The Lymphatic Vascular System: Does Nonuniform Lymphangion Length Limit Flow-Rate?

A previously developed model of a lymphatic vessel as a chain of lymphangions was investigated to determine whether lymphangions of unequal length reduce pumping relative to a similar chain of equal-length ones. The model incorporates passive elastic and active contractile properties taken from ex vivo measurements, and intravascular lymphatic valves as transvalvular pressure-dependent resistances to flow with hysteresis and transmural pressure-dependent bias to the open state as observed experimentally. Coordination of lymphangion contractions is managed by marrying an autonomous transmural pressure-dependent pacemaker for each lymphangion with bidirectional transmission of activation signals between lymphangions, qualitatively matching empirical observations. With eight lymphangions as used here and many nonlinear constraints, the model is capable of complex outcomes. The expected flow-rate advantage conferred by longer lymphangions everywhere was confirmed. However, the anticipated advantage of uniform lymphangions over those of unequal length, compared in chains of equal overall length, was not found. A wide variety of dynamical outcomes was observed, with the most powerful determinant being the adverse pressure difference, rather than the arrangement of long and short lymphangions. This work suggests that the wide variation in lymphangion length which is commonly observed in collecting lymphatic vessels does not confer disadvantage in pumping lymph.

Modelling the coupling of the M-clock and C-clock in lymphatic muscle cells.

Chronic dysfunction of the lymphatic vascular system results in fluid accumulation between cells: lymphoedema. The condition is commonly acquired secondary to diseases such as cancer or the associated therapies. The primary driving force for fluid return through the lymphatic vasculature is provided by contractions of the muscularized lymphatic collecting vessels, driven by electrochemical oscillations. However, there is an incomplete understanding of the molecular and bioelectric mechanisms involved in lymphatic muscle cell excitation, hampering the development and use of pharmacological therapies. Modelling in silico has contributed greatly to understanding the contributions of specific ion channels to the cardiac action potential, but modelling of these processes in lymphatic muscle remains limited. Here, we propose a model of oscillations in the membrane voltage (M-clock) and intracellular calcium concentrations (C-clock) of lymphatic muscle cells. We modify a model by Imtiaz and colleagues to enable the M-clock to drive the C-clock oscillations. This approach differs from typical models of calcium oscillators in lymphatic and related cell types, but is required to fit recent experimental data. We include an additional voltage dependence in the gating variable control for the L-type calcium channel, enabling the M-clock to oscillate independently of the C-clock. We use phase-plane analysis to show that these M-clock oscillations are qualitatively similar to those of a generalised FitzHugh-Nagumo model. We also provide phase plane analysis to understand the interaction of the M-clock and C-clock oscillations. The model and methods have the potential to help determine mechanisms and find targets for pharmacological treatment of lymphoedema.

Modelling secondary lymphatic valves with a flexible vessel wall: how geometry and material properties combine to provide function.

A three-dimensional finite-element fluid/structure interaction model of an intravascular lymphatic valve was constructed, and its properties were investigated under both favourable and adverse pressure differences, simulating valve opening and valve closure, respectively. The shear modulus of the neo-Hookean material of both vascular wall and valve leaflet was varied, as was the degree of valve opening at rest. Also investigated was how the valve characteristics were affected by prior application of pressure inflating the whole valve. The characteristics were parameterised by the volume flow rate through the valve, the hydraulic resistance to flow, and the maximum sinus radius and inter-leaflet-tip gap on the plane of symmetry bisecting the leaflet, all as functions of the applied pressure difference. Maximum sinus radius on the leaflet-bisection plane increased with increasing pressure applied to either end of the valve segment, but also reflected the non-circular deformation of the sinus cross section caused by the leaflet, such that it passed through a minimum at small favourable pressure differences. When the wall was stiff, the inter-leaflet gap increased sigmoidally during valve opening; when it was as flexible as the leaflet, the gap increased more linearly. Less pressure difference was required both to open and to close the valve when either the wall or the leaflet material was more flexible. The degree of bias of the valve characteristics to the open position increased with the inter-leaflet gap in the resting position and with valve inflation pressure. The characteristics of the simulated valve were compared with those specified in an existing lumped-parameter model of one or more collecting lymphangions and used to estimate a revised value for the constant in that model which controls the rate of valve opening/closure with variation in applied pressure difference. The effects of the revised value on the lymph pumping efficacy predicted by the lumped-parameter model were evaluated.

