Acta Biomaterialia 12 (2015) 207–215

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Acta Biomaterialia

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To what extent can cortical bone millimeter-scale elasticity be predictedby a two-phase composite model with variable porosity?

http://dx.doi.org/10.1016/j.actbio.2014.10.0111742-7061/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Vanderbilt Orthopaedic Institute, Medical Center East,South Tower, Suite 4200, Nashville, TN 37232-8774, USA. Tel.: +1 615 554 3035.

E-mail address: mathilde.granke@gmail.com (M. Granke).

Mathilde Granke a,b,h,⇑, Quentin Grimal a,b,h, William J. Parnell c, Kay Raum d, Alf Gerisch e,Françoise Peyrin f,g, Amena Saïed a,b,h, Pascal Laugier a,b,h

a Sorbonne Universités, UPMC Université Paris 06, UMR 7371, UMR_S 1146, Laboratoire d’Imagerie Biomédicale, F-75005, Paris, Franceb CNRS, UMR 7371, Laboratoire d’Imagerie Biomédicale, F-75005, Paris, Francec School of Mathematics, Alan Turing Building, University of Manchester, Manchester M139PL, UKd Julius Wolff Institute & Berlin-Brandenburg School for Regenerative Therapies, Charité-Universitätsmedizin Berlin, 13353 Berlin, Germanye Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germanyf ESRF, 6 rue Jules Horowitz, 38043 Grenoble, Franceg CREATIS INSERM U1044, CNRS 5220, INSA Lyon, Université de Lyon, 69621 Villeurbanne, Franceh INSERM, UMR_S 1146, Laboratoire d’Imagerie Biomédicale, F-75005, Paris, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 April 2014Received in revised form 1 September 2014Accepted 9 October 2014Available online 20 October 2014

Keywords:Mechanical modelAnisotropic elasticityCortical boneEffective propertiesPorosity

An evidence gap exists in fully understanding and reliably modeling the variations in elastic anisotropythat are observed at the millimeter scale in human cortical bone. The porosity (pore volume fraction) isknown to account for a large part, but not all, of the elasticity variations. This effect may be modeled by atwo-phase micromechanical model consisting of a homogeneous matrix pervaded by cylindrical pores.Although this model has been widely used, it lacks experimental validation. The aim of the present workis to revisit experimental data (elastic coefficients, porosity) previously obtained from 21 cortical bonespecimens from the femoral mid-diaphysis of 10 donors and test the validity of the model by proposinga detailed discussion of its hypotheses. This includes investigating to what extent the experimentaluncertainties, pore network modeling, and matrix elastic properties influence the model’s predictions.The results support the validity of the two-phase model of cortical bone which assumes that the essentialsource of variations of elastic properties at the millimeter-scale is the volume fraction of vascular poros-ity. We propose that the bulk of the remaining discrepancies between predicted stiffness coefficients andexperimental data (RMSE between 6% and 9%) is in part due to experimental errors and part due to smallvariations of the extravascular matrix properties. More significantly, although most of the models thathave been proposed for cortical bone were based on several homogenization steps and a large numberof variable parameters, we show that a model with a single parameter, namely the volume fraction ofvascular porosity, is a suitable representation for cortical bone. The results could provide a guide to buildspecimen-specific cortical bone models. This will be of interest to analyze the structure–function rela-tionship in bone and to design bone-mimicking materials.

� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction standing as well as a good representation of the elastic properties

Bone is a multiscale biocomposite whose structure andmechanical properties at one level determine the properties ofthe subsequent one. Despite numerous studies dedicated to theassessment of cortical bone mechanical properties, some questionsremain open regarding the determinants of cortical bone elasticproperties which are known to vary with, among other things,age, anatomical location, disease or drug treatment. A clear under-

and their variations is needed for the modeling of the macroscopic(organ-scale) behavior of bones, the investigation of structural–function relationships (remodeling) or the design of new in vivotechniques to monitor bone properties.

