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

  1. 1. Internal strains and stresses measured in cortical bone via high-energy X-ray diffraction J.D. Almer a , S.R. Stock b,* a XOR, Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA b 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 2005 Available 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 transverse cross-section of a 12–14 year old beagle fibula was studied with 80.7 keV radiation, and the transmission geometry was used to quantify internal 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 imperfect 00.1 fiber texture expected along the axis of a long bone. Residual compressive stress along the boneÕs longitudinal axis was observed in the specimen prior to testing: for 22.2 this stress equaled À95 MPa and for 00.2/00.4 was between À160 and À240 MPa. Diffraction pat- terns were collected for applied compressive stresses up to À110 MPa, and, up to about À100 MPa, internal stresses rose proportionally with applied stress but at a higher rate, corresponding to stress concentration in the mineral of 2.8 times the stress applied. The widths of the 00.2 and 00.4 diffraction peaks indicated that crystallite size perpendicular to the 00.1 planes increased from t = 41 nm before stress was applied to t = 44 nm at À118 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 collagen D-period of 66.4 nm (along the bone axis). Ó 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 Fratzl et al., 2004; Weiner and Wagner, 1998). The composite pre- serves, to a large degree, the desirable properties of the individual components: high toughness of the bone matrix, collagen fibrils stabilized by water, and high stiffness of the reinforcing phase, crystallites (i.e., single crystal nano- particles) of carbonated apatite (Currey, 2002; Fritton and Rubin, 2001). Mechanical properties and fracture susceptibility differ between healthy bone and bone with impaired functionality (as in osteoporosis), and reliable discrimination between the two remains a challenge, not only in the clinic but also in basic research employing ani- mal models. Factors other than decreased bone mineral density must be invoked to explain predisposition of bone to 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 subtractive processes (Heaney, 2003). Understanding is hindered by inherent variability from position to position in a single bone, over time at the same position in a given bone from one individual, between different bones from the same individual and between the same bone and same position in different individuals. Biewener (1993) described fracture propensity of bone in terms of distributions of bone fracture strength and loading spectra, and, considered in this light, it is clear that it is as important to quantify mechanical input to bone as it is to 1047-8477/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jsb.2005.08.003 * Corresponding author. Fax: +312 503 2544. E-mail address: (S.R. Stock). Journal of Structural Biology 152 (2005) 14–27 Journal of Structural Biology
  2. 2. identify ‘‘weak-link’’ microstructure(s) or changes in global parameters characterizing microarchitecture. Numerous investigators (Fritton and Rubin, 2001) have quantified this mechanical input in vivo with strain gages attached to cortical bone, typically at mid-diaphyses where installa- tion can be done with minimal disruption to muscles and tendons. With attached strain gages, however, the mechan- ical response of volumes beneath the boneÕs surface can only be inferred indirectly. This limitation hinders under- standing of effects such as that of mechanical input on bone remodeling: some sort of mechanotransduction must trans- late the forces applied to the whole bone to instructions at the cellular level directing adaptation of the bone (Knothe, 2001). It is surprising, therefore, that, except for (Borsato and Sasaki, 1997), the authors of this paper have found no re- ports in the literature on the use of the nanometer-sized strain gages intrinsic to bone, namely the unit cell of the mineral (carbonated apatite) reinforcement of the compos- ite material bone. That is not to say that X-ray scattering has not been applied to bone (see Section 2), but rather that X-ray diffraction measurement of the distortion of unit cells by applied loads (and its conversion to stress), a standard subject in undergraduate textbooks used in materials sci- ence (Cullity and Stock, 2001), has not been widely applied. Therefore, the study reported below examined whether stresses in the mineral phase of bone could be quantified as a function of applied load using transmission, high-ener- gy synchrotron X-ray diffraction. Unlike (Borsato and Sasaki, 1997), which examined thin samples cut from larger bones and measured stress only within 100–200 lm of the surface, the present work probed the entire, unaltered bone cross-section. The characteristics of synchrotron X-radia- tion and the general principles of stress measurement by X-ray diffraction are briefly introduced to provide back- ground for the specifics of the transmission synchrotron scattering geometry and the bone loading experiment. The details of analysis are presented in conjunction with the 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-energy X-ray absorptometry (DEXA), for example, is widely used to assess patientsÕ bone mineral loss in osteoporosis (Bolo- tin, 2001). X-ray CT (computed tomography) whole body scanners are present in most hospitals, and peripheral quantitative CT scanners enjoy increasing use in clinical re- search. Microarchitecture of bones is more and more fre- quently quantified using laboratory microCT systems (see Majumdar and Bay, 2001). Commercial availability of in vi- vo scanners (Scanco Medical; Skyscan) allows increasing use of microCT in longitudinal studies of individual ani- mals before, during and after treatment; while such studies were performed previously at synchrotron radiation sources (Kinney et al., 1995, 2000), Herculean effort was required. In the life sciences, X-ray diffractionÕs familiar avatar is structure determination of single-crystal biological macro- molecules such as proteins and viruses. A very different X-ray diffraction method is required to study the discontin- uous reinforcement phase in bone (carbonated apatite): polycrystalline X-ray diffraction techniques developed for materials engineering. These polycrystalline techniques are relatively unfamiliar outside of the physical sciences and engineering and require treatment before the specifics of high-energy X-ray diffraction in the transmission setting and of stress determination can be reviewed. 