ORIGINAL PAPER

I. Jasiuk Æ M. Ostoja-Starzewski

Modeling of bone at a single lamella level

Received: 29 September 2003 / Accepted: 15 May 2004 / Published online: 28 July 2004� Springer-Verlag 2004

Abstract This paper focuses on the ultrastructure ofbone at a single lamella level. At this scale, collagenfibrils reinforced with apatite crystals are aligned pref-erentially to form a lamella. At the next structural level,such lamella are stacked in different orientations to formeither osteons in cortical bone or trabecular pockets intrabecular bone. We use a finite element model, whichtreats small strain elasticity of a spatially random net-work of collagen fibrils, and compute anisotropic effec-tive stiffness tensors and deformations of such a singlelamella as a function of fibril volume fractions (orporosities), prescribed microgeometries, and fibril geo-metric and elastic properties.

1 Introduction

Our study focuses on the mechanics of a spatially ran-dom network of mineralized fibrils in bone, which arealigned in a preferential direction and form a singlelamella. More specifically, our goal is a passage from asuch random fibril network to an approximating con-tinuum representing properties of a lamella.

As background information, structurally, bone is acomposite material with a complex hierarchical structure(e.g., Parks and Lakes 1992; Lakes 1993; Weiner andTraub 1992; Rho et al. 1998). The two main componentsof bone are soft and ductile collagen fibrils, and stiff and

brittle apatite crystals. The collagen fibrils are about100–200 nm in diameter, of a circular or ellipsoidal crosssection, and made of triple helix molecules about 1 nmin diameter. They are reinforced with apatite crystals,which have irregular platelet-like shapes with an averagesize of 50·25·2 nm (Robinson 1952). The crystals arelocated both within and outside collagen fibrils, but theexact fibril–crystal interaction is not fully understood.The effective properties of a mineralized fibril (rein-forced with crystals), which serve as input for ouranalysis, have not yet been addressed explicitly in thebone literature. However, since the scope of the presentanalysis is linear elastic behavior of fibrils, the solution isscalable with the input at the single fibril level.

At the next hierarchical level, collagen fibrils, rein-forced with crystals, are arranged in a roughly parallelmanner and form lamellae. Our analysis focuses oneffective properties of a single lamella. At the next scale,these lamellae are stacked at different orientations toform laminated structures. These include orthogonal(perpendicular) arrangements and/or twisted (rotated)arrangements (Giraud-Guille 1988). In cortical (alsocalled compact) bone, these plywood-like structuresform composite cylinders called osteons, while in tra-becular (also called cancellous) bone, they form cres-cent-like laminated structures called trabecular pockets.The collection of osteons and interstitial bone (old ost-eons) make up cortical bone. The trabecular pocketsalong with interstitial bone make up randomly orientedstruts or plates called trabeculae, which, at the nextstructural scale, form a random porous network, withpores being filled with bone marrow. This complexhierarchical composite structure gives bone its uniqueproperties: high stiffness and strength due to stiff crys-tals, high fracture toughness due to ductile collagen, andlight weight due to porosities at several scales.

Thus, the following structural levels can be distin-guished (e.g., Rho et al. 1998): (a) nanostructural (fromseveral nanometers to one micron) involving collagenfibrils and apatite crystals, (b) sub-microstructural (1–10 lm) representing a single lamella, (c) microstructural

I. Jasiuk (&)Department of Mechanical & Industrial Engineering,Concordia University, Montreal, Quebec,H3G 1M8, CanadaE-mail: ijasiuk@me.concordia.ca

M. Ostoja-Starzewski (&)Department of Mechanical Engineering,McGill University, Montreal, Quebec,Canada, H3A 2 K6E-mail: martin.ostoja@mcgill.caTel.: +1-514-3987394Fax: +1-514-3987365

Biomechan Model Mechanobiol (2004) 3: 67–74DOI 10.1007/s10237-004-0048-5

(10–500 lm) involving a single osteon or trabecula, and(d) mesostructural, representing the overall cortical ortrabecular bone structure. One can also introduce thefifth level, (e) the macrostructural level, which representsthe overall bone, including both trabecular and corticalbone types.