At the mesoscale (2–10 mm [1]), cortical bone can be describedas a two-phase composite material consisting of a dense mineral-ized matrix and a soft phase, i.e. Haversian canals, Volkmann’scanals and resorption cavities (referred to as vascular porosity) con-taining fluid and soft tissues. The porosity has been established to bean important determinant of mesoscopic bone properties [2–4]. Onthe other hand, considering only published experimental studies inhuman cortical bone, the impact of the bone matrix elastic

208 M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215

properties (i.e. at the microscopic level) on bone mesoscale elastic-ity is a matter of debate in the literature [3,5]. In a previous exper-imental study [6], we addressed the question of the respectivecontributions of the variations of porosity and bone matrix elasticity(reflected by acoustical impedance) to changes of mesoscopic elas-tic properties. We found that the elastic properties of the matrixonly undergo small variations among different specimens (coeffi-cients of variation of matrix impedance values were <6%) and thatvariations in porosity account for most of the variations of meso-scopic elasticity, at least when the analyzed porosity range is large(3–27%). These results suggest that, in a first approach, the varia-tions of mesoscale cortical stiffness could be modeled by a simplemicromechanical model where the matrix would be the same forall bone specimens (i.e. fixed matrix stiffness coefficients) and theporosity would be the only specimen-dependent parameter. A rea-sonable model, already proposed by several authors [7–10], consistsof a two-phase micromechanical model: a homogeneous matrixwith transversely isotropic stiffness pervaded by cylindrical poresaligned with the direction of highest matrix stiffness. This two-phase model, when implemented with fixed matrix properties(referred to hereinafter as the reference model), correctly predictsthe trend in the variation of each elastic coefficient as a functionof the porosity [6]. However, there remain unexplained discrepan-cies between the predicted and measured stiffness coefficients formost of the specimens. These discrepancies may originate from dif-ferent sources. On the one hand, the two-phase model is only arough idealization of bone: the modeled porosity is uniformly dis-tributed and the pores are circular, regular and infinitely long; onthe other hand, the experimental data (evaluation of stiffness andvascular porosity) is subject to several measurement errors.

The objective of the present paper is to test the validity of thereference model (matrix pervaded by cylindrical pores) by propos-ing a detailed discussion of its hypotheses and to determine towhat extent cortical bone millimeter-scale anisotropic elasticitycan be predicted based on the sole knowledge of porosity. Oneadded value of this study is the systematic quantification of allthe potential sources of discrepancies that could be modeled andthe discussion of their relative contributions. One further original-ity of our work is that we compare the predictions of a popular cat-egory of micromechanical models accounting only for pore volumefraction and the predictions of a finite-element (FE) model whichaccounts for the distribution of the pore volume fraction withinthe cortical specimen.

This paper is organized as follows. Section 2 briefly presents theexperimental findings of Granke et al. [6] and the reference two-phase model. Section 3 quantifies the discrepancies between dataand model predictions. We then clarify how far the experimentaldata can be trusted (Section 4) before revisiting the hypothesesof the model to search for factors, besides changes in pore volumefraction, that would explain the discrepancies between data andmodel predictions (Section 5).

1 Code available online from www.labos.upmc.fr/lip/spip.php?rubrique133.

2. Experimental data and reference model

The specimen preparation and measurement methods, whichwere described in detail in Granke et al. [6], are summarizedbelow. The data used in the present study was obtained on 21 par-allelepiped specimens (nominal size 5 � 5 � 7 mm3) from 10female donors (mean age 81 years, range 66–98 years). The facesof the specimens were oriented according to the radial (1), circum-ferential (2), and longitudinal (3) axes defined by the anatomicalshape of the femoral diaphysis. The diagonal terms of the apparent(i.e. mesoscopic) stiffness tensor – longitudinal (C11, C22, C33) andshear (C44, C55, C66) elastic coefficients – were determined fromthe apparent mass density and wave velocity measurements using

a well-established pulse transmission method [11]. The vascularporosity P was obtained from 50 MHz scanning acoustic micros-copy (SAM [12,13]) with a resolution of 30 lm. The 3-D pore net-work was imaged with a resolution of 10 lm for a subset of 10specimens using synchrotron radiation microcomputed tomogra-phy (SR-lCT; ESRF, Grenoble, France).

The reference two-phase model predicts mesoscopic elasticproperties exclusively accounting for variations of the pore volumefraction. Various homogenization schemes have been used by dif-ferent authors to calculate the predictions of such a model: asymp-totic homogenization [7], the Mori–Tanaka (MT) method [9],generalized self-consistent method [8]. Our implementation ofthe reference model uses asymptotic homogenization (AH)[14,15]. The model hypothesizes that cortical bone can be regardedas a homogeneous transversely isotropic (TI) matrix pervaded byinfinite cylindrical pores, periodically distributed within the matrixmaterial (specifically on a hexagonal lattice). An orthonormalCartesian frame (x1, x2, x3) is attached to the model, where x3 isaligned with the axis of the cylindrical pores. The plane (x1, x2) isthe plane of isotropy for the matrix properties. Given a stiffnesstensor cm describing the matrix elasticity, a stiffness tensor cp

