A Graph-based Approach for Computational Modelof Bone Microstructure

Taehyong Kim∗

Department of ComputerScience and Engineering,

State University of New Yorkat Buffalo, USA

thkim7@buffalo.edu

Murali RamanathanDepartment of PharmaceuticalSciences, State University of

New York at Buffalo, USAmurali@buffalo.edu

Aidong ZhangDepartment of ComputerScience and Engineering,

State University of New Yorkat Buffalo, USA

azhang@buffalo.edu

ABSTRACTOsteoporosis, a condition in which bones become fragile andmore likely to break, can greatly affect our health. Nearlyhalf of all women now age 50 will someday have a brokenbone due to osteoporosis. The diagnosis of osteoporosis iscommonly done by tests that measure the amount of bonemass and predict the risk for fracture. However, the cur-rently available techniques for the diagnosis of osteoporosisare limited due to the lack of good measurements of bonequality.

In this paper, we develop a graph approached bone mi-crostructure model, which is capable of enabling quantita-tive assessment of bone mineral density and bone micro-architecture. Our paper focuses on microstructural bonemodeling and quantitative characterization of the qualityand fractural risk based on the microscopic bone structure.First, we illustrate the process of development on a three-dimensional bone structure model whose properties are sim-ilar to those of bone microarchitecture. Next, we list prob-lems of the current measurements of bone strength and qual-ity based on bone mineral density (BMD). Last, we intro-duce our measurements to quantitatively assess bone qualitybased on the bone model.

1. INTRODUCTIONOsteoporosis - a systemic skeletal disease characterized bylow bone mass and micro-architectural deterioration of bonetissue leading to enhanced bone fragility and a consequentincrease in fracture risk - afflicts an estimated 25 millionpeople in the United States [27]. In addition, osteoporosisaccounts for approximately 1.5 million new fractures eachyear, with associated medical charges including rehabilita-tion and extended treatment facilities of an estimated $10billion, according to the National Osteoporosis Foundation.Because osteoporosis affects primarily the elderly, the Na-

∗corresponding author

tional Osteoporosis Foundation estimates that these costswill increase to $200 billion by the year 2040, as the numberof Americans over the age of 65 years grows [2].

Traditional thinking on bone’s deterioration has focused onbone quantity - described by the bone mass or bone mineraldensity - as a predictor of fracture risk. Measurement ofbone mineral density (BMD) has been the central compo-nent of any provision that arises from osteoporosis [5]. Thediagnosis of osteoporosis thus centers on assessment of bonemass and quality. Except this method, there is no satisfiedclinical mean to assess bone quality. Diagnosis of osteoporo-sis, therefore, mainly depends on the measurement of bonemass until now.

As the efforts on diagnosis of osteoporosis, relationships be-tween BMD and fracture risks have been studied. A studyshows BMD of the low hip was an important predictor offracture risk based on the estimation in a meta-analysis ofdata from 12 cohort studies of 39,000 men and women [14].Techniques to accurately assess bone mineral mass in vivoare well established [21]. Current clinical prediction of frac-ture risk is therefore based primarily on bone mass alone.Nevertheless, a clearer understanding of the role played bybone microarchitecture in conjunction with bone mass is im-portant for ultimately dealing, as effectively as possible, withbone loss disorders, particularly osteoporosis.

Many evidences show that low BMD is not the sole factorresponsible for the fracture risk. A study by Sui Hui pro-vided that aging causes 10 times increase in fracture riskregardless of BMD [12]. Garnero also showed that the in-crease in BMD might not always give sufficient informationto treat patients [6]. This result supports that BMD alonecannot explain therapeutic benefits of antiresorptive agentsin treating osteoporosis.

Bone structure is highly complex even thought bone is asimple composite of a mineral phase. As the result of that,the microstructure of bone have been studied to understandstrength and quality of bone. Several structural approacheshave been done to overcome the shortcoming of BMD. Sincecomputing power has been doubled by every one and ahalf years, computer simulations of bone remodeling are at-tempted to correlate discrete physiological events with ob-served changes in bone morphology [26]. Although produc-ing useful results, these previous simulations are essentially

Permission to make digital or hard copies of all or part of this workfor personal or classroom use is granted without fee provided thatcopies are not made or distributed for profit or commercial advantageand that copies bear this notice and the full citation on the firstpage. To copy otherwise, to republish, to post on servers or toredistribute to lists, requires prior specific permission and/or a fee.