In this paper, we focus on the sub-microstructurallevel involving a single lamella made of collagen fibrilsreinforced with apatite crystals. Within each lamella,collagen fibrils are predominantly oriented in the samedirection (Ascenzi et al. 1978), as illustrated by our SEMmicrograph (Fig. 1). This preferential arrangement offibrils allows a high density of collagen per unit volume.The collagen fibrils form bundles 1–2 lm in diameterlying in the plane of the lamella. Each lamella containsellipsoidal cavities, called lacunae, that are typically0.1 lm in diameter and 1–3 lm in long axis; they liewith their long axis parallel to the lamella long axis(Fig. 1).

We represent a single lamella as a collection of min-eralized fibrils aligned in a preferential direction and, forsimplicity, we do not account for lacunae. The analysis isconducted by using a specially written finite elementcomputer program that was originally developed forstudying paper from the standpoint of mechanics of arandom network of cellulose fibers (Ostoja-Starzewskiet al. 1999; Ostoja-Starzewski and Stahl 2000). Thistime, instead of a small (say 2·2 mm) piece of paper, wehave a single lamella of fibrils, and instead of cellulosefibers, we have collagen fibrils reinforced with crystals,all treated as homogeneous and linear elastic rods inthree-dimensions (3D), rigidly bonded among them-selves according to the actual connectivity of the net-work.

Given this system of rigidly connected rods, we thushave a 3D frame with all the rods exhibiting axial, tor-

sional, and beam bending (Timoshenko type) responses.For each particular realization of the network, the rodsare placed according to a 3D random process in a testbox. Once the connectivity of all the fibrils is established,the network is subjected to affine displacement boundaryconditions, and the overall elastic stiffness tensor, as wellas the deformation modes in the context of linear elas-ticity, are obtained explicitly. Indeed, various parametricstudies may be conducted to study these as functions ofthe fibril volume fraction (or porosity), fibril alignment,fibril cross-sectional aspect ratio, and type of fibrilconnections. We report such a parametric study here,aiming at an investigation of a normal bone versus abone weakened by osteoporosis.

2 Problem formulation

From the mechanics standpoint, there are four basicmodels for constructing a spatial fibril structure (Ostoja-Starzewski 2002):

1. Two-force members (carrying axial forces only)connected by frictionless pins, resulting in a truss.Such a model would clearly be too simplistic for thelamella.

2. Rods (carrying axial and shear forces, plus bendingand torsional moments) connected rigidly, resultingin a 3D frame. That is the model adopted here.

3. Rods same as in (2), but connected elastically, alsoresulting in a frame, albeit of softer response than theone in (2). When elasticity of connections betweenrods tends to infinity, the stiffness of this structuretends to that of (2).

4. Fibrils treated as 3D bodies, rather than slender rods.

Now, given the spatially disordered geometry of thefibril structure, we simply prefer to work with thecomputational mechanics model which we already havefrom our research in paper physics, than with an effec-tive medium model like that of Cox, and its offshoot inthe shear-lag model. Note here that the Cox model is ill-posed (Ostoja-Starzewski et al. 1999): it is really amechanism (an underconstrained structure below therigidity percolation). The shear-lag model compensatesfor other shortcomings of the Cox model, but it providesonly rough estimates of effective moduli, and thus,would be useless in studies of mesoscale deformationpatterns of fibril networks.

Adopting our 3D finite element model (Ostoja-Starzewski and Stahl 2000) to the fibril network at hand,our assumptions are as follows:

– The test box is quasi-planar of size 15·15·0.5 lm.– Fibrils are straight, of length 10 lm, and have a rect-

angular cross section (thickness·width): 0.1·0.5 lm.– Fibrils are placed in the cell via a 3D homogeneous

Boolean model (e.g., Jeulin 2001), which involves aPoisson point process for germ points and aslight clustering of fibrils at each germ point, with

Fig. 1 Scanning electron microscopy image of trabecular bone at asingle lamella level. Fibrils are aligned in the preferential direction,forming bundles. A void in the center is called lacunae

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the in-plane scatter of fibrils having a uniformdistribution between �9� and 9�.

– The fibril angular directions a1 and a2 in the (x1, x2)and (x1, x3) planes, respectively, are sampled fromGaussian probability distributions with zero meanand standard deviation r.

