journal of orthopaedic research volume 13 issue 3 1995 [doi 10.1002_jor.1100130303] bruce martin --...

8/13/2019 Journal of Orthopaedic Research Volume 13 Issue 3 1995 [Doi 10.1002_jor.1100130303] Bruce Martin -- Mathemati

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Journal of Orthopaedic Research13309-316 The Journal of Bone and Joint Surgery, Inc.1995 Orthopaedic Research Society

Mathematical Model for Repair of Fatigue Damageand Stress Fracture in Osteonal Bone

Bruce MartinOrthopaedic Research Laboratories, University of California at Davis, Davis, California, U X A .

Summary: This paper assembles current concepts about bone fatigue and osteonal remodeling into a mathe-matical theory of the repair of fatigue damage and the etiology of stress fracture. The model was used toaddress three questions. (a) How does the half-life of fatigue damage compare with the duration of the re-modeling cycle? (b) Do es the porosity associated with the remodeling response contribu te to stress fracture?(c) To what extent is a periosteal callus response necessary t o augm ent repair by remodeling? To develop thetheory, existing experime ntal data were used to fo rmu late mathematical relationships betw een loading, dam -age, periosteal bone form ation. osteon al remodeling, porosity, an d elastic modulus. The resulting nonlinearrelationships were numerically solved in a n iterative fashion using a com puter, and the behavior of the m odelwas studied for various loading conditions and values of system param eters . The model adapted to increasedloading by increasing remodeling to repair the additional damage and by adding new bone periosteally toreduce strain. However, if too m uch loading was encou nter ed, the porosity associated with increased re mod -eling caused the system to becom e unstable; i.e., damage, porosity, and strain increased at a very high rate andwithout limit. It is proposed that this phen om enon is the equivalent of a stress fracture an d that its biologicaland mechanical ele men ts are significant in the etiology of stress fractures. Additional experiments must bedon e to test the m odel and provide b etter values for its parameters. How ever, the instability characteristic isrelatively insensitive to changes in model parameters.

Stress fractures are an important medical problemin the training of soldiers; in ballet dancers, racehorses, and high-performance athletes; and in recrea-tional athletes. Beyond this obvious area of signifi-cance, fatigue damage may play an important role inthe fragility of bone among the elderly 18,32).Knowl-edge of the means by which fatigue damage is control-led by remodeling is a prerequisite for understandingstress fractures in these populations.Stress fractures are the clinical manifestation of theaccumulation of fatigue damage in bone 5 ) . Engi-neering analyses of fatigue in bone have establishedthat the number of load cycles required for failure(NfaiJ s inversely proportional to the applied strainrange (s) (7,9-11).These studies have shown that

Nf,,,= k s 4 (1)where k and q are coefficients. It is of particular sig-nificance that q may be as large as 15, so that smallincreases in strain greatly reduce Nfa,,.Fatigue damage is not a well defined entity for any

Received January 13.1994; accepted August 17,1994.Address correspondence and reprint requests to R. B. Martinat Orthopaedic Research Laboratories, Research Facility, Room2000,4815 Second Avenue, University of California Davis Medi-cal Center, Sacramento, CA 95817, U.S.A.

material. In bone, microcracks normally are presentand can be distinguished from artifactual cracks by enbloc staining with basic fuchsin (5,15). These cracksare thought to be fatigue cracks because their num-bers increase following repetitive loading (3,8,29).Many investigators have hypothesized that internalbone remodeling serves to repair fatigue microcracks(14,16,23,33). Several of these authors have suggestedthat fatigue damage itself stimulates increased remod-eling, so that the repair process involves a negativefeedback loop.Burr e t al. (3) established an animal model forstudy-ing the connection between fatigue damage and re-modeling. They loaded canine forelimbs in three-pointbending for lo4cycles, applied at 2 Hz on one day andproducing 1,500 microstrain (pz) on the surface ofthe radius. When the dogs were killed 1-4 days later,microcrack density (cracks/mm2) was increased sig-nificantly. Also, osteonal resorption spaces, the firststage of osteonal remodeling, were located adjacent tomicrocracks more often than would be expected bychance alone. Mori and Burr (29) similarly loaded theleft limbs of dogs for lo4cycles at 2,500 p&.The rightlimb was loaded in the same manner 8 days later,immediately before the dogs were killed. Other dogsserved as controls. There was a 5-fold increase in crackdensity in the loaded limbs. There also was a signifi-

