Int. J. Data Mining and Bioinformatics, Vol. x, No. x, xxxx
Department of Computer Science and Engineering,
State University of New York at Buffalo,
201 Bell Hall, Buffalo, New York, USA 14260-2000
E-mail: thkim7@buffalo.edu
*Corresponding author
Department of Orthopedics,
State University of New York at Buffalo,
462 Grider Street, Buffalo, New York, USA 14215
E-mail: bone@buffalo.edu
Department of Pharmaceutical Sciences,
State University of New York at Buffalo,
517 Hochstetter Hall, Buffalo, New York, USA 14260-1200
E-mail: murali@buffalo.edu
Department of Computer Science and Engineering,
State University of New York at Buffalo,
201 Bell Hall, Buffalo, New York, USA 14260-2000
E-mail: azhang@buffalo.edu
Abstract: We develop a network model of bone microstructure and dynamics, BoneNET, which is capable of quantitative assessment of bone mineral density and bone remodeling dynamics. The study focuses on bone microstructure and remodeling dynamics based on a bone network model.
First, we introduce a network model of bone microstructure by describing structural properties and process of bone network modeling. Second, we explain a mathematical model of bone microstructure by analyzing the density for mineralized fibers of bone microstructure and then apply the mathematical model to develop a computational bone network model. Next, we show a bone remodeling dynamics of autocrine and paracrine interactions among osteoblast and osteoclast. It allows us to calculate cell population dynamics and changes in bone mass at multiple sites of bone remodeling. Last, study bone networks by proposing serval measurements to calculate bone strength and identify critical elements in bone microstructure. Our study provides an initial framework of bone remodeling simulation that can explain experimental observations in bone biology to explore bone diseases such as osteoporosis.
Copyright c 2011 Inderscience Enterprises Ltd.
1

2 T. Kim et al.
Keywords: Computational model of bone microstructure; Bone strength;
Bone mineral density (BMD); Bone remodeling dynamics; Osteoporosis.
Reference to this paper should be made as follows: Taehyong Kim
‘BoneNET: A Network Model of Bone Microstructure and Dynamics’, Int. J.
Data Mining and Bioinformatics , Vol. x, No. x, pp.xxx–xxx.
Biographical notes: Taehyong Kim is currently a PhD candidate in the
Department of Computer Science and Engineering at the State University of New York at Buffalo. His research interests include bioinformatics, data mining, stochastic simulation, dynamics modeling and software engineering.
Dr. Lawrence Bone received his MD in Orthopaedics from at the State
University of New York at Buffalo. Currently, he is a full professor and chair in the department of Orthopaedic Surgery at the State University of New York at Buffalo as well as the Residency Program Director. He has served as a reviewer for Clinical Orthopaedics and Related Research,
Journal of Bone and Joint Surgery, Bone and Joint Diseases, and Journal of Trauma. In past, he worked in various committees and groups including
Orthopaedic Trauma Association, American Academy of Orthopaedic Surgery and American College of Surgeons.
Dr. Murali Ramanathan received his PhD in Bioengineering from the
University of California-San Francisco and University of California-Berkeley
Joint Program in Bioengineering. He is a full professor of Pharmaceutical
Sciences at the State University of New York at Buffalo. His research focus is the treatment of multiple sclerosis (MS), an inflammatory-demyelinating disease of the central nervous system that affects over 1 million patients worldwide.
Dr. Aidong Zhang received her PhD in Computer Science from Purdue
University. She is a full professor and chair in the department of
Computer Science and Engineering at the State University of New York at
Buffalo and the principal investigator and program director of the Buffalo
Center for Biomedical Computing (BCBC). Her research interests include bioinformatics, data mining, database systems, content-based retrieval, and multimedia systems.
Over the past few years, empirical and theoretical studies of a network 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 (Newman, M.E.J., 2004). 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 understand strength and quality of bone.