2.1. X-ray diffraction The well-known equation k = 2dhkl sinh (BraggÕs 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 a polycrystalline sample (here carbonated apatite crystallites dispersed in a water-stabilized collagen matrix), only a very small fraction of the crystallites are oriented to diffract for a given incident beam orientation. If one employs a suit- ably collimated monochromatic X-ray beam, the properly oriented crystallites form a Debye cone of diffracted inten- sity. Multiple hkl diffract simultaneously, and each of the cones is from a different subset of crystallites. In the trans- mission geometry used in this study, an area detector is placed normal to the beam transmitted through the sample, and a circular ring of diffracted intensity is recorded where the detector intercepts each cone of diffracted radiation. If all 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 the rings 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 crystalline sample (here a specimen containing discrete single-crystal nanoparticles of carbonated apatite in a collagen matrix), the unit cells, however they are oriented, compress along the 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 produces elliptical diffraction rings: the circular rings of the un- stressed sample expand along the applied compressive stress axis y (azimuthal angles g = 90° and 270° in the detector plane) and contract normal to this direction, i.e., along x (azimuthal angles g = 0° and 180°). Quantifying the amount of diffraction ring distortion is the physical ba- sis of X-ray diffraction stress analysis. The measurements require very precise determination of changes in d-spacing, i.e., in the ring radii as a function of g; see (Almer et al., 2003) for details. If one considers the J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 15
  3. 3. magnitudes of stresses applied and of YoungÕs modulus (for purposes of illustration, take a low value, ry = 50 MPa, for the yield stress for compact bone and a typical value E = 20 GPa, Currey, 1990), the relationship r = Ee shows that ey $ 2.5 · 10À3 . The entire elliptical diffraction ring 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 phase to stresses necessitates more detailed consideration of the elastic constants than was employed above. Stress rij and strain ekl are second rank tensors related through the fourth rank elastic constants Cijkl, i.e., rij = Cijklekl. For a single crystal, numerical calculation of stress components from measured strains is straightfor- ward. For data obtained from crystallites with different ori- entations (i.e., a polycrystalline specimen), the strains measured by X-rays are averages, and to determine an average stress from these strains requires average elastic constants derived from Cijkl according to one of several approximations. The Reuss approximation assumes that all crystallites experience the same stress; the Voigt approx- imation assumes all grains within the sample are subjected to the same uniform strain. The Kro¨ner–Eshelby limit, cal- culated for anisotropic precipitates coupled to an isotropic matrix, yields values of elastic moduli close to those ob- served experimentally, that is, near the mean of the Reuss and Voigt limits (Noyan and Cohen, 1987). Table 1 gives elastic constants from the literature for a crystal of hydroxyapatite (Gardner et al., 1992). Note that Cijkl are listed using the abbreviated notation cmn = Cijkl (m,n = 1,. . .,6; i,j,k,l = 1,2,3, see (Nye, 1976)). These cij values were then used to calculate average X-ray elastic constants S1(hk.l), S2/2(hk.l) using the Voigt, Reuss, and Kro¨ner–Eshelby models (Computer program Hauk; Hauk, 1997) using definitions S1 = (Àm/E)average and S2/2 = ((1 + m)/E)average (Noyan and Cohen, 1987). All models indicate that the c-axis of the hexagonal unit cell is the elastically stiffest direction, with maximum deviations of 20% in different crystallographic directions. This indi- cates that hydroxyapatite is only weakly elastically aniso- tropic, in agreement with results from the fluoroapatite phase (Sha et al., 1994). Measured strains exx and eyy (where x and y are along the horizontal and vertical axes of the 2D diffraction pattern, respectively, and the X-ray beam is along z) are used to calculate internal stresses along the loading (i.e., y) direction using (Hauk, 1997): ryy ¼ 1 S2=2 eyy À S1 S2=2 À 3S1 eyy þ 2exx À Á ! ; ð1Þ where the transverse strains exx and ezz are assumed to be equal. The change of notation (exx for e11, etc.) is made to emphasize the coordinate axes of the detector and specimen. 