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310 B. M A R T I N

cant increase in resorption spaces 8 days after loadingbut not immediately afterward; this eliminates the pos-sibility that cracks were caused by resorption cavitiesrather than the other way around. Again, resorptionspaces were found adjacent to cracks more often thanwould be predicted by random associations.When a bone experiences a fatigue challenge, itsmechanical properties may be changed not only bydamage but also by increased remodeling space (re-sorption cavities and refilling osteons) produced bythe induced remodeling. It has been postulated thatthis may accelerate progression to a stress fracture(31)-increased remodeling introduces more poros-ity, which decreases the elastic modulus, elevates thestrains produced by the applied loads, and further in-creases damage formation. Thus, internal remodelingis required to repair fatigue damage, but in the processit may make the bone more susceptible to damage.Sufficient fatigue damage can stimulate the prolif-eration of woven bone from the periosteal surface(21,35). Presumably, this external modeling responsereduces strains in the weakened structure by increas-ing its cross section, counteracting the negative effectsof internal remodeling. It is therefore postulated to bean equally important mechanism for preventing stressfracture.With the assumption that these observations and hy-potheses summarize the essential elements of bonesresponse to fatigue damage, what implications do theyhave, taken together, for the control of such damage

and the etiology of stress fractures? This paper is anattempt to assemble these observations into a unifiedtheoretical model to gain a better understanding ofthe interactions between loading, damage, internal re-modeling, and formation of periosteal callus in healthypeople, in patients with stress fractures, and in variousexperimental situations. Specifically, three questionsare addressed. (a) How does the half-life of fatiguedamage (the time required for half of a bolus of dam-age to be removed) compare with the duration of theremodeling cycle? (b) Does the porosity associatedwith the remodeling response contribute to stress frac-ture even as damage is repaired? (c) To what extentis the periosteal response necessary to augment repairby remodeling?

THEORETICALMODELThe model melds Martins mathematical formulation of this

problem (28) with a computer model for osteonal remodeling 2 5 ) .The analysis considers fatigue damage and remodeling as seen ina representative cross section of the diaphysis of a long bone. Forthe sake of simplicity, the present model assumes that the crosssection and adjacent regions of the diaphysis are uniformly loadedin simple axial compression. A sequence of calculations simulatesdaily mechanical and remodeling changes in the bone. Much of themodel is based on simple geometric characteristics of osteonalremodeling by basic multicellular units. These characteristics havebeen histomorphometrically established for various species (26).

For convenience, fatigue damage is defined as millimeter of ob-servable crack length per square millimeter of cross section ofbone, although the theory would work as well for any definitionof damage that can be expressed per unit area of cross section. Therate of darnage formation is assumed to be a function of theloading rate (R L, n cycles per day) and the resulting strain range(s). On the basis of Eq. 1, the rate of damage formation is assumedto be (28):

DIF = k n sqRL= k,@ 2 )where kD is a damage rate coefficient that depends on loadingconditions and bone structure and QU = s ~ R , .s defined as thedamage potential. (Here, q may or may not be equal to that in Eq.1.) Equation 2 can be thought of as representing a situation inwhich only one kind of loading exists (as in most in vitro experi-ments). Alternatively, s may represent an effective strain rangeresulting from a mixture of loads that repeat with frequency R,and produce the same QD as the actual loads.

