Internal strains and stresses measured in cortical bone via ...

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Journal of Structural Biology 152 (2005) 14–27

StructuralBiology

Internal strains and stresses measured in cortical bone viahigh-energy X-ray diffraction

J.D. Almer a, S.R. Stock b,*

a XOR, Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USAb Institute for BioNAnotechnology in Medicine, Northwestern University, 303 E. Chicago Ave., Chicago, IL 60611-3008, USA

Received 15 June 2005; received in revised form 8 August 2005; accepted 9 August 2005Available online 8 September 2005

Abstract

High-energy synchrotron X-ray diffraction was used to study internal stresses in bone under in situ compressive loading. A transversecross-section of a 12–14 year old beagle fibula was studied with 80.7 keV radiation, and the transmission geometry was used to quantifyinternal strains and corresponding stresses in the mineral phase, carbonated hydroxyapatite. The diffraction patterns agreed with tabu-lated patterns, and the distribution of diffracted intensity around 00.2/00.4 and 22.2 diffraction rings was consistent with the imperfect00.1 fiber texture expected along the axis of a long bone. Residual compressive stress along the bone�s longitudinal axis was observed inthe specimen prior to testing: for 22.2 this stress equaled �95 MPa and for 00.2/00.4 was between �160 and �240 MPa. Diffraction pat-terns were collected for applied compressive stresses up to �110 MPa, and, up to about �100 MPa, internal stresses rose proportionallywith applied stress but at a higher rate, corresponding to stress concentration in the mineral of 2.8 times the stress applied. The widths ofthe 00.2 and 00.4 diffraction peaks indicated that crystallite size perpendicular to the 00.1 planes increased from t = 41 nm before stresswas applied to t = 44 nm at �118 MPa applied stress and that rms strain erms rose from 2200 le before loading to 4600 le at the max-imum applied stress. Small angle X-ray scattering of the unloaded sample, recorded after deformation was complete, showed a collagenD-period of 66.4 nm (along the bone axis).� 2005 Elsevier Inc. All rights reserved.

Keywords: X-ray diffraction; X-ray scattering; Synchrotron radiation; Stress; Bone

1. Introduction

Bone is a highly adaptive, particulate-reinforced com-posite that, through a complex hierarchical structure,achieves remarkable mechanical performance (see Fratzlet al., 2004; Weiner and Wagner, 1998). The composite pre-serves, to a large degree, the desirable properties of theindividual components: high toughness of the bone matrix,collagen fibrils stabilized by water, and high stiffness ofthe reinforcing phase, crystallites (i.e., single crystal nano-particles) of carbonated apatite (Currey, 2002; Frittonand Rubin, 2001). Mechanical properties and fracturesusceptibility differ between healthy bone and bone withimpaired functionality (as in osteoporosis), and reliable

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doi:10.1016/j.jsb.2005.08.003

* Corresponding author. Fax: +312 503 2544.E-mail address: [email protected] (S.R. Stock).

discrimination between the two remains a challenge, notonly in the clinic but also in basic research employing ani-mal models. Factors other than decreased bone mineraldensity must be invoked to explain predisposition of boneto fracture and appear to include degraded bone microar-chitecture and other aspects of bone quality; current think-ing focuses on imbalances in bone additive and subtractiveprocesses (Heaney, 2003). Understanding is hindered byinherent variability from position to position in a singlebone, over time at the same position in a given bone fromone individual, between different bones from the sameindividual and between the same bone and same positionin different individuals.

Biewener (1993) described fracture propensity of bone interms of distributions of bone fracture strength and loadingspectra, and, considered in this light, it is clear that it is asimportant to quantify mechanical input to bone as it is to

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J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 15

identify ‘‘weak-link’’ microstructure(s) or changes in globalparameters characterizing microarchitecture. Numerousinvestigators (Fritton and Rubin, 2001) have quantifiedthis mechanical input in vivo with strain gages attachedto cortical bone, typically at mid-diaphyses where installa-tion can be done with minimal disruption to muscles andtendons. With attached strain gages, however, the mechan-ical response of volumes beneath the bone�s surface canonly be inferred indirectly. This limitation hinders under-standing of effects such as that of mechanical input on boneremodeling: some sort of mechanotransduction must trans-late the forces applied to the whole bone to instructions atthe cellular level directing adaptation of the bone (Knothe,2001).

It is surprising, therefore, that, except for (Borsato andSasaki, 1997), the authors of this paper have found no re-ports in the literature on the use of the nanometer-sizedstrain gages intrinsic to bone, namely the unit cell of themineral (carbonated apatite) reinforcement of the compos-ite material bone. That is not to say that X-ray scatteringhas not been applied to bone (see Section 2), but rather thatX-ray diffraction measurement of the distortion of unit cellsby applied loads (and its conversion to stress), a standardsubject in undergraduate textbooks used in materials sci-ence (Cullity and Stock, 2001), has not been widely applied.

Therefore, the study reported below examined whetherstresses in the mineral phase of bone could be quantifiedas a function of applied load using transmission, high-ener-gy synchrotron X-ray diffraction. Unlike (Borsato andSasaki, 1997), which examined thin samples cut from largerbones and measured stress only within 100–200 lm of thesurface, the present work probed the entire, unaltered bonecross-section. The characteristics of synchrotron X-radia-tion and the general principles of stress measurement byX-ray diffraction are briefly introduced to provide back-ground for the specifics of the transmission synchrotronscattering geometry and the bone loading experiment.The details of analysis are presented in conjunction withthe data obtained at different applied stresses, and the re-sults corresponded well with prior strain gage studies.

2. Background

Some, but not all, X-ray clinical and basic science meth-ods are familiar to most bone researchers. Dual-energyX-ray absorptometry (DEXA), for example, is widely usedto assess patients� bone mineral loss in osteoporosis (Bolo-tin, 2001). X-ray CT (computed tomography) whole bodyscanners are present in most hospitals, and peripheralquantitative CT scanners enjoy increasing use in clinical re-search. Microarchitecture of bones is more and more fre-quently quantified using laboratory microCT systems (seeMajumdar and Bay, 2001). Commercial availability of in vi-vo scanners (Scanco Medical; Skyscan) allows increasinguse of microCT in longitudinal studies of individual ani-mals before, during and after treatment; while such studieswere performed previously at synchrotron radiation

sources (Kinney et al., 1995, 2000), Herculean effort wasrequired.

In the life sciences, X-ray diffraction�s familiar avatar isstructure determination of single-crystal biological macro-molecules such as proteins and viruses. A very differentX-ray diffraction method is required to study the discontin-uous reinforcement phase in bone (carbonated apatite):polycrystalline X-ray diffraction techniques developed formaterials engineering. These polycrystalline techniquesare relatively unfamiliar outside of the physical sciencesand engineering and require treatment before the specificsof high-energy X-ray diffraction in the transmission settingand of stress determination can be reviewed.

