Radial and Axial Displacement of the Initially-Tensioned Orthotropic Arterial Wall Under the Influence of Harmonics and Wave Reflection

Blood circulation in the cardiovascular (CV) system is essentially pulsatile wave propagation in arteries [1]. Clinical studies have identified the importance of radial displacement of the arterial wall in pulsatile wave propagation and established its clinical value for detection of atherosclerosis. Yet, radial-displacement-based clinical measures remain limited in early prediction of atheroscelerosis [2–7]. In recent years, axial displacement of the arterial wall is found to be comparable in amplitude to radial displacement of the arterial wall [2–7], be a more sensitive measure of subclinical atherosclerosis, and correlate with CV risk factors differently, as compared with radial-displacement-based clinical measures [6]. Meanwhile, studies on excised arteries find that axial initial tension and axial stress of the arterial wall affect arterial wall remodeling and growth and vascular homeostasis [8,9], but their measurement on live subjects is not established. Additionally, circumferential initial tension (i.e., diastolic blood pressure) of the arterial wall is found to affect the wave propagation velocity [10]. Yet, in recent decades, the theories and related numerical models on arterial pulsatile wave propagation are focused on the governing equations of blood flow in an artery and include only radial displacement of the arterial wall as a boundary condition [1], and thus are unsuitable to examine axial displacement and axial initial tension of the arterial wall for their role in pulsatile wave propagation.

The problem with blood circulation in the CV system has a long history. With radial and axial displacement of the arterial wall being considered, the problem of pulsatile wave propagation in arteries was extensively examined by many researchers in the 1950–1960s [11–15]. By adopting a free wave propagation approach, Womersley [11] laid out the theoretical framework and derived the wave expressions for three pulsatile parameters in blood flow, with the arterial wall modeled as a thin-walled isotropic tube with no initial tension and blood flow treated as an incompressible, Newtonian fluid. Later on, many factors arising from the physiological reality in the CV system were added to this framework [12–15]. For instance, Atabek [12] modeled the arterial wall as a thin-walled orthotropic tube with two initial tensions, and Mirsky [13] modeled the arterial wall as a thick-walled orthotropic tube with no initial tension but considered the influence of harmonics and wave reflection. Regardless of the tube model used, the wave expressions for three pulsatile parameters in blood flow derived by Womersley remain the same [14]. However, these theoretical studies were never applied to clinical studies, since axial displacement of the arterial wall was considered to be negligible at that time, as compared with radial displacement of the arterial wall. The theories on arterial pulsatile wave propagation assume that the mechanical properties of the arterial wall do not vary along with its thickness [1,12,13]. This assumption is consistent with clinical studies, in which the arterial wall displacement is measured at one point along the wall thickness to represent the collective behavior of the arterial wall at an artery site [1–7]. It should be noted that various constitutive models of the arterial wall are also developed that consider three layers and different biological components in the arterial wall along its thickness [16,17], but these models are aimed to examine the relation of mechanical function to biological function in the arterial wall and are unpractical for clinical studies yet.

In this study, the related theories from Atabek [12] and Mirsky [13] are combined and improved for revealing the role of axial displacement and two initial tensions of the arterial wall in pulsatile wave propagation under the influence of harmonics and wave reflection. The arterial wall is modeled as an initially-tensioned thin-walled orthotropic tube for examining the influence of harmonics on the wave velocity and pulsatile parameters with mathematical clarity. Afterward, the influence of wave reflection is added to pulsatile parameter expressions. With the harmonic values of measured pulsatile pressure and blood flow rate at the ascending aorta (AA) in the literature [13], the waveforms of radial wall displacement, axial wall displacement, and wall shear stress are calculated under different orthotropicity and axial initial tensions. The results are compared with Atabek and Mirky's studies, the author's previous associated study [18], as well as the related numerical studies [19,20]. The results are utilized to explain some related clinical findings and their physiological implications and possible clinical applications are further discussed.

The arterial wall has a layered anatomical structure and is initially tensioned in the axial direction, due to its anatomy [9], and in the circumferential direction, due to diastolic blood pressure (DBP). As such, the arterial wall is modeled as an initially-tensioned thin-walled orthotropic tube. As shown in Fig. 1(a), the arterial wall geometry includes the inner radius a at DBP and thickness h [12]. The arterial wall has elasticity E_θ and initial tension per unit length T_θ0 in the circumferential direction, and elasticity E_x and initial tension per unit length T_x0 in the axial direction (x-axis). During the time t of a pulse cycle, the arterial wall at a fixed axial-position x undergoes radial and axial motion, and thus has two pulsatile parameters: radial wall displacement η(x, t) and axial wall displacement ξ(x, t). The blood flow in the artery is assumed to be an incompressible Newtonian fluid and has three pulsatile parameters: radial blood flow velocity w(r, x, t), axial blood flow velocity u(r, x, t), and pulsatile blood pressure Δp(r, x, t), where the radial coordinate r varies within r ∈ (0, a).

