Abstract
This chapter presents the state of art in the field of nonlinear ultrasound applied to bone micro-damage assessment. An increasing number of groups have been conducting research in the past years on this particular topic, motivated by the particular sensitivity shown by nonlinear ultrasound methods in industrial materials and geomaterials. Some of the results obtained recently on bone damage assessment in vitrousing various nonlinear ultrasound techniques are presented. In particular, results obtained with higher harmonic generation, Dynamic Acousto-Elastic Testing (DAET), Nonlinear Resonant Ultrasound Spectroscopy (NRUS), and Nonlinear Wave Modulation Spectroscopy (NWMS) techniques are detailed. All those results show a very good potential for nonlinear ultrasound techniques for bone damage assessment. They should benefit from a proper quantification of the relationship between micro-damage and nonlinear ultrasound parameters. This could be obtained through a thorough statistical study which remains to be achieved. A practical implementation of an in vivosetup also remains to be conducted.
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Keywords
- Bone micro-damage
- Fracture risk assessment
- Nonlinear ultrasound
- Harmonic generation
- Dynamic Acousto-Elastic Testing
- Nonlinear Resonant Ultrasound Spectroscopy
- Nonlinear Wave Modulation Spectroscopy
15.1 Introduction
Cracks in solids were identified as sources of acoustic nonlinearity in industrial materials and geomaterials [1, 2]. These acoustical nonlinearities must be distinguished from elastic nonlinearity arising during irreversible plastic macroscopic deformation for strain of order 1% (see Chap. 1 ). Nonlinear acoustical techniques employ elastic waves with maximum strain amplitude of order 10− 5. Moreover it was suggested that microdamage initiates in bone tissue at strains of about 0.3%, below the apparent macroscopic yield strain of order 0.7% [3]. Consequently nonlinear acoustical techniques are non-destructive.
Promising results obtained in industrial non-destructive testing and geophysics motivated some research groups to apply these nonlinear acoustical methods to assess the level of microdamage in bone. On top of that, this motivation was supported by the growing interest in the role of microdamage in bone remodeling and bone biomechanics [4]. Finally the development of nonlinear acoustical techniques was also motivated by the failure of linear quantitative ultrasound to detect mechanical damage induced in trabecular bone [5].
Contrary to linear acoustics (see Chap. 11 and Chap. 14), in the framework of nonlinear acoustics, the propagation velocity and the attenuation (or dissipation) of acoustic waves are amplitude dependent. Those peculiarities give rise to various phenomena called nonlinear acoustical effects. Some of those nonlinear phenomena were measured in bone and are presented in Sect. 15.2. Section 15.3is dedicated to an introduction of the basic concepts employed to model the effect of cracks on the propagation of elastic waves.
15.2 Application of Nonlinear Acoustics to Experimental Assessment of Damage in Bone
In the context of nonlinear elasticity, the Hooke’s law relating stress σ to strain ε is no more linear and additional terms are introduced to model nonlinear elastic phenomena. The following equation of state (15.1) is assumed in this experimental section to model nonlinear elasticity in bone, based on the results of both experimental and theoretical studies in micro-inhomogeneous media [1, 6]:
where M 0is the linear elastic modulus, β and δ account for classical quadratic and cubic nonlinear elasticity, respectively, and α refers to nonclassical hysteretic quadratic nonlinearity. Δε is the maximum strain excursion, equal to the strain amplitude in the case of a constant-amplitude sinusoidal wave.
This section is devoted to the presentation of four experimental techniques applied to bone tissue in order to measure those three nonlinear elastic parameters.
15.2.1 Higher Harmonic Generation
The existence of elastic nonlinearity in a material induces a progressive distortion of an acoustic temporal waveform during propagation. When elastic nonlinearity can be modeled by a Taylor expansion of the elastic modulus, \(M = {M}_{0} + {M}_{1}\epsilon + {M}_{2}{\epsilon }^{2} + \cdots \),Footnote 1if the initial wave is monochromatic, the first order nonlinear interaction in the wave propagation equation gives rise to a solution containing new terms whose frequencies are higher multiples of the initial frequency ω ∕ (2π) usually named higher harmonics [7].
Hence a classical experiment consists in propagating a monochromatic acoustic wave in the probed material over a distance Land measure the amplitude of second harmonic (at twice the fundamental or initial frequency) in the received acoustic wave.Footnote 2The ratio between the amplitude of the second harmonic over the squared amplitude at the fundamental frequency quantifies the extent of the distortion of acoustic waveform, and consequently the magnitude of elastic nonlinearity in the medium. If diffraction and absorption (viscous and thermal) effects can be neglected over the propagation distance Land if phase velocity dispersion is sufficiently weak, a simple formula is obtained to evaluate the quadratic nonlinear elastic parameter β [7]:
if the acoustic displacement Ucan be measured, where k, U ω, and U 2ωare the wavenumber, the displacement amplitudes at the fundamental frequency and at twice the fundamental frequency, respectively,
if the acoustic pressure pis measured, where ρ0, c 0, p ω, and p 2ωare the linear density, the linear propagation velocity, the pressure amplitudes at the fundamental frequency and at twice the fundamental frequency, respectively. Note that M 0= ρ0 c 0 2. This parameter includes two sources of nonlinearity: the kinematic or geometric nonlinearity, related to the small deviation from the linear relation between the strain and the particle displacement gradient, and the physical nonlinearity, associated with the small deviation from the linear Hooke’s lawFootnote 3(15.1). In ordinary cases, for most of homogeneous fluids and solids, both sources of mechanical nonlinearity generally play a comparable role [8]. For fluids, β is usually expressed by:
where Band Aare homogeneous to M 1and M 0, respectively.
