Am J Physiol Cell Physiol 299: C922–C929, 2010.
First published July 21, 2010; doi:10.1152/ajpcell.00465.2009.
Numerical modeling of oxygen distributions in cortical and cancellous bone:
oxygen availability governs osteonal and trabecular dimensions
Adam M. Zahm,1 Michael A. Bucaro,2 Portonovo S. Ayyaswamy,3 Vickram Srinivas,1 Irving M. Shapiro,1
Christopher S. Adams,4 and Karthik Mukundakrishnan5
1
Department of Orthopaedic Surgery, Thomas Jefferson University, Philadelphia, Pennsylvania; 2School of Engineering
and Applied Sciences, Harvard University, Cambridge, Massachusetts; 3Department of Mechanical Engineering
and Applied Sciences, University of Pennsylvania, Philadelphia, Pennsylvania; 4Department of Anatomy, Philadelphia
College of Osteopathic Medicine, Philadelphia, Pennsylvania; and 5Dassault Systemes Simulia, Providence,
Rhode Island
Submitted 19 October 2009; accepted in final form 20 July 2010
osteon; bone trabeculum; kinetics; Krogh; H; aversian
governs the form and function of both soft
and hard tissues (25). In the bone, a decrease in the rate of
oxygen transport can severely impact skeletal health causing
delays in bone healing, a change in the rate of bone turnover,
and a raised incidence of vertebral and cranial malformations
(7, 26, 31, 33, 40). Most commonly, oxemic stress is thought
to be caused by a decrease in fluid flow during biomechanical
unloading (8, 13, 22, 34); when this occurs, there is a reduction
in the oxygen transport rate in the lacunocanalicular system
and impaired osteocyte function (29). While it is clear that the
local oxygen tension (PO2) is a potent regulator of skeletal cell
metabolism and viability (1, 38, 41, 43), the oxygen distribution profile in the bone is currently unknown.
The viability and function of cells in complex tissues is
dependent on the delivery of oxygen and nutrients. Several
numerical studies have attempted to model nutrient exchange
by stress-induced fluid flow and diffusion within the lacunocanalicular system of the bone (22, 29, 34, 45). While these
studies have provided values for low-molecular-weight solutes,
oxygen transport in the bone has not been addressed. In an
earlier investigation, we have employed the Krogh cylinder
assumption to model the oxygen supply to avian femoral
growth cartilage (16); this same model has been used to predict
the oxygen tension in skeletal muscle (24), brain (21), and bone
marrow (3, 4).
The Krogh cylinder is a geometrical construct that is composed of steady-state diffusion equations that can be radially
averaged and axially distributed across both cortical and cancellous bone (24). Since the osteon structure meets the geometric requirements of the Krogh cylinder, we have used this
model to evaluate the oxygen supply to the basic tissue unit of
cortical bone. While the geometry of the trabeculum, the basic
structural unit of cancellous bone, is less well defined than the
osteon, it has a cylindrical structure as its basis; we have used
the cylindrical construct as a basis for exploring the O2 gradient in this type of bone tissue. Results of the study predict that
gradients of oxygen exist both within the cortical and trabecular structures. Based on the magnitude of the predicted gradients, O2 availability may impact both bone formation and
osteocyte physiology.
MATHEMATICAL MODEL
OXYGEN DELIVERY
Address for reprint requests and other correspondence: K. Mukundakrishnan, Dassault Systemes Simulia Corp., Rising Sun Mills, 166 Valley St.,
Providence, RI 02909 (e-mail: [email protected]).
C922
A Kroghian model has been employed to simplify the mathematical
formulation under the following assumptions: 1) Oxygen tension
distribution is at a steady-state condition. 2) Bone structures are
axially symmetric. 3) The oxygen distribution obeys Krogh’s model in
each layer (24). 4) The oxygen consumption rate is described by
saturation-type Michaelis-Menten kinetics.
With the above conditions, the multiple layer oxygen consumption
may be mathematically described by
Ki
1 d
r dr
冉 冊
r
dPi
dr
⫽
Qmaxi Pi
Kmi ⫹ Pi
(1)
where Ki is the effective oxygen permeability, Pi is the oxygen partial
pressure (mmHg), Qmaxi is the maximal oxygen consumption rate, and
Kmi is the oxygen affinity rate of the ith layer under consideration. The
value of i ranges from 1 to N, where N is the number of layers. The
0363-6143/10 Copyright © 2010 the American Physiological Society
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Zahm AM, Bucaro MA, Ayyaswamy PS, Srinivas V, Shapiro
IM, Adams CS, Mukundakrishnan K. Numerical modeling of
oxygen distributions in cortical and cancellous bone: oxygen
availability governs osteonal and trabecular dimensions. Am J
Physiol Cell Physiol 299: C922–C929, 2010. First published July
21, 2010; doi:10.1152/ajpcell.00465.2009.—Whereas recent work
has demonstrated the role of oxygen tension in the regulation of
skeletal cell function and viability, the microenvironmental oxemic
status of bone cells remains unknown. In this study, we have
employed the Krogh cylinder model of oxygen diffusion to predict
the oxygen distribution profiles in cortical and cancellous bone.
