Relating osteon diameter to strain.

Abstract

Osteon diameter is generally smaller in bone regions that experience larger strains. A mechanism relating osteon diameter to strain is as yet unknown. We propose that strain-induced osteocyte signals inhibit osteoclastic bone resorption. This mechanism was previously shown to produce load-aligned osteons in computer simulations. Now we 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 moving down the loading axis. Such 'double-ended osteons' were reported in literature as a common type of osteon development. Further, we find that a steep gradient in strain magnitude can result in an osteonal tunnel with continuous resorption along the less strained side, which corresponds to 'drifting osteons' reported in literature.

Full text

Page 1

Relating osteon diameter to strain
René F.M. van Oers⁎, Ronald Ruimerman, Bert van Rietbergen, Peter A.J. Hilbers, Rik Huiskes
Department of Biomedical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
a b s t r a c ta r t i c l ei n f o
Article history:
Received 14 December 2007
Revised 24 April 2008
Accepted 10 May 2008
Available online 28 May 2008
Edited by: David Burr
Keywords:
Osteon diameter
Osteoclasts
BMU
Osteon diameter is generally smaller in bone regions that experience larger strains. A mechanism relating
osteon diameter to strain is as yet unknown. We propose that strain-induced osteocyte signals inhibit
osteoclastic bone resorption. This mechanism was previously shown to produce load-aligned osteons in
computer simulations. Nowwefind that it also predicts smallerosteondiameter forhigherloads. Additionally,
we find that our model predicts osteon development with two cutting cones, one moving up and one moving
down the loading axis. Such ‘double-ended osteons’ were reported in literature as a common type of osteon
development. Further, we find that a steep gradient in strain magnitude can result in an osteonal tunnel with
continuous resorption along the less strained side, which corresponds to ‘drifting osteons’ reported in
literature.
© 2008 Elsevier Inc. All rights reserved.
Introduction
Osteons are the basic structures of cortical bone. An osteon is a
tube-like structure with a central canal. It is formed when a ‘cutting
cone’ of osteoclasts excavates a tunnel, and a ‘closing cone’ of
osteoblasts fills the tunnel with new bone [40]. The osteoblasts do
not completely ‘close’ the tunnel, as blood vessels remain in the
middle. The team of osteoclasts and -blasts is known as a basic
multicellular unit (BMU). In humans, osteons range in diameter from
150 to 350 μm [1,4,7]. Osteon diameter is obviously determined by the
size of the osteoclast cutting cone. What determines the size of the
cutting cone is less clear. Frost [14] suggested an inverse relation
between osteon diameter and strain magnitude, since osteons are
usually wider in the endosteal side of the cortex, while strains are
larger in the periosteal side. Skedros et al. [47,48,50] analyzed the heel
bone of hoofed animals, in which the tension cortex habitually
experiences 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 inverse
relation comes from a study on immobilized primates [62]. A 7-month
period of immobilization resulted in cortical resorption cavities of
500–1500μmdiameter.Whentheimmobilizationperiodwasfollowed
by a recovery period, new bone was formed in these cavities and the
resulting structures appeared as “unusuallylarge osteons”. Thus, there
appearstobeaninverserelationbetweenstrainmagnitudeandosteon
diameter. The question then is: what mechanism relates osteon
diameter to strain?
A relation between strain and osteon diameter implies a relation
between strain and osteoclastic bone resorption. It is evident that
mechanical loads influence bone remodeling, as both osteons [41] and
trabeculae [59] are aligned to the principal stress directions. It is also
well established that bone mass is increased with increasing mechan-
ical loading and vice-versa. Muscle-building exercise was shown to
increase bone density at weight-bearing sites in young men [9].
Exercisethatinvolvesimpactloadingincreasedthebonedensityinthe
femurandradiusofprepubertalgirls[10].Immobilizationduetospinal
cord injury decreased the bone density at weight-bearing sites in
adults [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 osteoclast
activity and an increase in osteoblast activity [20]. The opposite was
observed in immobilized rat hind limbs [58]. In immobilized primates
[36] researchers observed an increase in resorptive surfaces and a
decrease in formative surfaces in trabecular bone, and impairment of
bone formation in cortical remodeling spaces. These effects of
mechanical loading on osteoclasts and osteoblasts may be regulated
by osteocytes. It is believed that osteocytes can sense mechanical
deformation of the bone [5,11,23,30,60]. In response they could signal
to 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 emit
biochemical signals that inhibit osteoclasts and stimulate osteoblasts.
This mechanism was previously used to explain osteonal [51,56] and
trabecular load alignment [23,44,56]. Because our model entails a
Bone 43 (2008) 476–482
⁎ 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. All rights reserved.
doi:10.1016/j.bone.2008.05.015
Contents lists available at ScienceDirect
Bone
journal homepage: www.elsevier.com/locate/bone