ACM-BCB 2010, Niagara Falls, NY, USA

Copyright © 2010 ACM ISBN 978-1-4503-0192-3... $10.00

ACM-BCB 2010 563

based on one-dimensional models, and therefore do not ac-count for the interconnected trabecular structure of cancel-lous bone or the distribution of remodeling sites that can beonly shown in the two-/three-dimensional structure. So sev-eral studies have worked on developing a simulation that isbased on a two-/three-dimensional structural model of can-cellous bone which allows for the occurrence of trabecularperforation, a naturally occurring event. Extensive studieson structural approaches are done [24, 15, 1]; however, thesestudies have not incorporated both biochemical reactions ofbone remodeling mechanism and BMD of previous measure-ments into a computer simulation model.

2. BACKGROUNDNetworks are encountered in a wide variety of contexts, suchas social networks, circuit networks, networks of neurons,and terrorist networks. Most of the real world relationshipsand cooperations among any kinds of objects could be rep-resented by networks. Specifically, many types of biologicalnetworks exist, which could be depicted as networks. Bonemodeling is one of the most exciting but challenging researcharea in terms of network modeling.

(a)

(b)

Figure 1: (a) The image of human femur bone and the

microscopic view of a bone structure are shown. (b) The

two-dimension structural bone network model and the

microscopic view of the bone model are shown.

As shown in Figure 1(a), bone is not a uniformly solid mate-rial, but rather has some spaces between its hard elements.There are two different bone types, one is cortical bone andthe other is trabecular bone. Cortical bone is the hard outerlayer of bones which is composed of compact bone tissue, so-called due to its minimal gaps and spaces. This tissue givesbones their smooth, white, and solid appearance, and ac-counts 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 spongybone), which is composed of a network of rod- and plate-likeelements that make the overall organ lighter and allowingroom for blood vessels and marrow. Trabecular bone ac-counts for the remaining 20% of total bone mass but hasnearly ten times the surface area of compact bone. Figure1(b) shows our network model which mimics the propertiesof cortical bone and the trabecular bone based on the dif-ferent densities from the center to edge.

The standard and routine approach for the diagnosis of os-teoporosis is to assess bone mineral density (BMD) usingeither dualenergy X-ray absorptiometry (DXA) or quantita-tive computed tomography (QCT). The estimated BMD hasshown to be a suitable predictor of fracture risk. However,major limitations of bone mineral densitometry are that itincompletely reflects variation in bone strength and thatdifferentiation of patients with and patients without verte-bral fractures is inaccurate [16, 23]. Other factors like bonemicroarchitecture contribute substantially to bone strengthand their evaluation can improve determination of bone qual-ity and strength [23, 13]; yet structural assessment has notbeen implemented in clinical routine because of a lack ofcomputational bone model. Furthermore, although recentstudies have demonstrated the relative importance of archi-tecture and mass as determinants of bone strength, architec-tural assessment techniques are significantly less developed[25]. Several studies show that accurate prediction of bonestrength using BMD alone is very challenging [17].

3. MODEL OF BONE MICROSTRUCTURETo overcome the lack of measurements of bone quality, westudy the problem in a new direction. We develop a mi-croscopic graph-based approach of a bone structure, a com-putational network model of bone microstructure which iscapable of enabling quantitative assessment of bone min-eral density, bone microarchitecture, and fracture risk. Thisstudy will find valuable methods and measurements on thequantification of bone quality and prediction of bone frac-ture risks. In what follows, we describe the process of mi-crostructural bone modeling in details.

Based on the bone microstructure, we model a three dimen-sional bone microarchitecture network as shown in Figure2. When we develop our abstracted bone network model,important components of the bone structure are consideredto build a bone model enabling computational analysis ina timely manner without losing the critical bone structuralproperties. We focused mineralized collagen fibers as impor-tant components from the bone cellular structure point ofview.

