Mathematical Network Model for Bone Mineral Density
(BMD) and Bone Quality Assessment
Taehyong Kim
∗
Lawrence Bone
Department of Computer
Science and Engineering,
State University of New York
at Buffalo, USA
Department of Orthopedics,
State University of New York
at Buffalo, USA
Department of Pharmaceutical
Sciences, State University of
New York at Buffalo, USA
Department of Computer
Science and Engineering,
State University of New York
at Buffalo, USA
bone@buffalo.edu
thkim7@buffalo.edu
Murali Ramanathan
murali@buffalo.edu
ABSTRACT
The bone fractures can appear in a patient as a result of an
exceeding stress level at a specific anatomic site, depending
the failure on fatigue phenomena, high loads or on a low
value of bone density. Indeed, independently from physiological conditions or specific pathologies such as osteoporotic
ones, bone mineral density (BMD) constitutes the main responsible of the strength of a selected bone region. In this
respect, the standard and routine approach for the diagnosis
of osteoporosis is to assess BMD shown to be a possible indicator of fracture risk. However, a major limitation of BMD
is that it incompletely reflects variation in bone strength.
Other factors like bone microarchitecture contribute substantially to bone strength and their evaluation can improve
determination of bone quality and strength; yet, structural
assessment has not been implemented in clinical routine because of the lack of a mathematical model for analyzing bone
structure.
In this paper, we develop a mathematical network model for
bone microstructure which is capable of quantitative assessment of bone mineral density and bone micro-architecture.
First, we design a bone continuum model by analyzing bone
image profile of dual-energy X-ray absorptiometry (DXA)
scan. Next, we introduce a mathematical network model of
bone microstructure which allows us to calculate BMDmodel
as well as the density distribution of bone microstructure for
patients. Last, we present realizations of the mathematical
network model based on DXA scan images of two differ∗corresponding author
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ACM-BCB ' 11, August 1-3, Chicago, IL, USA
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ACM-BCB 11
Aidong Zhang
azhang@buffalo.edu
ent patients as a representative example of bone network
model. Our study provides an initial framework of mathematical network bone model along with BMDmodel that
can enhance the diagnosis ability of bone disease such as osteoporosis. Eventually, it would be useful for a theoretical
testing framework of bone remodeling dynamics to leverage
new drug development for future treatments.
1.
INTRODUCTION
Over the past few years, empirical and theoretical analysis of network-based approaches have been one of the most
popular subjects of recent researches in many areas including technological, social, and biological fields. As a proven
fact, network theories have been applied with good success
to these real world systems [22]. With network modeling, a
complex form of real world systems could be transferred to
the simplest form with which important knowledge in real
world systems is efficiently attained. As one of good applications of network modeling, microstructure of bone is studied
to assess bone strength and bone quality.
Although bone is a simple composite of a mineral phase,
bone 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. In addition, bone is a dynamic, living tissue whose structure and shape continuously adjusts to
mainly provide structural framework. A firm skeleton makes
it possible to support weight and ensures protection for the
muscles and organs. Bone also participates in the maintenance of serum-mineral metabolism, and is considered an
important component of the immune system.
In this paper, we focus on developing the spatial component
of such a mathematical modeling framework and demonstrate how the density distribution and BMD of DXA scan
images can be incorporated to develop mathematical network bone model. In addition, we describe how the model
can possibly be delivered for the clinical application of the
69
model and the further assessment of bone quality.
This paper is organized as follows. In the following background section, we describe properties of bone and current
issues on analyzing bone strength. In the next section, we
introduce our continuum model with mathematical analysis based on DXA scan images as the first subsection of
the methods section followed by the bone network model
describing the process of microstructural bone modeling in
details. In the third subsection, we present two different realizations of the bone network model based on DXA scan
images of two patients as a representative example for the
clinical application of the model. We summarize our study
in the discussion and future work section.
