Introduction

The number of patients suffering from arteriosclerosis is rapidly increasing with the aging of society and dietary westernization. It is important to make an early diagnosis of arteriosclerosis to prevent serious diseases such as myocardial and cerebral infarction.

Recently, there are many methods for diagnosis of arteriosclerosis, such as angiography, intra-vascular ultrasound (IVUS), optical coherence tomography (OCT), computed tomography (CT), magnetic resonance imaging (MRI), and positron emission tomography (PET) [1–3]. These methods are invasive and/or not easily applicable. Therefore, they are unsuitable for repetitive diagnosis to monitor progression of arteriosclerosis. On the other hand, diagnosis with ultrasound is noninvasive, easy to use at bedside, and suitable for repetitive diagnosis.

Ultrasound B-mode imaging is widely used for the morphological diagnosis of the arterial wall, particularly for measurement of intima-media thickness [4–6]. In addition, methods for evaluating the elasticity and viscosity of the arterial wall have recently been developed [7–13]. For diagnosis of early-stage arteriosclerosis, it is useful to evaluate the elastic property of the arterial wall because arterial-wall elasticity changes with progression of arteriosclerosis [14].

For noninvasive clinical evaluation of arterial elasticity, the pulse wave velocity (PWV) method was developed for diagnosis of arteriosclerosis nearly 100 years ago [15, 16]. This method measures the pulse wave, which is the pressure wave generated by the heartbeat, that propagates in the longitudinal direction of the artery. Since PWV c PWV becomes faster with progression of arteriosclerosis, it is used as a diagnostic index of arteriosclerosis and is expressed by the Moens–Korteweg equation as follows [16]:

$$ c_{\rm PWV}=\sqrt{\frac{E_{\theta}h}{\rho D}}, $$
(1)

where \(E_{\theta}\) is the Young’s modulus in the circumferential direction of the artery, h is the thickness of the arterial wall, ρ is the blood density, and D is the diameter of the artery. The assumptions underlying the Moens–Korteweg equation are that the artery has a thin wall and is filled with an incompressible inviscid liquid. Since PWV is traditionally measured between the carotid artery and femoral artery (cfPWV (carotid-femoral PWV) method) [17, 18] it is not suitable for early diagnosis of arteriosclerosis because lesions of early-stage arteriosclerosis are several millimeters in size. In addition, PWV depends on a part of the arterial tree, i.e., PWV in the distal arteries is faster than that in the proximal arteries. Thus, measurement of regional PWV would be useful for diagnosis of early-stage arteriosclerosis.

We previously studied regional PWVs by measuring small vibration velocities of the arterial wall at two points along the arterial longitudinal direction [19]. To estimate regional PWV more accurately, in the present study, vibration velocities at multiple points in a region of about 14 mm along the arterial longitudinal direction were measured at a high temporal resolution of 3472 Hz. Vibration velocities of the arterial wall have frequency components mostly up to 30 Hz. However, a high temporal resolution is desirable to estimate a small time delay between vibration velocities due to pulse wave propagation, i.e., the pulse wave propagates a region of about 14 mm in 2.8 ms when PWV is 5 m/s.

Materials and methods

Measurement of small vibration velocity for the carotid arterial wall

In the present study, as illustrated in Figs. 1 and 2, minute vibration velocities of the human carotid arterial wall were measured by the phased-tracking method [20] at 72 points in the arterial longitudinal direction with intervals of 0.2 mm (total length of 14.4 mm) at a high temporal resolution of 3472 Hz. Modified ultrasonic diagnostic equipment (Aloka, α-10) was used to acquire ultrasonic RF echoes received by each transducer element [21, 22]. The center frequency and fractional bandwidth of an emitted ultrasound pulse were 8 MHz and 61 %, respectively. The received RF echo was sampled at 40 MHz with 16-bit resolution. The method for high frame rate imaging developed by our group [21] was used to obtain beamformed ultrasonic RF echoes at 72 beam positions at a very high frame rate of 3472 Hz.

Fig. 1
figure 1

Illustration of measurement of minute vibration velocity waveforms of the human carotid arterial wall at 72 points at intervals of 0.2 mm

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Fig. 2
figure 2

Illustration of the method for estimation of time delay between acceleration waveforms a(x m ; t) and {a(x m+M/2; t + τ i )} using their analytic signals

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Typical PWV is several m/s and passes through a small region of 14.4 mm in a very short period of a few milliseconds. In such a situation, a small random error in estimation of the time delay between velocity waveforms would lead to a significant error in estimation of PWV when PWV is estimated using velocities measured at only two points along the arterial longitudinal direction. Therefore, in the present study, vibration velocities were measured at multiple points to reduce random errors, such as electrical noise and arterial motion estimation error due to random interference of echoes from scatterers in the arterial wall [13].

