Abstract

Given the wide utility of radial vibration of the arterial wall for clinical values, this paper presents a theoretical study on the relations of radial vibration of the arterial wall to pulsatile parameters in blood flow. Pulse wave propagation in an artery is formulated as a combination of the governing equations of blood flow and the arterial wall and no-slip conditions at the blood-wall interface and is analyzed to obtain the wave velocity and the theoretical expressions for blood flow rate and radial wall displacement in terms of pulsatile pressure. With the harmonics of a pulse signal, theoretical relations of radial vibration of the arterial wall to pulsatile parameters in blood flow are derived under two conditions: without and with wave reflection. These theoretical relations identify the assumptions for the simplified relations employed in the utility of radial vibration of the arterial wall for clinical values. With the arterial wall treated as a unit-mass vibration system, these simplified relations are utilized for extraction of arterial indices from radial vibration of the arterial wall. Other applications of such relations for clinical values are discussed, and the interaction between the arterial wall and blood flow is further revealed from the perspective of energy and one-dimensional wave equations. With harmonics and wave reflection considered, the derived theoretical expressions for radial wall vibration, pulsatile parameters in blood flow, and the relations between them provide theoretical guidance for improving their interpretation of clinical values with clearly defined physiological implications and assumptions.

1 Introduction

Arterial pulse signals carry physiological and pathological information about the cardiovascular (CV) system and are interpreted to extract various arterial indices for detection and diagnosis of CV disease [15]. Based on the pulse wave propagation theory, the arterial pulse signals measured at the carotid and femoral artery sites using dual tonometry are utilized to extract the carotid-femoral pulse wave velocity (PWV) as an assessment of the global arterial stiffness (or elasticity) [1,2]. According to force equilibrium of the arterial wall, the local arterial elasticity is extracted from a combination of the arterial radial displacement signal and the pulsatile pressure signal at an artery site, which are measured simultaneously by an imaging instrument and a tonometer, respectively [2]. These two interpretations utilize only the measured arterial pulse signals themselves but involve high cost and operation complexity of the medical instruments, rendering them unsuitable for routine and at-home use.

Due to its simplicity and low cost, photoplethysmography (PPG) sensors are widely used to acquire arterial pulse signals, commonly referred to as PPG signals [410]. A PPG signal essentially captures the blood volume change in an artery. Its first-order and second-order derivatives have been utilized to extract arterial indices [410]. Since the blood volume change in an artery is caused by the radial displacement of the arterial wall and this displacement is only a small percentage of the arterial radius, it is reasonable to treat the PPG signal as the radial displacement of the arterial wall (or arterial vasomotion) [5,6,810]. Since the radial displacement of the arterial wall facilitates the blood flow velocity (or rate) in an artery, the first-order derivative (velocity) of a PPG signal is intuitively considered to represent the blood flow velocity [612]. Then, the second-order derivative (acceleration) of a PPG signal is considered to represent the acceleration of blood flow [410]. Yet, a clear theoretical basis for such interpretation of the two derivatives is still missing [612]. Without relating to pulsatile parameters in blood flow, different ratios of peaks and valleys in the second-order derivative of a PPG signal were found to carry clinical values via statistical analysis [4,7], but physiological implications of these ratios are unclear. In recent years, great similarity in waveform between a PPG signal and a pulsatile pressure signal has been identified [9,10,13]. Owing to their unparalleled convenience, PPG sensors have been increasingly studied for continuous, cuffless blood pressure measurement. Without theoretical guidance, different features of a PPG signal and its derivatives are combined for estimation of blood pressure via statistical analysis and machine-learning techniques [9,14,15]. With the goal of developing an at-home solution to arterial health assessment, the author's group has developed a microfluidic-based tactile sensor for arterial pulse signal measurement, which features ease of use by a layperson and low cost [2,3]. The measured arterial pulse signal is treated as the radial displacement of the arterial wall. Consequently, its first-order and second-order derivatives represent the velocity and acceleration of the arterial wall, respectively. Due to its time-harmonic nature, the pulsatile radial motion of the arterial wall at an artery site is radial vibration. Then, the arterial wall is modeled as a unit-mass vibration system, and its system parameters are extracted from the measured pulse signal and its two derivatives. A scaling analysis is conducted to relate the system parameters to pulsatile parameters in blood flow for extraction of arterial indices. Although this vibration-model-based analysis of an arterial pulse signal has been validated by experimental data [3], similar to PPG signals, a clear theoretical basis for the relations derived from the scaling analysis is still lacking.

Given the above-mentioned utility of radial vibration (displacement, velocity, and acceleration) of the arterial wall for extraction of arterial indices, this paper presents a theoretical study on the relations of radial vibration of the arterial wall to pulsatile parameters in blood flow. Previously, the author developed a pulse wave propagation theory for obtaining theoretical expressions for various pulsatile parameters in terms of pulsatile pressure, without harmonics in a pulse signal and wave reflection [16]. Later on, harmonics in a pulse signal and wave reflection were included in the theory for calculating the waveforms of various pulsatile parameters [17]. In this paper, the developed theory is tailored to investigate the relations of the radial displacement and its two derivatives of the arterial wall to blood flow rate and pulsatile pressure. It is worth mentioning that axial displacement of the arterial wall [16,17] is omitted in this paper, due to its negligible influence on the pulsatile parameters considered in this paper. The pulse wave propagation theory is first presented for the wave velocity and the theoretical expressions for a radial displacement of the arterial wall and blood flow rate in terms of pulsatile pressure. With the harmonics of a pulse signal, theoretical relations of radial vibration of the arterial wall to pulsatile parameters in blood flow are derived under two conditions: without and with wave reflection. These derived theoretical relations identify the assumptions that are needed to simplify the relations of radial vibration of the arterial wall to pulsatile parameters in blood flow for clinical values [2,3,6]. With the arterial wall modeled as a unit-mass vibration system, these simplified relations are utilized to extract arterial indices from radial vibration of the arterial wall. Finally, some other applications of such relations for clinical values are discussed, and new insights into the interaction between the arterial wall and blood flow are revealed from the perspective of energy and one-dimensional (1D) wave equations.

