An experimental study on the ultrasonic wave propagation in cancellous bone: Waveform changes during propagation Fuminori Fujita Laboratory of Ultrasonic Electronics, Research Center for Wave Electronics, Doshisha University, Kyotanabe, 610-0321 Kyoto, Japan
Katsunori Mizuno Underwater Technology Research Center, The University of Tokyo, Meguro-ku, Tokyo, 153-8505 Japan
Mami Matsukawaa) Laboratory of Ultrasonic Electronics, Research Center for Wave Electronics, Doshisha University, Kyotanabe, 610-0321 Kyoto, Japan
(Received 27 September 2012; revised 20 June 2013; accepted 29 July 2013) Wave propagation in a trabecular bone was experimentally investigated using an acoustic tube. For the purposes of this study, a cubic sample was gradually filed so the waveform change due to the sample thickness could be observed. The initial sample showed clear two-wave separation. As the sample became thinner, the fast and slow waves gradually overlapped. The apparent frequencies and amplitudes of the fast waves obtained from the time domain data decreased significantly for the smaller thicknesses. This indicates an increase in the apparent attenuation at the initial stage of the propagation. Next the authors investigated the distribution of the ultrasonic field after the transmission through the cancellous bone sample. In addition to a large aperture receiver, a needle-type ultrasonic transducer was used to observe the ultrasonic field. Within an area of the same size of the large transducer, the waveforms retrieved with the needle sensor exhibited high spatial variations; however, the averaged waveform in the plane was similar to the waveform obtained with the large aperture receiver. This indicates that the phase cancellation effect on the surface of the large aperture receiver can be one of the reasons for the strong apparent attenuation observed at the initial C 2013 Acoustical Society of America. stages of the propagation. V [http://dx.doi.org/10.1121/1.4824970] PACS number(s): 43.80.Cs, 43.80.Sh, 43.80.Ev, 43.80.Qf [ADP] I. INTRODUCTION
Cancellous bone is composed of a trabecular bone filled with bone marrow, and is an interesting poroelastic tissue in the human body.1 Because the bone marrow is a viscous liquid at body temperature, the cancellous bone contains both solid and liquid parts. One theory of the wave propagation in such porous media filled with liquid was first suggested by Biot. Since then, several researchers have discussed and challenged the application of this theory on the wave propagation in cancellous bone, especially concerning the two longitudinal wave propagation.2–4 The first experimental observation of two wave phenomenon was achieved by Hosokawa and Otani using a simple water-immersion technique in the megahertz range.5,6 The problem of this two wave phenomenon in cancellous bone is, however, the strong anisotropy and heterogeneity of the sample, which results in the difficulty to describe it in a theoretical model as pointed by Kaufman et al.7 Two wave phenomenon, therefore, has been extensively investigated experimentally or by numerical approaches. As for the numerical approach, after the first success of Bossy et al.8 and Nagatani et al.,9 several studies have confirmed the two wave propagation by the two-dimensional and three-dimensional (3D) a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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Pages: 4775–4781
finite-difference time-domain methods with reconstructed 3D CT images of actual or artificially constructed bones.10–13 These numerical studies have brought us clear images of complicated wave propagation and helped the understanding of this phenomenon. Actually, the heterogeneous and anisotropic structure of cancellous bone can result in the propagation of complex ultrasonic waves that make measurements and their interpretation of difficult, although the evaluation of cancellous bone structure can provide a good insight of the degree of osteoporosis.1 The wave characteristics of both fast and slow waves are critical to obtain information about structure and bone volume ratio. Because the fast wave mainly propagates in the solid trabecular frame, the wave properties reflect the state of the cancellous structure of the bone.14–16 In addition, Otani has reported a good relation between the slow wave amplitude and bone volume and introduced a new bone densitometry apparatus for clinical studies. The apparatus can give us data which have a high correlation with bone mineral density by peripheral quantitative computed tomography.17–19 Though in other circumstances the two waves substantially overlap and appear as only a single wave, in some conditions the two waves separate in the time-domain. The conventional analysis methods of experimental data, therefore, may suggest potentially misleading wave properties. Previous studies have demonstrated that interfering fast wave and slow wave modes can account for the apparent negative dispersion
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sometimes observed in the measurements of cancellous bone.20,21 The degree to which the fast waves and slow waves overlap depends on a number of factors including porosity, structural anisotropy, and the angle of insonification relative to the predominant trabecular orientation. Several studies therefore have focused on the interesting techniques to extract information of each wave from the mixed waves.22–24 The numerical approach has also given a good prediction to understand the complicated experimental data. The simulations made by Nagatani et al. have shown good agreements between experimental and numerical data of wave propagation in the identically simulated cancellous bone.13 They also pointed out the interesting initial wave propagation process in the cancellous bone, especially due to the spatial distribution of wave front. The complicated solidliquid structure of cancellous bone generates a fluctuation of wave fronts, especially the fast wave fronts, due to the multiple paths in the complex trabecular structure. Finally, the fluctuating waves are experimentally observed after the integration at the surface of the receiver. This integration may result in the very weak and highly attenuated fast wave during wave propagation and makes the fast wave measurements very difficult. The objective of this study is to investigate the propagation mechanism of the fast wave in the initial stage. Through the precise ultrasonic measurements using a pulse immersion technique, the spatial distribution of the fast wave fronts was experimentally investigated in detail. Discussions also include the apparent high attenuation of fast wave and the necessity of a separation of the fast wave from the slow wave; this is very important information for the clinical application of two wave phenomenon in the cancellous bone.
