Formation of a boundary layer in steady-state blood flow

This paper is devoted to a study of boundary layer formation in the steady flow of blood through the human aorta. Blood is treated as an incompressible fluid. Consideration is given to a flat-top velocity profile which combines the potential flow with the boundary layer; expressions for the displacement thickness, skin-friction and pressure in the entry region are derived.     

The resolution of the ultrasound pulsed Doppler for blood velocity measurements

Pulsed ultrasound Doppler velocity meters (PUDVM) permit noninvasive blood velocity measurements. The emitted ultrasound beam characteristics primarily determine the resolution of the instrument when recording velocity profiles. The sample volume, the small region over which velocity information data are detected, was found to be > 2·3 mm³ depending on the transducer disk dia., distance in front of the disk, sampling time increment, and pulse length. The shape of the sample volume approximates a cylinder in the near field and a frustrum of a cone in the far field. The end surfaces of the sample volume were affected by the emitted pulse shape. Ultrasonic beam cross-sections were found to be smaller than predicted by theory due to the finite threshold levels of the PUDVM. The variation of the sample volume with range was illustrated by steady laminar flow velocity profile measurements in rigid tubes. The accuracy of velocity measurements was within 5 per cent with slightly larger deviations occurring near the walls due to the finite sample volume.     

Influence of pulsatility on the development of intracardiac jets: an in vitro laser Doppler study.

So far, it has been hypothesized that numerical data obtained in steady flow conditions apply to pulsatile flows. In order to study the modifications of the velocity fields due to pulsatility, jets were produced by 8 orifices (with a diameter "D" of 4.4 to 11.3 mm) included in a chamber of 50 mm. The velocity was measured using laser Doppler anemometry with a pulsatile flow ("pf") and compared to the values obtained in steady ("sf"): at maximum velocity, the longitudinal velocity profile is qualitatively similar to this observed in steady flow: it is made of a plateau followed by an hyperbolic velocity decay in the turbulent area. The length of the core ("Lpf") is strongly related to "D" (Lpf = 3.72 D + 5.49, r = .99) and the velocity decay depends on the ratio between the distance "x" from the orifice and "D" (V/Vo = 2.83D/x + 3.46, r = .85, where V is the velocity at "x" and Vo the initial velocity). During the acceleration and the deceleration, the laminar core is disturbed by turbulences. The comparison of "pf" data with "sf" data demonstrated similar diameters at the origin of the jets (Dpf = 0.96 Dsf + .12, r = .99), but significant (p less than .0001) differences both for "L" and "V/Vo": Lpf = .91Lsf + 6.58, r = .97, V/Vopf = .63 V/Vosf + .34, r = .76. Thus, pulsatility modifies velocity fields and the results obtained in steady flow conditions do not apply to pulsatile jets.     

Analysis of the Velocity Distribution in Partially-Filled Circular Pipe Employing the Principle of Maximum Entropy

The flow velocity distribution in partially-filled circular pipe was investigated in this paper. The velocity profile is different from full-filled pipe flow, since the flow is driven by gravity, not by pressure. The research findings show that the position of maximum flow is below the water surface, and varies with the water depth. In the region of near tube wall, the fluid velocity is mainly influenced by the friction of the wall and the pipe bottom slope, and the variation of velocity is similar to full-filled pipe. But near the free water surface, the velocity distribution is mainly affected by the contractive tube wall and the secondary flow, and the variation of the velocity is relatively small. Literature retrieval results show relatively less research has been shown on the practical expression to describe the velocity distribution of partially-filled circular pipe. An expression of two-dimensional (2D) velocity distribution in partially-filled circular pipe flow was derived based on the principle of maximum entropy (POME). Different entropies were compared according to fluid knowledge, and non-extensive entropy was chosen. A new cumulative distribution function (CDF) of partially-filled circular pipe velocity in terms of flow depth was hypothesized. Combined with the CDF hypothesis, the 2D velocity distribution was derived, and the position of maximum velocity distribution was analyzed. The experimental results show that the estimated velocity values based on the principle of maximum Tsallis wavelet entropy are in good agreement with measured values.     

