(Lebedev et al.,2003). For bovine cortical bone,Lee et al.(2002) measuredQin the range 10-30; however they did not use any particular method to resolve overlapped peaks and therefore were unable to retrieve more than three resonant frequencies.

In this work we take advantage of recent developments of RUS to overcome the difficulties associated to the high damping of bone, which have until now prevented the measurement of the stiffness tensor of bone samples with RUS. These develop-ments include the processing method for the resolution of overlapping peaks already mentioned, which is completed here by a non-linear least-square optimization step to improve its precision. Another development used here is the iterative procedure for the estimation of the elastic constants developed byLanda et al.(2009) for highly anisotropic materials. Our objective is to show that RUS is suitable for the measure-ment of anisotropic elastic properties of cortical bone, despite the strong damping of bone. We evaluate the precision of the determination of the elastic constants and discuss the applicability of the method for a routine assessment of cortical bone elastic properties.

### 2.2 Materials and Methods

2.2.1 Specimen

The human cortical bone specimen used in this study was taken from a specimen collection prepared for a previous study (Granke et al.,2011). Ethical approval for the collection of specimens was granted by the Human Ethics Committee of the Centre du don des Corps at the University Paris Descartes (Paris, France). The tissue donors or their legal guardians provided informed written consent to give their tissue for investigation, in accord with legal clauses stated in the French Code of Public Health.

The specimen was a rectangular parallelepiped (dimensions:

5.44x6.58x6.58 mm^{3}) extracted from the lateral quadrant of a transverse
cross-section cut in the mid-diaphysis of a female donor femur (age 78 years). The
faces of the specimen were oriented according to the radial (axis 1), circumferential
(axis 2) and axial (axis 3) directions of bone. The specimen was defatted for
12 h in a chemical bath of diethylether and methanol (1:1). The six faces of the
specimen were polished with a hard synthetic cloth using 3 µm polycrystalline
diamond abrasive particles followed by a 0.05 µm aluminum oxide suspension
(Metadi Supreme and Master-prep, BuehlerR GmbH, Düsseldorf, Germany). The
specimen was stored in saline solution at ambient temperature and taken out of
the solution only for the time of a measurement. Therefore, all measurements were
made on a fully hydrated specimen. The specimen mass density was deduced to

3

1 2

2 mm

Figure 2.1: 3D reconstruction of the cortical bone specimen from SR-µCT. The sample’s faces are oriented according to the radial (axis 1), circumferential (axis 2) and axial (axis 3) directions of bone.

be ρ = 1948 kg/m^{3} from a mass measurement (M = 0.4588 g). Dimensions were
measured using a digital caliper (accuracy: ±0.02 mm).

The accuracy of RUS critically depends on the specimen shape which should
be close to a perfect parallelepiped. The faces of our specimen were parallel and
perpendicular within0.5^{◦}. (Migliori and Sarrao,1997) gave a detailed discussion of
the acceptable tolerance on sample shape; they found that errors of a few tenth of
a degree usually have a negligible influence on the resonant frequencies.

The specimen was imaged using 3D synchrotron radiation microtomography
(SR-µCT) with a 10 µm spatial resolution (Granke et al., 2011). The specimen
porosity (4.3%) and tissue bone mineral density (1.05 g/cm^{3}) were evaluated from
the reconstructed 3D image (Fig. 2.1). Qualitatively, it could be seen on the 3D
image that the distribution of the porosity in the specimen is homogeneous (no
porosity gradient, no big pores).

