A study of the bifurcation behaviour of a model of flow through a collapsible tube

Most of the elastic tubes found in the mammalian body will collapse from a distended circular cross section and when collapsed may undergo flow-induced oscillations. A mathematical model describing fluid flow in a collapsible tube is analysed using the software package AUTO-86. AUTO-86 is used for continuation and bifurcation problems in systems of non-linear ordinary differential equations. The model is a third-order lumped-parameter type and is based on the classical “Starling resistor”; it describes the unsteady flow behaviour and, in particular, the experimentally observed self-excited oscillations, in a way which is simple enough to give physical understanding, yet still firmly based on fluid mechanical principles. Some of the bifurcation types found in this model bear close resemblance to the types suggested by experimental observations of self-excited oscillations in collapsible tubes; they thus shed some light on the various topological changes which occur in practice, particularly in view of the fact that some of the points found numerically are diffcult to achieve experimentally, while the existence of others can only be inferred indirectly and uncertainly from experiment.     

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.     

Upper airway pressure-flow relationships in obstructive sleep apnea

We examined the pressure-flow relationships in patients with obstructive sleep apnea utilizing the concepts of a Starling resistor. In six patients with obstructive sleep apnea, we applied incremental levels of positive pressure through a nasal mask during non-rapid-eye-movement sleep. A positive critical opening pressure (Pcrit) of 3.3 +/- 3.3 (SD) cmH2O was demonstrated. As nasal pressure was raised above Pcrit, inspiratory airflow increased in proportion to the level of positive pressure applied until apneas were abolished (P less than 0.01). However, at pressures greater than Pcrit, esophageal pressures either did not correlate or correlated inversely with inspiratory airflow provided that esophageal pressure was less than Pcrit. When pressure was applied to a full face mask, inspiratory airflow did not occur and Pcrit could not be obtained at pressures well above Pcrit demonstrated with the nasal mask. These results are consistent with the view that the upper airway functions as a Starling resistor with a collapsible segment in the oropharynx. These findings offer a unifying construct for the association of sleep apnea, periodic hypopnea, and snoring.     

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.     

Nature and distribution of vascular resistance in hypoxic pig lungs

We used the vascular occlusion technique in pig lungs isolated in situ to describe the effects of hypoxia on the distribution of vascular resistance and to determine whether the resistive elements defined by this technique behaved as ohmic or Starling resistors during changes in flow at constant outflow pressure, changes in outflow pressure at constant flow, and reversal of flow. During normoxia, the largest pressure gradient occurred across the middle compliant region of the vasculature (delta Pm). The major effect of hypoxia was to increase delta Pm and the gradient across the relatively noncompliant arterial region (delta Pa). The gradient across the noncompliant venous region (delta Pv) changed only slightly, if at all. Both delta Pa and delta Pv increased with flow but delta Pm decreased. The pressure at the arterial end of the middle region was independent of flow and, when outflow pressure was increased, did not increase until the outflow pressure of the middle region exceeded 8.9 Torr during normoxia and 18.8 Torr during hypoxia. Backward perfusion increased the total pressure gradient across the lung, mainly because of an increase in delta Pm. These results can be explained by a model in which the arterial and venous regions are represented by ohmic resistors and the middle region is represented by a Starling resistor in series and proximal to an ohmic resistor. In terms of this model, hypoxia exerted its major effects by increasing the critical pressure provided by the Starling resistor of the middle region and the ohmic resistance of the arterial region.     

A Starling resistor regulates cerebral venous outflow in dogs

This investigation was undertaken to determine whether a Starling resistor or venous waterfall effect exists between the sagittal sinus and the cerebral veins such that increases in sagittal sinus pressure (Pss) do not abolish cerebral venous outflow and to examine two possible contributions of extracranial venous valves in regulating outflow. Anesthetized dogs were subjected to positive end-expiratory pressure (PEEP) before and after intracranial pressure (Pic) was elevated by inflation of an epidural balloon. PEEP raised Pss equally in all animals, but Pic and cerebral venous pressure (Pcv) increased less in the presence of intracranial hypertension. When Pss was low, passage of a catheter in the cerebral vein in and out of the sagittal sinus demonstrated an abrupt drop in pressure as the sinus was entered. When Pss was raised and lowered independently of superior vena caval pressure (Psvc) the changes in Pic and Pcv were less when Pss was decreased than when it was increased. Sustained increases and decreases in Psvc caused increases and decreases in Pcv, Pic, Pss, and external jugular venous pressure (Pejv) regardless of whether external jugular venous valves were present or absent. We conclude that a Starling resistor between the sagittal sinus and the cerebral veins regulates cerebral venous outflow when Pss is increased by PEEP and other maneuvers that raise Psvc. The waterfall maintains Pcv and Pic at normal levels when Psvc and Pss are reduced. Extracranial venous valves are not essential to this mechanism.     

