Blood flow distribution within the airway wall.

Altered perfusion of the bronchial mucosal plexus relative to the adventitial plexus may contribute to geometric changes in the airway wall and lumen. We studied bronchial perfusion distribution in sheep by using fluorescent microspheres at baseline and during intrabronchial artery challenge with methacholine chloride (MCh; n = 7). Additionally, we measured airway resistance (Raw) during MCh with control or increased perfusion (n = 9). Raw with MCh was significantly greater for high than control flow. Microspheres in histological sections lodged predominantly in the mucosa (60%), and this was not altered by MCh. However, more microspheres lodged in airways >1-mm in diameter during MCh and increased perfusion than MCh and control flow. In airways < or =1 mm in diameter, fewer microspheres lodged during control than increased flow. If the number of microspheres represents regional agonist access to airway smooth muscle, then the differences observed in Raw can be explained by the distribution of agonist. During challenge, there was greater MCh delivery to larger airways during increased flow and less delivery to smaller airways during control flow. The results demonstrate the effects of axial perfusion distribution on Raw.     

Viscous energy dissipation for steady flow in models of arterial bifurcations

In this paper the viscous energy dissipation in a series of glass model symmetric bifurcations—typical of human vascular branching—was studied. The bifurcations studied have included angles of 75°, 100° and 125° and total output/input area ratios of 0.73, 1.07 and 1.33. The flowrate range studied corresponded to parent tube Reynolds numbers in the range 100–1000.Pressure and flow measurements were made using a highly sensitive variable reluctance pressure transducer and electromagnetic flowmeter. The measurements were made in such a way as to indicate the net effect of the bifurcation.It was found that a dimensionless form of the viscous dissipation had a constant value for all the geometries investigated up to a Reynolds number of 800. Above this, for an angle of 125°, the measured energy dissipation increased. An analytical model based on entry flow principles showed good agreement with measured values except at an area ratio of 0.73. The reasons for this are discussed with the physiological implications of the results.     

A numerical study of the shape of the surface separating flow into branches in microvascular bifurcations.

The shape of the separating surface formed by the streamlines entering the branches of microvascular bifurcations plays a major role in determining the distribution of red blood cells and other blood constituents downstream from the bifurcation. Using the finite element method, we determined the shape of the surface through numerical solution of three dimensional Navier-Stokes equations for fluid flow at low Reynolds numbers in a T-type bifurcation of circular tubes. Calculations were done for a wide range of daughter branch to parent vessel diameter ratios and flow ratios. The effect of Reynolds number was also studied. Our numerical results are in good agreement with previously reported experimental data of Rong and Carr (Microvascular Research, Vol. 39, pp. 186-202, 1990). The numerical results of this study will be used to predict the concentration of blood constituents downstream from microvascular bifurcations providing that the inlet concentration profile is known.     

Endothelial cell morphometry of atherosclerotic lesions and flow profiles at aortic bifurcations in cholesterol fed rabbits.

Observations on shapes of endothelial cells both in sudanophilic and nonsudanophilic regions at bifurcations of the brachiocephalic (BC) and left subclavian (SA) arteries in hyperlipidemic rabbits were performed under a SEM. The stagnation point of flow and leading edges of flow dividers were nonsudanophilic and covered by round and long fusiform endothelial cells, respectively. The hips of flow dividers of both branchings, proven to be relatively low shear stress regions, by movement of microspheres in steady flow, were sudanophilic and covered by ellipsoidal cells. Similar studies were carried out in normolipidemic rabbits. It might be concluded that lipid deposition in hyperlipidemic rabbits occurs in relatively low shear stress regions, where endothelial cells are functionally activated, rather than in laminar high shear stress regions at the flow divider.     

The effect of unsteadiness on the flow through stenoses and bifurcations

This paper is concerned with the influence of a stenosis or a bifurcation on the flow through a tube. In particular the effect of unsteadiness is investigated using simple pulsatile and physiological type flows (Fig. 1). The experimental investigations reported herein are concerned with velocity measurements and flow visualizations. (see formula in text) These measurements, performed in a 60 degrees bifurcation, have permitted the reconstruction of the three-dimensional velocity profiles. The importance of the secondary flow in the branching is analyzed for various values of the flow parameters. Results of tests show a strong influence of unsteadiness on flow characteristics and then on hemodynamic factors. One conclusion is the following: if hemodynamic factors play an important role in the problems of atherosclerosis, then, for macrocirculation studies, it is necessary to take into account unsteadiness and, in particular, the actual shape of the flow-time forcing function.     