2.3. High energy diffraction with synchrotron X-radiation Brilliant hard X-ray beams available at third generation synchrotron radiation sources offer important advantages for many scientific and engineering applications. Shorter wavelength X-rays allow thicker samples to be studied non- invasively: 80 keV X-rays have $40% transmissivity through 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 Ewald sphere, i.e., to the geometry of diffraction in reciprocal space. The shorter the wavelength of the diffracting radia- tion, the greater the Ewald sphere radius and the more closely the Ewald sphere approximates a plane of the reci- procal lattice. For 100 keV electrons used in transmission electron microscopy (TEM) imaging of materials, the Ewald sphere radius is 27.0 A˚ À1 , and the diffraction pat- tern, at least near 000 appears, to be a plane through the reciprocal lattice (calcite d104 = 3.00 A˚ and reciprocal lat- tice period of 0.33 A˚ À1 ). Diffraction with 0.154 A˚ X-rays (6.5 A˚ À1 Ewald sphere radius) is closer to the TEM case than that of Cu Ka (the most commonly used wavelength with X-ray tube diffraction, 1.54 A˚ wavelength, 0.65 A˚ À1 Ewald sphere radius) (Stock et al., 2003). Therefore, more reflections are recorded simultaneously with high energy X-ray diffraction than with lower energies. The use of an area detector and forward scattering geometry 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 additional sources of error) is required to measure different crystal- lite orientations and where the orientation range is more limited. Table 1 Elastic constants (in GPa) from (Gardner et al., 1992) converted to X-ray elastic compliances S1 and S2/2 (in 10À6 MPaÀ1 ) for hydroxyapatite (c/a = 0.73) for Voigt, Reuss and Kro¨ner–Eshelby models Elastic stiffness coefficients: C11 C12 C13 C33 C44 137.0 42.5 54.9 172.0 39.6 hk.l Voigt Reuss Kro¨ner–Eshelby S1 S2/2 S1 S2/2 S1 S2/2 00.2/00.4 À2.38 11.14 À4.64 3.49 À2.27 10.2 22.2 À2.38 11.14 À2.86 10.0 À2.48 11.5 16 J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27
  4. 4. 2.4. Diffraction studies of bone X-ray and neutron scattering have contributed to under- standing of the mineral and matrix phases of bone. Several investigators 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/a ratio 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 the mineral 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 and Heidelbach, 1999) and with neutrons (Bacon et al., 1977, 1979, 1980), and preferred orientation of crystallite c-axes was interpreted in terms of principal applied stress axes in the structure. X-ray microbeam mapping has also been applied to bone using wide angle diffraction (Nakano et al., 2002; Rindby et al., 1998; Wenk and Heidelbach, 1999) and small angle X-ray scattering (SAXS) (Roschger et al., 2001; Kinney et al., 2001; Zizak et al., 2003) in order to study spatial variation of crystallite orientation. The 66–67 A˚ collagen axial periodicity has also been measured in bone samples using SAXS (Ascenzi et al., 1978, 1998; Kinney et al., 2001). Except for the neutron diffraction studies, the studies cited above (as well as the single prior X-ray diffraction stress measurement study (Borsato and Sasaki, 1997)) were performed on thin sections of bone. Those familiar only with the ubiquitous copper-target tube as sources of X-radiation might not realize that, as is indi- cated in Section 2.3, X-ray diffraction can be performed through entire bone cross-sections. 3. Materials and methods The sample used for the study described was cut from the mid-diaphysis of a canine fibula; the beagle was be- tween 12 and 14 years old. Two parallel surfaces 5.0 mm apart were cut from the fibula using a Buehler Isomet saw and a diamond wafering blade. The cuts were as close as possible to perpendicular to the long axis of the bone. The microCT slice in Fig. 1 (Scanco MicroCT-40 (Scanco Medical) operated at 45 kVp, data collected in the highest sensitivity 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 porosity and appeared to consist entirely of compact bone; the mi- croCT systemÕs 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-driven load frame, and applied load was determined with a 2000 lb. load cell, located below the specimen. Hardened steel platens (5 mm thickness, 1 cm diameter) were placed on either side of the bone specimen, and a hemispherical steel ball was placed between the drive screw and the upper platen for alignment. The sample was loaded stepwise in displacement control, and diffraction patterns were collected after the load gage revealed only gradual stress relaxation. During the exposures, the applied stress relaxed no more than 15%, and applied stress values given below are mean values during this minor relaxation. In this proof-of-principle experiment the bone was not irrigated with water, so some drying must have occurred. Transmission X-ray diffraction was performed at station 1-ID of the Advanced Photon Source, Argonne National Laboratory (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 beam passed through the sample (as measured with a PIN diode embedded in the tungsten beamstop after the sample). A MAR 345 image plate detector recorded the wide-angle dif- fraction patterns out to a d-spacing of 1 A˚ . The detector was perpendicular to the X-ray beam and was centered with respect to the beam stop. The detector area was 345 mm in diameter, and it was read with 150 lm pixels, with a readout time of $80 s. Exposures were recorded for 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 powder sample of ceria (CeO2, NIST Standard Reference Material SRM-674a, 1 mm diameter glass capillary tube) was used to correct for detector tilts, determine the sample-detector separation (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 the canine fibula was anticipated to be in the range of 50– 200 MPa. Diffraction patterns were recorded at 0, À14.5, À36.7, À94.7, and À118 MPa applied stress (r < 0 indi- cates compression). The 2D diffraction patterns record Fig. 1. MicroCT slice of the bone oriented transverse to the loading direction (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 absorbing than darker voxels. The arrow is parallel to the incident beam direction. J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 17
