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

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

Surfactant effects in model airway closure experiments.

The capillary instability that occurs on an annular film lining a tube is studied as a model of airway closure. Small waves in the film can amplify and form a plug across the tube. This dynamical behavior is studied using theoretical models and bench-top experiments. Our model predicts the initial growth rate of the instability and its dependence on surfactant effects. In experiments, an annular film is formed by infusion of water into an initially oil-filled glass capillary tube. The thickness of the oil film varies with the infusion flow rate. The instability growth rate and closure time are measured for a range of film thicknesses. Our theory predicts that a thinner film and higher surfactant activity enhance stability; surfactant can decrease the growth rate to 25% of its surfactant-free value. In experiments, we find that surfactant can decrease the growth rate to 20% and increase the closure time by a factor of 3.8. Functional values of a critical film thickness for closure support the theory that it increases in the presence of surfactant.     

Criteria for mucus transport in the airways by two-phase gas-liquid flow mechanism

The critical conditions for mucous layer transport in the respiratory airways by two-phase gas-liquid flow mechanism were investigated by using 0.5- and 1.0-cm-ID tube models. Several test liquids with rheological properties comparable to human sputum were supplied continuously into the vertically positioned tube models in such a way that the liquid could form a uniform layer while traveling upward through the tube with a continuous upward airflow. The critical airflow rate and critical liquid layer thickness required for the upward transport of the liquids were determined. The critical airflow rate was in the Reynolds number (Re) range of 142-1,132 in the 0.5-cm-ID tube model and 708-2,830 in the 1.0-cm-ID tube model depending on the types of liquids tested. In both models, the critical airflow rate was lower with viscoelastic liquids than with viscous oils. The critical liquid layer thickness ranged from 0.2 to 0.5 mm in the 0.5-cm-ID tube model and 0.8 to 1.4 mm in the 1.0-cm-ID tube model at Re of 2,800. These values decreased rapidly with increasing airflow rate. The critical thickness relative to the tube diameter ranged from 3 to 15% of the respective tube diameter and was lower by approximately 30-50% in the 0.5-cm-ID tube model than in the 1.0-cm-ID tube model over the entire Re range tested. The results indicate that the critical conditions for the mucus transport by two-phase gas-liquid flow mechanism are within the range that can be achieved in patients with bronchial hypersecretions during normal breathing.     

A nonlinear axisymmetric model with fluid-wall interactions for steady viscous flow in stenotic elastic tubes.

Arteries with high-grade stenoses may compress under physiologic conditions due to negative transmural pressure caused by high-velocity flow passing through the stenoses. To quantify the compressive conditions near the stenosis, a nonlinear axisymmetric model with fluid-wall interactions is introduced to simulate the viscous flow in a compliant stenotic tube. The nonlinear elastic properties of the tube (tube law) are measured experimentally and used in the model. The model is solved using ADINA (Automatic Dynamic Incremental Nonlinear Analysis), which is a finite element package capable of solving problems with fluid-structure interactions. Our results indicate that severe stenoses cause critical flow conditions such as negative pressure and high and low shear stresses, which may be related to artery compression, plaque cap rupture, platelet activation, and thrombus formation. The pressure filed near a stenosis has a complex pattern not seen in one-dimensional models. Negative transmural pressure as low as -24 mmHg for a 78 percent stenosis by diameter is observed at the throat of the stenosis for a downstream pressure of 30 mmHg. Maximum shear stress as a high as 1860 dyn/cm2 occurs at the throat of the stenoses, while low shear stress with reversed direction is observed right distal to the stenosis. Compressive stresses are observed inside the tube wall. The maximal principal stress and hoop stress in the 78 percent stenosis are 80 percent higher than that from the 50 percent stenosis used in our simulation. Flow rates under different pressure drop conditions are calculated and compared with experimental measurements and reasonable agreement is found for the prebuckling stage.     