Under the assumption of saturation-type Michaelis-Menten kinetics, our numerical modeling has indicated that, under steady-state
conditions, there would be oxygen gradients across mature osteons
and trabeculae. In Haversian bone, the calculated oxygen tension
decrement ranges from 15 to 60%. For trabecular bone, a much
shallower gradient is predicted. We note that, in Haversian bone,
the gradient is largely dependent on osteocyte oxygen utilization
and tissue oxygen diffusivity; in trabecular bone, the gradient is
dependent on oxygen utilization by cells lining the bone surface.
The Krogh model also predicts dramatic differences in oxygen
availability during bone development. Thus, during osteon formation, the modeling equations predict a steep oxygen gradient at the
initial stage of development, with the gradient becoming lesser as
osteonal layers are added. In contrast, during trabeculum formation, the oxygen gradient is steepest when the diameter of the
trabeculum is maximal. Based on these results, it is concluded that
significant oxygen gradients exist within cortical and cancellous
bone and that the oxygen tension may regulate the physical
dimensions of both osteons and bone trabeculae.
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NUMERICAL MODELING OF OXYGEN IN BONE
at the boundary of intermediate layers. These conditions are represented as
dP1
Central bone :
dr
⫽ 0 at r ⫽ R1
Pi ⫽ Pi⫹1
Intermediate layers :
Ki
dPi
dr
⫽ Ki⫹1
dPi⫹1
dr
冧
at r ⫽ Ri⫹1
Outer bone : P1⫽Pmarrow at r ⫽ R1 ⫽ RN
(5)
(6)
(7)
System Parameters
effective oxygen permeability Ki is equal to the product of oxygen
diffusivity Di (cm2/s) and the oxygen solubility coefficient ␣O2
(mol · cm⫺3 · mmHg⫺1). The associated boundary conditions are stated
below.
Cortical bone. The osteon, the unit structure of compact bone, has
been modeled as consisting of radially symmetric alternating layers of
osteocytes and bone matrix surrounding a Haversian canal containing
a capillary lined with osteoblasts (Fig. 1A). Because of the onedimensional nature of the Krogh model, each layer is modeled as
homogenous, thereby excluding considerations of cellular processes.
The value of the oxygen partial pressure at the capillary wall (r ⫽
Rcap ⫽ R1, where Rcap is capillary radius) is assumed to be known and
constant (Pcap). The oxygen flux at the boundary of the tissue cylinder
(r ⫽ RN) is assumed to be zero. Continuity of oxygen flux and of
partial pressure of oxygen Pi (also denoted by PO2) are enforced at the
boundary of intermediate layers. These conditions are mathematically
represented by
Capillary wall : P1 ⫽ Pcap at r ⫽ R1 ⫽ Rcap
Pi ⫽ Pi⫹1
dPi⫹1
Intermediate layers dPi
⫽ Ki⫹1
Ki
dr
dr
Outer bone :
dPN
dr
冧
at r ⫽ Ri⫹1
⫽ 0 at r ⫽ RN
(2)
(3)
Table 1. Cortical and trabecular bone measurements
(4)
Trabecular bone. The trabeculum, the unit structure of cancellous
bone, has been modeled as consisting of axially symmetric alternating
layers of osteocytes and bone matrix surrounded by a layer of
osteoblasts (Fig. 1B). At the center of the trabeculum (r ⫽ R1) the
oxygen flux is assumed to be zero. The value of the oxygen partial
pressure at the capillary wall (r ⫽ Rcap ⫽ R1) is assumed to be
constant (Pmarrow). Continuity of oxygen flux and of PO2 are enforced
AJP-Cell Physiol • VOL
Rcap, ␮m
ROP, ␮m
Rcap, ␮m
ROB␣ ⫺ ROB␤, ␮m
ROC␣ ⫺ ROC␤, ␮m
5
105
300
10
5
Dimensions of the capillary radius (Rcap), radius of the osteon perimeter
(ROP), inner radius of capillary layer (Rcap␤), thickness of osteoblast layer
(ROB␣ ⫺ ROB␤), and the thickness of osteocyte layer (ROC␣ ⫺ ROC␤) are
shown.