Correction to: A computational model of a network of initial lymphatics and pre-collectors with permeable interstitium.

In the published paper, we argued that, although there appear to be no data available on the subject, it is inherently unlikely that lymph having traversed a network of initial lymphatics and pre-collectors then encounters an abrupt transition to vessels with all of the typical properties of collecting lymphatics.

A computational model of a network of initial lymphatics and pre-collectors with permeable interstitium.

Initial lymphatic vessels are made up of overlapped endothelial cells that act as unidirectional valves enabling one-way drainage of tissue fluid into the lumen of the initial lymphatics when there is a favourable pressure gradient. Initial lymphatics subsequently drain this fluid into the collecting lymphatics. This paper describes a computational model for a network of passive rat mesenteric lymphatic vessels with sparse secondary valves. The network was simulated with the secondary valves both operational and non-operational. The effects on the cycle-mean outflow-rate from the network of both inflammation and the resistance of the surrounding interstitium were considered. The cycle-mean outflow-rate is sensitive to vessel stiffness. If the influence of primary-valve resistance is reduced relative to that of interstitial resistance and intravascular resistance, there is no absolute advantage of extrinsic pumping, since maximum outflow-rate occurs when vessels are rigid. However, there is relative advantage, in that the outflow-rate at intermediate stiffness is higher with the secondary valves functioning than when they are deactivated. If primary-valve resistance dominates, then extrinsic pumping of non-rigid vessels provides absolute advantage. The nonlinear relation between pressure drop and flow-rate of the endothelial primary valves, combined with downstream compliance and pulsatile external pressure, constitutes a separate mechanism of pumping. By enabling the consideration of interactions between multiple phenomena (primary valves, secondary valves, a real network geometry with multiple branches, deformable vessel walls, interstitial resistance and external pressures), the model offers a perspective for delineating physiological phenomena that have not yet been fully linked to the biomechanics of fluid flow through initial lymphatic networks.

Inhibition of Contraction Strength and Frequency by Wall Shear Stress in a Single-Lymphangion Model.

The phasic contractions of collecting lymphatic vessels are reduced in strength and occur at diminished frequency when a favorable pressure difference and the resulting antegrade flow create large fluid shear stresses at the luminal surface. This paper describes a minimal phenomenological model of this mechanism that is applied to a previously validated numerical model of a phasically contracting lymphangion. The parameters of the inhibition model are quantitatively matched to observations in isolated segments of rat lymphatic vessel, first for mesenteric lymphatics then for thoracic duct, and outcomes from the numerical model are then qualitatively compared with recent observations in isolated segments of rat thoracic duct.

Contraction of collecting lymphatics: organization of pressure-dependent rate for multiple lymphangions.

The paper describes the extension of a previously developed model of pressure-dependent contraction rate to the case of multiple lymphangions. Mechanical factors are key modulators of active lymphatic pumping. As part of the evolution of our lumped-parameter model to match experimental findings, we have designed an algorithm whereby the time until the next contraction depends on lymphangion transmural pressure in the contraction just completed. The functional dependence of frequency on pressure is quantitatively matched to isobaric contraction experiments on isolated lymphatic segments. When each of several lymphangions is given this ability, a scheme for their coordination must be instituted to match the observed synchronization. Accordingly, and in line with an experiment on an isolated lymphatic vessel segment in which we measured contraction sequence and conduction delay, we took the fundamental principle to be that local timing can be overridden by signals to initiate contraction that start in adjacent lymphangions, conducted with a short delay. The scheme leads to retrograde conduction when the lymphangion chain is pumping against an adverse pressure difference, but antegrade conduction when contractions occur with no or a favourable pressure difference. Abolition of these conducted signals leads to chaotic variation of cycle-mean flow-rate from the chain, diastolic duration in each lymphangion, and inter-lymphangion delays. Chaotic rhythm is also seen under other circumstances. Because the model responds to increasing adverse pressure difference by increasing the repetition rate of contractions, it maintains time-average output flow-rate better than one with fixed repetition rate.

