Ultrasonics 54 (2014) 1245–1250
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Two-wave behavior under various conditions of transition area from cancellous bone to cortical bone Yoshiki Nagatani a,⇑, Katsunori Mizuno b, Mami Matsukawa c a
Department of Electronics, Kobe City College of Technology, Kobe 651-2194, Japan Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan c Laboratory of Ultrasonic Electronics, Doshisha University, Kyotanabe, Kyoto 610-0321, Japan b
a r t i c l e
i n f o
Article history: Available online 12 November 2013 Keywords: Cancellous bone Two-wave phenomenon FDTD simulation
a b s t r a c t The two-wave phenomenon, the wave separation of a single ultrasonic pulse in cancellous bone, is expected to be a useful tool for the diagnosis of osteoporosis. However, because actual bone has a complicated structure, precise studies on the effect of transition conditions between cortical and cancellous parts are required. This study investigated how the transition condition influenced the two-wave generation using three-dimensional X-ray CT images of an equine radius and a three-dimensional simulation technique. As a result, any changes in the boundary between cortical part and trabecular part, which gives the actual complex structure of bone, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. The results led us to the possibility of using the two-wave phenomenon in a diagnostic system for osteoporosis in cases of a complex boundary. Ó 2013 Elsevier B.V. All rights reserved.
1. Background In porous media, propagation of two kinds of compressional wave, called the two-wave phenomenon, has been reported [1–4]. Since the wave separation behavior of a single ultrasonic pulse in cancellous bone was first reported by Hosokawa and Otani [5], this phenomenon has been considered a useful tool for the diagnosis of osteoporosis. The two waves propagate mainly in solid portion or liquid portion respectively; thus, the received wave contains information of their respective propagation media in addition to that obtained by the conventional speed of sound (SOS)/broadband ultrasound attenuation (BUA) method [5–7]. Because cancellous bone has a complicated porous structure, the detailed behavior of ultrasound in cancellous bone, including the effect of structural anisotropy, has been investigated [8–11]. Supporting these experimental studies, a simulation technique making use of the actual bone structure revealed the mechanism of the two-wave phenomenon [9,12–17]. In the in vivo assessment of bone, cancellous bone is always surrounded by cortical bone. Typically, the transition region between cortical part and trabecular part is not sharp [18,19]. Valentinitsch et al. assessed the spatial distribution and connectivity of different regions of human distal radius [19]. Around the transition region, ⇑ Corresponding author. Tel.: +81 78 795 3247. E-mail addresses: [email protected] (Y. Nagatani), [email protected]. ac.jp (K. Mizuno), [email protected] (M. Matsukawa). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.10.016
the bone volume fraction gradually decreases from the cortical side to the trabecular side, and it is mostly difficult to decide a clear border line. In previous studies, some simple methods to separate the cortical part and the trabecular part using CT or MR images were proposed [20,21]. Although the influence of the transition region is not negligible for the use in vivo measurement because the ultrasonic wave propagation in the transition region is considered to be very complicated, few quantitative evaluations of geometrical properties of the transition region was reported and only a few investigations using the specimen sealed by cortical bone have been performed [22–24]. Some studies have claimed that the two waves were rarely detected in the case of sealed pores. However we have found that the two-wave phenomenon occurs under artificial closed pore conditions in a comparative study of simulation and experimental measurements using a single specimen, suggesting that the sharp transition at the boundary did not cause extinction of the slow wave [25,26]. In addition, Hosokawa et al. showed that the transition condition between the cancellous and cortical bone regions affect both the fast and slow waves [27]. Because Hosokawa et al. investigated the wave propagation using simplified geometrical models of bone, more precise studies on the effect of gradual transition using more realistic bone models are required for a detailed understanding of the two-wave phenomenon. In this study, therefore, using three dimensional X-ray CT images of actual trabecular bone specimen, we investigated the effects of the transition
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condition between cortical and cancellous parts by simulating the detailed wave propagation in the model. 2. Materials and methods This study used the same cancellous bone, obtained from the distal part of the left radius of a racehorse, as was used in a recent study [26]. The bone volume fraction (BV/TV) of the equine bone sample was similar with human bone, however, the degree of
anisotropy (DA) was higher than that of human bone [7,26]. The high DA values result in clear two wave separation. It helps us to investigate the effects of transition conditions on the two wave behavior. In the simulation part of the study, the boundaries were sealed with cortical bone plates. The size of the ROI (region of interest) of the specimen was 15 15 13 mm3, and the thicknesses of the plate-like cortical bones were set to 2.0 mm [28,29]. The spatial resolution of the 3-D computed tomography (CT) images was 48.0 lm, and the size of simulation field was
(a)
(b)
Fig. 1. (a) The three parameters used to define changes in the transition condition between cortical and cancellous parts. The black portion in the figure indicates the trabeculae and cortical parts. The incident wave is transmitted from the top area. (b) Example of the effect of the each parameter P1, P2, and P3. The figures show the crosssection diagrams of the models when P1, P2, and P3 were fixed at certain values, respectively.
