7 Damage and Repair 19

have not yet adapted to increased activity can fail when repeated loads, such as thosedue to marching or running, cause fatigue damage and can cause ultimate fracture ofthe bone. '

Fracture of bone can also occur under normal loads if the strength of the bonehas decreased. Bone. tumors may cause defects in the bone that reduce its strength,and diseases such as osteoporosis may decrease the structural and material propertiesof bone tissue and increase the risk of fracture. The strength and structural charac-teristics of bone may also be decreased as a natural consequence of bone's remark-able ability to adapt to its mechanical environment. When loads on the bones areincreased or decreased, remodeling of the bone occurs. Increased loads can produceincreased bone mass (changes in geometry and density). Reduced loading on a bonecan occur when loads due to gravity are reduced during space flight or when func-tionalloading is reduced severely such as in cases of paralysis or prolonged bed rest.This produces a potentially detrimental decrease in the ability of the bone to sustainnormal loads. The loading on a portion of a bone may be reduced when the totalload applied to the bone is shared with an implant. Inthis case, the adaptive changesin geometty and material properties of bone tissue may be limited to a relativelysmall portion of the bone. Mathematical modeling of adaptive changes in bone isbecoming part of the design process for orthopaedic implants.

ArthritisDeterioration of the joints over time occurs to some degree in nearly all individuals.Osteoarthritis occurs when some combination of mechanical wear and biochemicaldegradation, erodes the articular cartilage. It is a localized effect 'and is most com-mon.jn the knee and hip since these joints bear th~ largest loads in the skeleton.

Damage to articular cartil;tg~ m;tYbe caused by normal wear and teru; disease, ortta~a. Osteoarthritis may result from the deterioration of joint surfaces that occurswith long-term use. It may occur due to primary damage to !he cartilage or cartilagechanges that are secondary to damage to subcondral bone. Osteoarthritis may also re-sult from. isolated trauma. For example, serious injuries to the knee may damage the ar-ticular cartilage directly or indirectly by causing changes' in th~ mechanics of kneefunction. Repeated insults to joint surfaces can also produce osteoarthritis. For exam-pie, using a jackhammer can lead to osteoarthritis in the wrists and elbows.

Damage to articular. cartilage may also be caused by systemic disease.Rheumatoid arthritis affects multiple joints and results.in damage to the articulatingsurfaces, changes in bone properties, and damage to soft tissue structures around thejoints. The articular cartilage swells, deteriorates, limits mobility, and puts pressureon the nerve endings in the underlying bone. Typically, this disease starts in thehands and spreads to the back and limbs. When ligamentous structures and tendonsheaths are damaged or destroyed, the normal configuration. of the bones with re-spect to each other may be lost, and abnormal loading on the already damaged ar-ticular surfaces may occur. . ,

II

tI•iIJ

I

Soft Tissue DamageSoft tissue structures such as ligaments, tendons, and the menisci in the knee maytear or rupture under traumatic loads. Damage to ligaments and tendons may occur


in the structures themselves or where they are attached to the bone. In the lattercase the failures can sometimes be entirely in the bone at the fracture site. The loca-tion' of the failure depends on the rate of loading because these materials are vis-coelastic-the strength and stiffness of the structures depend upon how fast theloads are applied.

RepairOne of the most extraordinary features of the musculoskeletal machine is that itrepairs itself if the damage is not too severe. In fact, bone damage mayobe whatprimarily initiates bone adaptation processes. This clearly sets the musculos~~letalsystem apart from other machines ~e engi~ee7 normally enco~ters_. The ability ofthe machine to repair itself varies with the site and nature of the injury, As previouslynoted, bone is particularly robust in its ability to repair itself from f~actures andadapt to changes in the loading environment. Muscles, tendons: a~d ligaments ~~ecapable of repair, although arguably not a,s0 capable of self-repair as bone. ,Hyali~ecartilage is at thebottom of the list in terms of ability to repair itself. If the damage ISsevere enough, and increasingly so as this machine ages, certain changes occur that arenot reversible; and if sufficient damage occurs, medical or surgical intervention maybe needed to restore normal function. Treatment is also usually needed if the partsbreak. Trauma can result in bone fractures, soft tissue failure, and other' damage.While healing from such trauma might successfully occur without treatment, thepain, morbidity, and other consequences to the health of the individual often makeintervention a necessity.

The goal 'of the subsequent chapters M this book is toprovide ,th~ engineeringbasis for dealing with the treatment of musculoskeletal disorders. The first task IStounderstand the basic rnechanicsofrhe normal.damaged, orrepaired musculoskele-tal system; the second is to design procedures or bone-implant systems to tre~t.thedisorders. The chapters will-deal with determining forces and moments atthe JOintsof the 'skeleton, the loading and deformation" of bones, and the various muscu-loskeletal soft tissues': analysis of whole bone behavior, and analysis and design ofvarious bone-implan; systems for fracture healing and joint replacement. Along theway we will .also discuss the properties of implant materials and pro~ide m:o~ede-tailed considerations of particular joints, such as the hip and knee. Although It ISnotpossible to exhaustively consider all the applications in musculoskeletal mechanicsand implant design, these chapters illustrate principles and methods that can be ap-plied universally to such problems.

1.8 I SummaryThe musculoskeletal system has four main functions: hematopoiesis; mineral stor-age, protection of the vital organs, and support and motion. The first hVo are phys-iological in nature, the second two mechanical. The focus ,of thls,bQo~_1s,on,supp?rtand motion-the skeletal system as a machine. It is a combination of rigid or resist-ant bodies having definite motions and capable of performing useful work. Thelinks of this machine are the bones, along with the soft tissue structures .associated

r Section 1.9 Exercises 21

with them. These links ace connected to each other at joints that enable the body tomove quickly in a relatively agile manner and transmit large forces from link to link.

It is important to distinguish between bones, the structures, and bone tissue, thematerial from which bones are made. There are two basic types of bone tissue. Cortical,or compact bone, is the most dense bone in the skeleton. Cancellous, or trabecular orspongy bone, is much less dense than cortical bone and is found in the epiphyseal re-gions at the ends of long bones, within the very thin cortices of vertebral bodies, andbetween the mote dense outer layees of bones such as the skull and pelvis.

Joints may be classified functionally or structurally. The functional classificationis determined by the amount of motion at the joint. A joint that allows large relativemotions is called a diarthrosis or a diartheodial joint. The structural classification isdetermined by how the bones at a joint are connected. Synovial joints are a subset ofdiarthrodial joints and are stabilized and constrained by a fibrous joint capsule, whichmay contain' connective ligaments. The articulating surfaces of the bones are coveredwith a thin layer of articular, cartilage, and the joint contains highly viscous synovialfluid. 'Synovial joints have extremely low friction because of the complex lubricationmechanisms that occur between the deformable surfaces of natural joints. Musclesare the actuators of the musculoskeletal machine. MusCles are coimected to bone by tendons; ligaments connect bone to bone.

Because the skeletal system functions as a machine, it 'can be damaged. Bonesbreak; joints wear out; and tendons; ligaments, and other soft-tissue structures rup-ture. In addition, diseases may independently cause skeletal elements to fail or maycontributeto the occurrence of failures under normal use. One of the most extra or-dinary features of the musculoskeletal machine is that it repairs itself if the damage isnot too severe; In fact, bone, damage may be what primarily initiates bone adaptation

,~processes. The ability ofthe inachine to repair itself varies with the site and nature of'the injury, and with aging. Bone is particularly robust in its ability to-repair itselffrom fractures and adapt to changes in the loading environment. If the damage is se-vere enough, medical or surgical intervention may be needed to restore normal func-tion. Our goal is to study the fundamental biomechanical principles of themusculoskeletal system and its repair using a variety of orthopaedic implants.

1.9 EXERCISES I

1.1 Identify the primary plane or planes of motion(frontal, sagittal, coronal) for the following motions:

(i) walking(ii) "jumping jacks" in calisthenics

(iii) baseball throw(iv) shot-put throw(v) golf club swing(vi) baseball bat swing

(vii) ice skater's spin

1.2 Various human joints are sometimes regarded asapproximations of common mechanical joints such asthe spherical or ball and socket joint (e.g., the hip with

three rotational degrees of freedom) or the hinge (e.g., theknee with one rotational degree of freedom). How wouldyou describe the motion of the following joints in termsof common mechanical joints (the hinge, the ball andsocket, the universal joint)?