Page 2

relation between strain and osteoclastic bone resorption, we now
investigate how it relates osteon diameter to mechanical loads.
Methods: the model
The model used in this study is similar to the model used in an
earlier study [56]. A bone structure is mapped onto a finite element
mesh consisting of square elements of uniform size Δx [m]. We
introduce,foreachelement,arelativebonedensitym(x,t),wherevector
x [m] denoteselement position and t [day] denotes time. Thedensitym
(x,t) ranges from a minimal value mminto 1. At mminthe element is
considered to be a marrow element, above mminit is considered to be a
bone element. The time t is represented by increments Δt [day], during
which the bone density of the elements can change. Hence, the m(x,t)-
valuesoftheelementsconstituteachangeablebonemorphology.Them
(x,t)-values also determine the stiffness of the elements, according to
[12]:
E x;t
ð Þ ¼ Emaxd m x;t
ðÞ3;
where Emax[Pa] is the Young's modulus for elements at maximal bone
density. The structure is subjected to external loads. We use a static
load to represent a cyclic load of given amplitude σ′ and frequency f.
The magnitude of the static load σ′ is chosen such that the resulting
strain-energy-density (SED) equals the peak SED rate of the dynamic
load:
ffiffiffiffi
For a derivation of this formula, see Huiskes [22] or Ruimerman
et al. [45]. Load transfer through the structure is evaluated by finite
element analysis (FEA), assuming isotropic and linear-elastic material
behavior. For 2D simulations we assume plane-strain conditions. FEA
is performed at the start of each increment to correct for the gradual
morphological changes in the bone.
σ V¼ 2:02σf:
p
Osteocytes
Osteocytes, located within the bone tissue, are assumed to sense a
mechanical stimulus R [J m−3s−1], a typical SED rate experienced in a
recent loading history at its location. Based on this sensation, the
osteocytes 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 a
signal molecule, where synthesis and decay are in balance [29]. Each
element receives an accumulated signal S from nearby osteocytes,
according to [22]:
S x;t
ð
where μ [J−1m3s] is the osteocyte mechanosensitivity and D [m] is a
diffusion-decay constant, xiis the position of osteocyte i and n are the
number of osteocytes less than dinfl[m] removed from x, where dinflis
the truncation distance for the osteocyte signal. Note that the signal
doesnot accumulate over the increments; each increment the signalis
calculated anew from the current SED distribution. Numerical settings
for the parameters were given and discussed in our previous paper
[56].
Þ ¼ ∑n
i¼1R xi;t
ðÞ:μ:e−d x;xi
ðÞ=D;
Osteoclasts
When the osteocyte signal S is strong enough, it inhibits osteoclast
attachment to the bone surface and their subsequent resorption
activities. Osteoclasts are explicitly modeled, using a cell simulation
method based on the cellular Potts model (CPM) [17,33]. This
simulation method was extensively described in our previous paper
[56], and will only be summarized here. An osteoclastoccupies several
adjacent elements and may change position by entering and leaving
elements. An osteoclast tries − by preferring certain moves over
others − to minimize an energy H:
H ¼ Hvolþ Hsurf¼ λd V−V0
ðÞ2þ
Z
surf
h A
ð Þd dA:
The volume energy Hvolensures 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. V0is based on a typical osteoclast
diameter of 50 μm [13]. The surface energy Hsurfof an osteoclast is
determined by its surroundings, as the contact energy h [m−2] of a
surface patchdA [m2] depends onthe type of the neighboringelement
(Fig. 2). That contact energies differ for different substrates is central
to the CPM. The cell attaches to substrates with low h and detaches
from substrates with high h, in order to minimize its surface energy.
The contact energy to bone hbis a function of the osteocyte signal S
in the neighboring bone element (Fig. 3). What it amounts to is that an
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 widest
osteons were observed in the tension endosteal region, the thinnest in the compression periosteal region. Images are from a skeletally mature elk calcaneus sectioned transverselyat
60% of length. (Adapted from Skedros et al. [50].)
477
R.F.M. van Oers et al. / Bone 43 (2008) 476–482

