Flutter in collapsible tubes: a theoretical model of wheezes

A mathematical analysis of flow through a flexible channel is examined as a model of flow-induced flutter oscillations that pertain to the production of wheezing breath sounds. The model provides predictions for the critical fluid speed that will initiate flutter waves of the wall, as well as their frequency and wavelength. The mathematical results are separated into linear theory (small oscillations) and nonlinear theory (larger oscillations). Linear theory determines the onset of the flutter, whereas nonlinear theory determines the relationships between the fluid speed and both the wave amplitudes and frequencies. The linear theory predictions correlate well with data taken at the onset of flutter and flow limitation during experiments of airflow in thick-walled collapsible tubes. The nonlinear theory predictions correlate well with data taken as these flows are forced to higher velocities while keeping the flow rate constant. Particular ranges of the parameters are selected to investigate and discuss the applications to airway flows. According to this theory, the mechanism of generation of wheezes is based in the interactions of fluid forces and friction and wall elastic-restoring forces and damping. In particular, a phase delay between the fluid pressure and wall motion is necessary. The wave speed theory of flow limitation is discussed with respect to the specific data and the flutter model.     

Critical pressures required for generation of forced expiratory wheezes

Flow limitation (FL) has recently been shown to be a necessary condition for the generation of forced expiratory wheezes (FEW) in normal subjects. The present study was designed to investigate whether it is also a sufficient condition. To do so we studied the effects of varying expiratory effort on generation of FEW. Six normal subjects exhaled with varying force into an orifice in line with a high-impedance suction pump. Esophageal (Pes), airway opening, and transpulmonary (Ptp) pressures were measured alongside flow rate, lung volume, and tracheal lung sounds. In each subject a certain critical degree of effort had to be attained before FEW were generated. This effort, measured as Pes at the onset of wheezes, varied among the subjects (range -11 to 45 cmH2O). Similarly, a minimal Ptp had to be reached for FEW to evolve (mean +/- SD -34 +/- 12 cmH2O, range -18 to -50 cmH2O). These critical Pes and Ptp values were significantly higher than those required for FL. It was concluded that, in addition to the requirement for FL, sufficient levels of effort and negative Ptp must exist before FEW can be generated. By analogy to experimental and theoretical results from studies on flow-induced oscillations in self-supporting collapsible tubes, it was further concluded that these pressures are required to induce flattening of the intrathoracic airways downstream from the choke point. It is this configurational change that causes air speed to become equal to or exceed the critical gas velocity needed to induce oscillations in soft-walled tubes.     

Unstable equilibrium behaviour in collapsible tubes

Thick-walled silicone rubber tube connected to rigid pipes upstream and downstream was externally pressurised (pe) to cause collapse while aqueous fluid flowed through propelled by a constant upstream head. Three types of equilibrium were found: stable equilibria (steady flow) at high downstream flow resistance R2, self-excited oscillations at low R2, and 'unattainable' (by varying external pressure) or exponentially unstable equilibria at intermediate R2. The self-excited oscillations were highly non-linear and appeared in four, apparently discrete, frequency bands: 2.7 Hz, 3.8-5.0 Hz, 12-16 Hz and 60-63 Hz, suggesting that the possible oscillation modes may be harmonically related. Stable, intermediate 'two-in-every-three-beats' oscillation was also observed, with a repetition frequency in the 3.8-5.0 Hz band. As pe was increased, self-excited oscillations were eventually suppressed, leaving internal fluid pressure varying with no single dominant frequency as a result of turbulent jet dissipation at the downstream rigid pipe connection. Comparison of pressure-wave velocity calculated from the local pressure-area relation for the tube with fluid velocity indicated that supercritical velocities were attained in the course of the self-excited oscillations.     

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.     

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.     

General tube law for collapsible thin and thick-wall tubes

Modeling the complex deformations of cylindrical tubes under external pressure is of interest in engineering and physiological applications. The highly non-linear post-buckling behavior of cross-section of the tube during collapse attracted researchers for years. Major efforts were concentrated on studying the behavior of thin-wall tubes. Unfortunately, the knowledge on post-buckling of thick-wall tubes is still incomplete, although many experimental and several theoretical studies have been performed. In this study we systematically studied the effect of the wall thickness on post-buckling behavior of the tube. For this purpose, we utilized a computational model for evaluation of the real geometry of the deformed cross-sectional area due to negative transmural (internal minus external) pressure. We also developed an experimental method to validate the computational results. Based on the computed cross-sections of tubes with different wall thicknesses, we developed a general tube law that accounts for thin or thick wall tubes and fits the numerical data of computed cross-sectional areas versus transmural pressures.     

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.     

