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Pulsatile pressure at an artery is a collection of harmonics of the heartbeat. This study examines harmonics of pulsatile pressure at different ages and its effect on other pulsatile parameters and waveform-based clinical indices. Based on a vibrating-string model of the arterial tree, wave velocity and characteristic impedance are related to arterial stiffness and radius. Blood velocity, wall shear stress (WSS), and driving force on the left ventricle (LV) are related to pulsatile pressure. Reflection magnitude and return time are related to input impedance. These relations are applied to pulsatile pressure and blood velocity at the ascending aorta (AA) and the carotid artery (CA) at different ages in a database to calculate harmonics of all the pulsatile parameters and reflection magnitude and return time at each harmonic. Harmonics of pulsatile pressure varies with aging and between the two arteries. Reflection magnitude and return time vary between harmonics. While wave reflection manifests the arterial tree (i.e., arterial stiffness and radius) and termination, harmonics of pulsatile pressure is a combination of the LV, the arterial tree, and termination. Harmonics of pulsatile pressure dictates harmonics of WSS and affects endothelial function. Harmonics of pulsatile pressure needs to serve as an independent clinical index indicative of the LV function and endothelial function. Reflection magnitude and return time of the 1st harmonic of pulsatile pressure serve as clinical indices indicative of arterial stiffness and radius.

 

Abstract A tube-load model is used to reconstruct aortic pressure waveform from peripheral pressure waveform. Yet, the reconstructed aortic pressure waveform is greatly affected by load impedance used. In this work, a vibrating-string model for closed-loop wave transmission and reflection between the aorta and periphery is developed to examine the roles of all the parameters involved in aortic pressure waveform. The arterial pulsatile wave theory gives rise to the standard one-dimensional wave equation for a vibrating string. A vibrating-string model based on radial displacement of the arterial wall is developed to relate aortic pressure waveform to peripheral pressure waveform, relate load impedance to input impedance, and derive theoretical expressions for associated clinical indices. The vibrating-string model is extended to incorporate blood velocity and is further connected to the left ventricle (LV) to study the role of the LV in aortic pressure waveform. The difference between the vibrating-string model and the tube-load model is also examined. Load impedance is identified as an indispensable independent parameter for reconstruction of aortic pressure waveform with accuracy, and its physiologically realistic harmonic dependence can only be obtained from the measured input impedance. The derived expressions for clinical indices interpret some clinical findings and underscore the role of harmonics in clinical indices. Some misconceptions in the tube-load model are revealed, including load impedance and characteristic impedance. This work clarifies the role of harmonics-dependence of load impedance and harmonics of aortic pressure waveform in determining clinical indices. 

Based on a one-dimensional (1D) uniform model of the arterial tree, various machine-learning techniques have been explored to reconstruct aortic pressure waveform (APW) from peripheral pressure waveform (PPW). This study aims to examine the feasibility of such reconstruction. Based on a 1D uniform vibrating-string model, transfer function (TF) of PPW to APW contains four harmonics-dependent parameters: value and phase of reflection coefficient (i.e., load impedance) at periphery and transmission parameter and transmission loss in the aorta-periphery section. Pressure waveforms and blood velocity waveforms at the ascending aorta (AA), the carotid artery (CA), and the radial artery (RA) of virtual health subjects at different ages in a prevalidated database are analyzed to calculate (1) reflection coefficient at the CA and the RA as two peripheries, (2) TF for the AA-CA and AA-RA sections, and (3) transmission parameter and transmission loss in the two sections. Harmonics-dependence of the four parameters varies with aging in both sections, and arterial nonuniformity makes it unpractical to configure any mathematical model for their harmonics-dependence. Instead of fluid-loading, arterial nonuniformity greatly affects transmission loss. Compared with higher harmonics, transmission loss dramatically alters reconstructed APW. A 1D uniform model allows accurate reconstruction of APW from PPW, with a caveat that baseline values of the four parameters at different harmonics under different cardiovascular (CV) conditions need to be established a priori. Alternatively, based on the baseline values, PPW can be directly utilized for inferring CV conditions.

 

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. 

Abstract This study examines radial and axial displacement of the arterial wall under the influence of harmonics and wave reflection for the role of axial wall displacement in pulsatile wave propagation. The arterial wall is modeled as an initially-tensioned thin-walled orthotropic tube. In conjunction with three pulsatile parameters in blood flow, a free wave propagation analysis is conducted on the governing equations of the arterial wall and no-slip conditions at the blood-wall interface to obtain the frequency equation and pulsatile parameter expressions under different harmonics. The influence of wave reflection is then added to pulsatile parameter expressions. With the harmonic values of measured pulsatile pressure and blood flow rate at the ascending aorta in the literature, the waveforms of radial wall displacement, axial wall displacement, and wall shear stress are calculated under different orthotropicity and axial initial tension. The developed theory and calculated results indicate that (1) difference in waveform between blood flow rate, wall shear stress, and axial wall displacement is caused by harmonics, rather than wave reflection; (2) Axial wall displacement does not affect blood flow rate, radial wall displacement, and wall shear stress; (3) Besides wall shear stress, radial wall displacement gradient also contributes to axial wall displacement and its contribution is adjusted by axial initial tension; (4) different wave reflections only noticeably affect the maximum and minimum values of wall shear stress; and (5) The amplitude and waveform of axial wall displacement are predominantly dictated by axial elasticity and axial initial tension, respectively. 