Steady flow and wall compression in stenotic arteries: a three-dimensional thick-wall model with fluid-wall interactions.

Severe stenosis may cause critical flow and wall mechanical conditions related to artery fatigue, artery compression, and plaque rupture, which leads directly to heart attack and stroke. The exact mechanism involved is not well understood. In this paper a nonlinear three-dimensional thick-wall model with fluid-wall interactions is introduced to simulate blood flow in carotid arteries with stenosis and to quantify physiological conditions under which wall compression or even collapse may occur. The mechanical properties of the tube wall were selected to match a thick-wall stenosis model made of PVA hydrogel. The experimentally measured nonlinear stress-strain relationship is implemented in the computational model using an incremental linear elasticity approach. The Navier-Stokes equations are used for the fluid model. An incremental boundary iteration method is used to handle the fluid-wall interactions. Our results indicate that severe stenosis causes considerable compressive stress in the tube wall and critical flow conditions such as negative pressure, high shear stress, and flow separation which may be related to artery compression, plaque cap rupture, platelet activation, and thrombus formation. The stress distribution has a very localized pattern and both maximum tensile stress (five times higher than normal average stress) and maximum compressive stress occur inside the stenotic section. Wall deformation, flow rates, and true severities of the stenosis under different pressure conditions are calculated and compared with experimental measurements and reasonable agreement is found.     

Measurement of oscillatory flow pressure gradient in an elastic artery model

In vitro experiments were conducted to measure the oscillatory flow pressure gradient along an elastic tube in order to assess the recent nonlinear theory of Wang and Tarbell. According to this theory, in an elastic tube with oscillatory flow, the mean (time-averaged) pressure gradient cannot be calculated using Poiseuille's law. The effect of wall motion creates a nonlinear convective acceleration, and an induced mean pressure gradient is required to balance the convective acceleration. The induced mean pressure gradient depends on the diameter variation over a cycle, the pulsatility and unsteadiness of the flow, and the phase difference between the pressure wave form and the flow wave form. The amplitude of the pressure gradient also depends on these parameters and may deviate significantly from Womersley's rigid tube theory. A flow loop was constructed to produce oscillatory flow in an elastic tube. Flow wave forms were measured with an ultrasonic flow probe, and ultrasonic diameter crystals were used to measure wall movement. A special device for pressure drop measurement was constructed using Millar catheter tip transducers to obtain both forward and backward pressure drops that were then averaged. This averaging method eliminated the static error of the pressure transducers. The pressure-flow phase angle was varied by clamping a distal elastic section at various locations. Pressure gradients were obtained for a range of phase angles between −55 ° and +49 °. The mean and amplitude of the measured pressure gradient were compared to theoretical values. Both positive and negative induced mean pressure gradients were measured over the range of phase angles. The measured pressure gradient amplitudes were always lower than predicted by Womersley's rigid tube theory. The experimental means and amplitudes are in good agreement with the elastic tube theoretical values. Thus, the experiments verify the theory of Wang and Tarbell.     

One-dimensional steady continuum model of retraction of pseudopod in leukocytes

A one-dimensional steady state continuum mechanics model of retraction of pseudopod in leukocytes is developed. The retracting pseudopod is assumed to move bodily toward the main cell body, the bulk motion of which can be represented by cytoplasmic flow within a typical stream tube through the leukocyte. The stream tube is approximated by a frictionless tube with prescribed geometry. The passive rheological properties of cytoplasm in the main cell body and in the pseudopod are modeled, respectively, by Maxwell fluid and Hookean solid. The two regions are assumed to be separated by a sharp interface at which actin gel solates and thereby changes its rheological properties as it flows from the pseudopod to the main cell body. The driving mechanism responsible for the active retraction motion is hypothesized to be a spontaneous deformation of the actin gel, analogous but not necessarily equal to the well known actin-myosin interaction. This results in an active contractile stress being developed in the pseudopod as well as in the cell cortex. The transverse traction pulls against the inclined wall of the stream tube and is transduced into an axial stress gradient, which in turn drives the flow. The tension on the tube wall is picked up by the prestressed cortical shell. Governing equations and boundary conditions are derived. A solution is obtained. Sample data are computed. Comparison of the theory with experiments shows that the model is compatible to the observations.     

