Time-domain representation of ventricular-arterial coupling as a windkessel and wave system

The differences in shape between central aortic pressure (P(Ao)) and flow waveforms have never been explained satisfactorily in that the assumed explanation (substantial reflected waves during diastole) remains controversial. As an alternative to the widely accepted frequency-domain model of arterial hemodynamics, we propose a functional, time-domain, arterial model that combines a blood conducting system and a reservoir (i.e., Frank's hydraulic integrator, the windkessel). In 15 anesthetized dogs, we measured P(Ao), flows, and dimensions and calculated windkessel pressure (P(Wk)) and volume (V(Wk)). We found that P(Wk) is proportional to thoracic aortic volume and that the volume of the thoracic aorta comprises 45.1 +/- 2.0% (mean +/- SE) of the total V(Wk). When we subtracted P(Wk) from P(Ao), we found that the difference (excess pressure) was proportional to aortic flow, thus resolving the differences between P(Ao) and flow waveforms and implying that reflected waves were minimal. We suggest that P(Ao) is the instantaneous summation of a time-varying reservoir pressure (i.e., P(Wk)) and the effects of (primarily) forward-traveling waves in this animal model.     

Systemic venous circulation. Waves propagating on a windkessel: relation of arterial and venous windkessels to systemic vascular resistance

Compared with arterial hemodynamics, there has been relatively little study of venous hemodynamics. We propose that the venous system behaves just like the arterial system: waves propagate on a time-varying reservoir, the windkessel, which functions as the reverse of the arterial windkessel. During later diastole, pressure increases exponentially to approach an asymptotic value as inflow continues in the absence of outflow. Our study in eight open-chest dogs showed that windkessel-related arterial resistance was approximately 62% of total systemic vascular resistance, whereas windkessel-related venous resistance was only approximately 7%. Total venous compliance was found to be 21 times larger than arterial compliance (n = 3). Inferior vena caval compliance (0.32 +/- 0.015 ml x mmHg(-1) x kg(-1); mean +/- SE) was approximately 14 times the aortic compliance (0.023 +/- 0.002 ml x mmHg(-1) x kg(-1); n = 8). Despite greater venous compliance, the variation in venous windkessel volume (i.e., compliance x windkessel pulse pressure; 7.8 +/- 1.1 ml) was only approximately 32% of the variation in aortic windkessel volume (24.3 +/- 2.9 ml) because of the larger arterial pressure variation. In addition, and contrary to previous understanding, waves generated by the right heart propagated upstream as far as the femoral vein, but excellent proportionality between the excess pressure and venous outflow suggests that no reflected waves returned to the right atrium. Thus the venous windkessel model not only successfully accounts for variations in the venous pressure and flow waveforms but also, in combination with the arterial windkessel, provides a coherent view of the systemic circulation.     

Wave intensity analysis of left ventricular filling: application of windkessel theory

We extend our recently published windkessel-wave interpretation of vascular function to the wave intensity analysis (WIA) of left ventricular (LV) filling dynamics by separating the pressure changes due to the windkessel from those due to traveling waves. With the use of LV compliance, the change in pressure due solely to LV volume changes (windkessel pressure) can be isolated. Inasmuch as the pressure measured in the cardiovascular system is the sum of its windkessel and wave components (excess pressure), it can be substituted into WIA, yielding the isolated wave effects on LV filling. Our study of six open-chest dogs demonstrated that once the windkessel effects are removed from WIA, the energy of diastolic suction is 2.6 times greater than we previously calculated. Volume-related changes in pressure (i.e., the windkessel or reservoir effect) must be considered first when wave motion is analyzed.     

Wave propagation in a model of the arterial circulation

The propagation of the arterial pulse wave in the large systemic arteries has been calculated using a linearised method of characteristics analysis to follow the waves generated by the heart. The model includes anatomical and physiological data for the 55 largest arteries adjusted so that the bifurcating tree of arteries is well matched for forward travelling waves. The peripheral arteries in the model are terminated by resistance elements which are adjusted to produce a physiologically reasonable distribution of mean blood flow. In the model, the pressure and velocity wave generated by the contraction of the left ventricle propagates to the periphery where it is reflected. These reflected waves are re-reflected by each of the bifurcations that they encounter and a very complex pattern of waves is generated. The results of the calculations exhibit many of the features of the systemic arteries, including the increase of the pulse pressure with distance away from the heart as well as the initial decrease and then the large increase in the magnitude of back flow during late systole going from the ascending aorta to the abdominal aorta to the arteries of the leg. The model is then used to study the effects of the reflection or absorption of waves by the heart and the mechanisms leading to the incisura are investigated. Calculations are carried out with the total occlusion of different arterial segments in order to model experiments in which the effects of the occlusion of different arteries on pressure and flow in the ascending aorta were measured. Finally, the effects of changes in peripheral resistance on pressure and velocity waveforms are also studied. We conclude from these calculations that the complex pattern of wave propagation in the large arteries may be the most important determinant of arterial haemodynamics.     