  5. 5. diffracted intensity I(r,g), where r is the magnitude of the vector between the incident beam position on the detector and the pixel of interest and g is the azimuthal angle be- tween r and the horizontal reference direction (see Fig. 2). Using the programs FIT2D (Hammersley, 1998; Hammersley et al., 1994a,b, 1996) and MATLAB (Math- works and MATLAB), the patterns were converted from polar into Cartesian coordinates (5° azimuthal and 150 lm radial bins), producing 72 1D radial patterns. Plots of the integrated intensity of the 00.2, 00.4, and 22.2 reflec- tions as a function of azimuthal angle (see Fig. 3) revealed the apatite texture. For comparison with reference patterns of hydroxyapatite, the Cartesian representation was trans- formed into a 1D diffraction pattern I(d) using BraggÕs law and azimuthal integration. The diffraction peaks in the 72 1D radial patterns were fit with pseudo-Voigt functions to provide radial values rg that were converted into absolute lattice plane spacings dg using geometric values determined from the (known) ceria d-spacings. Estimates of absolute errors in dg are be- low 10À4 for the experimental set-up (Almer et al., 2003). A key feature in the strain analysis was comparison of radius 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 a single radius, the invariant radius r*, and the correspond- ing orientation value was g* (see Fig. 4). Measured radii were referenced to this r* value to provide orientation-de- pendent (deviatoric) strain values: eg = (rg À r*)/r*. These strain profiles eg were fit to a biaxial strain model (He Fig. 2. (A) Two-dimensional diffraction pattern I (r,g): the darker the pixel the higher the diffracted intensity. The dashed line shows the orientation of g = 0° (g increasing clockwise), the 00.2 reflection is labeled and the faint white shadows running near g = 90° are caused by the electrical wires running to/ from the beam stop detector. (B) Cartesian plot of radius vs azimuth for the diffraction pattern in (A) and using the same gray scale as in (A). (C) Azimuthally integrated diffraction pattern converted to I (d) from (B) and compared with a reference pattern for synthetic hydroxyapatite (Ca5(PO4)3OH, ICDD Powder Diffraction File pattern 09-432). The vertical axis is in units of square root of intensity, and the height of the vertical lines indicates the intensity of each reflection identified by indices hkl. 18 J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27
  6. 6. and Smith, 1998) to account for sample geometry, and pro- vided values of the deviatoric strain components exx and eyy. These strain components, along with the X-ray elastic constants, were then used to calculate stress along the load- ing direction, ryy, via Eq. (1). Strictly speaking, the r* values are not necessarily equal to the strain-free value ro due to the potential presence of hydrostatic strains. This hydrostatic contribution can be determined 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 bone deformation. 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 can be 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 width at half maximum (FWHM) of each peak and consisted of an instrumental contribution DRinst as well as a contribu- tion DR from the crystallites themselves. DRinst for different peaks were determined using the ceria standard. Assuming Gaussian peak shapes, the mineral phaseÕs peak widths were DR ¼ p ðDR2 meas À DR2 instÞ, 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 rms strain (erms) contribute to peak width according to (Noyan and Cohen, 1987) D2h cos h ¼ 180 p k D þ 2 Dd d 2 * +1=2 sin h 2 4 3 5; ð2Þ where Æxæ denotes the mean of x and erms is represented by Æ(Dd/d)2 æ1/2 . Note that the first term inside the square brackets is the Scherrer equation and erms is sometimes termed microstrain (Cullity and Stock, 2001). Small angle scattering provides data on collagen period- icities and on mineral crystallite shapes and dimensions in bone (Fratzl et al., 1992, 1996). In order to record SAXS from the bone specimen, the MAR 345 detector (near the middle of the hutch) was translated out of the X-ray beam and 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 was recorded after the specimen was unloaded, and the expo- sure was 10 s. The resulting 2D pattern was integrated about g = 270° ± 10° to obtain the projection along the vertical (loading) direction and about g = 180° ± 10° to obtain the variation of scattered intensity normal to the boneÕs 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 azimuth as the ordinate. Several features are readily apparent in the transformed data. First, the intensity for a given reflec- tion (three of those used later in this work are identified) is not constant as a function of azimuth but varies in a sys- tematic manner, which indicates crystallographic texture. Fig. 4. Measured radial peak position vs azimuth for the 00.4 reflection as a function of applied compressive stress. The applied stress increased from 0 (A), to À14.5 (B), to À36.7 (C), to À94.7 (D) to À118 MPa (E), see Fig. 5. The invariant radial position r* (horizontal line between positions 577 and 577.5) and corresponding orientation g* are indicated (see text for definition of these two quantities). Fig. 3. Variation of peak intensity (normalized to one) vs azimuth for the 00.2, 22.2, and 00.4 peaks. The 00.2 and 00.4 variations are nearly identical, as expected, and indicate preferential alignment of the crystallite c-axis with the load axis (at g = 90° and 270°). Outliers at g = 90° arise from the beamstop wires (see Fig. 2). 1 This relationship is only valid in the small h limit, sush as in this study. J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 19
  7. 7. 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 the radius converted to d using experimental parameters (wavelength and sample-detector separation) in order to produce 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 parameters a = 9.418 A˚ and c = 6.884 A˚ . Fig. 3 plots the normalized intensities of 00.2, 00.4, and 22.2 reflections. The azimuthal variation of the 00.2 and 00.4 peak intensities is virtually identical (as expected) be- cause these are second and fourth order reflections of Bragg 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 the bone (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 azimuthal intensity distribution (compared to two for 00.2/00.4) are centered at angles 38°, 140°, 217°, and 323°, which is a con- sequence of the broad 00.1 fiber texture (see the second paragraph of Section 5). The azimuthal full-width at half maximum (FWHM) intensity of each reflection is a measure of the sharpness of the texture. For 00.2, FWHM were between 57° and 59°, 