Steady flow development past valve prostheses in a model human aorta—I. Centrally occluding valves

In this paper, laser-Doppler anemometry measurement of steady flow development in a model human aorta has been reported. Studies were made with uniform entry flow at the root of the aorta and our measurements showed the establishment of a pair of Dean vortices in the mid-arch region. Subsequently, the nature of flow development past centrally occluding caged ball valves in the model aorta was investigated. Our studies showed that in the ascending aorta, an asymmetric velocity profile is obtained with larger velocity gradients towards the inner wall of tertiary curvature (anatomically the left lateral wall) with centrally occluding valves. The peripheral flow past these valves prevented the development of Dean vortices in the mid-arch region. The caged ball valves at the root of the aorta had no discernible effect on the velocity profiles in the brachio-cephalic artery.     

Non-Newtonian Blood Flow in Left Coronary Arteries with Varying Stenosis: A Comparative Study

This paper presents Computational fluid dynamic (CFD) analysis of blood flow in three different 3-D models of left coronary artery (LCA). A comparative study of flow parameters (pressure distribution, velocity distribution and wall shear stress) in each of the models is done for a non-Newtonian (Carreau) as well as the Newtonian nature of blood viscosity over a complete cardiac cycle. The difference between these two types of behavior of blood is studied for both transient and steady states of flow. Additionally, flow parameters are compared for steady and transient boundary conditions considering blood as non-Newtonian fluid. The study shows that the highest wall shear stress (WSS), velocity and pressure are found in artery having stenosis in all the three branches of LCA. The use of Newtonian blood model is a good approximation for steady as well as transient blood flow boundary conditions if shear rate is above 100 s-1. However, the assumption of steady blood flow results in underestimating the values of flow parameters such as wall shear stress, pressure and velocity.     

Pulsed magnetohydrodynamic blood flow in a rigid vessel under physiological pressure gradient

Blood flow in a steady magnetic field has been of great interest over recent years. Many researchers have examined the effects of magnetic fields on velocity profiles and arterial pressure, and major studies have focused on steady or sinusoidal flows. In this paper, we present a solution for pulsed magnetohydrodynamic blood flow with a somewhat realistic physiological pressure wave obtained using a Windkessel lumped model. A pressure gradient is derived along a rigid vessel placed at the output of a compliant module which receives the ventricle outflow. Then, velocity profile and flow rate expressions are derived in the rigid vessel in the presence of a steady transverse magnetic field. As expected, results showed flow retardation and flattening. The adaptability of our solution approach allowed a comparison with previously addressed flow cases and calculations presented a good coherence with those well established solutions.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model.

Laser Doppler anemometry experiments and finite element simulations of steady flow in a three dimensional model of the carotid bifurcation were performed to investigate the influence of non-Newtonian properties of blood on the velocity distribution. The axial velocity distribution was measured for two fluids: a non-Newtonian blood analog fluid and a Newtonian reference fluid. Striking differences between the measured flow fields were found. The axial velocity field of the non-Newtonian fluid was flattened, had lower velocity gradients at the divider wall, and higher velocity gradients at the non-divider wall. The flow separation, as found with the Newtonian fluid, was absent. In the computations, the shear thinning behavior of the analog blood fluid was incorporated through the Carreau-Yasuda model. The viscoelastic properties of the fluid were not included. A comparison between the experimental and numerical results showed good agreement, both for the Newtonian and the non-Newtonian fluid. Since only shear thinning was included, this seems to be the dominant non-Newtonian property of the blood analog fluid under steady flow conditions.     

Basis of tetrodotoxin's selectivity in blockage of squid axons

The blockage of nerve activity by tetrodotoxin is unusually potent and specific. Our experiments were designed to distinguish whether its specificity of action was based on the identification of ions, the direction of cation flow, or differences in the early transient and late steady conductance pathways. Alkali cations were substituted for sodium in the sea water, bathing an "artificial node" in a voltage-clamped squid axon. When tetrodotoxin was added to the artificial sea waters at a concentration of 100 to 150 mM, it was found to always block the flow of cations through the early transient channel, both inward and outward, but it never blocked the flow of ions using the late steady pathway. We conclude that the selectivity of tetrodotoxin is based on some difference in these two channels.     

Prevention of blockage of partially obstructed coronary arteries with prostacyclin correlates with inhibition of platelet aggregation.