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24 24 26 mm3, including the ROI. The time step of the simulation was 5.0 ns. The trabeculae were well aligned along the direction of wave propagation. The mean bone volume fraction (trabecular volume (TV)–bone volume (BV) ratio) of the model was about 20%. A clear distribution of BV/TV inside this model was not observed. Based on this original model, the structure of the transition region between cortical and cancellous bone was changed in this simulation study; the diameter of trabecula in the transition region was gradually varied, which results in the gradation of BV/TV in the direction of wave propagation. While Hosokawa et al. assumed three simple parameters to describe the artificial transition conditions [27], it is not suitable in our investigation because we use actual bone models, where the trabecular size is not constant, instead of simple artificial models. Therefore, we newly defined following three parameters to control the transition conditions, which can be interpreted as a different shape of trabeculae near the cortical layer, as shown in Fig. 1a: P1: thickness of transition region (mm) (distance from the contact surface between transition region and cortical part), P2: BV/TV value at the contact surface between transition region and cortical part (% (or mm2/mm2)),
(a)
P1 = 0.0mm
Table 1 Parameters used in the FDTD simulation. Material 3
3
Density (10 kg/m ) Velocity (m/s)
Longitudinal Shear k
Lame’ s constant (GPa)
l
Water
Trabecular bone
1.0 1483 – 2.2 –
2.0 4405 2168 20.0 9.4
P3: spatial distribution of trabecular radius along the direction of wave propagation (mm/mm). The parameter P1 corresponds to d3 (the thickness within which the porosity varies in the cancellous bone region) in the report by Hosokawa and Nagatani [27], where they varied the parameter from 0 to 8 mm. The parameters P2 and P3 are expected to have the similar effects of hs1 (trabecular size in cancellous bone part) and hs2 (trabecular size at the boundary interfacial surface between transition region of trabecular and the cortical layer nearby) in the paper. As can easily be inferred, these three parameters P1, P2, and P3 are directly connected: the combination of P1 and P2 determines P3. Fig. 1b shows the examples of the effect of the each
2.4mm
4.8mm
P1 [mm] Arrival time of primary wave [µs]
(b)
0.0 -0.1 -0.2 -0.3 -0.4 -0.5
Amplitude of primary wave [dB]
90% 70% 50% 0
(c)
1
2
3
4
5
2
3
4
5
2
3
4
5
P2 = 90% 70% 50%
10 5 0 0
(d) Amplitude of secondary wave [dB]
P2 =
1
0 -5 P2 =
-10
90% 70% 50%
-15 0
1
P2 [%] P1 [mm]
Fig. 2. The effects of P1 and P2 on the peak amplitudes of slow waves. (a) A cross-section of the 3-D model; (b) the peak value of the slow wave, whose amplitude was normalized to the results at P1 = 0.0 mm. The error bar indicates the standard deviation of the data at 5 points.
