On the volume-flow mechanism of phloem transport

Summary A steady-state model of solution flow in a tubular semipermeable membrane is developed for an arbitrary distribution of solute sources and sinks along the translocation path. It is demonstrated that the volume-flow mechanism of phloem transport depends only on the two assumptions: 1. that the plasmalemma of the sieve tube is a differentially permeable membrane, and 2. that sugars are actively secreted into and absorbed from the lumen of the sieve tube. It is shown that in the absence of a pressure gradient, there is a negligible concentration gradient over most of the translocation path. However, in the presence of a pressure gradient a small concentration gradient develops as a result of the continually changing chemical potential of water along the direction of solution flow. For Poiseuille flow the concentration gradient is approximately proportional to the mean stream velocity.     

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.     

A computational study on the influence of catheter-delivered intravascular probes on blood flow in a coronary artery model

Catheter-delivered intravascular probes are widely used in clinical practice to measure coronary arterial velocity and pressure, but the artefactual effect of the probe on the variables being measured is not well characterised. A coronary artery was simulated with a 180 degrees curved tube 3mm in diameter and the effect of catheters of different diameters was modelled numerically under pulsatile flow conditions. The presence of a catheter increased pressure by 1.3-4.3 mmHg depending on its diameter, and reduced velocity-pressure phase-lag. For an ultrasound sample volume 5mm downstream from the probe tip, the underestimation in velocity measurement attributed to catheter blockage is approximately 15-21% for an average inlet velocity of 0.1m/s. The velocity measurement error is lower at higher mean flow velocity. Accuracy of clinical velocity measurements could be improved by moving the sample volume farther downstream from the probe tip, because the centrifugal pressure gradient intrinsic to the curvature promotes re-development of flow.     

Non-Newtonian flow patterns associated with an arterial stenosis.

A non-Newtonian constitutive equation for blood has been introduced in this paper. Using this equation, blood flow attributes such as velocity profiles, flowrate, pressure gradient, and wall shear stress in both straight and stenotic (constricted) tubes have been examined. Results showed that compared with Newtonian flow at the same flowrate, the non-Newtonian normally features larger pressure gradient, higher wall shear stress, and different velocity profile, especially in stenotic tube. In addition, the non-Newtonian stenotic flow appears to be more stable than Newtonian flow.     

Linear and nonlinear analyses of pulsatile blood flow in a cylindrical tube

A non-Newtonian shear-thinning constitutive relation is proposed to study pulsatile flow of whole blood in a cylindrical tube. The constitutive relation, which satisfies the principle of material frame indifference, is derived from viscometric data obtained from whole blood over a range of hematocrits. Assuming axisymmetric flow in a rigid cylindrical tube of constant diameter, a second-order, nonlinear partial differential equation governing the axial velocity component is obtained. Imposing a periodic pressure gradient, the governing equation was solved numerically using finite difference methods over a range of Stokes values and hematocrits. For a forcing frequency of 1 Hz, results are presented over tube diameters ranging between 0.1 and 2 cm and over hematocrits ranging between 10 and 80%. For a given hematocrit, velocity profiles predicted for the non-Newtonian model under sinusoidal forcing reveal attenuated volume flow rate and enhanced vorticity transport over the tube cross-section relative to a Newtonian fluid having a viscosity corresponding to the high shear-rate limit. For moderate to high Stokes numbers, consistent with flow in large arteries, our results revealed a viscosity distribution that was nearly time invariant. An analytic solution was obtained for a fluid having arbitrarily prescribed radially varying, temporally invariant viscosity and density distributions under arbitrary periodic pressure forcing. Close agreement was observed between our numerical and analytical results when the imposed viscosity distribution was chosen to approximate the time-averaged viscosity distribution predicted by the shear-thinning non-Newtonian model. For St > or approximately= 100, the disparity between our results and those of a Newtonian fluid of constant viscosity grows with a decreasing ratio of the DC to AC components of the pressure-gradient amplitude below 50%. In particular, for any purely oscillatory pressure-gradient (vanishing DC component), the Womersley solution is a particularly poor predictor of the amplitude and phase of wall shear rate for over half of the flow cycle. Under such circumstances, the analytical models presented here provide a simple and accurate means of estimating instantaneous wall shear rate, knowing only the pressure gradient and hematocrit.     