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)     

Experimental analysis of pulsatile flow through elastic collapsible tubes: application to cardiac assist device

This study presents a simulated analysis of Phased Compression Cardiac Assist Device (PCCAD) and evaluation of its applicability as a non-invasive temporary assist for a failing heart. The new technique is based on the chest pump mechanism for blood flow augmentation during external massage by phased compression of the abdominal and thoracic cavities. A semi-closed hydraulic system to simulate the systemic circulation was constructed; the system includes a left ventricle which functions according to the Starling principle and a pneumatic system which controls the pressures applied to the thoracic and abdominal cavities, in complete synchronization with the beating normal or failing heart. The possibility of manipulating the three pumps in series (venous, heart, and arterial) has been checked, and the principal parameters which effect the efficiency of the PCCAD were evaluated. This in vitro analysis shows the high potential of a non-invasive temporary cardiac assist device. It points to the necessary measures one has to take in order to achieve good synchronization and to interfere externally with the augmentation of cardiac output or with the augmentation of root aortic pressure.     

Tensioning the helix: a mechanism for force generation in twining plants

Twining plants use their helical stems to clasp supports and to generate a squeezing force, providing stability against gravity. To elucidate the mechanism that allows force generation, we measured the squeezing forces exerted by the twiner Dioscorea bulbifera while following its growth using time-lapse photography. We show that the development of the squeezing force is accompanied by stiffening of the stem and the expansion of stipules at the leaf base. We use a simple thin rod model to show that despite their small size and sparse distribution, stipules impose a stem deformation sufficient to account for the measured squeezing force. We further demonstrate that tensioning of the stem helix, although counter-intuitive, is the most effective mechanism for generating large squeezing forces in twining plants. Our observations and model point to a general mechanism for the generation of the twining force: a modest radial stem expansion during primary growth, or the growth of lateral structures such as leaf bases, causes a delayed stem tensioning that creates the squeezing forces necessary for twining plants to ascend their supports. Our study thus provides the long-sought answer to the question of how twining plants ascend smooth supports without the use of adhesive or hook-like structures.     

Regulated transport as a mechanism for pattern generation: Capabilities for phyllotaxis and beyond

Large-scale pattern formation is a frequently occurring phenomenon in biological organisms, and several local interaction rules for generating such patterns have been suggested. A mechanism driven by feedback between the plant hormone auxin and its polarly localized transport mediator PINFORMED1 has been proposed as a model for phyllotactic patterns in plants. It has been shown to agree with current biological experiments at a molecular level as well as with respect to the resulting patterns. We present a thorough investigation of variants of models based on auxin-regulated polarized transport and use analytical and numerical tools to derive requirements for these models to drive spontaneous pattern formation. We find that auxin concentrations in neighboring cells can feed back either on exocytosis or endocytosis and still produce patterns. In agreement with mutant experiments, the active cellular efflux is shown to be more important for pattern capabilities as compared to active influx. We also find that the feedback must originate from neighboring cells rather than from neighboring walls and that intracellular competition for the transport mediator is required for patterning. The importance of model parameters is investigated, especially regarding robustness to perturbations of experimentally estimated parameter values. Finally, the regulated transport mechanism is shown to be able to generate Turing patterns of various types.     

Forced expiratory wheezes are a manifestation of airway flow limitation

To study the mechanism of generation of respiratory wheezes we examined the relationships between forced expiratory wheezes (FEW) and flow limitation in the lung. Tracheal lung sounds were measured in six healthy subjects during forced expiration through a flow-limiting valve in series with a high-impedance suction pump. Mouth pressure, esophageal pressure, transpulmonary pressure (Ptp), flow (V), and volume were also measured. For any flow rate, V was constant until the subject became flow limited. The onset of flow limitation was documented by a small change in V and a sudden change in Ptp, which was previously found by Olafsson and Hyatt to correspond to the beginning of the flow plateau of the isovolume pressure-flow curve (J. Clin. Invest. 48: 564-573, 1969). FEW started 107 +/- 45 ml (SD) after the onset of flow limitation. Additional 79 +/- 65 ml were exhaled between the onset of FEW to the final sharp drop in V. The frequency spectra of FEW were the same as those of respiratory wheezes found in obstructive airway diseases. Administration of inhaled bronchodilator (isoproterenol) did not eliminate the FEW, nor did it change their relationship to flow limitation. The sequence of events around the onset of FEW, and the tight correlation with the onset of flow limitation correspond well to recent experimental observations on the onset of flutter in collapsible, thick-walled latex tubes.     