Three fundamental assumptions for arterial pulsatile wave propagation are that: (1) The five pulsatile parameters are axisymmetric and small perturbations; (2) Change of the inner radius of the arterial wall is negligible during arterial wall motion so that the conditions and stresses at the blood-wall interface are calculated at r = a; and 3). The inner radius of the arterial wall is much smaller than the pulse wavelength λ (a≪λ) and thus Δp(r, x, t) is independent of r and becomes Δp(x, t) [11–15].

where $α02=i3α2$ with $α=aρbnω/μ$ being Womersely number [11] and $β0=ianω/c=iβ$ with n being the n-th harmonics of the heart rate; ω and k = nω/c are the angular frequency of the heart rate and the nth wave number, respectively, where c is the wave velocity of the nth harmonic term of a pulse signal.

where ρ, υ_θ, and υ_x denote the density, circumferential Poisson's ratio, and axial Poisson's ratio of the arterial wall, respectively; and γ = E_x/E_θ. The first term on the right side of Eq. (3b) is wall shear stress (wss) τ_w.

Note that τ_x0 and τ_θ0 are normalized initial tensions, and F₁₀ is due to fluid-loading and takes complex values.

The two roots of Eq. (8) correspond to the Young wave (c₁) and the Lamb wave (c₂). The wave velocity (or phase velocity) of the Young wave and the Lamb wave becomes real(c₁) and real(c₂), respectively. The wave transmission per wavelength is then calculated as exp(−2πY/X), with c = X+Yi (X and Y take real values). The influence of harmonics on the wave velocity arises from F₁₀.

Note that c₁ is reduced to the Moens–Korteweg formula for pulse wave velocity (PWV), which is commonly used in clinical studies [1–7], and c₂ is reduced to longitudinal elastic wave velocity c_L [15].

Comparison of Eqs. (10a) and (11) implies that circumferential initial tension (or DBP) lowers the Young wave velocity and fluid-loading causes wave attenuation.

Wave reflection augments Δp(x, t) and η(x, t), due to their radial direction, but reduces Q(x, t), ξ(x, t), and τ_w(x, t), due to their axial direction. Note that the amplitudes of axial wall displacement and wall shear stress reveal different explicit dependence on harmonics (or n). In contrast, the amplitudes of blood flow rate and radial wall displacement do not show explicit dependence on harmonics (or n).

Although ξ(t) is measured at the carotid artery (CA) [2–7], the harmonic values of measured pulsatile pressure and blood flow rate waveforms are only available at the ascending aorta (AA) in the literature [13]. As such, we calculate the wave velocity and above-derived pulsatile parameters at the AA. The values for the AA are E_θ = 400 kPa, a  =  13 mm, and h  =  2 mm, and the fundamental frequency of the heart rate is 1.25 Hz (or 75 beats per minute) [13,24]. Circumferential initial tension T_θ0  = DBP×a is assumed at τ_θ0 = 0.1. The values of T_x0 and E_x in vivo are unknown. Based on Eq. (4b), τ_x0 = 0.1 is utilized to examine the situation where only τ_w causes ξ(t), and τ_x0 = 0.2 is utilized to examine the situation where both τ_w and the ∂η/∂x term with a factor of (T_x0−T_θ0) cause ξ(t). The value of γ = E_x/E_θ is varied to examine how orthotropicity affects the pulsatile parameters. Unless otherwise specified, the calculated pulsatile parameters are for the Young wave.

As shown in Table 1, wave velocity in the Young wave and the Lamb wave does not vary with harmonics, and the wave transmission in the two waves increases with the higher harmonics. The wave velocity and wave transmission in the Young wave are not affected by the values of E_x and τ_x0. The wave velocity and wave transmission in the Young wave with ξ(x, t) = 0 are also calculated and summarized in Table 2, showing that the inclusion of axial wall displacement does not affect the wave velocity and transmission in the Young wave.