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Fig. 1. Diagram of cortical and trabecular models. A: mature osteon model. The
osteon model is composed of a central blood vessel within a Haversian canal
that is surrounded by a single layer of osteoblasts and bone containing several
concentric layers of osteocytes. B: mature trabeculum model. The trabeculum
model is a solid cylinder of bone lined by a layer of osteoblasts and surrounded
by bone marrow and containing concentric layers of osteocytes. See MATHEMATICAL MODEL for details.
Although bone is a complex tissue, two unit structures predominate; namely, osteons and trabeculae. Osteons can be considered as
individual cylindrical systems containing concentric rings of osteocytes surrounded by lamellae composed of hydroxyapatite and proteins (mainly collagen type I). Trabeculae of cancellous bone are rodor plate-shaped struts of interconnected bone also containing osteocytes surrounded by a mineralized matrix (Fig. 1). Defining characteristics of osteonal and trabecular bone are presented in the following.
Osteon model. In the mature osteon, the diameter across the bone
cylinder may range from 200 to 300 ␮m (14, 46). Despite the range
of diameters, the thickness of bone between the central canal and the
osteon perimeter does not exceed 80 ␮m (15, 19). The central canal,
typically 30 –70 ␮m (18), contains a single capillary, which characteristically averages 7–9 ␮m in diameter but can range from 3 to 20
␮m (2, 5). At the arteriole end of the capillary supplying the osteon,
the PO2 is 95 mmHg; here we use a value of 40 mmHg as this is the
PO2 at the venous end of the capillary (35, 44). Since the diffusivity
of oxygen in bone has not been reported, this value is set to the
reported oxygen diffusion coefficient value for connective tissue as
1 ⫻ 10⫺5 cm2/s (44). Based on this information, the osteon has been
modeled as a 30-␮m central canal, with a 10-␮m capillary at its center
(Fig. 1A). The osteon wall thickness is set at 80 ␮m, producing an
osteon of 210 ␮m in diameter. Lining the wall of the canal is a
uniform, 10-␮m wide layer of osteoblasts. Within the bone, we
consider oxygen consumption by osteocytes at three locations from
the central canal. Table 1 shows the parameter values used in the
mature and developing osteon models.
Trabeculum model. Cancellous bone consists of numerous interconnected trabeculae contained within the bone marrow. The shortest
dimension of a trabeculum is typically no greater than 200 ␮m (18).
We consider the fully developed trabeculum to be a 200-␮m diameter
cylinder and model three evenly spaced osteocyte layers distributed
within trabeculum (Fig. 1B). The oxygen distribution during the
development of a primary trabeculum (see Fig. 5A) has also been
considered in the present study. Primary trabeculae develop from a
central core of calcified cartilage originally laid down in the growth
plate. The diameter of the calcified cartilage core is ⬃20 ␮m at its
smallest width (as measured from images in Ref. 37). Therefore, a
developing trabeculum has been modeled as a cylinder beginning with
an initial diameter of 20 ␮m and progressing to 200 ␮m as additional
C924
NUMERICAL MODELING OF OXYGEN IN BONE
RESULTS
Mature Osteon
A gradient in oxygen would exist in the afferent capillary as
the osteonal cells metabolize the available oxygen. For this
reason, we have considered capillary oxygen tensions ranging
from 100 mmHg at the capillary branch to 20 mmHg at the
capillary terminus (Fig. 2A). A PO2 of 100 mmHg would result
in a PO2 of 85.1 mmHg at the outermost osteocyte layers,
respectively. When the capillary PO2 is 40 mmHg, then the
oxygen concentration would be 35.1 mmHg at the bone lining
cell layer and 29.5, 27.1, and 26.1 mmHg at the first, second,
and third osteocyte layers, respectively. A capillary oxygen
value of 20 mmHg would generate a PO2 of 8.3 mmHg at the
osteon periphery. When the initial PO2 is lower than 20 mmHg,
anoxic conditions may exist for those osteocytes that are
located furthest from the capillary.
Since the diffusion coefficient of oxygen in connective tissue
(1 ⫻ 10⫺5 cm2/s) is likely a high estimate for mineralized
bone, we have also considered a lower limit case by setting the
coefficient to that of water in bone (7.8 ⫻ 10⫺7 cm2/s) (9).