Abstracted

Figure 2: From the bone microscopic structures, a

three-dimensional rod-like structure network is gener-

ated as an abstracted bone network. Attributes of bone

microscopic structures are calculated and applied to a

bone network.

A bone network is defined by a weighted undirected space-sensitive graph in a 1x1 unit circular region; G = {(V, E) | V

ACM-BCB 2010 564

is a set of nodes and E is a set of edges, E ⊆ V × V , anedge 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 networkrepresents a rod-like bone mineralized fiber and a node ina bone network represents a fiber binding point with whichbone cells move and interact with neighboring bone tissues.

(a) (b) (c)

(d)(e)(f )

Figure 3: (a)-(f) The six steps of generating process on

a two-dimensional bone structure network are shown.

Voronoi tessellations have been used to model the geometricarrangement of cells in morphogenetic or cancerous tissues.The interactions between cells are represented by adhesiveand repelling force densities on the cell contact borders.In addition, protrusive locomotion forces are implementedalong the cell boundaries at the tissue margin, and stochas-tic perturbations allow for non-deterministic motility effects.Simulations on the emerging system of stochastic differentialequations for position and velocity of cell centers have shownthe feasibility of this Voronoi method to generate realisticcell shapes [4].

(a) (b)

(c) (d)

l

r

node

edge

Figure 4: (a) A horizontal plane image of three-

dimensional bone structure network is shown, which is

equivalent to a two-dimensional bone structure network.

(b) Three-dimensional bone structure network is created

by 3D Voronoi tessellation. (c) The microscopic view of

the three dimensional bone structure model is shown.

(d) The 10x magnified view of the red rectangular re-

gion in (c) is shown. r and l are the radius and the

length of a cylinder-like edge respectively.

As the first step of our modeling process shown in Figure3(a), Voronoi sites are randomly generated, which are the

center position of rod-like structures as representatives ofmineralized fibers. The number of sites could vary depend-ing on the size and density of a targeted bone structure.Then, Voronoi sites are redeployed by the relative densitybetween cortical bone and trabecular bone shown in Figure3(b). The density of points in our model follows an expo-

nential distribution E(x) =∫ 1.5

−1.5λe−λx−1

dx, where λ is adensity slope coefficient and x is the density of mineralizedfibers in a 1mm2 area of bone sample images. Next, a bonemicrostructural network is created shown in Figure 3(c) as arepresentative of mineralized fibers which are one of the im-portant structural components for bone strength. Based onVoronoi sites, we calculate expected area occupied by eachVoronoi site using Voronoi tessellation.

In the next two steps shown in 3(d) and (e), Voronoi sitesare removed and randomly selected rods are also removedto pattern properties of bone microstructures. The methodof rods deletion step can be adjusted by different study pur-poses. In this study, two dimensional bone structures aregenerated by the model whose edges are randomly removedby 0%, 6% and 20% of edges, respectively. In Figure 3(f),different strengths of mineralized fibers are applied to rep-resent the characteristics of trabecular and cortical bone.For simplification of the model, we use a linear function togenerate differences on two types of bone structure.

Base on the two dimensional bone model, we can generatea three dimensional bone structure. To generate a three di-mensional bone structure, the same processes of two dimen-sional bone model are applied except 3D Voronoi tessellationprocess. Instead of generating Voronoi sites in two dimen-sional circular region, Voronoi sites are generated in threedimensional cylinder area shown in Figure 4(a) and (b). Anode and an edge in three dimensional bone structure areshown in Figure 4(c) and (d).

Since this bone model is developed based on image analysisof microscopic figures of bone, properties we have used inthis model would not be reliably measured. To make thismodel more realistic, accurate input data are required, suchas density of fibers, average thickness of fibers, and averagelength of fibers. However, well-understood knowledge withthis type of a study would be valuable in the prediction ofthe bone fracture risk. In the following section, we addressexisting measurements of the bone quality evaluation andwe formulate new measurements of bone quality based onthe microstructural bone network.

