Figure 4 (a) shows the observed waves that propagated in water only, i.e., without the bone model. The arrival time of each wavefront was defined at 5% of the maximum amplitude of the first positive peak. The highest amplitude was observed at the center of the receiver array (x = 0). Furthermore, as seen in the figure, the observed waveforms showed axis-symmetrical characteristics. The amplitudes decreased with the increase in the distance from the center. The arrival times at x = ±10 were the shortest because the wave that radiated from each end of the transmitter array reached these positions first.
Figure 4 (b) and (c) shows the observed waves that propagated through the empty model and the uniform model, respectively. In the empty model, the wave was focused mostly near the artery (x = 0); however, the maximum amplitude was observed at x = −2 in the uniform model, showing a small shift. Additionally, the waveforms were not axis-symmetric, and the time of the maximum amplitude observation was delayed. This indicated that ultrasound focusing was clearly affected by the complex bone shape.
Figure 5 (a) and (b) shows examples of the observed waves that had propagated in sample A (BV/TV = 59% and 49%, respectively). Compared with the waves propagating in water only, the amplitudes were decreased and the arrival time of the wavefronts changed in this sample. Changes in the observed waves could be attributed to the BV/TV. Additionally, small maximum amplitudes were not observed near the wavefronts; they were delayed, similar to the waves of the uniform model. Figure 5 (c) shows the relationship between BV/TV and the maximum amplitude observed at the receiver array. The maximum amplitude decreased rapidly at a BV/TV of approximately 50% and subsequently fluctuated due to the increase in BV/TV. The maximum amplitudes increased at a BV/TV of >70%. There was a wide area with large amplitudes. However, the positions of the receiver that showed the maximum amplitudes were not centered (x = 0) and were shifted to a slight extent at a BV/TV of 55%–65%.
Figure 6 (a) shows the observed waves that had propagated in sample B (BV/TV = 49%). Similarly, the arrival time of the wavefronts had changed in sample B. Figure 6 (b) shows the relationship between BV/TV and the maximum amplitude observed at the receiver array. Around a BV/TV of 55%, the maximum amplitude showed the minimum value. At a BV/TV of 50%–55%, the positions of the receivers that showed the maximum amplitudes shifted by approximately 10 mm in the direction of the x-axis from the center (x = 0).
In this study, the maximum amplitudes changed due to the BV/TV in samples A and B. Larsson et al. reported a decrease in BV/TV in cancellous bones with age . They showed that BV/TV in the skull bone of a 70-year-old woman was approximately 62%, whereas that of a 50-year-old woman was approximately 82%. These results indicate that decreasing BV/TV with age may prevent ultrasounds from reaching the arteries. This may make it difficult to obtain TCD measurements in elderly women. In the skull, the main alignment of the trabeculae is considered to be in the direction of the thickness . Therefore, the condition might be similar to that found in sample B. In case of sample B, there was a clear decrease in the maximum amplitude at a BV/TV of approximately 50%–55%, which are similar to the values reported in elderly patients.
The observed wave can be analyzed to easily reveal that the waves accompanied several small waves after passing through the bone. The small waves may originate from the multiple reflections and scattering. Subsequently, we checked the effects of scattering and reflections from the observed waveforms. Notably, as opposed to the waves that passed through water only, several late small waves were observed in the data of samples A and B. In addition, increase in back scattering and scattering in the other directions may result in a decrease in the total energy of the observed waves. Figure 7 shows the relationship between the sum of the squared signals [p2 shown in Eq. (5)] observed at each receiver and BV/TV in sample B.
Here, the observation time T was 50.5 μs. The horizontal axis shows the position of observing the array transducer and the vertical axis shows the BV/TV of the bone sample. At a BV/TV of approximately 55%, there is a decrease in the values of p2. This implies that the energy passing through the bone decreases due to scattering and multiple reflections.
In sample B, there was a minor shift in the maximum amplitude position to the positive direction of x. This could be attributed to the slight counterclockwise tilted in the trabeculae from the thickness. Consequently, ultrasound may be refracted in the bone model. Yamashita et. al. reported that the bone trabecular alignment affected the direction of ultrasound propagation . In this model, the bone trabecular alignment tilted to the positive direction of x. The ultrasound subsequently did not focus strongly near the artery, and the maximum amplitude positions shifted. Similar scattering phenomena and refraction of the waves may occur in an actual skull bone.