describing the elasticity of the material in pores, and the volumefraction of pores, a homogenized stiffness tensor C⁄ is calculatedusing a custom MatLab code1 (The MathWorks, Natick, MA). Sincethe specimens were kept moist during the measurements, the mate-rial in pores (undrained) is assumed to behave like bulk water, i.e.bulk modulus and Poisson ratio are set to 2.3 GPa and 0.49, respec-tively. Preliminary calculations indicated that computed effectiveproperties are not sensitive to small variations of the elastic proper-ties of the fluid material in pores (cp). In contrast, they are sensitiveto the elastic properties of the matrix (cm), which must be carefullychosen. In the reference model, we assign the same elastic propertiescm and cp to all of the specimens. This amounts to assuming the exis-tence of a ‘‘universal’’ matrix which has yet to be defined. We previ-ously determined [6] the optimal fixed matrix elastic coefficients forthe reference model by minimizing the distance between the exper-imental (Cii) and homogenized (C⁄ii) mesoscopic elastic coefficients.Precisely, cm is the tensor which minimizes the objective functiondefined as:

H0ðcmÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

k¼1

X6

i¼1

Cii;k � C�ii;kðcm; cp; PkÞCii;k

� �2s

; ð1Þ

where N is the number of bone specimens, Pk refers to the estimateof porosity of specimen k assessed from impedance maps, and Cii;k

and C�ii;k to the experimental and homogenized elastic coefficientsof k, respectively. Considering all the specimens (i.e. N = 21), theTI stiffness tensor cm which minimizes H0 was found to be:

cm11 ¼ 26:8GPa; cm

33 ¼ 35:1GPa; cm44 ¼ 7:3GPa;

cm66 ¼ 5:8GPa; and cm

13 ¼ 15:3GPa: ð2Þ

(These correspond to the following values of engineering moduliEm

T ¼ 16:5GPa, EmL ¼ 24:0GPa, Gm

T ¼ 5:8GPa, GmL ¼ 7:3GPa.) This

dataset is referred to hereinafter as the reference matrix elasticity.Values assigned to the bone matrix are consistent with the litera-ture [16–19].

In Granke et al. [6], elasticity values predicted with the refer-ence model were calculated to help interpret the data. However,this was without a detailed analysis of the general adequacy ofthe model, which is the purpose of the present paper.

M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215 209

3. Discrepancies between data and model predictions

The above values of the mineralized matrix (cm (Eq. (2)) havebeen obtained with one specific dataset. In order to ensure thatthe values are not critically dependent on the dataset, we appliedthe leave-one-out cross-validation (LOOCV) [20,21]. Ten datasetswere formed by excluding the specimens from the nth femur(n = 1,. . .,10) and pooling the specimens from the nine remaininghuman femurs. For each of these datasets, the elastic tensorcm{n} was computed using the objective function defined in Eq.(1) (here, N = 18, 19 or 20 depending on the excluded femur n).The optimized stiffness tensors cm{n} for the matrix calculatedfor the ten datasets were found to be close to the reference matrixas evidenced by the average relative distance between cm

ii {n} andcm

ii , which was less than 1.5% (Table 1), thereby confirming thatthe reference matrix properties (Eq. (2)) are not biased by the par-ticular set of specimens considered in this study.

The adequacy of the fit between the reference model and theexperiments was evaluated by means of the root mean squareerror (RMSE), i.e. the standard deviation of the residuals betweenthe experimental and predicted elastic coefficients. Here, thehomogenized stiffness tensor predicted by the reference modelfor a given specimen harvested from the femur n is computedusing cm{n}, cp, and the porosity of the specimen. Note that thehomogenized elasticity C⁄ is thus strictly independent of the meso-scale experimental data C. The RMSE absolute and correspondingrelative errors were found to be C11: 1.5 GPa (7.3%), C22: 1.6 GPa(8.7%), C33: 2.0 GPa (6.6%), C44: 0.4 GPa (6.3%); C55: 0.5 GPa (8.5%),C66:0.3 GPa (7.9%).

4. Quantification of experimental uncertainties

In this section, we assess whether measurement errors canexplain the deviation between experimental observations andmodel predictions.

4.1. Porosity and elasticity

Longitudinal (C11, C22, C33) and shear (C44, C55, C66) elasticcoefficients were obtained by processing longitudinal and shearultrasound velocity measurements, which lead to different experi-mental errors for longitudinal and shear coefficients. The measure-ment relative error EC (repeatability) is 3.2% and 4.7% for thelongitudinal and shear elastic coefficients, respectively [6]. Thestandard deviations corresponding to these errors were calculatedfor each coefficient and exhibited the following maximal values:0.7 GPa for C11 and C22, 1.1 GPa for C33, 0.3 GPa for C44 and C55,and 0.2 GPa for C66.

