Journal of Healthcare Engineering

Volume 2018 (2018), Article ID 1364185, 9 pages

https://doi.org/10.1155/2018/1364185

## A Novel Interpretation for Arterial Pulse Pressure Amplification in Health and Disease

^{1}Facultad Regional Buenos Aires, Grupo de Investigación en Bioingeniería (GIBIO) and Escuela de Estudios Avanzados en Ciencias de la Ingeniería (EEACI), Universidad Tecnológica Nacional, Medrano 951, C1179AAQ Buenos Aires, Argentina^{2}UMR 7190, Institut Jean Le Rond ∂'Alembert, CNRS and UPMC, Sorbonne Universités, 4 Place Jussieu, Boîte 162, 75005 Paris, France^{3}Universidad Tecnológica Nacional, Medrano 951, C1179AAQ Buenos Aires, Argentina^{4}Facultad Regional Buenos Aires, Centro de Procesamiento de Señales e Imagenes (CPSI) and Escuela de Estudios Avanzados en Ciencias de la Ingeniería (EEACI), Universidad Tecnológica Nacional, Medrano 951, C1179AAQ Buenos Aires, Argentina

Correspondence should be addressed to Manuel R. Alfonso

Received 22 May 2017; Revised 18 October 2017; Accepted 29 October 2017; Published 16 January 2018

Academic Editor: Andreas Maier

Copyright © 2018 Manuel R. Alfonso et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Arterial pressure waves have been described in one dimension using several approaches, such as lumped (Windkessel) or distributed (using Navier-Stokes equations) models. An alternative approach consists of modeling blood pressure waves using a Korteweg-de Vries (KdV) equation and representing pressure waves as combinations of solitons. This model captures many key features of wave propagation in the systemic network and, in particular, pulse pressure amplification (PPA), which is a mechanical biomarker of cardiovascular risk. The main objective of this work is to compare the propagation dynamics described by a KdV equation in a human-like arterial tree using acquired pressure waves. Furthermore, we analyzed the ability of our model to reproduce induced elastic changes in PPA due to different pathological conditions. To this end, numerical simulations were performed using acquired central pressure signals from different subject groups (young, adults, and hypertensive) as input and then comparing the output of the model with measured radial artery pressure waveforms. Pathological conditions were modeled as changes in arterial elasticity (*E*). Numerical results showed that the model was able to propagate acquired pressure waveforms and to reproduce PPA variations as a consequence of elastic changes. Calculated elasticity for each group was in accordance with the existing literature.

#### 1. Introduction

Pulse pressure amplification (PPA) is conventionally understood in clinical practice as the increase of pulse pressure (PP) amplitude as pressure waves propagate distally in the systemic network. Yet, PPA should rather be described as a distortion rather than an amplification of PP waves, represented by morphological alterations of pressure waveforms. Moreover, changes in PPA are associated with traditional cardiovascular risk factors, such as aging and hypertension [1, 2]. Indeed, a substantial decrease in mean diastolic pressure (perfusion) and a systolic central pressure increase (afterload) are observed in patients over 60 years old as a result of a progressive increase in arterial stiffness [3]. Consequently, a greater myocardial oxygen demand in the left ventricle and an impaired coronary perfusion are observed due to the decrease in mean arterial diastolic pressure [3]. Furthermore, in hypertensive patients, the decrease in large artery compliance (i.e., high values of arterial stiffness) is considered one of the major causes of PP increase. Additionally, hypertension is responsible for an increase in pulse wave velocity (PWV).

The study of PP propagation is therefore a major medical challenge and is essential to understand the dynamics of the circulatory system under normal or pathological conditions. Two different approaches have been used to efficiently describe the hemodynamics in the systemic network [4]. On the one hand, lumped parameter or 0D models [5–7] are particularly relevant when modeling interactions between the systemic network and other major systems (nervous, respiratory, and digestive) but are unable to describe pulse wave propagation. On the other hand, distributed 1D models [4] enable an efficient description of pulse wave propagation without the computational cost associated to 2D or 3D models.