Noninvasive determination of upper airway resistance and flow limitation.

We have shown that a polynomial equation, FP = AP3 + BP2 + CP + D, where F is flow and P is pressure, can accurately determine the presence of inspiratory flow limitation (IFL). This equation requires the invasive measurement of supraglottic pressure. We hypothesized that a modification of the equation that substitutes time for pressure would be accurate for the detection of IFL and allow for the noninvasive measurement of upper airway resistance. The modified equation is Ft = At3 + Bt2 + Ct + D, where F is flow and t is time from the onset of inspiration. To test our hypotheses, data analysis was performed as follows on 440 randomly chosen breaths from 18 subjects. First, we performed linear regression and determined that there is a linear relationship between pressure and time in the upper airway (R2 0.96 +/- 0.05, slope 0.96 +/- 0.06), indicating that time can be a surrogate for pressure. Second, we performed curve fitting and found that polynomial equation accurately predicts the relationship between flow and time in the upper airway (R2 0.93 +/- 0.12, error fit 0.02 +/- 0.08). Third, we performed a sensitivity-specificity analysis comparing the mathematical determination of IFL to manual determination using a pressure-flow loop. Mathematical determination had both high sensitivity (96%) and specificity (99%). Fourth, we calculated the upper airway resistance using the polynomial equation and compared the measurement to the manually determined upper airway resistance (also from a pressure-flow loop) using Bland-Altman analysis. Mean difference between calculated and measured upper airway resistance was 0.0 cmH2O x l(-1) x s(-1) (95% confidence interval -0.2, 0.2) with upper and lower limits of agreement of 2.8 cmH2O x l(-1) x s(-1) and -2.8 cmH2O x l(-1) x s(-1). We conclude that a polynomial equation can be used to model the flow-time relationship, allowing for the objective and accurate determination of upper airway resistance and the presence of IFL.     

Breaking symmetry in non-planar bifurcations: distribution of flow and wall shear stress

Non-planarity in blood vessels is known to influence arterial flows and wall shear stress. To gain insight, computational fluid dynamics (CFD) has been used to investigate effects of curvature and out-of-plane geometry on the distribution of fluid flows and wall shear stresses in a hypothetical non-planar bifurcation. Three-dimensional Navier-Stokes equations for a steady state Newtonian fluid were solved numerically using a finite element method. Non-planarity in one of the two daughter vessels is found to deflect flow from the inner wall of the vessel to the outer wall and to cause changes in the distribution of wall shear stresses. Results from this study agree to experimental observations and CFD simulations in the literature, and support the view that non-planarity in blood vessels is a factor with important haemodynamic significance and may play a key role in vascular biology and pathophysiology.     

A simplified noninvasive method to measure airway blood flow in humans.

Our laboratory has previously developed and validated a noninvasive soluble gas uptake method to measure airway blood flow (Qaw) in humans (Onorato DJ, Demirozu MC, Breitenbücher A, Atkins ND, Chediak AD, and Wanner A. Am J Respir Crit Care Med 149: 1132-1137, 1994; Scuri M, McCaskill V, Chediak AD, Abraham WM, and Wanner A. J Appl Physiol 79: 1386-1390, 1995). The method has the disadvantage of requiring eight breath-hold maneuvers for a single Qaw measurement, a complicated data analysis, and the inhalation of a potentially explosive gas mixture containing dimethylether (DME) and O2. Because of these shortcomings, the method thus far has not been used in other laboratories. We now simplified the method by having the subjects inhale 500 ml of a 10% DME-90% N2 gas mixture to fill the anatomical dead space, followed by a 5- or 15-s breath hold, and measuring the instantaneous DME and N2 concentrations and volume at the airway opening during the subsequent exhalation. From the difference in DME concentration in phase 1 of the expired N2 wash-in curve multiplied by the phase 1 dead space volume and divided by the mean DME concentration and the solubility coefficient for DME in tissue, Qaw can be calculated by using Fick's equation. We compared the new method to the validated old method in 10 healthy subjects and found mean +/- SE Qaw values of 34.6 +/- 2.3 and 34.6 +/- 2.8 microl.min(-1).ml(-1), respectively (r = 0.93; upper and lower 95% confidence limit +2.48 and -2.47). Using the new method, the mean coefficient of variation for two consecutive measurements was 4.4% (range 0-10.4%); inhalation of 1.2 mg albuterol caused a 53 +/- 14% increase in Qaw (P = 0.02) and inhalation of 2.4 mg methoxamine caused a 32 +/- 7% decrease in Qaw (P = 0.07). We conclude that the new method provides reliable values of and detects the expected changes in Qaw with vasoactive drugs. The simplicity and improved safety of the method should improve its acceptability for the noninvasive assessment of Qaw in clinical research.     

