Arterial system configuration and wave reflection

A fast computational model for circulatory dynamics: effects of left ventricle-aorta coupling

The course of diseases such as hypertension, systolic heart failure and heart failure with a preserved ejection fraction is affected by interactions between the left ventricle (LV) and the vasculature. To study these interactions, a computationally efficient, biophysically based mathematical model for the circulatory system is presented. In a four-chamber model of the heart, the LV is represented by a previously described low-order, wall volume-preserving model that includes torsion and base-to-apex and circumferential wall shortening and lengthening, and the other chambers are represented using spherical geometries. Active and passive myocardial mechanics of all four chambers are included. The cardiac model is coupled with a wave propagation model for the aorta and a closed lumped-parameter circulation model. Parameters for the normal heart and aorta are determined by fitting to experimental data. Changes in the timing and magnitude of pulse wave reflections by the aorta are demonstrated with changes in compliance and taper of the aorta as seen in aging (decreased compliance, increased diameter and length), and resulting effects on LV pressure-volume loops and LV fiber stress and sarcomere shortening are predicted. Effects of aging of the aorta combined with reduced LV contractile force (failing heart) are examined. In the failing heart, changes in aortic properties with aging affect stroke volume and sarcomere shortening without appreciable augmentation of aortic pressure, and the reflected pressure wave contributes an increased proportion of aortic pressure.

Optimised design of an arterial network model reproduces characteristic central and peripheral hemodynamic waveform features in young adults

Abstract

The arterial network in healthy young adults is thought to be structured to optimize wave reflection in the arterial system, producing an ascending aortic pressure waveform with three key features: early systolic peak, negative systolic augmentation and diastolic hump. One‐dimensional computer models have provided significant insights into arterial haemodynamics, but no previous models of the young adult have exhibited these three features. Given that this issue was likely to be related to unrepresentative or non‐optimized impedance properties of the model arterial networks, we developed a new ‘YoungAdult’ model that incorporated the following features: (i) a new and more accurate empirical equation for approximating wave speeds, based on area and relative distance to elastic–muscular arterial transition points; (ii) optimally matched arterial junctions; and (iii) an improved arterial network geometry that eliminated ‘within‐segment’ taper (which causes wave reflection in conduit arteries) whilst establishing ‘impedance‐preserving’ taper. These properties of the model led to wave reflection occurring predominantly at distal vascular beds, rather than in conduit arteries. The model predicted all three typical characteristics of an ascending aortic pressure waveform observed in young adults. When compared with non‐invasively acquired pressure and velocity measurements (obtained via tonometry and Doppler ultrasound in seven young adults), the model was also shown to reproduce the typical waveform morphology observed in the radial, brachial, carotid, temporal, femoral and tibial arteries. The YoungAdult model provides support for the concept that the arterial tree impedance in healthy young adults is exquisitely optimized, and it provides an important baseline model for investigating cardiovascular changes in ageing and disease states.

Key points

The origin of wave reflection in the arterial system is controversial, but reflection properties are likely to give rise to characteristic haemodynamic features in healthy young adults, including an early systolic peak, negative systolic augmentation and diastolic hump in the ascending aortic pressure waveform, and triphasic velocity profiles in peripheral arteries. Although computational modelling provides insights into arterial haemodynamics, no previous models have predicted all these features.

An established arterial network model was optimized by incorporating the following features: (i) a more accurate representation of arterial wave speeds; (ii) precisely matched junctions; and (iii) impedance‐preserving tapering, thereby minimizing wave reflection in conduit arteries in the forward direction. Comparison with in vivo data (n = 7 subjects) indicated that the characteristic waveform features in young adults were predicted accurately.

Our findings strongly imply that a healthy young arterial system is structured to optimize wave reflection in the main conduit arteries and that reflection of forward waves occurs primarily in the vicinity of vascular beds.