2.1. X-ray diffraction

The well-known equation k = 2dhkl sinh (Bragg�s law)describes the necessary relationship between X-ray wave-length k, Bragg hkl plane spacing dhkl and 2h, the angle be-tween incident and diffracted beam directions. In apolycrystalline sample (here carbonated apatite crystallitesdispersed in a water-stabilized collagen matrix), only a verysmall fraction of the crystallites are oriented to diffract fora given incident beam orientation. If one employs a suit-ably collimated monochromatic X-ray beam, the properlyoriented crystallites form a Debye cone of diffracted inten-sity. Multiple hkl diffract simultaneously, and each of thecones is from a different subset of crystallites. In the trans-mission geometry used in this study, an area detector isplaced normal to the beam transmitted through the sample,and a circular ring of diffracted intensity is recorded wherethe detector intercepts each cone of diffracted radiation. Ifall crystallite orientations are equally likely (random pow-der), the intensity around each diffraction ring will be con-stant. If certain crystallite orientations are more likely, i.e.,preferred, the intensity can vary azimuthally around therings and indicate the presence of crystallographic texture.

2.2. X-ray diffraction measurement of stress and strain

When a compressive stress is applied to a crystallinesample (here a specimen containing discrete single-crystalnanoparticles of carbonated apatite in a collagen matrix),the unit cells, however they are oriented, compress alongthe applied stress direction and, via the Poisson effect, ex-pand along directions normal to the applied stress axis.The directionally-dependent change in d-spacings produceselliptical diffraction rings: the circular rings of the un-stressed sample expand along the applied compressivestress axis y (azimuthal angles g = 90� and 270� in thedetector plane) and contract normal to this direction, i.e.,along x (azimuthal angles g = 0� and 180�). Quantifyingthe amount of diffraction ring distortion is the physical ba-sis of X-ray diffraction stress analysis.

The measurements require very precise determination ofchanges in d-spacing, i.e., in the ring radii as a function ofg; see (Almer et al., 2003) for details. If one considers the

16 J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27

magnitudes of stresses applied and of Young�s modulus(for purposes of illustration, take a low value, ry = 50MPa, for the yield stress for compact bone and a typicalvalue E = 20 GPa, Currey, 1990), the relationship r = Eeshows that ey � 2.5 · 10�3. The entire elliptical diffractionring is fit to determine strain, i.e., Dd/d (Cullity and Stock,2001; Noyan and Cohen, 1987), to the required precision.However, converting internal strains in the mineral phaseto stresses necessitates more detailed consideration of theelastic constants than was employed above.

Stress rij and strain ekl are second rank tensors relatedthrough the fourth rank elastic constants Cijkl, i.e.,rij = Cijklekl. For a single crystal, numerical calculation ofstress components from measured strains is straightfor-ward. For data obtained from crystallites with different ori-entations (i.e., a polycrystalline specimen), the strainsmeasured by X-rays are averages, and to determine anaverage stress from these strains requires average elasticconstants derived from Cijkl according to one of severalapproximations. The Reuss approximation assumes thatall crystallites experience the same stress; the Voigt approx-imation assumes all grains within the sample are subjectedto the same uniform strain. The Kroner–Eshelby limit, cal-culated for anisotropic precipitates coupled to an isotropicmatrix, yields values of elastic moduli close to those ob-served experimentally, that is, near the mean of the Reussand Voigt limits (Noyan and Cohen, 1987).

Table 1 gives elastic constants from the literature for acrystal of hydroxyapatite (Gardner et al., 1992). Note thatCijkl are listed using the abbreviated notation cmn = Cijkl

(m,n = 1, . . ., 6; i, j,k, l = 1,2,3, see (Nye, 1976)). These cijvalues were then used to calculate average X-ray elasticconstants S1(hk.l), S2/2(hk.l) using the Voigt, Reuss, andKroner–Eshelby models (Computer program Hauk;Hauk, 1997) using definitions S1 = (�m/E)average andS2/2 = ((1 + m)/E)average (Noyan and Cohen, 1987). Allmodels indicate that the c-axis of the hexagonal unit cellis the elastically stiffest direction, with maximum deviationsof 20% in different crystallographic directions. This indi-cates that hydroxyapatite is only weakly elastically aniso-tropic, in agreement with results from the fluoroapatitephase (Sha et al., 1994).

Measured strains exx and eyy (where x and y are alongthe horizontal and vertical axes of the 2D diffractionpattern, respectively, and the X-ray beam is along z) are

Table 1Elastic constants (in GPa) from (Gardner et al., 1992) converted to X-ray elastifor Voigt, Reuss and Kroner–Eshelby models

Elastic stiffness coefficients:C11 C12 C13 C3

137.0 42.5 54.9 17

hk.l Voigt Re

S1 S2/2 S1

00.2/00.4 �2.38 11.14 �22.2 �2.38 11.14 �

used to calculate internal stresses along the loading (i.e.,y) direction using (Hauk, 1997):

ryy ¼1

S2=2eyy �

S1

S2=2� 3S1

eyy þ 2exx� ��

; ð1Þ

where the transverse strains exx and ezz are assumed to beequal. The change of notation (exx for e11, etc.) is madeto emphasize the coordinate axes of the detector andspecimen.

2.3. High energy diffraction with synchrotron X-radiation

Brilliant hard X-ray beams available at third generationsynchrotron radiation sources offer important advantagesfor many scientific and engineering applications. Shorterwavelength X-rays allow thicker samples to be studied non-invasively: 80 keV X-rays have �40% transmissivitythrough 20 mm of calcite (CaCO3), the biomineral com-prising the skeleton of echinoderms such as sea urchins.A second advantage relates to the radius of the Ewaldsphere, i.e., to the geometry of diffraction in reciprocalspace. The shorter the wavelength of the diffracting radia-tion, the greater the Ewald sphere radius and the moreclosely the Ewald sphere approximates a plane of the reci-procal lattice. For 100 keV electrons used in transmissionelectron microscopy (TEM) imaging of materials, theEwald sphere radius is 27.0 A�1, and the diffraction pat-tern, at least near 000 appears, to be a plane through thereciprocal lattice (calcite d104 = 3.00 A and reciprocal lat-tice period of 0.33 A�1). Diffraction with 0.154 A X-rays(6.5 A�1 Ewald sphere radius) is closer to the TEM casethan that of Cu Ka (the most commonly used wavelengthwith X-ray tube diffraction, 1.54 A wavelength, 0.65 A�1

Ewald sphere radius) (Stock et al., 2003). Therefore, morereflections are recorded simultaneously with high energyX-ray diffraction than with lower energies.

The use of an area detector and forward scatteringgeometry allows a large range of sample orientations(encompassing the two principal directions of the bone)to be measured simultaneously, without sample move-ment. This differs from traditional methods where sam-ple movement (tilting, which introduces additionalsources of error) is required to measure different crystal-lite orientations and where the orientation range is morelimited.

c compliances S1 and S2/2 (in 10�6 MPa�1) for hydroxyapatite (c/a = 0.73)

3 C44

2.0 39.6

uss Kroner–Eshelby

S2/2 S1 S2/2

4.64 3.49 �2.27 10.22.86 10.0 �2.48 11.5

Fig. 1. MicroCT slice of the bone oriented transverse to the loadingdirection (i.e., the loading axis is perpendicular to the plane of the slice).The vertical field of view is 3.2 mm, and lighter voxels are more absorbingthan darker voxels. The arrow is parallel to the incident beam direction.


