Reflection coefficients in pulsatile flow through converging junctions and the pressure distribution in a simple loop

Analytical expressions for the reflection coefficients in pulsatile flow through converging junctions are derived by two independent methods and are used to study the effects of wave reflections on the pressure distribution in a simple vascular loop. A simulated physiological situation is used as an example in which the loop is formed by the combination of a bypass and a bypassed vessel, the relative diameter of the latter being varied in order to simulate a narrowing. The results demonstrate how, in the case of a converging junction, the effects of wave reflections on the pressure distribution in one vessel depend on conditions within the vessel itself as well as in the other. The new reflection coefficients take into account this interdependence of flow in the two vessels forming a converging junction, and are shown to be consistent with reflection coefficients commonly used in diverging junctions.     

Refilling of a hydraulically isolated embolized xylem vessel: model calculations

When they are hydraulically isolated, embolized xylem vessels can be refilled, while adjacent vessels remain under tension. This implies that the pressure of water in the refilling vessel must be equal to the bubble gas pressure, which sets physical constraints for recovery. A model of water exudation into the cylindrical vessel and of bubble dissolution based on the assumption of hydraulic isolation is developed. Refilling is made possible by the turgor of the living cells adjacent to the refilling vessel, and by a reflection coefficient below 1 for the exchange of solutes across the interface between the vessel and the adjacent cells. No active transport of solutes is assumed. Living cells are also capable of importing water from the water-conducting vessels. The most limiting factors were found to be the osmotic potential of living cells and the ratio of the volume of the adjacent living cells to that of the embolized vessel. With values for these of 1.5 MPa and 1, respectively, refilling times were in the order of hours for a broad range of possible values of water conductivity coefficients and effective diffusion distances for dissolved air, when the xylem water tension was below 0.6 MPa and constant. Inclusion of the daily pattern for xylem tension improved the simulations. The simulated gas pressure within the refilling vessel was in accordance with recent experimental results. The study shows that the refilling process is physically possible under hydraulic isolation, while water in surrounding vessels is under negative pressure. However, the osmotic potentials in the refilling vessel tend to be large (in the order of 1 MPa). Only if the xylem water tension is, at most, twice atmospheric pressure, the reflection coefficient remains close to 1 (0.95) and the ratio of the volume of the adjacent living cells to that of the embolized vessel is about 2, does the osmotic potential stay below 0.4 MPa.     

Some contributions to the mathematical biology of blood circulation. Reflections of pressure waves in the arterial system

If, along the length of a blood vessel, there is a discontinuous change of diameter, the pressure waves will be reflected at the discontinuity. The transmission and reflection coefficients of the waves are calculated in terms of the ratio of the diameters and the ratio of the propagation velocities of the waves in the parts of the blood vessel. We obtain also the reflection coefficient in terms of the ratios of the diameters, of the elasticity moduli, and of the wall thicknesses. The possible clinical importance of the results is pointed out for cases of coarctation, arteriosclerosis, and other pathological conditions which might effect local changes in the structure of the blood vessel wall.     

Smaller, stiffer coronary bypass can moderate or reverse the adverse effects of wave reflections.

The question of whether the mechanical stiffness of a coronary bypass or that of a diseased coronary artery can have a significant effect on the hemodynamics in these vessels is addressed analytically, with emphasis on the effects of wave reflections. The analysis is based on a model of the vessels involved, and the results show the essential hemodynamic effects in each vessel. It is found that in the absence of a bypass graft, wave reflections resulting from a narrowing and stiffening of a diseased coronary artery have the effect of actually aiding the flow in the diseased vessel. In the presence of a bypass graft, however, the effects of wave reflections are reversed and become adverse to flow in both the bypass graft and the diseased coronary artery. A stiffer bypass moderates these effects and is therefore preferable to a more elastic bypass. The adverse effects also depend critically on the relative diameter of the bypass. Here the results indicate that a bypass of smaller diameter than that of the native coronary artery can moderate and even reverse the adverse effects of wave reflections resulting from the presence of the bypass.     