As shown in Table 3, ξ(x, t) is not negligible in the Young wave, and the ratio of η₀/ξ₀ increases dramatically with the higher harmonics. This ratio increases with E_x and also varies with τ_x0. As shown in Table 4, the ratio of η₀/ξ₀ in the Lamb wave also varies with harmonics, but this ratio is about ∼10⁻³, indicating that η(x, t) can be considered to be solely from the Young wave. Based on the calculated values and phases of (1+mF₁₀) in the expressions for Q(x, t) and η(x, t) in Eqs. (15b) and (15c) in Table 5, Q(x, t) and η(x, t) are overall harmonics-independent, in the sense that they follow the same harmonic distribution of the pulsatile pressure. Without wave reflection, the phase shifts of Q(x, t) and η(x, t) relative to Δp(x, t) are negligible. As shown in Table 6(a), the characteristic impedance does not vary with harmonics and can be treated as a real value. Since the characteristic impedance is a quantitative measure of wave reflection, Table 6(a) indicates that γ and τ_x0 have no influence on wave reflection.

The characteristic impedance based on Eq. (20) can be calculated without knowing the mechanical properties of the arterial wall and can be treated as the measured wave reflection, as opposed to its theoretical counterpart in Eq. (18). As shown in Table 6(b), the calculated values based on Eq. (20) reveal random variation with harmonics. The calculated values based on Eqs. (18) and (20) are relatively close to each other for the rest harmonics, except for the 2nd and 3rd harmonics. This may indicate that the mechanical properties of the arterial wall are embedded in the relation of measured Δp(t) and Q(t). Based on Eqs. (17) and Table 6, the forward and backward waves of Δp(t) are calculated and consequently the forward and backward waves of Q(t) are also obtained, as shown in Fig. 2. The difference in waveform between Δp(t) and Q(t) is solely due to wave reflection, irrelevant to the harmonics. Evidently, different wave reflections from Eqs. (18) and (20) lead to different forward and backward waves of Δp(t) and Q(t), but the sum of the two pulsatile parameters remains the same for different wave reflections, because of Eq. (17).

Together with their forward and backward waves, the calculated η(t), ξ(t), and τ_w(t) waveforms under isotropic conditions (γ = 1) are plotted in Figs. 3 and 4, with τ_x0 = 0.1 and τ_x0 = 0.2, respectively. The η(t) waveform is the same as the Δp(t) waveform, due to the same influence of wave reflection and harmonics-independence. While ξ(t) is solely caused by τ_w(t) in Fig. 3, τ_w(t) and the ∂η/∂x term with a factor of (T_x0-T_θ0) both contribute to ξ(t) in Fig. 4. Note that the η(t) and τ_w(t) waveforms remain the same, regardless of the value of τ_x0. Yet, the ξ(t) waveform is completely altered by the ∂η/∂x term with a factor of (T_x0-T_θ0), due to that the value of 1+m depends on harmonics under τ_x0 = 0.1, but remains unchanged with harmonics under τ_x0 = 0.2. Although the influence of wave reflection on τ_w(t) and ξ(t) and Q(t) is the same, the different harmonics-dependence of τ_w(t) and ξ(t) and the harmonics-independence of Q(t) makes the three waveforms different. Different wave reflections have no effect on the η(t) waveform, but slightly affect the ξ(t) waveform. The wave reflection based on Eq. (20) (small wave reflection) noticeably reduces the maximum and minimum values of τ_w(t), as compared with the wave reflection based on Eq. (18) (large wave reflection).

The influence of orthotropicity of the arterial wall on the η(t), ξ(t), and τ_w(t) waveforms is examined in Figs. 5 and 6, under τ_x0 = 0.1 and τ_x0 = 0.2, respectively. The η(t) waveform with no axial wall displacement is also plotted in Figs. 5(a) and 6(a) for comparison. The η(t) waveform at different values of γ and τ_x0 and different wave reflections is exactly the same as the one with no axial wall displacement. Note that the wave reflection based on Eq. (18) is utilized for the τ_w(t) waveforms in Figs. 5(c) and 6(c), and the wave reflection based on Eq. (20) is used for the τ_w(t) waveforms in Figs. 5(d) and 6(d). The values of τ_x0 and γ have negligible effect on the τ_w(t) waveform and amplitude. In contrast, the wave reflection based on Eq. (20) noticeably reduces the maximum and minimum values of τ_w(t) at different values of τ_x0 and γ, as compared with the wave reflection based on Eq. (18). The value of γ has no effect on the ξ(t) waveform but greatly affects the ξ(t) amplitude at different values of τ_x0. Similarly, the ξ(t) waveform is dictated by the value of τ_x0 at different values of γ. Different wave reflections slightly adjust the ξ(t) waveform at different γ values.