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parameter P1, P2, and P3. Here, the shape of the cortical part was always same. A simulation algorithm of 3-D elastic finite-difference timedomain (FDTD) method was used to calculate wave propagation in bone specimens. The FDTD simulation software was written by our group, and its reliability was confirmed by a comparison with experimental data [13–15]. The parameters used in this simulation are shown in Table 1. A single sinusoidal pulse wave at 1 MHz was transmitted from a concave transducer, whose focal point was fixed to the frontal surface of the specimen [25]. Using these models, wave propagation was simulated by changing the focal point (the center of specimen and 4 other points (±5 mm distant from the center in x or y directions)). The plane receiver was placed opposite the transmitter.
received waveform of the model, which passed through the sharp (rapid) transition, did not show a perfect separation into fast wave and slow wave. Therefore, it is difficult to discuss the effects of the individual parameters on the secondary wave explicitly, because of a possible interference of two discriminative waves although some trials to separate the two waves have been performed [30–32]. In addition, the amplitude of the secondary wave may be interfered by the tail part of the primary wave which has lower frequency components [33]. On the contrary, the front part of the primary wave is less affected by waves propagated in various paths. We confirmed that there was no significant interference of other waves before the first peak of the primary arriving waves using instantaneous frequency analysis [32,34]. Fig. 2a shows the cross-sectional diagram of the processed 3-D model. The thickening trabeculae can be seen. Fig. 2b–d show relationships between P1 and the difference of the arrival time of the primary wave, the peak amplitude of the primary arriving wave, which includes so-called fast wave, and secondary arriving wave, which includes so-called slow wave, at P2 values of 90%, 70%, and 50%. The values were normalized to the results at P1 = 0.0 mm
3. Results and discussions First, the effect of P1 (gradation length [mm]) and P2 (BV/TV [%] at the interface) on the peak amplitude of the secondary wave, which includes the slow wave, was investigated. Note that the
P3 = 0.1
(a)
0.2
0.4
P1 = 2.4mm (fixed) P2 [%] Arrival time of primary wave [µs]
(b)
0.0 -0.1 -0.2 -0.3 -0.4 0.0
(c)
0.1
0.2
0.3
0.4
Amplitude of waves [dB]
5
P2 [%]
(d)
0 -5 -10
Primary Wave Secondary Wave
-15 0.0
0.1
0.0
0.1
0.2
0.3
0.4
0.3
0.4
100 80 60 40 20 0 0.2 P3 [mm/mm]
Fig. 3. The effect of P3 with a fixed P1 value of 2.4 mm. (a) The models; (b) the peak amplitude of slow waves; (c) the P2 value and (d) the received waveforms.
