Periodic flow at airway bifurcations. III. Energy dissipation

We measured the energy dissipation associated with large-amplitude periodic flow through airway bifurcation models. Each model consisted of a single asymmetric bifurcation with a different branching angle and area ratio, with each branch terminated into an identical elastic load. Sinusoidal volumetric oscillations were applied at the parent duct so that the upstream Reynolds number (Re) varied from 30 to 77,000 and the Womersley parameter (alpha) from 4 to 30. Pressures were measured continuously at the parent duct and at both terminals, and instantaneous branch flow rates were calculated. Time-averaged energy dissipation in the bifurcation was computed from an energy budget over a control volume integrated over a cycle and was expressed as a friction factor, F. We found that when tidal volume was small [ratio of tidal volume to resident (dead space) volume, VT/VD less than 1], F was independent of branching angle and fell with increasing alpha and VT/VD. When tidal volume was large (VT/VD greater than 1), F increased with increasing branching angle and varied less strongly with alpha and VT/VD. No simple benchmark flow represented the data well over the entire experimental range. This study demonstrates that only two nondimensional parameters, alpha and VT/VD, are necessary and are sufficient to describe time-averaged energy dissipation in a given bifurcation geometry during sinusoidal flow.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.     

A three-dimensional model of the human airway tree.

A three-dimensional (3D) model of the human airway tree is proposed using a deterministic algorithm that can generate a branching duct system in an organ. The algorithm is based on two principles: 1) the amount of fluid delivery through a branch is proportional to the volume of the region it supplies; and 2) the terminal branches are arranged homogeneously within the organ. These principles define the basic process of branching: generation of the dimensions and directionality of two daughter branches is governed by the properties of the parent branch and the region the parent supplies. The algorithm is composed of nine basic rules and four complementary rules. When the contour of an organ and the position of the trunk are specified, branches are successively generated by the algorithm. Applied to the human lung, the algorithm generates an airway tree that consists of approximately 54,000 branches. Its morphometric characteristics are in good agreement with those reported in the literature. The algorithm and the 3D airway model are useful for studying the structure-function relationship in the lung.     

Pressure excursions during oscillatory flow in a branching network of tubes

The pressure difference across individual branches of a four-generation network of branching tubes was measured with the objective of obtaining general laws to describe the pressure drop in the airways under conditions of oscillatory flow. Fourier decomposition showed that the pressure signals consisted of a dominant component at the excitation frequency ("fundamental") and a "first harmonic" of smaller magnitude. For values of the ratio Re/alpha less than 200, the fundamental mainly represented fluid acceleration, whereas the first harmonic reflected the effects both of viscous dissipation and the change in total cross-sectional area between parent and daughter generations. For values of Re/alpha greater than 200, the magnitude of the fundamental was considerably larger than that due to fluid acceleration alone, suggesting the possibility of onset of turbulence in the branching network. These pressure measurements were applied to a simple model of the dog lung to predict total airway resistance. The results are found to be in substantial agreement with physiological measurements.     

Heterogeneity of mean alveolar pressure during high-frequency oscillations

Mean alveolar pressure may exceed mean airway pressure during high-frequency oscillations (HFO). To assess the magnitude of this effect and its regional heterogeneity, we studied six excised dog lungs during HFO [frequency (f) 2-32 Hz; tidal volume (VT) 5-80 ml] at transpulmonary pressures (PL) of 6, 10, and 25 cmH2O. We measured mean pressure at the airway opening (Pao), trachea (Ptr), and four alveolar locations (PA) using alveolar capsules. Pao was measured at the oscillator pump, wherein the peak dynamic head was less than 0.2 cmH2O. Since the dynamic head was negligible here, and since these were excised lungs, Pao thus represented true applied transpulmonary pressure. Ptr increasingly underestimated Pao as f and VT increased, with Pao - Ptr approaching 8 cmH2O. PA (averaged over all locations) and Pao were nearly equal at all PL's, f's, and VT's, except at PL of 6, f 32 Hz, and VT 80 ml, where (PA - Pao) was 3 cmH2O. Remarkably, mean pressure in the base exceeded that in the apex increasingly as f and VT increased, the difference approaching 3 cmH2O at high f and VT. We conclude that, although global alveolar overdistension assessed by PA - Pao is small during HFO under these conditions, larger regional heterogeneity in PA's exists that may be a consequence of airway branching angle asymmetry and/or regional flow distribution.     

Airway branching morphology of mature and immature rabbit lungs.