(i) the elbow joint between the humerus andulna

(ii) the elbow joint between the humerus andradius (pronation-supination of the forearm)

(iii) the wrist(iv) the ankle(v) the shoulder

I


1.3 What are the advantages and disadvantages ofhaving skeletal muscles (e.g., the muscles that flex theelbow) lie roughly along the axis of the long bones?1.4 It has been suggested that humans have a finitemaximum possible height due to their particular bodyproportions; that is, it is not possible for a human beingto actually survive if one were to be, say, 6 meters talland have the mass distribution that normal-height hu-mans have. Why might this be true?1.5 The joints are typically identified by the two bonesadjacent to the joint; for example, the joint between themetacarpal and the proximal phalange of the finger (seeFigure 1.4) is called the metacarpal-phalangeal (MP)joint, and the joint between the proximal and middlephalanges is called the proximal interphalangeal (PIP)joint. Describe the motions at the various joints in eachdigit when executing a pinch between the .thumb tip andfifth fingertip.1.6 Most muscles activating the limbs have broad ori-gins (proximal attachments) and relatively small inser-tions (distal attachments) via tendons. What advantagesmight this arrangement offer?1.7 The trabecular architecture of cancellous bone. isquite specific in any particular bone. In the end of along bone like the proximal tibia, the trabeculae areroughly obliquely oriented arcades, while in a flat bone

f.,

like the scapula the trabeculae are oriented perpendi-cular and parallel to the surface. Why·might this beadvantageous?1.8 When we stand with our weight 'evenly balancedon our two feet, it turns out that the contact force actingacross each hip joint is nearly our body weight, insteadof about half our body weight. Without actual quantita-tive analysis of the problem, discuss why this should betrue. (The mechanics of the loads across joints will beaddressed in Chapter 2.)1.9 Our bones are really quite stiff and deflect verylittle under loads. However; they do 'deform some. Asyou will see in .later 'chapters, bone can be stretched tosomething on die order of 2 percent before fracturing.Discuss the advantage this deforrnability might pro-vide. What'wouldbe the advantages and disadvantagesof having greater deformabiliry, say twice as muchstrain to failure? '1.10 How are the extensor tendons 'constrained topass over the knee joint and the first metacarpal joint?1.11 . Use a.scale on Figure 1:1, 'a frontal 'view of theadult skeleton, to' 'estimate the ·ratio of the length to

. midshaft diameter for the femur, tibia, ulna, radius,and:humerus. Assume that the midshaft cross section is

. a hollow.. circular cylinder, Discuss how these ratiosvary from site to -site in the skeleton.

r

Ir

Loads and Motion inthe MusculoskeletalSystem

CHAPTER

Adopting the view of the musculoskeletal system as a mechanism or a macruhe letsus use common tools of mechanics to analyze the mechanical behavior of the sys-tem. the ana:lyses generally fall into one of two categories, those where we are in-terested in 'the motion of elements of the system, or those where we are interested inthe deformatioris and stresses in the elements Qf the system: For the first problem,we view themusculoskeletal system as a mechanism consisting of-rigid links, actua-tors, and coristraint elements and 'use the dynamics or statics methods of rigid bodymechanics to analyze the behavior. The stress analysis problems usually require thatrigid 'body analyses precede or accompany them to define the loads that are neces-sary for boundary conditions. Therefore, regardless of whether our interest is in therigid body dynamics problems themselves or stress analysis problems, the rigid bodymecharfics problem will probably have to be addressed.

In applications of rigid body mechanics to study the mechanical function ofthe musculoskeletal system, the common ingredients are (1) the description of theelements of the system in mechanical terms; (2) thedescription ofthe external con-straints on the System; and (3) the application of the laws of motion. The elementsofthe musculoskeleial system range from bones to nerves, and their correspondinganalogs in the model vary from rigid links to active force generators and controllers(Table 2.1). The elements to be included in the model and the complexity oftheirdescriptions in the model are the first choices to be made in rigid body analyses.

The literature that addresses issues regarding the elements listed in Table 2.1 isextensive; 'Much of itis relevant to biomechanics, including the. biological processesinvolved in the formation, growth, aging, disease, and repair of each anatomical el-emerit. Because our 'goals are limited primarily to describingthe skeleton as a mech-anisin or machine, we will forgo consideration of many of these issues. However,because the goals of any individual analysis may impinge directly on some biologicalbehavior or some more detailed issue of mechanical behavior; it may benecessary toincorporate them in a particular analysis.Here we briefly consider some of the pri-macy musculoskeletal characteristics that enter into rigid body analyses.

23

24 Chapter 2 Loads and Motion in the Musculoskeletal System

TABLE 2.1

Model Element

Bones or limb segmentsJoints

Rigid linksStandard joints: spherical, revolute, cardan, etc.Rigid contact surfaces (kinematic constraints)Deformable contact surfaces (force constraints)ActuatorsActuators + elastic + viscous elementsControllersElastic or viscoelastic springs

Muscles + tendonsNervesLigaments + joint capsules

2.1 I Basic ConceptsBonesIn this chapter, we· regard body segments as individual rigid links with dimensionsand mass properties determined by the combination' of tissues that make up. the seg-ments. Bones are organs that primarily make it. possible to .take such an approach,being. relatively rigid frameworks to which all the other tissues are directly or indi-rectly attached. Obviously, bones are not rigid, and in fact we will be very much in-terested in the way they act as deformable structures when we address their structuralbehavior. The mechanical properties of bone tissue as a structural material will be ad-dressed .at that point. However, for the present problem of describing forces in theskeletal system, we need not be concerned with thedeformabiliry of bone.

JointsJoints are also discussed in more detail subsequently. from the perspective of a dy-namics problem, the joints represent kinematic constraints. The two 'basic care-gories of skeletal joints are articulating joints, found at most appendicular skeletaljoints, and deformable joints, particularly the intervertebral discs between adjacentvertebral bodies in the spine. Anatomists might add joints such as the pubic sym-physis, which 'is essentially a fibrous tissue union. For our purposes, we need notdeal with such joints, since the relative motion is well within our allowance fordefining rigid bodies.

Some anatomical joints are readily modeled in dynamicor static.analyses assimple joints, particularly spherical (ball-socket) or revolute (hinge). The hipjoint and the scapulo-humeral joint at the shoulder are examples of the former,and rhe humero-ulnar joint at the elbow is an example of the latter. However, mostjoints are not so simply defined. For example, the wrist is a complex joint, andsupination-pronation motion of the hand involves the relative rotation of theradius about the ulna. The shoulder complex invol~es motion of the scapula withrespect to the rib cage, as well as motion of the humerus with respect to the scapula.Even the knee, which is often regarded as having a cam-like, planar motion, is infact three-dimensional. One of the major issues in developing appropriate kinematic

models for link dynamics problems in musculoskeletal motion is defining appropri-ate kinematics at the joints.

The direct measurement of motion of individual links in theory allows the. con-sideration of each element as an unconstrained rigid link. In this case, the nature ofthe joint can be inferred by the methods of kinematics. A number of studies havebeen carried outfor such joints as the knee and the wrist to establish the character-istics of the joints. These provide some guidance in making choices about the natureof the model forthe joint in a given application. A recurring theme in this book ismatching the choice of the model with the nature of.the question being asked.

Treating joints as kinematic constraints (where the relative motion is pre-scribed in some sense) ignores any deformations and .determines contact forces. Thereality, of course, is that even the articulating joints are not rigid connections.Articulating surfaces are covered with hyaline cartilage, which distributes the con-tact forces and provides remarkably low-friction bearings. In many problems of in-terest, it can be assumed that contact.is frictionless and cartilage deformations arenegligible. However, the detailed mechanics of the joints are often of interest, andthis willbe addressed in more detail later.

Muscles and Tendons.r-

In works focusing OJl. biomechanics of tissues, it is customary to consider the prop-erties of tendons along with those of ligaments because of their-similarities in struc-ture and load-carrying function. However, for our purposes, it is better to considertendons and muscles in combination. The crucial feature of muscle-tendon units asorgans.is their ability to actively convert-chemical energyro mechanical work. Thisis.accomplished by a process initiated by a neural-input at the cellular ·level. The gen-eration ~f an active force involves complex dynamics in both the activation and con-traction. processes. The total muscle force is the combination of the. active forcegeneration and the viscoelastic behaviors of the tendon and the passive muscle tissues./I. simple representation of amuscle is shown in Figure 2.1. The force generated bythe.active element is a function of. the level of activation, the length of the muscle,and the speed of contraction, or-Iengthening, The general nature of these. functionscan', be represented mathematically, but the model parameters are muscle specific.The.passive elements in the force model are also muscle specific, but can be muchmore easily estimated experimentally.

In order to incorporate such a model into a musculoskeletal dynamics prob-lem, it would be necessary to include some representation of the activation-contractiondynamics. A commonly used alternative is to apply the link dynamics equations to

CE SE

PE

DE

Section 2.1 Basic Concepts 25

FIGURE 2.1 One model ofa skeletal muscle as an actua-tor includes four compo-nents: a contractile elementthat generates a force basedon a neural stimulus throughan activation arid contractiondynamic process; an elasticelement in' series with thecontractile element; a parallelelastic element; 'and a parallelviscous 'element. The passiveelements represent the com-bined elastic and viscous ef-fects of the muscle and thetendon.

26 Chapter 2 j.oads and Motion in the Musculoskeletal System

FlCURE 2.2 The typicalforce-length ~e for a Iiga-'ment tested to failure shQWSaregion of relatively low forceand 'then a regiqn where theforces increase substantially,In addition to the nonlinearelasticity, th~re ca~ be signifi-Cant rate effects due to viscos-ity at sufficiently high speeds.

determine the muscle actions that must necessarily be generated. While this avoidsthe explicit incorporation of muscle dynamics, it must also be recognized as an ide-alization that may lead to errors.

In order to incorporate muscle forces into models, it is necessary to know theirl!nes of action. The simplest approach is to assume that the muscles act along straightlines from their origin to their insertions. Many muscles have broad attachments,particularly at their origins, or have multiple heads. This may necessitate representingthe muscles as several subunits in order to accurately represent the muscle action.Further; most muscles do. not, really act along straight lines, since they lie over deeperstructures that restrict their tendency to straighten under tension. Nonetheless; usefulresults have been obtained with the use of straight-line models of muscles.

Ligaments and Joint CapsulesIn addition to the actions of muscles, ,the ligaments that cross any joint may carryloads. In some cases, such as for the collateral and cruciare ligaments at the knee, theligaments are quite discrete. In other cases, a complex array.of ligaments surroundsthe joint, which is intimately integrated with the joint capsule. The hip joint and theposterior capsule of the knee are examples.

Ligamentous structures may be modeled as nonlinear springs that have alowstiffness rver some length range-Irhe toe region) and-then substantially increasedstiffness upon further-lengthening (FigureZ.Z). This behaviorlimits the range ofmotion ofthe joint. Because the stiffness remain's quite low over a normal range ofmotion for many ligaments; they are often omitted from models, In some casesthey are included as single spring elements; However, this is inadequate for 'manyligaments that have' sufficiently large cross sections. For example, different portionsof the anterior cruciate: ligament' are' tightened (reach the stiffer portion of theforce-length curvelar different lengths. If ligament behavior is important to themodel, it may be necessaryto represent aparticular ligament- as an assembly of Iig-amentous structures. ' " , '. !.