2.4. Diffraction studies of bone

X-ray and neutron scattering have contributed to under-standing of the mineral and matrix phases of bone. Severalinvestigators have used X-ray diffraction to examine chang-es in actual or synthetic bone mineral chemistry and crys-tallographic parameters such as carbonate content, c/aratio and crystallite size vs microstrain (Baig et al., 1999;Dalconi et al., 2003; Handschin and Stern, 1992, 1995;Smith and Smith, 1976). Pole figure measurements of themineral crystallites in bone have been performed with X-rays (Ascenzi et al., 1979; Nightingale and Lewis, 1971;Rindby et al., 1998; Sasaki and Sudoh, 1997; Wenk andHeidelbach, 1999) and with neutrons (Bacon et al., 1977,1979, 1980), and preferred orientation of crystallite c-axeswas interpreted in terms of principal applied stress axesin the structure. X-ray microbeam mapping has also beenapplied to bone using wide angle diffraction (Nakanoet al., 2002; Rindby et al., 1998; Wenk and Heidelbach,1999) and small angle X-ray scattering (SAXS) (Roschgeret al., 2001; Kinney et al., 2001; Zizak et al., 2003) in orderto study spatial variation of crystallite orientation. The66–67 A collagen axial periodicity has also been measuredin bone samples using SAXS (Ascenzi et al., 1978, 1998;Kinney et al., 2001). Except for the neutron diffractionstudies, the studies cited above (as well as the single priorX-ray diffraction stress measurement study (Borsato andSasaki, 1997)) were performed on thin sections of bone.Those familiar only with the ubiquitous copper-target tubeas sources of X-radiation might not realize that, as is indi-cated in Section 2.3, X-ray diffraction can be performedthrough entire bone cross-sections.

3. Materials and methods

The sample used for the study described was cut fromthe mid-diaphysis of a canine fibula; the beagle was be-tween 12 and 14 years old. Two parallel surfaces 5.0 mmapart were cut from the fibula using a Buehler Isometsaw and a diamond wafering blade. The cuts were as closeas possible to perpendicular to the long axis of the bone.The microCT slice in Fig. 1 (Scanco MicroCT-40 (ScancoMedical) operated at 45 kVp, data collected in the highestsensitivity mode, reconstruction with 6 lm voxels or vol-ume elements) shows the cross-section of the sample paral-lel to the cut surfaces. The sample contained little porosityand appeared to consist entirely of compact bone; the mi-croCT system�s software was used to measure the cross-sec-tional area for use in computation of the applied stress.

The sample was deformed in situ using a screw-drivenload frame, and applied load was determined with a2000 lb. load cell, located below the specimen. Hardenedsteel platens (5 mm thickness, 1 cm diameter) were placedon either side of the bone specimen, and a hemisphericalsteel ball was placed between the drive screw and the upperplaten for alignment. The sample was loaded stepwisein displacement control, and diffraction patterns were

collected after the load gage revealed only gradual stressrelaxation. During the exposures, the applied stress relaxedno more than 15%, and applied stress values given beloware mean values during this minor relaxation. In thisproof-of-principle experiment the bone was not irrigatedwith water, so some drying must have occurred.

Transmission X-ray diffraction was performed at station1-ID of the Advanced Photon Source, Argonne NationalLaboratory (APS); the monochromatic X-ray beam(80.715 keV, Au K-edge) was approximately 300 lm ·300 lm in size. Approximately 90% of the incident beampassed through the sample (as measured with a PIN diodeembedded in the tungsten beamstop after the sample). AMAR 345 image plate detector recorded the wide-angle dif-fraction patterns out to a d-spacing of 1 A. The detectorwas perpendicular to the X-ray beam and was centeredwith respect to the beam stop. The detector area was345 mm in diameter, and it was read with 150 lm pixels,with a readout time of �80 s. Exposures were recordedfor an incident flux of 4 · 1012 photons, which required�60 s and nearly saturated the most heavily exposed detec-tor pixels. A diffraction pattern from a reference powdersample of ceria (CeO2, NIST Standard Reference MaterialSRM-674a, 1 mm diameter glass capillary tube) was usedto correct for detector tilts, determine the sample-detectorseparation (968.0 mm) and give a reference (instrumental)peak breadth.

Based on compact bone data presented by Currey (1990)for a range of vertebrates and bone types, yielding of thecanine fibula was anticipated to be in the range of 50–200 MPa. Diffraction patterns were recorded at 0, �14.5,�36.7, �94.7, and �118 MPa applied stress (r < 0 indi-cates compression). The 2D diffraction patterns record


diffracted intensity I(r,g), where r is the magnitude of thevector between the incident beam position on the detectorand the pixel of interest and g is the azimuthal angle be-tween r and the horizontal reference direction (seeFig. 2). Using the programs FIT2D (Hammersley, 1998;Hammersley et al., 1994a,b, 1996) and MATLAB (Math-works and MATLAB), the patterns were converted frompolar into Cartesian coordinates (5� azimuthal and150 lm radial bins), producing 72 1D radial patterns. Plotsof the integrated intensity of the 00.2, 00.4, and 22.2 reflec-tions as a function of azimuthal angle (see Fig. 3) revealedthe apatite texture. For comparison with reference patternsof hydroxyapatite, the Cartesian representation was trans-formed into a 1D diffraction pattern I(d) using Bragg�s lawand azimuthal integration.

Fig. 2. (A) Two-dimensional diffraction pattern I (r,g): the darker the pixel thg = 0� (g increasing clockwise), the 00.2 reflection is labeled and the faint whitefrom the beam stop detector. (B) Cartesian plot of radius vs azimuth for thAzimuthally integrated diffraction pattern converted to I (d) from (B) and compICDD Powder Diffraction File pattern 09-432). The vertical axis is in units ofintensity of each reflection identified by indices hkl.

The diffraction peaks in the 72 1D radial patterns werefit with pseudo-Voigt functions to provide radial values rgthat were converted into absolute lattice plane spacingsdg using geometric values determined from the (known)ceria d-spacings. Estimates of absolute errors in dg are be-low 10�4 for the experimental set-up (Almer et al., 2003).