2 Pulse Wave Propagation Theory

As shown in Fig. 1, the arterial wall is treated as a thin-walled tube with two geometrical parameters: a as the inner radius at diastolic blood pressure (DBP) and h as the thickness. It is assumed that both a and h do not vary in a pulse cycle. Due to DBP, the arterial wall contains an initial tension T0 = DBP × a per unit length in the circumferential direction. With no axial displacement, the arterial wall undergoes solely the pulsatile radial displacement ur(x, t) to facilitate blood flow in the artery. Blood in the artery is assumed to be an incompressible, Newtonian fluid and contains three pulsatile parameters: radial velocity w(x, r, t), axial (x-axis) velocity u(x, r, t), and pulsatile pressure Δp(x, t). Note that blood is known to be a non-Newtonian fluid, but it behaves as a Newtonian fluid in large arteries [18]. In small arteries (i.e., a 0.3 mm), blood behaves as a non-Newtonian fluid and causes a noticeable reduction in wall shear stress [19]. Since all the pulse signals measured at the skin surface [115] are from large arteries, it is reasonable to treat blood as a Newtonian fluid. It is further assumed that (1) the four pulsatile parameters are axisymmetric and have small perturbations and (2) the arterial radius a is much smaller than the wavelength λ of the pulse wave (a ≪ λ) [16,17,20,21].

Fig. 1
Schematic of the arterial wall and blood flow in it and the associated geometries and pulsatile parameters
Fig. 1
Schematic of the arterial wall and blood flow in it and the associated geometries and pulsatile parameters
Close modal
Based on the above assumptions, the governing equations of blood flow in the artery include the continuity equation and the Navier–Stokes equations in the axial (x-axis) and radial (r-axis) directions [16,17,20,21]
wr+wr+ux=0
(1a)
ρbwt=Δpr+μ(2wr2+1rwrwr2+2wx2)
(1b)
ρbut=Δpx+μ(2ur2+1rur+2ux2)
(1c)

where ρb and μ denote the blood density and viscosity, respectively.

Womersley derived the solution to Eq. (1)—the wave expressions for w, u, and Δp [20] at the nth harmonic of the heart rate
w=[Δp0β02r2μα02+Bβ0α0J0(α0)J1(α0r/a)]ei(nωtknx)
(2a)
u=[Δp0β0aμα02+BJ0(α0r/a)J0(α0)]ei(nωtknx)
(2b)
Δp=Δp0ei(nωtknx)
(2c)

where α02=i3α2 with α=aρbnω/μ being the Womersley number and β0=ianω/c=iβ [20]; ω and kn = nω/c are the angular frequency of the heart rate and the nth wave number, respectively. Note that Δp0 and B are two constant unknowns.

The governing equation of the arterial wall is given as
ρh2urt2=Δp[(E+ηnωi)hT0]ura2
(3a)
where E and η denote the elasticity and viscosity, respectively, of the arterial wall in the circumferential direction. No-slip conditions at the blood-wall interface demand the blood flow velocities being equal to the arterial wall velocities in the radial and axial directions at the blood-wall interface
wr=a=urt
(3b)
ur=a=0
(3c)
The wave expression for ur(x,t) at the nth harmonic of the heart rate can be written as
ur=ur0ei(nωtknx)
(4)
Substituting Eqs. (2) and (4) into Eq. (3) give rise to the following 3 × 3 matrix equation with a vector of the three constant unknowns of Δp0, B, and ur0:
[β0aμα0210β02a2μα0212β0F10inω10[τ0(1+ηnωi/E)]Eha2ρh(nω)2]{Δp0Bur0}={000}
(5)
where
τ0=T0/(Eh)andF10=2J1(α0)α0J0(α0)
(6)
Note that F10 arises from fluid-loading and takes a complex value, and τ0 is the normalized initial tension. A nonzero solution to the vector demands that the determinant of the 3 × 3 matrices in Eq. (5) be equal to zero, giving rise to the wave velocity
c=c0(1F10){[(1+ηnωi/E)τ0]ρa2(nω)2E}
(7a)
where
c0=PWV=Eh2ρba
(7b)

where c0 is the same as the PWV commonly used in clinical studies [13]. Evidently, F10, τ0, and viscous and inertial terms of the arterial wall are all neglected in the estimation of PWV. The inertial term and viscous term in Eq. (7a) are both small quantities, as compared with the rest terms. Moreover, 1-F10 does not vary much with the harmonics (or n) of the heart rate. Thus, c remains the same for different harmonics in a pulse signal.

Based on Eq. (5), B and ur0 can be expressed in terms of Δp0
B=1cρbΔp0
(8a)
ur0=(1F10)a2c2ρbΔp0
(8b)
Based on Eq. (2b), the amplitude of axial blood velocity varies along the r-axis and is written as
u0(r)=1cρb[1J0(α0r/a)J0(α0)]Δp0
(8c)
Consequently, the blood flow rate amplitude Q0 is expressed as
Q0=0au0(r)2πrdr=πa2cρb(1F10)Δp0=2πacur0
(9)

The relations of Q0 and Δp0 to ur0 remain the same for different harmonics in a pulse signal.

