Identification of Critical Location on A Microstructural Bone Network
Taehyong Kim1
Jaehan Koh1
Kang Li1
1
Department of Computer Science
and Engineering, State University of
New York at Buffalo, USA
Murali Ramanathan2
Aidong Zhang1
2
Department of Pharmaceutical
Sciences, State University of New
York at Buffalo, USA
Email: {thkim7, jkoh, kli22, murali, azhang}@buffalo.edu
Abstract—Identification of a critical location in complex structures is one of the most important issues that affects the quality
of safety in human life. Specifically, finding high fracture spots
of a bone microstructure in our body is a very important
research topic; however, it is still not well understood. In this
paper, we study on identifying critical locations in a bone
microstructure with our bone network model. In fact, about
25 million people in the United States suffer from osteoporosis,
which is a systemic skeletal disease characterized by low bone
mass and micro-architectural deterioration of bone tissue leading
to enhanced bone fragility and a consequent increase in fracture
risk. However, currently available techniques for the diagnosis
of osteoporosis and the identification of critical locations of bone
microstructure are limited.
We create a bone network model based on properties of a
bone microstructure and we develop a method, called information
propagation, to identify critical locations in a bone network. Our
paper focuses on detecting important edges as critical locations
for the strength of bone microstructure when there are external
forces applied to the bone network. Then, we evaluate results
of the method in comparison with existing methods including
the weighted betweenness centrality and the weighted bridge
coefficient. We conclude with the discussion on advantages and
disadvantages among those methods.
I. I NTRODUCTION
Networks are encountered in a wide variety of contexts,
such as social networks, neuron networks, and molecular networks. Most of the real world relationships and cooperations
among any kinds of objects could be represented by networks.
Specifically, many types of biological networks exist, which
could be depicted as networks. Bone network modeling is one
of the most exciting but challenging research areas in terms
of network modeling. Although bone is a simple composite
of a mineral phase, its structure is highly complex. Bone is
not a uniformly solid material, but rather has some space
between its hard elements. Microstructural properties, e.g.,
cortical porosity and the presence of microcracks, contribute
to bone’s mechanical competence.
Osteoporosis, which is a systemic skeletal disease characterized by low bone mass and microarchitectural deterioration
of bone tissue leading to enhanced bone fragility, is mainly
due to abnormal bone remodeling cycles by a hormone imbalance [15]. As the efforts on the diagnosis of osteoporosis,
correlations between bone mineral density (BMD) and fracture
risk have been extensively studied; however, many studies
show that low BMD is not the sole factor for the fracture
risk [9], [3]. Other factors like bone microstructure also
contribute substantially to bone strength [6], [12]. Since the
microstructure of bone mainly consists of mineralized fibers
whose structure is similar to geometric arrangement of cells
in morphogenetic or cancerous tissues, we tried to create a
network model representing a bone microstructure to identify
critical parts in bone network structure.
This paper is organized as follows. In the following background section, we describe properties of bone and current
issues on analyzing bone strength. Then, we introduce our
bone network model with describing procedures of bone
network modeling in the first subsection of the methods section
and we explain the methods to find critical locations in a
bone network in the second subsection of the methods section.
In the section three, we analyze the result of the method,
information propagation, and compare with other methods,
such as the weighted betweenness centrality and the weighted
bridge coefficient, on finding critical locations in a sample
bone network. We close with the conclusion section.
II. BACKGROUND
As shown in Figures 1(a) and (b), bone is not a uniformly
solid material, but rather has some spaces between its hard
elements. There are two different bone types: one is cortical
bone and the other is trabecular bone. Cortical bone is the
hard outer layer of bones which is composed of compact bone
tissue. Filling the interior of the organ is the trabecular bone
tissue, which is composed of a network of rod- and plate-like
elements that make the overall organ lighter and allow room
for blood vessels and marrow.
Fig. 1. (a) The image of a bone microstructure is shown [10]. (b)
The microscopic view of a bone microstructure is shown [1].