With the use of this low diffusion coefficient, the PO2 would
decrease precipitously in the bone (Fig. 2B). Oxygen tensions
Table 2. Standard values of parameters used in simulations
Layer (i)
Ki,
mol · cm⫺1 · s · mmHg
Qmaxi,
mol · cm⫺3 · s⫺1
Kmi, mmHg
Bone
Haversian canal
Active osteoblast
Inactive osteoblast
Osteocyte
Marrow
1.8 ⫻ 10
6.0 ⫻ 10⫺14
3.0 ⫻ 10⫺14
3.0 ⫻ 10⫺14
3.0 ⫻ 10⫺14
3.0 ⫻ 10⫺14
1.06 ⫻ 10⫺7
1.95 ⫻ 10⫺8
1.95 ⫻ 10⫺8
2.2 ⫻ 10⫺9
3.0
3.0
3.0
3.0
⫺14
at the three osteocyte layers would be reduced from 29.5 to
11.4 mmHg, 27.1 to 2.5 mmHg, and from 26.1 to 0.5 mmHg,
for oxygen diffusion coefficients of 1 ⫻ 10⫺5 and 7.8 ⫻ 10⫺7
cm2/s, respectively. At the same time, by slowing the diffusion
of oxygen into the bone, osteoblasts surrounding the Haversian
canal would be exposed to a slightly higher PO2. Finally, we
have investigated the influence of cellular oxygen consumption
rates on oxygen distribution in the osteon by varying Qmax.
Minimal changes in the oxygen distribution are seen when the
Qmax values for bone lining cells are varied from 9.75 ⫻ 10⫺9 to
3.9 ⫻ 10⫺8 mol·cm⫺3 ·s⫺1 (Fig. 2C). In contrast, by varying osteocyte Qmax from 9.75 ⫻ 10⫺9 to 3.9 ⫻ 10⫺8 mol·cm⫺3 ·s⫺1, there
would be a profound alteration in the oxygen distribution
profile. At twice the rate, the PO2 at the outer osteocyte layer
would be 11.4 mmHg (Fig. 2D). The data suggests that osteocyte oxygen consumption has a more significant influence on
the PO2 distribution in osteons than that of the bone lining cells
on the Haversian surface. The results are not sensitive to
variations in the KM over the range reported for human cells
(data not shown).
Mature Trabeculum
In the trabecular model, we have evaluated the oxygen
distribution for a range of distances between the capillary layer
and the outer edge of the bone. The PO2 distribution across the
bone shifted with changes in the capillary distance, although
the gradient in the bone is consistent for all simulations (Fig. 3A).
When the capillary is adjacent to the bone surface (0 ␮m), the
PO2 at the inner osteocyte layer would be 35.2 mmHg. A
capillary distance of 200 ␮m would result in a PO2 of 18.9
mmHg at the first layer of osteocytes, 17.7 mmHg at the second
layer, and 17.0 mmHg at the inner osteocyte layer. These data
suggest that osteocytes of trabeculae at a distance from the
oxygen source may reside in a hypoxic environment. We have
also considered a lower limit case of oxygen diffusivity in the
bone by setting the bone oxygen diffusion coefficient to that of
water. Changing the oxygen diffusion coefficient to 7.8 ⫻ 10⫺7
cm2/s would reduce the PO2 at the osteocyte layers from 18.9
to 7.9 mmHg, 17.7 to 2.0 mmHg, and 17.0 to 0.6 mmHg (Fig.
3B). We have examined the effect of oxygen consumption on
PO2 distributions in the trabeculum by varying osteoblast and
osteocyte Qmax values. Unlike the osteon model in which there
was minimal effect of bone lining cells oxygen consumption on
the oxygen distribution, when we varied the osteoblast Qmax
value, appreciable changes in trabecular PO2 distributions are
observed (Fig 3C). Reducing osteocyte Qmax from 1.95 ⫻ 10⫺8
to 9.75 ⫻ 10⫺9 mol · cm⫺3 · s⫺1 increases the PO2 at the innermost osteocyte layers by 3.8 mmHg (Fig. 3D). These data
suggest that trabecular PO2 distributions are sensitive to modest
changes in both bone lining cell and osteocyte consumption
rates.
Developing Osteon
The effective oxygen permeability (Ki) is equal to the product of oxygen diffusivity
Di (cm2/s) and the oxygen solubility coefficient ␣i (mol·cm⫺3 ·mmHg⫺1). Qmaxi is the
maximal oxygen consumption rate, and Kmi is the oxygen affinity rate of the ith layer
under consideration.
AJP-Cell Physiol • VOL
To predict PO2 distributions during osteon formation, four
stages of osteon development have been modeled (Fig. 4A).