(SMX-100CT, Shimadzu, Kyoto, Japan). The spatial resolution of the 3D CT images was 41 lm. B. Ultrasonic measurements
During measurements at room temperature, the bone sample was immersed in degassed water and placed in an acoustical tube as shown in Fig. 2. The bone sample was placed at a 70 mm distance from the transmitter, because the amplitude of the ultrasonic wave was stable at this position in the acoustic tube. We set the wave propagation direction parallel to the bones main axis. Before each measurement, the specimens were degassed for 60 min to remove air bubbles trapped inside. A plane wideband polyvinylidene fluoride (PVDF) transmitter (rectangle active area 15 15 mm in size, self-made) was used. This comparatively large transducer was able to achieve a good signal-to-noise ratio of the observed waves.13 Two types of plane PVDF receivers were prepared to observe the waveforms which passed through the bone sample: (1) Rectangular type, 15 15 mm in size, self-made [Fig. 2(a)] same as the transmitter, (2) needle type, 1.0 mm in diameter (Toray Engineering, Tokyo, Japan) [Fig. 2(b)]. The latter was used to measure acoustic field in the case of a thin specimen. The distance between the two transducers was 100 mm for the rectangle receiver and 73 mm for the needle type receiver. In the case of the rectangular type transducer, we set it at the position of 100 mm to
II. MATERIALS AND METHODS A. Bone specimen preparation
A rectangular cancellous bone sample, 22.4 22.4 8.7 mm3 in size, was cut from a 36-month-old equine left radius using an oscillating saw. The size of the specimen was measured using a caliper (precision: 0.01 mm). To remove the bone marrow, the cubic samples were defatted using a water pick (HT j202, OMRON, Kyoto, Japan).5 Figure 1 shows the cross-sectional image of x-ray micro focus CT images of a sample in the bone axis direction
FIG. 1. (Color online) Trabecular specimen preparation and the scan image of the sample by x-ray micro CT. 4776
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FIG. 2. Ultrasonic measurement configurations. The wall and bottom of the acoustic tube were covered by a foam polystyrene plate to avoid the invasion of the ultrasonic wave. The receiver was a rectangular PVDF transducer or a needle type transducer. In the case of the needle type transducer, the receiver scanned in the plane normal to the wave propagation direction. Fujita et al.: Wave propagation in cancellous bone
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avoid the multiple reflection between the sample and the receiver. In order to observe the fluctuations of an ultrasonic wave front after the bone sample, the needle type receiver was located right behind the bone sample. A single cycle sinusoidal wave electric pulse at 0.7, 1.0, 1.2, or 1.5 MHz with an amplitude of 5 Vp–p (peak to peak voltage) was generated using a function generator (33250A, Agilent, CO, USA). It was amplified by 20 dB using a power amplifier (4055, NF Corporation, Kanagawa, Japan), and applied to the transmitter. The waves that passed through the sample in an acoustic tube were converted into electrical signals by the receiver and then retrieved by a digital oscilloscope (TDS 524A, Tektronix Inc., OR, USA, 500MS/S and 11 bit for the averaging mode) with a 20 dB preamplifier (5307, NF Corp., Kanagawa, Japan). The measurements were carried out using samples with different thicknesses ranging from 8.7 to 1.1 mm. The thickness was precisely controlled by polishing the sample using a polishing machine (Speed Lap, Maruto, Tokyo, Japan). In the case of spatial distribution measurements, the position of the needle receiver was moved by a 0.5 mm step in a 10 10 mm area centered on the acoustic axis to estimate the acoustic field in the tube [Fig. 2(b)]. A total of 441 measurements were performed in the plane.
amplitude. This is a simple but precise measurement which is applicable to the actual in vivo measurement system using two waves, where rapid evaluation is necessary.18,19 Following our former study, the apparent attenuations of fast and slow waves were also estimated from the observed waveforms in the time domain. To obtain the attenuation, we have used the data measured by the rectangular transducer. Here, to avoid the effects of overlapping, the amplitudes of the first small peaks were adopted to estimate the apparent fast wave attenuations. For the slow wave attenuation, the last positive peak amplitude was adopted. The method used to choose the peaks is illustrated in Fig. 3, showing how to estimate the attenuation. These amplitudes were measured from a series of samples with different thicknesses. The apparent attenuation was defined by a ¼ 20 logðVn =Vnþ1 Þ=Dd;
(1)
where Vn and Vnþ1 are the amplitudes of the peaks in the received radio frequency waveforms. The indices n and n þ 1 correspond to successive sample thicknesses differing by Dd of about 0.5 mm. The obtained value is the attenuation in the small portion of the sample. III. RESULTS AND DISCUSSION
C. Estimation of apparent wave properties from the time domain data
A. Observation of two wave phenomenon
Because fast and slow waves mostly overlap, we cannot obtain wave properties of each wave. Then, we estimated the apparent frequency and amplitude of two waves in the time domain as shown in Figs. 3(a) and 3(b). We have selected time intervals and peaks which seem to be less affected by the other wave to estimate the apparent frequency and
The observed waves which propagated in the sample have varied according to the sample thickness. At first we have confirmed the two wave propagation in the sample following the process of our former study13 using a rectangle PVDF transducer as a receiver. The observed waveforms are shown in Fig. 4. Two waves were clearly observed in the thicker sample, and
FIG. 3. The peaks and intervals (4(Tp Tt) and 2(Tz2 Tz1)) used for the estimation of apparent frequency and apparent attenuation. J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
FIG. 4. Changes in observed waveforms as a function of sample thickness. Fujita et al.: Wave propagation in cancellous bone
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gradually overlapped in the thinner samples. This means a certain thickness of the trabecular bone is required to observe the two wave phenomenon under an in vivo situation. In our present case, for thicknesses under 5 mm, it seemed difficult to observe two clearly separated waves. However, the wave front or the first peak of the fast wave can still be identified in the samples with thicknesses of 1.6 or 1.1 mm, although most parts of the fast wave is superposed with the following slow wave. This characteristic change of waveform was in good accordance with our past data using bovine cancellous bone.13 B. Apparent wave properties—effect of time domain analysis
The wave properties of these two waves have already been discussed by several researchers.25,26 Recently, new techniques to separate waves have been applied to the overlapped waves and have given us properties of each wave. Especially, the application of the Bayesian technique has almost succeeded in the estimation of fast and slow wave attenuation properties.27 The analysis of separated waves showed that the attenuation coefficients at a fixed frequency were constant during wave propagation. The separation of waves, however, is still difficult during in vivo clinical measurements, because complicated and long processes are necessary. In the in vivo measurements, therefore, the time domain data are often used to analyze the characteristics of two waves.18,19
FIG. 5. Apparent frequencies of observed waves as a function of sample thickness. (a) Fast wave and (b) slow wave. 4778
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During propagation, the wave amplitude in the time domain decreased rapidly in the initial state of wave propagation, together with an increase in the wave period. As a result, the apparent frequency decreased as illustrated in Figs. 5(a) and 5(b). One can clearly see that the decrease in the apparent frequency (increase of apparent wave period) of the fast wave varied together with the thickness of the sample, whereas the apparent slow wave frequency did not seem to display high variations. This tendency was similar in all measurement data with different input signal frequencies. Here, the small decrease in the slow wave attenuation with thickness lower than 2.5 mm possibly comes from the effects of overlapping fast waves, because the overlap of fast and slow waves was strong with thicknesses lower than 4 mm. The effects of the overlap might also affect the apparent frequency of the slow wave. Therefore, a precise understanding of the initial propagation of the slow wave seemed difficult with these time domain data. The apparent fast wave attenuation data also exhibits an interesting behavior: the fast wave strongly attenuates in the initial stage of the propagation as shown in Fig. 6(a). This is in good accordance with our former bovine data and consistent with the results obtained by Nelson et al.27 In this study, a broadband pulse wave was used as the input signal. Because the attenuation coefficients of both fast and slow waves increase with frequency, the amplitudes of higher frequency components are more attenuated than the lower frequency components. As the wave propagation proceeds, the nominal frequency of the pulse wave becomes lower, plus
FIG. 6. Apparent attenuation coefficients at each portion of the sample. (a) Fast wave and (b) slow wave. Fujita et al.: Wave propagation in cancellous bone
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In addition to the problem of time domain analysis, we should also take into account the effects of the complex trabecular structure of the sample. Especially in very thin trabecular samples, the propagating wave front is generally misaligned due to the wave propagating in the trabecular structure and the wave velocity difference between hard trabeculae (assuming approximately 3600 m/s from the cortical bone studies)1 and liquid (in vivo: Marrow; present experiments: Water). Actually, the initial irregular wave front has clearly been shown by several studies.11,13,28 Because the aperture of the receiver used in the ultrasonic system is large compared to the wavelength, the spatial fluctuation of the phase at the complex wave front is likely to be canceled at the receiver face. This phase cancellation is another mechanism which is responsible for the apparent drop in the wave amplitude. This kind of cancellation has been pointed out by Wear29 and Anderson et al.27 In the present data, the decrease in the apparent fast wave attenuation in each portion of the sample was prominent when the sample thickness was less than 4 mm. We measured the fluctuations of the wave front using a needle type receiver, whose diameter of active area was only 1 mm.