2. BACKGROUND
Bone is not a uniformly solid material, but rather has some
spaces between its hard elements as shown in Figure 1(a)
and (b). 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, so-called due to its minimal gaps and spaces. This
tissue gives bones their smooth, white, and solid appearance,
and accounts for 80% of the total bone mass of an adult
skeleton. Filling the interior of the organ is the trabecular
bone tissue (an open cell porous network also called cancellus
or spongy bone), which is composed of a network of rod- and
plate-like elements that make the overall organ lighter and
allowing room for blood vessels and marrow. Trabecular
bone accounts for the remaining 20% of total bone mass but
has nearly ten times the surface area of compact bone.
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 variation in bone strength and that differentiation of
patients with and patients without vertebral fractures is inaccurate [1, 5, 14, 25]. Other factors like bone microarchitecture contribute substantially to bone strength and their
evaluation can improve determination of bone quality and
strength [12, 20, 28]; yet structural assessment has not been
implemented in clinical routine because of a lack of computational bone model. Furthermore, although recent studies
have demonstrated the relative importance of architecture
and mass as determinants of bone strength, architectural assessment techniques are significantly less developed [26]. A
study shows that accurate prediction of bone strength using
BMD alone is very challenging [15]. In this circumstance,
we tried to study this problem in different direction with
mathematical network modeling of bone microstructure to
overcome those obstacles.
3. METHODS
In this section, we will show the mathematical analysis and
the process of the bone network modeling. Our modeling
approach involves two steps: i) develop a continuum model
by analyzing the density for mineralized fibers of bone microstructure and then ii) apply the continuum model to develop a computational network model for bone microstructure which is capable of quantitative assessment of bone mineral density.
3.1 The Continuum Model
(a)
(b)
Figure 1: (a) The cross section image of a femur bone
is shown [6]. (b) The microscopic view of a bone microstructure is shown [29].
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 [27]. As the efforts on the diagnosis of osteoporosis, correlations between BMD and fracture risk have
been extensively studied; however, many studies show that
low BMD is not the sole factor for the fracture risk [10, 9,
19, 4, 23]. Other factors like bone microstructure also contribute substantially to bone strength [13, 23]. 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.
The standard and routine approach for the diagnosis of osteoporosis is to assess BMD using either dual-energy X-ray
ACM-BCB 11
As the first step of our modeling process, we describe the
continuum model of bone microstructure which offers an effective method of calculating bone density. In general, theoretical models can not fully represent the total complexity
of the real world, but can only include certain aspects that
are important for solving a certain question or problem. For
example, mechanical properties of a bone are defined by the
masses of, and interactions between, minerals on a molecular scale; however, it is not always possible to employ every
particle or molecule to analyze a bone microstructure. To
overcome this problem, we employ continuum methodology
by exploiting the repetitive pattern of lattice structures and
developing an equivalent continuum model of a bone microstructure.
We provide the mathematical definition of the density distribution of bone mineral and the expression for bone mineral
mass relative to DXA scan image in this step. Since DXA
scan is broadly used to calculate BMD, important properties
of bone strength can be studied by analyzing bone density
distribution and BMD of DXA scan images [24, 2]. We use a
human femur bone image of DXA scan to analyze properties
of the bone density distribution and BMD.
Figure 2 shows the process of DXA scan and results of scan
images. From the DXA scan, x-ray images and BMD can be
obtained, which can be used for inputs of our mathematical
bone network model. Figure 2(c) shows a femur bone image
of a patient with osteoporosis by DXA scanning with which
we can identify the density distribution of the femur bone.
By image profiling on the black line section in Figure 2(d),
70
we can have the bone density distribution as a function of
the radius from the left side of the bone to the right side
of the bone. Figure 2(e) is shown as a sample image of the
cross section on the femur bone.
OˆP
O‰P
OŠP
where Ap is the projected area of the bone segment.
