The pulse wave velocity (PWV) is defined as the velocity of travel of flow velocity, pres-sure, and diameter waves generated by ventricular ejection. PWV has been proclaimed as an important parameter in the analysis of the arterial tree. Moreover, it is considered the gold standard for assessment of arterial elastance and directly related to direct mea-surement of arterial stiffness. PWV is one of the easiest meamea-surements for stiffness, but there is some uncertainty in the results obtained from the various methods. Although the value can be obtained well from the transit time between two different measured sites of the flow wave or diameter wave, it is much easier to measure pressure technically than others. Most of the older previous studies only considered propagation velocity of the invasively measured pressure wave. However, several recent studies have used blood flow velocity wave to measure PWV in humans non-invasively by the ultrasound [11]
and magnetic resonance imaging (MRI) [12]. Moreover, recent studies can obtain very precise values of PWV over short distance (less than 10 cm) with the transit-time MRI method. The method requires measurement of the flow pulse over multiple cycles to that distinct ensemble-averaged flow pulse can be generated.
The method that has been used most commonly to measure PWV is to measure the time of travel of the foot of the pulse wave over a known distance. Although locating the pressure waveform foot is subjective and less than accurate, the advantage of foot-to-foot PWV measurement is the simplicity of measurement [13]. The foot of the wave means the point at the end of diastole, when the steep rise of the wavefront starts. There are two reasons that the foot point is used to measure the velocity. First, by analogy with the interaction of a transient wave with reflected waves that it creates, the early part of the
wave will not be interfered by the reflections. Second, the wave velocity in each period of cardiac cycle is different, and the velocity to be measured is the velocity generated by ventricular ejection. Therefore, it is reasonable to treat the early wavefront at a region which maintains the identity in the propagated wave.
In this chapter, we use the method of measuring time difference between two wave-form. Like the most common used methods, we estimate the foot points of each signal and then calculate the propagation time. Because the distance between sensors is very short (7.5 mm) compared to recent methods, several technique are applied to increase the precision and stability of the result. For instance, the ensemble-average transit time is calculated over multiple beats. The concept of Theil-Sen estimator, also known as slope selection method, is also used to reduce the effect of outlier. The detail of Theil-Sen esti-mator is introduced in Section 4.1. In addition, we propose a novel method in which the transit time is estimated from the upstroke waveforms, which is inspired by the concept of timing alignment.
Before starting to calculate the pulse wave velocity, some notations and the signal model should be described. The diagram is shown in Figure 4.1. The sensor array is composed of 3x4 sensors, and the distance between each sensor is 7.5 mm, which is denoted as d. Let the direction of 4 sensors be the y-axis. The measure signals by the sensors in i column, j row is denoted as xi,j(t), where i = 1, ..., 3, j = 1, ..., 4, and the location of the sensor is at the position ~p = (i · d, j · d). The measured vessel is assumed a line crossing the region of the sensors, and the wavefront is orthogonal to the direction of the vessel. The velocity of the wavefront is denoted as ~v = (vx, vy). Assume the wave signal at the reference point (0, 0) is s(t). Then, the measured signals are assumed to be
Figure 4.1: Illustration of relative position between the sensor array and the artery
characterized by
xi,j(t) = hi,j· s(t ti,j ni,j(t)) + w(t), (4.1)
where hi,j is the channel gain from the vessel to sensors, ti,j is the transit time of wave from the reference point (0, 0) to the point (i · d, j · d), ni,j(t)is the transit time deviation and w(t) is a white Gaussian noise. The farther the location of the vessel from the sensor is, the larger the transit time deviation is. To calculate the transit time, the effective distance in the direction of the velocity is needed first. The effective distance is
effective distance = ~p· ~v
kvk = d(i· vx+ j· vy)
pv2x+ v2y , (4.2)
and the transit time is
ti,j = effective distance
kvk = d(i· vx+ j· vy)
vx2+ vy2 . (4.3)
4.1 Concept of Theil-Sen Estimator
The Theil-Sen estimator, also known as Sen’s slope estimator is one of the robust linear regression. Compared to the simple linear regression, which is the least square error estimator, the Theil-Sen estimator is much insensitive to outliers. The origin version of the estimator is to estimate linear model y = ax + b for the 2-dimension points (xi, yi) [14]. The slope a is the median of the slopes (yj yi)/(xj xi)determined by all pairs of sample points. After the slope is determined, the y-intercept b is the median of the values yi axi. In [15], the Theil-Sen estimator is extended to handle multi-dimension data points. For example, the 3-dimension Theil-Sen estimator, also called Oja-Niinimaa estimator, estimate a linear model z = ax + by + c for a set of the sample data. The estimation of the slope coefficient a and b is the median of the coefficient of the plane crossing any combination of three points. That is, given a set of the points pi = (xi, yi, zi)
The reason why we choose the Theil-Sen estimator is about its robustness. A robust linear estimator means it is able to decrease the effect of outliers. The breakdown point is a measurement of assessing the robustness of estimators. Intuitively, the breakdown point of an estimator represents the ability to tolerate the outliers. More specifically, it is the proportion of outlier the estimator can handle before giving an incorrect result(e.g.
