Bone 43 (2008) 476–482

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Relating osteon diameter to strain

René F.M. van Oers ⁎, Ronald Ruimerman, Bert van Rietbergen, Peter A.J. Hilbers, Rik HuiskesDepartment of Biomedical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

⁎ Corresponding author. Fax: +31 40 247 3744.E-mail address: r.f.m.v.oers@tue.nl (R.F.M. van Oers)

8756-3282/$ – see front matter © 2008 Elsevier Inc. Aldoi:10.1016/j.bone.2008.05.015

a b s t r a c t

a r t i c l e i n f o

Article history:

Osteon diameter is generall Received 14 December 2007Revised 24 April 2008Accepted 10 May 2008Available online 28 May 2008

Edited by: David Burr

Keywords:Osteon diameterOsteoclastsBMU

y smaller in bone regions that experience larger strains. A mechanism relatingosteon diameter to strain is as yet unknown. We propose that strain-induced osteocyte signals inhibitosteoclastic bone resorption. This mechanism was previously shown to produce load-aligned osteons incomputer simulations. Nowwe find that it also predicts smaller osteon diameter for higher loads. Additionally,we find that our model predicts osteon development with two cutting cones, one moving up and one movingdown the loading axis. Such ‘double-ended osteons’ were reported in literature as a common type of osteondevelopment. Further, we find that a steep gradient in strain magnitude can result in an osteonal tunnel withcontinuous resorption along the less strained side, which corresponds to ‘drifting osteons’ reported inliterature.

© 2008 Elsevier Inc. All rights reserved.

Introduction

Osteons are the basic structures of cortical bone. An osteon is atube-like structure with a central canal. It is formed when a ‘cuttingcone’ of osteoclasts excavates a tunnel, and a ‘closing cone’ ofosteoblasts fills the tunnel with new bone [40]. The osteoblasts donot completely ‘close’ the tunnel, as blood vessels remain in themiddle. The team of osteoclasts and -blasts is known as a basicmulticellular unit (BMU). In humans, osteons range in diameter from150 to 350 μm [1,4,7]. Osteon diameter is obviously determined by thesize of the osteoclast cutting cone. What determines the size of thecutting cone is less clear. Frost [14] suggested an inverse relationbetween osteon diameter and strain magnitude, since osteons areusually wider in the endosteal side of the cortex, while strains arelarger in the periosteal side. Skedros et al. [47,48,50] analyzed the heelbone of hoofed animals, in which the tension cortex habituallyexperiences a lower strain magnitude than the compression cortex[52]. They found larger osteon diameters in the tension cortex (Fig. 1),thus supporting Frost's hypothesis. Further evidence for this inverserelation comes from a study on immobilized primates [62]. A 7-monthperiod of immobilization resulted in cortical resorption cavities of500–1500 μmdiameter.When the immobilizationperiodwas followedby a recovery period, new bone was formed in these cavities and theresulting structures appeared as “unusually large osteons”. Thus, thereappears to be an inverse relation between strainmagnitude and osteon

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diameter. The question then is: what mechanism relates osteondiameter to strain?

A relation between strain and osteon diameter implies a relationbetween strain and osteoclastic bone resorption. It is evident thatmechanical loads influence bone remodeling, as both osteons [41] andtrabeculae [59] are aligned to the principal stress directions. It is alsowell established that bone mass is increased with increasing mechan-ical loading and vice-versa. Muscle-building exercise was shown toincrease bone density at weight-bearing sites in young men [9].Exercise that involves impact loading increased the bone density in thefemur and radius of prepubertal girls [10]. Immobilization due to spinalcord injury decreased the bone density at weight-bearing sites inadults [27]. 17 weeks of bed rest had a similar effect in adult men [31].Both osteoclasts and osteoblasts are involved in this relation.Mechanical stimulation of rat ulnae caused a decrease in osteoclastactivity and an increase in osteoblast activity [20]. The opposite wasobserved in immobilized rat hind limbs [58]. In immobilized primates[36] researchers observed an increase in resorptive surfaces and adecrease in formative surfaces in trabecular bone, and impairment ofbone formation in cortical remodeling spaces. These effects ofmechanical loading on osteoclasts and osteoblasts may be regulatedby osteocytes. It is believed that osteocytes can sense mechanicaldeformation of the bone [5,11,23,30,60]. In response they could signalto inhibit osteoclast activity [6,19] and stimulate osteoblast activity[5,8,18].

