Flow In Collapsible Tubes Research Articles

Experimental analysis of aspect ratio in iliac vein stenosis

BackgroundVeins are thin-walled tubes. Their lumen is roughly circular with an aspect ratio close to 1:1 under physiologic pressures. When they collapse owing to decreased internal pressure or external compression, the aspect ratio changes. The vertical diameter is usually diminished more than the transverse, with a considerable decrease in the lumen area. The recent emergence of stent correction of many venous compression syndromes, particularly iliac vein stenosis, has brought attention to the importance of the aspect ratio, quite apart from an overall decrease in caliber. The iliac vein pressure is influenced not only by stenosis, but also intra-abdominal pressure, right atrial pressure, and collaterals. We investigated the impact of aspect ratio in an experimental model incorporating these factors. MethodsInflow was provided from a header tank at 25 mm Hg pressure into a Penrose tubing enclosed in a polyethylene cylinder pressurized (Starling pressure) to simulate intra-abdominal pressures of 5 and 10 mm Hg. The Penrose drained into an outflow tank with a pressure of 7 mm Hg, simulating right atrial pressure. Stenosis was simulated with a series of three-dimensional, printed plastic nozzles with caliber areas of 50, 100, and 200 mm2 and varying in aspect ratios of 1:1 to 1:4. The flow and pressure in this system was monitored with the use of overflow collaterals in some experiments. ResultsFree flow from the header tank through the Penrose (zero Starling pressure) with a 200 mm2 circle nozzle into the outflow tank with zero pressure resulted in flow pressure of approximately 1.5 mm Hg. Using nozzles of a smaller caliber or an increased aspect ratio resulted in an increase of flow pressures of up to approximately 3.7 mm Hg. Flow into an outflow tank of 7 mm Hg simulating right atrial pressure further increased flow pressures by approximately 7 mm Hg. The addition of Starling pressures of 5 and 10 mm Hg simulating abdominal pressure increased flow pressure even further to the 10 to 17 mm Hg range. When the Starling pressure was dominant, the additional contribution of nozzle caliber stenosis or aspect ratio reduction to the overall flow pressure ranged from 2 to 6 mm Hg. Collateral overflow varied inversely with collateral resistance. Some experiments yielded an anomalous flow/pressure phenomena known to occur in collapsible tube flows. ConclusionsA decrease in the caliber or the aspect ratio of iliac vein stenosis was among several other factors that generate peripheral venous hypertension in an experimental model. Increased intra-abdominal pressure is a major influence that amplifies the pressure effects of aspect ratio or caliber reduction.

Starling resistors, autoregulation of cerebral perfusion and the pathogenesis of idiopathic intracranial hypertension.

Two critical functions for the control of intracranial fluids dynamics are carried on the venous side of the perfusion circuit: the first is the avoidance of cortical veins collapse during the physiological increases of cerebrospinal fluid (CSF) pressure in which they are immersed. The second, is the generation of an abrupt venous pressure drop at the confluence of the cortical veins with the dural sinuses that is required to allow a CSF outflow rate balanced with its production. There is evidence that both of these effects are ensured by a Starling resistor mechanism (a fluid dynamic construct that governs the flow in collapsible tubes exposed to variable external pressure) acting at the confluence of cortical veins in the dural sinus. This implies that, in normal circumstances of perfusion balance, a certain degree of venous collapse physiologically occurs at the distal end of the cortical vein. This is passively modulated by the transmural pressure of the venous wall (i.e. the difference between internal blood pressure and external CSF pressure). The mechanism provides that the blood pressure of the cortical vein upstream the collapsed segment is dynamically maintained a few mmHg higher than the CSF pressure, so as to prevent their collapse during the large physiological fluctuations of the intracranial pressure. Moreover, the partial collapse of the vein confluence also generates a sharp pressure drop of the blood entering into the sinus. The CSF is drained in dural sinus through arachnoid villi proportionally to its pressure gradient with the sinus blood. The venous pressure drop between cortical veins and dural sinus is therefore needed to ensure that the CSF can leave the cranio-spinal space with the same speed with which it is produced, without having to reach a too high pressure, which would compress the cortical veins. Notably, the mechanism requires that the walls of the dural sinuses are rigid enough to avoid the collapse under the external cerebrospinal fluid pressure, and predicts that in the presence of excessively flexible dural sinuses, the system admits a second point of balance between cerebral fluid pressure and dural sinus pressure, at higher values. The second balance state is due to the triggering of a self-limiting venous collapse feedback loop between the CSF pressure, that compresses the sinus, and the subsequent increase of the dural sinus pressure, that further raises the intracranial pressure. The loop may stabilize only when the maximum stretching allowed by the venous wall is reached. Then, a new relatively stable and self-sustaining balance state is achieved, at the price of a higher CSF and dural sinus pressure values. We propose that this model is crucially involved in Idiopatic Intracranial Hypertension pathogenesis with and without papilledema, a condition that could be described as a pathological new balance state, relatively stable, between intracranial and dural venous pressure, at higher absolute values.

