Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy

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TRABECULAR BONE DENSITY AND LOADING HISTORY: REGULATION OF CONNECTIVE TISSUE BIOLOGY BY MECHANICAL ENERGY D. R. CARTER, D. P. FYHRIE and R. T. WHALES Mechanical Engineering Department. Stanford University, Stanford. CA and Rehabilitation Research and Development Center, Veterans Administration Medical Center, Palo Alto, CA, U.S.A. Abstract-The method of considering a single loading condition in the study of stress/morphology relationships in trabecular bone is expanded to include the multiple loading conditions experienced by bone in ciro. The bone daily loading histories are characterized in terms of stress magnitudes or cyclic strain energy density and the number of loading cycles. Relationships between local bone apparent density and loading history are developed which assume that bone mass is adjusted in response to strength or energy considerations. Three different bone maintenance criteria are described which are formulated based upon: (I) continuum model effective stress, (2) continuum model fatigue damage accumulation density, and (3) bone tissue strain energy density. These approaches can be applied to predict variations in apparent density within bone and among bones. We show that all three criteria have similar mathematical forms and may be related to the density (or concentration) of bone strain energy which is transferred (dissipated) in the mineralized tissue. The loading history and energy transfer concepts developed here can be applied to many different situations of growth, functional adaptation, injury, and aging of connective tissues.

Large, heavy individuals who are physically active tend to have denser and stronger bones than frail, sedentary

KOhlENCLATURE stress tensor, uT= [u,, uI, u,. u4, Us. u,] (contracted notation) compliance matrix angular orientation of principal material directions w.r.t. principal stress er < [apt] continuum model effective stress continuum model energy (effective) stress continuum model ultimate efTective stress continuum model cyclic effective stress range continuum model strain energy density continuum model elastic modulus function relating 17~to 5 true tissue effective stress true tissue ultimate eITective stress true tissue strain energy density true tissue strain energy density dissipated per loading cycle bone apparent density cortical bone apparent density bone volume fraction daily bone maintenance stimulus fatigue damage fraction number of loading cycles number of cycles to fatigue failure fatigue constants stress exponent constant energy exponent constant subscript designating a specific loading case INTRODUCTION

It has long been recognized that a relationship exists between the severity of mechanical loading to which the skeleton is exposed during daily activities and the mass and strength of the bones (Thompson. 1919).

Received 23 May

1986;

in revisedform31 December 1986. 785

individuals. In addition to the density variations which occur among bones of different individuals, there are also complicated distributions of bone apparent density within bones of specific individuals. Roentgenograms of the proximal femora of four individuals are shown in Fig. 1. The general patterns of trabecular orientations and distributions of apparent density are similar for each of the bones. The magnitudes of apparent density at specific locations within the bones, however, are quite different as are the apparent densities of bone at the same site in different bones. Most researchers agree that the variations of apparent density within specific bones and among bones of different individuals are influenced by the loading histories to which the bone tissue is exposed. Previous investigators who have tried to explain the trabecular orientation and distribution of trabecular bone apparent density within a bone have considered a single ‘typical’ loading condition to which the body is exposed on a repetitive basis (Wolff, 1892; Meyer, 1867; Roux, 1880, 1881, 1895; Culmann, 1875; Hayes and Snyder, 198 1; Stone et al., 1984: Fyhrie and Carter, 1986). In the proximal femur this hypothetical load is often assumed to be that which is encountered during the single-limb-stance phase of gait. Such a loading condition creates a complicated stress distribution within the trabecular bone which correlates with trabecular morphology. Most have explicitly or implicitly assumed that bone apparent density distributions within a bone are optimized with respect to some measure of the local ultimate strength. That is, it is assumed that the ratio of stress to ultimate strength (stress ratio) is the same throughout the entire region