Branch angle and flow into a symmetric bifurcation

Arterial branches are found to be a major site for formation of arterial plaque. In this study, we investigate the role of the bifurcation angle on the flow into a symmetric bifurcation. Specially, how the changes in the bifurcation angle influences the distribution of axial wall shear in the bifurcation model. The flow in a range of branch opening half-angle of pi/25< or =theta< or =pi/4 are numerically simulated. The flow in the above models is calculated for the inlet flow Reynolds numbers of 250, 500, 1000, and 2000. It is found that at higher values of the opening angle of the bifurcation, the possibility and severity of flow separation at the appropriate wall location increases.     

Experimental flow studies in exact-replica phantoms of atherosclerotic carotid bifurcations under steady input conditions

Extensive flow studies are conducted in two carotid bifurcation flow phantoms. These phantoms exactly replicate the lumen of the plaque excised intact from two patients with severe carotid atherosclerosis. The input flow into the phantom's common carotid artery is steady. Novel scanning techniques for flow visualization and particle image velocimetry are used. In addition, a novel boundary treatment technique is employed in velocimetry to extract first order accurate velocity gradients at walls. The data show that the flow fields are highly three-dimensional. Numerous separation and recirculation zones dominate the flow domain, except at the lowest Reynolds numbers. The separation regions are often so severe that highly directed internal jets form. At high Reynolds numbers, the flows become unsteady and chaotic, even though the input flow is steady. Flow fields have large regions of energetic flow and almost stagnant recirculation zones. These recirculation zones range in size from the full size of the arteries to zones within crevasses smaller than 1 mm. Velocity field and streamline patterns conform well to the lumen geometry. The streamlines are highly tortuous. Stagnation points correlate well with the topological features of the stenosis. Vorticity maps confirm the highly complex and three dimensional nature of the flow. Wall shear stresses at the stenoses are estimated to be on the order of 10 Pa. These studies conclusively show that the nature of the flow in the diseased bifurcation is primarily dictated by the lumen geometry.     

Modeling the bifurcating flow in a human lung airway

The inspiratory flow characteristics in a three-generation lung airway have been numerically investigated using a control volume method to solve the fully three-dimensional laminar Navier-Stokes equations. The three-generation airway is extracted from the fifth to seventh branches of the model of Weibel (Morphometry of the Human Lung, Academic Press, New York, Springer, Berlin, 1963) with in-plane and 90 degrees off-plane configurations. Computations are carried out in the Reynolds number range of 200-1600, corresponding to mouth-air breathing rates ranging from 0.27 to 2.16l/s, or an averaged height of a man breathing from quiet to vigorous state. Particular attention is paid to establishing relations between the Reynolds number and the overall flow characteristics, including flow patterns and pressure drop. The ratio of airflow rate through the medial branch to that of the lateral branch for an in-plane airway increases as Re(0.227). However, the total pressure drop coefficient varies as Re(-0.497) for an in-plane airway and as Re(-0.464) for an off-plane airway. These pressure drop results are in good agreement with the experimentally measured behavior of Re(-0.5) and are more accurate than the numerically determined behavior of Re(-0.61) assuming the airways to be approximated by two-dimensional channels.     