B∕ Ais proportional to the first derivative of sound velocity with respect to the pressure under isentropic conditions [9]. This ratio B∕ Ais indeed a measurement of the so-called acoustoelastic effect which is addressed in Sect. 15.2.2. Thus the quadratic elastic nonlinearity leads to a modulation of the propagation velocity: \(c = {c}_{0} + \beta {v}_{ac}\), where v ac is the acoustic particle velocity. B∕ Avaries between 2 and 15 for homogeneous liquids and solidsFootnote 4and ranges from 0.2 to 0.7 for gases [7]. Nonetheless, micro-inhomogeneous media like granular rocks, unconsolidated granular media (sand or sediment), cracked solids or liquids with gas bubbles, exhibit anomalously high acoustic nonlinearity and B∕ Acan reach values up to 105[1, 7, 10]. For these peculiar materials, kinematic nonlinearity can be neglected with respect to the nonlinearity of the equation of state.
Interestingly, the value of B∕ Aassessed by the measurement of the second harmonic amplitude was shown to increase with the level of damage in metals. Thermal or mechanical damage was found to increase the value of β up to a few times [11–16].
The only reported in vivostudy on the acoustic nonlinearity exhibited by bone tissue was precisely performed by the measurement of the second harmonic amplitude [17, 18]. An ultrasonic (US) burst containing 20 periods is transmitted through the heel bone. Because of huge ultrasonic attenuation in trabecular bone,Footnote 5the fundamental frequency was chosen around 200 KHz. The acoustic pressure amplitude was of order a few hundreds of kPa. The authors conducted the experiment on five healthy volunteers and two osteopenic patients. For each subject, the T-score (see definition in Chap. 1) was measured by DXA (Dual energy X-ray Absorptiometry). Finally a substantial correlation was found out between the T-score and the ratio between the second harmonic amplitude and the fundamental amplitude. Thus this study suggests the ability of this nonlinear acoustical technique to distinguish between healthy and osteopenic subjects.
Nonetheless the physical origins of the increase of acoustic nonlinearity have to be clarified. Such a variation could be firstly attributed to an increase of the micro-crack density and/or the mean length of micro-cracks in osteopenic bone. Indeed a positive correlation between crack density and porosity was reported in other cortical and trabecular skeletal sites than heel but with various coefficients of determination (\({R}^{2} = 0.1 - 0.7\)) [19, 20]. Secondly the increase in the porosity also means the increase in the marrow volume fraction and consequently a decrease in the solid bone volume fraction. Calcaneal pores are mainly filled with yellow marrow which is in fact mostly fat. The nonlinear parameter B∕ Aof fat is close to 10, in the same order of magnitude as the value for an homogeneous undamaged solid. Hence, for healthy solid bone tissue, B∕ Amay be of order 10. Nevertheless, though this biphasic medium is made of a liquid and a solid whose B∕ Aare similar, the nonlinear elastic effects will be more important in the fluid phase than in the solid phase. Indeed the relative importance of this acoustic nonlinearity can be evaluated by:
where M ac is the acoustic Mach number. This allows to figure out that the relative magnitude is governed by β and M ac . As a conclusion, for a given acoustic pressure amplitude, the nonlinear phenomena generated in healthy solid bone tissue may be weaker than in marrow because solids are denser and stiffer than liquids. Finally a simple increase in the marrow volume fraction (or porosity) may alternatively increase the level of acoustic nonlinearity. However if the presence of micro-cracks sufficiently increase the value of B∕ Ain solid bone tissue, this could also lead to an increase of the global acoustic nonlinearity.
Besides, a rising number of experiments shows that the amplitude of the third harmonic (at three times the fundamental frequency) is more sensitive to the level of damage than the second harmonic. In an homogeneous undamaged material, the third harmonic is weaker than the second harmonic and its amplitude is proportional to the cube of the fundamental amplitude. On the contrary, the existence of inter-grain contacts or cracks in a solid enhances the third harmonic whose amplitude can even exceed the second harmonic amplitude [21]. Moreover the amplitude of the third harmonic is proportional to the square of the fundamental amplitude. Consequently the presence of “soft inclusions” embedded in a more rigid matrix modify the acoustic nonlinearity in qualitative and quantitative manners [2].
These effects were recently observed in vitroin trabecular human heel bone [22]. In this experiment, a 400 KHz burst is emitted by a focused transducer and received by a needle hydrophone after 45 mm of propagation. The measurement was performed when propagation occurs in water only and when a 24 mm-thick slice of defatted (and saturated with water) calcaneal trabecular bone is inserted on the propagation path near the hydrophone (Fig. 15.1). The maximum acoustic amplitude used in the experiment is 110 kPa.
As expected, the propagation through water produces a third harmonic weaker than the second harmonic and whose amplitude is proportional to the cube of the fundamental amplitude (Fig. 15.2). On the contrary, despite high attenuation induced by trabecular bone at the third harmonic frequency (30 dB/cm), the third harmonic amplitude exceeds the second harmonic amplitude and is proportional to the square of the fundamental amplitude (Fig. 15.2) when the bone sample is inserted on the ultrasonic path. This anomalously high third harmonic amplitude may originate from the presence of cracks in the solid bone tissue, as reported for damaged solids [21].