4. MEASUREMENTS OF BONE QUALITYA BMD test measures the density of minerals (such as cal-cium) in your bones, which has been the most importantcomponent estimating the quality of your bones. There areseveral ways to measure BMD. First, dual-energy X-ray ab-sorptiometry (DEXA) is the most accurate way to measureBMD. It uses two different X-ray beams to estimate bonedensity in your spine and hip. Strong, dense bones allow lessof the X-ray beam to pass through them. The amounts ofeach X-ray beam that are blocked by bone and soft tissue arecompared to each other [3]. Second, peripheral dual-energyX-ray absorptiometry (P-DEXA) is used as a type of DEXAtest. P-DEXA is not as useful as DEXA for finding out howwell a medicine used to treat osteoporosis is working [7].

ACM-BCB 2010 565

Third, dual photon absorptiometry (DPA) test uses a ra-dioactive substance to measure bone density [8]. Fourth,ultrasound test is generally used to look for problems. Ul-trasound uses sound waves to measure BMD, usually in yourheel. Ultrasound is quick, painless, and does not use po-tentially harmful radiation like X-rays. One disadvantageof ultrasound is that it cannot measure the density of thebones most likely to fracture (the hip and spine) from osteo-porosis. It is not used to keep track of how well a medicineused to treat osteoporosis is working [20]. Last, quantita-tively computed tomography (QCT) is a type of CT scanthat measures the density of a bone in the spine (vertebra).QCT is not usually used because it is expensive, uses higherradiation doses, and is less accurate than DEXA, P-DEXA,or DPA [10].

The previous measurements of bone strength based on BMDhave been extensively investigated [9, 20]. However, majorlimitations of BMD are that it incompletely reflects varia-tion in bone strength and bone microstructure. In addition,measurements based on bone structural assessment has beenpoorly studied in clinical routine because of the lack of com-putational bone models. Thus, we introduce a quantita-tive bone quality measurement using our three-dimensionalbone structure network model. With three dimensional viewof a bone network, it consists of nodes and edges whichare represented by microscopic sphere shapes and micro-scopic cylinder shapes respectively shown in Figure 4(c) and(d). Because the bone network model is developed based ongraph theory, we can quantitatively calculate the structuralstrength and the density of the bone network by the topo-logical information of network such as the number of nodesand edges and the relationship among nodes. The follow-ing method gives us a measurement that possibly analyzesthe condition of the bone network which reflects variationin bone strength and bone microstructure. Based on thismodel, we are able to assess the bone strength by the quan-titative BMD (QBMD) via

QBMD =π4

∑ni=1 r2

i liρ

Γ, (1)

where r and l are the radius and the length of each cylinder-like edge shown in Figure 4(d). Γ, n and ρ are the total areaof the annular region of the bone structure, the number ofedges, and the density coefficient, respectively. Since everysingle rod-like microstructure in bone affects the strength ofbone, the measurement of thickness, length and density onbone network, such as characteristics of mineralized fibers,are important to understand the bone condition. Thus, thismeasurement will be an important indicator of overall BMDbased on our model.

Existing measurements are mostly focused on measuringBMD and predicting overall bone health, which cannot pro-vide the strength of the certain area of bone. Based on ourbone model, we can profile the condition of a bone networkstructure by the following algorithm with which we can an-alyze a high fractural risk area of the bone structure. Weintroduce Connectivity-first Scan (CFS) algorithm, modi-fying the best-first search graph theory algorithm [11]. Itis the greedy network path search algorithm which worksgreatly for finding high probability fracture area in our bonenetwork. It is adapted from the best-first search algorithm

(a) (b) (c)

(d)(e)(f )

ν

ν ν

ν

ν

ν

Figure 5: A diagram for the algorithm of Connectivity-

first Scan (CFS) is illustrated. (a) CFS starts from the

node ν (b) CFS visits the next node with Max(∑l

i=1 ει)

among neighboring nodes. (c), (d) and (e) Execute (a)

and (b) until a priority queue (the candidate node list)

is empty. (f) Stop node visits and return the scanned

path as a result.