– Two cases of r are considered: r=p/10 and r=p/100.In the plots of Fig. 2a–c, these are referred to asSeries-1 and Series-2, respectively. They representcases of highly and very highly oriented fibers.

– Fibrils are generated so as to fill the cell according to apredetermined fibril volume fraction of 90%. At thatvolume fraction, there are about 550 fibrils, eachhaving an average of 16 bonds with the other fibrils.

– Elastic properties of collagen fibrils reinforced withapatite crystals are as follows: Young’s modulusE=24 GPa and Poisson’s ratio m=0.3. These arecalculated using the analytical model of Wagner andWeiner (1992), based on the model of Lusis et al.(1973). This model assumes that crystals are plateletsand a collagen fibril reinforced with crystals is repre-sented as a platelet-reinforced composite materialwith collagen being a matrix. The inputs for ourmodel are elastic properties E=1.5 GPa and m=0.3for a collagen fibril and E=150 GPa and m=0.3 foran apatite crystal, with 0.75 volume fraction of

crystals and the width-to-thickness ratio of plateletsequal to 10. Collagen and crystal properties weretaken as representative of those reported in literature(Currey 1969; Katz and Ukrainic 1971; Geoffrey 1972;Cowin 1989; Mammone and Hudson 1993). In ourmodel, we use a longitudinal modulus of such aplatelet-reinforced composite.

– All the connections between fibrils are formed asoutlined in the sequence of Fig. 2a–c. The first sketch(Fig. 2a) illustrates two fibrils, modeled by rectangu-lar cross-section rods, in contact because they overlapin space. This overlap is indicated by the volumeintersected by both rods in the second sketch(Fig. 2b). A shortest distance between the axes of bothfibrils is next identified, and in the final sketch(Fig. 2c), a quite short finite element is introduced tomodel the rigid bond contact between two fibrils. Ateach of its ends, this finite element is connected to twofinite elements along the axis of either fibril.

– An example of a fully connected network is shown inFig. 2d. This is a photograph of a 20·20·2 cm net-work physically produced by rapid prototyping froma computer model generated by our procedure. Thereare only two features which make this a little differentfrom our model of a lamella: the fibrils are of a cir-cular rather than rectangular cross section, and the

Fig. 2 a Tworods—representing twofibrils—in contact. b Volumeintersected by both rods.c Introduction of a finiteelement to model the rigidbond contact between two rods.d A 3D view of a fibril networkgenerated by rapid prototyping

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fibrils are not very strongly oriented in the x1 direc-tion.

– Once the entire connectivity is determined, each fibrilis treated as a sequence of 3D rod elements with axialextension, torsion, and bending in two mutuallyorthogonal planes. There are also short elements be-tween two consecutive bonds with other fibrils. Our3D test box thus contains a 3D frame of a complex,disordered geometry.

The components of the stiffness tensor, Cijkl, of thecell, relative to axes (x1, x2, x3), are computed by con-ducting displacement-controlled tests under affineboundary conditions (¶B is the test box boundary andstrain �ij

0 is uniform):

ui ¼ e0ijxj x 2 @B ð1Þ

The stiffness, Cijkl, is found by equating the elastic strainenergy stored in our discrete fibril network with theelastic strain energy of an approximating continuummaterial (Ostoja-Starzewski 2002), namely:

1

2

XNf

f¼1kuiuið Þ fð Þ ¼ e0ijCijkle

0kl ð2Þ

Here, ui is a generalized fibril displacement (such aselongation, torsion, shear, bending) and k is its corre-sponding spring constant. Also, the superscript f standsfor the fth fibril, and Nf stands for the total number offibrils. It should be added that, in Eq. 2, the averagestrain theorem was used, i.e., the average strain in anetwork is equal to the applied strain.