Computer-aided medicine and surgery

Marc Thiriet' was educated at Medicine Faculty of Lille and University Pierre and Marie Curie ([UPMC] MD), and then at Technology University of Compiègne (3 rd cycle Doctorate in Biomechanics), and Physics College of University Denis Diderot (Accreditation to Supervise Research). He was assistant physician in the lung disease department of Pontoise hospital. He is currently a member of the INRIA-UPMC-CNRS team REO in Laboratory Jacques-Louis Lions (applied math.) of UPMC. He worked in flows in collapsible tubes applied to airways and veins, 3D unsteady developing laminar flows in bend and branchings, both experimentally and numerically, as well as in models derived from 3D reconstruction of human anatomy. He is now involved in mathematical modeling of biological processes. Marc Thiriet is the author of Biology and Mechanics of Blood Flows (2˜Vols.) and the book series “Biomathematical and Biomechanical Modeling of the Circulatory and Ventilatory Systems'' (7˜published books). He was the associate editor for the medical encyclopedia “Pan Vascular Medicine”. He also wrote the chapter 30. Biofluid Flow and Heat Transfer of Handbook of Fluid Dynamics. He is the President of the French Committee for Intensive Computation in Biology and Medicine.

Steady analysis of transcritical flows in collapsible tubes with discontinuous mechanical properties: implications for arteries and veins

AbstractWe study the conditions under which discontinuous mechanical properties of a collapsible tube can induce transcritical flows, i.e. the transition through the critical state where the speed index (analogous to the Mach or the Froude numbers for compressible and free surface flows, respectively) is one. Such a critical transition may strongly modify the flow properties, cause a significant reduction in the cross-sectional area of the tube, and limit the flow. General relationships are obtained for a short segment using a one-dimensional model under steady flow conditions. Marginal curves delimiting the transcritical regions are identified in terms of the speed index and the cross-sectional area ratio. Since there are many examples of such flows in physiology and medicine, we also analyse the specific case of prosthesis (graft or stent) implantation in blood vessels. We then compute transcritical conditions for the case of stiffness and reference area variations, considering a collapsible tube characterized by physiological parameters representative of both arteries and veins. The results suggest that variations in mechanical properties may induce transcritical flow in veins but is unrealistic in arteries.

Flow in Collapsible Tubes with Discontinuous Mechanical Properties: Mathematical Model and Exact Solutions

AbstractWe formulate a one-dimensional time-dependent non-linear mathematical model for some types of physiological fluid flow in collapsible tubes with discontinuous material properties. The resulting 6 × 6 hyperbolic system is analysed and the associated Riemann problem is solved exactly. Although the solution algorithm deals with idealised cases, it is nonetheless uniquely well-suited for assessing the performance of numerical methods intended for simulating more general situations. Moreover, our model may be a useful starting point for numerical calculations of realistic flows involving rapid and discontinuous material property variations. One important example in mind is the simulation of blood flow in medium-to-large veins in humans. Finally, we also discuss some peculiarities of the model regarding the loss of strict hyperbolicity and uniqueness. In particular we show an example in which the solution of the Riemann problem is non unique.