Figure 7 shows waves observed in water at different positions in the plane perpendicular to the wave propagation direction. We can confirm that waveforms are similar at points ‹ and . This proves the acoustic field in the tube was symmetrical. Figure 8 shows waves observed after passing through a thin sample (thickness 1.1 mm). We can observe clear changes in the waveforms due to the position, and also see the fluctuations of the wave front. These fluctuations become much clearer when looking at the distribution of arrival times of the waves in the measurement plane (Fig. 9). Here, the arrival time of an individual wave is obtained by taking the initial rising time of the wave front at the amplitude of 10% of the slow wave maximum. Considering the diameter of the needle type receiver (1 mm) and the diameter of a trabecula (100 to 200 lm), the spatial resolution of the measurement is not yet perfect; however, there were dramatic fluctuations of the arrival time at the initial stage of wave propagation. The standard deviations of the arrival time in one plane (441 measurement points) were 0.12 and 0.09 ls, for the sample with thicknesses of 1.1 and 1.6 mm, respectively. Figure 9 also shows x-ray micro CT data of the sample. We then divided the images in Fig. 9 into nine parts and obtained the total area of trabeculae and averaged arrival time. The correlation between the data obtained from the two images was around R2 ¼ 0.73 (p < 0.01), telling that the fluctuation depends on the trabecular structure. The summation of waves measured in one plane also brings us interesting results. Figure 10 shows the comparison of observed waves by the rectangular receiver and the summation of all measured waves in one plane by the needle type receiver. In both cases of 3.5 and 1.1 mm thicknesses, the waveform obtained by the rectangular transducer and the
FIG. 7. Ultrasonic waves observed at different positions in the plane perpendicular to the wave propagation direction in water.
FIG. 8. Ultrasonic waves observed at different positions in the plane perpendicular to the wave propagation direction, after passing through the thin sample (thickness 1.1 mm).
the apparent attenuation seems to decrease because the attenuation coefficient becomes smaller. This is directly reflected in the first peak amplitude of the fast wave and often results in a apparent attenuation decrease in the initial stage of the propagation as Nelson et al. have already discussed.27 In addition, multiple reflections in a thin sample possibly affects the waveforms. C. Effects of rectangular aperture of the receiver
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summation of the waveforms obtained by the needle sensor was very similar. Compared with the observed waves at each position by the needle type receiver, the apparent nominal frequencies of the summation waves seemed to decrease, which came from the superposition of different waves at different positions in the plane. This data actually proves the effect of the large aperture of the receiver on the measurement and shows that the observed wave properties also depend on the spatial averaging effect in the measurement system. This effect should be taken into account in the in vivo time domain system to clearly define the properties of the waves which propagated in the cancellous bone.
IV. CONCLUSION
FIG. 9. (Color online) Distribution of the wave front in a plane after passing through the thin sample (thickness 1.1 mm) with micro CT data.