A key feature in the strain analysis was comparison ofradius vs orientation profiles as a function of applied stress.For each reflection studied, the (radius vs orientation) pro-files obtained at different applied stresses intersected at asingle radius, the invariant radius r*, and the correspond-ing orientation value was g* (see Fig. 4). Measured radiiwere referenced to this r* value to provide orientation-de-pendent (deviatoric) strain values: eg = (rg � r*)/r*. Thesestrain profiles eg were fit to a biaxial strain model (He

e higher the diffracted intensity. The dashed line shows the orientation ofshadows running near g = 90� are caused by the electrical wires running to/e diffraction pattern in (A) and using the same gray scale as in (A). (C)ared with a reference pattern for synthetic hydroxyapatite (Ca5(PO4)3OH,square root of intensity, and the height of the vertical lines indicates the

Fig. 4. Measured radial peak position vs azimuth for the 00.4 reflection asa function of applied compressive stress. The applied stress increased from0 (A), to �14.5 (B), to �36.7 (C), to �94.7 (D) to �118 MPa (E), seeFig. 5. The invariant radial position r* (horizontal line between positions577 and 577.5) and corresponding orientation g* are indicated (see text fordefinition of these two quantities).

Fig. 3. Variation of peak intensity (normalized to one) vs azimuth for the00.2, 22.2, and 00.4 peaks. The 00.2 and 00.4 variations are nearlyidentical, as expected, and indicate preferential alignment of the crystallitec-axis with the load axis (at g = 90� and 270�). Outliers at g = 90� arisefrom the beamstop wires (see Fig. 2).

1 This relationship is only valid in the small h limit, sush as in this study.


and Smith, 1998) to account for sample geometry, and pro-vided values of the deviatoric strain components exx andeyy. These strain components, along with the X-ray elasticconstants, were then used to calculate stress along the load-ing direction, ryy, via Eq. (1).

Strictly speaking, the r* values are not necessarily equalto the strain-free value ro due to the potential presence ofhydrostatic strains. This hydrostatic contribution can bedetermined using a strain-free powder of the same material,but this study reports only deviatoric strains and stresses,as these are the components primarily involved in bonedeformation. It is important to emphasize the use of r*

values allowed the deviatoric strain to be computed with-out the need of assuming tabulated values for (carbonated)hydroxyapatite lattice constants as references, which canbe a significant source of error due to the often non-stoichi-ometric compositions of bone (Handschin and Stern, 1992,1995).

Radial broadening DRmeas was defined as the full widthat half maximum (FWHM) of each peak and consisted ofan instrumental contribution DRinst as well as a contribu-tion DR from the crystallites themselves. DRinst for differentpeaks were determined using the ceria standard. AssumingGaussian peak shapes, the mineral phase�s peak widthswere DR ¼ pðDR2

meas � DR2instÞ, and DR were converted

to D2h (peak width) using D2h = DR/zs-d,1 where zs-d is

the sample-detector distance. Crystallite size (t) and rmsstrain (erms) contribute to peak width according to (Noyanand Cohen, 1987)

D2h� cos h ¼ 180

pkDþ 2

Ddd

� �2* +1=2

sin h

24

35; ð2Þ

where Æxæ denotes the mean of x and erms is represented byÆ(Dd/d)2æ1/2. Note that the first term inside the squarebrackets is the Scherrer equation and erms is sometimestermed microstrain (Cullity and Stock, 2001).

Small angle scattering provides data on collagen period-icities and on mineral crystallite shapes and dimensions inbone (Fratzl et al., 1992, 1996). In order to record SAXSfrom the bone specimen, the MAR 345 detector (near themiddle of the hutch) was translated out of the X-ray beamand a second detector, already positioned at the down-stream end of the hutch, was used. The SAXS detector(PI-CCD, 1152 · 1242 pixels, {22.5 lm}2 pixels) was locat-ed 4900 ± 1 mm from the sample, with the distance calcu-lated by a silver behenate standard. The SAXS pattern wasrecorded after the specimen was unloaded, and the expo-sure was 10 s. The resulting 2D pattern was integratedabout g = 270� ± 10� to obtain the projection along thevertical (loading) direction and about g = 180� ± 10� toobtain the variation of scattered intensity normal to thebone�s longitudinal axis.

4. Results

A 2D transmission diffraction pattern from the sample(acquired at maximum applied stress) appears in Fig. 2A,together with the definitions of radius and azimuth.Fig. 2B shows the pattern converted from polar to Carte-sian coordinates, with radius as the abscissa and azimuthas the ordinate. Several features are readily apparent inthe transformed data. First, the intensity for a given reflec-tion (three of those used later in this work are identified) isnot constant as a function of azimuth but varies in a sys-tematic manner, which indicates crystallographic texture.


Second, the radii from different reflections vary as a func-tion of azimuth, and this indicates directionally dependent(deviatoric) strain.

The data in Fig. 2B were azimuthally integrated, and theradius converted to d using experimental parameters(wavelength and sample-detector separation) in order toproduce a powder-like diffraction pattern, Fig. 2C. The dif-fraction peaks are all identified as belonging to hydroxyap-atite (ICDD Powder Diffraction File pattern 09-432,hexagonal axial system, space group P63/m) with approxi-mate composition Ca5(PO4)3(OH) and lattice parametersa = 9.418 A and c = 6.884 A.

Fig. 3 plots the normalized intensities of 00.2, 00.4, and22.2 reflections. The azimuthal variation of the 00.2 and00.4 peak intensities is virtually identical (as expected) be-cause these are second and fourth order reflections ofBragg plane type 00.1 and are from nearly identically-ori-ented crystallites. The peaks in this distribution are at 90�and 270� azimuths, i.e., along the longitudinal axis of thebone (which is also the compression axis in this experi-ment), and show that the bone has a 00.1 fiber texture.The four symmetry-related peaks in the 22.2 azimuthalintensity distribution (compared to two for 00.2/00.4) arecentered at angles 38�, 140�, 217�, and 323�, which is a con-sequence of the broad 00.1 fiber texture (see the secondparagraph of Section 5).

The azimuthal full-width at half maximum (FWHM)intensity of each reflection is a measure of the sharpnessof the texture. For 00.2, FWHM were between 57� and59�, and, for 22.2, FWHM ranged between 49� and 59�.The ratio of the 00.2 intensity at 90� to that at 180� pro-vides a measure of the strength of preferred orientation.After correction for background, the intensities at 90�and 180� azimuths were 4.5 · 104 and 5 · 103 counts,respectively. Therefore, approximately nine times morecrystallite volume had c-axes parallel to the bone�s longitu-dinal axis than perpendicular to it. For 22.2, the ratio ofmaximum to minimum azimuthal intensity was muchsmaller, namely, 1.3.

The measured radial peak position (in detector pixels)for the 00.4 reflection is plotted as a function of azimuthfor five different applied stresses in Fig. 4. As noted above,radii at all loads, including the unloaded state (curve A),vary as a function of azimuth. These data show that the00.4 diffraction rings are ellipses elongated along the load-ing direction (g � 90� and 270�) and compressed in the per-pendicular direction. Because the rings were elliptical, evenunder zero applied stress, compressive residual strain existsalong the vertical/loading (y) direction and tensile residualstrain along the in-plane (x) direction. As applied loadincreases, the ellipses deviate further from circularity(Curves B–E in Fig. 4), demonstrating larger strains arebeing produced.

