Effects of the cancellous bone structure in the skull on ultrasonic wave propagation

The Transcranial Doppler method (TCD) enables the measurement of cerebral blood ow velocity and detection of emboli by applying the ultrasound probe to the temporal bone window, orbital, or greater occipital foramen. TCD is widely used for the evaluation of cerebral vasospasm after subarachnoid hemorrhage, early detection of patients with arterial stenosis and the check of brain death. However, measurements often become dicult in elderly women. Among various factors for this problem, we focused on the effect of the skull bone on the ultrasound penetration into the brain. Especially, the effect of the temporal bone structure was investigated. Using a 2D digital bone model, wave propagation through the skull bone was investigated by the Finite-Difference Time-Domain (FDTD) method. We create bone models which have different BV/TV (Bone Volume/Total Volume) in diploe. Around BV/TV about 60% (similar to elderly women), the observed maximum amplitude decrease due to multiple reection and scattering. The results suggest that effects of osteoposis on the skull make TCD measurement dicult.


Introduction
The vertebrate skull has a relatively complicated structure; it mainly consists of the following three layers: the outer (cortical bone), diploe (cancellous bone), and inner (cortical bone) layers. The heterogeneous and anisotropic characteristics of these layers facilitate complex ultrasound propagation; however, recent advances in transcranial ultrasound transmission have provided new noninvasive therapeutic and diagnostic techniques for brain diseases [1,2].
However, most ultrasonic studies have assumed isotropic plate models, which do not accurately represent the actual structural and material characteristics of the skull [3]. Eguchi et al. reported that lowintensity pulsed ultrasound therapy improves cognitive impairment in mouse models of Alzheimer's disease and vascular dementia [4]. For the precise irradiation of ultrasound to the affected part, it is important to control ultrasound radiation, considering the complicated characteristics of skull. Among many diagnostic techniques for brain diseases, the transcranial Doppler (TCD) method is an approach that involves the application of an ultrasonic probe to the temporal bone window or the orbital or greater occipital foramen to evaluate cerebral blood ow velocity and detect emboli. It was rst reported by Aaslid et al. in 1982 [5]. Since then, the TCD method has been widely used in clinical practices for the evaluation of cerebral vasospasms after subarachnoid hemorrhage, early detection of thrombus in patients with arterial stenosis or left ventricular assistive heart [6], and determination of brain death [7].
However, it has recently been reported that measurements often become di cult in elderly Japanese women [8]. One possible reason may be the morphological changes in the skull that accompany aging.
Kaito reported that the thickness of the skull increases in the elderly [9]. In addition, primary and secondary osteoporosis may affect ultrasonic wave propagation. It is a well-known fact that the bone volume fraction in the cancellous bone decreases with the progression of osteoporosis [10]. Excessive decrease in bone volume fraction may result in strong ultrasound scattering and considerable attenuation of weak signals of the observed ultrasonic waves. There has been no discussion in the literature on how osteoporosis affects the TCD method.
In this study, we focused on the effects of the cancellous structure on ultrasonic wave propagation by analyzing the propagation phenomenon in a human skull model. Considering bone structural changes due to age, 2D digital partial skull models were created. Finite-difference time-domain (FDTD) simulation was applied to the digital models to assess ultrasonic wave propagation in the thickness direction, which is commonly used with the TCD method.