Teaching the effects of gravity and intravascular and alveolar pressures on the distribution of pulmonary blood flow using a classic paper by West et al

"Distribution of blood flow in isolated lung; relation to vascular and alveolar pressures" by J. B. West, C. T. Dollery, and A. Naimark (J Appl Physiol 19: 713-724, 1964) is a classic paper, although it has not yet been included in the Essays on the American Physiological Society Classic Papers Project (http://www.the-aps.org/publications/classics/). This is the paper that originally described the "zones of the lung." The final figure in the paper, which synthesizes the results and discussion, is now seen in most textbooks of physiology or respiratory physiology. The paper is also a model of clear, concise writing. The paper and its final figure can be used to teach or review a number of physiological concepts. These include the effects of gravity on pulmonary blood flow and pulmonary vascular resistance; recruitment and distention of pulmonary vessels; the importance of the transmural pressure on the diameter of collapsible distensible vessels; the Starling resistor; the interplay of the pulmonary artery, pulmonary vein, and alveolar pressures; and the vascular waterfall. In addition, the figure can be used to generate discovery learning and discussion of several physiological or pathophysiological effects on pulmonary vascular resistance and the distribution of pulmonary blood flow.     

One-dimensional steady inviscid flow through a stenotic collapsible tube

A one-dimensional inviscid solution for flow through a compliant tube with a stenosis is presented. The model is used to represent an artery with an atherosclerotic plaque and to investigate a range of conditions for which arterial collapse may occur. The coupled equations for flow through collapsible tubes are solved using a Runge-Kutta finite difference scheme. Quantitative results are given for specific physiological parameters including inlet and outlet pressure, flow rate, stenosis size, length and stiffness. The results suggest that high-grade stenotic arteries may exhibit collapse with typical physiological pressures. Critical stenoses may cause choking of flow at the throat followed by a transition to supercritical flow with tube collapse downstream. Greater amounts of stenosis produced a linear reduction of flow rate and a shortening of the collapsed region. Changes in stenosis length created proportional changes in the length of collapse. Increasing the stiffness of the stenosis to a value greater than the nominal tube stiffness caused a greater amount of flow limitation and more negative pressures, compared to a stenosis with constant stiffness. These findings assist in understanding the clinical consequences of flow through atherosclerotic arteries.     

Longitudinal tension variation in collapsible channels: a new mechanism for the breakdown of steady flow.

There are several mechanisms potentially involved in the breakdown of steady fluid flow in a collapsible tube under external pressure. Here we investigate one that has received little attention in the past: the fact that the longitudinal tension in the tube wall, T, decreases with distance downstream as a consequence of the viscous shear stress exerted by the fluid. If the tube is long enough, or the initial tension small enough, T may fall to zero before the end of the collapsible tube, and unsteady motion would presumably then ensue; this is what we mean by "breakdown." We study the phenomenon theoretically, when the flow Reynolds number is of order one, using lubrication theory in a symmetric two-dimensional channel in which the collapsible tube is replaced by membranes occupying a segment of each wall. The resulting nonlinear ordinary differential equations are solved numerically for values of the dimensionless parameters that cover all the qualitatively different types of solution (e.g., in which the channel is distended over all its length, collapsed over all its length, or distended in the upstream part and collapsed downstream). Reducing the longitudinal tension has a marked effect on the shape of the collapsible segment, causing it to become much more deformed for the same flow rate and external pressure. Indeed, the wall slope is predicted to become very large when the downstream tension is very small, so the model is not self-consistent then. Nevertheless, the parameter values for which T becomes zero are mapped out and are expected to be qualitatively useful.(ABSTRACT TRUNCATED AT 250 WORDS)     