This formulation is consistent with the recentlydeveloped ‘‘homogenization theory of random media’’(e.g., Ostoja-Starzewski 2001), in which Eq. 1 is one ofthree possible boundary conditions satisfying the Hillcondition. (The latter requires the mechanically definedHooke’s law to be equivalent to the energetically definedHooke’s law.) The other two boundary conditions in-clude a uniform traction,

ti ¼ r0ijnj x 2 @B ð3Þ

and an orthogonal mixed one, representing a scalarcomposition of Eqs. 1 and 3,

ui � e0ijxj

� �ti � r0

ijnj

� �¼ 0 x 2 @B ð4Þ

Let us note here that, given the fact that the uni-form tractions in Eq. 3 cannot be assigned to the do-main boundary of a disordered network in anunambiguous way (e.g., Ostoja-Starzewski and Wang1989), the only effectively applicable condition is givenby Eq. 1. Fortunately, however, recent studies onporous materials show elastic moduli obtained fromkinematic boundary conditions to be much closer toeffective (i.e., macroscopic) moduli than those obtainedusing traction boundary conditions (e.g., Jiang et al.2001).

Since the cell is effectively anisotropic, the followingdisplacement-controlled boundary conditions (withspecified uniform strain eij

0) were applied to obtainrespective in-plane stiffness components:

1. e110 to find the axial stiffness component C1111

2. e220 to find the transverse stiffness component C2222

3. e110=e22

0 to find the transverse stiffness componentC1122; note that C1111 and C2222 need to be found first

4. e120 to find the shear stiffness component C1212

In the above notation, the x1 and x2 axes are co-oriented, respectively, with the average axial, andtransverse, directions of fibrils. The lamella’s thicknesscoincides with the x3 direction. The moduli investigatedare planar (i.e., in the (x1, x2) plane).

3 Results and discussion

The axial C1111, transverse C2222, and shear C1212 stiff-ness components are displayed in three plots in Fig. 3for the entire ranges of volume fractions and for bothfibril orientations: r=p/10 and r=p/100. The fibril’sthickness-to-width aspect ratio is 1:5 and the fibril’sthickness-to-length ratio is 1:100. The results are given inGPa. They illustrate a decrease from the single fibril’smodulus of 24 GPa, due to a porous, disordered spatialarchitecture.

The following observations can be made about theC1111 stiffness component of the network: it goes up to�10 at 90% fibril volume fraction, as might be expected;there is a crossing of both linear trends of C1111 at �40%volume fractions; all three stiffness components are lin-ear in the function of volume fraction; the scatter isstrongest at high, rather than low, volume fractions. TheC2222 and C1212 moduli also go up for both fibril ori-entations with increasing fibril volume fraction, butthose trends, while linear, are less clean. Thus, the spa-tial disorder of the network has a stronger effect on thevariability of effective moduli under axial loading par-allel to the mean fibril direction (e11

0) than under the onetransverse to it (e22

0) or under shear loading (e120).

We next examine deformation patterns of networksunder e11

0 and e220 loadings. We first start with the

original network of Fig. 4a and subject it to two tests:

– straining in the x1 direction: Fig. 4b–c– straining in the x2 direction: Fig. 4d–e

While Fig. 4b, d show the deformed network, Fig. 4c,e show the differences in displacement of all the nodesfrom the uniform motions that would take placeaccording to Eq. 1. Thus, in the first place, we note thatthe deformations are highly non-uniform. However, ourprincipal observation is that, during axial stretching inthe x1 direction, one observes a ‘‘splitting’’ of the net-work into elongated ‘‘clusters’’ that carry the load. Bycontrast, and perhaps counter to the usual intuition, theaxial loading in the x2 direction leads to more uniformdeformation fields.

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As the fibril volume fraction decreases by a factor oftwo (45%), we have the network shown in Fig. 5a. InFig. 5b, c we conduct the same two tests as in Fig. 4b, c.Note that this new network, being more sparsely con-nected, splits into more pronounced clusters than before.Additionally, some fibrils tend to partially move/jump

outside of the original test domain; this suggests that thenetwork is close to the so-called rigidity percolationdiscussed in (Ostoja-Starzewski et al. 1999). Note,however, that the same network still deforms in a veryuniform manner when stretched in the x2 direction.

In Fig. 6a, we study the case of fibrils being two timesnarrower than in the first network, while keeping thesame number of fibrils and their identical spatial place-ment, as in Fig. 4. Evidently, due to the fibrils’ smallercross sections, the network connectivity worsens relativeto Fig. 4a. This poorer connectivity then leads to sig-nificant splitting of the network into clusters, which isagain more pronounced than in Fig. 4.