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 more crystallite volume had c-axes parallel to the boneÕs longitu- dinal axis than perpendicular to it. For 22.2, the ratio of maximum to minimum azimuthal intensity was much smaller, namely, 1.3. The measured radial peak position (in detector pixels) for the 00.4 reflection is plotted as a function of azimuth for 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 the 00.4 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, even under zero applied stress, compressive residual strain exists along the vertical/loading (y) direction and tensile residual strain along the in-plane (x) direction. As applied load increases, the ellipses deviate further from circularity (Curves B–E in Fig. 4), demonstrating larger strains are being produced. Table 2 lists applied stress, transverse strain exx and longitudinal strain eyy in the mineral particles and ryy cal- culated for the Reuss, Voigt, and Kro¨ner–Eshelby approx- imations. Note that strains are in units of le (microstrain, that is, in 10À6 strain). For 00.2, 570 le exx 2410 le and À1440 le eyy À6050 le, and 00.4 gives similar magni- tudes as expected. The strain magnitudes from 22.2 are smaller 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 calculated from the measured strains and the Kro¨ner–Eshelby model. The residual stress in the mineral, that is, the internal stress measured with zero applied stress, was À95 MPa for 22.2 (i.e., along the direction perpendicular to {22.2} in the crys- tallites) and was between À160 and À230 MPa for 00.1 (i.e., perpendicular to {00.1}). As expected, the 00.2 and 00.4 data in Fig. 5 were nearly coincident. Over much of the applied stress range, internal stress in the mineral crys- tallites rose linearly with applied stress. In analogy with Borsato and Sasaki (1997), the stress concentration factor X (defined as the stress in the mineral divided by the stress in 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 of X = 2.8 (dotted line) fits both the 22.2 and 00.2/00.4 reflec- tions for most of the loading curve. The linear relationship breaks down, however, above À94.7 MPa applied stress: the internal stress in the mineral particles rises at a greater rate. Fig. 6 shows experimental radial peak widths DR for the 00.2, 22.2, and 00.4 diffraction peaks as a functional of azi- muthal angle g. These peak widths were corrected for instrumental broadening DRinst, and the plots show DR(g) for the unloaded state (A), the intermediate stress (C, À36.7 MPa) and the maximum compressive stress (E, À118 MPa). In the unloaded condition, considerable anisotropy is seen for 00.2 and 22.2 (data are too incom- plete to draw a conclusion for 00.4). For 00.2, the radial peak 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° than along g = 0°, 180° (Table 3). At the maximum load, the dif- ference for 00.2 decreased to about one-half pixel and for 22.2 to one pixel (Table 3). Separation of rms strain and crystallite size contribu- tions to broadening requires values of peak width for two orders of the same hk.l. (Alternatively, approaches such as 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 size t was 41 nm without load vs 44 nm at À118 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 was attributed to crystallite size, these values t0 are depressed 20% or more from those in the more realistic determination including erms (Table 3). For 22.2, t0 = 15 nm at g = 0° and t0 = 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. There were two distinct peaks along the bone longitudinal direc- tion (indicated by arrows superimposed on the 2D pattern); 20 J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27
  8. 8. this is shown numerically in the plot of intensity I plotted vs q (Fig. 7, right), where q = 2p/d. Therefore, these peaks corresponded to the first- and third-orders of periodicity equal to 66.4 ± 0.5 nm. The second order peak was not ob- served because its structure factor is much smaller than that of the third-order peak (Ascenzi et al., 1985). The first-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 from the 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 agreed well with that from the powder diffraction file, confirming that the crystalline portion of the bone was carbonated apatite. While structure refinement with the Rietveld meth- od (a` la Lonardelli et al., 2005) could have been performed to provide more precise lattice parameters, etc., this was not done because the present studyÕs primary goal was investigating peak position changes (i.e., internal stresses) vs applied stress. The azimuthal FWHM of 00.2 and 00.4 averaged nearly 60° and indicated the spread in texture (Fig. 3). The much smaller strength of the 22.2 texture (ratio of maximum to minimum azimuthal intensity equaled 1.3) compared to the 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 stereographic projection, i.e., at the correct relative azimuthal positions and correct 2h (data not shown). The longitudinal bone axis was plotted at the Wulff netÕs north pole, and the en- trance and exit beams corresponded to directions at or near, respectively, the center of the net. The locus of 11.1 pole directions was those points 55° from 00.1, and the 11.1 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 poles corresponding to the half maximum intensities in the 22.2 azimuthal plot lay within the FWHM of the distribution Table 2 Experimental 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 rA a exx eyy rKE yy rR yy rV yy 00.2 0 570 À1440 À160 À380 À140 À14.5 850 À2140 À240 À560 À210 À36.7 1190 À2990 À330 À780 À290 À94.5 1850 À4630 À520 À1210 À450 À117.7 2410 À6050 À673 À1580 À590 22.2 0 340 À850 À94 À220 À80 À14.5 480 À1200 À130 À310 À120 À36.7 700 À1770 À200 À460 À170 À94.5 1300 À3260 À360 À850 À320 À117.7 1740 À4360 À490 À1140 À420 00.4 0 760 À1920 À210 À500 À190 À14.5 960 À2400 À270 À630 À230 À36.7 1160 À2920 À320 À760 À280 À94.5 1730 À4350 À480 À1130 À420 À117.7 2216 À5560 À620 À1450 À540 Longitudinal stresses rn yy corresponding to the observed strains are also given: the superscript n indicates the elastic constants model used to relate stress and strain, with R denoting the Reuss limit, V the Voigt limit, and KE the Kro¨ner–Eshelby approximation. Strains are given in le, and stresses in MPa. a Curves (A)–(E) in Fig. 4 correspond to applied stresses 0 to À117.7 MPa, respectively. Fig. 5. Internal stress in the carbonated apatite mineral phase calculated using measured strains (Table 2) and the Kro¨ner–Eshelby model vs applied stress for three reflections. Dotted lines show two stress concen- tration factors (1 = no stress concentration, 2.8 = best fit for 22.2) and symbols A–E label the loads corresponding to the data shown in Fig. 4. Uncertainties in internal stress values approximately equal the symbol sizes. J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 21