A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree

Using a frequency-domain Womersley-type model, we previously simulated pulsatile blood flow throughout the coronary arterial tree. Although this model represents a good approximation for the smaller vessels, it does not take into account the nonlinear convective energy losses in larger vessels. Here, using Womersley's theory, we present a hybrid model that considers the nonlinear effects for the larger epicardial arteries while simulating the distal vessels (down to the 1st capillary segments) with the use of Womersley's Theory. The main trunk and primary branches were discretized and modeled with one-dimensional Navier-Stokes equations, while the smaller-diameter vessels were treated as Womersley-type vessels. Energy losses associated with vessel bifurcations were incorporated in the present analysis. The formulation enables prediction of impedance and pressure and pulsatile flow distribution throughout the entire coronary arterial tree down to the first capillary segments in the arrested, vasodilated state. We found that the nonlinear convective term is negligible and the loss of energy at a bifurcation is small in the larger epicardial vessels of an arrested heart. Furthermore, we found that the flow waves along the trunk or at the primary branches tend to scale (normalized with respect to their mean values) to a single curve, except for a small phase angle difference. Finally, the model predictions for the inlet pressure and flow waves are in excellent agreement with previously published experimental results. This hybrid one-dimensional/Womersley model is an efficient approach that captures the essence of the hemodynamics of a complex large-scale vascular network. The present model has numerous applications to understanding the dynamics of coronary circulation.     

Pulse Wave Propagation in Elastic Tubes Having Longitudinal Changes in Area and Stiffness

The behavior of both step waves and sinusoidal waves in fluid-filled elastic vessels whose area and distensibility vary with distance is explored theoretically. It is shown that the behavior of these waves may be explained, to a large extent, by considering the effect of the continuous stream of infinitesimal reflections that is set up whenever any wave travels in a region of vessel where the local impedance, (that is, the ratio of elastic wavespeed to tube area) is not constant. It is found that in such vessels the behavior of sinusoidal waves over distances which are a fraction of a wavelength can be quite different from their average behavior over several wavelengths. Both behaviors are described analytically. The results are applied to the mammalian circulatory system, one of the most interesting results being that a longitudinal variation in the pressure and velocity amplitudes which has a wavelength roughly one-half that of standing waves is predicted. The treatment is essentially a linearized quasi-one-dimensional one, the major assumptions being that the fluid is inviscid, the mean flow is zero, and the vessel is perfectly elastic and constrained from motion in the longitudinal direction. As in the physiological situation, the ratio of fluid velocity to pulse propagation speed is assumed small. For comparison with the analytical results, the linearized equations are also solved numerically by computer.     

Wave reflection effects in the central circulation of American alligators (Alligator mississippiensis): what the heart sees

A large central compliance is thought to dominate the hemodynamics of all vertebrates except birds and mammals. Yet large crocodilians may adumbrate the avian and mammalian condition and set the stage for significant wave transmission (reflection) effects, with potentially detrimental impacts on cardiac performance. To investigate whether crocodilians exhibit wave reflection effects, pressures and flows were recorded from the right aorta, carotid artery, and femoral artery of six adult, anesthetized American alligators (Alligator mississippiensis) during control conditions and after experimentally induced vasodilation and constriction. Hallmarks of wave reflection phenomena were observed, including marked differences between the measured profiles for flow and pressure, peaking of the femoral pressure pulse, and a diastolic wave in the right aortic pressure profile. Pulse wave velocity and peripheral input impedance increased with progressive constriction, and thus changes in both the timing and magnitude of reflections accounted for the altered reflection effects. Resolution of pressure and flow waves into incident and reflected components showed substantial reflection effects within the right aorta, with reflection coefficients at the first harmonic approaching 0.3 when constricted. Material properties measured from isolated segments of blood vessels revealed a major reflection site at the periphery and, surprisingly, at the junction of the truncus and right aorta. Thus, while our results clearly show that significant wave reflection phenomena are not restricted to birds and mammals, they also suggest that rather than cope with potential negative impacts of reflections, the crocodilian heart simply avoids them because of a large impedance mismatch at the truncus.     