Travelling wave patterns in a model of the spinal pattern generator using spiking neurons

The aim of this study is to produce travelling waves in a planar net of artificial spiking neurons. Provided that the parameters of the waves – frequency, wavelength and orientation – can be sufficiently controlled, such a network can serve as a model of the spinal pattern generator for swimming and terrestrial quadruped locomotion. A previous implementation using non-spiking, sigmoid neurons lacked the physiological plausibility that can only be attained using more realistic spiking neurons. Simulations were conducted using three types of spiking neuronal models. First, leaky integrate-and-fire neurons were used. Second, we introduced a phenomenological bursting neuron. And third, a canonical model neuron was implemented which could reproduce the full dynamics of the Hodgkin–Huxley neuron. The conditions necessary to produce appropriate travelling waves corresponded largely to the known anatomy and physiology of the spinal cord. Especially important features for the generation of travelling waves were the topology of the local connections – so-called off-centre connectivity – the availability of dynamic synapses and, to some extent, the availability of bursting cell types. The latter were necessary to produce stable waves at the low frequencies observed in quadruped locomotion. In general, the phenomenon of travelling waves was very robust and largely independent of the network parameters and emulated cell types.     

Determination of wave speed and wave separation in the arteries

Considering waves in the arteries as infinitesimal wave fronts rather than sinusoidal wavetrains, the change in pressure across the wave front, dP, is related to the change in velocity, dU, that it induces by the "water hammer" equation, dP=+/-rhocdU, where rho is the density of blood and c is the local wave speed. When only unidirectional waves are present, this relationship corresponds to a straight line when P is plotted against U with slope rhoc. When both forward and backward waves are present, the PU-loop is no longer linear. Measurements in latex tubes and systemic and pulmonary arteries exhibit a linear range during early systole and this provides a way of determining the local wave speed from the slope of the linear portion of the loop. Once the wave speed is known, it is also possible to separate the measured P and U into their forward and backward components. In cases where reflected waves are prominent, this separation of waves can help clarify the pattern of waves in the arteries throughout the cardiac cycle.     

A two-stage regulatory system for pressure-constant perfusion of coronary vessels

This paper describes a double-loop servo-controlled pump system for the constant-pressure perfusion of a coronary artery. Due to the transient nature of changes in coronary vasomotor tone, such a perfusion system must have a fast regulatory response. In the first stage, a servo-controlled pump primes a windkessel having a volume of 35 ml with blood. The pumping rate is electronically controlled to maintain a constant pressure within the windkessel max. 700 mmHg. The maximal flow rate is 300 ml/min. To reduce the high pressure in the windkessel to the desired coronary perfusion pressure, a variable flow resistance, comprising a clamped thin-walled silicone tube, is provided in the output line of the system. A fast servo-motor drives the clamp and is controlled by an electronic regulator, using a second feedback loop from the pressure signal measured at the tip of the perfusion cannula. The system stabilizes the coronary perfusion pressure within 300 ms. An additional modulation of the setpoint signal in synchrony with the cardiac cycle improves the phasic pattern of the blood flow, and thus prevents changes in transmural blood flow distribution. The dead volume of the overall system is about 60 ml. Hemolysis caused by this system during five hours of perfusion in vivo is negligible.     

Peristaltic transport as the travelling deformation waves

Geometry approach to the theoretical and experimental investigations of peristaltic waves based on the travelling deformation waves and wave mass transfer theory (Dobrolyubov, 1991) is presented. The theory of travelling deformation waves is employed to determine uniformed expressions for mass transfer capability parameters of peristalsis. Slow (quasi-static) wave motion is considered which permits not to take into account dynamic phenomena.     

The effect of ion pumps on the speed of travelling waves in the fire-diffuse-fire model of Ca2+ release

The fire-diffuse-fire model provides an idealized model of Ca²⁺ release within living cells. The effect of calcium pumps, which drive Ca²⁺ back into internal stores, is often neglected for mathematical simplicity. Here we show how to explicitly analyse such effects by extending the work of Keizer et al. [Keizer, J. E., G. D. Smith, S. Ponce Dawson and J. Pearson (1998). Saltatory propagation of Ca²⁺ waves by Ca²⁺ sparks. Biophys. J. 75, 595–600.]. For travelling waves, in which release events occur sequentially, we construct the speed of waves in terms of the time-scale at which pumps operate. An immediate consequence of this analysis is that the inclusion of calcium pumps leads to multiple solutions. A linear stability analysis determines those solution branches in parameter space which are stable. Numerical continuation is used to provide explicit examples of the bifurcation diagrams of the speed of waves as a function of physiologically significant system parameters.     