BoneNET: A Network Model of Bone Microstructure and Dynamics 3
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.
Bone maintains its ability to change its internal structure by removal of old bone and its replacement with newly formed bone in localized processes called remodeling.
Remodeling is a fundamental property of bone that allows adaptation to a changing mechanical environment. Micro mineral structures of bone are removed where the mechanical demand of the skeleton is low and new bone is formed at those sites where mechanical strains are repeatedly detected. Remodeling also enables recoveries from micro damages caused by fatigue and shock. This continuous care of the bone matrix prevents its premature collapse and keeps its overall strength. Remodeling is a complex process performed by the coordinated activities of osteoblast and osteoclast. A bone dynamics modeling is based on the idea that the relative proportions of immature and mature osteoblast control the degree of osteoclastic activity.
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 (WHO Scientific Group, 2003). In addition, osteoporosis accounts for approximately 1.5 million new fractures each year, with associated medical charges including rehabilitation and extended treatment facilities of an estimated $10 billion, according to the National Osteoporosis Foundation. Because osteoporosis affects primarily the elderly, the National Osteoporosis Foundation estimates that these costs will increase to $200 billion by the year 2040, as the number of Americans over the age of 65 years grows (Barefield, E., 1996).
The standard and routine approach for the diagnosis of osteoporosis is to assess bone mineral density (BMD) using either dual energy X-ray absorptiometry (DXA) or quantitative computed tomography (QCT), which is shown to be a possible indicator of fracture risk. However, a major limitation of BMD is that it incompletely reflects variation in bone strength (Ammann, P. et al., 2003; Cummings, S.R. et al., 1995; Kanis,
J.A. et al., 2005; Taylor, B.C. et al., 2004). Other factors like bone microarchitecture contribute substantially to bone strength and their evaluation can improve determination of bone quality and strength (Ito, M. et al., 2005; Link, T.M. et al., 1998; Wigderowitz,
C.A. et al., 2000); however structural assessment has not been implemented in clinical routine because of the lack of a computational framework for analyzing bone structure and dynamics.
Bone dynamics modeling is one of the interesting but challenging research topics.
In this paper, we study the problem in a new direction. We develop a network model for a bone microstructure and dynamics, BoneNET, which is capable of quantitative assessment of bone mineral density and bone remodeling dynamics. This paper focuses on microstructural bone modeling, bone remodeling dynamics and evaluation of the quality of microscopic bone structure. The three main components of the paper are described as follows.

4 T. Kim et al.
We first introduce a network model of bone microstructure by describing structural properties and process of bone network modeling. Second, we simulate bone remodeling dynamics using our differential equations on a bone network model, which can give understanding of the metabolic bone reaction under the condition of uncertain biochemical or cellular interactions. Last, we study bone networks by proposing several measurements to calculate bone strength and identify crucial elements in bone microstructure.
Measurement of bone mineral density (BMD) is the central component of any provision that arises from osteoporosis (European Foundation for Osteoporosis, 1991); however, mounting evidence indicates that low BMD is not the sole factor responsible for the fracture risk (Johnell, O. et al., 2005; OTT, S.M. et al., 1987). Several structural approaches have been studied to overcome the shortcoming of BMD. Since computing power has been doubled by every one and a half years, computer simulations of bone remodeling are attempts to correlate discrete physiological events with observed changes in bone morphology (Kimmel, D.B., 1983, 1985; Reeve, J., 1986a,b; Weinhold, P.S. et al., 1994).
Although producing useful results, these previous simulations are essentially based on one-dimensional models, and therefore do not account for the interconnected trabecular structure of cancellous bone or the distribution of remodeling sites that can be only shown in the two-/three-dimensional structure. So several studies have worked on developing a simulation that is based on a two-/three-dimensional structural model of cancellous bone which allows for the occurrence of trabecular perforation, a naturally occurring event. Extensive studies of structural approaches are done (MacNeil, J.A. et al., 2008; Mueller, T.L. et al., 2008; Tsubota, K. et al., 2009; Kameo, Y. et al., 2008;
Pham, T.D., 2008; Fontaine, A., 2009; Adachi, T. et al., 2006; Arbenz, P. et al., 2008;
Adachi, T. et al., 2001; Muller, R., 2007; Adachi, T. et al., 2008; Boutroy, S. et al., 2008;
Tayyar, S. et al., 1999); however, these studies have not incorporated both biochemical reactions of bone remodeling mechanism and quantitative measurements of bone quality into a computer simulation model.
Modeling, simulation and visualization techniques based on network concepts can help address many of the challenges brought forth by the need for understanding and preparedness on unknown viruses and living tissues (Wu, F.X., 2008). A recent study indicates that a number of modeling and simulation applications analyze epidemics and diseases spreads (Kim, T. et al., 2009, 2008). Networks are encountered in a wide variety of contexts, such as social networks, circuit networks, networks of neurons, and terrorist 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. Many of these networks are dynamic (or not static) in that they grow and evolve as time goes by. Thus the prediction on biological networks is the subject of dynamic modeling. Bone dynamics is one of the most exciting but challenging research area in terms of network dynamic modeling.
Several ordinary differential equation (ODE) systems for bone remodeling cycle were introduced (Lemaire, V. et al., 2004; Komarova, S.V. et al., 2003). Since then the