In this work, we choose an alternative approach where long wave and perturbation theories allow us to derive a nonlinear dispersive and/or diffusive equation, like Korteweg-de Vries (KdV) equation, starting from the Navier-Stokes equations [8–11]. Behind this model is the idea that blood pressure (BP) waves can be considered as combinations of solitons. Laleg et al. [9] described this combination in details, through the nonlinear overlapping of two or three solitons. This model captures many of the phenomena observed in BP propagation, such as peaking (increase in amplitude), steepening (decrease in width), and changes in wave propagation velocity. Furthermore, McDonald found that an amplitude increase of arterial pulse is concomitant with a decrease in pulse width during the propagation of flow and pressure waveforms from the aorta to the saphenous artery in dogs [3], indicating a nonlinear rather than a linear behavior.

In previous works of our group [12, 13], a 1D arterial network was constructed in order to simulate the behavior of synthesized pulse pressure waveforms as a combination of solitons throughout the arterial tree. To this end, the pressure in each segment was computed using the KdV equation (KdVe), where vascular dimensions and elastic constants were obtained from the existing literature [14].

The main objective of this work is to use the arterial network previously described and the KdVe to compute the propagation of acquired pulse pressure waves through a human-like arterial tree and quantify the ability of our model to capture changes in PPA due to variations in arterial elasticity. To this end, numerical simulations will be performed, using a set of previously acquired central blood pressure (CBP) and peripheral blood pressure (PBP) waveforms from several individuals from four different groups: young, adult, hypertensive type I, and II.

#### 2. Materials and Methods

In this section, we present the simple nonlinear model (KdVe) describing blood pulse pressure propagation in an artery. We then introduce a computational framework allowing us to use acquired CBP-PBP as inputs-outputs of our model. Next, we design a numerical experiment to assess the ability of our model to reproduce changes in PPA due to changes in vessel elasticity (*E*). Finally, using a set of CBP/PBP-acquired signals, we perform a global parameter estimation of the arterial elasticity (*E*) value for each patient and then perform a statistical analysis.

##### 2.1. 1D KDV-Based Model Formulation

To explain BP waveforms and interpret the different phenomena that arise as they propagate along the arterial network, like the increase in amplitude and the decrease in width called “peaking” and “steepening” phenomena, respectively, this work introduces BP waves as a soliton combination. To understand the main behavior of soliton propagation, it is important to point out the following:
(i)Solitons have a bell shape and maintain their shape as they propagate.(ii)When solitons interact, they remain unchanged after the “collision,” except possibly for a phase shift.(iii)During interaction, the resulting shape is wider, and the amplitude is between the peak of the taller and the smaller one.(iv)Wave velocity and amplitude are dependent of *E*.(v)Each soliton has its own velocity, because of this, the waves separate as they propagate to the periphery.

With the above attributes considered, it is then easy to explain phenomena like peaking, steepening, and PPA. Due to the different velocities (which depends on *E*), when the waves arrive at the periphery, the initial separation has changed. The different solitons are now more separated from each other creating a taller waveform. In young’s (low *E*), this separation is bigger, adopting their respective original form: a taller and thinner one and a smaller and wider one, like the two typical bell shapes observed in the femoral artery.

In order to obtain the equations describing the dynamics of BP propagation along an elastic arterial segment, several authors [8, 11, 15, 16] propose:
(a)That large arteries are to be considered as elastic tubes and the fluid as incompressible(b)For large arteries, the continuum approach for blood is valid and that viscosity can be neglected [8, 17, 18]. Following these authors, the evolution of pressure (*P*) in an arterial segment can be described as follows:
where and are the corresponding space and time variables and the subscripts of and indicate spatial and temporal derivatives. The equation coefficients are defined as follows:
where the constant determines the typical Moens-Korteweg velocity of a wave propagating in an elastic tube, when all nonlinear terms are neglected [19, 20], is the elastic modulus (arterial stiffness), is the wall thickness, is the mean tube radius, is the blood density, is the wall density, and is the moment flux correction coefficient.

##### 2.2. Arterial System Model

In this work, a previously arterial network model was used [12]. This model consists of one long tapering *artery, composed of constant parameter vessels*, placed in a simple *cascading* order. In each of these segments, *the pulse pressure wave dynamics were modelled by* (1) *describing only forward soliton interactions*. At the inlet of the network (aorta), an acquired CBP is imposed. The computed PBP at the outlet of the final segment constitutes the output of the model (Figure 1).