A novel porous media-based approach to outflow boundary resistances of 1D arterial blood flow models

In this paper we introduce a novel method for prescribing terminal boundary conditions in one-dimensional arterial flow networks. This is carried out by coupling the terminal arterial vessel with a poro-elastic tube, representing the flow resistance offered by microcirculation. The performance of the proposed porous media-based model has been investigated through several different numerical examples. First, we investigate model parameters that have a profound influence on the flow and pressure distributions of the system. The simulation results have been compared against the waveforms generated by three elements (RCR) Windkessel model. The proposed model is also integrated into a realistic arterial tree, and the results obtained have been compared against experimental data at different locations of the network. The accuracy and simplicity of the proposed model demonstrates that it can be an excellent alternative for the existing models.

Impact of coronary bifurcation morphology on wave propagation

The branching pattern of the coronary vasculature is a key determinant of its function and plays a crucial role in shaping the pressure and velocity wave forms measured for clinical diagnosis. However, although multiple scaling laws have been proposed to characterize the branching pattern, the implications they have on wave propagation remain unassessed to date. To bridge this gap, we have developed a new theoretical framework by combining the mathematical formulation of scaling laws with the wave propagation theory in the pulsatile flow regime. This framework was then validated in multiple species using high-resolution cryomicrotome images of porcine, canine, and human coronary networks. Results demonstrate that the forward well-matchedness (no reflection for pressure/flow waves traveling from the coronary stem toward the microcirculation) is a salient feature in the coronary vasculature, and this result remains robust under many scenarios of the underlying pulse wave speed distribution assumed in the network. This result also implies a significant damping of the backward traveling waves, especially for smaller vessels (radius, <0.3 mm). Furthermore, the theoretical prediction of increasing area ratios (ratio between the area of the mother and daughter vessels) in more symmetric bifurcations found in the distal circulation was confirmed by experimental measurements. No differences were observed by clustering the vessel segments in terms of transmurality (from epicardium to endocardium) or perfusion territories (left anterior descending, left circumflex, and right coronary artery).

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.

On the mechanics underlying the reservoir-excess separation in systemic arteries and their implications for pulse wave analysis

Several works have separated the pressure waveform p in systemic arteries into reservoir p(r) and excess p(exc) components, p = p(r) + p(exc), to improve pulse wave analysis, using windkessel models to calculate the reservoir pressure. However, the mechanics underlying this separation and the physical meaning of p(r) and p(exc) have not yet been established. They are studied here using the time-domain, inviscid and linear one-dimensional (1-D) equations of blood flow in elastic vessels. Solution of these equations in a distributed model of the 55 larger human arteries shows that p(r) calculated using a two-element windkessel model is space-independent and well approximated by the compliance-weighted space-average pressure of the arterial network. When arterial junctions are well-matched for the propagation of forward-travelling waves, p(r) calculated using a three-element windkessel model is space-dependent in systole and early diastole and is made of all the reflected waves originated at the terminal (peripheral) reflection sites, whereas p(exc) is the sum of the rest of the waves, which are obtained by propagating the left ventricular flow ejection without any peripheral reflection. In addition, new definitions of the reservoir and excess pressures from simultaneous pressure and flow measurements at an arbitrary location are proposed here. They provide valuable information for pulse wave analysis and overcome the limitations of the current two- and three-element windkessel models to calculate p(r).

Modelling pulse wave propagation in the rabbit systemic circulation to assess the effects of altered nitric oxide synthesis

Pulse wave propagation in the mature rabbit systemic circulation was simulated using the one-dimensional equations of blood flow in compliant vessels. A corrosion cast of the rabbit circulation was manufactured to obtain arterial lengths and diameters. Pulse wave speeds and inflow and outflow boundary conditions were derived from in vivo data. Numerical results captured the main features of in vivo pressure and velocity pulse waveforms in the aorta, brachiocephalic artery and central ear artery. This model was used to elucidate haemodynamic mechanisms underlying changes in peripheral pulse waveforms observed in vivo after administering drugs that alter nitric oxide synthesis in the endothelial cells lining blood vessels. According to our model, these changes can be explained by single or combined alterations of blood viscosity, peripheral resistance and compliance, and the elasticity of conduit arteries.