In the phased-tracking method, ultrasonic pulses at angular frequency \({\omega_0}= 2{\pi}{f_0}\) were transmitted from an ultrasonic transducer placed on the skin surface at a time interval of \(\Updelta T=1/f_{\rm FR}\) (f FR: frame rate) at each ultrasonic beam position. The instantaneous distance between the moving arterial wall and the ultrasonic transducer is denoted by \(z(t) = c_0\tau_{\rm u}(t)/2\), where c 0 is the velocity of ultrasound, and \(\tau_{\rm u}(t)\) is the time delay required for two-way transmission between the ultrasonic transducer and the arterial wall. The phase θ(z;t) of the quadrature demodulated signal of the received ultrasonic echo is given by \(\theta(z;t) = \omega_0\tau_{\rm u}(t) =2\omega_0 z(t)/{c_0}\). The phase difference \(\Updelta \theta(z;t)\) between time t and \((t+ \Updelta T)\) is given by

$$ \Updelta \theta(z;t) = \theta(z;t+\Updelta T)-\theta(z;t) = \frac{2\omega_0\Updelta z(t)}{c_0}, $$
(2)

where \(\Updelta z(t)=z(t+\Updelta T)-z(t)\) is the displacement of an object during the period \(\Updelta T\). By dividing the displacement \(\Updelta z\) by the period \(\Updelta T\), the average velocity, denoted by \(v(t+\Updelta T/2)\), of the object during the period \(\Updelta T\) is given by the phase difference \(\Updelta \theta(z;t)\) between echoes as follows:

$$ v\left(t+\frac{\Updelta T}{2}\right)=\frac{\Updelta z(t)}{\Updelta T}=-\frac{c_0}{2\Updelta T}\frac{\Updelta \theta(z;t)}{\omega_0}, $$
(3)

In the present study, parallel beam forming (PBF) [21] was applied to realize a high frame rate of 3472 Hz. The minute vibration velocity waveforms were measured at 72 points on the human carotid artery wall at intervals of 0.2 mm.

Estimation of PWV by complex correlation approach

In the present study, the acceleration waveforms {a(x m ; t)} were obtained by applying time differentiation to the vibration velocity waveforms {v(x m ; t)}, where x m is the mth ultrasonic beam position, to enhance high frequency components contained in the measured velocity waveforms because the use of high frequency components improves the time resolution in estimation of time delays among the vibration waveforms due to the propagation of the pulse wave. To estimate PWV, the phase \(\phi(x_{m}; t_{0})\) of acceleration waveform \(a(x_{m}; t_{0})\) at a time of interest t 0 (=frame) measured at the mth ultrasonic beam position x m is obtained by applying the Hilbert transform to the acceleration waveforms {a(x m ; t)}. The Hilbert transform was applied to the acceleration waveforms during just an entire cardiac cycle. The phase \(\phi(x_{m}; t)\) is determined from the complex analytic signal g(x m ; t) of a(x m ; t).

To determine the PWV from the phases of the multiple analytic signals, a method based on the complex cross correlation function was developed in the present study. The complex cross correlation function \(\gamma(t_{0}, \tau_{i})\) at t 0 at lag \(\tau_{i}=i\cdot\Updelta T\), where i = −N l ,  −N l  + 1, …, −1, 0, 1, …, N l  − 1, N l (N l denotes the number of lags used in estimation of PWV), is defined as follows:

$$ \widehat{\gamma}(t_{0}, \tau_{i})=\sum_{m=0}^{M/2}\sum_{i=-N_{\rm c}}^{N_{\rm c}}g^{*}(x_{m}; t_{0})\cdot g(x_{m+M/2}; t_{0}+\tau_{i}), $$
(4)

where \(\tau_{i}=i\cdot \Updelta T\), M is the number of ultrasonic beam positions, and N c determines the number of frames used for calculation of the complex correlation function.