If the length-force curve for a ligamentous structure is known, and the coordi-nates of the attachments are known 'in each' corresponding bone, then the forces inthe ligaments can be predicted.from knowledge ofthe joint' motions. That is;' if-theposition of one bone is known with respect to the other, the 'ligament lengths can becalculated and the length-force curve used to calculate the forces. In this' way, the

[ 2000:

1'1&. 1000

5 10 15 20 25 30Deformation (mm)

IIi

Section 2.1 Basic Concepts 27

ligaments can be incorporated into the link dynamics models of the musculoskeletalsystem without conceptual difficulty. '

Link Dynamics ModelsFor modeling of any engineering system, the choices we make about elements inthe model should reflect the goals of the analysis. For example, if the purpose ofthe analysis is to estimate the steady-state maximum bending moment that OCcursin the mid-diaphysis of the femur during single-legged stance, 'then it is not necessaryto include details in the model such as ligament deformation or muscle activation-contraction dynamics since they are unlikely to significantly influence the soiu-tion, By contrast; if the 'goal is to estimate the forces in the knee ligamentsdue toa lateral impact force, ligament models with ioad versus deformation behaviorwould certainly 'be required. '

The external boundary conditions on the system are the external forces andkinematic constraints. Again, the choices we make in modeling these boundary con-ditions should be determined by the goals of our analysis. For example, it may be ac-ceptable in one situation to describe the interaction of the foot with the floor as akinematic constraint, while it may be better in another situation to model the con-tact as a force constraint and describe it as though there are springs between the footand floor. '

There are also a number of choices to be made within the context of applyingNewton's three laws. In the absence of motion, the problem reduces from a dy-namics problem to a static problem: In'addition, there' is the so-called quasi-staticproblem, where the system' does not maintain a fixed configuration (tile" staticsproblem), but the accelerations are small enough tlht'the inertial terms in the equa-tions of motion can be ignored. Tliis is a fairly common circumstance ,in skeletalmotion problems, and in this ca~ the analysis is effectively a statics problem withvariable geometry. In addition to incorporating the chan~ng'georrietriof the prob-lem, this approach carr capture' relevant dynamic effects, since it can reflect the dy-namic nature of the external loads .acting on the system for. each static analysis. Forexample, in a quasi-static analysis of gait, the ground reaction forces acting on thefoot include dynamic effects generated by.the impact 'forces that develop at the footon contact. '

In this chapter we present the application of rigid body mechanics to analyzeforces and motion of the skeletal system. Our goal is to develop modeling ap-proaches that permit us to characterize the internal forces in the system-the forcesacting on individual bones, ligaments, and tendons due to external loads and mo-tion. We will see that it is relatively straightforward to arrive at the resultant forcesand moments that represent the combined actions of all the load-carrying elementsacross individual joints. However, we will find that it is not even possible for us toarrive at the forces in the individual elements by using only the equations of dy-namics or statics for the general circumstance. We "fill explore some ways to dealwith this dilemma. We will also examine the problem of determining the kiriematicinput often required for these problems. Our most important goal is to presentways of thinking about such problems, not to catalog specific methods of solution.The latter task would be huge indeed, given the large and ever-growing literatureon these methods. '

70 Chapter 2 Loads and Motion in the Musculoskeletal System

10cmr:JtlOON1700 N

8\--FIGURE P 2.14

FIGURE P 2.13

"' = 0.8 rad/sec

a = .2 md/sec"

"loot = 10N

...." = 30N

Sr= fa

r.

Tissue Mechanics I:Bone

CHAPTER

'~,:

The material properties of musculoskeletal tissues depend upon their underlying mi-. crosrructures, which must be known, to understand the anatomicsite-, age-, and

disease-related variations that occur in the tissue and organ level mechanicalproperties. It .isimportanr at the onset to 'distinguish betweeri material and struc-niralbehavior; 'since terms such as "muscle," "bone," "cartilage," and "tendon"can be. used to' describe either tissues or-organs. The femur/for example, is anorgan 'composed of both cortical :and trabecular bone tissues, as well as marrow,blood vessels, periosteum,' and' nerves, all of' which contain a .variery of cells.Biomechanically, the organ' behaves as a' structure, whereas: the tissue behaves .as a'

'materia.l. Due to the hierarchical composite nature of all biological tissues, at smallerscales the tissue itself can be viewed as a structure. Thus, the real distinction betweenstructures and materials depends upon the geometric scale at which the microstruc-ture becomes blurred and can be considered to be a continuum. When we say "rna-terial" 'behavior, we mean that the tissue can be considered to be a continuum for aparticular analysis, but the distinction between material and microstructural be-havior is problem specific. For example, in some analyses, trabecular bone can betreated as a continuum; for others, its heterogeneous microstructure must be modeledexplicitly.":", Material behavior, by definition, is independent of the geometry of the testspecimen from which the properties were measured; Material properties are deter-milied 'from experiments, such as -the uniaxial tension test, which are performedoil.standardized specimens under controlled mechanical and environmental condi-ti-ons. These tests are designed to eliminate any behavior associated with specimengeometry and to produce uniform states of stress and strain from which the mate-rial properties can easily be calculated. One challenge in testing musculoskeletal tissuesis-measuring deformations, particularly for soft and porous tissues such as tendon andtrabecular bone. This, along with inter- and intra-individual heterogeneity, results in

71

72 Chapter 3 TIssue Mechanics I:Bone

considerable variation in the reported material properties for most tissues. Theseuncertainties must be considered in the development of an appropriate model forthe particular analysis at hand. .

Since material properties measured from a particular set of mechanical tests areindependent of geometry, in principle they can be used to describe the behavior of asmall unit of the equivalent tissue anywhere in the skeleton. Thus, we can talk aboutthe material properties of cortical bone tissue, which can be applied to cortical bone indifferent bones such as the femur, tibia, fibula, humerus, ulna, or radius. Strictlyspeaking, however, the data are valid only for tissue with the same microstructure andin the same environment as the test specimens.

Because biological tissues are heterogeneous, we usually attempt to relate themeasured variations in material properties with corresponding measures of the tissuemicrostructure and other parameters such as apparent density, mineralization, ororganic composition. Whether reported material properties for one anatomic site orindividual of a particular age can be extrapolated to other sites or ages depends uponthe nature of the analysis objectives and the precision and accuracy required for thematerial property data in the modeL

In contrast to material properties, structural properties describe the overall be-havior of the entire (or partial) organ. This behavior includes the combined effectsof material and geometric properties:

STRUCTURE (ORGAN.) = MATERIAL (TISSUE) X GEOMETRY

FUrther, because of biological heterogeneity, the material properties are gener-ally not uniform throughout the organ, and spatial distribution within the. organ isoften important. This can complicate determination of-the structural properties. Forexample, the maximum.force-carrying capacity of the proximal femur is a structur-al .properry that depends upon the material properties oi both the cortical. bone andthe highly-heterogeneous trabecular bone within the; proximal femur, as well as the.relevant geometric properties of the. bone, such as its.length.vcross-secrional area,and area moment of inertia.' Thus; structural properties, are highly specific to, the organunder analysis and 'rarely can be, extrapolated. to other. anatomic sites, whereasmaterial properties are more generic.

The inter- and intra-individual heterogeneity of various tissues and organs is afundamental characteristic of biological tissues. Consequently, structural analyses ofbiological systems, differ from many engineered systems for which material proper-ties are often uniform and homogeneous. As a result, stress analysis of the muscu-loskeletal system is challenging because one has to first determine the appropriatematerial model to use by answering important questions. Is it necessary to accountfor heterogeneity, or is it acceptable. to assume the simpler 'case of homogeneousproperties? Is linear modeling adequate, or is it necessary to account for complexnonlinear or time-dependent behaviors? To address these issues, we will describe themicrostructure of 'musculoskeletal tissues, describe their material properties, anddiscuss how these properties are used in different types of biomechanical analyses.We will also address. structural analysis of some selected organs as an example ofgeneral approaches that can be taken to perform stress analyses of any muscu-loskeletal structure.

Section 3.2 Composition of Bone 73

We will also consider briefly in this chapter one characteristic that makes muscu-loskeletal ~terials ~y unique, w~ch is their ability to adapt to both their biologicaland ~echamcal environments, This has been known in general for a long time, butouly m recent years have we begun to understand the details of the processes thatoccur at the .cellular and molecular levels. This emerging field of study has come tobe known as mechanobiology. It embodies the extremely important concept that, atthe cellular and molecular levels, we cannot understand the mechanics without un-derstanding the biology, and we cannot understand the biology without understand-ing the mechanics. .. Although we can learn much about the fundamental mechanics of a bone-impl~nt system at a particular time by using the basic concepts of structural me-chanics alone, the mechanobiology of these systems must be included if we are tounderst~nd how ~one-~plant structures change over- time in response to alteredmecharucal and biological environments. Such an approach is the basis of whatsome: have term~d "preclinical testing" of implants-the process by which candi-date ~plant designs can be evaluated in silico (i.e., by means of computer models)fOJ th~lr long-t~rm effects on bone morphology, long before ever inserting the im-plant Into a patient. .;

.. Regarding the adaptive behavior of bone, inthe current chapter we have limitedq\lC~~I\(est.o briefly discussing and illustrating the mechanics of bone adaptation in'keepingwirh the scope and-thrust of this book. The description does demonstrateth~general concept. and i~lustra~es ~e process by which one can "engineer" adaptiveresponses to~tress into biornechanical analysis and,design., ', , WIth this backgro~nd; we can now turn our attention to a description of themajor muscuI?,k(!letaltt~sues. In this chapter we focuson bone; in Chapter 4 wecqnsl~e, soft tISSues-articular cartilage,. meniscus, tendon, ligament, the interverte-bral disc, and-muscle, The. q:>mI!osition and basic structural. characteristics of bonetissue will be described first. Then the material properties of cortical;nd cancellousbone will be presented. Finally, we will address the structural consequences of het-erogeneity Qll tissue behavior.