Phloem ultrastructure and pressure flow: Sieve-Element-Occlusion-Related agglomerations do not affect translocation

Since the first ultrastructural investigations of sieve tubes in the early 1960s, their structure has been a matter of debate. Because sieve tube structure defines frictional interactions in the tube system, the presence of P protein obstructions shown in many transmission electron micrographs led to a discussion about the mode of phloem transport. At present, it is generally agreed that P protein agglomerations are preparation artifacts due to injury, the lumen of sieve tubes is free of obstructions, and phloem flow is driven by an osmotically generated pressure differential according to Münch's classical hypothesis. Here, we show that the phloem contains a distinctive network of protein filaments. Stable transgenic lines expressing Arabidopsis thaliana Sieve-Element-Occlusion-Related1 (SEOR1)-yellow fluorescent protein fusions show that At SEOR1 meshworks at the margins and clots in the lumen are a general feature of living sieve tubes. Live imaging of phloem flow and flow velocity measurements in individual tubes indicate that At SEOR1 agglomerations do not markedly affect or alter flow. A transmission electron microscopy preparation protocol has been generated showing sieve tube ultrastructure of unprecedented quality. A reconstruction of sieve tube ultrastructure served as basis for tube resistance calculations. The impact of agglomerations on phloem flow is discussed.     

Interaction forces between red cells agglutinated by antibody. I. Theoretical.

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutrally buoyant spheres suspended in viscous liquids undergoing a uniform shear flow has been given by Arp and Mason (1977). The method is based on the matrix formulation of hydrodynamic resistances in creeping flow by Brenner and O'Neill (1972). We describe the solution of the Brenner-O'Neill force-torque vector equation in terms of the particle and external flow field coordinates and derive expressions for the normal force acting along, and the shear force acting perpendicular to, the axis of the doublet of spheres, the latter explicitly given for the first time. The equations consist of a term comprising force and torque coefficients obtained from the matrices of the hydrodynamic resistances (functions of the distance h between sphere surfaces which have been computed), and terms comprising the orientation of the doublet axis relative to the coordinates of the external flow field and the shear stress (which can be experimentally determined). We have applied the theory to a system of doublets of sphered, hardened human red cells of group A or B antigenic type cross-linked by the corresponding antibody at a fixed interparticle distance. Working from studies of the breakup of doublets of red cells in an accelerating Poiseuille flow, given in the succeeding paper, we are able to compute the hydrodynamic force required to separate the two spheres. Previous work has shown that the theory can be applied to doublets in a variable shear, Poiseuille flow, provided the ratio of particle to tube diameter is small. In calculating the force-torque coefficients it was assumed that the cells are crosslinked by antibody with h = 20 nm.     

Determinants of valve gating in collecting lymphatic vessels from rat mesentery

Secondary lymphatic valves are essential for minimizing backflow of lymph and are presumed to gate passively according to the instantaneous trans-valve pressure gradient. We hypothesized that valve gating is also modulated by vessel distention, which could alter leaflet stiffness and coaptation. To test this hypothesis, we devised protocols to measure the small pressure gradients required to open or close lymphatic valves and determine if the gradients varied as a function of vessel diameter. Lymphatic vessels were isolated from rat mesentery, cannulated, and pressurized using a servo-control system. Detection of valve leaflet position simultaneously with diameter and intraluminal pressure changes in two-valve segments revealed the detailed temporal relationships between these parameters during the lymphatic contraction cycle. The timing of valve movements was similar to that of cardiac valves, but only when lymphatic vessel afterload was elevated. The pressure gradients required to open or close a valve were determined in one-valve segments during slow, ramp-wise pressure elevation, either from the input or output side of the valve. Tests were conducted over a wide range of baseline pressures (and thus diameters) in passive vessels as well as in vessels with two levels of imposed tone. Surprisingly, the pressure gradient required for valve closure varied >20-fold (0.1-2.2 cmH(2)O) as a passive vessel progressively distended. Similarly, the pressure gradient required for valve opening varied sixfold with vessel distention. Finally, our functional evidence supports the concept that lymphatic muscle tone exerts an indirect effect on valve gating.     