Y. Nagatani et al. / Ultrasonics 54 (2014) 1245–1250
(the condition in the original structure). The error bar indicates the standard deviation of the data at five points. The Fig. 2b shows a clear monotonous decrease of arrival time which means the increase of the primary wave speed. Here, the amplitudes of the primary wave were not greatly changed regardless of the conditions of gradation length P1. In contrast, the amplitude of the secondary waves decreased monotonously. These characteristics of two different kinds of waves are similar to the results of the simulation data and experimentally observed data reported by our group [27]. We pointed that this tendency came from the nature that fast and slow waves respectively propagate mainly in the trabecular part and pore part. Although the detailed behavior of the waves in the complex condition of the transition region should be confirmed carefully in the future work, our results support the previous understandings of the two wave propagation mechanism. In addition, even when the gradation length was very long (P1 = 4.8 mm), very small secondary waves could be seen (< 10 to 15 dB). Next, the effect of P3 (spatial distribution of BV/TV) was investigated. Fixing P1 (the thickness of transition region) at 2.4 mm, P3 was varied from 0.0 to 0.4 mm/mm. The condition at P3 = 0.0 mm/mm indicates the original model (i.e. without processing). The results when the focal point was set on the center of specimen are shown in Fig. 3. Fig. 3b and c shows the relationship between P3 and the arrival time of primary wave, the normalized peak amplitude of primary waves and secondary waves, respectively. When P3 is about 0.1 mm, the amplitude of primary wave shows a maximum and the amplitude of secondary wave shows a minimum. Then the amplitudes of primary wave and secondary wave are restored to the original values with an increase of P3. Regardless of this interesting behavior, the arrival time of the primary wave monotonously decreases, telling the increase in the speed. This discriminative tendency was also seen in the report by Hosokawa et al. using simple stratified models. Fig. 3d shows the relationship between P3 and P2. The approximate P2 value at the peak point was about 55%. The waveforms under each condition, shown in Fig. 4, clearly reveal that the speed of the secondary wave monotonously becomes higher with the increase of P3, in contrast to the changes in the amplitude. The primary wave also speeds up with increasing P3. In these simulations, where the trabeculae in the specimen were aligned clearly, any changes in the parameters did not
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eliminate the generation of either the primary wave or the secondary wave. In addition, in case of the thick cortical area of 2.4 mm (whole P1 area was cortical), the amplitude of secondary wave was 2.3 dB smaller than that of the original model (the gray dotted line in Fig. 4). As many literatures have pointed, regardless of the behavior that the fast longitudinal wave propagates stably in dense media (high BV/TV) and the slower wave propagates dominantly in pore media (low BV/TV), the two waves can be seen when the wave propagates parallel to the trabecular alignment. The results of the model of this paper, which have gradual change of trabecular thickness, also showed the two waves. Fig. 4 shows that the amplitude of the primary wave was small and the amplitude of the secondary wave was large when the model had sharp boundary, and vice versa. This interesting phenomenon can be interpreted as the result of the behavior that the wave in liquid portion gradually enters into solid portion while propagating in the transition region. This behavior may cause the smaller amplitude of the secondary wave. On the contrary, when the structure in the transition is sharp, which is essentially equivalent to the situation of the rapid growth of the trabecular thickness toward the interface (i.e. P3 is large), the propagation of the secondary wave is less interfered by the wave diffusion in the transition region. These observations emphasize the relevance of the two-wave measurement and encourage the use of the two-wave phenomenon in practical in vivo applications. However, the authors are aware of the limitations of this report because the results came from only one specimen and the validity of the modeling algorithm was not confirmed quantitatively. Therefore, the generality should be confirmed in the future work using a number of specimens including the one which have weak or unclear alignment and more realistic geometrical form of human bones. Moreover, the symmetric property of two waves possibly brings us novel information about the transition region, which might be useful for the diagnosis of osteoporosis because not only the bone density but also the trabecular structure changes as the progression of osteoporosis [19,21]. The interesting behavior of two waves including amplitudes, frequency characteristics, propagation speeds, and scattering patterns also should be investigated carefully in the forthcoming study. 4. Conclusion We have confirmed the two-wave phenomenon in bone with smooth transition between the cortical and cancellous parts using three dimensional X-ray CT images of a radius of a racehorse. Any change in the transition region between cortical part and trabecular part, which has the actual complex structure of bone specimen, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. This result leads us to the possibility of using the two-wave phenomenon as a diagnostic system for osteoporosis in the in vivo situation where the transition condition may be complex. For practical use, a precise investigation based on a quantitative estimation of real human bone is required. Acknowledgments
Fig. 4. The received waveforms with various P3 values with the fixed P1 value of 2.4 mm. The gray dotted line indicates the result when the whole P1 area was cortical bone without any trabeculae.