  9. 9. 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 along the axis of this long bone agreed with the bulk of the data in the literature. Bacon et al. (1977, 1979, 1980) examined three types of bovine bone: the scapula, lumbar vertebra and mandible. The c-axes of apatite crystallites lay along the 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 direction along which attached muscles pulled). Nightingale and Lewis (1971) and Sasaki and Sudoh (1997) both examined bovine femora and found the c-axes were biaxially orient- ed: the larger subset with c parallel to the bone axis and a 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 boneÕs surface. Wenk and Heidelbach (1999) found the apatite c-axes in bovine metacarpal cortical bone had rotation freedom within the plane parallel to the cortical surface, this difference from other long bone observations being due perhaps to com- plex loading experienced by this bone in vitro (Martin et al., 1998). The mineral particles in human L4 vertebrae were oriented perpendicular to the outer shell in growth Fig. 6. Instrumentally-corrected radial peak width vs azimuthal orientation g at three applied stresses (A, unloaded; C = À36.7 MPa, E = À118 MPa) for reflections (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 over the angular ranges 135° to 225° and 315° to 45°. Table 3 Radial 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 3.6 00.4 2.4 577.2 4.1 — 3.5 — 4.8 22.2 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 44 erms (le) 2200 4600 00.2 t0 (nm) 33 28 22.2 t0 (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 values determined from peak broadening along the vertical direction (g = 90°, 270°) are shown (i.e., 00.2/00.4 analysis) as well as t0 calculated assuming rms strain is not present (00.2 and 22.2 analyses). 22 J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27
  10. 10. cartilage while they were oriented parallel to this surface in the cortical bone shell, this presumably reflecting the colla- gen structural differences between cartilage and bone (Ros- chger et al., 2001). Rindby et al. (1998) performed microbeam diffraction of thin sections of human femora and found two populations of crystallites: one set with c- axes parallel to Haversian canals and one set tangential to the canals. Ascenzi et al. (1979) in studies of individual osteons found that the number of longitudinally oriented crystallites (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 aligned perpendicular to the direction of the trabeculae (Roschger et al., 2001; Fratzl et al., 1997; Rinnarthaler et al., 1999; Jaschouz et al., 2003). The combination of scanning SAXS and wide angle diffraction showed a fiber texture exists within the trabeculae, and the plate-shaped nanoparticles are 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 residual internal deviatoric stresses were present in the mineral phase (Fig. 5). The difference in internal stress ryy of 22.2 and 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 any X-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 of osteon lamellae to show that such residual stresses exist in osteons. Recently Ascenzi (1999) converted the lamellae distortions, using the well-known Volterra dislocation equations, 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 lamellae to a net macrostress in mineral in a bone cross-section, but this link is quite plausible. The stresses, residual or applied, within a volume of a specimen must satisfy the equations of mechanical equilib- rium. The existence of compressive residual stresses in the mineral that are on the order of À200 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 in collagen, stresses required to balance that in the mineral phase. One might view this latter point as a significant inconsistency in the analysis presented in Section 4. Set aside for the moment the question of whether mechanical properties of monolithic collagen tissues (i.e., non-mineral- ized tendon) apply to collagen in bone. Consider first an engineered composite whose components have well-defined properties and whether the notion of equal-and-opposite stresses 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. The deformation/stress levels in the composite matrix are well known to differ considerably from what is typical of the matrix 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 tensile stresses in excess of 1400 MPa despite the fact that the monolithic matrix (aluminum alloy AA 6061-0) would have yielded at about 85 MPa and have had an ultimate tensile strength of about 155 MPa (Boyer, 1985). There- fore, if the notion of equal-and-opposite stresses applied, the compositeÕs 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. 10 100 1000 10000 0.05 0.15 0.25 0.35 0.45 q (1/A) Intensity(arb.units) 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 located behind 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., the boneÕs 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 integrated along 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. J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 23