Thus, we propose ‘‘network splitting and clusterformation’’ as a possible mechanism of softening

Fig. 3 a Computation of C1111. b Computation of C2222.c Computation of C1212

Fig. 4 a Undeformed network geometry. b Network subjected toan axial strain e11

0. c Differences, shown as red line segments,between true node displacements and those that would result froman affine displacement field due to overall uniform straine11

0=constant. d Network subjected to an axial strain e220.

e Differences, shown as red line segments, between true nodedisplacements and those that would result from an affinedisplacement field due to overall uniform strain e22

0=constant.Note that the red line segments in c are much longer and moreclustered than those in e

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(i.e., lowering of elastic moduli) of a disordered fibrilnetwork, which possibly are operational in osteoporosis.Clearly, the mechanical phenomena displayed in Figs. 5and 6 could be responsible for material failure, througha buckling-type instability, at a lower stress level thanwould be the case in a more ‘‘healthy’’ network ofFig. 4.

For an osteoporotic bone, one generally uses eitherone or both hypotheses: lower volume fraction of fibrilsand lower moduli of fibrils due to lower mineral (apatitecrystal) content or degradation of collagen propertiesdue to osteoporosis. In our simulations, we have focusedon the first hypothesis by taking either a somewhatdepleted network (Fig. 5) or a network with narrowerfibrils (Fig. 6).

4 Discussion of challenges

The experimental investigation of bone at the lamellalevel and a possible experimental verification of ournumerical results presented in ‘‘Results and discussion’’would pose numerous challenges. These include thefollowing:

1. Knowledge of the mechanical properties of mainconstituents of bone, collagen, and apatite crystals, isessential in modeling bone. However, the accurateexperimental determination of their properties is verydifficult due to their small size. The data in literaturefor collagen ranges from 1.2–1.5 GPa for elasticmodulus and 0.22–0.35 for Poisson’s ratio, and for

Fig. 5 a Undeformed network geometry at twice lower fibervolume fraction. b Network subjected to an axial strain e11

0.c Differences, shown as red line segments, between true nodedisplacements and those that would result from an affinedisplacement field due to overall uniform strain e11

0=constant.d Network subjected to an axial strain e22

0. e Differences, shown asred line segments, between true node displacements and those thatwould result from an affine displacement field due to overalluniform strain e22

0=constant. Note that the red line segments inc are much longer and more clustered than those in e, and those inFig. 4c and e, respectively

Fig. 6 a Undeformed network geometry. b Network subjected toan axial strain e11

0. c Differences, shown as red line segments,between true node displacements and those that would result froman affine displacement field due to overall uniform straine11

0=constant. d Network subjected to an axial strain e220.

e Differences, shown as red line segments, between true nodedisplacements and those that would result from an affinedisplacement field due to overall uniform strain e22

0=constant.Note that the red line segments in c are much longer and moreclustered than those in e, and those in Fig. 4c and e, respectively

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apatite crystals 62.5–170 GPa for elastic modulus and0.27–0.35 for Poisson’s ratio (Bonfield and Grynpas1977; Currey 1969; Katz and Ukrainic 1971; Geoffrey1972; Cowin 1989; Mammone and Hudson 1993). Inthis paper, we choose E=1.5 GPa and m=0.3 forcollagen and E=150 GPa and m=0.3 for apatitecrystals, and use the model of Wagner and Weiner(1992), following Lusis et al. (1973).

2. Characterization of the interface between collagenfibrils and crystals is also important. For example, isthe bonding perfect or does some slip or debondingtake place? There are very limited experimental re-ports on that issue. In modeling studies, Mammoneand Hudson (1993) are the only ones to assume agiven debonding threshold, but the actual value is notavailable from experiments.

3. The prediction of effective properties of a fibril rein-forced with crystals must be verified experimentally.We used a continuum approach involving a mi-cromechanics model of Wagner and Weiner (1992)based on Lusis et al.’s (1973) approach, which rep-resented bone as a composite material with polymermatrix reinforced with platelet-shaped crystals.However, because a collagen fibril reinforced withcrystals and its components (protein triple helices andapatite crystals) have nanosized dimensions, appli-cability of the continuum assumption needs to beverified. Moreover, since the crystal is comparable insize to the collagen fibril diameter, the condition ofseparation of scales may not be reached. More spe-cifically, the fibril size may be too small with respectto apatite crystals and the size of representative vol-ume element (RVE), which is needed for a continuumapproximation, may not be reached. In that case,there will be the effect of surfaces and boundaryconditions. The model of Wagner and Weiner (1992)assumes the existence of the RVE.

