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.     

On the energy dissipation in a tank-treading human red blood cell.

The energy dissipation in the membrane (ED mem) and in the cytoplasm (ED cyt) of tank-treading human red blood cells is estimated. The tank-tread motion of the membrane occurs when the cells in a sheared suspension assume a steady-state of orientation (Fischer et al., 1978, Science [Wash. D. C.], 202:894). The kinematic data used are from red cells suspended either in a dextran-saline solution at a low hematocrit, or in plasma at a hematocrit of 45%. The viscosities of the cytoplasm and the membrane are taken from the literature. The cell in dextran was subjected to seven different shear rates. Both ED mem and ED cyt showed a strong increase with shear rate. Their ratio, however, was always of the order of 1. From this value and the value which was given by Hochmuth et al. (1979, Biophys. J., 26:101) for a shape recovery of a red cell, it is concluded that the range of ED mem/ED cyt for all possible geometries is 1-100.     

Measurement of human fetal blood flow.

Real-time B-mode ultrasonography was combined with a pulsed Doppler ultrasound technique for transcutaneous measurement of human fetal blood flow in the aorta and intra-abdominal part of the umbilical vein. The target vessel was located and its diameter measured in the two-dimensional real-time image. The pulsed Doppler transducer was attached to the real-time transducer at a fixed angle. By processing the Doppler shift signals the instrument estimated the mean and maximum blood velocities and the integral under the velocity curves. This permitted calculation of the blood flow. The method was applied to 26 fetuses in normal late pregnancies. Mean blood flow in the descending part of the fetal aorta based on maximum velocity was 191 ml/kg/min. Mean flow in the intra-abdominal part of the umbilical vein was 110 ml/kg/min. This method of measurement is non-invasive and opens new perspectives in studying fetal haemodynamics.     

Erythrocyte distribution in arterial blood flow: V. Flow with minimal energy dissipation

Flow of blood is analyzed under the assumptions of Deakin (1967b, 1968). It is found that for physiological values of hematocrit, there is little variation of erythrocyte concentration with radial distance. This conclusion is compared with those of other analyses, and an experiment is proposed which would decide between the various theories.     

Erythrocyte distribution in arterial blood flow: II. The hypothesis of minimal energy dissipation

In a previous paper (Bull. Math. Biophysics,29, 549–563, 1967) the author derived equations to represent the flow of blood in an artery. It was pointed out that these did not completely characterize the system and that an additional hypothesis was required. The hypothesis of minimal energy dissipation had been thought to imply a central tendency on the part of suspended particles (erythrocytes). It is here shown that if the fluid is non-Newtonian this may not be so.     

Pulsatile blood flow, shear force, energy dissipation and Murray's Law

Background  

Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow.     

Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen et al. (Ann Biomed Eng 28:1281–1299, 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of ‘large’ arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the ‘smaller’ arteries and veins of radii \(\ge \) 50  \(\upmu \) m. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment for circulatory diseases within the lung.     

Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model.

Numerical analysis of flow phenomena and wall shear stresses in the human carotid artery bifurcation has been carried out using a three-dimensional geometrical model. The primary aim of this study is the detailed discussion of non-Newtonian flow velocity and wall shear stress during the pulse cycle. A comparison of non-Newtonian and Newtonian results is also presented. The applied non-Newtonian behavior of blood is based on measured dynamic viscosity. In the foreground of discussion are the flow characteristics in the carotid sinus. The investigation shows complex flow patterns especially in the carotid sinus where flow separation occurs at the outer wall throughout the systolic deceleration phase. The changing sign of the velocity near the outer sinus wall results in oscillating shear stress during the pulse cycle. At the outer wall of the sinus at maximum diameter level the shear stress ranges from -1.92 N/m2 to 1.22 N/m2 with a time-averaged value of 0.04 N/m2. At the inner wall of the sinus at maximum diameter level the shear stress range is from 1.16 N/m2 to 4.18 N/m2 with a mean of 1.97 N/m2. The comparison of non-Newtonian and Newtonian results indicates unchanged flow phenomena and rather minor differences in the basic flow characteristics.     

Viscous energy dissipation for steady flow in models of arterial bifurcations

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

Pulsatile non-newtonian blood flow in three-dimensional carotid bifurcation models: a numerical study of flow phenomena under different bifurcation angles

Flow and stress patterns in human carotid artery bifurcation models, which differ in the bifurcation angle, are analysed numerically under physiologically relevant flow conditions. The governing Navier-Stokes equations describing pulsatile, three-dimensional flow of an incompressible non-Newtonian fluid are approximated using a pressure correction finite element method, which has been developed recently. The non-Newtonian behaviour of blood is modelled using Casson's relation, based on measured dynamic viscosity. The study concentrates on flow and stress characteristics in the carotid sinus. The results show that the complex flow in the sinus is affected by the angle variation. The magnitude of reversed flow, the extension of the recirculation zone in the outer sinus region and the duration of flow separation during the pulse cycle as well as the resulting wall shear stress are clearly different in the small angle and in the large angle bifurcation. The haemodynamic phenomena, which are important in atherogenesis, are more pronounced in the large angle bifurcation.     