Our theory is based on a mechanosensory function of osteocytes.Upon sensing a mechanical stimulus, osteocytes are assumed to emitbiochemical signals that inhibit osteoclasts and stimulate osteoblasts.This mechanism was previously used to explain osteonal [51,56] andtrabecular load alignment [23,44,56]. Because our model entails a

Fig. 1. A bone subjected to both axial compression and bending (left) has higher strain magnitudes in the compression than in the tension side of the cortex (right, top). The widestosteons were observed in the tension endosteal region, the thinnest in the compression periosteal region. Images are from a skeletally mature elk calcaneus sectioned transversely at60% of length. (Adapted from Skedros et al. [50].)

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relation between strain and osteoclastic bone resorption, we nowinvestigate how it relates osteon diameter to mechanical loads.

Methods: the model

The model used in this study is similar to the model used in anearlier study [56]. A bone structure is mapped onto a finite elementmesh consisting of square elements of uniform size Δx [m]. Weintroduce, for each element, a relative bone densitym(x,t),where vectorx [m] denotes element position and t [day] denotes time. The densitym(x,t) ranges from a minimal value mmin to 1. At mmin the element isconsidered to be a marrow element, abovemmin it is considered to be abone element. The time t is represented by increments Δt [day], duringwhich the bone density of the elements can change. Hence, them(x,t)-values of the elements constitute a changeable bonemorphology. Them(x,t)-values also determine the stiffness of the elements, according to[12]:

E x; tð Þ ¼ Emaxd m x; tð Þ3;

where Emax [Pa] is the Young's modulus for elements at maximal bonedensity. The structure is subjected to external loads. We use a staticload to represent a cyclic load of given amplitude σ′ and frequency f.The magnitude of the static load σ′ is chosen such that the resultingstrain-energy-density (SED) equals the peak SED rate of the dynamicload:

σ V¼ 2:02σffiffiffiffif :

p

For a derivation of this formula, see Huiskes [22] or Ruimermanet al. [45]. Load transfer through the structure is evaluated by finiteelement analysis (FEA), assuming isotropic and linear-elastic materialbehavior. For 2D simulations we assume plane-strain conditions. FEAis performed at the start of each increment to correct for the gradualmorphological changes in the bone.

Osteocytes

Osteocytes, located within the bone tissue, are assumed to sense amechanical stimulus R [J m−3 s−1], a typical SED rate experienced in arecent loading history at its location. Based on this sensation, theosteocytes emit a biochemical signal. This signal decreases exponen-tially in strength with increasing distance d [m] from the osteocyte.

The exponential function represents the steady-state distribution of asignal molecule, where synthesis and decay are in balance [29]. Eachelement receives an accumulated signal S from nearby osteocytes,according to [22]:

S x; tð Þ ¼ ∑ni¼1 R xi; tð Þ:μ:e−d x;xið Þ=D;

where μ [J−1 m3 s] is the osteocyte mechanosensitivity and D [m] is adiffusion-decay constant, xi is the position of osteocyte i and n are thenumber of osteocytes less than dinfl [m] removed from x, where dinfl isthe truncation distance for the osteocyte signal. Note that the signaldoes not accumulate over the increments; each increment the signal iscalculated anew from the current SED distribution. Numerical settingsfor the parameters were given and discussed in our previous paper[56].

Osteoclasts

When the osteocyte signal S is strong enough, it inhibits osteoclastattachment to the bone surface and their subsequent resorptionactivities. Osteoclasts are explicitly modeled, using a cell simulationmethod based on the cellular Potts model (CPM) [17,33]. Thissimulation method was extensively described in our previous paper[56], andwill only be summarized here. An osteoclast occupies severaladjacent elements and may change position by entering and leavingelements. An osteoclast tries − by preferring certain moves overothers − to minimize an energy H:

H ¼ Hvol þ Hsurf ¼ λd V−V0ð Þ2þZ

surf

h Að Þd dA:

The volume energy Hvol ensures that the osteoclast volume V [m3]remains close to a target volume V0, where the inelasticity λ [m−6]sets the strength of this constraint. V0 is based on a typical osteoclastdiameter of 50 μm [13]. The surface energy Hsurf of an osteoclast isdetermined by its surroundings, as the contact energy h [m−2] of asurface patch dA [m2] depends on the type of the neighboring element(Fig. 2). That contact energies differ for different substrates is centralto the CPM. The cell attaches to substrates with low h and detachesfrom substrates with high h, in order to minimize its surface energy.

