The compression and expansion waves of the forward and backward flows: an in-vitro arterial model

Measurement, Analysis and Interpretation of Pressure/Flow Waves in Blood Vessels

The optimal performance of the cardiovascular system, as well as the break-down of this performance with disease, both involve complex biomechanical interactions between the heart, conduit vascular networks and microvascular beds. ‘Wave analysis’ refers to a group of techniques that provide valuable insight into these interactions by scrutinizing the shape of blood pressure and flow/velocity waveforms. The aim of this review paper is to provide a comprehensive introduction to wave analysis, with a focus on key concepts and practical application rather than mathematical derivations. We begin with an overview of invasive and non-invasive measurement techniques that can be used to obtain the signals required for wave analysis. We then review the most widely used wave analysis techniques—pulse wave analysis, wave separation and wave intensity analysis—and associated methods for estimating local wave speed or characteristic impedance that are required for decomposing waveforms into forward and backward wave components. This is followed by a discussion of the biomechanical phenomena that generate waves and the processes that modulate wave amplitude, both of which are critical for interpreting measured wave patterns. Finally, we provide a brief update on several emerging techniques/concepts in the wave analysis field, namely wave potential and the reservoir-excess pressure approach.

Using wave intensity analysis to determine local reflection coefficient in flexible tubes

It has been shown that reflected waves affect the shape and magnitude of the arterial pressure waveform, and that reflected waves have physiological and clinical prognostic values. In general the reflection coefficient is defined as the ratio of the energy of the reflected to the incident wave. Since pressure has the units of energy per unit volume, arterial reflection coefficient are traditionally defined as the ratio of reflected to the incident pressure. We demonstrate that this approach maybe prone to inaccuracies when applied locally. One of the main objectives of this work is to examine the possibility of using wave intensity, which has units of energy flux per unit area, to determine the reflection coefficient. We used an in vitro experimental setting with a single inlet tube joined to a second tube with different properties to form a single reflection site. The second tube was long enough to ensure that reflections from its outlet did not obscure the interactions of the initial wave. We generated an approximately half sinusoidal wave at the inlet of the tube and took measurements of pressure and flow along the tube. We calculated the reflection coefficient using wave intensity (R_dI and R_dI^0.5) and wave energy (R_I and R_I^0.5) as well as the measured pressure (R_dP) and compared these results with the reflection coefficient calculated theoretically based on the mechanical properties of the tubes. The experimental results show that the reflection coefficients determined by all the techniques we studied increased or decreased with distance from the reflection site, depending on the type of reflection. In our experiments, R_dP, R_dI^0.5 and R_I^0.5 are the most reliable parameters to measure the mean reflection coefficient, whilst R_dI and R_I provide the best measure of the local reflection coefficient, closest to the reflection site. Additional work with bifurcations, tapered tubes and in vivo experiments are needed to further understand, validate the method and assess its potential clinical use.

A mathematical model of pressure and flow waveforms in the aortic root

The differences in the pressure and flow waveforms in the aortic root have not been explained so far in a satisfactory mathematical way. It is a generally accepted idea that the existence of the reflected wave causes the differences in shapes of pressure and flow. In this paper, a mathematical model is proposed that explains the blood pressure and flow waveforms based on changes in left ventricular volume during blood ejection into the aorta. According to the model, a change in volume of the left ventricle during contraction can be mathematically presented with solutions of differential equations that describe the behavior of a second-order system. The proposed mathematical equations of pressure and flow waveforms are derived from left ventricular volume change and basic equations of fluid dynamics. The position of the reflected wave depends on the age and elasticity of arteries, and has an effect on the flow and pressure waveforms. The model is in acceptable agreement with the experimental data available.

Wave intensity analysis in air-filled flexible vessels

Wave intensity analysis (WIA) is an analytical technique generally used to investigate the propagation of waves in the cardiovascular system. Despite its increasing usage in the cardiovascular system, to our knowledge WIA has never been applied to the respiratory system. Given the analogies between arteries and airways (i.e. fluid flow in flexible vessels), the aim of this work is to test the applicability of WIA with gas flow instead of liquid flow. The models employed in this study are similar to earlier studies used for arterial investigations. Simultaneous pressure (P) and velocity (U) measurements were initially made in a single tube and then in several flexible tubes connected in series. Wave speed was calculated using the foot-to-foot method (cf), which was used to separate analytically the measured P and U waveforms into their forward and backward components. Further, the data were used to calculate wave intensity, which was also separated into its forward and backward components. Although the measured wave speed was relatively high, the results showed that the onsets and the nature of reflections (compression/expansion) derived with WIA, corresponded well to those anticipated using the theory of waves in liquid-filled elastic tubes. On average the difference between the experimental and theoretical arrival time of reflection was 6.1% and 3.6% for the single vessel and multivessel experiment, respectively. The results suggest that WIA can provide relatively accurate information on reflections in air-filled flexible tubes, warranting further studies to explore the full potential of this technique in the respiratory system.