A model for the mechanism of tip extension in pollen tubes

Three main mechanisms are proposed to account for the tip growth of pollen tubes. (1) The tip region is supported against the internal osmotic pressure of the cell by a fibrillar network, composed mainly of microfilaments, that is stabilized by calcium ions. Tip extension is promoted by a lowering of the local cytoplasmic calcium ion concentration, through uptake by the mitochondria and/or endoplasmic reticulum, which leads to a weakening of the fibrillar network. (2) Vesicles, derived from dictyosomes in the main body of the tube, fuse with the apical plasma membrane, providing new membrane and further carbohydrate for the wall. The rate of fusion is proportional to the rate of diffusion of calcium ion across the plasma membrane at the tip. (3) The callose lining present in the pollen tube wall, except at the tip, renders the wall impermeable and restricts entry of calcium ions to the apical plasma membrane. This restriction limits the rate of vesicle fusion, and tube growth, to the tip.This model is discussed in the light of previous observations on the growth and structure of pollen tubes under normal and experimental conditions.     

Hemoglobin: A mechanism for the generation of hydroxyl radicals

Oxyhemoglobin (HbO₂) reduces Fe(III) NTA aerobically to become methemoglobin (metHb) and Fe(II)NTA. These conditions are favorable for the generation via Fenton chemistry of the hydroxyl radical that was measured by HPLC using salicylate as a probe. The levels of hydroxyl radicals generated are a function of both the percent metHb formed and the chemical nature of the buffer. The rates of formation of both metHb and hydroxyl radicals were dependent upon the concentration of Fe(III)NTA. Of the buffers tested, HEPES was the most effective scavenger of hydroxyl radicals while the other buffers scavenged in the order: HEPES > Tris > MOPS > NaCl ≈ unbuffered. The addition of catalase to remove H₂0₂ or bathophenanthroline to chelate Fe(II) inhibited virtually all hydroxyl radical formation. Carbonyl formation from free radical oxidation of amino acids was found to be 0.1 mol/mol of hemoglobin. These experiments demonstrate the ability of hemoglobin to participate directly in the generation of hydroxyl radicals mediated by redox metals, and provide insight into potential oxidative damage from metals released into the blood during some pathologic disorders including iron overload.     

Free radical generation by early glycation products: a mechanism for accelerated atherogenesis in diabetes

Non-enzymatic glycation of reactive amino groups in model proteins increased the rate of free radical production at physiologic pH by nearly fifty-fold over non-glycated protein. Superoxide generation was confirmed by electron paramagnetic resonance measurements with the spin-trap phenyl-t-butyl-nitrone. Both Schiff base and Amadori glycation products were found to generate free radicals in a ratio of 1:1.5. Free radicals generated by glycated protein increased peroxidation of membranes of linoleic/arachidonic acid vesicles nearly 2-fold over control, suggesting that the increased glycation of proteins in diabetes may accelerate vascular wall lipid oxidative modification.     

Flow limitation and wheezes in a constant flow and volume lung preparation

To facilitate the study of respiratory wheezes in an animal lung model, an isovolume, constant-flow excised dog lung preparation was developed. Dog lungs were inflated to 26 +/- 4 cmH2O and coated with layers of epoxy glue and polyester compound. A rigid shell 2 mm thick was obtained around the entire pleural surface and the extra-pulmonary airways. The adhesive forces between the pleura and the shell were strong enough to hold the lung distended after the inflation pressure was removed. Holes 2 mm diam were drilled through the shell over one of the lung lobes in an array, 4 cm across. The holes penetrated the pleural surface, so that constant flow could be maintained in the expiratory direction by activating a suction pump connected to the trachea. Downstream suction pressure and flow rate were measured with a mercury manometer and a rotameter, respectively. Sounds were recorded by a small (0.6 cm OD) microphone inserted into the trachea. When suction pressure was increased, flow initially increased to 31 +/- 3 l/min. Further increase of suction pressure caused only very slight additional increase in flow (i.e., flow limitation). During this plateau of flow, a pure tone was generated with acoustic properties similar to respiratory wheezes. Both the flow plateau and the wheezing sounds could be eliminated by freezing the lungs. It is concluded that wheezing sounds were associated with flow limitation in this preparation. It is suggested that the stable acoustic properties obtained by this preparation may become useful in the analysis of mechanisms of wheezing lung sounds generation.     

Unraveling the mechanism for respiratory rhythm generation

Breathing is generated by a neuronal network located within the caudal brainstem. One area of particular significance for respiratory rhythm generation is the pre-B?tzinger (preBotC) complex in the ventrolateral medulla. An important step towards understanding the cellular and network basis by which neurons within this region generate the respiratory rhythm was made in a recent study by Koshiya and Smith.(1) Using simultaneous image analysis and electrophysiological techniques these authors identified a discrete population of synaptically-coupled pacemaker neurons within the preBotC. They postulated that these neurons constitute the minimal essential network component (kernel) for generating the respiratory rhythm. BioEssays 22:6-9, 2000.