The standard and routine approach for the diagnosis of
osteoporosis is to assess BMD using either dualenergy X-ray
absorptiometry (DXA) or quantitative computed tomography
(QCT). The estimated BMD has shown to be a suitable
predictor of fracture risk. However, major limitations of bone
mineral densitometry are that it incompletely reflects variations
in bone strength and that differentiation between patients with
and without vertebral fractures is inaccurate. Other factors
like bone microarchitecture contribute substantially to bone
strength and their evaluation can improve determination of
bone quality and strength [5].
III. M ETHODS
In the first subsection, we provide a network model of a
bone microstructure and we explain our method to find critical
locations in a bone network in the following subsection.
A. Bone Network Model
To overcome the lack of measurements of bone quality,
we study the problem in a new direction. We develop a
microscopic graph-based approach of a bone structure, a
computational network model of bone microstructure which is
capable of enabling quantitative assessment of bone strength
and fracture risk. Based on the bone network model, we devise
a method to find a critical location as a fragile point of a bone
microstructure. In what follows, we describe the process of
microstructural bone modeling in details.
Based on properties of a bone microstructure, we design
a bone microstructure network model. When we develop our
abstracted bone network model, important components of the
bone structure are considered to form a bone model enabling
computational analysis in a timely manner without losing the
critical bone structural properties. We consider mineralized
collagen fibers as important components from the bone cellular
structure point of view.
(a)
(b)
(c)
120
1.0
Density
Density
0.8
100
0.6
0.4
80
0.2
0
60
-0.4
-0.2
0
0.2
-0.4
0.4
Position
(f )
(e)
-0.2
0
0.2
0.4
Position
(d)
(a) A femur bone image for a patient with osteoporosis
by DXA scan is shown. The stick-like white object in the center
of the figure is an implant by orthopedic surgery. (b) The plot for
bone density from the red line section of Figure (a) is shown with
image profiling process. (c) The plot for bone density is shown after
the geometrical correction process is applied. (d) A two-dimensional
bone microstructure network is constructed based on bone density.
(e) An example of a three-dimensional bone microstructure network
is shown. (f) A projection image of a three-dimensional bone microstructure network is shown.
Fig. 2.
First, we obtained the density of bone and the properties of
bone microstructure by image processing techniques since the
most important factors of bone properties are understood by
analyzing bone density and bone structural characteristics. We
used four human femur bone images of DXA scan to analyze
properties of bone microstructure since DXA scan is broadly
used to calculate BMD [13], [2]. Figure 2(a) shows a femur
bone image of a female patient with osteoporosis by DXA
scan with which we can identify the density differences of
a femur bone. By image profiling on the red line section in
Figure 2(a), we can have the bone density data as a function of
bone position sequence from left to right shown in Figure 2(b).
Since the DXA image shown in 2(a) is the projection image
of a cylinder-shape bone, a geometrical correction process is
needed to accurately measure the density per bone area from
the projection image. We define the geometrical correction
formula, ς, as follows.
½
2(Ro sinθo − Ri sinθi ), if x < Ri
ς=
,
(1)
2Ro sinθo ,
if x ≥ Ri
where Ro is the radius of annular region of bone, Ri is the
radius of cortical region of bone, θo is the angle to form a right
triangle which consists of perpendicular to projection source
and line for radius to Ro , θi is the angle to form a right triangle
which consists of perpendicular to projection source and line
for radius to Ri and x is the location of projection source on
the radius of bone.
Figures 2(d) and (e) are examples of a two-dimensional
bone microstructure network and of a three-dimensional bone
microstructure network, respectively. The detailed process of
constructing the bone microstructure network can be found
in [7], [8]. Figure 2(f) shows the sample projection image
of our three-dimensional bone structure network generated by
millions of 0.3 um2 artificial beams to estimate bone density in
our bone model. Like DXA, thick and dense structural parts
of our bone model allow less of the artificial beam to pass
through them. The projection image is generated by millions
of pixels containing the level on the remaining strength of
artificial beam calculated via e−λω such that λ is a projection
contrast coefficient and ω is the density of each cylinder-like
microstructure. The amount of artificial beams that is blocked
by bone microstructures can be compared to each other by
this projection process.