Three-dimensional non-linear buckling of thick-walled elastic tubes under pressure

This paper is concerned with numerical simulations of three-dimensional finite deformation of a thick-walled circular elastic tube subject to internal or external pressure and zero displacement on its ends. We formulate the system of equations that can accommodate large strain and displacement for the incompressible isotropic neo-Hookean material. The fully non-linear governing equations are solved using the C++ based object-oriented finite element library libMesh. A Lagrangian mesh is used to discretize the governing equations, and a weighted residual Galerkin method and Newton iteration solver are used in the numerical scheme. To overcome the sensitivity of the fully non-linear system to small changes in the iterations, the analytical form of the Jacobian matrix is derived, which ensures a fast and better numerical convergence than using a numerically approximated Jacobian matrix.Results are presented for different parameters in terms of wall thickness/radius ratio, and length/radius ratio, as well as internal/external pressure. Validation of the model is achieved by the excellent agreement with the results obtained using the commercial package Abaqus. Comparison is also made with the previous work on axisymmetric version of the same system (Zhu et al., 2008 [34]; Zhu et al. 2010 [43]), and interesting fully three-dimensional post-buckling deformations are highlighted. The success of the current approach paves the way for fluid–structure interaction studies with potential application to collapsible tube flows and modeling of complex physiological systems.

A Rational Derivation of a Tube Law from Shell Theory

We consider small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross section. The tube is subject to an axial pre-stress, and the deformations result from an applied transmural pressure. An approximate tube law (linking the transmural pressure with the cross-sectional area and its axial derivatives) is derived from shell theory in the distinguished asymptotic limit in which the tube's behaviour is dominated by the restoring forces from the axial pre-stress and azimuthal bending. This is possible because the deformations of the tube induced by both the transmural pressure and the axial forces can be described, to very good approximation, by a single azimuthal mode of deformation of axially varying amplitude. The resulting tube law is compared with numerical solutions of the full shell equations and good agreement is found (provided the tube is sufficiently long and the wall not too thin so that in-plane shearing is negligible). We discuss the applications of our results to the modelling of flow in collapsible tubes.

Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes

We present a theoretical description of flow-induced self-excited oscillations in the Starling resistor—a pre-stretched thin-walled elastic tube that is mounted on two rigid tubes and enclosed in a pressure chamber. Assuming that the flow through the elastic tube is driven by imposing the flow rate at the downstream end, we study the development of small-amplitude long-wavelength high-frequency oscillations, combining the results of two previous studies in which we analysed the fluid and solid mechanics of the problem in isolation. We derive a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillations, and determine the slow growth or decay of the normal modes by considering the system’s energy budget. We compare the theoretical predictions for the mode shapes, frequencies and growth rates with the results of direct numerical simulations, based on the solution of the three-dimensional Navier–Stokes equations, coupled to the equations of shell theory, and find good agreement between the results. Our results provide the first asymptotic predictions for the onset of self-excited oscillations in three-dimensional collapsible tube flows.

The energetics of flow through a rapidly oscillating tube. Part 1. General theory

We examine the effect of prescribed wall-driven oscillations of a flexible tube of arbitrary cross-section, through which a flow is driven by prescribing either a steady flux at the downstream end or a steady pressure difference between the ends. A large-Womersley-number large-Strouhal-number regime is considered, in which the oscillations of the wall are small in amplitude, but sufficiently rapid to ensure viscous effects are confined to a thin boundary layer. We derive asymptotic expressions for the flow fields and evaluate the energy budget. A general result for the conditions under which there is zero net energy transfer from the flow to the wall is provided. This is presented as a critical inverse Strouhal number (a dimensionless measure of the background flow rate) which is expressed only in terms of the tube geometry, the fluid properties and the profile of the prescribed wall oscillations. Our results identify an essential component of a fundamental mechanism for self-excited oscillations in three-dimensional collapsible tube flows, and enable us to assess how geometric and flow properties affect the stability of the system.