By denoting the original time delay of g(x m+M/2; t 0) from g(x m ; t 0) due to the propagation of the pulse wave in the length of \(\delta x\cdot M/2\) (δx: lateral spacing of ultrasonic beams) by δτ and denoting the dominant frequency of {g(x m ; t 0)} at t 0 by f a, δτ is expressed as follows:

$$ \delta\tau=-\frac{\angle\gamma(t_{0}, 0)}{ 2\pi f_{\rm a}}. $$
(5)

For other lag τ i , the model \(\angle\widehat{\gamma}(t_{0}, \tau_{i})\) of the phase of γ(t 0, τ i ) is expressed as follows:

$$ \begin{aligned} \angle\widehat{\gamma}(t_{0}, \tau_{i})&=2\pi f_{\rm a}(\tau_{i}-\delta\tau)\\ &=2\pi f_{\rm a}\cdot(i\cdot\Updelta T)-2\pi f_{\rm a}\delta\tau\\ &\equiv p\cdot i+q. \end{aligned}. $$
(6)

To estimate the dominant frequency f a and time delay δτ due to propagation of the pulse wave, the coefficients p and q are determined by fitting a linear function defined by Eq. (6) to measured \(\{\angle\gamma(t_{0}, \tau_{i})\}\) using the least squares method as follows: the difference α(t 0) between the angle of the measured complex correlation function \(\angle\gamma(t_{0}, \tau_{i})\) and the model \(\angle\widehat{\gamma}(t_{0}, \tau_{i})\) is expressed as follows:

$$ \alpha(t_{0})=\sum_{i=-N_{l}}^{N_{l}}|\angle\gamma(t_{0}, \tau_{i})-(p\cdot i+q)|^{2}. $$
(7)

To determine the coefficients \(\widehat{p}\) and \(\widehat{q}\), which minimize α(t 0), partially differentiated α(t 0) with respect to p and q are set to zero. By solving the resultant simultaneous equations, the coefficients p and q are determined as follows:

$$ \begin{aligned} \widehat{p}&=&\frac{\left(2N_{\rm c} +1\right) \left\{\sum\nolimits_{i=-N_{\rm c}}^{N_{\rm c}}i\cdot\angle\gamma(t_{0}, \tau_{i})\right\}-\left(\sum\nolimits_{i=-N_{\rm c}}^{N_c}i\right) \left\{\sum\nolimits_{i=-N_{\rm c}}^{N_{\rm c}}\angle\gamma(t_{0}, \tau_{i})\right\}}{\left(2N_{\rm c} +1\right) \left(\sum\nolimits_{i=-N_{\rm c}}^{N_{\rm c}}i^{2}\right)-\left(\sum\nolimits_{i=-N_{\rm c}}^{N_{\rm c}}i\right)^{2}}. \end{aligned} $$
(8)
$$ \begin{aligned} \widehat{q}=\frac{\widehat{p}\sum\nolimits_{i=-N_{\rm c}}^{N_{\rm c}}i-\sum\nolimits_{i=-N_{\rm c}}^{N_{\rm c}} \angle\gamma(t_{0}, \tau_{i})}{\left(2N_{\rm c} +1\right)}. \end{aligned} $$
(9)

Based on Eq. (6), the dominant frequency f a is determined using the estimated coefficient \(\widehat{p}\) as follows:

$$ \widehat{f}_{\rm a}=\frac{ \widehat{p}}{ 2\pi\Updelta T}. $$
(10)

Finally, PWV c PWV(t 0) is estimated based on Eq. (6) using the estimated dominant frequency \(\widehat{f}_{\rm a}\) and coefficient \(\widehat{q}\) as follows:

$$ \begin{aligned} c_{\rm PWV}(t_{0})&=\frac{\frac{M}{2}\cdot\delta x}{\widehat{\delta\tau}}\nonumber\\ &=-\frac{\pi \widehat{f}_{\rm a}\cdot M\cdot\delta x}{\widehat{q}}. \end{aligned} $$
(11)

In Eq. (4), the complex correlation function between analytic signals {g(x m ; t 0)} and {g(x m+M/2; t 0)} was used because the distance between each pair of the analytic signals should be as large as possible to increase the time delay between the analytic signals and also all the M analytic signals can be used to reduce random errors.

Results

Figure 3a–c shows the B-mode images of 24-(subject A), 23-(subject B), and 38-(subject C) year-old males, respectively. Figures 4, 5, and 6 show in vivo experimental results of subjects A, B, and C, respectively. The short segments of their common carotid arteries of 14.4 mm ((72 beam positions) × (0.2 beam spacing)) were measured. In each of Figs. 4, 5, and 6, electrocardiogram (ECG) (a); the displacement waveform (b), which is obtained by integrating the vibration velocity in (c) with respect to time; vibration velocity measured at the 0th beam position (the 0th beam was nearest to the heart) (c); and the acceleration waveform (d), which is obtained by temporal differentiation of the vibration velocity in (c), are shown. Velocities at the media-adventitia boundaries of the posterior walls were measured as the vibration velocities of the arterial walls.