The contact energy to bone hb is a function of the osteocyte signal Sin the neighboring bone element (Fig. 3). What it amounts to is that an

Fig. 2. Sketch of the CPM. The contact energy of an osteoclast (OCL1) to marrow hm andother osteoclasts hocl are constants. The contact energy to bone hb(S) depends on theosteocyte signal S.

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osteoclast adheres to bone surfaces where the osteocyte signal isweak. Two signal thresholds are used, S0 and S1. If the osteocyte signalis below S0, osteoclast-bone adhesion is strong. Between S0 and S1adhesion weakens and above S1 there is no adhesion. Once settled, anosteoclast can resorb adjacent bone elements. As the osteoclast entersa bone element, its bone density is reduced to mmin. The probabilityfor resorption depends on the local adhesion between the osteoclastand the bone surface, which is determined by the osteocyte signal.

In our previous study [56], we used a fixed number of osteoclastsin simulations of osteon development. This limits the width of theosteoclast cutting cone and the resulting osteon. Since these are thetopics of investigation, we now allow addition and removal ofosteoclasts from the cutting cone. We assume that osteoclastorigination can occur on exposed bone surfaces, where adhesionconditions are favorable. The rationale for this assumption ispresented in the discussion of this paper. At every increment newosteoclasts can originate, with origination probability OP [m−2 d−1],on exposed (i.e., not covered by osteoclasts or-blasts) bone surfaceswith low osteocyte signal (SbS0). Themodel also describes osteoclastdeath. It is assumed that osteoclasts detached from the bone surfaceremain viable for only a short period. The relevance of thisassumption is also addressed in the discussion. If an osteoclastdoes not resorb −because it is not adjacent to bone or does not adhereto adjacent bone− for a period Td [d], it is removed.

Fig. 3. The osteocyte signal S at the bone surface determines osteoclast adhesion. A) Thecontact energy to bone hb is a function of S. B) Configurations of minimal surface energyfor different hb.

Osteoblasts

Osteoblasts are recruited to exposed bone surfaces where theosteocyte signal exceeds a threshold Sobl for a period Tr [d]. They thenform bone according to

Δmobl ¼ τd S x; tð Þ−Soblð Þd ΔtΔx

where the change in m(x,t) due to osteoblast activity is denoted withthe index obl, and τ [m day−1] determines the bone formation rate.The newly formed bone is assumed to have the same osteocytedensity as pre-existing bone. It is covered with a layer of osteoblasts.Osteoclasts do not adhere to or originate on these surfaces.

Methods: the simulations

Osteon development is simulated in a 4×4 mm2 piece of compactbone tissue, subjected to compressive loads in the vertical direction(Fig. 4A). Osteocytes are positioned in the bone tissue at a density of1600 mm−2 [39]. Osteoclasts start from an initial resorption cavity of180 μm diameter (Fig. 4B). For the origination probability OP of newosteoclasts in the cutting cone we use a value of 5000 m−1·d−1. Thistranslates to 1 new osteoclast per day on a 200 μmwide cutting cone ifit happens to be completely exposed. The period Td during which adetached osteoclast remains viable is set to 1 day, as recent studieshave demonstrated that detachment induces apoptosis in osteoclastswithin a matter of hours [46,63].

In our previous study [56] we simulated osteon development usingan 18 MPa load. Because we want to investigate the influence of theload on osteon diameter, we compare simulations at five differentloading magnitudes: 17.5, 18.0, 18.5, 19.0 and 19.5 MPa. All otherparameter settings are as in [56]. Because osteoclast origination andmovement are stochastic, we performed each simulation five timeswith different random seeds (a ‘random seed’ is a settable startingpoint for a random number generator). The simulations ran for 100model increments, representing 25 day remodeling periods.

In addition, we perform 3 simulations where loading is also com-pressive in the vertical direction, but its magnitude increases from left toright,with anaverageof 18MPa. This ramp loadingwas chosenasvaryingfrom9 to27, 6 to 30, and from0 to36MPa. This loading ismeant to reflectthe increasing strain magnitude from endosteum to periosteum (Fig. 1).