3 Relations of Radial Vibration of the Arterial Wall to Pulsatile Parameters in Blood Flow

An arterial pulse signal is a collection of the harmonics of the heart rate and experiences wave reflection in the CV system [22,23]. In this section, with the harmonics of a pulse signal considered, the relations of radial vibration of the arterial wall to pulsatile parameters in blood flow are derived under two conditions: without and with wave reflection.

3.1 Without Wave Reflection.

Pulsatile pressure is the driving force for the pulse wave propagation in the artery [16,17,20,21]. When there is no wave reflection, the pulsatile pressure input at an artery site (or at a fixed-x position) is expressed as
Δp(t)=nAnei(nωtknx)
(10)
where An denotes the amplitude of the nth harmonic in a pulsatile pressure signal. According to Sec. 2, the radial displacement of the arterial wall and the blood flow rate are
ur(t)=(1F10)a2c2ρbnAnei(nωtknx)
(11)
Q(t)=1F10cρbπa2nAnei(nωtknx)
(12)
As the time-derivative of the blood flow rate, the acceleration of blood flow ∂Q(t)/∂t is expressed as
Q(t)t=1F10ρbπa2niknAnei(nωtknx)
(13)
The pulsatile pressure gradient Δpx(t) is given by
Δpx(t)=niknAnei(nωtknx)
(14)
Comparison of Eqs. (13) and (14) leads to
Q(t)t=1F10ρbπa2Δpx(t)
(15)

Thus, the acceleration of blood flow can represent the pulsatile pressure gradient.

Now, we relate the above pulsatile parameters in blood flow to radial vibration of the arterial wall. The first-order derivative of the radial wall displacement is the velocity vr(t) of the arterial wall
vr(t)=durdt=(1F10)a2cρbniknAnei(nωtknx)
(16)
Comparison of Eqs. (12) and (16) gives rise to the following relation:
vr(t)=durdt=12πaniknAnei(nωtknx)nAnei(nωtknx)Q(t)
(17a)
By only keeping the first harmonic (n = 1) in the equation, the following relation is obtained:
vr1(t)=ik12πaQ1(t)
(17b)

Thus, the first harmonic of the radial velocity can represent the first harmonic of the blood flow rate.

Based on Eq. (16), the second-order derivative of the radial displacement of the arterial wall is the acceleration of the arterial wall is
ar(t)=d2urdt2=(1F10)a2ρbnkn2Anei(nωtknx)
(18)
Then, a combination of Eqs. (18), (13), and (14) gives rise to
ar(t)=d2urdt2=12πankn2Anei(nωtknx)niknAnei(nωtknx)Q(t)t=(1F10)a2ρbnkn2Anei(nωtknx)niknAnei(nωtknx)Δpx(t)
(19a)
Similarly, by only keeping the first harmonic in the equation, the above equation is simplified as
ar1(t)=ik12πaQ1t=ik1(1F10)a2ρbΔpx1(t)
(19b)
where Δpx1(t) is the first harmonic of the pulsatile pressure gradient
Δpx1(t)=ik1Δp1(t)
(20)

Based on Eq. (19b), the first harmonic of the radial acceleration can represent the first harmonic of the acceleration of blood flow and the first harmonic of the pulsatile pressure gradient.

3.2 With Wave Reflection.

When wave reflection is considered, the pulsatile pressure input at an artery site becomes
Δp(t)=n(Aneiknx+Aneiknx)einωt
(21)
where the first and the second terms denote the forward and backward waves of the pulsatile pressure, respectively. Then, the radial displacement of the arterial wall becomes
ur(t)=(1F10)a2c2ρbn(Aneiknx+Aneiknx)einωt
(22)

Since both Δp(t) and ur(t) are in the radial direction, wave reflection augments them. Based on Eqs. (21) and (22), the waveform of ur(t) is the same as the waveform of Δp(t).

The blood flow rate is written as
Q(t)=1F10cρbπa2n(AneiknxAneiknx)einωt
(23)
Wave reflection reduces the blood flow rate, due to its axial direction. The acceleration of blood flow ∂Q(t)/∂t is expressed as
Q(t)t=1F10ρbπa2nikn(AneiknxAneiknx)einωt
(24)
The pulsatile pressure gradient Δpx(t) is given by
Δpx(t)=nikn(AneiknxAneiknx)einωt
(25)
Comparison of Eqs. (24) and (25) leads to
Q(t)t=1F10ρbπa2Δpx(t)
(26)

Equations (15) and (26) imply that the waveform of ∂Q(t)/∂t can represent the waveform of Δpx(t), regardless of wave reflection.

The velocity of the arterial wall is expressed as
vr(t)=(1F10)a2cρbnikn(Aneiknx+Aneiknx)einωt
(27)
By keeping only the first harmonic in Eqs. (23) and (27), the following relations are obtained:
Q1(t)=1F10cρbπa2(A1eik1xA1eik1x)eiωt=Qf1(t)Qb1(t)
(28a)
vr1(t)=(1F10)a2cρbik1(A1eik1x+A1eik1x)eiωt
(28b)
where Qf1(t) and Qb1(t) are the first-harmonic forward and backward waves of the blood flow rate, respectively. A comparison of the above two equations shows
vr1(t)=ik12πa{Qf1(t)+Qb1(t)}
(29)

Due to the opposite effects of wave reflection on vr(t) and Q(t), the waveform of vr(t) fails to represent the waveform of Q(t), even when only their first harmonics are considered.