Seasonal changes in wood formation of Ulmus pumila and U. minor and its relation with Dutch elm disease

Elms containing narrow and scattered vessels have been reported to be more resistant to Ophiostoma novo-ulmi (Dutch elm disease pathogen) than elms with large and contiguous vessels. However, recent measurements in Ulmus pumila and U. minor showed a contrary trend. The pin method was applied to 4-yr-old branches of eight clones planted in Madrid. During 2002, radial growth increments and vessel diameters were measured monthly, and beetle trapping was undertaken weekly. U. minor formed larger vessels at the beginning of the season, coinciding with a peak of captured beetles, but, up to June 15, vessels were larger for U. pumila. The number of vessels per group, the transversal area per vessel group, and the mean theoretical hydraulic conductances were significantly higher for U. minor on most dates. Researchers should take into consideration the seasonal changes in vessel size. The results highlight that seasonal variation of vessel diameters and hydraulic parameters, in combination with beetle abundance, are the main factors that could explain the different susceptibility of both elm species to O. novo-ulmi.     

The effect of commercial and recreational traffic on the resuspension of sediment in Navigation Pool 9 of the Upper Mississippi River

This study was conducted to determine some effects of commercial and recreational traffic on the resuspension of sediment in Navigation Pool No. 9 of the Upper Mississippi River. Fifty commercial vessel passages were examined at five different main channel locations and at side channels that were adjacent to each of the main channel locations. Sixteen recreational vessel passages were examined at one main channel location and its adjacent side channel and at a channel located in the backwaters. The backwater channel was not directly influenced by navigation in the main channel. Changes in total non-filterable residue (TNFR) and average particle size of suspended silts were used to assess some effects of navigation. Seventy-eight percent of the commercial vessel passages resulted in significant increases of TNFR and/or average particle size in the water column. In the main channel, TNFR increased from 3.4% to 15% above ambient levels; in the side channels, increases ranged from 2.5% to 21.7%. The average diameter of the resuspended silts increased by 0.21 to 2.34 m. In the main channel, 50% of the recreational vessel passages caused increased TNFR and all passages increased average particle size. In the backwater channel, all of the recreational passages caused increases in TNFR and average particle size. Total resuspended sediment transported downstream ranged from an estimated 0.82 to 1015.7 mTons/passage in the main channel for commercial vessels, 0.39 to 0.64 mTons/passage in the main channel for recreational vessels, 0.22 to 28.12 mTons/passage in side channels due to commercial vessels in the main channel, and 0.54 to 2.08 mTons/ passage in the backwater channel for recreational vessels. Bed-sediment composition, location of the vessels in the channel, channel geometry, the number of successive passages, and vessel speed were identified as factors that affected the magnitude of the resuspension.     

Readdressing the issue of thermally significant blood vessels using a countercurrent vessel network

A physiologically realistic arterio-venous countercurrent vessel network model consisting of ten branching vessel generations, where the diameter of each generation of vessels is smaller than the previous ones, has been created and used to determine the thermal significance of different vessel generations by investigating their ability to exchange thermal energy with the tissue. The temperature distribution in the 3D network (8178 vessels; diameters from 10 to 1000 microm) is obtained by solving the conduction equation in the tissue and the convective energy equation with a specified Nusselt number in the vessels. The sensitivity of the exchange of energy between the vessels and the tissue to changes in the network parameters is studied for two cases; a high temperature thermal therapy case when tissue is heated by a uniformly distributed source term and the network cools the tissue, and a hypothermia related case, when tissue is cooled from the surface and the blood heats the tissue. Results show that first, the relative roles of vessels of different diameters are strongly determined by the inlet temperatures to those vessels (e.g., as affected by changing mass flow rates), and the surrounding tissue temperature, but not by their diameter. Second, changes in the following do not significantly affect the heat transfer rates between tissue and vessels; (a) the ratio of arterial to venous vessel diameter, (b) the diameter reduction coefficient (the ratio of diameters of successive vessel generations), and (c) the Nusselt number. Third, both arteries and veins play significant roles in the exchange of energy between tissue and vessels, with arteries playing a more significant role. These results suggest that the determination of which diameter vessels are thermally important should be performed on a case-by-case, problem dependent basis. And, that in the development of site-specific vessel network models, reasonable predictions of the relative roles of different vessel diameters can be obtained by using any physiologically realistic values of Nusselt number and the diameter reduction coefficient.     