4. The lamellar properties can be measured using ananoindentation technique, yet, such a technique isstill not well understood for anisotropic materials andit predicts an isotropic elastic modulus that is ob-tained by assuming a Poisson’s ratio. Measuredproperties reported in literature also refer to mea-surements of bone tissue that represent the propertiesof laminated structure, which is one structural levelhigher, and not measurements for a single lamella(Hoffler 2000; Rho et al. 1993; Roy et al. 1998; Zyssetet al. 1998, 2004; Turner et al. 1999). The measure-ment of properties of a single lamella could possiblybe made using a nanoindentation by applying loadssmaller than used previously to ensure that indentdepths are much smaller than lamellar thickness.This, however, would give high scatter in propertiesdue to the heterogeneity of ultrastructure at that le-vel. Alternately, single lamella properties could pos-sibly be inferred from the measurements of laminatedstructure if the information on the plywoodarrangement of lamellar layers were available at thelocation of nanoindentation. This could be done

using scanning electron microscopy or transmissionelectron microscopy on the tissue directly below thenanoindentation site.

5. Other experimental techniques such as ultrasound ormicrobending tests have also been used to measurebone tissue properties (Ashman and Rho 1988;Rho et al. 1993; Choi et al. 1990), but they were usedat the next scale, the microstructural level represent-ing bone’s laminated structure, not the sub-micro-structural level, which represents the single lamellalevel.

The above five points refer mainly to the effectiveelastic properties of a single lamella and the challenges inobtaining such information experimentally. The nextpoint addresses the deformation of single lamella.

There may exist microtest devices that could capturethe deformation of fibrils at a single lamella level, butsuch experiments would be very complex. To ourknowledge, there are no published experimental resultsaddressing deformations in bone at this level.

Thus, the approach presented in this paper provides anumerical experiment which can give information onproperties and deformations at the single lamella thatare extremely challenging to study experimentally. Thislevel is very important because it is where fracture ini-tiates (crack initiation at single fibril and propagationacross neighboring fibrils). Information on deformationsat single lamella level can thus help in understanding thefracture mechanisms in such heterogeneous and complexmaterials like bone. The model presented can addressboth normal and osteoporotic bone.

5 Conclusions

We adapted a computational mechanics model, origi-nally developed in paper physics, to study the singlelamella level in bone. To our knowledge, this is the firstmicromechanics-based study incorporating spatiallydisordered geometry of bone at that structural scale. Theoutputs of this study include effective anisotropic elasticmoduli of a single lamella and deformation mechanismsof fibrils at a single lamella level. The model displays thetendency of fibril networks to undergo non-uniformstrains and form clusters of fibrils under axial loadingco-aligned with fibrils’ preferential direction. This ten-dency is amplified when networks are depleted by havingeither a lower volume fraction or thinner fibrils, and thisin turn suggests bone failure, through a buckling-typeinstability, at a much lower stress level than would bethe case in a denser network. This may representdeformation responses of osteoporotic bone.

Acknowledgments The first author acknowledges support of thisresearch by the NSF (grant CMS-0085137). The work of the secondauthor was made possible through support by the Canada Re-search Chairs program and the NSERC. SEM image in Fig. 1 wastaken in the Integrated Microscopy and Microanalytical Facility(Dr. R.P. Apkarian, director).

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References

Ascenzi A, Bonucci E, Ripamonti A, Rovery N (1978) X-ray dif-fraction and electron microscope of osteons during calcifica-tion. Calcif Tissue Int 25:133–143

Ashman RB, Rho JY (1988) Elastic modulus of trabecular bonematerial. J Biomech 21:177–181

Bonfield W, Grynpas MD (1977) Anisotropy of the Young’smodulus of bone. Nature 270:453–454

Choi K, Kuhn JL, Ciarelli MJ, Golstein SA (1990) The elasticmoduli of human subchondral, trabecular and cortical bonetissue and the size-dependency of cortical bone modulus.J Biomech 23:1103–1113

