Unstable equilibrium behaviour in collapsible tubes

Thick-walled silicone rubber tube connected to rigid pipes upstream and downstream was externally pressurised (pe) to cause collapse while aqueous fluid flowed through propelled by a constant upstream head. Three types of equilibrium were found: stable equilibria (steady flow) at high downstream flow resistance R2, self-excited oscillations at low R2, and 'unattainable' (by varying external pressure) or exponentially unstable equilibria at intermediate R2. The self-excited oscillations were highly non-linear and appeared in four, apparently discrete, frequency bands: 2.7 Hz, 3.8-5.0 Hz, 12-16 Hz and 60-63 Hz, suggesting that the possible oscillation modes may be harmonically related. Stable, intermediate 'two-in-every-three-beats' oscillation was also observed, with a repetition frequency in the 3.8-5.0 Hz band. As pe was increased, self-excited oscillations were eventually suppressed, leaving internal fluid pressure varying with no single dominant frequency as a result of turbulent jet dissipation at the downstream rigid pipe connection. Comparison of pressure-wave velocity calculated from the local pressure-area relation for the tube with fluid velocity indicated that supercritical velocities were attained in the course of the self-excited oscillations.     

Chaotic oscillations in a simple collapsible-tube model.

A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.     

Numerical analysis for stability and self-excited oscillation in collapsible tube flow.

This paper describes numerical analysis of collapsible tube flow based on the one-dimensional distributed parameter model of Hayashi. In the present model the effect of flow separation at the collapsed part is replaced with simple viscous friction along the tube, so no ad-hoc modeling for flow separation in former studies is required. A stable semi-implicit numerical procedure based on the SIMPLE method is developed for the problem of flow and tube interaction. The numerical result for a characteristic self-excited oscillation agrees qualitatively with the experimental result. Nonlinear stability of the steady state dependent on the amplitude of the disturbance is numerically investigated and the result is compared with the linear stability analysis based on the former lumped parameter model. Finally, initiation of the self-excited oscillation is examined by applying the initial disturbance at the upstream end of the tube. The disturbance propagates in the downstream direction and is amplified to the self-excited oscillation.     

Modulation analysis of forced non-linear oscillators for biological modelling.

The synchronization of self-sustained oscillations by a forcing oscillation is of interest in a number of biological models. It has been considered for circadian rhythm modelling, heart-rate variability studies and forced breathing experiments. Outside the range of synchronization, conditions of almost-entrainment occur in which changes in amplitude and/or frequency are apparent. It is shown in this paper that such conditions can be analysed as modulation phenomena using the analytical method of harmonic balance. The degree of non-linearity in the self-sustained oscillation affects the nature of modulation, in that increasing distortion gives a trend towards frequency rather than amplitude modulation. The analytical results compare favourably with spectral analysis of simulated oscillators.     

Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.     

Mechanical properties of the upper airway

The series and shunt components of the impedance of the upper airway (Zuaw) were evaluated from measurements obtained during a Valsalva maneuver by means of a modified forced oscillation technique. When the cheeks are supported, the upper airway can be represented by a single distributed transmission line. The homogeneity of this line was confirmed by measuring separately Zuaw and the impedance of the mouth. Correction of the impedance of the respiratory system, determined by means of the forced oscillations technique, for the shunt properties of Zuaw results in some modifications of the frequency dependence of resistance (Rrs) in healthy adults and in marked changes of the absolute values of Rrs in children and in patients with obstructive lung disease.     

Dynamic model of the bronchial tree

We present a distributed model of the bronchial tree which simulates the global dynamic characteristics of the lung. Local mechanical characteristics of each airway are represented by RCL circuits and parameters of the electrical components are determined from local physiological data. The bronchi geometry is described by Weibel's symmetric model, the flow in each airway is assumed laminar and mixing effects at the bifurcations are neglected; the transpulmonary pressure is assumed to be sinusoidal. In simulations of quiet breathing the resistance to airflow is found to be dominant, the flow amplitude decreasing as breathing frequency increases, but remaining almost constant in all the generations. Simulations of ventilation through obstructed lungs show frequency dependence of the dynamic characteristics in very compliant lungs. The global resistance to airflow and the dynamic compliance of the bronchi decrease as the forced oscillation frequency increases in a pattern similar to in vivo measurements in diseased lungs. This may be an outcome of the RCL properties of the network rather than due to uneven distribution of mechanical properties of the lung.     