With the arterial wall modeled as an initially-tensioned thin-walled orthotropic tube, this study aims to analyze radial and axial motion of the arterial wall and thereby reveal the role of axial motion and two initial tensions of the arterial wall in arterial pulse wave propagation. By incorporating related clinical findings into the pulse wave theory in the literature, a theoretical study is conducted on arterial pulse wave propagation with radial and axial wall motion. Since the Young wave is excited by pulsatile pressure and is examined in clinical studies, commonly measured pulsatile parameters in the Young wave are expressed in terms of pulsatile pressure and their values are calculated with the well-established values of circumferential elasticity (E) and initial tension (T0) and assumed values of axial elasticity (Ex) and initial tension (Tx0) at the ascending aorta and the carotid artery. The corresponding values with exclusion of axial wall motion are also calculated. Comparison of the calculated results between inclusion and exclusion of axial wall motion indicates that 1) axial wall motion does not affect radial wall motion and other commonly measured pulsatile parameters, except wall shear stress; 2) axial wall motion is caused by wall shear stress and radial wall displacement gradient with a factor of (Tx0T0), and enables axial power transmission through the arterial wall; and 3) while radial wall motion reflects E and T0, axial wall motion reflects Ex and (Tx0T0). 

In this work, we demonstrate an adjustable microfluidic tactile sensor for measurement of post-exercise response of local arterial parameters. The sensor entailed a polydimethylsiloxane (PDMS) microstructure embedded with a 5×1 resistive transducer array. The pulse signal in an artery deflected the microstructure and registered as a resistance change by the transducer aligned at the artery. PDMS layers of different thicknesses were added to adjust the microstructure thickness for achieving good sensor-artery conformity at the radial artery (RA) and the carotid artery (CA). Pulse signals of nine (n=9) young healthy male subjects were measured at-rest and at different times post-exercise, and a medical instrument was used to simultaneously measure their blood pressure and heart rate. Vibration-model-based analysis was conducted on a measured pulse signal to estimate local arterial parameters: elasticity, viscosity, and radius. The arterial elasticity and viscosity increased, and the arterial radius decreased at the two arteries 1min post-exercise, relative to at-rest. The changes in pulse pressure (PP) and mean blood pressure (MAP) between at-rest and 1min post-exercise were not correlated with that of heart rate and arterial parameters. After the large 1min post-exercise response, the arterial parameters and PP all went back to their at-rest values over time post-exercise.Clinical Relevance— The study results show the potential application of an affordable, user-friendly device for a more comprehensive arterial health assessment. 

With consideration of a full set of mechanical properties: elasticity, viscosity, and axial and circumferential initial tensions, and radial and axial motion of the arterial wall, this paper presents a theoretical study of pulse wave propagation in arteries and evaluates pulse wave velocity and transmission at the carotid artery (CA) and the ascending aorta (AA). The arterial wall is treated as an initially-tensioned, isotropic, thin-walled membrane, and the flowing blood in the artery is treated as an incompressible Newtonian fluid. Pulse wave propagation in arteries is formulated as a combination of the governing equations of radial and axial motion of the arterial wall, the governing equations of flowing blood in the artery, and the interface conditions that relate the arterial wall variables to the flowing blood variables. We conduct a free wave propagation analysis of the problem and derive a frequency equation. The solution to the frequency equation indicates two waves: Young wave and Lamb wave, propagating in the arterial tree. With the related values at the CA and the AA, we evaluate the influence of arterial wall properties on their wave velocity and transmission, and find the opposite effects of axial and circumferential initial tensions on transmission of both waves. Physiological implications of such influence are discussed. 

This work is aimed to establish engineering theories of the coupled longitudinal and radial motion of the arterial wall. By treating the arterial wall as a piano string in the longitudinal direction and as a viscoelastic material in the circumferential direction, and considering pulsatile pressure and wall shear stress from axial blood flow in an artery, the fully-formed governing equations of the coupled motion of the arterial wall are obtained and are related to the engineering theories of axial blood flow for a unified engineering understanding of blood circulation in the cardiovascular (CV) system. The longitudinal wall motion and the radial wall motion are essentially a longitudinal elastic wave and a transverse elastic wave, respectively, traveling along the arterial tree, with their own propagation velocities dictated by the physical properties and geometrical parameters of the arterial wall. The longitudinal initial tension is essential for generating a transverse elastic wave in the arterial wall to accompany the pulsatile pressure wave in axial blood flow. Under aging and subclinical atherosclerosis, propagation of the two elastic waves and coupling of the two elastic waves weakens and consequently might undermine blood circulation.