Wood anatomy and vessel characteristics of spiny monkey orange (Strychnos spinosa) in Benin (West Africa)

The response of plant species to varying climate conditions in tropical Africa remains poorly understood but can be assessed using wood anatomical traits. These traits play an important role for the adaptive capacity of a species to environmental stress, since environmental conditions can modify the proportion, size, and morphology of wood anatomical elements. This study reports quantitative data on vessel characteristics of the diffuse porous angiosperm Spiny monkey orange (Strychnos spinosa Lam.) in Benin in tropical West Africa. The vessel-related anatomical traits varied with high amplitude (coefficient of variation CV ˃ 25%) between different sites located in different climate zones. The variability of the traits is higher within one climatic zone than between climatic zones, and even more pronounced within trees. Consequently, the climatic zones have less influence on the studied features than local site conditions. However, the study showed that S. spinosa individuals that have numerous vessels also have a high lumen fraction and total ring area. On the other hand, individuals presenting a high vessel density also display vessels of smaller size. The correlation between vessel number and total ring area on the one hand, and between vessel size and lumen fraction on the other hand are highly significant and positive. In Benin, S. spinosa wood anatomical traits are likely linked to local site factors rather than to regional climatic factors.     

Earlywood vessel size of oak as a potential proxy for spring precipitation in mesic sites

Aim In this study, we evaluate the importance of the mean earlywood vessel size of oaks as a potential proxy for climate in mesic areas. Location The study was conducted in Switzerland at three forest sites dominated by oak (Quercus petraea and Q. pubescens). The three sites were in different climatic zones, varying mainly in terms of precipitation regime. Methods Three 50‐year‐long site chronologies of mean earlywood vessel size and tree‐ring widths were obtained at each site and related to monthly meteorological records in order to identify the main variables controlling growth. The responses of mean vessel size to climate were compared with those of the width variables to evaluate the potential climatic information recorded by the earlywood vessels. Results The results show that the mean vessel size has a different and stronger response to climate than ring‐width variables, although its common signal and year‐to‐year variability are lower. This response is better in particular at mesic sites, where it is linked to precipitation during spring, i.e. at the time of vessel formation, and is probably related to the occurrence of only a few processes controlling vessel growth, whereas radial increment is controlled by multiple and varying factors. Main conclusions The mean earlywood vessel size of oak appears to be a promising proxy for future climate reconstructions of mesic sites, where radial growth is not controlled by a single limiting factor.     

Presence of paratracheal water storage tissue does not alter vessel characters in cactus wood

This research tested hypotheses that the presence of water storage tissues immediately adjacent to vessels would protect vessels from cavitation and would result in evolution of broader vessels that occur in fewer, smaller clusters relative to vessels surrounded by a matrix of fibers. We examined 21 species that have dimorphic wood, that is, at one stage in their life they produce a wood with a fibrous matrix surrounding the vessels and at another stage they produce wood with abundant paratracheal parenchyma or wide-band tracheids. In only one species were vessels in the water storage matrix broader than those in the fibrous matrix of the same plant. In most specimens, fibrous wood had smaller clusters of vessels than water storage wood, and a greater percentage of vessels in fibrous wood were solitary. Presence of abundant paratracheal water storage tissue was not correlated with a reduced number or size of rays. Axial masses in fibrous wood were not consistently narrower than those of water storage wood, consequently their vessels were not consistently closer to water stored in rays. Wood strength may be more important than conduction safety in determining vessel cluster size and widths of rays and axial masses.     