Each iteration period (A t = 1 day). a number of new basicmulticellular units ar e acl ivated within the section: i.e.. a numberof tunnelinl: basic multicellular units reach the plane of the sectionand begin to open resorption cavities in it. The activation fre-quency (I,) is the number of basic multicellular units that reachthe section per square millimeter per day. As each basic multicel-Mar unit passes through the section, its resorption cavity expandsto the diameter of an osteonal cement line. removing a portion o fthe cortical area (which is subsequently refilled) and any damageit contains. Assuming tha t fatigue damage can be removed only bythis process and that the rate of removal is proportional to theamount of existing damage (because the more there is. the greaterthe probabiility that a basic multicellular unit will hit it), the rateof damage removal (or repair) is

DK= D f,nr,F, 3 )where D is the existing damage and ru s the radius of the osieonalcement line (fa.nr,2 s the amount of cross section removed by a l lof the basic multicellular units created i n 1 day). FS is a damagerepair specificity factor, which accounts for any mechanisms thatserve to direct remodeling to sites of fatigue damage as opposedto strictly random remodeling. Fs is estimated to be about 5 onthe basis of the ohserved frequency with which cracks are associ-ated with new resorpt ion cavities (3,6,29).The net ra te of accretionof fatigue damage is D = D, - DK. n the equilibrium state. DR=DF and thee corresponding burden of damage waiting to be re-paired is

Do = (ku@ncl)/(rrr,2f,,FS.) ( 4 )where the 0 subscripts indicate equilibrium values.

Initially. a normal pattern of daily loads is assumed io beapplied to ithe section, producing an equilibrium strain range (so)and load rate (RLO), nd the corresponding equilibrium damagepotential db,,,) is assumed to maintain the resulting damage at anequilibrium level (D,,).To simulate a fatigue challenge, the modelallows R,, cycles per day of additional loading, producing a strainrange (sE) to be superimposed on the normal activity. The rate ofdamage formation predicted by Eq. 2 for this additional loading isadded to thmt produced by the equilibrium loading.

5 ) b = k ~ ( s o ~ R ~ i ,S I , ~ R L E )k u ( @ n o + @ r w )Values for k,, and q were obtained from the t w o fatigue dose-

response experiments by Burr and co-workers (3.29) (Appendlx).Then, allowing F, = 5 and f, ,, = 0.0064immiday. Eq . 4 w s rear-ranged to obtain @ = soqRLo 9.83 x lo-$ cycles per day. This isequivalent, for example, to RLo= 1,448 cycles per day and s,)= 300p .These values, or any equivalent set. can be used as equilibriumloading conditions in the model, and it will approximate thr dam-age in the t:xperiments of Burr and co-workers (3,29). (An exper-

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M O D E L I N G S T R E S S F R A CT U R E D Y N A M I C S 311

x0< 0.8E

0.000 0.005 0.010DAMAGE, /rnm

FIG. 1. Graph of Eq. 7, the hypothetical relationship between theactivation frequency of basic multicellular units and fatigue dam-age. The two data points represent values for the normal controls(left) and the experimental dogs (right) in the study of Mori andBurr (29).

iment with many different values for sF must be done to determinemore accurately such variables as q and k .)A sigmoidal dose-response relationship is assumed between theactivation frequency (f,,) and the current damage (D) (Fig. I .Mathematically. this is expressed as