15.2.2 Dynamic Acoustoelastic Testing (DAET)
From the end of the nineteenth century, the primary measurements of elastic nonlinearity were performed by static methods leading to the thermodynamic p-v-T diagram, from which the pressure and temperature dependences of the bulk elastic modulus were deduced for fluids and solids [23]. In the beginning of the 20th century, dynamic resonant or interferometric techniques were developed to measure elastic moduli or sound velocity as functions of temperature and hydrostatic pressure [24–26]. Finally, with theoretical developments of the effect of a static stress field on the propagation of elastic waves [27, 28], followed by the possibility of generating an ultrasound (US) pulse [29] and thus of measuring the sound velocity by the time-of-flight (TOF) determination, acoustoelastic testing became an alternative way to measure elastic nonlinearity. This technique consists in measuring changes in the US velocity induced by a hydrostatic [30, 31] or uniaxial [32] stress. For metals and polymers, the relative variation in US velocity is of the order of 0.001 and 0.01%/MPa of the applied stress, respectively. Interestingly, in damaged or granular media, the presence of cracks or intergrain contacts can give rise to variations in US velocity exceeding 1%/MPa of applied stress [33], some orders of magnitude higher than in undamaged solids. Moreover, in these peculiar media, US attenuation is also affected by the application of a static stress as a result of the progressive closing of cracks when the external stress is increased [34, 35]. Finally, the induced variations in the US velocity and attenuation can be related to acoustic nonlinear elasticity and dissipation, respectively.
15.2.2.1 Principle of Dynamic Acoustoelastic Testing
Close to the work of Ichida et al. [36] and Gremaud et al. [37], dynamic acoustoelastic testing (DAET) was firstly developed in a remotely manner [38, 39] for liquids, gels as well as porous and non-porous rather soft solids. In a water tank, the probed sample is simultaneously crossed by two acoustic waves propagating in perpendicular directions (Fig. 15.3):
-
The probing wave: US pulses emitted from one side of the sample by an immersion transducer and received by another US transducer at the other side of the sample.
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The pumping wave: a low-frequency (LF) wave generated in water by a vibrating disk and received by a hydrophone placed near the sample.
Similar to quasi-static acoustoelastic testing, the LF acoustic pressure is expected to modulate the TOF and the attenuation of US pulses.
DAET needs the LF wave to be quasi-static over a US TOF and quasi-uniform in the probed volume. As a result, the LF period must be at least ten times higher than the US TOF. A quasi-uniform LF pressure amplitude in the probed volume is obtained if the LF wavelength is much higher than the characteristic size of the investigated volume. Moreover the diameter of the LF radiating disk is chosen so that the LF pressure amplitude is almost constant over the US propagation path. The diameter of the LF disk being smaller than the LF wavelength the LF diffraction pattern offers a soft decreasing profile for the LF pressure amplitude along the disk axis (Fig. 15.4). Consequently the pressure in the water surrounding the sample is quasi-uniform and sinusoidally modulated in time, inducing successively quasi-hydrostatic compression and expansion of the sample, when the LF acoustic pressure takes positive and negative values, respectively. Typically, the distance between US transducers equals a few centimeters, then the US TOF is of order 10 μs, so that the frequency of the pumping wave equals a few KHz. The LF pressure amplitude can not exceed 100 kPa because cavitation may occur during the expansion phase as soon as the acoustic pressure amplitude exceeds the ambient pressure.
Note that the dimensions of the water tank must be larger than the characteristic LF diffraction length \({L}_{d} = k{a}^{2}/2 \approx 3\,\mathrm{cm}\), where kand aare the LF wavenumber and the LF disk radius, respectively. Indeed the walls must be approximately 30 cm away from the LF disk so that reflections from the wall are negligible. Plane or focused US immersion transducers are used to generate and receive the US pulses. For a US frequency of order 1 MHz and 13 mm diameter plane transducers used to obtain the following results, the US beam is collimated over a few centimeters.
Thus DAET is capable of noninvasive (acoustic transducers are not bounded on the sample) and regional (region of interest is a cylindrical volume whose diameter equals the lateral resolution of the US beam) measurements of elastic and dissipative acoustic nonlinearities induced by dynamic tensile/compressive quasi-hydrostatic loading. It is worth noticing that if the sample is porous, it has to be saturated with water before performing DAET.
For each US pulse, the time-of-flight modulation (TOFM) and the relative energy modulation (REM), related to elastic and dissipative acoustic nonlinearities respectively, are computed [38, 39]. TOFM is determined by the time position of the interpolated maximum of the cross-correlation function between the current US pulse with index iand the first pulse (i= 1) which propagates through the medium with no LF perturbation [38]:
Furthermore, the Fourier transform of each US pulse with index iis computed to calculate its energy E(i) as the integral of the power spectrum in the frequency bandwidth defined at − 10 dB of the maximum amplitude. For the US pulse with index i, REM is given by:
where E(1) is the energy of the first US pulse that propagates through the medium without LF loading. Finally, each US pulse is associated with the mean value of the LF pressure during its TOF (Fig. 15.5).
The synchronization of the LF and US signals allows to plot TOFM and REM as functions of the LF pressure. Figure 15.6represents the two diagrams obtained for a distance of 57 mm between the US transducers, without any sample (only water) and with a 52 mm thickness sample of PMMA (polymethyl methacrylate) inserted between the US transducers.
For water without any sample and with a PMMA sample inserted in the interaction area, no nonlinear dissipative effects are measured whereas the acoustoelastic effect leads to a linear relation between TOFM and the LF pressure (Fig. 15.6).