BoneNET: A Network Model of Bone Microstructure and Dynamics 5 differential equation methods have been refined and improved in various ways (Ryser,
M.D. et al., 2009; Komarova S.V., 2006); however, there is no computational framework of bone microstructure and dynamics model enabling quantitative assessments of bone quality.
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.
We will 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.
For the first step of the continuum model, we describe the mathematical definition of the density distribution of bone mineral and the expression for bone mineral mass relative to DXA scan image. 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 (Pludowski, P. et al., 2004; Barnes, C.,
2005). We use a human femur bone image of DXA scan to analyze properties of the bone density distribution and BMD.
Figure 1 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 1(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 1(d), 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 1(e) is shown as a sample image of the cross section on the femur bone.
Model for Distribution of Bone Mineral : The mean concentration of bone mineral c ( r ) expressed as area fraction varied in the radial direction r according to a cumulative normal distribution:
∫ c ( r ) =
√ 0
2 π r
− r e
− ( z − µ )2
2 σ
2 dz.
(1)
The mean and standard deviation of the underlying normal distribution are denoted by
µ and σ , respectively, whereas c
0 is a constant of proportionality. The values of µ and σ

6 T. Kim et al.
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OŒP pˆ”ŽŒšG–™”G››—aVVžžžUŒ‹‰U
œ›ŒŸˆšUŒ‹œV›V‰–‹ Š–”—›U——
O‹P
Figure 1 : (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.
can be estimated from a DXA scan image using parameter estimation by Nelder-Mead algorithm (Nelder, J.A. et al., 1965). 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:
∫
R
0 m = w
0 rc ( r ) dr.
(2) r =0
The radius outside of bone is denoted by R
0 and the density coefficient of bone mineral is denoted by w
0
. 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:
BM D = m
A p
= m
πR
0
2
= w
0
πR
0
2
∫
R
0 r =0 rc ( r ) dr, (3) where A p 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 (Ingle, J.D.J. et al., 1988). 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, (4)

BoneNET: A Network Model of Bone Microstructure and Dynamics 7 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.
dy = d ( r sin θ ) = d (
√ r 2
− x 2 ) (5)
Combining the Beer’s law with the path length expression, we obtain:
A ( x ) = e
−
αc ( r )(
√ r 2 − x 2 )
.
(6)
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 (Dougherty, G, 2009).
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 ) = i =1
[ y ( d, µ i
, σ i
, R i
)
− x ( µ i
, σ i
, R i
)]
2
, (8) such that d in the objective function represents the vector containing the model parameters that are to be estimated.
The quantity x ( µ i
, σ i
, R i
) represents the data point corresponding to each observation i wherein µ i is the mean of the observed image density distribution corresponding to the data point, σ i is the standard deviation of the observed image density distribution to the data point and R i is the outside radius of bone. The y ( d, µ i
, σ i
, R i
) is the calculated value from the model that corresponds to µ i
, σ i
, and
R i
. Simplex minimization with the Nelder-Mead algorithm was employed to determine the parameter estimates for minimum discrepancy (Nelder, J.A. et al., 1965). Java code for the minimization algorithm was obtained from (Flanagan, M.T., 2009).
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, R
0
, 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

8 T. Kim et al.
mineralized collagen fibers as important components from the bone cellular structure point of view.
A bone network is defined by a weighted undirected space-sensitive 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 , 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.
Figure 2 : (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 = 1
− e
− β wi 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 2(a) described below.
Distribution of Voronoi Vertexes : As an initial model, 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:
λ ( r ) =
λ
0
2 π
∫
− r e
−
( z
−
µ )2
2 σ
2
−∞ dz.
(9)
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 λ
0 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 , 2 π ) .
These vertexes provide the Voronoi sites or centers.
As the next process of the model, bone microstructure is created shown in Figure
2(b) as a representative of mineralized fibers which are one of the important structural

