C. D. Bertram

A mathematical model of unsteady collapsible tube behaviour

A simple, third-order lumped parameter model is presented to describe unsteady flow in a short segment of collapsible tube held between two rigid segments and contained in a pressurised chamber. Equilibrium states and their stability are analysed in detail, as is fully non-linear time dependent behaviour, including in particular the excitation and sustenance of limit--cycle oscillations. The model explicitly neglects both wave propagation (and hence the possibility of choking) and the influence on the elastic properties of the tube of longitudinal tension, but it is otherwise firmly based on fluid mechanical principles. The results emphasise the profound importance of (a) the unsteady head loss (but with some pressure recovery) in the separated flow at the oscillating throat, and (b) the mechanical properties of the parts of the system both downstream and upstream of the collapsible segment. The nature of the upstream segment in particular determines whether it is an upstream pressure head or the inflow to the collapsible segment that is held constant during oscillations. The results are discussed in the context both of other models and of experiment.

Transitional Flow at the Venous Anastomosis of an Arteriovenous Graft: Potential Activation of the ERK1/2 Mechanotransduction Pathway

We present experimental and computational results that describe the level, distribution, and importance of velocity fluctuations within the venous anastomosis of an arteriovenous graft. The motivation of this work is to understand better the importance of biomechanical forces in the development of intimal hyperplasia within these grafts. Steady-flow in vitro studies (Re = 1060 and 1820) were conducted within a graft model that represents the venous anastomosis to measure velocity by means of laser Doppler anemometry. Numerical simulations with the same geometry and flow conditions were conducted by employing the spectral element technique. As flow enters the vein from the graft, the velocity field exhibits flow separation and coherent structures (weak turbulence) that originate from the separation shear layer. We also report results of a porcine animal study in which the distribution and magnitude of vein-wall vibration on the venous anastomosis were measured at the time of graft construction. Preliminary molecular biology studies indicate elevated activity levels of the extracellular regulatory kinase ERK1/2, a mitogen-activated protein kinase involved in mechanotransduction, at regions of increased vein-wall vibration. These findings suggest a potential relationship between the associated turbulence-induced vein-wall vibration and the development of intimal hyperplasia in arteriovenous grafts. Further research is necessary, however, in order to determine if a correlation exists and to differentiate the vibration effect from that of flow related effects.

Application of nonlinear dynamics concepts to the analysis of self-excited oscillations of a collapsible tube conveying a fluid

Several different types of oscillation were observed during flow through a thick-walled silicone rubber tube when the external pressure was large enough to cause collapse. The Reynolds number was above 4,400. With upstream head, and transmural pressure at the downstream end of the tube, as control variables, control-space diagrams exhibited well-defined regions of low (2–6 Hz), intermediate, and high frequency (over 60 Hz) oscillation, and of small noise-like fluctuations. The data, including aperiodic oscillatory operating points which may indicate the presence of chaos, are analyzed by dynamical systems methods. Transitions between different regions of control space are discussed in terms of topological bifurcation types. Spectral analysis is used to distinguish between quasi-periodic and aperiodic waveforms. Although the dimension of the dynamical system is unknown, phase planes are plotted, both as one transduced signal versus another and as one versus itself delayed. Return maps and Poincare sections are plotted, the latter using three-dimensional phase portraits in which the third coordinate axis was produced by further delay of the one signal. Coordinates for higher-dimensional phase portraits are also defined, using the eigenvectors of covariance matrices constructed from sequences of the recorded data points for one signal. Poincare sections are plotted for such three-dimensional portraits, using the lowest-frequency-component coordinates. Singular value decomposition of “local neighbourhood” matrices is used to define the local dimension of the system in a small region of the high-dimension phase space. Despite the use of these sophisticated techniques, one cannot unequivocally conclude from these data sets that the system is chaotic. The applicability of such methods to complex experiments that yield data which are nonoptimal for these purposes are discussed.

The effects of wall thickness, axial strain and end proximity on the pressure-area relation of collapsible tubes

Segments of silicone rubber tube were suspended between rigid pipes and subjected to slowly varying transmural pressure covering a range from slight distension to collapse with osculation. The local inside cross-sectional area at a chosen axial site was simultaneously measured via catheter by an electrical impedance method. Pressure-area relations were recorded thus at various axial sites, under varying conditions of axial tube wall tension, in tubes of two different wall thickness (0.3 and 0.4 of mean radius). Unsupported tube segment length was also varied by means of an insert device. The relations were used to calculate the variation of wave velocity with area according to Youngs equation. First opposite wall contact during collapse was shown to occur at a smaller fraction of undistended circular cross-sectional area than in the thin-walled tubes investigated previously by others.

The relation between arterial viscoelasticity and wave propagation in the canine femoral artery in vivo.