Page 3

osteoclast adheres to bone surfaces where the osteocyte signal is
weak. Two signal thresholds are used, S0and S1. If the osteocyte signal
is below S0, osteoclast-bone adhesion is strong. Between S0and S1
adhesion weakens and above S1there is no adhesion. Once settled, an
osteoclast can resorb adjacent bone elements. As the osteoclast enters
a bone element, its bone density is reduced to mmin. The probability
for resorption depends on the local adhesion between the osteoclast
and the bone surface, which is determined by the osteocyte signal.
In our previous study [56], we used a fixed number of osteoclasts
in simulations of osteon development. This limits the width of the
osteoclast cutting cone and the resulting osteon. Since these are the
topics of investigation, we now allow addition and removal of
osteoclasts from the cutting cone. We assume that osteoclast
origination can occur on exposed bone surfaces, where adhesion
conditions are favorable. The rationale for this assumption is
presented in the discussion of this paper. At every increment new
osteoclasts can originate, with origination probability OP [m−2d−1],
on exposed (i.e., not covered by osteoclasts or-blasts) bone surfaces
with low osteocyte signal (SbS0). The model also describes osteoclast
death. It is assumed that osteoclasts detached from the bone surface
remain viable for only a short period. The relevance of this
assumption is also addressed in the discussion. If an osteoclast
does not resorb −because it is not adjacent to bone ordoes not adhere
to adjacent bone− for a period Td[d], it is removed.
Osteoblasts
Osteoblasts are recruited to exposed bone surfaces where the
osteocyte signal exceeds a threshold Soblfor a period Tr[d]. They then
form bone according to
Δmobl¼ τd S x;t
ðÞ−Sobl
ðÞdΔt
Δx
where the change in m(x,t) due to osteoblast activity is denoted with
the index obl, and τ [m day−1] determines the bone formation rate.
The newly formed bone is assumed to have the same osteocyte
density 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 mm2piece of compact
bone tissue, subjected to compressive loads in the vertical direction
(Fig. 4A). Osteocytes are positioned in the bone tissue at a density of
1600 mm−2[39]. Osteoclasts start from an initial resorption cavity of
180 μm diameter (Fig. 4B). For the origination probability OP of new
osteoclasts in the cutting cone we use a value of 5000 m−1·d−1. This
translates to 1 new osteoclast per day on a 200 μmwide cutting cone if
it happens to be completely exposed. The period Tdduring which a
detached osteoclast remains viable is set to 1 day, as recent studies
have demonstrated that detachment induces apoptosis in osteoclasts
within a matter of hours [46,63].
In our previous study [56] wesimulated osteon development using
an 18 MPa load. Because we want to investigate the influence of the
load on osteon diameter, we compare simulations at five different
loading magnitudes: 17.5, 18.0, 18.5, 19.0 and 19.5 MPa. All other
parameter settings are as in [56]. Because osteoclast origination and
movement are stochastic, we performed each simulation five times
with different random seeds (a ‘random seed’ is a settable starting
point for a random number generator). The simulations ran for 100
model increments, representing 25 day remodeling periods.
In addition, we perform 3 simulations where loading is also com-
pressive in the verticaldirection, but its magnitude increases from left to
right,withanaverageof18MPa.Thisramploadingwaschosenasvarying
from9to27,6to30,andfrom0to36MPa.Thisloadingismeanttoreflect
the increasing strain magnitude from endosteum to periosteum (Fig.1).
Results
Fig. 5A shows the strain-energy-density (SED) around the initial
cavity, for the 19.5 MPa simulation. Strains are low along the loading
Fig. 3. The osteocyte signal S at the bone surface determines osteoclast adhesion. A) The
contact energy to bone hbis a function of S. B) Configurations of minimal surface energy
for different hb.
Fig. 4. A) The initial configuration consists of a 4×4 mm2piece of compact bone with a
resorption 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. 2. Sketch of the CPM. The contact energy of an osteoclast (OCL1) to marrow hmand
other osteoclasts hoclare constants. The contact energy to bone hb(S) depends on the
osteocyte signal S.
478
R.F.M. van Oers et al. / Bone 43 (2008) 476–482