Respiratory resistance with histamine challenge by single-breath and forced oscillation methods

Relaxed expirations were obtained from five anesthetized dogs under control conditions and during various rates of intravenous infusion of histamine. All volume vs. time curves obtained from 20 ms to 2 s after the start of expiration were poorly described by a single exponential function but were fitted very well by a biexponential function. The resistance of the respiratory system as a function of frequency from 2 to 26 Hz was also determined by the forced oscillation method in the same dogs. Three two-compartment models of the respiratory system were identified from the exponentials fitted to the relaxed expiration data, and the one that had the most plausible parameter values under control conditions consisted of a homogeneous lung compartment connected to a viscoelastic compartment. Although a two-compartment model is arguably appropriate for describing relaxed expirations in normal dogs, physiological considerations suggest that there should be more than two interacting components with histamine infusion. We cannot identify all these components from our data, however. The equivalent complex impedance of the respiratory system was also calculated from the biexponential curves and showed significant variation in resistance over the frequency range from 0 to 2 Hz and negligible variation above 2 Hz. The calculated resistances at 2 Hz were consistently higher than those obtained by the forced oscillation method, which may be due to the nonlinear behavior of the respiratory system during relaxed expiration. We conclude that the single-breath and forced oscillation methods should be viewed as providing complimentary information about respiratory resistance.     

Lung hyperinflation in isolated dog lungs during high-frequency oscillation

To study the phenomenon of lung hyperinflation (LHI), i.e., an increase in lung volume without a concomitant rise in airway pressure, we measured lung volume changes in isolated dog lungs during high-frequency oscillation (HFO) with air, He, and SF6 and with mean tracheal pressure controlled at 2.5, 5.0, and 7.5 cmH2O. The tidal volume and frequency used were 1.5 ml/kg body wt and 20 Hz, respectively. LHI was observed during HFO in all cases except for a few trials with He. The degree of LHI was inversely related to mean tracheal pressure and varied directly with gas density. Maximum expiratory flow rate (Vmax) was measured during forced expiration induced by a vacuum source (-150 cmH2O) at the trachea. Vmax was consistently higher than the peak oscillatory flow rate (Vosc) during HFO, demonstrating that overall expiratory flow limitation did not cause LHI in isolated dog lungs. Asymmetry of inspiratory and expiratory impedances seems to be one cause of LHI, although other factors are involved.     

Measurement of total respiratory impedance in infants by the forced oscillation technique

The forced oscillation technique according to Làndsér et al. (J. Appl. Physiol. 41:101-106, 1976) was modified for use in infants. Adaptations, including a flexible tube to connect the infant to the measuring system and a bias flow to avoid rebreathing, did not influence impedance values. The linearity of the respiratory system was assessed and confirmed by 1) applying pseudo-random noise oscillations at three different amplitudes to 7 infants and 2) comparing in 12 infants impedance values obtained with pseudo-random noise and with sinusoidal oscillations at 12 and 32 Hz. Intersubject variability, averaged for all frequencies, was 6%. In 17 infants the relative error (+/- SD) between two series of five measurements within a time interval of 15 min was 0.5 +/- 5.7%. No statistically significant difference was found between impedance values before and after repositioning of the infant's head, whereas rotation resulted in a decrease in resistance and no effect on reactance. Our results indicate that the infant-adapted forced pseudo-random noise oscillation technique has the potential to give valuable information about ventilatory lung function in infants.     

Low-frequency pulmonary impedance in rabbits and its response to inhaled methacholine.