2.2.2 Resonant Frequencies Calculation

RUS is based on a comparison of calculated and measured resonant frequencies. We used a well-documented method to calculate the frequencies of mechanical resonance for a solid of given elasticity and rectangular parallelepiped geometry (Migliori and Sarrao,1997). Briefly, the resonant angular frequencies ω were found by searching the stationary points of the Lagrangian

L= 1 2

Z

V

ρω^{2}u^{2}_{i} −C_{ijkl}∂u_{i}

∂x_{j}

∂u_{k}

∂x_{l}

dV, (2.1)

where V and ρ are respectively the specimen’s volume and mass density, C_{ijkl} are
the stiffness constants and u is the displacement field. By expanding the
displace-ment field in a set of polynomial functions (Rayleigh-Ritz method), the stationary

2.2. Materials and Methods 25

5 mm Receiver

Emitter

Charge Amplifier

Network

Analyser Computer

In

Out

Figure 2.2: Photograph of the bone specimen inserted between the two transducers and block diagram of the experimental setup.

equation∂L= 0 was written as a generalized eigenvalue problem

ω^{2}Ma=Ka, (2.2)

where M and K denote respectively the mass and stiffness matrices of the
vibra-tion problem. Eq. (2.2) was solved numerically, giving the eigenvalues ω^{2} and the
eigenvectorsa, which contain the coefficients of the polynomial expansion.

2.2.3 Resonant Frequencies Measurement

Transducers were built from small piezoceramic elements (diameter of 3 mm and length of 2 mm) polarized in compression and suspended to copper thin films. This construction reduces the transmission of energy to the mechanical holder and then prevents any unwanted system resonance (Migliori and Sarrao,1997). Support films were glued to steel cylinders connected to the ground, which provided isolation from electromagnetic perturbation and cross-talking between transducers. With these transducers the measurement system was non-resonant in the frequency range of interest.

The specimen was placed between the two transducers, held by two opposite corners to be as close as possible to stress-free boundaries conditions (Fig.2.2). A Vectorial Network Analyzer (Bode 100, Omicron electronics GmbH, Klaus, Austria) was used to record the frequency response of the specimen between 100 kHz and 280 kHz with 50 Hz resolution. The receiver output signal was preconditioned before recording by a broadband charge amplifier (HQA-15M-10T, Femto Messtechnik GmbH, Berlin, Germany). Phase and magnitude of the frequency response were saved for further signal processing.

Preliminary measurements showed that the number of resonant frequencies which can be retrieved may vary from one measurement to another on a same speci-men. This is expected due to the difficulty to excite the vibrational modes which are

associated to a motion of the specimen’s corner nearly parallel to the surface of the transducers. This difficulty is well known for low damping material (Stekel et al., 1992) and is even more critical in the case of high damping, where the signal is weak and the modal overlap is strong. Although in previous preliminary tests we ob-served that it was possible to determine the full stiffness tensor of a bone-mimicking material based on a single positioning of the specimen (Bernard et al., 2011), the quality of the measurement can be increased by combining a few measurements of the specimen. Indeed, intermediate repositioning introduces small variations of the transducer-specimen coupling which in practice increases the probability of exciting and detecting weakly excited modes. Accordingly, the measurements were repeated, removing and replacing the specimen six times. This limited the number of missed modes in the frequency range of interest. Repeated measurements also provided a way to estimate uncertainty on the resonant frequencies.

2.2.4 Signal Processing

When RUS is applied to low damping materials, resonant frequencies can simply be picked as the sharp local maxima of the spectrum. Each resonant peak can also be fitted to a Lorentzian lineshape if the damping factors are to be studied (Leisure et al., 2004). In the context of bone, peaks are broad and strongly overlap each others. In this case, resonant frequencies rarely correspond to local maxima of the spectrum and peaks cannot be fitted individually. A dedicated signal processing method must be used for the estimation of resonant frequencies from the spectrum.

We used a method initially developed for the ultrasonic spectroscopy of large rocks specimens, which show an important damping, described in (Lebedev, 2002) and (Lebedev et al.,2003) and summarized below.

The frequency response F R of the specimen in the frequency range of interest can be modelized as a sum of M Lorentzian lineshapes:

F R(f) =

M

X

k=1

ak

(f_{k}^{2}−f^{2}) +i(f_{k}f /Q_{k}) (2.3)
with ak the complex amplitudes, fk the resonant frequencies and Qk the resonant
quality factors. Equivalently, the response can be represented in time domain by
the impulse response, which is a sum of damped exponentials involving the same
parameters. The problem of estimating the signal parameters from discrete samples
of the impulse response has been addressed in the signal processing literature. A
solution using a linear prediction filter proposed by Kumaresan and Tufts (1982)
has been introduced by Lebedevin the context of RUS and was used in this study.