Table 2 lists applied stress, transverse strain exx andlongitudinal strain eyy in the mineral particles and ryy cal-culated for the Reuss, Voigt, and Kroner–Eshelby approx-imations. Note that strains are in units of le (microstrain,

that is, in 10�6 strain). For 00.2, 570 le < exx < 2410 le and�1440 le < eyy < �6050 le, and 00.4 gives similar magni-tudes as expected. The strain magnitudes from 22.2 aresmaller than those of 00.2/00.4.

Fig. 5 plots internal stress in the mineral phase as a func-tion of applied stress; these internal stresses were calculatedfrom the measured strains and the Kroner–Eshelby model.The residual stress in the mineral, that is, the internal stressmeasured with zero applied stress, was �95 MPa for 22.2(i.e., along the direction perpendicular to {22.2} in the crys-tallites) and was between �160 and �230 MPa for 00.1(i.e., perpendicular to {00.1}). As expected, the 00.2 and00.4 data in Fig. 5 were nearly coincident. Over much ofthe applied stress range, internal stress in the mineral crys-tallites rose linearly with applied stress. In analogy withBorsato and Sasaki (1997), the stress concentration factorX (defined as the stress in the mineral divided by the stressin the bone) for both the 22.2 and 00.2/00.4 data was deter-mined from this slope. As seen in Fig. 5, a single value ofX = 2.8 (dotted line) fits both the 22.2 and 00.2/00.4 reflec-tions for most of the loading curve. The linear relationshipbreaks down, however, above �94.7 MPa applied stress:the internal stress in the mineral particles rises at a greaterrate.

Fig. 6 shows experimental radial peak widths DR for the00.2, 22.2, and 00.4 diffraction peaks as a functional of azi-muthal angle g. These peak widths were corrected forinstrumental broadening DRinst, and the plots showDR(g) for the unloaded state (A), the intermediate stress(C, �36.7 MPa) and the maximum compressive stress(E, �118 MPa). In the unloaded condition, considerableanisotropy is seen for 00.2 and 22.2 (data are too incom-plete to draw a conclusion for 00.4). For 00.2, the radialpeak width is a full pixel smaller along the loading direc-tion compared to the transverse direction, and for 22.2,the width is two pixels smaller along g = 90�, 270� thanalong g = 0�, 180� (Table 3). At the maximum load, the dif-ference for 00.2 decreased to about one-half pixel and for22.2 to one pixel (Table 3).

Separation of rms strain and crystallite size contribu-tions to broadening requires values of peak width for twoorders of the same hk.l. (Alternatively, approaches suchas that of Williamson and Hall, see Cullity and Stock,2001, can be utilized.) For the loading direction (g = 90�,270�), 00.2 and 00.4 data were available. The crystallite sizet was 41 nm without load vs 44 nm at �118 MPa, and erms

was 2180 le in the unloaded state and 4630 le at the high-est compressive applied stress (Table 3). If the erms contri-bution were ignored and all sample broadening wasattributed to crystallite size, these values t 0 are depressed20% or more from those in the more realistic determinationincluding erms (Table 3). For 22.2, t

0 = 15 nm at g = 0� andt 0 = 20 nm at g = 90�, and there was little change from (A)to (E), i.e., with increased load.

The 2D SAXS pattern is shown in Fig. 7, left. Therewere two distinct peaks along the bone longitudinal direc-tion (indicated by arrows superimposed on the 2D pattern);

Table 2Experimental transverse exx and longitudinal eyy strains from apatite 00.2, 22.2 and 00.4 reflections as a function of applied stress rA

hk.l rAa exx eyy rKE

yy rRyy rVyy

00.2 0 570 �1440 �160 �380 �140�14.5 850 �2140 �240 �560 �210�36.7 1190 �2990 �330 �780 �290�94.5 1850 �4630 �520 �1210 �450�117.7 2410 �6050 �673 �1580 �590

22.2 0 340 �850 �94 �220 �80�14.5 480 �1200 �130 �310 �120�36.7 700 �1770 �200 �460 �170�94.5 1300 �3260 �360 �850 �320�117.7 1740 �4360 �490 �1140 �420

00.4 0 760 �1920 �210 �500 �190�14.5 960 �2400 �270 �630 �230�36.7 1160 �2920 �320 �760 �280�94.5 1730 �4350 �480 �1130 �420�117.7 2216 �5560 �620 �1450 �540

Longitudinal stresses rnyy corresponding to the observed strains are also given: the superscript n indicates the elastic constants model used to relate stressand strain, with R denoting the Reuss limit, V the Voigt limit, and KE the Kroner–Eshelby approximation. Strains are given in le, and stresses in MPa.a Curves (A)–(E) in Fig. 4 correspond to applied stresses 0 to �117.7 MPa, respectively.

Fig. 5. Internal stress in the carbonated apatite mineral phase calculatedusing measured strains (Table 2) and the Kroner–Eshelby model vsapplied stress for three reflections. Dotted lines show two stress concen-tration factors (1 = no stress concentration, 2.8 = best fit for 22.2) andsymbols A–E label the loads corresponding to the data shown in Fig. 4.Uncertainties in internal stress values approximately equal the symbolsizes.


this is shown numerically in the plot of intensity I plottedvs q (Fig. 7, right), where q = 2p/d. Therefore, these peakscorresponded to the first- and third-orders of periodicityequal to 66.4 ± 0.5 nm. The second order peak was not ob-served because its structure factor is much smaller thanthat of the third-order peak (Ascenzi et al., 1985). Thefirst-order peak had an azimuthal FWHM of 42� ± 2�,but the third order peak was too weak for reliable azimuth-al FWHM measurement.

Also present in the SAXS pattern was an elliptically-shaped halo of intensity near the direct beam (labeled‘‘E’’ in Fig. 7, left), with the major axis along the x

(transverse) direction. The intensity in this halo exhibited

no distinct peaks, but rather decreased monotonically fromthe direct beam, as shown in the I vs q plot (Fig. 7, right).

5. Discussion

The 1D diffraction pattern I (d) shown in Fig. 2C agreedwell with that from the powder diffraction file, confirmingthat the crystalline portion of the bone was carbonatedapatite. While structure refinement with the Rietveld meth-od (a la Lonardelli et al., 2005) could have been performedto provide more precise lattice parameters, etc., this wasnot done because the present study�s primary goal wasinvestigating peak position changes (i.e., internal stresses)vs applied stress.