The first represents the osteon at the earliest remodeling
stage in which osteoblasts line the bone surface but have not
yet deposited new osteoid (Fig. 4A, I). Subsequently, these
osteoblasts form bone and, as they are entombed in the
mineralized matrix, differentiate into osteocytes (II). The
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layers of bone and osteocyte layers are added. The oxygen tension of
human bone marrow has been reported to be 51.8 ⫾ 14.5 mmHg. A
bone marrow oxygen tension at the low end of this range of 40 mmHg
has been selected for the trabeculum studies.
Table 2 shows the parameter values used in the mature developing
trabeculum models.
Cellular parameters. The values of representative standard parameters for each layer used in the simulation are listed in Tables 1 and
2. Osteoblasts typically range from 8 to 14 ␮m in diameter, whereas
osteocyte diameters range from 3 to 7 ␮m. In our model, we define
osteoblast and osteocyte diameters as 10 and 5 ␮m, respectively. The
reported maximum oxygen consumption rate (Qmax) for osteoblasts is
200 nmol · h⫺1 · million cells⫺1 (23), which corresponds to 1.06 ⫻
10⫺7 mol·cm⫺3 ·s⫺1, assuming each osteoblast occupies a spherical
volume of 10 ␮m in diameter. Osteocyte Qmax is ⬃42 nmol·h⫺1 ·million
cells⫺1 (derived from Refs. 23 and 39), which corresponds to 1.95 ⫻
10⫺8 mol·cm⫺3 ·s⫺1. Osteoblast layers in the mature osteon and trabeculum models, representing bone lining cells, are assigned a Qmax equal
to that of the osteocyte. The mature granulocyte Qmax value of 2.2 ⫻
10⫺9 mol·cm⫺3 ·s⫺1 was obtained from Chow et al. (3). Osteoblast and
osteocyte KM values for oxygen have not been reported in the literature.
Since the reported range of KM values for human cell types is 0.17–5.9
mmHg (12), the KM value for all cells was set to 3.0 mmHg. The value
of effective oxygen permeability in all cells was set to 3 ⫻ 10⫺14
mol·cm·s⫺1 ·mmHg, the value reported for mature granulocytes (3).
NUMERICAL MODELING OF OXYGEN IN BONE
C925
three stages shown in Fig. 4A, II–IV, depict the developing
osteon as progressive layers of bone and emdedded osteocytes are formed. When the modeling equations are applied
to this structure, the results indicate that at the earliest stage
of formation (I), the oxygen supply to the osteonal periphery
would be most severely limited. For a capillary PO2 of 40
mmHg, the PO2 at the outer edge of the developing osteon
would be 7.6 mmHg in the first stage (Fig. 4B). As successive bone and osteocyte layers are added, the peripheral PO2
would increase to 16.1, 20.4, and 22.5 mmHg in stages II,
III, and IV, respectively. This effect is due in large part to
the decreasing circumference of the anabolic osteoblast
layer, as well as the reduction in the distance of this cell
layer to the oxygen source.
matrix (Fig. 5A, vI). Subsequent stages represent the growing
trabeculum as osteocyte and bone layers are sequentially added
(Fig. 5A, II–IV). This model predicts that the PO2 at the center
of the trabeculum decreases with increasing trabeculum diameter. In the initial stage, the PO2 at the core of the trabeculum
would be 12.9 mmHg, a significant reduction from the capillary PO2 of 40 mmHg (Fig. 5B). As osteocyte and bone layers
are added to the trabeculum stages II, III, and IV, the PO2 at the
core would be expected to decrease to 5.7, 3.6, and 2.5 mmHg,
respectively. These data suggest that continued matrix deposition at the trabecular surface eventually leads to hypoxic stress
at the trabeculum core.
Developing Trabeculum
The osteon has been modeled as an axially symmetric
system consisting of a central blood vessel surrounded by an
osteoblast layer and multiple concentric alternating layers of
bone matrix and osteocytes. In the region of the osteon that is
closest to the afferent arteriolar capillary, the PO2 is predicted
The PO2 distributions during trabeculum formation have
been simulated using the four-stage model shown in Fig. 5A.