Valve-related modes of pump failure in collecting lymphatics: numerical and experimental investigation.

Lymph is transported along collecting lymphatic vessels by intrinsic and extrinsic pumping. The walls have muscle of a type intermediate between blood-vascular smooth muscle and myocardium; a contracting segment between two valves (a lymphangion) constitutes a pump. This intrinsic mechanism is investigated ex vivo in isolated, spontaneously contracting, perfused segments subjected to controlled external pressures. The reaction to varying afterload is probed by slowly ramping up the outlet pressure until pumping fails. Often the failure occurs when the contraction raises intra-lymphangion pressure insufficiently to overcome the outlet pressure, open the outlet valve and cause ejection, but many segments fail by other means, the mechanisms of which are not clear. We here elucidate those mechanisms by resort to a numerical model. Experimental observations are paired with comparable findings from computer simulations, using a lumped-parameter model that incorporates previously measured valve properties, plus new measurements of active contractile and passive elastic properties, and the dependence of contraction frequency on transmural pressure, all taken from isobaric twitch contraction experiments in the same vessel. Surprisingly, the model predicts seven different possible modes of pump failure, each defined by a different sequence of valve events, with their occurrence depending on the parameter values and boundary conditions. Some, but not all, modes were found experimentally. Further model investigation reveals routes by which a vessel exhibiting one mode of failure might under altered circumstances exhibit another.

A Poroelastic Fluid/Structure-Interaction Model of Cerebrospinal Fluid Dynamics in the Cord With Syringomyelia and Adjacent Subarachnoid-Space Stenosis.

An existing axisymmetric fluid/structure-interaction (FSI) model of the spinal cord, pia mater, subarachnoid space, and dura mater in the presence of syringomyelia and subarachnoid-space stenosis was modified to include porous solids. This allowed investigation of a hypothesis for syrinx fluid ingress from cerebrospinal fluid (CSF). Gross model deformation was unchanged by the addition of porosity, but pressure oscillated more in the syrinx and the subarachnoid space below the stenosis. The poroelastic model still exhibited elevated mean pressure in the subarachnoid space below the stenosis and in the syrinx. With realistic cord permeability, there was slight oscillatory shunt flow bypassing the stenosis via the porous tissue over the syrinx. Weak steady streaming flow occurred in a circuit involving craniocaudal flow through the stenosis and back via the syrinx. Mean syrinx volume was scarcely altered when the adjacent stenosis bisected the syrinx, but increased slightly when the syrinx was predominantly located caudal to the stenosis. The fluid content of the tissues over the syrinx oscillated, absorbing most of the radial flow seeping from the subarachnoid space so that it did not reach the syrinx. To a lesser extent, this cyclic swelling in a boundary layer of cord tissue just below the pia occurred all along the cord, representing a mechanism for exchange of interstitial fluid (ISF) and cerebrospinal fluid which could explain recent tracer findings without invoking perivascular conduits. The model demonstrates that syrinx volume increase is possible when there is subarachnoid-space stenosis and the cord and pia are permeable.

Pump function curve shape for a model lymphatic vessel.