With the knowledge of bone microstructure and density, we
designed a bone network model as follows. A bone network
is defined by a weighted undirected space-sensitive graph in
a circular region; G = {(V, E) | V is a set of nodes and E
is a set of edges, E ⊆ V × V , an edge e = (i, j) connects
two nodes i and j, i, j ∈ V , W (e) is a weight of edge e,
e ∈ E}. An edge in a bone network represents a rod-like bone
mineralized fiber and a node in a bone network represents a
fiber binding point with which bone cells move and interact
with neighboring bone tissues.
Since this bone model is developed based on image analysis
of microscopic figures of bone, properties we have used in this
model would not be reliably measured. To make this model
more realistic, accurate input data are required, such as density
of fibers, average thickness of fibers, and average length of
fibers. However, well-understood knowledge with this type
of study would be valuable on understanding bone strength
and critical locations of bone microstructure. In the following
subsection, we introduce the method, information propagation,
to find critical locations and other methods for comparison.
B. Identification of Critical Locations in Bone Network
In this section, we introduce our new method to identify critical locations of a bone network. First, we address properties
of the sample bone network and explain our new method by
illustrating the process of the algorithm. Then, we calculate
results with our method and compare it with other existing
methods, including the weighted betweenness centrality and
the weighted bridge coefficient. Finally, we analyze our results
and discuss advantages and disadvantages in the following
results section.
Our motivation for the algorithm is based on the energy
flow in a network. Since energy does not just randomly flow
over a network, but has a certain direction from sources to
destinations at any given time. Likewise, bone is stressed by
certain external force at any specific locations and directions.
With these characteristics of the structural stress in a bone
network, we tried to find critical locations in a sample bone
network under a certain stress.
(a)
(b)
(c)
Fig. 3. The illustration of the crack propagation in a lattice of atoms
is shown when there are external forces applied to the left and right
[16]. (a) Four micro-cracks in the lattice of atoms are shown as
the initial status. (b) The areas which have micro-cracks have more
stretch energy than any other areas. (c) The area with the most stretch
energy starts to break and other areas with micro-cracks continue to
break until the lattice is broken into two parts.
Figure 3 illustrates the crack propagation in a lattice of
atoms as the method to find critical locations in a bone
network. A break starts from a microscopic crack, which
would be an imperfection in the material when it was made, or
created by repeated flexing “fatigue.” A crack can grow longer
and larger when a force is continuously applied. Likewise, a
break in human bone starts from microscopic flaws created
by external forces of daily life or a bone disease such as
osteoporosis.
Base on this idea, we designed an algorithm to find critical
edges against a bone network. The method measures the
quantity of stress energy in each edge and selects the edge
that has the most quantity of stress energy in a bone network.
In the real world, bone is not just broken by any forces, but
broken by the external force that attack a certain weak location
in a certain direction. As shown in Figure 4(a), we suppose that
there are external force applied to the right side of the bone
network and every node in the outermost of the right side in
the bone network (red colored nodes) receives the same energy
from the force. Then, we set the left side of the bone network
fixed to ground. In this setup, we try to identify the edge which
contains the most energy from the force as a critical location
of the bone network.
(a)
(b)
T1
x0
θ
φ y0
F0
ψ
T2
Fig. 4. (a) 1 × 1 unit (1 unit ≈ 5mm) square from a two dimensional
sample bone network is shown, consisting of 686 nodes and 943
edges. The red nodes in the right side of the network are sources
for external stress and the blue nodes in the left side of the network
are grounded as destinations for external stress. (b) The method to
transfer stress energy from a source node to two destination nodes is
illustrated. F0 is a source node for external force and T1 and T2 are
the destination nodes for external force.
Figure 4(b) illustrates how we calculate the force energy
transformation in the bone network. Suppose, there is no
energy leak in this model, then the following method is
defined based on the information equivalence between source
and destination nodes. Transferred horizontal force, x0 , and
vertical force, y0 , shown in Figure 4(b) are defined and
calculated by following equations.