Relationship to Imaging Data: We used Beer’s law to
relate the continuum model to medical quantitative imaging
findings obtained by DXA scanning and x-ray imaging methods. Beer’s law states that the logarithm of the fraction of
electromagnetic energy absorbed is a linear function of the
concentration of the absorbing species and the path length
[11]. Integration of Beer’s law over a differential element dy
is necessary for obtaining the image intensity at position x
from the center because the concentration of bone mineral
varies along the path length of the x-ray beam:
d ln A(x) = −αc(r)dy,
pˆ”ŽŒšG–™”G››—aVVžžžUŒ‹‰U
œ›ŒŸˆšUŒ‹œV›V‰–‹ Š–”—›U——
O‹P
OŒP
(4)
where A(x) is fraction of the input x-ray intensity that is
absorbed by bone mineral at position x from the center, α
is the extinction coefficient, and dy is the differential path
length.
p
dy = d(r sin θ) = d( r2 − x2 )
(5)
Combining the Beer’s law with the path length expression,
we obtain:
√
2
2
A(x) = e−αc(r)( r −x ) .
(6)
Figure 2: (a) A DXA machine. (b) An DXA scan image
for whole human body is shown. (c) The femur bone
is shown as the major component to measure BMD. (d)
The magnified image of femur bone. (e) A cross section
image of femur bone is shown.
Model for Distribution of Bone Mineral: The mean
concentration of bone mineral c(r) expressed as an area fraction varying in the radial direction r according to a cumulative normal distribution:
Z r −(z−µ)2
c0
(1)
e 2σ2 dz.
c(r) = √
2π −r
The mean and standard deviation of the underlying normal
distribution are denoted by µ and σ, respectively, whereas
c0 is a constant of proportionality. The values of µ and σ
can be estimated from a DXA scan image using parameter
estimation by Nelder-Mead algorithm [21]. Parameter estimation by Nelder-Mead algorithm will be explained at the
end of this subsection.
Expression for Bone Mineral Mass: The mean mass of
bone mineral m was obtained from the expression for c(r)
by integrating over an annular differential element:
Z R0
m = w0
rc(r)dr.
(2)
r=0
The radius outside of bone is denoted by R0 and the density
coefficient of bone mineral is denoted by w0 . Bone mineral
mass can be viewed as the first moment over the concentration distribution.
Expression for Bone Mineral Density: The bone mineral density, BMD, is a projected density. It was obtained
from the bone mineral mass m as:
Z R0
m
w0
m
=
rc(r)dr,
(3)
=
BM D =
Ap
πR0 2
πR0 2 r=0
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The image intensity is proportional to the transmitted energy T (x) that reaches the imaging device:
T (x) = 1 − A(x).
(7)
The continuum model predicts that imaging data will be a
linear function of T (x).
Image Analysis and Regression: We conducted image
analysis for DXA scan images of the femur bone from a
patient to identify macroscopic radial variations in BMD.
The imaging data were analyzed using ImageJ software [7].
Parameters, µ and σ, were estimated by minimizing the
squared difference between the density distribution from the
DXA scan image and the density distribution of the mathematical network bone model. The model calculations were
averaged over n = 100 realizations. The minimization objective function F (d) is defined as:
F (d) =
n
X
i=1
[y(d, µi , σi , Ri ) − x(µi , σi , Ri )]2 ,
(8)
such that d in the objective function represents the vector
containing the model parameters that are to be estimated.
The quantity x(µi , σi , Ri ) represents the data point corresponding to each observation i wherein µi is the mean of
the observed image density distribution corresponding to
the data point, σi is the standard deviation of the observed
image density distribution to the data point and Ri is the
outside radius of bone. The y(d, µi , σi , Ri ) is the calculated
value from the model that corresponds to µi , σi , and Ri .
Simplex minimization with the Nelder-Mead algorithm was
employed to determine the parameter estimates for minimum discrepancy [21]. Java code for the minimization algorithm was obtained from [8].