In a first step, the samples of the experimental frequency response were converted in time domain by inverse Fourier transform. Then, the best linear filter predicting the

2.2. Materials and Methods 27

130 140 150 160 170 180

10^{−2}

Frequency (kHz)

Amplitude (a.u)

Figure 2.3: Typical measured spectrum (plain line) and reconstruction from the lin-ear prediction method (dot line). The spectrum reconstructed after the non-linlin-ear optimization step is indistinguishable from the measured spectrum. •: Resonant fre-quencies obtained from linear prediction. +: Resonant frequencies after non-linear optimization.

time domain data from the L≥M first samples was constructed. In the particular
case L = M, the frequencies f_{k} and quality factors Q_{k} are given by the positions
of the zeros of the constructed predictive filter. Finally, the complex amplitudesak

were obtained by the resolution of a linear system in the frequency domain. Note that the numberM does not need to be known a priori, but was determined from the inspection of the eigenvalues spectrum of the autocorrelation matrix of the time domain signal.

Because we found slight discrepancies between the experimental response and the response reconstructed from Eq. (2.3) using the estimated parameters, we add a non-linear optimization step to the signal processing procedure presented above. The determined parameters were used as initial parameters of a non-linear optimization algorithm (Levenberg-Marquardt), searching the set of parameters for which the difference between Eq. (2.3) and the experimental frequency response was minimal in a least-square sense. In order to reduce the number of independent parameters and then the complexity of the optimization step, signal processing was performed in small frequency bands containing only a few resonant frequencies (typically 6-8).

In practice, we found that the agreement between experimental and reconstructed spectra was improved by the non-linear optimization step (Fig. 2.3) and that the repeatability of the parameters estimation was improved.

2.2.5 Elastic Constants Estimation

Evaluation of the elastic constants in RUS is based on an optimization process, which usually consists in minimization of the function

F(C_{k}) =X

where Ck stands for the set of independent elastic constants (the number of Ck

depends on the assumed elastic symmetry). This optimization process requires association of each measured resonant frequency to the corresponding calculated frequency. This association can be difficult for two reasons:

1. Elastic constants used as initial values for the calculation of the resonant frequencies may be far from the actual values, so that measured and calculated frequencies do not appear in the same order (frequency crossing).

2. Some of the resonant frequencies can be missed during the measurement, as already mentioned.

These two difficulties were overcome by the use of an iterative procedure for
simultaneous association of frequencies and constants estimation, as proposed by
Landa et al.(2009) for application of RUS to ferroelastic materials. Because modal
density is low for the first vibration modes, the probability of frequency crossing
is low and a missed frequency can easily be identified. Thus, for a given
reason-able initial guess of the elastic properties, the n first measured frequencies could
be confidently associated with their corresponding calculated frequencies. A new
set of independent elastic constant C_{l}^{∗} was then defined, which has an orthogonal
relationship with the original set:

C_{k}=β_{kl}C_{l}^{∗}, (2.5)

where [β_{1l}· · ·β_{ql}] is the lth eigenvector of the sensitivity matrix G^{t}G, computed
from the partial derivatives of thenassociated resonant frequencies with respect to
each independent constant:

The combinations C_{l}^{∗} could be sorted by their sensitivities
S_{l}=

2.2. Materials and Methods 29

where λ_{l} denote the lth eigenvalue of the matrix G^{t}G. The sensitivitiesS_{l} reflect
the quantity of information contained in the n associated frequencies about each
combination C_{l}^{∗} , and then sort the combinations from the most determinable to
the less determinable. A minimization of