)
x0 : F0 cosψ = T1 cosθ + T2 cosφ
,
(2)
y0 : F0 sinψ = T1 sinθ − T2 sinφ
where F0 is a source node for external force, the T1 and T2
are the destination nodes for transferred force, ψ is the angle
from x axis for the source force from F0 , and θ and φ is
the angle from x axis for the destination forces to T1 and
T2 . From this formula, the quantity of force energy from any
sources is equivalent to the quantity of force energy to any
destinations regardless how many sources and destinations are
involved in energy transformation. We calculate the quantity
of force energy transferred to two destination nodes, T1 and
T2 , from a source of force energy, F0 , via
¯
¯
¯ −(F0 cosψ × sinφ) − (F0 sinψ × cosφ) ¯
¯
¯ , (3)
T1 = ¯
−(cosθ × sinφ) − (sinθ × cosφ) ¯
¯
¯
¯ (cosθ × F0 sinψ) − (sinθ × F0 cosψ) ¯
¯,
T2 = ¯¯
(4)
−(cosθ × sinφ) − (sinθ × cosφ) ¯
where F0 is the quantity of force energy in the source, ψ is
the angle from the x axis of the source edge, θ is the angle
from the x axis of the first destination edge and φ is the angle
from the x axis of the second destination edge. Algorithm 1
explains the step by step processes of the method.
To compare our result, we find critical locations of the
sample bone network using existing network analysis methods
including betweenness centrality and bridging coefficient. The
betweenness centrality for edge e, Cbt (e), is calculated by
P
(e)
method, Cbt (e) = s6=v6=t∈V σst
σst , where σst is the number
Algorithm 1 Information Propagation(S, D, G)
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:
G: a bone network
S: start node stack which is on the right most side of graph G
D: ground node list which is on the left most side of graph G
Nν : a neighbor node list of ν with hop(ν → i) = 1 and x
coordinate xi < xν
n: size of Nν
Fi : energy in node i
ei→j : quantity of energy flow of an edge from i to j
repeat
while S is not empty do
Pop a start node s from S
for i = 1 to n do
Calculate T1 , T2 , ...Ti from Fs
Move to Ti
Calculate current Ti = Ti + previous Ti
Calculate current ei→j = ei→j + previous ei→j
end for
Push T1 , T2 , ...Ti into S
end while
Add M ax(ei→j ) into R
Remove M ax(ei→j ) from G
until No path on S → D
Recognize node list, CR , linked to R
Add directed linked edges of CR into R
print R
features to an edge. We defined the weight of an edge e with
l(j)
, where l(j) is
the length of an edge e calculated via max(l(j))
the length of an edge e and max(l(j)) is the maximum length
of an edge in a bone network.
The weighted bridging coefficient of a node v is defined as
the average weight of leaving the direct neighbor sub-graph
of a node v. The weighted bridging coefficient of a node v is
defined via
X
1
ρ(i)
Cwbc (v) =
·
,
(6)
d(v)
wi − w0
i∈N (v)
where d(v) is the degree of a node v, node i is directly
connected to node v, ρ(i) is the average weight of edges
leaving the direct neighbor sub-graph of node i, wi is the
average weight of edges directly connected to node i and w0
is the weight of the edges connected to node v from node i.
The weighted bridging coefficient of an edge e is defined
as the product of the weighted average of bridging coefficient
of two incident nodes i and j for an edge e and the reciprocal
of the number of common direct neighbor nodes of nodes i
and j. The bridging coefficient of an edge e is defined by:
Cwbc (e) =
of shortest paths from the node s to the node t, and σst (e) is
the number of shortest paths from s to t that pass through an
edge e [11].
Since betweenness centrality does not consider spatial characteristics of the bone network, we modified betweenness
centrality to be sensitive on those information by limiting
source and destination nodes on measuring shortest paths to
make a reasonable comparison. In addition, we incorporated
the length of edges as a weight value for the weighted
betweennees centrality via
X γst (e)
Cwbt (e) =
,
(5)
γst
ρ(i)Cwbc (i) + ρ(j)Cwbc (j)
,
(ρ(i) + ρ(j))(|C(i, j)| + 1)
e(i, j) ∈ E
(7)
where nodes i and j are the two incident nodes to edge e, ρ(i)
is the average weight of edges leaving the direct neighbor subgraph of node i, Cwbc (i) is the weighted bridging coefficient
of node i, C(i, j) is the set of common direct neighbor nodes
of nodes i and j.