The relation between TOFM and the LF pressure can be related to elastic nonlinear parameters βFootnote 6and δ, associated with quadratic and cubic nonlinearity, respectively [38, 39]. Because the propagation velocity equals \(\sqrt{M/\rho }\), where Mis the elastic modulus corresponding to the type of US propagation and ρ the density, TOFM is proportional to small variations of the elastic modulus ΔM sample induced by the LF pressure p LF Footnote 7:
where L sample , ρ sample , c sample and M 0= ρ sample c sample 2are the propagation length in the probed medium, the density, the propagation velocity and the elastic modulus with no LF perturbation.
Using (15.8) and extracting the slope \(\frac{\partial TOFM} {\partial {p}_{LF}}\)by linear fitting of the relation measured between TOFM and the LF pressure (Fig. 15.6), β equals 4. 9 ± 0. 3 and 11 ± 1 for water and PMMA, respectively. These values are in agreement with the literature [7].
After illustration and validation of the method in water and with a PMMA sample, results obtained for calcaneal human trabecular bone are now presented.
15.2.2.2 Results in Human Calcaneal Trabecular Bone
The acoustic nonlinearity exhibited by a human calcaneus whose lateral faces were sliced to obtain parallel surfaces is presented. The same 24 mm-thick slice of trabecular bone was investigated with the harmonic distortion method (Sect. 15.2.1). The marrow was removed by immersion in hot water and in trichloroethylene. Then the sample was saturated with water and placed in the experimental setup. Figure 15.7illustrates the two investigated regions of interest (ROI): The upper part of the calcaneus (ROI 1) where the porosity is relatively low (75%± 5) and trabeculae are plate-like shaped, the posterior part (ROI 2) where the porosity is higher (89%± 2) and trabeculae are rather rod-like shaped [40].
Figures 15.8and 15.9show that the acoustic nonlinearities measured in the ROI 1 are an order of magnitude higher than the ROI 2. Whereas the ROI 2 does not change significantly the TOFM diagram measured in water without the sample, weak dissipative nonlinearities are observed in the ROI 2 while only noise is measured in the relation REM vs.LF pressure without the sample (Fig. 15.9). The corresponding quadratic nonlinear elastic parameter β equals 10. Interestingly the ROI 1 exhibits huge acoustic nonlinearity with tension-compression asymmetries and hysteresis for both TOFM and REM (Fig. 15.8). Using a quadratic fit, we obtain β = 150 and δ = 4. 106for ROI 1. Consequently β is an order of magnitude higher than for undamaged solids and the value is in agreement with a previous study [41]. The fact that δ ≫ β2is another manifestation of non-classical elastic nonlinearity, was attributed to the presence of intergrain and/or cracks in granular rocks [42]. Tension-compression asymmetry and hysteresis were also observed by DAET in cracked pyrex [39], cracks are thereby again pointed out as being the source of acoustic nonlinearity investigated by DAET in trabecular bone. Note however that the bottom/anterior region of the calcaneus, which is highly porous (95.5 ± 1.5%) [40], was also investigated but does not change the TOFM and REM measured in water without the sample, certainly because of too low solid bone volume fraction.
Aware that the treatment used to defat this bone sample induces a denaturation of the solid bone tissue, regional DAET scanning was conducted on 8 whole human calcanei defatted using the Supercrit{ ©}(BIOBank, France) techniqueFootnote 8(supercritical CO 2delipidation) ensuring minimum denaturation of bone tissue. Their lateral faces were also sliced to obtain parallel surfaces. The age of donors ranges 70–90 years old.
Two samples out of 8 exhibited high acoustic nonlinearities in ROI 1 and weaker effects in ROI 2. The other six bone samples do not change significantly the results obtained in water without any sample. Figure 15.10shows the DAET diagrams obtained for the sample exhibiting the highest acoustic nonlinearity. Quadratic nonlinear elasticity is high (\(\beta = -100\)) and a large hysteresis is observed in the relation between TOFM and the LF pressure. The anomalous negative sign of β requires further investigation to understand the responsible physical phenomenon. Moreover US energy modulation is also measured, REM reaches − 3% in tension and 2% in compression with hysteresis as well. The reason why only two out of eight calcanei exhibited important acoustic nonlinearities may be the dispersion in the level of microdamage reported in histological studies [43].
In order to test the hypothesis that heterogeneity of the level of microdamage is responsible for weaker acoustic nonlinearities in ROI 2 than in ROI 1 in the “supposedly most damaged” sample, a histological quantification was recently reported by Moreschi et al. [44]. The sample exhibiting the highest acoustic nonlinearities was firstly bulk stained with 0.02% alizarin complexone (chelating fluorochrome) which bind to free calcium links so that cracks are labelled [45]. Secondly the sample is embedded in a polymeric resin and cut in 300 μm thick slices using a low-speed diamond saw. Cracks and split trabeculae were then counted under laser confocal microscopy in regions 1 and 2 (Fig. 15.11). Interestingly the crack density equals 0.2 ± 0.015 crack/mm2in ROI 1Footnote 9and is half this value in ROI 2. Similarly the split trabeculae density equals 0.26 ± 0.047 crack/mm2in ROI 1 and reaches only 0.11 ± 0.039 in ROI 2.
Consequently, this preliminary histological study supports the idea that the overall number of cracks is higher in ROI 1 than in ROI 2 for two reasons:
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The crack density is higher in region 1 than in region 2.