一种测定葡萄木质部压力梯度的新方法

通过同时测定葡萄木质部导管液流平均流量和流速,应用变形的哈根-泊萧叶方程,研究了测定葡萄木质部压力梯度的新方法。结果表明,在直径分别为7mm的葡萄茎段和4mm的葡萄叶柄上,用此新方法测得的木质部压力梯度与实际加在其上的压力梯度吻合的很好,说明此新方法可用于连续、实时测定葡萄木质部压力梯度。用此新方法在离体葡萄枝上测定发现,随光照强度增强,其木质部压力梯度成线性正相关增加。     

Computational models evaluating the impact of sieve plates and radial water exchange on phloem pressure gradients

The sugar conducting phloem in angiosperms is a high resistance pathway made up of sieve elements bounded by sieve plates. The high resistance generated by sieve plates may be a trade‐off for promoting quick sealing in the event of injury. However, previous modeling efforts have demonstrated a wide variation in the contribution of sieve plates towards total sieve tube resistance. In the current study, we generated high resolution scanning electron microscope images of sieve plates from balsam poplar and integrated them into a mathematical model using Comsol Multiphysics software. We found that sieve plates contribute upwards of 85% towards total sieve tube resistance. Utilizing the Navier–Stokes equations, we found that oblong pores may create over 50% more resistance in comparison with round pores of the same area. Although radial water flows in phloem sieve tubes have been previously considered, their impact on alleviating pressure gradients has not been fully studied. Our novel simulations find that radial water flow can reduce pressure requirements by half in comparison with modeled sieve tubes with no radial permeability. We discuss the implication that sieve tubes may alleviate pressure requirements to overcome high resistances by regulating their membrane permeability along the entire transport pathway.     

Determination of erythrocyte transit times through micropores. II-- Influence of experimental and physicochemical factors

A new red blood cell filtration system, termed the Cell Transit Time Analyzer (CTTA), has been developed in order to measure the individual transit times of a large number of cells through cylindrical micropores in special "oligopore" filters: the system operates on the electrical conductometric principle and employs special computer software to provide several measures of the resulting transit time histogram. Using this system with filters having pore diameters of 4.5 or 5.0 cm and length to diameter ratios of 3.0 to 4.7, we have evaluated the effects of several experimental factors on the flow behavior of normal and modified human RBC. Our results indicate : 1) linear PBC pressure - flow behavior over a driving pressure range of 2 to 10.5 cm H2O with zero velocity intercepts at delta P = 0, thus suggesting the Poiseuille - like nature of the flow; 2) resistance to flow or "apparent viscosities" for normal RBC which are between 3.1 to 3.9 cPoise and are independent of driving pressure and pore geometry; 3) increased flow resistance (i.e., increased transit times) for old versus young RBC and for RBC made less deformable by DNP-induced crenation or by heat treatment at 48 degrees C; 4) increased mean transit time and poorer reproducibility when using EDTA rather than heparin as the anticoagulant agent. Further, using mixtures of heat-treated and normal RBC and various percentile values of the transit time histogram. We have been able to demonstrate the presence of sub-populations of rigid cells and thus the value of measurements which allow statistical analyses of RBC populations.     

Pressure flow patterns in a cylinder with reabsorbing walls

The problem of fluid motion in renal tubules, in contrast to ordinary flow through cylinders with impermeable walls, is complicated by the existence of radial velocities generated by reabsorption processes. As a first approach to this problem, the Navier Stokes equations for axially symmetric, slow flow in an infinite cylinder whose walls reabsorb fluid are integrated. If the rate of reabsorption is constant, the solutions resemble the conventional Poiseuille flow, i.e., the longitudinal velocity profile is parabolic. In addition the drop in mean pressure is proportional to the mean axial flow, the length of tube between reference points, and inversely proportional to the fourth power of the radius. If the rate of reabsorption is a linear function of the distance from the origin, the presence of an additive term alters these relations. If, for example, the gradient in reabsorption is positive, the axial velocity profile tends to flatten and when the gradient is sufficiently large, the maximum velocity moves from the center of the stream toward the periphery, leaving a relative minimum at the center. In passing from the center of the tube to the walls, the radial velocity passes through a miximum, regardless of the reabsorption properties of the wall.     