Coronary arteries (circumflex or left anterior descending) of anesthetized dogs were partially obstructed to approximately 5% of the normal lumen size by fitting a plastic cylinder around the vessel. Under these conditions, blood flow in the artery was not maintained but, instead, gradually declined over a few minutes until the vessel was completely blocked. Shaking the plastic obstructor restored blood flow temporarily, however, flow gradually declined again to zero. Sometimes flow was spontaneously restored by immediate increases that occurred at irregular intervals while, on other occasions, blood flow had to be restored by shaking the obstructor every time the rate declined to near zero. Intravenous infusion of prostacyclin (PGI2) at 15 to 150 ng/kg/min reversed and prevented the blockage of the coronary arteries. The efficacy of PGI2 in preventing blockage correlated with inhibition of ADP-induced platelet aggregation in platelet rich plasma prepared from blood samples withdrawn from the dogs during PGI2 infusion. Other coronary vasodilators, nitroglycerin and PGE2, that have no antiaggregatory effects, failed to prevent blockage whereas PGE1 and indomethacin, which do block aggregation, also prevented blockage of the vessels. PGI2 or its precursor, PGH2, dripped topically on the obstructed site prevented the blockage of the artery. This local effect of IGI2 could be obtained with amounts too small to cause systemic inhibition of platelet aggregation. The results show that PGI2 prevents blockage of partially obstructed coronary arteries and this effect correlates with inhibition of platelet aggregation. Furthermore, the data suggest that locally produced PGI2 may have a local antiaggregatory effect without inhibiting platelet aggregation in the general circulation.     

A model for blood flow through a stenotic tube

This article deals with the introduction of the modified Casson's fluid model as the true representation for the blood for the steady laminar flow through a small diameter artery with axi-symmetric identical double stenoses in series. The governing equations are solved by using the finite element method. The results for the velocity profiles, the pressure and the wall shear stress distributions in addition to the location and length of the flow reversal zones have been brought out and discussed in reference to the severity of the disease. It has been observed that the non-Newtonian nature of the blood helps in reducing the magnitude of the peak wall shear stress at the throat and the length of the reversed flow regions in the post stenotic dilatation.     

Self-sustained Oscillations in Blood Flow Through a Honeycomb Capillary Network

Numerical simulations of unsteady blood flow through a honeycomb network originating at multiple inlets and terminating at multiple outlets are presented and discussed under the assumption that blood behaves as a continuum with variable constitution. Unlike a tree network, the honeycomb network exhibits both diverging and converging bifurcations between branching capillary segments. Numerical results based on a finite difference method demonstrate that as in the case of tree networks considered in previous studies, the cell partitioning law at diverging bifurcations is an important parameter in both steady and unsteady flow. Specifically, a steady flow may spontaneously develop self-sustained oscillations at critical conditions by way of a Hopf bifurcation. Contrary to tree-like networks comprised entirely of diverging bifurcations, the critical parameters for instability in honeycomb networks depend weakly on the system size. The blockage of one or more network segments due to the presence of large cells or the occurrence of capillary constriction may cause flow reversal or trigger a transition to unsteady flow.     

Optimum kinetic energy dissipation in blood flow in glass capillaries by flow field determination: Analysis of curvature effect by axial tomographic and image velocimetry techniques

Based on the variation in the optical density due to erythrocyte concentration and movement, the axial tomographic and image velocimetry techniques are respectively applied to determine the flow field, i.e., the distribution of erythrocytes and axial and radial velocity components, in steady blood flow through a curved glass capillary with a diameter of 180 μm. The data at four positions (two straight and two curved segments of the capillary) are recorded by a video-microscopic system on a video cassette. The erythrocyte and velocity distribution profiles change from symmetric at the straight position to an asymmetric shape at the curved sections. These profiles become symmetric again at the straight section of the capillary. The increase in the radial velocity component at curved portions is attributed to the secondary flow. The tomograms obtained by concentration profiles show respective changes in the cellular population at various cross-sectional positions. The kinetic energy dissipation, as calculated based on a determination of the flow field, is the minimum for the observed profiles. Any deviation towards parabolic form leads to the dissipation of a higher amount of energy.     