786

D. R. CARTER. D.

P. FYHRIE~II~R. T. WHALEN

of cancellous bone. Trabecular bone alignment has been shown to correspond to principal stress directions and consequently reduces the apparent density required to achieve a specific stress ratio. Fyhrie and Carter (1986) developed a formal expression of these assumptions in a single optimization principle in which bone anisotropy (trabecular orientation) and apparent density are adjusted to optimize some objective function. They also showed that optimization of objective functions other than stress ratio (such as strain energy density) could be used to predict trabecular morphology. Studies which consider a single load application have proved to be very useful in understanding the relationships between the form and function of trabecular bone. However, a better understanding of these relationships requires a more complete consideration of the nature of intermittent, repeated loading of bone during daily activities. One must consider the fact that during daily activities the bone is repeatedly loaded under many different loading conditions with a widely varying number of loading cycles. In expanding our considerations of bone stresses to include repeated loading conditions, it is appropriate to consider bone damage accumulation and fatigue safety factors as well as ‘static’ stress ratio in the bone. The goals of this study are to introduce approaches to predict bone apparent density using, (1) a genera1 method ofdefining loading history which encompasses the single loading strength optimization; (2)a technique based on fatigue damage accumulation; and (3) a bone strain energy density interpretation of loading history. We also show how the density of strain energy transfer (dissipation) in the mineralized tissue over some time period may relate to the morphology predictions made using these three methods. We hypothesize that bone functional adaptation is the consequence of a cellular response to the density of strain energy transferred to the biological tissues. The approaches introduced in this study may beconvenient methods for representing the density of energy which is transferred in an ‘average’day. The magnitude of this strain energy transfer density may be an important parameter in the maintenance of local bone apparent density.

QUANTIFYING

THE LOADIXC

HISTORY

In the analyses presented here, we assume that some relatively stable (homeostatic) situation exists between bone stress and local bone density. That is, the daily activity level of the individual has been constant for long enough that the skeleton is not experiencing significant bone loss or gain, Our goal is then to determine the relationship between the stress history and the current trabecular bone apparent density at a specific site. To make our analysis tractable, we must choose some characteristic time period over which we can summarize the relevant loading history. The time

period required to achieve significant alterations in bone mass is on the order of a day or a few days. We choose to accumulate loading effects over one day to calculate what we call the ‘average daily history’,. Since daily activities are assumed to be roughly equal from day to day, and bone homeostasis exists, the concept of an average daily loading history is appropriate. We assume that the loading history at a specific site provides a stimulus to bone metabolism. If a homeostatic situation exists, the magnitude of that stimulus will favor neither an increase nor a decrease in bone apparent density. Therefore, each region of bone has associated with it a bone maintenance stimulus which is necessary for homeostasis. Stress approach

The stimulus for bone maintenance can conceivably be influenced by many different mechanical factors including the cyclic stress (or strain) magnitudes, strain rates during loading, and the number of loading cycles. Because of the nature of normal activities and exercise patterns, the loading strain rates tend to correlate with the magnitudes of the imposed stresses. As a first approximation we will assume that the most important parameters of the loading history to consider are the magnitudes of the cyclic stresses and the number of applied loading cycles. We will further assume that the daily activities can be decomposed into histograms of load histories which consist of i discrete loading conditions which are each associated with ni cycles of load applications per day, For example, i = 1 may represent the activity of single-limb-stance during walking with n, = 5000 cycles. The activity of getting out of a chair may be represented by i = 2, n2 = 37 cycles. We make the simplifying assumption that the relative magnitudes and direction of the principal stresses are constant during the loading cycle (proportional loading) but may vary with each ith loading case. We further assume that the severity of the loading state created during each ith load pattern can be expressed as a single scalar ‘effective stress’ parameter, ci, which indicates the ‘magnitude’ of the cyclic stress tensor, oil for a certain theory of failure (or in our case, bone maintenance theory) (Fyhrie and Carter, 1986). One way that the effective stress, ci, is used is to compare it to the effective stress for failure, r?“,,.An example of this is the use of the von Mises effective stress for plastic yield analyses of ductile metals or the maximum principal stress in analyses of brittle materials. Using the effective stress, a summary of the average daily stress history can be assembled. One application of the daily stress history approach could be realized using finite element techniques to solve for bone stresses under various loading conditions. For example, Fig. 2 shows assumed loading conditions on the proximal femur during three different states (i = 1, 2, 3). The stress state Ui at any particular location can be calculated and the effective stress ci determined. This effective stress is associated with ni loading repetitions.