3.2 I Composition of BoneFrom .an engine~ring perspe~tive, bone is a remarkable material having uniquemateria] properties and has, like aImost all biological tissues, the ability to repair it-self and ada~t to it~ ~ech~nical environment by biological remodeling and turnover,As a tissue, It e~lublts WIde variations in morphology, ranging from the delicate,open-celled architecture of the trabecular physes, to the dense, fiber-reinforceda~rangemen~ of the dia~hyseal cortex. Bone adjusts to changing environmental con-ditions by highly orgaruzed structural adaptations. For instance, with heavy exercise,whole ~ones can change both material and geometric properties to provide struc-tures of mcreasedJoad-bearing capacity. With aging and disuse, bone tissue is resorbed,resultmg in substanttallosses of tissue stiffness and strength. To compensate, boththe periosteal and ehd?steal diameters of the cortices can expand, thereby maintaininga structure of approximately constant stiffness and strength.

74 Chapter 3 Tissue Mechanics I: Bone

FIGURE 3;1 Diagrammaticrepresentation of the hierar-chical molecular structure ofcollagen, Starting with a sin-gle helical protein chain con-sisting of a variety of aminoacids connected by peptidebonds (top), three of thesepolypeptide chains are com-bined to form the triple helixtropocollagen molecule (sec-ond down), referred to mostoften as simply the collagenmolecule, The different typesof collagen have differenttypes of polypeptide chains,The collagen molecules arearranged in parallel in a regu-lar quarter-stagger arrange-ment to comprise the collagenfibril (bottom).

Bone is composed of inorganic and organic phases and, like all biological ma-terials, water. On a weight basis, hone is approximately 60 percent inorganic, 30percent organic, and Iflpercent water. On a volumebasis, these proportions areabout 40 percent, 35 percent, .and 25 percent, respectively. The inorganic phase of boneis a ceramic crystalline type mineral, whichis an impure form of naturally occurringcalcium phosphate.most.oftenreferred to as hydroxyapatite: Ca,!O(P04)6(OH)z: Theapatite crystals are-tiny (2':'5 run thick X 15 run wide X 20-50 ruri long plates) andcontain impurities such as potassium, magnesium, Strontium, and sodium (in place ofthe calcium ions); carbonate (in place of the phosphate ions); and chloride or fluoride(in place of the hydroxyl ions). Thus, bone mineral is best considered as an impureform of hydroxyapatite.' ' ,', ,

The organic phase of bone consists primarily of type I collagen (90 percent byweight), some other minot collagen types (Ill and ,VI);and a variety of noncollage-nousproteins (mostly osreocalcin, osreonectin, osteopontin, and bone sialoprotein).The molecular structure of collagen (Figure 3.1), the.strongest and most abundantproteiri in the body; 'starts with a right-handed triple helix of three left-hand helicalpolypeptide "alpha" chains. The-resulting "tropocollagen" moleculeis.rod shaped,with alength of about 300nmand diameter of 1.5 run. These collagenmoleeulesare then arranged in parallel-with ,each other head 'to, tail; but with a gap or "holezone" of about 35 ,hID between' each molecule, They are-arranged with surroundingcollagen molecules in- a.quarter-staggered fashiorf,'Additibnal, pores exist along-thesidesof the collagen molecules between.neighbors. Mineralization 'is'thought to startwithin the hole zones and then progress to the pores; the noncollagenous proteinsate also found within these spaces. The result ofthis is the mineralized fibril, which,as we will seenext, is arranged in a number of ways-to produce the overall compose,ire. Collagen fibrils in bone range fiom20-40 hen in diameter, suggesting there-are200-800 collagen molecules in the cross section, of a fibril.

/'/_ "_/ _

'-'-' ----__ -'_'-__'

_' -"-' '-'-'_, ---' '_---_', _,_'_

Section 3.3 Bone as a Hierarchical Composite Material 75

3.3 , Bone as a Hierarchical Composite MaterialtJierarchical LevelsBone tissue is 'a hierarchical composite at many levels (Figure 3.2). At the lowest level(",0; 1 micron scale), it is a composite of mineralized collagen fibrils. At the next level(",10 micron scale), these fibrilsare arranged in two forms, either as stacked thinsheets called lamellae (about, 7 microns thick) 'that contain unidirectional fibrils in al-ternate angles between layers or a; a block of randomly oriented "woven" fibrils.Laffiellar bone is most, common, although woven bone is found in situations of rapidgrowth in children and large animals and also during the initial stages of bone fracturehealing. Laminar or "plexiform" bone consists ofsmaller sandwich-typeconstructionsof layered lamellae, arranged around pperungs fo~ blood v'essels. This type of bone isoften interspersed with woven bone and is commoninlarge animals, such as cows, thatgrow rapidly. ','" ,

Lamellar bone can take various forms at the next hierarchical level (0.5-1.0 mmscale). Primary lamellar bone is new bone that consists of the large concentric rings oflamellae that circle the outer 2-3 mm of the diaphysis, similar to growth rings on a tree.The most common type of cortical bone in adult humans is called osteonal orHaversian bone, where about 10-15 lamellae are arranged in concentric cylindersabout a central Haversian canal (a canal about 50 microns in diameter), which con-tiliis blood vessel capillaries, nerves, and a variety of bone cells' (Figure 3.3). The sub-'structure of concentric lamellae, including the Haversian canal, is termed an osteon,which has a diameter of about 200 microns and lengths of 1-3 mm. Other channels,called Volkmann's canals, about the same diameter as Haversian canals, run perpendi-'ciilar to ,the Haversian canals; providing radial paths for blood' vessels. ,.-

. Osteons represent.the primary discrete 'unit of human adtilt cortical bone, andare' continually being tom downand' replaced by the various types of Done cells in abiological process called borie remodeling. Over time, the osteon can be completely

1. MineralizedCollagen Fibril '[0,1 micron]

2. LamellarWoven[10 micronl

3. Primary LamellarHaversian··.LBminarWoven(500 microns]

4, TrabecularCortical{>1000 microns]

RGURE 3.2 The four levelsof bone microstructure, fromthe level of mineralized colla-gen fibrils to cortical and tra-becular bone. It is generallyassumed that, at the formerlevel, all bone is equal, al-though there can be subtle dif-ferences in the nature of thelamellar architecture anddegree of mineralization be-tween cortical and trabecu-lar bone. (Adapted fromWainwright et aI., MechanicalDesign in Organisms. HalstedPress, New York, 1976.)

76 •Chapter 3 Tissue Mechanics I: Bone

FIGURE 3.3 Diagram of asector of the shaft of a longbone, showing the differenttypes of cortical bone, trabec-ular bone, and the variouschannels. The osteons are 10-eated between the outer andinner circumferential lamellae,(Tortora, G. J. Principles ofHuman Anatomy. Harper andRon, New York, 1983.)

Lyinphlitiq vessel inHaverslah canal

BloOd vessel I,i-:Volkmann's canal

Volkmann's canal

removed, leaving behind a hole (resorption cavity, about 200 microns in diameter),which is then filled in .by a new osteon. Typically.rhere are about 1()"'15 Haversiancanals per mm2 cross section of adult human cortical bone. II. cement line, which ,isa thin, Iow-mineral-contenr layer of calcified mucopolysaccharides with verylittlecollagen, surrounds each newly formed osteon, The cement line .is a weak interfacebetween the osteon and the surrounding interstitial.bone, This may.actually improve'the fatigue properties of cortical bone by providing avenues for dissipating ,energyduring crack propagation, , ,", ' , '

, Since mineralization of a new osteon is a slow process that can take months,the distribution of the degree of mineralization can be large at any time in a particu-lar whole bone cross section. The remodeling process occurs in part to repair the fa-tigue damage that can occur in bone during strenuous repetitive activities.

Within this structure, there is an underlying level of porosity at the scale of5-10 microns or less that is associated with the bone cells. 'Osteocytes, the mostcommon type of bone cell, reside alone, surrounded by a thin layer of extra cellularfluid, in small ellipsoidal holes (about 5 microns minor diameter; 7-:8 microns majordiameter) called-lacunae, of which there are about 25,00a-per riim3 in bone tissue(Figures 3.3 and 3.4). The lacunae are generally arranged along the interfaces betweenlamellae. Each osteocyte has arms or processes that extend from the cell throughtiny (""0.5 micron, diameter, 3-7 microns long) channels called canaliculi and meetwith the processes of surrounding cells at cellular gap junctions. Gap junctions arearrays of small pores in the cell membrane that make connections between the inte-riors of neighboring cells, allowing direct passage of small molecules such as ionsfrom one cell to another. There are about 50-100 canaliculi per single lacuna(Figure 3.4) and about one million per mnr' of 'bone. The resulting intercommunica-tion between osteocytes provides an important mechanism by which bone cells arethought to sense mechanical loading or deformation and transmit signals elsewhereto the osteoclast bone cells that remove bone tissue. In this way, the remodeling

r Section 3.3 Bone as a Hierarchical Composite Material 77

FIGURE 3.4 Fnvironmentalscanning electron microscopepicture of a fracture surface ofa pieceof cortical bone, show-ing a fracrured lacuna at low(left)and high (right)magnifi-cations. Note the elliptiealshape of.the lacuna and themultiple smaller pores. Theseare the canaliculi, whichhave a diameter of about 0.5micron.

sequence is thought to be, be initiated when loading' becomes too different fromsome physiological set point or if the bone becomes damaged.

