Oscillations in a collapsed-tube analog of the brachial artery under a sphygmomanometer cuff

To determine whether self-excited oscillations in a Starling resistor are relevant to physiological situations, a collapsible tube conveying an aqueous flow was externally pressurized along only a central segment of its unsupported length. This was achieved by passing the tube through a shorter and wider collapsible sleeve which was mounted in Starling resistor fashion in a pressure chamber. The tube size and material, and all other experimental parameters, were as used in our previous Starling resistor studies. Both low- and high-frequency self-excited oscillations were observed, but the low-frequency oscillations were sensitive to the sleeve type and length relative to unsupported distance. Pressure-flow characteristics showed multiple oscillatory modes, which differed quantitatively from those observed in comparable Starling resistors. Slow variation of driving pressure gave differing behavior according to whether the pressure was rising or falling, in accord with the hysteresis noted on the characteristics and in the tube law. The results are discussed in terms of the various possible mechanisms of collapsible tube instability, and reasons are presented for the absence of the low-frequency mode under most physiological circumstances.     

A mathematical model of flow through a collapsible tube--I. Model and steady flow results

A one-dimensional model is presented to describe the flow through a collapsible tube whose ends are tethered to rigid tubes and which is enclosed in a pressurized chamber. Results are presented for the special case of steady flow. Predicted pressure drop versus flow rate (delta P-Q) characteristics agree qualitatively with available experimental data. The significance of the model and of various physical parameters, in regard to the shape of these characteristics, is discussed.     

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)     

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.     

Flow through a Collapsible Tube: Experimental Analysis and Mathematical Model

Flow through thin-wall axisymmetric tubes has long been of interest to physiologists. Analysis is complicated by the fact that such tubes will collapse when the transmural pressure (internal minus external pressure) is near zero. Because of the absence of any body of related knowledge in other sciences or engineering, previous workers have directed their efforts towards experimental studies of flow in collapsible tubes. More recently, some attention has been given towards analytical studies. Results of an extensive series of experiments show that the significant system parameter is transmural pressure. The cross-sectional area of the tube depends upon the transmural pressure, and changes in cross-section in turn affect the flow geometry. Based on experimental studies, a lumped parameter system model is proposed for the collapsible tube. The mathematical model is simulated on a hybrid computer. Experimental data were used to define the functional relationship between cross-sectional area and transmural pressure as well as the relation between the energy loss coefficient and cross-sectional area. Computer results confirm the validity of the model for both steady and transient flow conditions.     

Possible sources of discrepancy between sphygmomanometer cuff pressure and blood pressure quantified in a collapsible-tube analogue.

This paper examines the assumption that the audible events detected as Korotkov sounds in sphygmomanometry occur when blood pressure equals arm-cuff pressure. Several effects that contribute to discrepancy between these pressures are quantified using an idealised arm-and-cuff system consisting of a thick-walled collapsible tube subject to external compression along a central part of its length. The effects studied are (1) transverse pressure difference, resulting from tissues sustaining a part of the external compression through (a) circumferential bending stiffness and (b) longitudinal curvature of the tensed localised neck at the site of initial collapse, (2) longitudinal pressure difference between upstream pressure and pressure at the collapse point due to both (a) viscous and (b) inertial pressure drop. These effects are found to compensate partially for each other; the pressure within the vessel at the collapse point is less than the cuff pressure, but is also less than the blood pressure at the upstream end of the cuff. All four of the contributing terms increase proportionally to the flow-rate raised to a power greater than one, except the viscous pressure drop. Owing to a progressive shortening of the collapsed neck as flow-rate increases, the viscous term is almost independent of the flow-rate. The overall discrepancy displays less flow-rate dependency and is smaller than some of the terms which contribute to it. This means that considerable accuracy is needed if measurements of the effects are to be used to correct the raw data on cuff pressure at the time of Korotkov sound emission so as to obtain an improved estimate of the blood pressure.     