Landscape geometry and travelling waves in the larch budmoth

Travelling waves in cyclic populations refer to temporal shifts in peak densities moving across space in a wave‐like fashion. The epicentre hypothesis states that peak densities begin in specific geographic foci and then spread into adjoining areas. Travelling waves have been confirmed in a number of population systems, begging questions about their causes. Herein we apply a newly developed statistical technique, wavelet phase analysis, to historical data to document that the travelling waves in larch budmoth (LBM) outbreaks arise from two epicentres, both located in areas with high concentrations of favourable habitat. We propose that the spatial arrangement of the landscape mosaic is responsible for initiating the travelling waves. We use a tri‐trophic model of LBM dynamics to demonstrate that landscape heterogeneity (specifically gradients in density of favourable habitat) alone, is capable of inducing waves from epicentres. Our study provides unique evidence of how landscape features can mould travelling waves.     

Calcium wave propagation by calcium-induced calcium release: An unusual excitable system

We discuss in detail the behaviour of a model, proposed by Goldbeteret al. (1990.Proc. natn. Acad. Sci. 87, 1461–1465), for intracellular calcium wave propagation by calcium-induced calcium release, focusing our attention on excitability and the propagation of waves in one spatial dimension. The model with no diffusion behaves like a generic excitable system, and threshold behaviour, excitability and oscillations can be understood within this general framework. However, when diffusion is included, the model no longer behaves like a generic excitable system; the fast and slow variables are not distinct and previous results on excitable systems do not necessarily apply. We consider a piecewise linear simplification of the model, and construct travelling pulse and periodic plane wave solutions to the simplified model. The analogous behaviour in the full model is studied numerically. Goldbeter's model for calciuminduced calcium release is an excitable system of a type not previously studied in detail.     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

Percutaneous Coronary Intervention Enhances Accelerative Wave Intensity in Coronary Arteries

Background

The systolic forward travelling compression wave (sFCW) and diastolic backward travelling decompression waves (dBEW) predominantly accelerate coronary blood flow. The effect of a coronary stenosis on the intensity of these waves in the distal vessel is unknown. We investigated the relationship between established physiological indices of hyperemic coronary flow and the intensity of the two major accelerative coronary waves identified by Coronary Wave Intensity analysis (CWIA).

Methodology / Principal Findings

Simultaneous intracoronary pressure and velocity measurement was performed during adenosine induced hyperemia in 17 patients with pressure / Doppler flow wires positioned distal to the target lesion. CWI profiles were generated from this data. Fractional Flow Reserve (FFR) and Coronary Flow Velocity Reserve (CFVR) were calculated concurrently. The intensity of the dBEW was significantly correlated with FFR (R = -0.70, P = 0.003) and CFVR (R = -0.73, P = 0.001). The intensity of the sFCW was also significantly correlated with baseline FFR (R = 0.71, p = 0.002) and CFVR (R = 0.59, P = 0.01). Stenting of the target lesion resulted in a median 178% (interquartile range 55–280%) (P<0.0001) increase in sFCW intensity and a median 117% (interquartile range 27–509%) (P = 0.001) increase in dBEW intensity. The increase in accelerative wave intensity following PCI was proportionate to the baseline FFR and CFVR, such that stenting of lesions associated with the greatest flow limitation (lowest FFR and CFVR) resulted in the largest increases in wave intensity.

Conclusions

Increasing ischemia severity is associated with proportionate reductions in cumulative intensity of both major accelerative coronary waves. Impaired diastolic microvascular decompression may represent a novel, important pathophysiologic mechanism driving the reduction in coronary blood flow in the setting of an epicardial stenosis.     