Algorithm 1 Connectivity-first Scan(s, ν, G)

1: G: a space-sensitive graph G of a bone network2: ν: a node as a conjunction point among rod-like microstruc-

tures3: s: ν with the minimum value of

∑li=1 ει such that

4: l is the number of edges directly connected to ν,5: ε is the thickness of edge, and the ι is the length of edge6: n: the neighbor node directly connected to ν7: ν = s8: repeat

9: Add every n of ν into a priority queue, C by∑l

i=1 ει10: Pop n from C11: ν = n and set visit(n) = true12: Add ν into the visited list, V13: until !isEmpty(C)14: Return V

that explores a graph by expanding the most promising nodechosen according to a certain rule [22].

The CFS process starts from the least dense microstructurenode and then it visits the most dense node among neigh-bor nodes by the rule shown in Figure 5 and Algorithm 1.The CFS selects the first-visit node based on MIN(

∑li=1 ει)

where l is the number of edges directly connected to ν, ε isthe thickness of edge, and ι is the length of edge.

∑li=1 ει

of every neighboring node n is calculated and the node nwith Max(

∑li=1 ει) among all neighboring nodes is set to ν

which is the next visit node. A priority queue was used for(unvisited) nodes sorted by

∑li=1 ει.

Sample 1 2 3QBMD 11.65 10.38 6.79

Table 1: QBMD of three samples

To evaluate QBMD and CSF, we generated the three differ-ent bone networks based on the bone model shown in Figure6. For the computational simplicity, two-dimensional bonenetworks were created on the 1x1 annular region consistingof 6248 nodes and 9509 edges. Figure 6(a), (b) and (c) repre-

ACM-BCB 2010 566

Figure 6: The bone network is created which consists

of 6248 nodes and 9509 edges. (a) The healthy condi-

tion of bone model is shown. There is no glitch on the

plot for the normal status of bone model. The sudden

increments at around 5500 of V are due to the outside

concrete area of the bone network. (b) 6% of bone struc-

ture is deteriorated comparing the healthy condition of

bone model. The plot shows glitches at the beginning of

the result and spikes at the end of the result. The sud-

den increments at around 4500 of V are the same reason

of (a). (c) 23% of bone structure is deteriorated compar-

ing with the healthy condition of bone model. The plot

clearly shows glitches and spikes at any points.

sent the bone structural network of the healthy bone (Sam-ple 1), the bone with osteopenia (Sample 2) and the bonewith osteoporosis (Sample 3) respectively. First we calcu-lated QBMD of three samples shown in Table 1. As we ex-pected, the healthy bone shows the highest value of QBMD.The bone with osteopenia follows the healthy bone and thebone with osteoporosis has the lowest value of QBMD. Ac-cording to the result of QBMD, we can quantitatively eval-uate the overall quality of bone.

Next, we ran CFS algorithm against three different samplebone networks. With the result of the CFS, the dense areasand the sparse areas of the bone network are easily distin-guished as shown in Figure 6, which can be useful for thebone fracture predictions. The plot in Figure 6(a) shows thesmooth increments of the density from the trabecular bone(spongy bone) to the cortical bone (compact bone) withoutshowing the any glitches, which means that there is no weakpart of the bone network. The plot in Figure 6(b) starts toshow glitches indicating the weak area of the bone networkand the plot in Figure 6(c) shows clear glitches and spikesindicating the bone is already in the the osteoporosis stage.

In this study, no evaluation method is introduced for com-

paring between our measurements, QBMD and CFS, andthe density measurement in practice, BMD. However, itwould be an useful theoretical testing framework of bonemicrostructure to analyze bone structural properties if ac-curate input data and evaluation processes are available. Inaddition, this study could be an initial step for further stud-ies on quantitative analysis of bone microstructure and bonequality.

5. CONCLUSION AND FUTURE WORKWe have presented a network model of bone microarchi-tecture and developed a three-dimensional bone structuremodel. We introduced the measurement of bone qualitybased on the bone network model, QBMD, with which theoverall bone strength and quality can be analyzed. We alsoprovided the profiling method of bone network, CFS, to an-alyze a high fractural risk area of the bone structure.