The transport capacity of a contractile segment of lymphatic vessel is defined by its pump function curve relating mean flow-rate and adverse pressure difference. Numerous system characteristics affect curve shape and the magnitude of the generated flow-rates and pressures. Some cannot be varied experimentally, but their separate and interacting effects can be systematically revealed numerically. This paper explores variations in the rate of change of active tension and the form of the relation between active tension and muscle length, factors not known from experiment to functional precision. Whether the pump function curve bends toward or away from the origin depends partly on the curvature of the passive pressure-diameter relation near zero transmural pressure, but rather more on the form of the relation between active tension and muscle length. A pump function curve bending away from the origin defines a well-performing pump by maximum steady output power. This behaviour is favoured by a length/active-tension relationship which sustains tension at smaller lengths. Such a relationship also favours high peak mechanical efficiency, defined as output power divided by the input power obtained from the lymphangion diameter changes and active-tension time-course. The results highlight the need to pin down experimentally the form of the length/active-tension relationship.

Incorporating measured valve properties into a numerical model of a lymphatic vessel.

An existing lumped-parameter model of multiple lymphangions (lymphatic vascular segments) in series is adapted for the incorporation of recent physiological measurements of lymphatic vascular properties. The new data show very marked nonlinearity of the passive pressure-diameter relation during distension, relative to comparable blood vessels, and complex valve behaviour. Since lymph is transported as a result of either the active contraction or the passive squeezing of vascular segments situated between two one-way valves, the performance of these valves is of primary importance. The valves display hysteresis (the opening and closing pressure drop thresholds differ), a bias to staying open (both state changes occur when the trans-valve pressure drop is adverse) and pressure-drop threshold dependence on transmural pressure. These properties, in combination with the strong nonlinearity that valve operation represents, have in turn caused intriguing numerical problems in the model, and we describe numerical stratagems by which we have overcome the problems. The principal problem is also generalised into a relatively simple mathematical example, for which solution detail is provided using two different solvers.

Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values.

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure-diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds and (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve-closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.

Simulation of a chain of collapsible contracting lymphangions with progressive valve closure.

The aim of this investigation was to achieve the first step toward a comprehensive model of the lymphatic system. A numerical model has been constructed of a lymphatic vessel, consisting of a short series chain of contractile segments (lymphangions) and of intersegmental valves. The changing diameter of a segment governs the difference between the flows through inlet and outlet valves and is itself governed by a balance between transmural pressure and passive and active wall properties. The compliance of segments is maximal at intermediate diameters and decreases when the segments are subject to greatly positive or negative transmural pressure. Fluid flow is the result of time-varying active contraction causing diameter to reduce and is limited by segmental viscous and valvular resistance. The valves effect a smooth transition from low forward-flow resistance to high backflow resistance. Contraction occurs sequentially in successive lymphangions in the forward-flow direction. The behavior of chains of one to five lymphangions was investigated by means of pump function curves, with variation of valve opening parameters, maximum contractility, lymphangion size gradation, number of lymphangions, and phase delay between adjacent lymphangion contractions. The model was reasonably robust numerically, with mean flow-rate generally reducing as adverse pressure was increased. Sequential contraction was found to be much more efficient than synchronized contraction. At the highest adverse pressures, pumping failed by one of two mechanisms, depending on parameter settings: either mean leakback flow exceeded forward pumping or contraction failed to open the lymphangion outlet valve. Maximum pressure and maximum flow-rate were both sensitive to the contractile state; maximum pressure was also determined by the number of lymphangions in series. Maximum flow-rate was highly sensitive to the transmural pressure experienced by the most upstream lymphangions, suggesting that many feeding lymphatics would be needed to supply one downstream lymphangion chain pumping at optimal transmural pressure.

Evaluation by fluid/structure-interaction spinal-cord simulation of the effects of subarachnoid-space stenosis on an adjacent syrinx.