At stage I, an osteoblast layer lines a small cylinder of calcified
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DISCUSSION
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Fig. 2. Predicted steady-state oxygen distributions across the mature osteon. A: effect of capillary PO2 on the oxygen distribution. The initial capillary
pressure values (Pcap) are 100, 80, 60, 40, and 20 mmHg. As the capillary PO2 are reduced from 100 to 20 mmHg, there is a concomitant reduction in
oxygen distribution across the osteon. B: effect of diffusivity on the oxygen distribution. The model evaluated the effect of diffusivity on oxygen
distribution across the osteon; values chosen were 7.8 ⫻ 10⫺7 cm2/s (H2O in bone) and 1 ⫻ 10⫺5 cm2/s (oxygen in connective tissue). Substituting the
oxygen diffusivity in the connective tissue matrix for water results in a drastic decrease in the PO2 across the osteon. C: effect of oxygen consumption
rate for bone lining cells/inactive osteoblasts on the oxygen distribution. The Qmax values are 9.75 ⫻ 10⫺9, 1.97 ⫻ 10⫺8, and 3.9 ⫻ 10⫺8 mol · cm⫺3 · s⫺1.
Note bone lining cell consumption exerts a minimal effect on osteon oxygen distribution. D: effect of the oxygen consumption rate of osteocytes on the
oxygen distribution. The osteocyte Qmax values are 9.75 ⫻ 10⫺9, 1.95 ⫻ 10⫺8, and 3.9 ⫻ 10⫺8 mol · cm⫺3 · s⫺1. In contrast to the bone lining cells/inactive
osteoblasts (C), osteocyte oxygen consumption markedly alters the oxygen distribution in the osteon. The radial position of each layer is shown along
the x-axis.
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NUMERICAL MODELING OF OXYGEN IN BONE
to be 100 mmHg; osteocytes most radially removed from the
capillary would experience a 15% reduction in PO2. In the
region of osteon most distant from the afferent blood supply,
there would be a fourfold drop in capillary PO2. Thus, in
Haversian bone, an oxygen gradient would exist across the
osteon and along its length. Since the gradients would be
severe, it would likely exert an effect on the architecture of the
osteon. We predict it would limit the length of the osteon and
the number of the cells that can be supported metabolically
within each radius. Whereas the literature on this topic is
sparse, a number of workers have commented on the constancy
of osteonal parameters. Jowsey (20) reported that for animals
the size of Rhesus monkeys or larger, despite tremendous
variations in body mass and skeletal load, the reported average
osteon diameter ranges from 216 –250 ␮m (human, 223 ␮m).
Even in compact dinosaur bone, the mean osteon diameter was
246 ␮m. The apparent upper limit to osteon size may be a
consequence of the limitations in oxygen supply predicted by
our model.
The values presented above are based on the assumption that
the diffusivity of oxygen across the osteon is governed by the
diffusivity of oxygen in connective tissues. However, the
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diffusion of molecules such as glucose is ⬃50% greater in
connective tissue than in the bone (30), and hence, the connective tissue value represents an upper limit of oxygen diffusion. As a lower limit, we have chosen the diffusion coefficient
of oxygen in bone to be equal to that of water in bone. Unlike
oxygen, water molecules readily bind to the bone matrix (9)
and thus diffusion is slowed. Another reason for choosing a
lower bound for the diffusion coefficient is that the oxygen
diffusion path in actual bone is very complex. This is due to the
fact that bone matrix largely comprises the cytoplasmic processes of the osteocytes and the sleeve of fluid external to these
processes but contained within the canaliculi. The presence of
such processes may offer increased resistance to diffusion that
as been indirectly accounted here by decreasing the oxygen
diffusion coefficient to that of water (aqueous phase). In
assuming a uniform diffusivity equal to that of water, we have
treated the canalicular porosity with the contained cells, the
cell processes, and the surrounding fluid as a single homogeneous aqueous phase since cells mainly contain water with
dissolved salts. When the value of the diffusivity of oxygen in
connective tissue is replaced by that of water in bone, a drastic
change in the oxygen gradient across the osteon is evident.
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Fig. 3. Predicted steady-state oxygen distributions in the mature trabeculum. A: effect of capillary distance on the oxygen distribution. The capillary PO2 is set
to 40 mmHg and the predicted PO2 at 0 –200 ␮m from the capillaries from either side of the trabeculum are depicted. Note the trabeculum oxygen supply is most
severely limited when the capillary is furthest removed from the trabeculum. B: effect of diffusivity on the oxygen distribution. Values chosen were 7.8 ⫻ 10⫺7
cm2/s (H2O in bone) and 1 ⫻ 10⫺5 cm2/s (oxygen in connective tissue). The data predicts that if water diffusivity in bone is used, the cells in bone would
experience a marked decrease in PO2. C: effect of oxygen consumption rate for bone lining cells/inactive osteoblasts on oxygen distribution. Bone lining
cells/inactive osteoblasts serve to dramatically alter oxygen distribution across the trabeculum. D: effect of oxygen consumption rates for osteocytes on the
oxygen distribution. Osteocytes serve to dramatically alter oxygen distribution across the trabeculum. The radial position of each layer is shown along the x-axis.