f a = (faofmax)/(fao [ fa t tm - a 0 1 exp[k,f,,,,(D - DoYDol) 6 )where fa is the maximum allowable activation frequency, takento be l.0/mm2/day, a value slightly greate r than the largest ob-served value of 0.38/mm2/day (30). k K s a coefficient. which wasadjusted so that the numbers of newly activated basic multicellularunits per square millimeter approximated those in the study byMori and Burr (29) (Fig. 1).The model uses the history of f; tocalculate porosity as previously described (25) (Appendix).The elastic modulus (E ) is calculated from porosity (P) usingthe relationship (24.27):E = 15(1 - P)' GPa. Because the model isnormalized to an equilibrium condition. i t is insensitive to themodulus at zero porosity (15 GPa here). It is recognized that Ealso may be diminished by fatigue damage. but this effect is no tincluded in the present model.Damage also is assumed to stimulate formation of periostealbone, which increases the area of the cross section. No dose-response data have been found relating the apposition rate orperiosteal bone (Mp) to fatigue damage, but Turner el al. (34)found that when rat tibias were bent 36 times a day for 12 days.surface bone formation was lamellar at lower loads and convertedto woven hone at higher loads. Apposition rates of lamellar bonewere linearly proportional to the magnitude of load or strain, butproduction of woven bone was an all or nothing response (therate of bone formation was not correlated with load). Mathemat-ically, this may be represcnted by

M p = k p ( D - Do)/D,,M P = M,

for D < DClor D Dc. ( 7 )

where k, is a coefficient and M, is the apposition rate for wovenbone formation. If M,* is the maximum apposition rate for lamellarbone formation, achieved at D = Dc, then k P = M,*/([D, /Do]- 1).The data of Turner et al. (34) suggested using Mw = 0.030mm/day and Mp* = 0.004 mm/day. Finding no information on Dc.

1 studied the behavior of the model as this parameter was varied,starting with O.OOS/mm-a value slightly larger than the 0.003/mmmeasured in the experiment of Mori and Burr. which produced nosignificant periosteal response. The daily change in periosteal ra-dius is calculated as Arp = MpAt,and cortical area is calculated withthe assumption that the endosteal radius is constant at 3 mm. Todetermine strain, the applied load is divided by the model's cross-sectional area and elastic modulus.All of the aforementioned calculations were incorporated in acompiled BASIC program (Fig. 2) .The behavior of the system wastested extensively. but. for brevity, only the results relevant to thethree questions posed at the end of the introductory section arepresented here.

RESULTSWhen the model 's p aram eters were adjusted to sim-ulate the experiments of Burr and co-workers (3,29),as already described, the model reproduced data onresorption spaces and dam age (Fig. 3).This is not p re-sented as validation of the model but simply to show

how, when porosity, stra in, activation frequency, anddamage are plotted as functions of time, the modelprovides insight regarding the overall behavior of thesystem. Th e mode l could be validated by additionalexperiments of a similar nature.Using this version of the m odel, I addressed the firstquestion regarding the com parison of the half-life offatigue damage with the duration of the remodelingcycle. When 10' cycles were applied on a single day, atsE = 2,500 o r 3,500 p Eq. 3 caused the model toremove dam age faster in proportio n to ( a ) the amoun tof damage introduced and (b) the increase in FS (Fig.4). Th e du ration of the remodeling cycle was 71 days.For sE= 2.500 p he damage half-life was consider-

L OA D I N G

\ = FiFAtI P I I I H I . IC O R T A R E A . AFIG. 2 Flow chart for the model. The calculations are keyed t oequation numbers in the text. Integrations are over the iterationperiod ( A t = 1 day) unless otherwise indicated. The calculation ofQ (the net bone added daily by all active basic multicellular units)from f, (activation frequency) history is shown in abbreviatedform. Sec the Appendix for details.)


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0.10 [ 05

b- 0.010 . 0 2 I I I I 2 0 . 0 0

.04 r-----l0 .00

250 0.0000 2 0 4 0 60 80 100 0 20 40 6 0 8 0 100T IME, d a y s T IME, d a y s

FIG. 3. Simulation of the experiments by Burr and co-workers (3,29), showing plots of porosity, strain, activation frequency ACT F), anddamage (DAM) as functions of time, with sE = 2,500 p (broken line) and sE = 1.500F (solid line). Note that the strain graph shows thestrains being produced by the normal daily loading (soand Rm), ot those produced by the experimental loading, which lasted only 1 day(arbitrarily set at day 5). Data points show values for damage, which were measured in the experiments, and activation frequency, whichwas deduced from measurement of resorption spaces. There are transient increases in porosity and strain due to increased remodeling,and the rate at which damage is removed is relatively slow. The half-life of the bolus of damage is several months.ably longer than this unless the repair specificity fac-tor (F,) was quite large (greater than 10).When sE=3,500 ~ L E ,he half-life was about half the remodelingperiod for Fs = 5. The cost of removing more damageor removing it less efficiently (with a smaller F,)wasa greater transient increase in porosity due to thegreater numbers of basic multicellular units activated(left panels in Fig. 4).