BoneNET: A Network Model of Bone Microstructure and Dynamics 9 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 (Bock, M. et al., 2009). 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 2(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 2(d).
The bone edges were pruned by the weight of edges to provide a geometric model 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 t i
, l i to calculate edge weight value, edge. Edges are pruned by the probability of the survival function, p = 1
− e
− β where β > 0 w e
, where t i is thickness of edge and l i is the rate parameter of the survival function, w i is t i
× l i is length of and w wi wmax max is
, the maximum value among w i
.
OˆP itkGdGXU_[^\
O‰P 1
0.8
0.6
0.4
0.2
0
0
µ dWUY\SG σ dWUW\
20 40
Sequence
60 80
O‹P itk
”–‹Œ“
GdGWU`[^\
1
OŠP
0.8
0.6
0.4
0.2
0
0
µ dWUY\SG σ dWUW\
200 400 600
Sequence
800 1000
Figure 3 : (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.
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, BMD model to be quantitatively measured and evaluated.
in which BMD and BMD model are shown

10 T. Kim et al.
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
′
= i =1 w i
, (10) where n is the number of edges and w i is the weight of each edge.
Expression for Bone Mineral Density : The bone mineral density BMD model for a realization of the network model was obtained from the preceding formula for bone mineral mass combined with the pruning probability of edges as:
BMD model
= i =0 w i
(1
− e
−
β wi wmax ) , where w max is the maximum value of edge weights and β is a rate constant.
(11)
As a second component of our computational model, we develop an ODE for bone remodeling dynamics working on our bone network extended from a mathematical model (Komarova, S.V. et al., 2003).
(a) (b)
100
95
90
85
0 50 100
Time (days)
150
Figure 4 : (a) The process of bone formation is shown. (b) A typical bone remodeling cycle is shown with the percentage of bone mass as a function of time. There is a 2% underfilling of each remodeling site.
An osteoblast is a mononucleate cell that is responsible for bone formation, and an osteoclast is a type of bone cell that removes bone tissue by removing its mineralized matrix. The interactions between osteoblast and osteoclast, which guarantee a proper balance between bone gain and loss, is known as coupling. Metabolic bone diseases appear when a biochemical or cellular link of this finely organized network is chronically disrupted (Marathe, A. et al., 2008; Lemaire, V. et al., 2004). To understand the metabolic bone reaction on unbalanced biochemical or cellular interactions, we develop bone remodeling dynamics (as shown in Figures 4) on our bone network model.
The red edge in a partial bone network contains a basic multicellular unit (BMU) shown in the left side of Figure 4(a) and the remodeling cycle of a BMU, which consists of four distinct phases: activation, resorption, reversal and formation, shown in the right side of Figure 4(a). When a BMU is proliferated of osteoclast and osteoblast, a

BoneNET: A Network Model of Bone Microstructure and Dynamics 11 resorption cavity in a rectangular shaped edge is created in the simulation to approximate the semilunar structure. In the model, we assumed that every edge can retain a BMU in which osteoclast and osteoblast interact to reform bone tissues. Both osteoclast and osteoblast can produce local effectors capable of activating or inhibiting themselves or the other cell type. In addition, bone cells are transferred to neighbor edges in this model because they are living tissues and moving around the bone structure. We define a bone remodeling dynamics on our bone network model as follows: dC c
( e ) dt
= α
1
C c
( e ) g
11 C b
( e ) g
21
−
β
1
C c
( e )
− npC c
( e ) + pC c
( i ) , i =0 dC b
( e ) dt
= α
2
C c
( e ) g
12 C b
( e ) g
22
−
β
2
C b
( e ) ,
(12)
(13) where C c
( e ) and C b
( e ) are the number of osteoclast and osteoblast in an edge e ; α i and β i are activities of cell production and removal; parameters g ij represent the net effectiveness of osteoclast- or osteoblast-derived autocrine or paracrine factors; p is the probability of osteoclast transportation to neighboring edges; n is the number of directly connected edges on pair nodes of an edge e in a bone network. Figure 5 illustrates the flow process of osteoclast among neighboring edges on a part of a bone network.
(a) (b)
(d) (c)
Figure 5 : (a) The edge with red color stands for a BMU containing osteoclast (b) The edges with blue color interact with the edge with red color. 1% osteoclast from the edge with red color are sent to neighboring edges, edges with blue color. (c) and (d) Every edge interact with neighboring edges, in which 1% of osteoclast from each eage are sent to neighboring edges.
Figure 4(b) shows a typical cycle of bone remodeling dynamics when BMU is proliferated of bone cells. In Figure 4(b), the resorption phase is represented as 37 days in length, with a maximum resorption of 14% bone mass. The reversal phase which follows the resorption phase lasts for 12 days. The subsequent formation phase is 67 days long.
The number of osteoclast and osteoblast under stable state conditions were assumed to consist of less differentiated cells that were unable to resorb or build bone, but able to participate in autocrine and paracrine signaling. Increases in cell numbers above stable state levels were attributed to the proliferation and differentiation of precursors into mature cells enabling to remove or build bone. We assumed that the rates of bone