Ultrasonic wave propagation in an equine cancellous bone was experimentally investigated using an acoustic tube. First, using a rectangular large aperture receiver, the twowave phenomenon was clearly observed. The propagation was then followed by changing the sample thickness and it was found that the two waves were clearly separated in the samples with thicknesses larger than 5 mm. Although the structure and bone characteristics of the equine bone are possibly different from the human bone, these data hint that a certain thickness of the trabecular bone might be required in order to clearly observe and exploit the two wave phenomenon in vivo. The characteristic decrease in the apparent fast wave attenuation was obvious in the initial stage of the wave propagation. This tendency was in agreement with our past bovine data. The decrease of the attenuation in very thin samples was then discussed. One possible reason is the effect of the large aperture receiver. We experimentally showed the strong fluctuations of the wave front in the initial stage of the propagation, which were then canceled at the surface of the comparatively large aperture receiver. ACKNOWLEDGMENTS
Part of this work was supported by the Regional Innovation Strategy Support Program of Ministry of Education, Culture, Sports, Science and Technology, and Japan and Grant-in-Aid for Scientific Research (B) of the Japan Society for the Promotion of Science. The authors would like to thank Professor Yoshiki Nagatani at Kobe City College of Technology and Kevann Esmaeili at Doshisha University for their valuable scientific advice. 1
FIG. 10. Comparison of ultrasonic waves obtained by the different receivers: (a) The waves measured by the rectangular large receiver, and (b) the summation of all measured waves in the plane by the needle type receiver. 4780
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P. Laugier and G. Haiat, editors, Bone Quantitative Ultrasound (Springer, Dordrecht, Netherlands, 2011), pp. 1–6. 2 T. J. Plona, “Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies,” Appl. Phys. Lett. 36, 259–261 (1980). 3 J. G. Berryman, “Confirmation of Biot’s theory,” Appl. Phys. Lett. 37, 382–384 (1980). 4 R. Lakes, H. S. Yoon, and J. L. Katz, “Slow compressional wave propagation in wet human and bovine cortical bone,” Science 220, 513–515 (1983). 5 A. Hosokawa and T. Otani, “Ultrasonic wave propagation in bovine cancellous bone,” J. Acoust. Soc. Am. 101, 558–562 (1997). 6 A. Hosokawa and T. Otani, “Acoustic anisotropy in bovine cancellous bone,” J. Acoust. Soc. Am. 103, 2718–2722 (1998). Fujita et al.: Wave propagation in cancellous bone
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J. J. Kaufmann, G. Luo, and R. S. Siffert, “Ultrasound simulation in bone,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1205–1218 (2008). 8 E. Bossy, F. Padilla, F. Peyrin, and P. Laugier, “Three dimensional simulation of ultrasound propagation through trabecular bone structure measured by synchrotron microtomography,” Phys. Med. Biol. 50, 5545–5556 (2005). 9 Y. Nagatani, H. Imaizumi, T. Fukuda, M. Matsukawa, Y. Watanabe, and T. Otani, “Applicability of finite-difference time-domain method to simulation of wave propagation in cancellous bone,” Jpn. J. Appl. Phys. 45(1), 7186–7190 (2006). 10 E. Bossy, P. Laugier, F. Peyrin, and F. Padilla, “Attenuation in trabecular bone: A comparison between numerical simulation and experimental results in human femur,” J. Acoust. Soc. Am. 22, 2469–2475 (2007). 11 G. Ha€ıat, F. Padilla, F. Peyrin, and P. Laugier, “Fast wave ultrasonic propagation in trabecula bone: Numerical study of the influence of porosity and structural anisotropy,” J. Acoust. Soc. Am. 123, 1694–1705 (2008). 12 A. Hosokawa, “Development of a numerical cancellous bone model for finite-difference time-domain simulations of ultrasound propagation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1219–1233 (2008). 13 Y. Nagatani, K. Mizuno, T. Saeki, M. Matsukawa, T. Sakaguchi, and H. Hosoi, “Numerical and experimental study on the wave attenuation in bone—FDTD simulation of ultrasound propagation in cancellous bone,” Ultrasonics 48, 607–612 (2008). 14 A. Hosokawa, T. Otani, T. Suzaki, Y. Kubo, and S. Takai, “Influence of trabecular structure on ultrasonic wave propagation in bovine cancellous bone,” Jpn. J. Appl. Phys. 36, 3233–3237 (1997). 15 K. Mizuno, M. Matsukawa, T. Otani, M. Takada, I. Mano, and T. Tsujimoto, “Effects of structural anisotropy of cancellous bone on speed of ultrasonic fast waves in the bovine femur,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1480–1487 (2008). 16 K. Mizuno, H. Somiya, T. Kubo, M. Matsukawa, T. Otani, and T. Tsujimoto, “Influence of cancellous bone microstructure on two ultrasonic wave propagations in bovine femur: An in vitro study,” J. Acoust. Soc. Am. 128, 3181–3189 (2010). 17 T. Otani, “Quantitative estimation of bone density and bone quality using acoustic parameters of cancellous bone for fast and slow waves,” Jpn. J. Appl. Phys. 44, 4578–4582 (2005). 18 T. Otani, I. Mano, T. Tsujimoto, T. Yamamoto, R. Teshima, and H. Naka, “Estimation of in vivo cancellous bone elasticity,” Jpn. J. Appl. Phys. 48, 1–5 (2009).
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H. Sai, G. Iguchi, T. Tobimatsu, K. Takahashi, T. Otani, K. Horii, I. Mano, I. Nagai, H. Iio, T. Fujita, K. Yoh, and H. Baba, “Novel ultrasonic bone densitometry based on two longitudinal waves: Significant correlation with pQCT measurement values and age-related changes in trabecular bone density, cortical thickness, and elastic modulus of trabecular bone in a normal Japanese population,” Osteoporosis Int. 21, 1781–1790 (2010). 20 P. H. Nicholson, G. Lowet, C. M. Langton, J. Dequeker, and G. Van del Perre, “A comparison of time-domain and frequency-domain approaches to ultrasonic velocity measurement in trabecular bone,” Phys. Med. Biol. 41, 2421–2435 (1996). 21 K. A. Wear, “Group velocity, phase velocity, and dispersion in human calcaneus in vivo,” J. Acoust. Soc. Am. 121, 2431–2437 (2007). 22 K. A. Wear, “Cancellous bone analysis with modified least rectangles Prony’s method and chirp filter: Phantom experiments and simulation,” J. Acoust. Soc. Am. 128, 2191–2203 (2010). 23 J. J. Hoffman, A. M. Nelson, M. R. Holland, and J. G. Miller, “Cancellous bone fast and slow waves obtained with Bayesian probability theory correlate with porosity from computed tomography,” J. Acoust. Soc. Am. 132, 1830–1837 (2012). 24 S. Hasegawa, Y. Nagatani, K. Mizuno, and M. Matsukawa, “Wavelet transform analysis of ultrasonic wave propagation in cancellous bone,” Jpn. J. Appl. Phys. 49, 1–5 (2010). 25 K. I. Lee and S. W. Yoon, “Comparison of acoustic characteristics predicted by Biot’s theory and the modified Biot-Attenbrough model in cancellous bone,” J. Biomech. 39, 364–368 (2006). 26 L. Cardoso, F. Teboul, L. Sedel, C. Oddou, and A. Meunier, “In vitro acoustic waves propagation in human and bovine cancellous bone,” J. Bone Min. Res. 18, 1803–1812 (2003). 27 A. M. Nelson, J. J. Hoffman, C. C. Anderson, M. R. Holland, Y. Nagatani, K. Mizuno, M. Matsukawa, and J. G. Miller, “Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone,” J. Acoust. Soc. Am. 130, 2233–2240 (2011). 28 A. Q. Bauer, K. R. Marutyan, M. R. Holland, and J. G. Miller, “Is the Kramers-Kronig relationship between ultrasonic attenuation and dispersion maintained in the presence of apparent losses due to phase cancellation?,” J. Acoust. Soc. Am. 122, 222–228 (2007). 29 K. A. Wear, “The effect of phase cancellation on estimates of calcaneal broadband ultrasound attenuation in vivo,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 1352–1359 (2007).