To address the other two questions regarding the

negative effects of the remodeling space and the im-portance of the periosteal response, I first examinedthe behavior of the model when no periosteal re-sponse was present. Suppose, in contrast to Burr'sstudies, a constant amount of additional daily loadingis superimposed on CDD0using load control. That is,RLEcycles of a load are applied each day such that theresulting strain range is sk= sE,on the first day. Later,the strain produced by the load may change due to

0.30 I 3 I 0.008 I i i I

::::i .g 0.15 0.0040 0.100 .05

0.006 F = 2 5 0 0 p E- - - _ - - _- - - - 0 002 - - - - - _ _ _ _ _ _- - - - - - _ _ _ _ _ _

g 0.20g 0.15.25 i J 0.004 10.00 .000

- _0 . 2 0cn .155 0.0020.05

0.00 0.0000 2 0 40 6 0 8 0 100 0 20 4 0 6 0 8 0 100TIME, d a y s TIME, days

F= 15_ - F= 1 F= 5 . ___FIG. 4. The effects of the magnitude of damage and repair specificity factor (Fs)on the transient increase in porosity (left) and the rateof damage removal (right). The top and bottom panels simulate experiments for which R L ~lo4 cycles on a single day and sE = 2,500and 3,500p espectively. (Fs = 5 was used in all other parts of the study.)

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M D E L I N G S T R E S S F R A C T U R E D Y N A M I C S 3130 3

g 0.2g 0.1LnLT

0.0

7 0 0 06000500040003000

0 . 37 J;;.E 0 .2E\k 0.1toQ 0.00.01 0

0 5 10 15 20 25 30TIME, rno.

E 0 008E.006

W2 0.0040.0020 0000 5 0 15 20 2 5 30

TIME, mo.90 109 ...... 185 ............ c p dLE

FIG. 5. Graphs of porosity, strain, activation frequency (ACT F), and damage (DAM). The strain plotted is due to the continuingexperimental loads and shows the behavior of the model without a periosteal response. As described in the text, ongoing experimentalloading initially producing sEI = 2,500 FLEs superimposed on normal loading. If RI.F s 100 cycles per day (cpd) or less, the system is stableover a lifetime; when RLE lightly exceeds this critical value, instability is precipi tated.remodeling-induced changes in porosity, but the loadremains the same, as it would, for example, if an armyrecruit began to march 5 km with the same pack everyday. When this situation is pertained and the experi-mental loading had a damage potential QDE = sErqRLE,which was less than a critical value cDDc the additionaldamage increased the remodeling rate and the poros-ity and the bone became less stiff, but a new equilib-rium was reached (Fig.5) .However, when the damage

0 . o L6000

z2 4000t-vLT 2000

0 3 6 9 12 1 5TIME, rno.

potential exceeded QDc, remodeling never caughtup with the increased rate of damage formation. Po-rosity, strain, and damage increased at an ever increas-ing rate, without limit. It is reasonable to consider thisinstability to be fatigue failure of the model (i-e.,a complete stress fracture). If aDDas only slightlysuper-critical, this instability could be quite insidious.Damage leveled off at what seemed to be a new equi-librium level and then, suddenly, after what could be

\k 0.2

0.10 0

0.01 00.008

.0062 0.004I

Q

w

0.0020.000p- j- _..........0 3 6 9 1 2 1 5

TIME, mo.R = 185 ........... 225 21 - - 322- CPdLE

FIG. 6 Behavior of the system when a periosteal response is present. This is similar to the behavior depicted in Fig. 5,except that greatervalues of RLE an be tolerated before failure. When instability occurred. strain values quickly became greater than the values shown here,exceeding failure strains for bone (approximately 2-3 ).cpd = cycles per day.