Results

Fig. 5A shows the strain-energy-density (SED) around the initialcavity, for the 19.5 MPa simulation. Strains are low along the loading

Fig. 4. A) The initial configuration consists of a 4×4 mm2 piece of compact bone with aresorption cavity. The structure is loaded compressively in the vertical direction.B) Detail showing initial osteoclasts (red dots) in the cavity and osteocytes (black dots)in the bone.

Fig. 5. A) Around the cavity, strains are low in the loading direction (vertical) and highin transverse direction. B) The model osteocytes translate this into corresponding bio-chemical signals.

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axis and high in the transverse direction. The osteocyte signal (Fig. 5B)induced by these strains has a similar distribution. Fig. 6 shows thedevelopment of the 19.5 MPa simulation. The eight initial osteoclasts,one element in size at first, quickly expand to the target volume. Theosteoclasts located at top and bottom of the cavity start to resorb bone,while those located on the left and right side of the cavity detach fromthe surface and die. The top and bottom osteoclasts proceed to resorba tunnel along the loading axis. Thus, there are two cutting cones, onemoving up and one moving down the loading axis. During the courseof the simulation, the osteocyte signal around the cutting conesremains low in loading direction and high in the transverse directions.Osteoblasts are recruited to the eroded surface where they form aclosing cone of new bone.

In all simulations, the strains −and consequently the osteocytesignal− around the initial cavity are also high in the loading directionand high at the sides. However, the ‘lateral signal’ is not as strong asin the 19.5 MPa simulation, and lateral resorption is initially notinhibited. The cavity widens, and more osteoclasts are recruited to thecutting-cone surface. As the cutting cone expands, the lateral strainsincrease, until the lateral signal becomes high enough to inhibit furtherlateral resorption. From then on, the cutting cone proceeds in thevertical direction, while remaining constant in size. Fig. 7 shows thefinal configurations for all 5 simulations. The osteons in the 17.5 MPasimulations are clearly wider than the 19.5 MPa osteons. Thesimulations were performed 5 times with different random seeds.Fig. 8 shows the resulting osteon widths for all simulations. For the19.5 MPa simulation only two data points are given, because in threesimulations the osteons ceased to grow: the cutting cones ‘died out’when detaching osteoclasts were not timely replaced by originatingones.

The simulations where the strain magnitude increased left toright from 9 to 27 MPa and from 6 to 30 MPa, produced osteonssimilar to the ones described above. The simulation at a steepergradient, from 0 to 36 MPa, produced a different cavitation pattern

Fig. 6. Starting from the initial cavity, osteoclasts (OCLs) excavate a tunnel along the loading abrown).

(Fig. 9). In this simulation resorption is not only confined to the topand bottom, but it continues along the entire left side of the tunnel,although at a slower pace than the top and bottom osteoclasts. Thetop and bottom osteoclasts proceed more or less in the verticaldirection, but slowly move towards the left. As a result, all osteoclastsin Fig. 9 are on the left half of the simulation volume, whereas theinitial cavity was precisely in the middle. The strain distributionthroughout the simulation shows an asymmetric strain distributionwith higher strains on the right side than on the left side. The strainconcentration to the left slows down the osteoclasts there, but doesnot stop them, because this strain concentration continuouslyweakens and moves to the left, as the top and bottom tips proceedin a direction skewed to the left.

Discussion

We present a bone-remodeling theory according to which strain-induced osteocyte signals restrain cutting cones. Because strainsaround a cavity are low in loading and high in transverse directions[51], this mechanism orients BMU's in the loading direction [56]. It canexplain why osteons generally run parallel to the main loadingdirection [41]. The restraints on the cutting cones increase withloading magnitude. As our simulations demonstrate, this reducescutting-cone size, which results in smaller osteon diameters. Hence,the theory provides an explanation for the inverse relationshipbetween loading magnitude and osteon diameter.

Figs. 7 and 8 show that this relationship is quite strong in oursimulations. Relatively small changes in the loading magnitude causelarge variations in osteon diameter. We do not want to suggest thatthis relationship is equally strong in actual bone. In our simulationsthe osteocyte signal depends linearly on the strain-energy-density in a2-dimensional architecture, whereas in actual bone osteocytic sig-nalling might be a nonlinear response to strain-induced fluid flow in3-dimensional architectures. We suggest that any mechanism, inwhich strain-induced signals inhibit osteoclasts, will produce an in-verse relationship between loading magnitude and osteon diameter.