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The bone volume fraction is higher in region 1 than in region 2 (solid bone surface fraction equals 41.6% and 29% in ROI 1 and 2, respectively).
These may be the causes that give rise to acoustic nonlinearities higher in ROI 1 than in ROI 2. However the results indicate that DAET may not be able to detect microdamage if simultaneously the porosity is too high ( > ≈ 85%) and the crack density is too weak ( < ≈ 0. 1 crack/mm2).
All these experimental findings put together suggest that DAET is a sensitive tool to assess the level of microdamage in trabecular bone. Moreover DAET allows noninvasive and regional measurement of elastic and dissipative acoustic nonlinearities, whose instantaneous effects can be plotted as a function of the LF pressure. Consequently a possible tension-compression asymmetry and hysteresis, which are signatures of the presence of cracks, can be observed.
Finally Moreschi et al. recently performed a study testing the correlations between acoustic nonlinearity measured by DAET, mechanical damage induced by fatigue [46] and histological quantification of microdamage by sequential labelling of the cracks using chelating fluorochromes of two different colors [47]. The quantification of the crack density before and after damaging mechanical testing is expected to provide quantitative relations between the level and the type of microdamage and the level and type of acoustic nonlinearity, as previously performed for rocks [48, 49] and carbon fiber reinforced plastic [50]. This current research is the necessary step before an in vivoapplication can be considered. Furthermore, DAET could also be applied to cortical bone using axial US transmission instead of transverse US transmission as presented here for trabecular bone.
15.2.3 Nonlinear Resonant Ultrasound Spectroscopy (NRUS)
Nonlinear Resonant Ultrasound Spectroscopy (NRUS) is a technique that has been developed as an extension of the Resonant Ultrasound Spectroscopy (RUS) technique [51], originally designed to assess the full linear elastic tensor of materials from their resonant behavior. The NRUS technique also exploits the resonant behavior of samples, but to retrieve the nonlinear elastic behavior of the material. As has been developed in the previous section, knowledge on the nonlinear elastic behavior of a sample, and especially on the nonlinear hysteretic behavior, can bring a relevant insight on its damage state. The NRUS technique has proved useful in various materials [21, 51–54], for non-destructive evaluation, and has been applied to bone [55, 56].
The purpose of this technique is precisely to assess the hysteretic behavior of a material sample. Micro-crack accumulation in a material sample is responsible for a softening of the material for increasing excitation amplitudes, leading to a decrease of the resonance frequency when excitation amplitude increases. From the expression of the nonlinear modulus in (15.1), it can be shown that, for a resonant mode, the resonance frequency shift expresses as a function of strain [57]:
The parameter α is called the nonlinear hysteretic parameter, and conveys information about the amount of hysteretic nonlinearity in a material. δ is the parameter describing the classical cubic nonlinearity. In bone, and particularly in damaged bone, the classical cubic nonlinearity is negligible compared to the hysteretic nonlinearity. Therefore, only the linear term of (15.10) remains, and the observed frequency shift is directly proportional to the nonlinear hysteretic parameter α. The NRUS technique is based on this approximation, and provides a very useful tool for the measurement of the parameter α.
As the classical nonlinearity is neglected compared to the hysteretic nonlinearity in bone, a linear decrease of the resonance frequency can be observed for increasing strain amplitudes. Therefore, measurement of the resonance frequency shift as a function of increasing strain amplitudes gives access to the hysteretic parameter in a straightforward manner.
The nonlinear hysteretic behavior of bone samples has been investigated using the NRUS method in a few studies in human and bovine femur [55, 56]. Frequency sweeps were applied to the bone samples for gradually increasing drive amplitudes, and the modal peak frequencies were measured at each drive amplitude (Fig. 15.12). The resonance modes were determined experimentally, and through finite elements modeling methods, using a CT scan 3D image of the sample as an input. The nonlinear parameter was then simply derived from the resonance frequency shift as a function of strain, according to (15.10).
Samples were gradually damaged using compressive mechanical testing (with an INSTRON mechanical testing machine) and the nonlinear parameter α was assessed for each damage step using NRUS.
15.2.3.1 nfluence of Damage Accumulation
Figure 15.13shows typical resonance responses as an example from one sample at three different damage steps. It can be observed that, as damage accumulates, the resonance frequency shift for increasing excitation amplitude increases. For all the samples tested (around 30 samples), the nonlinear parameter derived from NRUS increased with the number of mechanical testing cycles and it started to increase as for the first few mechanical testing cycles. A similar behavior could have been expected for the hysteresis of the load/displacement curve, measured by the mechanical testing device, since it is the quasistatic equivalent of the dynamic nonlinear hysteretic parameter α [58, 59]. However, such behavior could not be observed in the quasistatic regime, where significant changes of the slope and hysteresis of the load/displacement curve were observable only just before failure, as can be seen on Fig. 15.14.
Figure 15.15shows the evolution of \((\alpha /{\alpha }_{0} - 1)\)derived from NRUS as a function of fatigue cycles, as an example for three samples. Here, α0is the nonlinear parameter α in the undamaged state, before mechanical testing. On the same figure are shown the behavior of the slope of the load/displacement curve, and of the damage parameter D, the parameter \((h/{h}_{0} - 1)\), h being the hysteresis of the load/displacement curve obtained during mechanical testing. The nonlinear parameter α increases much more with accumulating damage than any other parameter that could be measured, which suggests a much stronger sensitivity. In addition, the measured nonlinear parameter α shows change immediately, after the first cycling in most cases, and changes significantly over the duration of cycling in most cases, when other parameters start to change significantly just before failure. This suggests that the nonlinear parameter could be a potential tool for early damage detection in bone.