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a long time, the damage exploded, rising rapidly ata very high rate. When the model param eters were setto simulate those in Burrs experiments, QDc. = 3.05 xlo- cycles per day (eg., RLE= 101cycles per day andsE,= 2,500 PE). When 102 cycles per day were applied,it took 20 years for failu re to occur, and w ith 103 cyclesper day, i t took about 4 years. Slightly higher loadingrates resulted in failure after a few months or weeks(not shown).Clearly, without a periosteal response, the modelcan withstand only relatively sm all fatigue challengesbecause of the inherent instability caused by the re-modeling space. Now, consider th e third qu estion: themodels behavior with a periosteal response in placeand Dc set at 0.005/mm. Ag ain, sEIwas set at 2,500 pand R L Ewas incremented to increase the experimen-tal damage potential. When RLE 185 cycles per day(which previously produced early failure, Fig. 5), theaddition of external bone acted to reduce strain, andthe system returned toward a new equilibrium levelafter a transient respo nse lasting abo ut a year (Fig. 6 ) .Increasing R L Ecaused larger transients that endedmore quickly. until another critical damage potentialwas reached, and instability implying failure ensued.Again, the inherent instability of the system was re-markable. As R E increased, remodeling respondedever m ore vigorously, and t he challenge was overcom emo re and m ore quickly. The n, when one m ore cycleper day was added, instead of dam age not decreasing,the system precipitously failed.

The corresponding changes in cortical area wereSO-lSO%, depending on how long bone was formed.Woven bone formation only began after 3 monthswhen R, = 321 or 322 cycles per day an d began afte r6 months when RL.E 225 cycles per day. Once theperiosteum rcturncd to lamellar bone form ation, thechallenge had been met successfully; no oscillationsbetween these two modes w ere observed.When the parameter Dc was inc reased, less loadingwas required to produce failure because the periostealresponse was retarded. A very small change in Dcconverted a stable system to an unstable one. Insta-bility also was observed when the periosteal dose-response function was mo deled as a sigmoidal curve.

DISCUSSIONThe goal of this analysis was to continue the devel-opment of a mathematical model for the repair offatigue dam age by remodeling, incorporating temp o-ral aspects of internal remodeling and the effects ofa periosteal response. Predictions were obtained re-garding thr ee specific questions. First, the m odel pre-dicted that the half-life of a bolus of fatigue damageis several months and is substantially reduced if the

amoun t of damage is increased or the remodeling iswell directed at foci of damage. Second, the model