787

Trabecular

loading i=l

loading i=2

loading i=3

(i = 1. daily activities.

Fig. 2. Example of three different loading conditions

The stimulus required for bone maintenance is a function of the stress magnitudes and loading cycles as well as the current apparent density. In a homeostatic situation, we will assume (as a first approximation) that the daily stimulus to maintain a certain bone mass is related by a linear superposition of the stimuli created by each ith loading conditions. To calculate the stimulus in each ith loading condition, we do not know at this point the relative importance of cyclic stress magnitude and thenumber of loading cycles. We thus assume that a region of bone which is experiencing neither a net loss or gain of bone apparent density will be exposed to a constant daily stimulus S* that can be expressed as

or

(1)

where m is a constant. The value of m is a weighting factor for the relative dependence of the stimulus on stress ratio and loading cycles. If m = I, the importance of the stress ratio and the number of cycles are of the same order. Increasing values of m indicate increasing dependence of the stimulus on the activities which are associated with high stress magnitudes. One would expect the relations expressed in equation (1) to be valid only in some physiologic range of stress magnitudes encountered during normal activities and would not apply with impact stresses close to the failure stress. If we assume that the bone maintenance stimulus is constant everywhere, the bone ultimate strength is s?I20:8-3

789

bone density and loading hlstory

2, 3) which might be encountered

in different

given as c?“,,cc [Qlia:]““.

(2)

For a single loading to failure, the bone strength, culrr is approximately proportional to the square of the apparent density, p (Carter and Hayes, 1977b). The bone apparent density is therefore approximated as p a [CI~~~]“~“.

(3)

Notice that in the case of a single typical load analysis of previous investigators (i = I, n, = arbitrary constant), equation (3) degenerates to p a (6m)‘i2m,

(4)

p x .Z?’‘2.

(5)

or This is the same result as derived using the ‘typical load’ strength optimization principle introduced by Fyhrie and Carter (1986). Fatigue damage approach

Mechanical testing has demonstrated that devitalized cortical bone tissue has very poor resistance to fatigue fracture and that repeated loading causes the accumulation of diffuse material damage (Carter and Hayes, 1977a; Carter and Caler, 1985). Evidence of microscopic accumulations of trabecular fatigue damage have been well documented in clinical and experimental osteoarthritis (Harrison et al., 1953; Radin et al., 1973) and in osteoporotic individuals (Freeman et al., 1974). It has been hypothesized that fatigue damage accumulation may control or influence functional adaptation in bone (Frost, 1960; Radin, 1972; Carter and Hayes, 1977~; Burr et al., 1985; Carter,

790

D. R. CARTER. D. P. FYHRIEand R. T.

1984; Carter and Caler, 1985; Fyhrie, 1986). The fatigue damage approach developed in this section assumes that the daily damage fraction accumulated in trabecular bone is constant throughout the bone structure. The local apparent density has therefore been established to ensure that the safety margin for fatigue fracture is equal at all locations. The number of cycles to failure (Ni) for bone tissue exposed to a cyclic stress range, Aai, can be approximated by the relationship (Carter and Hayes, 1977c; Lafferty, 1978):

where A,6 = experimentally determined constants. During the course of a single day we will have many different loading magnitudes, ABi, each applied for a different number of loading cycles, q. Following the approach of Carter and Cater (1985), we use the Miner-Palmgren theory to calculate the daily fatigue damage fraction D, DF=x

0 :

day

N

i’

where Ni is the number of loading cycles required to cause failure at the ith stress level. The value of D,can vary between 0 and 1.0. When D, equals 1.0, fatigue fracture of the devitalized bone will occur. Combining equations (6) and (7), and assuming that the bone maintenance stimulus, S*, is the daily fatigue damage fraction, D,, we find S* a (I/&)

effective stress. With strain energy density, the apparent density was found to be proportional to the cube root of an effective stress squared. This difference can be reconciled when one considers that the continuum model for trabecular bone misrepresents the true stress and energy values of the mineralized bone tissue. Cancellous bone consists of both mineralized bone and non-mineralized (mostly marrow) spaces. the volume fraction, c, can be defined as the ratio of bone volume to total bulk volume (including nonmineralized components). In a continuum model of bone, the apparent stiffness matrix (or apparent elastic moduli) are approximately proportional to the cube of the apparent density (Carter and Hayes, 1977b; Fyhrie and Carter, 1986). The stored strain energy density, U, of the continuum model under moderate (approximately linear elastic) loading is in reality an ‘apparent’ strain energy density and is calculated as u = fu%(p,