As for the error on the porosity estimate P, the comparison onten specimens between P and the volumetric porosity obtained

Table 1Optimized stiffness tensor for the matrix properties calculated after considering different

Excluded femur (matrix elastic tensor) cm11 [GPa] cm

33 [G

None (cm, reference model) 26.8 35.1#218 (cm{1}) 26.6 35.2#227 (cm{2}) 26.8 35.2#228 (cm{3}) 26.9 35.3#245 (cm{4}) 26.6 34.9#251 (cm{5}) 26.9 34.4#260 (cm{6}) 26.8 34.6#263 (cm{7}) 26.6 35.2#267 (cm{8}) 27.3 37.5#268 (cm{9}) 26.9 34.9#271 (cm{10}) 26.6 34.8Mean ± SD 26.8 ± 0.2 35.2 ±

from SR-lCT (taken as a reference) led to an average error ofEP = 0.8% point of porosity [6].

When taking into account the measurement errors, we consid-ered that: (i) the actual experimental elastic coefficient lies withinDC = [(1 � EC) � Cii , (1 + EC) � Cii], where Cii is the experimentallymeasured elastic coefficient and EC = 0.032 and 0.047 for the longi-tudinal and shear elastic coefficients respectively; and (ii) the pre-dicted elastic coefficient, for a specimen with estimated porosity P,lies within DC⁄ = [C⁄ii(cm, cp, P + EP) , C⁄ii(cm, cp, P � EP)], whereEP = 0.008. We found that the ranges DC and DC⁄ overlapped for72 out of the 126 measured coefficients (Fig. 1). For 19 out of the21 investigated specimens, there was at least one elastic coefficientfor which DC and DC⁄ did not overlap. Based on these results, it canbe concluded that the measurement errors cannot account for theobserved discrepancies between experimental and predicted elasticcoefficients.

4.2. Misalignment of the specimen during cutting

In the model, the pores and the axis of symmetry of the matrixstiffness are aligned with direction 3. Thus it is assumed that the 1–2-plane defined from the specimen faces after cutting is actuallyperpendicular to the pores and is the plane of isotropy. However,the specimen faces may not be well aligned with the anatomicalaxes due to inaccuracy of the anatomical landmarks used for thecut. We evaluated the degree of possible misalignment based onthe pores orientation observed in longitudinal sections cut fromthe ten specimens imaged with SR-lCT (see online material). Themaximum misalignment was estimated to be 10�. The conse-quence of misalignment is that the stiffness coefficients measuredare not precisely the coefficients Cii on the diagonal of the tensormatrix expressed in the natural basis of the specimen material sup-posed to be TI. To quantify the error on the experimental assess-ment of the latter, we compared the diagonal stiffnesscoefficients of the reference model with a 10� off-axis deviationof axis 3 to the diagonal stiffness coefficients of the referencemodel tensor in the natural basis (Fig. 1). The maximum valuesof the relative variations were DC⁄11 = 1.0%, DC⁄33 = 1.3%,DC⁄44 = 1.7%, and DC⁄66 = 1.0%. These values are significantly less thanthe observed discrepancies between the reference model and exper-imental points.

5. Revisiting the model hypotheses

5.1. Porosity distribution

In the reference model, any variability of pore shape, size anddistribution was disregarded. The influence of these factors wasinvestigated for the subset of ten specimens imaged by SR-lCT.We proceeded in two steps:

sets of specimens.

Pa] cm44 [GPa] cm

66 [GPa] cm13 [GPa]

7.3 5.8 15.47.2 5.7 15.77.3 5.8 15.67.3 5.8 15.67.2 5.7 15.47.2 5.8 13.77.3 5.8 14.37.3 5.7 15.77.3 5.8 21.07.4 5.8 13.57.3 5.8 13.6

0.9 7.3 ± 0.04 5.8 ± 0.03 15.4 ± 2.2

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r ela

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]

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C11 C22 C33

experimental errorreference10° off-axis deviat.

C44 C55 C66

experimental errorreference10° off-axis deviat.

Fig. 1. Longitudinal (left) and shear (right) elastic coefficients vs. porosity. The solid lines display the elastic coefficients computed with the reference model. The dotted linesshow the influence of a 10� off-axis deviation of axis 3. Boxes represent the experimental errors on the measurement of elastic coefficients Cii and the evaluation of porosity.The boxes highlighted in red indicate those measurement errors which cannot entirely explain the distance between the experimental and predicted elastic coefficients (54out of 126 measured coefficients).