predicted that the porosity associated with remodel-ing to remove fatigue damage can produce a highlyunstable state in which strain and damage increaserapidly and without limit under continued loading.This condition may be interpreted as representing astress fracture. Finally, it was show n th at a periostealresponse substantially increases the amount ot load-ing that can be tolerated but does not remove thetendency of the system to be unstable.The most unique feature of this theory is its inclu-sion of the details of osteonal remodeling. Severalinvestigators have mod eled t he effects of internal re-modeling on b one density and mechanical properties(12,13,19). In each of these studies, bone density wasassumed to be a function of a strain-related variable(strain energy density), and that strain was in turn afunction of density through the elastic modulus, crea t-ing a feedback loop. Ca rter e t al . (12) suggested for-mu lating the effects of fatigue damage on remodelingas in Eqs. 1 and 2, but no previous models have ac-counted for the details of basic multicellular unit-based remodeling that could significantly affect themechanical beh avior of the system, such as the remo d-eling space and the time required for the remodelingto occur. BeauprC et al. (2) accounted for the varia-tions in the internal surface are a of bone that accom-pany changes in density; this helped t o account for thetime-dependency of mechanically adaptive remodel-ing, but the transient increase in porosity charac-teristic of remodeling was not modeled. This is theprimary strength of the present mod el, and the featuretha t acclounts for its unstable behavior.Turning to the limitations of th e m odel, the first, andmost easily remedied, is the simple stress state thatwas assumed (uniaxial c omp ression). This simplifica-tion allowed the other aspects of the model to beisolated and explored. The next version of the modelshould use finite element analysis to calculate morerealistic stress states for a diaphyseal region. Manystress fra ctures scem to occur in characteristic, local-ized regions, consistent with inhomogeneous stressstates (22). How ever , there is no reason to assume thatsuch inhomogeneities would eliminate the instabilityeffects o f the pres ent m odel: ind eed , they might exac-e rba te them.A second limitation is the assumption that wovenbone added by the periosteal response has the samemechanical pro per ties as lamellar bo ne when strain iscalculated. Altho ugh this clearly is not the case (andshould be corrected in subsequent models), the effectof the curre nt approximation is to reduce rather thanincrease the instability problem. Woven bone is lessstiff than lamellar bo ne and would thus protect againstremodeling-induced increases in strain less effectivelythan mlodeled here. Another l imitation is that thepresent model does n o t account for the mechanical


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effects of damage on the modulus of the bone. It isanticipated that adding this effect also would en han cerather than diminish the instability behavior.A more difficult limitation is the lack of knowledgeabout the relationships between loading and damage.This is a very difficult problem, which can only be

solved experimentally. I t is, however. entirely analo-gous to the general problem of defining and mea-suring a "remodeling stimulus" in other theories ofmechanically adaptive remo deling. Similarly, the esti-mate of F, and the assumed relationships betweendamage and activation frequency or periosteal appo-sition in the present model are analogous to the as-signment of arbitrary coefficients relating changes indensity to strain energy in oth er m odels. Param etricstudies of the mod el suggest that its general prediction(that remodeling is capable of repairing damage dur-ing a fatigue challenge bu t produces instability whenloading is too sev ere) is insensitive to such details ofthe loading-damage-remodeling interactions. Further-more, the present model can be used t o design in vivoexperiments to obtain more information about suchdetails and to test predictions concerning dam age, ac-tivation frequency. porosity, and strain.The question arises as to whether the instabilitysee n in the mod el is truly indicative of fatigue failure.I propose that it is. because calculated strains quicklyexceeded the failure strain of the bone. It is possiblethat an unknown biological process halts the increasein remodeling space before failure occurs, but thatalso would halt th e repair of damage. The fund amenta lproblem is that damage can be repaired only by re-modeling and remodeling requires an increase in po-rosity. Th ese increases in rem odeling sp ace ar e readilyobservable when sites of stress frac ture are exam inedhistologically (21,31). Therefore, i t is reasonable topostulate that the instability observed in the m odel isindicative of stress fracture.It is far too early for the model to provide conclu-sive insight con cerning th e v agaries of stress fractures,but it does sup por t the clinical impression that remod -eling intended to repair damage can instead causedam age to increase dramatically w hen th e capabilitiesof the system are exceeded (31). The unpredictablenatu re of stress fractures may stem from the fact thatthe system tha t produces and controls fatigue dam ageis nonlinear and small changes in the conditions ofsuch systems can result in vastly different outcomes.This may explain why stress fractures can occur with-out previous symptoms or in ways that are difficult topredict.Recently, there has been growing interest in the

modeling process a re potentially important sources ofinstability not only in repair of fatigue damage butalso in ot her m echanically adaptive remo deling. Anal-yses that only assume relationships between strainand density, for exam ple, and ign ore the processes th atactually connect these variables. may miss clinicallysignificant elements of the problem.