Flutter in flow-limited collapsible tubes: a mechanism for generation of wheezes

We studied flutter in collapsible tubes as a possible mechanism for the generation of respiratory wheezes. The pressure-flow relationships and the wall oscillations of thick-walled [wall thickness (h)-to-lumen radius (r) ratio 1:1.7 to 1.3] self-supporting latex and Silastic tubes mounted between rigid pipes were measured. A high-impedance vacuum pump was connected to the downstream end. Upstream and downstream valves were used to control corresponding resistances. We found loud honking sounds and tube wall oscillations that occurred only when the tubes were buckled and flow limiting, i.e., when the flow became constant and independent of downstream driving pressure. The overall range of oscillatory frequencies was 260-750 Hz for airflow, presenting as sharp peaks of power on the frequency spectrum. The oscillatory frequencies (f) were higher at higher fluid velocities (u) and with narrower distance between opposing flattened walls (2b), resulting from increasing downstream suction pressure and the transmural pressure becoming more negative. The effect of u and b on f for a latex tube (h-to-r ratio 1:1.7) were found to be f = 228 + 0.021 (u/b). These relationships were valid throughout the range of oscillations in this tube (283-720 Hz) and with flow rates of 12-64 l/min. The experimental data were compared with predictions of the fluid dynamic flutter theory and the vortex-induced wall vibrations mechanism. We conclude that viscid flutter in soft tubes is the more probable mechanism for the generation of oscillations in the soft tube model and is a possible mechanism for the generation of respiratory wheezes.     

The flow field downstream of an oscillating collapsed tube

A two-component laser Doppler anemometer was used to determine the velocity of aqueous flow in the region from 0.25 to 2.5 diameters downstream of a collapsible tube while the tube was executing vigorous repetitive flow-induced oscillations. The Reynolds number for the time-averaged flow was 10,750. A simultaneous measurement of the pressure at the downstream end of the tube was used to align all the results in time at sixty locations in each of the two principal planes defined by the axes of collapse of the flexible tube upstream. The raw data of seed-particle velocity were used to create a periodic waveform for each measured velocity component at each location by least-squares fitting of a Fourier series. The results are presented as both velocity vectors and interpolated contours, for each of ten salient instants during the cycle of oscillation. In the plane of the collapse major axis, the dominant feature is the jet which emerges from each of the two tube lobes when it collapses, but transient retrograde flow is observed on both the central and lateral edges of this jet. In the orthogonal, minor-axis plane, the dominant feature is the retrograde flow, which during part of the cycle extends over the whole plane. All these features are essentially confined to the first 1.5 diameters of the rigid pipe downstream of the flexible tube. These data map the temporal and spatial extent of the highly three-dimensional reversing flow just downstream of an oscillating collapsed tube.     

Effects of abdominal pressure on venous return: abdominal vascular zone conditions

The effects of changes in abdominal pressure (Pab) on inferior vena cava (IVC) venous return were analyzed using a model of the IVC circulation based on a concept of abdominal vascular zone conditions analogous to pulmonary vascular zone conditions. We hypothesized that an increase in Pab would increase IVC venous return when the IVC pressure at the level of the diaphragm (Pivc) exceeds the sum of Pab and the critical closing transmural pressure (Pc), i.e., zone 3 conditions, but reduce IVC venous return when Pivc is below the sum of Pab and Pc, i.e., zone 2 conditions. The validity of the model was tested in 12 canine experiments with an open-chest IVC bypass. An increase in Pab produced by phrenic stimulation increased the IVC venous return when Pivc-Pab was positive but decreased the IVC venous return when Pivc - Pab was negative. The value of Pivc - Pab that separated net increases from decreases in venous return was 1.00 +/- 0.72 (SE) mmHg (n = 6). An increase in Pivc did not influence the femoral venous pressure when Pivc was lower than the sum of Pab and a constant, 0.96 +/- 0.70 mmHg (n = 6), consistent with presence of a waterfall. These results agreed closely with the predictions of the model and its computer simulation. The abdominal venous compartment appears to function with changes in Pab either as a capacitor in zone 3 conditions or as a collapsible Starling resistor with little wall tone in zone 2 conditions.     

Numerical simulation of noninvasive blood pressure measurement

In this paper, a simulation model based on the partially pressurized collapsible tube model for reproducing noninvasive blood pressure measurement is presented. The model consists of a collapsible tube, which models the pressurized part of the artery, rigid pipes connected to the collapsible tube, which model proximal and distal region far from the pressurized part, and the Windkessel model, which represents the capacitance and the resistance of the distal part of the circulation. The blood flow is simplified to a one-dimensional system. Collapse and expansion of the tube is represented by the change in the cross-sectional area of the tube considering the force balance acting on the tube membrane in the direction normal to the tube axis. They are solved using the Runge-Kutta method. This simple model can easily reproduce the oscillation of inner fluid and corresponding tube collapse typical for the Korotkoff sounds generated by the cuff pressure. The numerical result is compared with the experiment and shows good agreement.     