Evidence for solution flow in the phloem of willow

Sucrose specific mass transfer measurements were made in a translocating willow shoot (Salix viminalis L.) by a steady state labelling technique and the translocate sucrose specific activity, concentration and velocity monitored by analysis of the honeydew from two colonies of the willow aphid Tuberolachnus salignus Gmelin. The values of sucrose SMT obtained were related to the simultaneous measurements of translocate concentration and velocity and to the gradients of sucrose concentration within the stem transport path to determine if transport was a bulk flow or a diffusional analogue. Estimates of potassium ion concentration in the sieve tubes were made, using aphid honeydew, and related to the sucrose SMT measured simultaneously. Correlations were found between translocate concentration, velocity and SMT which suggested that solution flow was occurring rather than a process analogous to diffusion. Evidence was obtained that velocity of flow was a valid concept and that the measured velocity was being lowered by leakage of tracer from the sieve tubes. The analysis of potassium concentration suggested that if solution flow was occurring then potassium must be very exchangeable down the transport path. A good correlation was observed between the SMT of sucrose and the combined gradient of sucrose and potassium concentration, though this gradient was in the opposite direction to transport in some cases.Abbreviations SMT Sucrose specific mass transfer rate - SAR Specific activity ratio - OP Osmotic pressure     

A Mathematical Treatment of Munch's Pressure-Flow Hypothesis of Phloem Translocation

The steady state solutions of two mathematical models are used to evaluate Münch's pressure-flow hypothesis of phloem translocation. The models assume a continuous active loading and unloading of translocate but differ in the site of loading and unloading and the route of water to the sieve tube. The dimensions of the translocation system taken are the average observed values for sugar beet and are intended to simulate translocation from a mature source leaf to an expanding sink leaf. The volume flow rate of solution along the sieve tube, water flow rate into the sieve tube, hydrostatic pressure, and concentration of sucrose in the sieve tube are obtained from a numerical computer solution of the models. The mass transfer rate, velocity of translocation, and osmotic and hydrostatic pressures are consistent with empirical findings. Owing to the resistance to water flow offered by the lateral membranes, the hydrostatic pressure generated by the osmotic pressure can be considerably less than would be predicted by the solute concentration. These models suggest that translocation at observed rates and velocities can be driven by a water potential difference between the sieve tube and surrounding tissue and are consistent with the pressure-flow hypothesis of translocation.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Mass and Energy Transport Through Slit Pores: Application to Planar Poiseuille Flow

Abstract

We develop a simple, efficient and general statistical mechanical technique for calculating the pressure tensor and the heat flux vector in atomic fluids. The method is applied to the case of planar Poiseuille flow through a narrow slit pore and the results indicate that our technique is accurate and relatively efficient. A second method to calculate shear stress is derived from the momentum continuity equation. This mesoscopic method again is seen to be accurate with good computational efficiency.

We also find that the commonly used approximation to the Irving-Kirkwood expression for the heat flux and the pressure tensor (where the Irving-Kirkwood O_ij operator is set equal to unity-the so-called IK1 approximation), leads to incorrect results for highly inhomogeneous fluids. In such cases the pressure tensor and heat flux vector display spurious oscillations.

We calculate the spatially dependent viscosity across a narrow pore and find that it exhibits real but weak oscillations, a consequence of oscillations in the number density. Finally we point out that if the heat flux vector is coupled to the gradient of the square of the strain rate tensor such an effect will only affect the shape of the temperature profile. For planar Poiseuille flow, the temperature profile should deviate from the classical quartic form and include an additional quadratic component. The actual magnitude and shape of the heat flux vector remain exactly as they would if such a coupling did not exist.     

Measurement of turgor pressure and its gradient in the Phloem of oak

A direct method is described for measuring the pressure in secondary phloem sieve tubes of oak trees. One end of a 26-gauge stainless steel tube was shaped such that when it penetrated the outer bark and transected a few sieve elements, it was stopped by the xylem so that small openings in the end allowed phloem sap to enter the tube. The other end of the stainless tube (phloem needle) was joined to a long glass capillary sealed at its other end to form a manometer for measuring phloem sap pressure. A method for measuring the average osmotic and turgor pressures in cells of leaves is also described. Phloem turgor pressures varied greatly in a series of phloem punctures around the trunk at 1.5 and at 6.3 meters. The variation in turgor pressure was always greater than the variation in osmotic pressure. In a series of turgor pressures arranged in descending order, the values in a sequence for the upper level was usually a little (0-3 atm) larger than the values for the lower level. These results may suggest that translocation of assimilate is favored by a small turgor pressure gradient, but they do more to emphasize the complications in measuring gradients in an elastic low resistance distribution system composed of contiguous longitudinal conduits. The results also imply that the sieve tubes are inflated with assimilate fluid under high pressure which can readily move longitudinally and with less pressure drop than would be necessary if the sieve tubes were rigid.     