A finite-element numerical model was constructed of the spinal cord, pia mater, filum terminale, cerebrospinal fluid in the spinal subarachnoid space (SSS), and dura mater. The cord was hollowed out by a thoracic syrinx of length 140 mm, and the SSS included a stenosis of length 30 mm opposite this syrinx. The stenosis severity was varied from 0% to 90% by area. Pressure pulse excitation was applied to the model either at the cranial end of the SSS, simulating the effect of cranial arterial pulsation, or externally to the abdominal dura mater, simulating the effect of cough. A very short pulse was used to examine wave propagation; a pulse emulating cardiac systole was used to examine the effects of fluid displacement. Additionally, repetitive sinusoidal excitation was applied cranially. Bulk fluid flow past the stenosis gave rise to prominent longitudinal pressure dissociation ("suck") in the SSS adjacent to the syrinx. However, this did not proportionally increase the longitudinal motion of fluid in the syrinx. The inertia of the fluid in the SSS, together with the compliance of this space, gave a resonance capable of being excited constructively or destructively by cardiac or coughing impulses. The main effect of mild stenosis was to lower the frequency of this resonance; severe stenosis damped out to-and-fro motions after the end of the applied excitation. Syrinx fluid motion indicated the fluid momentum and thus the pressure developed when the fluid was stopped by the end of the syrinx; however, the tearing stress in the local cord material depended also on the instantaneous local SSS pressure and was therefore not well predicted by syrinx fluid motion. Stenosis was also shown to give rise to a one-way valve effect causing raised SSS pressure caudally and slight average cord displacement cranially. The investigation showed that previous qualitative predictions of the effects of suck neglected factors that reduced the extent of the resulting syrinx fluid motion and of the cord tearing stress, which ultimately determines whether the syrinx lengthens.

Fluid flow in distensible vessels.

1. Flow in single vascular conduits is reviewed, divided into distended and deflated vessels. 2. In distended vessels with pulsatile flow, wave propagation and reflection dominate the spatial and temporal distribution of pressure, determining the shape, size and relative timing of measured pressure waveforms, as well as the instantaneous pressure gradient everywhere. Considerable research has been devoted to accessing the information on pathological vascular malformations contained in reflected waves. Slow waves of contraction of vessel wall muscle, responsible for transport of oesophageal, ureteral and gut contents, have also been modelled. 3. The pressure gradient in a vessel drives the flow. Flow rate can be predicted both analytically and numerically, but analytical theory is limited to idealized geometry. The complex geometry of biological system conduits necessitates computation instead. Initially limited to rigid boundaries, numerical methods now include fluid-structure interaction and can simultaneously model solute transport, thus predicting accurately the environment of the mechanosensors and chemosensors at vessel surfaces. 4. Deflated vessels display all phenomena found in distended vessels, but have additional unique behaviours, especially flow rate limitation and flow-induced oscillation. Flow rate limitation is widespread in the human body and has particular diagnostic importance in respiratory investigation. Because of their liquid lining, the pulmonary airways are also characterized by important two-phase flows, where surface tension phenomena create flows and determine the patency and state of collapse of conduits. 5. Apart from the vital example of phonation, sustained self-excited oscillation is largely avoided in the human body. Where it occurs in snoring, it is implicated in the pathological condition of sleep apnoea.

PIV measurements of the flow field just downstream of an oscillating collapsible tube.

We probed the time-varying flow field immediately downstream of a flexible tube conveying an aqueous flow, during flow-induced oscillation of small amplitude, at time-averaged Reynolds numbers (Re) in the range 300-550. Velocity vector components in the plane of a laser sheet were measured by high-speed ("time-resolved") particle image velocimetry. The sheet was aligned alternately with both the major axis and the minor axis of the collapsing tube by rotating the pressure chamber in which the tube was mounted. The Womersley number of the oscillations was approximately 10. In the major-axis plane the flow fields were characterized by two jets that varied in lateral spacing. The rapid deceleration of flow at maximal collapse caused the jets momentarily to merge about one diameter into the downstream pipe, and strengthened and enlarged the existing retrograde flow lateral to each jet. Collapse also spread the jets maximally, allowing retrograde flow between them during the ascent from its minimum of the pressure at the end of the flexible tube. The minor-axis flow fields showed that the between-jet retrograde flow at this time extended all the way across the pipe. Whereas the retrograde flow lateral to the jets terminated within three diameters of the tube end at Re=335 at all times, it extended beyond three diameters at Re=525 for some 25% of the cycle including the time of maximal flow deceleration. Off-axis sheet positioning revealed the lateral jets to be crescent shaped. When the pressure outside the tube was increased, flattening the tube more, the jets retained a more consistent lateral position. These results illuminate the flows created by collapsible-tube oscillation in a laminar regime accessible to numerical modeling.