We assessed pulmonary mechanics in six open-chest rabbits (3 young and 3 adult) by the forced oscillation technique between 0.16 and 10.64 Hz. Under control conditions, pulmonary resistance (RL) decreased markedly between 0.16 and 4 Hz, after which it became reasonably constant. Measurements of alveolar pressure from two alveolar capsules in each rabbit showed that the large decrease of RL with increasing frequency below 4 Hz was due to lung tissue rheology and that tissue resistance was close to zero above 4 Hz. Estimates of resistance and elastance, also obtained by fitting tidal ventilation data at 1 Hz to the equation of the linear single-compartment model, gave values for RL motion that were slightly higher than those obtained by forced oscillations at the same frequency, presumably because of the flow dependence of airways resistance. After treatment with increasing doses of aerosolized methacholine, RL and pulmonary elastance between 0.16 and 1.34 Hz progressively increased, as did the point at which the pulmonary reactance crossed zero (the resonant frequency). The alveolar pressure measurements showed the lung to become increasingly inhomogeneously ventilated in all six animals, whereas in the three younger rabbits lobar atelectasis developed at high methacholine concentrations and the alveolar capsules ceased to communicate with the central airways. We conclude that the low-frequency pulmonary impedance of rabbits exhibits the same qualitative features observed in other species and that it is a sensitive indicator of the changes in pulmonary mechanics occurring during bronchoconstriction.     

Experimentation of Fish Swimming Based on Tracking Locomotion Locus

There are many kinds of swimming mode in the fish world, and we investigated two of them, used by cyprinids and bulltrout. In this paper we track the locomotion locus by marks in different flow velocity from 0.2 m·s^-1 to 0.8 m·s^-1. By fit the data above we could find out the locomotion mechanism of the two kinds of fish and generate a mathematical model of fish kine- matics. The cyprinid fish has a greater oscillation period and amplitude compared with the bulltrout, and the bulltrout changes velocity mainly by controlling frequency of oscillation.     

Frequency encoding in renal blood flow regulation

With a model of renal blood flow regulation, we examined consequences of tubuloglomerular feedback (TGF) coupling to the myogenic mechanism via voltage-gated Ca channels. The model reproduces the characteristic oscillations of the two mechanisms and predicts frequency and amplitude modulation of the myogenic oscillation by TGF. Analysis by wavelet transforms of single-nephron blood flow confirms that both amplitude and frequency of the myogenic oscillation are modulated by TGF. We developed a double-wavelet transform technique to estimate modulation frequency. Median value of the ratio of modulation frequency to TGF frequency in measurements from 10 rats was 0.95 for amplitude modulation and 0.97 for frequency modulation, a result consistent with TGF as the modulating signal. The simulation predicted that the modulation was regular, while the experimental data showed much greater variability from one TGF cycle to the next. We used a blood pressure signal recorded by telemetry from a conscious rat as the input to the model. Blood pressure fluctuations induced variability in the modulation records similar to those found in the nephron blood flow results. Frequency and amplitude modulation can provide robust communication between TGF and the myogenic mechanism.     

Gas transport during oscillatory flow in a network of branching tubes

A transport coefficient was measured for a range of oscillatory flow conditions in a branching network of tubes. Measurements were made both across the first generation of a three-generation network and the second generation of a four-generation network. The results for these two series of tests were similar, indicating that there was no significant effect due to the system boundaries. The results are cast in terms of an effective axial diffusion coefficient of the form (Formula: see text) where kappa is the molecular diffusivity, Vt is the local stroke volume (cc); and f is the oscillation frequency (Hz). These results are compared to those obtained by other investigators in branching systems of similar geometry. At low frequency, this result is found to be in approximate agreement with the steady flow result of Scherer, et al. [15]. This expression differs from the oscillatory flow results of Tarbell, et al. [19] for liquids, primarily in terms of the effects of oscillation frequency.     

Changes of respiratory input impedance during breathing in humans.

Changes of total respiratory resistance (Rrs) and reactance (Xrs) were studied between 8 and 32 Hz at five moments during the respiratory cycle in healthy adults (group A) and children (group B) and in patients with chronic obstructive lung disease (group C) and with upper airway obstruction (group D). Two forced oscillation techniques were used: the conventional one and the head generator, with the oscillations applied at the mouth and around the head of the subject, respectively. Both techniques yielded similar results. Rrs is lowest during the transition from inspiration to expiration and highest in the course of expiration, except in group D. Mean Xrs is highest at the transitions from inspiration to expiration or vice versa and lowest during expiration, except in group D. In groups C and D, the increases of Rrs are accompanied by a more pronounced negative frequency dependence of Rrs. The variations of Rrs and Xrs appear to be markedly flow dependent and may be a consequence of the interaction of breathing with oscillatory flows.     

