Non-linear separation of pressure, velocity and wave intensity into forward and backward components

The contribution of upper and lower body arterial vessels to the aortic root reflections: A one-dimensional computational study

BACKGROUND AND OBJECTIVES

Reflections measured at the aortic root are of physiological and clinical interest and thought to be composed of the superimposed reflections arriving from the upper and lower parts of the circulatory system. However, the specific contribution of each region to the overall reflection measurement has not been thoroughly examined. This study aims to elucidate the relative contribution of reflected waves arising from the upper and lower human body vasculature to those observed at the aortic root.

METHODS

We utilised a one-dimensional (1D) computational model of wave propagation to study reflections in an arterial model that included 37 largest arteries. A narrow Gaussian-shaped pulse was introduced to the arterial model from five distal locations: carotid, brachial, radial, renal, and anterior tibial. The propagation of each pulse towards the ascending aorta was computationally tracked. We calculated the reflected pressure and wave intensity at the ascending aorta in each case. The results are presented as a ratio of the initial pulse.

RESULTS

The findings of this study indicates that pressure pulses originated at the lower body can hardly be observed, while those originated from the upper body account for the largest portion of reflected waves seen at the ascending aorta.

CONCLUSIONS

Our study validates the findings of earlier studies, which demonstrated that human arterial bifurcations have a significantly lower reflection coefficient in the forward direction as compared to the backward direction. The results of this study underscore the need for further in-vivo investigations to provide a deeper understanding of the nature and characteristics of reflections observed in the ascending aorta, which can inform the development of effective strategies for the management of arterial diseases.

Inter-arm differences in regional arterial stiffness and geometry lead to inter-arm systolic blood pressure differences: A modelling study

An inter-arm systolic blood pressure difference (ISBPD), if substantial in magnitude (typically defined as ≥10 mmHg), is a potential cardiovascular risk factor in adults, due to its association with cardiovascular events/mortality. A substantial ISBPD occurs in approximately 10% of the adult population, and, although associations with vascular disease and elevated stiffness have been reported, the mechanisms underlying ISBPD remain unknown. The aim of this study was to investigate whether inter-arm differences in segmental pulse wave velocity, cross-sectional area, or vascular bed compliance/resistance could give rise to substantial differences in brachial pressures between arms; for example, due to differences in pulse wave transmission and reflection. Using an established one-dimensional model of the major systemic arteries, pulse wave velocity (PWV) was uniformly increased or decreased in arteries of 1) the supra-aortic region leading up to the arm, 2) the brachial region, 3) the forearm, and 4) all of these (entire arm pathway); for the left arm, right arm, and both arms. Cross-sectional area and vascular bed compliance and resistance of the arms were similarly varied. Inter-arm differences in segmental PWV and cross-sectional area (but not bilateral changes) led to associated substantial inter-arm SBP differences, which were observed with changes to brachial, forearm and/or entire arm pathways and were related to altered transmission of forward waves and amplitude/timing of reflected waves. Vascular bed compliance and resistance had minimal influence. We conclude that inter-arm differences in arterial stiffness and geometry may contribute to inter-arm systolic blood pressure differences, warranting further investigation.

Impact of tapering of arterial vessels on blood pressure, pulse wave velocity, and wave intensity analysis using one-dimensional computational model

The angle of arterial tapering increases with ageing, and the geometrical changes of the aorta may cause an increase in central arterial pressure and stiffness. The impact of tapering has been primarily studied using frequency-domain transmission line theories. In this work, we revisit the problem of tapering and investigate its effect on blood pressure and pulse wave velocity (PWV) using a time-domain analysis with a 1D computational model. First, tapering is modelled as a stepwise reduction in diameter and compared with results from a continuously tapered segment. Next, we studied wave reflections in a combination of stepwise diameter reduction of straight vessels and bifurcations, then repeated the experiments with decreasing the length to physiological values. As the model's segments became shorter in length, wave reflections and re-reflections resulted in waves overlapping in time. We extended our work by examining the effect of increasing the tapering angle on blood pressure and wave intensity in physiological models: a model of the thoracic aorta and a model of upper thoracic and descending aorta connected to the iliac bifurcation. Vessels tapering inherently changed the ratio between the inlet and outlet cross-sectional areas, increasing the vessel resistance and reducing the compliance compared with non-tapered vessels. These variables influence peak and pulse pressure. In addition, it is well established that pulse wave velocity increases in an ageing arterial tree. This work provides confirmation that tapering induces reflections and offers an additional explanation to the observation of increased peak pressure and decreased diastolic pressure distally in the arterial tree.