12 T. Kim et al.
resorption and formation are proportional to the numbers of osteoclast and osteoblast respectively exceeding stable state levels. Thus, bone mass changes according to the numbers of osteoclast and osteoblast as follows: dM y i
=
− k
1 y
1
+ k
2 y
2 dt
=
{
C c,b
0 ,
− C c,b
, if C c,b if C c,b
> C c,b
≤ ¯ c,b
,
(14)
(15) where M formation; is total bone mass; y i k i is normalized activity of bone resorption and are the numbers of cells actively resorbing or forming bone; and C c,b are the numbers of osteoclast and osteoblast at steady state respectively defined by
C g c
12
= ( g
21
β
1
/α
1
)
(1 − g
22
−
(1
− g
11
/γ )
( β
2
)(1
− g
22
/α
) .
2
) g
21
/γ
, C b
= ( β
1
/α
1
) g
12
/γ
( β
2
/α
2
)
1 − ( g
11
/γ ) where γ =
In this section, we firstly create a bone network representing a microstructure of healthy bone according to the process of bone network model. Then, we conduct the simulation of the bone remodeling dynamics for the case of estrogen deficiency using the bone network. Finally, we provide four measurements to quantitatively evaluate bone quality of a bone network.
As we illustrate the generation process of a bone network in previous sections, we created a bone network consisting of 6248 nodes and 9509 edges. To obtain a proper number of nodes and edges for a bone network, the number of mineral fibers of several microstructural bone images are studied (Khosla, S. et al., 2006). The initial edge weight set to one for every edge and the node weight is set to the average weight of edges directly connected to node. Figure 6(a) shows the bone network for the microstructure of healthy bone. As the next step of our experimentation, we simulate the bone remodeling dynamics of estrogen deficiency. Estrogen deficiency is the principal cause of postmenopausal osteoporosis and it results in an acceleration of the rate of bone remodeling with a net increase in both osteoblastogenesis and osteoclastogenesis.
The consequences are a profound bone mineral loss (Manolagas, S.C., 2000). In order to mimic bone mineral loss, we set the number of osteoblast, osteoclast and other parameters shown in Table 1. In this experimental study, parameters we have used would not be reliably checked; however we consider that the result obtained from our simulation fairly shows osteoporosis progression caused by estrogen deficiency. Since bone remodeling occurs asynchronously at multiple sites and involves resorption and formation by the number of osteoclast and osteoblast, respectively, we randomly selected
10% edges of a bone network and 6 osteoclasts and 150 osteoblasts are set as the initial number of bone cells on those selected edges. We ran the simulation 10000 time units
(equivalent to 10000 days or 27.4 years) to analyze the effect of the bone remodeling dynamics when estrogen deficiency presents. Figure 7 shows a representative example of the bone remodeling cycle in the edge with the initial number of bone cells. Figure
6 shows the visualization of initial condition of a bone network shown in the left figure and the bone network after the simulation shown in the right figure. We consider that the node lost more than 50% of the node weight is deleted from the bone network.