The acceleration of the arterial wall is expressed as
ar(t)=(1F10)a2ρbnkn2(Aneiknx+Aneiknx)einωt
(30)
By keeping only the first harmonics of the pulsatile parameters in Eqs. (24), (25), and (30), the following relations are obtained:
Q1(t)t=1F10ρbπa2ik1(A1eik1xA1eik1x)eiωt=Qf1(t)tQb1(t)t
(31a)
Δpx1(t)=ik1(A1eik1xA1eik1x)eiωt=Δpxf1(t)+Δpxb1(t)
(31b)
ar1(t)=(1F10)πa2cρbk12(A1eik1x+A1eik1x)eiωt
(31c)
where Δpxf1 and Δpxb1 denote the first-harmonic forward and backward waves of Δpx(t), respectively. A comparison of the above three equations gives rise to
ar1(t)=ik12πa{Qf1(t)t+Qb1(t)t}=ik1a2ρb(1F10){Δpxf1(t)Δpxb1(t)}
(32)
Due to wave reflection, the waveform of ar(t) neither represents the waveform of ∂Q(t)/∂t nor the waveform of Δpx(t), even when only their first harmonics are considered. Note that Δpx1(t) is related to Δp1(t) by
Δpx1(t)=ik1(A1eik1xA1eik1x)(A1eik1x+A1eik1x)Δp1(t)
(33)

4 Extraction of Arterial Indices With the Arterial Wall as a Unit-Mass Vibration System

4.1 Relation of Radial Vibration of the Arterial Wall to Its Mechanical Properties and Geometries.

At an artery site, the pulsatile radial motion of the arterial wall is essentially radial vibration. As such, the arterial wall at an artery site can be modeled as a unit-mass vibration system with pulsatile pressure as the driving force [2,3]
Md2urdt2+Ddurdt+Kur=Δp(t)ρh
(34)
where the damping coefficient D and the spring stiffness K are related to the elasticity and viscosity of the arterial wall by
D=ηρa2,K=E(1τ0)ρa2
(35)

To simplify the measurement method for extraction of arterial indices, the following theory is developed for extracting arterial indices solely from one single measured arterial radial displacement signal with no calibration [2,3]. Figure 2 shows a measured arterial radial displacement signal and its first-order and second-order derivatives. The key features labeled in the figure can be easily obtained by using built-in functions in matlab.

Fig. 2
The radial displacement, velocity, and acceleration of the arterial wall in one pulse cycle, labeled with the associated key features (adapted from Ref. [3])
Fig. 2
The radial displacement, velocity, and acceleration of the arterial wall in one pulse cycle, labeled with the associated key features (adapted from Ref. [3])
Close modal
The pulsatile pressure amplitude Δp0 can be approximated by the difference between the maximum and minimum acceleration of the arterial wall
Δp0ρh(ar_maxar_min)Δt/T
(36)
Note that Δt/T is utilized to factor in difference in heart rate between different conditions (e.g., before versus postexercise) and subjects. Consequently, K and D can be estimated by
Kar_maxar_minur0ΔtT,Dar_maxar_minvr0ΔtT
(37)

where ur0 and vr0 are the radial displacement and velocity amplitudes of the arterial wall, respectively.

4.2 Extraction of Arterial Indices.

As analyzed in Sec. 2, when wave reflection is neglected and only the first harmonics of the pulsatile parameters are considered, vr(t) can represent Q(t) and Δp(t) can represent Δpx(t). The following analysis for extraction of arterial indices from the key features in Fig. 2 is based on these two assumptions. Note that subscript 1 for the first harmonic is dropped for simplicity here, and subscript 0 denotes the amplitude of a pulsatile parameter. According to Eq. (17b), the blood flow rate amplitude Q0 can be approximated by the velocity amplitude vr0 of the arterial wall
Q0=2πavr0ik
(38)
Based on Eq. (33), the pulsatile pressure gradient amplitude Δpx0 is associated with the pulsatile pressure amplitude Δp0 by
Δpx0=(ik)Δp0
(39)
The peripheral vascular resistance (PVR) at an artery site is given by [2,3,23]
PVR=Δpx0Q0=8μπa4
(40)
By substituting Eqs. (38) and (39) into Eq. (40), the following relation is obtained:
k2Δp02πavr0=8μπa4
(41)
Note that the wave number k is related to the wavelength λ by
k=ωc=2πλ,λ=c60HR=cT
(42)
where HR denotes the heart rate. It is assumed that λ and μ do not vary between subjects and different conditions, substituting Eqs. (36) and (37) into Eq. (41) leads to the following relation:
a(vr0Δp0)1/3D1/3
(43a)
Consequently, based on Eq. (35), the elasticity and viscosity of the arterial wall and PWV can be estimated by
ηD1/3,E(1τ0)KD2/3
(43b)
PWV1τ0KD1/3
(43c)

5 Discussion

With harmonics and wave reflection, the theoretical expressions for radial wall vibration and pulsatile parameters in blood flow better represent the pulsatile parameters measured in clinical studies, and the derived theoretical relations between radial wall vibration and pulsatile parameters in blood flow better manifest the intrinsic relations between them in an artery. Measurement complexity and high-cost prevent from measuring all the pulsatile parameters in an artery for best assessing the cardiovascular system. As such, theoretically solid relations between different pulsatile parameters become more important for interpreting only one measured pulse signal for clinical values with clearly defined physiological implications and assumptions.