Page 4

axis and high in the transverse direction. The osteocyte signal (Fig. 5B)
induced by these strains has a similar distribution. Fig. 6 shows the
development of the 19.5 MPa simulation. The eight initial osteoclasts,
one element in size at first, quickly expand to the target volume. The
osteoclastslocated at top andbottom of the cavitystart toresorb bone,
while those located on the left and right side of the cavity detach from
the surface and die. The top and bottom osteoclasts proceed to resorb
a tunnel along the loading axis. Thus, there are two cutting cones, one
moving up and one moving down the loading axis. During the course
of the simulation, the osteocyte signal around the cutting cones
remains low in loading direction and high in the transverse directions.
Osteoblasts are recruited to the eroded surface where they form a
closing cone of new bone.
In all simulations, the strains −and consequently the osteocyte
signal− around the initial cavity are also high in the loading direction
and high at the sides. However, the ‘lateral signal’ is not as strong as
in the 19.5 MPa simulation, and lateral resorption is initially not
inhibited. The cavity widens, and more osteoclasts are recruited to the
cutting-cone surface. As the cutting cone expands, the lateral strains
increase,untilthelateralsignalbecomeshighenoughtoinhibitfurther
lateral resorption. From then on, the cutting cone proceeds in the
vertical direction, while remaining constant in size. Fig. 7 shows the
final configurations for all 5 simulations. The osteons in the 17.5 MPa
simulations are clearly wider than the 19.5 MPa osteons. The
simulations were performed 5 times with different random seeds.
Fig. 8 shows the resulting osteon widths for all simulations. For the
19.5 MPa simulation only two data points are given, because in three
simulations the osteons ceased to grow: the cutting cones ‘died out’
when detaching osteoclasts were not timely replaced by originating
ones.
The simulations where the strain magnitude increased left to
right from 9 to 27 MPa and from 6 to 30 MPa, produced osteons
similar to the ones described above. The simulation at a steeper
gradient, from 0 to 36 MPa, produced a different cavitation pattern
(Fig. 9). In this simulation resorption is not only confined to the top
and bottom, but it continues along the entire left side of the tunnel,
although at a slower pace than the top and bottom osteoclasts. The
top and bottom osteoclasts proceed more or less in the vertical
direction, but slowly move towards the left. As a result, all osteoclasts
in Fig. 9 are on the left half of the simulation volume, whereas the
initial cavity was precisely in the middle. The strain distribution
throughout the simulation shows an asymmetric strain distribution
with higher strains on the right side than on the left side. The strain
concentration to the left slows down the osteoclasts there, but does
not stop them, because this strain concentration continuously
weakens and moves to the left, as the top and bottom tips proceed
in a direction skewed to the left.
Discussion
We present a bone-remodeling theory according to which strain-
induced osteocyte signals restrain cutting cones. Because strains
around a cavity are low in loading and high in transverse directions
[51], this mechanism orients BMU's in the loading direction [56]. It can
explain why osteons generally run parallel to the main loading
direction [41]. The restraints on the cutting cones increase with
loading magnitude. As our simulations demonstrate, this reduces
cutting-cone size, which results in smaller osteon diameters. Hence,
the theory provides an explanation for the inverse relationship
between loading magnitude and osteon diameter.
Figs. 7 and 8 show that this relationship is quite strong in our
simulations. Relatively small changes in the loading magnitude cause
large variations in osteon diameter. We do not want to suggest that
this relationship is equally strong in actual bone. In our simulations
theosteocyte signaldepends linearlyonthe strain-energy-densityin a
2-dimensional architecture, whereas in actual bone osteocytic sig-
nalling might be a nonlinear response to strain-induced fluid flow in
3-dimensional architectures. We suggest that any mechanism, in
which 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 cutting
cones ‘died out’ since detaching osteoclasts were not timely replaced
by originating ones. This would suggest that habitually overstressed
cortical bone may have no intracortical remodeling. It is questionable,
however, if this would be expected in real bone as well. Habitually
overstressed bone would accumulate microdamage, which would
likely disrupt osteocyte signaling. Several studies indicate that
microdamage induces osteocyte apoptosis [3,42,57]. The reduction
of osteocyte signal due to apoptosis would enable osteoclast to enter
these regions and create a new osteon. These effects of microdamage
and 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
Fig. 6. Starting from the initial cavity, osteoclasts (OCLs) excavate a tunnel along the loading axis. Osteoblasts (OBLs) are recruited to the tunnel wall, where they form new bone (light
brown).
Fig. 5. A) Around the cavity, strains are low in the loading direction (vertical) and high
in transverse direction. B) The model osteocytes translate this into corresponding bio-
chemical signals.
479
R.F.M. van Oers et al. / Bone 43 (2008) 476–482