BoneNET: A Network Model of Bone Microstructure and Dynamics
Parameter Description Value
C
C g g g g c b
α
1
α
2
β
1
β
2
11
21
12
22 p k k
1
2
Number of osteoclasts
Number of osteoblasts
6 cell/edge
150 cell/edge
Rate constants of osteoclast formation 3 cell/day
Rate constants of osteoblast formation 4 cell/day
Rate constants of osteoclast death
Rate constants of osteoblast death
Effectiveness of osteoclast-derived
0.2 cell/day
0.02 cell/day
1.089
autocrine regulation
Effectiveness of osteoblast-derived paracrine regulation
Effectiveness of osteoclast-derived
1.0
-0.5
paracrine regulation
Effectiveness of osteoblast-derived autocrine regulation
Probability of osteoclast movement
0
1%
Rate constant for osteoclast by RANKL/RANK binding
Rate constant for RANKL removal by RANKL/RANK binding
0.093 cell/day
0.0008 cell/day
Table 1 Parameter descriptions and values
(a) (b)
13
Figure 6 : The visualization of bone network is shown. The left figure, network 1, is the initial status of bone network and the right figure, network 2, is the bone network after 10000 days of simulation.
As the first measurement for evaluating overall bone network strength, we devised a measurement of the bone structural strength using entropy of the bone network. In information theory, entropy is a measure of the uncertainty associated with a random variable. High entropy means that the target structure or data is flexible and it can be easily reformed, which is not desirable for bone structure. We characterize entropy ξ of bone strength via
ξ =
∑ i ∈ V ( v )
− pd i log pd i
, (16) where V ( v ) is the node list in a bone network; p is the degree probability of a node in a bone network; and d i is the degree of a node i . Comparing the healthy bone

14 T. Kim et al.
100
90
80
70
60
100
95
90
0 500
Time (days)
1000
50
0 2000 4000 6000
Time (days)
8000 10000
Figure 7 : The percentage of bone mass as a function of time on bone dynamics is shown. The grayed oval area of the Figure 7 is shown in the inset plot. The resorption on bone mass is about 8% and underfilling of each remodeling cycle is about 2%.
network shown in Figure 6(a), the bone network representing osteoporosis shown in
Figure 6(b) has many loosely coupled structural spots. This measurement indicate overall bone strength by the bone structural characteristics.
Second, we evaluated the overall strength of bone network with moment of inertia
(MOI), which is a measure of an object’s resistance to changes in its rotation rate. Since the nature of bone strength is the resistance to changes from outer forces, MOI is a useful measurement for understanding of bone strength. We define MOI of bone strength
ρ via
ρ =
∑ i ∈ V ( v ) w i
δ i
2
, (17) where V ( v ) is the node list in a bone network; w i is the weight of a node i ; and δ i is the distance from center of the bone network to the node i . The weight of a node i , w i
, which represents the strength of a node i is defined as the average weight of edges directly connected to a node connected to the node v and i , u i w i
= i
∈
E ( v ) u i
/n , where E ( v ) is the edge list directly is the weight of an edge i . The weight of an edge i , u i
, is defined by the thickness of an edge i and the length of an edge i , u i
= h
−
1 i l i
, where h i is the thickness of an edge i and l i is the length of an edge i .
Third, we used average shortest path length (ASPL) as a measurement of a bone network. ASPL in a bone network explains whether nodes are strongly connected among neighboring nodes or not; thus the network structural strength can be analyzed. ASPL,
τ , is defined as
τ =
2
∑ i
∈
I ( v )
∑ j
∈
O ( v ) n ( n
−
1) min ( i
→ j )
, (18) where I ( v ) is the node list containing 10% of nodes near to the center of a bone network; O ( v ) is the node list containing 10% of nodes near to the annular region of a bone network; and min ( i
→ j ) is the shortest path length from node i to node j .

BoneNET: A Network Model of Bone Microstructure and Dynamics 15
To evaluate that our metrics is effective on understanding bone quality, we compare two sample bone networks. Bone network 1 represents a healthy bone network and bone network 2 represents a bone network with osteoporosis shown in Figure 6(a) and (b), respectively.
Method Network 1 Network 2
Entropy 0.092
MOI 33572.0
ASPL 46.82
0.340
23755.6
65.56
Table 2 Results of three evaluation methods on bone networks
80
70
60
50
40
0 20 40 60
I(v)
80 100 120 140
Figure 8 : ASPL as a function of I ( v ) sorted by ASPL is shown. The solid line stands for network 1 and the dotted line stands for network 2.
The results of three measurements on two sample networks are shown in Table 2. As we expected, results of three measurements clearly show that the structure of network 1 is stronger than network 2 in overall network characteristics. For the case of entropy test, network 1 has lower value of entropy than network 2 has, which means that network
1 has less diversity on the node degree than network 2 has. On the contrary, network
2 has more diversity on the node degree which indicates that structure of the network can be easily reformed which is not desirable for a bone structure. From the result of
MOI, we can learn that bone network 1 has better resistance from outer forces than network 2 has. By the result of ASPL, we can analyze the connectivity from the center of a bone network to the annular region of a bone network. Figure 8 shows ASPL as a function of I ( v ) sorted by ASPL. ASPL of network 1 slightly decreases from 48.9 to
46.1 comparing that ASPL of network 2 decreases from 78.1 to 61.8.
Previous measurements are mostly focused on measuring overall bone strength, which cannot show crucial parts for bone quality. Based on our bone model, we can detect a high fractural risk part of a bone network by crack propagation properties (Kim,