Emission of alfvén waves from a nonuniform MHD waveguide

The efficiency of the wave energy loss from a nonuniform MHD waveguide due to the conversion of the trapped magnetosonic waveguide modes into runaway Alfvén waves is estimated theoretically. It is shown that, if the waveguide parameters experience a jumplike change along the waveguide axis, the interaction between the waveguide modes and Alfvén waves occurs precisely at this “jump.” This effect is incorporated into the boundary conditions. A set of coupled integral equations with a singular kernel is derived in order to determine the transmission and reflection coefficients for the waveguide modes. The poles in the kernels of the integral operators correspond to the surface waves. When the jump in the waveguide parameters is small, analytic expressions for the frequency dependence of the transformation coefficients are obtained by using a model profile of the Alfvén velocity along the magnetic field. For the jump characterized by the small parameter value ε=0.3, the wave-amplitude transformation coefficient can amount to 5–10%. Under the phase synchronization condition (when the phase velocities of the waveguide modes on both sides of the jump are the same), the wave-energy transformation coefficient is much higher: it increases from a fraction of one percent to tens of percent. The transformation of fast magnetosonic waves into Alfvén waves is resonant in character, which ensures the frequency and wavelength filteringof the emitted Alfvén perturbations.     

A circle of Willis simulation using distensible vessels and pulsatile flow

The development of a one-dimensional numerical (finite-difference) model of the arterial network surrounding the circle of Willis is described based on the full Navier-Stokes and conservation of mass equations generalized for distensible vessels. The present model assumes an elastic wall defined by a logarithmic pressure-area relation obtained from the literature. The viscous term in the momentum equation is evaluated using the slope of a Karman-Pohlhausen velocity profile at the vessel boundary. The afferent vessels (two carotids and two vertebrals) are forced with a canine physiologic pressure signature corresponding to an aortic site. The network associated with each main efferent artery of the circle is represented by a single vessel containing an appropriate amount of resistance so that the mean flow through the system is distributed in accordance with the weight of brain irrigated by each vessel as determined from a steady flow model of the same network. This resistance is placed a quarter wave-length downstream from the heart to insure proper reflection from the terminations, where the quarter wavelength is determined using the frequency corresponding to the first minimum on an input impedance-frequency diagram obtained at the heart. Computer results are given as time histories of pressure and flow at any model nodal point starting from initial conditions of null flow and constant pressure throughout the model. Variations in these pressure and flow distributions caused by the introduction of pathologic situations into the model illustrate the efficacy of the simulation and of the circle in equalizing and redistributing flows in abnormal situations.     

Influence of the outer-magnetospheric magnetohydrodynamic waveguide on the reflection of hydromagnetic waves from a shear flow at the magnetopause

The coefficient of reflection of a fast magnetosonic wave incident on the magnetosphere from the solar wind is studied analytically in the framework of a plane-stratified model of the medium with allowance for the transverse inhomogeneity of the magnetosphere and a jump of the plasma parameters at the magnetopause. Three factors decisively affecting the properties of reflection are taken into account: the shear flow of the solar wind plasma relative to the magnetosphere; the presence of a magnetospheric magnetohydrodynamic waveguide caused by the transverse plasma inhomogeneity; and the presence of an Alfvén resonance deep in the magnetosphere, where the oscillation energy dissipates. If the solar wind velocity exceeds the wave phase velocity along the magnetopause, then the wave energy in the solar wind is negative and such a wave experiences overreflection. In the opposite case, the wave energy is positive and the wave is reflected only partially. The wave reflection has a pronounced resonant character: the reflection coefficient has deep narrow minima or high narrow maxima at the eigenfrequencies of the magnetospheric waveguide. For other frequencies, the reflection coefficient only slightly differs from unity. The wave energy influx into the magnetosphere is positive for waves with both positive and negative energies. For waves with a negative energy, this is a consequence of their overreflection, because the flux of negative energy carried away by the reflected wave exceeds the incident flux of negative energy.     