  11. 11. That the mechanical behavior of collagen in bone differs from that in unmineralized collagen was postulated by McCutchen (1975). Landis et al. (1995) provided direct evi- dence of the aforementioned difference: they observed that the characteristic low modulus toe region (preceding the linear portion of the load-elongation curve in unmineral- ized turkey tendon) disappeared when samples containing mineral were tested. The very process of nucleation and growth 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 changed mechanical behavior of collagen in bone has been explicitly acknowledged in the Ja¨ger–Fratzl mechanical model of bone (Ja¨ger and Fratzl, 2000; Gao et al., 2003). Under an applied tensile load, the mineral platelets carry the tensile load with the protein matrix transferring the load between mineral crystallites via shear. In essence, the large aspect ratio of the mineral platelets compensates for the low mod- ulus of the protein phase, and the nanometer size of the mineral platelets in bone may be the result of fracture strength optimization. The very intimate relationship be- tween mineral crystallite and its collagen substrate (TEM investigations 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 present study was 2.8, somewhat larger than the value of 2.2 ± 0.1 found in Borsato and Sasaki (1997). Borsato and Sasaki (1997) used a laboratory h/2h diffractometer and in situ uniaxial tensile loading of a bone wafer to mea- sure peak shifts of the apatite 00.4 reflection as a function of applied stress. Differences in loading (here compression, tensile loading in Borsato and Sasaki (1997)) may be the reason for the difference: unequal compressive and tensile responses 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 data reflect an average over 1–2 mm of sample thickness while (Borsato and Sasaki, 1997) reported results from material within 0.1 mm of the surface. Both studies indicate, howev- er, that the mineral phase carries a higher fraction of the applied load during deformation. This is, of course, consis- tent with the mineral phase being stiffer than the collagen phase 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 strain gage attached to the bone) was linear up to the maximum applied 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 À94.7 MPa (in agreement with the results of (Borsato and Sasaki, 1997)) and rose more rapidly between À94.7 and À118 MPa applied compressive stress. Values of the yield 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 À4030 to À6050 le upon increasing applied stress from À94.7 to À118 MPa. Compact bone fails in longitudinal compression at strains as high as À14,000 to À21,000 le and begins to yield at À6000 to À8000 le (Biewener, 1993). Therefore, one is inclined to interpret the increasing rise in mineral phase internal stress between À94 and À118 MPa as evidence of yielding in the bone. In order to firmly establish this, additional specimens need to be tested with diffraction patterns recorded at smaller stress increments. Further, a strain gage should be attached to the bone so that the macroscopic stress–strain relationship can be recorded for correlation with the mineral phase internal stress. Use of strain gages cemented to bone is a standard procedure (Fritton and Rubin, 2001), and such strain gages are transparent to X-rays of the energy used in this study. The magnitude of the internal strains eyy and exx in the mineral crystallites (00.2, 00.4, and 22.2 reflections, see Table 2) were consistent with strains measured in vivo with strain gages during vigorous activity. In various mammals and avians, in vivo peak principal compressive strains at the surfaces of cortical bone range between À1700 and À2500 le during strenuous exercise (Biewener, 1993). In general, caudal midshafts show large principal compressive strain aligned with the long axis of the bone, and cranial surfaces show large tensile principal strains. Peak strains measured during the fastest running were À2000 le in the dog tibia, À2400 le in dog radii, À3200 le in horse tibiae and À2900 le in horse radii (Rubin and Lanyon, 1982). Using implanted strain gages on a tibia, principal strains during human zigzag running (uphill and downhill, respec- tively) were À1226 and À1147 le compressive and 743 and 707 le tensile (Burr et al., 1996). In a human tibia, jumping from 1.7 m height produced tensile strain of 2100 le and compressive strains À840 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 were somewhat lower, however, than other values in the litera- ture. The present values (for 00.l) were t = 41 and 44 nm and erms = 2200 le and 4600 le with no applied load and the highest load measured, respectively. Neglecting erms yields tÕ = 33 nm and 28 nm, respectively. Weiner and Wagner (Weiner and Wagner, 1998) summarize TEM- and SAXS-derived crystallite sizes as 1.5-4.0 nm by $25 nm by $50 nm. X-ray diffraction of desiccated, ground bone (human crista iliaca) showed t increased up to 20% along the c-axis (25 nm for younger bone to 35 nm for older) and decreased up to 20% perpendicular to this axis (20–15 nm, respectively); microstrain decreased for all reflections (the amount of decrease Æe2 æ002 from 10,000 le to 6000 le, respectively, is typical) over the range of ages studied (Handschin and Stern, 1992, 1995) ; the 24 J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27