The influence of arterial dimensions and viscoelasticity on pulse wave propagation has been expressed in many theoretical models of blood flow in arteries, but few experimental tests of these theories in vivo have been reported. The measurements required for such tests include not only the arterial viscoelasticity, diameter, and wall thickness, but also the true propagation coefficients and impedances, for comparison with the values ‘predicted’ by solution of the model equations. We made such measurements in 16 experiments on the femoral artery in nine anesthetized dogs. A two-point pressure and flow technique was used to measure wave propagation, and an ultrasonic micrometer was used to measure vessel diameter as a function of time and pressure. Measured attenuation constants ranged from 0.010 at 1.3 Hz to 0.075 at 12.7 Hz, and were more than twice as large as those predicted by two representative linear models. True phase velocity, which increased from 6.71 m/sec at 1.3 Hz to 10.54 m/sec at 12.7 Hz, agreed closely with the values computed by the Cox model but were lower than those given by the Jager model. The resistive, but not the reactive, component of longitudinal impedance was significantly greater than predicted by the models at all frequencies. The experiments do not identify the source of these discrepancies. The use of linear models to calculate pulsatile blood flow from pressure gradients in relatively small vessels, or to calculate attenuation and characteristic impedance from arterial viscoelasticity in vessels of any size, produces significant errors.

Flux enhancement in crossflow microfiltration using a collapsible-tube pulsation generator

Abstract A dilute suspension of silica particles in water was microfiltered in an unbaffled tubular ceramic membrane. The crossflow was varied periodically at frequencies inducing fluid-dynamic unsteadiness using a novel device for this application, namely a flexible tube compressed to non-circular cross-section by external pressure. This arrangement is unstable and leads to self-sustained oscillations which can be used to add a pulsatile component to a steady flow. With this arrangement, and a time-averaged crossflow sufficient for turbulence in the membrane filter, filtrate flux increases due to pulsation of up to 60% were demonstrated, without systematic optimisation of the pulse parameters. The gain persisted over the entire range of mean transmembrane pressure and crossflow velocity investigated.

A novel textured surface for blood-contact

Blood-contacting surface modifications aimed at reduction of thromboembolic complications have included the texturing of surfaces so as to promote the formation of a stable pseudo-neointima. A technique has been developed whereby a textured surface consisting of regularly spaced micro-fibres was produced on a smooth base plane. Polyurethane vascular patches with and without the textured luminal surface were fabricated and implanted bilaterally in ovine carotid arteries for 1- and 3-week implantation periods (n = 6 per period). One of 6 arteries with textured patches in the 1-week group was occluded. All other arteries were patent. At 1 week, all patent textured patches had adherent thrombus covering the entire patch surface. By 3 weeks, the thrombus had organised to form a stable pseudo-neointima. Non-textured patches at 1 week had only partial surface coverage of thrombus. At 3 weeks, 4 of 6 non-textured patches had significant red thrombus in the lumen. At 3 weeks, there was also evidence of cellular migration from artery onto both textured and non-textured patches. These findings suggest that the major role of the textured surface was as a promoter of a stabilised thrombus base onto which subsequent cellular migration and tissue healing occurred more rapidly than onto a smooth polyurethane surface.

The Origins of Syringomyelia: Numerical Models of Fluid/Structure Interactions in the Spinal Cord

A two-dimensional axi-symmetric numerical model is constructed of the spinal cord, consisting of elastic cord tissue surrounded by aqueous cerebrospinal fluid, in turn surrounded by elastic dura. The geometric and elastic parameters are simplified but of realistic order, compared with existing measurements. A distal reflecting site models scar tissue formed by earlier trauma to the cord, which is commonly associated with syrinx formation. Transients equivalent to both arterial pulsation and percussive coughing are used to excite wave propagation. Propagation is investigated in this model and one with a central canal down the middle of the cord tissue, and in further idealized versions of it, including a model with no cord, one with a rigid cord, one with a rigid dura, and a double-length untapered variant of the rigid-dura model. Analytical predictions for axial and radial wave-speeds in these different situations are compared with, and used to explain, the numerical outcomes. We find that the anatomic circumstances of the spinal cerebrospinal fluid cavity probably do not allow for significant wave steepening phenomena. The results indicate that wave propagation in the real cord is set by the elastic properties of both the cord tissue and the confining dura mater, fat, and bone. The central canal does not influence the wave propagation significantly.

Flow-induced oscillation of collapsed tubes and airway structures

The self-excited oscillation of airway structures and flexible tubes in response to flow is reviewed. The structures range from tiny airways deep in the lung causing wheezing at the end of a forced expiration, to the pursed lips of a brass musical instrument player. Other airway structures that vibrate include the vocal cords (and their avian equivalent, the syrinx) and the soft palate of a snorer. These biological cases are compared with experiments on and theories for the self-excited oscillation of flexible tubes conveying a flow on the laboratory bench, with particular reference to those observations dealing with the situation where the inertia of the tube wall is dominant. In each case an attempt is made to summarise the current state of understanding. Finally, some outstanding challenges are identified.

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.