Study of pulse transit time oscillations during obstructive sleep apnoea by using a distributed model

The study of arterial compliance is useful in understanding the geometrical and mechanical properties of a systemic arterial tree. Numerous mathematical models have shown their potential in relating the physical phenomena in the arterial tree to properties of the wall itself. However, limited model is available that describes the pulse transit time (PTT) oscillations of a sleeping child during tidal breathing and obstructive sleep apnoea (OSA). Data from 20 children (17 male; aged 6.4 +/- 4.1 yr) whom were recruited for overnight polysomnography (PSG) were used. A modified windkessel model with related physiological parameters was utilised to describe PTT fluctuations due to the cardiovascular system during sleep. Verification with the recorded PSG data showed similar trends with the model for both types of respiratory events. For tidal breathing, undamped PTT oscillations of 3.89 s were predicted by the model while actual data yielded a mean value of 3.72 +/- 0.79 s. Conversely, under-damping PTT responses were expected based on the model for OSA. The model estimated a Q factor of 4.23 and actual mean data were 3.86 +/- 0.64. Hence, the findings herein suggest that the proposed model has the potential to illustrate tidal breathing and OSA events in sleeping children.     

A Resonance Approach to Cochlear Mechanics

Background

How does the cochlea analyse sound into its component frequencies? In the 1850s Helmholtz thought it occurred by resonance, whereas a century later Békésy''s work indicated a travelling wave. The latter answer seemed to settle the question, but with the discovery in 1978 that the cochlea emits sound, the mechanics of the cochlea was back on the drawing board. Recent studies have raised questions about whether the travelling wave, as currently understood, is adequate to explain observations.

Approach

Applying basic resonance principles, this paper revisits the question. A graded bank of harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and it is found that resonance gives rise to similar frequency responses, group delays, and travelling wave velocities as observed by experiment. The overall effect of the group delay gradient is to produce a decelerating wave of peak displacement moving from base to apex at characteristic travelling wave speeds. The extensive literature on chains of coupled oscillators is considered, and the occurrence of travelling waves, pseudowaves, phase plateaus, and forced resonance in such systems is noted.

Conclusion and significance

This alternative approach to cochlear mechanics shows that a travelling wave can simply arise as an apparently moving amplitude peak which passes along a bank of resonators without carrying energy. This highlights the possible role of the fast pressure wave and indicates how phase delays and group delays of a set of driven harmonic oscillators can generate an apparent travelling wave. It is possible to view the cochlea as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and travelling wave theories.     

Estimation of changes in instantaneous aortic blood flow by the analysis of arterial blood pressure

The purpose of this study was to introduce and validate a new algorithm to estimate instantaneous aortic blood flow (ABF) by mathematical analysis of arterial blood pressure (ABP) waveforms. The algorithm is based on an autoregressive with exogenous input (ARX) model. We applied this algorithm to diastolic ABP waveforms to estimate the autoregressive model coefficients by requiring the estimated diastolic flow to be zero. The algorithm incorporating the coefficients was then applied to the entire ABP signal to estimate ABF. The algorithm was applied to six Yorkshire swine data sets over a wide range of physiological conditions for validation. Quantitative measures of waveform shape (standard deviation, skewness, and kurtosis), as well as stroke volume and cardiac output from the estimated ABF, were computed. Values of these measures were compared with those obtained from ABF waveforms recorded using a Transonic aortic flow probe placed around the aortic root. The estimation errors were compared with those obtained using a windkessel model. The ARX model algorithm achieved significantly lower errors in the waveform measures, stroke volume, and cardiac output than those obtained using the windkessel model (P < 0.05).     

Wave dissipation in flexible tubes in the time domain: in vitro model of arterial waves

Earlier work of wave dissipation in flexible tubes and arteries has been carried out predominantly in the frequency domain and most of the studies used the measured pressure waveform for presenting the results. In this work we investigate the pattern of wave dissipation in the time domain using the separated forward and backward travelling waves in flexible tubes. We tested four sizes of latex tubes of 2m in length each, where a single semi-sinusoidal in shape, pressure wave, was produced at the inlet of each tube. Simultaneous measurements of pressure and flow waveforms were recorded every 5cm along the tubes and wave speed was determined using the pressure-velocity loop method (PU-loop). The measured data and wave speed were used to separate the pressure waveform and wave intensity, into their forward and backward directions, using wave intensity analysis (WIA). Also, the energy carried by the wave was calculated by integrating the relevant area under the wave intensity curve. The peak of the measured pressure waveform increased downstream, however, the peak of the separated forward pressure waveform decreased exponentially along the tube. Wave intensity and energy also dissipated exponentially along the travelling distance. The peaks of the separated pressure and wave intensity decreased in the forward in a similar exponential way to that in the backward direction in all four tube sizes. Also, the smaller the size of the tube the greater wave dissipation it caused. We conclude that wave separation is useful in studying wave dissipation in elastic tubes, and WIA provides a convenient method for determining the dissipation of the energy carried by the wave along the travelled distance. The separated pressure waveform, wave intensity and wave energy dissipate exponentially with the travelling distance, and wave dissipation varies conversely with the diameter of elastic tubes.