IV. R ESULTS
In this section, we provide results of identifying critical
locations in the sample bone network when there are external
forces applied to the right side of the bone network. Then, we
evaluate the results of three different approaches.
s6=v6=t∈V
where γst is the number of shortest paths multiplied by the
edge length from the node s to the node t, and γst (e) is the
number of shortest paths multiplied by the edge length from
s to t that pass through an edge e.
As the second method for comparison, bridge coefficient
is employed. A bridge is a node or an edge that is located
between and connects modules in a network. In other words,
a bridge is a node v or an edge e that has high bridging coefficient value. The bridging coefficient of a node v is defined
as the average probability of leaving the direct neighbor subgraph of a node v. The bridging
coefficient of a node v is
P
δ(i)
1
defined by Cbc (v) = d(v)
i∈N (v) d(i)−1 , where d(v) is the
degree of a node v and δ(v) is the number of edges leaving
the direct neighbor sub-graph of node v. In the sample bone
network, we can find critical locations by finding a bridge
node or a bridge edge containing the highest value of bridge
coefficient [4].
We also modified bridging coefficient to be sensitive on
spatial characteristics of bone network by adding weight
(a) The two-dimension structural bone network model is
shown (b) The microscopic view of the bone model are shown.
Fig. 5.
We created a two dimensional 1 × 1 unit (1 unit ≈ 5mm)
square bone network extracted from our bone network as a
representative of bone microstructure shown in Figure 5. The
network consists of 686 nodes and 943 edges as representatives of bone microstructure. Specifically, we assume that a
node stands for an intersection point among mineral fibers
and an edge stands for a group of mineral fibers in bone
microstructure, which is an important element for maintaining
structural strength of a bone network. In this experimentation,
Fig. 6. Figures illustrate steps for identifying critical locations in the
sample 1 × 1 square unit bone network. Red edges are the identified
critical edges in each step, which are removed in the next step to find
the next critical edge.
To compare the result of the information propagation algorithm, we calculated critical locations with the method of
the weighted edge betweenness centrality and the weighted
bridge coefficient in the sample bone network with the same
condition to our method. We iteratively ran each algorithm to
find the first 30 critical edges as representatives of important
locations in the bone network. Figure 7(a) shows average
shortest path length (ASPL) as a function of 30 critical edge
cuts by each method.
We
P
P calculated average shortest path
i∈R(v)
j∈L(v) min(i→j)
length via τ =
, where R(v) is the
mn
node list in the right most side of the bone network; L(v)
is the node list in the left most side of the bone network;
min(i → j) is the shortest path length from node i to node
j; m is the number of node in R(v); and n is the number of
node in L(v) [7].
The plot in Figure 7(a) implies the number of the isolated
bone network blocks since the sharp increment of τ indicates
the segmentation of the network. Information propagation and
weighted betweenness centrality method in Figure 7(a) show
one and three sharp increments of τ , which indicates that the
bone network breaks into two and four isolated bone network
blocks, respectively. Other two methods, including weighted
bridge coefficient and random cut, do not create any separated
blocks of the bone network even after 30 edge cuts.
We also employed moment of inertia (MOI), which is a
measure of an object’s resistance to changes in its rotation rate,
(b) 2430
600
2400
450
2370
ρ
(a) 750
τ
we suppose that there are external forces applied to the right
side of the sample bone network.
We calculated critical locations in the sample bone network
shown in Figure 6(a) with the information propagation algorithm. Figure 6 shows steps to find critical edges as fragile
points in the sample bone network. In Figure 6(b), the first
three critical edges are found as the most fragile points by
external forces. Figures 6(c), (d), and (e) show each step to
find next critical edges after removing previous critical edges.