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Katsunori Mizuno Underwater Technology Research Center, The University of Tokyo, Meguro-ku, Tokyo, 153-8505 Japan
Mami Matsukawaa) Laboratory of Ultrasonic Electronics, Research Center for Wave Electronics, Doshisha University, Kyotanabe, 610-0321 Kyoto, Japan
(Received 27 September 2012; revised 20 June 2013; accepted 29 July 2013) Wave propagation in a trabecular bone was experimentally investigated using an acoustic tube. For the purposes of this study, a cubic sample was gradually filed so the waveform change due to the sample thickness could be observed. The initial sample showed clear two-wave separation. As the sample became thinner, the fast and slow waves gradually overlapped. The apparent frequencies and amplitudes of the fast waves obtained from the time domain data decreased significantly for the smaller thicknesses. This indicates an increase in the apparent attenuation at the initial stage of the propagation. Next the authors investigated the distribution of the ultrasonic field after the transmission through the cancellous bone sample. In addition to a large aperture receiver, a needle-type ultrasonic transducer was used to observe the ultrasonic field. Within an area of the same size of the large transducer, the waveforms retrieved with the needle sensor exhibited high spatial variations; however, the averaged waveform in the plane was similar to the waveform obtained with the large aperture receiver. This indicates that the phase cancellation effect on the surface of the large aperture receiver can be one of the reasons for the strong apparent attenuation observed at the initial C 2013 Acoustical Society of America. stages of the propagation. V [http://dx.doi.org/10.1121/1.4824970] PACS number(s): 43.80.Cs, 43.80.Sh, 43.80.Ev, 43.80.Qf [ADP] I. INTRODUCTION
Cancellous bone is composed of a trabecular bone filled with bone marrow, and is an interesting poroelastic tissue in the human body.1 Because the bone marrow is a viscous liquid at body temperature, the cancellous bone contains both solid and liquid parts. One theory of the wave propagation in such porous media filled with liquid was first suggested by Biot. Since then, several researchers have discussed and challenged the application of this theory on the wave propagation in cancellous bone, especially concerning the two longitudinal wave propagation.2–4 The first experimental observation of two wave phenomenon was achieved by Hosokawa and Otani using a simple water-immersion technique in the megahertz range.5,6 The problem of this two wave phenomenon in cancellous bone is, however, the strong anisotropy and heterogeneity of the sample, which results in the difficulty to describe it in a theoretical model as pointed by Kaufman et al.7 Two wave phenomenon, therefore, has been extensively investigated experimentally or by numerical approaches. As for the numerical approach, after the first success of Bossy et al.8 and Nagatani et al.,9 several studies have confirmed the two wave propagation by the two-dimensional and three-dimensional (3D) a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
J. Acoust. Soc. Am. 134 (6), Pt. 2, December 2013
Pages: 4775–4781
finite-difference time-domain methods with reconstructed 3D CT images of actual or artificially constructed bones.10–13 These numerical studies have brought us clear images of complicated wave propagation and helped the understanding of this phenomenon. Actually, the heterogeneous and anisotropic structure of cancellous bone can result in the propagation of complex ultrasonic waves that make measurements and their interpretation of difficult, although the evaluation of cancellous bone structure can provide a good insight of the degree of osteoporosis.1 The wave characteristics of both fast and slow waves are critical to obtain information about structure and bone volume ratio. Because the fast wave mainly propagates in the solid trabecular frame, the wave properties reflect the state of the cancellous structure of the bone.14–16 In addition, Otani has reported a good relation between the slow wave amplitude and bone volume and introduced a new bone densitometry apparatus for clinical studies. The apparatus can give us data which have a high correlation with bone mineral density by peripheral quantitative computed tomography.17–19 Though in other circumstances the two waves substantially overlap and appear as only a single wave, in some conditions the two waves separate in the time-domain. The conventional analysis methods of experimental data, therefore, may suggest potentially misleading wave properties. Previous studies have demonstrated that interfering fast wave and slow wave modes can account for the apparent negative dispersion
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sometimes observed in the measurements of cancellous bone.20,21 The degree to which the fast waves and slow waves overlap depends on a number of factors including porosity, structural anisotropy, and the angle of insonification relative to the predominant trabecular orientation. Several studies therefore have focused on the interesting techniques to extract information of each wave from the mixed waves.22–24 The numerical approach has also given a good prediction to understand the complicated experimental data. The simulations made by Nagatani et al. have shown good agreements between experimental and numerical data of wave propagation in the identically simulated cancellous bone.13 They also pointed out the interesting initial wave propagation process in the cancellous bone, especially due to the spatial distribution of wave front. The complicated solidliquid structure of cancellous bone generates a fluctuation of wave fronts, especially the fast wave fronts, due to the multiple paths in the complex trabecular structure. Finally, the fluctuating waves are experimentally observed after the integration at the surface of the receiver. This integration may result in the very weak and highly attenuated fast wave during wave propagation and makes the fast wave measurements very difficult. The objective of this study is to investigate the propagation mechanism of the fast wave in the initial stage. Through the precise ultrasonic measurements using a pulse immersion technique, the spatial distribution of the fast wave fronts was experimentally investigated in detail. Discussions also include the apparent high attenuation of fast wave and the necessity of a separation of the fast wave from the slow wave; this is very important information for the clinical application of two wave phenomenon in the cancellous bone.