15.2.3.2 Influence of Donor Age
Figure 15.16shows the evolution of the nonlinear parameter α with the age of donors. This curve can be approximated by an exponential, or second order relationship.
This second order behavior of the nonlinear parameter as a function of donor age is similar to direct measurements of damage in bone described in [60]. A larger scatter of α was observed in the region of the curve corresponding to older ages, and values tend to be larger with age in general. This observation is consistent with the distribution of damage accumulation across ages reported in [60]. The similarity of the behaviors of the nonlinear parameter and micro-damage accumulation as a function of age is an additional, qualitative indication that the nonlinear hysteretic parameter provides an relevant insight on the damage state of bone.
15.2.4 Nonlinear Wave Modulation Spectroscopy (NWMS)
The nonlinear wave modulation spectroscopy technique is based on the interaction of two waves with different frequencies f 0and f 1(typically f 0≪ f 1) with amplitudes A 0and A 1. Due to the presence of damage in the bone sample, the two waves interact, creating harmonic frequencies (e.g., 2f 0, 3f 0, etc. f 0being the fundamental frequency) as well as sum and difference frequencies (sidebands), f 1± f 0and proportional in amplitude to the product of the primary wave amplitudes. This technique has been applied broadly in industrial materials and geomaterials [21, 54, 61, 62], and has proved to be efficient even in the presence of elastically linear scatterers [63]. The first attempt to use this technique is human bone was reported by Donskoy and Sutin (1997) [41], who retrieved the nonlinear parameter in trabecular bone in the 100 KHz frequency range, showing that it was an order of magnitude stronger in bone than in nonporous media. Further NWMS measurements were performed, respectively in human cortical and trabecular femoral bone, by Ulrich et al. (2007) and Zacharias et al. (2009) [64, 65]. In the latter study, the technique has been used in both healthy and osteoporotic human trabecular bone, in the 50 KHz range, and the results exhibit a higher level of sidebands in the osteoporotic bone, potentially more micro-damaged. In the study by Ulrich et al., the technique was applied in the 200 KHz frequency range, on human cortical bone, progressively damaged using a mechanical testing device. Low-frequency vibrational modes of a human femur sample were simultaneously excited by a mechanical impulse (induced by a light tap on the sample) and a high frequency, continuous wave tone, in this case 223 KHz (Fig. 15.17). This frequency was selected as it was the frequency for which highest amplitude could be applied with the source transducers used. The vibrational modes mix (multiply) with the pure tone, producing multiple sidebands.
In Fig. 15.18are shown some results obtained using the Nonlinear Wave Modulation Spectroscopy technique in vitroin human femur. The bone samples were subjected successively, to 45000 and 75000 mechanical testing cycles, inducing micro-damage to accumulate. The sidebands (f 1± f 0and f 1± 2f 0) energy was found to increase with accumulating damage, visible on Fig. 15.18, left.
A new dynamic nonlinear parameter Γ, taking into account both the first order classical nonlinear parameter β and the hysteretic nonlinear parameter α (see (15.1)) was defined as the area below the linear frequency spectrum of the sample response, containing the first order (f 1± f 0) and second order (f 1± 2f 0) sidebands, here from 215–231 KHz, in order to include the effects of multiple sidebands simultaneously (Fig. 15.18, left). The evolution of this nonlinear parameter Γis shown in Fig. 15.18(center), along with the evolution of a quasistatic damage parameter D, derived from the slope of the stress-strain curves, obtained during the quasistatic mechanical testing experiments. Figure 15.18shows that the dynamic nonlinear parameter Γchanges by about 700%, when the change in slope from the quasistatic experiment remains roughly the same until the last damage step, where a change of about 10% only is observable. This confirms the observations made by Muller et al. using the Nonlinear Resonant Spectroscopy technique [56]: the dynamic nonlinear parameters are far more sensitive than the quasistatic linear parameters, and are sensitive to earlier damage.
It is also possible to isolate the two nonlinear parameters β and α by calculating the ratio of the first or second order sideband amplitudes respectively, to the drive amplitude. Figure 15.18(right) shows the separate evolutions of the nonlinear parameters β and α as a function of accumulating damage (i.e. as a function of fatigue cycles), normalized to their respective values measured in the undamaged state. This normalization is important here, considering the fact that no absolute value could be obtained for the nonlinear parameters, since no calibration measurements have been performed that would link quantitatively the nonlinear parameters values to an amount of damage. It appears that both parameters significantly increase with increasing damage. However, the nonlinear parameter α seems to be more sensitive than the nonlinear parameter β, since the values of β are in a range of 0–60, while the values of α are in a range of 0–120, for the same sample.
15.3 Theoretical Modeling of Damage-induced Nonlinearity, Limitations of the Technique
15.3.1 Physical Origins of Nonlinearity in Damaged Bone
The previous section reviews different experimental nonlinear acoustic techniques that can be used for a noninvasive assessment of bone mechanics. Although ultrasound measurements give access to mechanical parameters strongly related to stress and strain, and to fracture risk, they do not allow their direct measurement. Experimental studies described in the previous section showed that the measurement of nonlinear ultrasound parameters can provide a straightforward access to damage amount in bone. However, the use of appropriate models is required to fully characterize the nonlinear relationship between stress and strain in damaged bone. This full characterization could be useful for a better understanding of the nonlinearity induced by damage in bone. Indeed, the damage-induced nonlinearity that can be measured acoustically at the macroscopic level results from some nonlinear phenomena at the crack level and below, on a microscopic scale. A large amount of research has been conducted to establish the link between microscopic and macroscopic scales in terms of damage and nonlinearity. Different types of models are studied and used in the literature: phenomenological models and theoretical models.