The azimuthal FWHM of 00.2 and 00.4 averaged nearly60� and indicated the spread in texture (Fig. 3). The muchsmaller strength of the 22.2 texture (ratio of maximum tominimum azimuthal intensity equaled 1.3) compared tothe 00.2/00.4 texture (ratio of 9) is expected for an imper-fect (broadened) fiber texture. Using methods outlined else-where (Cullity and Stock, 2001; Haase et al., 1999),diffracted beam directions for the maxima and half-maxi-ma points of 00.2 and 22.2 were plotted on a stereographicprojection, i.e., at the correct relative azimuthal positionsand correct 2h (data not shown). The longitudinal boneaxis was plotted at the Wulff net�s north pole, and the en-trance and exit beams corresponded to directions at ornear, respectively, the center of the net. The locus of 11.1pole directions was those points 55� from 00.1, and the11.1 pole corresponding to the maxima at azimuth 140�was within 5–10� of the locus of poles 55� from the longi-tudinal axis of the bone (i.e., the 00.1 pole direction corre-sponding the maxima in 00.2 intensity). Further, the polescorresponding to the half maximum intensities in the 22.2azimuthal plot lay within the FWHM of the distribution

Fig. 6. Instrumentally-corrected radial peak width vs azimuthal orientation g at three applied stresses (A, unloaded; C = �36.7 MPa, E = �118 MPa) forreflections (A) 00.2, (B) 22.2, and (C) 00.4. Texture and a high slope of background intensity prevented accurate peak width determination for 00.4 overthe angular ranges 135� to 225� and 315� to 45�.

Table 3Radial peak broadening for different reflections and selected azimuthal orientations

hk.l h (�) R0 DRinst DR

Load: A E

g = 0� 90� g = 0� 90�

00.2 1.2 288.6 3.4 3.75 3.0 3.85 3.600.4 2.4 577.2 4.1 — 3.5 — 4.822.2 2.2 511.4 4.0 7.0 5.0 7.0 5.8

Analysis Parameter Load

A E

00.2/00.4 t (nm) 41 44erms (le) 2200 4600

00.2 t 0 (nm) 33 2822.2 t 0 (nm) 15–20

Radial values (R0, DR) are in pixels, and DR are values corrected for instrumental broadening DRinst. Crystallite size t and rms strain erms valuesdetermined from peak broadening along the vertical direction (g = 90�, 270�) are shown (i.e., 00.2/00.4 analysis) as well as t 0 calculated assuming rmsstrain is not present (00.2 and 22.2 analyses).


of 00.1 poles revealed by Fig. 3. The distribution of azi-muthal intensity for 22.2 and 00.2/00.4 was consistent,therefore, with a broadened 00.1 fiber texture, with the fi-ber axis parallel to the longitudinal axis of the bone (i.e.,the loading axis of the bone).

The observation of the imperfect 00.1 fiber texture alongthe axis of this long bone agreed with the bulk of the datain the literature. Bacon et al. (1977, 1979, 1980) examinedthree types of bovine bone: the scapula, lumbar vertebraand mandible. The c-axes of apatite crystallites lay alongthe length of the trabeculae in cancellous bone, and in com-pact bone the c-axes tended to lie along the loading direc-tion (i.e. along the axes of long bones or along the directionalong which attached muscles pulled). Nightingale and

Lewis (1971) and Sasaki and Sudoh (1997) both examinedbovine femora and found the c-axes were biaxially orient-ed: the larger subset with c parallel to the bone axis anda smaller, but still quite significant fraction, oriented per-pendicular to the bone long axis. Nightingale and Lewis(1971) also found the main crystallite c-orientation in bo-vine frontal bone was normal to the bone�s surface. Wenkand Heidelbach (1999) found the apatite c-axes in bovinemetacarpal cortical bone had rotation freedom within theplane parallel to the cortical surface, this difference fromother long bone observations being due perhaps to com-plex loading experienced by this bone in vitro (Martinet al., 1998). The mineral particles in human L4 vertebraewere oriented perpendicular to the outer shell in growth

10

100

1000

10000

0.05 0.15 0.25 0.35 0.45

q (1/A)

Inte

nsit

y (a

rb. u

nits

)

1

3

T

L

1000

950

900

850

800

50 100 150 200 250

Fig. 7. (left) 2D SAXS pattern from the sample center after unloading. The lighter the pixel, the higher the scattered intensity. The direct beam is locatedbehind a cross-shaped tungsten beamstop and is noted with an asterisk. The arrows show the distinct peaks in intensity along the vertical direction (i.e., thebone�s longitudinal axis). The elliptical halo ‘‘E’’ of scattered intensity near the direct beam is strongest in the in-plane direction. X–Y units are pixels(22.5 lm size), and intensity is on a logarithmic scale. One of the integration regions is indicated by the dashed lines. (right) SAXS intensity integratedalong both the vertical (u = 270�) and horizontal (u = 0�) directions (labeled by ‘‘L’’ and ‘‘T’’, respectively, for the longitudinal and tranverse bone axes).Numbers ‘‘1’’ and ‘‘3’’ indicate the two peaks in scattered intensity identified by arrows in the 2D pattern.


cartilage while they were oriented parallel to this surface inthe cortical bone shell, this presumably reflecting the colla-gen structural differences between cartilage and bone (Ros-chger et al., 2001). Rindby et al. (1998) performedmicrobeam diffraction of thin sections of human femoraand found two populations of crystallites: one set with c-axes parallel to Haversian canals and one set tangentialto the canals. Ascenzi et al. (1979) in studies of individualosteons found that the number of longitudinally orientedcrystallites (c-axes parallel to the canal axes) increased pro-gressively from transversely oriented osteons to alternative-ly and longitudinally oriented ones.

Scanning SAXS of bone thin sections have revealed apa-tite platelets in human trabeculae have normals alignedperpendicular to the direction of the trabeculae (Roschgeret al., 2001; Fratzl et al., 1997; Rinnarthaler et al., 1999;Jaschouz et al., 2003). The combination of scanning SAXSand wide angle diffraction showed a fiber texture existswithin the trabeculae, and the plate-shaped nanoparticlesare aligned with the lamellae within the trabeculae (Jasc-houz et al., 2003).

The diffraction rings (Fig. 4) were always elongated,even with no applied stress, and this indicated residualinternal deviatoric stresses were present in the mineralphase (Fig. 5). The difference in internal stress ryy of 22.2and 00.2/00.4 presumably reflects crystallographic differ-ences relative to the apatite crystallite particle morphology.While the authors of the present work are unaware of anyX-ray diffraction measurements that show residual macro-stresses (different from the stresses associated with micro-strains discussed below, see Cullity and Stock, 2001;Noyan and Cohen, 1987; Nye, 1976), Ascenzi and Benve-nuti (1977) used dissection and subsequent distortion ofosteon lamellae to show that such residual stresses existin osteons. Recently Ascenzi (1999) converted the lamellaedistortions, using the well-known Volterra dislocationequations, to estimate that compressive residual stresses

in one particular lamella reached 110 MPa. It is, of course,a major jump from macrostress in a single osteon lamellaeto a net macrostress in mineral in a bone cross-section, butthis link is quite plausible.