71
3.2 The Network Model
In this subsection, we will show the process of our mathematical bone network modeling in details and two representative
realizations of the bone model on DXA scan images. The
network model was developed based on the parameters of the
continuous model, µ, σ, R0 , described in the previous subsection. From the bone continuum model section, we describe
the creation process of a mathematical network bone model.
When we develop our mathematical bone network model,
important components of the bone structure are considered
to build a bone model enabling computational analysis in
a timely manner without losing the critical bone structural
properties. We focused mineralized collagen fibers as important components from the bone cellular structure point of
view.
A bone network is defined by a weighted undirected spacesensitive graph in a 1x1 unit 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 , we 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.
The mean and standard deviation of the underlying normal
distribution denoted by µ and σ, respectively, are the same
as those for the concentration on bone mineral in the continuum model, whereas λ0 is a constant of proportionality.
The actual number of vertexes n(r) at a radial location r,
was obtained as a Poisson random variate, P oisson(λ(r)).
These points were randomly positioned in the angular direction by drawing random variates from a uniform distribution, U nif orm(0, 2π). These vertexes provide the Voronoi
sites or centers.
As the next process of the model, bone microstructure is created shown in Figure 3(b) as a representative of mineralized
fibers which are one of the important structural components
for bone strength. Based on Voronoi vertexes, we calculate
expected area occupied by each Voronoi vertex based on
Voronoi tessellation method.
Voronoi Tessellation: The Voronoi tessellation method
proposed in the following can be used regardless of the exact functional form selected for the point density function
λ(r). The point distribution was converted to a network
using the Voronoi tessellation method [3]. Voronoi tessellation converts a set of points into discrete regions containing of points that are closer to that point than any other.
The boundaries of the discrete polygonal Voronoi regions
are straight lines. We interpreted the Voronoi boundaries as
network edges representing the mineral matrix comprising
bone.
In the next step shown in Figure 3(c), weight of edges are
assigned based on the cumulated normal distribution function, λ(r), with the same parameter values of µ and σ. Then,
bone edges are pruned by the proportion of the edge weight
to pattern bone microstructures shown in Figure 3(d).
Figure 3: (a) Voronoi vertexes are generated based on
the distribution of c(r). (b) Model structure is created
by Voronoi tessellation. (c) Weight of edges are assigned
by λ(r). (d) Edges are pruned by the probability p =
−β
wi
1 − e wmax where β > 0 is the rate parameter of the
survival function.
As the first step of our modeling process, Voronoi vertexes
are generated by the rule of Distribution of Voronoi Centers, which are the center position of rod-like structures as
representatives of mineralized fibers, shown in Figure 3(a)
described below.
Distribution of Voronoi Vertexes: As an initial framework, we generated a field of vertexes that were randomly
distributed in the angular direction. The vertexes can be
interpreted as the center of the bone mineral in the mathematical network model. The mean number of vertexes per
unit area (i.e., point density) r varied in the radial direction
according to a cumulative normal distribution:
Z −r −(z−µ)2
λ0
e 2σ2 dz.
(9)
λ(r) = √
2π −∞
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The bone edges were pruned by the weight of edges to provide a geometric framework that approximated the bone
mineralization. Each edge e of the bone network was assumed to have different thickness and firmly connected to
the other inter-connected edges at every corner point in the
network. Each edge was associated with weight factors ti , li
to calculate edge weight value, we , where ti is thickness of
edge and li is length of edge. Edges are pruned by the probwi
−β
ability of the survival function, p = 1 − e wmax , where
β > 0 is the rate parameter of the survival function, wi is
ti × li and wmax is the maximum value among wi .
As one of methods to verify the bone network realization of
our model, a mathematical evaluation is employed to compare BMD of a DXA scan image and BMD of a realization
of the bone network model. We defined the bone mineral
mass, m′ , and BMD of our mathematical model, BMDmodel
in which BMD and BMDmodel are shown to be quantitatively measured and evaluated.