&a,

(11)

where u = continuum model apparent stress; S = continuum model apparent compliance matrix; 8 = orientation of principle material directions relative to principle stress directions; p = bone apparent density. All of the strain energy is, however, stored in the mineralized bone tissue. Therefore the ‘true’ strain energy density in the bone tissue is much greater than the ‘apparent’ strain energy density calculated using the continuum model. To better reflect the energy stored in the mineralized bone tissue, we introduce the parameter

c n,Ac$.

u, = u/v,

dv

If the daily fatigue damage fraction is constant at all locations then (jut, a [Zr~b~:]*‘~.

WHALEN

(9)

If we again employ the fact that bone strength, a,,,, is approximately proportional to the square of the apparent density, equation (9) then implies that

Strain energy approach Fyhrie and Carter (1986) suggested that the trabeculae are aligned and the apparent density adjusted to optimize some objective function. Using a continuum model for the trabecular bone, two special forms of this objective function were considered for a single typical load application. By applying an optimization based on (I) strain energy density and (2) strength, relationships between the local stress and bone apparent density were established. Although the mathematical forms of these two special optimization approaches were identical, a different relationship between local apparent density and stress resulted due to the different influences ofapparent density on strength and energy. With strength optimization, the apparent density is proportional to the square root of an

or

(12) ub

=

$7Ts(p,

@a.

where Ub= average true bone tissue strain energy density; v = volume fraction of mineralized bone. We now believe that it is more reasonable to expect that bone optimization will be achieved with respect to the average true bone tissue strain energy density, ub, rather that the continuum representation of strain energy density U. The argument for using ub as the objective function is that one would expect cellular reactions to stress to be a result of mineralized tissue strain energy only rather than an average of strain energy over mineralized tissue and marrow spaces. The true density of mineralized bone can be considered, in the first approximation, asa constant (Gong et al., 1964, Carter and Hayes, 1977b). The volume fraction is therefore approximately proportional to the apparent density. Equation (12) can therefore be rewritten as &, = +(p,/p)u5(p,

‘)a>

(13)

where pr is the density of compact bone. Following a loading history approach comparable to that used in the previous section, we assume that the

791

Trabecular bone density and loading histork daily stimulus to maintain bone mass, S*, can be directly associated with the stimulus to the mineralized tissue and is given by S* X x fl,(Ci,i)‘* d”y or in terms density

of the continuum

model

(14) strain

energy

s*c&,,ut. Pk *OS

(15)

If we again assume that the bone maintenance stimulus is constant everywhere, the local bone apparent density is given as p r [ErrIU”]“k.

(16)

In the case of a single, ‘typical load’ analysis (i = 1, n, = arbitrary constant), equation (16) degenerates to (17)

PK u,

and the apparent density is directly proportional to the continuum model strain energy density. This means that the average true bone tissue strain energy density, lJ,, is constant throughout the bone. We (Fyhrie and Carter, 1986) have previously introduced the concept of an energy stress, 5Pnr,,,yr where c?&~

= 2E,,,U,

(18)

and E,,, is the average elastic modulus in the three principal material directions in an orthotropic material. Since the elastic moduli are approximately proportional to the cube of the apparent density, (19)

u a &JP’. Substitution yields

of this expression

into equation

(16)

Pa

C~ni~:~,,,lp'kl"k,

cw

P

Cxni6&-rgy,l"4kt

(21)

x

and if we let m = 2k, p cc [Zni6~~;rergy~‘i2m.

(22)

If we assume that our stress, (T,,,gp,, is the effective stress ci then equation (22) is identical to equation (3). The dependence of apparent density on loading history is therefore the same using either the stress ratio approach (equation 3) or the strain energy density approach (equation 16).