Mucus clearance by two-phase gas-liquid flow mechanism: asymmetric periodic flow model

Mucus transport by two-phase gas-liquid flow mechanism was investigated with in vitro flow models under asymmetric periodic airflow conditions with nine different liquid solutions with rheological properties similar to human sputum. The flow model was made with 1.0-cm-ID glass tube and positioned either vertically or horizontally. With a constant supply of the test liquids into the model tube (0.5 ml/min), the liquid layer transport speed (LLTS) as well as the mean liquid layer thickness at steady-state condition (hs) was measured in conjunction with various airflow patterns of different expiratory and inspiratory flow rate, breathing frequency (f), and tidal volume (VT). The flow patterns were maintained within the range of normal breathing. In the horizontal tube model, LLTS ranged from 1.14 +/- 0.02 to 3.39 +/- 0.04 cm/min at the peak expiratory flow rate (VEp) of 30-60 l/min. The inspiratory flow rate, as well as f and VT did not affect LLTS. However, LLTS increased with increasing VEp, and at the same VEp LLTS was higher with viscoelastic than with viscous liquid. In the vertical tube model, the upward transport of mucus could not be achieved at VEp lower than 30 l/min particularly with low viscosity and low elasticity fluid. However, at high values of VEp, LLTS was comparable to that in the horizontal tube model with viscoelastic fluid, whereas LLTS of viscous liquid showed 26-40% lower than that in the horizontal tube model. The value of hs was 5-20% of the tube diameter at VEp of 30-60 l/min in both models. These results indicate that effective mucus clearance can be achieved by two-phase gas-liquid flow mechanism in patients with excessive bronchial secretions with biased tidal breathing favoring the expiratory flow and that the clearance can be further promoted by changing rheological properties of mucus.     

Bubble splitting in bifurcating tubes: a model study of cardiovascular gas emboli transport.

The transport of long gas bubbles, suspended in liquid, through symmetric bifurcations, is investigated experimentally and theoretically as a model of cardiovascular gas bubble transport in air embolism and gas embolotherapy. The relevant dimensionless parameters in the models match the corresponding values for arteries and arterioles. The effects of roll angle (the angle the plane of the bifurcation makes with the horizontal), capillary number (a dimensionless indicator of flow), and bubble volume (or length) on the splitting of bubbles as they pass through the bifurcation are examined. Splitting is observed to be more homogenous at higher capillary numbers and lower roll angles. It is shown that, at nonzero roll angles, there is a critical value of the capillary number below which the bubbles do not split and are transported entirely into the upper branch. The value of the critical capillary number increases with roll angle and parent tube diameter. A unique bubble motion is observed at the critical capillary number and for slightly slower flows: the bubble begins to split, the meniscus in the lower branch then moves backward, and finally the entire bubble enters the upper branch. These findings suggest that, in large vessels, emboli tend to be transported upward unless flow is unusually strong but that a more homogeneous distribution of emboli occurs in smaller vessels. This corresponds to previous observations that air emboli tend to lodge in the upper regions of the lungs and suggests that relatively uniform infarction of tumors by gas embolotherapy may be possible.     