Viscous energy dissipation in a model of the human bronchial tree

The viscous energy dissipation in a two generation model of the human bronchial tree is determined from inspiratory velocity and static pressure data obtained for large Reynolds numbers (10⁴ < Re < 10⁵). This dissipation is found to be an increasing function of both Re and distance downstream from the inlet of the model. The ratio of the dissipation in the model to the energy dissipation in an equivalent straight pipe system is determined. This ratio, Z^*, for the model is compared to values in the literature for lower (laminar) Re. There is more dissipation in the branched model than in a straight pipe (Z^* > 1) and turbulence keeps Z^* at roughly a fixed value for large Reynolds numbers (10⁴ < Re < 10⁵). Z^* values for curved pipes are also compared to the branching system values. It is found that the energy dissipation for the branched model behaves similarly to that in curved pipes.     

Effects of reducing uterine blood flow on fetal blood flow distribution and oxygen delivery.

We examined the effect of graded reduction in uterine blood flow on distribution of cardiac output and oxygen delivery to fetal organs and venous blood flow patterns in 9 fetal sheep using the radionuclide-labeled microsphere technique. We reduced uterine blood flow in two steps, decreasing fetal oxygen delivery to 70% and 50% of normal, and compared the results with those from a similar study from our laboratory on graded umbilical cord compression. With 50% reduction in fetal oxygen delivery, blood flow and the fraction of the cardiac output distributed to the brain, heart, and adrenal gland increased and that to the lungs, carcass, skin, and scalp decreased. Oxygen delivery to the brain and myocardium was maintained, while that to the adrenal doubled, and that to the brain stem increased transiently. The decrease in oxygen delivery to both carcass and lower body segment correlated linearly with oxygen consumption (P less than 0.001). The proportion of umbilical venous blood passing through the ductus venosus increased from 44.6% to 53% (P less than 0.05). The preferential distribution of ductus venosus blood flow through the foramen ovale to the heart and brain increased, but that to the upper carcass decreased so that ductus venosus-derived blood flow to the upper body did not change. Hence, the oxygen delivered to the brain from the ductus venosus was maintained, and that to the heart increased 54% even though ductus venosus-derived oxygen delivery to the upper body fell 34%. Abdominal inferior vena caval blood flow and its contribution to cardiac output decreased, but the proportion of the abdominal inferior vena caval blood distributed through the foramen ovale also increased from 23.0 to 30.9%. However, the actual amount of inferior vena caval blood passing through the foramen ovale did not change. There was a 70% fall in oxygen delivery to the upper body segment from the inferior vena cava. A greater portion of superior vena caval blood was also shunted through the foramen ovale to the upper body, but the actual amounts of blood and oxygen delivered to the upper body from this source were small. Thus, graded reduction of uterine blood flow causes a redistribution of fetal oxygen delivery and of venous flow patterns, which is clearly different from that observed previously during graded umbilical cord occlusion.     

Pulsatile non-Newtonian blood flow simulation through a bifurcation with an aneurysm

Blood flow is analysed by means of computer simulation in an idealized arterial bifurcation model which is pathologically altered by a saccular aneurysm. The theoretical study of the flow pattern and the paths of fluid particles is carried out under pulsatile Newtonian and non-Newtonian flow conditions. The governing equations are solved numerically with the use of the finite element method. The results show the disturbed blood flow in the bifurcation and the relatively low intra-aneurysmal flow circulation. In addition to the study of basic flow patterns in the segment, a comparison of non-Newtonian and Newtonian results is carried out. This comparison proves that for the considered large artery model under physiological flow conditions where the yield number is relatively low there is no essential difference in the results.     

Measurement of umbilical blood flow in fetal lambs in utero.

We continuously measured umbilical blood flow in fetal lambs in utero by placing an electromagnetic flow transducer around the common umbilical artery. Umbilical arteries originate from a short common segment as the terminal branches of the descending aorta. This segment was isolated by a retroperitoneal surgical approach and encircled with a specially constructed electromagnetic flow transducer. Catheters were also placed in fetal vessles to monitor pressure and derive flow values by the radionuclide-labeled microsphere technique. The fetus and ewe were allowed to recover for two days before studies were performed. Average umbilical blood flow obtained in 11 animals with the transducer was 199 ml/kg per min. In seven animals flow measurements obtained with the transducer were compared with those derived from microsphere injections. Paired measurements varied by an average of only 5.3%. This technique makes possible the accurate and instantaneous measurement of umbilical blood flow in fetal lambs in utero over a prolonged period.     

Stabilization of pressure drop in the feeder arteries of rats as affected by an increase in blood flow

The dependence of the pressure drop (PD) along conducting vessels between the aorta and the distal end of a. saphena on the blood flow in the artery has been studied in rats. The PD was shown to react to rapid blood flow increase from 0.6 to 1.2 ml/min, with a drastic upstroke followed by a gradual decrease to the initial value within 20 s. When the blood flow was returned to the initial level the PD was recovered during 40 s. A rapid flow increase from 0.1 to 1.0 ml/min in 3 s was accompanied by proportional changes in PD. However, a slow blood flow increase from 0.1 to 1.5 ml/min in 600 s did not induce any marked changes in PD in the range of the blood flow from 0.5 to 1.5 ml/min. The observed stabilization of PD may be attributed to the property of conducting arteries to increase their internal diameter in response to blood flow increase.     

A software tool for fetal blood flow analysis

Doppler ultrasonography is a widely used technique for determination of the fetal blood flow pattern. Determination of the waveform qualities was done manually, with considerable inter- and intraobserver variations. In order to limit the variations and the time-consuming data entry, a Fetal Blood Flow Analysis software program was developed to facilitate ease of determination of Doppler signals. This article describes the development and unique features of the software program, made specifically to meet the obstetric and gynecology department's needs.     

Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier–Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau–Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.