15.3.1.1 Phenomenological Model
Phenomenological models used to describe micro-cracked solids are based on the observation that micro-damage induces a hysteresis in the stress-strain relation (Fig. 15.19). This phenomenon has been observed in a large class of materials, as well as in bone. In this paradigm, a microdamaged material is described as an ensemble of hysteretic units called hysterons, small structures at the microscopic level that are responsible for the hysteretic behavior of the stress-strain relationship. The strain response of each hysteron is modeled by the combination of a classical nonlinear term, and a nonclassical nonlinear contribution attributed to hysteretic behavior [1, 66]. In order to describe this phenomenon, a model has been established by Guyer and McCall [6, 67], inspired by the work of Preisach and Mayergoyz in magnetism [68, 69]. The nonclassical nonlinear contribution is obtained by stating that the hysteretic units can only be found in two equilibrium states: open or closed. Therefore, each hysteron can be fully described by two sets of parameter pairs: (σ o , σ c ) and (ε o , ε c ), which respectively describe the stresses and strains necessary for the hysterons to be in open or closed states. Particular rheological relationships can be attributed to hysterons [70]. The hysterons can be arranged in the Preisach-Mayergoyz space (PM space, Fig. 15.19), that allows to keep track of the state (open or closed) of a whole distribution of hysterons. This arrangement in the PM space constitutes the link between microscopic and mesoscopic scales. An equation of state (the stress/strain relation) is derived from the hysterons distribution density in the PM space. Note that it is possible to simplify this phenomenological model, in the case of small strains, by assuming a uniform density of the hysterons in the PM space. In this case, the nonclassical hysteretic nonlinearity can be described as having a quadratic dependence on strain, which qualitatively agrees with experimental results obtained with NRUS, and is shown in (15.1).
This phenomenological model can be numerically derived using various processes. Among them, the Local Interaction Simulation Approach (LISA) considers the hysteretic elements at the microscopic levels and models wave propagation in such media [70]. This modeling approach provides good results but is extremely demanding in terms of computational resources. Another numerical approach that has been used considers mesoscopic units, representing a statistical ensemble of hysterons (as opposed to the LISA method that considers each hysteron individually). These new units have a characteristic size smaller than the wavelength, exhibit hysteretic strain responses at the mesoscopic scale, and are used as elementary cells for finite differences simulations, using an elastodynamic finite integration technique (EFIT) [71, 72].
15.3.1.2 Theoretical Models
Some drawbacks can be pointed out regarding the phenomenological model described in the previous section. First, the numerical implementations of the model requires an a prioriassumption about the distribution of hysteretic units, used as an input to the model. Secondly, the model does not describe an accurate picture of the physical phenomena responsible for the very typical nonlinear behavior of damaged solids. Some theoretical and experimental work has been conducted in various fields such as geophysics, non destructive evaluation, and granular materials physics, in order to depict the physical phenomena at stake. Although crack-induced nonlinearity is probably not fully understood and modeled yet, it appears that the observed phenomena (nonlinear modulation, hysteresis in the dynamic stress/strain relation...) are the consequences of various causes. One of these causes has been thoroughly studied over the years, and corresponds to the purely elastic nonlinearity, due for instance to Hertzian contacts, at the cracks inner contacts scale [73]. For this particular cause of crack-induced nonlinearity, the magnitude of the nonlinear effects is sensitive to the presence of weak, damaged regions in the material [74, 75]. In addition to these classical nonlinear elastic effects, more particular effects have been observed such as hysteretic nonlinearity. Some models have been derived [10, 62], based on the description of hysteresis as amplitude dependent dissipation. In these models, dissipation can be attributed to friction/adhesion hysteresis at the crack interface [76], or to locally enhanced thermoelastic coupling at the inner crack contacts. Note that in the case of friction/adhesion, a strain threshold is required to allow the phenomenon to occur. This threshold has never been accurately calculated for bone, but corresponds to the ratio of the interatomic distance to the crack diameter [62], which would lead in bone to strains around 10− 5, in most cases higher than the strains used in the various experimental studies (from 10− 6to 10− 5). This could lead, as a first approximation, to the conclusion that the friction adhesion mechanisms in bone are unlikely to contribute significantly to the nonlinear hysteretic observed phenomenon, but only a thorough theoretical study of the possible mechanisms responsible for bone nonlinear behavior, taking into account bone peculiarities (heterogeneous, multiscale, micro-damaged with crack filled with a viscous fluid), could provide answers to these questions.