Reservoir-wave separation and wave intensity analysis applied to carotid arteries: a hybrid 1D model to understand haemodynamics

Pressure waveforms measured at different locations in the cardiovascular system present a very similar diastolic decay. Previous work has shown the cardiovascular system can be modelled as a Windkessel and wave system. This concept has been extended to any arbitrary location in the cardiovascular system. We suggest that it is possible to calculate a time-varying reservoir pressure P(t) and a distance- and time-varying wave pressure p(x, t) by fitting an exponential function to the diastolic decay of the measured pressure P; defining that the measured pressure P(x, t) = P(t)+p(x, t). Velocity waveforms U can also be separated into its reservoir, U , and wave, u,components as U(x, t) = U (x, t) + u(x, t).In this study we explore the implications of applying are servoir-wave separation and wave intensity analysis techniques to understand the haemodynamics of in-vivo, noninvasive measurements of P and U in the carotid arteries of normal human subjects. Wave intensity analysis reveals a particular wave pattern where reflections can be estimated easily, but foremost, it shows that reflections are a lot smaller than previously thought.We suggest through the use of this model that the heart is the main wave generator of the cardiovascular system. The arterial system instead of impeding the flow, it stores it and distributes it throughout the arteries towards the tissue during diastole. There are some wave reflections, mainly during systole,that contribute to the changes in the pressure and velocity waveforms, however, they are small and are more evident as the measurements get further away from the ascending aorta.The application of wave intensity analysis to non-invasively measured data can provide a good insight on the physiology and the local and global properties of the cardiovascular system in health and disease in the clinical setting. This study shows preliminary results and the potential of the technique for analysing non-invasive measures, and could be particularly useful to understand and quantify the effects of therapeutic drugs in the cardiovascular system.

Separation of the reservoir and wave pressure and velocity from measurements at an arbitrary location in arteries

Previous studies based on measurements made in the ascending aorta have demonstrated that it can be useful to separate the arterial pressure P into a reservoir pressure P* generated by the windkessel effect and a wave pressure p generated by the arterial waves: P = P*+p. The separation in these studies was relatively straightforward since the flow into the arterial system was measured. In this study the idea is extended to measurements of pressure and velocity at sites distal to the aortic root where flow into the arterial system is not known. P* is calculated from P at an arbitrary location in a large artery by fitting the pressure fall-off in diastole to an exponential function and assuming that p is proportional to the flow into the arterial system. A local reservoir velocity U* that is proportional to P* is also defined. The separation algorithm is applied to in vivo human and canine data and to numerical data generated using a one-dimensional model of pulse wave propagation in the larger conduit arteries. The results show that the proposed algorithm is reasonably robust, allowing for the separation of the measured pressure and velocity into reservoir and wave pressures and velocities. Application to data measured simultaneously in the aorta of the dog shows that the reservoir pressure is fairly uniform along the aorta, a test of self-consistency of the assumptions leading to the algorithm. Application to data generated with a validated numerical model indicates that the parameters derived by fitting the pressure data are close to the known values which were used to generate the numerical data. Finally, application to data measured in the human thoracic aorta indicates the potential usefulness of the separation.

Can the modified Allen's test always detect sufficient collateral flow in the hand? A computational study

Blood flow in the largest arteries of the arm up to the digital arteries is numerically modelled using the one-dimensional equations of pressure and flow wave propagation in compliant vessels. The model can be applied to different anatomies of arterial networks and can simulate compression of arteries, these allowing us to simulate the modified Allen's test (MAT) and to assess its suitability for the detection of sufficient collateral flow in the hand if radial blood supply is interrupted. The test measures blood flow in the superficial palmar arch before and during compression of the radial artery. The absence of reversal flow in the palmar arch with the compression indicates insufficient collateral flow and is referred to as a positive MAT. This study shows that small calibres of the superficial palmar arch and insufficient compression of the radial artery can lead to false-positive results. Measurement of the drop in digital systolic pressures with compression of the radial artery has proved to be a more sensitive test to predict the presence of sufficient ulnar collateral flow in networks with small calibres of the superficial palmar arch. However, this study also shows that digital pressure measurements can fail in detecting enough collateral flow if the radial artery is insufficiently compressed.