Laser-Doppler measurements of velocities just downstream of a collapsible tube during flow-induced oscillations

The flow field less than one diameter downstream of the end of a collapsible tube executing self excited oscillations was examined using a two-component fiber-optic laser-Doppler anemometer. The time-averaged Reynolds number of the flow was 11,000. With the tube oscillating periodically, results obtained during many cycles of oscillation were combined to yield surface plots of the axial component over the cross section at 16 phases of the cycle. By combining measurements obtained with the laser probe in two different orientations, secondary flow vectors over the cross section were likewise constructed for 16 phases. The measurements showed strongly phasic turbulence intensity, with the phase of high intensity coinciding with the time of maximal tube collapse. Reverse flow occurred during much of the cycle, at places in the cross section that agree with our previous observations of laminar and turbulent steady flow through a rigid simulated collapsed tube.     

Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress

A mathematical model of the arterial baroreflex was developed and used to assess the stability of the reflex and its potential role in producing the low-frequency arterial blood pressure oscillations called Mayer waves that are commonly seen in humans and animals in response to decreased central blood volume. The model consists of an arrangement of discrete-time filters derived from published physiological studies, which is reduced to a numerical expression for the baroreflex open-loop frequency response. Model stability was assessed for two states: normal and decreased central blood volume. The state of decreased central blood volume was simulated by decreasing baroreflex parasympathetic heart rate gain and by increasing baroreflex sympathetic vaso/venomotor gains as occurs with the unloading of cardiopulmonary baroreceptors. For the normal state, the feedback system was stable by the Nyquist criterion (gain margin = 0.6), but in the hypovolemic state, the gain margin was small (0.07), and the closed-loop frequency response exhibited a sharp peak (gain of 11) at 0.07 Hz, the same frequency as that observed for arterial pressure fluctuations in a group of healthy standing subjects. These findings support the theory that stresses affecting central blood volume, including upright posture, can reduce the stability of the normally stable arterial baroreflex feedback, leading to resonance and low-frequency blood pressure waves.     

Effect of inspiratory nasal loading on pharyngeal resistance

Nasal obstruction has been shown to increase the number of apneas during sleep in normal subjects and in some may actually cause the sleep apnea syndrome. We postulated that the pharynx may act as a Starling resistor, where increases in negative inspiratory pressure result in elevated resistance across a collapsible pharyngeal segment. To test this theory in normal subjects we studied 10 men and 10 women during wakefulness. Pharyngeal resistance (the resistance across the airway segment between the choanae and the epiglottis) was determined in the normal state and with three inspiratory loads added externally. Flow was measured using a pneumotachometer and a sealed face mask; epiglottic pressure by a latex balloon placed just above the epiglottis and choanal pressure by anterior rhinometry. Pharyngeal resistance (measured at 300 ml/s) could thus be determined. Base-line inspiratory pharnygeal resistance was 1.6 +/- 0.2 cmH2O . l-1 . s. This increased to 2.3 +/- 0.3, 2.8 +/- 0.4, and 2.9 +/- 0.4 cmH2O . l-1 . s, respectively, with the addition of 1.3, 2.7, and 6.7 cmH2O . l-1 . s inspiratory load. The resistance at each level of load was significantly different from the base-line resistance determination (P less than 0.05) but not different from each other. We conclude that added nasal resistive loads during inspiration cause an increase in pharyngeal resistance during wakefulness but that this resistance does not increase further with additional increments of load.     

Cardiac MyBP-C phosphorylation regulates the Frank–Starling relationship in murine hearts

The Frank–Starling relationship establishes that elevated end-diastolic volume progressively increases ventricular pressure and stroke volume in healthy hearts. The relationship is modulated by a number of physiological inputs and is often depressed in human heart failure. Emerging evidence suggests that cardiac myosin-binding protein-C (cMyBP-C) contributes to the Frank–Starling relationship. We measured contractile properties at multiple levels of structural organization to determine the role of cMyBP-C and its phosphorylation in regulating (1) the sarcomere length dependence of power in cardiac myofilaments and (2) the Frank–Starling relationship in vivo. We compared transgenic mice expressing wild-type cMyBP-C on the null background, which have ∼50% phosphorylated cMyBP-C (Controls), to transgenic mice lacking cMyBP-C (KO) and to mice expressing cMyBP-C that have serine-273, -282, and -302 mutated to aspartate (cMyBP-C t3SD) or alanine (cMyBP-C t3SA) on the null background to mimic either constitutive PKA phosphorylation or nonphosphorylated cMyBP-C, respectively. We observed a continuum of length dependence of power output in myocyte preparations. Sarcomere length dependence of power progressively increased with a rank ordering of cMyBP-C KO = cMyBP-C t3SA < Control < cMyBP-C t3SD. Length dependence of myofilament power translated, at least in part, to hearts, whereby Frank–Starling relationships were steepest in cMyBP-C t3SD mice. The results support the hypothesis that cMyBP-C and its phosphorylation state tune sarcomere length dependence of myofibrillar power, and these regulatory processes translate across spatial levels of myocardial organization to control beat-to-beat ventricular performance.     