Expression for Bone Mineral Mass: The mean mass of
bone mineral m′ for a realization of the network model was
obtained as the summation over all the edges:
m′ =
n
X
(10)
wi ,
i=1
72
where n is the number of edges and wi is the weight of each
edge.
Expression for Bone Mineral Density: The bone mineral density BMDmodel for a realization of the network model
was defined from the preceding formula for bone mineral
mass combined with the pruning probability of edges as:
BMDmodel =
n
X
i=0
wi (1 − e
w
−β w i
max
),
(11)
where wmax is the maximum value of edge weights and β is
a pruning rate constant.
4. EXPERIMENTAL STUDY
In the previous section, we have provided mathematical methods to calculate bone quality on the bone network model by
BMDmodel . By the calculation of BMDmodel , a realization of
the bone network model can be possibly compared to BMD
of a DXA scan image on a given femur bone with analyzing
the value of BMD with the value of BMDmodel .
As a representative example, we applied the bone modeling
method to create a bone network based on a given DXA scan
image. We calculated BMD of a bone model realization,
BMDmodel , according to the given formula in the previous
section. Then, we compared the BMD of a given DXA scan
image and BMDmodel of the realization of the bone model.
BMDmodel of the realization by bone network model was
calculated as 0.9475 if BMD of a given DXA scan image is
0.9643 (g cm−2 ), which is BMD ≈ BMDmodel .
DXA Image
Mathematical Model
BMD= 0.9643
BMDmodel = 0.9475
Table 1: BMD comparison between DXA scanned image
and mathematical bone network model.
It explains that the realization of the bone network model
properly reflects characteristics of bone structural density
shown in Figures 4(a) and (d), and Table 1. Slight difference
of the value between BMD and BMDmodel is caused by the
randomness of the probability p to prune bone edges in the
network. Moreover, we incorporated Nelder-Mead algorithm
to fit µ and σ of the bone network, which also conserves the
characteristics of the density distribution on the bone image
with µ = 0.25 and σ = 0.05 shown in Figures 4(b) and (c).
In addition to the mathematical evaluation method, the application of mathematical bone network model was exemplified by creating two different realizations of the bone network model based on given DXA scan images of two patients
shown in Figure 5. With the different DXA scan images between patient one in the first column and patient two in the
second column shown in Figures 5(a) and (b), respectively,
we applied our mathematical bone network model approach
to create bone networks for two different DXA scan images.
Although patient one and patient two would have similar
BMD values, the characteristic of the density distribution on
each bone could be quite different from each other. As shown
in the first row of Figures 5(a) and (b), patient one has
the thinner area of the high density region of bone whereas
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Figure 4: (a) A cross section image of a femur bone is
shown. (b) The density distribution of the projection
image on the femur bone is shown. (c) The density distribution of the projection image on a bone network is
shown. (d) A visualization of the realization on bone
network model is shown.
patient two has the wider area of the high density region.
On the contrary, patient two has the smaller area of the bone
core region but the denser than those of patient one.
The second row in Figures 5(a) and (b) shows the density
distribution of the cross section on each DXA scan image
by profiling the perpendicular (black) line in the first row of
Figures 5(a) and (b). Then, we calculated values of µ and
σ by fitting to the cumulated normal distribution function,
λ(r), to apply characteristics of the density distribution on
each bone image into bone network realizations. The third
row shows the density distribution on the realization of bone
network model for each DXA scan image after minimizing
the squared difference between the density distribution of
the DXA scan image and the density distribution of the
mathematical network bone model with the proposed function in the previous section, F (d), and Nelder-Mead algorithm. The density distribution of patient one was fitted
to µ = 0.35 and σ = 0.05 and the density distribution of
patient two was fitted to µ = 0.25 and σ = 0.05.