A mathematical model for the peristaltic flow of chyme movement in small intestine

A mathematical model based on viscoelastic fluid (fractional Oldroyd-B model) flow is considered for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic flow of chyme is modeled more realistically by assuming that the peristaltic rush wave is a sinusoidal wave, which propagates along the tube. The governing equations are simplified by making the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions of problem are obtained by using homotopy analysis method and convergence of the obtained series solution is properly checked. For the realistic values of the emerging parameters such as fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube, the numerical results for the pressure difference and the frictional force across one wavelength are computed and discussed the roles played by these parameters during the peristaltic flow. On the basis of this study, it is found that the first fractional parameter, relaxation time and Froude number resist the movement of chyme, while, the second fractional parameter, retardation time, Reynolds number and inclination of tube favour the movement of chyme through the small intestine during pumping. It is further revealed that size of trapped bolus reduces with increasing the amplitude ratio whereas it is unaltered with other parameters.     

Microcontinuum model for pulsatile blood flow through a stenosed tube

The effects of polar nature of blood and pulsatility on flow through a stenosed tube have been analysed by assuming blood as a micropolar fluid. Linearized solutions of basic equations are obtained through consecutive applications of finite Hankel and Laplace transforms. The analytical expressions for axial and particle angular velocities, wall shear stress, resistance to flow and apparent viscosity have been obtained. The axial velocity profiles for Newtonian and micropolar fluids have been compared. The interesting observation of this analysis is velocity, in certain parts of cycle, for micropolar fluid is higher than Newtonain fluid. Variation of apparent viscosity eta a with tube radius shows both inverse Fahraeus-Lindqvist and Fahraeus-Lindqvist effects. Finally, the resistance to flow and wall shear stress for normal and diseased blood have been computed and compared.     

A semi-empirical model for flow of blood and other particulate suspensions through narrow tubes

A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. It envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. It has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematocrit. The wall layer thickness is found to decrease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the case of blood may be attributed to the deformability of the erythrocytes.     

Influence of bifurcations on forced oscillations in an airway model.

Forced oscillations is a technique to determine respiratory input impedance from small amplitude sinusoidal pressure excursions introduced at the airway opening. Models used to predict respiratory input impedance typically ignore the direct effect of bifurcations on the flow, and treat airway branches as individual straight tubes placed appropriately in parallel and series. The flow within the individual tubes is assumed equivalent to that which would occur in infinitely long tubes. In this study we examined the influence of bifurcations on impedance for conditions of the forced oscillatory technique. We measured input impedance using forced oscillations in straight tubes and in an anatomically-relevant, four generation physical model of a human airway network. The input impedance measured experimentally compared well to that obtained theoretically using model predictions. The predictive scheme was based on appropriate parallel and series combinations of theoretically computed individual tube impedances, which were computed from solutions to oscillatory flow of a compressible gas in an infinitely long rigid tube. The agreement between experimental measurements and predictions indicates that bifurcations play a relatively minor direct role on the flow impedance for conditions of the forced oscillations technique. These results are explained in terms of the small tidal volumes used, whereby the axial distance traveled by a fluid particle during an oscillation cycle is appreciably smaller than branch segment lengths. Accordingly, only a small fraction of fluid particles travel through the bifurcation region, and the remainder experience an environment approaching flow in an infinite straight tube. The relevance of the study to the prediction of impedances in the human lung during forced oscillations is discussed.     

Wall shear rate measurements in an elastic curved artery model

Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40–56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (36–55% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from −20° to −66°, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer wall decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role.     

Effects of helium-oxygen mixtures on endotracheal tubes: an in vitro study

QUESTION: To determine flow pattern and critical Reynolds numbers in endotracheal tubes submitted to different helium-oxygen mixtures under laboratory conditions. MATERIALS AND METHODS: Flow-pressure relationships were performed for seven endotracheal tubes (rectilinear position, entry length applied) with distal end open to atmosphere (predicted internal diameters: 6-9 mm). Nine helium-oxygen mixtures were tested, with FIHe varying from zero to 0.78 (increment: 10%). Nine flows were tested, with rates varying from 0.25 to 1.60 l s(-1) (increment: 0.15 l s(-1)). Gas flow resistance was calculated, and for each endotracheal tube, a Moody diagram was realised. Flow regime and critical Reynolds numbers were then determined (fully established laminar, nonestablished laminar, smooth turbulent, or rough). RESULTS: Even low concentration of helium in inspiratory mixture reduces endotracheal tubes resistance. Effect is maximal for high flows, small tube and high FIHe. Critical Reynolds numbers are inversely correlated to tube diameter. ANSWER: Under laboratory conditions, flow pattern in endotracheal tubes varies from fully established laminar to rough. Knowledge of the critical Reynolds numbers allows correct application of fluid mechanic formula when studying tube or gaseous mixture effects on respiratory mechanisms.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Chaotic oscillations in a simple collapsible-tube model.