(SMX-100CT, Shimadzu, Kyoto, Japan). The spatial resolution of the 3D CT images was 41 lm. B. Ultrasonic measurements
During measurements at room temperature, the bone sample was immersed in degassed water and placed in an acoustical tube as shown in Fig. 2. The bone sample was placed at a 70 mm distance from the transmitter, because the amplitude of the ultrasonic wave was stable at this position in the acoustic tube. We set the wave propagation direction parallel to the bones main axis. Before each measurement, the specimens were degassed for 60 min to remove air bubbles trapped inside. A plane wideband polyvinylidene fluoride (PVDF) transmitter (rectangle active area 15 15 mm in size, self-made) was used. This comparatively large transducer was able to achieve a good signal-to-noise ratio of the observed waves.13 Two types of plane PVDF receivers were prepared to observe the waveforms which passed through the bone sample: (1) Rectangular type, 15 15 mm in size, self-made [Fig. 2(a)] same as the transmitter, (2) needle type, 1.0 mm in diameter (Toray Engineering, Tokyo, Japan) [Fig. 2(b)]. The latter was used to measure acoustic field in the case of a thin specimen. The distance between the two transducers was 100 mm for the rectangle receiver and 73 mm for the needle type receiver. In the case of the rectangular type transducer, we set it at the position of 100 mm to
II. MATERIALS AND METHODS A. Bone specimen preparation
A rectangular cancellous bone sample, 22.4 22.4 8.7 mm3 in size, was cut from a 36-month-old equine left radius using an oscillating saw. The size of the specimen was measured using a caliper (precision: 0.01 mm). To remove the bone marrow, the cubic samples were defatted using a water pick (HT j202, OMRON, Kyoto, Japan).5 Figure 1 shows the cross-sectional image of x-ray micro focus CT images of a sample in the bone axis direction
FIG. 1. (Color online) Trabecular specimen preparation and the scan image of the sample by x-ray micro CT. 4776
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FIG. 2. Ultrasonic measurement configurations. The wall and bottom of the acoustic tube were covered by a foam polystyrene plate to avoid the invasion of the ultrasonic wave. The receiver was a rectangular PVDF transducer or a needle type transducer. In the case of the needle type transducer, the receiver scanned in the plane normal to the wave propagation direction. Fujita et al.: Wave propagation in cancellous bone
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avoid the multiple reflection between the sample and the receiver. In order to observe the fluctuations of an ultrasonic wave front after the bone sample, the needle type receiver was located right behind the bone sample. A single cycle sinusoidal wave electric pulse at 0.7, 1.0, 1.2, or 1.5 MHz with an amplitude of 5 Vp–p (peak to peak voltage) was generated using a function generator (33250A, Agilent, CO, USA). It was amplified by 20 dB using a power amplifier (4055, NF Corporation, Kanagawa, Japan), and applied to the transmitter. The waves that passed through the sample in an acoustic tube were converted into electrical signals by the receiver and then retrieved by a digital oscilloscope (TDS 524A, Tektronix Inc., OR, USA, 500MS/S and 11 bit for the averaging mode) with a 20 dB preamplifier (5307, NF Corp., Kanagawa, Japan). The measurements were carried out using samples with different thicknesses ranging from 8.7 to 1.1 mm. The thickness was precisely controlled by polishing the sample using a polishing machine (Speed Lap, Maruto, Tokyo, Japan). In the case of spatial distribution measurements, the position of the needle receiver was moved by a 0.5 mm step in a 10 10 mm area centered on the acoustic axis to estimate the acoustic field in the tube [Fig. 2(b)]. A total of 441 measurements were performed in the plane.
amplitude. This is a simple but precise measurement which is applicable to the actual in vivo measurement system using two waves, where rapid evaluation is necessary.18,19 Following our former study, the apparent attenuations of fast and slow waves were also estimated from the observed waveforms in the time domain. To obtain the attenuation, we have used the data measured by the rectangular transducer. Here, to avoid the effects of overlapping, the amplitudes of the first small peaks were adopted to estimate the apparent fast wave attenuations. For the slow wave attenuation, the last positive peak amplitude was adopted. The method used to choose the peaks is illustrated in Fig. 3, showing how to estimate the attenuation. These amplitudes were measured from a series of samples with different thicknesses. The apparent attenuation was defined by a ¼ 20 logðVn =Vnþ1 Þ=Dd;
(1)
where Vn and Vnþ1 are the amplitudes of the peaks in the received radio frequency waveforms. The indices n and n þ 1 correspond to successive sample thicknesses differing by Dd of about 0.5 mm. The obtained value is the attenuation in the small portion of the sample. III. RESULTS AND DISCUSSION
C. Estimation of apparent wave properties from the time domain data
A. Observation of two wave phenomenon
Because fast and slow waves mostly overlap, we cannot obtain wave properties of each wave. Then, we estimated the apparent frequency and amplitude of two waves in the time domain as shown in Figs. 3(a) and 3(b). We have selected time intervals and peaks which seem to be less affected by the other wave to estimate the apparent frequency and
The observed waves which propagated in the sample have varied according to the sample thickness. At first we have confirmed the two wave propagation in the sample following the process of our former study13 using a rectangle PVDF transducer as a receiver. The observed waveforms are shown in Fig. 4. Two waves were clearly observed in the thicker sample, and
FIG. 3. The peaks and intervals (4(Tp Tt) and 2(Tz2 Tz1)) used for the estimation of apparent frequency and apparent attenuation. J. Acoust. Soc. Am., Vol. 134, No. 6, Pt. 2, December 2013
FIG. 4. Changes in observed waveforms as a function of sample thickness. Fujita et al.: Wave propagation in cancellous bone
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gradually overlapped in the thinner samples. This means a certain thickness of the trabecular bone is required to observe the two wave phenomenon under an in vivo situation. In our present case, for thicknesses under 5 mm, it seemed difficult to observe two clearly separated waves. However, the wave front or the first peak of the fast wave can still be identified in the samples with thicknesses of 1.6 or 1.1 mm, although most parts of the fast wave is superposed with the following slow wave. This characteristic change of waveform was in good accordance with our past data using bovine cancellous bone.13 B. Apparent wave properties—effect of time domain analysis
The wave properties of these two waves have already been discussed by several researchers.25,26 Recently, new techniques to separate waves have been applied to the overlapped waves and have given us properties of each wave. Especially, the application of the Bayesian technique has almost succeeded in the estimation of fast and slow wave attenuation properties.27 The analysis of separated waves showed that the attenuation coefficients at a fixed frequency were constant during wave propagation. The separation of waves, however, is still difficult during in vivo clinical measurements, because complicated and long processes are necessary. In the in vivo measurements, therefore, the time domain data are often used to analyze the characteristics of two waves.18,19
FIG. 5. Apparent frequencies of observed waves as a function of sample thickness. (a) Fast wave and (b) slow wave. 4778
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During propagation, the wave amplitude in the time domain decreased rapidly in the initial state of wave propagation, together with an increase in the wave period. As a result, the apparent frequency decreased as illustrated in Figs. 5(a) and 5(b). One can clearly see that the decrease in the apparent frequency (increase of apparent wave period) of the fast wave varied together with the thickness of the sample, whereas the apparent slow wave frequency did not seem to display high variations. This tendency was similar in all measurement data with different input signal frequencies. Here, the small decrease in the slow wave attenuation with thickness lower than 2.5 mm possibly comes from the effects of overlapping fast waves, because the overlap of fast and slow waves was strong with thicknesses lower than 4 mm. The effects of the overlap might also affect the apparent frequency of the slow wave. Therefore, a precise understanding of the initial propagation of the slow wave seemed difficult with these time domain data. The apparent fast wave attenuation data also exhibits an interesting behavior: the fast wave strongly attenuates in the initial stage of the propagation as shown in Fig. 6(a). This is in good accordance with our former bovine data and consistent with the results obtained by Nelson et al.27 In this study, a broadband pulse wave was used as the input signal. Because the attenuation coefficients of both fast and slow waves increase with frequency, the amplitudes of higher frequency components are more attenuated than the lower frequency components. As the wave propagation proceeds, the nominal frequency of the pulse wave becomes lower, plus
FIG. 6. Apparent attenuation coefficients at each portion of the sample. (a) Fast wave and (b) slow wave. Fujita et al.: Wave propagation in cancellous bone
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In addition to the problem of time domain analysis, we should also take into account the effects of the complex trabecular structure of the sample. Especially in very thin trabecular samples, the propagating wave front is generally misaligned due to the wave propagating in the trabecular structure and the wave velocity difference between hard trabeculae (assuming approximately 3600 m/s from the cortical bone studies)1 and liquid (in vivo: Marrow; present experiments: Water). Actually, the initial irregular wave front has clearly been shown by several studies.11,13,28 Because the aperture of the receiver used in the ultrasonic system is large compared to the wavelength, the spatial fluctuation of the phase at the complex wave front is likely to be canceled at the receiver face. This phase cancellation is another mechanism which is responsible for the apparent drop in the wave amplitude. This kind of cancellation has been pointed out by Wear29 and Anderson et al.27 In the present data, the decrease in the apparent fast wave attenuation in each portion of the sample was prominent when the sample thickness was less than 4 mm. We measured the fluctuations of the wave front using a needle type receiver, whose diameter of active area was only 1 mm.