The scheme of Horsfield et al. for describing the pulmonary airway tree (J Appl Physiol 52: 21-26, 1982) catalogs each airway according to its order and the difference in order of its two daughters (denoted Delta). Although this scheme captures the natural asymmetry in the airway tree, it is still deterministic, because it assumes that all airways of a given order are the same; yet such variability is extremely important in determining the overall behavior of the lungs. We therefore analyzed complete lung lobes from three mature and two immature rabbits and determined the Horsfield order and Delta of every airway down to the terminal bronchioles. We also measured the diameter of each airway. This allowed us to determine the average structure of the rabbit airway tree, the variation about this average, and also how the structures of mature and immature airway trees compare. We found some variation in branching asymmetry and airway diameter at a given order between animals but no evidence of systematic differences in structure between mature and immature lungs. We found evidence of a difference in the branching structure of the peripheral vs. the central part of the airway tree (the break point being around order 20). We also determined the nature of the variation in Delta and diameter as a function of order, which should be valuable for the development of computer models seeking to encapsulate the naturally occurring regional variation in airway geometry in the normal rabbit lung.     

Shear stress distribution in arterial tree models, generated by constrained constructive optimization.

Models of arterial trees are generated by the algorithm of Constrained Constructive Optimization (CCO). Straight cylindrical, binary branching tubes are arranged in an optimized fashion so as to convey blood to the terminal sites of the tree, which are distributed over a predefined area, representing the tissue to be perfused. All terminal segments supply equal flows at a unique terminal pressure, and the radii of parent and daughter segments are related via a bifurcation law. The connective structure and geometry of the model are optimized according to a target function such as total intravascular volume. The shear rate between blood and the vessel walls is computed in each segment and a new method is presented for rescaling a given CCO tree to a desired value of shear rate in the root segment. The effect of viscosity varying with shear rate is evaluated and a new method is presented for rescaling a CCO-tree segment by segment to consistent values of radii and variable viscosity. Shear stress is evaluated for its deviation from being proportional to shear rate and then subjected to various types of analyses. Usually both, shear stress and its variability, are found to be larger in the smaller than in the larger segments of the CCO-model trees. However, it is shown how the shear-stress distribution can be reshuffled between small and large segments when rescaling a CCO tree to obey a different bifurcation law, while its whole geometry remains unchanged and all boundary conditions remain fulfilled. The selection of optimization target is found to drastically affect shear-stress variability within bifurcations, which reaches a distinct minimum if the model is optimized according to intravascular volume. Finally, a rank-analysis of shear stress within each bifurcation shows that only two out of six possible rank patterns actually occur: the parent segment always experiences medium shear stress while minimum shear stress resides mostly in the larger, less frequently in the smaller daughter.     

Upper airway extraluminal tissue pressure fluctuations during breathing in rabbits.

Transmural pressure at any level in the upper airway is dependent on the difference between intraluminal airway and extraluminal tissue pressure (ETP). We hypothesized that ETP would be influenced by topography, head and neck position, resistive loading, and stimulated breathing. Twenty-eight male, New Zealand White, anesthetized, spontaneously breathing rabbits breathed via a face mask with attached pneumotachograph to measure airflow and pressure transducer to monitor mask pressure. Tidal volume was measured via integration of the airflow signal. ETP was measured with a pressure transducer-tipped catheter inserted in the tissues of the lateral (ETPlat, n = 28) and anterior (ETPant, n = 21) pharyngeal wall. Head position was controlled at 30, 50, or 70 degrees, and the effect of addition of an external resistor, brief occlusion, or stimulated breathing was examined. Mean ETPlat was approximately 0.7 cmH2O greater than mean ETPant when adjusted for degree of head and neck flexion (P < 0.05). Mean, maximum, and minimum ETP values increased significantly by 0.7-0.8 cmH2O/20 degrees of head and neck flexion when adjusted for site of measurement (P < 0.0001). The main effect of resistive loading and occlusion was an increase in the change in ETPlat (maximum - minimum ETPlat) and change in ETPant at all head and neck positions (P < 0.05). Mean ETPlat and ETPant increased with increasing tidal volume at head and neck position of 30 degrees (all P < 0.05). In conclusion, ETP was nonhomogeneously distributed around the upper airway and increased with both increasing head and neck flexion and increasing tidal volume. Brief airway occlusion increased the size of respiratory-related ETP fluctuations in upper airway ETP.     

Reversible and irreversible components of central-airway flow resistance

The flow energy loss (head loss) through a cast of canine central airways is found to be nearly independent of flow direction. By contrast, head loss in geometrically-simpler branching sections at comparable flow conditions is highly irreversible, with inspiratory loss being greater by nearly two units of dynamic pressure (2 X 1/2 rho V2). In these branching sections head loss appears to be independent of important geometric parameters such as the branch length/diameter ratio and the exit/inlet flow-area ratio. An analysis of these observations suggests that kinetic energy factors, not shear stresses, account for most of the energy dissipated in central airways and in simple bifurcating sections. Inspiratory loss in bifurcations is greatly increased by the onset of flow separation: irreversibility is minimal in central airways, where separation either is absent or else is much less pronounced.     