210 M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215

Step 1: Each specimen 3-D volume was divided into N adjacentsubvolumes svk (k = 1,. . .,N) of approximately 1.5 � 1.5 � 1.5 mm3

(Fig. 2a and b). The 3-D porosity of each subvolume was calculatedfrom the SR-lCT segmented data. The reference matrix elastic andpore tensors cm and cp were assigned to the bone matrix and mate-rial in pore phases, respectively. The homogenized elastic tensor C⁄kwas then calculated on each subvolume svk using the analytical AHscheme. The procedure yields a representation of the distributionof millimeter-scale elasticity within the specimen. The elastic fluc-tuations are entirely due to fluctuations of porosity within thespecimen (Fig. 2c).

Step 2: The second step involved solving the homogenizationproblem for the whole volume: the homogenized elastic propertiesof each bone specimen were obtained using FE computations as inGrimal et al. [1] using a classical procedure [22]. Briefly, the mate-rial properties at all points M(x,y,z) that belong to subvolume svk

were set to be the same constant C(x,y,z) = C⁄k (Fig. 2d). Six sets ofkinematic uniform (KUBC) and stress uniform (SUBC) boundary con-ditions were applied successively. Stress and strain fields were calcu-lated with a commercial FE code (COMSOL Multiphysics� 3.5) in theframework of linear elasticity. The computed apparent stiffness(KUBC) or compliance (SUBC) tensors were obtained by dividingcomponents of strain and stress fields. CSUBC and CKUBC provide lower

Fig. 2. (a and b) The bone volume is divided into subvolumes of approximately 1.5 mmdata. The homogenized elastic tensor of each subvolume is computed using asymptotic hdependent material properties are retrieved from the homogenized elastic tensors calculto assess the lower and upper bounds of the apparent elastic coefficients.

and upper bounds of the apparent tensor. Note that the computa-tional cost to calculate these bounds without step 1, i.e. a computa-tion conducted on the entire volume with a mesh so fine as to matchthe resolution of the SR-lCT images, would have been prohibitivelyhigh. With our approach, the convergence of the apparent stiffnesstensor computed using an unstructured mesh of tetrahedral ele-ments was obtained for a mesh composed of about 7000 quadraticLagrange elements with a characteristic size of 700 lm.

The homogenization procedure (steps 1 and 2) was validated bycomputing the bounds CSUBC and CKUBC on an artificial data set(Fig. 3) corresponding to the reference model, i.e. made of a homo-geneous matrix and cylindrical pores organized on a hexagonalperiodic pattern for a porosity of 12.5%. The bounds computed byFE modelling were within 10�3 GPa of the theoretical value givenby the reference model.

The CSUBC and CKUBC bounds computed for the ten bone speci-mens were found to be very close to the predictions of the refer-ence model: the maximum relative error for the differentcoefficients were DC11 = 1.3%, DC22 = 1.2%, DC33 = 0.8%,DC44 = 1.3%, DC55 = 1.1% and DC66 = 2.4%. This was in spite of thelarge variations of porosity that are present within some of thespecimens, which are typically caused by the presence of largeresorption cavities. A striking example is the specimen ‘‘06’’ for

. (c) The 3-D porosity of each subvolume is calculated from the SR-lCT segmentedomogenization. (d) Finite-element modeling on the bone specimen. The coordinate-ated on each subvolume. Applying a set of uniform boundary conditions allows one

Fig. 3. Homogenized elastic properties: reference model (pore volume fraction) vs.finite-element model (real pore shape and distribution taken into account).Illustration of the coefficient C11 for an artificial dataset that corresponds to thereference model (i.e. made of a homogeneous matrix and cylindrical poresorganized on a hexagonal periodic pattern) and 10 human bone specimens.

Aspect ratio δ = length/diameter0 5 10 15 20 25 30 35 40

30

25

20

15

10

5

0

C11 = C22

C33

C44 = C55

C66

C12C13 = C23

Cortical bone

Elas

tic c

oeffi

cien

ts [G

Pa]

Fig. 4. Homogenized elastic coefficients as obtained from the Mori–Tanaka (MT)model (solid lines) vs. the pore aspect ratio (=length/diameter) for a given porosityof 15%. The effective elastic coefficients as obtained with the reference model(dotted lines) have been superimposed. The grey zone represents the range ofaspect ratios of Haversian canals in femoral human cortical bone.

M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215 211

which sub-volumes porosities range from 4 to 37%. In general, thelargest discrepancies were found for C11, C22 and C66, and weremore pronounced for those specimens that display large variationsof porosity (e.g. specimens ‘‘03’’, ‘‘06’’, ‘‘08’’) (Fig. 3). These resultssuggest that the spatial distribution of pore size and shape as wellas the variations between different specimens, independently ofvariations of pore volume fraction, are not likely the principalcause of the discrepancies between the reference model predic-tions and experimental results.