Hydraulic properties of individual xylem vessels of Fraxinus americana

Studies of the hydraulic properties of xylem vessels have been limited to measurements of whole plant or whole stem segments. This approach allows the longitudinal transport properties of the ensemble of vessels within a stem to be determined, but provides little information on radial transport. Here the xylem of Fraxinus americana L. has been examined using a new method that allows the transport properties of individual vessels to be examined. One goal of this study was to quantify transport parameters relevant to embolism repair. The longitudinal conductivity of vessel segments open at both ends (i.e. no end walls) agreed with values predicted by the Poiseuille equation. Radial specific conductance (conductance per unit area) was approximately six orders of magnitude lower than the longitudinal conductance of the vessel segment normalized by the cross-sectional area of the vessel lumen. There was a step increase in the radial specific conductance of previously gas-filled vessels when the delivery pressure exceeded 0.4 MPa. This is consistent with the idea that positive pressure, required for embolism repair, can be compartmentalized within a vessel if the bordered pit chambers are gas-filled. The diffusion coefficient for the movement of gas from a pressurized air-filled vessel was of the same order of magnitude as that for air diffusing through water (1.95 e(-9) m(2) s(-1)). Estimates of the time needed to displace all of the gas from an air-filled vessel were in the order of 20 min, suggesting that gas removal may not be a major limitation in embolism repair.     

Induction of air embolism in xylem conduits of pre-defined diameter.

A new method is presented that enables the induction of embolisms in a fraction of all xylem vessels, based on diameter, at one cut end of a stem segment. The method is based on the different capillary characteristic of xylem vessels of different cross-sectional size. To verify the method, air embolisms were induced in cut xylem vessels of chrysanthemum (Dendranthemaxgrandiflorum Tzvelev cv. Cassa) stem segments at different xylem tensions and compared with the distribution of gas-filled vessels as visualized by cryo-scanning electron microscopy (Cryo-SEM). At -6 kPa xylem pressure, air-entrance was only induced in large diameter vessels (>30 microm), while at -24 kPa embolisms were induced in almost all xylem vessels (>10 microm). Although the principle of the embolization method worked well, smaller diameter vessels were observed to be embolized than was expected according to the calculations. The role of cross-sectional shape and contact angle between xylem sap and vessel wall at the menisci are discussed. After correction for the observed (diameter independent) deviation from circularity of the cross-sectional vessel shape the contact angle was calculated to be approximately 55 degrees. Hydraulic resistance (Rh) measurements before and after embolization showed that the effect of embolizing only large diameter cut xylem vessels had only a small influence on overall Rh of a stem segment. Embolizing all cut xylem vessels at one cut end almost trebled overall Rh. The difference was discussed in the light of the networking capacity of the xylem system.     

The Dissipation and Dispersion of Small Waves in Arteries and Veins with Viscoelastic Wall Properties

Theoretical and experimental evidence suggests that the dissipation of high frequency pressure waves in blood vessels is caused primarily by the viscoelastic behavior of the vessel wall. In this theoretical analysis the vessels are considered as fluid-filled circular cylindrical shells whose walls have isotropic and homogeneous viscoelastic properties and are subjected to an initial axial stretch and a transmural pressure. If the wall material is incompressible and behaves as a Voigt solid in shear, the results predict a decrease in wave amplitude per wavelength which is essentially independent of frequency over a wide range. This finding is in qualitative agreement with recent experiments on anesthetized dogs. A parametric study also shows a great sensitivity of the dissipation to changes in transmural pressure and axial stretch. Axisymmetric waves are only mildly dispersive, while all nonaxisymmetric waves are highly dispersive and exhibit much stronger damping per wavelength at low frequencies than do axisymmetric waves.