In Atabek's study [12], the ratio of η₀/ξ₀ was examined only at the fundamental frequency of the heart rate. Upon finding that ξ₀ was unpractially large, as compared with η₀, neighboring tissue at the outer side of the arterial wall was then added to Eq. (4(b)) as three mechanical constraints (inertial, elastic, and damping) to lower ξ₀. Yet, the addition of the mechanical constraints alters longitudinal elastic wave propagation as the nature of axial wall displacement. Moreover, addition of these constraints is inconsistent with the governing equation of radial wall displacement, where circumferential elasticity has factored in the constraints of neighboring tissue in the radial direction. Atabek's study examined wave velocity and transmission and the ratio of η₀/ξ₀, but did not examine blood flow rate and wall shear stress, and the influence of harmonics and wave reflection on these pulsatile parameters.

In Mirsky's study [13], the arterial wall was modeled as a thick-walled tube with no initial tension and the influence of harmonics and wave reflection on pulsatile parameters was considered. With the harmonic values for Δp(t) and Q(t) at the AA in Table 7, η(t) was found to be irrelevant to axial wall displacement under isotropic and orthotropic conditions, but τ_w(t) was not examined in that study. The ξ(t) waveform in Fig. 5 is a little bit similar to the ξ(t) waveform with Poisson's ratio being zero in Mirsky's study, and the difference between the two might be due to the difference in the tube models used. Due to a lack of mathematical clarity in the derived pulsatile parameter expressions, harmonics-dependence of the ξ(t) waveform was not identified in Mirsky's study. The influence of axial initial tension on the ξ(t) waveform was not considered.

The previous theoretical study conducted by the author [18] derived the mathematical expressions of various pulsatile parameters in terms of pulsatile pressure, but the influence of harmonics and wave reflection on the pulsatile parameters was not considered and the waveforms of different pulsatile parameters could not be plotted. Thus, how harmonics and wave reflection affects the waveforms of pulsatile parameters were not obtained. Meanwhile, the related numerical studies [19,20] on radial and axial displacement of the arterial wall examined pulsatile wave propagation in time domain, and thus were incapable of identifying the influence of harmonics and wave reflection of pulsatile parameters with mathematical clarity. Moreover, these numerical studies did not consider the two initial tensions and thus did not identify the effect of τ_θ0 on wave velocity and the effect of τ_x0 on the ξ(t) waveform.

In this study, pulsatile wave propagation is analyzed in the frequency domain for mathematical relations between pulsatile parameters under different harmonics, which allows identifying the influence of harmonics on pulsatile parameters. The influence of wave reflection on the pulsatile parameters is determined by their directions. Afterward, the pulsatile parameters with harmonics and wave reflection are expressed in the time domain, and consequently, their waveforms can be calculated, based on the harmonic values of measured pulsatile pressure and blood flow rate. The influence of wave reflection and harmonics on the five pulsatile parameters is summarized in Table 8. This table clearly explains the difference in waveform between ξ(t), τ_w(t), and Q(t), and predicts the identical waveform for Δp(t) and η(t). Table 9 summarizes the influence of harmonics on wave velocity and the related terms in Eq. (16). The harmonics-independence of η(t) and Q(t) arises from the harmonics-independence of 1+mF₁₀. The value of τ_x0 affects the dependence of 1+m on harmonics and thus alters the ξ(t) waveform, as shown in Figs. 3(b) and 4(b). The influence of harmonics on mF₁₀ remains the same under the two values of τ_x0, and thus the τ_w(t) waveform remains unchanged, as shown in Figs. 3(c) and 4(c). Different wave reflections only noticeably affect the maximum and minimum values of τ_w(t). Thus, characteristic impedance might not be a very effective indicator of arterial health.

As the input, Δp(t) is expected to determine the harmonic values for all the rest pulsatile parameters at an artery. However, the measured Δp(t) cannot separate its forward wave from its backward wave by itself, as shown in Eq. (16a). The harmonic values of the measured Δp(t) are from the forward and backward waves of Δp(t). To obtain the forward and backward waves of Δp(t), the measured Q(t) is needed, as shown in Eq. (17). As such, Δp(t) and Q(t) are both needed to determine the harmonic values at an artery. The waveforms of η(t), ξ(t), and τ_w(t) can then be calculated. Although it is well recognized that an arterial pulse signal is a collection of harmonics of the heart rate [13], there are no studies on what factors determine the harmonic values at an artery, to the author's best knowledge.

Due to its ease access, the carotid artery (CA) is focused on for ξ(t) measurement [2–7]. The measured ξ(t) waveform is found to be different from the τ_w(t) waveform at the CA, although τ_w(t) is commonly considered as the source for causing ξ(t) [4,5]. As shown above, this difference can be explained by the different harmonics-dependence of τ_w(t) and ξ(t) and the contribution of the ∂η/∂x term with a factor of (T_x0−T_θ0) to ξ(t). Due to its difficult access, the AA is not measured for the ξ(t) waveform yet, and thus no comparison can be made for the calculated ξ(t) waveform.