APPENDIXValues of q and kD were o btain ed from the d ata inBurr's experim ents (3.29). as follows. D am age was cal-culated as the prod uct of crack density and length foreach experiment. (C rack lengths were not measuredin th e first cxperime nt [ 3 ]but were found to be about0.052 mm in both loaded and control regions in thesecond experiment [29]. The refore, this num ber wasused for cra ck length in all cases.) D,, was take n as themean damage in the control bones of both experi-

ments (0.000754/mm). It was assumed th at the differ-ence between th e damage in the experimental bonesand that in the controls (D o) was the damage pr o-duced by the loJ cycles of experimental loading: 1.e..ADE= DF:- Do.With Eq. 2 written just for the exper-imental loading, D'F = ADE/At = knsEqRLE.his equa-tion was written for each of the two experiments. andthe two eq uations were solved for q and k,, with At =1day and RLE lo4cycles per day. For t he first exper-iment 3 ) . E= 0.0015and ADE= 0.000494/mm and forthe second experiment (29), sF = 0.0025 and ADE=0.002158imm. Th e result was q = 2.89and kn = 7.2/mm.

The calculations for porosity are based on the ide-alized geom etry of a basic multicellular unit as seen inlongitudinal and cross sections (Fig. 7). as well as itslongitudinal advancement rate (v) and mean refillingra te (MF).Th e times required for the resorptive (TK).reversal (T,),and refilling (T,) phases of the remod-eling cycle are estimated as T, = LR/v,T, = L,/v, and2= IT rc - r,A, = nr;

FIG. 7. Diagram showing the geometry of a typical basic multicel-lular unit in longitud inal and cro ss sections. The data fo r osteonalremodel ing that are used in the model are from prev ious s tud ies(1.20). The values of the var iab les are rad ius of the cement l ineIpossibility that instability may be present in variousaspects of skeletal control involving bone modelingand remodeling (17). I believe that the transient in-creases in porosity and delays inherent in the re-

r < . ) . 0 .095 mm ; rad ius of the haversian canal r H ) . 0.020 mm; tun-neling rate (v). 0 .039 mm lday: refilling rate (MF).0.00121 mm lday:leng th of reversal reg ion (L,), 0.143 mm; and leng th of r e so rp t io ncavity ( L ~ ) , .200 mm. L~ = "(1, - r,)/MF is the length of t h eref i l l ingreg ion .

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TF=(rC- H)/MF.Also estimated from these geometricdata are the mean daily amounts of bone removed byeach resorbing basic multicellular unit (Qc = Ac/TRmm2 resorbed per basic multicellular unit per day)and added to the section by each refilling basic multi-cellular unit (a, = AB/TFmm2 formed per basic mul-ticellular unit per day).The history of fa is used to determine how manybasic multicellular units are resorbing (NR) and refill-ing (NF)on the current day.NR s found by numericallyintegrating f a from TR days ago to the present time,and NF,by integrating from (TR + TI+ TF) o (TR+ TI)days ago.

Porosity has two components: PHC,due to com-pleted haversian canals, and PRs,due to the remod-eling space, defined as the combined areas of allresorbing, reversing, and refilling basic multicellularunits. As remodeling continues. PHCncreases at a de-creasing rate because some new osteons overlap thecanals of old ones. Consequently, PHC pproaches anequilibrium value, which is the ratio of the area of thehaversian canal to the area inside the cement line (24).To avoid the distraction of this secondary effect, whenthe porosity due to active basic multicellular units isthe primary concern, PHCs assumed to be constant atits equilibrium value of 4.43%. The model keeps trackof total porosity by calculating Q, the net amount ofbone added daily per square millimeter of section byall active basic multicellular units:

The term (1- PHC) erves to correct for new basicmulticellular units that overlap an existing haversiancanal (26). The daily change in total porosity is calcu-lated as AP = QAt. These calculations are explainedfurther in Martin (25).

Acknowledgment:This work was supported by National Insti-tutes of Health Grant AR41644.

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