Cowin SC (1989) Bone mechanics. CRC Press, Boca Raton,Florida

Currey JD (1969) The relationship between the stiffness and themineral content of bone. J Biomech 2:477–480

Geoffrey HB (1972) The biochemistry and physiology of bone, vol1. Academic Press, New York

Giraud-Guille MM (1988) Twisted plywood architecture of colla-gen fibrils in human compact bone osteons. Calcif Tissue Int42:202–209

Hoffler CE, Moore KE, Kozloff K, Zysset PK, Brown MB,Goldstein SA (2000) Heterogeneity of bone lamellar-levelelastic moduli. Bone 26:603–609

Jeulin D (2001) Random structure models for homogenization andfracture statistics. In: Jeulin D, Ostoja-Starzewski M (eds)Mechanics of random and multiscale microstructures. CISMcourses and lectures, vol 430, Springer, Berlin Heidelberg NewYork, pp 33–91

Jiang M, Alzebdeh K, Jasiuk I, Ostoja-Starzewski M (2001) Scaleand boundary conditions effects in elastic properties of randomcomposites. Acta Mech 148:63–78

Katz JL, Ukrainic K (1971) On the anisotropy of elastic propertiesof hydroxyapatite. J Biomech 4:221–227

Lakes R (1993) Materials with structural hierarchy. Nature361:511–515

Lusis J, Woodhams RT, Xhantos M (1973) The effect of flakeaspect ratio on the flexural properties of mica reinforced plas-tics. Polym Eng Sci 13:139–145

Mammone JF, Hudson SM (1993) Micromechanics of bonestrength and fracture. J Biomech 26:439–446

Ostoja-Starzewski M (2001) Mechanics of random materials:stochastics, scale effects, and computation. In: Jeulin D,

Ostoja-Starzewski M (eds) Mechanics of random and multiscalemicrostructures. CISM courses and lectures, vol 430, Springer,Berlin Heidelberg New York, pp 93–161

Ostoja-Starzewski M (2002) Lattice models in micromechanics.Appl Mech Rev 55(1):35–60

Ostoja-Starzewski M, Stahl DC (2000) Random fiber networks andspecial elastic orthotropy of paper. J Elasticity 60(2):131–149

Ostoja-Starzewski M, Wang C (1989) Linear elasticity of planarDelaunay networks: random field characterization of effectivemoduli. Acta Mech 80:61–80

Ostoja-Starzewski M, Quadrelli MB, Stahl DC (1999) Kinematicsand stress transfer in quasi-planar random fiber networks. CRAcad Sci Paris Serie IIb 327:1223–1229

Park JB, Lakes RS (1992) Biomaterials. Plenum, New YorkRho JY, Ashman RB, Turner CH (1993) Young’s modulus of

trabecular and cortical bone mineral: ultrasonic and microten-sile measurements. J Biomech 26:111–119

Rho JY, Kuhn-Spearing L, Ziuopos P (1998) Mechanical proper-ties and the hierarchical structure of bone. Med Eng Phys20:92–102

Robinson R (1952) An electron microscopy study of the crystallineinorganic component of bone and its relationship to the organicmatrix. J Bone Joint Surg 34a:389–435

Roy M, Rho JY, Tsui TY, Pharr GM (1998) Variation of Young’smodulus and hardness in human lumbar vertebrae measured bynanoindentation. In: Advances in bioengineering, BED vol 33,ASME publication

Turner CH, Rho J, Takano Y, Tsui TY, Pharr GM (1999) Theelastic properties of trabecular and cortical bone tissues aresimilar: results from two microscopic measurement techniques.J Biomech 32:437–441

Wagner HD, Weiner S (1992) On the relationship between themicrostructure of bone and its mechanical stiffness. J Biomech24:1311–1320

Weiner S, Traub W (1992) Bone structure: from angstroms tomicrons. FASEB 6:879–885

Zysset PK, Guo XE, Hoffler CE, Moore KE, Goldstein SA (1998)Mechanical properties of human trabecular bone quantified bynanoindentation. Technol Health Care 6:429–432

Zysset PK, Guo XE, Hoffler CE, Moore KE, Goldstein SA (2004)Elastic modulus and hardness of human cortical and trabecularlamellae measured by nanoindentation (in press)

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