Influence of bifurcations on forced oscillations in an airway model.

Forced oscillations is a technique to determine respiratory input impedance from small amplitude sinusoidal pressure excursions introduced at the airway opening. Models used to predict respiratory input impedance typically ignore the direct effect of bifurcations on the flow, and treat airway branches as individual straight tubes placed appropriately in parallel and series. The flow within the individual tubes is assumed equivalent to that which would occur in infinitely long tubes. In this study we examined the influence of bifurcations on impedance for conditions of the forced oscillatory technique. We measured input impedance using forced oscillations in straight tubes and in an anatomically-relevant, four generation physical model of a human airway network. The input impedance measured experimentally compared well to that obtained theoretically using model predictions. The predictive scheme was based on appropriate parallel and series combinations of theoretically computed individual tube impedances, which were computed from solutions to oscillatory flow of a compressible gas in an infinitely long rigid tube. The agreement between experimental measurements and predictions indicates that bifurcations play a relatively minor direct role on the flow impedance for conditions of the forced oscillations technique. These results are explained in terms of the small tidal volumes used, whereby the axial distance traveled by a fluid particle during an oscillation cycle is appreciably smaller than branch segment lengths. Accordingly, only a small fraction of fluid particles travel through the bifurcation region, and the remainder experience an environment approaching flow in an infinite straight tube. The relevance of the study to the prediction of impedances in the human lung during forced oscillations is discussed.     

One-dimensional computer analysis of oscillatory flow in rigid tubes.

The dynamic characteristics of catheter-transducer systems using rigid tubes with compliance lumped in the transducer and oscillatory flow of fluid in rigid tubes were analyzed. A digital computer model based on one dimensional laminar oscillatory flow was developed and verified by exact solution of the Navier-Stokes Equation. Experimental results indicated that the damping ratio and resistance is much higher at higher frequencies of oscillation than predicted by the one dimensional model. An empirical correction factor was developed and incorporated into the computer model to correct the model to the experimental data. Amplitude of oscillation was found to have no effect on damping ratio so it was concluded that the increased damping ratio and resistance at higher frequencies was not due to turbulence but to two dimensional flow effects. Graphs and equations were developed to calculate damping ratio and undamped natural frequency of a catheter-transducer system from system parameters. Graphs and equations were also developed to calculate resistance and inertance for oscillatory flow in rigid tubes from system parameters and frequency of oscillation.     

Calcium oscillations induced by ATP in human umbilical cord smooth muscle cells

Arterial smooth muscle cells exhibit vasomotion, related to oscillations in intracellular Ca(2+) concentration, but the origin and function of these has not yet been fully determined. We measured intracellular Ca(2+) using conventional fluorescent methods in primary cultured, human umbilical cord artery smooth muscle cells (HUCASMC). Spontaneous oscillations in Ca(2+) was found in only 1% of all cells but exogenous, micromolar concentrations of ATP could induce Ca(2+) oscillations in 70% of cells with the most common pattern being one of regular amplitude and frequency with a return to basal levels between each peak. The P2Y agonist, UTP, but not the P2X agonist alphabeta-methylene ATP, could also induce Ca(2+) oscillations. Once induced, these oscillations could not be blocked by G-protein, PLC, VGCC or TRP channel antagonists applied individually, but could be prevented when antagonists were applied together. In the presence of EGTA, micromolar concentrations of ATP induced an elevation in intracellular Ca(2+) but did not induce Ca(2+) oscillations. The oscillation frequency induced by ATP was affected by bath Ca(2+) concentration. Taken together, these data suggest that external Ca(2+) entry maintains the Ca(2+) oscillation induced by activation of P2Y receptors. Once induced, multiple mechanisms are involved to maintain the oscillation and the oscillation frequency is determined by the speed of Ca(2+) refilling. Chronic hypoxia enhanced the Ca(2+) response and altered the oscillation frequency. We suggest that these oscillations may play a role in the maintenance of umbilical blood flow during situations in which GPCR are activated.