Figure 7 shows waves observed in water at different positions in the plane perpendicular to the wave propagation direction. We can confirm that waveforms are similar at points ‹ and . This proves the acoustic field in the tube was symmetrical. Figure 8 shows waves observed after passing through a thin sample (thickness 1.1 mm). We can observe clear changes in the waveforms due to the position, and also see the fluctuations of the wave front. These fluctuations become much clearer when looking at the distribution of arrival times of the waves in the measurement plane (Fig. 9). Here, the arrival time of an individual wave is obtained by taking the initial rising time of the wave front at the amplitude of 10% of the slow wave maximum. Considering the diameter of the needle type receiver (1 mm) and the diameter of a trabecula (100 to 200 lm), the spatial resolution of the measurement is not yet perfect; however, there were dramatic fluctuations of the arrival time at the initial stage of wave propagation. The standard deviations of the arrival time in one plane (441 measurement points) were 0.12 and 0.09 ls, for the sample with thicknesses of 1.1 and 1.6 mm, respectively. Figure 9 also shows x-ray micro CT data of the sample. We then divided the images in Fig. 9 into nine parts and obtained the total area of trabeculae and averaged arrival time. The correlation between the data obtained from the two images was around R2 ¼ 0.73 (p < 0.01), telling that the fluctuation depends on the trabecular structure. The summation of waves measured in one plane also brings us interesting results. Figure 10 shows the comparison of observed waves by the rectangular receiver and the summation of all measured waves in one plane by the needle type receiver. In both cases of 3.5 and 1.1 mm thicknesses, the waveform obtained by the rectangular transducer and the
FIG. 7. Ultrasonic waves observed at different positions in the plane perpendicular to the wave propagation direction in water.
FIG. 8. Ultrasonic waves observed at different positions in the plane perpendicular to the wave propagation direction, after passing through the thin sample (thickness 1.1 mm).
the apparent attenuation seems to decrease because the attenuation coefficient becomes smaller. This is directly reflected in the first peak amplitude of the fast wave and often results in a apparent attenuation decrease in the initial stage of the propagation as Nelson et al. have already discussed.27 In addition, multiple reflections in a thin sample possibly affects the waveforms. C. Effects of rectangular aperture of the receiver
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summation of the waveforms obtained by the needle sensor was very similar. Compared with the observed waves at each position by the needle type receiver, the apparent nominal frequencies of the summation waves seemed to decrease, which came from the superposition of different waves at different positions in the plane. This data actually proves the effect of the large aperture of the receiver on the measurement and shows that the observed wave properties also depend on the spatial averaging effect in the measurement system. This effect should be taken into account in the in vivo time domain system to clearly define the properties of the waves which propagated in the cancellous bone.
IV. CONCLUSION
FIG. 9. (Color online) Distribution of the wave front in a plane after passing through the thin sample (thickness 1.1 mm) with micro CT data.