5.1 Applicability of the Assumptions in Interpretation of the Radial Vibration of the Arterial Wall for Clinical Values.

As shown in Sec. 4, to extract the arterial radius a from radial vibration of the arterial wall, two assumptions are necessary: (1) wave reflection is neglected; and (2) only the first harmonics of the pulsatile parameters are considered. Given that the early time duration of a pulse cycle is considered to be free of wave reflection and the first harmonics of the pulsatile parameters are dominant [22,23]; and vr0 and Δp0 are extracted from the key features in the early time duration of a pulse cycle, these two assumptions are reasonably realistic. Additionally, vr0 and Δp0 are calculated from Eqs. (36) and (37) include both wave reflection and all the harmonics, with wave reflection having the same augmenting effect in vr0 and Δp0. Then, according to Eq. (43a), the inclusion of wave reflection and all the harmonics in both vr0 and Δp0 might alleviate the effect of wave reflection and harmonics on the extraction of a. The estimation of arterial indices based on Eq. (43) has been experimentally validated by the measured difference of arterial indices on healthy subjects between at-rest and immediately after postexercise with statistical significance [3].

For measurement simplicity, PPG signals have been utilized to represent other pulsatile parameters. As pointed out earlier, A PPG signal essentially represents the radial displacement of the arterial wall [5,6,810]. The velocity of the PPG signal (or velocity of PPG) is considered to represent the blood flow rate in an artery; the acceleration of the PPG signal is considered to represent the acceleration of blood flow in an artery and consequently the pulsatile pressure gradient in the literature [410]. As derived in Sec. 3, interpretation of the velocity and acceleration of a PPG signal as blood flow rate and acceleration of blood flow, respectively, is only valid, when wave reflection is not considered and only the first harmonics of the pulsatile parameters are considered. This may explain the algorithms for adjusting the velocity of a PPG signal to match the waveform of the blood flow rate measured using an imaging instrument in the literature [11,12]. Due to the technical difficulty in measuring Δpx, the first-order derivative of measured blood flow rate Q(t) was utilized to represent the waveform of Δpx(t) [24]. It is interesting to note that only this representation is valid, regardless of wave reflection and harmonics. Taken together, wave reflection and harmonics modify the relations of different pulsatile parameters and their time-derivatives that are obtained under the assumptions of no wave reflection and one harmonic term [25].

Without relating to pulsatile parameters in blood flow, a PPG signal and its derivatives are also utilized for clinical values via statistical analysis [4,710]. Without relating to forward and backward waves of a pulse signal, peaks and valleys in the second-order derivative of a PPG signal were arbitrarily labeled as a, b, c, d, e waves, and different ratios of these peaks and valleys were explored for their clinical values [4,710]. Among them, b/a ratio (|ar_min/ar_max| ratio in Fig. 2) was most studied. It was positively correlated with age and arterial stiffness but was negatively correlated with arterial distensibility [4]. When the second-order derivative of a PPG signal is interpreted as radial wall acceleration, a large b/a ratio implies reduced radial wall displacement and then increased arterial elasticity. With aging, arterial elasticity is increased and arterial distensibility is reduced. As such, the physiological implication of the b/a ratio becomes clear, when it is interpreted in terms of the radial wall acceleration. Similarly, without physiological implications, different features (i.e., peaks and valleys and their time intervals) and their combinations of a PPG signal and its two derivatives have been utilized for estimating blood pressure [9,14,15] in recent years. Given the complexity of the combination of different features [15], the physiological implication of such a combination is not discussed here.

5.2 Characteristic Impedance.

Characteristic impedance is an important clinical marker for the CV system and is defined as [23]
Zc=ΔpQπa2=ρbc1F10
(44)
where Δp and Q denote the forward waves of the two parameters, and according to the above equation, Zc manifests the wave velocity c and is also affected by F10. The majority of the studies on Zc neglect F10 [22,23,26]. To extract the value of Zc from the measured Q and Δp, the following equation is utilized:
ρbQt=πa2Δpx2τwπa
(45)
where τw is wall shear stress (wss). When wave reflection is neglected, the pulsatile pressure gradient and wall shear stress [23] can be represented by
Δpx=1cΔpt
(46a)
τw(x,t)=aF102cΔpt
(46b)
Substituting Eq. (46) into Eq. (45) gives rise to
Zc=ρbc(1F10)=πa2Δpt/Qt
(47)

Consequently, the value of Zc can be extracted from the time derivatives of the measured Q and Δp. It is worth emphasizing that the extracted value of Zc is based on Eq. (47) factors in τw and harmonics. In the study of Lucas et al. [23], the authors did not include the influence of F10 on Zc and thus failed to recognize the inclusion of τw in the extracted value of Zc from Eq. (47). Table 1 lists the calculated values of Zc for the first ten harmonics at the carotid artery, based on the related mechanical properties and geometries [27]. The calculated values for the two parameters without F10: c0 = 8.29 m/s, ρbc0=8741 Pa⋅s/m, are independent of harmonics. Evidently, F10 (or τw) reduces wave velocity but introduces wave attenuation. Meanwhile, the influence of F10 on Zc is overall negligible.

Table 1

The calculated values of wave velocity and characteristic impedance for the first ten harmonics at the carotid artery (a =3.3 mm, h =0.62 mm, heart rate = 75 bpm, and E =771 kPa [27])

c (m/s)Zc=ρbc1F10
nRealImaginaryValue (Pa⋅s/m)Phase (deg)
16.680.339607.8–9.9
27.030.489209.9–6.7
37.180.559035.0–5.4
47.270.608932.3–4.6
57.340.648862.7–4.1
67.380.678811.7–3.7
77.420.698772.2–3.4
87.450.718740.5–3.2
97.470.728714.4–3.0
107.490.738692.3–2.8
c (m/s)Zc=ρbc1F10
nRealImaginaryValue (Pa⋅s/m)Phase (deg)
16.680.339607.8–9.9
27.030.489209.9–6.7
37.180.559035.0–5.4
47.270.608932.3–4.6
57.340.648862.7–4.1
67.380.678811.7–3.7
77.420.698772.2–3.4
87.450.718740.5–3.2
97.470.728714.4–3.0
107.490.738692.3–2.8

