Pulse Wave Decomposition Analysis for Arterial Stiffness Evaluation

In the time domain, features can be calculated from the timing and amplitude of several fiducial points. The starting point of the pulse wave indicates the beginning of a pulse cycle and the end of the previous one. The time of the inflection point marks the arrival of the P_b (O'Rourke and Yaginuma, 1984 (link)). The notch is caused by aortic valve closure and blood reflux, representing the transition between the systolic and diastolic phases (Hartmann et al., 2019 (link)). The pulse wave systolic period is the duration between the starting point and the dicrotic notch point of the pulse wave, followed by the pulse wave diastolic period. Usually, the local maxima of the second derivative of the pulse waveforms are utilized to extract inflection points and dicrotic notch points (as in Figure 2 (Vlachopoulos et al., 2011 (link))).
For some participants (e.g., those with severe atherosclerosis), the inflection point of the aortic pulse wave is difficult or even impossible to extract. In order to make this pulse wave decomposition method more practical, it has been proposed to use 30% of the systolic time as the location of the inflection point (Miyashita et al., 1994 (link); Westerhof et al., 2006 (link)). In this paper, for pulse wave with inconspicuous inflection point, 30% of ejection time (ET) is used as the location of the inflection point to calculate the relevant features of pulse wave decomposition. The beginning of the pulse wave systole indicates the time of aortic valve opening and the start of ejection, and the notch time of the pulse wave is the time of aortic valve closure and the end of ejection. ET represents the systolic time of the pulse wave, which is determined by subtracting the beginning time from the end time of aortic flow (as in Figure 3).
In the arterial system, both aortic pressure and flow waveforms consist of forward waves (P_f, Q_f) and backward waves (P_b, Q_b). The CAPW mainly comprises forward and lower limb reflection waves (Westerhof et al., 1972 (link)). As shown in Figure 4, CAPW equals the sum of the P_f and P_b; and the flow wave equals the difference between the Q_f and Q_b, (as shown in Eq. 1, 2). $\mathrm{P}={\mathrm{P}}_{\mathrm{f}}+{\mathrm{P}}_{\mathrm{b}}$
$\mathrm{Q}={\mathrm{Q}}_{\mathrm{f}}+{\mathrm{Q}}_{\mathrm{b}}$
The basic principle of pulse wave decomposition is as follows (Westerhof et al., 1972 (link)): ${\mathrm{P}}_{\mathrm{f}}=\frac{\mathrm{P}+{\mathrm{Z}}_{\mathrm{c}}\times \mathrm{Q}}{2}$
${\mathrm{P}}_{\mathrm{b}}=\frac{\mathrm{P}-{\mathrm{Z}}_{\mathrm{c}}\times \mathrm{Q}}{2}$ where, Q = U*A represents aortic flow; U is the flow velocity; A is blood vessels cross-sectional area; Z_c is the characteristic impedance.
Since the pulse waveform is not affected by the P_b in the early systolic phase, Z_c equals the ratio of blood pressure to flow (Li, 1986 (link); Khir et al., 2001 (link)), and Z_c can also be calculated by high-frequency input impedance (Murgo et al., 1981 (link); Miyashita et al., 1994 (link)). The input impedance (Z_in) is defined as follows: ${\mathrm{Z}}_{\mathrm{i}\mathrm{n}}\left(\mathrm{w}\right)=\mathrm{P}\left(\mathrm{w}\right)/\mathrm{Q}\left(\mathrm{w}\right)$ where P(w) and Q(w) are pressure and flow frequency components.
RI is the amplitude ratio of P_b to the sum of P_b and P_f, and the amplitude ratio of P_b to P_f is RM (Hametner et al., 2013 (link)). RM and RI are defined as follows: $\mathrm{R}\mathrm{M}=\frac{\left|{\mathrm{P}}_{\mathrm{b}}\right|}{\left|{\mathrm{P}}_{\mathrm{f}}\right|}$
$\mathrm{R}\mathrm{I}=\frac{\left|{\mathrm{P}}_{\mathrm{b}}\right|}{\left|{\mathrm{P}}_{\mathrm{b}}\right|+\left|{\mathrm{P}}_{\mathrm{f}}\right|}$
PTT can be determined by pulse wave decomposition, an important index to assess arterial stiffness in the young and old (Qasem and Avolio, 2008 (link)). PTT can be calculated as half the time difference between P_f and P_b (T_fb), as in Eq. 8. $\mathrm{P}\mathrm{T}\mathrm{T}={\mathrm{T}}_{\mathrm{f}\mathrm{b}}/2$
Qasem and Avolio calculated the cross-correlation coefficient of P_f and P_b to determine T_fb (Qasem and Avolio, 2008 (link)). The time of maximum cross-correlation coefficient is the T_fb between P_f and P_b (as in Figure 5).