Ultrasonic wave propagation in an equine cancellous bone was experimentally investigated using an acoustic tube. First, using a rectangular large aperture receiver, the twowave phenomenon was clearly observed. The propagation was then followed by changing the sample thickness and it was found that the two waves were clearly separated in the samples with thicknesses larger than 5 mm. Although the structure and bone characteristics of the equine bone are possibly different from the human bone, these data hint that a certain thickness of the trabecular bone might be required in order to clearly observe and exploit the two wave phenomenon in vivo. The characteristic decrease in the apparent fast wave attenuation was obvious in the initial stage of the wave propagation. This tendency was in agreement with our past bovine data. The decrease of the attenuation in very thin samples was then discussed. One possible reason is the effect of the large aperture receiver. We experimentally showed the strong fluctuations of the wave front in the initial stage of the propagation, which were then canceled at the surface of the comparatively large aperture receiver. ACKNOWLEDGMENTS
Part of this work was supported by the Regional Innovation Strategy Support Program of Ministry of Education, Culture, Sports, Science and Technology, and Japan and Grant-in-Aid for Scientific Research (B) of the Japan Society for the Promotion of Science. The authors would like to thank Professor Yoshiki Nagatani at Kobe City College of Technology and Kevann Esmaeili at Doshisha University for their valuable scientific advice. 1
FIG. 10. Comparison of ultrasonic waves obtained by the different receivers: (a) The waves measured by the rectangular large receiver, and (b) the summation of all measured waves in the plane by the needle type receiver. 4780
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P. Laugier and G. Haiat, editors, Bone Quantitative Ultrasound (Springer, Dordrecht, Netherlands, 2011), pp. 1–6. 2 T. J. Plona, “Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies,” Appl. Phys. Lett. 36, 259–261 (1980). 3 J. G. Berryman, “Confirmation of Biot’s theory,” Appl. Phys. Lett. 37, 382–384 (1980). 4 R. Lakes, H. S. Yoon, and J. L. Katz, “Slow compressional wave propagation in wet human and bovine cortical bone,” Science 220, 513–515 (1983). 5 A. Hosokawa and T. Otani, “Ultrasonic wave propagation in bovine cancellous bone,” J. Acoust. Soc. Am. 101, 558–562 (1997). 6 A. Hosokawa and T. Otani, “Acoustic anisotropy in bovine cancellous bone,” J. Acoust. Soc. Am. 103, 2718–2722 (1998). Fujita et al.: Wave propagation in cancellous bone
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J. J. Kaufmann, G. Luo, and R. S. Siffert, “Ultrasound simulation in bone,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1205–1218 (2008). 8 E. Bossy, F. Padilla, F. Peyrin, and P. Laugier, “Three dimensional simulation of ultrasound propagation through trabecular bone structure measured by synchrotron microtomography,” Phys. Med. Biol. 50, 5545–5556 (2005). 9 Y. Nagatani, H. Imaizumi, T. Fukuda, M. Matsukawa, Y. Watanabe, and T. Otani, “Applicability of finite-difference time-domain method to simulation of wave propagation in cancellous bone,” Jpn. J. Appl. Phys. 45(1), 7186–7190 (2006). 10 E. Bossy, P. Laugier, F. Peyrin, and F. Padilla, “Attenuation in trabecular bone: A comparison between numerical simulation and experimental results in human femur,” J. Acoust. Soc. Am. 22, 2469–2475 (2007). 11 G. Ha€ıat, F. Padilla, F. Peyrin, and P. Laugier, “Fast wave ultrasonic propagation in trabecula bone: Numerical study of the influence of porosity and structural anisotropy,” J. Acoust. Soc. Am. 123, 1694–1705 (2008). 12 A. Hosokawa, “Development of a numerical cancellous bone model for finite-difference time-domain simulations of ultrasound propagation,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1219–1233 (2008). 13 Y. Nagatani, K. Mizuno, T. Saeki, M. Matsukawa, T. Sakaguchi, and H. Hosoi, “Numerical and experimental study on the wave attenuation in bone—FDTD simulation of ultrasound propagation in cancellous bone,” Ultrasonics 48, 607–612 (2008). 14 A. Hosokawa, T. Otani, T. Suzaki, Y. Kubo, and S. Takai, “Influence of trabecular structure on ultrasonic wave propagation in bovine cancellous bone,” Jpn. J. Appl. Phys. 36, 3233–3237 (1997). 15 K. Mizuno, M. Matsukawa, T. Otani, M. Takada, I. Mano, and T. Tsujimoto, “Effects of structural anisotropy of cancellous bone on speed of ultrasonic fast waves in the bovine femur,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55, 1480–1487 (2008). 16 K. Mizuno, H. Somiya, T. Kubo, M. Matsukawa, T. Otani, and T. Tsujimoto, “Influence of cancellous bone microstructure on two ultrasonic wave propagations in bovine femur: An in vitro study,” J. Acoust. Soc. Am. 128, 3181–3189 (2010). 17 T. Otani, “Quantitative estimation of bone density and bone quality using acoustic parameters of cancellous bone for fast and slow waves,” Jpn. J. Appl. Phys. 44, 4578–4582 (2005). 18 T. Otani, I. Mano, T. Tsujimoto, T. Yamamoto, R. Teshima, and H. Naka, “Estimation of in vivo cancellous bone elasticity,” Jpn. J. Appl. Phys. 48, 1–5 (2009).
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H. Sai, G. Iguchi, T. Tobimatsu, K. Takahashi, T. Otani, K. Horii, I. Mano, I. Nagai, H. Iio, T. Fujita, K. Yoh, and H. Baba, “Novel ultrasonic bone densitometry based on two longitudinal waves: Significant correlation with pQCT measurement values and age-related changes in trabecular bone density, cortical thickness, and elastic modulus of trabecular bone in a normal Japanese population,” Osteoporosis Int. 21, 1781–1790 (2010). 20 P. H. Nicholson, G. Lowet, C. M. Langton, J. Dequeker, and G. Van del Perre, “A comparison of time-domain and frequency-domain approaches to ultrasonic velocity measurement in trabecular bone,” Phys. Med. Biol. 41, 2421–2435 (1996). 21 K. A. Wear, “Group velocity, phase velocity, and dispersion in human calcaneus in vivo,” J. Acoust. Soc. Am. 121, 2431–2437 (2007). 22 K. A. Wear, “Cancellous bone analysis with modified least rectangles Prony’s method and chirp filter: Phantom experiments and simulation,” J. Acoust. Soc. Am. 128, 2191–2203 (2010). 23 J. J. Hoffman, A. M. Nelson, M. R. Holland, and J. G. Miller, “Cancellous bone fast and slow waves obtained with Bayesian probability theory correlate with porosity from computed tomography,” J. Acoust. Soc. Am. 132, 1830–1837 (2012). 24 S. Hasegawa, Y. Nagatani, K. Mizuno, and M. Matsukawa, “Wavelet transform analysis of ultrasonic wave propagation in cancellous bone,” Jpn. J. Appl. Phys. 49, 1–5 (2010). 25 K. I. Lee and S. W. Yoon, “Comparison of acoustic characteristics predicted by Biot’s theory and the modified Biot-Attenbrough model in cancellous bone,” J. Biomech. 39, 364–368 (2006). 26 L. Cardoso, F. Teboul, L. Sedel, C. Oddou, and A. Meunier, “In vitro acoustic waves propagation in human and bovine cancellous bone,” J. Bone Min. Res. 18, 1803–1812 (2003). 27 A. M. Nelson, J. J. Hoffman, C. C. Anderson, M. R. Holland, Y. Nagatani, K. Mizuno, M. Matsukawa, and J. G. Miller, “Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone,” J. Acoust. Soc. Am. 130, 2233–2240 (2011). 28 A. Q. Bauer, K. R. Marutyan, M. R. Holland, and J. G. Miller, “Is the Kramers-Kronig relationship between ultrasonic attenuation and dispersion maintained in the presence of apparent losses due to phase cancellation?,” J. Acoust. Soc. Am. 122, 222–228 (2007). 29 K. A. Wear, “The effect of phase cancellation on estimates of calcaneal broadband ultrasound attenuation in vivo,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 1352–1359 (2007).
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