Numerical solutions for steady and unsteady flow in a model of the pulmonary airways

A computational model is presented for unsteady flow through a collapsible tube with variable wall stiffness. The one-dimensional flow equations are solved for inlet, outlet and external conditions that vary with time and for a tube with time-dependent, spatially-distributed local properties. In particular, the effects of nonuniformities and local perturbations in stiffness distribution in the tube are studied. By allowing the flow to evolve in time, asymptotically steady flows are calculated. When simulating a quasi-steady reduction in downstream pressure, the model demonstrates critical transitions, the phenomena of wave-speed limitation and the sites of flow limitation. It also exhibits conditions for which viscous flow limitation occurs. Computations of rapid, unsteady changes of the exit pressure illustrate the phenomena occurring at the onset of a cough, and the generation and propagation of elastic jumps.     

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.     

Model analogues in the study of cephalic circulation

Simple laboratory models are useful to demonstrate cardiovascular principles involving the effects of gravity on the distribution of blood flow to the heads of animals, especially tall ones like the giraffe. They show that negative pressures cannot occur in collapsible vessels of the head, unless they are protected from collapse by external structures such as the cranium and cervical vertebrae. Negative pressures in the cerebral-spinal fluid (CSF) can prevent cerebral circulation from collapsing, and the spinal veins of the venous plexus can return blood to the heart in essentially rigid vessels. However, cephalic vessels outside the cranium are collapsible, so require positive blood pressures to establish flow; CSF pressure and venous plexus flow are irrelevant in this regard. Pressures in collapsible vessels reflect pressures exerted by surrounding tissues, which may explain the observed pressure gradient in the giraffe jugular vein. Tissue pressure is distinct from interstitial fluid pressure which has little influence on pressure gradients across the walls of major vessels.     

Cerebral perfusion pressure in giraffe: modelling the effects of head-raising and -lowering

Loss of consciousness caused by positional changes of the head results from reduced cerebral blood flow (CBF). CBF is related to cerebral perfusion pressure (CPP). CPP is the difference between mean arterial pressure (MAP) at the head and intracranial pressure (ICP). The positional change of the giraffe head between ground level and standing upright is the largest of all animals yet loss of consciousness does not occur. We have investigated the possibility that an increase in CPP protects giraffe from fainting, using a mechanical model that functioned as an anatomical U-tube. It consisted of a rigid ascending “carotid” limb, a collapsible “brain” tube drained by a rigid, “vertebral venous plexus” (VVP) tube, and a collapsible “head” tube drained by a collapsible tube representing the “jugular vein”. The descending tubes could be rotated relative to the “carotid” tube to be horizontal, or at 30°, 45°, and 60° to the vertical to simulate changes in head position. Pressure at the top of the “carotid” tube was intracranial MAP, at the top of the “VVP” tube was ICP, and the difference CPP. In the simulated “head-up” position and a fluid flow rate of 4 L min⁻¹, CPP was ∼170 mmHg. With the VVP tube horizontal, CPP fell from ∼170 to 45 mmHg, but increased to ∼67 mmHg at 30° “down”, to ∼70 mmHg at 45° “down” and to ∼75 at 60° “down”. The fall in CPP in the head-down positions resulted from a decrease in viscous resistance in, and dissipation of pressure to, the “head” and “jugular” tubes. These data provide an estimate of cranial pressure changes in giraffe during positional changes of the head, and suggest that an increase in CPP plays a significant role in maintaining CBF during head-raising and that it may be an important mechanism for preventing fainting in giraffe.     

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.     

A one-dimensional unsteady separable and reattachable flow model for collapsible tube-flow analysis

Taking into account both flow separation and reattachment observed in available experimental results on flows in a quasi-two-dimensional channel, we present a one-dimensional unsteady flow model, which is applicable to a flow in a collapsible tube. The flow model has been derived from the two-dimensional Navier-Stokes equations by introducing the concept of a dividing streamline, which divides a separated flow into a jet and a dead-water zone. We also present a criterion for the determination of a separation point. Numerical results show that the locations of the predicted separation points agree well with the experimental data. The predicted static pressure of the separated flow is almost constant downstream of the separation point and increases quickly just before the reattachment point as observed in the experiment. Finally, using the present flow model and the separation criterion, we examine the oscillatory behavior of an unsteady flow in a symmetric channel whose walls move sinusoidally.     