NUMERICAL MODELING OF OXYGEN IN BONE
C927
The Krogh cylinder permitted us to model the oxygen
distribution in the developing osteon. In this case, osteon
formation is initiated along the circumference of the cutting
cone and continued toward the central capillary as alternating
layers of bone matrix and osteocytes are added. The steepest
drop in oxygen is observed during the initial stage of osteon
formation. At this stage, the distant osteoblast layer is at
maximum circumference, and hence the oxygen gradient
would be predicted to be steep. As layers of bone and osteocytes are added, the circumference of the osteoblast cell layer
decreases and the PO2 gradient becomes shallower. Since
biosynthetic activity depends on adequate supplies of oxygen,
the volume of the cutting cone and the capillary oxygen tension
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Fig. 4. Predicted steady-state oxygen distributions during osteon formation.
A: schematic of the developing osteon architectures. Osteoblasts are initially
bound to the surface of the cutting cone (I). After osteoid mineralization, a
second and third layer of cells secrete and mineralize the extracellular matrix
(II–IV). Radial distances from the capillary layer to the outer edge of the three
osteocyte layers are 46.25, 67.5, and 88.75 ␮m. B: oxygen distribution during
stages I–IV of osteon development. Osteon perimeter PO2 values for developmental stages were as follows: I ⫽ 7.6, II ⫽ 16.2, III ⫽ 20.4, and IV ⫽ 22.5
mmHg. The data indicate that as the osteon develops, there is a progressive
decrease in oxygen stress.
Thus within 50 ␮m of the capillary there is a dramatic fall in
PO2. At the third osteocyte layer (100 ␮m) there is almost total
anoxia in the radially distributed osteocytes. It is noteworthy
that in vivo the majority of dead osteocytes and empty lacunae
are found at the periphery of the osteon and interstitial lamellar
bone (11, 27) where our model predicts oxygen availability is
most severely limited. The maximum distance from a capillary
at which osteocytes receive adequate oxygen may therefore
limit osteon architecture.
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Fig. 5. Predicted steady-state oxygen distributions during trabeculum formation. A: schematic of the developing trabeculum architectures. Unlike Haversian bone, trabeculum bone grows by apposition (I). Subsequently, these cells
lay down mineralized osteoid and become osteocytes (II). Further layers of
osteoblasts are recruited to the surface that synthesize new bone and differentiate into osteocytes (III an IV). Trabeculum diameters for developmental
stages I–IV were 20, 80, 140, and 200 ␮m. Radial distances from the outer
capillary to the three osteocyte layers were 17.1, 46.4, and 75.7 ␮m. B: oxygen
distribution during each stage of trabeculum development. Corresponding
trabecular core PO2 for each stage were 12.9 (I), 5.7 (II), 3.6 (III), and 2.5 (IV)
mmHg. Note unlike Haversian bone, with the progressive growth of the bone
trabeculum there is a profound stepwise decrease in PO2 and a corresponding
increase in oxemic stress.
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NUMERICAL MODELING OF OXYGEN IN BONE
AJP-Cell Physiol • VOL
marrow and blood capillaries. As the trabeculum diameter is
increased, the PO2 at its center would be expected to substantially decrease (12.9 –2.5 mmHg). At this PO2 range, despite the
proximity of vascular elements, cells in the interior of the
trabeculum would be expected to generate energy by glycolysis. In contrast, on the outside of the trabeculum, anabolic
metabolism would be enhanced as cells generated energy
through oxidative phosphorylation.
Model Limitations
The simplicity of the Krogh cylinder model has been employed in several anatomical problems. However, this simplicity presents several limitations in modeling oxygen gradients in
bone. For instance, the matrix of bone is not solid but rather
contains an extensive canalicular network housing the cellular
processes of osteocytes. As the layers in Krogh model have
been assumed to be axially symmetric, the presence of such
osteocyte processes have not been considered. In addition, the
symmetrical nature of the Krogh cylinder does not consider the
packet-like nature of trabecular bone structures observed in
vivo. These structural inconsistencies may affect the transport
of nutrients within trabeculae. Finally, the Krogh cylinder
considers transport of oxygen as being governed by passive
diffusion alone. However, mechanical loading of the skeleton
is known to facilitate increased solute transfer within the bone
matrix through convective transport of nutrients. Although we
have indirectly included the effects of convection by varying
the oxygen consumption rates of both the osteoblasts and
osteocytes, the current predictions may differ from solving the
convective-diffusive transport equation. Nonetheless, oxygen
transport via diffusion is still likely to dominate during periods
of inactivity or weightlessness, when the skeleton is unloaded.