For high loaded bone, our simulations predict that the cuttingcones ‘died out’ since detaching osteoclasts were not timely replacedby originating ones. This would suggest that habitually overstressedcortical bone may have no intracortical remodeling. It is questionable,however, if this would be expected in real bone as well. Habituallyoverstressed bone would accumulate microdamage, which wouldlikely disrupt osteocyte signaling. Several studies indicate thatmicrodamage induces osteocyte apoptosis [3,42,57]. The reductionof osteocyte signal due to apoptosis would enable osteoclast to enterthese regions and create a new osteon. These effects of microdamageand apoptosis, however, are not included in the model.

The proposed mechanism, inhibition of resorption by strain-induced osteocyte signals, is not really controversial. It is widely

xis. Osteoblasts (OBLs) are recruited to the tunnel wall, where they form new bone (light

Fig. 8. Osteon widths at different loading magnitudes.

Fig. 7. Final configurations of the 5 simulations. The mean width of each osteon is given below.

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believed that osteocytes serve as sensors of mechanical stimuli withinbone tissue and there are also indications that osteocytic signals caninhibit bone resorption by osteoclasts [6,19]. The actual osteocyticsignaling molecules, expressed upon mechanical stimulation toinhibit osteoclastic bone resorption, remain to be identified. Nitricoxide (NO) has been proposed as a likely candidate [6,28,54]. It isreleased by osteocytes after fluid flow stimulation [2] and causesosteoclasts to detach form the bone surface [32]. The RANKL/OPGsystem could also be involved. Receptor activator of NF-κB ligand(RANKL) is an important factor for osteoclast differentiation andsurvival, and osteoprotegerin (OPG) is its antagonist [53]. Both havebeen shown to be expressed by osteocytes [64], suggesting thatosteocytes can either stimulate or inhibit osteoclasts, depending onthe expressed RANKL/OPG ratio. Recently, You et al. [61] found thatthis ratio decreases after mechanical stimulation of osteocytes,implying a shift toward inhibition of osteoclasts.

In our model the coupling between osteoclast and osteoblastactivity is purely regulated by mechanotransduction regulated byosteocytes. No direct coupling between osteoclast and osteoblast cellstypes is assumed. Although a direct coupling between these cell typescould be included in the model, we demonstrate in this and earlierpapers [44,56] that this is not necessary to explain typical morpho-logical phenomena related to bone loading. Related to this point, itshould be noted that the diameter of the osteon is determined solelyby the osteoclasts. The osteoblasts determine only to what extent andhow quickly the tunnel will be filled.

The model incorporates osteoclast origination. Osteoclasts aremultinucleated cells that originate from mononuclear precursors.These precursors tightly adhere to bone, and fuse with each other toform multinucleated osteoclasts [37]. After fusion, the nuclei have amean lifespan of 12.5 days [24], which is much shorter than theduration of BMU progression. Hence, the multinucleated osteoclastteam in the cutting cone requires a continuous supply of new pre-osteoclasts. This supply is provided by a developing blood vessel thatclosely follows the cutting cone [40]. It is unclear to what extent therecruited pre-osteoclasts fuse with existing osteoclasts or merge toform new osteoclasts. We assume that where pre-osteoclasts canadhere to exposed bone surface, the formation of new osteoclasts ispossible.

The model also incorporates osteoclast death, assuming thatdetachment from the bone surface induces apoptosis. This assumptionwas first based on the observation by Fukushima et al. [15] that pre-osteoclasts only fuse with osteoclasts attached to the bone surface.Thus, considering the short lifespan of osteoclast nuclei [24], osteo-clasts would not survive an extended period of detachment. Further-more, recent studies have clearly demonstrated that detachmentinduces apoptosis in osteoclasts [46,63].

The resorption spaces in our simulations develop in two oppositedirections along the loading axis. Although most schematic repre-sentations in literature depict the BMU as developing in one direction,it was shown by Tappen [55] that simultaneous ‘proximal–distaltunneling’ is very common. While tracing developing osteons in serialbone slices, he often found resorption spaces at both their distal andproximal ends, indicating that these osteons were advancing in bothdirections. Note that the proximal–distal axis generally corresponds tothe loading axis in human long bones. Johnson [26] used the term‘double-ended osteons’ for proximal–distal tunneling, and decribedthem as: “cavity formation may start at one point and extendsimultaneously in opposite directions along the axis, with two cuttingcones”.