16 T. Kim et al.
T. et al., 2010c). 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)
Figure 9 : The illustration of the crack propagation in a lattice of atoms is shown when there are external forces applied to the left and right (Xie, C., 2008). (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 9 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 10(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.
Figure 10(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, x
0
, and vertical force, y
0
, shown in Figure 10(b) are defined and calculated by following equations.
x
0 y
0
: F
0 cosψ = T
1 cosθ + T
2 cosϕ
}
,
: F
0 sinψ = T
1 sinθ
−
T
2 sinϕ
(19)

BoneNET: A Network Model of Bone Microstructure and Dynamics
(a) (b)
T
1
y
0 x
0
F
0
17
T
2
Figure 10 : (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.
F
0 is a source node for external force and T
1 and T
2 are the destination nodes for external force.
where F
0 is a source node for external force, the T
1 and T
2 are the destination nodes for transferred force, ψ is the angle from x axis for the source force from F
0
, and θ and ϕ is the angle from x axis for the destination forces to T
1 and T
2
. 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, T
1 and T
2
, from a source of force energy, F
0
, via
T
1
=
−
( F
0 cosψ
× sinϕ )
−
( F
0 sinψ
× cosϕ )
− ( cosθ × sinϕ ) − ( sinθ × cosϕ )
,
T
2
=
( cosθ × F
0 sinψ ) − ( sinθ × F
0 cosψ )
−
( cosθ
× sinϕ )
−
( sinθ
× cosϕ )
,
(20)
(21) where F
0 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
∑
C bt
( e ) = s = v = t
∈
V to the node t , and σ st
( e )
, where σ st e , C bt
( e ) , is calculated by method,
σ st
σ st is the number of shortest paths from the node s
( e ) is the number of shortest paths from s to t that pass through an edge e (Newman, M.E.J., 2005).
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
C wbt
( e ) =
∑ s = v = t
∈
V
γ st
( e )
,
γ st
(22)

18 T. Kim et al.
Algorithm 1 Information Propagation( S , D , G )
15:
16:
17:
18:
19:
11:
12:
13:
14:
20:
21:
22:
23:
24:
8:
9:
10:
4:
5:
6:
7:
1:
2:
3:
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
N
ν
: a neighbor node list of ν with hop ( ν
→ i ) = 1 and x coordinate x i n : size of N
ν
F i
: energy in node i
< x
ν e i
→ 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 T
1
, T
2
, ...T
i from F s
Move to T i
Calculate current T i
= T i
+ previous T i
Calculate current e i → j
= e i → j
+ previous e i → j end for
Push T
1
, T
2
, ...T
i into S end while
Add M ax ( e i → j
) into R
Remove M ax ( e i → j
) from G until No path on S
→
D
Recognize node list, C
R
, linked to R
Add directed linked edges of C
R print R into R where γ st is the number of shortest paths multiplied by the edge length from the node s to the node t , and γ st
( 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 sub-graph of a node v . The bridging coefficient of a node v is defined by C bc
( v ) =
1 d ( v ) i
∈
N ( v )
δ ( i ) 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 (Hwang, W. et al., 2008).
We also modified bridging coefficient to be sensitive on spatial characteristics of bone network by adding weight features to an edge. We defined the weight of an edge e with the length of an edge e calculated via l ( j ) max ( l ( j ))
, where l ( j ) is 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
C wbc
( v ) =
1 d ( v )
·
∑ i ∈ N ( v ) w i
ρ ( i )
− w
0
, (23)