  12. 12. crystallite size and microstrain changes were significant at the 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 increasing t0 with age (note microstrain contributions were neglected) (Dalconi et al., 2003). In human L4 vertebrae (15 weeks postconception to 97 years postnatal), Roschger et al. (Roschger et al., 2001) found structures at the tissue level continued to alter throughout life, but, contrary to what was described above, properties at the material level (min- eral density, crystallite dimensions, mineral crystallograph- ic texture) developed essentially during the first years of age and remained constant thereafter. In thin sections of a hu- man femora (perpendicular to and parallel to Haversian canal axes, respectively), Rindby et al. (1998) found t = 50 ± 5 nm and erms = 15,000 le and 25–40 nm and 15,000 le, respectively, in transverse cuts; perpendicular to the crystallite c-axes, t $ 10–20 nm. With neutron dif- fraction, Bacon et al. (1977, 1979, 1980) determined t0 = 10 nm and 33 nm along the a and c directions, respectively. The FWHM for 22.2 is substantially larger than that of 00.4, and the values of t0 are smaller, as expected from the tabular crystallite geometry and the angles between the normals 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 of 22.2 is much more pronounced than that of 00.2. It is also interesting that the azimuthal variation in radial peak width fades with increasing applied stress; the data are too 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 22.2/00.1 broadening ratios. Studies on metals show certain types of dislocations give anisotropic broadening of 00h vs all hkl (Klimanek and Kuzel, 1988). The 66.4 ± 0.5 nm periodicity observed along the fibulaÕs longitudinal axis after compressive loading (Fig. 7) corre- sponds to the collagen D-period (Weiner and Wagner, 1998). As noted by other investigators, SAXS is sensitive to 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 of collagen spacing (Ascenzi et al., 1985). This value can be compared 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 bonesÕ axes and 67.6 nm by Kinney et al. (2001) in dentin. The different D-period observed in the pres- ent work may reflect residual stresses present in vitro, resid- ual stresses from the prior (inelastic) loading of the specimen and/or stresses produced by specimen dehydration. X-ray scattering 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). In mineralized collagen like bone, the constraint of the reinforcing phase limits changes in D-period, but it still may be possible to observe changes with increased tensile load and to correlate these with strain gage measurements. Recording SAXS patterns during in situ loading of bone will no doubt provide very interesting results. The elliptical halo at lowest angle in the SAXS pattern (Fig. 7) is attributed to scattering from individual mineral crystallites. Recalling that the pattern represents dimen- sions in reciprocal space, the distribution indicates that crystallite size is, on the average, largest along the boneÕs longitudinal axis. This is consistent with the expected mor- phology of the apatite crystallites (rods or plates with their longest dimensions parallel to the c-axes) and the strong c- axis texture (nine times more crystallite volume aligned with c-axes along the boneÕs longitudinal axis than along transverse directions). The azimuthal dependence of line broadening in wide angle diffraction patterns (Table 3) also indicates smaller crystallite dimensions along transverse vs longitudinal directions, exactly what would be expected for rod- or plate-shaped, intrafibrillar crystals. We note that taking SAXS patterns at different sample rotations about the vertical direction (i.e., the boneÕs longitudinal axis) may allow the in-plane ordering/shape of the particles to be 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 controlled experiments. The observation of a compressive residual stress is interesting, whatever its origin. If dehydration can 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 and fracture. If, on the other hand, the residual stress is due solely to dehydration, then controlled drying of cortical bone samples combined with high energy wide angle dif- fraction and small angle scattering and with loading will provide an additional avenue for studying collagen-apatite interactions. 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 mineral phase were measured with transmission diffraction as a function of applied stress, and the data showed that period- icities in the collagen phase could be quantified (via small angle scattering) in the same specimen, with the same beam and essentially simultaneously. The measured internal stress in the mineral phase rose nearly three times more rapidly than that applied to the sample. A compressive residual (deviatoric) stress was observed in the mineral at zero applied stress, and SAXS revealed a D-period in the collagen which was slightly smaller than that expected. Whether either or both were produced by sample dehydra- tion requires further investigation. J.D. Almer, S.R. Stock / Journal of Structural Biology 152 (2005) 14–27 25
  13. 13. 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 limbs or vertebrae. In vivo strain/stress measurements in anesthe- tized animal models should also be practical using the experimental geometry of this study. It should also be pos- sible to couple the wide angle diffraction studies (providing mineral phase strain) with SAXS observations of changes of collagen D-period spacing with applied stress (providing collagen phase strain). Observation of a bone as it dried would illuminate certain aspects of the role of collagen in load transfer between mineral particles. Strain gages could easily be attached to the bone gage section, allowing stress– strain curves to be plotted, and use of two or more strain gages would allow possible strain gradients in the sample to be investigated. Such extensions of the present study should lead to a better understanding of load sharing in bone as a function of deformation. Acknowledgments The authors thank Professor R. Sumner, Rush Medical College, Chicago, for providing the bone sample. The re- search was performed at station 1-ID of XOR-APS. Use of the Advanced Photon Source was supported by the US 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 FacilityÕs Scanco MicroCT-40 is also acknowledged. References Almer, J., Lienert, U., Peng, R.L., Schlauer, C., Ode´n, M., 2003. Strain and texture analysis of coatings using high-energy X-rays. J. Appl. Phys. 94, 697–702. Ascenzi, M.G., 1999. A first estimation of prestress in so-called circularly fibered osteonic lamellae. J. Biomech. 32, 935–942. Ascenzi, A., Benvenuti, A., 1977. Evidence of a state of initial stress in osteonic lamellae. J. Biomech. 10, 447–453. 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