The stresses, residual or applied, within a volume of aspecimen must satisfy the equations of mechanical equilib-rium. The existence of compressive residual stresses in themineral that are on the order of �200 MPa would seem,at first glance, to imply that more-or-less equal and oppo-site stresses would exist in the collagen phase of bone, i.e.,tensile stresses substantially larger than the failure stress incollagen, stresses required to balance that in the mineralphase. One might view this latter point as a significantinconsistency in the analysis presented in Section 4. Setaside for the moment the question of whether mechanicalproperties of monolithic collagen tissues (i.e., non-mineral-ized tendon) apply to collagen in bone. Consider first anengineered composite whose components have well-definedproperties and whether the notion of equal-and-oppositestresses in the matrix should apply.

One class of engineering composites reinforces weak,compliant matrices with stiff second phases, and the pri-mary role of the matrix is to transfer loads between adja-cent reinforcement particles, plates or fibers. Thedeformation/stress levels in the composite matrix are wellknown to differ considerably from what is typical of thematrix as a monolithic material. Consider, for example,the results of Breunig et al. (1991) on SiC/Al composites.Specimens of the composite failed under uniaxial tensilestresses in excess of 1400 MPa despite the fact that themonolithic matrix (aluminum alloy AA 6061-0) wouldhave yielded at about 85 MPa and have had an ultimatetensile strength of about 155 MPa (Boyer, 1985). There-fore, if the notion of equal-and-opposite stresses applied,the composite�s aluminum matrix experienced stresses sub-stantially higher than it should be able to endure, a situa-tion like that of collagen in the present sample.


That the mechanical behavior of collagen in bone differsfrom that in unmineralized collagen was postulated byMcCutchen (1975). Landis et al. (1995) provided direct evi-dence of the aforementioned difference: they observed thatthe characteristic low modulus toe region (preceding thelinear portion of the load-elongation curve in unmineral-ized turkey tendon) disappeared when samples containingmineral were tested. The very process of nucleation andgrowth of apatite crystallites on the collagen fibril sub-strates strongly suggests that collagen in bone cannot re-spond as it would in the absence of mineral. The changedmechanical behavior of collagen in bone has been explicitlyacknowledged in the Jager–Fratzl mechanical model ofbone (Jager and Fratzl, 2000; Gao et al., 2003). Under anapplied tensile load, the mineral platelets carry the tensileload with the protein matrix transferring the load betweenmineral crystallites via shear. In essence, the large aspectratio of the mineral platelets compensates for the low mod-ulus of the protein phase, and the nanometer size of themineral platelets in bone may be the result of fracturestrength optimization. The very intimate relationship be-tween mineral crystallite and its collagen substrate (TEMinvestigations such as, Landis, 1995; Landis et al., 1995)provides nanostructural basis to expect collagen mechani-cal properties in bone to differ substantially from proper-ties in monolithic collagen.

The stress concentration factor found in the presentstudy was 2.8, somewhat larger than the value of2.2 ± 0.1 found in Borsato and Sasaki (1997). Borsatoand Sasaki (1997) used a laboratory h/2h diffractometerand in situ uniaxial tensile loading of a bone wafer to mea-sure peak shifts of the apatite 00.4 reflection as a functionof applied stress. Differences in loading (here compression,tensile loading in Borsato and Sasaki (1997)) may be thereason for the difference: unequal compressive and tensileresponses of materials, in general and in bone specifically,are well known. It may be that bovine femora respond dif-ferently than canine fibulae. Additionally, the present datareflect an average over 1–2 mm of sample thickness while(Borsato and Sasaki, 1997) reported results from materialwithin 0.1 mm of the surface. Both studies indicate, howev-er, that the mineral phase carries a higher fraction of theapplied load during deformation. This is, of course, consis-tent with the mineral phase being stiffer than the collagenphase and with a reinforcing function of the discontinuous‘‘filler’’ phase.

In Borsato and Sasaki (1997) the macroscopic stress–strain curve (applied stress vs strain measured by a straingage attached to the bone) was linear up to the maximumapplied tensile stress (just in excess of 100 MPa), and inter-nal stress in the mineral crystallites vs applied stress also re-mained linear up to the maximum applied stress. In Fig. 5,internal stress varied linearly with applied stress up to�94.7 MPa (in agreement with the results of (Borsatoand Sasaki, 1997)) and rose more rapidly between �94.7and �118 MPa applied compressive stress. Values of theyield stress for canine fibulae are not available, but, from

a wide range of animal bones, yielding is expected at ap-plied stresses between 50 and 200 MPa (Currey, 1990).The internal strain eyy in the mineral phase rose from�4030 to �6050 le upon increasing applied stress from�94.7 to �118 MPa. Compact bone fails in longitudinalcompression at strains as high as �14,000 to �21,000 leand begins to yield at �6000 to �8000 le (Biewener,1993). Therefore, one is inclined to interpret the increasingrise in mineral phase internal stress between �94 and�118 MPa as evidence of yielding in the bone. In orderto firmly establish this, additional specimens need to betested with diffraction patterns recorded at smaller stressincrements. Further, a strain gage should be attached tothe bone so that the macroscopic stress–strain relationshipcan be recorded for correlation with the mineral phaseinternal stress. Use of strain gages cemented to bone is astandard procedure (Fritton and Rubin, 2001), and suchstrain gages are transparent to X-rays of the energy usedin this study.

The magnitude of the internal strains eyy and exx in themineral crystallites (00.2, 00.4, and 22.2 reflections, seeTable 2) were consistent with strains measured in vivo withstrain gages during vigorous activity. In various mammalsand avians, in vivo peak principal compressive strains atthe surfaces of cortical bone range between �1700 and�2500 le during strenuous exercise (Biewener, 1993). Ingeneral, caudal midshafts show large principal compressivestrain aligned with the long axis of the bone, and cranialsurfaces show large tensile principal strains. Peak strainsmeasured during the fastest running were �2000 le in thedog tibia, �2400 le in dog radii, �3200 le in horse tibiaeand �2900 le in horse radii (Rubin and Lanyon, 1982).Using implanted strain gages on a tibia, principal strainsduring human zigzag running (uphill and downhill, respec-tively) were �1226 and �1147 le compressive and 743 and707 le tensile (Burr et al., 1996). In a human tibia, jumpingfrom 1.7 m height produced tensile strain of 2100 le andcompressive strains �840 le (Currey, 2003 citing Hillam,1996).