At the highest hierarchicallevel,(l-2 mm), there are two types of bone: (1) cor-tical bone, which comes as tightly packed lamellaeHaversian, laminar, or wovenbone; and (2) trabecular bone, which is a highly porous cellular solid. In the latter,the lamellae are arranged in lesswell-organizedvpackers" to form a series of rodsand plates, about 200-300 microns thick, interspersed with large marrow spaces(Figure 3.5). Sometimes, when the rods and plates in trabecular bone are very thick,osreons can be found, but this is rare in human bone.

(a)

(e)

(b)

(d)

FIGURE 3.5 Various archi-tectures of trabecular bone,as demonstrated by smallsamples (3 X 3 X 1mnr')of trabecular bone from(a) bovine proximal tibia;(b) human proximal tibia;(c) human femoral head; and(d) human vertebral body. Ineach case, the main trabecu-lar orientarion is in the verti-cal direction. (Keaveryet. al.,Annu Rev. Biomed, Eng.2001 3:307-333.)

Chapter 3 Tissue Mechanics I: Bone

Cortica1 and Trabecular BoneThe maim difference, therefore, between cortical and trabecular bone is the opencellular st:ructure of the latter. The actual bone 'tissue at the underlying hierarchicallevel is very similar, being made of lamellar bone arranged as previously describedfor cortical bone or in the more irregularly shaped packets for trabecular bone.However, one difference does exist. There is more bone remodeling on the freesurfaces of the rods and plates within trabecular bone. than on the internal sur-faces of Haversian canals within the 'cortical bone, and newly formed bone is lessmineralized than older bone. Therefore, trabecular bone tends to be less mineral-ized than cortical bone, although the difference is subtle; This difference in miner-alization and the different arrangements of the lamellar bone are thought toproduce slightly lower material properties of the tissue that makes up the struts oftrabecular bone-trabecular rissue-scompared with the tissue that makes up cor-tical bone. It is difficult to measure these properties directly due to the small sizeand irregular shapes of individual trabeculae .. The resulting uncertainties in thetrabecular tissue-level properties complicate micromechanical analysis of trabecu-lar bone. . , ,

.Both cortical and trabecular tissues are porous ..Cortical bone has a porosity Pless than about 30 percent, or equivalently, a volume fraction "f greater than about0.70 ("f = 1 - P/IOO). Volume fraction is-the ratio of the volume of actual bone tis-sue to the bulk volume of the specimen, which includes the volume associated withthe vascular pore spaces, but iguores the presence of lacunae and canaliculi.Trabecular bone has a volume fraction rarely greater than about 0.60, so the differ-ence between cortical and trabecular bone is fairly distinct. Porosity ..of adult humanfemoral cortical bone, foi'e~3JD.P\e, can vary from as low <\s 5 percent.at age 20 up toalmost 30 perfCIlt at age·8o;:,.Poi6sity of trabecularbone y:ujes· from about 50 per-cent in the YOUJIgadult tempeal head up to about'9y~pe~c!!rit In the-elderly vertebra.,'. '," ...", ".~,- .'.',',- .. :'~~~f;:-<- ~t... ." . ~<'i,'~~:' .;.

Volume Ftactiol;1and pensityMaterial prope.rties ~f bone, particularly stiffness an:'a:sqength, afe strongly dependenton the volume fraction and density. A variety of measures are used to describe bonedensity, some of which describe the volume fraction.andothers of which do not.The terminology can be confusing. The most common measures of bone density aretissue density and apparent density, which can be obtained for hydrated, dehydrat-ed, or deorganified' bone. Tissue density, Ptiss, is defined as me ratio of mass to vol-ume of the actual bone tissue, not including any vascular porosity. It is similar forhydratedcorticaland trabecularbone, varies little, and is about 2.0 g1cm3• The den-siry measuresfor dehydratedbone, bone that is heated for 172 day:Sat 6Q:.,'lQ~C toremove all water, are termed ~drY" densities. . . . ' .. ' .. c· . /"""

Apparent den~ity, P'~P'is defined as the ratio of the mass of bOn~~tissue to thebulk volume of the' specimen, including the volume associated with the vascularpore spaces. It is easier to. measure than volume '~aCtion, since it does not 'requireuse of thin sections or Archimedes' principle; for thisreasonitis a commonly usedindicator of boneporosity, Volume fraction, tissue density, and apparent densitiesare related as follows:

Papp = Ptiss "f

.[

Section 3.4 Elastic Anisotropy 79

. -,» The apparent density of hydrated human cortical bone is about 1.85 g1cm3

-and vanes Iittle from site to site. In contrast, the average apparent density of trabec-ular bone depends grea~ly on anatomic sit~'.It is as low as 0.10 g/crrr' for the elderly.spine, ab~ut 0.30g/cm for the human tibia, and up to about 0.:50 g/cm ' for theioad-bearmg portions of the proximal femur, After skeletal maturity (around age.QS};'trabecular bone apparent density decreases with aging at a rate of about

. 32percent per decade..Mineral content (or·minerallorgartic content ratio) also substantially affects me-

charucal properties ?~bone. Ash densities (tissue and apparent) can be obtained when.the bone ISdeorganified by ~eating it in a furnace for 24 hours at 700°C; thereby re-"!ovmg anw~ter and orgaruc ma~erial. .Th,; ratio of ash weight to dry weight is ofren:used to de~be. ~e perc,entage mineralization, or ash content, of bone tissue, regard-.ll~s of wh~er It IScorticalor trabecular bone. The percent mineralization increases~th ag~ during skeletal growth (through age 20-25),' but does not appear to changelWlthagmg for adults.

The wider variation in density. for trabecular than' cortical bone results in afm~ch gr~ater heterogene~ty in its material properties compared with cortical bone.:G~ven·thls heterogen';lty, It is rarely adequate to discuss the biornechanical properties;0£ trabecular bo~e wI~out..reference to Its apparent density. (or volume fraction).[8; ',"': Another differentiaror between cortical and trabecular bone is the scale of the,dd~mant por~sity. Cortical bone is a solid-like structure that contains a series ofJV0Ids.havmg,d.lmenslO~s about 200 microns or less (Haversian canals, Volkmann's:1)3nals,.resorption cavities, lacunae, and canaliculi). Trabecular bone is a network of'small; mterconnected rods and plates, ca~led trabeculae, with relatively large spaces. ~tween th~. Individual trabeculae contain few'of the voids (lacunae and cartalicuIi-:o,hly.sometun~s do. they contain Haversian .canals) .that .are contained in cortical

: :bon~ .. How~ver, typical t~~knesses of individual trabeculae are in the range' of4~~300 nucrons, and typlcalmtertrabecular spacing is on the order oiSOO-lSOO:m~crons. Both of these parameters depend.heavily uponage and anatomic Site~~~fgufe.3.5}. Therefore, the porosity oftrabecular bone (typically' 50. to 9.5 percent)~s,dommated by the spaces between individual trabeculae, not the voids within the~abecuIae. As the volume fraction increases, the architecture tends to be more plate-''!ke; It I~more rod-like at lower volume fractions, The space between trabeculae is,-?Ue?With marrow in vivo, which has a negligible mechanical role; except perhapsin high energy loading.

3.4 I Elastic Anisotropy

There is a preferred orientation m the microstructure o{bOne: In cortical bone, thepreferred.onentaoon ISdete~ed by the generally parallel assembly of the osteons,whereas 10 trabecular bone It IS determined by a predominant aligument of meplates and rods. As result' of this orientation, bone is an anisotropic material. Likemost bIOlogical materials-for example, wood-the preferred orientation is calledthe gr Ani . . .res am. sotropic material properties depend on the direction ofloading with

. peer to the material or grain axes. Common engineered materials such as met-als and plastics can often be considered isotropic (their mechanical properties are

•80 Chapter 3 TIssueMechanics I: Bone

FIGURE 3.6 Principal ma-terial coordinate system foran orthotropic material. Thiscoordinate system (right) isaligned with the mutuallyorthogonal "grain" axes ofthe material's microstructure(left).As a class of anisotropicmaterials, orthotropic materi-als have their grain along threemutually perpendicular axes.

independent of direction), but even for these materials anisotropy arises in some cir-cumstances. For example, rolled metals have different properties in different direc-tions, and composites are often designed to take advantage of directionality. Formusculoskeletal tissues, anisotropy cannot be avoided. Unfortunately, descriptionof the properties for anisotropic materials is much more complex than it is forisotropic materials, since the mathematical description of the anisotropic propertiesare specific to the coordinate system. We will discuss some overall aspects ofanisotropy before we present the general theory in Section 3.7.

Principal Matenal Coordinate SystemThe first step in understanding anisotropic material behavior is to describe the de-gree of anisotropy. Many biological materials are orthotropic, for which the materialbehavior can be described with reference to three mutually perpendicular materialaxes (principal material axes) at each point (Figure 3.6). We will restrict our attentionto this degree of anisotropy or less.

The description of the principal material coordinate system typically variesfrom point to point within a structure (Figure3.7). In many cases,.it is convenient todescribe such spatial variations by using a curvilinear coordinate system. The lattercan be considered as a local coordinate system that accounts for the unique spatialvariation in microstructure orientation. If there is symmetry about anyone of theprincipal material coordinate axes, then the other two axes can be interchanged, andisotropy exists in the corresponding plane 'of these axes. This is called transverseisotropy, since the "transverse" plane is isotropic (Figure 3.8). For isotropic materi-als, all three axes can be interchanged; there is no preferred orientation.

One important issue in biomechanical analysis of tissues is to decide on the de-gree of anisotropy. This decision should be made in the Context of the analysis ob-jectives and the required accuracy of the solution. Oftentimes, it is necessary toperform a complete orthotropic analysis (see Section.3.8) in order to quantify theerror associated with making simplifying assumptions of isotropy; If the error iswithin the precision of the overall. analysis or.is otherwise considered to be 'suffi-ciently small, then it may be reasonable .ro ignore the anisotropy in subsequentanalyses. The advantage, of course, of assuming isotropy is that ·analysis is much ,simpler mathematically, and results are easier to interpret. One example of this is the

3

2

·'i.';

m·'·

. 1 2

'Ll.