Accurate modelling of unsteady flows in collapsible tubes

The context of this paper is the development of a general and efficient numerical haemodynamic tool to help clinicians and researchers in understanding of physiological flow phenomena. We propose an accurate one-dimensional Runge–Kutta discontinuous Galerkin (RK-DG) method coupled with lumped parameter models for the boundary conditions. The suggested model has already been successfully applied to haemodynamics in arteries and is now extended for the flow in collapsible tubes such as veins. The main difference with cardiovascular simulations is that the flow may become supercritical and elastic jumps may appear with the numerical consequence that scheme may not remain monotone if no limiting procedure is introduced. We show that our second-order RK-DG method equipped with an approximate Roe's Riemann solver and a slope-limiting procedure allows us to capture elastic jumps accurately. Moreover, this paper demonstrates that the complex physics associated with such flows is more accurately modelled than with traditional methods such as finite difference methods or finite volumes. We present various benchmark problems that show the flexibility and applicability of the numerical method. Our solutions are compared with analytical solutions when they are available and with solutions obtained using other numerical methods. Finally, to illustrate the clinical interest, we study the emptying process in a calf vein squeezed by contracting skeletal muscle in a normal and pathological subject. We compare our results with experimental simulations and discuss the sensitivity to parameters of our model.

Flow in collapsible tubes or over compliant surfaces for biomedical applications

AbstractNine papers in the area of ‘Flow in collapsible tubes or over compliant surfaces for biomedical applications’ are collated and presented in this special issue. Copyright © 2009 John Wiley & Sons, Ltd.

A GROUP CLASSIFICATION OF A SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS MODELING FLOW IN COLLAPSIBLE TUBES

The purpose of this work is to perform group classification of a coupled system of partial differential equations (PDEs) modeling a flow in collapsible tubes. This system of PDEs contains unknown functions of the dependent variables whose forms are specified via the classification with respect to subalgebras of real three and four-dimensional Lie algebras.

How rapidly oscillating collapsible tubes extract energy from a viscous mean flow

We present a combined theoretical and computational analysis of three-dimensional unsteady finite-Reynolds-number flows in collapsible tubes whose walls perform prescribed high-frequency oscillations which resemble those typically observed in experiments with a Starling resistor. Following an analysis of the flow fields, we investigate the system's overall energy budget and establish the critical Reynolds number,Recrit, at which the wall begins to extract energy from the flow. We conjecture thatRecritcorresponds to the Reynolds number beyond which collapsible tubes are capable of performing sustained self-excited oscillations. Our computations suggest a simple functional relationship betweenRecritand the system parameters, and we present a scaling argument to explain this observation. Finally, we demonstrate that, within the framework of the instability mechanism analysed here, self-excited oscillations of collapsible tubes are much more likely to develop from steady-state configurations in which the tube is buckled non-axisymmetrically, rather than from axisymmetric steady states, which is in agreement with experimental observations.

Call for Papers: Special Issue on ‘Flow in Collapsible Tubes or Over Compliant Surfaces for Biomedical Applications’ Communications in Numerical Methods in Engineering (CNM)

Call for Papers: Special Issue on ‘<i>Flow in Collapsible Tubes or Over Compliant Surfaces for Biomedical Applications</i>’ <i>Communications in Numerical Methods in Engineering</i> (<i>CNM</i>)

Call for Papers: Special Issue on ‘Flow in collapsible tube or over compliant surface for biomedical applications’ Communications in Numerical Methods in Engineering (CNM)

Call for Papers: Special Issue on ‘<i>Flow in collapsible tube or over compliant surface for biomedical applications</i>’ <i>Communications in Numerical Methods in Engineering</i> (<i>CNM</i>)

Call for Papers: Special Issue on ‘Flow in collapsible tubes or over compliant surfaces for biomedical applications’ Communications in Numerical Methods in Engineering