5.2. Pore length

The reference model assumes that the pores, representingHaversian channels and resorption cavities, are infinitely long.The fact that the pores are actually finite may be a source of dis-crepancy between the model predictions and the experiments.Models based on Eshelby’s solution for ellipsoidal inclusions in amatrix [23] allow for the consideration of the shape of the pores,i.e. their aspect ratio. Among the possible formulations based onEshelby’s solution, the MT appears to be the most relevant [24].Note that the MT method has been used by several authors to rep-resent bone at the millimeter scale [9,10,25–27]. When the ellip-soid in the MT scheme is cylindrical (i.e. the pores are infinitelylong), the AH (reference) and MT models yield very close resultsfor the entire range of porosity of cortical bone. (However, it isnoteworthy that the AH method offers the advantage of being sta-ble, even at high porosity [24].) Accordingly, we considered a MTmodel of cortical bone mesoscopic elasticity: the elastic propertiesof the matrix and pores were defined by the same tensors as for thereference model, respectively cm and cp, the inclusions were spher-oids, and the aspect ratio d (major semi-axis over minor semi-axis)was chosen with regard to the general shape of the pores. Inhuman femoral mid-diaphysis, the osteon length is 4 mm on aver-age [28]. The diameter of the Haversian canals in women is(mean ± SD (min–max)) 150 ± 119 (57–457) lm [29]. Accordingly,we assumed that d resides in the range 10–70. Computationsshowed that the MT effective elastic properties change only veryslightly when increasing the aspect ratio beyond 10 (solid line inFig. 4), suggesting that aspect ratios of ellipsoidal inclusions assmall as 10–20 can be considered of infinite extent.

6. Discussion

Validating models of bone tissue elasticity should consist in acomparison of measured stiffness tensors and specimen-specific

model predictions. In practice, it is difficult to measure all theterms of the stiffness tensor so that only a few elasticity constantsare used for the validation: Dong and Guo [8] used two shear andtwo longitudinal coefficients; Deuerling et al. [30] and Baumanet al. [31] used only longitudinal coefficients. In previous works,the specimen-specific model predictions have used a variety ofspecimen data such as porosity [8,31], elasticity and areal fractionsof osteonal and interstitial tissues [8] and average orientation ofmineral crystals [31]. Here, we have investigated a popular two-phase composite model of cortical bone which predicts the depen-dency of the mesoscopic elastic coefficients on porosity. Thestrength of the present study lies in the number of subjects (21specimens from ten female donors), the number of measured andpredicted elastic coefficients (three longitudinal and three shearcoefficients, although the transverse isotropic model only predictsfour different coefficients) and the assessment of Haversian poros-ity for each specimen.

We first examined the experimental uncertainties. Although theprecision of the experimental data was acceptable, we recognizethat it could be improved. The precision of the vascular porosityestimate would increase if calculated from the 3-D volume data,e.g. from a SR-lCT scan. Regarding the measurement of elasticproperties in human cortical bone, Bernard et al. [32] recentlydemonstrated the suitability for resonant ultrasound spectroscopyto bring the precision of Young and shear moduli down to �0.5%.Even though the experimental uncertainties were suboptimal inthe present work, they did not account for the observed differencesbetween the measured and predicted elastic coefficients, confirm-ing the need for a close examination of the model assumptions.

In the reference model, the vascular porosity was idealized asinfinite cylinders of circular cross-section aligned along the bonelong axis. We found that, for realistic aspect ratios (length of thepore/diameter of the pore) of the Haversian canal, i.e. in the range10–70, modeling the pores as infinite cylinders yields a very goodapproximation of pores of finite length. Cortical bone is character-ized by a gradient of porosity from the endosteal to the periostealregion [12,33] as well as changes in the pores size [29,34], and thepresence of large resorption cavities [28,35,36]. We combined AHand FEM in a two-step scheme to account for the spatial heteroge-neity of the pores distribution which results in fluctuations of mil-limeter-scale elastic properties within the measured specimens ofnominal dimensions 5 � 5 � 7 mm3. The results indicated that thedetails of the distribution of the porosity play a negligible role inthe averaged strain and stress distribution at the specimen scale,hence on the values of apparent elastic properties. Using a

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Experiments C66

Model reference matrix ±10%

Experiments C11 C22

Model reference matrix ±10%

Experiments C33

Model reference matrix ±10%

Experiments C44 C55

Model reference matrix ±10%

Fig. 5. Relative differences between experimental data and the predictions of the reference model vs. porosity. The solid horizontal lines correspond to the reference model.The dotted lines correspond to the predictions of the model with modified matrix elastic coefficients (±10% starting from the reference model values).