15.3.2 Limitations of the Techniques and Perspectives
The first limitation of the nonlinear ultrasonic techniques conducted until now for bone damage detection is that they still lack of a comparison to independent, quantitative evaluation of damage. A growing number of research groups are currently working on this subject and it appears clearly now that the next step in this research area will be to perform these validation measurements. A comparison to quasistatic measurements obtained using mechanical testing machine would have to be conducted carefully. In particular, one would have to keep in mind that time scales are a very important issue in this problem, since it deals with dissipation mechanisms. Therefore, a comparison between quasistatic and dynamic measurements would necessarily have some limitations. For instance, the thermal fluctuations, very likely to contribute to the hysteretic behavior of damaged bone subjected to a mechanical solicitation, will certainly be different for different strain rates. At very low frequency, for quasistatic solicitations, one can expect the system to have enough time to relax to its thermodynamic equilibrium within a period, whereas the characteristic time of the system could be longer than an excitation period at higher frequencies, leading to an increased hysteretic behavior [66]. A good candidate for the validation of the nonlinear acoustic measurements for damage characterization would be histological measurements. Histology is now considered as the gold standard for damage assessment in vitroin bone [77]. A quantitative comparison between nonlinear ultrasound parameters and the histologically measured damage could provide the quantitative relationship between damage and nonlinearity that is still needed, as long as it is statistically valid, i.e.conducted on a significant amount of samples. The derivation of this empirical relationship would be a huge progress in the field, especially in the context where a model still has to be developed for the damage-induced nonlinear behavior of bone. In terms of modeling, a lot of work has still to be done. The first step would be a proper identification of the different physical origins for the nonlinearity in damaged bone. Finer research has to be conducted in order to understand some of the observed phenomena such as the linear decrease of the resonance frequency shift as a function of amplitude in the NRUS measurements, or the fact that the measured nonlinear parameter β in trabecular bone is significantly different in traction and in compression, when the Young’s modulus does not change much in the two situations [78].
Another work that still remains to be conducted is the design of an in vivosetup, able to evaluate the nonlinear parameters β and α through a layer of soft tissue, in a clinical context. The choice of the anatomical site should correspond to an effective clinical fracture site such as the hip or the vertebrae. Again, some modeling, and some experimental trials would have to be conducted in order to circumvent the potential difficulties related to the presence of soft tissue around the bone.
15.4 Conclusion
A rising number of research groups has now reported essentially in vitroobservations of acoustic nonlinearity in bone tissue. Four experimental techniques were applied to bone: harmonic distortion of a monochromatic wave, dynamic acoustoelastic testing, nonlinear resonant ultrasound spectroscopy and nonlinear wave modulation spectroscopy. It is worth noticing that these experimental findings attest the existence of non-classical acoustic nonlinearity because of anomalously high values of classical elastic nonlinear parameters β and δ and qualitative peculiarities like tension/compression asymmetry, hysteresis and resonance linear frequency shift, as was observed in damaged industrial materials and geomaterials.
The current research efforts now focus on the production of evidences that cracks embedded in the solid bone tissue are the origin of acoustic nonlinearity. Very few studies were conducted on a significant set of bone samples originating from different donors. Among these, NRUS and DAET showed a large dispersion in the level of acoustic nonlinearity, corroborating, if cracks are assumed to be sources of acoustic nonlinearity, the large dispersion in the level of microdamage reported in terms of crack density by histological studies [44, 47, 56].
The very recent trend in the research field is to conduct both progressively damaging mechanical testing and histological quantification of microdamage by fluorescence microscopy together with the measurement of acoustic nonlinearity [44, 47]. Controlled mechanical testings are used as a tool to increase progressively the level of microdamage in bone samples and to monitor the associated reduction of the macroscopic mechanical integrity. This in vitroobjectivation of the relation between the actual level of microdamage and the level and type of acoustic nonlinearity is essential before the development of an in vivononlinear acoustical technique is addressed. Depending on the in vivosensitivity of these nonlinear acoustical methods, the measurement of the acoustic nonlinearity exhibited by bone tissue may be a powerful non-invasive tool to assess the level of microdamage generated in bone and to improve our comprehension of the role of cracks in bone remodeling and bone biomechanics.
Notes
- 1.
M 0is the linear elastic modulus or the second-order (in energy) elastic constant, whereas M 1 M 2are the third-order and fourth-order elastic constants, respectively, which account for nonlinear elasticity. M 1and M 2are negative for most of the materials.
- 2.
Practically, a burst containing at least ten acoustic periods is emitted to facilitate the extraction of the second harmonic amplitude in the frequency domain after the computation of the Fourier transform of the received acoustic signal.
- 3.
In other words, the nonlinearity in the equation of state of the material, relating stress to strain.
- 4.
In an isotropic solid, for a compressional plane wave, \(\beta = -(3/2 + {C}_{111}/(2{C}_{11}))\), where C 111and C 11are elastic constants homogeneous to M 1and M 0, respectively [7]. The value 3 ∕ 2 instead of 1 in the expression of β is related to the difference between Lagrangian and Eulerian descriptions of particle motion. Moreover the negative sign arises from the difference in the definitions of the pressure and the stress. Besides, the reader has to pay attention to the definition of β when comparing values obtained by different studies. Indeed the parameter of quadratic nonlinear elasticity is sometimes defined as \(\beta = -(3 + {C}_{111}/{C}_{11})\), twice the value usually employed in the “fluid” community.
- 5.
The heel bone or calcaneus contains 95% of trabecular surrounded by a thin cortical shell.
- 6.
In the DAET configuration, the convective effect cannot occur because the LF and US beams propagate in perpendicular directions. In this section, we redefine β as \(\beta = B/A\).
- 7.
For most of materials, small relative variations of the density can be neglected compared to small relative variations of the elastic modulus.
- 8.
- 9.
The crack density is expressed in cracks number per square mm of bone tissue.
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Muller, M., Renaud, G. (2011). Nonlinear Acoustics for Non-invasive Assessment of Bone Micro-damage. In: Laugier, P., Haïat, G. (eds) Bone Quantitative Ultrasound. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0017-8_15
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