Chronic measurement, using a Doppler probe, of uterine artery flow in the gravid guinea-pig

A chronic animal model is described which permits for the first time the continuous measurement of uterine artery blood flow velocity in the pregnant guinea-pig by using a miniaturized Doppler flow probe. Preliminary validation revealed that alterations in actual blood flow are directly and proportionally related to the change in the Doppler shift (r = 0.984) from 0 to 100 ml/h. The velocity signal baseline was as stable as that of systemic blood pressure. Depending upon the individual animal's flow velocity, a deviation of 2-5% from baseline was statistically significant. With experience, greater than 90% of preparations were successful and a 30-day interval was often available for study. Uterine artery flow velocity increased steadily between 45 and 55 days of gestation. Instrumentation did not result in fetal growth retardation. A reduction in flow velocity occurred during general anaesthesia using ketamine and the antianxietal xylazine. In agreement with the reports of other investigators using a different model, both hydralazine and angiotensin II increased uterine blood velocity and adrenaline reduced it.     

Differential blood flow changes between the future dominant and subordinate follicles precede diameter changes during follicle selection in mares

Diameter deviation during a follicular wave is characterized by the continued growth of the developing dominant follicle and reduced growth and regression of the subordinate follicles. This study considered the hypothesis that reduced blood flow in the future largest subordinate follicle precedes the beginning of diameter deviation. The hypothesis was tested by quantifying the daily changes in blood-flow velocities and blood-flow area within the wall of follicles before and during diameter deviation in mares (n = 7). The blood-flow end points were quantified daily by transrectal color Doppler ultrasonography. Follicles were identified retrospectively by rank as F1 (largest) and F2 according to the maximum attained diameter. Follicles were grouped into nine F1 diameter ranges of 3.0 mm each (equivalent to 1 day's growth) centered on 6.5, 9.5, 12.5, 15.5, 18.5, 21.5, 24.5, 27.5, and 30.5 mm. Diameter deviation began in the 24.5-mm group, as indicated by a smaller (P < 0.05) difference between F1 and F2 in the 24.5-mm group than in the 27.5-mm group. Based on a similar approach, peak systolic velocity and time-averaged maximum velocity of blood flow began to deviate between F1 and F2 in the 18.5-mm group (P < 0.04) and blood flow area began to deviate in the 21.5-mm group (P < 0.009). Thus, differential blood flow area between F1 and F2 began an average of 3.0 mm (equivalent to 1 day) and differential blood-flow velocities began an average of 6.0 mm before the beginning of diameter deviation. The results demonstrated that deviation between F1 and F2 in the blood flow of the follicle walls occurred 1 or 2 days before deviation in follicle diameter during follicle selection in mares.     

Relationship between erythrocyte aggregate size and flow rate in skeletal muscle venules

In previous studies we showed that intravenous infusion of Dextran 500 in the rat causes blunting of the velocity profile of red blood cells in venules at low shear rates. To determine whether this blunting is associated with the formation of red blood cell aggregates, we measured the length and width of particles in the venular flow stream at systemic hematocrits up to 20% with a high-speed video camera and a new image analysis technique. Data were obtained at various shear rates under normal (nonaggregating) conditions as well as after infusion of Dextran 500. Under normal conditions, particle length (parallel to the vessel axis) was 6.5 +/- 2.7 microm and width (perpendicular to the axis) was 6.1 +/- 1.7 microm, in agreement with published dimensions of individual red blood cells for this species. After Dextran 500 infusion, particle length and width increased significantly to 8.7 +/- 5.1 and 10.4 +/- 4.4 microm, respectively. Particle dimensions were greater in the central region of the flow stream for both normal and dextran-treated blood and increased at low flow rates with dextran-treated blood. This study provides direct confirmation of aggregate formation at low shear in venules with high-molecular-weight dextran as well as an estimate of aggregate size and range.     

A model investigation of the velocity and pressure spectra in vascular murmurs

A physical model consisting of an axisymmetrical jet in a rigid plexiglass pipe was used to study the flow and pressure fluctuations downstream from an aortic stenosis. The fluctuating velocity components, u and v, at several locations in the steady liquid jet were measured using a laser Doppler anemometer system. Simultaneous wall pressure fluctuations were monitored by an array of nine miniature pressure transducers wall mounted in the axial direction. This paper presents the detailed measurements of mean velocity profiles, turbulent intensity distributions and RMS pressure fluctuations. The energy spectra obtained for the pressure fluctuations and the u and v velocity components are compared. Contrary to earlier works, we found that the differences between peak frequencies of the pressure spectra and the characteristic frequencies of the velocity spectra vary with positions downstream from the nozzle. These differences are discussed in light of pseudosound generation by the eddy structures in the stenotic flow field.     