Call for Papers: Special Issue on ‘<i>Flow in collapsible tubes or over compliant surfaces for biomedical applications</i>’ <i>Communications in Numerical Methods in Engineering</i>

Call for Papers: Special Issue on ‘Flow in Collapsible Tubes or Over Compliant Surfaces for Biomedical Applications’ Communications in Numerical Methods in Engineering (CNM)

Call for Papers: Special Issue on ‘<i>Flow in Collapsible Tubes or Over Compliant Surfaces for Biomedical Applications</i>’ <i>Communications in Numerical Methods in Engineering (CNM)</i>

A One Dimensional Model to Predict Steady Flow Through a Collapsible Tube

Flow through tubes that can collapse during normal operation characterizes virtually every bodily fluid-carrying vessel. A MATLAB program was developed to solve a one-dimensional numerical model for steady state collapsible tube flow. The numerical results were then compared with experimental findings of Bertram [10]. The model included a novel polynomial-defined relationship between the Reynolds number and skin friction coefficient. The results are in strong agreement with experiment (r > 0.95). Further correlation analysis showed significance at the 0.025 level [r 14 = 0.986; r 17 = 0.975; r 15 = 0.990; r 15 = 0.956; r 19 = 0.943; p ≪ 0.001]. An average error of less than ten percent existed between computations and experiment.

Contribution of Inertia on Venous Flow in the Lower Limb during Stationary Gait

Blood flows toward the heart through collapsible vessels, the veins. The equations of flow in collapsible tubes in motion show a strong dependence on body forces resulting from gravity and acceleration. This paper analyzes the contribution of body forces to venous blood flow during walking on level ground. It combines the biomechanics of gait and theory of collapsible tubes to point out that body forces due to gravity and limb acceleration cannot be overlooked when considering the determinants of venous blood flow during locomotion. The study involved the development of a kinematic model of the limb as a multi-pendulum arrangement in which the limb segments undergo angular displacements. Angular velocities and accelerations were determined and the body forces were calculated during various phases of the gait cycle. A vascular model of the leg's major venous system was also constructed, and the accelerations due to body and gravity forces were calculated in specific venous segments, using the data from the kinematic model. The results showed there were large, fast variations in the axial component (Gx–Mx) of the body forces in veins between the hip and the ankle. Acceleration peaks down to –2G were obtained at normal locomotion. At fast locomotion, a distal vein in the shank displayed values of (Gx–Mx)/G equal to –3.2. Given the down-to-up orientation of the x-axis, the axial component Mx was usually positive in the axial veins, and Mx could shift from positive to negative during the gait cycle in the popliteal vein and the dorsal venous arch.

Mathematical study of the role of non-linear venous compliance in the cranial volume-pressure test.

The role of the cerebral venous bed in the cranial volume-pressure test was examined by means of a mathematical model. The cerebral vascular bed was represented by a single arterial compartment and two venous compartments in series. The lumped-parameter formulation for the vascular compartments was derived from a one-dimensional theory of flow in collapsible tubes. It was assumed in the model that the cranial volume is constant. The results show that most of the additional volume of cerebrospinal fluid (deltaV(CSF)) was accommodated by collapse of the cerebral venous bed. This profoundly altered the venous haemodynamics and was reflected in the cranial pressure P(CSF). The cranial volume-pressure curve obtained from the model was consistent with experimental data; the curve was flat for 0 < or = deltaV(CSF) < or = 20 ml and 35 < or = deltaV(CSF) < or = 40 ml, and steep for 20 < or = deltaV(CSF) < or = 35 ml and deltaV(CSF) > or = 40 ml. For deltaV(CSF) > 25 ml and P(CSF) > 5.3 kPa (40 mmHg), cerebral blood flow dropped. When P(CSF) was greater than the mean arterial pressure, all the veins collapsed. The conclusion of the study was that the shape of the cranial volume-pressure curve can be explained by changes in the venous bed caused by various degrees of collapse and/or distension.