Properties we have incorporated in this model, such as thenumber of nodes and the weight of edges representing differ-ent densities of trabecular and cortical bone, would not bereliably ascertained. Prompt and proper information ob-taining from high-quality orthopaedic data is critical foraccurate predictions to reduce these uncertainties. How-ever, well-understood knowledge with this type of the modelwould be extremely valuable in the prediction of the bonefracture risk and the analysis of the bone quality. A feasi-ble strategy on diagnosing the bone using a computationalbone model could offer potential for better understandingthe bone quality.

The analysis of dynamics on network models has been broadlyused for understanding many structural connections and re-lationships. A study based on the network model shows thedefensive effectiveness on network dynamics by damage iso-lation of strategic self-destruction [19]. Another example ofstudies on dynamics is the evolution and the hypotheticalcontainment of Avian Influenza outbreak [18]. A networkdynamics model on bone remodeling is also one of the in-teresting but challenging research area because bone is adynamic and living tissue whose structure and shape con-tinuously adjusts to mainly provide structural framework.

In the future work, we will develop a mathematical model ofbone dynamics based on this bone network model. This willbe enabling understanding of bone related diseases, such asosteoporosis, as well as bone remodeling processes. More-over, it will be useful for a theoretical testing framework ofbone dynamics to conduct analysis of drug effects.

Eventually, this study will provide new ways to the treat-ment of bone diseases, such as osteoporosis. By simulatingosteoporosis effects, our model can show the microarchitec-tural bone development of osteoporosis and the effect of ex-isting and new drugs, which has a direct impact on women’shealth. Moreover, the simulation paradigm developed in thisproject may serve as guides in identifying pertinent animalexperiments and developing timely clinical preventive andtherapeutic programs. Finally, our model can be potentialto lend insights into other medical problems, such as neu-rological disorders and leukemia, by identifying the diseaseprogression and prevention.

ACM-BCB 2010 567

6. REFERENCES[1] Adachi, T., Suzuki, Y., Tsuhota, K., and Hojo, M.

Computer simulation of tarabecular remodeling inproximal femur using large-scale fe-models. Journal ofBiomechanics, 41:S124, 2008.

[2] Barefield, E. Osteoporosis-related hip fractures cost$13 billion to $18 billion yearly. Food Review, 1996.

[3] Barnes, C., Newall, F., Ignjatovic, V., Wong, P.,Cameron, F., Jones, G., and Monagle, P. Reducedbone density in children on long-term warfarin.Pediatric Research, 57:578–581, 2005.

[4] Bock, M., Tyagi, A.K., Kreft, J.U., and Alt, W.Generalized voronoi tessellation as a model oftwo-dimensional cell tissue dynamics. BiologicalPhysics, 0901:4469, 2009.

[5] E. F. for Osteoporosis. Consensus developmentconference: Prophylaxis and treatment of osteoporosis.Osteonorosis International, 1:114–117, 1991.

[6] Garnero, P., Delmas, P.D. Noninvasive techniques forassessing skeletal changes in inflammatory arthritis:bone biomarkers. Curr Opin Rheumatol., 16:428–434,2004.

[7] Gilsanz, V. Bone density in children: A review oftechniques available and indications. Euro J Rad,26:177–182, 1998.

[8] Harrison, J.E., Krishnan, S.S., Muller, C., Strauss, A.,Mukherjee, S., and Sturtridge, W.C. How well doesdual photon absorptiometry predict osteoporosis? acomparison between neutron activation analysis anddexa. Journal of Radioanalytical and NuclearChemistry, 169:301–305, 1993.

[9] Heijckmann, A.C., Dumitrescu, B., Kruseman,A.C.N., Geusens, P., Wolffenbuttel, B.H.R., Vries,J.D., Drent, M., and Huijberts, M.S.P. Quantitativeultrasound does not identify patients with aninflammatory disease at risk of vertebral deformities.BMC Musculoskeletal Disorders, 9:1471–, 2008.

[10] Hernandez, E.R., Seco-Durban, C., Revilla, M.,Gonzalez-Riola, J., and Rico, H. Evaluation of bonedensity with peripheral quantitative computedtomography in healthy premenopausal,perimenopausal, and postmenopausal women. Age andAgeing, 24:447–450, 1995.