2..trabec.ular ori~nt~tion in the proximal femur, which is highly complex, and s atialv~natlons exist ill the ~rienta.tion of the principal material coordinate s;stemb gure 3.7). The assumption of Isotropy maybe reasonable for preliminary analysesut ill some cases the elastic and failure behaviors of the structure m d d cri "

cally on anisotropic characteristics. ay epen CCItl-

Section 3.4 Elastic Anisotropy 81

FIGURE 3.7 Spatial varia-tions in the orientation of theprincipal material coordinatesystem can occur. In manycases, a local coordinate sys-tem in cylindrical coordinatescan be used to describe suchspatial variations.

E.

FIGURE 3.8 Transverseisotropy. The 1 and' 2 axes inthe "transverse" plane canbe interchanged. The 3 axisis the stiffest direction and isreferred to as the "longitudi-nal" direction. Adult humancortical bone (bottom right),for example, is often con-sidered to be transverselyisotropic, due to the longitu-dinal orientation of the os-reons (surrounded by cementlines, CL) andlor Haversiancanals (HC). The lacunae (L)do not have a longitudinalorientation.


3.5 I Material Properties of Cortical BoneAs a consequence of its compositevosreonal microstructure, the material propertiesof cortical bone are anisotropic and inhomogeneous (vary with spatial location).These properties depend on loading rate, and when loaded past the yield point, cor-tical bone shows behavior characteristic of plasticity, damage accumulation, creep,and fatigue. Thus, under typicalloa:ding, its behavior is viscoelastic or viscoplasticand can be further modified if damage occurs. For many practical stress analysis ap-plications, however, conical bone can be regarded as an elastic 'material over somerange of loading.

AnisotropyHuman cortical bone is generally assumed to be transversely isotropic; meaning thatit has one primary material axis (the longitudinal direction) and is isotropic in theplane perpendicular to this axis (the transverse plane, Figure 3.8). The longitudinalaxis is generally aligned with the diaphyseal axis of long bones. Conical bone isboth stronger and stiffer when loaded in the longitudinal direction, compared with ~the radial or circumferential directions. This structure efficiently resists the largelyuniaxial stresses that develop along the diaphyseal 'axis during habitual activitiessuch as walking and running. , ,,' .' . '

As shown in Table 3.1, a total of five independentmaterial constants are re-quired to describe the transversely isotropic elastic properties of conical bone (forexample, two Young's modulus values, one shear modulus, and two Poisson's ratios),'whereas only two independent material constants ate required for isotropic behavior.The longitudinal Young's modulus of human cortical bone is in the range of about10-22 GPa and is about one-fourth that of aluminum. Its strength properties arealso anisotropic (Table 3.2). ' ':'

, In addition to having anisotropic elastic and strength' properties, corticalbonealso has asymmetric strengths; it is stronger in compression than: tension for eachprincipal material direction (Table 3.2). The longitudinal compressive strength of170-210 MPifar exceedsthat of-engineering construction materials 'such as con-crete. The.strength to modulus ratio forcortical bone is about 1.12 percent and 0.78percent for longitudinalcompression and tension, respectively. COmpared with high:performance engineering JIleial:alloyrfsuch as Aluminum 6061- T6 and Titanium 6AI-4V, with corresponding ratios of about 0.45 percent and 0.7J percent, respectively, it

Longitudinal modulus (MPa)Transverse modulus (MPa)Shearmodulus (MPa)Longitudinal Poisson's ratioTransverse Poisson's ratio

17,00011,500 '

3,3000.46 ,0.58

INore that these properties are referred to the principal material coordinate system;Source: ReiUy and Burstein (1975) ] Biomechanics 8:393-405.

TABLE 3.2

Section 3.5 Material Properties of Cortical Bone 83

Transverse' (MPa)

TensionCompreSSionTension

133193,51

13368

.,~is seen that cortical bone has a relatively large strength to modulus ratio. In that,:i·, sense, it is a high-performance material, particularly in compression. ,, 21',> When loaded to failure in a monotonic test, human cortical bone exhibits an

,,;'0' ihiti.al linear elastic behavio~ a, marked yield point, and failure at a relatively lowc') s~raln le~el (FIgure. 3.9). Unlike the ultimate stresses, which are.higher in compres-, ; slpn;.ulttmate strains ~e higher in tension for longitudinalloadipg. These strains

,(i;!nibe ••u~ to 5 percentin.young adults and fall to less than 1 percent in the very eld-'" ;erly: cortlcal.bone b~comes m~re bri~le with aging. In contrast to its longitudinal, '. tensile .beh~vlOr, cortl~l bone IS relatively brittle in, tension for transverse loading, and bnttle m compression for all loading directions. Cortical bone is weakest when

loaded transversely in tension and is also weak in shear.The asymmetry and anisotropy of strength have' practical clinical relevance.

For example, transverse tension can be generated when' large intramedullary im-plants, such as taper~d uncemented hip stems are driven too far during surgery intothe, femoral dIaphYSISand produce circumferential or hoop stresses. If these stressesbecome too great, then the bone cracks longitudinally. When this occurs, it is treatedby S~PP?~g. the bone externally. with circumferential tensioned wires: and weightbearing IS ~lted until the crack IS healed. Development of large transverse tensilestressrarely, If ever, occurs during normal physiological behavior.

Shear (MPa)Compression

INote that these properties are referred to the principal material coordinate system.Source: ReiUy and Burstein (1975) ] Biomechanics 8:393-405.

150

100

'i'SO

~ 0.,.,-SOeg;

-100

-1SO

-200-2 -1

------

tension

--- Longitudinal

-- Transverse

o 2 3 4Strain ('¥o)

FIGURE ).9 Typical stress-srrain behavior for humancortical bone. The bone isstiffer in the longitudinal di-rection, indicative of its elasticanisorropy. It is also srrongerin compression than tension,indicative of its strength asym-metry (modulus is the samein tension and compression).Cortical bone is relatively duc-tile for longirudinal tension,but is brittle in aU other load-ing modes.

84 Chapter 3 1issue Mechanics I: Bone

Because of the complex microstructure of cortical bone, it is clear that ,simpleisotropic and symmetric criteria such as the von Mises criterion cannot, descnb~ th.eanisotropic and asymmetric strength properties of this tis~ue. The Tsal-_Wu cnten-on commonly used for fiber-reinforced composite materials, works quite well forcortical bone although 'it requires 7-12 constants, depending on the degree ofanisotropy assumed (transversely isotropic vs. orthotropic). This criterion takes intoaccount the difference in tensile and compressive strengths, as well as the low shearstrength with respect to the tensile strength, and the anisotropy. It is the most suit-able criterion available for cortical bone.

For an orthotropic material loaded in its principal material coordinate system,the three-dimensional quadratic Tsai-Wu multiaxial failure criterion can be ex-pressed as

F,-<Ti + F;,-<T,-<T; = 1, j = 1,6; i = 1,6 ("contracted" tensor notation)

where a, are the stress components and 1) and F;; are the experimentally ~eri~edTsai-Wu coefficients. For the principal material coordinate system, the a, indicesj = 1-3 correspond to normal stresses: 0'11> 0'22, 0'33" respectively; j ~ 4-6 corte-spond to shear stresses: 0'12, 0'23, 0'13' When the coe~icients of odd-P?wer terms inshear are set to.zero to ensure sign-independent solutions for shear failure--and re-alizing the symmetry of the 1); coeffiCients-this expression is expanded as follows:

1)0'1 + ~0'2 + F.J0'3+1)101 + ~01 + F.J3~+21)20"10'2 + 2,1)30'10'3 + 2~30'2O:3

+F44<ri + ~5c?s + F660'~ = 1" '

Mathematically, this is a .quadratic equation, ellipsoidal to' be specific. The.nine coefficients, are found from uniaxial tests (tension, compressionj .and torsion,performed on longitudinally and transversely oriented specimens); and the re.ma_in-ing three ,~strength interaction" coefficients (1)2,1'13, and ~3) :ue found from biaxial,triaxial, or off-axis uniaxial tests. Let's derive an expression for the ,Pi 'and 1)1coefficients.

Section 3.5 Material Properties of Cortical Bone 85

HeterogeneityWhile it is often appropriate to assume average properties for cortical bone, asshown in Table 3.1 and Table ~.2.it may be necessary in s,ome cases to account forthe heterogeneity that can arise from variations, in microstructural parameters suchas porosity. For example, for stem stress analysis in bone-implant systems usingcomposite beam theory, since the modulus of the metal is,much greater than that ofthe cortical bone, ±20 percent variations in the modulus .of the cortical bone willnot affect substantially the calculated stem stresses. However, if there is no implantand the, focus is on bone stress=for example, in a study of bone fracture mechanics--errors in theassumed material properties of the bone become more important, ",,' As described earlier, cortical porosity, which is due primarily to the, variationsin the number, length, and diameter of Haversian and Volkmann's canals, can varyfrom less than 5 percent to almost 30 percent and is positively co~related with agebecause the bone becomes more porous ~ith aging. Both modulus and ultimatestress can be reduced by 50 percent when porosity is increased from 5 percent to 30percent (Figure 3.10). Thus, cortical bone properties for specific individuals depend

j :;;~i'

FIGURE 3.10 Dependenceof the ultimate tensile stressof human cortical bone onvolume fraction (expressedas a percentage). Ages of thespecimens were in the range20-100. Redrawn fromMcCaldenetal. (1993)J BoneJt Surg 75A:1193-1205.


FIGURE 3.11 Modulus vs.calcium content (in mwg ofdehydrated bone tissue) forcortical bone taken from 18different species. Redrawnfrom Currey ·(1988) J Bio-mech 21:131-139.