Pulmonary hemodynamics and gas exchange properties during progressive edema

In this investigation we have studied the effect of increments of pulmonary edema on pulmonary hemodynamics, and physiological and hemodynamic shunt in an isolated lung preparation. Hemodynamic shunt was defined by the slope of the relationship between pulmonary arterial and airway pressures; when the slope decreases, there is a greater degree of shunt. Cardiovascular changes were analyzed using a Starling resistor model of the pulmonary circulation where the effective downstream pressure to flow as seen from the pulmonary artery exceeds the pulmonary venous outflow pressure. This effective downstream pressure is referred to as the critical pressure (Pc), and at low lung inflation the locus of this critical pressure is in extra-alveolar vessels. With 3-4 h of progressive edema to an average of 185% initial lobe weight we found a progressive rise in pulmonary arterial pressure (Ppa) from 12.1 to 21.5 cmH2O. About one-third of this increase in Ppa resulted from an increased Pc and the remainder resulted from an increased resistance upstream from the locus of Pc. These results are consistent with the hypothesis that the interstitial accumulation of fluid creates enough of an increase in interstitial pressure to compress extra-alveolar vessels. There was no significant correlation between the amount of edema and the measured physiologic shunt, but the hemodynamic shunt showed a highly significant correlation. The hemodynamic shunt theoretically measures the extent of obstructed airways and may be a useful index of the degree of pulmonary edema.     

Numerical schemes for unsteady fluid flow through collapsible tubes

The study of fluid flow through compliant tubes is a fluid-structure type problem, in which a dynamic equilibrium is maintained between the fluid and the tube wall. The analogy between this flow and gas dynamics initiated the use of a number of numerical methods which were originally developed to solve compressible flow in rigid ducts. In this study we investigate the solutions obtained by applying the Lax-Wendroff and MacCormack schemes to one-dimensional incompressible flow through a straight collapsible tube. The time-evolving numerical results were compared with exact steady-state solutions. For boundary conditions which were held fixed after a prescribed rise time, the unsteady numerical solution converges to the exact steady-state solution with very good accuracy. The stability and accuracy of all the methods depend on the amount of viscous pressure loss dictated by wall friction. Flows with undamped oscillations cannot, however, be solved with these techniques.     

Three-dimensional flows in a hyperelastic vessel under external pressure

We study the collapsible behaviour of a vessel conveying viscous flows subject to external pressure, a scenario that could occur in many physiological applications. The vessel is modelled as a three-dimensional cylindrical tube of nonlinear hyperelastic material. To solve the fully coupled fluid–structure interaction, we have developed a novel approach based on the Arbitrary Lagrangian–Eulerian (ALE) method and the frontal solver. The method of rotating spines is used to enable an automatic mesh adaptation. The numerical code is verified extensively with published results and those obtained using the commercial packages in simpler cases, e.g. ANSYS for the structure with the prescribed flow, and FLUENT for the fluid flow with prescribed structure deformation. We examine three different hyperelastic material models for the tube for the first time in this context and show that at the small strain, all three material models give similar results. However, for the large strain, results differ depending on the material model used. We further study the behaviour of the tube under a mode-3 buckling and reveal its complex flow patterns under various external pressures. To understand these flow patterns, we show how energy dissipation is associated with the boundary layers created at the narrowest collapsed section of the tube, and how the transverse flow forms a virtual sink to feed a strong axial jet. We found that the energy dissipation associated with the recirculation does not coincide with the flow separation zone itself, but overlaps with the streamlines that divide the three recirculation zones. Finally, we examine the bifurcation diagrams for both mode-3 and mode-2 collapses and reveal that multiple solutions exist for a range of the Reynolds number. Our work is a step towards modelling more realistic physiological flows in collapsible arteries and veins.