Model And Method
2D digital temporal bone model In this study, four types of models were fabricated and are shown in Figure 2. In the rst model (empty model), the diploe was lled with water to mimic bone marrow (BV/TV = 0%). In the second model (uniform model), the diploe was lled with bone (BV/TV = 100%). In the third model (sample A), the trabecular bone was aligned nearly parallel to the cortical bone layer, and the diploe was lled with water and the trabecular bone was created from a CT image of the actual cancellous bone in an equine femur.
The trabeculae were highly aligned and showed a high degree of anisotropy (DA; 2.5). In the fourth model (sample B), the trabeculae were mostly aligned perpendicular to the cortical layer. This was created from a CT image of the cancellous bone of a swine skull. Figure 2 shows examples with the BV/TV of bone (50%-60%) that was similar to the BV/TV of an elderly human skull [12]. Because the CT images are shown using grayscale (0-255), the models were created by choosing several different grayscale thresholds to change BV/TV. However, control of the grayscale did not have a strong in uence on the alignment direction of the trabeculae. Here in the case of sample A, 15 thresholds in a 256-scale gray image were selected. The BV/TV values of the models from sample A are shown in Table 1 (a). In the case of sample B, 11 thresholds were selected. The BV/TV of the models are shown in Table 1

FDTD method
An elastic 2D FDTD simulation was performed to simulate the longitudinal and shear wave propagation in the model [13,14]. Similar to the previous studies by our group [15,16], an in-house FDTD source code was used. This simulation did not consider attenuation during propagation. The relevant equations for the x direction are Eqs. (1)- (3): whereσ xx , σ xy , v, and ρ represent the normal stress, shear stress, particle velocity, and density of the medium, respectively. Next, we estimated the associated elastic constants. The matrix of elastic constants is shown in Eq. (4): Here, direction 1 is the front-back direction of the skull and direction 2 is the thickness direction. The constants of the model were assumed to be homogeneous. c11 was estimated from the speed of sound in the axial direction of the bovine femur cortical bone, referring to a study by Yamato et al. c44 was obtained assuming the Poisson's ratio of 0.33 [17][18][19].
These equations were calculated digitally using the central difference method. The stress and particle velocity were both calculated alternately in the spatial and time domains, which is called "the leapfrog method." The Higdon's second-order absorption boundary condition was implemented at the edges of the simulation area as an absorption layer [20]. Figure 3 shows the simulation model that was used. A transmitter array (length = 20 mm) was placed outside the bone model and in front of the thinnest part of the bone. The transmitter was an array of 100 transducers that controlled the phase to focus the wave near the artery. The bone model was placed 2 mm from the transmitter after considering skin thickness. The receiver array (length of each receiver was 1 mm) was placed 60 mm from the transmitter. The receiver and artery positions were similar to the expected positions of middle cerebral artery or posterior cerebral artery, which are usually measured from the temporal bone window by TCD [21]. The position of the artery is expected to be at the center of the receivers. The radiated waveform from the transmitter was a single sinusoidal wave at 2 MHz with a Hann window.

Method (Condition)
The spatial and time resolutions in the simulations were 14 μm and 2.5 ns, respectively. These resolutions satis ed the Courant stability condition [22]. In this simulation, the model was immersed in water to mimic the surrounding soft tissues. The longitudinal wave velocity and density of water were 1500 m/s and 1000 kg/m 3 , respectively.

Results And Discussion
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 de ned at 5% of the maximum amplitude of the rst positive peak. The highest amplitude was observed at the center of the receiver array (x = 0). Furthermore, as seen in the gure, 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 rst. 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 uctuated 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 (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).

Discussion
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 [12]. 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 di cult 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 [23]. 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 re ections and scattering. Subsequently, we checked the effects of scattering and re ections 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 [p 2 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 p 2 . This implies that the energy passing through the bone decreases due to scattering and multiple re ections.
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 [24]. 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.

Conclusion
In this study, the effect of skull bone structural changes on ultrasound propagation was investigated by FDTD simulation. By changing the trabecular BV/TV in the 2D human temporal bone models, we found a decrease in ultrasound amplitudes owing to the presence of scattering and re ections. Ultrasound attenuated well in the bone with a BV/TV of 60%, which was similar to the BV/TV value of the diploe in the skull of elderly women. The structural changes in the skull may affect ultrasonic penetration into the brain and may be one factor that can explain the di culty in measuring TCD in elderly Japanese women. This study also indicates that the effect of skull structure on the ultrasound propagation is also an important factor for future brain therapy and diagnosis.