212 M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215

voxel-based FE model, Baumann et al. [31] found that a non-uni-form spatial distribution of intracortical porosity results in anorthotropic behavior (weaker stiffness in the radial direction ascompared to the circumferential, especially towards the epiphy-ses), which they mainly attributed to the endosteal resorption.We did not observe this phenomenon, likely because we harvestedbone specimens that covered the entire cortex but did not includethe trabecularized areas (e.g. Fig. 4A in Ref. [37]), i.e. the largeresorption cavities typically present on the endosteal surface.

Interestingly, the model gives a satisfactory prediction of thevariations of millimeter-scale elastic coefficients by assuming thatthe porosity variations between samples are due to changes ofeither diameter or number of cylindrical pores aligned with thebone axis (the analytical model does not make any distinctionbetween these two options). It should be noted that the analyticalmodel disregards the network of pores perpendicular to the boneaxis (Volkmann’s canals), while the FE model takes the real shapeand distribution of the vascular pores and resorption lacunae intoaccount. The question why it is possible to obtain satisfactory pre-dictions without explicitly modeling the Volkmann’s canals wasnot in the scope of this study. However, one reason could be thatcortical tissue in long bones contains many more Haversian canals

than Volkmann canals. Therefore, the variability in overall porevolume fraction can be assumed to be dominated by variations inthe Haversian canals network. It is of course possible to build amodel that accounts separately for pores aligned and perpendicu-lar to the bone axis, which will allow this question to be addressedin future studies.

The predictions of the model critically depend on the assumedvalues of the mineralized matrix stiffness. Using data from the lit-erature is questionable as bone matrix elastic properties can be sig-nificantly different depending on the cortical site [38], thespecimen preparation [39–41] or the spatial resolution of the prob-ing technique (e.g. the penetration depth in nanoindentation test-ing [42,43] or the lateral dimension of the ultrasound beam in SAM[44]). Moreover, most of experimental bone studies do not providethe full stiffness tensor but only elastic properties in one direction(along the osteons’ axial direction) and usually discriminatebetween osteonal and interstitial tissue instead of providing aver-age elastic properties for the bone matrix. To the best of our knowl-edge, there is no study reporting the anisotropic elastic propertiesfor native matrix tissue from a human femoral mid-diaphysis.

In the present work, the model assumes fixed stiffness coeffi-cients (cm) for the bone matrix. However, the elastic properties of

M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215 213

the matrix in human femoral bone are susceptible to change with,among other factors, age [45] and anatomical location [19]. Physi-ological variations of elasticity at the microscopic level have beendocumented to range between 5 and 15% [19,45]. Note that thiswas the case for the specimens used in this study [6]: precisely,the conversion of the acoustic impedance values into elastic coef-ficients [46] led to intra-specimen elastic variations of 8–10.5%.

We found that the discrepancies between model predictionsand experimental data can be explained neither by experimentalerrors nor by the detailed shape and distribution of the pores. Indi-vidual variation of matrix elasticity is one factor which warrantsfurther studies. As a first step, we propose to assess the sensitivityof effective stiffness coefficients to matrix stiffness controlled vari-ations. We computed solutions for two sets of matrix coefficientsdefined as a ±10% variation of the reference values (Eq. (2)). Theassociated variations of the predicted effective elasticity (averagedover the entire measured porosity range) are DC⁄11 = ± 1.8 GPa,DC⁄33 = ± 2.8 GPa, DC⁄44 = ± 0.6 Pa, DC⁄66 = ± 0.4 GPa. Hence, the effec-tive elasticity variations due to small (±10%) matrix property varia-tions are likely larger than the experimental uncertainties andconsistent with the range of fluctuations of the experimental data(Fig. 5). This result suggests that a precise specimen-specific modelof a cortical bone specimen should account for specimen-specificmatrix elastic properties, which are likely to vary with, for example,changes in tissue mineral content [47–49] and average orientation ofmineral fibrils [30,50].

While a simple scaling of all elastic coefficients was sufficient totest for the influence of the matrix elasticity, this approach remainstoo simplistic for an accurate specimen-specific model (e.g. anincrease in the axial stiffness cm

33 may not necessarily be associatedwith an increase in cm

11). Hence, the validation of a proper model ofthe matrix properties and their variations appears as a natural per-spective of this work.