Page 5

believed that osteocytes serve as sensors of mechanical stimuli within
bone tissue and there are also indications that osteocytic signals can
inhibit bone resorption by osteoclasts [6,19]. The actual osteocytic
signaling molecules, expressed upon mechanical stimulation to
inhibit osteoclastic bone resorption, remain to be identified. Nitric
oxide (NO) has been proposed as a likely candidate [6,28,54]. It is
released by osteocytes after fluid flow stimulation [2] and causes
osteoclasts to detach form the bone surface [32]. The RANKL/OPG
system could also be involved. Receptor activator of NF-κB ligand
(RANKL) is an important factor for osteoclast differentiation and
survival, and osteoprotegerin (OPG) is its antagonist [53]. Both have
been shown to be expressed by osteocytes [64], suggesting that
osteocytes can either stimulate or inhibit osteoclasts, depending on
the expressed RANKL/OPG ratio. Recently, You et al. [61] found that
this ratio decreases after mechanical stimulation of osteocytes,
implying a shift toward inhibition of osteoclasts.
In our model the coupling between osteoclast and osteoblast
activity is purely regulated by mechanotransduction regulated by
osteocytes. No direct coupling between osteoclast and osteoblast cells
types is assumed. Although a direct coupling between these cell types
could be included in the model, we demonstrate in this and earlier
papers [44,56] that this is not necessary to explain typical morpho-
logical phenomena related to bone loading. Related to this point, it
should be noted that the diameter of the osteon is determined solely
by the osteoclasts. The osteoblasts determine only to what extent and
how quickly the tunnel will be filled.
The model incorporates osteoclast origination. Osteoclasts are
multinucleated cells that originate from mononuclear precursors.
These precursors tightly adhere to bone, and fuse with each other to
form multinucleated osteoclasts [37]. After fusion, the nuclei have a
mean lifespan of 12.5 days [24], which is much shorter than the
duration of BMU progression. Hence, the multinucleated osteoclast
team in the cutting cone requires a continuous supply of new pre-
osteoclasts. This supply is provided by a developing blood vessel that
closely follows the cutting cone [40]. It is unclear to what extent the
recruited pre-osteoclasts fuse with existing osteoclasts or merge to
form new osteoclasts. We assume that where pre-osteoclasts can
adhere to exposed bone surface, the formation of new osteoclasts is
possible.
The model also incorporates osteoclast death, assuming that
detachment from thebone surface inducesapoptosis. This assumption
was 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 detachment
induces apoptosis in osteoclasts [46,63].
The resorption spaces in our simulations develop in two opposite
directions 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–distal
tunneling’ is very common. While tracing developing osteons in serial
bone slices, he often found resorption spaces at both their distal and
proximal ends, indicating that these osteons were advancing in both
directions. Note that the proximal–distal axis generallycorresponds to
the loading axis in human long bones. Johnson [26] used the term
‘double-ended osteons’ for proximal–distal tunneling, and decribed
them as: “cavity formation may start at one point and extend
simultaneously in opposite directions along the axis, with two cutting
cones”.
The double-ended osteon development in our simulations is a
straightforward result of the strain environment around the cavity.
This initial cavity, however, is a simplified representation of BMU
origination: there are no other porosities nearby, which simplifies the
strain environment, and its entire internal surface is exposed to
osteoclastic resorption. Few studies treat BMU origination in cortical
bone. Tappen [55] reported that most BMU's originate from the walls
of haversian canals, from a so-called ‘breakout zone’, where they first
resorb laterally through the canal wall, before turning proximally and
distally. It is unknown what causes this initial breakout, where
resorption occurs perpendicular to the main loading direction.
Possibly it is triggered by microdamage near a haversion canal of
origin [35]. BMU origination may also start on the surface of a
Volkmann'scanal[25].TheBMUisthenalreadyorientedeitherdistally
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.
480
R.F.M. van Oers et al. / Bone 43 (2008) 476–482