We gratefully acknowledge helpful discussions with Professor Emeritus Takahiko Otani at Doshisha University. This study was supported in part by KAKENHI Grant Number 24360161 and 25871038 from the Japan Society for the Promotion of Science (JSPS) and the Regional Innovation Strategy Support Program of the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Two-wave behavior under various conditions of transition area from cancellous bone to cortical bone Yoshiki Nagatani a,⇑, Katsunori Mizuno b, Mami Matsukawa c a
Department of Electronics, Kobe City College of Technology, Kobe 651-2194, Japan Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan c Laboratory of Ultrasonic Electronics, Doshisha University, Kyotanabe, Kyoto 610-0321, Japan b
a r t i c l e
i n f o
Article history: Available online 12 November 2013 Keywords: Cancellous bone Two-wave phenomenon FDTD simulation
a b s t r a c t The two-wave phenomenon, the wave separation of a single ultrasonic pulse in cancellous bone, is expected to be a useful tool for the diagnosis of osteoporosis. However, because actual bone has a complicated structure, precise studies on the effect of transition conditions between cortical and cancellous parts are required. This study investigated how the transition condition influenced the two-wave generation using three-dimensional X-ray CT images of an equine radius and a three-dimensional simulation technique. As a result, any changes in the boundary between cortical part and trabecular part, which gives the actual complex structure of bone, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. The results led us to the possibility of using the two-wave phenomenon in a diagnostic system for osteoporosis in cases of a complex boundary. Ó 2013 Elsevier B.V. All rights reserved.
1. Background In porous media, propagation of two kinds of compressional wave, called the two-wave phenomenon, has been reported [1–4]. Since the wave separation behavior of a single ultrasonic pulse in cancellous bone was first reported by Hosokawa and Otani [5], this phenomenon has been considered a useful tool for the diagnosis of osteoporosis. The two waves propagate mainly in solid portion or liquid portion respectively; thus, the received wave contains information of their respective propagation media in addition to that obtained by the conventional speed of sound (SOS)/broadband ultrasound attenuation (BUA) method [5–7]. Because cancellous bone has a complicated porous structure, the detailed behavior of ultrasound in cancellous bone, including the effect of structural anisotropy, has been investigated [8–11]. Supporting these experimental studies, a simulation technique making use of the actual bone structure revealed the mechanism of the two-wave phenomenon [9,12–17]. In the in vivo assessment of bone, cancellous bone is always surrounded by cortical bone. Typically, the transition region between cortical part and trabecular part is not sharp [18,19]. Valentinitsch et al. assessed the spatial distribution and connectivity of different regions of human distal radius [19]. Around the transition region, ⇑ Corresponding author. Tel.: +81 78 795 3247. E-mail addresses: [email protected] (Y. Nagatani), [email protected]. ac.jp (K. Mizuno), [email protected] (M. Matsukawa). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.10.016
the bone volume fraction gradually decreases from the cortical side to the trabecular side, and it is mostly difficult to decide a clear border line. In previous studies, some simple methods to separate the cortical part and the trabecular part using CT or MR images were proposed [20,21]. Although the influence of the transition region is not negligible for the use in vivo measurement because the ultrasonic wave propagation in the transition region is considered to be very complicated, few quantitative evaluations of geometrical properties of the transition region was reported and only a few investigations using the specimen sealed by cortical bone have been performed [22–24]. Some studies have claimed that the two waves were rarely detected in the case of sealed pores. However we have found that the two-wave phenomenon occurs under artificial closed pore conditions in a comparative study of simulation and experimental measurements using a single specimen, suggesting that the sharp transition at the boundary did not cause extinction of the slow wave [25,26]. In addition, Hosokawa et al. showed that the transition condition between the cancellous and cortical bone regions affect both the fast and slow waves [27]. Because Hosokawa et al. investigated the wave propagation using simplified geometrical models of bone, more precise studies on the effect of gradual transition using more realistic bone models are required for a detailed understanding of the two-wave phenomenon. In this study, therefore, using three dimensional X-ray CT images of actual trabecular bone specimen, we investigated the effects of the transition
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condition between cortical and cancellous parts by simulating the detailed wave propagation in the model. 2. Materials and methods This study used the same cancellous bone, obtained from the distal part of the left radius of a racehorse, as was used in a recent study [26]. The bone volume fraction (BV/TV) of the equine bone sample was similar with human bone, however, the degree of
anisotropy (DA) was higher than that of human bone [7,26]. The high DA values result in clear two wave separation. It helps us to investigate the effects of transition conditions on the two wave behavior. In the simulation part of the study, the boundaries were sealed with cortical bone plates. The size of the ROI (region of interest) of the specimen was 15 15 13 mm3, and the thicknesses of the plate-like cortical bones were set to 2.0 mm [28,29]. The spatial resolution of the 3-D computed tomography (CT) images was 48.0 lm, and the size of simulation field was
(a)
(b)
Fig. 1. (a) The three parameters used to define changes in the transition condition between cortical and cancellous parts. The black portion in the figure indicates the trabeculae and cortical parts. The incident wave is transmitted from the top area. (b) Example of the effect of the each parameter P1, P2, and P3. The figures show the crosssection diagrams of the models when P1, P2, and P3 were fixed at certain values, respectively.