Comparisons The effective stress history approach to predicting apparent density (equation 3) incorporates the variable m which determines the relative dependence of the stimulus on stress magnitude and the number of loading cycles. The effective stress measure which is employed in the stress history approach is dictated by the failure criterion which is assumed. In general, such a criterion can be anisotropic and allow differences in

tensile and compressive strength; although previous studies have revealed either no differences (Carter et al., 1980; Bensusan et al.. 1983) or small differences (Kaplan et al., 1985) in cancellous bone tensile and compressive strength. Due to the porous nature of trabecular bone, however, it is imperative that the assumed failure surface be closed in stress space (Fyhrie and Carter, 1986). The use of von Mises‘ effective stress is thus inappropriate. The fatigue damage accumulation method (equation 10) can be considered a specicll cuse of the elfective stress approach (equation 3). Both methods result in identical mathematical forms. The value of the variable, m, in the effective stress method is unknown and must be determined from in riro experiments or observations. The parameter b in the fatigue method must be established from laboratory fatigue tests of devitalized bone specimens. The value of h is simply the slope of the S-N curve (stress vs cycles to failure) for devitalized trabecular bone. The in aitro fatigue characteristics of trabecular bone are not well understood. Difficulties in obtaining a large number of specimens with equal bone density and trabecular morphology mean that such fatigue tests will be associated with a large amount of data scatter. Kolbel et al. (1976) tested the tensile fatigue properties of the trabecular bone interface with acrylic bone cement. Their data suggest that the slope of the S-N (log-log)curve may lie between values of 6 and 32. The complicated morphology of trabecular bone insures that either tensile or compressive loading of specimens will cause bending of the constituent trabeculae. It is also likely that within many trabeculae, both tensile and compressive stresses will be created irt uivo during different loading cycles. Rotating bending tests of cortical bone specimens have resulted in an S-N slope of approximately 7.8 (Carter and Hayes, 1976). Fully reversed (tension-compression) fatigue tests of cortical bone have resulted in S-N slopes of about 5.3 (Carter et al., 198 1). As a first guess, one may choose to use b values of 5-8 for trabecular bone exposed to various loading conditions. The strain energy history approach as developed above differs from the other two methods since it initially considers the bone maintenance stimulus at the level of the true mineralized tissue (Cl,,)rather than at the level of the continuum representation of trabecular bone. The effective stress approach can be developed from the tissue level also. The apparent ultimate stress, (3,,,, is related to an effective failure stress of cortical bone tissue, bbu,,, by a structurally dependent relationship, R(p), as (Gibson, 1984) eulr = E(P)~b”I,.

(23)

Since, c?,,, a p2, we must have, R a p2. Assuming linear behavior for cancellous bone, and a loading proportional to that of the strength test, the same function will relate the apparent effective stress, 5, to the true bone effective stress, a,, in any ith loading case.

792

D. R. CARTER. D. P.

FYHRIE and R. T. WHALEN

This gives, 5 = R(p)e, x p*&,

(24)

Or Cb x

_

ci

p2.

(25)

_

Since, + a &,

equation (25) can be replaced by -

-_

ob a

0, d.lr.

(26)

The effective stress continuum approach to represent the bone maintenance stimulus (equation 1) can thus be written in terms of a tissue level effective stress as (27) TRANSDUCTION

OF MECHANICAL TO CHEMICAL ENERGY

Energy dissipation and fatigue damage of non-lioing

mareriuls Considerations of bone loading histories have direct analogies to the study of fatigue in non-livingengineering materials. Two different approaches are used to develop criteria for the initiation of a fatigue crack in smooth