Measurement and analysis of the flow field in a pulsatile ventricularassist device.

A pneumatically driven ventricular assist device (VAD) of unconventional geometry, designed to mitigate adverse haemostatic phenomena by optimization of flow patterns, was investigated using two-component laser Doppler anemometry (LDA). Data were taken on a rectilinear grid over two orthogonal traverses, allowing synthesis of three-dimensional velocity vectors over most of the grid. The resulting data were examined by two-dimensional and three-dimensional static and animated computer graphic visualizations of the time-varying vector fields. This analysis revealed illuminating and previously unobserved features of the complex flow fields within pulsatile VADs and identified design considerations that bear upon minimization of turbulence and of flow recirculation and stasis. The findings of this study suggest that, while total abolition of undesirable flow phenomena in a diaphragm-type blood pump is probably impossible, comprehensive investigation in vitro can engender considerable improvements in the efficacy of a device.

Tensile radial stress in the spinal cord related to arachnoiditis or tethering: a numerical model.

Spinal arachnoiditis comprises fibrous scarring of the subarachnoid space, following spinal trauma or inflammation, and is often associated with syringomyelia. We hypothesised that cord-to-dura attachments could cause transient tensile cord radial stress, as pressure waves propagate. This was tested in a fluid-structure interaction model, simulating three types of cord tethering, with 'arachnoiditis' confined to a short mid-section of the cord. The annular system was excited abdominally with a short transient, and the resulting Young and Lamb waves and reflections were analysed. Radial mid-section tethering was less significant than axial tethering, which gave rise to tensile radial stress locally when the cord was not fixed cranially. Simulated as inextensible string connections to the dura, arachnoiditis caused both localised tensile radial stress and localised low pressure in the cord as the transient passed. The extent of these effects was sensitive to the relative stiffness of the dura and cord. Tensile radial stress may create a syrinx in previously normal cord tissue, and transiently lowered pressure may draw in interstitial fluid, causing the syrinx to enlarge if fluid exit is inhibited. The suggested mechanism could also explain the juxtaposition of syrinxes to regions of arachnoiditis.

Measurement for implantable rotary blood pumps.

Rotary blood pumps offer a cost-effective way to assist the failing heart. Relative to their pulsatile cousins, they can consist of remarkably few moving parts, with attendant advantages in reliability. These advantages are realized in full only if the entire assist system is kept maximally simple. Control of the pump must therefore be based on a minimum number of measurement devices. This paper reviews the measurements that are made in the wide range of implantable rotary blood pump designs that are in development for ventricular assist. In a number of these, fluid-mechanical variables are estimated indirectly from measurements of motor speed and current or power. The introduction explains the goals of rotary blood pump control by comparison to the innate properties of the natural heart. Then motor and fluid-mechanical variables that may be transduced are discussed. Methods of indirect estimation of pressure drop and flow-rate are dealt with, followed by ways of detecting unusual states such as inflow obstruction. It is found that detection of these alone can be the basis of an adequate control strategy. Some groups have estimated variables pertaining to the heart that is being assisted, and there has also been work on monitoring the ongoing health of the assist system itself. The review concludes with a brief look at the wider measurement context for the intensive-care facility that proposes to use such devices to provide circulatory support.

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.