Based on Eqs. (4a) and (4b), pulsatile pressure causes radial wall displacement, while wall shear stress and the ∂η/∂x term cause axial wall displacement. Given that pulsatile pressure (a few kPa) is three order of magnitude higher than wall shear stress (a few Pa), it is understandable that radial wall displacement is coupled into axial wall displacement, but axial wall displacement does not affect radial wall displacement and blood flow rate. This may validate that clinical studies can neglect axial wall displacement for these common clinical measures and their relations. Thus, in terms of facilitating blood flow in an artery, axial wall displacement is trivial, as compared with radial wall displacement.

According to Secs. 2 and 3, radial wall displacement reflects circumferential elasticity and circumferential initial tension, while axial wall displacement amplitude and waveform manifest axial elasticity and axial initial tension, respectively. This may explain the reason why axial wall displacement and radial wall displacement correlate with CV risk factors differently. Although axial wall displacement at the CA has been extensively measured in recent years, its physical mechanism is still unclear. One study found that other than wall shear stress, there should be other pulsatile forces in the axial direction [25]. This work identifies the ∂η/∂x term with a factor of (T_x0-T_θ0) as another pulsatile force causing axial wall displacement. Since axial wall displacement is more sensitive to subclinical atherosclerosis, it might suggest that axial elasticity and axial initial tension are also indicators of arterial health, and are affected by subclinical atherosclerosis to a larger extent, as compared with circumferential elasticity.

Although axial wall displacement is trivial in facilitating blood flow in an artery, its identified clinical values suggest its importance to arterial health. Moreover, axial wall displacement allows axial strain/strain to occur in the arterial wall. According to the tissue-level studies on vascular homeostasis [8,9], other than wall shear stress, circumferential stress and axial stress of the arterial wall is also important for maintaining vascular homeostasis. As analyzed here, clinical values and vascular homeostasis of axial wall displacement are derived from axial elasticity and aixal initial tension of the arterial wall, given that these two mechanical properties predominantly dictate axial wall displacement. Currently, axial elasticity and axial initial tension cannot be measured in vivo, due to a lack of the related theories [2–7,25]. The influence of axial elasticity on axial displacement amplitude and the influence of axial initial tension on axial displacement waveform may provide a theoretical basis for their measurement and study of vascular homeostasis in vivo.

There are four major limitations to this study. First, there is no measured ξ(t) waveform at the AA for comparison with the calculated one. The ξ(t) waveform is commonly measured at the CA. Since the harmonic values of pulsatile pressure and blood flow rate at the CA are not available, no calculation of the ξ(t) waveform at the CA can be conducted. Second, strictly speaking, the arterial wall at the AA is a thick-walled tube, given its ratio of h/a is 0.15. For examining the influence of harmonics and wave reflection on pulsatile parameters with mathematical clarity, a thin-walled tube is used to model the arterial wall. Third, the study neglects the curved and tapering geometrical nature of the arterial wall. Lastly, this study does not consider a possible contribution to ξ(t) and τ_w(t) from the Lamb wave. Nevertheless, these limitations may affect the quantitative relations between pulsatile parameters and the arterial wall mechanical properties, but are not expected to alter the roles and physiological implications of axial wall displacement and two initial tensions in the Young wave revealed in this study.

In this paper, a theoretical study is presented on radial and axial displacement of the arterial wall for deriving wave velocity and pulsatile parameter expressions under the influence of harmonics and wave reflection. With the harmonic values of measured pulsatile pressure and blood flow rate at the AA in the literature, the waveforms of radial wall displacement, axial wall displacement, and wall shear stress are calculated with different values of orthotropicity and axial initial tension for the influence of axial elasticity and axial initial tension on these pulsatile parameters. While blood flow rate and radial wall displacement follow the harmonic distribution of pulsatile pressure, axial wall displacement and wall shear stress show different harmonics-dependence. The amplitude and waveform of axial wall displacement are predominantly dictated by axial elasticity and axial initial tension, respectively. Although axial wall displacement is trivial in terms of facilitating blood flow in an artery, its clinical values for the detection of subclinical atherosclerosis and its role in vascular homeostasis suggest the importance of axial elasticity and axial initial tension to arterial health. The obtained results provide a theoretical basis for the measurement of axial elasticity and axial initial tension in vivo and the study of vascular homeostasis in vivo.

• National Science Foundation (NSF) Division of Chemical, Bioengineering, Environmental, and Transport Systems (Grant No. 1936005; Funder ID: 10.13039/100000146).