35 •30.. • ·a.. 25 ·Q_ •

••• 0

'" ....." 20 .'.....:; .....,'"0 .."::;: 15 .... ,.'"

."0> .. - •c

10e . .....

" .... -.-.. •§? . ..... • •5 • e •. •....0150 200 250 300

Ca (mglg)

upon porosity. The average values shown in Table 3.1 and Tabie 3.2 provide rea-sonable properties for "typical" individuals. .

It is-well established that, over multiple species in which there is a farge'varia-rion of mineralization;stiffue:ss and-strength increase with increasing mineralization(the ratio of ash to dry weightsin Figure 3.11). However, mineralization does notvary much in adult humans, and therefore normal variations in mineralization donot appear to play an important role in modulus or strength ofadult human' corticalbone. This illustrates an important observation, Even though a tissue constituentcan substantially affecfrnaierial properties over it range ofindividuals, 'it may havelittle effect onthe subpopulation ofinterest. . .

Section 3.5 .Material Properties of Cortical Bone 87

Fatigue, Creep, and ViscoelasticityCortical bone also exhibits fatigue (Figure 3.12a) and creep (Figure 3.12b) and has agre.ater resistance to failure in these modes in compression than tension. Fatigue prop-e~es are ~ormally.eXpressed on a traditional S-N curve (stress vs. number of cycles tofa~l~re!, J~st as wlt.h metals, but stress is usually divided by Young's modulus tommirmze mterspecimen scatter of the data. Interestingly, for cortical bone, whenthe fatigue and creep behaviors are expressed as functions of stress/modulus vs.time-to-failure, experimental scatter is reduced, fatigue life is independent of frequency(O.2-~.O Hz.range), and substantial similarities appear between the fatigue and creepbehaviors (Figure 3.13). This suggests that levels of strain (=stress/modUlus) determine


'i?Q.e 0',

b.;.,i!!

US 0'2

N, N2Number ofCycles, log N

(a) (b)

RGURE 3,.12 (~) Typicalfatigue ~N curve.TIlls is usuaUyplotted with the ordinate on-a I,OSscale,where the relationship is linear, but which corresponds to the foll?~ng type of nonlin-ear relationship between number of cycles t() failure; (Nf) and ~pph~d ~onstant stressa: l't = A.,-l'. According to this model of mat~rialbehaV1o~the mate~ will fail a; 1>1; cy~es.ofstress U'. (b) Typicalcreep response showing the three classicalstages. FIrSt,there IS a rapid in-

crease ;;. strain immediatelyafter the constant stress is applied; second~there is ~ steady r~teof increase of strain (an approximately constant "creep-rate"); 'ami-third, there IS a rapid In-

crease in strain just before the specimenfractures.

'0 Comp - fatigueI] 'Comp - creep

0.010

I].[]01]

.' Tens - fatigue• ' 'Tens - creep

Miles (fatigue)10 100 1000

t t t..:::I'sIl 0.006

~i0.004

0.002.

~ []. . i ~. D

• .' ~... . 0° ~ 9 C D..rigorous~'~-;,.~;.~,I~it.~,~,=.-••~.~o'---~exerciSe .. 0 0

running ----------------" •walking --------------

Time 10 Failure (seconds)

R~URE 3,13' Fatigue and creep behaviors of human cortical bone vs, time to failure',Forfatigue loading, the ordinate 011 this grapb 'can be converted to number of cycles by multiply-ing the jime to failure by the frequency, which is typically one. cycle.per seco~d for ?ormalwalking. Note that both creep and fatigue resistances are lower m ~~on, consistent WIthrno-notonic behavior.Data from Carter et aI. (1981) Acta Orthop ScanJ 52:481-490 and Caler andCarter (1989) ] Biomechanics22:625-635.

I]

Section 3.5 Material Properties of Cortical Bone

these failure properties and that the underlying fatigue and creep mechanisms arerelated.

The fatigue and creep properties shown in Figure 3.13 were obtained from de-vitalized bone specimens, in which, obviously, no biological healing could occur.Because bone cells repair fatigue damage in vivo, these fatigue life values are lowerbounds on the in vivo fatigue life. Therefore, it is unlikely that high cycle (low stress)fangue failure occurs in vivo, since the resulting fatigue damage would be healed bio-logically before large enough cracks could develop that would cause overt fractureof the bone. Low 'cycle fatigue (stress fractures), however, can occur when higherlevels of repetitive stress are applied over shorter time intervals, such as in marching

, military recruits and marathon runners. 'When cortical bone is loaded to its monotonic yield point, but not fractured,

and is then unloaded, permanent residual strains develop (Figure 3.14) similar toductile metal behavior. When cortical bone is loaded -beyond its yield point, un-loaded, and reloaded, its modulus is reduced (Figure 3:14a). This is evidence ofdamage, something that does not occur in metals where the modulus after plasticyielding is the same as the initial modulus. This complex viscoplastic, damaged mate-rial behavior is difficult to account for in stress analyses, but it probably has signifi-cant biological consequences. As the surrounding bone matrix permanentlydeformsand sustains damage, 'cells may be altered and' a biological response may be induced;Which prompts the bone cells to repair the daniage done to, the bone matrix.

Cortical bone is a viscoelastic material (Figure 3.15). Its modulus and strengthincrease as the rate of loading is increased. Over a six-order-of-magnitude increasein strain rate, modulus only changes by a factor of two, and strength by a factor ofthree. Thus, for the majority of physiological activities, which tend to occur in a rel-atively narrow range of strain rates (0.01-1.0 percent strain per second), cortical

160

[Time

Sirain(%) Load withconstant stnJss

Unload

(a) (b) ,

RGURE 3: 14 ., (a) Damage behavior of cortical bone. Evidence of microstructural damageis seen by ·the reduction in modulus that occurs when the specimen is reloaded after initialyielding. 'From MT Fondrk (1989), Ph.D. dissertation, Case Western Reserve University.(b)Viscoplastic behavior of cortical bone. When a low stress is applied to the bone, the strainremains constant over time, and there is no permanent deformation after unloading. As themagnitude of the stress is increased, the rate of creep increases, and a larger permanent defor-mation exis~ after unloading. From Fondrk et al. (1988) ] Biomechanics 21:623-630.

89


400FIGURE 1.15 Strain ratesensitivity of cortical bone forlongitudinal tensile loading.This is evidence of viscoelasticbehavior, although the effectis not substantial. Typically,modulus and strength in-crease only by factors twoand three, respectively, as theloading rate is increased by sixorders .of magnitude. FromMcElhaney (1966) J App/Physio/21:1231-1236.

FIGURE 1.16 (a) Reduc-tions of human cortical bonemechanical properties withage. Modulus is not reducedmuch, if at all, whereasstrength is reduced more, at arate of about 2 percent perdecade. From Burstein et al.(1976) J Bone It Surg 58A:82-86. (b) Ultimate strain de-creases markedly with age, at arate of about 10 percent of itsyoung value per decade. FromMcCalden et a1. (1993) I BoneIt Surg 75A:1193-1205.

300

100

l/sec__ --O.I/sec

0.5 1.0Strain (%)

1.5 2.0

Section 3.6 Material Properties of Trabecular Bone

bone can reasonably be assumed to behave elastically. H0;-vever, theincreased stiffnessand strength properties and the tendency .toward more brirtle behavior are importantin high strain-rate situations such as high-speed trauma and perhaps during falls.Analysis methods for viscoelastic materials are discussed in Chapter 4.

Aging and DiseaseAging and disease also affect the mechanical properties of cortical bone. Modulusvaries little, if at all, with age, but tensile ultimate stress decreases at a rate of approxi-mately 2 percent per decade (Figure 3.16a). Perhaps most importantly, tensile ultimatestrain decreases by about 10 percent per decade, from a high of almost 5 percentstrain at age 20-30 years to a low of less than 1 percent strain beyond about 80 years(Figure 3.16b). Old bone is more brittle than young bone. Thus, the energy to frac-ture, given by the total area under the stress-strain curve before fracture.Js much lessfor old bone than for younger bone. Fracture toughness also decreases with aging.

180........ modulus 25

f

-<>- strength

t f20_

• t I ~I 15 !2...rr t ::J

t I-t--, 10~·5 ::;: .

'lel60.s::;

g>140

~ 120

~E .1005

80 020 30 40 50 60 70 80 90

Age (years)

(a)

5 Y =423 - 0.033 X, R" = 0.58

4

O+--+--+-~~'_~~~o 20 40 60 80 100 120

Age (years)

(b)

~.6 I Material Properties of Trabecular Bone.,

As discussed earlier, trabecular bone is a highlyheterogeneous material, and its ma-terial properties vary accordingly. From a biomechanical perspective, the most im-portanr microstructural parameter for trabecular bone is its apparent. density, incontrast to the tissue density of the individual trabeculae. Similarly, we distinguishbetween apparent {continuum} and tissue properties of trabecular bone. The contin-uum properties of trabecular bone primarily depend upon the architecture and, to alesser extent, on the tissue properties.

To treat trabecular bone as a continuum, its dimensions must be on the orderof 5-10 mm or greater. Smaller specimens of trabecular bone may have to be treatedas structures for which concepts of material properties such as Young's modulus donot apply, since they depend upon the size of the specimen. This concept of scale andcontinuum V5. microstructural behavior is critical when analyzing the mechanicalbehavior of trabecular bone and most other biological tissues. In the discussion thatfollows, the trabecular bone properties ate continuum values, which have been de-

•termined with the use of appropriately sized test specimens.