Multistep homogenization schemes can be used to derive thestiffness tensor of matrix elasticity, starting from the physical prop-erties of bone constituents (collagen, water, mineral) [9,25,26,51].Upon assuming certain composition and organization rules forthe different phases, it may be possible to obtain a transversely iso-tropic stiffness tensor with less than five degrees of freedom[10,52]. Modeling of cortical bone material properties at the milli-meter scale with a two-phase model is a framework that was usedhere for elastic modeling. It is worth noting that strength [53], vis-coelasticity [54] and poroelasticity [55] of cortical bone may also beexplained in this framework.

A limitation of the study is the unique anatomical origin of ourspecimens which were all harvested in the femoral mid-diaphysis,and therefore exhibit transversely isotropic elastic properties, inagreement with previous studies [11,56]. The application of theproposed model to anatomical sites which can reveal an apparentorthotropic elastic behavior is not straightforward (e.g. near thefemoral or tibial epiphyses [57,58]). Further investigations areneeded to clarify the respective contributions of the matrix elastic-ity symmetry and the pore network to the orthotropic behaviorand consequently adapt the model. Finally, future studies shouldinclude a larger number of specimens and/or a higher precisionto distinguish between the discrepancies that can be attributedto either experimental noise or matrix elasticity.

7. Conclusion

In this work we compared model predictions of effective stiff-ness with experimental data on human cortical bone specimens.Although most of the models that have been proposed for corticalbone were based on several steps of homogenization and a largenumber of variable parameters, the careful comparison conducted

here between experimental data and model predictions supportour hypothesis that a relatively simple model, namely a two-phasecomposite material, is a suitable representation for cortical bone.Several factors may in principle have an effect on millimeter-scaleelastic properties: relative fractions of osteonal and interstitial tis-sues, osteon types associated with different patterns of fibril orien-tations, volume fractions and shapes of porosities at the differenthierarchy levels, quality and volume fractions of mineral and colla-gen molecules, etc. The results presented in this paper support thevalidity of the two-phase composite material model of corticalbone which assumes that the essential source of variations of elas-tic properties at the millimeter-scale is the volume fraction ofHaversian porosity. We propose that the bulk of the remaining dis-crepancies between predicted stiffness coefficients and experi-mental data (RMSE between 6% and 9%) is in part due toexperimental errors and in part to small variations in the extravas-cular matrix properties.

The outcome of this study provides valuable insights for pre-dicting the variations of bone elasticity at the millimeter scale.Ultimately, a simple and accessible model that can reliably predictchanges in anisotropic elasticity would be a useful tool for the bonecommunity, e.g. to feed the FE models commonly used in fracturerisk assessment or orthopedics (implant development, preopera-tive planning) or to investigate structure–function relationships(effect of bone remodeling on local elasticity).

Future in vitro studies may consider including an individualizedmatrix elasticity in order to obtain a model specific to a given cor-tical bone specimen. For in vivo applications, there is, to date, noclinical tool allowing for the assessment of matrix elasticity froma patient’s bone. However, implementing the proposed model(with fixed matrix properties) in subject-specific FE analyseswould be straightforward. This could be done directly from CT datain a similar manner to that described by Hellmich et al. [59], i.e. byconverting the pore volume fraction of each voxel (deduced fromits Hounsfield unit value) into the corresponding anisotropic elas-tic tensor using the two-phase micromechanical model presentedin this work.

Such implementation would constitute a step forward inimproving bone mechanical behavior predictions as it overcomesone of the main flaws of current subject-specific FE models, i.e.material properties are frequently assumed to be isotropic [60].

Additionally, another class of problems that can benefit fromthe present work are finite-difference time-domain simulationsaiming at elucidating the interaction mechanisms between ultra-sound and bone structures [61].

Acknowledgements

This work has been conducted within the European AssociatedLaboratory ‘‘Ultrasound Based Assessment of Bone’’ (ULAB) fundedby CNRS (France). It was also supported by the ESRF Long TermProposal MD431 and the Deutsche Forschungsgemeinschaft withinthe priority program SPP1420 ‘Biomimetic Materials Research:Functionality by Hierarchical Structuring of Materials’ (grantsRa1380/7-1 and Ge1894/3-1). W.P. is grateful to the Engineeringand Physical Sciences Research Council (EPSRC) for funding viagrant EP/H01011/1.

Appendix A. Figures with essential colour discrimination

Certain figures in this article, particularly Figures 1 and 2, aredifficult to interpret in black and white. The full colour imagescan be found in the on-line version, at http://dx.doi.org/10.1016/j.actbio.2014.10.011.

214 M. Granke et al. / Acta Biomaterialia 12 (2015) 207–215

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