BoneNET: A Network Model of Bone Microstructure and Dynamics 19 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 , w i average weight of edges directly connected to node i and w
0 is the 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:
C wbc
( e ) =
ρ ( i ) C wbc
( i ) + ρ ( j ) C wbc
( j )
( ρ ( i ) + ρ ( j ))(
|
C ( i, j )
|
+ 1)
, e ( i, j )
∈
E (24) 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 sub-graph of node i , C wbc
( 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 .
Figure 11 : (a) The two-dimension structural bone network model is shown (b) The microscopic view of the bone model are shown.
For the experimentation, 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 11. 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, 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 12(a) with the information propagation algorithm. Figure 12 shows steps to find critical edges as fragile points in the sample bone network. In Figure 12(b), the first three critical edges are found as the most fragile points by external forces. Figures 12(c), (d), and (e) show each step to find next critical edges after removing previous critical edges. Finally, figure 12(f) shows the bone network broken into two pieces after 30 critical edges are removed, which represents the fracture of a bone.
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 13(a) shows average

20 T. Kim et al.
Figure 12 : 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.
shortest path length (ASPL) as a function of 30 critical edge cuts by each method. We calculated average shortest path length via τ =
∑ i
∈
R ( v ) j
∈
L ( v ) mn min ( i
→ j )
, where R ( v ) is the 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 )
(Kim, T. et al., 2010a).
The plot in Figure 13(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 13(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.
(a) 750
600
(b) 2430
2400
450
300
2370
ρ
2340
2310
150
0
0 10
Edge Cut
20 30
2280
0 10
Edge Cut
20 30
Figure 13 : (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.

BoneNET: A Network Model of Bone Microstructure and Dynamics 21
We also employed MOI, ρ , to evaluate the overall strength of bone network. Figure
13(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 13(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 14 shows the visualization of results after 30 critical edge cuts by each method. Figure 14(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 13(a), weighted betweenness centrality method created four isolated bone network blocks by deleting critical edges shown in Figure
14(b). Next, in Figure 14(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 14(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 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 14(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.
In this paper, we presented a network model for a bone microstructure and dynamics,
BoneNET, a computational model of bone dynamics which is capable of quantitative assessment of bone strength and bone quality. As the first component of the model for bone dynamics, we introduced a bone network model representing bone microstructure of mineral fibers. Second, we utilized a mathematical model to simulate bone remodeling dynamics working on our bone network. Last, we devised measurements to calculate bone strength and identity crucial elements in bone microstructure.
The results obtained from our simulated model corroborate behaviors of the bone remodeling system, including the tight coupling between osteoblast and osteoclast.
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 extremely valuable in the analysis of the bone quality. This study would help find valuable methods and measurements on the prediction of bone fracture risks as well as the quantification of bone quality.

22 T. Kim et al.
(a)
(b)
(c)
(d)
Figure 14 : 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 (Wending, Z., 2007).
The future work of this study is the design and implementation of a computational model of bone microarchitecture to analyze short and long-term effect of disease progression. The results of this study would provide new ways to the treatment of bone diseases. As a start of bone dynamics modeling, this study might reveal broad insights into how it may be possible to combat and control the spreading of problematic effects across a structural network model.
Also, this type of modeling would be critical in many applications. For example, it would become a useful mean for understanding of new disease progression and for modeling of treatment plan for new diseases. The model could elucidate selected aspects of microarchitecture bone adaptation and the interrelationships of bone mass, biomechanical stiffness, and strength. The model simulating osteoporosis could show the microarchitectural bone development of osteoporosis and the effect of existing and new drugs, which has a direct impact on women’s health. The model could also demonstrate quantitatively that stiffness of bone quality is a function of both mass and architectural structure.

BoneNET: A Network Model of Bone Microstructure and Dynamics 23
As a long term impact, the computational paradigm developed in this study would serve as guides in identifying pertinent animal experiments and developing timely clinical preventive and therapeutic programs. Finally, our model could be potential to lend insights into other clinical problems, such as neurological disorders and leukemia, by identifying the disease progression and prevention. Our computational model has the potential for substantial health care savings in the short term and with the prevention of future problems, the rate of savings is in the multi-billions. In conclusion, a feasible strategy on diagnosing the bone using a computational bone model could offer potential for better understanding bone disease progression and prevention.
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