Fig. 3
figure 3

B-mode images of carotid arteries of 23-(subject A), 24-(subject B), and 38-(subject C) year-old healthy males

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Fig. 4
figure 4

In vivo experimental results of subject A. a Electrocardiogram, b displacement, c velocity v(x m ; t), and d acceleration a(x m ; t) of the posterior wall measured at the 0th ultrasonic beam

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Fig. 5
figure 5

In vivo experimental results of subject B. a Electrocardiogram, b displacement, c velocity v(x m ; t), and d acceleration a(x m ; t) of the posterior wall measured at the 0th ultrasonic beam

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Fig. 6
figure 6

In vivo experimental results of subject C. a Electrocardiogram, b displacement, c velocity v(x m ; t), and d acceleration a(x m ; t) of the posterior wall measured at the 0th ultrasonic beam

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The vibration of the arterial wall is generated by the ejection of blood from the left ventricle of the heart, and expansion of the artery corresponds to the acceleration waveform around 0.1 s. For example, the enlarged view of the acceleration waveforms of subject A measured at the 0th (proximal) and 71st (distal) beams during a period 0–0.3 s is shown in Fig. 7. As can be seen in Fig. 7, it is recognized that the positive peak around 0.07 s at the 0th beam precedes that of the 71st beam very slightly. In the present study, the large positive peak around t = 0.1 s (t 0 for estimation of PWV is shown by the vertical dashed line in Fig. 4), which was caused by ejection of blood from the left ventricle, was analyzed for estimation of PWV.

Fig. 7
figure 7

Enlarged view of the acceleration waveforms {a(x m ; t)} of subject A during 0–0.3 s measured at the 0th (solid line) and 71th (dashed line) ultrasonic beams

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To estimate PWVs, the Hilbert transform was applied to the acceleration waveforms. The complex cross correlation function defined in Eq. (4) was obtained by setting t 0 to be the time of the peak in the acceleration waveform measured at the 0th beam position around 0.1 s. The assigned t 0 is shown by the vertical dashed lines in Figs. 4, 5, and 6. The length of the correlation window was set at 10 ms (corresponding to N c = 17), which roughly corresponded to the width of the peak in the acceleration waveform. Plots in Fig. 8 show the measured phase \(\angle\gamma(t_{0}, \tau_{i})\) of the complex correlation function plotted as a function of the lag τ i . The PWV was estimated from the slope \(\widehat{p}\) and intercept \(\widehat{q}\) of the regression line (shown by the dashed line in Fig. 8) based on Eq. (11). By applying the same procedure to the acceleration waveforms of subjects B and C, regional PWVs of subjects A, B, and C were estimated to be 5.6, 6.4, and 6.7 m/s, respectively. The estimated pulse wave velocities were similar to those reported in the literature [23–25].

Fig. 8
figure 8

Phase of complex correlation function \(\angle\gamma(t_{0}, \tau_{i})\) plotted as a function of lag τ i . Plots and the dashed line show measured phases and the regression line

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By observing the acceleration waveform of subject A (shown in Fig. 4d), there is a small but distinct peak around the time of 0.2 s. In the literature, such a peak detected in the acceleration waveform is reported to be a pulse wave component reflected in the distal position of the arterial tree [26–28]. As can be seen in Fig. 7, the acceleration waveform at the 71st beam (nearer to the head) precedes that at the 0th beam very slightly. The PWV for this component was estimated by setting t 0 to the time of the peak in the acceleration waveform measured at the 0th beam position around 0.2 s (t 0 for this reflection component and is shown by the vertical dashed line in Fig. 4). The estimated PWV for the reflection component was −8.4 m/s, where the negative velocity means the propagation from the periphery to the heart. As described above, the regional PWV of the reflection component could also be estimated separately using the proposed method.

Furthermore, for evaluation of the reproducibility of measurements using the proposed method, five additional measurements were performed with respect to subject C on the same day. Figure 9a shows acceleration waveforms in the 0th scan lines for the five measurements. Using the phase \(\angle\gamma(t_{0}; \tau_{i})\) of the complex correlation function shown in Fig. 9b, the regional pulse wave velocities were estimated. The mean and standard deviation for the five measurements were 7.0 and 0.72 m/s (10.4 % of the mean value), respectively.