Pendelluft flow in symmetric airway bifurcations.

We propose a mathematical model for pendelluft flow in a single airway bifurcation. The model is motivated by an apparatus used in an experimental study of the pendelluft by Ultman et al. (1988). We derive differential equations governing the fluid flow, which directly connect physiological parameters to the variables determining the pendelluft; this approach allows us to include nonlinearity in the model. If nonlinearity is neglected, our model is identical to the R-I-C circuits used by previous investigators. If nonlinearity is retained, we show that pendelluft can occur even in perfectly symmetric airway bifurcations. For the specific apparatus used in the experiments of High et al. (1991), we demonstrate that two qualitatively different pendelluft flows can occur in the system.     

Expiration rate drives human airway design

Analyses of human airway architecture based on calculations of airflow resistance or energy dissipation suggest that the branching pattern is not optimized for minimizing energy loss by flow dissipation during respiration. Airway flow dissipates only a few percent of the total body work during normal breathing, so branching patterns deviate from minimum energy loss to also optimize other physiological needs. Studies of airway performance often record some measure of expiration, such as FEV1 (Forced Expiratory Volume in 1 s), because airway constriction during expiration limits the rate of rapid respiration. We posit that lung structure is optimized for the rate of expiration as well as minimum energy loss. By increasing the daughter-to-parent airway diameter ratio (h) from 0.794 (corresponding to the energy minimum for symmetrically branching airways) to 0.85 (the observed value in humans) luminal pressures at airway generations 4-15 were substantially increased during exercise (a 4.5 and 15 cmH₂O increase during moderate and heavy exercise, respectively). Values of h somewhat larger than 0.794 help airways remain open during expiration by increasing both viscous pressure drop and convective acceleration pressure drop. Asymmetric bifurcations also exhibit higher proximal airway pressures than symmetric ones, but the improvement was not large.     

Effect of hypoproteinemia on lung fluid balance in awake newborn lambs

To study the influence of plasma protein concentration on fluid balance in the newborn lung, we measured pulmonary arterial and left atrial pressures, lung lymph flow, and concentrations of protein in lymph and plasma of eight lambs, 2-3 wk old, before and after we reduced their plasma protein concentration from 5.8 +/- 0.3 to 3.6 +/- 0.6 g/dl. Each lamb underwent two studies, interrupted by a 3-day period in which we drained protein-rich systemic lymph through a thoracic duct fistula and replaced fluid losses with feedings of a protein-free solution of electrolytes and glucose. Each study consisted of a 2-h control period followed by 4 h of increased lung microvascular pressure produced by inflation of a balloon in the left atrium. Body weight and vascular pressures did not differ significantly during the two studies, but lung lymph flow increased from 2.6 +/- 0.1 ml/h during normoproteinemia to 4.1 +/- 0.1 ml/h during hypoproteinemia. During development of hypoproteinemia, the average difference in protein osmotic pressure between plasma and lymph decreased by 1.6 +/- 2 Torr at normal left atrial pressure and by 4.9 +/- 2.2 Torr at elevated left atrial pressure. When applied to the Starling equation governing microvascular fluid balance, these changes in liquid driving pressure were sufficient to account for the observed increases in lung fluid filtration; reduction of plasma protein concentration did not cause a statistically significant change in calculated filtration coefficient. Protein loss did not influence net protein clearance from the lungs nor did it accentuate the increase in lymph flow associated with left atrial pressure elevation.(ABSTRACT TRUNCATED AT 250 WORDS)     

Interstitial fluid pressure gradient measured by micropuncture in excised dog lung

We have directly measured lung interstitial fluid pressure at sites of fluid filtration by micropuncturing excised left lower lobes of dog lung. We blood-perfused each lobe after cannulating its artery, vein, and bronchus to produce a desired amount of edema. Then, to stop further edema, we air-embolized the lobe. Holding the lobe at a constant airway pressure of 5 cmH2O, we measured interstitial fluid pressure using beveled glass micropipettes and the servo-null method. In 31 lobes, divided into 6 groups according to severity of edema, we micropunctured the subpleural interstitium in alveolar wall junctions, in adventitia around 50-micron venules, and in the hilum. In all groups an interstitial fluid pressure gradient existed from the junctions to the hilum. Junctional, adventitial, and hilar pressures, which were (relative to pleural pressure) 1.3 +/- 0.2, 0.3 +/- 0.5, and -1.8 +/- 0.2 cmH2O, respectively, in nonedematous lobes, rose with edema to plateau at 4.1 +/- 0.4, 2.0 +/- 0.2, and 0.4 +/- 0.3 cmH2O, respectively. We also measured junctional and adventitial pressures near the base and apex in each of 10 lobes. The pressures were identical, indicating no vertical interstitial fluid pressure gradient in uniformly expanded nonedematous lobes which lack a vertical pleural pressure gradient. In edematous lobes basal pressure exceeded apical but the pressure difference was entirely attributable to greater basal edema. We conclude that the presence of an alveolohilar gradient of lung interstitial fluid pressure, without a base-apex gradient, represents the mechanism for driving fluid flow from alveoli toward the hilum.     