Crystallite size (c-axis) determined in this study was con-sistent with values reported in the literature; rms strains(microstrains) determined here from 00.2/00.4 peaks weresomewhat lower, however, than other values in the litera-ture. The present values (for 00.l) were t = 41 and 44 nmand erms = 2200 le and 4600 le with no applied loadand the highest load measured, respectively. Neglectingerms yields t� = 33 nm and 28 nm, respectively. Weinerand Wagner (Weiner and Wagner, 1998) summarizeTEM- and SAXS-derived crystallite sizes as 1.5-4.0 nmby �25 nm by �50 nm. X-ray diffraction of desiccated,ground bone (human crista iliaca) showed t increased upto 20% along the c-axis (25 nm for younger bone to35 nm for older) and decreased up to 20% perpendicularto this axis (20–15 nm, respectively); microstrain decreasedfor all reflections (the amount of decrease Æe2æ002 from10,000 le to 6000 le, respectively, is typical) over the rangeof ages studied (Handschin and Stern, 1992, 1995) ; the


crystallite size and microstrain changes were significant atthe p < 0.05 level. Microbeam diffraction of lumbar verte-brae of human foeti (16–26 weeks), of adult bone (33 years)and of synthetic hydroxyapatite (HAP) revealed increasingt 0 with age (note microstrain contributions were neglected)(Dalconi et al., 2003). In human L4 vertebrae (15 weekspostconception to 97 years postnatal), Roschger et al.(Roschger et al., 2001) found structures at the tissue levelcontinued to alter throughout life, but, contrary to whatwas described above, properties at the material level (min-eral density, crystallite dimensions, mineral crystallograph-ic texture) developed essentially during the first years of ageand remained constant thereafter. In thin sections of a hu-man femora (perpendicular to and parallel to Haversiancanal axes, respectively), Rindby et al. (1998) foundt = 50 ± 5 nm and erms = 15,000 le and 25–40 nm and15,000 le, respectively, in transverse cuts; perpendicularto the crystallite c-axes, t � 10–20 nm. With neutron dif-fraction, Bacon et al. (1977, 1979, 1980) determinedt 0 = 10 nm and 33 nm along the a and c directions,respectively.

The FWHM for 22.2 is substantially larger than that of00.4, and the values of t 0 are smaller, as expected from thetabular crystallite geometry and the angles between thenormals to the two planes. Both 22.2 and 00.2 show azi-muthal variations in radial peak broadening (Table 3),and, in both cases, the widths are larger along g = 0�(transverse bone direction) than along g = 90� (longitudi-nal bone direction). Azimuthal variation in FWHM of22.2 is much more pronounced than that of 00.2. It is alsointeresting that the azimuthal variation in radial peakwidth fades with increasing applied stress; the data aretoo limited, however, to assay an explanation. We specu-late that (i) compositional effects (imperfect crystal chemis-try), (ii) different crystallite populations and/or (iii)dislocation effects may have produced the high 22.2/00.1broadening ratios. Studies on metals show certain typesof dislocations give anisotropic broadening of 00h vs allhkl (Klimanek and Kuzel, 1988).

The 66.4 ± 0.5 nm periodicity observed along the fibula�slongitudinal axis after compressive loading (Fig. 7) corre-sponds to the collagen D-period (Weiner and Wagner,1998). As noted by other investigators, SAXS is sensitiveto electron density differences and thus primarily to the min-eral phase in bone, but this mineral phase follows the colla-gen spacing, so SAXS from bone is an indirect measure ofcollagen spacing (Ascenzi et al., 1985). This value can becompared to 67.0 ± 0.3 nm (peaks up to the sixth-order)reported by Nakano et al. (2002) and Roschger et al.(2001) along long bones� axes and 67.6 nm by Kinney et al.(2001) in dentin. The differentD-period observed in the pres-ent work may reflect residual stresses present in vitro, resid-ual stresses from the prior (inelastic) loading of the specimenand/or stresses produced by specimen dehydration. X-rayscattering of in situ loaded tendon (tension) has shown sub-stantial collagen D-period changes, up to several percent(Sasaki and Odajima, 1996; Sasaki et al., 1999; Puxkandl

et al., 2002). Inmineralized collagen like bone, the constraintof the reinforcing phase limits changes inD-period, but it stillmay be possible to observe changes with increased tensileload and to correlate these with strain gage measurements.Recording SAXS patterns during in situ loading of bone willno doubt provide very interesting results.

The elliptical halo at lowest angle in the SAXS pattern(Fig. 7) is attributed to scattering from individual mineralcrystallites. Recalling that the pattern represents dimen-sions in reciprocal space, the distribution indicates thatcrystallite size is, on the average, largest along the bone�slongitudinal axis. This is consistent with the expected mor-phology of the apatite crystallites (rods or plates with theirlongest dimensions parallel to the c-axes) and the strong c-axis texture (nine times more crystallite volume alignedwith c-axes along the bone�s longitudinal axis than alongtransverse directions). The azimuthal dependence of linebroadening in wide angle diffraction patterns (Table 3) alsoindicates smaller crystallite dimensions along transverse vslongitudinal directions, exactly what would be expected forrod- or plate-shaped, intrafibrillar crystals. We note thattaking SAXS patterns at different sample rotations aboutthe vertical direction (i.e., the bone�s longitudinal axis)may allow the in-plane ordering/shape of the particles tobe deduced, but this remains to be done.

Determination of the contribution of specimen dehydra-tion to the present results (residual stress, rms strain, colla-gen D-period) will require further, better controlledexperiments. The observation of a compressive residualstress is interesting, whatever its origin. If dehydrationcan be shown to have no effect on the magnitude of resid-ual stress in the canine fibula, then the results suggest an as-of-yet unconsidered dimension to bone deformation andfracture. If, on the other hand, the residual stress is duesolely to dehydration, then controlled drying of corticalbone samples combined with high energy wide angle dif-fraction and small angle scattering and with loading willprovide an additional avenue for studying collagen-apatiteinteractions.

6. Conclusions and outlook

This high energy X-ray study combined wide angle dif-fraction and small angle scattering of cortical bone. Devia-toric strains (and corresponding stresses) in the mineralphase were measured with transmission diffraction as afunction of applied stress, and the data showed that period-icities in the collagen phase could be quantified (via smallangle scattering) in the same specimen, with the same beamand essentially simultaneously. The measured internalstress in the mineral phase rose nearly three times morerapidly than that applied to the sample. A compressiveresidual (deviatoric) stress was observed in the mineral atzero applied stress, and SAXS revealed a D-period in thecollagen which was slightly smaller than that expected.Whether either or both were produced by sample dehydra-tion requires further investigation.


The results demonstrate that it should be straightfor-ward to measure mineral phase internal strain and the cor-responding stress in situ in intact, artificially loaded limbsor vertebrae. In vivo strain/stress measurements in anesthe-tized animal models should also be practical using theexperimental geometry of this study. It should also be pos-sible to couple the wide angle diffraction studies (providingmineral phase strain) with SAXS observations of changesof collagen D-period spacing with applied stress (providingcollagen phase strain). Observation of a bone as it driedwould illuminate certain aspects of the role of collagen inload transfer between mineral particles. Strain gages couldeasily be attached to the bone gage section, allowing stress–strain curves to be plotted, and use of two or more straingages would allow possible strain gradients in the sampleto be investigated. Such extensions of the present studyshould lead to a better understanding of load sharing inbone as a function of deformation.

Acknowledgments

The authors thank Professor R. Sumner, Rush MedicalCollege, Chicago, for providing the bone sample. The re-search was performed at station 1-ID of XOR-APS. Useof the Advanced Photon Source was supported by theUS Department of Energy, Office of Science, Office of Ba-sic Energy Sciences, under Contract No. W-31-109-Eng-38.Use of the Northwestern University MicroCT Facility�sScanco MicroCT-40 is also acknowledged.

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