A new approach for the evaluation of the severity of coarctation of the aorta using Doppler velocity index and effective orifice area: in vitro validation and clinical implications

Early detection and accurate estimation of COA severity are the most important predictors of successful long-term outcome. However, current clinical parameters used for the evaluation of the severity of COA have several limitations and are flow dependent. The objectives of this study are to evaluate the limitations of current existing parameters for the evaluation of the severity of coarctation of the aorta (COA) and suggest two new parameters: COA Doppler velocity index and COA effective orifice area. Three different severities of COAs were tested in a mock flow circulation model under various flow conditions and in the presence of normal and stenotic aortic valves. Catheter trans-COA pressure gradients and Doppler echocardiographic trans-COA pressure gradients were evaluated. COA Doppler velocity index was defined as the ratio of pre-COA to post-COA peak velocities measured by Doppler echocardiography. COA Doppler effective orifice area was determined using continuity equation. The results show that peak-to-peak trans-COA pressure gradient significantly increased with flow rate (from 83% to 85%). Peak Doppler pressure gradient also significantly increased with flow rate (80-85%). A stenotic or bicuspid aortic valve increased peak Doppler pressure gradient by 20-50% for a COA severity of 75%. Both COA Doppler velocity index and COA effective orifice area did not demonstrate significant flow dependence or dependence upon aortic valve condition. As a conclusion, COA Doppler velocity index and COA effective orifice area are flow independent and do not depend on aortic valve conditions. They can, then, more accurately predict the severity of COA.     

The Influence of Drag on Nonlinear Oscillatory Flow through Concentric Annulus

A mathematical model has been developed to study the effect of particle drag parameter and frequency parameter on velocity and pressure gradient in nonlinear oscillatory two phase flow. The main purpose is to apply the model to study the combined effect of introduction of the catheter and elastic properties of the arterial wall on the pulsatile nature of the blood flow. We model the artery as an isotropic thin walled elastic tube and the catheter as a coaxial flexible tube. Blood is modeled as an incompressible particulate viscous Newtonian fluid. Perturbation technique has been applied to find the approximations for velocity and pressure gradient up to second order. Numerical solutions are investigated with graphical presentations to understand the effects of drag parameter, frequency parameter and phase angle on velocity along radial direction and pressure gradient along axial directions. As the drag parameter increases, mean pressure gradient and mean velocity will be decreased. As frequency parameter increases mean velocity profile bends near the outer wall. Due to elastic nature of artery wall, a thin catheter experience small oscillations and a thick catheter remains stationary inside the artery. Finally, the effect of catheterization on various physiologically important flow rate characteristics—mean velocity, mean pressure gradient are studied for a range of different catheter sizes, particle drag parameter and frequency parameters.     

Effects of aggregation on the flow properties of red blood cell suspensions in narrow vertical tubes

The flow properties of aggregating red cell suspensions flowing at low rates through vertical tubes with diameters from 30 microns to 150 microns are analyzed using a theoretical model. Unidirectional flow is assumed, and the distributions of velocity and red cell concentration are assumed to be axisymmetric. A three-layer approximation is used for the distribution of red cells, with a cylindrical central core of aggregated red cells moving with uniform velocity, a cell-free marginal layer near the tube wall, and an annular region located between the core and the marginal layer containing suspended non-aggregating red cells. This suspension is assumed to behave approximately as a Newtonian fluid whose viscosity increases exponentially with red cell concentration. Physical arguments concerning the mechanics of red cell attachment to, and detachment from the aggregated core lead to a kinetic equation for core formation. From this kinetic equation and the equation for conservation of red cell volume flux, a relationship between core radius and pressure gradient is obtained. Then the relative viscosity is calculated as a function of pseudo-shear rate. At low flow rates, it is shown that the relative viscosity decreases with decreasing flow and that the dependence of relative viscosity on shear rates is more pronounced in larger tubes. It is also found that the relative viscosity decreases with increasing aggregation tendency of suspension. These theoretical predictions are in good qualitative and quantitative agreement with experimental results.