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24 24 26 mm3, including the ROI. The time step of the simulation was 5.0 ns. The trabeculae were well aligned along the direction of wave propagation. The mean bone volume fraction (trabecular volume (TV)–bone volume (BV) ratio) of the model was about 20%. A clear distribution of BV/TV inside this model was not observed. Based on this original model, the structure of the transition region between cortical and cancellous bone was changed in this simulation study; the diameter of trabecula in the transition region was gradually varied, which results in the gradation of BV/TV in the direction of wave propagation. While Hosokawa et al. assumed three simple parameters to describe the artificial transition conditions [27], it is not suitable in our investigation because we use actual bone models, where the trabecular size is not constant, instead of simple artificial models. Therefore, we newly defined following three parameters to control the transition conditions, which can be interpreted as a different shape of trabeculae near the cortical layer, as shown in Fig. 1a: P1: thickness of transition region (mm) (distance from the contact surface between transition region and cortical part), P2: BV/TV value at the contact surface between transition region and cortical part (% (or mm2/mm2)),
(a)
P1 = 0.0mm
Table 1 Parameters used in the FDTD simulation. Material 3
3
Density (10 kg/m ) Velocity (m/s)
Longitudinal Shear k
Lame’ s constant (GPa)
l
Water
Trabecular bone
1.0 1483 – 2.2 –
2.0 4405 2168 20.0 9.4
P3: spatial distribution of trabecular radius along the direction of wave propagation (mm/mm). The parameter P1 corresponds to d3 (the thickness within which the porosity varies in the cancellous bone region) in the report by Hosokawa and Nagatani [27], where they varied the parameter from 0 to 8 mm. The parameters P2 and P3 are expected to have the similar effects of hs1 (trabecular size in cancellous bone part) and hs2 (trabecular size at the boundary interfacial surface between transition region of trabecular and the cortical layer nearby) in the paper. As can easily be inferred, these three parameters P1, P2, and P3 are directly connected: the combination of P1 and P2 determines P3. Fig. 1b shows the examples of the effect of the each
2.4mm
4.8mm
P1 [mm] Arrival time of primary wave [µs]
(b)
0.0 -0.1 -0.2 -0.3 -0.4 -0.5
Amplitude of primary wave [dB]
90% 70% 50% 0
(c)
1
2
3
4
5
2
3
4
5
2
3
4
5
P2 = 90% 70% 50%
10 5 0 0
(d) Amplitude of secondary wave [dB]
P2 =
1
0 -5 P2 =
-10
90% 70% 50%
-15 0
1
P2 [%] P1 [mm]
Fig. 2. The effects of P1 and P2 on the peak amplitudes of slow waves. (a) A cross-section of the 3-D model; (b) the peak value of the slow wave, whose amplitude was normalized to the results at P1 = 0.0 mm. The error bar indicates the standard deviation of the data at 5 points.