The double-ended osteon development in our simulations is astraightforward result of the strain environment around the cavity.This initial cavity, however, is a simplified representation of BMUorigination: there are no other porosities nearby, which simplifies thestrain environment, and its entire internal surface is exposed toosteoclastic resorption. Few studies treat BMU origination in corticalbone. Tappen [55] reported that most BMU's originate from the wallsof haversian canals, from a so-called ‘breakout zone’, where they firstresorb laterally through the canal wall, before turning proximally anddistally. It is unknown what causes this initial breakout, whereresorption occurs perpendicular to the main loading direction.Possibly it is triggered by microdamage near a haversion canal oforigin [35]. BMU origination may also start on the surface of aVolkmann's canal [25]. The BMU is then already oriented either distally

Fig. 10. Sketches from Johnson [26] of longitudinal sections of human cortex showingvarious cavitation patterns. Number 4 is described as an eccentric cavitation withresorption along one side facing the marrow cavity and refill on the opposite side.

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or proximally from the start, while the opposite side of the Volkmann'scanal may still be covered by lining cells. It is conceivable that themode of BMU origination largely determines whether an osteon willbe single-or double-ended.

The simulation with the steep gradient in strain magnituderesulted in a tunnel with resorption not only confined to the tips,but also along the entire lesser-strained side (Fig. 9). This type ofcavitation strongly resembles the fourth sketch in Fig. 10, whichwas described by Johnson [26] as: “Cavities may develop excentricallyby osteoclast resorption along one side of the length of the canal(the side facing toward older bone near themarrow cavity) while refillis active on the opposite side.” The description that the resorbing sidefaces the marrow cavity reveals another similarity: strain magnitudesdecrease towards the marrow cavity. This type of BMU is called a‘drifting osteon’ and was extensively investigated and reviewed byRobling and Stout [43]. They implicate strain gradients as a cause forthe drift towards the endosteum. Our simulations indeed suggestdrifting osteons may arise as a result of steep gradients in the strainmagnitude. We can also speculate that in an idealized cortex withoutany further porosities, a drifting osteon would drift uninterruptedlytoward the marrow cavity.

Our theory provides an explanation, but no function, for theinverse relationship between loadingmagnitude and osteon diameter.It is interesting to note that, from a mechanical point of view, itis beneficial to have smaller osteons in highly loaded regions forthree reasons. First, the strength of the bone will be less affectedduring the generation of a new osteon if it creates a smaller cavity.Second, a large number of smaller osteonswill bemore efficient than asmall number of larger osteons to reduce the effect of damageaccumulation since cement lines effectively serve as barriers to crackpropagation [16,38]. Third, it was suggested by Skedros et al. [49] thatsmaller osteon diameter in high-strain bone serves to enhance pulloutresistance. Osteon pullout occurs when an osteon that bridges a crack,is pulled out from the bone on one side of the crack [21]. This requiresdebonding along its cement line surface. Osteon diameter affects theratio between shear stress in the cement line and tensile stress on theosteon. Given a tensile force F on an osteon with diameter d andpullout length L, the tensile stress σ on the osteon and the shear stressτ in the cement line are given by σ=4F/πd2 and τ=F/πdL. The ratiobetween τ and σ thus is τ/σ=d/4L. Theoretically, pullout is thus morelikely when osteons are larger in diameter, because this increases

Fig. 9.With the imposed load steeply increasing from left to right, resorption continuesnot only in the tips but also along the entire left side of the tunnel.

shear stress in the cement line boundary relative to the tensile stresson the osteon [34].

Although we mention these benefits from strain-controlled osteondiameter, they should be seen in a wider perspective. In our currentand previous paper, we hypothesize that one mechanism (strain-induced osteocytes inhibit osteoclasts) causes several phenomena,such as osteonal load alignment, trabecular load alignment, resorptionof dead osteocytes, smaller osteon diameter in high-strain regions,double-ended osteons and drifting osteons. The functional meaning ofthis regulatory mechanism thus extends far beyond its effects onosteon diameter.

Acknowledgments

This work was supported by the Netherlands Organization forScientific Research, section Computational Life Sciences (NWO/CLS,grant number 635.100.014).

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