Developing steady laminar flow through uniform straight tubes with varying wall cross curvature

Numerical calculations are used to determine not only the wall shear stress but also the entry length in a laminar steady flow of an incompressible Newtonian fluid. The fluid is conveyed through rigid straight tubes with axially uniform cross sections, which mimic collapsed vessels. For each tube configuration, the "Navier-Stokes" equations are solved using the finite element method. The numerical tests are performed with the same value of the volume flow-rate whatever the tube configuration for three "Reynolds numbers". The wall shear stress is computed and determined along the axis of the tube, then the entry length is estimated by introducing two indexes by using: (i) the axial fluid velocity, and (ii) the wall shear stress. The results are analysed in order to exhibit the mechanical environment of cultured endothelial cells in the flow chamber for which the test conditions will be well-defined. For example, in a tube configuration where the opposite walls are in contact for which the inner perimeter and the area of the cross section are respectively given by 45 mm and 37.02 mm(2), the computed entry lengths with the criteria defined by (i) and (ii) are equals to about 118 and 126 mm, respectively for R(e0) = 500.     

Prevention of blockage of partially obstructed coronary arteries with prostacyclin correlates with inhibition of platelet aggregation

Coronary arteries (circumflex or left anterior descending) of anesthetized dogs were partially obstructed to approximately 5% of the normal lumen size by fitting a plastic cylinder around the vessel. Under these conditions, blood flow in the artery was not maintained but, instead, gradually declined over a few minutes until the vessel was completely blocked. Shaking the plastic obstructor restored blood flow temporarily, however, flow gradually declined again to zero. Sometimes flow was spontaneously restored by immediate increases that occurred at irregular intervals while, on other occasions, blood flow had to be restored by shaking the obstructor every time the rate declined to near zero. Intravenous infusion of prostacyclin (PGI₂) at 15 to 150 ng/kg/min reversed and prevented the blockage of the coronary arteries. The efficacy of PGI₂ in preventing blockage correlated with inhibition of ADP-induced platelet aggregation in platelet rich plasma prepared from blood samples withdrawn from the dogs during PGI₂ infusion. Other coronary vasodilators, nitroglycerin and PGE₂, that have no antiaggregatory effects, failed to prevent blockage whereas PGE₁ and indomethacin, which do block aggregation, also prevented blockage of the vessels. PGI₂ or its precursor, PGH₂, dripped topically on the obstructed site prevented the blockage of the artery. This local effect of PGI₂ could be obtained with amounts too small to cause systemic inhibition of platelet aggregation. The results show that PGI₂ prevents blockage of partially obstructed coronary arteries and this effect correlates with inhibition of platelet aggregation. Furthermore, the data suggest that locally produced PGI₂ may have a local antiaggregatory effect without inhibiting platelet aggregation in the general circulation.     

Spatial velocity profile in mouse embryonic aorta and Doppler-derived volumetric flow: a preliminary model

Characterizing embryonic circulatory physiology requires accurate cardiac output and flow data. Despite recent applications of high-frequency ultrasound Doppler to the study of embryonic circulation, current Doppler analysis of volumetric flow is relatively crude. To improve Doppler derivation of volumetric flow, we sought a preliminary model of the spatial velocity profile in the mouse embryonic dorsal aorta using ultrasound biomicroscopy (UBM)-Doppler data. Embryonic hematocrit is 0.05-0.10 so rheologic properties must be insignificant. Low Reynolds numbers (<500) and Womersley parameters (<0.76) suggest laminar flow. UBM demonstrated a circular dorsal aortic cross section with no significant tapering. Low Dean numbers (<100) suggest the presence of minimal skewing of the spatial velocity profile. The inlet length allows for fully developed flow. There is no apparent aortic wall pulsatility. Extrapolation of prior studies to these vessel diameters (300-350 microm) and flow velocities (~50-200 mm/s) suggests parabolic spatial velocity profiles. Therefore, mouse embryonic dorsal aortic blood flow may correspond to Poiseuille flow in a straight rigid tube with parabolic spatial velocity profiles. As a first approximation, these results are an important step toward precise in utero ultrasound characterization of blood flow within the developing mammalian circulation.