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parameter P1, P2, and P3. Here, the shape of the cortical part was always same. A simulation algorithm of 3-D elastic finite-difference timedomain (FDTD) method was used to calculate wave propagation in bone specimens. The FDTD simulation software was written by our group, and its reliability was confirmed by a comparison with experimental data [13–15]. The parameters used in this simulation are shown in Table 1. A single sinusoidal pulse wave at 1 MHz was transmitted from a concave transducer, whose focal point was fixed to the frontal surface of the specimen [25]. Using these models, wave propagation was simulated by changing the focal point (the center of specimen and 4 other points (±5 mm distant from the center in x or y directions)). The plane receiver was placed opposite the transmitter.
received waveform of the model, which passed through the sharp (rapid) transition, did not show a perfect separation into fast wave and slow wave. Therefore, it is difficult to discuss the effects of the individual parameters on the secondary wave explicitly, because of a possible interference of two discriminative waves although some trials to separate the two waves have been performed [30–32]. In addition, the amplitude of the secondary wave may be interfered by the tail part of the primary wave which has lower frequency components [33]. On the contrary, the front part of the primary wave is less affected by waves propagated in various paths. We confirmed that there was no significant interference of other waves before the first peak of the primary arriving waves using instantaneous frequency analysis [32,34]. Fig. 2a shows the cross-sectional diagram of the processed 3-D model. The thickening trabeculae can be seen. Fig. 2b–d show relationships between P1 and the difference of the arrival time of the primary wave, the peak amplitude of the primary arriving wave, which includes so-called fast wave, and secondary arriving wave, which includes so-called slow wave, at P2 values of 90%, 70%, and 50%. The values were normalized to the results at P1 = 0.0 mm
3. Results and discussions First, the effect of P1 (gradation length [mm]) and P2 (BV/TV [%] at the interface) on the peak amplitude of the secondary wave, which includes the slow wave, was investigated. Note that the
P3 = 0.1
(a)
0.2
0.4
P1 = 2.4mm (fixed) P2 [%] Arrival time of primary wave [µs]
(b)
0.0 -0.1 -0.2 -0.3 -0.4 0.0
(c)
0.1
0.2
0.3
0.4
Amplitude of waves [dB]
5
P2 [%]
(d)
0 -5 -10
Primary Wave Secondary Wave
-15 0.0
0.1
0.0
0.1
0.2
0.3
0.4
0.3
0.4
100 80 60 40 20 0 0.2 P3 [mm/mm]
Fig. 3. The effect of P3 with a fixed P1 value of 2.4 mm. (a) The models; (b) the peak amplitude of slow waves; (c) the P2 value and (d) the received waveforms.
Y. Nagatani et al. / Ultrasonics 54 (2014) 1245–1250
(the condition in the original structure). The error bar indicates the standard deviation of the data at five points. The Fig. 2b shows a clear monotonous decrease of arrival time which means the increase of the primary wave speed. Here, the amplitudes of the primary wave were not greatly changed regardless of the conditions of gradation length P1. In contrast, the amplitude of the secondary waves decreased monotonously. These characteristics of two different kinds of waves are similar to the results of the simulation data and experimentally observed data reported by our group [27]. We pointed that this tendency came from the nature that fast and slow waves respectively propagate mainly in the trabecular part and pore part. Although the detailed behavior of the waves in the complex condition of the transition region should be confirmed carefully in the future work, our results support the previous understandings of the two wave propagation mechanism. In addition, even when the gradation length was very long (P1 = 4.8 mm), very small secondary waves could be seen (< 10 to 15 dB). Next, the effect of P3 (spatial distribution of BV/TV) was investigated. Fixing P1 (the thickness of transition region) at 2.4 mm, P3 was varied from 0.0 to 0.4 mm/mm. The condition at P3 = 0.0 mm/mm indicates the original model (i.e. without processing). The results when the focal point was set on the center of specimen are shown in Fig. 3. Fig. 3b and c shows the relationship between P3 and the arrival time of primary wave, the normalized peak amplitude of primary waves and secondary waves, respectively. When P3 is about 0.1 mm, the amplitude of primary wave shows a maximum and the amplitude of secondary wave shows a minimum. Then the amplitudes of primary wave