Apparent DensityThe compressive stress-strain behavior of trabecular bone (Figure 3.17) is typicalof a class of porous materials called cellular solids (recall Figure 3.5). It displays

91

92 Chapter 3 Ti~ue Mechanics I: Bone

40 -- Compression---- Tension

0.5 1.0 1.5 2.0' 2.5 3.0Strain (%)

(b)

1.0 2.0Strain (%)

(a)

3.0

FIGURE 3_17 Srress-strain behavior for compressive and tensile loading of bovine (left)and human vertebral (right) trabecular bone, showing the wide range in strength that is typi-cal for trabecular bone. The bovine bone here is 10 times stronger and.stiffer than-the' humanvertebralbone. X indicates tensile fracture. For compression, the curves have been truncated.Left: Keaveny et al. (1994) J Biomechanics, 27:1137-1146; and right: Kopperdahl andKeaveny (1998)J Biomechanics, 31:901~08.

an approximately linearly elastic region followed by a local peak, and then astrain-softening or plateau region 'of near constant stress with increasing strain.Tensile behavior is much more brittle, with fracture occurring at relatively lowstrains. Most importantly, for trabecular bone, the 'stiffness' and strength dependon its apparent density and can vary by two orders of magnitude within the samemetaphyseal region.

Becauseof the large variation-in apparent density for trabecular bone, its me-chanical properties cannot generally be described by average values. This makesanalysis of structural problems with trabecular bone more difficult, than those forcortical bone, since it is usually necessary to account for variations in apparent den,sity within a region. of trabecular bone. As usual, however, the reqnired accuracy ofthe properties depends on the precision of the analysis at hand. A two-fold variationin trabecular modulus from 200 to 400 MPa may have. little effect on calculatedstem stresses in a bone-implant system, but would be crucial in an analysis of osteo-porotic spine fracture for which the trabecular bone can be the primary load-bearingcomponent. Regardless, it should be realized that there 'are substantial variations inthe. mechanical properties of trabecular bone across anatomic sites, and effortsshould be made to use site-specific average values if available. Table 3.3 gives suchproperties for the tibia, femur, and vertebra.

Depending on the trabecular architecture (the general arrangement, shape, anddimensions of the individual trabeculae for a particular specimen), modulus or strengthvary with apparent density.by a linear relationship or a power law relationship with anexponent of 1-3. For example, linear relations between strength and volume fractionoccur for different sites, the slope typically being lower for the lower density site(Figure 3.18). This is due to underlying differences in architectural structure: Theplates for high-density .bone are more efficient than the rods of lower density bone.But a squared power law fits the pooled data well (beware statistical artifacts!).

"''''00 r-,00

00

'"oM

'"o

o"!,....

\0['.. 0 0v)rr)" vi ti')NN N M

00r-,o

ojI

'"....,...;I

'",...;'"

93


FIGURE 3.18 Dependenceof compressive on-axisstrength on apparent densityfor trabecular bone for twodifferent sites: bovine tibial(BPT) and human vertebral(HVB) trabecular bone. Thedifference in slopes in the lin-ear relations for each site isdue to the different architec-tures, being mainly plate-likein the bovine bone and rod-like in the human vertebralbone. When all the data arepooled, there is a strongsquared power law relation-ship, with ,2 = 0.94. (BoneMechanics Handbook, EditorSC Cowin, CRC Press BocaRaton, 2001.)

FIGURE 3.19 Age-relatedreductions in compressivestrength of human femoraland vertebral trabecular bone.Note that for any given age,there remains much scatter inthe data; that is, age is not agood determinant of behavior.Data from Mosekilde et al.(1987) Bone 8:79-85 andMcCalden et aI. (1997) JBoneJt Surg 79A:421-427.

-0- Human vertebra: Y = -1.46 + 21.9X. ,2 = 0.71Bovine 1ilia: Y= -45.9 + 137.7X;,2 =·0.76

Y= 215.5x"-31.,2 = CP6Pooled: Y= 100.4)(2-'·.,2 = 0.94

40

'ie30U>

"!!!W20~E.,5 10

Apparent Density (glcm"l

Vertebra: Y =·5.34 - O.054X, ,. = 0.66

Femur: Y= 15.1 - O.I09X." = 0.51

16 0

o14

l'2e 10

B

Age (years)

Both architecture and apparent density change with aging and disease, resultingin large age-related reductions in strength (Figure 3.19). Table 3.3 gives relationshipsberween strength and density for the tibia, femur, and vertebra. The relationships areslightly different berween sites because of the·different microstructural architectures ofeach site, although reasonable performance can be obtained by using relationships forthe pooled data.

In general, average modulus values' .per anatomic site for trabecular bone varyfrom as low as about 3000 MPa in the elderly spine to over 3000 MPa in the load-bearing portions of the femoral head. Ultimate strength values for individual specimensvary from 0.1-40 MPa, typically being a factor '1001ess than modulus (Table 3.4). Bycontrast, the yield strains are relatively uniform. For higher density trabecular bone,

'D .,.,'D"

99a.....,0 ....00

r-, 'D

.... '"99............ ....00

o:3±o

a..",

"''''00.... ""' ....00

o.,.,o

95

96 Chapter 3' Tissue Mechanics I: Bone

*RGURE 3.20 Compressiveand tensile yield strains vs.human anatomic site for tra-becular bone specimens test-ed on-axis. Compressive yieldstrains were always greaterthan tensile yield strains. Barsdenote ±1 SD. The numberswithin the bars show thesignificant (p < 0.05) cor-relation coefficients againstapparent density, confirmingthat in most sites' there was nodependence on density, and atmost only a weak correlation.From Morgan and Keaveny(2001) J Biomechanics 34:569-577.

1.0• Compression D Tension

ce.00-ca;>=

Vertebra Trochanter FemoralNeck

ProximalTibia

compressive and tensile values are about 0.8 percent and 0:6 percent, respectively(Figure 3.20): For lower density bone, such as in the vertebral body, compressiveyield strains are lower at about 0.7 percent, due presumably to underlying large de-formation bending or buckling-type failure mechanisms of the individual trabeculaethat cat! occur when the density is low. In this situation, the individual trabecular tendto be longer and thinner than for high-density bone, and' according to engineeringbuckling theory, this may promote buckling as a failure mode; compared with mate-rial yielding. Buckling or excessive bending do not appear t<>occur much for high-density bone (volume fraction> 0.25) and do not occur, for tensile loading. It ismost interesting, however, that the variation in yield strain Within sites is very smalland is even small across sites. As seen next, this greatly facilitates statistical analysisof trabecular bone failure properties.

AnisotropyLike cortical bone, trabecular bone is an anisotropic material; Modulus and strength canvary by as much as eight times, .depending on the direction of loading compared withthe principal material direction, which coincides With the main trabecular orienta-tion. For some sites, such as the vertebral body of the spine, the general anatomicdirections (e.g., the inferior-superior axis) coincide with themain trabecular orien-tation. However, for most sites this is not the case, and the main trabecular orienta-tion is oblique to the anatomic axes, The classic example of this is in the proximalfemur, where the trabecular orientation varies with location and is never alignedwith the anatomic axes (recall Figure 3.7). In these cases, a local coordinate systemcan be used to describe the principal material directions. In the neck region of theproximal femur, for example, the trabecular orientation tends to follow the orienta-tion of the neck axis, but it doesn't do so exactly. Thus, -it can be technicallychal-Ienging to describe the oftentimes spatially varying principa] material directions fortrabecular bone, and this complicates whole-bone structural analysis. Interestingly

Section 3.6 Material Properties of Trabecular Bone

~20006 1500'":0:J 1000'8::;: 500

oTension Compression

Loading ModeTension Compression

Loading Mode

e- FIGURE 3.21 Dependence of yield strain (left) and Young's modulus (right) on specimenorientation in tension and compression, for dense bovine trabecular bone. For the off-axisorientation, the specimen axes were offset 30-40· from the principal trabecular (on-axis) di-re~ion. Error ba~s show ±1 SD. In contrast to the yield strains, which were isotropic, butasymrnernc, elastic modulus and yield stress (not shown) were clearly anisotropic. FromChang et a!. (1999) J Orthop Res 17:582-585.

enough, the yield strains for trabecular bone appear to be isotropic despite the substan-tial anisotropy of the elastic and strength properties (Figure 3.21). This can simplifyfailure analyses, as described below.

Due to the substantial heterogeneity in modulus and strength of trabecularbone, it should now be clear that the question "What is the modulus (or strength) oftrabecular bone?" is not easily answered. Details such as anatomic site, species, ·age101l~ingdirection, and disease' state must be specified; in structural analyses of wholebones, one s?o~d use average trabecula~ properties that are appropriate to the spe-cific anatomic site of the patient population urider analysis. ..

Failure, Fatigue, .and CreepThe multiaxial failure behavior of trabecular bone is an important aspect of its 'in vivofailure behavior, because failure usually occurs during some type of trauma where loadswill be multiaxial or oblique to the principal material direction. Criteria such as vonMise~ stress failure theory should not be used when shear stresses are high, since thesecntena do not account for the relatively low shear strength of trabecular bone, com-pared with its tensile and compressive strengths. Instead, a criterion based on maxi-mum strains should be used. Such a criterion is advantageous because it is relativelyindependent of variations in apparent density-at least within an anaromic.site=and is'also isotropic. The criterion works as follows: Strains for any state of loading are con-verted into principal strains. When either themaximum or minimum principal strainexceeds the maximum allowable on-axis uniaxial strains, then failure is assumed tooccur..The on-axis data shown in Figure 3.20 can be used to estimate the maximum al-lowable tensile and compressive failure strains for the anatomic site under analysis.Note that, despite this yield strain criterion being both homogeneous and isotropic,strength remains asymmetric, since yield strains are lower in tension than compression.

. Like cortical bone, trabecular bone exhibits the phenomena of creep and fatigue(Figure 3.22). Trabecular bone also undergoes modulus and strength reductions if it isloaded beyond its yield point, unloaded, and reloaded (compare Figure 3.23a with

97

The Masculoskeletal System & Tissue Mechanics

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