Blood flow multiscale phenomena

The cardiovascular disease is one of most frequent cause deaths in modern society. The objective of this work is analyse the effect of dynamic vascular geometry (curvature, torsion, bifurcation) and pulsatile blood nature on secondary flow, wall shear stress and platelet deposition. The problem was examined as multi-scale physical phenomena using perturbation analysis and numerical modelling. The secondary flow determined as influence pulsatile pressure, vascular tube time-dependent bending and torsion on the main axial flow. Bifurcation and branching phenomena are analysed experimentally through, blood-like fluid pulsatile flow across elastic rubber-like Y-model model. The problem complex geometry near branching in platelet deposit modelling is resolved numerically as Falker-Skan flow.     

Inspiratory airway obstruction does not affect lung fluid balance in lambs

The purpose of this study was to examine the effects of inspiratory airway obstruction on lung fluid balance in newborn lambs. We studied seven 2- to 4-wk-old lambs that were sedated with chloral hydrate and allowed to breathe 30-40% O2 spontaneously through an endotracheal tube. We measured lung lymph flow, lymph and plasma protein concentrations, pulmonary arterial and left atrial pressures, mean and phasic pleural pressures and airway pressures, and cardiac output during a 2-h base-line period and then during a 2- to 3-h period of inspiratory airway obstruction produced by partially occluding the inspiratory limb of a nonrebreathing valve attached to the endotracheal tube. During inspiratory airway obstruction, both pleural and airway pressures decreased 5 Torr, whereas pulmonary arterial and left atrial pressures each decreased 4 Torr. As a result, calculated filtration pressure remained unchanged. Inspiratory airway obstruction had no effect on steady-state lung lymph flow or the lymph protein concentration relative to that of plasma. We conclude that in the spontaneously breathing lamb, any decrease in interstitial pressure resulting from inspiratory airway obstruction is offset by a decrease in microvascular hydrostatic pressure so that net fluid filtration remains unchanged.     

Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature.

The branching systems in our body (vascular and bronchial trees) and those in the environment (plant trees and river systems) are characterized by a fractal nature: the self-similarity in the bifurcation pattern. They increase their branch density toward terminals according to a power function with the exponent called fractal dimension (D). From a stochastic model based-on this feature, we formulated the fractal-based integrals to calculate such morphological parameters as aggregated branch length, surface area, and content volume for any given range of radius (r). It was followed by the derivation of branch number and cross-sectional area, by virtue of the logarithmic sectioning of the r axis and of the branch radius-length relation also given by a power function of r with an exponent (alpha). These derivatives allowed us to quantify various hydrodynamic parameters of vascular and bronchial trees as fluid conduit systems, including the individual branch flow rate, mean flow velocity, wall shear rate and stress, internal pressure, and circumferential tension. The validity of these expressions was verified by comparing the outcomes with actual data measured in vivo in the vascular beds. From additional analyses of the terminal branch number, we found a simple equation relating the exponent (m) of the empirical power law (Murray's so-called cube law) to the other exponents as (m=D+alpha). Finally, allometric studies of mammalian vascular trees revealed uniform and scale-independent distributions of terminal arterioles in organs, which afforded an infarct index, reflecting the severity of tissue damage following arterial infarction.     

Light microscopic detection of sugar residues in glycoconjugates of salivary glands and the pancreas with lectin-horseradish peroxidase conjugates. II. Rat

Summary Salivary glands and pancreases from male rats were stained with a battery of ten different lectin-horseradish peroxidase conjugates. Qualitative and quantitative differences were observed in the content of terminal sugar residues in stored secretory glycoproteins in parenchymal cells of glands having a similar histological structure. Heterogeneity in the content of secretory glycoconjugates was also found between cells in the same exocrine glands, which were previously thought to be identical on the basis of classical morphological and histochemical staining studies. Similar differences were observed in the structure of glycoconjugates associated with the apical surface of epithelial cells lining glandular excretory ducts. Intercalated ducts presented a gland specific staining pattern different from that of the glandular secretory cell population, whereas striated duct and interlobular duct epithelial cells stained similarly in all major rat exocrine glands. A comparison of lectin binding patterns in identical histological sites in the mouse, reported in a companion paper, is provided, and the similarities and differences between these two rodent species are discussed. In addition to providing valuable information concerning the localization and structure of tissue complex carbohydrates, a comparison of staining in the same tissue sites with labelled lectins reported biochemically to have similar binding specificity has revealed interesting differences in the binding specificity of these macromolecules.