5.3 Evidence of Harmonics in the Measured Acceleration Signal of the Arterial Wall.

Due to their small size and low cost, accelerometers have also been utilized to measure the acceleration signal caused by radial vibration of the arterial wall [28]. The measured acceleration signal is then integrated once to obtain the velocity and twice to obtain the displacement of the arterial wall. It was found [28] that the velocity amplitude of the arterial wall is well above ω×ur0, and the maximum acceleration of the arterial wall is well above ω×vr0, manifesting the existence of the harmonics in a pulse signal, given the same effect of wave reflection on them. Accordingly, the inclusion of the harmonics in both vr0 and Δp0 in Eq. (43a) might alleviate the effect of harmonics on the extraction of a.

5.4 New Insights on the Interaction Between the Arterial Wall and Blood Flow

5.4.1 Energy Associated With the Arterial Wall and Blood Flow.

As analyzed in Sec. 2, the pulse wave propagation in an artery is a coordinated response of blood flow and the arterial wall. The maximum potential elastic energy and the maximum kinetic energy in the arterial wall per unit length (along the x-axis) are given by
PEwall=12E(ur0a)22πah,KEwall=12ρ2πahvr02
(48)
Due to a high E and a small a, KEwallPEwall. The maximum kinetic energy of blood flow per unit length is given by
KEblood=12ρbQ0πa22
(49)

It is interesting to note that equating PEwall to KEblood gives rise to c0 in Eq. (7b).

Now, it is tempted to conclude that elastic energy of the arterial wall and kinetic energy of blood flow keep exchanging to facilitate blood flow. Yet, according to Eq. (9), since the imaginary part of the wave velocity is much smaller than its real part, Q(t) and ur(t) reach their maximum values at very similar times in a pulse cycle, suggesting that the blood flow and the arterial wall reach their peak values at similar times. This is opposite to a mass-spring vibration system, in which the kinetic energy of the mass is out of phase with the elasticity energy of the spring. As such, it is unreasonable to treat the blood as the mass and the arterial wall as the spring for an equivalent vibration model, which would further lead to an extremely high angular frequency (or heart rate). Then, the unit-mass vibration model of the arterial wall with the pulsatile pressure from blood flow as the driving force is more realistic to relate the pulsatile radial motion of the arterial wall to pulsatile parameters in blood flow for extraction of arterial indices. Note that the discussion on the energy involved in an artery here is focused on large healthy arteries with the assumptions in Sec. 2. Whether CV disease affects the energy relation depends on how CV disease affects the relation between radial vibration of the arterial and blood flow rate.

5.4.2 One-Dimensional Wave Equation for the Interaction Between the Arterial Wall and Blood Flow.

The pulse wave propagation in an artery results from the interaction between the arterial wall and blood flow, but Eqs. (1) and (3) are totally different from the 1D wave equations. Here, we attempt to derive a 1D wave equation that manifests the interaction between the arterial wall and blood flow. To this end, average axial blood velocity u¯(x,t) is chosen as the pulsatile parameter for deriving a 1D wave equation, given that this parameter directly represents blood circulation. As shown in Fig. 3, for a segment of dx in an artery, its mass m is from the blood flow
m=ρbπa2dx
(50)
Fig. 3
The interaction between the arterial wall and the blood flow is equivalent to longitudinal wave propagation with average axial blood flow velocity u¯(x,t) as the pulsatile parameter in an elastic rod with elasticity Ex-axis, density ρb, and cross section area πa2
Fig. 3
The interaction between the arterial wall and the blood flow is equivalent to longitudinal wave propagation with average axial blood flow velocity u¯(x,t) as the pulsatile parameter in an elastic rod with elasticity Ex-axis, density ρb, and cross section area πa2
Close modal
The 1D wave equation for u¯(x,t) is written as
2u¯t2=c022u¯x2
(51)
Note that u¯(x,t) propagates with the same wave velocity as pulsatile pressure and radial displacement of the arterial wall. By substituting Eq. (7b) into Eq. (51) and multiplying m on both sides of Eq. (51), the following equation is obtained:
ρbπa22u¯t2=Exaxisπa22u¯x2withExaxis=Eh2a
(52)

where Ex-axis arises from the arterial wall.

The wave equation for an elastic rod is
ρA2ut2=EA2ux2
(53)

where E, ρ, and A denote the elasticity, density, and cross section area of the elastic rod, respectively, and the pulsatile parameter u(x, t) is the axial displacement of the elastic rod. In an elastic rod, the same medium provides mass and the elasticity. In contrast, in an artery, the arterial wall provides the elasticity Ex-axis and the blood provides the mass m. Comparison of Eqs. (52) and (53) reveals that the interaction between the arterial wall and blood flow is equivalent to longitudinal wave propagation with u¯(x,t)as the pulsatile parameter in an elastic rod with elasticity Ex-axis, density ρb, and cross section area πa2.

6 Conclusion

With two physiological realities: harmonics and wave reflection, being considered, the theoretical expressions for radial vibration of the arterial wall and pulsatile parameters in blood flow are first derived. Afterward, the theoretical relations between them are derived, revealing the effect of harmonics and wave reflection on these relations. The derived theoretical relations identify the assumptions that are needed in the utility of radial vibration of the arterial wall for extraction of arterial indices. With the arterial wall modeled as unit-mass vibration system, the obtained relations under the assumptions are utilized to relate radial wall vibration to arterial indices, providing a solid theoretical basis for the previous related experimental work [3]. These theoretical relations further identify some misconceptions in the interpretation of a PPG signal and its two derivatives for clinical values. New insights on the interaction between the arterial wall and blood flow are further revealed from the perspective of energy and 1D wave equations. Given the ever-increasing need for measurement simplicity for routine and at-home monitoring of the cardiovascular system, interpretation of only a PPG signal or a pulse signal measured by a tactile sensor for clinical values becomes more important. The derived theoretical expressions for all the pulsatile parameters and their relations in this paper provide theoretical guidance for improving their interpretation of clinical values with clearly defined physiological implications and assumptions.