F(C_{l}^{∗}) =X

n

fn^{exp}−f_{n}^{cal}(C_{l}^{∗})
fn^{exp}

^{2}

, (2.8)

was done (Levenberg-Marquardt method), with the combinationsC_{l}^{∗} presenting an
important sensitivity (S >1 kHz/GPa) as free parameters, while the other
combi-nations were kept fixed. Updated constantsC_{k}^{u} were computed back from Eq. (2.5)
at the end of the fit and used to calculate updated resonant frequencies. Because the
updated constantsC_{k}^{u} obtained by processing the firstnfrequencies are closer to the
actual elastic properties of the specimen than the initial guess, calculated
frequen-cies are also closer to experimental frequenfrequen-cies and the association can be extended
to then^{0} > n first frequencies. The procedure was repeated iteratively, using more
combinations in the minimization as more resonant frequencies could be associated
without ambiguity, until all measured frequencies were associated. Finally, all the
combinationsC_{l}^{∗} were optimized simultaneously.

The iterative process was completed assuming an orthotropic symmetry (nine
independent constants), which is the most general elastic symmetry usually assumed
for bone. Note however that the method can assume material symmetries of lower
order. The initial values for the optimization problem were mean values of bone
elasticity of aged women from (Granke et al., 2011): C_{11} = 19.3, C_{22} = 19.8,
C_{33} = 29.2, C_{44} = 5.8, C_{55} = 5.6 and C_{66} = 4.2 (all values in GPa). Since
this last study did not provide the values of off-diagonal elastic constants, we used
C_{12}=C_{13}=C_{23}=C_{11}−2×C_{66}= 10.9 GPa.

2.2.6 Uncertainty on the elastic constants

FollowingMigliori et al.(1993), we used the curvature of the cost functionF around the minimum to estimate the uncertainty on each particular constant. The method consists in finding the largest possible changes on the constants which lead to an increase of2%inF.

The uncertainty on the measured resonant frequencies also provide a way to es-timate uncertainty on the elastic constants by the mean of Monte-Carlo simulation.

We generated 100 random sets of frequencies from independent normal distributions centered on the experimental mean values and using experimental standard devia-tions. This 100 sets were used to estimate the distribution of the elastic constants values. The mean and standard deviation on each constant was then calculated.

Table 2.1: Elastic constant of the bone specimen evaluated from RUS and BWV, with their uncertainties (all values in GPa).

C11 C22 C33 C12 C13 C23 C44 C55 C66

RUS 20.3 20.2 31.7 10.7 13.4 13.4 6.38 6.32 4.80

(0.6) (0.6) (0.8) (0.6) (0.7) (0.7) (0.02) (0.03) (0.02)

BWV 21.6 21.4 31.3 - - - 6.5 6.5 4.8

(0.7) (0.7) (1.0) - - - (0.3) (0.3) (0.2)

2.2.7 Engineering moduli

From the measured elastic constants the 6×6stiffness matrix was constructed and numerically inverted to obtain the compliance matrix (Auld,1990). From the latter, engineering moduli (Young’s moduli, shear moduli and Poisson’s ratios) were calcu-lated (Bower,2009). This was repeated for the 100 sets of constants obtained from Monte-Carlo simulation, allowing to estimate the uncertainties (standard deviation) on the engineering moduli.

2.2.8 Ultrasonic Velocities Measurement

For comparison purpose, we repeated BWV measurements as described by (Granke et al., 2011). Briefly, the elastic constants on the diagonal of the stiffness tensor were deduced from velocities of longitudinal and shear ultrasonic waves propagating in the principal directions of the specimen. Velocities were obtained from the time-of-flight of an ultrasonic pulse propagating trough the specimen between a pair of transducers contacting two opposite faces. Central frequencies of 2.25 MHz and 1 MHz were used for longitudinal and shear wave velocities measurements respectively (V105RM and V152RM, Panametrics Inc., Waltham, MA). Signals were acquired using an oscilloscope (TDS 2012, Tektronix Inc., Beaverton, OR) and processed with MatLab.

Repeatability of the method, evaluated to be 3.2% and 4.7% for longitudinal and shear elastic constants respectively (Granke et al.,2011), was used to evaluate uncertainty on the elastic constants measured from BWV.