metallic

specimens

subjected

to

multiaxial

stresses. The first approach is based on an extension of static yield theories to the case of repeated loading. An effective stress is defined and calculated for each loading cycle and the number of cycles to crack initiation is calculated as the cumulative effect of the inIh.rence of each loading cycle of the effective stresses (Garud, 1981). The second technique is based on the observation that fatigue failure will not occur unless energy is dissipated during cyclic loading. The number of cycles to crack initiation is therefore calculated by a consideration of the cumulative density of energy dissipated by repeated loading. It is significant that the effective stress and energy dissipation methods are roughly equivalent in their ability to predict fatigue failure in metals. The hysteresis energy approach to fatigue was introduced by lnglis (1927). Enomoto (1955) developed a theoretical S-N curve (stress vs number of cycles to failure) by employing the hypothesis that failure occurs when the energy dissipated in each cycle in excess of a certain non-damaging amount accumulates to a critical total value. Implicit in his derivations was the assumption that the energy dissipation during each loading cycle was a power function of the stress amplitude and independent of the number of loading cycles. This assumption was later shown to be approximately true after an initial ‘shakedown’ period for medium and high cycle fatigue failure (Feltner and Morrow, 1961; Halford, 1966; Chang et al., 1968). Leis (1977) proposed the use of an energy-based fatigue and creep-fatigue damage parameter to account for both viscous and non-viscous energy losses during cyclic loading. He also assumed that the energy losses were constant in each loading cycle. Such

cyclic

analyses may be pertinent to bone tissue since it shows both cycle-dependent and time-dependent damage accumulation (Carter and Caler, 1985). Energy transfer in biological tissues

Biological growth, maintenance and remodelling regulated by mechanical forces are accomplished via the transfer of mechanical energy into energy required for chemical reactions. The concept of energy transfer, therefore, must be a central issue in any effort to understand functional adaptation in biology. When loads are applied to bones. energy is imparted to the tissues. Most of this energy is stored in the form of intermolecular forces and is recoverable when the loads are removed. Some of the energy, however, is dissipated or transferred to the tissues in the form of heat or a change in internal energy. Heat generated in cyclic loading is attributed to frictional effects. Energy transferred in ciro during cyclic loading is represented by the hysteresis observed in the loading curve. In oitro cyclic testing of devitalized bone has demonstrated substantial hysteresis in bone which increases dramatically with increasing stress magnitudes and also increases with a fatigue accumulation. Most investigators have implicitly assumed that only a portion of the energy transferred (or dissipated in hysteresis) in uico is in a form which the cells utilize in bone metabolic regulation. Thus there are many theories that have been proposed for the specific mechanical control mechanism of bone mass. These theories include: stress generated electrical potentials, fatigue microdamage, direct load effects on cell membranes, and changes in mineral solubility under load. The thermodynamic considerations of energy dissipation discussed above suggest that heat energy created during cyclic loading may also affect bone metabolism. Since we do not yet know the mechanism for cell regulation. it may be prudent at this stage to consider all of the energy dissipated in an ‘average’ day as potentially important in bone maintenance. The relationships developed in the previous section which relate bone apparent density to loading history (equations 3, IO, 16 and 23) do not explicitly include a measure of daily energy dissipation. The implementation of these methods of calculating loading history, however, can be considered as methods for representing a parameter which will reflect some measure of daily energy density transferred to the tissues. In the strain energy density method for representing the load history, we assume that the stimulus for bone maintenance is governed by the number of cycles and the cyclic average true tissue energy density, CJ,, imposed upon the mineralized tissue. We further assume that the stimulus is constant throughout the bone tissue. In each load cycle in which the energy ub is imposed, we can further assume that a small portion of that energy, ub#, is transferred or dissipated. This assumption allows us to hypothesize yet another formulation for bone maintenance. If we conjecture that the average true tissue energy transfer density per

793

Trabecular bone density and loading history day

is

constant

throughout

the mineralized

tissue, then

S* r C n,(L:,,), = constant. d”Y

(28)

This statement is consistent with the previously derived energy formulation (equations 14, 15) only if u,, x r;!,

(29)

C’b,x (L/p)‘.

(30)

and therefore

Relating the strain energy transfer density continuum stress (equation 19). we find u,, cc (~,..,.,,,iPZYk.

to the

(31)

The continuum model effective stress can be related to the energy transfer idea if, u*, K (o/a,,,)“, or

(32) U,, cc (e/pZ)m.

On the tissue level (equation 26). we can express the hysteresis energy per cycle in terms of the tissue elfective stress as U,, a 5:.