Funding Data

  • National Science Foundation (NSF) (Grant No. 1936005; Funder ID: 10.13039/100000146).

References

1.
Tanaka
,
H.
,
2017
, “
Various Indices of Arterial Stiffness: Are They Closely Related or Distinctly Different?
,”
Pulse (Basel)
,
5
(
1–4
), pp.
1
6
.10.1159/000461594
2.
Wang
,
D.
,
Reynolds
,
L.
,
Alberts
,
T.
,
Vahala
,
L.
, and
Hao
,
Z.
,
2019
, “
Model-Based Analysis of Arterial Pulse Signals for Tracking Changes in Arterial Wall Parameters: A Pilot Study
,”
Biomech. Model Mechanobiol.
,
18
(
6
), pp.
1629
1638
.10.1007/s10237-019-01165-x
3.
Rahman
,
M. M.
,
Twiddy
,
H.
,
Reynolds
,
L.
, and
Hao
,
Z.
,
2021
, “
Measurement of Post-Exercise Response of Local Arterial Parameters Using an Adjustable Microfluidic Tactile Sensor
,”
Annual International Conference of the IEEE Engineering in Medicine and Biology Society
, Mexico, Nov. 1–5, pp.
1284
1287
.10.1109/EMBC46164.2021.9630432
4.
Elgendi
,
M.
,
2012
, “
On the Analysis of Fingertip Photoplethysmogram Signals
,”
Curr. Cardiol. Rev.
,
8
(
1
), pp.
14
25
.10.2174/157340312801215782
5.
Wang
,
C. Z.
, and
Zheng
,
Y. P.
,
2010
, “
Comparison Between Reflection-Mode Photoplethysmography and Arterial Diameter Change Detected by Ultrasound at the Region of Radial Artery
,”
Blood Press. Monit.
,
15
(
4
), pp.
213
219
.10.1097/MBP.0b013e328338aada
6.
Shin
,
H.
, and
Min
,
S. D.
,
2017
, “
Feasibility Study for the Non-Invasive Blood Pressure Estimation Based on PPG Morphology: Normotensive Subject Study
,”
Biomed. Eng. Online
,
16
(
1
), p.
10
.10.1186/s12938-016-0302-y
7.
Elgendi
,
M.
,
Liang
,
Y.
, and
Ward
,
R.
,
2018
, “
Toward Generating More Diagnostic Features From Photoplethysmogram Waveforms. Diseases
,”
Diseases
,
6
(
1
), p.
20
.10.3390/diseases6010020
8.
Ding
,
X.
,
Yan
,
B. P.
,
Zhang
,
Y. T.
,
Liu
,
J.
,
Zhao
,
N.
, and
Tsang
,
H. K.
,
2017
, “
Pulse Transit Time Based Continuous Cuffless Blood Pressure Estimation: A New Extension and a Comprehensive Evaluation
,”
Sci. Rep.
,
7
(
1
), p.
11554
.10.1038/s41598-017-11507-3
9.
Xing
,
X.
,
Ma
,
Z.
,
Zhang
,
M.
,
Zhou
,
Y.
,
Dong
,
W.
, and
Song
,
M.
,
2019
, “
An Unobtrusive and Calibration-Free Blood Pressure Estimation Method Using Photoplethysmography and Biometrics
,”
Sci. Rep.
,
9
(
1
), p.
8611
.10.1038/s41598-019-45175-2
10.
Martínez
,
G.
,
Howard
,
N.
,
Abbott
,
D.
,
Lim
,
K.
,
Ward
,
R.
, and
Elgendi
,
M.
,
2018
, “
Can Photoplethysmography Replace Arterial Blood Pressure in the Assessment of Blood Pressure?
,”
J. Clin. Med.
,
7
(
10
), p.
316
.10.3390/jcm7100316
11.
Cook
,
L. B.
,
2001
, “
Extracting Arterial Flow Waveforms From Pulse Oximeter Waveforms Apparatus
,”
Anaesthesia
,
56
(
6
), pp.
551
555
.10.1046/j.1365-2044.2001.01986.x
12.
Wisely
,
N. A.
, and
Cook
,
L. B.
,
2001
, “
Arterial Flow Waveforms From Pulse Oximetry Compared With Measured Doppler Flow Waveforms Apparatus
,”
Anaesthesia
,
56
(
6
), pp.
556
561
.10.1046/j.1365-2044.2001.01987.x
13.
Wang
,
H.
,
Wang
,
L.
,
Sun
,
N.
,
Yao
,
Y.
,
Hao
,
L.
,
Xu
,
L.
, and
Greenwald
,
S. E.
,
2020
, “
Quantitative Comparison of the Performance of Piezoresistive, Piezoelectric, Acceleration, and Optical Pulse Wave Sensors
,”
Front. Physiol.
,
10
, p.
1563
.10.3389/fphys.2019.01563
14.
Xing
,
X.
,
Ma
,
Z.
,
Zhang
,
M.
,
Gao
,
X.
,
Li
,
Y.
,
Song
,
M.
, and
Dong
,
W. F.
,
2020
, “
Robust Blood Pressure Estimation From Finger Photoplethysmography Using Age-Dependent Linear Models
,”
Physiol. Meas.
,
41
(
2
), p.
025007
.10.1088/1361-6579/ab755d
15.
Natarajan