Physical characteristics in the new model of the cerebrospinal fluid system

It is unknown which factors determine the changes in cerebrospinal fluid (CSF) pressure inside the craniospinal system during the changes of the body position. To test this, we have developed a new model of the CSF system, which by its biophysical characteristics and dimensions imitates the CSF system in cats. The results obtained on a model were compared to those in animals observed during changes of body position. A new model was constructed from two parts with different physical characteristics. The "cranial" part is developed from a plastic tube with unchangeable volume, while the "spinal" part is made of a rubber baloon, with modulus of elasticity similar to that of animal spinal dura. In upright position, in the "cranial" part of the model the negative pressure appears without any measurable changes in the fluid volume, while in "spinal" part the fluid pressure is positive. All of the observed changes are in accordance to the law of the fluid mechanics. Alterations of the CSF pressure in cats during the changes of the body position are not significantly different compared to those observed on our new model. This suggests that the CSF pressure changes are related to the fluid mechanics, and do not depend on CSF secretion and circulation. It seems that in all body positions the cranial volume of blood and CSF remains constant, which enables a good blood brain perfusion.     

Steady flow through collapsible tubes: measurements of flow and geometry.

Compliant tubes attain a complex three-dimensional geometry when the external pressure exceeds the internal pressure and the tube is partially collapsed. A new technique for remote measurement of dynamic surfaces was applied to classical experiments with collapsible tubes. This work presents measurements of the three-dimensional structure of the tube as well as pressure and flow measurements during static loading and during steady-state fluid flow. Results are shown for two tubes of the same material and internal diameter but with different wall thicknesses. The measured tube laws compare well with previously published data and suggest the possible existence of a similarity tube law. The steady flow measurements did not compare well with the one-dimensional theoretical predictions.     

The role of venous valves in pressure shielding

Background

It is widely accepted that venous valves play an important role in reducing the pressure applied to the veins under dynamic load conditions, such as the act of standing up. This understanding is, however, qualitative and not quantitative. The purpose of this paper is to quantify the pressure shielding effect and its variation with a number of system parameters.

Methods

A one-dimensional mathematical model of a collapsible tube, with the facility to introduce valves at any position, was used. The model has been exercised to compute transient pressure and flow distributions along the vein under the action of an imposed gravity field (standing up).

Results

A quantitative evaluation of the effect of a valve, or valves, on the shielding of the vein from peak transient pressure effects was undertaken. The model used reported that a valve decreased the dynamic pressures applied to a vein when gravity is applied by a considerable amount.

Conclusion

The model has the potential to increase understanding of dynamic physical effects in venous physiology, and ultimately might be used as part of an interventional planning tool.     

Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals. This research was partially supported by the National Science Foundation; it was done in part by D. K. Whirlow in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Carnegie Institute of Technology.     

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.     

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.     

Analysis of the collapsibility of the upper airway in a spectrum of sleep-disordered breathing: a modelling approach

Using a simplified model of the upper airways with two independent collapsible elements (nostrils and hypo-pharynx), we calculated the cross-sectional area of these two elements, taking into account pressure drops. We experimentally measured flow and pressure in the fossa and hypo-pharynx in various syndromes. This allowed us to compare the behaviour of the area supplied by our model with the aerodynamic resistance that is often used to analyse upper airway flow limitation events. We showed that nostril and hypo-pharyngeal areas are better correlated than the resistance values and thus concluded that the pressure divided by the square of the flow is a better parameter for analysing flow limitation in upper airways than resistance. Owing to its simplicity, our model is able to supply the area of the collapsible element in real time, which is impossible with more sophisticated models.     

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.