Finally, figure 6(f) shows the bone network broken into two
pieces after 30 critical edges are removed, which represents
the fracture of a bone.
2340
300
2310
150
0
2280
0
10
20
Edge Cut
30
0
10
20
30
Edge Cut
(a) Average shortest path length (ASPL) as a function of 30
critical edge cuts by each method is shown. Black dots, gray dots,
white dots and white squares represent the method of information
propagation, weighted edge betweenness centrality, weighted bridge
coefficient, and random cut, respectively. (b) Moment of inertia
(MOI) as a function of 30 critical edge cuts by each method is shown.
Fig. 7.
to evaluate the overall strengthP
of bone network. We defined
MOI of bone strength via ρ = i∈V (v) wi δi 2 , where V (v) is
the node list in a bone network; wi is the weight of a node i;
and δi is the distance from the center of the bone network to
the node i. The weight of a node i, wi , which represents the
strength of a node i is defined as the average
P weight of edges
directly connected to a node i, wi = n1 · i∈E(v) ui , where
E(v) is the edge list directly connected to the node v, n is the
number of edges in E(v) and ui is the weight of an edge i.
The weight of an edge i, ui , is set to 0 (cut) or 1 (connected)
[7].
Figure 7(b) shows the plot for ρ as a function of 30
critical edge cuts by each method. High ρ value means that a
possible crack line of the bone network is identified without
damaging the overall strength of the bone network. ρ value
for information propagation method in Figure 7(b) indicates
that critical edges of the network are effectively identified after
30 critical edge cuts. Weighted betweenness centrality method
also shows a comparable performance, but its performance
worsen after 22nd edge cuts. Weighted bridge coefficient and
random cut method do not effectively find critical edges since
ρ value of each method almost linearly decreases.
Figure 8 shows the visualization of results after 30 critical
edge cuts by each method. Figure 8(a) shows the bone
network after critical edges are identified by the information
propagation method against the sample bone network. Two
isolated bone networks are created by removing critical edges
found by information propagation method. As we expected
from Figure 7(a), weighted betweenness centrality method
created four isolated bone network blocks by deleting critical
edges shown in Figure 8(b). Next, in Figure 8(c), the sample
bone network remains in the same even after removing 30
critical edges found by the weighted bridge coefficient method.
Random cut method also failed to create any isolated bone
network block. Last, figure 8(d) depicts crack patterns in a
real bone as an example of a possible crack line.
From these results, we found that the method of weighted
edge betweenness centrality created more isolated bone network blocks than any other methods by removing 30 critical
(a)
not be reliably measured. In addition, a verification step on
our results with real bone break experimentations would be
necessary to apply this model to patients. Nevertheless, wellunderstood knowledge with this type of study would have a
great impact on identifying fragile points and understanding
fracture process in a bone microstructure. By applying the basic ideas of this study, we would develop a testing framework
of a crack propagation process for various materials composing
of interacting atoms among which the interactions govern the
material’s behavior under different conditions.
As a future work, we will continue to work on mathematical
modeling for identifying critical locations on other structural
networks, such as a molecular structural network and a road
system network. This would enable us to understand critical
locations of various networks as well as prepare for uncertain
damages.
(b)
(c)
R EFERENCES
(d)
Figure shows three results for critical locations detected
by information propagation, weighted betweenness centrality and
weighted bridge coefficient, respectively. Critical edges detected by
methods are colored by red. (a) Two isolated bone networks are
created by deleting critical edges. (b) Four isolated bone networks
are created by deleting critical edges. (c) There is no isolated bone
network created by deleting critical edges. (d) An example of cracks
on a real bone is shown [14].
Fig. 8.
edges. However, it is not always the case that bone is broken
into many pieces in the event of bone fracture process unless
extremely powerful external forces are applied. Instead, a
small break in a bone starts from a microscopic crack created
by an impact or repeated flexing “fatigue.” Then, a crack grows
longer and larger when a force is continuously applied shown
in Figure 8(d). We believe that finding crack line by identifying
critical locations of the bone network in a certain condition is
very useful not only for understanding fracture risk of a bone
microstructure but also for diagnosing bone diseases, such as
osteoporosis.