A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.     

Computational model for the transition from peristaltic to pulsatile flow in the embryonic heart tube

Early in development, the heart is a single muscle-wrapped tube without formed valves. Yet survival of the embryo depends on the ability of this tube to pump blood at steadily increasing rates and pressures. Developmental biologists historically have speculated that the heart tube pumps via a peristaltic mechanism, with a wave of contraction propagating from the inflow to the outflow end. Physiological measurements, however, have shown that the flow becomes pulsatile in character quite early in development, before the valves form. Here, we use a computational model for flow though the embryonic heart to explore the pumping mechanism. Results from the model show that endocardial cushions, which are valve primordia arising near the ends of the tube, induce a transition from peristaltic to pulsatile flow. Comparison of numerical results with published experimental data shows reasonably good agreement for various pressure and flow parameters. This study illustrates the interrelationship between form and function in the early embryonic heart.     

Mucus transport in the airways by two-phase gas-liquid flow mechanism: continuous flow model

Mucus transport speed induced by two-phase gas-liquid interaction was measured in the continuous two-phase annular flow tube models, and factors influencing the transport speed were assessed in conjunction with rheological properties of mucus. The flow model was made with 1.0-cm-ID glass tubes and positioned either vertically or horizontally. During a continuous passage of airflow through the model tube, mucus stimulants were supplied into the tube at a rate of 0.5-2.0 ml/min. The advancing speed of the leading edge of the mucous layer and mean mucous layer thickness were then measured. The transport speed in the vertical tube model ranged from 1.1 to 3.1 cm/min with a mucus feed rate of 0.5 ml/min at airflow rates of 0.33-1.17 l/s and increased with increasing airflow rates but decreased rapidly with increasing viscosity of mucus. The transport speed increased almost proportionally with increasing mucus feed rate. Elasticity of mucus did not affect the transport speed itself. However, more elastic mucus caused lower flow resistance and thereby could be transported with a much reduced work load. The transport speed in the horizontal tube model was 5-60% faster than that in the vertical tube model. The mean mucous layer thickness in the vertical tube model was found to be in the range of 0.5-1.5 mm in the experimental conditions used, and decreased rapidly with increasing airflow rate and decreasing viscosity of mucus. From these data the transport speed could be functionally related to airway diameter, mucous layer thickness, and mucus production rate.     

Pulsatile pressure and flow in arterial stenoses simulated in a mathematical model

This paper presents the detailed pulsatile pressure and flow velocity patterns inside an axis symmetric stenosis model with 75% constriction. The pressure and velocities have been calculated by solving the Navier-Stokes equations by the finite element method, the velocity profile in a straight tube caused by a pulsating driving pressure has been calculated first and then used as a boundary condition for the stenosis calculations. The results of the mathematical simulations of the stenosis model have been obtained in terms of velocity vectors, streamlines and isobars at 16 different instances in time, each 15 degrees apart during a cardiac cycle. The calculated velocity field shows that a vortex is developed at the wall distal to the stenosis as the velocity decreases from the peak systolic value. At the site of the vortex, a local pressure minimum is found due to the conversion of pressure to kinetic energy. When the flow is reversed, the reversal occurs first along the wall, thus forcing the vortex toward the the centre of the tube. As the reverse flow velocity increases, a vortex is also developed at the proximal site of the stenosis.