arbitrarily large error). For example, given n random variables (X1, ..., Xn) ⇠ N (µ, 2) and we use X = X1+...+Xn n to estimate the mean. Then the breakdown point of the es-timator is 0 because the result can be arbitrarily large if one of X1, ..., Xn is arbitrarily large. Back to the Theil-Sen estimator, the breakdown point of it is 29.3%, and the break-down point of the 3-dimension case is 20.6%. Compared to the least-square estimator which has breakdown point of 0, the Theil-Sen estimator is much more robust because the breakdown point is larger.
4.2 Foot Point Estimation
In this section a PWV estimation method using foot points is described. After calculating the foot points of each signals, the velocity are estimated by the time difference. To calculate the foot points, first we have to locate the timing of each heart beat. Then the upstroke regions of each signal are searched near these timing. At last, the foot points are estimated in the region of upstroke by several definitions. In the following paragraph, the procedure of finding foot point is introduced.
At the beginning, we locate the peaks of the wave, which means the systolic time.
Because the distance between sensors is very small, the time of the systole are close.
After finding the systolic time, foot points of each signal are searched near these reference time. The method to locate the peaks is described in follows. The heart rate is estimated with the summation of signals from all sensors. Here we assume that the signal to noise ratio of the summation of signals is large. For normal people, the resting heart rate is in range between 50 and 90 per minute, so the range of corresponding time period is 60/90 s to 60/50 s. Therefore, we calculate the auto-correlation and find the position of the
maximum to estimate the period and heart rate, that is
Finding the period of the heart beat, we can roughly know the time difference between peaks, and we can determine whether a local maximum is the systolic peak or not based on the information. Because the value of the systolic peak should be the largest in a period, so we check if every local maximum is larger than its neighborhood or not. We denote a parameter window size, which value is equal to 0.6 times of the period, as the size of the neighborhood. If a local maximum is larger than its neighbor data which the distance is lower than window size, then this local maximum is determined as the peak caused by systole, illustrated as Figure 4.2. Similarly, we can use the same way to locate the timing of the end diastole, checking the local minimum is smaller than its neighbor or not.
Second, we find the upstroke region of each channel using the location of peaks as reference points. The systolic peaks of each signal are searched in the neighbor of the reference points. In the result, the upstroke region is between the systolic peak and its previous local minimum. If there is noise corrupting the signal heavily and there is no peak found in the neighborhood of the reference point, we just ignore this point. If the ratio of ignoring is higher than 20%, the signal is determined to be highly interfered and it will be ignored in the following procedure.
Last, the foot point for each upstroke region is estimated by several ways. In [4], the common definitions of the foot point include intersecting tangent, maximum of 2nd derivative, and 10% upstroke, which are shown in Figure 4.3. Intersecting tangent is the
Figure 4.2: The concept of finding the systolic peak
point of the intersection of the tangent to the systolic upstroke with the horizontal line through the local minimum. Maximum of 2nd derivative is the point of maximum first derivative of upstroke. The 10% upstroke is the point that the value is 10% of the upstroke amplitude. These three kinds of foot points are experimented the performance separately.
The foot point of each heart beat of each signal is calculated by the procedure men-tioned above. Afterward, the normalized transit time is estimated using the concept of the Theil-Sen estimator. In the next two subsections two methods are introduced, including 2-dimension and 3-dimension estimator.