(33)

It is interesting to note that power relationships between hysteresis energy and stress magnitude (equations 31, 32) have been demonstrated in repeated loading of metal specimens where the density is constant. Such a relationship is yet to be demonstrated for bone. Although during low cycle fatigue tests increasing energy dissipation per cycle has been observed, medium and high cycle fatigue tests ofcompact bone, in our experience, show a more constant energy loss per cycle throughout most of the loading history (Carter rf al., 1981). Bonfield and Li (1967) found that in moderate cyclic torsional loading, the hysteresis energy was higher in the first few loadings but then became approximately constant with repeated cycles. This is similar to the ‘shakedown’ phenomenon observed in metal specimens. Further studies on the influence of stress magnitude and the number of loading cycles on bone hysteresis energy are in order.

DISCUSSIOS ASD SUM.MARY

The derived relationships between apparent density and stress history or strain energy density history (equations 3. 16 or 22) are more general than the relationship based on fatigue damage since the power exponents are, as of yet, undetermined values. The application of the multiple loading criteria represented by equations (3). (16) and (22), are currently hampered by our lack of knowledge concerning the value of m (or k). Future quantitative studies of the distribution of apparent densities within bones subjected to different loading histories could conceivably lead to an estimate of m(k) values. Similarly. studies which relate physical activity levels of different individuals to bone density

may be useful in estimating values for these parameters. If the value of m is found to be in the range of 5-8, then the strain energy and effective stress history approaches may give results that are indistinguishable from those based on bone mass regulation by the accumulation of fatigue damage. This would not necessarily mean that fatigue damage controls bone mass but would imply that the mechanism(s) which control the local biology result in a distribution of bone mass in which comparable amounts of fatigue damage are accumulated at all sites. The considerations of loading history and energy transfer density in biological tissues introduced here were made in the context of the functional adaptation of trabecular bone. It should be realized, however, that these concepts can be applied to any situation in which repeated mechanical loading may influence the growth. adaptation, injury, or aging of living tissue. Carter et crl. (in press a, b) used a linear superposition of the strain energy density patterns created during three different loading conditions (Fig. 2, for example) to simulate the loading history in a finite element model of the femoral cartilage anlage. The technique used was an application of the strain energy history method which is formally presented here. The application can also be shown to be equivalent to a von Mises effective stress (or shear stress) history method (due to the incompressibility of cartilage). The hypothesis that evolved from the analysis was that the cartilage strain energy density history and therefore the density of energy transfer drives the biochemical reactions involved with the ossification of the anlage and the formation of the secondary ossilic nucleus. The continued application of the same loading history approaches will, we propose, predict the apparent density distribution in the mineralized skeleton (Carter et al., in press a). Analogous applications to other connective tissue structures are appropriate. Studies which apply the techniques described in this paper are ongoing and will report on the applications of the loading history and energy transfer concepts to investigations of the growth and development of the chondro-osseous skeleton and within bone and among bone variations in bone apparent density. Ackno&dgement--This work was supported by NIH grants AM 32377 and AM 01163 (RCDA for D. R. Carter) and the Veterans Administration. We thank Professor Charles Steele, William Caler and Marcy Wong for their assistance and

suggestions.

REFERESCES Bensusan, J. S., Davy, D. T.. Heiple, K. G. and Verdin, P. T. (1983) Tensile,compressive and torsional testing ofcancellous bone. Trans. 29th Orrhop. Res. Sot. 8, 132. Bontield and Li (1967) Anisotropy of nonelastic flow in bone. J. appl. Phys. 38, 2450-2455.

Burr, D. B., Martin, R. B, SehatRer, M. B. and Radin, E. L. (1985) Bone remodeling in response to in riro fatigue microdamage.

J. Eiomechanics 18, 189-200.

79-t

D. R. CARTER.D. P. FYHRIE and R. T. WHALEN

Carter. D. R. ( 1983) Mechanical loading histories and cortical bone remodelling. Cnlcij: Tissue Inr. 36. S19-S24.

Gong, J. K.. Arnold, J. S. and Cohn, S. H. (196-t) Composition of trabecular and cortical bone. Anut. Rec. 119, 325-331.

Carter, D. R. and Caler. W. E. (1985) A cumulative damage model for bone fracture. J. Orthop. Res. 3, 84-90.

Halford, G.

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