Page 6

orproximallyfromthestart,whiletheoppositesideof theVolkmann's
canal may still be covered by lining cells. It is conceivable that the
mode of BMU origination largely determines whether an osteon will
be single-or double-ended.
The simulation with the steep gradient in strain magnitude
resulted in a tunnel with resorption not only confined to the tips,
but also along the entire lesser-strained side (Fig. 9). This type of
cavitation strongly resembles the fourth sketch in Fig. 10, which
was described by Johnson [26] as: “Cavities may develop excentrically
by osteoclast resorption along one side of the length of the canal
(the side facing towardolder bone near the marrowcavity) while refill
is active on the opposite side.” The description that the resorbing side
faces the marrow cavity reveals another similarity: strain magnitudes
decrease towards the marrow cavity. This type of BMU is called a
‘drifting osteon’ and was extensively investigated and reviewed by
Robling and Stout [43]. They implicate strain gradients as a cause for
the drift towards the endosteum. Our simulations indeed suggest
drifting osteons may arise as a result of steep gradients in the strain
magnitude. We can also speculate that in an idealized cortex without
any further porosities, a drifting osteon would drift uninterruptedly
toward the marrow cavity.
Our theory provides an explanation, but no function, for the
inverse relationship between loading magnitude and osteon diameter.
It is interesting to note that, from a mechanical point of view, it
is beneficial to have smaller osteons in highly loaded regions for
three reasons. First, the strength of the bone will be less affected
during the generation of a new osteon if it creates a smaller cavity.
Second,a largenumberof smallerosteonswill be more efficientthan a
small number of larger osteons to reduce the effect of damage
accumulation since cement lines effectively serve as barriers to crack
propagation [16,38]. Third, it was suggested by Skedros et al. [49] that
smallerosteon diameter in high-strain bone serves to enhance pullout
resistance. Osteon pullout occurs when an osteon that bridges a crack,
is pulled out from the bone on one side of the crack [21]. This requires
debonding along its cement line surface. Osteon diameter affects the
ratio between shear stress in the cement line and tensile stress on the
osteon. Given a tensile force F on an osteon with diameter d and
pullout length L, the tensile stress σ on the osteon and the shear stress
τ in the cement line are given by σ=4F/πd2and τ=F/πdL. The ratio
between τ and σ thus is τ/σ=d/4L. Theoretically, pullout is thus more
likely when osteons are larger in diameter, because this increases
shear stress in the cement line boundary relative to the tensile stress
on the osteon [34].
Although we mention these benefits from strain-controlled osteon
diameter, they should be seen in a wider perspective. In our current
and previous paper, we hypothesize that one mechanism (strain-
induced osteocytes inhibit osteoclasts) causes several phenomena,
such as osteonal load alignment, trabecularloadalignment, resorption
of dead osteocytes, smaller osteon diameter in high-strain regions,
double-ended osteons and drifting osteons. The functional meaningof
this regulatory mechanism thus extends far beyond its effects on
osteon diameter.
Acknowledgments
This work was supported by the Netherlands Organization for
Scientific Research, section Computational Life Sciences (NWO/CLS,
grant number 635.100.014).
References
[1] Agerbaek MO, Eriksen EF, Kragstrup J, Mosekilde L, Melsen F. A reconstruction of
the remodelling cycle in normal human cortical iliac bone. Bone Miner
1991;12:101–12.
[2] Bacabac RG, Smit TH, Mullender MG, Dijcks SJ, Van Loon JJ, Klein-Nulend J. Nitric
oxide production by bone cells is fluid shear stress rate dependent. Biochem
Biophys Res Commun 2004;315:823–9.
[3] Bentolila V, Boyce TM, Fyhrie DP, Drumb R, Skerry TM, Schaffler MB. Intracortical
remodeling in adult rat long bones after fatigue loading. Bone 1998;23:275–81.
[4] Brockstedt H, Bollerslev J, Melsen F, Mosekilde L. Cortical bone remodeling
in autosomal dominant osteopetrosis: a study of two different phenotypes. Bone
1996;18:67–72.
[5] Burger EH, Klein-Nulend J. Mechanotransduction in bone — role of the lacuno-
canalicular network. FASEB J 1999;13(Suppl):S101–12.
[6] Burger EH, Klein-Nulend J, Smit TH. Strain-derived canalicular fluid flow regulates
osteoclastactivityinaremodelingosteon—aproposal.JBiomech2003;36:1453–9.
[7] Cattaneo C, Dimartino S, Scali S, Craig OE, Grandi M, Sokol RJ. Determining the
human origin of fragments of burnt bone: a comparative study of histological,
immunological and DNA techniques. Forensic Sci Int 1999;102:181–91.
[8] Chow JW, Chambers TJ. Indomethacin has distinct early and late actions on bone
formation induced by mechanical stimulation. Am J Physiol Endocrinol Metab
1994;267:E287–92.
[9] Colletti LA, Edwards J, Gordon L, Shary J, Bell NH. The effects of muscle-building
exercise on bone mineral density of the radius, spine, and hip in young men. Calcif
Tissue Int 1989;45:12–4.
[10] Courteix D, Lespessailles E, Peres SL, Obert P, Germain P, Benhamou CL. Effects of
physical training on bone mineral density in prepubertal girls: a comparative
study between impact-loading and non-impact-loading sports. Osteoporos Int
1998;8:152–8.
[11] Cowin SC, Moss-Salentijn L, Moss ML. Candidates for the mechanosensory system
in bone. J Biomech Eng 1991;113:191–7.
Fig. 9. With the imposed load steeply increasing from left to right, resorption continues
not only in the tips but also along the entire left side of the tunnel.
Fig. 10. Sketches from Johnson [26] of longitudinal sections of human cortex showing
various cavitation patterns. Number 4 is described as an eccentric cavitation with
resorption along one side facing the marrow cavity and refill on the opposite side.
481
R.F.M. van Oers et al. / Bone 43 (2008) 476–482