Hemodynamic correlates of late systolic flow velocity augmentation in the carotid artery

Background. The contour of the common carotid artery (CCA) blood flow velocity waveform changes with age; CCA flow velocity increases during late systole, and this may contribute to cerebrovascular disease. Late systolic flow velocity augmentation can be quantified using the flow augmentation index (FAIx). We examined hemodynamic correlates of FAIx to gain insight into determinants of CCA flow patterns. Methods. CCA Doppler ultrasound and wave intensity analysis (WIA) were used to assess regional hemodynamics in 18 young healthy men (age 22 ± 1 years). Forward waves (W₁) and backward waves (negative area, NA) were measured and used to calculate the reflection index (NA/W₁ = RIx). Additional parameters included W₂ which is a forward travelling expansion/decompression wave of myocardial origin that produces suction, CCA single-point pulse wave velocity (PWV) as a measure of arterial stiffness, and CCA pressure augmentation index (AIx). Results. Primary correlates of FAIx included W₂ (r = − 0.52, P < 0.05), _logRIx (r = 0.56, P < 0.05), and AIx (r = 0.60, P < 0.05). FAIx was not associated with CCA stiffness (P > 0.05). Conclusions. FAIx is a complex ventricular-vascular coupling parameter that is associated with both increased expansion wave magnitude (increased suction from the left ventricle) and increased pressure from wave reflections.

Manipulation of arterial stiffness, wave reflections, and retrograde shear rate in the femoral artery using lower limb external compression

Exposure of the arterial wall to retrograde shear acutely leads to endothelial dysfunction and chronically contributes to a proatherogenic vascular phenotype. Arterial stiffness and increased pressure from wave reflections are known arbiters of blood flow in the systemic circulation and each related to atherosclerosis. Using distal external compression of the calf to increase upstream retrograde shear in the superficial femoral artery (SFA), we examined the hypothesis that changes in retrograde shear are correlated with changes in SFA stiffness and pressure from wave reflections. For this purpose, a pneumatic cuff was applied to the calf and inflated to 0, 35, and 70 mmHg (5 min compression, randomized order, separated by 5 min) in 16 healthy young men (23 ± 1 years of age). Doppler ultrasound and wave intensity analysis was used to measure SFA retrograde shear rate, reflected pressure wave intensity (negative area [NA]), elastic modulus (Ep), and a single-point pulse wave velocity (PWV) during acute cuff inflation. Cuff inflation resulted in stepwise increases in retrograde shear rate (P < 0.05 for main effect). There were also significant cuff pressure-dependent increases in NA, Ep, and PWV across conditions (P < 0.05 for main effects). Change in NA, but not Ep or PWV, was associated with change in retrograde shear rate across conditions (P < 0.05). In conclusion, external compression of the calf increases retrograde shear, arterial stiffness, and pressure from wave reflection in the upstream SFA in a dose-dependent manner. Wave reflection intensity, but not arterial stiffness, is correlated with changes in peripheral retrograde shear with this hemodynamic manipulation.

Clinical benefits of integrating cardiac and vascular models

INTRODUCTION

Recent advances in computational methods and medical imaging techniques have enabled non-invasive exploration of cardiovascular pathologies, from cardiac level to complex arterial networks. The potential of cardiac and vascular modeling in guiding and monitoring therapies could be further extended through the integration of the two systems.

AREAS COVERED

This review includes advances in methods for cardiac electromechanics and vascular flow simulations. The results of a literature search depicting the state of the art in cardiac and vascular modeling are reviewed. The paper goes on to address the benefits and challenges of combined cardiovascular modeling, highlighting the relevance of specific cardiovascular features and implementation. Various alternative approaches and insights on future directions are presented and analyzed with respect to their applicability to clinical practice.

EXPERT OPINION

The article has emerged from the exploration of currently available cardiac and vascular mathematical tools and their corresponding clinical application. The summarized analysis suggests that future efforts should be aimed at developing more accurate and patient-specific mathematical models integrating cardiac and vascular functions to enhance the knowledge of cardiovascular pathologies.

One-dimensional computational model of pulse wave propagation in the human bronchial tree

Airflow in the respiratory system has been predominantly studied in rigid ducts. Three-dimensional simulations are computationally expensive. One-dimensional (1-D) modelling offers a good compromise between accuracy and computational cost. In this work we described the propagation of air pulse in a model of human airways using the 1-D equations of flow in compliant vessels. Seven generations of bifurcations, starting from the trachea, were studied. Peripheral airways (from the 8(th) to 23(rd) generation) were modelled using lumped parameter models. Peripheral resistance values for normal and emphysematous lungs were taken from the literature. An acceleration pulse, very short in time, was enforced at the inlet of trachea. The results suggest that compression (positive pressure peaks) and expansion (negative pressure peaks) waves are generated according to the reflection coefficients of the corresponding reflection sites (bifurcations and terminal reflections). Different values for peripheral bronchial resistance generate three different terminal reflections, all negative with different wave amplitudes. The sensitivity of the code to different peripheral resistances suggests that the 1-D formulation is a promising tool for a better understanding of the impact of disease on the velocity and pressure waveforms in the first generations of airway vessels.