,
K.
,
Block
,
R. C.
,
Yavarimanesh
,
M.
,
Chandrasekhar
,
A.
,
Mestha
,
L. K.
,
Inan
,
O. T.
,
Hahn
,
J. O.
, and
Mukkamala
,
R.
,
2022
, “
Photoplethysmography Fast Upstroke Time Intervals Can Be Useful Features for Cuff-Less Measurement of Blood Pressure Changes in Humans
,”
IEEE Trans. Biomed. Eng.
,
69
(
1
), pp.
53
62
.10.1109/TBME.2021.3087105
16.
Smith
,
S. M.
,
Marin
,
J.
,
Adams
,
A.
,
West
,
K.
, and
Hao
,
Z.
,
2022
, “
Radial and Axial Motion of the Initially Tensioned Orthotropic Arterial Wall in Arterial Pulse Wave Propagation
,”
ASME J. Med. Diag.
,
5
(
2
), p.
021004
.10.1115/1.4053863
17.
Hao
,
Z.
,
2022
, “
Radial and Axial Displacement of the Initially-Tensioned Orthotropic Arterial Wall Under the Influence of Harmonics and Wave Reflection
,”
ASME J. Med. Diagn.
,
5
(
4
), p.
041007
.10.1115/1.4054883
18.
Thomas
,
B.
, and
Sumam
,
K. S.
,
2016
, “
Blood Flow in Human Arterial System-A Review
,”
Procedia Technol.
,
24
, pp.
339
346
.10.1016/j.protcy.2016.05.045
19.
Anastasiou
,
A. D.
,
Spyrogianni
,
A. S.
,
Koskinas
,
K. C.
,
Giannoglou
,
G. D.
, and
Paras
,
S. V.
,
2012
, “
Experimental Investigation of the Flow of a Blood Analogue Fluid in a Replica of a Bifurcated Small Artery
,”
Med. Eng. Phys.
,
34
(
2
), pp.
211
218
.10.1016/j.medengphy.2011.07.012
20.
Womersley
,
J. R.
,
1955
, “
XXIV. Oscillatory Motion of a Viscous Liquid in a Thin-Walled Elastic Tube—I: The Linear Approximation for Long Waves
,”
London Edinburgh Dublin Philos. Mag. J. Sci.
,
46
(
373
), pp.
199
221
.10.1080/14786440208520564
21.
Atabek
,
H. B.
, and
Lew
,
H. S.
,
1966
, “
Wave Propagation Through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube
,”
Biophys. J.
,
6
(
4
), pp.
481
503
.10.1016/S0006-3495(66)86671-7
22.
Qureshi
,
M. U.
,
Colebank
,
M. J.
,
Schreier
,
D. A.
,
Tabima
,
D. M.
,
Haider
,
M. A.
,
Chesler
,
N. C.
, and
Olufsen
,
M. S.
,
2018
, “
Characteristic Impedance: Frequency or Time Domain Approach?
,”
Physiol. Meas.
,
39
(
1
), p.
014004
.10.1088/1361-6579/aa9d60
23.
Lucas
,
C. L.
,
Wilcox
,
B. R.
,
Ha
,
B.
, and
Henry
,
G. W.
,
1988
, “
Comparison of Time Domain Algorithms for Estimating Aortic Characteristic Impedance in Humans
,”
IEEE Trans. Biomed. Eng.
,
35
(
1
), pp.
62
68
.10.1109/10.1337
24.
Tasu
,
J. P.
,
Mousseaux
,
E.
,
Delouche
,
A.
,
Oddou
,
C.
,
Jolivet
,
O.
, and
Bittoun
,
J.
,
2000
, “
Estimation of Pressure Gradients in Pulsatile Flow From Magnetic Resonance Acceleration Measurements
,”
Magn. Reson. Med.
,
44
(
1
), pp.
66
72
.10.1002/1522-2594(200007)44:1<66::AID-MRM11>3.0.CO;2-#
25.
Bernstein
,
D. P.
,
2009
, “
Impedance Cardiography: Pulsatile Blood Flow and the Biophysical and Electrodynamic Basis for the Stroke Volume Equations
,”
J. Electr. Bioimped.
,
1
(
1
), pp.
2
17
.10.5617/jeb.51
26.
Willemet
,
M.
, and
Alastruey
,
J.
,
2015
, “
Arterial Pressure and Flow Wave Analysis Using Time-Domain 1-D Hemodynamics
,”
Ann. Biomed. Eng.
,
43
(
1
), pp.
190
206
.10.1007/s10439-014-1087-4
27.
Jagielska
,
K.
,
Trzupek
,
D.
,
Lepers
,
M.
,
Pelc
,
A.
, and
Zieliński
,
P.
,
2007
, “
Effect of Surrounding Tissue on Propagation of Axisymmetric Waves in Arteries
,”
Phys. Rev. E
,
76
(
6 Pt 2
), p.
066304
.10.1103/PhysRevE.76.066304
28.
Di Lascio
,
N.
,
Gemignani
,
V.
,
Bruno
,
R. M.
,
Bianchini
,
E.
,
Stea
,
F.
,
Ghiadoni
,
L.
, and
Faita
,
F.
,
2015
, “
Noninvasive Assessment of Carotid Pulse Pressure Values: An Accelerometric-Based Approach
,”
IEEE Trans. Biomed. Eng.
,
63
(
4
), pp.
869
875
.10.1109/TBME.2015.2477538