V. C ONCLUSION AND F UTURE W ORK
In this paper, we introduced a bone network model based
on the properties of a bone microstructure, and we developed
the method, information propagation to identify a critical
location in the bone network. We also incorporated existing
methods including the weighted betweenness centrality and
the weighted bridge coefficient into our method, and compared advantages and disadvantages among those methods to
evaluate our results.
Properties we have incorporated in this model, such as the
density of bone microstructure and the weight of edges would
[1] Archimorph. Microscopy bone, 2010. [Online; accessed 28-Jun-2010].
[2] Barnes, C., Newall, F., Ignjatovic, V., Wong, P., Cameron, F., Jones,
G., and Monagle, P. Reduced bone density in children on long-term
warfarin. Pediatric Research, 57:578–581, 2005.
[3] Bruyere, O. et al. Relationship between bone mineral density changes
and fracture risk reduction in patients treated with strontium ranelate.
Journal of Clinical Endocrinology and Metabolism, 92:3076–3081,
2007.
[4] Hwang, W., Kim, T., Ramanathan, M., and Zhang, A. Bridging centrality: Graph mining from element level to group level. In International
Conference on Knowledge Discovery and Data Mining (ACM SIGKDD),
pages 336–344, 2008.
[5] Ito, M., Ikeda, K., Nishiguchi, M., Shindo, H., Uetani, M., Hosoi, T.,
and Orimo, H. Multi-detector row ct imaging of vertebral microstructure
for evaluation of fracture risk. J Bone Miner Res, 20:1828–1836, 2005.
[6] Johnell, O., Kanis, J.A., Oden, A., Johansson, H., De Laet, C., Delmas,
P., Eisman, J.A., Fujiwara, S., Kroger, H., Mellstrom, D., Meunier,
P.J., Melton, L.J. 3rd, O’Neill, T., Pols, H., Reeve, J., Silman, A., and
Tenenhouse, A. Predictive value of bmd for hip and other fractures.
Journal of bone and mineral research, 20:1185–94, 2005.
[7] Kim, T., Hwang, W., Zhang, A. and Ramanathan, M. Computational
framework for microstructural bone dynamics model and its evaluation. In International Conference on Bioinformatics and Bioengineering
(BIBE), pages 92–98, 2010.
[8] Kim, T., Ramanathan, M., and Zhang, A. A graph-based approach
for computational model of bone microstructure. In ACM Workshop
(IWBNA) in conjuction with International Conference on Bioinformatics
and Computational Biology (ACM-BCB), 2010.
[9] Kumasaka, S., Asa, K., Kawamata, R., Okada, T., Miyake, M. and
Kashima, I. Relationship between bone mineral density and bone
stiffness in bone fracture. Oral Radiology, 21:38–40, 2005.
[10] Margerie, E.d., Robin, J.P., Verrier, D., Cubo, J., Groscolas, R., and
Castanet, J. Assessing a relationship between bone microstructure and
growth rate: a fluorescent labelling study in the king penguin chick.
Journal of Experimental Biology, 27:869–879, 2004.
[11] Newman, M.E.J. A measure of betweenness centrality based on random
walks. Social Networks, 27:39–54, 2005.
[12] OTT, S.M., Kilcoyne, R.F., and Chesnut, C.H. 3rd. Ability of four
different techniques of measuring bone mass to diagnose vertebral
fractures in postmenopausal women. Bone Miner Res, 2:201–210, 1987.
[13] Pludowski, P., Lebiedowski, M., and Lorenc, R.S. Dual energy x-ray
absorptiometry. Osteoporos Int, 14:317–322, 2004.
[14] Wending, Z. Micro-structure of gigantoraptor bone showing the annual
growth lines, 2007. [Online; accessed 20-July-2010].
[15] WHO Scientific Group. Prevention and management of osteoporosis.
WHO Technical Report Series, World Health Organization, 921:1–192,
2003.
[16] Xie, C. An atomic look at why things break, 2008. [Online; accessed
12-July-2010].