Fig. 9
figure 9

a Acceleration waveforms in the 0th scan lines measured for the five additional measurements with respect to subject C. b Phases of complex correlation functions \(\{\angle\gamma(t_{0}, \tau_{i})\}\) plotted as a function of lag τ i . Plots and lines show measured phases and the regression lines

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Discussion

For one of the subjects in the present study, a pulse wave component, which was considered to be reflected by the distal arterial tree, was found. By applying the Hilbert transform and the proposed complex cross correlation technique to the acceleration waveforms obtained by differentiating the measured velocity waveforms, the forward and reflection components were separately analyzed. The estimated propagation speed of the forward component was lower (5.6 m/s) than that (8.4 m/s in the opposite direction) of the reflection component. One of the reasons for this difference was considered to be the difference between blood pressures at the arrival times of the forward and reflection components in the carotid artery.

It was reported that the arterial wall distension waveform can be assumed to be the waveform of blood pressure [29–31]. Figure 4b shows the displacement of the posterior arterial wall obtained by integrating the measured velocity waveform with respect to time. At the time (around 0.1 s) of the peak of the forward component in the acceleration waveform shown in Fig. 4d, the magnitude of the displacement is nearly zero. This fact suggests that blood pressure at the arrival of the forward component corresponds to the diastolic blood pressure. On the other hand, the displacement at the time of the arrival of the reflection component is nearly maximum, which suggests that blood pressure at the time of arrival of the reflection component is nearly the systolic blood pressure. It is also reported that the stress–strain relationship of the arterial wall is nonlinear [32–35], and the elastic modulus of the arterial wall at higher blood pressure is larger than that at lower blood pressure. Therefore, the PWV of the reflection component (presumably at higher blood pressure) was considered to be larger than that at the forward component.

In the present study, the propagation of pulse waves in a local region in the carotid artery was analyzed in detail by measuring vibration velocities of the arterial wall at a very high temporal resolution of 3472 Hz. To measure the regional PWV in an arterial longitudinal segment of about 15 mm, a small time delay of a few milliseconds between vibration waveforms of the arterial wall needs to be estimated. For this purpose, in the present study, acceleration waveforms, which contain higher frequency components compared with displacement waveforms, were measured at a high temporal resolution, and a method for estimation of a small time delay using the phase of the acceleration waveform was developed.

Figures 10b–d show the displacement, velocity, and acceleration waveforms sampled at a frame rate of 35 Hz (waveforms in Fig. 4 were down sampled), which is a typical frame rate of conventional ultrasonic diagnostic equipment. As shown in Fig. 10, it is difficult to measure the velocity and acceleration waveforms at a temporal resolution of 35 Hz. Using the displacement waveform shown in Fig. 10d, which was sampled at 35 Hz, the pulse wave velocity was also estimated by the proposed method described in “Estimation of PWV by a complex correlation approach”. The estimated PWV was −37.2 m/s, which was significantly different from that reported in the literature. Therefore, it is necessary to measure the vibration waveform at a high temporal resolution.

Fig. 10
figure 10

Vibration waveforms of subject A (waveforms in Fig. 4 are down sampled at 35 Hz). a Electrocardiogram, b displacement, c velocity v(x m ; t), and d acceleration a(x m ; t) of the posterior wall measured at the 0th ultrasonic beam

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As described above, in the present study, regional pulse wave velocities were noninvasively measured in vivo by the proposed method and compared with those reported in the literature. In future studies, it will be necessary to evaluate the accuracy of estimation of regional PWV using the proposed method by basic experiments using phantoms. In addition, the conditions of the spacing and number of scan lines and frame rate, etc., required for measurement of regional PWV also need to be investigated by phantom experiments. However, the results obtained in the present study show the possibility of the proposed method for measurement of regional PWV, and such a method would be useful for diagnosis of arteriosclerosis.

Conclusion

For detailed analyses of pulse wave propagation in a local region of 14.4 mm in the carotid artery, in the present study, small vibrations on the arterial wall were measured with ultrasound at a very high frame rate of 3472 Hz. By applying the Hilbert transform to the acceleration waveforms of the arterial wall, which were obtained by differentiating the measured vibration velocity waveforms, pulse wave velocities of three healthy subjects were estimated using the proposed complex cross correlation approach. Furthermore, in one of the subjects, there was a component propagating from the heart to the periphery and from the periphery to the heart. Even in such a case, pulse wave velocities for these forward and reflection components were separately estimated, and the magnitudes of the estimated pulse wave velocities were similar to those reported in the literature. Such a method for measurement of regional PWV would be useful for more sensitive evaluation of the change in elasticity due to progression of arteriosclerosis.