[11] Huckenbeck, U., and Ruland, D. Graph-TheoreticConcepts in Computer Science., volume 484. SpringerBerlin / Heidelberg, 1991.

[12] Hui, S.L., Slemenda, C.W. and Johnston, C.C. Ageand bone mass as predictors of fracture in aprospective study. The Journal of ClinicalInvestigation, 81:1804–1809, 1988.

[13] Ito, M., Ikeda, K., Nishiguchi, M., Shindo, H., Uetani,M., Hosoi, T., and Orimo, H. Multi-detector row ctimaging of vertebral microstructure for evaluation offracture risk. J Bone Miner Res, 20:1828–1836, 2005.

[14] Johnell, O., Kanis, J.A., Oden, A., Johansson, H., DeLaet, 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., andTenenhouse, A. Predictive value of bmd for hip andother fractures. Journal of bone and mineral research,20:1185–94, 2005.

[15] Kameo, Y., Adachi, T., and Hojo, M. Estimation of

trabecular bone premeability using fluorescent imageof lacuno canalicular system. Journal of Biomechanics,2008.

[16] Kanis, J.A., Borgstrom, F., De Laet, C., Johansson,H., Johnell, O., Jonsson, B., Oden, A., Zethraeus, N.,Pfleger, B., and Khaltaev, N. Assessment of fracturerisk. Osteoporos Int, 164:581–589, 2005.

[17] Kaufman, J.J., Nasser, P., Mont, M., Hakim, N., Pilla,A.A. and Siffert, R.S. Texture analysis of vertebralcomputed tomography scans improve trabecularstrength estimation. Trans Orthopaed Res Soc, 14:264,1989.

[18] Kim, T., Hwang, W., Zhang, A., Sen, S., andRamanathan, M. Multi-agent model analysis of thecontainment strategy for avian influenza (ai) in southkorea. In IEEE International Conference onBioinformatics and Biomedicine, pages 335–338, 2008.

[19] Kim, T., Hwang, W., Zhang, A., Ramanathan, M. andSen, S. Damage isolation via strategic self-destruction:A case study in 2d random networks. EurophysicsLetters, 2009.

[20] Muller, M., Mitton, D., Talmant, M., Johnson, P., andLaugier, P. Nonlinear ultrasound can detectaccumulated damage in human bone. Journal ofBiomechanics, 41:1062–1068, 2008.

[21] OTT, S.M., Kilcoyne, R.F., and Chesnut, C.H. 3rd.Ability of four different techniques of measuring bonemass to diagnose vertebral fractures inpostmenopausal women. Bone Miner Res, 2:201–210,1987.

[22] Russell, S.J. and Norvig, P. Artificial Intelligence:Modern Approach. Prentice Hall, 1995.

[23] Taylor, B.C., Schreiner, P.J., Stone, K.L., Fink, H.A.,Cummings, S.R., Nevitt, M.C., Bowman, P.J., andEnsrud, K.E. Long-term prediction of incident hipfracture risk in elderly white women: study ofosteoporotic fractures. J Am Geriatr Soc Int,52:1479–1486, 2004.

[24] Tsubota, K., Suzuki, Y., Yamada, T., Hojo, M.,Makinouchi, A., and Adachi, T. Computer simulationof trabecular remodeling in human proximal femurusing large-scale voxel fe models: Approach tounderstanding wolffSs law. Journal of Biomechanics,42:1088–1094, 2009.

[25] Turner, C.H., and Cowin S.C. . Dependence of elasticconstants of an anisotropic porous material uponporosity and fabric. J Mat Sci, 22:3178–3184, 1987.

[26] Weinhold, P.S., Gilbert, J.A., and Woodard, J.C. Thesignificance of transient changes in trabecular boneremodeling activation. Bone, 15:577–584, 1994.

[27] WHO Scientific Group. Prevention and managementof osteoporosis. WHO Technical Report Series, WorldHealth Organization, 921:1–192, 2003.

ACM-BCB 2010 568