and secondary wave are restored to the original values with an increase of P3. Regardless of this interesting behavior, the arrival time of the primary wave monotonously decreases, telling the increase in the speed. This discriminative tendency was also seen in the report by Hosokawa et al. using simple stratified models. Fig. 3d shows the relationship between P3 and P2. The approximate P2 value at the peak point was about 55%. The waveforms under each condition, shown in Fig. 4, clearly reveal that the speed of the secondary wave monotonously becomes higher with the increase of P3, in contrast to the changes in the amplitude. The primary wave also speeds up with increasing P3. In these simulations, where the trabeculae in the specimen were aligned clearly, any changes in the parameters did not
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eliminate the generation of either the primary wave or the secondary wave. In addition, in case of the thick cortical area of 2.4 mm (whole P1 area was cortical), the amplitude of secondary wave was 2.3 dB smaller than that of the original model (the gray dotted line in Fig. 4). As many literatures have pointed, regardless of the behavior that the fast longitudinal wave propagates stably in dense media (high BV/TV) and the slower wave propagates dominantly in pore media (low BV/TV), the two waves can be seen when the wave propagates parallel to the trabecular alignment. The results of the model of this paper, which have gradual change of trabecular thickness, also showed the two waves. Fig. 4 shows that the amplitude of the primary wave was small and the amplitude of the secondary wave was large when the model had sharp boundary, and vice versa. This interesting phenomenon can be interpreted as the result of the behavior that the wave in liquid portion gradually enters into solid portion while propagating in the transition region. This behavior may cause the smaller amplitude of the secondary wave. On the contrary, when the structure in the transition is sharp, which is essentially equivalent to the situation of the rapid growth of the trabecular thickness toward the interface (i.e. P3 is large), the propagation of the secondary wave is less interfered by the wave diffusion in the transition region. These observations emphasize the relevance of the two-wave measurement and encourage the use of the two-wave phenomenon in practical in vivo applications. However, the authors are aware of the limitations of this report because the results came from only one specimen and the validity of the modeling algorithm was not confirmed quantitatively. Therefore, the generality should be confirmed in the future work using a number of specimens including the one which have weak or unclear alignment and more realistic geometrical form of human bones. Moreover, the symmetric property of two waves possibly brings us novel information about the transition region, which might be useful for the diagnosis of osteoporosis because not only the bone density but also the trabecular structure changes as the progression of osteoporosis [19,21]. The interesting behavior of two waves including amplitudes, frequency characteristics, propagation speeds, and scattering patterns also should be investigated carefully in the forthcoming study. 4. Conclusion We have confirmed the two-wave phenomenon in bone with smooth transition between the cortical and cancellous parts using three dimensional X-ray CT images of a radius of a racehorse. Any change in the transition region between cortical part and trabecular part, which has the actual complex structure of bone specimen, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. This result leads us to the possibility of using the two-wave phenomenon as a diagnostic system for osteoporosis in the in vivo situation where the transition condition may be complex. For practical use, a precise investigation based on a quantitative estimation of real human bone is required. Acknowledgments
Fig. 4. The received waveforms with various P3 values with the fixed P1 value of 2.4 mm. The gray dotted line indicates the result when the whole P1 area was cortical bone without any trabeculae.
We gratefully acknowledge helpful discussions with Professor Emeritus Takahiko Otani at Doshisha University. This study was supported in part by KAKENHI Grant Number 24360161 and 25871038 from the Japan Society for the Promotion of Science (JSPS) and the Regional Innovation Strategy Support Program of the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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