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations II. Singularity theoretic approach and highly degenerate bifurcations

In the Hodgkin-Huxley equations (HH), we have identified the parameter regions in which either two stable periodic solutions with different amplitudes and periods and an equilibrium point or two stable periodic solutions coexist. The global structure of bifurcations in the multiple-parameter space in the HH suggested that the bistabilities of the periodic solutions are associated with the degenerate Hopf bifurcation points by which several qualitatively different behaviors are organized. In this paper, we clarify this by analyzing the details of the degenerate Hopf bifurcations using the singularity theory approach which deals with local bifurcations near a highly degenerate fixed point. Received: 23 April 1999 / Accepted in revised form: 24 September 1999     

Nonsymmetrical bifurcations in arterial branching

The results of optimality studies of the branching angles of arterial bifurcations are extended to nonsymmetrical bifurcations. Predicted nonsymmetrical bifurcations are found to be not unlike those observed in the cardiovascular system.     

Prediction of maximum expiratory flow rate from area-transmural pressure curve of compressed airway.

The site of greatest airway deformation in dog lungs was located during maximum expiratory flow by use of tantalum bronchography, fiberoptic bronchoscopy, and airway pressure measurements. A series of area vs. transmural pressure curves for each of these segments of the airway was produced after stepwise changes in transmural pressure. Measurements of area were made using cinephotography to elucidate the effect of time on airway compliance. The maximum flow rate was calculated using the t = 0.1 s compliance curve of the airway. An equation was derived so that maximum flow (V) could be calculated from the area (A) and transmural pressure (Ptm) of the flow-limiting segment. This equation, V = K-A square root of Ptm, implied that if V were constant then A must vary as Ptm-1/2. It was demonstrated that the area-transmural pressure curve of the flow-limiting segment showed this relationship between A and Ptm and that the flow calculated from this equation and the data from the A-Ptm curve gave flows identical to those measured during maximum expiration. The phenomena of effort-independent flow and negative effort dependence are also explained in terms of the area-transmural pressure curve of the flow-limiting segment.     

Transient bifurcations in neural error correction

This paper presents an investigation into the responses of neurons to errors in presynaptic spike trains. Errors are viewed, in nonlinear dynamical terms, as brief-duration changes in stationary presynaptic spike trains which induce transient responses in the postsynaptic cell. As these are generally large-magnitude transients, linearized neural models are not helpful. Instead, the responses of a full, nonlinear physiological model of a neuron that includes the recognized living prototype of an inhibitory synapse are analyzed. More specifically, the transients are examined in the context of the stationary behaviors that precede and succeed each error. It is shown that one and two dimensional bifurcation diagrams can be constructed from the transient responses--that there are marked changes in the transient responses at points that correspond to bifurcations in the stationary responses, qualitative changes in transients on either side of bifurcations, and only quantitative changes in transients between bifurcations.     

The principle of upper airway unidirectional flow facilitates breathing in humans

Upper airway unidirectional breathing, nose in and mouth out, is used by panting dogs to facilitate heat removal via water evaporation from the respiratory system. Why some humans instinctively employ the same breathing pattern during respiratory distress is still open to question. We hypothesized that 1) humans unconsciously perform unidirectional breathing because it improves breathing efficiency, 2) such an improvement is achieved by bypassing upper airway dead space, and 3) the magnitude of the improvement is inversely proportional to the tidal volume. Four breathing patterns were performed in random order in 10 healthy volunteers first with normal breathing effort, then with variable tidal volumes: mouth in and mouth out (MMB); nose in and nose out (NNB); nose in and mouth out (NMB); and mouth in and nose out (MNB). We found that unidirectional breathing bypasses anatomical dead space and improves breathing efficiency. At tidal volumes of approximately 380 ml, the functional anatomical dead space during NMB (81 +/- 31 ml) or MNB (101 +/- 20 ml) was significantly lower than that during MMB (148 +/- 15 ml) or NNB (130 +/- 13 ml) (all P < 0.001), and the breathing efficiency obtained with NMB (78 +/- 9%) or MNB (73 +/- 6%) was significantly higher than that with MMB (61 +/- 6%) or NNB (66 +/- 3%) (all P < 0.001). The improvement in breathing efficiency increased as tidal volume decreased. Unidirectional breathing results in a significant reduction in functional anatomical dead space and improvement in breathing efficiency. We suggest this may be the reason that such a breathing pattern is preferred during respiratory distress.     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.