Page 7

[12] Currey JD. The effect of porosity and mineral content on the Young's modulus of
elasticity of compact bone. J Biomech 1988;21:131–9.
[13] EriksenEF,KassemM.Thecellularbasisofboneremodeling.Triangle1992;31:45–57.
[14] Frost HM. Skeletal structural adaptations to mechanical usage (SATMU):
2. Redefining Wolff's Law: the remodeling problem. Anat Rec 1990;226:414–22.
[15] Fukushima O, Bekker PJ, Gay CV. Characterization of the functional stages of
osteoclasts by enzyme histochemistry and electron microscopy. Anat Rec 1991;231:
298–315.
[16] Gibson VA, Stover SM, Gibeling JC, Hazelwood SJ, Martin RB. Osteonal effects on
elastic modulus and fatigue life in equine bone. J Biomech 2006;39:217–25.
[17] Glazier JA, Graner F. Simulation of the differential adhesion driven rearrangement
of biological cells. Phys Rev E 1993;47:2128–54.
[18] Heino TJ, Hentunen TA, Väänänen HK. Conditioned medium from osteocytes
stimulates the proliferation of bone marrow mesenchymal stem cells and their
differentiation into osteoblasts. Exp Cell Res 2004;294:458–68.
[19] Heino TJ, Hentunen TA, Väänänen HK. Osteocytes inhibit osteoclastic bone
resorption through transforming growth factor-β: enhancement by estrogen.
J Cell Biochem 2002;85:185–97.
[20] Hillam RA, Skerry TM. Inhibition of bone resorption and stimulation of formation by
mechanicalloadingofthemodelingratulnainvivo.JBoneMinerRes1995;10:683–9.
[21] Hiller LP, Stover SM, Gibson VA, Gibeling JC, Prater CS, Hazelwood SJ, et al. Osteon
pullout in the equine third metacarpal bone: effects of ex vivo fatigue. J Orthop Res
2003;21:481–8.
[22] Huiskes R. If bone is the answer, then what is the question? J Anat 2000;197:
145–56.
[23] Huiskes R, Ruimerman R, van Lenthe GH, Janssen JD. Effects of mechanical forces
on maintenance and adaptation of form in trabecular bone. Nature 2000;405:
704–6.
[24] Jaworski ZFG, Duck B, Sekaly G. Kinetics of osteoclasts and their nuclei in evolving
secondary haversian systems. J Anat 1981;133:397–405.
[25] Jaworski ZFG. Haversian systems and haversian bone. In: Hall BK, editor. Bone
metabolism and mineralization, vol. 4. Boca Raton, FL: CRC Press; 1992. p. 21–45.
[26] Johnson LC. Morphologic analysis of pathology. In: Frost HM, editor. Bone bio-
dynamics. Boston, MA: Little, Brown, and Company; 1964. p. 543–654.
[27] KiratliBJ,SmithAE,NauenbergT,KallfelzCF,PerkashI.Bonemineralandgeometric
changesthroughthe femur withimmobilization duetospinalcordinjury.JRehabil
Res Dev 2000;37:225–33.
[28] Klein-Nulend J, Nijweide PJ, Burger EH. Osteocyte and bone structure. Curr
Osteoporos Rep 2003;1:5–10.
[29] Lancaster J. Diffusion of free nitric oxide. Methods Enzymol 1996;268:31–50.
[30] Lanyon LE. Osteocytes, strain detection, bone modeling and remodeling. Calcif
Tissue Int 1993;53(Suppl):S102–7.
[31] Leblanc AD, Schneider VS, Evans HJ, Engelbretson DA, Krebs JM. Bone mineral loss
and recovery after 17 weeks of bedrest. J Bone Miner Res 1990;5:843–50.
[32] Mancini L, Moradi-Bidhendi N, BrandiML, MacIntyre I.Nitric oxide superoxide and
peroxynitrite modulate osteoclast activity. Biochem Biophys Res Commun
1998;243:785–90.
[33] Marée AFM, Hogeweg P. How amoeboids self-organize into a fruiting body:
multicellular coordination in Dictyostelium discoideum. Proc Natl Acad Sci U S A
2001;98:3879–83.
[34] Martin RB, Burr DB, Sharkey NA, editors. Skeletal tissue mechanics. New York:
Springer; 1998. p. 204.
[35] Martin RB. Is all cortical bone remodeling initiated by microdamage? Bone
2002;30:8–13.
[36] Mathews CHE, Aswani SP, Parfitt AM. Hypogravitational effects of hypodynamia on
bone cell function and the dynamics of bone remodeling. In: Kazarian L, Cann C,
Parfitt M, Simmons D, Morey-Holton E, editors. A 14-day ground-based
hypokinesia study in nonhuman primates: a compilation of results. NASA Tech.
Mem. 81268. Springfield, VA: NTIS; 1981. p. 16–28.
[37] Miyamoto T, Suda T. Differentiation and function of osteoclasts. Keio J Med
2003;52:1–7.
[38] Mohsin S, O'Brien FJ, Lee TC. Osteonal crack barriers in ovine compact bone. J Anat
2006;208:81–9.
[39] Mullender MG, Huiskes R. Proposal for the regulatory mechanism of Wolff's law.
J Orthop Res 1995;13:503–12.