The last row in Figures 5(a) and (b) clearly shows the different density patterns reflecting the difference of DXA scan
images of two patients, which provides further support on
our mathematical bone network model. It explains that the
different characteristics of the density distribution on DXA
images can be applied into realizations of our mathematical
model. Since such structural differences would be an important aspect on measuring bone strength, this approach
could be a valuable step to accurately evaluate bone quality.
With the capability of a mathematical bone network model,
personalized assessments of bone strength would possibly be
employed into a diagnosis process of bone disease by creating
different realizations of the bone model for particular DXA
scan images among patients. In addition, by utilizing existing computational methods and mathematical measurements with this model, the structural evaluation of bone
and the estimation of fracture risk would be quantitatively
73
region in DXA scan image reflects the low-density region in
bone network model. To verify this hypothesis, we analyzed
femur DXA scan images from patients and demonstrated
that similarities of BMD and BMDmodel as well as fitting of
the density distribution by Nelder-Mead algorithm between
the DXA scan images and realizations of the bone model.
The motivation on exploring image profiling for fracture risk
estimation was its potential to capture bone fragility features
related to spatial similarity that have not been captured by
conventional bone densitometry. The mathematical bone
network model with image profiling approaches would be a
good method for providing better expectation on fracture
risk compared to the BMD approach alone. In particular,
when BMD was combined with the mathematical model approaches, such as the estimation of density distribution and
the spatial model of bone microarchitecture, the fracture
risk and bone strength would be well measured comparing
to using BMD alone.
Figure 5: Two DXA images are shown for different realization examples of our mathematical bone network
model. (a) A DXA scan image with thin cortical bone
area and thick trabecular bone area is shown. (b) A DXA
scan image with thick cortical bone and thin trabecular
bone is shown.
Our next step of this study is to develop and evaluate a
novel mathematical bone model which is capable of providing a more effective platform for evaluating fracture risk
and bone strength. In addition, this model would be a fundamental framework of bone-modulating drugs and dosing
regimens during drug development. A well-designed mathematical model could potentially yield clinical insights by
providing: i) structurally derived quantitative measures of
bone microarchitecture, ii) a quantitative framework for understanding patterns in the bone imaging results from DXA
scan, iii) the differential impact of drug treatment on trabecular and cortical bone, and iv) a mechanistic understanding
of the relationship of bone microarchitecture to fracture risk.
measured [16, 18, 17]. However, parameters we have incorporated in this model would not be reliably ascertained.
Prompt and proper information obtaining from high-quality
orthopaedic data is critical for accurate analysis to reduce
these uncertainties. Well-understood knowledge with this
type of the model would be valuable in the analysis of the
bone quality. We believe that our mathematical bone network model could be a useful tool not only for understanding properties of bone microstructure but also for diagnosing
bone strength and bone diseases, such as osteoporosis.
In the future work, we will also develop a mathematical
framework of bone remodeling dynamics by extending our
mathematical bone network model. This will be enabling
understanding of bone related diseases, such as osteoporosis,
as well as bone remodeling processes. Moreover, it would be
useful for a theoretical testing framework of bone dynamics
to conduct analysis of drug effects. Eventually, this study
would provide new ways to the treatment of bone diseases,
such as osteoporosis.
5. DISCUSSION AND FUTURE WORK
6.
We presented a mathematical network model of bone microstructure and procedures to realize bone network model
based on DXA scan images. We also provided a mathematical method and a representative example for different realizations of bone network model based on DXA scan images
of two patients.
In this study, we adapted the image profiling method that
is routinely used in image registration to develop a networkbased bone model as a fundamental framework for measuring bone strength. In contrast to conventional bone densitometry which measures BMD, our mathematical bone network modeling approach aims to capture information about
the bone microstructure density distribution of bone mineral in a given femur bone. Our mathematical model is
based on the hypothesis that properties of the high-density
region in DXA scan image reflects the high-density region
in bone network model, and properties of the low-density
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