Future modeling of oxygen gradients within skeletal structures
will address these limitations of the Krogh cylindrical model.
CONCLUSION
The data presented here predict that significant oxygen
gradients exist across the basic unit structures of mature bone.
These gradients may explain why the radial measurements of
both osteonal and trabecular structures are constant across a
wide range of animal species. Our data suggests that the
oxemic state regulates not just the overall dimensions of bone
microcomponents but also plays a role in development and
replacement of Haversian and trabecular bone systems. The
oxemic state of the bone strut or osteon would be expected to
markedly influence the expression of the transcription factor
HIF-1. Aside from promoting anaerobic glycolysis, inhibiting
the TCA cycle, and suppressing oxidative phosphorylation, this
transcription factor upregulates vascular endothelial growth
factor, a key molecular regulator of cell survival and angiogenesis (10, 48). Hence, the imposition of the hypoxic state
would serve to maintain the function of osteoblasts and osteocytes, promote bone turnover, and, if extreme, trigger vascular
invasion. Experiments are in progress to examine these relationships.
ACKNOWLEDGMENTS
This work was supported by National Aeronautics and Space Administration Grant NAG9-1400, National Institutes of Health Grants DE-13319,
DE-10875, and DE-05748, and the Department of Defense Grant DAMD1703-1-0713.
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would therefore determine bone architecture. Jaworski and
others (17, 42) have suggested that the oxygen supply regulates
cutting cone formation and diameter. We, and others, have
previously demonstrated that the oxygen supply regulates the
phenotype of both osteoblasts and preosteocytes (6, 28, 32, 36,
41, 47). From this perspective, the oxygen supply may limit the
maximal diameter of functional osteons by regulating cell
activity and differentiation in both the cutting and closing
cones.
We are cognizant of the fact that trabeculae have an irregular
architecture with a preponderance of bone struts being tabletlike or cylindrical in shape. Since the Krogh equations are best
applied to a uniform cylinder, we have assumed that the
underlying architecture is cylindrical and that subsequent appositional growth formed the conventionally recognized bone
trabeculae. Thus, for the Krogh equations to be useful, it is best
to consider the initial 1–200 ␮m of the trabeculum structure.
Although the architecture of the trabeculum is the reverse of
the Haversian system (blood vessels on the outside of the
trabeculum versus a central capillary at the center of the
osteon), the change in oxygen distribution across the trabeculum is similar. However, some differences do exist. First, at all
predicted oxygen tensions, the gradient is much shallower;
second, unlike the Haversian system, where one would predict
that there would be a substantial decrease in available oxygen
along the length of the osteon, the external surfaces of the
trabeculum, enveloped in the bone marrow, would be well
supplied with oxygen.
When the diffusion coefficient of oxygen in connective
tissue is replaced by the lower limit value (coefficient of water
in bone), a precipitous fall in the PO2 is observed for those cells
residing 20 –50 ␮m within the trabecular structure. Our model
predicts that the osteocytes at the core of the trabeculum would
be exposed to a PO2 between 0.6 and 17 mmHg. The model
also predicts differences between osteonal and trabecular bone
when Qmax is considered. Numerical modeling indicates that
the trabecular oxygen gradient is very sensitive to osteoblast
and osteocyte Qmax values. Thus, when osteocyte Qmax is
doubled, within the trabecular core there is a marked decrease
in oxygen distribution. In contrast to the osteon model, the
predicted steady-state oxygen distribution across the trabeculum is greatly influenced by the oxygen consumption of the
bone lining cell/osteoblast layer. The enhanced gradient likely
results from the increased area of the osteoblast layer and the
greater distance from the capillary. In this context, it may be
noted that physiological loading affects the distribution of
nutrients across the bone due to convective transport of nutrients to osteocytes in the canaliculi network. Such convective
transport causes an increase in the oxygen (nutrients, in general) consumption rate of osteocytes (increase in Qmax). While
the convective transport of oxygen has not been directly
considered in the present study, its effect on increasing the
oxygen transport to osteocytes has been modeled by varying
the Qmax values. Our model results predict that with an increase
in the Qmax of osteocytes (i.e., under physiological loading
conditions) significant oxygen gradients may exist (Fig. 2D).
Unlike the Haversian system, development of the trabeculum is by appositional growth and is similar to that of many
tissues. The data predict that the highest PO2 would be experienced by cells at the initial stage of strut formation when the
majority of osteocytes were closest to the enveloping bone
NUMERICAL MODELING OF OXYGEN IN BONE
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
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