Novel wave power analysis linking pressure-flow waves, wave potential, and the forward and backward components of hydraulic power

Wave intensity analysis provides detailed insights into factors influencing hemodynamics. However, wave intensity is not a conserved quantity, so it is sensitive to diameter variations and is not distributed among branches of a junction. Moreover, the fundamental relation between waves and hydraulic power is unclear. We, therefore, propose an alternative to wave intensity called “wave power,” calculated via incremental changes in pressure and flow (dPdQ) and a novel time-domain separation of hydraulic pressure power and kinetic power into forward and backward wave-related components (ΠP± and ΠQ±). Wave power has several useful properties: 1) it is obtained directly from flow measurements, without requiring further calculation of velocity; 2) it is a quasi-conserved quantity that may be used to study the relative distribution of waves at junctions; and 3) it has the units of power (Watts). We also uncover a simple relationship between wave power and changes in ΠP± and show that wave reflection reduces transmitted power. Absolute values of ΠP± represent wave potential, a recently introduced concept that unifies steady and pulsatile aspects of hemodynamics. We show that wave potential represents the hydraulic energy potential stored in a compliant pressurized vessel, with spatial gradients producing waves that transfer this energy. These techniques and principles are verified numerically and also experimentally with pressure/flow measurements in all branches of a central bifurcation in sheep, under a wide range of hemodynamic conditions. The proposed “wave power analysis,” encompassing wave power, wave potential, and wave separation of hydraulic power provides a potent time-domain approach for analyzing hemodynamics.

Numerical Method of Characteristics for One-Dimensional Blood Flow

Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant.

An effective model of blood flow in capillary beds

In this article we derive applicable expressions for the macroscopic compliance and resistance of microvascular networks. This work yields a lumped-parameter model to describe the hemodynamics of capillary beds. Our derivation takes into account the multiscale nature of capillary networks, the influence of blood volume and pressure on the effective resistance and compliance, as well as, the nonlinear interdependence between these two properties. As a result, we obtain a simple and useful model to study hypotensive and hypertensive phenomena. We include two implementations of our theory: (i) pulmonary hypertension where the flow resistance is predicted as a function of pulmonary vascular tone. We derive from first-principles the inverse proportional relation between resistance and compliance of the pulmonary tree, which explains why the RC factor remains nearly constant across a population with increasing severity of pulmonary hypertension. (ii) The critical closing pressure in pulmonary hypotension where the flow rate dramatically decreases due to the partial collapse of the capillary bed. In both cases, the results from our proposed model compare accurately with experimental data.

A computational study of pressure wave reflections in the pulmonary arteries

Experiments using wave intensity analysis suggest that the pulmonary circulation in sheep and dogs is characterized by negative or open-end type wave reflections, that reduce the systolic pressure. Since the pulmonary physiology is similar in most mammals, including humans, we test and verify this hypothesis by using a subject specific one-dimensional model of the human pulmonary circulation and a conventional wave intensity analysis. Using the simulated pressure and velocity, we also analyse the performance of the P-U loop and sum of squares techniques for estimating the local pulse wave velocity in the pulmonary arteries, and then analyse the effects of these methods on linear wave separation in the main pulmonary artery. P-U loops are found to provide much better estimates than the sum of squares technique at proximal locations, but both techniques accumulate progressive error at distal locations away from heart, particularly near junctions. The pulse wave velocity estimated using the sum of squares method also gives rise to an artificial early systolic backward compression wave. Finally, we study the influence of three types of pulmonary hypertension viz. pulmonary arterial hypertension, chronic thromboembolic pulmonary hypertension and pulmonary hypertension associated with hypoxic lung disease. Simulating these conditions by changing the relevant parameters in the model and then applying the wave intensity analysis, we observe that for each group the early systolic backward decompression wave reflected from proximal junctions is maintained, whilst the initial forward compression and the late systolic backward compression waves amplify with increasing pathology and contribute significantly to increases in systolic pressure.

Wave potential and the one-dimensional windkessel as a wave-based paradigm of diastolic arterial hemodynamics

Controversy exists about whether one-dimensional wave theory can explain the "self-canceling" waves that accompany the diastolic pressure decay and discharge of the arterial reservoir. Although it has been proposed that reservoir and wave effects be treated as separate phenomena, thus avoiding the issue of self-canceling waves, we have argued that reservoir effects are a phenomenological and mathematical subset of wave effects. However, a complete wave-based explanation of self-canceling diastolic expansion (pressure-decreasing) waves has not yet been advanced. These waves are present in the forward and backward components of arterial pressure and flow (P ± and Q ±, respectively), which are calculated by integrating incremental pressure and flow changes (dP ± and dQ ±, respectively). While the integration constants for this calculation have previously been considered arbitrary, we showed that physiologically meaningful constants can be obtained by identifying "undisturbed pressure" as mean circulatory pressure. Using a series of numeric experiments, absolute P ± and Q ± values were shown to represent "wave potential," gradients of which produce propagating wavefronts. With the aid of a "one-dimensional windkessel," we showed how wave theory predicts discharge of the arterial reservoir. Simulated data, along with hemodynamic recordings in seven sheep, suggested that self-canceling diastolic waves arise from repeated and diffuse reflection of the late systolic forward expansion wave throughout the arterial system and at the closed aortic valve, along with progressive leakage of wave potential from the conduit arteries. The combination of wave and wave potential concepts leads to a comprehensive one-dimensional (i.e., wave-based) explanation of arterial hemodynamics, including the diastolic pressure decay.