[40] Parfitt AM. Osteonal and hemi-osteonal remodeling: the spatial and temporal
framework for signal traffic in adult human bone. J Cell Biochem 1994;55:273–86.
[41] Petrtyl M, Hert J, Fiala P. Spatial organization of the haversian bone in man.
J Biomech 1996;29:161–9.
[42] Qiu SJ, Hoshaw SJ, Gibson GJ, Lundin-Cannon KD, Schaffler MB. Osteocyte
apoptosis in reaction to matrix damage in compact bone. Trans Orthop Res Soc
1997;22:89.
[43] Robling AG, Stout SD. Morphology of the drifting osteon. Cells Tissues Organs
1999;164:192–204.
[44] RuimermanR,HilbersPAJ,vanRietbergenB,HuiskesR.Atheoreticalframeworkfor
strain-related trabecular bone maintenance and adaptation. J Biomech 2005;38:
931–41.
[45] Ruimerman R, Huiskes R, van Lenthe GH, Janssen JD. A computer-simulation
model relating bone-cell metabolism to mechanical adaptation of trabecular
architecture. Comput Methods Biomech Biomed Engng 2001;4:433–48.
[46] Sakai H, Kobayashi Y, Sakai E, Shibata M, Kato Y. Cell adhesion is a prerequisite for
osteoclast survival. Biochem Biophys Res Commun 2000;270:550–6.
[47] Skedros JG, Mason MW, Bloebaum RD. Differences in osteonal morphology be-
tweentensileand compressivecortices of a bending skeletal system: indications of
potential strain-specific differences in bone microstructure. Anat Rec 1994;239:
405–13.
[48] Skedros JG, Mason MW, Bloebaum RD. Modeling and remodeling in a developing
artiodactyl calcaneus: a model for evaluating Frost's Mechanostat hypothesis and
its corollaries. Anat Rec 2001;263:167–85.
[49] Skedros JG, Sorenson SM, Jenson NH. Are distributions of secondary osteon
variants useful for interpreting load history in mammalian bones? Cells Tissues
Organs 2007;185:285–307.
[50] Skedros JG, Su SC, Bloebaum RD. Biomechanical implications of mineral content
and microstructural variations in cortical bone of horse, elk, and sheep calcanei.
Anat Rec 1997;249:297–316.
[51] Smit TH, Burger EH. Is BMU-coupling a strain-regulated phenomenon? A finite
element analysis. J Bone Miner Res 2000;15:301–7.
[52] Su SC, Skedros JG, Bachus KN, Bloebaum RD. Loading conditions and cortical bone
construction of an artiodactyl calcaneus. J Exp Biol 1999;202:3239–54.
[53] Suda T, Takahashi N, Udagawa N, Jimi E, Gillespie MT, Martin TJ. Modulation of
osteoclast differentiation and function by the new members of the tumor necrosis
factor receptor and ligand families. Endocr Rev 1999;20:345–57.
[54] Tan SD, de Vries TJ, Kuijpers-Jagtman AM, Semeins CM, Everts V, Klein-Nulend J.
Osteocytes subjected to fluid flow inhibit osteoclast formation and bone re-
sorption. Bone 2007;41:745–51.
[55] Tappen NC. Three-dimensional studies of resorption spaces and developing
osteons. Am J Anat 1977;149:301–32.
[56] van Oers RFM, Ruimerman R, Tanck E, Hilbers PAJ, Huiskes R. A unified theory for
osteonal and hemi-osteonal remodeling. Bone 2008;42:250–9.
[57] Verborgt O, Gibson GJ, Schaffler MB. Loss of osteocyte integrity in association
with microdamage and bone remodeling after fatigue in vivo. J Bone Miner Res
2000;15:60–7.
[58] Weinreb M, Rodan GA, Thompson DD. Osteopenia in the immobilized rat hind
limb is associated with increased bone resorption and decreased bone formation.
Bone 1989;10:187–94.
[59] Wolff J. Das gesetz der transformation der knochen. Berlin: Hirschwald; 1892.
translated as: The law of bone remodeling. Berlin: Springer; 1986.
[60] You L, Cowin SC, Schaffler MB, Weinbaum S. A model for strain amplification in the
actin cytoskeleton of osteocytes due to fluid drag on pericellular matrix. J Biomech
2001;34:1375–86.
[61] You L, Temiyasathit S, Lee P, Kim C, Tummala P, Yao W, et al. Osteocytes as
mechanosensors in the inhibition of bone resorption due to mechanical loading.
Bone 2008;42:172–9.
[62] Young DR, Niklowitz WJ, Brown RJ, Jee WSS. Immobilization-associated osteo-
porosis in primates. Bone 1986;7:109–17.
[63] Zhao H, Ross FP, Teitelbaum SL. Unoccupied αvβ3 integrin regulates osteoclast
apoptosis bytransmitting a positive death signal. Mol Endocrinol 2005;19:771–80.
[64] Zhao S, Kato Y, Zhang Y, Harris S, Ahuja SS, Bonewald LF. MLO-Y4 osteocyte-like
cells support osteoclast formation and activation. J Bone Miner Res 2002;17:
2068–79.
482
R.F.M. van Oers et al. / Bone 43 (2008) 476–482